diff --git a/parse/train/B1x5KiCcFX/B1x5KiCcFX.md b/parse/train/B1x5KiCcFX/B1x5KiCcFX.md new file mode 100644 index 0000000000000000000000000000000000000000..8d232bfa2d7f81216c27cf1f53000531916a99b7 --- /dev/null +++ b/parse/train/B1x5KiCcFX/B1x5KiCcFX.md @@ -0,0 +1,689 @@ +# 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. + +# REFERENCES + +Martin Arjovsky and Léon Bottou. 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Proceedings of International Conference on Learning Representations, 2018. + +Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016. + +Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017. + +# 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 $\beta = 1$ . \ No newline at end of file diff --git a/parse/train/B1x5KiCcFX/B1x5KiCcFX_content_list.json b/parse/train/B1x5KiCcFX/B1x5KiCcFX_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6ea0e8dddae341b4c1240bcab795dde70b82eeb0 --- /dev/null +++ b/parse/train/B1x5KiCcFX/B1x5KiCcFX_content_list.json @@ -0,0 +1,3223 @@ +[ + { + "type": "text", + "text": "UNDERSTANDING GANS VIA GENERALIZATION ANALYSIS FOR DISCONNECTED SUPPORT ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 233, + 263, + 764, + 458 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 483, + 336, + 498 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 512, + 825, + 625 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 631, + 825, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 763, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 152, + 825, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The contributions of this paper are summarized as follows: ", + "bbox": [ + 174, + 257, + 558, + 272 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We show that GANs perform better than other standard methods of estimating probability measures when the measure satisfies the disconnected support property. \n2. 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. \n3. Based on the generalization bound, we provide a theoretical guideline for selecting architectures of generators and discriminators. ", + "bbox": [ + 210, + 284, + 825, + 381 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "All the proofs are given in Supplementary materials. ", + "bbox": [ + 173, + 393, + 516, + 409 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 428, + 339, + 444 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 NOTATION ", + "text_level": 1, + "bbox": [ + 174, + 459, + 289, + 473 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 484, + 826, + 660 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 GENERAL FRAMEWORK OF GANS ", + "text_level": 1, + "bbox": [ + 176, + 676, + 452, + 691 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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 $x$ . ", + "bbox": [ + 174, + 703, + 825, + 775 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 781, + 825, + 839 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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), ", + "bbox": [ + 174, + 845, + 825, + 887 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/244cc3b646fd5ce25bfd87530f5bce0261ab7fa6e0a4061047d9305688e29ce1.jpg", + "text": "$$\nd _ { \\mathcal { F } } ( P , P ^ { \\prime } ) : = \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\mathbb { E } _ { X \\sim P } [ f ( X ) ] - \\mathbb { E } _ { X \\sim P ^ { \\prime } } [ f ( X ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 330, + 893, + 666, + 921 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "between probability measures $P$ and $P ^ { \\prime }$ . ", + "bbox": [ + 174, + 103, + 439, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 125, + 825, + 190 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2889892051aaebf92d10f96d4f4372712eacc49eecc36b7f3b6d573c8e5c6f31.jpg", + "text": "$$\n{ \\widehat { g } } \\in \\mathop { \\mathrm { a r g m i n } } _ { g \\in { \\mathcal { G } } } d _ { { \\mathcal { F } } } \\left( P _ { n } , P _ { g , m } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 195, + 591, + 222 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 234, + 825, + 319 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We assume that $\\mathcal { F }$ is large enough to contain functions which can work as a discriminator, namely, we assume that the following holds: ", + "bbox": [ + 173, + 325, + 825, + 354 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/748c55c4048b7427fe6cec25a9587408d3de30c99ffcffff84626d6cab0055bc.jpg", + "text": "$$\nd _ { \\mathcal { F } } ( P , P ^ { \\prime } ) = 0 \\Leftrightarrow P = P ^ { \\prime } .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 359, + 591, + 377 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A sufficient condition for (3) is investigated in Zhang et al. (2018). ", + "bbox": [ + 173, + 382, + 611, + 398 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 DEEP NEURAL NETWORKS FOR GENERATORS AND DISCRIMINATORS ", + "text_level": 1, + "bbox": [ + 174, + 414, + 694, + 429 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 171, + 439, + 825, + 469 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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 $\\eta ( x ) = ( \\operatorname* { m a x } \\{ x _ { d } , 0 \\} ) _ { d \\in [ D ^ { \\prime } ] }$ . ", + "bbox": [ + 173, + 474, + 826, + 601 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We define functions of DNNs with an architecture $\\Theta$ as $\\xi [ \\Theta ] : \\mathbb { R } ^ { D ^ { \\prime } } \\mathbb { R } ^ { D ^ { \\prime \\prime } }$ by ", + "bbox": [ + 174, + 607, + 691, + 623 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c1bff21106541b56e28f7e91281647e125e485cb28a99d1e4cd96af35c7c9245.jpg", + "text": "$$\n\\xi [ \\Theta ] ( x ) = x ^ { ( L + 1 ) } , x ^ { ( 1 ) } : = x , x ^ { ( \\ell + 1 ) } : = \\eta ( A _ { \\ell } x ^ { ( \\ell ) } + b _ { \\ell } ) , \\mathrm { f o r } \\ell \\in [ L ] .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 630, + 730, + 648 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The function class of DNNs is thus given by ", + "bbox": [ + 176, + 654, + 465, + 670 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f418b78952b41d50b1e3e5bd160ee7d100699572e4b4dfb1a600a4cac20d581f.jpg", + "text": "$$\n\\Xi ( S , B , L ) : = \\Big \\{ \\xi [ \\Theta ] : I ^ { D } \\to \\mathbb { R } \\ | \\ \\| \\Theta \\| _ { 0 } \\leqslant S , \\| \\Theta \\| _ { \\infty } \\leqslant B , | \\Theta | \\leqslant L \\Big \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 272, + 674, + 722, + 702 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 707, + 825, + 750 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 DISCONNECTED SUPPORT PROPERTY", + "text_level": 1, + "bbox": [ + 174, + 768, + 514, + 785 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 INTRODUCTION AND EXAMPLE ", + "text_level": 1, + "bbox": [ + 176, + 800, + 431, + 814 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/fe32edfd04a7a392e42a9668e55374015bbdbbcd473787ac6a8e24e86e9e385a.jpg", + "image_caption": [ + "Figure 1: Plot of the MNIST data. " + ], + "image_footnote": [], + "bbox": [ + 232, + 204, + 470, + 332 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/9e1bdfe866576746ea80f727f2a4ff3e3c05341c6ba33fc8c5e775ffcb4bc9c3.jpg", + "image_caption": [ + "Figure 2: Plot of the animal data. " + ], + "image_footnote": [], + "bbox": [ + 522, + 207, + 763, + 330 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 MATHEMATICAL FORMULATION OF DISCONNECTED SUPPORTS ", + "text_level": 1, + "bbox": [ + 176, + 401, + 648, + 415 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 426, + 828, + 483 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We first prepare a family of subsets as a component in the disconnected supports: ", + "bbox": [ + 176, + 489, + 705, + 505 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { S _ { \\alpha , J } : = \\left\\{ S \\subset I ^ { D } \\ \\right| } \\end{array}$ A boundary of $S$ is $J$ combination of $\\alpha$ -smooth hyper surfaces . ", + "bbox": [ + 194, + 511, + 774, + 527 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The supplementary material will provide a more rigorous definition. ", + "bbox": [ + 174, + 532, + 619, + 547 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 554, + 825, + 611 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 1. (Disconnected Supports / Global Support) \nLet $M \\geqslant 2$ . A probability measure $P$ on $( I ^ { D } , \\bar { \\Sigma } )$ has $M$ disconnected supports, if there exist \nnonempty disjoint sets ${ \\cal S } _ { 1 } , . . . , { \\cal S } _ { M } \\in { \\cal S } _ { \\alpha , J }$ such that ", + "bbox": [ + 173, + 613, + 826, + 656 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d49ac0ba410889d90988a4f12f3a2458848edb0bb875626527da7747e70f70f3.jpg", + "text": "$$\n\\operatorname { S u p p } ( P ) = \\bigcup _ { m \\in [ M ] } S _ { m } .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 661, + 578, + 696 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A probability measure $P$ on $( I ^ { D } , \\Sigma )$ has a global support, if $\\operatorname { S u p p } ( P ) = I ^ { D }$ . ", + "bbox": [ + 173, + 702, + 681, + 719 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 3 illustrates the disconnected support property. ", + "bbox": [ + 176, + 728, + 526, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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 $P$ . ", + "bbox": [ + 174, + 750, + 825, + 779 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2. (Local Smoothness) ", + "bbox": [ + 174, + 782, + 400, + 796 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 796, + 825, + 828 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e19e218c7e6abdd12d08ca3642dcd8a2421289fe3137a9961e593a6121f77109.jpg", + "text": "$$\nP ( B ) = P _ { Z } ( \\gamma _ { m } ^ { - 1 } ( B ) ) , \\forall B \\in \\sigma ( S _ { m } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 833, + 620, + 852 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "for $m \\in [ M ]$ ", + "bbox": [ + 173, + 854, + 264, + 871 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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 $S _ { m }$ . ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 1. (Locally Smooth Density Functions) \nIf a probability measure $P$ with $M$ disconnected support is locally $\\beta$ -smooth, then there exists a \nfunction $p _ { m } : S _ { m } \\to \\mathbb { R } _ { + }$ such that ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b4a21b019ccbff3e9458c9cfc3e1e5df6feff71e8912130595a22ac9d678e394.jpg", + "text": "$$\nP ( B ) = \\int _ { B } p _ { m } ( x ) d \\lambda , B \\in \\sigma ( S _ { m } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 154, + 614, + 188 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\lambda$ is the Lebesgue measure, and $p _ { m }$ is $\\beta$ -smooth for all $m \\in [ M ]$ . ", + "bbox": [ + 174, + 195, + 643, + 210 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We call $p _ { m }$ as a local density function. ", + "bbox": [ + 174, + 222, + 426, + 237 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 243, + 826, + 289 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/c3dfa9a97b096a21af5f6a1718c1eb7930b940550b40573c085a60e6e567de44.jpg", + "image_caption": [ + "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 $S _ { 2 }$ . " + ], + "image_footnote": [], + "bbox": [ + 245, + 321, + 444, + 435 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/cfd77cfd712870f7426ee318b45107ce716b4b22206e74bddb9e99ccc7ae3428.jpg", + "image_caption": [ + "Figure 4: Illustration of a generator $g$ . To represent discontinuous $S _ { 1 }$ and $S _ { 2 }$ , $g$ should be discontinuous. " + ], + "image_footnote": [], + "bbox": [ + 547, + 324, + 753, + 436 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 DIFFICULTY WITH DISCONNECTED SUPPORTS ", + "text_level": 1, + "bbox": [ + 174, + 537, + 532, + 551 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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 $P ^ { \\prime }$ . ", + "bbox": [ + 173, + 564, + 825, + 719 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 724, + 825, + 768 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0c52694113ff095830b769476fa4a6696c17d1d7b569dace3312c727f13a10cf.jpg", + "text": "$$\nd _ { 2 } ( P ^ { * } , \\widehat { P } ) = O \\left( n ^ { - \\beta / ( 2 \\beta + D ) } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 775, + 607, + 803 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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 $\\beta$ ). ", + "bbox": [ + 173, + 809, + 825, + 839 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "On the other hand, we consider a case in which $P ^ { * }$ has the disconnected support property. ", + "bbox": [ + 174, + 844, + 764, + 861 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 1. (Deterioration of other standard methods) There exists $P ^ { * }$ of the disconnected support property and locally $\\beta$ -smooth such that ", + "bbox": [ + 173, + 864, + 728, + 893 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b9730f5b9fd0d563aeb76358ab989af6dc4cf556065c4cfb43470574c4e23436.jpg", + "text": "$$\nd _ { 2 } ( P ^ { * } , \\widehat { P } ) = { \\cal O } \\left( n ^ { - 1 / ( 2 + D ) } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 395, + 901, + 602, + 928 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 173, + 103, + 823, + 146 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 152, + 826, + 209 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Lemma 2. (Discontinuous Generators for Disconnected Supports) \nIf $P ^ { * }$ has disconnected supports and $P ^ { * } = P _ { g ^ { * } }$ with a generator $g ^ { * }$ , then $g ^ { * }$ is not uniformly continuous. ", + "bbox": [ + 173, + 213, + 825, + 255 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 265, + 825, + 352 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 GENERALIZATION BY GANS ", + "text_level": 1, + "bbox": [ + 176, + 371, + 444, + 388 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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 $S _ { g } , B _ { g } , L _ { g }$ . ", + "bbox": [ + 173, + 402, + 826, + 523 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A standard line of discussing generalization, we should consider statistical errors and approximation errors. We define a measure of the complexity of $\\mathcal { F }$ ", + "bbox": [ + 173, + 527, + 825, + 556 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f0f88abbaefa9738a0a29630202f8f301c05bb4e4e4f420ebdcc285caa321e65.jpg", + "text": "$$\n\\Upsilon _ { n } ( \\mathcal { F } ) : = \\operatorname* { i n f } _ { \\eta > 0 } 4 \\eta + 1 2 n ^ { - 1 / 2 } \\int _ { \\eta } ^ { c } \\log \\mathcal { N } ( ( L _ { 1 } + 1 ) ^ { - 1 } \\epsilon , \\mathcal { F } , \\| \\cdot \\| _ { n } ) ^ { 1 / 2 } d \\epsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 269, + 561, + 725, + 597 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 603, + 823, + 635 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/e8004170a92a32429e0f943a2b2340f2c2b08e56351771bb26df4b194635a501.jpg", + "text": "$$\n\\Upsilon _ { n } ( \\mathcal { F } ) \\geqslant \\mathbb { E } \\left[ \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\frac { 1 } { n } \\left| \\sum _ { i \\in [ n ] } \\tau _ { i } f ( X _ { i } ) \\right| \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 369, + 638, + 625, + 690 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 694, + 825, + 737 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9bf9edf5e6753d43f6d5c96ac98ecd3bcea8f5845a0a611ab96d3d97575558b2.jpg", + "text": "$$\n\\Upsilon _ { n } ( \\mathcal { F } ) \\leqslant C _ { \\mathcal { F } } n ^ { - 1 / \\kappa } ,\n$$", + "text_format": "latex", + "bbox": [ + 428, + 742, + 568, + 762 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "with some constants $C _ { \\mathcal { F } } > 0$ and $\\kappa \\geqslant 2$ . ", + "bbox": [ + 174, + 767, + 439, + 781 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Regarding approximation errors, we need to consider approximation of a discontinuous function; \nsince Lemma 2 shows that a discontinuous generator is necessary to represent disconnected supports. \nTo approximate such generators, DNNs in GANs has an advantage. ", + "bbox": [ + 174, + 789, + 826, + 830 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 834, + 825, + 878 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/437207fb9e38c506a61f082a97c240114373981539eedb3ef9cea0172d2b0c65.jpg", + "text": "$$\n\\lVert \\dot { \\boldsymbol g } _ { d } - \\boldsymbol g _ { d } ^ { * } \\rVert _ { L ^ { 2 } } \\leqslant c _ { g } M S _ { g } ^ { - \\beta / D } , \\forall d \\in [ D ] .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 883, + 630, + 904 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Furthermore, if $P ^ { * } = P _ { g ^ { * } }$ has a global support and it is $\\beta$ -smooth, (4) holds with $M = 1$ ", + "bbox": [ + 169, + 909, + 763, + 925 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 103, + 826, + 146 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Based on Lemma 3, we obtain the main theorem for generalization analysis. ", + "bbox": [ + 173, + 152, + 673, + 167 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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, ", + "bbox": [ + 174, + 170, + 823, + 228 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/55f6ba0c10d1b2920d2f98ec941c4c93f252d262de471fecb5cc785a79409abf.jpg", + "text": "$$\nd _ { \\mathcal { F } } \\big ( P ^ { * } , P _ { \\widehat { g } } \\big ) \\leqslant \\underbrace { \\Upsilon _ { m } ( \\widetilde { \\mathcal { F } } ) + c _ { 1 } \\frac { \\sqrt { S _ { g } } + \\sqrt { S _ { f } } } { \\sqrt { m } } } _ { = : I } + \\underbrace { c _ { 2 } M D S _ { g } ^ { - \\beta / D } } _ { = : I I } + \\underbrace { \\Upsilon _ { n } ( \\widetilde { \\mathcal { F } } ) + c _ { 3 } \\sqrt { \\frac { S _ { f } } { n } } } _ { = : I I I } .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 231, + 766, + 286 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Furthermore, $i f P ^ { * }$ has a global support and it is $\\beta$ -smooth, (5) holds with $M = 1$ ", + "bbox": [ + 173, + 290, + 715, + 306 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 171, + 315, + 821, + 344 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof Outline: By the definition of $P _ { \\hat { g } }$ in (2) and standard calculation, we obtain the inequality ", + "bbox": [ + 176, + 351, + 797, + 366 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/93deb4439d4b2492a38423a6f0889a0166b5890088440a7df3df0970d525765a.jpg", + "text": "$$\nd _ { \\mathcal { F } } ( P ^ { * } , P _ { \\hat { g } } ) \\leqslant \\underbrace { 2 \\operatorname* { s u p } _ { g \\in \\mathcal { G } } \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\big | \\mathbb { E } _ { P _ { g , m } } [ f ( X ) ] - \\mathbb { E } _ { P _ { g } } [ f ( X ) ] \\big | } _ { = : i } + \\underbrace { \\operatorname* { i n f } _ { g \\in \\mathcal { G } } d _ { \\mathcal { F } } ( P _ { g } , P ^ { * } ) } _ { = : i i } + \\underbrace { 2 d _ { \\mathcal { F } } ( P _ { n } , P _ { 0 } ) } _ { = : i i i } .\n$$", + "text_format": "latex", + "bbox": [ + 199, + 371, + 799, + 417 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. □ ", + "bbox": [ + 173, + 421, + 826, + 506 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 511, + 825, + 569 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/e48c2040dd109246ebb3ccd8ac6d4973e4a527c42643d9df0e6c9dc500daf35c.jpg", + "text": "$$\nS _ { g } = c _ { g } m ^ { D / ( 2 \\beta + D ) } , ~ \\mathrm { a n d } ~ S _ { f } = c _ { f } n ^ { ( \\kappa - 2 ) / \\kappa } ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 573, + 643, + 593 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "for optimizing the bound (5). We then obtain the following corollary. ", + "bbox": [ + 173, + 595, + 625, + 611 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 1. (Convergence Rate of GANs) \nMake the same assumptions as Theorem $^ { l }$ , and set $S _ { f }$ and $S _ { g }$ as in (6). Then, with high probability \nconverging to 1, we obtain ", + "bbox": [ + 173, + 614, + 826, + 656 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/0242a21b28cf208c69fa84d2b9696407b82dac3667dc7686f4f04ee4dba30595.jpg", + "text": "$$\nd _ { \\mathcal { F } } ( P ^ { \\ast } , P _ { \\hat { g } } ) = O \\left( n ^ { - 1 / \\kappa } \\right) + O \\left( m ^ { - 1 / \\kappa } + m ^ { - \\beta / ( 2 \\beta + D ) } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 660, + 694, + 688 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 171, + 700, + 826, + 733 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Remark 1. (Heterogeneous Smoothness) ", + "bbox": [ + 173, + 736, + 449, + 750 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 750, + 825, + 791 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/e498f4cc78a2f525e2d9205ab99029d6060970965bc8e0760699ef2d389644a8.jpg", + "text": "$$\nd _ { \\mathcal { F } } ( P ^ { \\ast } , P _ { \\hat { g } } ) = O \\left( n ^ { - 1 / \\kappa } \\right) + O \\left( m ^ { - 1 / \\kappa } + m ^ { - \\widetilde { \\beta } / ( 2 \\widetilde { \\beta } + D ) } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 795, + 692, + 823 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\widetilde { \\beta } : = \\operatorname* { m i n } _ { m \\in \\left[ M \\right] } \\beta _ { m }$ ", + "bbox": [ + 174, + 828, + 352, + 847 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 863, + 312, + 881 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 200 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 208, + 825, + 338 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.1 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 174, + 356, + 338, + 369 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 382, + 825, + 465 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 472, + 825, + 542 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 549, + 825, + 592 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 NUMERICAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 613, + 434, + 628 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 643, + 825, + 770 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 777, + 823, + 833 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/b7802bb8ea485d4b37d7f38150a4da78b2af66548ec317a150627253f99cdfd2.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 598, + 125, + 800, + 280 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/fdfdd8b80275cb2384c0f8e41eeca8edf4984999a69ba9efe06b8bc5ead17296.jpg", + "image_caption": [ + "Figure 5: Generalization errors. ", + "Figure 6: Estimated density functions with the case (B). " + ], + "image_footnote": [], + "bbox": [ + 181, + 117, + 566, + 286 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 368, + 318, + 385 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "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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", + "bbox": [ + 171, + 861, + 825, + 891 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Supplementary Materials for “Understanding GANs via disconnected Support Detection” ", + "text_level": 1, + "bbox": [ + 215, + 99, + 781, + 137 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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 $\\Omega$ . ", + "bbox": [ + 176, + 147, + 823, + 191 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A SOME ADDITIONAL INFORMATION ", + "text_level": 1, + "bbox": [ + 174, + 212, + 500, + 228 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A Rigorous Definition of $S _ { \\alpha , J }$ ", + "text_level": 1, + "bbox": [ + 174, + 243, + 380, + 258 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 265, + 825, + 308 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/18d13b9c5a1ed09135ab06b369a668b7c550abd7b74685b4d524b947b2a0bbb6.jpg", + "text": "$$\n\\Psi _ { h } = \\Psi ( x _ { 1 } , \\ldots , x _ { d - 1 } , x _ { d } \\pm h ( x _ { 1 } , \\ldots , x _ { d - 1 } , x _ { d + 1 } , \\ldots , x _ { D } ) , x _ { d + 1 } , . . . , x _ { D } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 246, + 314, + 748, + 332 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 176, + 338, + 821, + 367 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/5b93b4c4bcb78e0e0035e3fda4777f33dd4aedcca7443a3b23f69650a466ea48.jpg", + "text": "$$\nS _ { \\alpha , J } : = \\left\\{ S \\subset [ 0 , 1 ] ^ { D } \\mid { \\cal I } _ { S } = \\Psi _ { h _ { 1 } } \\otimes \\cdot \\cdot \\cdot \\otimes \\Psi _ { h _ { J } } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 375, + 665, + 395 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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 $\\Psi _ { h _ { 1 } } , . . . , \\Psi _ { h _ { J } }$ . ", + "bbox": [ + 173, + 402, + 821, + 433 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A Support of Probability Measures ", + "text_level": 1, + "bbox": [ + 174, + 438, + 419, + 453 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 458, + 823, + 488 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/d5877d293783b1b4ac1b59e5434a024795a56da044f05101f960fa2812038622.jpg", + "text": "$$\n\\mathrm { S u p p } ( P ) : = \\bigg \\{ x \\in I ^ { D } \\ | \\ P ( N _ { x } ) > 0 , \\forall N _ { x } \\in \\Sigma \\bigg \\} .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 492, + 660, + 527 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B PROOFS ", + "text_level": 1, + "bbox": [ + 174, + 542, + 276, + 559 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B.1 PROOF OF LEMMA 1 ", + "text_level": 1, + "bbox": [ + 174, + 574, + 357, + 590 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 176, + 598, + 821, + 617 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/a295eb6d8e6cfd958247c4b22dd94c3131e6e1c3f18de4163c37d6d41fa8d632.jpg", + "text": "$$\nP _ { X } ( B ) = P _ { Z } ( \\gamma _ { m } ^ { - 1 } ( B ) ) = \\int _ { \\gamma _ { m } ^ { - 1 } ( B ) } p _ { Z } ( z ) d z ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 622, + 647, + 659 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where $p _ { Z }$ is a density function of a uniform measure $P _ { Z }$ . By changing variables $x = \\gamma _ { m } ( z )$ , we have ", + "bbox": [ + 171, + 665, + 823, + 681 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/e8d8f30d672cb8d3b343149ba370671f7151204ba587d3f40fbbbb2296396ce6.jpg", + "text": "$$\n\\int _ { \\gamma _ { m } ^ { - 1 } ( B ) } p _ { Z } ( z ) d z = \\int _ { B } p _ { Z } ( \\gamma _ { m } ^ { - 1 } ( x ) ) J _ { \\gamma _ { m } } ( x ) d x ,\n$$", + "text_format": "latex", + "bbox": [ + 341, + 686, + 651, + 723 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 178, + 729, + 821, + 763 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/bdbd742866a02c89fdb99d4fdeab787de295196430695486a8d750e1b8aa4e04.jpg", + "text": "$$\nP _ { X } ( B ) = \\int _ { B } J _ { g _ { m } } ( x ) d x = : \\int _ { B } p _ { m } ( x ) d x ,\n$$", + "text_format": "latex", + "bbox": [ + 361, + 770, + 635, + 804 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "and $p _ { m }$ is $\\beta$ -smooth since $\\gamma _ { m }$ is $\\beta + 1$ -smooth and bijective. ", + "bbox": [ + 173, + 809, + 571, + 825 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B.2 PROOF OF LEMMA 2 ", + "text_level": 1, + "bbox": [ + 174, + 840, + 359, + 856 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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. □ ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 171, + 128, + 825, + 159 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 165, + 826, + 212 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/05f0697cf684236905416453966f6d511efecaa191a0fc8f5ee0046c182c3c5a.jpg", + "text": "$$\n\\widetilde { \\boldsymbol { g } } _ { m } ( \\boldsymbol { x } ) = ( \\gamma _ { m , 1 } ( \\boldsymbol { x } ) , . . . , \\gamma _ { m , D } ( \\boldsymbol { x } ) ) ^ { \\top } ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 213, + 612, + 232 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $\\gamma _ { m , d } \\in H ^ { \\beta } ( I ^ { D } )$ . Then, we obtain the form of $g ^ { * }$ as ", + "bbox": [ + 171, + 234, + 557, + 250 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/01b2b8268e655990e9ef44c14a15fadad47644d28b6dce3c0bd1a6c98e401062.jpg", + "text": "$$\ng ^ { * } = \\sum _ { m \\in [ M ] } \\widetilde { g } _ { m } \\otimes \\pmb { I } _ { \\widetilde { S } _ { m } } .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 252, + 576, + 286 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Also, by the definition of $\\mathcal { S } _ { 2 \\beta , J }$ which contains $\\widetilde { S } _ { m }$ , we obtain the form ", + "bbox": [ + 173, + 290, + 642, + 306 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7d8e1889d32be15346656dc11d6cd68d41113071f36fdbcdda8d16691c2ef324.jpg", + "text": "$$\n{ \\cal I } _ { { \\widetilde { \\cal S } } _ { m } } = \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } ,\n$$", + "text_format": "latex", + "bbox": [ + 433, + 309, + 562, + 340 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "with existing $\\psi _ { h _ { m , j } }$ . Then, we have ", + "bbox": [ + 174, + 342, + 408, + 358 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/391f427f735494794df2b95d0feebbeec2bb91f6cbf66d580748a3b604ae9631.jpg", + "text": "$$\ng ^ { * } = \\sum _ { m \\in [ M ] } \\widetilde { g } _ { m } \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 359, + 591, + 395 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "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)). ", + "bbox": [ + 173, + 402, + 826, + 479 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "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 $\\zeta \\bar { [ \\Theta _ { S , m } ] } ^ { - } = \\zeta [ \\Theta _ { \\times } ] ( \\bar { \\zeta } [ \\bar { \\Theta } _ { h , m , 1 } ] \\bar { ( \\cdot ) } , \\cdot . . . , \\zeta [ \\Theta _ { h , m , 1 } ] \\bar { ( \\cdot ) } )$ . ", + "bbox": [ + 173, + 484, + 825, + 534 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We consider a neural network ", + "bbox": [ + 173, + 537, + 369, + 551 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/5936dd95c637eaaf7181d955347f3c72533939cd09f33a147da0553a7d457560.jpg", + "text": "$$\n\\zeta [ \\Theta _ { d } ] = \\zeta [ \\Theta _ { + } ] ( \\zeta [ \\Theta _ { \\times } ] ( \\zeta [ \\Theta _ { \\gamma , d , 1 } ] ( \\cdot ) , \\zeta [ \\Theta _ { S , 1 } ] ( \\cdot ) ) , . . . , \\zeta [ \\Theta _ { \\times } ] ( \\zeta [ \\Theta _ { \\gamma , d , M } ] ( \\cdot ) , \\zeta [ \\Theta _ { S , M } ] ( \\cdot ) ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 202, + 554, + 792, + 571 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Then, an approximation error is evaluated as ", + "bbox": [ + 176, + 573, + 467, + 587 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/fc1a49e41c4f44640fa37ef07efc66fa7752bde0a694d76458b15e34c9dae3b9.jpg", + "text": "$$\n\\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}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 593, + 761, + 925 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where the last inequality follows the Hölder’s inequality. ", + "bbox": [ + 173, + 102, + 545, + 119 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "About $T _ { 1 , m }$ , there exists a corresponding $\\zeta [ \\Theta _ { \\gamma , d , m } ]$ such that ", + "bbox": [ + 173, + 125, + 578, + 141 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/cf9c02d78c82bf0ec4ea1cb90c186ae5c767f0d912f196c31dd60b47b8cd27fd.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\gamma _ { m , d } - \\zeta [ \\Theta _ { \\gamma , d , m } ] \\| _ { L ^ { 2 } } \\lesssim \\| \\Theta _ { \\gamma , d , m } \\| _ { 1 } ^ { - ( \\beta + 1 ) / D } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 145, + 647, + 166 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 170, + 821, + 200 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/77e994909b4a447157740b340865eb889db89e77490394d2184e59711f2b59bb.jpg", + "text": "$$\nT _ { 1 , m } \\lesssim \\| \\Theta _ { \\gamma , d , m } \\| _ { 1 } ^ { - ( \\beta + 1 ) / D } .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 204, + 591, + 227 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 236, + 823, + 266 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d7454f652b0711a7d3cb44c37c8f4cbc5d5bae5b1cfed4488ab166a10c8da4eb.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\gamma _ { m , d } - \\zeta [ \\Theta _ { \\gamma , d , m } ] \\| _ { L ^ { 2 } } } \\\\ & { \\leqslant \\| \\displaystyle \\bigoplus _ { j \\in [ J ] } \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { \\star } ] ( \\zeta [ \\Theta _ { h , m , 1 } ] ( \\cdot ) , \\ldots , \\zeta [ \\Theta _ { h , m , 1 } ] ( \\cdot ) ) \\| _ { L ^ { 2 } } } \\\\ & { \\leqslant \\| \\displaystyle \\bigoplus _ { j \\in [ J ] } \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { h , m , j } ] \\| _ { L ^ { 2 } } + \\| \\displaystyle \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { \\star } ] ( \\zeta [ \\Theta _ { h , m , 1 } ] ( \\cdot ) , \\ldots , \\zeta [ \\Theta _ { h , m , 1 } ] ( \\cdot ) ) \\| _ { L ^ { 2 } } } \\\\ & { \\leqslant \\displaystyle \\sum _ { j \\in [ J ] } \\prod _ { j \\in [ J ] } \\prod _ { l = \\delta , j } ( | \\psi _ { h _ { m , j } } , | \\| _ { L ^ { 2 } } \\forall [ \\Theta _ { h , m , j } ] \\| _ { L ^ { 2 } } ) \\| \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { h , m , j } ] \\| _ { L ^ { 2 } } + \\epsilon _ { \\times } } \\\\ & { \\leqslant \\displaystyle \\sum _ { j \\in [ J ] } \\| \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { h , m , j } ] \\| _ { L ^ { 2 } } + \\epsilon _ { \\times } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 179, + 270, + 815, + 452 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 457, + 825, + 502 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1d7f4f313626e55c967af4c3f3fb082d5c83258e1185fbdd428b02a5adbf150c.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { h , m , j } ] \\| _ { L ^ { 2 } } \\leqslant \\| \\Theta _ { h , m , j } \\| _ { 1 } ^ { - \\beta / ( D - 1 ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 344, + 506, + 650, + 529 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We apply the boundedness of $\\zeta [ \\Theta _ { h , m , j } ]$ , we have ", + "bbox": [ + 173, + 531, + 500, + 546 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1c462416da948e58732d16b5247713638a980ec75680ea3f0e9c8341a5ee690b.jpg", + "text": "$$\nT _ { 2 , m } \\lesssim \\sum _ { j \\in [ J ] } \\| \\Theta _ { h , m , j } \\| _ { 1 } ^ { - \\beta / ( D - 1 ) } + \\epsilon _ { \\times } .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 551, + 625, + 587 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Combining the bounds for $T _ { 1 , m }$ and $T _ { 2 , m }$ , we bound ", + "bbox": [ + 173, + 598, + 521, + 614 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/17ca5910080208ec8a052c5a4fb1c9a5d97427d8cc3cfb01cef837a1a950a61f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left\\| \\displaystyle \\sum _ { m \\in [ M ] } \\gamma _ { m , d } \\bigotimes \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { d } ] \\right\\| _ { L ^ { 2 } } } \\\\ & { \\leqslant \\displaystyle \\sum _ { m \\in [ M ] } \\| \\Theta _ { \\gamma , d , m } \\| _ { 1 } ^ { - ( \\beta + 1 ) / D } + \\displaystyle \\sum _ { m \\in [ M ] } \\sum _ { j \\in [ J ] } \\| \\Theta _ { h , m , j } \\| _ { 1 } ^ { - \\beta / ( D - 1 ) } + ( M + 1 ) \\epsilon _ { \\times } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 617, + 754, + 705 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 176, + 709, + 825, + 761 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/b8173faebdf3e1ca9bd6e6a523e04439b02b7d46c88eb2527f521f3b06174517.jpg", + "text": "$$\n\\left. \\sum _ { m \\in [ M ] } \\gamma _ { m , d } \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } - \\zeta [ \\Theta _ { d } ] \\right. _ { L ^ { 2 } } \\lesssim M J S ^ { - \\beta / D } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 763, + 671, + 814 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Since $J$ is finite, we obtain the result. ", + "bbox": [ + 174, + 818, + 419, + 833 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.4 PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 174, + 848, + 375, + 864 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By the definition of $\\widehat g$ in (2), the following inequality holds ", + "bbox": [ + 174, + 875, + 562, + 890 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d36f8f5386383d724a3e648a7f9f17852a6da57c033f7262bad182c15ebc8ad9.jpg", + "text": "$$\nd _ { \\mathcal { F } } ( P _ { n } , P _ { \\hat { g } , m } ) = \\operatorname* { s u p } _ { f \\in \\mathcal { F } } ( P _ { n } f - P _ { \\hat { g } , m } f ) \\leqslant d _ { \\mathcal { F } } ( P _ { n } , P _ { g , m } ) = \\operatorname* { s u p } _ { f \\in \\mathcal { F } } ( P _ { n } f - P _ { g , m } f ) ,\n$$", + "text_format": "latex", + "bbox": [ + 238, + 893, + 758, + 921 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "for arbitrary $g \\in { \\mathcal { G } }$ . ", + "bbox": [ + 173, + 103, + 300, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We consider a bound for $d _ { \\mathcal { F } } ( P ^ { \\ast } , P _ { \\hat { g } } )$ as ", + "bbox": [ + 174, + 125, + 437, + 140 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c017ea76d4ab6b949a131df430c696d0b1638127ebbaedaebf3598b63a2cd861.jpg", + "text": "$$\n\\begin{array} { r l } & { d _ { \\mathcal { F } } \\big ( P ^ { * } , P _ { \\hat { g } } \\big ) = \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\big ( P ^ { * } f - P _ { \\hat { g } } f \\big ) } \\\\ & { \\quad \\quad = \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\big ( P ^ { * } f - P _ { n } f + P _ { n } f - P _ { \\hat { g } , m } f + P _ { \\hat { g } , m } f - P _ { \\hat { g } } f \\big ) } \\\\ & { \\quad \\quad \\leqslant \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\big ( P ^ { * } f - P _ { n } f + P _ { \\hat { g } , m } f - P _ { \\hat { g } } f \\big ) + \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\big ( P _ { n } f - P _ { \\hat { g } , m } f \\big ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 145, + 743, + 231 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 171, + 236, + 825, + 265 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/1bb6c2fcfb35b64fa37f1b8f5bf2812682b81056e5cbe6d4a9843aa540c4f8d8.jpg", + "text": "$$\n\\begin{array} { r l } & { d _ { \\mathcal { F } } ( P ^ { * } , P _ { \\widehat { g } } ) } \\\\ & { \\leqslant \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } ( P ^ { * } f - P _ { n } f + P _ { \\widehat { g } , m } f - P _ { \\widehat { g } } f ) + \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } ( P _ { n } f - P ^ { * } f + P ^ { * } f - P _ { \\widehat { g } } f + P _ { \\widehat { g } } f - P _ { \\widehat { g } , m } f ) } \\\\ & { \\leqslant 2 \\underset { g \\in \\mathcal { G } } { \\operatorname* { s u p } } \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } | P _ { g , m } f - P _ { g } f | + \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } ( P ^ { * } f - P _ { \\widehat { g } } f ) + 2 \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } | P _ { n } f - P ^ { * } f | } \\\\ & { = : i + i i + i i i . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 270, + 802, + 366 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Here, $i$ denotes an effect from $m$ samplings, $\\romannumeral 2$ denotes an approximation error, and iii denotes an uncertainty with the $n$ observations. ", + "bbox": [ + 174, + 371, + 823, + 400 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 405, + 823, + 435 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 439, + 825, + 481 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/b70491ed7f2676d24d835ffb7afbbe4935d0c28236950d4bcae1ee4b243db677.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\operatorname* { s u p } _ { h \\in \\mathcal { H } } | \\frac { 1 } { n } \\sum _ { i \\in [ n ] } h ( X _ { i } ) - \\mathbb { E } _ { P } [ h ( X ) ] | } } \\\\ & { } & { \\leqslant \\operatorname* { i n f } _ { \\eta > 0 } \\{ 4 \\eta + 1 2 \\int _ { \\eta } ^ { C _ { h } } \\sqrt { \\frac { \\log \\mathcal { N } ( \\epsilon , \\mathcal { H } , \\| \\cdot \\| _ { L ^ { \\infty } } ) } { n } } d \\epsilon + \\sqrt { \\frac { 2 \\tau \\sigma ^ { 2 } + 4 C _ { \\theta } } { n } } + \\frac { \\tau C _ { h } } { n } ( \\frac { 2 } { 3 } + C _ { \\theta } ) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 486, + 790, + 580 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "with probability at least $1 - 2 \\exp ( - \\tau )$ for all $\\tau > 0$ . ", + "bbox": [ + 173, + 585, + 524, + 601 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 616, + 825, + 659 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "By applying the symmetrization and concentration techniques (Proposition 7.10 in Steinwart & Christmann (2008)), we obtain ", + "bbox": [ + 169, + 665, + 825, + 693 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/1ce850c8246d003b5b70992d14295a920619fbb39e6448d1988ed020cd0dd243.jpg", + "text": "$$\n\\mathbb { E } _ { P ^ { \\otimes n } } \\left[ \\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\left| \\frac { 1 } { n } \\sum _ { i \\in [ n ] } h ( X _ { i } ) - \\mathbb { E } [ h ( X ) ] \\right| \\right] \\leqslant 2 \\mathbb { E } _ { P ^ { \\otimes n } \\otimes \\nu ^ { \\otimes n } } \\left[ \\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\left| \\frac { 1 } { n } \\sum _ { i \\in [ n ] } u _ { i } h ( X _ { i } ) \\right| \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 222, + 698, + 772, + 750 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 176, + 753, + 825, + 797 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ba273dfeccef9a093f017d5f2c47c501e5a22d5329078364c74cdbc63b41aaff.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\left| \\frac { 1 } { n } \\sum _ { i \\in [ n ] } h ( X _ { i } ) - \\mathbb { E } _ { P } [ h ( X ) ] \\right| } \\\\ & { \\leqslant ( 1 + \\theta ) \\mathbb { E } _ { P ^ { \\otimes n } \\otimes \\nu ^ { \\otimes n } } \\left[ \\displaystyle \\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\left| \\frac { 1 } { n } \\sum _ { i \\in [ n ] } u _ { i } h ( X _ { i } ) \\right| \\right] + \\sqrt { \\frac { 2 \\tau \\sigma ^ { 2 } } { n } } + \\frac { \\tau C _ { h } } { n } \\left( \\frac { 2 } { 3 } + \\displaystyle \\frac { 1 } { \\theta } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 801, + 758, + 905 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "with probability at least $1 - \\exp ( - \\tau )$ for all $\\tau > 0$ and $\\theta > 0$ . ", + "bbox": [ + 173, + 909, + 581, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "About the term with the Rademacher variable, we also apply a similar strategy (Lemma A.4 in Bartlett et al. (2005)), then obtain ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d2a1a5fb9571b710f684a9df06c55cda3ac83623c2c7782d7816f1b6b7490a8a.jpg", + "text": "$$\n\\mathbb { E } _ { P \\hat { \\otimes } n } \\otimes _ { \\mathcal { V } } \\otimes n ^ { \\prime } \\left[ \\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\left| \\frac { 1 } { n } \\sum _ { i \\in [ n ] } u _ { i } h ( X _ { i } ) \\right| \\right] \\leqslant \\frac { 1 } { 1 - \\theta ^ { \\prime } } \\mathbb { E } _ { \\nu } \\otimes n \\left[ \\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\left| \\frac { 1 } { n } \\sum _ { i \\in [ n ] } u _ { i } h ( X _ { i } ) \\right| \\right] + \\frac { \\tau ^ { \\prime } C _ { h } } { n \\theta ^ { \\prime } ( 1 - \\theta ^ { \\prime } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 189, + 137, + 807, + 189 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "with probability at least $1 - \\exp ( - \\tau ^ { \\prime } )$ for all $\\tau ^ { \\prime } > 0$ and $\\theta ^ { \\prime } > 0$ . ", + "bbox": [ + 173, + 208, + 596, + 226 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 229, + 825, + 273 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/45ae6f45ab663afc8436f73d5e2a8897f7c1309dd82843745b2fd195bfc014b1.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\boldsymbol \\nu \\otimes \\boldsymbol n } \\left[ \\displaystyle \\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\left| \\frac { 1 } { n } \\sum _ { i \\in [ n ] } u _ { i } h ( X _ { i } ) \\right| \\right] \\leqslant \\displaystyle \\operatorname* { i n f } _ { \\boldsymbol \\theta ^ { \\prime \\prime } > 0 } \\left\\{ 4 \\boldsymbol \\theta ^ { \\prime \\prime } + 1 2 \\int _ { \\boldsymbol \\theta ^ { \\prime \\prime } } ^ { \\tilde { C } _ { h } } \\sqrt { \\frac { \\log \\mathcal { N } ( \\epsilon , \\mathcal { H } , \\| \\cdot \\| _ { n } ) } { n } } d \\epsilon \\right\\} } \\\\ { \\leqslant \\displaystyle \\operatorname* { i n f } _ { \\boldsymbol \\theta ^ { \\prime \\prime } > 0 } \\left\\{ 4 \\boldsymbol \\theta ^ { \\prime \\prime } + 1 2 \\int _ { \\boldsymbol \\theta ^ { \\prime \\prime } } ^ { C _ { h } } \\sqrt { \\frac { \\log \\mathcal { N } ( \\epsilon , \\mathcal { H } , \\| \\cdot \\| _ { L ^ { \\infty } } ) } { n } } d \\epsilon \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 279, + 764, + 373 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the last inequality follows a bound for an empirical norm and the boundedness of $\\mathcal { H }$ . ", + "bbox": [ + 171, + 378, + 771, + 393 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Combining (10), (11) and (12) and changing variables, we obtain the result. ", + "bbox": [ + 174, + 400, + 671, + 415 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "To bound $I$ with $\\widetilde { X } _ { 1 } , . . . , \\widetilde { X } _ { m } \\sim P _ { g }$ , we consider the following value ", + "bbox": [ + 173, + 429, + 619, + 449 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/1a10f31982fe724de76bb6fbe9ec64ef90d09ab59f0ad4a365e3a2424a18b029.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { g \\in \\mathcal { G } } { \\operatorname* { s u p } } \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\left| \\frac { 1 } { m } \\sum _ { i \\in [ m ] } f ( \\widetilde { X } _ { i } ) - \\mathbb { E } _ { P _ { g } } [ h ( X ) ] \\right| = \\underset { g \\in \\mathcal { G } } { \\operatorname* { s u p } } \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\left| \\frac { 1 } { m } \\sum _ { i \\in [ m ] } f \\circ g ( Z _ { i } ) - \\mathbb { E } _ { P _ { \\mathcal { Z } } } [ f \\circ g ( X ) ] \\right| } \\\\ & { \\qquad = : \\underset { h \\in \\mathcal { H } } { \\operatorname* { s u p } } \\left| \\frac { 1 } { m } \\sum _ { i \\in [ m ] } h ( Z _ { i } ) - \\mathbb { E } _ { P _ { \\mathcal { Z } } } [ h ( X ) ] \\right| , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 454, + 794, + 556 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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 $\\mathcal { N } ( \\epsilon , \\mathcal { H } , \\| \\cdot \\| _ { L ^ { \\infty } } )$ . ", + "bbox": [ + 173, + 560, + 823, + 592 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Lemma 5. Assume that $f \\in { \\mathcal { F } }$ is $L _ { 1 }$ -Lipschitz continuous. We obtain ", + "bbox": [ + 173, + 594, + 632, + 609 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/73c61559df1947ed3a944d7011c2ccba86211047d09e844a52dd7b5301606ee7.jpg", + "text": "$$\n\\log \\mathcal { N } ( \\epsilon , \\mathcal { H } , \\| \\cdot \\| _ { L ^ { \\infty } } ) \\leqslant \\log \\mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \\epsilon , \\mathcal { G } , \\| \\cdot \\| _ { L ^ { \\infty } } ) + \\log \\mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \\epsilon , \\mathcal { F } , \\| \\cdot \\| _ { L ^ { \\infty } } )\n$$", + "text_format": "latex", + "bbox": [ + 200, + 614, + 794, + 633 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 174, + 647, + 823, + 678 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/8a74c5daf2b3435f74bff3ae49944a3274c2c9c692b86f3dc16635403273f512.jpg", + "text": "$$\nH : = \\left\\{ h = f \\circ g \\vert g \\in G , h \\in F \\right\\} \\subset \\mathcal { H } ,\n$$", + "text_format": "latex", + "bbox": [ + 362, + 683, + 633, + 718 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "and we known $| H | = | G | \\times | F |$ . ", + "bbox": [ + 174, + 724, + 388, + 741 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 174, + 746, + 826, + 789 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/b79659cf5ccb7b50f7cde03b49e7679a36c1782e9b37ff25668f6e4bb05e83d3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| h - h ^ { \\prime } \\| _ { L ^ { \\infty } } \\leqslant \\| f \\circ g - f ^ { \\prime } \\circ g ^ { \\prime } \\| _ { L ^ { \\infty } } } \\\\ & { \\qquad = \\| f \\circ g - f \\circ g ^ { \\prime } + f \\circ g ^ { \\prime } - f ^ { \\prime } \\circ g ^ { \\prime } \\| _ { L ^ { \\infty } } } \\\\ & { \\qquad \\leqslant \\| f \\circ g - f \\circ g ^ { \\prime } \\| _ { L ^ { \\infty } } + \\| f \\circ g ^ { \\prime } - f ^ { \\prime } \\circ g ^ { \\prime } \\| _ { L ^ { \\infty } } } \\\\ & { \\qquad \\leqslant L _ { 1 } \\| g - g ^ { \\prime } \\| _ { L ^ { \\infty } } + \\| f - f ^ { \\prime } \\| _ { L ^ { \\infty } } } \\\\ & { \\qquad \\leqslant ( L _ { 1 } + 1 ) \\epsilon . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 794, + 689, + 891 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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. □ ", + "bbox": [ + 171, + 895, + 823, + 925 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Now, we have the following entropy bound ", + "bbox": [ + 174, + 103, + 457, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/314b08165b4d70c54db8c9640ce17ce8ce7dfb898fe871033f7ea4706feecd55.jpg", + "text": "$$\n\\begin{array} { r l } & { \\log \\mathcal { N } ( \\epsilon , \\mathcal { H } , \\| \\cdot \\| _ { L ^ { \\infty } } ) } \\\\ & { \\leqslant \\log { \\mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \\epsilon , \\mathcal { G } , \\| \\cdot \\| _ { L ^ { \\infty } } ) } + \\log \\mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \\epsilon , \\mathcal { F } , \\| \\cdot \\| _ { L ^ { \\infty } } ) } \\\\ & { \\leqslant ( S _ { g } + 1 ) \\log ( 2 ( L _ { 1 } + 1 ) \\epsilon ^ { - 1 } ( L _ { g } + 1 ) D _ { g } B _ { g } ) } \\\\ & { \\quad + \\operatorname* { m i n } \\Biggl \\{ ( S _ { f } + 1 ) \\log ( 2 ( L _ { 1 } + 1 ) \\epsilon ^ { - 1 } ( L _ { f } + 1 ) D _ { f } B _ { f } ) , C _ { \\kappa } ( 1 + L _ { 1 } ) \\epsilon ^ { - \\kappa } \\Biggr \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 121, + 746, + 217 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 178, + 218, + 823, + 268 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/c6a35d871a7ef2e5880ef4c018bfe3ccc8f73dba709f276734969c8158edf8ae.jpg", + "text": "$$\n\\begin{array} { r l r } { { 2 \\operatorname* { s u p } _ { j \\in \\mathcal { T } } \\big | P _ { n } f - P ^ { * } f \\big | } } \\\\ & { \\leqslant 4 \\eta + \\frac { 1 2 } { n ^ { 1 / 2 } } \\int _ { \\eta } ^ { \\infty } \\operatorname* { m i n } \\Big \\{ N _ { c } ( \\widetilde { \\mathcal { F } } ) , ( S _ { f } + 1 ) \\log ( \\Gamma _ { f } \\epsilon ^ { - 1 } ) \\Big \\} ^ { 1 / 2 } d \\epsilon } \\\\ & { } & { + \\frac { ( 2 \\tau \\sigma ^ { 2 } + C _ { \\theta } ) ^ { 1 / 2 } } { n ^ { 1 / 2 } } + \\frac { \\tau C _ { h } ( 2 / 3 + C _ { \\theta } ) } { n } } \\\\ & { \\leqslant 4 \\eta + \\frac { 1 2 } { n ^ { 1 / 2 } } \\int _ { \\eta } ^ { \\infty } N _ { c } ( \\widetilde { \\mathcal { F } } ) ^ { 1 / 2 } d \\epsilon + \\frac { 1 2 } { n ^ { 1 / 2 } } ( S _ { f } + 1 ) ^ { 1 / 2 } ( \\log ^ { 1 / 2 } \\Gamma _ { f } + C _ { \\widetilde { \\mathcal { F } } } \\log ^ { 1 / 2 } C _ { \\widetilde { \\mathcal { F } } } - \\eta \\log ^ { 1 / 2 } \\eta ) } \\\\ & { } & { + \\frac { ( 2 7 \\sigma ^ { 2 } + C _ { \\theta } ) ^ { 1 / 2 } } { n ^ { 1 / 2 } } + \\frac { \\tau C _ { h } ( 2 / 3 + C _ { \\theta } ) } { n } } \\\\ & { \\leqslant \\Upsilon _ { n } ( \\widetilde { \\mathcal { F } } ) + \\frac { 1 } { n ^ { 1 / 2 } } ( ( S _ { f } + 1 ) ^ { 1 / 2 } A _ { 1 } + A _ { 2 } ) + \\frac { A _ { 3 } } { n } , \\qquad \\mathrm { ~ o ~ } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 273, + 805, + 472 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 478, + 825, + 517 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/c0c27879c94e4bf24392d0c1926fc20f7689d5fba1da7188e0521300a4de67e5.jpg", + "text": "$$\ni i i \\leqslant \\Upsilon _ { n } ( \\widetilde { \\mathcal { F } } ) + \\frac { 1 } { n ^ { 1 / 2 } } \\left( ( S _ { f } + 1 ) ^ { 1 / 2 } A _ { 1 } + A _ { 2 } \\right) + \\frac { A _ { 3 } } { n } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 521, + 673, + 551 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "About $I$ , we define $\\Gamma _ { g } : = 2 ( L _ { f } + 1 ) D _ { g } B _ { g }$ and obtain a similar bound as ", + "bbox": [ + 174, + 561, + 655, + 579 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d37a9814e431549b5d0424f6689ca2ea0f7491e0b17d45aa964d3cd02017f6a8.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { i \\leqslant \\Upsilon _ { m } ( \\widetilde { \\mathcal { F } } ) + \\displaystyle \\frac { 1 } { m ^ { 1 / 2 } } \\left( ( S _ { g } + 1 ) ^ { 1 / 2 } A _ { 1 } ^ { \\prime } + A _ { 2 } ^ { \\prime } \\right) + \\displaystyle \\frac { A _ { 3 } ^ { \\prime } } { m } , } \\\\ & { } & { A _ { 1 } ^ { \\prime } = \\smash { 1 2 ( \\log ^ { 1 / 2 } \\Gamma _ { f } + \\log ^ { 1 / 2 } \\Gamma _ { g } ) } , A _ { 2 } = C _ { \\widetilde { \\mathcal { F } } } \\log ^ { 1 / 2 } C _ { \\widetilde { \\mathcal { F } } } + C _ { \\mathcal { G } } \\log ^ { 1 / 2 } C _ { \\mathcal { G } } + 2 ( 2 \\tau \\sigma ^ { 2 } + C _ { \\theta } ) ^ { 1 / 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 582, + 826, + 638 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where and $A _ { 3 } \\stackrel { - } { = } 2 \\tau C _ { h } ( 2 / 3 + \\check { C } _ { \\theta } )$ . ", + "bbox": [ + 174, + 617, + 823, + 648 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 655, + 823, + 684 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/87a1840845c38c2a8411f5cf730d1ca08ac2e938aa23bbbf3af0fde6ddecba99.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { i } i = P _ { g } \\ast f - P _ { g } f } \\\\ & { = \\displaystyle \\int \\big ( f o { g ^ { \\ast } } - f o \\bar { g } i \\big ) d P _ { Z } } \\\\ & { \\lesssim L \\displaystyle \\int \\big | \\displaystyle { g ^ { \\ast } } - \\bar { g } \\big | | d P _ { Z } } \\\\ & { \\lesssim L _ { 1 } \\left( \\displaystyle \\sum _ { i \\neq [ D ] } \\left\\lceil g _ { i } ^ { \\ast } ( x ) - \\bar { g } _ { d } ( x ) \\right\\rceil ^ { 2 } d P _ { Z } ( x ) \\right) ^ { 1 / 2 } } \\\\ & { - L _ { 1 } \\left( \\displaystyle \\sum _ { i \\neq [ D ] } \\| g _ { i } ^ { \\ast } - \\bar { g } _ { d } \\| _ { L ^ { 2 } } ^ { 2 } \\right) ^ { 1 / 2 } } \\\\ & { \\lesssim \\epsilon _ { g , L } \\displaystyle L _ { 1 } D M _ { S } e ^ { \\frac { 1 } { g } / D } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 707, + 656, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 169, + 103, + 826, + 132 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/f8c772a0ec8f554eb9da6fd8517f95537f30a8ed0b94fa5c49f046e5fc78d9ff.jpg", + "text": "$$\n\\begin{array} { r } { i i \\leqslant c _ { 2 } M D S _ { g } ^ { - \\beta / D } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 433, + 138, + 565, + 160 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Combining the result, we obtain the result of Theorem 1. ", + "bbox": [ + 173, + 171, + 547, + 188 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.5 PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 174, + 203, + 398, + 218 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 229, + 825, + 287 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 173, + 294, + 825, + 342 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/952c2c83ffeeba44a2c16663872e5c111863ef867e65db8ba4a7bd63cb194e57.jpg", + "text": "$$\ng _ { 2 } ( z ) = ( g _ { 2 , 1 } ( z _ { 1 } ) , . . . , g _ { 2 , D } ( z _ { D } ) ) ^ { \\top } ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 348, + 612, + 368 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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 $\\beta = 1$ . ", + "bbox": [ + 173, + 375, + 826, + 449 + ], + "page_idx": 17 + } +] \ No newline at end of file diff --git a/parse/train/B1x5KiCcFX/B1x5KiCcFX_middle.json b/parse/train/B1x5KiCcFX/B1x5KiCcFX_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..44f0c4d78d6a60cbb74d2a0b7434d0beae3e5470 --- /dev/null +++ b/parse/train/B1x5KiCcFX/B1x5KiCcFX_middle.json @@ -0,0 +1,62436 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "score": 1.0, + "content": "UNDERSTANDING GANS VIA GENERALIZATION", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 412, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 412, + 118 + ], + "score": 1.0, + "content": "ANALYSIS FOR DISCONNECTED SUPPORT", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 209, + 468, + 363 + ], + "lines": [ + { + "bbox": [ + 142, + 209, + 470, + 222 + ], + "spans": [ + { + "bbox": [ + 142, + 209, + 470, + 222 + ], + "score": 1.0, + "content": "This paper provides theoretical analysis of generative adversarial networks (GANs)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 220, + 470, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 220, + 470, + 234 + ], + "score": 1.0, + "content": "to explain its advantages over other standard methods of learning probability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 231, + 469, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 469, + 245 + ], + "score": 1.0, + "content": "measures. 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It is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "score": 1.0, + "content": "theoretically shown that, unlike other popular models, GANs do not suffer from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 470, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 470, + 321 + ], + "score": 1.0, + "content": "the decrease of generalization performance caused by the disconnected support", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 470, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 470, + 331 + ], + "score": 1.0, + "content": "property. We rigorously quantify the generalization performance of GANs of a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 469, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 469, + 342 + ], + "score": 1.0, + "content": "given architecture, and compare it with the performance of the other models. Based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 342, + 469, + 353 + ], + "spans": [ + { + "bbox": [ + 142, + 342, + 469, + 353 + ], + "score": 1.0, + "content": "on the theory, we also provide a guideline for selecting deep network architecture", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 351, + 466, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 351, + 466, + 365 + ], + "score": 1.0, + "content": "for GANs. We demonstrate some numerical examples which support our results.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5, + "bbox_fs": [ + 141, + 209, + 470, + 365 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 383, + 206, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 208, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 208, + 397 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) attract much attention as tech-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "nology for learning a distribution and generating data. The purpose of GANs is to learn a probability", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "measure from a given dataset and generate samples from the learned measure. It is often seen that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 451 + ], + "score": 1.0, + "content": "samples generated by GANs can extract effectively features in the real world; it is difficult, for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "instance, to distinguish real images and generated images. By practical successes, a countless number", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "of variations of GANs have been developed (Dziugaite et al., 2015; Arjovsky et al., 2017; Li et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "2015; Nowozin et al., 2016; Gulrajani et al., 2017; Zhao et al., 2016), and applied to a wide range of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 483, + 341, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 341, + 496 + ], + "score": 1.0, + "content": "tasks (Reed et al., 2016; Zhu et al., 2017; Gauthier, 2014).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 406, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Understanding the remarkable performance of GANs is, however, still a challenging problem. There", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "are active discussions on the role of generators in its learning scheme (Goodfellow, 2016; Arjovsky &", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "Bottou, 2017; Arora et al., 2018; Creswell et al., 2018), and adversarial structures with discriminators", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "are also a target of interest (Lotter et al., 2015; Zhang et al., 2018). A gaming structure between gen-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "erators and discriminators is also regarded as a useful factor in the mechanism of GANs (Mescheder", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "et al., 2017; Arora et al., 2017; Heusel et al., 2017). Generalization performance of GANs has been", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "investigated in several studies (Liang, 2017; Liu et al., 2017; Tolstikhin et al., 2017). In spite of these", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "studies, it is not yet clear why GANs can generate well-extracted data better than other standard", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 145, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 145, + 600 + ], + "score": 1.0, + "content": "methods.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 500, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "This paper introduces the disconnected support property, and explains an advantage of GANs in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "connection with this notion. The disconnected support property refers to a probability measures of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "which the support is divided into several disjoint sets, allowing non-differentiable density on the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "boundary. The property makes a probability measure be complex, hence it can be an obstacle for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "standard methods to learn the measure effectively. This property, however, is popularly seen in many", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 416, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 416, + 672 + ], + "score": 1.0, + "content": "real data, especially data with cluster structure, as demonstrated in Section 3.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 604, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "We investigate in detail the approximation and estimation ability of GANs and some other methods,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "and provide novel generalization analysis of probability measures with disconnected support. Firstly,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "we show that the other methods suffer worse generalization performance due to complex structures", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "of disconnected supports (Proposition 1 and Lemma 2). Secondly, our generalization analysis reveals", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "score": 1.0, + "content": "that GANs can learn the probability measure without loss of efficiency under the the disconnected", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "supports (Theorem 1, Corollary 1 and ??). Additionally, we derive a guideline for choosing the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "number of layers or connections of the generator and discriminator from the generalization analysis.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 309, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 309, + 118 + ], + "score": 1.0, + "content": "Numerical results support our theoretical findings.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 677, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "supports (Theorem 1, Corollary 1 and ??). Additionally, we derive a guideline for choosing the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "number of layers or connections of the generator and discriminator from the generalization analysis.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 309, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 309, + 118 + ], + "score": 1.0, + "content": "Numerical results support our theoretical findings.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "We remark that the disconnected support property is different from the low-dimensional supports", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "studied in Arjovsky & Bottou (2017), where the support of a measure generated by neural networks is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "disjoint to the measure of observations. In contrast, this paper considers the case in which the support", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "of the observation measure is divided into disjoint subsets. The problem of disconnected supports", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "is complement to the low-dimensionality, hence these two problems can be investigated separately.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "In this paper, to simplify the discussion, we assume that the support of a probability measure is not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 178, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 178, + 199 + ], + "score": 1.0, + "content": "low-dimensional.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 342, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 343, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 343, + 217 + ], + "score": 1.0, + "content": "The contributions of this paper are summarized as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 129, + 225, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 129, + 223, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 129, + 223, + 505, + 239 + ], + "score": 1.0, + "content": "1. We show that GANs perform better than other standard methods of estimating probability", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 236, + 430, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 236, + 430, + 249 + ], + "score": 1.0, + "content": "measures when the measure satisfies the disconnected support property.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 130, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 130, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "2. We provide a new generalization error bound under a general formulation of GANs by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "analyzing an approximation error. The result is thus applicable to a wide range of variations", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 269, + 183, + 280 + ], + "spans": [ + { + "bbox": [ + 142, + 269, + 183, + 280 + ], + "score": 1.0, + "content": "of GANs.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 128, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 128, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "3. Based on the generalization bound, we provide a theoretical guideline for selecting architec-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 291, + 299, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 299, + 303 + ], + "score": 1.0, + "content": "tures of generators and discriminators.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 316, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 317, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 317, + 325 + ], + "score": 1.0, + "content": "All the proofs are given in Supplementary materials.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 339, + 208, + 352 + ], + "lines": [ + { + "bbox": [ + 104, + 337, + 210, + 355 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 210, + 355 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 107, + 364, + 177, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 178, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 178, + 377 + ], + "score": 1.0, + "content": "2.1 NOTATION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 104, + 383, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 177, + 398 + ], + "score": 1.0, + "content": "We use notations", + "type": "text" + }, + { + "bbox": [ + 177, + 385, + 221, + 397 + ], + "score": 0.92, + "content": "I : = [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 383, + 235, + 398 + ], + "score": 1.0, + "content": ". 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Here,", + "type": "text" + }, + { + "bbox": [ + 364, + 560, + 372, + 570 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "bounds the number of non-zero", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 391, + 583 + ], + "score": 1.0, + "content": "parameters of DNNs, namely, it controls the sparseness of DNNs.", + "type": "text" + }, + { + "bbox": [ + 391, + 572, + 401, + 581 + ], + "score": 0.83, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "is a bound for scales of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 583, + 155, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 155, + 594 + ], + "score": 1.0, + "content": "parameters.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 560, + 506, + 594 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 609, + 315, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 608, + 317, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 317, + 624 + ], + "score": 1.0, + "content": "3 DISCONNECTED SUPPORT PROPERTY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 108, + 634, + 264, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 265, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 265, + 647 + ], + "score": 1.0, + "content": "3.1 INTRODUCTION AND EXAMPLE", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "It is often observed that data in real world data the support of its probability measure may not be", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "connected but a union of disjoint subsets. This is typical if the data has cluster structures, as seen in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "many data sets for classification tasks. Moreover, the density function of the probability measure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "may not be smooth at a boundary of the support. Figures 1 (MNIST, (LeCun et al., 1998)) and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "2 (Shelter Animal, Center) illustrate such examples in the real world. They are projected onto a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "2-dimensional Euclidean space by t-SNE (Maaten & Hinton, 2008) so that they preserve the original", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "distance structure among points. We can see that both of the data are concentrated on several disjoint", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "subsets and there are a clear gap or empty regions between some of the subsets. This observation", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "suggests that the disconnected property of probability measures should be addressed in discussing", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "estimation of probability measures, while standard analysis does not consider this phenomenon. In", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "fact, this paper will show that the disconnected supports property has an important role in showing an", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 326, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 326, + 138 + ], + "score": 1.0, + "content": "advantage of GANs over standard estimation methods.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 654, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "subsets and there are a clear gap or empty regions between some of the subsets. This observation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "suggests that the disconnected property of probability measures should be addressed in discussing", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "estimation of probability measures, while standard analysis does not consider this phenomenon. 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We consider popular nonparametric methods, for which", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 473, + 482 + ], + "score": 1.0, + "content": "the generalization performance is well-studied in the asymptotics of the observation size", + "type": "text" + }, + { + "bbox": [ + 474, + 471, + 480, + 479 + ], + "score": 0.63, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 470, + 505, + 482 + ], + "score": 1.0, + "content": ". 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(Deterioration of other standard methods)", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 695, + 447, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 156, + 709 + ], + "score": 1.0, + "content": "There exists", + "type": "text" + }, + { + "bbox": [ + 156, + 696, + 170, + 706 + ], + "score": 0.86, + "content": "P ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 695, + 367, + 709 + ], + "score": 1.0, + "content": "of the disconnected support property and locally", + "type": "text" + }, + { + "bbox": [ + 368, + 699, + 374, + 708 + ], + "score": 0.89, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 695, + 447, + 709 + ], + "score": 1.0, + "content": "-smooth such that", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 685, + 447, + 709 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 714, + 369, + 735 + ], + "lines": [ + { + "bbox": [ + 242, + 714, + 369, + 735 + ], + "spans": [ + { + "bbox": [ + 242, + 714, + 369, + 735 + ], + "score": 0.93, + "content": "d _ { 2 } ( P ^ { * } , \\widehat { P } ) = { \\cal O } \\left( n ^ { - 1 / ( 2 + D ) } \\right) .", + "type": "interline_equation", + "image_path": "b9730f5b9fd0d563aeb76358ab989af6dc4cf556065c4cfb43470574c4e23436.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 242, + 714, + 369, + 735 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 132, + 96 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 83, + 146, + 93 + ], + "score": 0.87, + "content": "P ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "has disconnected supports, the errors are worse than those for the global support, indepen-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 136, + 106 + ], + "score": 1.0, + "content": "dent of", + "type": "text" + }, + { + "bbox": [ + 137, + 94, + 144, + 105 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ". 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Intuitively, to make", + "type": "text" + }, + { + "bbox": [ + 291, + 222, + 343, + 235 + ], + "score": 0.92, + "content": "P _ { g ^ { * } } ( B ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 221, + 358, + 236 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 358, + 221, + 390, + 232 + ], + "score": 0.91, + "content": "B \\in I ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 221, + 412, + 236 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 412, + 222, + 453, + 234 + ], + "score": 0.93, + "content": "\\lambda ( B ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 221, + 506, + 236 + ], + "score": 1.0, + "content": ", the slope of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 118, + 246 + ], + "score": 0.87, + "content": "g ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 233, + 129, + 249 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 129, + 234, + 158, + 245 + ], + "score": 0.91, + "content": "z \\in I ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 233, + 278, + 249 + ], + "score": 1.0, + "content": "should be close to infinite for", + "type": "text" + }, + { + "bbox": [ + 279, + 235, + 334, + 247 + ], + "score": 0.94, + "content": "z \\in g ^ { * , - 1 } ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 233, + 363, + 249 + ], + "score": 1.0, + "content": ", hence", + "type": "text" + }, + { + "bbox": [ + 363, + 235, + 375, + 246 + ], + "score": 0.89, + "content": "g ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 233, + 506, + 249 + ], + "score": 1.0, + "content": "cannot be uniformly continuous", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "(see Figure 4). Because of the discontinuity, generative models with smooth functions, such as an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "score": 1.0, + "content": "adversarial generative model with kernel generators (Sinn & Rawat, 2018), cannot work well with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 200, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 200, + 281 + ], + "score": 1.0, + "content": "disconnected supports.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 294, + 272, + 308 + ], + "lines": [ + { + "bbox": [ + 104, + 294, + 274, + 309 + ], + "spans": [ + { + "bbox": [ + 104, + 294, + 274, + 309 + ], + "score": 1.0, + "content": "4 GENERALIZATION BY GANS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 506, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "We provide generalization analysis for GANs for probability measures with and without disconnected", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 305, + 343 + ], + "score": 1.0, + "content": "supports. 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} _ { 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 } .", + "type": "interline_equation", + "image_path": "93deb4439d4b2492a38423a6f0889a0166b5890088440a7df3df0970d525765a.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 122, + 294, + 489, + 306.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 122, + 306.3333333333333, + 489, + 318.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 122, + 318.66666666666663, + 489, + 330.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 507, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 146, + 347 + ], + "score": 1.0, + "content": "To obtain", + "type": "text" + }, + { + "bbox": [ + 147, + 334, + 171, + 345 + ], + "score": 0.9, + "content": "i \\leqslant I", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 333, + 188, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 189, + 334, + 229, + 345 + ], + "score": 0.9, + "content": "i i i \\leqslant I I I", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 333, + 507, + 347 + ], + "score": 1.0, + "content": ", we apply an empirical process technique (van der Vaart & Wellner,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "1996), especially convergence of integral probability measures (Sriperumbudur et al., 2012) and the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 453, + 369 + ], + "score": 1.0, + "content": "entropy control technique (Lemma 4 and 5 in the supplementary material). To show", + "type": "text" + }, + { + "bbox": [ + 453, + 356, + 487, + 366 + ], + "score": 0.9, + "content": "i i \\leqslant I I", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 356, + 506, + 369 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "employ recent results on approximation ability of DNNs (Yarotsky, 2017; Petersen & Voigtlaender,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 377, + 507, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 407, + 391 + ], + "score": 1.0, + "content": "2017; Imaizumi & Fukumizu, 2018) and obtain an approximation bound for", + "type": "text" + }, + { + "bbox": [ + 407, + 378, + 456, + 390 + ], + "score": 0.93, + "content": "d _ { \\mathcal { F } } ( P _ { g } , P ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 377, + 507, + 391 + ], + "score": 1.0, + "content": "(Lemma 3).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 365, + 401 + ], + "score": 1.0, + "content": "Combining these results, we obtain the statement of Theorem 1.", + "type": "text" + }, + { + "bbox": [ + 494, + 389, + 506, + 400 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 336, + 419 + ], + "score": 1.0, + "content": "Theorem 1 provides two trade-off relations with respect to", + "type": "text" + }, + { + "bbox": [ + 337, + 406, + 349, + 418 + ], + "score": 0.9, + "content": "S _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 406, + 366, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 367, + 406, + 379, + 418 + ], + "score": 0.88, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 406, + 462, + 419 + ], + "score": 1.0, + "content": ". The generator class", + "type": "text" + }, + { + "bbox": [ + 462, + 406, + 470, + 416 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "controls", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 346, + 430 + ], + "score": 1.0, + "content": "uncertainty by sampling and the approximation error, while", + "type": "text" + }, + { + "bbox": [ + 346, + 417, + 358, + 429 + ], + "score": 0.9, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "controls uncertainty of observations", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "and discrimination. For balancing the trade-offs, we select the number of parameters (connections of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 277, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 221, + 452 + ], + "score": 1.0, + "content": "DNNs) with some constants", + "type": "text" + }, + { + "bbox": [ + 221, + 439, + 263, + 451 + ], + "score": 0.92, + "content": "c _ { g } , c _ { f } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 438, + 277, + 452 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 454, + 394, + 470 + ], + "lines": [ + { + "bbox": [ + 216, + 454, + 394, + 470 + ], + "spans": [ + { + "bbox": [ + 216, + 454, + 394, + 470 + ], + "score": 0.92, + "content": "S _ { g } = c _ { g } m ^ { D / ( 2 \\beta + D ) } , ~ \\mathrm { a n d } ~ S _ { f } = c _ { f } n ^ { ( \\kappa - 2 ) / \\kappa } ,", + "type": "interline_equation", + "image_path": "e48c2040dd109246ebb3ccd8ac6d4973e4a527c42643d9df0e6c9dc500daf35c.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 216, + 454, + 394, + 470 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 383, + 484 + ], + "lines": [ + { + "bbox": [ + 104, + 470, + 385, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 385, + 488 + ], + "score": 1.0, + "content": "for optimizing the bound (5). We then obtain the following corollary.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 506, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 281, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 281, + 499 + ], + "score": 1.0, + "content": "Corollary 1. (Convergence Rate of GANs)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 269, + 511 + ], + "score": 1.0, + "content": "Make the same assumptions as Theorem", + "type": "text" + }, + { + "bbox": [ + 270, + 499, + 275, + 507 + ], + "score": 0.61, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 497, + 310, + 511 + ], + "score": 1.0, + "content": ", and set", + "type": "text" + }, + { + "bbox": [ + 311, + 498, + 322, + 510 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 497, + 342, + 511 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 342, + 498, + 353, + 510 + ], + "score": 0.89, + "content": "S _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "as in (6). Then, with high probability", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 510, + 215, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 215, + 521 + ], + "score": 1.0, + "content": "converging to 1, we obtain", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 523, + 425, + 545 + ], + "lines": [ + { + "bbox": [ + 186, + 523, + 425, + 545 + ], + "spans": [ + { + "bbox": [ + 186, + 523, + 425, + 545 + ], + "score": 0.91, + "content": "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) .", + "type": "interline_equation", + "image_path": "0242a21b28cf208c69fa84d2b9696407b82dac3667dc7686f4f04ee4dba30595.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 186, + 523, + 425, + 545 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 555, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 507, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 163, + 569 + ], + "score": 1.0, + "content": "A selection of", + "type": "text" + }, + { + "bbox": [ + 164, + 555, + 173, + 567 + ], + "score": 0.86, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 554, + 321, + 569 + ], + "score": 1.0, + "content": "determines the first term in (7), since `", + "type": "text" + }, + { + "bbox": [ + 321, + 559, + 328, + 567 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 554, + 376, + 569 + ], + "score": 1.0, + "content": "depends on ˘", + "type": "text" + }, + { + "bbox": [ + 376, + 555, + 385, + 567 + ], + "score": 0.86, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 554, + 479, + 569 + ], + "score": 1.0, + "content": ". For an example, when", + "type": "text" + }, + { + "bbox": [ + 479, + 557, + 488, + 567 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 554, + 507, + 569 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 565, + 470, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 283, + 582 + ], + "score": 1.0, + "content": "set of 1-Lipschitz functions, the first term is", + "type": "text" + }, + { + "bbox": [ + 284, + 568, + 349, + 582 + ], + "score": 0.91, + "content": "O \\left( n ^ { - 1 / \\left( 2 + 2 D \\right) } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 565, + 470, + 582 + ], + "score": 1.0, + "content": "(Sriperumbudur et al., 2012).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 275, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 276, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 276, + 597 + ], + "score": 1.0, + "content": "Remark 1. (Heterogeneous Smoothness)", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 243, + 606 + ], + "score": 1.0, + "content": "Corollary 1 can be extended when", + "type": "text" + }, + { + "bbox": [ + 243, + 594, + 257, + 604 + ], + "score": 0.87, + "content": "P ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 592, + 392, + 606 + ], + "score": 1.0, + "content": "has different smoothness for each", + "type": "text" + }, + { + "bbox": [ + 393, + 594, + 431, + 606 + ], + "score": 0.91, + "content": "m \\in [ M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 592, + 451, + 606 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 452, + 594, + 466, + 604 + ], + "score": 0.87, + "content": "P ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "is locally", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 107, + 606, + 120, + 616 + ], + "score": 0.88, + "content": "\\beta _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 605, + 187, + 618 + ], + "score": 1.0, + "content": "-smooth on a set", + "type": "text" + }, + { + "bbox": [ + 188, + 605, + 202, + 616 + ], + "score": 0.89, + "content": "S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 605, + 505, + 618 + ], + "score": 1.0, + "content": ". In this case, we can easily extend our analysis in Theorem 1 and Corollary", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 288, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 288, + 628 + ], + "score": 1.0, + "content": "1, and obtain the following convergence rate.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 630, + 424, + 652 + ], + "lines": [ + { + "bbox": [ + 186, + 630, + 424, + 652 + ], + "spans": [ + { + "bbox": [ + 186, + 630, + 424, + 652 + ], + "score": 0.93, + "content": "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) ,", + "type": "interline_equation", + "image_path": "e498f4cc78a2f525e2d9205ab99029d6060970965bc8e0760699ef2d389644a8.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 186, + 630, + 424, + 652 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 216, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 213, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 133, + 674 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 656, + 213, + 671 + ], + "score": 0.89, + "content": "\\widetilde { \\beta } : = \\operatorname* { m i n } _ { m \\in \\left[ M \\right] } \\beta _ { m }", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "title", + "bbox": [ + 107, + 684, + 191, + 698 + ], + "lines": [ + { + "bbox": [ + 104, + 683, + 192, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 683, + 192, + 701 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "We emphasize the theoretical results show that GANs do not suffer from the effect of disconnected", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 183, + 734 + ], + "score": 1.0, + "content": "supports. 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This implies an advantage of GANs in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 351, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 351, + 117 + ], + "score": 1.0, + "content": "comparison with the other standard methods (Proposition 1).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 505, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 412, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 412, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 412, + 135 + ], + "score": 1.0, + "content": "Based on Lemma 3, we obtain the main theorem for generalization analysis.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 119, + 412, + 135 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 135, + 504, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 265, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 265, + 147 + ], + "score": 1.0, + "content": "Theorem 1. 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To show", + "type": "text" + }, + { + "bbox": [ + 453, + 356, + 487, + 366 + ], + "score": 0.9, + "content": "i i \\leqslant I I", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 356, + 506, + 369 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "employ recent results on approximation ability of DNNs (Yarotsky, 2017; Petersen & Voigtlaender,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 377, + 507, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 407, + 391 + ], + "score": 1.0, + "content": "2017; Imaizumi & Fukumizu, 2018) and obtain an approximation bound for", + "type": "text" + }, + { + "bbox": [ + 407, + 378, + 456, + 390 + ], + "score": 0.93, + "content": "d _ { \\mathcal { F } } ( P _ { g } , P ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 377, + 507, + 391 + ], + "score": 1.0, + "content": "(Lemma 3).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 365, + 401 + ], + "score": 1.0, + "content": "Combining these results, we obtain the statement of Theorem 1.", + "type": "text" + }, + { + "bbox": [ + 494, + 389, + 506, + 400 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 333, + 507, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 336, + 419 + ], + "score": 1.0, + "content": "Theorem 1 provides two trade-off relations with respect to", + "type": "text" + }, + { + "bbox": [ + 337, + 406, + 349, + 418 + ], + "score": 0.9, + "content": "S _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 406, + 366, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 367, + 406, + 379, + 418 + ], + "score": 0.88, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 406, + 462, + 419 + ], + "score": 1.0, + "content": ". 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Then, with high probability", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 510, + 215, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 215, + 521 + ], + "score": 1.0, + "content": "converging to 1, we obtain", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + } + ], + "index": 31, + "bbox_fs": [ + 105, + 487, + 505, + 521 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 523, + 425, + 545 + ], + "lines": [ + { + "bbox": [ + 186, + 523, + 425, + 545 + ], + "spans": [ + { + "bbox": [ + 186, + 523, + 425, + 545 + ], + "score": 0.91, + "content": "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) .", + "type": "interline_equation", + "image_path": "0242a21b28cf208c69fa84d2b9696407b82dac3667dc7686f4f04ee4dba30595.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 186, + 523, + 425, + 545 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 555, + 506, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 507, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 163, + 569 + ], + "score": 1.0, + "content": "A selection of", + "type": "text" + }, + { + "bbox": [ + 164, + 555, + 173, + 567 + ], + "score": 0.86, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 554, + 321, + 569 + ], + "score": 1.0, + "content": "determines the first term in (7), since `", + "type": "text" + }, + { + "bbox": [ + 321, + 559, + 328, + 567 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 554, + 376, + 569 + ], + "score": 1.0, + "content": "depends on ˘", + "type": "text" + }, + { + "bbox": [ + 376, + 555, + 385, + 567 + ], + "score": 0.86, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 554, + 479, + 569 + ], + "score": 1.0, + "content": ". For an example, when", + "type": "text" + }, + { + "bbox": [ + 479, + 557, + 488, + 567 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 554, + 507, + 569 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 565, + 470, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 283, + 582 + ], + "score": 1.0, + "content": "set of 1-Lipschitz functions, the first term is", + "type": "text" + }, + { + "bbox": [ + 284, + 568, + 349, + 582 + ], + "score": 0.91, + "content": "O \\left( n ^ { - 1 / \\left( 2 + 2 D \\right) } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 565, + 470, + 582 + ], + "score": 1.0, + "content": "(Sriperumbudur et al., 2012).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 554, + 507, + 582 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 275, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 276, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 276, + 597 + ], + "score": 1.0, + "content": "Remark 1. (Heterogeneous Smoothness)", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 582, + 276, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 243, + 606 + ], + "score": 1.0, + "content": "Corollary 1 can be extended when", + "type": "text" + }, + { + "bbox": [ + 243, + 594, + 257, + 604 + ], + "score": 0.87, + "content": "P ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 592, + 392, + 606 + ], + "score": 1.0, + "content": "has different smoothness for each", + "type": "text" + }, + { + "bbox": [ + 393, + 594, + 431, + 606 + ], + "score": 0.91, + "content": "m \\in [ M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 592, + 451, + 606 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 452, + 594, + 466, + 604 + ], + "score": 0.87, + "content": "P ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "is locally", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 107, + 606, + 120, + 616 + ], + "score": 0.88, + "content": "\\beta _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 605, + 187, + 618 + ], + "score": 1.0, + "content": "-smooth on a set", + "type": "text" + }, + { + "bbox": [ + 188, + 605, + 202, + 616 + ], + "score": 0.89, + "content": "S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 605, + 505, + 618 + ], + "score": 1.0, + "content": ". In this case, we can easily extend our analysis in Theorem 1 and Corollary", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 288, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 288, + 628 + ], + "score": 1.0, + "content": "1, and obtain the following convergence rate.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 592, + 505, + 628 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 630, + 424, + 652 + ], + "lines": [ + { + "bbox": [ + 186, + 630, + 424, + 652 + ], + "spans": [ + { + "bbox": [ + 186, + 630, + 424, + 652 + ], + "score": 0.93, + "content": "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) ,", + "type": "interline_equation", + "image_path": "e498f4cc78a2f525e2d9205ab99029d6060970965bc8e0760699ef2d389644a8.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 186, + 630, + 424, + 652 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 216, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 213, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 133, + 674 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 656, + 213, + 671 + ], + "score": 0.89, + "content": "\\widetilde { \\beta } : = \\operatorname* { m i n } _ { m \\in \\left[ M \\right] } \\beta _ { m }", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 653, + 213, + 674 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 684, + 191, + 698 + ], + "lines": [ + { + "bbox": [ + 104, + 683, + 192, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 683, + 192, + 701 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "We emphasize the theoretical results show that GANs do not suffer from the effect of disconnected", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 183, + 734 + ], + "score": 1.0, + "content": "supports. A larger", + "type": "text" + }, + { + "bbox": [ + 184, + 721, + 191, + 732 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "improves performance of GANs even with disconnected supports, as shown", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "in Theorem 1 and Corollary 1. This phenomenon is different from the result of the other methods", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "discussed in Proposition 1. Hence, we can state that GANs have advantages over the other methods", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "which are deteriorated by the disconnected property (Section 3.3). In other words, when data are", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "generated from a probability measure with disconnected supports and sufficiently smooth in each", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "of the sets, only GANs can estimate the measure effectively and the other methdos cannot. This", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "advantage of GANs comes from the approximation power for discontinuous generators shown in", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 150, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 150, + 160 + ], + "score": 1.0, + "content": "Lemma 3.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 710, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "in Theorem 1 and Corollary 1. This phenomenon is different from the result of the other methods", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "discussed in Proposition 1. Hence, we can state that GANs have advantages over the other methods", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "which are deteriorated by the disconnected property (Section 3.3). In other words, when data are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "generated from a probability measure with disconnected supports and sufficiently smooth in each", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "of the sets, only GANs can estimate the measure effectively and the other methdos cannot. This", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "advantage of GANs comes from the approximation power for discontinuous generators shown in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 150, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 150, + 160 + ], + "score": 1.0, + "content": "Lemma 3.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 502, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 495, + 178 + ], + "score": 1.0, + "content": "The results (5) and (7) provide interpretation about performance of GANs. About convergence with", + "type": "text" + }, + { + "bbox": [ + 496, + 168, + 502, + 175 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 179, + 191 + ], + "score": 1.0, + "content": "the complexity of", + "type": "text" + }, + { + "bbox": [ + 179, + 176, + 189, + 188 + ], + "score": 0.86, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 177, + 206, + 191 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 178, + 219, + 190 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "control a trade-off between convergence and a power of discrimination.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 164, + 202 + ], + "score": 1.0, + "content": "While smaller", + "type": "text" + }, + { + "bbox": [ + 164, + 190, + 176, + 201 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 189, + 290, + 202 + ], + "score": 1.0, + "content": "reduce the errors in terms of", + "type": "text" + }, + { + "bbox": [ + 291, + 190, + 304, + 200 + ], + "score": 0.89, + "content": "d _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 189, + 346, + 202 + ], + "score": 1.0, + "content": ", too small", + "type": "text" + }, + { + "bbox": [ + 347, + 189, + 359, + 201 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "can lose the power of discrimination", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 225, + 212 + ], + "score": 1.0, + "content": "to satisfy (1). Hence, setting", + "type": "text" + }, + { + "bbox": [ + 226, + 200, + 238, + 212 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "as in (6) can keep the discrimination power and does not worsen", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 236, + 226 + ], + "score": 1.0, + "content": "the overall rate of convergence", + "type": "text" + }, + { + "bbox": [ + 237, + 212, + 277, + 225 + ], + "score": 0.93, + "content": "O ( n ^ { - 1 / \\kappa } )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 212, + 387, + 226 + ], + "score": 1.0, + "content": ". About convergence with", + "type": "text" + }, + { + "bbox": [ + 388, + 214, + 397, + 223 + ], + "score": 0.58, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 212, + 402, + 226 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 402, + 213, + 414, + 225 + ], + "score": 0.81, + "content": "S _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 212, + 506, + 226 + ], + "score": 1.0, + "content": "controls the trade-off", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 394, + 237 + ], + "score": 1.0, + "content": "between the bias and variance of the estimator. An optimal way to select", + "type": "text" + }, + { + "bbox": [ + 395, + 225, + 406, + 236 + ], + "score": 0.88, + "content": "S _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "is provided in (6) which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 154, + 247 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 155, + 235, + 162, + 246 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 234, + 180, + 247 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 235, + 190, + 245 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 234, + 318, + 247 + ], + "score": 1.0, + "content": ", and it is more important when", + "type": "text" + }, + { + "bbox": [ + 318, + 237, + 325, + 245 + ], + "score": 0.77, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 234, + 381, + 247 + ], + "score": 1.0, + "content": "is small (e.g.", + "type": "text" + }, + { + "bbox": [ + 381, + 236, + 407, + 245 + ], + "score": 0.88, + "content": "\\kappa = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "as MMD-GAN). Based", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "score": 1.0, + "content": "on the interpretation and the selection rule (6), our study can provide a guideline for a design of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 257, + 198, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 198, + 268 + ], + "score": 1.0, + "content": "architecture of DNNs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 282, + 207, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 209, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 209, + 295 + ], + "score": 1.0, + "content": "5.1 RELATED WORKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "Compared with studies for understanding GANs (Dziugaite et al., 2015; Liang, 2017; Liu et al., 2017;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "Zhang et al., 2018; Biau et al., 2018; Arora et al., 2017), we show that GANs can avoid influences", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "of disconnected supports, unlike the other methods, hence it is a source of the advantage of GANs.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "This paper is the first work to focus on the disconnected support property, while several discussions", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 361 + ], + "score": 1.0, + "content": "(Goodfellow, 2016; Liang, 2017; Creswell et al., 2018) focus on models and metrics of the learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 180, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 180, + 370 + ], + "score": 1.0, + "content": "scheme of GANs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 504, + 387 + ], + "score": 1.0, + "content": "It is important to compare our result with other studies for generalization analysis. Although some", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "existing studies (Dziugaite et al., 2015; Liang, 2017; Liu et al., 2017; Zhang et al., 2018; Biau et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 507, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 507, + 410 + ], + "score": 1.0, + "content": "2018; Arora et al., 2017) provide generalization analysis, they do not analyze an approximation effect,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 406, + 507, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 198, + 423 + ], + "score": 1.0, + "content": "namely, they evaluate", + "type": "text" + }, + { + "bbox": [ + 198, + 408, + 322, + 420 + ], + "score": 0.92, + "content": "\\begin{array} { r } { d _ { \\mathcal { F } } ( P ^ { * } , P _ { \\hat { g } } ) - \\operatorname* { i n f } _ { g \\in \\mathcal { G } } ( P ^ { * } , P _ { g } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 406, + 439, + 423 + ], + "score": 1.0, + "content": ". Since we analyze the term", + "type": "text" + }, + { + "bbox": [ + 439, + 408, + 502, + 420 + ], + "score": 0.92, + "content": "{ \\operatorname* { i n f } } _ { g \\in { \\mathcal { G } } } ( P ^ { * } , P _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 406, + 507, + 423 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 419, + 446, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 446, + 432 + ], + "score": 1.0, + "content": "we can provide a more general bound and discuss the effect of disconnected support.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "score": 1.0, + "content": "As we mentioned in the introduction, this paper is not along with studies for low-dimensional supports", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "(Arjovsky & Bottou, 2017). We also note that an optimization aspect and gaming aspect of GANs are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 457, + 507, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 507, + 472 + ], + "score": 1.0, + "content": "out of concerns of this paper. We focus on the statistical aspects of GANs such as sample complexity.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 486, + 266, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 268, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 268, + 500 + ], + "score": 1.0, + "content": "6 NUMERICAL EXPERIMENTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "We compare the numerical performance of GANs and the other methods with toy data with. We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "generate synthetic data from the following two settings: (A) Gaussian distribution restricted on a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "compact set (global support), and (B) a probability measure with two disconnected supports (density", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 336, + 558 + ], + "score": 1.0, + "content": "function is the black solid line in Figure 6). We generate", + "type": "text" + }, + { + "bbox": [ + 336, + 545, + 372, + 555 + ], + "score": 0.8, + "content": "n = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 543, + 506, + 558 + ], + "score": 1.0, + "content": ", 1000, ..., 5000 observations and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "estimate the true probability measures with Wasserstein GAN, MMD-GAN, KDE (the Gaussian", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "kernel and the Epanechnikov kernel), SDE (Fourier basis), and NB (Dirichlet process prior). Hyper-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 577, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 426, + 591 + ], + "score": 1.0, + "content": "parameters for these methods are selected by cross-validation. For GANs, we set", + "type": "text" + }, + { + "bbox": [ + 426, + 578, + 455, + 587 + ], + "score": 0.87, + "content": "m = n", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 577, + 490, + 591 + ], + "score": 1.0, + "content": ". We use", + "type": "text" + }, + { + "bbox": [ + 491, + 577, + 504, + 588 + ], + "score": 0.88, + "content": "d _ { \\mathcal { F } }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 416, + 601 + ], + "score": 1.0, + "content": "to evaluate errors by GANs, and a root of the expected squared errors with the", + "type": "text" + }, + { + "bbox": [ + 416, + 588, + 429, + 598 + ], + "score": 0.89, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "-norm for the other", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 600, + 313, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 313, + 612 + ], + "score": 1.0, + "content": "methods. The plots are the mean of 30 replications.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "Figure 5 shows generalization errors by the methods. With the disconnected support case (B), we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "plot the estimated density in Figure 6. The black line shows the true density, the dashed line is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "by estimated densities of the other methods, and bars are histograms by GANs. The results by", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 495, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 495, + 662 + ], + "score": 1.0, + "content": "Wasserstein-GAN and MMD-GAN are almost same, so we omit the result by Wasserstein-GAN.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "In Figure 5, we see that in the case of global support (A), the error of GANs and the other methods are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "comparable. In contrast, in the case of disconnected supports (B), the other standard methods show", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "worse generalization and only GANs keep the high performance. From Figure 6, we can see that", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "score": 1.0, + "content": "GANs can reveal the disconnected supports, while some of the other methods fail to fit. KDE with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "the Gaussian kernel represents the disconnected support by employing a small bandwidth. However,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 498, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 498, + 733 + ], + "score": 1.0, + "content": "the small bandwidth yields a too sharp density, tending to worsen the generalization performance.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 159 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 82, + 506, + 160 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 502, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 495, + 178 + ], + "score": 1.0, + "content": "The results (5) and (7) provide interpretation about performance of GANs. About convergence with", + "type": "text" + }, + { + "bbox": [ + 496, + 168, + 502, + 175 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 179, + 191 + ], + "score": 1.0, + "content": "the complexity of", + "type": "text" + }, + { + "bbox": [ + 179, + 176, + 189, + 188 + ], + "score": 0.86, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 177, + 206, + 191 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 178, + 219, + 190 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "control a trade-off between convergence and a power of discrimination.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 164, + 202 + ], + "score": 1.0, + "content": "While smaller", + "type": "text" + }, + { + "bbox": [ + 164, + 190, + 176, + 201 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 189, + 290, + 202 + ], + "score": 1.0, + "content": "reduce the errors in terms of", + "type": "text" + }, + { + "bbox": [ + 291, + 190, + 304, + 200 + ], + "score": 0.89, + "content": "d _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 189, + 346, + 202 + ], + "score": 1.0, + "content": ", too small", + "type": "text" + }, + { + "bbox": [ + 347, + 189, + 359, + 201 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "can lose the power of discrimination", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 225, + 212 + ], + "score": 1.0, + "content": "to satisfy (1). Hence, setting", + "type": "text" + }, + { + "bbox": [ + 226, + 200, + 238, + 212 + ], + "score": 0.89, + "content": "S _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "as in (6) can keep the discrimination power and does not worsen", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 236, + 226 + ], + "score": 1.0, + "content": "the overall rate of convergence", + "type": "text" + }, + { + "bbox": [ + 237, + 212, + 277, + 225 + ], + "score": 0.93, + "content": "O ( n ^ { - 1 / \\kappa } )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 212, + 387, + 226 + ], + "score": 1.0, + "content": ". About convergence with", + "type": "text" + }, + { + "bbox": [ + 388, + 214, + 397, + 223 + ], + "score": 0.58, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 212, + 402, + 226 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 402, + 213, + 414, + 225 + ], + "score": 0.81, + "content": "S _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 212, + 506, + 226 + ], + "score": 1.0, + "content": "controls the trade-off", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 394, + 237 + ], + "score": 1.0, + "content": "between the bias and variance of the estimator. An optimal way to select", + "type": "text" + }, + { + "bbox": [ + 395, + 225, + 406, + 236 + ], + "score": 0.88, + "content": "S _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "is provided in (6) which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 154, + 247 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 155, + 235, + 162, + 246 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 234, + 180, + 247 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 235, + 190, + 245 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 234, + 318, + 247 + ], + "score": 1.0, + "content": ", and it is more important when", + "type": "text" + }, + { + "bbox": [ + 318, + 237, + 325, + 245 + ], + "score": 0.77, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 234, + 381, + 247 + ], + "score": 1.0, + "content": "is small (e.g.", + "type": "text" + }, + { + "bbox": [ + 381, + 236, + 407, + 245 + ], + "score": 0.88, + "content": "\\kappa = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "as MMD-GAN). Based", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "score": 1.0, + "content": "on the interpretation and the selection rule (6), our study can provide a guideline for a design of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 257, + 198, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 198, + 268 + ], + "score": 1.0, + "content": "architecture of DNNs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 164, + 506, + 268 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 282, + 207, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 209, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 209, + 295 + ], + "score": 1.0, + "content": "5.1 RELATED WORKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "Compared with studies for understanding GANs (Dziugaite et al., 2015; Liang, 2017; Liu et al., 2017;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "Zhang et al., 2018; Biau et al., 2018; Arora et al., 2017), we show that GANs can avoid influences", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "of disconnected supports, unlike the other methods, hence it is a source of the advantage of GANs.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "This paper is the first work to focus on the disconnected support property, while several discussions", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 361 + ], + "score": 1.0, + "content": "(Goodfellow, 2016; Liang, 2017; Creswell et al., 2018) focus on models and metrics of the learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 180, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 180, + 370 + ], + "score": 1.0, + "content": "scheme of GANs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 303, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 504, + 387 + ], + "score": 1.0, + "content": "It is important to compare our result with other studies for generalization analysis. Although some", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "existing studies (Dziugaite et al., 2015; Liang, 2017; Liu et al., 2017; Zhang et al., 2018; Biau et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 507, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 507, + 410 + ], + "score": 1.0, + "content": "2018; Arora et al., 2017) provide generalization analysis, they do not analyze an approximation effect,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 406, + 507, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 198, + 423 + ], + "score": 1.0, + "content": "namely, they evaluate", + "type": "text" + }, + { + "bbox": [ + 198, + 408, + 322, + 420 + ], + "score": 0.92, + "content": "\\begin{array} { r } { d _ { \\mathcal { F } } ( P ^ { * } , P _ { \\hat { g } } ) - \\operatorname* { i n f } _ { g \\in \\mathcal { G } } ( P ^ { * } , P _ { g } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 406, + 439, + 423 + ], + "score": 1.0, + "content": ". Since we analyze the term", + "type": "text" + }, + { + "bbox": [ + 439, + 408, + 502, + 420 + ], + "score": 0.92, + "content": "{ \\operatorname* { i n f } } _ { g \\in { \\mathcal { G } } } ( P ^ { * } , P _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 406, + 507, + 423 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 419, + 446, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 446, + 432 + ], + "score": 1.0, + "content": "we can provide a more general bound and discuss the effect of disconnected support.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 374, + 507, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 449 + ], + "score": 1.0, + "content": "As we mentioned in the introduction, this paper is not along with studies for low-dimensional supports", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "(Arjovsky & Bottou, 2017). We also note that an optimization aspect and gaming aspect of GANs are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 457, + 507, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 507, + 472 + ], + "score": 1.0, + "content": "out of concerns of this paper. We focus on the statistical aspects of GANs such as sample complexity.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 434, + 507, + 472 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 486, + 266, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 268, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 268, + 500 + ], + "score": 1.0, + "content": "6 NUMERICAL EXPERIMENTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "We compare the numerical performance of GANs and the other methods with toy data with. We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "generate synthetic data from the following two settings: (A) Gaussian distribution restricted on a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "compact set (global support), and (B) a probability measure with two disconnected supports (density", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 336, + 558 + ], + "score": 1.0, + "content": "function is the black solid line in Figure 6). We generate", + "type": "text" + }, + { + "bbox": [ + 336, + 545, + 372, + 555 + ], + "score": 0.8, + "content": "n = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 543, + 506, + 558 + ], + "score": 1.0, + "content": ", 1000, ..., 5000 observations and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "estimate the true probability measures with Wasserstein GAN, MMD-GAN, KDE (the Gaussian", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "kernel and the Epanechnikov kernel), SDE (Fourier basis), and NB (Dirichlet process prior). Hyper-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 577, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 426, + 591 + ], + "score": 1.0, + "content": "parameters for these methods are selected by cross-validation. For GANs, we set", + "type": "text" + }, + { + "bbox": [ + 426, + 578, + 455, + 587 + ], + "score": 0.87, + "content": "m = n", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 577, + 490, + 591 + ], + "score": 1.0, + "content": ". We use", + "type": "text" + }, + { + "bbox": [ + 491, + 577, + 504, + 588 + ], + "score": 0.88, + "content": "d _ { \\mathcal { F } }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 416, + 601 + ], + "score": 1.0, + "content": "to evaluate errors by GANs, and a root of the expected squared errors with the", + "type": "text" + }, + { + "bbox": [ + 416, + 588, + 429, + 598 + ], + "score": 0.89, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "-norm for the other", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 600, + 313, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 313, + 612 + ], + "score": 1.0, + "content": "methods. The plots are the mean of 30 replications.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 510, + 507, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "Figure 5 shows generalization errors by the methods. With the disconnected support case (B), we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "plot the estimated density in Figure 6. The black line shows the true density, the dashed line is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "by estimated densities of the other methods, and bars are histograms by GANs. The results by", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 495, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 495, + 662 + ], + "score": 1.0, + "content": "Wasserstein-GAN and MMD-GAN are almost same, so we omit the result by Wasserstein-GAN.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 616, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "In Figure 5, we see that in the case of global support (A), the error of GANs and the other methods are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "comparable. In contrast, in the case of disconnected supports (B), the other standard methods show", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "worse generalization and only GANs keep the high performance. From Figure 6, we can see that", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "score": 1.0, + "content": "GANs can reveal the disconnected supports, while some of the other methods fail to fit. KDE with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "the Gaussian kernel represents the disconnected support by employing a small bandwidth. 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Then, we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 285, + 362, + 313 + ], + "lines": [ + { + "bbox": [ + 248, + 285, + 362, + 313 + ], + "spans": [ + { + "bbox": [ + 248, + 285, + 362, + 313 + ], + "score": 0.94, + "content": "g ^ { * } = \\sum _ { m \\in [ M ] } \\widetilde { g } _ { m } \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } .", + "type": "interline_equation", + "image_path": "391f427f735494794df2b95d0feebbeec2bb91f6cbf66d580748a3b604ae9631.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 248, + 285, + 362, + 313 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 506, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "Preliminarily, we apply sub-neural networks from Yarotsky (2017) and Petersen & Voigtlaenderř", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 328, + 507, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 157, + 347 + ], + "score": 1.0, + "content": "(2017). Let", + "type": "text" + }, + { + "bbox": [ + 157, + 331, + 185, + 343 + ], + "score": 0.93, + "content": "\\zeta [ \\Theta _ { + } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 328, + 345, + 347 + ], + "score": 1.0, + "content": "be a network for summation such that", + "type": "text" + }, + { + "bbox": [ + 346, + 330, + 484, + 344 + ], + "score": 0.9, + "content": "\\begin{array} { r } { { \\bf \\Pi } \\dot { \\zeta } [ \\Theta _ { + } ] ( x _ { 1 } , . . . , x _ { D ^ { \\prime } } ) = \\sum _ { d \\in [ D ^ { \\prime } ] } x _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 328, + 507, + 347 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 343, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 134, + 357 + ], + "score": 0.91, + "content": "\\zeta [ \\Theta _ { \\times } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 343, + 345, + 360 + ], + "score": 1.0, + "content": "be a network for approximate multiplication such as", + "type": "text" + }, + { + "bbox": [ + 346, + 344, + 504, + 358 + ], + "score": 0.89, + "content": "| \\zeta [ \\Theta _ { \\times } ] ( x _ { 1 } , . . . , x _ { D ^ { \\prime } } ) - \\prod _ { d \\in [ D ^ { \\prime } ] } x _ { d } | < \\epsilon", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 151, + 370 + ], + "score": 1.0, + "content": "with some", + "type": "text" + }, + { + "bbox": [ + 151, + 358, + 176, + 368 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 357, + 205, + 370 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 205, + 357, + 242, + 369 + ], + "score": 0.91, + "content": "x , x ^ { \\prime } \\in I", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 357, + 495, + 370 + ], + "score": 1.0, + "content": "(Proposition 3 in Yarotsky (2017) and Lemma 1 in Imaizumi", + "type": "text" + }, + { + "bbox": [ + 496, + 358, + 505, + 368 + ], + "score": 0.6, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 367, + 184, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 184, + 381 + ], + "score": 1.0, + "content": "Fukumizu (2018)).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 102, + 380, + 508, + 403 + ], + "spans": [ + { + "bbox": [ + 102, + 380, + 224, + 403 + ], + "score": 1.0, + "content": "We consider approximation", + "type": "text" + }, + { + "bbox": [ + 225, + 385, + 338, + 399 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\sum _ { m \\in [ M ] } \\gamma _ { m , d } \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 380, + 379, + 403 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 379, + 385, + 417, + 397 + ], + "score": 0.92, + "content": "d \\in \\mathsf { \\Gamma } [ D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 380, + 443, + 403 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 443, + 385, + 486, + 398 + ], + "score": 0.93, + "content": "\\zeta [ \\Theta _ { \\gamma , d , m } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 380, + 508, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 397, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 148, + 411 + ], + "score": 0.88, + "content": "\\zeta [ \\Theta _ { h , m , j } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 397, + 165, + 412 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 165, + 398, + 244, + 410 + ], + "score": 0.89, + "content": "d \\in [ D ] , m \\in [ M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 397, + 264, + 412 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 264, + 399, + 297, + 410 + ], + "score": 0.87, + "content": "j \\in [ J ]", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 397, + 506, + 412 + ], + "score": 1.0, + "content": ", and we will specify the networks later. Also, let", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 108, + 408, + 313, + 423 + ], + "spans": [ + { + "bbox": [ + 108, + 410, + 309, + 422 + ], + "score": 0.82, + "content": "\\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 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 408, + 313, + 423 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 226, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 227, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 227, + 438 + ], + "score": 1.0, + "content": "We consider a neural network", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 439, + 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approximation error is evaluated as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 470, + 466, + 733 + ], + "lines": [ + { + "bbox": [ + 124, + 470, + 466, + 733 + ], + "spans": [ + { + "bbox": [ + 124, + 470, + 466, + 733 + ], + "score": 0.9, + "content": "\\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}", + "type": "interline_equation", + "image_path": "fc1a49e41c4f44640fa37ef07efc66fa7752bde0a694d76458b15e34c9dae3b9.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 124, + 470, + 466, + 557.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 124, + 557.6666666666666, + 466, + 645.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 124, + 645.3333333333333, + 466, + 732.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + 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By Stein (2016), we can extend", + "type": "text" + }, + { + "bbox": [ + 266, + 143, + 330, + 156 + ], + "score": 0.92, + "content": "g _ { m } : \\widetilde { S } _ { m } S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 143, + 342, + 159 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 343, + 144, + 406, + 156 + ], + "score": 0.91, + "content": "\\tilde { g } _ { m } : I ^ { D } \\to S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 143, + 489, + 159 + ], + "score": 1.0, + "content": "since boundaries of", + "type": "text" + }, + { + "bbox": [ + 489, + 144, + 503, + 156 + ], + "score": 0.89, + "content": "\\widetilde { S } _ { m }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 156, + 388, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 388, + 169 + ], + "score": 1.0, + "content": "are Lipschitz continuous. Then, we provide the following formulation", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 129, + 506, + 169 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 169, + 375, + 184 + ], + "lines": [ + { + "bbox": [ + 233, + 169, + 375, + 184 + ], + "spans": [ + { + "bbox": [ + 233, + 169, + 375, + 184 + ], + "score": 0.92, + "content": "\\widetilde { \\boldsymbol { g } } _ { m } ( \\boldsymbol { x } ) = ( \\gamma _ { m , 1 } ( \\boldsymbol { x } ) , . . . , \\gamma _ { m , D } ( \\boldsymbol { x } ) ) ^ { \\top } ,", + "type": "interline_equation", + "image_path": "05f0697cf684236905416453966f6d511efecaa191a0fc8f5ee0046c182c3c5a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 233, + 169, + 375, + 184 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 186, + 341, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 343, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 133, + 201 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 186, + 199, + 199 + ], + "score": 0.93, + "content": "\\gamma _ { m , d } \\in H ^ { \\beta } ( I ^ { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 184, + 317, + 201 + ], + "score": 1.0, + "content": ". Then, we obtain the form of", + "type": "text" + }, + { + "bbox": [ + 317, + 187, + 329, + 198 + ], + "score": 0.89, + "content": "g ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 184, + 343, + 201 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 184, + 343, + 201 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 200, + 353, + 227 + ], + "lines": [ + { + "bbox": [ + 257, + 200, + 353, + 227 + ], + "spans": [ + { + "bbox": [ + 257, + 200, + 353, + 227 + ], + "score": 0.94, + "content": "g ^ { * } = \\sum _ { m \\in [ M ] } \\widetilde { g } _ { m } \\otimes \\pmb { I } _ { \\widetilde { S } _ { m } } .", + "type": "interline_equation", + "image_path": "01b2b8268e655990e9ef44c14a15fadad47644d28b6dce3c0bd1a6c98e401062.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 257, + 200, + 353, + 227 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 393, + 243 + ], + "lines": [ + { + "bbox": [ + 104, + 228, + 394, + 245 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 208, + 245 + ], + "score": 1.0, + "content": "Also, by the definition of", + "type": "text" + }, + { + "bbox": [ + 209, + 232, + 232, + 244 + ], + "score": 0.93, + "content": "\\mathcal { S } _ { 2 \\beta , J }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 228, + 296, + 245 + ], + "score": 1.0, + "content": "which contains", + "type": "text" + }, + { + "bbox": [ + 297, + 230, + 311, + 243 + ], + "score": 0.93, + "content": "\\widetilde { S } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 228, + 394, + 245 + ], + "score": 1.0, + "content": ", we obtain the form", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 228, + 394, + 245 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 245, + 344, + 270 + ], + "lines": [ + { + "bbox": [ + 265, + 245, + 344, + 270 + ], + "spans": [ + { + "bbox": [ + 265, + 245, + 344, + 270 + ], + "score": 0.95, + "content": "{ \\cal I } _ { { \\widetilde { \\cal S } } _ { m } } = \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } ,", + "type": "interline_equation", + "image_path": "7d8e1889d32be15346656dc11d6cd68d41113071f36fdbcdda8d16691c2ef324.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 265, + 245, + 344, + 270 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 250, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 250, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 160, + 286 + ], + "score": 1.0, + "content": "with existing", + "type": "text" + }, + { + "bbox": [ + 160, + 272, + 185, + 285 + ], + "score": 0.92, + "content": "\\psi _ { h _ { m , j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 270, + 250, + 286 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 270, + 250, + 286 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 285, + 362, + 313 + ], + "lines": [ + { + "bbox": [ + 248, + 285, + 362, + 313 + ], + "spans": [ + { + "bbox": [ + 248, + 285, + 362, + 313 + ], + "score": 0.94, + "content": "g ^ { * } = \\sum _ { m \\in [ M ] } \\widetilde { g } _ { m } \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } .", + "type": "interline_equation", + "image_path": "391f427f735494794df2b95d0feebbeec2bb91f6cbf66d580748a3b604ae9631.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 248, + 285, + 362, + 313 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 506, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "Preliminarily, we apply sub-neural networks from Yarotsky (2017) and Petersen & Voigtlaenderř", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 328, + 507, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 157, + 347 + ], + "score": 1.0, + "content": "(2017). Let", + "type": "text" + }, + { + "bbox": [ + 157, + 331, + 185, + 343 + ], + "score": 0.93, + "content": "\\zeta [ \\Theta _ { + } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 328, + 345, + 347 + ], + "score": 1.0, + "content": "be a network for summation such that", + "type": "text" + }, + { + "bbox": [ + 346, + 330, + 484, + 344 + ], + "score": 0.9, + "content": "\\begin{array} { r } { { \\bf \\Pi } \\dot { \\zeta } [ \\Theta _ { + } ] ( x _ { 1 } , . . . , x _ { D ^ { \\prime } } ) = \\sum _ { d \\in [ D ^ { \\prime } ] } x _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 328, + 507, + 347 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 343, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 134, + 357 + ], + "score": 0.91, + "content": "\\zeta [ \\Theta _ { \\times } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 343, + 345, + 360 + ], + "score": 1.0, + "content": "be a network for approximate multiplication such as", + "type": "text" + }, + { + "bbox": [ + 346, + 344, + 504, + 358 + ], + "score": 0.89, + "content": "| \\zeta [ \\Theta _ { \\times } ] ( x _ { 1 } , . . . , x _ { D ^ { \\prime } } ) - \\prod _ { d \\in [ D ^ { \\prime } ] } x _ { d } | < \\epsilon", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 151, + 370 + ], + "score": 1.0, + "content": "with some", + "type": "text" + }, + { + "bbox": [ + 151, + 358, + 176, + 368 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 357, + 205, + 370 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 205, + 357, + 242, + 369 + ], + "score": 0.91, + "content": "x , x ^ { \\prime } \\in I", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 357, + 495, + 370 + ], + "score": 1.0, + "content": "(Proposition 3 in Yarotsky (2017) and Lemma 1 in Imaizumi", + "type": "text" + }, + { + "bbox": [ + 496, + 358, + 505, + 368 + ], + "score": 0.6, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 367, + 184, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 184, + 381 + ], + "score": 1.0, + "content": "Fukumizu (2018)).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 319, + 507, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 102, + 380, + 508, + 403 + ], + "spans": [ + { + "bbox": [ + 102, + 380, + 224, + 403 + ], + "score": 1.0, + "content": "We consider approximation", + "type": "text" + }, + { + "bbox": [ + 225, + 385, + 338, + 399 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\sum _ { m \\in [ M ] } \\gamma _ { m , d } \\bigotimes _ { j \\in [ J ] } \\psi _ { h _ { m , j } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 380, + 379, + 403 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 379, + 385, + 417, + 397 + ], + "score": 0.92, + "content": "d \\in \\mathsf { \\Gamma } [ D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 380, + 443, + 403 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 443, + 385, + 486, + 398 + ], + "score": 0.93, + "content": "\\zeta [ \\Theta _ { \\gamma , d , m } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 380, + 508, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 397, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 148, + 411 + ], + "score": 0.88, + "content": "\\zeta [ \\Theta _ { h , m , j } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 397, + 165, + 412 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 165, + 398, + 244, + 410 + ], + "score": 0.89, + "content": "d \\in [ D ] , m \\in [ M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 397, + 264, + 412 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 264, + 399, + 297, + 410 + ], + "score": 0.87, + "content": "j \\in [ J ]", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 397, + 506, + 412 + ], + "score": 1.0, + "content": ", and we will specify the networks later. Also, let", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 108, + 408, + 313, + 423 + ], + "spans": [ + { + "bbox": [ + 108, + 410, + 309, + 422 + ], + "score": 0.82, + "content": "\\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 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 408, + 313, + 423 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 102, + 380, + 508, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 226, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 227, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 227, + 438 + ], + "score": 1.0, + "content": "We consider a neural network", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, 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[ + { + "bbox": [ + 106, + 453, + 286, + 466 + ], + "score": 1.0, + "content": "Then, an approximation error is evaluated as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 453, + 286, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 470, + 466, + 733 + ], + "lines": [ + { + "bbox": [ + 124, + 470, + 466, + 733 + ], + "spans": [ + { + "bbox": [ + 124, + 470, + 466, + 733 + ], + "score": 0.9, + "content": "\\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}", + "type": "interline_equation", + "image_path": "fc1a49e41c4f44640fa37ef07efc66fa7752bde0a694d76458b15e34c9dae3b9.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 124, + 470, + 466, + 557.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 124, + 557.6666666666666, + 466, + 645.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 124, + 645.3333333333333, + 466, + 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Then, we evaluate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 174, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 174, + 213 + ], + "score": 1.0, + "content": "the following as", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 214, + 499, + 358 + ], + "lines": [ + { + "bbox": [ + 110, + 214, + 499, + 358 + ], + "spans": [ + { + "bbox": [ + 110, + 214, + 499, + 358 + ], + "score": 0.94, + "content": "\\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}", + "type": "interline_equation", + "image_path": "d7454f652b0711a7d3cb44c37c8f4cbc5d5bae5b1cfed4488ab166a10c8da4eb.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 110, + 214, + 499, + 262.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 110, + 262.0, + 499, + 310.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 110, + 310.0, + 499, + 358.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 360, + 507, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 309, + 379 + ], + "score": 1.0, + "content": "Here, the last inequality follows the boundedness of", + "type": "text" + }, + { + "bbox": [ + 309, + 363, + 336, + 376 + ], + "score": 0.91, + "content": "\\psi _ { h _ { m , j ^ { \\prime } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 360, + 354, + 379 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 362, + 395, + 375 + ], + "score": 0.93, + "content": "\\zeta [ \\Theta _ { h , m , j } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 360, + 507, + 379 + ], + "score": 1.0, + "content": "by Theorem 3.1 in Petersen", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 325, + 387 + ], + "score": 1.0, + "content": "& Voigtlaender (2017). 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\\beta / ( D - 1 ) } + \\epsilon _ { \\times } .", + "type": "interline_equation", + "image_path": "1c462416da948e58732d16b5247713638a980ec75680ea3f0e9c8341a5ee690b.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 227, + 437, + 383, + 465 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 319, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 320, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 214, + 488 + ], + "score": 1.0, + "content": "Combining the bounds for", + "type": "text" + }, + { + "bbox": [ + 214, + 475, + 235, + 487 + ], + "score": 0.92, + "content": "T _ { 1 , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 473, + 253, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 475, + 273, + 487 + ], + "score": 0.92, + "content": "T _ { 2 , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 473, + 320, + 488 + ], + "score": 1.0, + "content": ", we bound", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 489, + 462, + 559 + ], + "lines": [ + { + "bbox": [ + 149, + 489, + 462, + 559 + ], + "spans": [ + { + "bbox": [ + 149, + 489, + 462, + 559 + ], + "score": 0.95, + "content": "\\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}", + "type": "interline_equation", + "image_path": "17ca5910080208ec8a052c5a4fb1c9a5d97427d8cc3cfb01cef837a1a950a61f.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 149, + 489, + 462, + 512.3333333333334 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 149, + 512.3333333333334, + 462, + 535.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 149, + 535.6666666666667, + 462, + 559.0000000000001 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 562, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 104, + 560, + 502, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 228, + 578 + ], + "score": 1.0, + "content": "Here, we consider a parameter", + "type": "text" + }, + { + "bbox": [ + 229, + 563, + 306, + 577 + ], + "score": 0.93, + "content": "\\begin{array} { r } { S = \\sum _ { d \\in [ D ] } \\| \\Theta _ { d } \\| _ { 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 560, + 345, + 578 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 345, + 563, + 502, + 576 + ], + "score": 0.92, + "content": "S = \\| \\Theta _ { \\gamma , d , m } \\| _ { 1 } \\asymp \\| \\Theta _ { h , m , j } \\| _ { 1 } \\asymp \\| \\Theta _ { \\times } \\| _ { 1 }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 574, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 462, + 590 + ], + "score": 1.0, + "content": "Also, following Yarotsky (2017) and Petersen & Voigtlaender (2017), a proper selection of", + "type": "text" + }, + { + "bbox": [ + 463, + 576, + 504, + 588 + ], + "score": 0.92, + "content": "L = | \\Theta _ { \\times } |", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 585, + 349, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 123, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 590, + 176, + 602 + ], + "score": 0.93, + "content": "B = \\| \\Theta _ { \\times } \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 587, + 214, + 606 + ], + "score": 1.0, + "content": "provides", + "type": "text" + }, + { + "bbox": [ + 214, + 587, + 283, + 603 + ], + "score": 0.93, + "content": "\\epsilon _ { \\times } \\lesssim \\| \\Theta _ { \\times } \\| _ { 0 } ^ { - 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\\beta / ( D - 1 ) } + \\epsilon _ { \\times } .", + "type": "interline_equation", + "image_path": "1c462416da948e58732d16b5247713638a980ec75680ea3f0e9c8341a5ee690b.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 227, + 437, + 383, + 465 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 319, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 320, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 214, + 488 + ], + "score": 1.0, + "content": "Combining the bounds for", + "type": "text" + }, + { + "bbox": [ + 214, + 475, + 235, + 487 + ], + "score": 0.92, + "content": "T _ { 1 , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 473, + 253, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 475, + 273, + 487 + ], + "score": 0.92, + "content": "T _ { 2 , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 473, + 320, + 488 + ], + "score": 1.0, + "content": ", we bound", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 473, + 320, + 488 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 489, + 462, + 559 + ], + "lines": [ + { + "bbox": [ + 149, + 489, + 462, + 559 + ], + "spans": [ + { + "bbox": [ + 149, + 489, + 462, + 559 + ], + "score": 0.95, + "content": "\\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}", + "type": "interline_equation", + "image_path": "17ca5910080208ec8a052c5a4fb1c9a5d97427d8cc3cfb01cef837a1a950a61f.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 149, + 489, + 462, + 512.3333333333334 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 149, + 512.3333333333334, + 462, + 535.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 149, + 535.6666666666667, + 462, + 559.0000000000001 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 562, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 104, + 560, + 502, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 228, + 578 + ], + "score": 1.0, + "content": "Here, we consider a parameter", + "type": "text" + }, + { + "bbox": [ + 229, + 563, + 306, + 577 + ], + "score": 0.93, + "content": "\\begin{array} { r } { S = \\sum _ { d \\in [ D ] } \\| \\Theta _ { d } \\| _ { 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 560, + 345, + 578 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 345, + 563, + 502, + 576 + ], + "score": 0.92, + "content": "S = \\| \\Theta _ { \\gamma , d , m } \\| _ { 1 } \\asymp \\| \\Theta _ { h , m , j } \\| _ { 1 } \\asymp \\| \\Theta _ { \\times } \\| _ { 1 }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 574, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 462, + 590 + ], + "score": 1.0, + "content": "Also, following Yarotsky (2017) and Petersen & Voigtlaender (2017), a proper selection of", + "type": "text" + }, + { + "bbox": [ + 463, + 576, + 504, + 588 + ], + "score": 0.92, + "content": "L = | \\Theta _ { \\times } |", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 585, + 349, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 123, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 590, + 176, + 602 + ], + "score": 0.93, + "content": "B = \\| \\Theta _ { \\times } \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 587, + 214, + 606 + ], + "score": 1.0, + "content": "provides", + "type": "text" + }, + { + "bbox": [ + 214, + 587, + 283, + 603 + ], + "score": 0.93, + "content": "\\epsilon _ { \\times } \\lesssim \\| \\Theta _ { \\times } \\| _ { 0 } ^ { - \\beta / D }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 585, + 349, + 604 + ], + "score": 1.0, + "content": "}´β{D0 . Then, we have", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 560, + 504, + 606 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 605, + 411, + 645 + ], + "lines": [ + { + "bbox": [ + 199, + 605, + 411, + 645 + ], + "spans": [ + { + "bbox": [ + 199, + 605, + 411, + 645 + ], + "score": 0.93, + "content": "\\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 } .", + "type": "interline_equation", + "image_path": "b8173faebdf3e1ca9bd6e6a523e04439b02b7d46c88eb2527f521f3b06174517.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 605, + 411, + 625.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 199, + 625.0, + 411, + 645.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 257, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 257, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 131, + 661 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 649, + 139, + 658 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 648, + 257, + 661 + ], + "score": 1.0, + "content": "is finite, we obtain the result.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 648, + 257, + 661 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 672, + 230, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 230, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 230, + 685 + ], + "score": 1.0, + "content": "B.4 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 693, + 344, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 692, + 344, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 186, + 707 + ], + "score": 1.0, + "content": "By the definition of", + "type": "text" + }, + { + "bbox": [ + 187, + 694, + 193, + 705 + ], + "score": 0.86, + "content": "\\widehat g", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 692, + 344, + 707 + ], + "score": 1.0, + "content": "in (2), the following inequality holds", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 692, + 344, + 707 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 708, + 464, + 730 + ], + "lines": [ + { + "bbox": [ + 146, + 708, + 464, + 730 + ], + "spans": [ + { + "bbox": [ + 146, + 708, + 464, + 730 + ], + "score": 0.92, + "content": "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 ) ,", + "type": "interline_equation", + "image_path": "d36f8f5386383d724a3e648a7f9f17852a6da57c033f7262bad182c15ebc8ad9.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 146, + 708, + 464, + 730 + ], + "spans": [], + "index": 29 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 184, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 185, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 157, + 96 + ], + "score": 1.0, + "content": "for arbitrary", + "type": "text" + }, + { + "bbox": [ + 158, + 83, + 181, + 94 + ], + "score": 0.9, + "content": "g \\in { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 81, + 185, + 96 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 268, + 111 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 269, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 206, + 113 + ], + "score": 1.0, + "content": "We consider a bound for", + "type": "text" + }, + { + "bbox": [ + 206, + 99, + 255, + 112 + ], + "score": 0.94, + "content": "d _ { \\mathcal { F } } ( P ^ { \\ast } , P _ { \\hat { g } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 97, + 269, + 113 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 115, + 455, + 183 + ], + "lines": [ + { + "bbox": [ + 155, + 115, + 455, + 183 + ], + "spans": [ + { + "bbox": [ + 155, + 115, + 455, + 183 + ], + "score": 0.95, + "content": "\\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 - 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This result follows a standard technique of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 333, + 438, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 438, + 346 + ], + "score": 1.0, + "content": "the empirical process theory and we provide its outline for a sake of completeness.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 168, + 361 + ], + "score": 1.0, + "content": "Lemma 4. Let", + "type": "text" + }, + { + "bbox": [ + 168, + 348, + 178, + 358 + ], + "score": 0.53, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 347, + 345, + 361 + ], + "score": 1.0, + "content": "be a some set of measurable functions and", + "type": "text" + }, + { + "bbox": [ + 345, + 348, + 411, + 360 + ], + "score": 0.91, + "content": "X _ { 1 } , . . . , X _ { n } \\sim P", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "be i.i.d. n observations.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 161, + 371 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 162, + 358, + 234, + 371 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { P } [ h ^ { 2 } ( X ) ] \\leqslant \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 359, + 254, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 359, + 307, + 371 + ], + "score": 0.92, + "content": "\\| h \\| _ { L ^ { \\infty } } < C _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 359, + 421, + 371 + ], + "score": 1.0, + "content": "hold with finite parameters", + "type": "text" + }, + { + "bbox": [ + 421, + 358, + 452, + 369 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 359, + 470, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 359, + 502, + 370 + ], + "score": 0.92, + "content": "C _ { h } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 359, + 506, + 371 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 370, + 295, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 222, + 382 + ], + "score": 1.0, + "content": "Then, there exists a constant", + "type": "text" + }, + { + "bbox": [ + 222, + 371, + 234, + 381 + ], + "score": 0.84, + "content": "C _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 370, + 295, + 382 + ], + "score": 1.0, + "content": "and we obtain", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 385, + 484, + 460 + ], + "lines": [ + { + "bbox": [ + 127, + 385, + 484, + 460 + ], + "spans": [ + { + "bbox": [ + 127, + 385, + 484, + 460 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "b70491ed7f2676d24d835ffb7afbbe4935d0c28236950d4bcae1ee4b243db677.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 127, + 385, + 484, + 410.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 127, + 410.0, + 484, + 435.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 127, + 435.0, + 484, + 460.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 321, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 322, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 204, + 478 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 204, + 464, + 265, + 476 + ], + "score": 0.87, + "content": "1 - 2 \\exp ( - \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 463, + 293, + 478 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 294, + 464, + 318, + 475 + ], + "score": 0.88, + "content": "\\tau > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 463, + 322, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 507, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 507, + 501 + ], + "score": 1.0, + "content": "Proof. At the beginning, we bound an expectation of the empirical process (Steinwart & Christmann,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "2008; Bartlett et al., 2005; Massart, 2000; Sriperumbudur et al., 2012). Afterward, we evaluate a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 510, + 356, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 356, + 523 + ], + "score": 1.0, + "content": "concentration of the empirical process around the expectation.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 104, + 527, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "score": 1.0, + "content": "By applying the symmetrization and concentration techniques (Proposition 7.10 in Steinwart &", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 538, + 232, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 232, + 549 + ], + "score": 1.0, + "content": "Christmann (2008)), we obtain", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 553, + 473, + 594 + ], + "lines": [ + { + "bbox": [ + 136, + 553, + 473, + 594 + ], + "spans": [ + { + "bbox": [ + 136, + 553, + 473, + 594 + ], + "score": 0.94, + "content": "\\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] ,", + "type": "interline_equation", + "image_path": "1ce850c8246d003b5b70992d14295a920619fbb39e6448d1988ed020cd0dd243.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 136, + 553, + 473, + 566.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 136, + 566.6666666666666, + 473, + 580.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 136, + 580.3333333333333, + 473, + 593.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 597, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 133, + 610 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 599, + 163, + 609 + ], + "score": 0.92, + "content": "u _ { i } \\sim \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "is the Rademacher variable which takes 0 or 1 with probability 0.5. 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This result follows a standard technique of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 333, + 438, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 438, + 346 + ], + "score": 1.0, + "content": "the empirical process theory and we provide its outline for a sake of completeness.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 321, + 506, + 346 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 168, + 361 + ], + "score": 1.0, + "content": "Lemma 4. Let", + "type": "text" + }, + { + "bbox": [ + 168, + 348, + 178, + 358 + ], + "score": 0.53, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 347, + 345, + 361 + ], + "score": 1.0, + "content": "be a some set of measurable functions and", + "type": "text" + }, + { + "bbox": [ + 345, + 348, + 411, + 360 + ], + "score": 0.91, + "content": "X _ { 1 } , . . . , X _ { n } \\sim P", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "be i.i.d. n observations.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 161, + 371 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 162, + 358, + 234, + 371 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { P } [ h ^ { 2 } ( X ) ] \\leqslant \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 359, + 254, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 359, + 307, + 371 + ], + "score": 0.92, + "content": "\\| h \\| _ { L ^ { \\infty } } < C _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 359, + 421, + 371 + ], + "score": 1.0, + "content": "hold with finite parameters", + "type": "text" + }, + { + "bbox": [ + 421, + 358, + 452, + 369 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 359, + 470, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 359, + 502, + 370 + ], + "score": 0.92, + "content": "C _ { h } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 359, + 506, + 371 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 370, + 295, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 222, + 382 + ], + "score": 1.0, + "content": "Then, there exists a constant", + "type": "text" + }, + { + "bbox": [ + 222, + 371, + 234, + 381 + ], + "score": 0.84, + "content": "C _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 370, + 295, + 382 + ], + "score": 1.0, + "content": "and we obtain", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 347, + 506, + 382 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 385, + 484, + 460 + ], + "lines": [ + { + "bbox": [ + 127, + 385, + 484, + 460 + ], + "spans": [ + { + "bbox": [ + 127, + 385, + 484, + 460 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "b70491ed7f2676d24d835ffb7afbbe4935d0c28236950d4bcae1ee4b243db677.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 127, + 385, + 484, + 410.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 127, + 410.0, + 484, + 435.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 127, + 435.0, + 484, + 460.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 321, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 322, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 204, + 478 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 204, + 464, + 265, + 476 + ], + "score": 0.87, + "content": "1 - 2 \\exp ( - \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 463, + 293, + 478 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 294, + 464, + 318, + 475 + ], + "score": 0.88, + "content": "\\tau > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 463, + 322, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 463, + 322, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 507, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 507, + 501 + ], + "score": 1.0, + "content": "Proof. At the beginning, we bound an expectation of the empirical process (Steinwart & Christmann,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "2008; Bartlett et al., 2005; Massart, 2000; Sriperumbudur et al., 2012). Afterward, we evaluate a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 510, + 356, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 356, + 523 + ], + "score": 1.0, + "content": "concentration of the empirical process around the expectation.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 487, + 507, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 527, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "score": 1.0, + "content": "By applying the symmetrization and concentration techniques (Proposition 7.10 in Steinwart &", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 538, + 232, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 232, + 549 + ], + "score": 1.0, + "content": "Christmann (2008)), we obtain", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 527, + 506, + 549 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 553, + 473, + 594 + ], + "lines": [ + { + "bbox": [ + 136, + 553, + 473, + 594 + ], + "spans": [ + { + "bbox": [ + 136, + 553, + 473, + 594 + ], + "score": 0.94, + "content": "\\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] ,", + "type": "interline_equation", + "image_path": "1ce850c8246d003b5b70992d14295a920619fbb39e6448d1988ed020cd0dd243.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 136, + 553, + 473, + 566.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 136, + 566.6666666666666, + 473, + 580.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 136, + 580.3333333333333, + 473, + 593.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 597, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 133, + 610 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 599, + 163, + 609 + ], + "score": 0.92, + "content": "u _ { i } \\sim \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "is the Rademacher variable which takes 0 or 1 with probability 0.5. 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(2005)), then obtain", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 109, + 494, + 150 + ], + "lines": [ + { + "bbox": [ + 116, + 109, + 494, + 150 + ], + "spans": [ + { + "bbox": [ + 116, + 109, + 494, + 150 + ], + "score": 0.94, + "content": "\\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 } ) } ,", + "type": "interline_equation", + "image_path": "d2a1a5fb9571b710f684a9df06c55cda3ac83623c2c7782d7816f1b6b7490a8a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 116, + 109, + 494, + 122.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 116, + 122.66666666666667, + 494, + 136.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 116, + 136.33333333333334, + 494, + 150.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 365, + 179 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 365, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 203, + 180 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 204, + 167, + 260, + 179 + ], + "score": 0.93, + "content": "1 - \\exp ( - \\tau ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 165, + 288, + 180 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 288, + 167, + 316, + 177 + ], + "score": 0.91, + "content": "\\tau ^ { \\prime } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 165, + 334, + 180 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 167, + 361, + 177 + ], + "score": 0.9, + "content": "\\theta ^ { \\prime } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 165, + 365, + 180 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 103, + 179, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 103, + 179, + 127, + 200 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 127, + 183, + 156, + 196 + ], + "score": 0.89, + "content": "\\| { \\bf \\nabla } \\cdot { \\bf \\nabla } \\| _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 179, + 285, + 200 + ], + "score": 1.0, + "content": "be an empirical norm as ˇ ˇı", + "type": "text" + }, + { + "bbox": [ + 286, + 182, + 414, + 198 + ], + "score": 0.9, + "content": "\\begin{array} { r l r } { \\| f \\| _ { n } ^ { 2 } } & { { } = } & { n ^ { - 1 } \\sum _ { i \\in [ n ] } f ( X _ { i } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 179, + 507, + 200 + ], + "score": 1.0, + "content": ". 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\\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}", + "type": "interline_equation", + "image_path": "1a10f31982fe724de76bb6fbe9ec64ef90d09ab59f0ad4a365e3a2424a18b029.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 125, + 360, + 486, + 387.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 125, + 387.0, + 486, + 414.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 125, + 414.0, + 486, + 441.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 104, + 442, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 176, + 460 + ], + "score": 1.0, + "content": "where we define", + "type": "text" + }, + { + "bbox": [ + 176, + 445, + 312, + 457 + ], + "score": 0.92, + "content": "\\mathcal { H } = \\left\\{ h = f \\circ g \\vert f \\in \\mathcal { F } , g \\in \\mathcal { G } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 442, + 506, + 460 + ], + "score": 1.0, + "content": ". To apply Lemma 4, we investigate a covering", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 456, + 211, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 139, + 469 + ], + "score": 1.0, + "content": "number", + "type": "text" + }, + { + "bbox": [ + 140, + 457, + 207, + 469 + ], + "score": 0.92, + "content": "\\mathcal { N } ( \\epsilon , \\mathcal { H } , \\| \\cdot \\| _ { L ^ { \\infty } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 456, + 211, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 387, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 386, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 205, + 484 + ], + "score": 1.0, + "content": "Lemma 5. Assume that", + "type": "text" + }, + { + "bbox": [ + 205, + 471, + 231, + 483 + ], + "score": 0.9, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 471, + 241, + 484 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 241, + 472, + 253, + 482 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 471, + 386, + 484 + ], + "score": 1.0, + "content": "-Lipschitz continuous. We obtain", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 487, + 486, + 502 + ], + "lines": [ + { + "bbox": [ + 123, + 487, + 486, + 502 + ], + "spans": [ + { + "bbox": [ + 123, + 487, + 486, + 502 + ], + "score": 0.89, + "content": "\\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 } } )", + "type": "interline_equation", + "image_path": "73c61559df1947ed3a944d7011c2ccba86211047d09e844a52dd7b5301606ee7.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 123, + 487, + 486, + 502 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 504, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 153, + 526 + ], + "score": 1.0, + "content": "Proof. Fix", + "type": "text" + }, + { + "bbox": [ + 153, + 514, + 179, + 524 + ], + "score": 0.84, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 513, + 201, + 526 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 202, + 514, + 233, + 524 + ], + "score": 0.91, + "content": "G \\subset { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 513, + 252, + 526 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 514, + 285, + 524 + ], + "score": 0.9, + "content": "F \\subset { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 513, + 445, + 526 + ], + "score": 1.0, + "content": "be covering sets as a set of centers of", + "type": "text" + }, + { + "bbox": [ + 446, + 516, + 451, + 524 + ], + "score": 0.78, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "-balls for the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 142, + 537 + ], + "score": 1.0, + "content": "covering", + "type": "text" + }, + { + "bbox": [ + 142, + 525, + 150, + 536 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 524, + 168, + 537 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 525, + 177, + 534 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 524, + 226, + 537 + ], + "score": 1.0, + "content": ". Obviously,", + "type": "text" + }, + { + "bbox": [ + 226, + 524, + 316, + 537 + ], + "score": 0.92, + "content": "| G | = \\mathcal { N } ( \\epsilon , \\mathcal { G } , \\| \\cdot \\| _ { L ^ { \\infty } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 524, + 334, + 537 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 524, + 426, + 537 + ], + "score": 0.92, + "content": "| F | = \\mathcal { N } ( \\epsilon , \\mathcal { F } , \\| \\cdot \\| _ { L ^ { \\infty } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 524, + 506, + 537 + ], + "score": 1.0, + "content": ". 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(2005)), then obtain", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 105 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 109, + 494, + 150 + ], + "lines": [ + { + "bbox": [ + 116, + 109, + 494, + 150 + ], + "spans": [ + { + "bbox": [ + 116, + 109, + 494, + 150 + ], + "score": 0.94, + "content": "\\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 } ) } ,", + "type": "interline_equation", + "image_path": "d2a1a5fb9571b710f684a9df06c55cda3ac83623c2c7782d7816f1b6b7490a8a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 116, + 109, + 494, + 122.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 116, + 122.66666666666667, + 494, + 136.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 116, + 136.33333333333334, + 494, + 150.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 365, + 179 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 365, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 203, + 180 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 204, + 167, + 260, + 179 + ], + "score": 0.93, + "content": "1 - \\exp ( - \\tau ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 165, + 288, + 180 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 288, + 167, + 316, + 177 + ], + "score": 0.91, + "content": "\\tau ^ { \\prime } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 165, + 334, + 180 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 167, + 361, + 177 + ], + "score": 0.9, + "content": "\\theta ^ { \\prime } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 165, + 365, + 180 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 165, + 365, + 180 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 103, + 179, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 103, + 179, + 127, + 200 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 127, + 183, + 156, + 196 + ], + "score": 0.89, + "content": "\\| { \\bf \\nabla } \\cdot { \\bf \\nabla } \\| _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 179, + 285, + 200 + ], + "score": 1.0, + "content": "be an empirical norm as ˇ ˇı", + "type": "text" + }, + { + "bbox": [ + 286, + 182, + 414, + 198 + ], + "score": 0.9, + "content": "\\begin{array} { r l r } { \\| f \\| _ { n } ^ { 2 } } & { { } = } & { n ^ { - 1 } \\sum _ { i \\in [ n ] } f ( X _ { i } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 179, + 507, + 200 + ], + "score": 1.0, + "content": ". About the term", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 196, + 433, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 250, + 217 + ], + "score": 0.93, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 196, + 433, + 218 + ], + "score": 1.0, + "content": ", we apply the chaining technique and obtain", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 103, + 179, + 507, + 218 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 221, + 468, + 296 + ], + "lines": [ + { + "bbox": [ + 125, + 221, + 468, + 296 + ], + "spans": [ + { + "bbox": [ + 125, + 221, + 468, + 296 + ], + "score": 0.94, + "content": "\\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}", + "type": "interline_equation", + "image_path": "45ae6f45ab663afc8436f73d5e2a8897f7c1309dd82843745b2fd195bfc014b1.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 125, + 221, + 468, + 246.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 125, + 246.0, + 468, + 271.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 125, + 271.0, + 468, + 296.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 300, + 472, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 473, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 459, + 313 + ], + "score": 1.0, + "content": "where the last inequality follows a bound for an empirical norm and the boundedness of", + "type": "text" + }, + { + "bbox": [ + 459, + 301, + 468, + 310 + ], + "score": 0.84, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 299, + 473, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 299, + 473, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 411, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 410, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 410, + 331 + ], + "score": 1.0, + "content": "Combining (10), (11) and (12) and changing variables, we obtain the result.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 316, + 410, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 379, + 356 + ], + "lines": [ + { + "bbox": [ + 104, + 339, + 380, + 358 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 146, + 358 + ], + "score": 1.0, + "content": "To bound", + "type": "text" + }, + { + "bbox": [ + 146, + 344, + 153, + 353 + ], + "score": 0.79, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 339, + 174, + 358 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 174, + 341, + 245, + 355 + ], + "score": 0.93, + "content": "\\widetilde { X } _ { 1 } , . . . , \\widetilde { X } _ { m } \\sim P _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 339, + 380, + 358 + ], + "score": 1.0, + "content": ", we consider the following value", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 339, + 380, + 358 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 360, + 486, + 441 + ], + "lines": [ + { + "bbox": [ + 125, + 360, + 486, + 441 + ], + "spans": [ + { + "bbox": [ + 125, + 360, + 486, + 441 + ], + "score": 0.94, + "content": "\\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}", + "type": "interline_equation", + "image_path": "1a10f31982fe724de76bb6fbe9ec64ef90d09ab59f0ad4a365e3a2424a18b029.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 125, + 360, + 486, + 387.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 125, + 387.0, + 486, + 414.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 125, + 414.0, + 486, + 441.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 104, + 442, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 176, + 460 + ], + "score": 1.0, + "content": "where we define", + "type": "text" + }, + { + "bbox": [ + 176, + 445, + 312, + 457 + ], + "score": 0.92, + "content": "\\mathcal { H } = \\left\\{ h = f \\circ g \\vert f \\in \\mathcal { F } , g \\in \\mathcal { G } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 442, + 506, + 460 + ], + "score": 1.0, + "content": ". 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Assume that", + "type": "text" + }, + { + "bbox": [ + 205, + 471, + 231, + 483 + ], + "score": 0.9, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 471, + 241, + 484 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 241, + 472, + 253, + 482 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 471, + 386, + 484 + ], + "score": 1.0, + "content": "-Lipschitz continuous. We obtain", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 471, + 386, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 487, + 486, + 502 + ], + "lines": [ + { + "bbox": [ + 123, + 487, + 486, + 502 + ], + "spans": [ + { + "bbox": [ + 123, + 487, + 486, + 502 + ], + "score": 0.89, + "content": "\\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 } } )", + "type": "interline_equation", + "image_path": "73c61559df1947ed3a944d7011c2ccba86211047d09e844a52dd7b5301606ee7.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 123, + 487, + 486, + 502 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 504, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 153, + 526 + ], + "score": 1.0, + "content": "Proof. Fix", + "type": "text" + }, + { + "bbox": [ + 153, + 514, + 179, + 524 + ], + "score": 0.84, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 513, + 201, + 526 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 202, + 514, + 233, + 524 + ], + "score": 0.91, + "content": "G \\subset { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 513, + 252, + 526 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 514, + 285, + 524 + ], + "score": 0.9, + "content": "F \\subset { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 513, + 445, + 526 + ], + "score": 1.0, + "content": "be covering sets as a set of centers of", + "type": "text" + }, + { + "bbox": [ + 446, + 516, + 451, + 524 + ], + "score": 0.78, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "-balls for the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 142, + 537 + ], + "score": 1.0, + "content": "covering", + "type": "text" + }, + { + "bbox": [ + 142, + 525, + 150, + 536 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 524, + 168, + 537 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 525, + 177, + 534 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 524, + 226, + 537 + ], + "score": 1.0, + "content": ". Obviously,", + "type": "text" + }, + { + "bbox": [ + 226, + 524, + 316, + 537 + ], + "score": 0.92, + "content": "| G | = \\mathcal { N } ( \\epsilon , \\mathcal { G } , \\| \\cdot \\| _ { L ^ { \\infty } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 524, + 334, + 537 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 524, + 426, + 537 + ], + "score": 0.92, + "content": "| F | = \\mathcal { N } ( \\epsilon , \\mathcal { F } , \\| \\cdot \\| _ { L ^ { \\infty } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 524, + 506, + 537 + ], + "score": 1.0, + "content": ". We define a subset", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 513, + 506, + 537 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 541, + 388, + 569 + ], + "lines": [ + { + "bbox": [ + 222, + 541, + 388, + 569 + ], + "spans": [ + { + "bbox": [ + 222, + 541, + 388, + 569 + ], + "score": 0.94, + "content": "H : = \\left\\{ h = f \\circ g \\vert g \\in G , h \\in F \\right\\} \\subset \\mathcal { H } ,", + "type": "interline_equation", + "image_path": "8a74c5daf2b3435f74bff3ae49944a3274c2c9c692b86f3dc16635403273f512.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 222, + 541, + 388, + 569 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 238, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 239, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 167, + 588 + ], + "score": 1.0, + "content": "and we known", + "type": "text" + }, + { + "bbox": [ + 167, + 574, + 235, + 587 + ], + "score": 0.92, + "content": "| H | = | G | \\times | F |", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 572, + 239, + 588 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 572, + 239, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 506, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 138, + 605 + ], + "score": 1.0, + "content": "For any", + "type": "text" + }, + { + "bbox": [ + 139, + 592, + 164, + 602 + ], + "score": 0.9, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 589, + 211, + 605 + ], + "score": 1.0, + "content": ", there exist", + "type": "text" + }, + { + "bbox": [ + 211, + 592, + 235, + 603 + ], + "score": 0.91, + "content": "g \\in { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 589, + 252, + 605 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 592, + 278, + 603 + ], + "score": 0.92, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 589, + 301, + 605 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 301, + 592, + 341, + 603 + ], + "score": 0.91, + "content": "f = f \\circ g", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 589, + 506, + 605 + ], + "score": 1.0, + "content": "holds. 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Let", + "type": "text" + }, + { + "bbox": [ + 453, + 603, + 502, + 614 + ], + "score": 0.92, + "content": "h ^ { \\prime } = f ^ { \\prime } \\circ g ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 602, + 506, + 616 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 613, + 223, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 223, + 626 + ], + "score": 1.0, + "content": "and we measure the distance", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 589, + 506, + 626 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 629, + 422, + 706 + ], + "lines": [ + { + "bbox": [ + 188, + 629, + 422, + 706 + ], + "spans": [ + { + "bbox": [ + 188, + 629, + 422, + 706 + ], + "score": 0.95, + "content": "\\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}", + "type": "interline_equation", + "image_path": "b79659cf5ccb7b50f7cde03b49e7679a36c1782e9b37ff25668f6e4bb05e83d3.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 188, + 629, + 422, + 644.4 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 188, + 644.4, + 422, + 659.8 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 188, + 659.8, + 422, + 675.1999999999999 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 188, + 675.1999999999999, + 422, + 690.5999999999999 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 188, + 690.5999999999999, + 422, + 705.9999999999999 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 342, + 722 + ], + "score": 1.0, + "content": "Here, the third inequality follows the Lipschitz property of", + "type": "text" + }, + { + "bbox": [ + 342, + 710, + 367, + 721 + ], + "score": 0.89, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 708, + 451, + 722 + ], + "score": 1.0, + "content": ". Here, we know that", + "type": "text" + }, + { + "bbox": [ + 452, + 710, + 461, + 720 + ], + "score": 0.84, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "is covered", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 119, + 733 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 721, + 160, + 732 + ], + "score": 0.93, + "content": "( L _ { 1 } + 1 ) \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 720, + 246, + 733 + ], + "score": 1.0, + "content": "-balls with the center", + "type": "text" + }, + { + "bbox": [ + 246, + 723, + 255, + 730 + ], + "score": 0.86, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 720, + 285, + 733 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 285, + 721, + 353, + 732 + ], + "score": 0.93, + "content": "| H | = | G | \\times | F |", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 720, + 423, + 733 + ], + "score": 1.0, + "content": ", the result holds.", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 506, + 732 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 708, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 280, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 281, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 281, + 96 + ], + "score": 1.0, + "content": "Now, we have the following entropy bound", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 96, + 457, + 172 + ], + "lines": [ + { + "bbox": [ + 155, + 96, + 457, + 172 + ], + "spans": [ + { + "bbox": [ + 155, + 96, + 457, + 172 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "314b08165b4d70c54db8c9640ce17ce8ce7dfb898fe871033f7ea4706feecd55.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 155, + 96, + 457, + 121.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 155, + 121.33333333333333, + 457, + 146.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 155, + 146.66666666666666, + 457, + 172.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 173, + 504, + 213 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 507, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 126, + 190 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 173, + 237, + 189 + ], + "score": 0.9, + "content": "\\begin{array} { r } { D _ { g } : = \\prod _ { \\ell \\in [ L _ { g } + 1 ] } ( D _ { \\ell } + 1 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 171, + 255, + 190 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 174, + 367, + 189 + ], + "score": 0.92, + "content": "\\begin{array} { r } { D _ { f } : = \\prod _ { \\ell \\in [ L _ { f } + 1 ] } ( D _ { \\ell } + 1 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 171, + 387, + 190 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 387, + 173, + 484, + 187 + ], + "score": 0.91, + "content": "\\Gamma _ { f } : = 2 ( L _ { f } + 1 ) D _ { f } B _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 171, + 507, + 190 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 187, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 225, + 202 + ], + "score": 0.89, + "content": "N _ { \\epsilon } ( \\widetilde { \\mathcal { F } } ) : = \\log \\mathcal { N } ( \\epsilon , \\widetilde { \\mathcal { F } } , \\| \\cdot \\| _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 187, + 506, + 204 + ], + "score": 1.0, + "content": "for brevity. Here, we apply Lemma 8 in Schmidt-Hieber (2017) for the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 200, + 424, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 181, + 214 + ], + "score": 1.0, + "content": "entropy bound for", + "type": "text" + }, + { + "bbox": [ + 181, + 202, + 189, + 212 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 200, + 207, + 214 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 202, + 216, + 211 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 200, + 424, + 214 + ], + "score": 1.0, + "content": ". Using the entropy bound and Lemma 4, we obtain", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 217, + 493, + 374 + ], + "lines": [ + { + "bbox": [ + 117, + 217, + 493, + 374 + ], + "spans": [ + { + "bbox": [ + 117, + 217, + 493, + 374 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "c6a35d871a7ef2e5880ef4c018bfe3ccc8f73dba709f276734969c8158edf8ae.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 117, + 217, + 493, + 269.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 117, + 269.3333333333333, + 493, + 321.66666666666663 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 117, + 321.66666666666663, + 493, + 373.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 149, + 395 + ], + "score": 1.0, + "content": "with some", + "type": "text" + }, + { + "bbox": [ + 149, + 381, + 173, + 393 + ], + "score": 0.61, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 378, + 178, + 395 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 179, + 379, + 411, + 393 + ], + "score": 0.89, + "content": "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 }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 378, + 432, + 395 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 433, + 380, + 505, + 393 + ], + "score": 0.91, + "content": "A _ { 3 } = \\tau C _ { h } ( 2 / 3 +", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 389, + 434, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 120, + 407 + ], + "score": 0.63, + "content": "C _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 389, + 180, + 414 + ], + "score": 1.0, + "content": "q. Also, we set", + "type": "text" + }, + { + "bbox": [ + 180, + 394, + 365, + 410 + ], + "score": 0.92, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 389, + 434, + 414 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 413, + 412, + 437 + ], + "lines": [ + { + "bbox": [ + 199, + 413, + 412, + 437 + ], + "spans": [ + { + "bbox": [ + 199, + 413, + 412, + 437 + ], + "score": 0.85, + "content": "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 } .", + "type": "interline_equation", + "image_path": "c0c27879c94e4bf24392d0c1926fc20f7689d5fba1da7188e0521300a4de67e5.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 199, + 413, + 412, + 437 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 401, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 401, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 133, + 460 + ], + "score": 1.0, + "content": "About", + "type": "text" + }, + { + "bbox": [ + 134, + 447, + 140, + 456 + ], + "score": 0.78, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 445, + 185, + 460 + ], + "score": 1.0, + "content": ", we define", + "type": "text" + }, + { + "bbox": [ + 185, + 446, + 280, + 459 + ], + "score": 0.93, + "content": "\\Gamma _ { g } : = 2 ( L _ { f } + 1 ) D _ { g } B _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 445, + 401, + 460 + ], + "score": 1.0, + "content": "and obtain a similar bound as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 461, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 132, + 461, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 132, + 461, + 506, + 506 + ], + "score": 0.87, + "content": "\\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}", + "type": "interline_equation", + "image_path": "d37a9814e431549b5d0424f6689ca2ea0f7491e0b17d45aa964d3cd02017f6a8.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 132, + 461, + 506, + 476.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 132, + 476.0, + 506, + 491.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 132, + 491.0, + 506, + 506.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 504, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 132, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 132, + 505 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 500, + 223, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 123, + 516 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 502, + 217, + 514 + ], + "score": 0.9, + "content": "A _ { 3 } \\stackrel { - } { = } 2 \\tau C _ { h } ( 2 / 3 + \\check { C } _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 500, + 223, + 516 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.0 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 504, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 133, + 532 + ], + "score": 1.0, + "content": "About", + "type": "text" + }, + { + "bbox": [ + 134, + 520, + 142, + 529 + ], + "score": 0.8, + "content": "\\romannumeral 2", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 519, + 505, + 532 + ], + "score": 1.0, + "content": ", we evaluate the error from approximation by constructing a specific deep neural network", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 529, + 495, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 277, + 543 + ], + "score": 1.0, + "content": "for generators. We apply Lemma 3 and let", + "type": "text" + }, + { + "bbox": [ + 278, + 530, + 343, + 542 + ], + "score": 0.93, + "content": "\\dot { \\boldsymbol g } = ( \\dot { g } _ { 1 } , . . . , \\dot { g } _ { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 529, + 495, + 543 + ], + "score": 1.0, + "content": "be a generator specified in Lemma 3.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 560, + 402, + 732 + ], + "lines": [ + { + "bbox": [ + 208, + 560, + 402, + 732 + ], + "spans": [ + { + "bbox": [ + 208, + 560, + 402, + 732 + ], + "score": 0.92, + "content": "\\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 ) - 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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}", + "type": "interline_equation", + "image_path": "314b08165b4d70c54db8c9640ce17ce8ce7dfb898fe871033f7ea4706feecd55.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 155, + 96, + 457, + 121.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 155, + 121.33333333333333, + 457, + 146.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 155, + 146.66666666666666, + 457, + 172.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 173, + 504, + 213 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 507, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 126, + 190 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 173, + 237, + 189 + ], + "score": 0.9, + "content": "\\begin{array} { r } { D _ { g } : = \\prod _ { \\ell \\in [ L _ { g } + 1 ] } ( D _ { \\ell } + 1 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 171, + 255, + 190 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 174, + 367, + 189 + ], + "score": 0.92, + "content": "\\begin{array} { r } { D _ { f } : = \\prod _ { \\ell \\in [ L _ { f } + 1 ] } ( D _ { \\ell } + 1 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 171, + 387, + 190 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 387, + 173, + 484, + 187 + ], + "score": 0.91, + "content": "\\Gamma _ { f } : = 2 ( L _ { f } + 1 ) D _ { f } B _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 171, + 507, + 190 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 187, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 225, + 202 + ], + "score": 0.89, + "content": "N _ { \\epsilon } ( \\widetilde { \\mathcal { F } } ) : = \\log \\mathcal { N } ( \\epsilon , \\widetilde { \\mathcal { F } } , \\| \\cdot \\| _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 187, + 506, + 204 + ], + "score": 1.0, + "content": "for brevity. Here, we apply Lemma 8 in Schmidt-Hieber (2017) for the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 200, + 424, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 181, + 214 + ], + "score": 1.0, + "content": "entropy bound for", + "type": "text" + }, + { + "bbox": [ + 181, + 202, + 189, + 212 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 200, + 207, + 214 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 202, + 216, + 211 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 200, + 424, + 214 + ], + "score": 1.0, + "content": ". Using the entropy bound and Lemma 4, we obtain", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 171, + 507, + 214 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 217, + 493, + 374 + ], + "lines": [ + { + "bbox": [ + 117, + 217, + 493, + 374 + ], + "spans": [ + { + "bbox": [ + 117, + 217, + 493, + 374 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "c6a35d871a7ef2e5880ef4c018bfe3ccc8f73dba709f276734969c8158edf8ae.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 117, + 217, + 493, + 269.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 117, + 269.3333333333333, + 493, + 321.66666666666663 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 117, + 321.66666666666663, + 493, + 373.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 149, + 395 + ], + "score": 1.0, + "content": "with some", + "type": "text" + }, + { + "bbox": [ + 149, + 381, + 173, + 393 + ], + "score": 0.61, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 378, + 178, + 395 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 179, + 379, + 411, + 393 + ], + "score": 0.89, + "content": "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 }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 378, + 432, + 395 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 433, + 380, + 505, + 393 + ], + "score": 0.91, + "content": "A _ { 3 } = \\tau C _ { h } ( 2 / 3 +", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 389, + 434, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 120, + 407 + ], + "score": 0.63, + "content": "C _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 389, + 180, + 414 + ], + "score": 1.0, + "content": "q. Also, we set", + "type": "text" + }, + { + "bbox": [ + 180, + 394, + 365, + 410 + ], + "score": 0.92, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 389, + 434, + 414 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 378, + 505, + 414 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 413, + 412, + 437 + ], + "lines": [ + { + "bbox": [ + 199, + 413, + 412, + 437 + ], + "spans": [ + { + "bbox": [ + 199, + 413, + 412, + 437 + ], + "score": 0.85, + "content": "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 } .", + "type": "interline_equation", + "image_path": "c0c27879c94e4bf24392d0c1926fc20f7689d5fba1da7188e0521300a4de67e5.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 199, + 413, + 412, + 437 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 401, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 401, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 133, + 460 + ], + "score": 1.0, + "content": "About", + "type": "text" + }, + { + "bbox": [ + 134, + 447, + 140, + 456 + ], + "score": 0.78, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 445, + 185, + 460 + ], + "score": 1.0, + "content": ", we define", + "type": "text" + }, + { + "bbox": [ + 185, + 446, + 280, + 459 + ], + "score": 0.93, + "content": "\\Gamma _ { g } : = 2 ( L _ { f } + 1 ) D _ { g } B _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 445, + 401, + 460 + ], + "score": 1.0, + "content": "and obtain a similar bound as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 445, + 401, + 460 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 461, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 132, + 461, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 132, + 461, + 506, + 506 + ], + "score": 0.87, + "content": "\\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}", + "type": "interline_equation", + "image_path": "d37a9814e431549b5d0424f6689ca2ea0f7491e0b17d45aa964d3cd02017f6a8.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 132, + 461, + 506, + 476.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 132, + 476.0, + 506, + 491.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 132, + 491.0, + 506, + 506.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 504, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 132, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 132, + 505 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 500, + 223, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 123, + 516 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 502, + 217, + 514 + ], + "score": 0.9, + "content": "A _ { 3 } \\stackrel { - } { = } 2 \\tau C _ { h } ( 2 / 3 + \\check { C } _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 500, + 223, + 516 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.0, + "bbox_fs": [ + 105, + 487, + 223, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 504, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 133, + 532 + ], + "score": 1.0, + "content": "About", + "type": "text" + }, + { + "bbox": [ + 134, + 520, + 142, + 529 + ], + "score": 0.8, + "content": "\\romannumeral 2", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 519, + 505, + 532 + ], + "score": 1.0, + "content": ", we evaluate the error from approximation by constructing a specific deep neural network", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 529, + 495, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 277, + 543 + ], + "score": 1.0, + "content": "for generators. 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Then, we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 110, + 346, + 127 + ], + "lines": [ + { + "bbox": [ + 265, + 110, + 346, + 127 + ], + "spans": [ + { + "bbox": [ + 265, + 110, + 346, + 127 + ], + "score": 0.92, + "content": "\\begin{array} { r } { i i \\leqslant c _ { 2 } M D S _ { g } ^ { - \\beta / D } . } \\end{array}", + "type": "interline_equation", + "image_path": "f8c772a0ec8f554eb9da6fd8517f95537f30a8ed0b94fa5c49f046e5fc78d9ff.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 265, + 110, + 346, + 127 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 136, + 335, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 136, + 336, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 336, + 150 + ], + "score": 1.0, + "content": "Combining the result, we obtain the result of Theorem 1.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 161, + 244, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 245, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 245, + 174 + ], + "score": 1.0, + "content": "B.5 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 132, + 196 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 133, + 183, + 142, + 192 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 181, + 278, + 196 + ], + "score": 1.0, + "content": "are globally smooth, we obtain a", + "type": "text" + }, + { + "bbox": [ + 279, + 183, + 286, + 194 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 181, + 401, + 196 + ], + "score": 1.0, + "content": "-smooth density function on", + "type": "text" + }, + { + "bbox": [ + 401, + 182, + 414, + 192 + ], + "score": 0.88, + "content": "I ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "by its definition. Due", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 193, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 104, + 193, + 506, + 206 + ], + "score": 1.0, + "content": "to the smoothness, the studies for nonparametric statistics (Nadaraya, 1964; Ghosal et al., 2007;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 239, + 217 + ], + "score": 1.0, + "content": "Efromovich, 2010; van der Vaart", + "type": "text" + }, + { + "bbox": [ + 239, + 205, + 248, + 214 + ], + "score": 0.25, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "van Zanten, 2008; Tsybakov, 2009) guarantees that the methods", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 214, + 480, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 267, + 230 + ], + "score": 1.0, + "content": "(KDE,NB,SDE, and GP) obtain the rate", + "type": "text" + }, + { + "bbox": [ + 268, + 215, + 332, + 228 + ], + "score": 0.92, + "content": "O ( n ^ { - \\beta / ( 2 \\beta + D ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 214, + 440, + 230 + ], + "score": 1.0, + "content": "with respect to the roof of", + "type": "text" + }, + { + "bbox": [ + 440, + 216, + 452, + 226 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 214, + 480, + 230 + ], + "score": 1.0, + "content": "norm.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 133, + 246 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 133, + 234, + 142, + 243 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 232, + 475, + 246 + ], + "score": 1.0, + "content": "have disconnected supports and locally smooth, we consider a following specific", + "type": "text" + }, + { + "bbox": [ + 475, + 234, + 484, + 243 + ], + "score": 0.78, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 232, + 506, + 246 + ], + "score": 1.0, + "content": ". 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\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad 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+ 417.0, + 296.0, + 417.0 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 17, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/Hkfmn5n6W/Hkfmn5n6W.md b/parse/train/Hkfmn5n6W/Hkfmn5n6W.md new file mode 100644 index 0000000000000000000000000000000000000000..b7200efa391090fb256ff579a59697e30d00db78 --- /dev/null +++ b/parse/train/Hkfmn5n6W/Hkfmn5n6W.md @@ -0,0 +1,1442 @@ +# EXPONENTIALLY VANISHING SUB-OPTIMAL LOCAL MINIMA IN MULTILAYER NEURAL NETWORKS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska et al. (2015)) suggest that local minima with high error are exponentially rare in high dimensions. However, to prove low error guarantees for Multilayer Neural Networks (MNNs), previous works so far required either a heavily modified MNN model or training method, strong assumptions on the labels (e.g., “near” linear separability), or an unrealistically wide hidden layer with $\Omega \left( N \right)$ units. + +Results: We examine a MNN with one hidden layer of piecewise linear units, a single output, and a quadratic loss. We prove that, with high probability in the limit of $N \to \infty$ datapoints, the volume of differentiable regions of the empiric loss containing sub-optimal differentiable local minima is exponentially vanishing in comparison with the same volume of global minima, given standard normal input of dimension $d _ { 0 } = \tilde { \Omega } \left( \sqrt { N } \right)$ , and a more realistic number of $d _ { 1 } = \tilde { \Omega } \left( N / d _ { 0 } \right)$ hidden units. We demonstrate our results numerically: for example, $0 \%$ binary classification training error on CIFAR with only $N / d _ { 0 } \approx 1 6$ hidden neurons. + +# 1 INTRODUCTION + +Motivation. Multilayer Neural Networks (MNNs), trained with simple variants of stochastic gradient descent (SGD), have achieved state-of-the-art performances in many areas of machine learning (LeCun et al., 2015). However, theoretical explanations seem to lag far behind this empirical success (though many hardness results exist, e.g., (Síma, 2002; Shamir, 2016)). For example, as a common rule-of-the-thumb, a MNN should have at least as many parameters as training samples. However, it is unclear why such over-parameterized MNNs often exhibit remarkably small generalization error (i.e., difference between “training error” and “test error”), even without explicit regularization (Zhang et al., 2017a). + +Moreover, it has long been a mystery why MNNs often achieve low training error (Dauphin et al., 2014). SGD is only guaranteed to converge to critical points in which the gradient of the expected loss is zero (Bottou, 1998), and, specifically, to local minima (Pemantle, 1990) (this is true also for regular gradient descent (Lee et al., 2016)). Since loss functions parameterized by MNN weights are non-convex, it is unclear why does SGD often work well – rather than converging to sub-optimal local minima with high training error, which are known to exist (Fukumizu & Amari, 2000; Swirszcz et al., 2016). Understanding this behavior is especially relevant in important cases where SGD does get stuck (He et al., 2016) – where training error may be a bottleneck in further improving performance. + +Ideally, we would like to quantify the probability to converge to a local minimum as a function of the error at this minimum, where the probability is taken with the respect to the randomness of the initialization of the weights, the data and SGD. Specifically, we would like to know, under which conditions this probability is very small if the error is high, as was observed empirically (e.g., (Dauphin et al., 2014; Goodfellow et al., 2015)). However, this seems to be a daunting task for realistic MNNs, since it requires a characterization of the sizes and distributions of the basins of attraction for all local minima. + +Previous works (Dauphin et al., 2014; Choromanska et al., 2015), based on statistical physics analogies, suggested a simpler property of MNNs: that with high probability, local minima with high error diminish exponentially with the number of parameters. Though proving such a geometric property with realistic assumptions would not guarantee convergence to global minima, it appears to be a necessary first step in this direction (see discussion on section 6). It was therefore pointed out as an open problem at the Conference of Learning Theory (COLT) 2015. However, one has to be careful and use realistic MMN architectures, or this problem becomes “too easy”. + +For example, one can easily achieve zero training error (Nilsson, 1965; Baum, 1988) – if the MNN’s last hidden layer has more neurons than training samples. Such extremely wide MNNs are easy to optimize (Yu, 1992; Huang et al., 2006; Livni et al., 2014; Haeffele & Vidal, 2015; Shen, 2016; Nguyen & Hein, 2017). In this case, the hidden layer becomes linearly separable in classification tasks, with high probability over the random initialization of the weights. Thus, by training the last layer we get to a global minimum (zero training error). However, such extremely wide layers are not very useful, since they result in a huge number of weights, and serious overfitting issues. Also, training only the last layer seems to take little advantage of the inherently non-linear nature of MNNs. + +Therefore, in this paper we are interested to understand the properties of local and global minima, but at a more practical number of parameters – and when at least two weight layers are trained. For example, Alexnet (Krizhevsky, 2014) is trained using about 1.2 million ImageNet examples, and has about 60 million parameters – 16 million of these in the two last weight layers. Suppose we now train the last two weight layers in such an over-parameterized MNN. When do the sub-optimal local minima become exponentially rare in comparison to the global minima? + +Main contributions. We focus on MNNs with a single hidden layer and piecewise linear units, optimized using the Mean Square Error (MSE) in a supervised binary classification task (Section 2). We define $N$ as the number of training samples, $d _ { l }$ as the width of the $l$ -th activation layer, and $g \left( x \right) \dot { < } h \left( x \right)$ as an asymptotic inequality in the leading order (formally: $\begin{array} { r } { \operatorname* { l i m } _ { x \infty } \frac { \log g ( x ) } { \log h ( x ) } < 1 ) } \end{array}$ . We examine Differentiable Local Minima (DLMs) of the MSE: sub-optimal DLMs where at least a fraction of $\epsilon > 0$ of the training samples are classified incorrectly, and global minima where all samples are classified correctly. + +Our main result, Theorem 10, states that, with high probability, the total volume of the differentiable regions of the MSE containing sub-optimal DLMs is exponentially vanishing in comparison to the same volume of global minima, given that: + +Assumption 1. The datapoints (MNN inputs) are sampled from a standard normal distribution. + +Assumption 2. $N \to \infty$ , $d _ { 0 } \left( N \right)$ and $d _ { 1 } \left( N \right)$ increase with $N$ , while $\epsilon \in ( 0 , 1 )$ is a constant1. + +Assumption 3. The input dimension scales as $\sqrt { N } \dot { < } d _ { 0 } \dot { \le } N$ . + +Assumption 4. The hidden layer width scales as + +$$ +\frac { N \log ^ { 4 } N } { d _ { 0 } } \dot { < } d _ { 1 } \dot { < } N . +$$ + +Importantly, we use a standard, unmodified, MNN model, and make no assumptions on the target function. Moreover, as the number of parameters in the MNN is approximately $d _ { 0 } d _ { 1 }$ , we require only “asymptotically mild” over-parameterization: $d _ { 0 } d _ { 1 } { \dot { > } } N \log ^ { 4 } N$ from eq. (1.1). For example, if $d _ { 0 } \propto N$ , we only require $d _ { 1 } \dot { > } \log ^ { 4 } N$ neurons. This improves over previously known results (Yu, 1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016; Nguyen & Hein, 2017) – which require an extremely wide hidden layer with $d _ { 1 } \geq N$ neurons (and thus $N d _ { 0 }$ parameters) to remove sub-optimal local minima with high probability. + +In section 5 we validate our results numerically. We show that indeed the training error becomes low when the number of parameters is close to $N$ . For example, with binary classification on CIFAR and ImageNet, with only 16 and 105 hidden neurons (about $N / d _ { 0 }$ ), respectively, we obtain less then $0 . 1 \%$ training error. Additionally, we find that convergence to non-differentiable critical points does not appear to be very common. + +Lastly, in section 6 we discuss our results might be extended, such as how to apply them to “mildly” non-differentiable critical points. + +Plausibility of assumptions. Assumption 1 is common in this type of analysis (Andoni et al., 2014; Choromanska et al., 2015; Xie et al., 2016; Tian, 2017; Brutzkus & Globerson, 2017). At first it may appear rather unrealistic, especially since the inputs are correlated in typical datasets. However, this no-correlation part of the assumption may seem more justified if we recall that datasets are many times whitened before being used as inputs. Alternatively, if, as in our motivating question, we consider the input to the our simple MNN to be the output of the previous layers of a deep MNN with fixed random weights, this also tends to de-correlate inputs (Poole et al., 2016, Figure 3). The remaining part of assumption 1, that the distribution is normal, is indeed strong, but might be relaxed in the future, e.g. using central limit theorem type arguments. + +In assumption 2 we use this asymptotic limit to simplify our proofs and final results. Multiplicative constants and finite (yet large) $N$ results can be found by inspection of the proofs. We assume a constant error $\epsilon$ since typically the limit $\epsilon 0$ is avoided to prevent overfitting. + +In assumption 3, for simplicity we have $d _ { 0 } { \dot { \le } } N$ , since in the case $d _ { 0 } \geq N$ the input is generically linearly separable, and sub-optimal local minima are not a problem (Gori & Tesi, 1992; Safran & Shamir, 2016). Additionally, we have $\sqrt { N } \dot { < } d _ { 0 }$ , which seems very reasonable, since for example, $d _ { 0 } / N \approx 0 . 0 1 6 , 0 . 0$ 61 and 0.055 MNIST, CIFAR and ImageNet, respectively. + +In assumption 4, for simplicity we have $d _ { 1 } \dot { < } N$ , since, as mentioned earlier, if $d _ { 1 } \geq N$ the hidden layer is linearly separable with high probability, which removes sub-optimal local minima. The other bound $N \log ^ { 4 } N { \dot { < } } d _ { 0 } d _ { 1 }$ is our main innovation – a large over-parameterization which is nevertheless asymptotically mild and improves previous results. + +Previous work. So far, general low (training or test) error guarantees for MNNs could not be found – unless the underlying model (MNN) or learning method (SGD or its variants) have been significantly modified. For example, (Dauphin et al., 2014) made an analogy with high-dimensional random Gaussian functions, local minima with high error are exponentially rare in high dimensions; (Choromanska et al., 2015; Kawaguchi, 2016) replaced the units (activation functions) with independent random variables; (Pennington & Bahri, 2017) replaces the weights and error residuals with independent random variables; (Baldi, 1989; Saxe et al., 2014; Hardt & Ma, 2017; Lu & Kawaguchi, 2017; Zhou & Feng, 2017) used linear units; (Zhang et al., 2017b) used unconventional units (e.g., polynomials) and very large hidden layers $( d _ { 1 } = \mathrm { p o l y } \left( d _ { 0 } \right)$ , typically $\gg N$ ); (Brutzkus & Globerson, 2017; Du et al., 2017; Shalev-Shwartz et al., 2017) used a modified convnet model with less then $d _ { 0 }$ parameters (therefore, not a universal approximator (Cybenko, 1989; Hornik, 1991)); (Tian, 2017; Soltanolkotabi et al., 2017; Li & Yuan, 2017) assume the weights are initialized very close to those of the teacher generating the labels; and (Janzamin et al., 2015; Zhong et al., 2017) use a non-standard tensor method during training. Such approaches fall short of explaining the widespread success of standard MNN models and training practices. + +Other works placed strong assumptions on the target functions. For example, to prove convergence of the training error near the global minimum, (Gori & Tesi, 1992) assumed linearly separable datasets, while (Safran & Shamir, 2016) assumed strong clustering of the targets (“near” linear-separability). Also, (Andoni et al., 2014) showed a $p$ -degree polynomial is learnable by a MNN, if the hidden layer is very large $( d _ { 1 } = \Omega \left( d _ { 0 } ^ { 6 p } \right)$ , typically $\gg N$ ) so learning the last weight layer is sufficient. However, these are not the typical regimes in which MNNs are required or used. In contrast, we make no assumption on the target function. Other closely related results (Soudry & Carmon, 2016; Xie et al., 2016) also used unrealistic assumptions, are discussed in section 6, in regards to the details of our main results. + +Therefore, in contrast to previous works, the assumptions in this paper are applicable in some situations (e.g., Gaussian input) where a MNN trained using SGD might be used and be useful (e.g., have a lower test error then a linear classier). + +# 2 PRELIMINARIES AND NOTATION + +Model. We examine a Multilayer Neural Network (MNN) with a single hidden layer and a scalar output. The MNN is trained on a finite training set of $N$ datapoints (features) $\textbf { X } \triangleq$ $\left[ \mathbf { x } ^ { ( 1 ) } , \ldots , \mathbf { x } ^ { ( N ) } \right] \ \in \ \mathbb { R } ^ { d _ { 0 } \times N }$ with their target labels $\mathbf { y } \ \triangleq \ \left[ y ^ { ( 1 ) } , \ldots , y ^ { ( N ) } \right] ^ { \intercal } \ \in \ \left\{ 0 , 1 \right\} ^ { N } -$ each datapoint-label pair $\left( \mathbf { x } ^ { ( n ) } , y ^ { ( n ) } \right)$ is independently sampled from some joint distribution $\mathbb { P } _ { X , Y }$ . We define $\mathbf { W } = \left[ \mathbf { w } _ { 1 } , \ldots , \mathbf { w } _ { d _ { 1 } } \right] ^ { \top } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ and $\mathbf { z } \in \mathbb { R } ^ { d _ { 1 } }$ as the first and second weight layers (bias terms are ignored for simplicity), respectively, and $f \left( \cdot \right)$ as the common leaky rectifier linear unit (LReLU (Maas et al., 2013)) + +$$ +\begin{array} { r } { f \left( u \right) \triangleq u a \left( u \right) \mathrm { ~ w i t h ~ } a \left( u \right) \triangleq \left\{ \begin{array} { l l } { 1 } & { \mathrm { , ~ i f ~ } , u > 0 } \\ { \rho } & { \mathrm { , ~ i f ~ } u < 0 } \end{array} \right. , } \end{array} +$$ + +for some $\rho \neq 1$ (so the MNN is non-linear) , where both functions $f$ and $a$ operate component-wise (e.g., for any matrix M: $\left( f \left( \mathbf { M } \right) \right) _ { i j } = f \left( M _ { i j } \right) )$ . Thus, the output of the MNN on the entire dataset can be written as + +$$ +{ f } \left( { { \mathbf { W } } { \mathbf { X } } } \right) ^ { \top } { \mathbf { z } } \in \mathbb { R } ^ { N } . +$$ + +We use the mean square error (MSE) loss for optimization + +$$ +\operatorname { M S E } \triangleq \frac { 1 } { N } \left. \mathbf { e } \right. ^ { 2 } \mathrm { w i t h } \mathbf { e } \triangleq \mathbf { y } - f \left( \mathbf { W } \mathbf { X } \right) ^ { \top } \mathbf { z } , +$$ + +where $\lVert \cdot \rVert$ is the standard euclidean norm. Also, we measure the empiric performance as the fraction of samples that are classified correctly using a decision threshold at $y = 0 . 5$ , and denote this as the mean classification error, or $\mathbf { M C E } ^ { 2 }$ . Note that the variables $\mathbf { e }$ , MSE, MCE and other related variables (e.g., their derivatives) all depend on $\mathbf { W } , \mathbf { z } , \mathbf { X } , \mathbf { y }$ and $\rho$ , but we keep this dependency implicit, to avoid cumbersome notation. + +Additional Notation. We define g (x) \epsilon > 0$ . We would like to compare the angular volume (definition 3) corresponding to both types of DLMs. Thus, we make the following definitions. + +Definition 4. We define3 $\mathcal { L } _ { \epsilon } \subset \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ as the union of differentiable regions containing sub-optimal DLMs with $\mathrm { M C E } > \epsilon$ , and $\mathcal { G } \subset \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ as the union of differentiable regions containing global minima with $\mathrm { M C E } = 0$ . + +Definition 5. We define the constant $\gamma _ { \epsilon }$ as $\begin{array} { r } { \gamma _ { \epsilon } \triangleq 0 . 2 3 \operatorname* { m a x } \left[ \operatorname* { l i m } _ { N \to \infty } \left( d _ { 0 } \left( N \right) / N \right) , \epsilon \right] ^ { 3 / 4 } } \end{array}$ if $\rho \neq$ $\{ 0 , 1 \}$ , and $\gamma _ { \epsilon } \triangleq 0 . 2 3 \epsilon ^ { 3 / 4 }$ if $\rho = 0$ . + +In this section, we use assumptions 1-4 (stated in section 1) to bound the angular volume of the region $\mathcal { L } _ { \epsilon }$ encapsulating all sub-optimal DLMs, the region $\mathcal { G }$ , encapsulating all global minima, and the ratio between the two. + +Angular volume of sub-optimal DLMs. First, in appendix section 8 we prove the following upper bound in expectation + +Theorem 6. Given assumptions 1-4, the expected angular volume of sub-optimal DLMs, with $\mathrm { M C E } > \epsilon > 0$ , is exponentially vanishing in $N$ as + +$$ +\begin{array} { r } { \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { \leq } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) . } \end{array} +$$ + +and, using Markov inequality, its immediate probabilistic corollary + +Corollary 7. Given assumptions $I$ -4, for any $\delta > 0$ (possibly a vanishing function of $N$ ), we have, with probability $1 - \delta$ , that the angular volume of sub-optimal DLMs, with $\mathrm { M C E } > \epsilon > 0 ,$ , is exponentially vanishing in $N$ as + +$$ +{ \mathcal { V } } \left( { \mathcal { L } } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { \leq } \frac { 1 } { \delta } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) +$$ + +Proof idea of Theorem 6: we first show that in differentiable regions with $\mathrm { M C E } > \epsilon > 0$ , the condition in Lemma 2, $\left( \mathbf { A } \circ \mathbf { X } \right) \mathbf { e } = 0$ , implies that $\mathbf { A } = a \left( \mathbf { W } \mathbf { X } \right)$ must have a low rank. Then, we show that, when $\mathbf { X } \sim { \mathcal { N } }$ and $\mathbf { W } \sim { \mathcal { N } }$ , the matrix $\mathbf { A } = a \left( \mathbf { W } \mathbf { X } \right)$ has a low rank with exponentially low probability. Combining both facts, we obtain the bound. + +Existence of global minima. Next, to compare the volume of sub-optimal DLMs with that of global minima, in appendix section 9 we show first that, generically, global minima do exist (using a variant of the proof of (Baum, 1988, Theorem 1)): + +Theorem 8. For any $\mathbf { y } \in \{ 0 , 1 \} ^ { N }$ and $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N }$ almost everywhere4 we find matrices $\mathbf { W } ^ { * } \in$ $\mathbb { R } ^ { d _ { 1 } ^ { * } \times d _ { 0 } }$ and $\mathbf { z } ^ { * } \in \mathbb { R } ^ { d _ { 1 } ^ { * } }$ , such that $\mathbf { y } = f \left( \mathbf { W } ^ { * } \mathbf { X } \right) ^ { \top } \mathbf { z } ^ { * }$ , where $d _ { 1 } ^ { * } \triangleq 4 \lceil N / ( 2 d _ { 0 } - 2 ) \rceil$ and $\forall i , n$ : $\mathbf { w } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \neq 0$ . Therefore, every MNN with $d _ { 1 } \geq d _ { 1 } ^ { * }$ has a DLM which achieves zero error $\mathbf { e } = 0$ . + +Recently (Zhang et al., 2017a, Theorem 1) similarly proved that a 2-layer MNN with approximately $2 N$ parameters can achieve zero error. However, that proof required $N$ neurons (similarly to (Nilsson, 1965; Baum, 1988; Yu, 1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016)), while Theorem 8 here requires much less: approximately $d _ { 1 } ^ { * } \approx 2 N / d _ { 0 }$ . Also, (Hardt & Ma, 2017, Theorem 3.2) showed a deep residual network with $N \log N$ parameters can achieve zero error. In contrast, here we require just one hidden layer with $2 N$ parameters. + +Note the construction in Theorem 8 here achieves zero training error by overfitting to the data realization, so it is not expected to be a “good” solution in terms of generalization. To get good generalization, one needs to add additional assumptions on the data ( $\mathbf { X }$ and y). Such a possible (common yet insufficient for MNNs) assumption is that the problem is “realizable”, i.e., there exist a small “solution MNN”, which achieves low error. For example, in the zero error case: + +Assumption 5. (Optional) The labels are generated by some teacher $\mathbf { y } = f \left( \mathbf { W } ^ { * } \mathbf { X } \right) ^ { \top } \mathbf { z } ^ { * }$ with weight matrices $\mathbf { W } ^ { * } \in \mathbb { R } ^ { d _ { 1 } ^ { * } \times d _ { 0 } }$ and $\mathbf { z } ^ { * } \in \mathbb { R } ^ { d _ { 1 } ^ { * } }$ independent of $\mathbf { X }$ , for some $d _ { 1 } ^ { * } { \dot { < } } N / d _ { 0 }$ . + +This assumption is not required for our main result (Theorem 10) – it is merely helpful in improving the following lower bound on $\mathcal { V } \left( \mathcal { G } \right)$ . + +Angular volume of global minima. We prove in appendix section 10: + +Theorem 9. Given assumptions 1-3, we set $\delta { \doteq } \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } / N$ and $d _ { 1 } ^ { * } = 2 N / d _ { 0 }$ , or if assumption 5 holds, we set $d _ { 1 } ^ { * }$ as in this assumption. Then, with probability $1 - \delta$ , the angular volume of global minima is lower bounded as, + +$$ +\mathscr { V } \left( \mathscr { G } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { > } \exp \left( - d _ { 1 } ^ { * } d _ { 0 } \log N \right) \dot { \geq } \exp \left( - 2 N \log N \right) . +$$ + +Proof idea: First, we lower bound $\mathcal { V } \left( \mathcal { G } \right)$ with the angular volume of a single differentiable region of one global minimum $( \mathbf { W } ^ { * } , \mathbf { z } ^ { * } )$ – either from Theorem 8, or from assumption 5. Then we show that this angular volume is lower bounded when $\mathbf { W } \sim { \mathcal { N } }$ , given a certain angular margin between the datapoints in $\mathbf { X }$ and the rows of $\mathbf { W } ^ { * }$ . We then calculate the probability of obtaining this margin when $\mathbf { X } \sim { \mathcal { N } }$ . Combining both results, we obtain the final bound. + +Main result: angular volume ratio. Finally, combining Theorems 6 and 9 it is straightforward to prove our main result in this paper, as we do in appendix section 11: + +Theorem 10. Given assumptions 1-3, we set $\delta \doteq \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } / N$ . Then, with probability $1 - \delta$ , the angular volume of sub-optimal DLMs, with $\mathrm { M C E } > \epsilon > 0$ , is exponentially vanishing in $N _ { : }$ , in comparison to the angular volume of global minima with $\mathrm { M C E } = 0$ + +$$ +\frac { \mathscr { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) } { \mathscr { V } \left( \mathcal { G } \left( \mathbf { X } , \mathbf { y } \right) \right) } \dot { \leq } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) \dot { \leq } \exp \left( - \gamma _ { \epsilon } N \log N \right) . +$$ + +# 5 NUMERICAL EXPERIMENTS + +Theorem 10 implies that, with “asymptotically mild” over-parameterization (i.e. in which #parameters $\mathrm { = } \tilde { \Omega } \left( N \right) )$ , differentiable regions in weight space containing sub-optimal DLMs (with high MCE) are exponentially small in comparison with the same regions for global minima. Since these results are asymptotic in $N \infty$ , in this section we examine it numerically for a finite number of samples and parameters. We perform experiments on random data, MNIST, CIFAR10 and ImageNetILSVRC2012. In each experiment, we used ReLU activations $( \rho = 0 )$ ), a binary classification target (we divided the original classes to two groups), MSE loss for optimization (eq. (2.3)), and MCE to determine classification error. Additional implementation details are given in appendix part III. + +![](images/1795ffcce4a4ec7991ec4c69666c8a8186811014eb6fb035a522afcfd4f8e18c.jpg) +Figure 5.1: Gaussian data: final training error (mean±std, 30 repetitions) in the overparameterized regime is low (right of the dashed black line). We trained MNNs with one and two hiddens layer (with widths equal to $d = d _ { 0 }$ ) on a synthetic random dataset in which $\forall n = 1 , \ldots , N$ , $\mathbf { x } ^ { ( n ) }$ was drawn from a normal distribution $\mathcal { N } ( 0 , 1 )$ , and $y ^ { ( n ) } = \pm 1$ with probability 0.5. + +Table 1: Binary classification of MNIST, CIFAR and ImageNet: 1-hidden layer achieves very low training error (MCE) with a few hidden neurons, so that #parameters $\approx d _ { 0 } d _ { 1 } \approx N$ . In ImageNet we downsampled the images to allow input whitening. + +
MCEdod1N#parameters/N
MNIST0%784897·1040.999
CIFAR0%3072165·1040.983
ImageNet (downsampled to 64 × 64)0.1%12288105128·1041.008
+ +First, on the small synthetic Gaussian random data (matching our assumptions) we perform a scan on various networks and dataset sizes. With either one or two hidden layers (Figure 5.1) , the error goes to zero when the number of non-redundant parameters (approximately $d _ { 0 } d _ { 1 } ,$ ) is greater than the number of samples, as suggested by our asymptotic results. Second, on the non-syntehtic datasets, MNIST, CIFAR and ImageNet (In ImageNet we downsampled the images to size $6 4 \times 6 4$ , to allow input whitening) we only perform a simulation with a single 1-hidden layer MNN for which #parameters $\approx N$ , and again find (Table 1) that the final error is zero (for MNIST and CIFAR) or very low (ImageNet). + +Lastly, in Figure 5.2 we find that, on the Gaussian dataset, the inputs to the hidden neurons converge to a distinctly non-zero value. This indicates we converged to differentiable critical points – since nondifferentiable critical points must have zero neural inputs. Note that occasionally, during optimization, we could find some neural inputs with very low values near numerical precision level, so convergence to non-differentiable minima may be possible. However, as explained in the next section, as long as the number of neural inputs equal to zero are not too large, our bounds also hold for these minima. + +# 6 DISCUSSION + +In this paper we examine Differentiable Local Minima (DLMs) of the empiric loss of Multilayer Neural Networks (MNNs) with one hidden layer, scalar output, and LReLU nonlinearities (section 2). We prove (Theorem 10) that with high probability the angular volume (definition 3) of sub-optimal DLMs is exponentially vanishing in comparison to the angular volume of global minima (definition 4), under assumptions 1-4. This results from an upper bound on sub-optimal DLMs (Theorem 6) and a lower bound on global minima (Theorem 9). + +![](images/e75b9ef834dd7be439435fa2e13f28f31f7a9f670be1cdff33d9185308bb07d8.jpg) +Figure 5.2: Gaussian data: convergence of the MSE to differentiable critical points, as indicated by the convergence of the neural inputs to distinctly non-zero values. We trained MNNs with one hidden layer on the Gaussian dataset from Figure 5.1, with various widths $d = d _ { 0 } = d _ { 1 }$ and $N = \lfloor d ^ { 2 } / 5 \rfloor$ for 1000 epochs, then decreased the learning rate exponentially for another 1000 epochs. This was repeated 30 times. For all $d$ and repeats, we see that (left) the final absolute value of the minimal neural input $( i . e . , \operatorname* { m i n } _ { i , n } \left| \mathbf { w } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right| )$ in the range of $1 0 ^ { - 3 } - 1 0 ^ { 0 }$ , which is much larger then $( \mathrm { r } i g h t )$ the final MSE error for all $d$ and all repeats – in the range $1 0 ^ { - 3 1 } - 1 0 ^ { - 7 }$ . + +Convergence of SGD to DLMs. These results suggest a mechanism through which low training error is obtained in such MNNs. However, they do not guarantee it. One issue is that sub-optimal DLMs may have exponentially large basins of attraction. We see two possible paths that might address this issue in future work, using additional assumptions on y. One approach is to show that, with high probability, no sub optimal DLM falls within the vanishingly small differentiable regions we bounded in Theorem 6. Another approach would be to bound the size of these basins of attraction, by showing that sufficiently large of number of differentiable regions near the DLM are also vanishingly small (other methods might also help here (Freeman & Bruna, 2016)). Another issue is that SGD might get stuck near differentiable saddle points, if their Hessian does not have strictly negative eigenvalues (i.e., the strict saddle property (Sun et al., 2015)). It should be straightforward to show that such points also have exponentially vanishing angular volume, similar to sub-optimal DLMs. Lastly, SGD might also converge to non-differentiable critical points, which we discuss next. + +Non-differentiable critical points. The proof of Theorem 6 stems from a first order necessary condition (Lemma 2): $\left( \mathbf { A } \circ \mathbf { \bar { X } } \right) \mathbf { e } = 0$ , which is true for any DLM. However, non-differentiable critical points, in which some neural inputs are exactly zero, may also exist (though, numerically, they don’t seem very common – see Figure 5.2). In this case, to derive a similar bound, we can replace the condition with $\mathbf { P } \left( \mathbf { A } \circ \mathbf { X } \right) \mathbf { e } = 0$ , where $\mathbf { P }$ is a projection matrix to the subspace orthogonal to the non-differentiable directions. As long as there are not too many zero neural inputs, we should be able to obtain similar results. For example, if only a constant ratio $r$ of the neural inputs are zero, we can simply choose $\mathbf { P }$ to remove all rows of $( \mathbf { A } \circ \mathbf { X } )$ corresponding to those neurons, and proceed with exactly the same proof as before, with $d _ { 1 }$ replaced with $( 1 - r ) d _ { 1 }$ . It remains a theoretical challenge to find reasonable assumptions under which the number of non-differentiable directions (i.e., zero neural inputs) does not become too large. + +Related results. Two works have also derived related results using the $\left( \mathbf { A } \circ \mathbf { X } \right) \mathbf { e } = 0$ condition from Lemma 2. In (Soudry & Carmon, 2016), it was noticed that an infinitesimal perturbation of $\mathbf { A }$ makes the matrix $\mathbf { A } \circ \mathbf { X }$ full rank with probability 1 (Allman et al., 2009, Lemma 13) – which entails that $\mathbf { e } = 0$ at all DLMs. Though a simple and intuitive approach, such an infinitesimal perturbation is problematic: from continuity, it cannot change the original MSE at sub-optimal DLMs – unless the weights go to infinity, or the DLM becomes non-differentiable – which are both undesirable results. An extension of this analysis was also done to constrain e using the singular values of $\mathbf { A } \circ \mathbf { X }$ (Xie et al., 2016), deriving bounds that are easier to combine with generalization bounds. Though a promising approach, the size of the sub-optimal regions (where the error is high) does not vanish exponentially in the derived bounds. More importantly, these bounds require assumptions on the activation kernel spectrum $\gamma _ { m }$ , which do not appear to hold in practice (e.g., (Xie et al., 2016, Theorems 1,3) require $m \gamma _ { m } \gg 1$ to hold with high probability, while $m \gamma _ { m } < \bar { 1 0 } ^ { - 2 }$ in (Xie et al., 2016, Figure 1)). + +Modifications and extensions. There are many relatively simple extensions of these results: the Gaussian assumption could be relaxed to other near-isotropic distributions (e.g., sparse-land model, (Elad, 2010, Section 9.2)) and other convex loss functions are possible instead of the quadratic loss. 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First, however, we define additional notation (some already defined in the main paper) and mention some known results, which we will use in our proofs. + +# EXTENDED PRELIMINARIES + +• The indicator function ${ \mathcal { T } } \left( { \mathcal { A } } \right) \triangleq { \left\{ \begin{array} { l l } { 1 } & { , { \mathrm { i f } } \ A } \\ { 0 } & { , { \mathrm { e l s e } } } \end{array} \right. }$ , for any event $\mathcal { A }$ . +• Kronecker’s delta $\delta _ { i j } \triangleq { \mathcal { T } } ( i = j )$ . +• The Matrix $\mathbf { I } _ { d }$ as the identity matrix in $\mathbb { R } ^ { d \times d }$ , and $\mathbf { I } _ { d \times k }$ is the relevant $\mathbb { R } ^ { d \times k }$ upper left sub-matrix of the identity matrix. +• $[ L ] \triangleq \{ 1 , 2 , \dots , L \}$ +The vector ${ \bf m } _ { n }$ as the $n$ ’th column of a matrix $\mathbf { M }$ , unless defined otherwise (then ${ \bf m } _ { n }$ will be a row of $\mathbf { M }$ ). +• $\mathbf M > 0$ implies that $\forall i , j : M _ { i j } > 0$ . +• $\mathbf { M } _ { S }$ is the matrix composed of the columns of $\mathbf { M }$ that are in the index set $S$ . +• A property holds “M-almost everywhere” (a.e. for short), if the set of entries of $\mathbf { M }$ for which the property does not hold has zero measure (Lebesgue). +• $\begin{array} { r } { \| \mathbf { v } \| _ { 0 } = \sum _ { i = 1 } ^ { d } \mathcal { T } \left( v _ { i } > 0 \right) } \end{array}$ is the $L _ { 0 }$ “norm” that counts the number of non-zero values in $\mathbf { v } \in \mathbb { R } ^ { d }$ . +• If $\mathbf { x } \sim \mathcal { N } ( \pmb { \mu } , \pmb { \Sigma } )$ the $\mathbf { x }$ is random Gaussian vector. +• $\begin{array} { r } { \phi \left( x \right) \triangleq \frac { 1 } { \sqrt { 2 \pi } } \exp \left( - \frac { 1 } { 2 } x ^ { 2 } \right) } \end{array}$ as the univariate Gaussian probability density function. • $\textstyle \Phi \left( x \right) \triangleq \int _ { - \infty } ^ { x } \phi \left( u \right) d u$ as the Gaussian cumulative distribution function. +• $B \left( x , y \right)$ as the beta function. + +Lastly, we recall the well known Markov Inequality: + +Fact 11. (Markov Inequality) For any random variable $X \geq 0$ , we have $\forall \eta > 0$ + +$$ +\mathbb { P } \left( X \geq \eta \right) \leq \frac { \mathbb { E } X } { \eta } . +$$ + +# Part I + +# Proofs of the main results + +7 FIRST ORDER CONDITION: PROOF OF LEMMA 2 + +Lemma 12. (Lemma 2 restated) At all DLMs in ${ \mathcal { D } } _ { \mathbf { A } } \left( \mathbf { X } \right)$ the residual error e is identical, and furthermore + +$$ +\left( \mathbf { A } \circ \mathbf { X } \right) \mathbf { e } = 0 . +$$ + +Proof. Let $\mathbf { W } = \left[ \mathbf { w } _ { 1 } , \ldots , \mathbf { w } _ { d _ { 1 } } \right] ^ { \top } \in \mathcal { D } _ { \mathbf { A } } \left( \mathbf { X } \right)$ , $\mathbf { G } \triangleq \mathbf { A } \circ \mathbf { X } \ \in \ \mathbb { R } ^ { d _ { 0 } d _ { 1 } \times N }$ , $\tilde { \mathbf { W } } = \mathrm { d i a g } \left( \mathbf { z } \right) \mathbf { W } =$ $\left[ \tilde { \mathbf { w } } _ { 1 } , \ldots , \tilde { \mathbf { w } } _ { d _ { 1 } } \right] ^ { \top }$ and $\tilde { \mathbf { w } } \triangleq \mathrm { v e c } \left( \tilde { \mathbf { W } } ^ { \top } \right) \in \mathbb { R } ^ { d _ { 0 } d _ { 1 } }$ , where $\operatorname { d i a g } \left( \mathbf { v } \right)$ is the diagonal matrix with $\mathbf { v }$ in its + +diagonal, and vec (M) is vector obtained by stacking the columns of the matrix $\mathbf { M }$ on top of one another. Then, we can re-write the MSE (eq. (2.3)) as + +$$ +\mathrm { M S E } = { \frac { 1 } { N } } \left\| \mathbf { y } - \mathbf { G } ^ { \top } { \tilde { \mathbf { w } } } \right\| ^ { 2 } = { \frac { 1 } { N } } \left\| \mathbf { e } \right\| ^ { 2 } , +$$ + +where $\mathbf { G } ^ { \top } \tilde { \mathbf { w } }$ is the output of the MNN. Now, if $( \mathbf { W } , \mathbf { z } )$ is a DLM of the MSE in eq. (2.3), then there is no infinitesimal perturbation of $( \mathbf { W } , \mathbf { z } )$ which reduces this MSE. + +Next, for each row $i$ , we will show that $\partial \mathrm { M S E } / \partial \tilde { \bf w } _ { i } = 0$ , since otherwise we can find an infinitesimal perturbation of $( \mathbf { W } , \mathbf { z } )$ which decreases the MSE, contradicting the assumption that $( \mathbf { W } , \mathbf { z } )$ is a local minimum. For each row $i$ , we divide into two cases: + +First, we consider the case $z _ { i } \neq 0$ . In this case, any infinitesimal perturbation $\mathbf { q } _ { i }$ in $\tilde { \mathbf { w } } _ { \mathbf { i } }$ can be produced by an infinitesimal perturbation in $\mathbf { w } _ { i }$ $\mathbf { \widetilde { w } } _ { i } + \mathbf { q } _ { i } = ( \mathbf { w } _ { i } + \mathbf { q } _ { i } / z _ { i } ) z _ { i }$ . Therefore, unless the gradient $\partial \mathrm { { M S E } } / \partial \tilde { \bf { w } } _ { i }$ is equal to zero, we can choose an infinitesimal perturbation $\mathbf { q } _ { i }$ in the opposite direction to this gradient, which will decrease the MSE. + +Second, we consider the case $z _ { i } ~ = ~ 0$ . In this case, the MSE is not affected by changes made exclusively to $\mathbf { w } _ { i }$ . Therefore, all $\mathbf { w } _ { i }$ derivatives of the MSE are equal to zero $( \partial ^ { \bar { k } } \mathrm { M S E } / \partial ^ { k } \mathbf { w } _ { i }$ , to any order $k$ ) . Also, since we are at a differentiable local minimum, $\partial \mathrm { M S E } / \partial z _ { i } = 0$ . Thus, using a Taylor expansion, if we perturb $\left( \mathbf { w } _ { i } , z _ { i } \right)$ by $( \hat { \mathbf { w } } _ { i } , \hat { z } _ { i } )$ then the MSE is perturbed by + +$$ +\hat { z } _ { i } \hat { \mathbf { w } } _ { i } ^ { \top } \frac { \partial } { \partial \tilde { \mathbf { w } } _ { i } } \frac { \partial } { \partial z _ { i } } \mathrm { M S E } + O ( \hat { z } _ { i } ^ { 2 } ) +$$ + +Therefore, unless $\partial ^ { 2 } \mathrm { M S E } / \left( \partial \mathbf { w } _ { i } \partial z _ { i } \right) = 0$ we can choose $\hat { \mathbf { w } } _ { i }$ and a sufficiently small $\hat { z } _ { i }$ such that the MSE is decreased. Lastly, using the chain rule + +$$ +\frac { \partial } { \partial z _ { i } } \frac { \partial } { \partial \mathbf { w } _ { i } } \mathrm { M S E } = \frac { \partial } { \partial z _ { i } } \left[ z _ { i } \frac { \partial } { \partial \tilde { \mathbf { w } } _ { i } } \mathrm { M S E } \right] = \frac { \partial } { \partial \tilde { \mathbf { w } } _ { i } } \mathrm { M S E } . +$$ + +Thus, $\partial \mathrm { M S E } / \partial \tilde { \bf w } _ { i } = 0$ . This implies that w˜ is also a $\mathrm { D L M } ^ { 5 }$ of eq. (7.2), which entails + +$$ +0 = - \frac { N } { 2 } \frac { \partial } { \partial \tilde { \bf w } _ { i } } \mathrm { M S E } = { \bf G } \left( { \bf y } - { \bf G } ^ { \top } \tilde { \bf w } \right) . +$$ + +Since $\mathbf { G } = \mathbf { A } \circ \mathbf { X }$ and $\mathbf { e } = \mathbf { y } - \mathbf { G } ^ { \top } \tilde { \mathbf { w } }$ this proves eq. (7.1). Now, for any two solutions $\tilde { \mathbf { w } } _ { 1 }$ and $\tilde { \mathbf { w } } _ { 2 }$ of eq. (7.3), we have + +$$ +0 = \mathbf { G } \left( \mathbf { y } - \mathbf { G } ^ { \top } \tilde { \mathbf { w } } _ { 1 } \right) - \mathbf { G } \left( \mathbf { y } - \mathbf { G } ^ { \top } \tilde { \mathbf { w } } _ { 1 } \right) = \mathbf { G } \mathbf { G } ^ { \top } \left( \tilde { \mathbf { w } } _ { 2 } - \tilde { \mathbf { w } } _ { 1 } \right) . +$$ + +Multiplying by $\big ( \tilde { \mathbf { w } } _ { 2 } - \tilde { \mathbf { w } } _ { 1 } \big ) ^ { \top }$ from the left we obtain + +$$ +\left\| \mathbf { G } ^ { \top } \left( \tilde { \mathbf { w } } _ { 2 } - \tilde { \mathbf { w } } _ { 1 } \right) \right\| ^ { 2 } = 0 \Rightarrow \mathbf { G } ^ { \top } \left( \tilde { \mathbf { w } } _ { 2 } - \tilde { \mathbf { w } } _ { 1 } \right) = 0 . +$$ + +Therefore, the MNN output and the residual error $\mathbf { e }$ are equal for all DLMs in ${ \mathcal { D } } _ { \mathbf { A } } \left( \mathbf { X } \right)$ . + +# 8 SUB-OPTIMAL DIFFERENTIABLE LOCAL MINIMA: PROOF OF THEOREM 6 AND ITS COROLLARY + +Theorem 13. (Theorem 6 restated) Given assumptions 1-4, the expected angular volume of suboptimal DLMs, with $\mathrm { M C E } > \epsilon > 0$ , is exponentially vanishing in $N$ as + +$$ +\begin{array} { r } { \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { \leq } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) , } \end{array} +$$ + +To prove this theorem we upper bound the angular volume of $\mathcal { L } _ { \epsilon }$ (definition 4), i.e., differentiable regions in which there exist DLMs with $\mathrm { M C E } > \epsilon > 0$ . Our proof uses the first order necessary condition for DLMs from Lemma 2, $\left( \mathbf { A } \circ \mathbf { X } \right) \mathbf { e } = 0$ , to find which configurations of $\mathbf { A }$ allow for a high residual error e with $\mathrm { M C E } > \epsilon > 0$ . In these configurations $\mathbf { A } \circ \mathbf { X }$ cannot have full rank, and therefore, as we show (Lemma 14 below), $\mathbf { A } = a \left( \mathbf { W } \mathbf { X } \right)$ must have a low rank. However, $\mathbf { A } = a \left( \mathbf { W } \mathbf { X } \right)$ has a low rank with exponentially low probability when $\mathbf x \sim \mathcal N$ and $\mathbf { W } \sim { \mathcal { N } }$ (Lemmas 15 and 16 below). Thus, we derive an upper bound on $\mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right)$ . + +Before we begin, let us recall some notation: $[ L ] \triangleq \{ 1 , 2 , \dots , L \} , \mathbf { M } > 0$ implies that $\forall i , j : M _ { i j } >$ 0, $\mathbf { M } _ { S }$ is the matrix composed of the columns of M that are in the index set $S$ , $\left\| \mathbf { v } \right\| _ { 0 }$ as the $L _ { 0 }$ “norm” that counts the number of non-zero values in $\mathbf { v }$ . First we consider the case $\rho \neq 0$ . Also, we denote $K _ { r } \triangleq \operatorname* { m a x } \left[ N \epsilon , r d _ { 0 } \right]$ . + +First we consider the case $\rho \neq 0$ . + +From definition 3 of the angular volume + +$$ +\begin{array} { r l } { \mathfrak { r } _ { \mathbf { x } \times \mathcal { N } } \mathcal { V } ( \mathcal { L } , ( \mathbf { X } , \mathbf { y } ) ) = \mathfrak { h } _ { ( \mathbf { x } , \mathbf { y } ) \sim \mathcal { N } _ { \mathbf { x } \times \mathbf { y } } } \times \mathbf { w } \times ( \mathbf { \overline { { B } } } \otimes \mathcal { L } , ( \mathbf { x } , \mathbf { x } , \mathbf { y } ) ) } & { } \\ { \overset { ( 1 ) } { \leq } \mathbf { R } _ { ( \mathbf { x } , \mathbf { y } ) \sim \mathcal { D } _ { \mathbf { x } \times \mathbf { y } } } \times \mathbf { w } \times \times ( \mathbf { a } \in \{ \rho _ { 1 } \} ) ^ { d _ { 1 } \times \mathbb { N } } , \ \mathbf { W } \in \mathcal { D } _ { \mathbf { A } } ( \mathbf { X } ) , \mathbf { v } \in \mathbb { R } ^ { N } \ ; \ ( \mathbf { A } \circ \mathbf { X } ) \times \mathbf { v } \times \mathbf { w } } & { } \\ { \overset { ( 2 ) } { \leq } \mathbf { R } _ { \mathbf { x } \times \mathcal { N } , \mathbf { w } \times \mathbf { N } } \times ( \mathbf { A } \in \{ \rho _ { 1 } \} ) ^ { d _ { 1 } \times \mathbb { N } } , \ \mathbf { W } \in \mathcal { D } _ { \mathbf { A } } ( \mathbf { X } ) , \mathbf { v } \in \mathbb { R } ^ { N } \ ; \ ( \mathbf { A } \circ \mathbf { X } ) \times \mathbf { v } = 0 , } & { } \\ { \overset { ( 4 ) } { \leq } \mathbf { R } _ { \mathbf { x } \times \mathcal { N } , \mathbf { w } \times \mathbf { N } } ( \mathbf { S } \subset [ \mathbf { N } ] \ \cdot \{ | \mathbf { S } | \ \ \geq \operatorname* { m a x } [ N _ { c , \mathrm { t r a l i k } } ( \alpha ( \langle \mathbf { W } \mathbf { X } ) ) d _ { 0 } + 1 ] \} } & { } \\ - \mathbf { B } _ { \mathbf { x } \times \mathbf { y } } [ \mathbb { W } _ { \mathbf { w } \times \mathbf { x } } ( \mathcal { S } \subset [ \mathbf { N } ] \ \cdot \{ | \mathbf { S } | \ \geq \operatorname* { m a x } [ \mathcal { N } _ { c , \mathrm { r a l i k } } ( \alpha ( \mathbf { W } \mathbf { X } ) ) d _ \end{array} +$$ + +where + +1. If we are at DLM a in $\mathcal { D } _ { \mathbf { A } } \left( \mathbf { X } \right)$ , then Lemma 2 implies $\left( \mathbf { A } \circ \mathbf { X } \right) \mathbf { e } = 0$ . Also, if $e ^ { ( n ) } = 0$ on some sample, we necessarily classify it correctly, and therefore $\mathrm { M C E } \leq \| \mathbf { e } \| _ { 0 } / N$ . Since $\mathrm { M C E } > \epsilon$ in $\mathcal { L } _ { \epsilon }$ this implies that $N \bar { \epsilon } < \| \mathbf { e } \| _ { 0 }$ . Thus, this inequality holds for ${ \bf v } = { \bf e }$ . + +2. We apply assumption 1, that $\mathbf { X } \sim { \mathcal { N } }$ . + +3. Assumption 4 implies $d _ { 0 } d _ { 1 } { \dot { > } } N \log ^ { 4 } N \geq N$ . Thus, we can apply the following Lemma, proven in appendix section 12.1: + +Lemma 14. Let $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N }$ , $\mathbf { A } \in \{ \rho , 1 \} ^ { d _ { 1 } \times N }$ , $S \subset [ N ]$ and $d _ { 0 } d _ { 1 } \geq N$ . Then, simultaneously for every possible A and $S$ such that + +$$ +\left. S \right. \leq \mathrm { r a n k } \left( { \bf A } _ { S } \right) d _ { 0 } , +$$ + +we have that, $\mathbf { X }$ -a.e., $\mathbf { \Delta } \nexists \mathbf { v } \in \mathbb { R } ^ { N }$ such that $v _ { n } \neq 0 \forall n \in S$ and $( \mathbf { A } \circ \mathbf { X } ) \mathbf { v } = 0$ . + +4. Recall that $K _ { r } \triangleq \operatorname* { m a x } \left[ N \epsilon , r d _ { 0 } \right]$ . We use the union bound over all possible ranks $r \geq 1$ we ignore the $r = 0$ case since for $\rho \neq 0$ (see eq. (2.1)) there is zero probability that rank $( a ( \mathbf { W } \mathbf { X } _ { S } ) ) = 0$ for some non-empty $S$ . For each rank $r \geq 1$ , it is required that $| S | > K _ { r } = \operatorname* { m a x } \left[ N \epsilon , r d _ { 0 } \right]$ , so $| S | = K _ { r }$ is a relaxation of the original condition, and thus its probability is not lower. + +5. We again use the union bound over all possible subsets $S$ of size $K _ { r }$ . + +Thus, from eq. (8.1), we have + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \mathbf { x } \times \mathbf { x } ^ { \beta } } \nu ( \xi , \alpha ( \mathbf { x } , \mathbf { x } ) ) } \\ & = \frac { \nu ( \mathbf { x } ) } { 2 } \sum _ { i = 1 } ^ { N } \sum _ { \substack { s \geq 0 , i \geq 1 } } \sum _ { \substack { [ \xi \in \mathcal { S } _ { \mathbf { x } } \times \mathcal { N } _ { i } ] ( \xi \in \mathcal { S } _ { \mathbf { x } } \times \{ i \} ) = ( \xi \cup \mathbf { X } _ { \mathbf { x } } ) ] = ( \xi \mid \mathbf { X } _ { \mathbf { x } } ) } } \\ & { \leq \frac { \nu ( \mathbf { x } ) } { 2 } \sum _ { i = 1 } ^ { N } \operatorname* { s u p } _ { ( \xi \in \mathcal { S } _ { \mathbf { x } } ) } [ \xi \operatorname* { s u p } _ { ( \xi \in \mathcal { S } _ { \mathbf { x } } ) ] } ( \operatorname* { s u p } _ { ( \xi \in \mathcal { S } _ { \mathbf { x } } ) \{ ( \xi \mathbf { X } _ { \mathbf { x } } ) \} = \{ \alpha \} } ] } \\ & \overset { ( a ) } { \geq } \sum _ { i = 1 } ^ { N } ( \frac { \nu } { L } ) \sum _ { \substack { ( \xi ) \leq ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ) = ( \xi ) } } \\ & \overset { ( a ) } { \leq } \sum _ { i = 1 } ^ { N } ( \frac { \nu } { L } ) \sum _ \substack { ( \xi ) \leq ( \xi ) ( \xi ) \leq ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) ( \xi ) \leq ( \xi ) \mathrm { h } } \{ \alpha \} \exp \{ ( \xi \} \sum _ \mathbf { x } \in \{ \xi \} \} \xi ) ( \xi ) ( \xi ) ( \end{array} +$$ + +1. Since we take the expectation over $\mathbf { X }$ , the location of $S$ does not affect the probability. Therefore, we can set without loss of generality $S = \left[ K _ { r } \right]$ . + +2. Note that $r \leq N / d _ { 0 } \dot { < } \operatorname* { m i n } \left[ d _ { 0 } , d _ { 1 } \right]$ from assumptions 3 and 4. Thus, with $k = K _ { r } \ge d _ { 0 }$ , we apply the following Lemma, proven in appendix section 12.2: + +Lemma 15. Let $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times k }$ be a random matrix with independent and identically distributed columns, and $\mathbf { W } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ an independent standard random Gaussian matrix. Then, in the limit m $\mathrm { i n } \left[ k , d _ { 0 } , d _ { 1 } \right] \dot { > } r$ , + +$$ +\begin{array} { r } { \mathbb { P } \left( \mathrm { r a n k } \left( a \left( \mathbf { W } \mathbf { X } \right) \right) = r \right) \dot { \leq } 2 ^ { k + r d _ { 0 } \left( \log d _ { 1 } + \log k \right) + r ^ { 2 } } \mathbb { P } \left( \mathbf { W } \mathbf { X } _ { [ \lfloor k / 2 \rfloor ] } > 0 \right) . } \end{array} +$$ + +3. Note that $K _ { r } \geq N \epsilon \dot { = } N > 2 d _ { 1 }$ , and $\operatorname* { m i n } \left[ K _ { r } , d _ { 0 } , d _ { 1 } \right] \dot { > } d _ { 0 } \underline { { d _ { 1 } } } / K _ { r } \dot { > } 1$ from assumptions 2 and 4. Thus, we apply the following Lemma (with $\mathbf { C } = \mathbf { X } ^ { \top }$ , $\mathbf { B } = \mathbf { W } ^ { \top }$ , $M = d _ { 0 }$ , $L = d _ { 1 }$ and $N = K _ { r } / 2$ ), proven in appendix section 12.3: + +Lemma 16. Let $\mathbf { C } \in \mathbb { R } ^ { N \times M }$ and $\mathbf { B } \in \mathbb { R } ^ { M \times L }$ be two independent standard random Gaussian matrices. Without loss of generality, assume $N \geq L ,$ , and denote $\alpha \triangleq M L / N$ . Then, in the regime $M \leq N$ and in the limit min $[ N , M , L ] \dot { > } \alpha \dot { > } 1$ , we have + +$$ +\begin{array} { r } { \mathbb { P } \left( \mathbf { C B } > 0 \right) \dot { \leq } \exp \left( - 0 . 4 N \alpha ^ { 1 / 4 } \right) . } \end{array} +$$ + +4. We use $r d _ { 0 } \leq N , \left( \begin{array} { c } { { N } } \\ { { K _ { r } } } \end{array} \right) \leq 2 ^ { N } , K _ { r } \leq N ,$ , and $d _ { 1 } \dot { < } N$ (from assumption 4) and $r ^ { 2 } \leq$ $N ^ { 2 } / d _ { 0 } ^ { 2 } { \dot { < } } N$ (from assumption (3)) to simplify the combintaorial expressions. + +5. First, note that $r = 1$ is the maximal term in the sum, so we can neglect the other, exponentially smaller, terms. Second, from assumption 3 we have $d _ { 0 } { \dot { \le } } N$ , so + +$$ +\operatorname* { l i m } _ { N \to \infty } 0 . 2 3 \operatorname* { m a x } \left[ \epsilon , d _ { 0 } \left( N \right) / N \right] ^ { 3 / 4 } = 0 . 2 3 \operatorname* { m a x } \left[ \epsilon , \operatorname* { l i m } _ { N \to \infty } d _ { 0 } \left( N \right) / N \right] ^ { 3 / 4 } = \gamma _ { \epsilon } . +$$ + +Third, from assumption 4 we have $N \log ^ { 4 } N { \dot { < } } d _ { 0 } d _ { 1 }$ , so the $2 ^ { N \log N }$ term is negligible. + +Thus, + +$$ +\begin{array} { r } { \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { \leq } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) . } \end{array} +$$ + +which proves the Theorem for the case $\rho \neq 0$ . + +Next, we consider the case $\rho = 0$ . In this case, we need to change transition (4) in eq. (8.1), so the sum starts from $r = 0$ , since now we can have rank $( a ( \mathbf { W } \mathbf { X } _ { S } ) \bar { ) } = 0$ . Following exactly the same + +logic (except the modification to the sum), we only need to modify transition (5)in eq. (8.2) – since now the maximal term in the sum is at $r = 0$ . This entails $\gamma _ { \epsilon } = 0 . 2 3 \epsilon ^ { 3 / 4 }$ . + +Corollary 17. (Corollary 7 restated) Given assumptions 1-4, for any $\delta > 0$ (possibly a vanishing function of $N _ { . }$ ), we have, with probability $1 - \delta$ , that the angular volume of sub-optimal DLMs, with $\mathrm { M C E } > \epsilon > 0$ , is exponentially vanishing in $N$ as + +$$ +{ \mathcal { V } } \left( { \mathcal { L } } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { \leq } \frac { 1 } { \delta } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) +$$ + +Proof. Since $\mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) \geq 0$ we can use Markov’s Theorem $( \mathrm { F a c t } 1 1 ) \forall \eta > 0$ : + +$$ +\mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left( \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) < \eta \right) > 1 - \frac { \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) } { \eta } +$$ + +denoting $\begin{array} { r } { \eta = \frac { 1 } { \delta } \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) } \end{array}$ , and using Theorem (6) we prove the corollary. + +$$ +\begin{array} { r l } & { 1 - \delta < \mathbb { P } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left( \displaystyle \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) < \frac { 1 } { \delta } \mathbb { E } _ { { \mathbf { X } } \sim { \mathcal { N } } } \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) \right) } \\ & { \quad \quad \quad < \mathbb { P } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left( \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) \dot { \leq } \frac { 1 } { \delta } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) \right) } \end{array} +$$ + +where we note that replacing a regular inequality $<$ with inequality in the leading order $\dot { \leq }$ only removes constraints, and therefore increases the probability. □ + +# 9 CONSTRUCTION OF GLOBAL MINIMA: PROOF OF THEOREM 8: + +Recall the LReLU non-linearity + +$$ +f \left( x \right) \triangleq { \left\{ \begin{array} { l l } { \rho x } & { , { \mathrm { i f ~ } } x < 0 } \\ { x } & { , { \mathrm { i f ~ } } x \geq 0 } \end{array} \right. } +$$ + +in eq. (2.1), where $\rho \neq 1$ + +Theorem 18. (Theorem 8 restated) For any $\mathbf { y } \in \{ 0 , 1 \} ^ { N }$ and $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N }$ almost everywhere we find matrices $\mathbf { W ^ { * } } ~ \in ~ \mathbb { R } ^ { d _ { 1 } ^ { * } \times d _ { 0 } }$ and $\mathbf { z } ^ { * } ~ \in ~ \mathbb { R } ^ { d _ { 1 } ^ { * } }$ , such that $\mathbf { y } ~ = ~ f \left( \mathbf { W } ^ { * } \mathbf { X } \right) ^ { \top } \mathbf { z } ^ { * }$ , where $d _ { 1 } ^ { * } \ \triangleq$ $4 \left\lceil N / \left( 2 d _ { 0 } - 2 \right) \right\rceil$ and $\forall i , n : \mathbf { w } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \neq 0$ . Therefore, every MNN with $d _ { 1 } \ \geq \ d _ { 1 } ^ { * }$ has a DLM which achieves zero error $\mathbf { e } = 0$ . + +We prove the existence of a solution $( \mathbf { W } ^ { * } , \mathbf { z } ^ { * } )$ , by explicitly constructing it. This construction is a variant of (Baum, 1988, Theorem 1), except we use LReLU without bias and MSE – instead of threshold units with bias and MCE. First, we note that for any $\epsilon _ { 1 } > \epsilon _ { 2 } > 0$ , the following trapezoid function can be written as a scaled sum of four LReLU: + +$$ +\begin{array} { l } { \displaystyle \tau \left( x \right) \triangleq \left\{ \begin{array} { l l } { 0 } & { \mathrm { , ~ i f ~ } | x | > \epsilon _ { 1 } } \\ { 1 } & { \mathrm { , ~ i f ~ } | x | \leq \epsilon _ { 2 } } \\ { \frac { \epsilon _ { 1 } - | x | } { \epsilon _ { 1 } - \epsilon _ { 2 } } } & { \mathrm { , ~ i f ~ } \epsilon _ { 2 } < | x | \leq \epsilon _ { 1 } } \\ { = \displaystyle \frac { 1 } { \epsilon _ { 1 } - \epsilon _ { 2 } } \frac { 1 } { 1 - \rho } \left[ f \left( x + \epsilon _ { 1 } \right) - f \left( x + \epsilon _ { 2 } \right) - f \left( x - \epsilon _ { 2 } \right) + f \left( x - \epsilon _ { 1 } \right) \right] . } \end{array} \right. } \end{array} +$$ + +Next, we examine the set of data points which are classified to 1: $S ^ { + } \triangleq \left\{ n \in [ N ] | y ^ { ( n ) } = 1 \right\}$ . Without loss of generality, assume $\vert S ^ { + } \vert \le \frac { N } { 2 }$ . We partition $S ^ { + }$ to + +$$ +K = \left\lceil { \frac { | S ^ { + } | } { d _ { 0 } - 1 } } \right\rceil \leq \left\lceil { \frac { N } { 2 \left( d _ { 0 } - 1 \right) } } \right\rceil +$$ + +subsets $\left\{ { \cal { S } } _ { i } ^ { + } \right\} _ { i = 1 } ^ { K }$ , each with no more than $d _ { 0 } - 1$ samples. For almost any dataset we can find K $K$ i=1hyperplanes passing through the origin, with normals such that each hyperplane contains all $d _ { 0 } - 1$ points in subset ${ S } _ { i } ^ { + }$ , i.e., + +$$ +\tilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { X } _ { S _ { i } ^ { + } } = 0 , +$$ + +but no other point, so $\forall n \notin S _ { i } ^ { + } : \tilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \neq 0$ , + +If $\epsilon _ { 1 } , \epsilon _ { 2 }$ in eq. (9.1) are sufficiently small $( \forall n \notin S _ { i } ^ { + } : \left| \tilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right| > \epsilon _ { 1 } )$ then we have + +$$ +\tau \left( \tilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right) = \left\{ \begin{array} { l l } { 1 } & { \mathrm { , ~ i f ~ } n \in S _ { i } ^ { + } } \\ { 0 } & { \mathrm { , ~ e l s e } } \end{array} \right. . +$$ + +Then we have + +$$ +\sum _ { i = 1 } ^ { K } \tau \left( \tilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right) = \left\{ \mathbf { 1 } \quad , \mathrm { i f } n \in \mathcal { S } ^ { + } \right. +$$ + +which gives the correct classification on all the data points. Thus, from eq. (9.1), we can construct a MNN with + +$$ +d _ { 1 } ^ { * } = 4 K +$$ + +hidden neurons which achieves zero error. This is straightforward to do if we have a bias in each neuron. To construct this MNN even without bias, we first find a vector $\hat { \mathbf { w } } _ { i }$ such that + +$$ +\hat { \mathbf { w } } _ { i } ^ { \top } \left[ { \bf X } _ { S _ { i } ^ { + } } , \tilde { \mathbf { w } } _ { i } \right] = \left[ 1 , \ldots , 1 , 1 , 0 \right] . +$$ + +Note that this is possible since $\left[ { \bf X } _ { S _ { i } ^ { + } } , \tilde { { \bf w } } _ { i } \right]$ has full rank $\mathbf { X }$ -a.e. (the matrix $\mathbf { X } _ { S _ { i } ^ { + } } \in \mathbb { R } ^ { d _ { 0 } \times d _ { 0 } - 1 }$ has, $\mathbf { X }$ -a.e., one zero left eigenvector, which is $\tilde { \mathbf { w } } _ { i }$ , according to eq. (9.2)). Additionally, we can set + +$$ +\begin{array} { r } { \| \tilde { \mathbf { w } } _ { i } \| = \| \hat { \mathbf { w } } _ { i } \| ~ , } \end{array} +$$ + +since changing the scale of $\mathbf { w } _ { i }$ would not affect the validity of eq. (9.2). Then, we denote + +$$ +\begin{array} { r l } & { { \mathbf w } _ { i } ^ { ( 1 ) } \triangleq \tilde { \mathbf { w } } _ { i } + \epsilon _ { 1 } \hat { \mathbf { w } } _ { i } ; { \mathbf w } _ { i } ^ { ( 2 ) } \triangleq \tilde { \mathbf { w } } _ { i } + \epsilon _ { 2 } \hat { \mathbf { w } } _ { i } } \\ & { { \mathbf w } _ { i } ^ { ( 3 ) } \triangleq \tilde { \mathbf { w } } _ { i } - \epsilon _ { 2 } \hat { \mathbf { w } } _ { i } ; { \mathbf w } _ { i } ^ { ( 4 ) } \triangleq \tilde { \mathbf { w } } _ { i } - \epsilon _ { 1 } \hat { \mathbf { w } } _ { i } . } \end{array} +$$ + +Note, from eqs. (9.2) and (9.4) that this choice satisfies + +$$ +\forall n \in S _ { i } ^ { + } : \mathbf { w } _ { i } ^ { ( j ) \top } \mathbf { x } ^ { ( n ) } = \left\{ \begin{array} { l l } { \epsilon _ { 1 } } & { , \mathrm { i f } j = 1 } \\ { \epsilon _ { 2 } } & { , \mathrm { i f } j = 2 } \\ { - \epsilon _ { 2 } } & { , \mathrm { i f } j = 3 } \\ { - \epsilon _ { 1 } } & { , \mathrm { i f } j = 4 } \end{array} \right. . +$$ + +Also, to ensure that $\forall n \notin S _ { i } ^ { + }$ the sign of $\mathbf { w } _ { i } ^ { ( j ) ^ { \top } } \mathbf { x } ^ { ( n ) }$ does not change for different $j$ , for some $\beta , \gamma < 1$ we define + +$$ +\epsilon _ { 1 } = \beta \frac { \operatorname* { m i n } _ { n \not \in { \cal S } _ { i } ^ { + } } \left| \tilde { \mathbf { w } } _ { i } ^ { \top } { \mathbf { x } } ^ { ( n ) } \right| } { \operatorname* { m a x } _ { n \not \in { \cal S } _ { i } ^ { + } } \left| \hat { \mathbf { w } } _ { i } ^ { \top } { \mathbf { x } } ^ { ( n ) } \right| } , \epsilon _ { 2 } = \gamma \epsilon _ { 1 } , +$$ + +where with probability $\begin{array} { r } { 1 , \operatorname* { m i n } _ { n \not \in \mathcal { S } _ { i } ^ { + } } \left| \tilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right| > 0 } \end{array}$ and $\mathrm { m a x } _ { n \notin { \cal S } _ { i } ^ { + } } \left| \hat { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right| > 0$ . Defining + +$$ +\begin{array} { r l } & { \mathbf { W } _ { i } \triangleq \left[ \mathbf { w } _ { i } ^ { ( 1 ) } , \mathbf { w } _ { i } ^ { ( 2 ) } , \mathbf { w } _ { i } ^ { ( 3 ) } , \mathbf { w } _ { i } ^ { ( 4 ) } \right] ^ { \top } \in \mathbb { R } ^ { 4 K \times d _ { 0 } } } \\ & { \mathbf { z } _ { i } \triangleq \left[ 1 , - 1 , - 1 , 1 \right] ^ { \top } \in \mathbb { R } ^ { 4 } } \end{array} +$$ + +and combining all the above facts, we have + +$$ +\begin{array} { l } { f \left( \mathbf { W } _ { i } \mathbf { x } ^ { ( n ) } \right) ^ { \top } \mathbf { z } _ { i } } \\ { \ = \displaystyle \frac { 1 } { \epsilon _ { 1 } - \epsilon _ { 2 } } \frac { 1 } { 1 - \rho } \left[ f \left( \mathbf { w } _ { i } ^ { ( 1 ) \top } \mathbf { x } ^ { ( n ) } \right) - f \left( \mathbf { w } _ { i } ^ { ( 2 ) \top } \mathbf { x } ^ { ( n ) } \right) - f \left( \mathbf { w } _ { i } ^ { ( 3 ) \top } \mathbf { x } ^ { ( n ) } \right) + f \left( \mathbf { w } _ { i } ^ { ( 3 ) \top } \mathbf { x } ^ { ( n ) } \right) \right] } \\ { \ = \displaystyle \frac { 1 } { \epsilon _ { 1 } - \epsilon _ { 2 } } \frac { 1 } { 1 - \rho } \left[ f \left( \widetilde { \mathbf { w } } _ { i } ^ { ( 1 ) } \mathbf { x } ^ { ( n ) } + \epsilon _ { 1 } \widehat { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right) - f \left( \widetilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } + \epsilon _ { 2 } \widehat { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right) \right. } \\ { \ - \left. f \left( \widetilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } - \epsilon _ { 2 } \widehat { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right) + f \left( \widetilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } - \epsilon _ { 1 } \widehat { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right) \right] } \\ { \ = \displaystyle \frac { 1 } { \left\{ 0 \right. \ } , \ \mathrm { i f } n \in \mathcal { S } _ { i } ^ { + } } . \end{array} +$$ + +Thus, for + +$$ +\begin{array} { c } { { \displaystyle { \bf W } ^ { * } = \left[ { \bf W } _ { 1 } ^ { \top } , \ldots , { \bf W } _ { K } ^ { \top } \right] ^ { \top } \in \mathbb { R } ^ { 4 \times d _ { 0 } } } } \\ { { \displaystyle { \bf z } ^ { * } = \frac { 1 } { \epsilon _ { 1 } - \epsilon _ { 2 } } \frac { 1 } { 1 - \rho } \cdot \left[ { \bf z } _ { 1 } , \ldots , { \bf z } _ { K } \right] \in \mathbb { R } ^ { 4 K } } } \end{array} +$$ + +we obtain a MNN that implements + +$$ +f \left( \mathbf { W } ^ { * } \mathbf { x } ^ { ( n ) } \right) ^ { \top } \mathbf { z } ^ { * } = { \left\{ \begin{array} { l l } { 1 } & { , { \mathrm { i f } } \ n \in S ^ { + } } \\ { 0 } & { , { \mathrm { e l s e } } } \end{array} \right. } +$$ + +and thus achieves zero error. Clearly, from this construction, if $\mathbf { w } _ { i }$ is a row of $\mathbf { W } ^ { * }$ , then $\forall n \in S _ { i } ^ { + } , \forall i :$ $| \mathbf { w } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } | \geq \epsilon _ { 2 }$ , and with probability $1 \forall n \notin S _ { i } ^ { + } , \forall i : \left| \mathbf { w } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right| > 0$ , so this construction does not touch any non-differentiable region of the MSE.  + +# 10 GLOBAL MINIMA: PROOF OF THEOREM 9 + +Theorem 19. (Theorem 9 restated). Given assumptions 1-3, we set $\delta \dot { = } \sqrt { { \textstyle \frac { 8 } { \pi } } } d _ { 0 } ^ { - 1 / 2 } \ +$ $2 d _ { 0 } ^ { 1 / 2 } \sqrt { { \log d _ { 0 } } } / { N }$ and $d _ { 1 } ^ { * } = 2 N / d _ { 0 }$ , or if assumption $^ { 5 }$ holds, we set $d _ { 1 } ^ { * }$ as in this assumption. Then, with probability $1 - \delta$ , the angular volume of global minima is lower bounded as, + +$$ +\mathscr { V } \left( \mathscr { G } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { > } \exp \left( - d _ { 1 } ^ { \ast } d _ { 0 } \log N \right) \dot { \geq } \exp \left( - 2 N \log N \right) . +$$ + +In this section we lower bound the angular volume of $\mathcal { G }$ (definition 4), i.e., differentiable regions in which there exist DLMs with $\mathrm { { M C E } = 0 }$ . We lower bound $\mathcal { V } \left( \mathcal { G } \right)$ using the angular volume corresponding to the differentiable region containing a single global minimum. + +From assumption 4, we have $d _ { 0 } d _ { 1 } \dot { > } N$ , so we can apply Theorem 8 and say that the labels are generated using a $( \mathbf { X } , \mathbf { y } )$ -dependent MNN: $\mathbf { y } ~ = ~ f \left( \mathbf { W } ^ { * } \mathbf { X } \right) ^ { \top } \mathbf { z } ^ { * }$ with target weights $\begin{array} { r l } { \mathbf { W } ^ { * } } & { { } = } \end{array}$ $\left[ { \bf w } _ { 1 } ^ { * \top } , \ldots , { \bf w } _ { d _ { 1 } ^ { * } } ^ { * \top } \right] ^ { \top } \in \mathbb { R } ^ { d _ { 1 } ^ { * } \times d _ { 0 } }$ and $\mathbf { z } ^ { * } \in \mathbb { R } ^ { d _ { 1 } }$ . If, in addition, assumption 5 holds then we can assume $\mathbf { W } ^ { * }$ and $\mathbf { z } ^ { \ast }$ are independent from $( \mathbf { X } , \mathbf { y } )$ . In both cases, the following differentiable region + +$$ +\begin{array} { r } { \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) \triangleq \left\{ \mathbf { W } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } } | \forall i \le d _ { 1 } ^ { * } : \ \mathrm { s i g n } \left( \mathbf { w } _ { i } ^ { \top } \mathbf { X } \right) = \mathrm { s i g n } \left( \mathbf { w } _ { i } ^ { * ^ { \top } } \mathbf { X } \right) \right\} , } \end{array} +$$ + +also contains a differentiable global minimum (just set $\mathbf { w } _ { i } = \mathbf { w } _ { i } ^ { * }$ , $z _ { i } = z _ { i } ^ { * } \forall i \leq d _ { 1 } ^ { * }$ , and $z _ { i } = 0$ $\forall i > d _ { 1 } ^ { * } ,$ ), and therefore $\forall \mathbf { X } , \mathbf { y }$ and their corresponding $\mathbf { W } ^ { * }$ , we have + +$$ +\mathcal { G } \left( \mathbf { X } , \mathbf { y } \right) \supset \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) +$$ + +Also, we will make use of the following definition. + +Definition 20. Let $\mathbf { X }$ have an angular margin $\alpha$ from $\mathbf { W } ^ { * }$ if all datapoints (columns in $\mathbf { X }$ ) are at an angle of at least $\alpha$ from all the weight hyperplanes (rows of $\mathbf { W } ^ { * }$ ) , i.e., $\mathbf { X }$ is in the set + +$$ +\mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \triangleq \left. \mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N } | \forall i , n : \left| \frac { \mathbf { x } ^ { ( n ) \top } \mathbf { w } _ { i } ^ { * } } { \left\| \mathbf { x } ^ { ( n ) } \right\| \left\| \mathbf { w } _ { i } ^ { * } \right\| } \right| > \mathrm { s i n } \alpha \right. . +$$ + +Using the definitions in eqs. (10.3) and (10.1), we prove the Theorem using the following three Lemmas. + +First, In appendix section 13.2 we prove + +Lemma 21. For any $\alpha$ , $i f \mathbf { W } ^ { * }$ is independent from W then, in the limit $N \to \infty$ , $\forall \mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right)$ with l $\mathrm { { o g } s i n } \alpha \dot { > } d _ { 0 } ^ { - 1 } \log d _ { 0 }$ + +$$ +\begin{array} { r } { \mathcal { V } \left( \tilde { \mathcal { G } } \right) = \mathbb { P } _ { \mathbf { W } \sim \mathcal { N } } \left( \mathbf { W } \in \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) \right) \dot { \geq } \exp \left( d _ { 0 } d _ { 1 } ^ { * } \log \sin \alpha \right) . } \end{array} +$$ + +Second, in appendix section 13.3 we prove + +Lemma 22. Let $\mathbf { W } ^ { * } \in \mathbb { R } ^ { d _ { 1 } ^ { * } \times d _ { 0 } }$ a fixed matrix independent of $\mathbf { X }$ . Then, in the limit $N \infty$ with $d _ { 1 } ^ { * } { \dot { \leq } } d _ { 0 } { \dot { \leq } } N$ , the probability of not having an angular margin $\sin \alpha = 1 / \left( d _ { 1 } ^ { * } d _ { 0 } N \right)$ (eq. (10.3)) is upper bounded by + +$$ +\mathbb { P } \left( \mathbf { X } \notin \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \dot { \leq } \sqrt { \frac { 2 } { \pi } } d _ { 0 } ^ { - 1 / 2 } +$$ + +Lastly, in appendix section 13.4 we prove + +Lemma 23. Let $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N }$ be a standard random Gaussian matrix of datapoints. Then we can find, with probability $I$ , $( \mathbf { X } , \mathbf { y } )$ -dependent matrices $\mathbf { W } ^ { * }$ and $\mathbf { z } ^ { \ast }$ as in Theorem 8 (where $d _ { 1 } ^ { * }$ , $4 \ : \lceil N / \left( 2 d _ { 0 } - 2 \right) \rceil $ . Moreover, in the limit $N \to \infty$ , where $N / d _ { 0 } { \dot { \le } } d _ { 0 } { \dot { \le } } N$ , for any y, we can bound the probability of not having an angular margin (eq. (10.3)) with $\sin \alpha = 1 / \left( d _ { 1 } ^ { * } d _ { 0 } N \right)$ by + +$$ +\mathbb { P } \left( \mathbf { X } \notin \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \dot { \leq } \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } +$$ + +Recall that $\forall \mathbf { X } , \mathbf { y }$ and their corresponding $\mathbf { W } ^ { * }$ , we have $\mathcal { G } \left( \mathbf { X } , \mathbf { y } \right) \subset \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right)$ (eq. (10.2)). Thus, combining Lemmas 21 with $\sin \alpha = 1 / \left( \bar { d } _ { 1 } ^ { * } d _ { 0 } N \right)$ together with either Lemma 22 or 23, we prove the first (left) inequality of Theorem 9: + +$$ +{ \mathcal { V } } \left( { \mathcal { G } } \left( \mathbf { X } , \mathbf { y } \right) \right) { \dot { \geq } } \exp \left( - d _ { 1 } ^ { * } d _ { 0 } \log N \right) +$$ + +Next, if $d _ { 1 } ^ { * } = 2 N / d _ { 0 }$ or $d _ { 1 } ^ { * } { \dot { < } } N / d _ { 0 }$ (is assumption 5 holds), we obtain the second (right) inequality + +$$ +\exp \left( - d _ { 1 } ^ { \ast } d _ { 0 } \log N \right) \dot { \geq } \exp \left( - 2 N \log N \right) . +$$ + +11 VOLUME RATIO OF GLOBAL AND LOCAL MINIMA: PROOF OF THEOREM 10 + +Theorem 24. (Theorem 10 restated) Given assumptions 1-3, we set $\delta \doteq \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } +$ $2 d _ { 0 } ^ { 1 / 2 } \sqrt { { \log d _ { 0 } } } / { N }$ . Then, with probability $1 - \delta$ , the angular volume of sub-optimal DLMs, with $\mathrm { M C E } > \epsilon > 0 ,$ , is exponentially vanishing in $N ,$ , in comparison to the angular volume of global minima with $\mathrm { M C E } = 0$ + +$$ +\frac { \mathscr { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) } { \mathscr { V } \left( \mathcal { G } \left( \mathbf { X } , \mathbf { y } \right) \right) } \dot { \leq } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) \dot { \leq } \exp \left( - \gamma _ { \epsilon } N \log N \right) . +$$ + +To prove this theorem we first calculate the expectation of the angular volume ratio given the $\mathbf { X }$ -event that the bound in Theorem 9 holds (given assumptions 1-3), i.e., $\mathcal { V } \left( \mathcal { G } \left( \mathbf { X } , \mathbf { y } \right) \right) \dot { \geq } \dot { \exp } \left( - 2 N \log N \right)$ . Denoting this even $ { \varepsilon } ^ { 6 }$ as $\mathcal { M }$ , we find: + +$$ +\begin{array} { r l } & { \mathbb { E } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left[ \frac { \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) } { \mathcal { V } \left( \mathcal { G } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) } | { \mathcal { M } } \right] \overset { ( 1 ) } { \leq } \frac { \mathbb { E } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left[ \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) \big | { \mathcal { M } } \right] } { \exp \left( - 2 N \log N \right) } \overset { ( 2 ) } { \leq } } \\ & { \frac { \mathbb { E } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left[ \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) \right] } { \mathbb { P } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left( { \mathcal { M } } \right) \exp \left( - 2 N \log N \right) } \overset { ( 3 ) } { \leq } \frac { \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) } { \mathbb { P } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left( { \mathcal { M } } \right) \exp \left( - 2 N \log N \right) } \overset { ( 4 ) } { \leq } } \\ & { \quad \quad \frac { \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) } { \exp \left( - 2 N \log N \right) } \overset { ( 5 ) } { \leq } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) } \end{array} +$$ + +where + +1. We apply Theorem 9. +2. We use the following fact + +Fact 25. For any variable $X \geq 0$ and event $\mathcal { A }$ (where $\bar { A }$ is its complement) + +$$ +\mathbb { E } \left[ X \right] = \mathbb { E } \left[ X | { \cal { A } } \right] \mathbb { P } \left( { \cal { A } } \right) + \mathbb { E } \left[ X | { \bar { \cal { A } } } \right] ( 1 - \mathbb { P } \left( { \cal { A } } \right) ) \geq \mathbb { E } \left[ X | { \cal { A } } \right] \mathbb { P } \left( { \cal { A } } \right) +$$ + +3. We apply Theorem 6. + +4. We apply Theorem 9. + +5. We use assumption 4, which implies $\gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \dot { > } 2 N \log N .$ + +For simplicity, in the reminder of the proof we denote + +$$ +R \left( \mathbf { X } \right) \triangleq \frac { \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( \mathbf { X } , \mathbf { y } \right) \right) } { \mathcal { V } \left( \mathcal { G } \left( \mathbf { X } , \mathbf { y } \right) \right) } . +$$ + +From Markov inequality (Fact 11), since $R \left( { \bf X } \right) \geq 0$ , we have $\forall \eta \left( N \right) > 0$ : + +$$ +\mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) \geq \eta \left( N \right) \vert \mathcal { M } \right] \leq \frac { \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) \vert \mathcal { M } \right] } { \eta \left( N \right) } +$$ + +On the other hand, from fact 25, we have + +$$ +\begin{array} { r } { 1 - \operatorname { \mathbb { P } } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) < \eta \left( N \right) \vert \mathcal { M } \right] \ge 1 - \frac { \operatorname { \mathbb { P } } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) < \eta \left( N \right) \right] } { \operatorname { \mathbb { P } } _ { \mathbf { X } \sim \mathcal { N } } \left( \mathcal { M } \right) } . } \end{array} +$$ + +Combining Eqs. (11.2)-(11.3) we obtain + +$$ +\frac { \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) \vert \mathcal { M } \right] } { \eta \left( N \right) } \geq 1 - \frac { \mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) < \eta \left( N \right) \right] } { \mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left( \mathcal { M } \right) } , +$$ + +and so + +$$ +\mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left( \mathcal { M } \right) - \mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left( \mathcal { M } \right) \frac { \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) \middle | \mathcal { M } \right] } { \eta \left( N \right) } \leq \mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) < \eta \left( N \right) \right] . +$$ + +We choose + +$$ +\eta \left( { \cal N } \right) = N \mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left( { \cal M } \right) \mathbb { E } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) \vert \mathcal { M } \right] \doteq \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) +$$ + +so that + +$$ +\mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left( \mathcal { M } \right) - \frac { 1 } { N } \le \mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left[ R \left( \mathbf { X } \right) \dot { \le } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) \right] . +$$ + +Then, from Theorem 9 we have + +$$ +1 - \mathbb { P } _ { \mathbf { X } \sim \mathcal { N } } \left( \mathcal { M } \right) \dot { \leq } \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } . +$$ + +so we obtain the first (left) inequality in the Theorem (10) + +$$ +\sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } \dot { \geq 1 } - \mathbb { P } _ { { \mathbf { X } } \sim { \mathcal { N } } } \left[ \frac { \mathcal { V } \left( \mathcal { L } _ { \epsilon } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) } { \mathcal { V } \left( \mathcal { G } \left( { \mathbf { X } } , { \mathbf { y } } \right) \right) } \dot { \leq } \exp \left( - \gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \right) \right] . +$$ + +Lastly, we note that assumption 4 implies $\gamma _ { \epsilon } N ^ { 3 / 4 } \left[ d _ { 1 } d _ { 0 } \right] ^ { 1 / 4 } \dot { > } N \log N$ , which proves the second (right) inequality of the theorem. + +# Part II + +# Proofs of technical results + +In this part we prove the technical results used in part I. + +12 UPPER BOUNDING THE ANGULAR VOLUME OF SUB-OPTIMAL DIFFERENTIABLE LOCAL MINIMA: PROOFS OF LEMMAS USED IN SECTION 8 + +# 12.1 PROOF OF LEMMA 14 + +In this section we will prove Lemma 14 in subsection 12.3.3. Recall the following definition Definition 26. Let + +$$ +{ \bf A } = [ { \pmb a } _ { 1 } , \ldots , { \pmb a } _ { N } ] \ ; \ { \bf X } = [ { \pmb x } _ { 1 } , \ldots , { \pmb x } _ { N } ] , +$$ + +where $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N }$ and $\mathbf { A } \in \mathbb { R } ^ { d _ { 1 } \times N }$ . The Khatari-Rao product between the two matrices is defined as + +$$ +\begin{array} { r l } { \mathbf { A } \circ \mathbf { X } } & { \triangleq } & { \left[ \pmb { a } _ { 1 } \otimes \mathbf { x } _ { 1 } , \pmb { a } _ { 2 } \otimes \mathbf { x } _ { 2 } , . . . \pmb { a } _ { N } \otimes \mathbf { x } _ { N } \right] } \\ & { = } & { \left( \begin{array} { c c c } { a _ { 1 1 } \mathbf { x } _ { 1 } } & { a _ { 1 2 } \mathbf { x } _ { 2 } } & { . . . } \\ { a _ { 2 1 } \mathbf { x } _ { 1 } } & { a _ { 2 2 } \mathbf { x } _ { 2 } } & { \ddots } \\ { \vdots } & { \ddots } & { \ddots } \end{array} \right) . } \end{array} +$$ + +Lemma 27. (Lemma 14 restated) Let $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N }$ , $\mathbf { A } \in \{ \rho , 1 \} ^ { d _ { 1 } \times N }$ , $S \subset [ N ]$ and $d _ { 0 } d _ { 1 } \geq N$ . Then, simultaneously for every possible A and $S$ such that + +$$ +\left. S \right. \leq \mathrm { r a n k } \left( { \bf A } _ { S } \right) d _ { 0 } , +$$ + +we have that, X-a.e., $\mathbf { \Delta } \nexists \mathbf { v } \in \mathbb { R } ^ { N }$ such that $v _ { n } \neq 0 \forall n \in S$ and $( \mathbf { A } \circ \mathbf { X } ) \mathbf { v } = 0$ + +Proof. We examine specific $\mathbf { A } \in \{ \rho , 1 \} ^ { d _ { 1 } \times N }$ and $S \subset [ N ]$ , and such that $| S | \le d _ { S } d _ { 0 }$ , where we defined $d _ { S } \triangleq \operatorname { r a n k } \left( \mathbf { A } _ { S } \right)$ . We assume that $d _ { S } \geq 1$ , since otherwise the proof is trivial. Also, we assume by contradiction that $\exists \mathbf { v } \in \mathbb { R } ^ { N }$ such that $v _ { i } \neq 0 \forall i \in S$ and $( \mathbf { A } \circ \mathbf { X } ) \mathbf { v } = 0$ . Without loss of generality, assume that $S = \{ 1 , 2 , . . . , | S | \}$ and that $\mathbf { } a _ { 1 } , \mathbf { } a _ { 2 } , . . . , \mathbf { } a _ { d _ { S } }$ are linearly independent. Then + +$$ +( \mathbf { A } \circ \mathbf { X } ) \mathbf { v } = \sum _ { n = 1 } ^ { | S | } v _ { n } a _ { k , n } \mathbf { x } _ { n } = 0 +$$ + +for every $1 \leq k \leq d _ { 1 }$ . From the definition of $S$ we must have $v _ { n } \neq 0$ for every $1 \leq n \leq | S |$ . Since $\mathbf { } a _ { 1 } , \mathbf { } a _ { 2 } , . . . , \mathbf { } a _ { d _ { S } }$ are linearly independent, the rows of $\mathbf { A } _ { d _ { S } } = [ \pmb { a } _ { 1 } , \pmb { a } _ { 2 } , . . . , \pmb { a } _ { d _ { S } } ]$ span a $d _ { S }$ -dimensional space. Therefore, it is possible to find a matrix $\mathbf { R }$ such that $\mathbf { R } \mathbf { A } _ { d _ { S } } = [ \mathbf { I } _ { d _ { S } \times d _ { S } } , 0 _ { d _ { S } \times ( d _ { 1 } - d _ { S } ) } ] ^ { \top }$ , where $0 _ { i \times j }$ is the all zeros matrix with $i$ columns and $j$ rows. Consider now $\mathbf { A } _ { S } \circ \mathbf { X } _ { S }$ , i.e., the matrix composed of the columns of $\mathbf { A } \circ \mathbf { X }$ in $S$ . Applying $\mathbf { R } ^ { \prime } = \mathbf { R } \otimes \mathbf { I } _ { d _ { 0 } }$ to $\mathbf { A } _ { S } \circ \mathbf { X } _ { S }$ , turns (12.2) into $d _ { 0 } d _ { S }$ equations in the variables $v _ { 1 } , . . . , v _ { | S | }$ , of the form + +$$ +v _ { k } \mathbf { x } _ { k } + \sum _ { n = d _ { S } + 1 } ^ { | S | } v _ { n } { \tilde { a } } _ { k , n } \mathbf { x } _ { n } = 0 +$$ + +for every $1 \leq k \leq d _ { S }$ . We prove by induction that for every $1 \leq d \leq d _ { S }$ , the first $d _ { 0 } d$ equations are linearly independent, except for a set of matrices $\mathbf { X }$ of measure 0. This will immediately imply $| S | > d _ { S } \dot { d } _ { 0 }$ , or else eq. 12.2 cannot be true for $\mathbf { v } \neq 0$ . which will contradict our assumption, as required. The induction can be viewed as carrying out Gaussian elimination of the system of equations described by (12.3), where in each elimination step we characterize the set of matrices $\mathbf { X }$ that for which that step is impossible, and show it has measure 0. + +For $d = 1$ , the first $d _ { 0 }$ equations read $\begin{array} { r } { v _ { 1 } \mathbf { x } _ { 1 } + \sum _ { n = d _ { S } + 1 } ^ { | S | } v _ { n } \tilde { a } _ { 1 , n } \mathbf { x } _ { n } = 0 } \end{array}$ , and since $v _ { 1 } \neq 0$ , we must have $\mathbf { x } _ { 1 } \in \mathrm { S p a n } \left\{ \widetilde { a } _ { 1 , d _ { S } + 1 } \mathbf { x } _ { d _ { S } + 1 } , . . . , \widetilde { a } _ { 1 , | S | } \mathbf { x } _ { | S | } \right\}$ . However, except for a set of measure 0 with respect to $\mathbf { x } _ { 1 }$ (a linear subspace of $\mathbb { R } ^ { d _ { 0 } }$ with dimension less than $d _ { 0 }$ ), this can only happen if dim Span $\left\{ \tilde { a } _ { 1 , d _ { S } + 1 } \mathbf { x } _ { d _ { S } + 1 } , . . . , \tilde { a } _ { 1 , | S | } \mathbf { x } _ { | S | } \right\} = d _ { 0 }$ , which implies $| S | \ge d _ { S } - 1 + d _ { 0 } > d _ { 0 }$ and also that the first $d _ { 0 }$ rows are linearly independent (since there are $d _ { 0 }$ independent columns). + +For a general $d$ , we begin by performing Gaussian elimination on the first $( d - 1 ) d _ { 0 }$ equations, resulting in a new set of $r _ { d }$ equations, such that every new equation contains one variable that appears in no other new equation. Let $C$ be the set of the indices (equivalently, columns) of these variables $r _ { d }$ variables. From (12.3) it is clear none of the variables $v _ { d } , v _ { d + 1 } , . . . , v _ { d _ { S } }$ appear in the first $( d - 1 ) d _ { 0 }$ equations, and therefore $C \subseteq S ^ { \prime } = S \setminus \{ d , d + 1 , . . . , d _ { S } \}$ . By our induction assumptions, except for a set of measure 0, the first $( d - 1 ) d _ { 0 }$ are independent, which means that $| C | = r _ { d } = ( d - 1 ) d _ { 0 }$ . We now extend the Gaussian elimination to the next $d _ { 0 }$ equations, and eliminate all the variables in $C$ from them. The result of the elimination can be written down as, + +$$ +v _ { d } \mathbf { x } _ { d } + \sum _ { n \in S ^ { \prime } \backslash C } v _ { n } \left( \tilde { a } _ { d , n } \mathbf { I } _ { d _ { 0 } } - \mathbf { Y } \right) \mathbf { x } _ { n } = { 0 } , +$$ + +where $\mathbf { Y }$ is a square matrix of size $d _ { 0 }$ whose coefficients depend only on $\{ \widetilde { a } _ { k , n } \} _ { n \in C , d > k \geq 1 }$ and on $\{ \mathbf { x } _ { n } \} _ { n \in C }$ , and in particular do not depend on $\mathbf { x } _ { d }$ and $\left\{ \mathbf { x } _ { n } \right\} _ { n \in S ^ { \prime } \backslash C }$ . + +Now set $\tilde { \mathbf { x } } _ { n } \ = \ ( \tilde { a } _ { d , n } \mathbf { I } _ { d _ { 0 } } - \mathbf { Y } ) \mathbf { x } _ { n }$ for $n \in S ^ { \prime } \setminus C$ . As in the case of $d \ : = \ : 1$ , since $v _ { d } \neq 0$ , $\mathbf { x } _ { d } \in \mathrm { S p a n } \{ \tilde { \mathbf { x } } _ { n } \} _ { n \in S ^ { \prime } \backslash C }$ . Therefore, for all values of $\mathbf { x } _ { d } \in \mathbb { R } ^ { d _ { 0 } }$ but a set of measure zero (linear subspace of with dimension less than $d _ { 0 }$ ), we must have di $\begin{array} { r } { \operatorname* { m } \mathrm { { S p a n } } \{ \tilde { \mathbf { x } } _ { n } \} _ { n \in S ^ { \prime } \backslash C } = d _ { 0 } } \end{array}$ . From the independence of $\{ \tilde { \mathbf { x } } _ { n } \} _ { n \in S ^ { \prime } \backslash C }$ on $\mathbf { x } _ { d }$ it follows that $\mathrm { d i m } \mathrm { S p a n } \{ \tilde { \mathbf { x } } _ { n } \} _ { n \in S ^ { \prime } \backslash C } = d _ { 0 }$ holds a.e. with respect to the Lebesgue measure over $\mathbf { x }$ . + +Whenever $\mathrm { d i m } \mathrm { S p a n } \{ \tilde { \mathbf { x } } _ { n } \} _ { n \in S ^ { \prime } \backslash C } = d _ { 0 }$ we must have $| S ^ { \prime } C | \geq d _ { 0 }$ and therefore + +$$ +| S | > | S ^ { \prime } | = | C | + | S ^ { \prime } \setminus C | \geq \left( d - 1 \right) d _ { 0 } + d _ { 0 } = d _ { 0 } d . +$$ + +Moreover, $\mathrm { d i m } \mathrm { S p a n } \{ \tilde { \mathbf { x } } _ { n } \} _ { n \in S ^ { \prime } \backslash C } = d _ { 0 }$ implies that the $d _ { 0 }$ equations $\begin{array} { r } { v _ { d } \mathbf { x } _ { d } + \sum _ { n \in S ^ { \prime } \backslash C } v _ { n } \tilde { \mathbf { x } } _ { n } = 0 } \end{array}$ are independent. Thus, we may perform another step of Gaussian elimination on these $d _ { 0 }$ equations, forming $d _ { 0 }$ new equations each with a variable unique to it. Denoting by $C ^ { \prime }$ the set of these $d _ { 0 }$ variables, it is seen from (12.4) that $C ^ { \prime } \subseteq ( S ^ { \prime } \cup \{ d \} ) \setminus C$ and in particular $C ^ { \prime }$ is disjoint from $C$ . Thus, considering the first $( d - 1 ) d _ { 0 }$ equations together with the new $d _ { 0 }$ equations, we see that there is a set $C \cup C ^ { \prime }$ of $d _ { 0 } d$ variables, such that each variable in $C \cup C ^ { \prime }$ appears only in one of the $d _ { 0 } d$ equations, and each of the $d _ { 0 } d$ contains only a single variable in $C \cup C ^ { \prime }$ . This means that the first $d _ { 0 } d$ must be linearly independent for all values of $\mathbf { X }$ except for a set of Lebesgue measure zero, completing the induction. + +Thus, we have proven, that for some $\mathbf { A } \in \{ \rho , 1 \} ^ { d _ { 1 } \times N }$ and $S \subset [ N ]$ such that $| S | \leq \mathrm { r a n k } \left( \mathbf { A } _ { S } \right) d _ { 0 }$ the event + +$$ +\mathcal E \left( \mathbf A , S \right) = \left\{ \mathbf X \in \mathbb R ^ { d _ { 0 } \times N } | \exists \mathbf v \in \mathbb R ^ { N } : \left( \mathbf A \circ \mathbf X \right) \mathbf v = 0 \mathrm { a n d } v _ { n } \neq 0 , \forall n \in S \right\} +$$ + +has zero measure. The event discussed in the theorem is a union of these events: + +$$ +\mathcal { E } _ { 0 } \triangleq \bigcup _ { \mathbf { A } \in \{ \rho , 1 \} ^ { d _ { 1 } \times N } } \left[ \bigcup _ { S \subset [ N ] : | S | \leq \operatorname { r a n k } ( \mathbf { A } _ { S } ) d _ { 0 } } \mathcal { E } \left( \mathbf { A } , S \right) \right] , +$$ + +and it also has zero measure, since it is a finite union of zero measure events. + +For completeness we note the following corollary, which is not necessary for a our main results. + +Corollary 28. If $N \leq d _ { 1 } d _ { 0 }$ , then rank $( \mathbf { A } \circ \mathbf { X } ) = N$ , X-a.e., if and only $i f ,$ + +$$ +\forall S \subseteq [ N ] : | S | \leq \mathrm { r a n k } \left( \mathbf { A } _ { S } \right) d _ { 0 } . +$$ + +Proof. We define $d _ { S } \triangleq \operatorname { r a n k } \left( \mathbf { A } _ { S } \right)$ and $\mathbf { A } \circ \mathbf { X }$ . The necessity of the condition $| S | \le d _ { 0 } d _ { S }$ holds for every $\mathbf { X }$ , as can be seen from the following counting argument. Since the matrix $\mathbf { A } _ { S }$ has rank $d _ { S }$ , + +there exists an invertible row transformation matrix $\mathbf { R }$ , such that $\mathbf { R A } _ { S }$ has only $d _ { S }$ non-zero rows. Consider now $\mathbf { G } _ { S } = \mathbf { A } _ { S } \circ \mathbf { X } _ { S }$ , i.e., the matrix composed of the columns of $\mathbf { G }$ in $S$ . We have + +$$ +\mathbf { G } _ { S } ^ { \prime } = ( \mathbf { R } \mathbf { A } _ { S } ) \circ \mathbf { X } _ { S } = \mathbf { R } ^ { \prime } \left( \mathbf { A } _ { S } \circ \mathbf { X } _ { S } \right) = \mathbf { R } ^ { \prime } \mathbf { G } _ { S } , +$$ + +where $\mathbf { R } ^ { \prime } = \mathbf { R } \otimes \mathbf { I } _ { d _ { 0 } }$ is also an invertible row transformation matrix, which applies $\mathbf { R }$ separately on the $d _ { 0 }$ sub-matrices of $\mathbf { G } _ { S }$ that are constructed by taking one every $d _ { 0 }$ rows. Since $\mathbf { G } _ { S } ^ { \prime }$ has at most $d _ { 0 } d _ { S }$ non-zero rows, the rank of $\mathbf { G } _ { S }$ cannot exceed $d _ { 0 } d _ { S }$ . Therefore, if $| S | > d _ { 0 } d _ { S }$ , $\mathbf { G } _ { S }$ will not have full column rank, and hence neither will $\mathbf { G }$ . To demonstrate sufficiency a.e., suppose $\mathbf { G }$ does not have full column rank. Let $S$ be the minimum set of columns of $\mathbf { G }$ which are linearly dependent. Since the columns of $\mathbf { G } _ { S }$ are assumed linearly dependent there exists $\mathbf { v } \in \mathbb { R } ^ { | S | }$ such $\| \mathbf { v } \| _ { 0 } = | S |$ and $\mathbf { G } _ { S } \mathbf { v } = 0$ . Using Lemma 28 we complete the proof. □ + +# 12.2 PROOF OF LEMMA 15 + +In this section we will prove Lemma 15 in subsection 12.3.3. This proof relies on two rather basic results, which we first prove in subsections 12.2.1 and 12.2.2. + +# 12.2.1 NUMBER OF DICHOTOMIES INDUCED BY A HYPERPLANE + +Fact 29. A hyperplane ${ \mathbf w } \in d _ { 0 }$ can separate a given set of points $\mathbf { X } = \left[ \mathbf { x } ^ { ( 1 ) } , \ldots , \mathbf { x } ^ { ( N ) } \right] \in \mathbb { R } ^ { d _ { 0 } \times N }$ into several different dichotomies, i.e., different results for $\mathrm { s i g n } \left( \mathbf { w } ^ { \top } \mathbf { X } \right)$ . The number of dichotomies is upper bounded as follows: + +$$ +\sum _ { \mathbf { h } \in \{ - 1 , 1 \} ^ { N } } \mathscr { T } \left( \exists \mathbf { w } : \mathrm { s i g n } \left( \mathbf { w } ^ { \top } \mathbf { X } \right) = \mathbf { h } ^ { \top } \right) \leq 2 \sum _ { k = 0 } ^ { d _ { 0 } - 1 } \left( \begin{array} { l } { N - 1 } \\ { k } \end{array} \right) \leq 2 N ^ { d _ { 0 } } . +$$ + +Proof. See (Cover, 1965, Theorem 1) for a proof of the left inequality as equality (the Schläfli Theorem) in the case that the columns of $\mathbf { X }$ are in “general position” (which holds $\mathbf { X }$ -a.e, see definition in (Cover, 1965)) . If $\mathbf { X }$ is not in general position then this result becomes an upper bound, since some dichotomies might not be possible. + +Next, we prove the right inequality. For $N = 1$ and $N = 2$ the inequality trivially holds. For $N \geq 3$ , we have + +$$ +2 \sum _ { k = 0 } ^ { d _ { 0 } - 1 } \left( \begin{array} { c } { { N - 1 } } \\ { { k } } \end{array} \right) \stackrel { \left( 1 \right) } { \leq } 2 \sum _ { k = 0 } ^ { d _ { 0 } - 1 } \left( N - 1 \right) ^ { k } \stackrel { \left( 2 \right) } { \leq } 2 \frac { \left( N - 1 \right) ^ { d _ { 0 } } - 1 } { N - 2 } \leq 2 N ^ { d _ { 0 } } . +$$ + +where in (1) we used the bound $\left( \begin{array} { c } { { N } } \\ { { k } } \end{array} \right) \le N ^ { k }$ , in (2) we used the sum of a geometric series. + +# 12.2.2 A BASIC PROBABILISTIC BOUND + +Lemma 30. Let $\mathbf { H } = \left[ \mathbf { h } _ { 1 } ^ { \top } , \ldots , \mathbf { h } _ { d _ { 1 } } ^ { \top } \right] ^ { \top } \in \{ - 1 , 1 \} ^ { d _ { 1 } \times k }$ be a deterministic binary matrix, ${ \textbf { W } } =$ $\left[ \mathbf { w } _ { 1 } ^ { \top } , \ldots , \mathbf { w } _ { d _ { 1 } } ^ { \top } \right] ^ { \top } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ be an independent standard random Gaussian matrix, and $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times k }$ be a random matrix with independent and identically distributed columns. + +$$ +\begin{array} { r } { \mathbb { P } \left( \operatorname { s i g n } \left( { \mathbf { W } } { \mathbf { X } } \right) = { \mathbf { H } } \right) \leq \left( \begin{array} { c } { k } \\ { \lfloor k / 2 \rfloor } \end{array} \right) \mathbb { P } \left( { \mathbf { W } } { \mathbf { X } } _ { [ \lfloor k / 2 \rfloor ] } > 0 \right) . } \end{array} +$$ + +Proof. By direct calculation + +$$ +\begin{array} { r l } { \displaystyle \widehat { \mathbf { \xi } } \displaystyle \geq ( \mathrm { s i g n } ( \mathbf { W X } ) = \mathbf { H } ) = } & { \displaystyle \mathbb { E } [ \mathbb { P } ( \mathrm { s i g n } ( \mathbf { W X } ) = \mathbf { H } \vert \mathbf { X } ) ] \stackrel { ( 1 ) } { = } \mathbb { E } [ \displaystyle \prod _ { i = 1 } ^ { d } \mathbb { P } ( \mathrm { s i g n } ( \mathbf { w } _ { i } ^ { \top } \mathbf { X } ) = \mathbf { h } _ { i } ^ { \top } \vert \mathbf { X } ) ] } \\ & { \stackrel { ( 2 ) } { \leq } \mathbb { E } [ \displaystyle \prod _ { i = 1 } ^ { d _ { 1 } } \mathbb { P } ( \mathbf { w } _ { i } ^ { \top } \mathbf { X } _ { \delta ( \mathbf { h } _ { i } ) } > 0 \vert \mathbf { X } ) ] \stackrel { ( 3 ) } { \leq } \mathbb { E } [ \displaystyle \prod _ { i = 1 } ^ { d _ { 1 } } \mathbb { P } ( \mathbf { w } _ { i } ^ { \top } \mathbf { X } _ { S _ { i } } > 0 \vert \mathbf { X } ) ] } \\ & { \stackrel { ( 4 ) } { = } \mathbb { E } [ \mathbb { P } ( \mathbf { W X } _ { S _ { s } } > 0 \vert \mathbf { X } ) ] \stackrel { ( 5 ) } { \leq } \mathbb { E } [ \displaystyle \sum _ { S \geq [ R _ { 1 } ] : S \mid = \lfloor k / 2 \rfloor } } \\ & { = \displaystyle \sum _ { S \leq [ k ] : S = \lfloor k / 2 \rfloor } \mathbb { E } [ \mathbb { P } ( \mathbf { W X } _ { S } > 0 \vert \mathbf { X } ) ] \stackrel { ( 6 ) } { = } ( \begin{array} { l } { k } \\ { \lfloor k / 2 \rfloor } \end{array} ) \mathbb { P } ( \mathbf { W } _ { \mathbf { X } _ { \lfloor k / 2 \rfloor } } > 0 ) . } \end{array} +$$ + +where + +1. We used the independence of the $\mathbf { w } _ { i }$ . + +2. We define $\hat { S } _ { \pm } \left( \mathbf { h } \right) \ \triangleq \ \left\{ S \subset \left[ k \right] : \pm \mathbf { h } _ { S } ^ { \top } > 0 \right\}$ as the sets in which $\mathbf { h }$ is always positive/negative, and $\hat { S } \left( \mathbf { h } \right)$ as the maximal set between these two. Note that $\mathbf { w } _ { i }$ has a standard normal distribution which is symmetric to sign flips, so $\forall S : \mathbb { P } \left( \mathbf { w } _ { i } ^ { \top } \mathbf { X } _ { S } > 0 | \mathbf { X } \right) =$ $\mathbb { P } \left( \mathbf { w } _ { i } ^ { \top } \mathbf { X } _ { S } < 0 | \mathbf { X } \right)$ . + +3. Note that $\left| \hat { S } \left( \mathbf { h } \right) \right| \ge \lfloor k / 2 \rfloor$ . Therefore, we define S∗ = argmax $\mathbb { P } \left( \mathbf { w } _ { i } ^ { \top } \mathbf { X } _ { S } > 0 | \mathbf { X } \right)$ . $S C [ k ] : \bar { | } S | = \lfloor k / 2 \rfloor$ + +4. We used the independence of the $\mathbf { w } _ { i }$ + +5. The maximum is a single term in the following sum of non-negative terms. + +6. Taking the expectation over $\mathbf { X }$ , since the columns of $\mathbf { X }$ are independent and identically distributed, the location of $S$ does not affect the probability. Therefore, we can set without loss of generality $S = [ \lfloor k / 2 \rfloor ]$ . + +# 12.2.3 MAIN PROOF: BOUND ON THE NUMBER OF CONFIGURATIONS FOR A BINARY MATRIX WITH CERTAIN RANK + +Recall the function $a \left( \cdot \right)$ from eq. (2.1): + +$$ +a \left( u \right) \triangleq \left\{ \begin{array} { l l } { 1 } & { , \mathrm { i f } , u > 0 } \\ { \rho } & { , \mathrm { i f } u < 0 } \end{array} \right. . +$$ + +where $\rho \neq 1$ + +Lemma 31. (Lemma 15 restated). Let $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times k }$ be a random matrix with independent and identically distributed columns, and $\mathbf { W } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ an independent standard random Gaussian matrix. Then, in the limit $\operatorname* { m i n } \left[ k , d _ { 0 } , d _ { 1 } \right] \dot { > } r$ , + +$$ +\begin{array} { r } { \mathbb { P } \left( \mathrm { r a n k } \left( a \left( \mathbf { W } \mathbf { X } \right) \right) = r \right) \dot { \leq } 2 ^ { k + r d _ { 0 } ( \log d _ { 1 } + \log k ) + r ^ { 2 } } \mathbb { P } \left( \mathbf { W } \mathbf { X } _ { [ \lfloor k / 2 \rfloor ] } > 0 \right) . } \end{array} +$$ + +Proof. We denote $\mathbf { A } = a \left( \mathbf { W } \mathbf { X } \right) \in \{ \rho , 1 \} ^ { d _ { 1 } \times k }$ . For any such A for which rank $( \mathbf { A } ) = r$ , we have a collection of $r$ rows that span the remaining rows. There are $\left( \begin{array} { c } { d _ { 1 } } \\ { r } \end{array} \right)$ possible locations for these $r$ spanning rows. In these rows there exist a collection of $r$ columns that span the remaining columns. There are $\left( \begin{array} { l } { k } \\ { r } \end{array} \right)$ possible locations for these $r$ spanning columns. At the intersection of the spanning rows and columns, there exist a full rank sub-matrix D. We denote $\tilde { \mathbf { A } }$ as the matrix A which rows and columns are permuted so that $\mathbf { D }$ is the lower right block + +$$ +\begin{array} { r } { \tilde { \mathbf { A } } \triangleq \left( \begin{array} { l l } { \mathbf { Z } } & { \mathbf { B } } \\ { \mathbf { C } } & { \mathbf { D } } \end{array} \right) = a \left( \begin{array} { l l } { \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } } & { \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } } \\ { \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } } & { \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } } \end{array} \right) , } \end{array} +$$ + +where $\mathbf { D }$ is an invertible $r \times r$ matrix, and we divided $\mathbf { X }$ and $\mathbf { W }$ to the corresponding block matrices + +$$ +\mathbf { W } \triangleq \left[ \mathbf { W } _ { 1 } ^ { \top } , \mathbf { W } _ { 2 } ^ { \top } \right] ^ { \top } , \mathbf { X } \triangleq \left[ \mathbf { X } _ { 1 } , \mathbf { X } _ { 2 } \right] , +$$ + +with $\mathbf { W } _ { 2 } \in \mathbb { R } ^ { r \times d _ { 0 } }$ rows and $\mathbf { X } _ { 2 } \in \mathbb { R } ^ { d _ { 0 } \times r }$ . + +Since $\mathrm { r a n k } \left( \tilde { \mathbf { A } } \right) = r$ , the first $d _ { 1 } - r$ rows are contained in the span of the last $r$ rows. Therefore, there exists a matrix $\mathbf { Q }$ such that $\mathbf { Q C } = \mathbf { Z }$ and $\mathbf { Q D } = \mathbf { B }$ . Since $\mathbf { D }$ is invertible, this implies that ${ \bf Q } = { \bf B } { \bf D } ^ { - 1 }$ and therefore + +$$ +{ \bf Z } = { \bf B } { \bf D } ^ { - 1 } { \bf C } , +$$ + +i.e., $\mathbf { B } , \mathbf { C }$ and $\mathbf { D }$ uniquely determine $\mathbf { Z }$ . + +Using the union bound over all possible permutations from $\mathbf { A }$ to $\tilde { \mathbf { A } }$ , and eq. (12.9), we have + +$$ +\begin{array} { r l r } & { \mathbb { P } ( \mathrm { r a n k } ( \mathbf { A } ) = r ) } & { ( 1 2 . 1 0 ) } \\ & { \leq ( \begin{array} { l } { d _ { 1 } } \\ { r } \end{array} ) ( \begin{array} { l } { k } \\ { r } \end{array} ) \mathbb { P } ( \mathrm { r a n k } ( \bar { \mathbf { A } } ) = r ) } \\ & { \leq ( \begin{array} { l } { d _ { 1 } } \\ { r } \end{array} ) ( \begin{array} { l } { k } \\ { r } \end{array} ) \mathbb { P } ( \mathbf { Z } = \mathbf { B } \mathbf { D } ^ { - 1 } \mathbf { C } ) } \\ & { = ( \begin{array} { l } { d _ { 1 } } \\ { r } \end{array} ) ( \begin{array} { l } { k } \\ { r } \end{array} ) \mathbb { P } ( a ( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } ) [ a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } ) = a ( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } ) ) } \\ & = ( \begin{array} { l } { d _ { 1 } } \\ { r } \end{array} ) ( \begin{array} { l } { k } \\ { r } \end{array} ) \underbrace { \mathrm { H } ( a ( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } ) [ a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } ) = a ( \mathbf { H } ) | \mathrm { s i g n } ( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } ) = \mathbf { H } ) \mathbb { P } ( \mathrm { s i g n } ( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } ) ) } \end{array} +$$ + +Using Lemma 30, we have + +$$ +\begin{array} { r } { \mathbb { P } \left( \mathrm { s i g n } \left( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } \right) = \mathbf { H } \right) \leq \left( \begin{array} { c } { k - r } \\ { \lfloor ( k - r ) / 2 \rfloor } \end{array} \right) \mathbb { P } \left( \mathbf { W } _ { 1 } \mathbf { X } _ { \lfloor \lfloor ( k - r ) / 2 \rfloor \rfloor } > 0 \right) , } \end{array} +$$ + +an upper bound which does not depend on $\mathbf { H }$ . So all that remains is to compute the sum: + +$$ +\begin{array} { r l } & { \qquad \displaystyle \sum _ { \mathbf { H } \in \{ - 1 , 1 \} ^ { ( d _ { 1 } - r ) \times ( k - r ) } } \mathbb { P } ( a ( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } ) [ a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } ) = a ( \mathbf { H } ) | \mathrm { s i g n } ( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } ) = \mathbf { H } ) } \\ { = } & { \qquad \displaystyle \sum _ { \mathbf { H } \in \{ - 1 , 1 \} ^ { ( d _ { 1 } - r ) \times ( k - r ) } } \mathbb { E } [ \mathbb { P } ( a ( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } ) [ a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } ) = a ( \mathbf { H } ) | \mathbf { W } _ { 1 } , \mathbf { X } _ { 1 } ) | \mathrm { s i g n } ( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } ) = \mathbf { H } } \\ & { \mathrm { H } _ { 1 } ^ { 1 } [ \displaystyle \sum _ { \mathbf { H } \in \{ - 1 , 1 \} ^ { ( d _ { 1 } - r ) \times ( k - r ) } } \mathbb { Z } ( \exists ( \mathbf { W } _ { 2 } , \mathbf { X } _ { 2 } ) : a ( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } ) [ a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } ) = a ( \mathbf { H } ) ) ] \mathrm { s i g n } ( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } ) = \mathbf { H } } \end{array} +$$ + +$$ +\begin{array} { r l } & { \overset { 2 } { \leq } \mathbb { R } [ 2 ^ { r ^ { 2 } } [ \displaystyle \sum _ { \mathbf { H } \in \{ - 1 , 1 \} ^ { ( d _ { \mathbf { X } } _ { 2 } : \mathrm { ~ s i g n } } } ( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } ) = \mathbf { H } ) ] [ \displaystyle \sum _ { \mathbf { H } \in \{ - 1 , 1 \} ^ { r \times ( k - r ) } } \mathbb { Z } ( \exists \mathbf { W } _ { 2 } : \mathrm { s i g n } ( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } ) = \mathbf { H } ) ] ] | \mathrm { s i g n } } \\ & { \leq \mathbb { R } [ 2 ^ { r ^ { 2 } } [ \displaystyle \sum _ { \mathbf { h } \in \{ - 1 , 1 \} ^ { ( d _ { \mathbf { X } } : \mathrm { s i g n } } } ( \mathbf { W } _ { 1 } \mathbf { x } ) = \mathbf { h } ) ] ^ { r } [ \displaystyle \sum _ { \mathbf { h } \in \{ - 1 , 1 \} ^ { ( k - r ) } } \mathbb { Z } ( \exists \mathbf { w } : \mathrm { s i g n } ( \mathbf { w } ^ { \mathsf { T } } \mathbf { X } _ { 1 } ) = \mathbf { h } ^ { \mathsf { T } } ) ] ^ { r } ] | \mathrm { s i g n } ( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } ) } \\ & { \overset { 3 } { \leq } \mathbb { R } [ 2 ^ { r ^ { 2 } } 2 ^ { r d _ { 0 } \log ( d _ { 1 } - r ) + r _ { 2 } r d _ { 0 } \log ( k - r ) + r } ] | \mathrm { s i g n } ( \mathbf { W } _ { 1 } \mathbf { X } _ { 1 } ) = \mathbf { H } ] } \\ & { = 2 ^ { r d _ { 0 } [ \log ( d _ { 1 } - r ) + \log ( k - r ) ] + r ^ { 2 } + 2 r } , } \end{array} +$$ + +where + +1. Given $( \mathbf { W } _ { 1 } , \mathbf { X } _ { 1 } )$ , and eq. (12.8), the indicator function in eq. (12.12) is equal to zero only if $\begin{array} { r } { \mathbb { P } \left( a \left( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } \right) \left[ a \left( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } \right) \right] ^ { - 1 } a \left( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } \right) = \mathbf { A } \mathbf { | } \mathbf { W } _ { 1 } , \mathbf { X } _ { 1 } \right) = 0 } \end{array}$ , and one otherwise. + +2. This sum counts the number of values of $\mathbf { H }$ consistent with $\mathbf { W } _ { 1 }$ and $\mathbf { X } _ { 1 }$ . Conditioned on $( \mathbf { W } _ { 1 } , \mathbf { X } _ { 1 } ) , \mathbf { D } = \left[ a \left( \mathbf { W } _ { 2 } \mathbf { X } _ { 2 } \right) \right] ^ { - 1 } , \mathbf { B } = a \left( \mathbf { W } _ { 1 } \mathbf { X } _ { 2 } \right)$ and $\mathbf { C } = a \left( \mathbf { W } _ { 2 } \mathbf { X } _ { 1 } \right)$ can have multiple values, depending on $\mathbf { W } _ { 2 }$ and $\mathbf { X } _ { 2 }$ . Also, any single value for $( { \bf D } , { \bf B } , { \bf C } )$ results in a single value of $\mathbf { H }$ . Therefore, the number of possible values of $\mathbf { H }$ in eq. (12.12) is upper bounded by the product of the number of possible values of $\mathbf { D }$ , $\mathbf { B }$ and $\mathbf { C }$ , which is product in the following equation. + +3. The function $\begin{array} { r } { \sum _ { \mathbf { h } \in \{ - 1 , 1 \} ^ { ( k - r ) } } \mathscr { T } \left( \exists \mathbf { w } : \mathrm { s i g n } \left( \mathbf { w } ^ { \top } \mathbf { X } _ { 1 } \right) = \mathbf { h } ^ { \top } \right) } \end{array}$  counts the number of dichotomies that can be induced by the linear classifier w on $\mathbf { X } _ { 1 }$ . Using eq. (12.7) we can bound this number by $2 \left( k - r \right) ^ { d _ { 0 } }$ . Similarly, the other sum can be bounded by $2 \left( d _ { 1 } - r \right) ^ { r }$ . + +Combining eqs. (12.10), (12.11) and (12.13) we obtain + +$$ +\begin{array} { r l } & { \mathbb { P } \left( \mathrm { r a n k } \left( \mathbf { A } \right) = r \right) \leq } \\ & { \left( \begin{array} { l } { d _ { 1 } } \\ { r } \end{array} \right) \left( \begin{array} { l } { k } \\ { r } \end{array} \right) \left( \begin{array} { c } { k - r } \\ { \lfloor ( k - r ) / 2 \rfloor } \end{array} \right) 2 ^ { r d _ { 0 } \left[ \log ( d _ { 1 } - r ) + \log ( k - r ) \right] + r ^ { 2 } + 2 r } \mathbb { P } \left( \mathbf { W } _ { 1 } \mathbf { X } _ { [ \lfloor ( k - r ) / 2 \rfloor ] } > 0 \right) . } \end{array} +$$ + +Next, we take the log. To upper bound $\left( \begin{array} { c } { { N } } \\ { { k } } \end{array} \right)$ , for small $k$ we use $\left( \begin{array} { c } { { N } } \\ { { k } } \end{array} \right) \le N ^ { k }$ , while for $k = N / 2$ , we use $\left( \begin{array} { c } { { N } } \\ { { N / 2 } } \end{array} \right) \le 2 ^ { N }$ . Thus, we obtain + +$$ +\begin{array} { r l r } & { } & { \log \mathbb { P } \left( \mathrm { r a n k } \left( \mathbf { A } \right) = r \right) \leq \left( r d _ { 0 } \left( \log \left( d _ { 1 } - r \right) + \log \left( k - r \right) \right) + r ^ { 2 } + 2 r \right) \log 2 \qquad ( 1 2 \triangle ( r \mathrm { ~ o ~ t ~ } r ) } \\ & { } & { + r \log d _ { 1 } + r \log k + ( k - r ) \log 2 + \log \mathbb { P } \left( \mathbf { W } _ { 1 } \mathbf { X } _ { [ \vert ( k - r ) / 2 ] ] } > 0 \right) . } \end{array} +$$ + +Recalling that $\mathbf { W } _ { 1 } \in \mathbb { R } ^ { ( d _ { 1 } - r ) \times d _ { 0 } }$ while $\mathbf { W } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ , we obtain from Jensen’s inequality + +$$ +\begin{array} { r } { \log \mathbb { P } \left( \mathbf { W } _ { 1 } \mathbf { X } _ { [ \lfloor ( k - r ) / 2 \rfloor ] } > 0 \right) \le \frac { \left\lfloor \left( k - r \right) / 2 \right\rfloor \left\lfloor d _ { 1 } - r \right\rfloor } { \left\lfloor k / 2 \right\rfloor \left\lfloor d _ { 1 } \right\rfloor } \log \mathbb { P } \left( \mathbf { W } \mathbf { X } _ { [ \lfloor k / 2 \rfloor ] } > 0 \right) . } \end{array} +$$ + +Taking the limit $\mathrm { n i n } \left[ k , d _ { 0 } , d _ { 1 } \right] \dot { > } r$ on eqs. (12.14) and (12.15) we obtain + +$$ +\begin{array} { r } { \mathbb { P } \left( \mathrm { r a n k } \left( \mathbf { A } \right) = r \right) \overset { * } { \leq } 2 ^ { k + r d _ { 0 } ( \log d _ { 1 } + \log k ) + t ^ { 2 } } \mathbb { P } \left( { \mathbf { W } } \mathbf { X } _ { [ \lfloor k / 2 \rfloor ] } > 0 \right) . } \end{array} +$$ + +# 12.3 PROOF OF LEMMA 16 + +In this section we will prove Lemma 16 in subsection 12.3.3. This proof relies on more elementary results, which we first prove in subsections 12.3.1 and 12.3.2. + +# 12.3.1 ORTHANT PROBABILITY OF A RANDOM GAUSSIAN VECTOR + +Recall that $\phi \left( x \right)$ and $\Phi \left( x \right)$ are, respectively, the probability density function and cumulative distribution function for a scalar standard normal random variable. + +Definition 32. We define the following functions $\forall x \geq 0$ + +$$ +\begin{array} { l } { \displaystyle { g \left( x \right) \triangleq \frac { x \Phi \left( x \right) } { \phi \left( x \right) } , } } \\ { \displaystyle { \psi \left( x \right) \triangleq \frac { \left( g ^ { - 1 } \left( x \right) \right) ^ { 2 } } { 2 x } - \log \left( \Phi \left( g ^ { - 1 } \left( x \right) \right) \right) , } } \end{array} +$$ + +where the inverse function $g ^ { - 1 } ( x ) : [ 0 , \infty ) [ 0 , \infty )$ is well defined since $g \left( x \right)$ monotonically increase from 0 to $\infty$ , for $x \geq 0$ . + +Lemma 33. Let $\mathbf { z } \sim \mathcal { N } ( 0 , \pmb { \Sigma } )$ be a random Gaussian vector in $\mathbb { R } ^ { K }$ , with a covariance matrix $\Sigma _ { i j } = \left( 1 - \theta K ^ { - 1 } \right) \delta _ { m n } + \theta K ^ { - 1 }$ where $K \gg \theta > 0$ . Then, recalling $\psi \left( \theta \right)$ in eq. (12.17), we have + +$$ +\begin{array} { r } { \log \mathbb { P } \left( \forall i : z _ { i } > 0 \right) \le - K \psi \left( \theta \right) + O \left( \log K \right) . } \end{array} +$$ + +Proof. Note that we can write $\mathbf { z } = \mathbf { u } + \eta$ , where $\mathbf { u } \sim { \mathcal { N } } \left( 0 , \left( 1 - \theta K ^ { - 1 } \right) \mathbf { I } _ { K } \right)$ , and $\eta \sim \mathcal { N } \left( 0 , \theta K ^ { - 1 } \right)$ . Using this notation, we have + +$$ +\begin{array} { r l } & { \quad \mathbb { P } \left( \overline { { \psi } } : z _ { i } > 0 \right) } \\ & { = \int _ { - \infty } ^ { \infty } d \eta \left[ \displaystyle \prod _ { i = 1 } ^ { K } \int _ { - \infty } ^ { \infty } d u _ { i } \mathcal { T } \left( \sqrt { 1 - \theta K ^ { - 1 } } u _ { i } + \sqrt { \theta K ^ { - 1 } } \eta > 0 \right) \phi \left( u _ { i } \right) \right] \phi \left( \eta \right) } \\ & { = \int _ { - \infty } ^ { \infty } d \eta \left[ \Phi \left( \sqrt { \frac { \theta K ^ { - 1 } } { 1 - \theta K ^ { - 1 } } } \eta \right) \right] ^ { K } \phi \left( \eta \right) } \\ & { \stackrel { ( 1 ) } { = } \sqrt { \displaystyle \frac { \theta } { 2 \pi \left( K - \theta \right) } } \int _ { - \infty } ^ { \infty } d \xi \left[ \Phi \left( \xi \right) \right] ^ { K } \exp \left( - \frac { \left( K - \theta \right) \xi ^ { 2 } } { 2 \theta } \right) } \\ & { = \sqrt { \displaystyle \frac { \theta } { 2 \pi \left( K - \theta \right) } } \int _ { - \infty } ^ { \infty } d \xi \exp \left( \frac { \xi ^ { 2 } } { 2 } \right) \exp \left[ K \left( \log \Phi \left( \xi \right) - \frac { \xi ^ { 2 } } { 2 \theta } \right) \right] , } \end{array} +$$ + +where in (1) we changed the variable of integration to $\xi = \sqrt { \theta / \left( K - \theta \right) } \eta$ . We denote, for a fixed $\theta$ + +$$ +\begin{array} { l } { { \displaystyle q \left( \xi \right) \triangleq \log \Phi \left( \xi \right) - \frac { \xi ^ { 2 } } { 2 \theta } } } \\ { { \displaystyle h \left( \xi \right) \triangleq \sqrt { \frac { \theta } { 2 \pi \left( K - \theta \right) } } \exp \left( \frac { \xi ^ { 2 } } { 2 } \right) } } \end{array} +$$ + +and $\xi _ { 0 }$ as its global maximum. Since $q$ is twice differentiable, we can use Laplace’s method (e.g., (Butler, 2007)) to simplify eq. (12.18) + +$$ +\log \int _ { - \infty } ^ { \infty } h \left( \xi \right) \exp \left( K q \left( \xi \right) \right) d \xi = K q \left( \xi _ { 0 } \right) + O \left( \log K \right) . +$$ + +To find $\xi _ { 0 }$ , we differentiate $q \left( \xi \right)$ and equate to zero to obtain + +$$ +q ^ { \prime } \left( \xi \right) = \frac { \phi \left( \xi \right) } { \Phi \left( \xi \right) } - \frac { 1 } { \theta } \xi = 0 . +$$ + +which implies (recall eq. (12.16)) + +$$ +g \left( \xi \right) \triangleq \frac { \xi \Phi \left( \xi \right) } { \phi \left( \xi \right) } = \theta . +$$ + +This is a monotonically increasing function from 0 to $\infty$ in the range $\xi \ge 0$ . Its inverse function can also be defined in that range $g ^ { - 1 } \check { ( \theta ) } : [ 0 , \infty ] [ 0 , \infty ]$ . This implies that this equation has only one solution, $\xi _ { 0 } = g ^ { - 1 } \left( \theta \right)$ . Since $\begin{array} { r } { \operatorname* { l i m } _ { \xi \to \infty } q \left( \xi \right) = - \infty } \end{array}$ , this $\xi _ { 0 }$ is indeed the global maximum of $q \left( \xi \right)$ Substituting this solution into $q \left( \xi \right)$ , we get (recall eq. (12.17)) + +$$ +\forall \theta > 0 : q \left( \xi _ { 0 } \right) = - \psi \left( \theta \right) = q \left( g ^ { - 1 } \left( \theta \right) \right) = \log \left( \Phi \left( g ^ { - 1 } \left( \theta \right) \right) \right) - \frac { \left( g ^ { - 1 } \left( \theta \right) \right) ^ { 2 } } { 2 \theta } . +$$ + +Using eq. (12.18), (12.21) and (12.24) we obtain: + +$$ +\begin{array} { l } { \displaystyle \log \mathbb { P } \left( \forall i : z _ { i } > 0 \right) } \\ { = \log \left[ \int _ { - \infty } ^ { \infty } d \xi \exp \left( \frac { \xi ^ { 2 } } { 2 } \right) \exp \left[ K \left( \log \Phi \left( \xi \right) - \frac { \xi ^ { 2 } } { 2 \theta } \right) \right] \right] + O \left( \log K \right) } \\ { = - K \psi \left( \theta \right) + O \left( \log K \right) . } \end{array} +$$ + +Next, we generalize the previous Lemma to a general covariance matrix. + +Corollary 34. Let $\mathbf { u } \sim \mathcal { N } ( 0 , \pmb { \Sigma } )$ be a random Gaussian vector in $\mathbb { R } ^ { K }$ for which $\forall n : \Sigma _ { n n } = 1$ , and $\theta \geq K \operatorname* { m a x } _ { n , m : n \neq m } \Sigma _ { n m } > 0$ . Then, again, for large $K$ + +$$ +\begin{array} { r } { \log \mathbb { P } \left( \forall i : u _ { i } > 0 \right) \le - K \psi \left( \theta \right) + O \left( \log K \right) . } \end{array} +$$ + +Proof. We define $\tilde { \mathbf { u } } \sim \mathcal { N } \left( 0 , \tilde { \Sigma } \right)$ , with $\tilde { \Sigma } _ { m n } = \left( 1 - \theta K ^ { - 1 } \right) \delta _ { m n } + \theta K ^ { - 1 }$ . Note that $\forall n : \Sigma _ { n n } =$ $\tilde { \Sigma } _ { n n } = 1$ and $\forall m \neq n$ : $\Sigma _ { m n } \leq \tilde { \Sigma } _ { m n }$ . Therefore, from Slepian’s Lemma (Slepian, 1962, Lemma 1), + +$$ +\mathbb { P } \left( \forall n : \tilde { u } _ { n } > 0 \right) \ge \mathbb { P } \left( \forall n : u _ { n } > 0 \right) . +$$ + +Using Lemma 33 on $\tilde { \mathbf { u } }$ completes the proof. + +# 12.3.2 MUTUAL COHERENCE BOUNDS + +Definition 35. We define the mutual coherence of the columns of a matrix $\mathbf { A } = [ \pmb { a } _ { 1 } , \cdots , \pmb { a } _ { N } ] \in$ $\mathbb { R } ^ { M \times N }$ as the maximal angle between different columns + +$$ +\gamma \left( { \bf A } \right) \triangleq \operatorname* { m a x } _ { i , j : i \neq j } \frac { \left| { \pmb a } _ { i } ^ { \top } { \pmb a } _ { j } \right| } { \left\| { \pmb a } _ { i } \right\| \left\| { \pmb a } _ { j } \right\| } . +$$ + +Note that $\gamma \left( \mathbf { A } \right) \leq 1$ and from (Welch, 1974), for $\begin{array} { r } { N \ge M , \gamma \left( { \mathbf { A } } \right) \ge \sqrt { \frac { N - M } { M \left( N - 1 \right) } } . } \end{array}$ + +Lemma 36. Let $\mathbf { A } = [ \pmb { a } _ { 1 } , \cdot \cdot \cdot , \pmb { a } _ { N } ] \in \mathbb { R } ^ { M \times N }$ be a standard random Gaussian matrix, and $\gamma \left( \mathbf { A } \right)$ is the mutual coherence of it columns (see definition 35). Then + +$$ +\mathbb { P } \left( \gamma \left( { \bf A } \right) > \epsilon \right) \le 2 N ^ { 2 } \exp \left( - \frac { M \epsilon ^ { 2 } } { 2 4 } \right) . +$$ + +Proof. In this case, we have from (Chen & Peng, 2016, Appendix 1): + +$$ +\mathbb { P } \left( \gamma \left( \mathbf { A } \right) > \epsilon \right) \leq N \left( N - 1 \right) \left[ \exp \left( - \frac { M a ^ { 2 } \epsilon ^ { 2 } } { 4 \left( 1 + \epsilon / 2 \right) } \right) + \exp \left( - \frac { M } { 4 } \left( 1 - a \right) ^ { 2 } \right) \right] , +$$ + +for any $a \in ( 0 , 1 )$ . Setting $a = 1 - \epsilon / 2$ + +$$ +\begin{array} { r l } & { \mathbb { P } \left( \gamma \left( \mathbf { A } \right) > \epsilon \right) \le N \left( N - 1 \right) \left[ \exp \left( - \frac { M \left( 1 - \epsilon / 2 \right) ^ { 2 } \epsilon ^ { 2 } } { 4 \left( 1 + \epsilon / 2 \right) } \right) + \exp \left( - \frac { M } { 1 6 } \epsilon ^ { 2 } \right) \right] } \\ & { \qquad \overset { \left( 1 \right) } { \le } N \left( N - 1 \right) \left[ \exp \left( - \frac { M \epsilon ^ { 2 } } { 2 4 } \right) + \exp \left( - \frac { M } { 1 6 } \epsilon ^ { 2 } \right) \right] } \\ & { \qquad \le 2 N ^ { 2 } \exp \left( - \frac { M \epsilon ^ { 2 } } { 2 4 } \right) , } \end{array} +$$ + +where in (1) we can assume that $\epsilon \leq 1$ , since for $\epsilon \geq 1$ , we have $\mathbb { P } \left( \gamma \left( \mathbf { A } \right) > \epsilon \right) = 0$ (recall $\gamma ( \mathbf { A } ) \leq 1 )$ . □ + +Lemma 37. Let $\mathbf { B } = [ \mathbf { b } _ { 1 } , \cdots , \mathbf { b } _ { L } ] \in \mathbb { R } ^ { M \times L }$ be a standard random Gaussian matrix and mutual coherence $\gamma$ as in definition 35. Then, $\forall \epsilon > 0$ and $\forall K \in [ L ]$ : + +$$ +\mathbb { P } \left( \operatorname* { m i n } _ { S \subset [ N ] : | S | = K } \gamma \left( \mathbf { B } _ { S } \right) > \epsilon \right) \leq \exp \left[ \left( 2 \log \left( 2 K \right) - \frac { M \epsilon ^ { 2 } } { 2 4 } \right) \left( \frac { L } { K } - 1 \right) \right] . +$$ + +Proof. We upper bound this probability by partitioning the set of column vectors into $\lfloor L / K \rfloor$ subsets $S _ { i }$ of size $| \bar { S _ { i } } \bar { | } = K$ and require that in each subset the mutual coherence is lower bounded by $\epsilon$ . + +Since the columns are independent, we have + +$$ +\begin{array} { r l } & { \mathbb { P } \left( \underset { S \subset [ N ] : | S | = K } { \operatorname* { m i n } } \gamma \left( \mathbf { B } _ { S } \right) > \epsilon \right) } \\ & { \le \displaystyle \prod _ { i = 1 } ^ { \lfloor L / K \rfloor } \mathbb { P } \left( \forall S = \{ 1 + \left( i - 1 \right) K , 2 + \left( 1 - i \right) K , \ldots , i K \} : \gamma \left( \mathbf { B } _ { S } \right) > \epsilon \right) } \\ & { \stackrel { \mathrm { ( 1 ) } } { \le } \displaystyle \prod _ { i = 1 } ^ { L / K - 1 } 2 K ^ { 2 } \exp \left( - \frac { M \epsilon ^ { 2 } } { 2 4 } \right) } \\ & { \le \exp \left[ \left( 2 \log \left( 2 K \right) - \frac { M \epsilon ^ { 2 } } { 2 4 } \right) \left( \frac { L } { K } - 1 \right) \right] , } \end{array} +$$ + +where in (1) we used the bound from Lemma 36. + +12.3.3 MAIN PROOF: ORTHANT PROBABILITY OF A PRODUCT GAUSSIAN MATRICES + +Lemma 38. (Lemma 16 restated). Let $\mathbf { C } = \left[ \mathbf { c } _ { 1 } , \cdots , \mathbf { c } _ { N } \right] ^ { \top } \in \mathbb { R } ^ { N \times M }$ and $\mathbf { B } \in \mathbb { R } ^ { M \times L }$ be two independent random Gaussian matrices. Without loss of generality, assume $N \geq L$ , and denote $\alpha \triangleq M L / N$ . Then, in the regime $M \leq N$ and in the limit $\operatorname* { m i n } \left[ N , M , L \right] \dot { > } \alpha \dot { > } 1$ , we have + +$$ +\begin{array} { r } { \mathbb { P } \left( \mathbf { C B } > 0 \right) \dot { \leq } \exp \left( - 0 . 4 N \alpha ^ { 1 / 4 } \right) . } \end{array} +$$ + +Proof. For some $\theta > 0$ , and subset $S$ such that $| S | = K < L$ , we have + +$$ +\begin{array} { r l } & { \quad \mathbb { P } \left( \mathbf { C B } > 0 \right) } \\ & { \le \mathbb { P } \left( \mathbf { C B } _ { S } > 0 | \gamma \left( \mathbf { B } _ { S } \right) \le \epsilon \right) \mathbb { P } \left( \gamma \left( \mathbf { B } _ { S } \right) \le \epsilon \right) + \mathbb { P } \left( \mathbf { C B } _ { S } > 0 | \gamma \left( \mathbf { B } _ { S } \right) > \epsilon \right) \mathbb { P } \left( \gamma \left( \mathbf { B } _ { S } \right) > \epsilon \right) } \\ & { \le \mathbb { P } \left( \mathbf { C B } _ { S } > 0 | \gamma \left( \mathbf { B } _ { S } \right) \le \epsilon \right) + \mathbb { P } \left( \gamma \left( \mathbf { B } _ { S } \right) > \epsilon \right) } \\ & { = \mathbb { E } \left[ \left[ \mathbb { P } \left( \mathbf { c } _ { 1 } ^ { \top } \mathbf { B } _ { S } > 0 | \mathbf { B } _ { S } , \gamma \left( \mathbf { B } _ { S } \right) \le \epsilon \right) \right] ^ { N } | \gamma \left( \mathbf { B } _ { S } \right) \le \epsilon \right] + \mathbb { P } \left( \gamma \left( \mathbf { B } _ { S } \right) > \epsilon \right) , } \end{array} +$$ + +where in the last equality we used the fact that the rows of $\mathbf { C }$ are independent and identically distributed. + +We choose a specific subset + +$$ +S ^ { * } = \mathrm { a r g m i n } _ { S C [ L ] : | S | = K } \gamma \left( \mathbf { B } _ { S } \right) +$$ + +to minimize the second term and then upper bound it using Lemma 37 with $\theta = K \epsilon$ ; additionally, we apply Corollary 34 on the first term with the components of the vector $\mathbf { u }$ being + +$$ +u _ { i } = \left( \mathbf { B } _ { S } ^ { \top } \mathbf { c } _ { 1 } \right) _ { i } / \sqrt { \left( \mathbf { B } _ { S } ^ { \top } \mathbf { B } _ { S } \right) _ { i i } } \in \mathbb { R } ^ { K } , +$$ + +which is a Gaussian random vector with mean zero and covariance $\pmb { \Sigma }$ for which $\forall i : \Sigma _ { i i } = 1$ and $\forall i \neq j : \Sigma _ { i j } \leq \epsilon = \theta K ^ { - 1 }$ . Thus, we obtain + +$$ +\mathbb { P } \left( \mathbf { C } \mathbf { B } > 0 \right) \leq \exp \left( - N K \psi \left( \theta \right) + O \left( N \log K \right) \right) + \exp \left[ \left( \log \left( 2 K \right) ^ { 2 } - \frac { M \theta ^ { 2 } } { 2 4 K ^ { 2 } } \right) \left( \frac { L } { K } - 1 \right) \right] , +$$ + +where we recall $\psi \left( \theta \right)$ is defined in eq. (12.17). + +Next, we wish to select good values for $\theta$ and $K$ , which minimize this bound for large $( M , N , L , K )$ Thus, keeping only the first order terms in each exponent (assuming $L \gg K \gg 1$ ), we aim to minimize the function as much as possible + +$$ +f \left( K , \theta \right) \triangleq \exp \left( - N K \psi \left( \theta \right) \right) + \exp \left( - \frac { M \theta ^ { 2 } L } { 2 4 K ^ { 3 } } \right) . +$$ + +Note that the first term is decreasing in $K$ , while the second term increases. Therefore, for any $\theta$ the minimum of this function in $K$ would be approximately achieved when both terms are equal, i.e., + +$$ +N K \psi \left( \theta \right) = \frac { M \theta ^ { 2 } L } { 2 4 K ^ { 3 } } , +$$ + +so we choose + +$$ +{ \cal K } \left( \theta \right) = \left( { \frac { \theta ^ { 2 } M L } { 2 4 \psi \left( \theta \right) N } } \right) ^ { 1 / 4 } . +$$ + +Substituting $K \left( \theta \right)$ into $f \left( K , \theta \right)$ yields + +$$ +f \left( K \left( \theta \right) , \theta \right) = 2 \exp \left( - N \left[ \frac { \psi ^ { 3 } \left( \theta \right) \theta ^ { 2 } M L } { 2 4 N } \right] ^ { 1 / 4 } \right) . +$$ + +To minimize this function in $\theta$ , we need to maximize the function $\psi ^ { 3 } \left( \theta \right) \theta ^ { 2 }$ (which has a single maximum). Doing this numerically gives us + +$$ +\theta _ { * } \approx 2 3 . 2 5 ; \psi \left( \theta _ { * } \right) \approx 0 . 1 0 6 2 ; \psi ^ { 3 } \left( \theta _ { * } \right) \theta _ { * } ^ { 2 } \approx 0 . 6 4 7 8 . +$$ + +Substituting eqs. (12.27) and (12.28) into eq. (12.25), we obtain + +$$ +\begin{array} { r l } & { \mathbb { P } \left( \mathbf { C } \mathbf { B } > 0 \right) } \\ & { \leq \exp \left( - N \left[ \frac { M L } { 3 7 . 0 5 N } \right] ^ { 1 / 4 } + O \left( N \log K \right) \right) } \\ & { + \exp \left[ - N \left[ \frac { M L } { 3 7 . 0 5 N } \right] ^ { 1 / 4 } + 2 L \frac { \log K } { K } + \frac { M \theta ^ { 2 } } { 2 4 K ^ { 2 } } - \log \left( 2 K ^ { 2 } \right) \right] } \\ & { \leq \exp \left( - N \left[ \frac { M L } { 3 7 . 0 5 N } \right] ^ { 1 / 4 } + O \left( N \log \left( \frac { M L } { N } \right) \right) \right) , } \end{array} +$$ + +where in the last line we used $N \geq L , N \geq M$ and $\operatorname* { m i n } \left[ N , M , L \right] \dot { > } \alpha \dot { > } 1$ . Taking the log, and denoting $\alpha \triangleq M L / N$ , we thus obtain + +$$ +\begin{array} { r } { \log \mathbb { P } \left( \mathbf { C B } > 0 \right) \le - 0 . 4 N \alpha ^ { 1 / 4 } + O \left( N \log \alpha \right) , } \end{array} +$$ + +Therefore, in the limit that $N \to \infty$ and $\alpha ( N ) \infty$ , with $\alpha \left( N \right) \dot { < } N$ , we have + +$$ +\begin{array} { r } { \mathbb { P } \left( \mathbf { C B } > 0 \right) \dot { \leq } \exp \left( - 0 . 4 N \alpha ^ { 1 / 4 } \right) . } \end{array} +$$ + +# 13 LOWER BOUNDING THE ANGULAR VOLUME OF GLOBAL MINIMA: PROOF OF LEMMAS USED IN SECTION 10 + +# 13.1 ANGLES BETWEEN RANDOM GAUSSIAN VECTORS + +To prove the results in the next appendix sections, we will rely on the following basic Lemma. + +Lemma 39. For any vector y and $\mathbf { x } \sim \mathcal { N } \left( 0 , \mathbf { I } _ { d _ { 0 } } \right)$ , we have + +$$ +\mathbb { P } \left( \left| \frac { \mathbf { x } ^ { \top } \mathbf { y } } { \left\| \mathbf { x } \right\| \left\| \mathbf { y } \right\| } \right| > \cos \left( \epsilon \right) \right) \geq \frac { 2 \sin \left( \epsilon \right) ^ { d _ { 0 } - 1 } } { \left( d _ { 0 } - 1 \right) B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } +$$ + +$$ +\mathbb { P } \left( \left| \frac { \mathbf { x } ^ { \top } \mathbf { y } } { \left\| \mathbf { x } \right\| \left\| \mathbf { y } \right\| } \right| < u \right) \leq \frac { 2 u } { B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } , +$$ + +where we recall that $B \left( x , y \right)$ is the beta function. + +Proof. Since $\mathcal { N } ( 0 , \mathbf { I } _ { d _ { 0 } } )$ is spherically symmetric, we can set $\mathbf { y } = \left[ 1 , 0 \ldots , 0 \right] ^ { \top }$ , without loss of generality. Therefore, + +$$ +\left| { \frac { \mathbf { x } ^ { \top } \mathbf { y } } { \left\| \mathbf { x } \right\| \left\| \mathbf { y } \right\| } } \right| ^ { 2 } = { \frac { x _ { 1 } ^ { 2 } } { x _ { 1 } ^ { 2 } + \sum _ { i = 2 } ^ { d _ { 0 } } x _ { i } ^ { 2 } } } \sim { \mathcal { B } } \left( { \frac { 1 } { 2 } } , { \frac { d _ { 0 } - 1 } { 2 } } \right) , +$$ + +the Beta distribution, since $x _ { 1 } ^ { 2 } \sim \chi ^ { 2 } ( 1 )$ and ${ \textstyle \sum _ { i = 2 } ^ { d _ { 0 } } x _ { i } ^ { 2 } \sim \chi ^ { 2 } \left( d _ { 0 } - 1 \right) }$ are independent chi-square random variables. + +Suppose $Z \sim B \left( \alpha , \beta \right)$ , $\alpha \in ( 0 , 1 )$ , and $\beta > 1$ + +$$ +\mathbb { P } \left( Z > u \right) = \frac { \int _ { u } ^ { 1 } x ^ { \alpha - 1 } \left( 1 - x \right) ^ { \beta - 1 } d x } { B \left( \alpha , \beta \right) } \geq \frac { \int _ { u } ^ { 1 } 1 ^ { \alpha - 1 } \left( 1 - x \right) ^ { \beta - 1 } d x } { B \left( \alpha , \beta \right) } = \frac { \int _ { 0 } ^ { 1 - u } x ^ { \beta - 1 } d x } { B \left( \alpha , \beta \right) } = \frac { \left( 1 - u \right) ^ { \beta } } { \beta B \left( \alpha , \beta \right) } . +$$ + +Therefore, for $\epsilon > 0$ , + +$$ +\mathbb { P } \left( \left| \frac { \mathbf { x } ^ { \mathsf { T } } \mathbf { y } } { \left\| \mathbf { x } \right\| \left\| \mathbf { y } \right\| } \right| ^ { 2 } > \cos ^ { 2 } \left( \epsilon \right) \right) \geq \frac { 2 \left( 1 - \cos ^ { 2 } \left( \epsilon \right) \right) ^ { \frac { d _ { 0 } - 1 } { 2 } } } { \left( d _ { 0 } - 1 \right) B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } = \frac { 2 \sin \left( \epsilon \right) ^ { d _ { 0 } - 1 } } { \left( d _ { 0 } - 1 \right) B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } , +$$ + +which proves eq. (13.1). + +Similarly, for $\alpha \in ( 0 , 1 )$ and $\beta > 1$ + +$$ +\mathbb { P } \left( Z < u \right) = \frac { \int _ { 0 } ^ { u } x ^ { \alpha - 1 } \left( 1 - x \right) ^ { \beta - 1 } d x } { B \left( \alpha , \beta \right) } \leq \frac { \int _ { 0 } ^ { u } x ^ { \alpha - 1 } 1 ^ { \beta - 1 } d x } { B \left( \alpha , \beta \right) } = \frac { u ^ { \alpha } } { \alpha B \left( \alpha , \beta \right) } . +$$ + +Therefore, for $\epsilon > 0$ + +$$ +\mathbb { P } \left( \left| \frac { \mathbf { x } ^ { \top } \mathbf { y } } { \left\| \mathbf { x } \right\| \left\| \mathbf { y } \right\| } \right| ^ { 2 } < u ^ { 2 } \right) \leq \frac { 2 u } { B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } , +$$ + +which proves eq. (13.2). + +# 13.2 PROOF OF LEMMA 21: + +Given three matrices: datapoints, $\begin{array} { r c l c l } { \mathbf { X } } & { = } & { \left[ \mathbf { x } ^ { ( 1 ) } , \ldots , \mathbf { x } ^ { ( N ) } \right] } & { \in } & { \mathbb { R } ^ { d _ { 0 } \times N } } \end{array}$ , weights $\begin{array} { r l } { \mathbf { W } } & { { } = } \end{array}$ $\left[ { \bf w } _ { 1 } ^ { \top } , \ldots , { \bf w } _ { d _ { 1 } } ^ { \top } \right] ^ { \top } \ \in \ \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } }$ , and target weights $\mathbf W ^ { * } ~ = ~ \left[ \mathbf w _ { 1 } ^ { * \top } , \ldots , \mathbf w _ { d _ { 1 } ^ { * } } ^ { * \top } \right] ^ { \top } ~ \in ~ \mathbb { R } ^ { d _ { 1 } ^ { * } \times d _ { 0 } }$ , with $d _ { 1 } ^ { * } \leq d _ { 1 }$ ,we recall the following definitions: + +$$ +\mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \triangleq \left\{ \mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N } | \forall i , n : \left| \frac { \mathbf { x } ^ { ( n ) \top } \mathbf { w } _ { i } ^ { * } } { \left\| \mathbf { x } ^ { ( n ) } \right\| \left\| \mathbf { w } _ { i } ^ { * } \right\| } \right| > \mathrm { s i n } \alpha \right\} +$$ + +and + +$$ +\begin{array} { r } { \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) \triangleq \left\{ \mathbf { W } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } } | \forall i \le d _ { 1 } ^ { * } : \ \mathrm { s i g n } \left( \mathbf { w } _ { i } ^ { \top } \mathbf { X } \right) = \mathrm { s i g n } \left( \mathbf { w } _ { i } ^ { * } { } ^ { \top } \mathbf { X } \right) \right\} . } \end{array} +$$ + +Using these definitions, in this section we prove the following Lemma. + +Lemma 40. (Lemma 21 restated). For any $\alpha$ , if $\mathbf { W } ^ { * }$ is independent from W then, in the limit $N \to \infty$ , $\forall \mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right)$ with $\log \sin \alpha { \dot { > } } { \dot { d } _ { 0 } } ^ { - 1 } \log { d _ { 0 } }$ + +$$ +\begin{array} { r } { \mathbb { P } _ { \mathbf { W } \sim \mathcal { N } } \left( \mathbf { W } \in \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) \right) \dot { \geq } \exp \left( d _ { 0 } d _ { 1 } ^ { * } \log \sin \alpha \right) . } \end{array} +$$ + +Proof. To lower bound $\mathbb { P } _ { \mathbf { W } \sim \mathcal { N } } \left( \mathbf { W } \in \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) \right) \forall \mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right)$ , we define the event that all weight hyperplanes (with normals $\mathbf { w } _ { i }$ ) have an angle of at least $\alpha$ from the corresponding target hyperplanes (with normals $\mathbf { w } _ { i } ^ { * }$ ). + +$$ +\begin{array} { r } { \tilde { \mathcal { G } } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) = \left\{ \mathbf { W } \in \mathbb { R } ^ { d _ { 1 } \times d _ { 0 } } \big | \left| \frac { \mathbf { W } _ { i } ^ { \top } \mathbf { w } _ { i } ^ { * } } { \left\| \mathbf { w } _ { i } \right\| \left\| \mathbf { w } _ { i } ^ { * } \right\| } \right| < \cos \left( \alpha \right) \right\} . } \end{array} +$$ + +In order that $\mathrm { s i g n } \left( \mathbf { w } _ { i } ^ { \top } \mathbf { x } ^ { ( n ) } \right) \neq \mathrm { s i g n } \left( \mathbf { w } _ { 1 } ^ { \ast \top } \mathbf { x } ^ { ( n ) } \right)$ , $\mathbf { w } _ { i }$ must be rotated in respect to $\mathbf { w } _ { i } ^ { * }$ by an angle greater then the angular margin $\alpha$ , which is the minimal the angle between $\mathbf { x } ^ { ( n ) }$ and the solution hyperplanes (with normals $\mathbf { w } _ { i } ^ { * }$ ). Therefore, we have that, given $\mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right)$ , + +$$ +\forall \alpha : \bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \tilde { \mathcal { G } } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \subset \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) . +$$ + +And so, $\forall \mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right)$ : + +$$ +\begin{array} { r l } & { \mathbb { P } _ { \mathbf { W } \sim \mathcal { N } } \left( \mathbf { W } \in \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) \right) \overset { ( 1 ) } { \geq } \mathbb { P } _ { \mathbf { W } \sim \mathcal { N } } \left( \mathbf { W } \in \displaystyle \bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \tilde { \mathcal { G } } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) } \\ & { \overset { ( 2 ) } { = } \displaystyle \prod _ { i = 1 } ^ { d _ { 1 } ^ { * } } \mathbb { P } _ { \mathbf { W } \sim \mathcal { N } } \left( \mathbf { W } \in \tilde { \mathcal { G } } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \overset { ( 3 ) } { \geq } \left[ \frac { 2 \sin \left( \alpha \right) ^ { d _ { 0 } - 1 } } { \left( d _ { 0 } - 1 \right) B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } \right] ^ { d _ { 1 } ^ { * } } , } \end{array} +$$ + +where in (1) we used eq. (13.5), in (2) we used the independence of $\{ { \bf w } _ { i } \} _ { i = 1 } ^ { d _ { 1 } ^ { * } }$ and in (3) we used eq. (13.1) from Lemma 39. Lastly, to simplify this equation we use the asymptotic expansion of the beta function $B \left( { \textstyle \frac { 1 } { 2 } } , x \right) = { \sqrt { \pi / x } } + O \left( x ^ { - 3 / 2 } \right)$ for large $x$ : + +$$ +\begin{array} { r } { \log \mathbb { P } _ { \mathbf { W } \sim \mathcal { N } } \left( \mathbf { W } \in \tilde { \mathcal { G } } \left( \mathbf { X } , \mathbf { W } ^ { * } \right) \right) \geq d _ { 0 } d _ { 1 } ^ { * } \log \sin \alpha + O \left( d _ { 1 } ^ { * } \log d _ { 0 } \right) . } \end{array} +$$ + +We obtain the Lemma in the limit $N \to \infty$ when $\log \sin \alpha { \dot { > } } d _ { 0 } ^ { - 1 } \log d _ { 0 }$ . + +# 13.3 PROOF OF LEMMA 22: + +Lemma 41. (Lemma 22 restated). Let $\mathbf { W } ^ { * } = \left[ \mathbf { w } _ { 1 } ^ { \top } , \ldots , \mathbf { w } _ { d _ { 1 } ^ { * } } ^ { \top } \right] ^ { \top } \in \mathbb { R } ^ { d _ { 1 } ^ { * } \times d _ { 0 } }$ a fixed matrix independent of $\mathbf { X }$ . Then, in the limit $N \infty$ with $d _ { 1 } ^ { * } { \dot { \leq } } d _ { 0 } { \dot { \leq } } N _ { \mathrm { ~ } }$ , the probability of not having an angular margin $\sin \alpha = 1 / \left( d _ { 1 } ^ { * } d _ { 0 } N \right)$ (eq. (13.3)) is upper bounded by + +$$ +\mathbb { P } \left( \mathbf { X } \notin \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \dot { \leq } \sqrt { \frac { 2 } { \pi } } d _ { 0 } ^ { - 1 / 2 } +$$ + +Proof. We define + +$$ +\mathcal M _ { n , i } ^ { \alpha } ( \mathbf W ^ { * } ) \triangleq \{ \mathbf X \in \mathbb R ^ { d _ { 0 } \times N } | | \frac { \mathbf x ^ { ( n ) \top } \mathbf w _ { i } ^ { * } } { \| \mathbf x ^ { ( n ) } \| \| \mathbf w _ { i } ^ { * } \| } | > \sin ( \alpha ) \} , +$$ + +and $\begin{array} { r } { \mathcal { M } _ { n } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \triangleq \bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \mathcal { M } _ { n , i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) } \end{array}$ . Since $\begin{array} { r } { \mathcal { M } \left( \mathbf { W } ^ { * } \right) = \bigcap _ { n = 1 } ^ { N } \mathcal { M } _ { n } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) } \end{array}$ , we have + +$$ +\begin{array} { r l } & { \displaystyle \mathbb { P } \left( \mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \stackrel { ( 1 ) } { = } \prod _ { n = 1 } ^ { N } \mathbb { P } \left( \mathbf { X } \in \mathcal { M } _ { n } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) = \prod _ { n = 1 } ^ { N } \left[ 1 - \mathbb { P } \left( \mathbf { X } \notin \mathcal { M } _ { n } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \right] } \\ & { \displaystyle \overset { ( 2 ) } { \geq } \prod _ { n = 1 } ^ { N } \left[ 1 - \sum _ { i = 1 } ^ { d _ { 1 } ^ { * } } \mathbb { P } \left( \mathbf { X } \notin \mathcal { M } _ { n , i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \right] \stackrel { ( 3 ) } { \geq } \left[ 1 - d _ { 1 } ^ { * } \frac { 2 \sin \left( \alpha \right) } { B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } \right] ^ { N } , } \end{array} +$$ + +where in (1) we used the independence of x(n) Nn= , in (2) we use the union bound, and in (3) we use eq. (13.2) from Lemma 39. Taking the log and we using the asymptotic expansion of the beta function $B \left( { \textstyle \frac { 1 } { 2 } } , x \right) = { \sqrt { \pi / x } } + O \left( x ^ { - 3 / 2 } \right)$ for large $x$ , we get + +$$ +\begin{array} { r l } & { \log \mathbb { P } \left( \mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \geq N \log \left[ 1 - \sqrt { \frac { 2 } { \pi } d _ { 0 } } d _ { 1 } ^ { * } \sin \alpha + O \left( d _ { 1 } ^ { * } d _ { 0 } ^ { - 1 / 2 } \sin \alpha \right) \right] } \\ & { \qquad = - \sqrt { \frac { 2 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + O \left( d _ { 0 } ^ { - 3 / 2 } / N + d _ { 0 } ^ { - 1 } N ^ { - 2 } \right) , } \end{array} +$$ + +where in the last line we recalled $\sin \alpha = 1 / N$ . Recalling that $d _ { 1 } ^ { * } { \dot { \le } } d _ { 0 } { \dot { \le } } N$ , we find + +$$ +\mathbb { P } \left( \mathbf { X } \notin \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \dot { \geq } 1 - \exp \left( - \sqrt { \frac { 2 } { \pi } } d _ { 0 } ^ { - 1 / 2 } \right) \geq \sqrt { \frac { 2 } { \pi } } d _ { 0 } ^ { - 1 / 2 } +$$ + +# 13.4 PROOF OF LEMMA 23: + +Lemma 42. (Lemma 23 restated). Let $\mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N }$ be a standard random Gaussian matrix of datapoints. Then we can find, with probability $^ { l }$ , $( \mathbf { X } , \mathbf { y } )$ -dependent matrices $\mathbf { W } ^ { * }$ and $\mathbf { z } ^ { \ast }$ as in Theorem 8 (where $d _ { 1 } ^ { * } \triangleq 4 \lceil N / \left( 2 d _ { 0 } - 2 \right) \rceil ,$ ). Moreover, in the limit $N \to \infty$ , where $N / d _ { 0 } { \dot { \le } } d _ { 0 } { \dot { \le } } N$ , for any y, we can bound the probability of not having an angular margin (eq. (13.3)) with $\sin \alpha =$ $1 / \left( d _ { 1 } ^ { * } d _ { 0 } N \right)$ by + +$$ +\mathbb { P } \left( \mathbf { X } \notin \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \dot { \leq } \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } +$$ + +Proof. In this proof we heavily rely on the notation and results from the proof of in appendix section 9. Without loss of generality we assume $S _ { 1 } ^ { + } = [ d _ { 0 } - 1 ]$ . Unfortunately, we can’t use Lemma $4 1 -$ this proof is significantly more complicated since the constructed solution $\mathbf { W } ^ { * }$ depends on $\mathbf { X }$ (we keep this dependence implicit, for brevity). Similarly to the proof of Lemma 41, we define, + +$$ +\mathcal { M } _ { i , n } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \triangleq \left\{ \mathbf { X } \in \mathbb { R } ^ { d _ { 0 } \times N } | \left| \frac { \mathbf { x } ^ { ( n ) \top } \mathbf { w } _ { i } ^ { * } } { \left\| \mathbf { x } ^ { ( n ) } \right\| \left\| \mathbf { w } _ { i } ^ { * } \right\| } \right| > \sin \left( \alpha \right) \right\} +$$ + +and $\begin{array} { r } { \mathcal { M } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \triangleq \bigcap _ { n = 1 } ^ { N } \mathcal { M } _ { i , n } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) , \operatorname { s o } \mathcal { M } \left( \mathbf { W } ^ { * } \right) = \bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \mathcal { M } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) . \operatorname { W e } } \end{array}$ have + +$$ +\mathbb { P } \left( \mathbf { X } \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) = 1 - \mathbb { P } \left( \mathbf { X } \not \in \mathcal { M } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \overset { ( 1 ) } { \geq } 1 - \sum _ { i = 1 } ^ { d _ { 1 } } \mathbb { P } \left( \mathbf { X } \not \in \mathcal { M } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) +$$ + +nd in (2) we used the fact that, from symmetr. Next, we examine the minimal angular margin in $\forall i :$ $\mathbb { P } \left( \mathbf { X } \notin \mathcal { M } _ { i } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) = \mathbb { P } \left( \mathbf { X } \notin \mathcal { M } _ { 1 } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right)$ $\mathcal { M } _ { 1 , n } ^ { \alpha }$ separately for $\forall n < d _ { 0 }$ and $\forall n \geq d _ { 0 }$ . Recalling the construction of in appendix section 9, we have, for $\forall n < d _ { 0 }$ : + +$$ +\operatorname* { m i n } _ { i , n < d _ { 0 } } \left. \frac { \mathbf { x } ^ { ( n ) \top } \mathbf { w } _ { i } ^ { * } } { \left\| \mathbf { x } ^ { ( n ) } \right\| \left\| \mathbf { w } _ { i } ^ { * } \right\| } \right. = \operatorname* { m i n } _ { n < d _ { 0 } , \pm } \frac { \left| \left( \tilde { \mathbf { w } } _ { 1 } \pm \epsilon _ { 2 } \hat { \mathbf { w } } _ { 1 } \right) ^ { \top } \mathbf { x } ^ { ( n ) } \right| } { \left\| \tilde { \mathbf { w } } _ { 1 } \pm \epsilon _ { 2 } \hat { \mathbf { w } } _ { 1 } \right\| \left\| \mathbf { x } ^ { ( n ) } \right\| } +$$ + +$$ +\begin{array} { r } { \underline { { \underline { { ( 1 ) } } } } _ { n < d _ { 0 } , \pm } \frac { \epsilon _ { 2 } } { \left\| \tilde { \mathbf { w } } _ { 1 } \pm \epsilon _ { 2 } \hat { \mathbf { w } } _ { 1 } \right\| \left\| \mathbf { x } ^ { ( n ) } \right\| } \overset { ( 2 ) } { = } \frac { \gamma \epsilon _ { 1 } / \sqrt { 1 + \gamma ^ { 2 } \epsilon _ { 1 } ^ { 2 } } } { \left\| \hat { \mathbf { w } } _ { 1 } \right\| \operatorname* { m a x } _ { n < d _ { 0 } } \left\| \mathbf { x } ^ { ( n ) } \right\| } , } \end{array} +$$ + +where in (1) we used $\forall n < d _ { 0 }$ : $\mathbf { x } ^ { ( n ) \top } \hat { \mathbf { w } } _ { 1 } = 1$ and $\mathbf { x } ^ { ( n ) \top } \tilde { \mathbf { w } } _ { 1 } = 0$ , from the construction of $\tilde { \mathbf { w } } _ { 1 }$ and $\hat { \mathbf { w } } _ { 1 }$ (eqs. (9.2), (9.5), and (9.4)), and in (2) we used the fact that $\hat { \mathbf { w } } _ { 1 } ^ { \top } \tilde { \mathbf { w } } _ { 1 } = 0$ from eq. (9.4) together with $\| \tilde { \mathbf { w } } _ { 1 } \| = \| \hat { \mathbf { w } } _ { 1 } \|$ from eq. (9.5), and $\epsilon _ { 2 } = \gamma \epsilon _ { 1 }$ from eq. (9.7). + +For $\forall n \geq d _ { 0 }$ : + +$$ +\operatorname* { m i n } _ { i , n \geq d _ { 0 } } \left. \frac { \mathbf { x } ^ { \left( n \right) \top } \mathbf { w } _ { i } ^ { * } } { \left. \mathbf { x } ^ { \left( n \right) } \right. \left. \mathbf { w } _ { i } ^ { * } \right. } \right. = \operatorname* { m i n } _ { n \geq d _ { 0 } , \pm } \frac { \left. \left( \tilde { \mathbf { w } } _ { 1 } \pm \epsilon _ { 1 } \hat { \mathbf { w } } _ { 1 } \right) ^ { \top } \mathbf { x } ^ { \left( n \right) } \right. } { \left. \tilde { \mathbf { w } } _ { 1 } \pm \epsilon _ { 1 } \hat { \mathbf { w } } _ { 1 } \right. \left. \mathbf { x } ^ { \left( n \right) } \right. } \geq \frac { \left( 1 - \gamma \beta \right) \epsilon _ { 1 } } { \gamma \beta \sqrt { 1 + \epsilon _ { 1 } ^ { 2 } } } \operatorname* { m i n } _ { n \geq d _ { 0 } } \frac { \left. \tilde { \mathbf { w } } _ { 1 } ^ { \top } \mathbf { x } ^ { \left( n \right) } \right. } { \left. \tilde { \mathbf { w } } _ { 1 } \right. \left. \mathbf { x } ^ { \left( n \right) } \right. } , +$$ + +where we used the fact that $\forall n \geq d _ { 0 } : \epsilon _ { 2 } \left| \hat { \mathbf { w } } _ { 1 } ^ { \top } \mathbf { x } ^ { ( n ) } \right| \leq \gamma \beta \left| \tilde { \mathbf { w } } _ { 1 } ^ { \top } \mathbf { x } ^ { ( n ) } \right|$ , from eq. (9.7), and also that $\hat { \mathbf { w } } _ { 1 } ^ { \top } \tilde { \mathbf { w } } _ { 1 } = 0$ from eq. (9.4). + +We substitute eqs. (13.8) and (13.9) into $\mathbb { P } \left( \mathbf { X } \in \mathcal { M } _ { 1 } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right)$ : + +$\mathbb { P } \left( \mathbf { X } \in \mathcal { M } _ { 1 } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right)$ + +$$ +\begin{array} { r l } & { \geq \mathbb { P } \left( \frac { \gamma \epsilon _ { 1 } / \sqrt { 1 + \gamma ^ { 2 } \epsilon _ { 1 } ^ { 2 } } } { \left\| \hat { \mathbf { w } } _ { 1 } \right\| \operatorname* { m a x } _ { n < d _ { 0 } } \left\| \mathbf { x } ^ { ( n ) } \right\| } > \sin \alpha , \frac { \left( 1 - \gamma \beta \right) \epsilon _ { 1 } } { \gamma \beta \sqrt { 1 + \epsilon _ { 1 } ^ { 2 } } } \underset { n \geq d _ { 0 } } { \operatorname* { m i n } } \frac { \left| \hat { \mathbf { w } } _ { 1 } ^ { \top } \mathbf { x } ^ { ( n ) } \right| } { \left\| \hat { \mathbf { w } } _ { 1 } \right\| \left\| \mathbf { x } ^ { ( n ) } \right\| } > \sin \alpha \right) } \\ & { \overset { ( 1 ) } { \geq } \mathbb { P } \left( \frac { \gamma \kappa } { \left\| \hat { \mathbf { w } } _ { 1 } \right\| \operatorname* { m a x } _ { n < d _ { 0 } } \left\| \mathbf { x } ^ { ( n ) } \right\| } > \sin \alpha , \frac { \left( 1 - \gamma \beta \right) } { \gamma \beta } \kappa \underset { n \geq d _ { 0 } } { \operatorname* { m i n } } \frac { x _ { 1 } ^ { ( n ) } } { \left\| \mathbf { x } ^ { ( n ) } \right\| } > \sin \alpha , \frac { \epsilon _ { 1 } } { \sqrt { 1 + \epsilon _ { 1 } ^ { 2 } } } > \kappa \right) } \end{array} +$$ + +$$ +\overset { \mathtt { ( 2 ) } } { \geq } \mathbb { P } \left( \frac { \gamma \kappa } { \eta \sin \alpha } > \left. \hat { \mathbf { w } } _ { 1 } \right. , \eta > \operatorname* { m a x } _ { n < d _ { 0 } } \left. \mathbf { x } ^ { ( n ) } \right. \right) \mathbb { P } \left( \frac { ( 1 - \gamma \beta ) } { \gamma \beta } \kappa \operatorname* { m i n } _ { n \geq d _ { 0 } } \frac { x _ { 1 } ^ { ( n ) } } { \left. \mathbf { x } ^ { ( n ) } \right. } > \sin \alpha , \frac { \epsilon _ { 1 } } { \sqrt { 1 + \epsilon _ { 1 } ^ { 2 } } } > \kappa \right) , +$$ + +where in (1) we rotate the axes so that $\hat { \mathbf { w } } _ { 1 } \propto [ 1 , 0 , 0 \ldots , 0 ]$ axes $\tilde { \mathbf { w } } _ { 1 } \propto [ 0 , 1 , 0 , 0 \ldots , 0 ] - \mathrm { t h i s }$ is possible due to the spherical symmetry of $\mathbf { x } ^ { ( n ) }$ , and the fact that $\hat { \mathbf { w } } _ { 1 }$ and $\tilde { \mathbf { w } } _ { 1 }$ are functions of $\mathbf { x } ^ { ( n ) }$ for $n < d _ { 0 }$ (from eqs. (9.4) and (9.2)), and as such, they are independent from $\mathbf { x } ^ { ( n ) }$ for $n \geq d _ { 0 }$ , in (2) we use that fact that $\left. \hat { \mathbf { w } } _ { 1 } \right.$ and $\operatorname* { m a x } _ { n < d _ { 0 } } \left\| \mathbf { x } ^ { ( n ) } \right\|$ are functions of $\mathbf { x } ^ { ( n ) }$ for $n < d _ { 0 }$ , and as such, they are independent from $\mathbf { x } ^ { ( n ) }$ for $n \geq d _ { 0 }$ . Thus, + +$$ +\begin{array} { r l } & { \mathbb { P } ^ { \nu } ( \mathbf { X } \times M _ { 1 } ^ { \nu } ( \mathbf { W } ^ { \nu } ) ) } \\ & { \geq \left( 1 - \mathbb { P } \left( \frac { \mathcal { X } } { \eta \sin \alpha } \leq \left. \mathbf { w } _ { 1 } \right. \textnormal { \texttt { O } } \eta \leq \operatorname* { m a x } \left. \mathbf { s } ^ { \nu } ( \mathbf { u } ) \right. \right) \right) } \\ & { \cdot \left( 1 - \mathbb { P } \left( \frac { \left\{ 1 - \eta \beta \right\} } { \gamma \beta } \right) \operatorname* { m i n } \frac { x _ { 1 } ^ { \nu \alpha } } { x _ { 2 } ^ { \nu \alpha } \left. \mathbf { s } ^ { \nu \alpha } \right. } \leq \sin \alpha \alpha \frac { \mathbb { E } _ { 1 } } { \sqrt { 1 + \epsilon _ { 1 } ^ { \alpha } } } \leq \kappa \right) \right) } \\ & { \overset { ( a ) } { \geq } \left( 1 - \mathbb { P } \left( \frac { \mathcal { X } } { \eta \sin \alpha } \leq \left. \mathbf { s } \right. _ { 1 } \right) - \mathbb { P } \left( \eta \leq \operatorname* { m a x } \left. \mathbf { s } ^ { \nu \alpha } \right. \right) \right) } \\ & { \cdot \left( 1 - \mathbb { P } \left( \frac { \left\{ 1 - \eta \beta \right\} } { \gamma \beta } \operatorname* { m i n } \frac { x _ { 1 } ^ { \nu \alpha } } { x _ { 2 } ^ { \nu \alpha } \left. \mathbf { s } ^ { \nu \alpha } \right. } \leq \sin \alpha \right) - \mathbb { E } \left( \frac { \mathcal { X } } { \sqrt { 1 + \epsilon _ { 1 } ^ { \alpha } } } \leq \kappa \right) \right) } \\ & { = \left( \mathbb { P } \left( \eta > \frac { \mathcal { X } } { \eta \sin \alpha } \left. \mathbf { s } ^ { \nu \alpha } \right. \right) - \mathbb { P } \left( \frac { \mathcal { Y } } { \eta \sin \alpha } \leq \left. \mathbf { s } \right. \right) \right) } \\ & \cdot \left( \mathbb { E } \left( \frac { \left\{ 1 - \gamma \beta \right\} } { \gamma \beta } \right) \operatorname* { m i n } \frac { x _ { 1 } ^ { \nu \alpha } } { x _ { 2 } ^ { \nu \alpha } \left. \mathbf { s } ^ { \nu \alpha } \right. } \leq \sin \alpha \right) - \mathbb { P } \left( \frac { \mathcal { Z } } { \sqrt { 1 + \epsilon _ { 1 } ^ { \alpha } } } \leq \kappa \end{array} +$$ + +where in (1) we use the union bound on both probability terms. + +All that remains is to calculate each remaining probability term in eq. (13.11). First, we have + +$$ +\begin{array} { r l } & { \mathbb { P } \left( \frac { \epsilon _ { 1 } } { \sqrt { 1 + \epsilon _ { 1 } ^ { 2 } } } \leq \kappa \right) = 1 - \mathbb { P } \left( \frac { \kappa } { \sqrt { 1 - \kappa ^ { 2 } } } < \epsilon _ { 1 } \right) } \\ & { \stackrel { \left( 1 \right) } { = } 1 - \mathbb { P } \left( \underset { n \geq d _ { 0 } } { \operatorname* { m i n } } \left| \tilde { \mathbf { w } } _ { i } ^ { \top } \mathbf { x } ^ { \left( n \right) } \right| > \frac { \kappa } { \sqrt { 1 - \kappa ^ { 2 } } } \frac { 1 } { \beta } \right) \stackrel { \left( 2 \right) } { = } 1 - \mathbb { P } \left( \underset { n \geq d _ { 0 } } { \operatorname* { m i n } } \left| \frac { x _ { 2 } ^ { \left( n \right) } } { x _ { 1 } ^ { \left( n \right) } } \right| > \frac { \kappa } { \sqrt { 1 - \kappa ^ { 2 } } } \frac { 1 } { \beta } \right) } \\ & { \stackrel { \left( 3 \right) } { = } 1 - \left[ \mathbb { P } \left( \left| \frac { x _ { 2 } ^ { \left( 1 \right) } } { x _ { 1 } ^ { \left( 1 \right) } } \right| > \frac { \kappa } { \sqrt { 1 - \kappa ^ { 2 } } } \frac { 1 } { \beta } \right) \right] ^ { N - d _ { 0 } - 1 } \stackrel { \left( 4 \right) } { \leq } 1 - \left[ 1 - \frac { 2 } { \pi } \arctan \left( \frac { \kappa } { \sqrt { 1 - \kappa ^ { 2 } } } \frac { 1 } { \beta } \right) \right] ^ { N } , } \end{array} +$$ + +where in (1) we used eq. (9.7), in (2) we recall that in eq. (13.10) we rotated the axes so that $\hat { \mathbf { w } } _ { 1 } \propto [ 1 , 0 , 0 \ldots , 0 ]$ axes $\tilde { \mathbf { w } } _ { 1 } \propto [ 0 , 1 , 0 , 0 \ldots , 0 ]$ , in (3) we used the independence of different $\mathbf { x } ^ { ( n ) }$ , and in (4) we used the fact that the ratio of two independent Gaussian variables is distributed according to the symmetric Cauchy distribution, which has the cumulative distribution function $\mathbb { P } \left( X > \breve { x } \right) = \textstyle \frac { 1 } { 2 } - \breve { \frac { } { \pi } }$ arctan $( x )$ , and therefore $\begin{array} { r } { \mathbb { P } \left( | X | > x \right) = 1 - \frac { 2 } { \pi } } \end{array}$ arctan $( x )$ . + +Second, we use eq. (13.2) + +$$ +\mathbb { P } \left( \operatorname* { m i n } _ { n \geq d _ { 0 } } \frac { x _ { 1 } ^ { ( n ) } } { \left\| \mathbf { x } ^ { ( n ) } \right\| } > \frac { \gamma \beta \sin \alpha } { \left( 1 - \gamma \beta \right) \kappa } \right) > \left[ 1 - \frac { 2 \gamma \beta \sin \alpha } { \left( 1 - \gamma \beta \right) \kappa B \left( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } \right) } \right] ^ { N } . +$$ + +Third, $\left\| \mathbf { x } ^ { ( n ) } \right\| ^ { 2 }$ is distributed according to the chi-square distribution of order $d _ { 0 }$ , so for $\eta ^ { 2 } > d _ { 0 }$ + +$$ +\begin{array} { r } { \mathbb { P } \left( \left\| \mathbf { x } ^ { ( n ) } \right\| ^ { 2 } \geq \eta ^ { 2 } \right) \leq \left( \eta ^ { 2 } \exp \left( 1 - \eta ^ { 2 } / d _ { 0 } \right) / d _ { 0 } \right) ^ { d _ { 0 } / 2 } . } \end{array} +$$ + +Therefore, + +$$ +\mathbb { P } \left( \operatorname* { m a x } _ { n < d _ { 0 } } \left\| \mathbf { x } ^ { ( n ) } \right\| ^ { 2 } < \eta ^ { 2 } \right) > \left[ 1 - \left( \eta ^ { 2 } \exp \left( 1 - \eta ^ { 2 } / d _ { 0 } \right) / d _ { 0 } \right) ^ { d _ { 0 } / 2 } \right] ^ { d _ { 0 } - 1 } . +$$ + +Lastly, we bound $\| \tilde { \mathbf { w } } _ { 1 } \| = \| \hat { \mathbf { w } } _ { 1 } \|$ (from eq. (9.5)). From eq. (9.4), we have + +$$ +\hat { \mathbf { w } } _ { 1 } ^ { \top } \mathbf { X } _ { [ d _ { 0 } - 1 ] } = \left[ 1 , \ldots , 1 , 1 \right] , +$$ + +where $\mathbf { X } _ { [ d _ { 0 } - 1 ] }$ has a singular value decomposition + +$$ +\mathbf { X } _ { [ d _ { 0 } - 1 ] } = \sum _ { i = 1 } ^ { d _ { 0 } } \sigma _ { i } \mathbf { u } _ { i } \mathbf { v } _ { i } ^ { \top } , +$$ + +with $\sigma _ { i }$ being the singular values, and $\mathbf { u } _ { i }$ and $\mathbf { v } _ { i }$ being the singular vectors. The singular values are ordered from smallest to largest, and $\sigma _ { 1 } = 0$ with $\mathbf { u } _ { 1 } = \tilde { \mathbf { w } } _ { 1 }$ , from eq. (9.2). With probability 1, the other $d _ { 0 } - 1$ singular value are non-zero: they are the square roots of the eigenvalues of the random matrix $\mathbf { \bar { X } } _ { [ d _ { 0 } - 1 ] } ^ { \top } \mathbf { \bar { X } } _ { [ d _ { 0 } - 1 ] } \in \mathbb { R } ^ { d _ { 0 } - 1 \times d _ { 0 } - 1 }$ . Taking the squared norm of eq. (13.15), we have + +$$ +d _ { 0 } - 1 = \hat { \mathbf { w } } _ { 1 } ^ { \top } \mathbf { X } _ { [ d _ { 0 } - 1 ] } \mathbf { X } _ { [ d _ { 0 } - 1 ] } ^ { \top } \hat { \mathbf { w } } _ { 1 } = \sum _ { i = 1 } ^ { d _ { 0 } } { \sigma _ { i } ^ { 2 } } \left( \mathbf { u } _ { i } ^ { \top } \hat { \mathbf { w } } _ { 1 } \right) ^ { 2 } \geq \sigma _ { 2 } ^ { 2 } \left. \hat { \mathbf { w } } _ { 1 } \right. ^ { 2 } , +$$ + +where the last inequality stems from the fact that $\mathbf { u } _ { 1 } ^ { \top } \hat { \mathbf { w } } _ { 1 } = \tilde { \mathbf { w } } _ { 1 } ^ { \top } \hat { \mathbf { w } } _ { 1 } = 0$ (from eq. (9.4)), so the minimal possible value is attained when $\mathbf { u } _ { 2 } ^ { \top } \hat { \mathbf { w } } _ { 1 } = \| \hat { \mathbf { w } } _ { 1 } \|$ . The minimal nonzero singular value, $\sigma _ { 2 }$ , can be bounded using the following result from (Rudelson & Vershynin, 2010, eq. (3.2)) + +$$ +\begin{array} { r } { \mathbb { P } \left( \underset { { \mathbf { r } } \in \mathbb { R } ^ { d _ { 0 } } } { \operatorname* { m i n } } \left. \mathbf { X } _ { [ d _ { 0 } ] } \mathbf { r } \right. \leq \eta d _ { 0 } ^ { - 1 / 2 } \right) \leq \eta . } \end{array} +$$ + +Since + +$$ +\sigma _ { 2 } = \operatorname* { m i n } _ { \mathbf { r } \in \mathbb { R } ^ { d _ { 0 } - 1 } } \left\| \mathbf { X } _ { [ d _ { 0 } - 1 ] } \mathbf { r } \right\| \geq \operatorname* { m i n } _ { \mathbf { r } \in \mathbb { R } ^ { d _ { 0 } } } \left\| \mathbf { X } _ { [ d _ { 0 } ] } \mathbf { r } \right\| +$$ + +we have, + +$$ +\begin{array} { r } { \mathbb { P } \left( \sigma _ { 2 } < \eta d _ { 0 } ^ { - 1 / 2 } \right) \le \eta . } \end{array} +$$ + +Combining this with eq. (13.16) we get + +$$ +\mathbb { P } \left( \frac { \beta \kappa } { \eta \sin \alpha } < \| \mathbf { w } _ { 1 } \| \right) \le \frac { \eta d _ { 0 } } { \beta \kappa } \sin \alpha . +$$ + +Lastly, combining eqs. (13.12), (13.13), (13.14) and (13.17) into eqs. (13.7) and (13.11), we get, for $\eta ^ { 2 } > d _ { 0 }$ , + +$$ +\begin{array} { r l } & { \mathbb { P } ( \mathbf { X } \in \mathcal { M } ^ { \alpha } ( \mathbf { W } ^ { \star } ) ) } \\ & { \geq 1 - d _ { 1 } ^ { \ast } ( 1 - ( [ 1 - ( \eta ^ { 2 } \exp ( 1 - \eta ^ { 2 } / d _ { 0 } ) / d _ { 0 } ) ^ { \hat { d } _ { 0 } / 2 } ] ^ { d _ { 0 } - 1 } - \frac { \eta d _ { 0 } } { \gamma \kappa } \sin \alpha ) } \\ & { \Bigg . \Bigg . \Bigg . } \\ & { \quad \Bigg . \Bigg . \Bigg . \Bigg . \Bigg . \Bigg . \Bigg . \Bigg . \Bigg \Gamma ( 1 - \frac { 2 \gamma \beta \sin \alpha } { ( 1 - \gamma \beta ) \kappa B ( \frac { 1 } { 2 } , \frac { d _ { 0 } - 1 } { 2 } ) } \Bigg ] ^ { N } - [ 1 - \frac { 2 } { \pi } \arctan ( \frac { \kappa } { \sqrt { 1 - \kappa ^ { 2 } } } \frac { 1 } { \beta } ) ] ^ { N } \Bigg ) \Bigg ) } \\ & { \geq 1 - d _ { 1 } ^ { \ast } ( 1 - ( [ 1 - ( \log d _ { 0 } \exp ( 1 - \log d _ { 0 } ) ) ^ { d _ { 0 } / 2 } ] ^ { d _ { 0 } - 1 } - \frac { 2 d _ { 0 } ^ { \hat { d } _ { 0 } ^ { \prime } / 2 } \sqrt { \log { d _ { 0 } } } } { d _ { 1 } ^ { \ast } N } ) \Bigg . } \\ & { \Bigg ( [ 1 - \sqrt { \frac { \kappa } { \pi } } \frac { 1 } { d _ { 1 } ^ { \ast } d _ { 0 } ^ { \dagger } N } + \mathcal { O } ( \frac { 1 } { N d _ { 1 } ^ { \ast } d _ { 0 } ^ { \dagger } } ) ] ^ { N } - 0 . 4 5 ^ { N } \Bigg ) \Bigg ) ~ , } \end{array} +$$ + +where in the last line we take $\beta = \gamma = \kappa = 1 / \sqrt { 2 }$ , $\eta = d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } }$ , $\sin \alpha = 1 / \left( d _ { 1 } ^ { * } d _ { 0 } N \right)$ . Using the asymptotic expansion of the beta function $B \left( { \textstyle { \frac { 1 } { 2 } } } , x \right) = { \sqrt { \pi / x } } + O \left( x ^ { - 3 / 2 } \right)$ for large $x$ , we obtain, + +$$ +\begin{array} { r l } & { \| \arcsin \alpha - \operatorname* { l i } ( \theta C \ln ( \theta ) ) } \\ & { 1 - \operatorname* { P } ( \mathbf { X } \in \mathcal { M } ^ { \alpha } ( \operatorname { W e V } ^ { * } ) ) } \\ & { \leq d _ { 1 } ^ { \alpha } ( 1 - ( [ 1 - \exp ( - \frac { d _ { 0 } } { 2 } \operatorname* { l i m } _ { | \phi | } ( \frac { d _ { 0 } } { \log ( d _ { 0 } \phi _ { 0 } ) } ) ) ] ^ { \theta | \alpha - 1 } - \frac { 2 d _ { 1 } ^ { \alpha } | ^ { 2 } \sqrt { \log ( \frac { d _ { 0 } } { \log ( d _ { 0 } ) } } ) } { d _ { 1 } ^ { \alpha } } ) } \\ & { \cdot ( [ 1 - \sqrt { \frac { \alpha } { \sqrt { \kappa } } \frac { 1 } { \sqrt { \kappa } \sqrt { \kappa } \frac { 1 } { \log ( d _ { 0 } ^ { \alpha } ) ^ { 2 } } } } + \mathcal { O } ( \frac { 1 } { \sqrt { \kappa } \frac { 1 } { \sqrt { \kappa } \phi _ { 0 } ^ { 2 } } \phi _ { 0 } ^ { 2 } } ) ] ^ { \mathbf { N } } - 2 ^ { - \kappa } ) ) } \\ & { = d _ { 1 } ^ { \alpha } ( 1 - ( 1 - \frac { 2 d _ { 1 } ^ { \alpha } / \sqrt { \kappa } \log ( \frac { d _ { 0 } } { \kappa } ) } { d _ { 1 } ^ { \alpha } \sqrt { \kappa } } + \mathcal { O } ( \frac { d _ { 0 } \tan \phi } { d _ { 0 } \sqrt { \kappa } } ( - \frac { d _ { 0 } } { 2 } \log ( \frac { d _ { 0 } } { \log ( d _ { 0 } \phi _ { 0 } ) } ) ) ) ) } \\ & \cdot ( 1 - \sqrt \frac { \kappa } { \delta } \frac { 1 } { \lambda _ { 0 } ^ { \alpha } \sqrt { \kappa } ( \frac { 1 } { \log ( d _ { 0 } ^ { \alpha } ) ^ { 2 } } } + \mathcal { O } ( \frac { 1 } { \kappa \phi _ { 0 } ^ { \alpha } \sqrt { \kappa } ^ { 2 } } + \frac { 1 } { d _ { 1 } ^ { \alpha } \sqrt { \kappa } ^ { 2 } \alpha ^ { 2 } } + \alpha _ { 1 } ^ { \alpha } \mathcal { A } _ { 0 } \exp \ \end{array} +$$ + +Thus, taking the log, and using $\log { ( 1 - x ) } = - x + O \left( x ^ { 2 } \right)$ , we obtain, forsin $\alpha = 1 / \left( d _ { 1 } ^ { * } d _ { 0 } N \right)$ + +log P (X ∈ Mα (W∗)) + +$$ +\begin{array} { r l } & { \log \mathfrak { s } : \mathrm { ~ } \ll \mathcal { N } : \mathrm { ~ } \ll \mathcal { N } : \mathrm { ~ } \ll \mathcal { N } } \\ & { \geq \log \left( 1 - \sqrt { \frac { 8 } { \pi } } \frac { 1 } { d _ { 0 } ^ { 1 / 2 } } - \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } + O \left( \frac { 1 } { d _ { 0 } ^ { 3 / 2 } } + \frac { d _ { 0 } ^ { 1 / 4 } } { d _ { 1 } ^ { * } N } + d _ { 1 } ^ { * } 2 ^ { - N } + d _ { 0 } \exp \left( - \frac { d _ { 0 } } { 2 } \log \left( \frac { d _ { 0 } } { \log d _ { 0 } } \right) \right) \right) \right) } \\ & { = - \sqrt { \frac { 8 } { \pi } } \frac { 1 } { d _ { 0 } ^ { 1 / 2 } } - \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } + O \left( \frac { 1 } { d _ { 0 } ^ { 3 / 2 } } + \frac { d _ { 0 } ^ { 1 / 4 } } { d _ { 1 } ^ { * } N } + d _ { 1 } ^ { * } 2 ^ { - N } + d _ { 0 } \exp \left( - \frac { d _ { 0 } } { 2 } \log \left( \frac { d _ { 0 } } { \log d _ { 0 } } \right) \right) \right) . } \end{array} +$$ + +Recall that $d _ { 1 } ^ { * } \triangleq 4 \lceil N / \left( 2 d _ { 0 } - 2 \right) \rceil \doteq N / d _ { 0 }$ . Taking the limit $N \to \infty$ , $d _ { 0 } \to \infty$ with $d _ { 1 } ^ { * } { \dot { \leq } } d _ { 0 } { \dot { \leq } } N$ , we have + +$$ +\mathbb { P } \left( \mathbf { X } \notin { \mathcal { M } } ^ { \alpha } \left( \mathbf { W } ^ { * } \right) \right) \dot { \leq } 1 - \exp \left( - \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } - \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } \right) \leq \sqrt { \frac { 8 } { \pi } } d _ { 0 } ^ { - 1 / 2 } + \frac { 2 d _ { 0 } ^ { 1 / 2 } \sqrt { \log d _ { 0 } } } { N } +$$ + +# Part III + +# Numerical Experiments - implementation details + +Code and trained models for CIFAR and ImageNet results is available here https://github. com/MNNsMinima/Paper. In MNIST, CIFAR and ImageNet we performed binary classification on between the original odd and even class numbers. In we performed this binary classification between digits $0 - 4$ and $5 - 9$ . Weights were initialized to be uniform with mean zero and variance $2 / d$ , where $d$ is fan-in (here the width of the previous neuron layer), as suggested in (He et al., 2015). In each epoch we randomly permuted the dataset and used the Adam (Kingma & Ba, 2014) optimization method (a variant of SGD) with $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 9 , \varepsilon = 1 0 ^ { - 8 }$ . Different learning rates and mini-batch sizes were selected for each dataset and architecture. In CIFAR10 and ImageNet we used a learning-rate of $\alpha = 1 0 ^ { - 3 }$ and a mini-batch size of 1024; also, ZCA whitening of the training samples was done to remove correlations between the input dimensions, allowing faster convergence. We define $L$ as the number of weight layers. For the random dataset we use a mini-batch size of $\left\lfloor \operatorname* { m i n } \left( N / 2 , d / 2 \right) \right\rfloor$ with learning rate $\alpha = 0 . 1$ and 0.05, for $L = 2$ and 3, respectively. In the random data parameter scans the training was done for no more than 4000 epochs – we stopped if $\mathbf { M C E } = 0$ was reached. \ No newline at end of file diff --git a/parse/train/Hkfmn5n6W/Hkfmn5n6W_content_list.json b/parse/train/Hkfmn5n6W/Hkfmn5n6W_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..67f40b988578db7cabe460a92037e97598dd1ce4 --- /dev/null +++ b/parse/train/Hkfmn5n6W/Hkfmn5n6W_content_list.json @@ -0,0 +1,6282 @@ +[ + { + "type": "text", + "text": "EXPONENTIALLY VANISHING SUB-OPTIMAL LOCAL MINIMA IN MULTILAYER NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 101, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska et al. (2015)) suggest that local minima with high error are exponentially rare in high dimensions. However, to prove low error guarantees for Multilayer Neural Networks (MNNs), previous works so far required either a heavily modified MNN model or training method, strong assumptions on the labels (e.g., “near” linear separability), or an unrealistically wide hidden layer with $\\Omega \\left( N \\right)$ units. ", + "bbox": [ + 232, + 265, + 764, + 348 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Results: We examine a MNN with one hidden layer of piecewise linear units, a single output, and a quadratic loss. We prove that, with high probability in the limit of $N \\to \\infty$ datapoints, the volume of differentiable regions of the empiric loss containing sub-optimal differentiable local minima is exponentially vanishing in comparison with the same volume of global minima, given standard normal input of dimension $d _ { 0 } = \\tilde { \\Omega } \\left( \\sqrt { N } \\right)$ , and a more realistic number of $d _ { 1 } = \\tilde { \\Omega } \\left( N / d _ { 0 } \\right)$ hidden units. We demonstrate our results numerically: for example, $0 \\%$ binary classification training error on CIFAR with only $N / d _ { 0 } \\approx 1 6$ hidden neurons. ", + "bbox": [ + 233, + 351, + 766, + 472 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 496, + 336, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Motivation. Multilayer Neural Networks (MNNs), trained with simple variants of stochastic gradient descent (SGD), have achieved state-of-the-art performances in many areas of machine learning (LeCun et al., 2015). However, theoretical explanations seem to lag far behind this empirical success (though many hardness results exist, e.g., (Síma, 2002; Shamir, 2016)). For example, as a common rule-of-the-thumb, a MNN should have at least as many parameters as training samples. However, it is unclear why such over-parameterized MNNs often exhibit remarkably small generalization error (i.e., difference between “training error” and “test error”), even without explicit regularization (Zhang et al., 2017a). ", + "bbox": [ + 173, + 526, + 825, + 638 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Moreover, it has long been a mystery why MNNs often achieve low training error (Dauphin et al., 2014). SGD is only guaranteed to converge to critical points in which the gradient of the expected loss is zero (Bottou, 1998), and, specifically, to local minima (Pemantle, 1990) (this is true also for regular gradient descent (Lee et al., 2016)). Since loss functions parameterized by MNN weights are non-convex, it is unclear why does SGD often work well – rather than converging to sub-optimal local minima with high training error, which are known to exist (Fukumizu & Amari, 2000; Swirszcz et al., 2016). Understanding this behavior is especially relevant in important cases where SGD does get stuck (He et al., 2016) – where training error may be a bottleneck in further improving performance. ", + "bbox": [ + 174, + 645, + 825, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ideally, we would like to quantify the probability to converge to a local minimum as a function of the error at this minimum, where the probability is taken with the respect to the randomness of the initialization of the weights, the data and SGD. Specifically, we would like to know, under which conditions this probability is very small if the error is high, as was observed empirically (e.g., (Dauphin et al., 2014; Goodfellow et al., 2015)). However, this seems to be a daunting task for realistic MNNs, since it requires a characterization of the sizes and distributions of the basins of attraction for all local minima. ", + "bbox": [ + 174, + 762, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Previous works (Dauphin et al., 2014; Choromanska et al., 2015), based on statistical physics analogies, suggested a simpler property of MNNs: that with high probability, local minima with high error diminish exponentially with the number of parameters. Though proving such a geometric property with realistic assumptions would not guarantee convergence to global minima, it appears to be a necessary first step in this direction (see discussion on section 6). It was therefore pointed out as an open problem at the Conference of Learning Theory (COLT) 2015. However, one has to be careful and use realistic MMN architectures, or this problem becomes “too easy”. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "For example, one can easily achieve zero training error (Nilsson, 1965; Baum, 1988) – if the MNN’s last hidden layer has more neurons than training samples. Such extremely wide MNNs are easy to optimize (Yu, 1992; Huang et al., 2006; Livni et al., 2014; Haeffele & Vidal, 2015; Shen, 2016; Nguyen & Hein, 2017). In this case, the hidden layer becomes linearly separable in classification tasks, with high probability over the random initialization of the weights. Thus, by training the last layer we get to a global minimum (zero training error). However, such extremely wide layers are not very useful, since they result in a huge number of weights, and serious overfitting issues. Also, training only the last layer seems to take little advantage of the inherently non-linear nature of MNNs. ", + "bbox": [ + 174, + 152, + 825, + 263 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Therefore, in this paper we are interested to understand the properties of local and global minima, but at a more practical number of parameters – and when at least two weight layers are trained. For example, Alexnet (Krizhevsky, 2014) is trained using about 1.2 million ImageNet examples, and has about 60 million parameters – 16 million of these in the two last weight layers. Suppose we now train the last two weight layers in such an over-parameterized MNN. When do the sub-optimal local minima become exponentially rare in comparison to the global minima? ", + "bbox": [ + 174, + 270, + 825, + 354 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Main contributions. We focus on MNNs with a single hidden layer and piecewise linear units, optimized using the Mean Square Error (MSE) in a supervised binary classification task (Section 2). We define $N$ as the number of training samples, $d _ { l }$ as the width of the $l$ -th activation layer, and $g \\left( x \\right) \\dot { < } h \\left( x \\right)$ as an asymptotic inequality in the leading order (formally: $\\begin{array} { r } { \\operatorname* { l i m } _ { x \\infty } \\frac { \\log g ( x ) } { \\log h ( x ) } < 1 ) } \\end{array}$ . We examine Differentiable Local Minima (DLMs) of the MSE: sub-optimal DLMs where at least a fraction of $\\epsilon > 0$ of the training samples are classified incorrectly, and global minima where all samples are classified correctly. ", + "bbox": [ + 173, + 369, + 825, + 473 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our main result, Theorem 10, states that, with high probability, the total volume of the differentiable regions of the MSE containing sub-optimal DLMs is exponentially vanishing in comparison to the same volume of global minima, given that: ", + "bbox": [ + 176, + 479, + 821, + 522 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Assumption 1. The datapoints (MNN inputs) are sampled from a standard normal distribution. ", + "bbox": [ + 178, + 525, + 799, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Assumption 2. $N \\to \\infty$ , $d _ { 0 } \\left( N \\right)$ and $d _ { 1 } \\left( N \\right)$ increase with $N$ , while $\\epsilon \\in ( 0 , 1 )$ is a constant1. ", + "bbox": [ + 178, + 544, + 790, + 560 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Assumption 3. The input dimension scales as $\\sqrt { N } \\dot { < } d _ { 0 } \\dot { \\le } N$ . ", + "bbox": [ + 174, + 564, + 576, + 580 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Assumption 4. The hidden layer width scales as ", + "bbox": [ + 176, + 584, + 496, + 598 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/309480aa6f3b645542d7d9fa450a1d45c09539fa5867092e47f66f400a8912c2.jpg", + "text": "$$\n\\frac { N \\log ^ { 4 } N } { d _ { 0 } } \\dot { < } d _ { 1 } \\dot { < } N .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 603, + 566, + 640 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Importantly, we use a standard, unmodified, MNN model, and make no assumptions on the target function. Moreover, as the number of parameters in the MNN is approximately $d _ { 0 } d _ { 1 }$ , we require only “asymptotically mild” over-parameterization: $d _ { 0 } d _ { 1 } { \\dot { > } } N \\log ^ { 4 } N$ from eq. (1.1). For example, if $d _ { 0 } \\propto N$ , we only require $d _ { 1 } \\dot { > } \\log ^ { 4 } N$ neurons. This improves over previously known results (Yu, 1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016; Nguyen & Hein, 2017) – which require an extremely wide hidden layer with $d _ { 1 } \\geq N$ neurons (and thus $N d _ { 0 }$ parameters) to remove sub-optimal local minima with high probability. ", + "bbox": [ + 173, + 643, + 825, + 744 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In section 5 we validate our results numerically. We show that indeed the training error becomes low when the number of parameters is close to $N$ . For example, with binary classification on CIFAR and ImageNet, with only 16 and 105 hidden neurons (about $N / d _ { 0 }$ ), respectively, we obtain less then $0 . 1 \\%$ training error. Additionally, we find that convergence to non-differentiable critical points does not appear to be very common. ", + "bbox": [ + 174, + 751, + 825, + 821 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Lastly, in section 6 we discuss our results might be extended, such as how to apply them to “mildly” non-differentiable critical points. ", + "bbox": [ + 173, + 828, + 825, + 856 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Plausibility of assumptions. Assumption 1 is common in this type of analysis (Andoni et al., 2014; Choromanska et al., 2015; Xie et al., 2016; Tian, 2017; Brutzkus & Globerson, 2017). At first it may appear rather unrealistic, especially since the inputs are correlated in typical datasets. However, this no-correlation part of the assumption may seem more justified if we recall that datasets are many times whitened before being used as inputs. Alternatively, if, as in our motivating question, we consider the input to the our simple MNN to be the output of the previous layers of a deep MNN with fixed random weights, this also tends to de-correlate inputs (Poole et al., 2016, Figure 3). The remaining part of assumption 1, that the distribution is normal, is indeed strong, but might be relaxed in the future, e.g. using central limit theorem type arguments. ", + "bbox": [ + 176, + 872, + 825, + 900 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In assumption 2 we use this asymptotic limit to simplify our proofs and final results. Multiplicative constants and finite (yet large) $N$ results can be found by inspection of the proofs. We assume a constant error $\\epsilon$ since typically the limit $\\epsilon 0$ is avoided to prevent overfitting. ", + "bbox": [ + 174, + 208, + 825, + 251 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In assumption 3, for simplicity we have $d _ { 0 } { \\dot { \\le } } N$ , since in the case $d _ { 0 } \\geq N$ the input is generically linearly separable, and sub-optimal local minima are not a problem (Gori & Tesi, 1992; Safran & Shamir, 2016). Additionally, we have $\\sqrt { N } \\dot { < } d _ { 0 }$ , which seems very reasonable, since for example, $d _ { 0 } / N \\approx 0 . 0 1 6 , 0 . 0$ 61 and 0.055 MNIST, CIFAR and ImageNet, respectively. ", + "bbox": [ + 174, + 256, + 825, + 314 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In assumption 4, for simplicity we have $d _ { 1 } \\dot { < } N$ , since, as mentioned earlier, if $d _ { 1 } \\geq N$ the hidden layer is linearly separable with high probability, which removes sub-optimal local minima. The other bound $N \\log ^ { 4 } N { \\dot { < } } d _ { 0 } d _ { 1 }$ is our main innovation – a large over-parameterization which is nevertheless asymptotically mild and improves previous results. ", + "bbox": [ + 174, + 321, + 825, + 380 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Previous work. So far, general low (training or test) error guarantees for MNNs could not be found – unless the underlying model (MNN) or learning method (SGD or its variants) have been significantly modified. For example, (Dauphin et al., 2014) made an analogy with high-dimensional random Gaussian functions, local minima with high error are exponentially rare in high dimensions; (Choromanska et al., 2015; Kawaguchi, 2016) replaced the units (activation functions) with independent random variables; (Pennington & Bahri, 2017) replaces the weights and error residuals with independent random variables; (Baldi, 1989; Saxe et al., 2014; Hardt & Ma, 2017; Lu & Kawaguchi, 2017; Zhou & Feng, 2017) used linear units; (Zhang et al., 2017b) used unconventional units (e.g., polynomials) and very large hidden layers $( d _ { 1 } = \\mathrm { p o l y } \\left( d _ { 0 } \\right)$ , typically $\\gg N$ ); (Brutzkus & Globerson, 2017; Du et al., 2017; Shalev-Shwartz et al., 2017) used a modified convnet model with less then $d _ { 0 }$ parameters (therefore, not a universal approximator (Cybenko, 1989; Hornik, 1991)); (Tian, 2017; Soltanolkotabi et al., 2017; Li & Yuan, 2017) assume the weights are initialized very close to those of the teacher generating the labels; and (Janzamin et al., 2015; Zhong et al., 2017) use a non-standard tensor method during training. Such approaches fall short of explaining the widespread success of standard MNN models and training practices. ", + "bbox": [ + 173, + 400, + 826, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Other works placed strong assumptions on the target functions. For example, to prove convergence of the training error near the global minimum, (Gori & Tesi, 1992) assumed linearly separable datasets, while (Safran & Shamir, 2016) assumed strong clustering of the targets (“near” linear-separability). Also, (Andoni et al., 2014) showed a $p$ -degree polynomial is learnable by a MNN, if the hidden layer is very large $( d _ { 1 } = \\Omega \\left( d _ { 0 } ^ { 6 p } \\right)$ , typically $\\gg N$ ) so learning the last weight layer is sufficient. However, these are not the typical regimes in which MNNs are required or used. In contrast, we make no assumption on the target function. Other closely related results (Soudry & Carmon, 2016; Xie et al., 2016) also used unrealistic assumptions, are discussed in section 6, in regards to the details of our main results. ", + "bbox": [ + 173, + 614, + 826, + 750 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Therefore, in contrast to previous works, the assumptions in this paper are applicable in some situations (e.g., Gaussian input) where a MNN trained using SGD might be used and be useful (e.g., have a lower test error then a linear classier). ", + "bbox": [ + 176, + 756, + 825, + 797 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 PRELIMINARIES AND NOTATION ", + "text_level": 1, + "bbox": [ + 176, + 824, + 470, + 839 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Model. We examine a Multilayer Neural Network (MNN) with a single hidden layer and a scalar output. The MNN is trained on a finite training set of $N$ datapoints (features) $\\textbf { X } \\triangleq$ $\\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] \\ \\in \\ \\mathbb { R } ^ { d _ { 0 } \\times N }$ with their target labels $\\mathbf { y } \\ \\triangleq \\ \\left[ y ^ { ( 1 ) } , \\ldots , y ^ { ( N ) } \\right] ^ { \\intercal } \\ \\in \\ \\left\\{ 0 , 1 \\right\\} ^ { N } -$ each datapoint-label pair $\\left( \\mathbf { x } ^ { ( n ) } , y ^ { ( n ) } \\right)$ is independently sampled from some joint distribution $\\mathbb { P } _ { X , Y }$ . We define $\\mathbf { W } = \\left[ \\mathbf { w } _ { 1 } , \\ldots , \\mathbf { w } _ { d _ { 1 } } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ and $\\mathbf { z } \\in \\mathbb { R } ^ { d _ { 1 } }$ as the first and second weight layers (bias terms are ignored for simplicity), respectively, and $f \\left( \\cdot \\right)$ as the common leaky rectifier linear unit (LReLU (Maas et al., 2013)) ", + "bbox": [ + 174, + 858, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 101, + 825, + 146 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7d14c788deba198f0ff60c39cd0408b4df4d1bb3ecf62504fd6993a55ad18adc.jpg", + "text": "$$\n\\begin{array} { r } { f \\left( u \\right) \\triangleq u a \\left( u \\right) \\mathrm { ~ w i t h ~ } a \\left( u \\right) \\triangleq \\left\\{ \\begin{array} { l l } { 1 } & { \\mathrm { , ~ i f ~ } , u > 0 } \\\\ { \\rho } & { \\mathrm { , ~ i f ~ } u < 0 } \\end{array} \\right. , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 154, + 656, + 189 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "for some $\\rho \\neq 1$ (so the MNN is non-linear) , where both functions $f$ and $a$ operate component-wise (e.g., for any matrix M: $\\left( f \\left( \\mathbf { M } \\right) \\right) _ { i j } = f \\left( M _ { i j } \\right) )$ . Thus, the output of the MNN on the entire dataset can be written as ", + "bbox": [ + 173, + 195, + 825, + 239 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/31d1af998fb84aae4abcba4d1bf0020e8eebacdbc76bd0f09db83f51d01ccacb.jpg", + "text": "$$\n{ f } \\left( { { \\mathbf { W } } { \\mathbf { X } } } \\right) ^ { \\top } { \\mathbf { z } } \\in \\mathbb { R } ^ { N } .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 236, + 566, + 256 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We use the mean square error (MSE) loss for optimization ", + "bbox": [ + 173, + 258, + 558, + 275 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e30e527d3a5961bf2ad196071dc9d2f7110c46f6ce76c43d60553cb325fef718.jpg", + "text": "$$\n\\operatorname { M S E } \\triangleq \\frac { 1 } { N } \\left. \\mathbf { e } \\right. ^ { 2 } \\mathrm { w i t h } \\mathbf { e } \\triangleq \\mathbf { y } - f \\left( \\mathbf { W } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 280, + 651, + 310 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\lVert \\cdot \\rVert$ is the standard euclidean norm. Also, we measure the empiric performance as the fraction of samples that are classified correctly using a decision threshold at $y = 0 . 5$ , and denote this as the mean classification error, or $\\mathbf { M C E } ^ { 2 }$ . Note that the variables $\\mathbf { e }$ , MSE, MCE and other related variables (e.g., their derivatives) all depend on $\\mathbf { W } , \\mathbf { z } , \\mathbf { X } , \\mathbf { y }$ and $\\rho$ , but we keep this dependency implicit, to avoid cumbersome notation. ", + "bbox": [ + 173, + 318, + 825, + 387 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Additional Notation. We define g (x) \\epsilon > 0$ . We would like to compare the angular volume (definition 3) corresponding to both types of DLMs. Thus, we make the following definitions. ", + "bbox": [ + 174, + 484, + 825, + 527 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 4. We define3 $\\mathcal { L } _ { \\epsilon } \\subset \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ as the union of differentiable regions containing sub-optimal DLMs with $\\mathrm { M C E } > \\epsilon$ , and $\\mathcal { G } \\subset \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ as the union of differentiable regions containing global minima with $\\mathrm { M C E } = 0$ . ", + "bbox": [ + 174, + 530, + 825, + 574 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 5. We define the constant $\\gamma _ { \\epsilon }$ as $\\begin{array} { r } { \\gamma _ { \\epsilon } \\triangleq 0 . 2 3 \\operatorname* { m a x } \\left[ \\operatorname* { l i m } _ { N \\to \\infty } \\left( d _ { 0 } \\left( N \\right) / N \\right) , \\epsilon \\right] ^ { 3 / 4 } } \\end{array}$ if $\\rho \\neq$ $\\{ 0 , 1 \\}$ , and $\\gamma _ { \\epsilon } \\triangleq 0 . 2 3 \\epsilon ^ { 3 / 4 }$ if $\\rho = 0$ . ", + "bbox": [ + 173, + 577, + 823, + 609 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we use assumptions 1-4 (stated in section 1) to bound the angular volume of the region $\\mathcal { L } _ { \\epsilon }$ encapsulating all sub-optimal DLMs, the region $\\mathcal { G }$ , encapsulating all global minima, and the ratio between the two. ", + "bbox": [ + 174, + 621, + 825, + 664 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Angular volume of sub-optimal DLMs. First, in appendix section 8 we prove the following upper bound in expectation ", + "bbox": [ + 171, + 678, + 823, + 708 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 6. Given assumptions 1-4, the expected angular volume of sub-optimal DLMs, with $\\mathrm { M C E } > \\epsilon > 0$ , is exponentially vanishing in $N$ as ", + "bbox": [ + 173, + 712, + 823, + 741 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f9fa76de6d9e0120508d973d5a1ef18a3e2f783e5e614e6039ab64a4e03c9b56.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 747, + 669, + 775 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "and, using Markov inequality, its immediate probabilistic corollary ", + "bbox": [ + 173, + 787, + 612, + 804 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Corollary 7. Given assumptions $I$ -4, for any $\\delta > 0$ (possibly a vanishing function of $N$ ), we have, with probability $1 - \\delta$ , that the angular volume of sub-optimal DLMs, with $\\mathrm { M C E } > \\epsilon > 0 ,$ , is exponentially vanishing in $N$ as ", + "bbox": [ + 173, + 808, + 826, + 851 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c6785e691f46a9da4a76dd01010670a877cd784df1e265a9f23c23c259459f39.jpg", + "text": "$$\n{ \\mathcal { V } } \\left( { \\mathcal { L } } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 348, + 857, + 647, + 887 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof idea of Theorem 6: we first show that in differentiable regions with $\\mathrm { M C E } > \\epsilon > 0$ , the condition in Lemma 2, $\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0$ , implies that $\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)$ must have a low rank. Then, we show that, when $\\mathbf { X } \\sim { \\mathcal { N } }$ and $\\mathbf { W } \\sim { \\mathcal { N } }$ , the matrix $\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)$ has a low rank with exponentially low probability. Combining both facts, we obtain the bound. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Existence of global minima. Next, to compare the volume of sub-optimal DLMs with that of global minima, in appendix section 9 we show first that, generically, global minima do exist (using a variant of the proof of (Baum, 1988, Theorem 1)): ", + "bbox": [ + 173, + 174, + 826, + 217 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 8. For any $\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }$ and $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ almost everywhere4 we find matrices $\\mathbf { W } ^ { * } \\in$ $\\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }$ and $\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } }$ , such that $\\mathbf { y } = f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }$ , where $d _ { 1 } ^ { * } \\triangleq 4 \\lceil N / ( 2 d _ { 0 } - 2 ) \\rceil$ and $\\forall i , n$ : $\\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0$ . Therefore, every MNN with $d _ { 1 } \\geq d _ { 1 } ^ { * }$ has a DLM which achieves zero error $\\mathbf { e } = 0$ . ", + "bbox": [ + 173, + 218, + 825, + 267 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Recently (Zhang et al., 2017a, Theorem 1) similarly proved that a 2-layer MNN with approximately $2 N$ parameters can achieve zero error. However, that proof required $N$ neurons (similarly to (Nilsson, 1965; Baum, 1988; Yu, 1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016)), while Theorem 8 here requires much less: approximately $d _ { 1 } ^ { * } \\approx 2 N / d _ { 0 }$ . Also, (Hardt & Ma, 2017, Theorem 3.2) showed a deep residual network with $N \\log N$ parameters can achieve zero error. In contrast, here we require just one hidden layer with $2 N$ parameters. ", + "bbox": [ + 173, + 275, + 825, + 361 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note the construction in Theorem 8 here achieves zero training error by overfitting to the data realization, so it is not expected to be a “good” solution in terms of generalization. To get good generalization, one needs to add additional assumptions on the data ( $\\mathbf { X }$ and y). Such a possible (common yet insufficient for MNNs) assumption is that the problem is “realizable”, i.e., there exist a small “solution MNN”, which achieves low error. For example, in the zero error case: ", + "bbox": [ + 173, + 366, + 825, + 436 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Assumption 5. (Optional) The labels are generated by some teacher $\\mathbf { y } = f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }$ with weight matrices $\\mathbf { W } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }$ and $\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } }$ independent of $\\mathbf { X }$ , for some $d _ { 1 } ^ { * } { \\dot { < } } N / d _ { 0 }$ . ", + "bbox": [ + 173, + 441, + 823, + 473 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This assumption is not required for our main result (Theorem 10) – it is merely helpful in improving the following lower bound on $\\mathcal { V } \\left( \\mathcal { G } \\right)$ . ", + "bbox": [ + 173, + 478, + 823, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Angular volume of global minima. We prove in appendix section 10: ", + "bbox": [ + 173, + 521, + 648, + 537 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 9. Given assumptions 1-3, we set $\\delta { \\doteq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } / N$ and $d _ { 1 } ^ { * } = 2 N / d _ { 0 }$ , or if assumption 5 holds, we set $d _ { 1 } ^ { * }$ as in this assumption. Then, with probability $1 - \\delta$ , the angular volume of global minima is lower bounded as, ", + "bbox": [ + 174, + 540, + 826, + 590 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/b1bb19380c59ff35fb86408a28e30cca957030b3e32dc7c2ba7e2b505f44b117.jpg", + "text": "$$\n\\mathscr { V } \\left( \\mathscr { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { > } \\exp \\left( - d _ { 1 } ^ { * } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 593, + 687, + 611 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof idea: First, we lower bound $\\mathcal { V } \\left( \\mathcal { G } \\right)$ with the angular volume of a single differentiable region of one global minimum $( \\mathbf { W } ^ { * } , \\mathbf { z } ^ { * } )$ – either from Theorem 8, or from assumption 5. Then we show that this angular volume is lower bounded when $\\mathbf { W } \\sim { \\mathcal { N } }$ , given a certain angular margin between the datapoints in $\\mathbf { X }$ and the rows of $\\mathbf { W } ^ { * }$ . We then calculate the probability of obtaining this margin when $\\mathbf { X } \\sim { \\mathcal { N } }$ . Combining both results, we obtain the final bound. ", + "bbox": [ + 173, + 621, + 825, + 691 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Main result: angular volume ratio. Finally, combining Theorems 6 and 9 it is straightforward to prove our main result in this paper, as we do in appendix section 11: ", + "bbox": [ + 171, + 705, + 823, + 734 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 10. Given assumptions 1-3, we set $\\delta \\doteq \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } / N$ . Then, with probability $1 - \\delta$ , the angular volume of sub-optimal DLMs, with $\\mathrm { M C E } > \\epsilon > 0$ , is exponentially vanishing in $N _ { : }$ , in comparison to the angular volume of global minima with $\\mathrm { M C E } = 0$ ", + "bbox": [ + 173, + 737, + 825, + 789 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/3af6fc16ee09e676267d0a9d7849e66e063aa542e2cd863187ff96b954d366e5.jpg", + "text": "$$\n\\frac { \\mathscr { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathscr { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N \\log N \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 791, + 717, + 825 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 NUMERICAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 839, + 436, + 857 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 10 implies that, with “asymptotically mild” over-parameterization (i.e. in which #parameters $\\mathrm { = } \\tilde { \\Omega } \\left( N \\right) )$ , differentiable regions in weight space containing sub-optimal DLMs (with high MCE) are exponentially small in comparison with the same regions for global minima. Since these results are asymptotic in $N \\infty$ , in this section we examine it numerically for a finite number of samples and parameters. We perform experiments on random data, MNIST, CIFAR10 and ImageNetILSVRC2012. In each experiment, we used ReLU activations $( \\rho = 0 )$ ), a binary classification target (we divided the original classes to two groups), MSE loss for optimization (eq. (2.3)), and MCE to determine classification error. Additional implementation details are given in appendix part III. ", + "bbox": [ + 174, + 869, + 825, + 901 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/1795ffcce4a4ec7991ec4c69666c8a8186811014eb6fb035a522afcfd4f8e18c.jpg", + "image_caption": [ + "Figure 5.1: Gaussian data: final training error (mean±std, 30 repetitions) in the overparameterized regime is low (right of the dashed black line). We trained MNNs with one and two hiddens layer (with widths equal to $d = d _ { 0 }$ ) on a synthetic random dataset in which $\\forall n = 1 , \\ldots , N$ , $\\mathbf { x } ^ { ( n ) }$ was drawn from a normal distribution $\\mathcal { N } ( 0 , 1 )$ , and $y ^ { ( n ) } = \\pm 1$ with probability 0.5. " + ], + "image_footnote": [], + "bbox": [ + 267, + 98, + 730, + 257 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/90f46a301115caa01b674399625ba334c01a765237cad7467e7e249166e66cfc.jpg", + "table_caption": [ + "Table 1: Binary classification of MNIST, CIFAR and ImageNet: 1-hidden layer achieves very low training error (MCE) with a few hidden neurons, so that #parameters $\\approx d _ { 0 } d _ { 1 } \\approx N$ . In ImageNet we downsampled the images to allow input whitening. " + ], + "table_footnote": [], + "table_body": "
MCEdod1N#parameters/N
MNIST0%784897·1040.999
CIFAR0%3072165·1040.983
ImageNet (downsampled to 64 × 64)0.1%12288105128·1041.008
", + "bbox": [ + 178, + 342, + 820, + 406 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 484, + 825, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "First, on the small synthetic Gaussian random data (matching our assumptions) we perform a scan on various networks and dataset sizes. With either one or two hidden layers (Figure 5.1) , the error goes to zero when the number of non-redundant parameters (approximately $d _ { 0 } d _ { 1 } ,$ ) is greater than the number of samples, as suggested by our asymptotic results. Second, on the non-syntehtic datasets, MNIST, CIFAR and ImageNet (In ImageNet we downsampled the images to size $6 4 \\times 6 4$ , to allow input whitening) we only perform a simulation with a single 1-hidden layer MNN for which #parameters $\\approx N$ , and again find (Table 1) that the final error is zero (for MNIST and CIFAR) or very low (ImageNet). ", + "bbox": [ + 174, + 574, + 825, + 686 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lastly, in Figure 5.2 we find that, on the Gaussian dataset, the inputs to the hidden neurons converge to a distinctly non-zero value. This indicates we converged to differentiable critical points – since nondifferentiable critical points must have zero neural inputs. Note that occasionally, during optimization, we could find some neural inputs with very low values near numerical precision level, so convergence to non-differentiable minima may be possible. However, as explained in the next section, as long as the number of neural inputs equal to zero are not too large, our bounds also hold for these minima. ", + "bbox": [ + 174, + 693, + 825, + 776 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 797, + 310, + 813 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this paper we examine Differentiable Local Minima (DLMs) of the empiric loss of Multilayer Neural Networks (MNNs) with one hidden layer, scalar output, and LReLU nonlinearities (section 2). We prove (Theorem 10) that with high probability the angular volume (definition 3) of sub-optimal DLMs is exponentially vanishing in comparison to the angular volume of global minima (definition 4), under assumptions 1-4. This results from an upper bound on sub-optimal DLMs (Theorem 6) and a lower bound on global minima (Theorem 9). ", + "bbox": [ + 174, + 828, + 825, + 912 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e75b9ef834dd7be439435fa2e13f28f31f7a9f670be1cdff33d9185308bb07d8.jpg", + "image_caption": [ + "Figure 5.2: Gaussian data: convergence of the MSE to differentiable critical points, as indicated by the convergence of the neural inputs to distinctly non-zero values. We trained MNNs with one hidden layer on the Gaussian dataset from Figure 5.1, with various widths $d = d _ { 0 } = d _ { 1 }$ and $N = \\lfloor d ^ { 2 } / 5 \\rfloor$ for 1000 epochs, then decreased the learning rate exponentially for another 1000 epochs. This was repeated 30 times. For all $d$ and repeats, we see that (left) the final absolute value of the minimal neural input $( i . e . , \\operatorname* { m i n } _ { i , n } \\left| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| )$ in the range of $1 0 ^ { - 3 } - 1 0 ^ { 0 }$ , which is much larger then $( \\mathrm { r } i g h t )$ the final MSE error for all $d$ and all repeats – in the range $1 0 ^ { - 3 1 } - 1 0 ^ { - 7 }$ . " + ], + "image_footnote": [], + "bbox": [ + 173, + 104, + 500, + 202 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Convergence of SGD to DLMs. These results suggest a mechanism through which low training error is obtained in such MNNs. However, they do not guarantee it. One issue is that sub-optimal DLMs may have exponentially large basins of attraction. We see two possible paths that might address this issue in future work, using additional assumptions on y. One approach is to show that, with high probability, no sub optimal DLM falls within the vanishingly small differentiable regions we bounded in Theorem 6. Another approach would be to bound the size of these basins of attraction, by showing that sufficiently large of number of differentiable regions near the DLM are also vanishingly small (other methods might also help here (Freeman & Bruna, 2016)). Another issue is that SGD might get stuck near differentiable saddle points, if their Hessian does not have strictly negative eigenvalues (i.e., the strict saddle property (Sun et al., 2015)). It should be straightforward to show that such points also have exponentially vanishing angular volume, similar to sub-optimal DLMs. Lastly, SGD might also converge to non-differentiable critical points, which we discuss next. ", + "bbox": [ + 514, + 103, + 825, + 435 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 436, + 622, + 450 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Non-differentiable critical points. The proof of Theorem 6 stems from a first order necessary condition (Lemma 2): $\\left( \\mathbf { A } \\circ \\mathbf { \\bar { X } } \\right) \\mathbf { e } = 0$ , which is true for any DLM. However, non-differentiable critical points, in which some neural inputs are exactly zero, may also exist (though, numerically, they don’t seem very common – see Figure 5.2). In this case, to derive a similar bound, we can replace the condition with $\\mathbf { P } \\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0$ , where $\\mathbf { P }$ is a projection matrix to the subspace orthogonal to the non-differentiable directions. As long as there are not too many zero neural inputs, we should be able to obtain similar results. For example, if only a constant ratio $r$ of the neural inputs are zero, we can simply choose $\\mathbf { P }$ to remove all rows of $( \\mathbf { A } \\circ \\mathbf { X } )$ corresponding to those neurons, and proceed with exactly the same proof as before, with $d _ { 1 }$ replaced with $( 1 - r ) d _ { 1 }$ . It remains a theoretical challenge to find reasonable assumptions under which the number of non-differentiable directions (i.e., zero neural inputs) does not become too large. ", + "bbox": [ + 174, + 469, + 825, + 622 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Related results. Two works have also derived related results using the $\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0$ condition from Lemma 2. In (Soudry & Carmon, 2016), it was noticed that an infinitesimal perturbation of $\\mathbf { A }$ makes the matrix $\\mathbf { A } \\circ \\mathbf { X }$ full rank with probability 1 (Allman et al., 2009, Lemma 13) – which entails that $\\mathbf { e } = 0$ at all DLMs. Though a simple and intuitive approach, such an infinitesimal perturbation is problematic: from continuity, it cannot change the original MSE at sub-optimal DLMs – unless the weights go to infinity, or the DLM becomes non-differentiable – which are both undesirable results. An extension of this analysis was also done to constrain e using the singular values of $\\mathbf { A } \\circ \\mathbf { X }$ (Xie et al., 2016), deriving bounds that are easier to combine with generalization bounds. Though a promising approach, the size of the sub-optimal regions (where the error is high) does not vanish exponentially in the derived bounds. More importantly, these bounds require assumptions on the activation kernel spectrum $\\gamma _ { m }$ , which do not appear to hold in practice (e.g., (Xie et al., 2016, Theorems 1,3) require $m \\gamma _ { m } \\gg 1$ to hold with high probability, while $m \\gamma _ { m } < \\bar { 1 0 } ^ { - 2 }$ in (Xie et al., 2016, Figure 1)). ", + "bbox": [ + 173, + 640, + 825, + 808 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Modifications and extensions. There are many relatively simple extensions of these results: the Gaussian assumption could be relaxed to other near-isotropic distributions (e.g., sparse-land model, (Elad, 2010, Section 9.2)) and other convex loss functions are possible instead of the quadratic loss. 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", + "bbox": [ + 171, + 98, + 828, + 853 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Supplementary information - Appendix ", + "text_level": 1, + "bbox": [ + 174, + 98, + 746, + 127 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The appendix is divided into three parts. In part I we prove all the main theorems mentioned in the paper. Some of these rely on other technical results, which we prove later in part II. Lastly, in part III we give additional numerical details and results. First, however, we define additional notation (some already defined in the main paper) and mention some known results, which we will use in our proofs. ", + "bbox": [ + 173, + 147, + 826, + 204 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "EXTENDED PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 224, + 408, + 241 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "• The indicator function ${ \\mathcal { T } } \\left( { \\mathcal { A } } \\right) \\triangleq { \\left\\{ \\begin{array} { l l } { 1 } & { , { \\mathrm { i f } } \\ A } \\\\ { 0 } & { , { \\mathrm { e l s e } } } \\end{array} \\right. }$ , for any event $\\mathcal { A }$ . \n• Kronecker’s delta $\\delta _ { i j } \\triangleq { \\mathcal { T } } ( i = j )$ . \n• The Matrix $\\mathbf { I } _ { d }$ as the identity matrix in $\\mathbb { R } ^ { d \\times d }$ , and $\\mathbf { I } _ { d \\times k }$ is the relevant $\\mathbb { R } ^ { d \\times k }$ upper left sub-matrix of the identity matrix. \n• $[ L ] \\triangleq \\{ 1 , 2 , \\dots , L \\}$ \nThe vector ${ \\bf m } _ { n }$ as the $n$ ’th column of a matrix $\\mathbf { M }$ , unless defined otherwise (then ${ \\bf m } _ { n }$ will be a row of $\\mathbf { M }$ ). \n• $\\mathbf M > 0$ implies that $\\forall i , j : M _ { i j } > 0$ . \n• $\\mathbf { M } _ { S }$ is the matrix composed of the columns of $\\mathbf { M }$ that are in the index set $S$ . \n• A property holds “M-almost everywhere” (a.e. for short), if the set of entries of $\\mathbf { M }$ for which the property does not hold has zero measure (Lebesgue). \n• $\\begin{array} { r } { \\| \\mathbf { v } \\| _ { 0 } = \\sum _ { i = 1 } ^ { d } \\mathcal { T } \\left( v _ { i } > 0 \\right) } \\end{array}$ is the $L _ { 0 }$ “norm” that counts the number of non-zero values in $\\mathbf { v } \\in \\mathbb { R } ^ { d }$ . \n• If $\\mathbf { x } \\sim \\mathcal { N } ( \\pmb { \\mu } , \\pmb { \\Sigma } )$ the $\\mathbf { x }$ is random Gaussian vector. \n• $\\begin{array} { r } { \\phi \\left( x \\right) \\triangleq \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp \\left( - \\frac { 1 } { 2 } x ^ { 2 } \\right) } \\end{array}$ as the univariate Gaussian probability density function. • $\\textstyle \\Phi \\left( x \\right) \\triangleq \\int _ { - \\infty } ^ { x } \\phi \\left( u \\right) d u$ as the Gaussian cumulative distribution function. \n• $B \\left( x , y \\right)$ as the beta function. ", + "bbox": [ + 214, + 251, + 826, + 599 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Lastly, we recall the well known Markov Inequality: ", + "bbox": [ + 173, + 609, + 519, + 625 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Fact 11. (Markov Inequality) For any random variable $X \\geq 0$ , we have $\\forall \\eta > 0$ ", + "bbox": [ + 174, + 628, + 710, + 645 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/ea33d643859be498473f709832490660f7942fb65f2fb63c65a2e5aecca6cdef.jpg", + "text": "$$\n\\mathbb { P } \\left( X \\geq \\eta \\right) \\leq \\frac { \\mathbb { E } X } { \\eta } .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 650, + 565, + 684 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Part I ", + "text_level": 1, + "bbox": [ + 173, + 710, + 236, + 729 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Proofs of the main results ", + "text_level": 1, + "bbox": [ + 173, + 741, + 545, + 768 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "7 FIRST ORDER CONDITION: PROOF OF LEMMA 2 ", + "bbox": [ + 174, + 790, + 601, + 809 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Lemma 12. (Lemma 2 restated) At all DLMs in ${ \\mathcal { D } } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)$ the residual error e is identical, and furthermore ", + "bbox": [ + 169, + 821, + 825, + 852 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/4a2db5c58e310b6fb8cb380966205883c913f5184f7326ec966df11ca7b27d1e.jpg", + "text": "$$\n\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 444, + 852, + 555, + 868 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Proof. Let $\\mathbf { W } = \\left[ \\mathbf { w } _ { 1 } , \\ldots , \\mathbf { w } _ { d _ { 1 } } \\right] ^ { \\top } \\in \\mathcal { D } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)$ , $\\mathbf { G } \\triangleq \\mathbf { A } \\circ \\mathbf { X } \\ \\in \\ \\mathbb { R } ^ { d _ { 0 } d _ { 1 } \\times N }$ , $\\tilde { \\mathbf { W } } = \\mathrm { d i a g } \\left( \\mathbf { z } \\right) \\mathbf { W } =$ $\\left[ \\tilde { \\mathbf { w } } _ { 1 } , \\ldots , \\tilde { \\mathbf { w } } _ { d _ { 1 } } \\right] ^ { \\top }$ and $\\tilde { \\mathbf { w } } \\triangleq \\mathrm { v e c } \\left( \\tilde { \\mathbf { W } } ^ { \\top } \\right) \\in \\mathbb { R } ^ { d _ { 0 } d _ { 1 } }$ , where $\\operatorname { d i a g } \\left( \\mathbf { v } \\right)$ is the diagonal matrix with $\\mathbf { v }$ in its ", + "bbox": [ + 174, + 885, + 825, + 928 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "diagonal, and vec (M) is vector obtained by stacking the columns of the matrix $\\mathbf { M }$ on top of one another. Then, we can re-write the MSE (eq. (2.3)) as ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/2030c62eb28304531a776e0e08483ac21c2f2e8d5d3f66eaf371a620dfdcbb7a.jpg", + "text": "$$\n\\mathrm { M S E } = { \\frac { 1 } { N } } \\left\\| \\mathbf { y } - \\mathbf { G } ^ { \\top } { \\tilde { \\mathbf { w } } } \\right\\| ^ { 2 } = { \\frac { 1 } { N } } \\left\\| \\mathbf { e } \\right\\| ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 137, + 629, + 167 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where $\\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } }$ is the output of the MNN. Now, if $( \\mathbf { W } , \\mathbf { z } )$ is a DLM of the MSE in eq. (2.3), then there is no infinitesimal perturbation of $( \\mathbf { W } , \\mathbf { z } )$ which reduces this MSE. ", + "bbox": [ + 174, + 174, + 825, + 203 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Next, for each row $i$ , we will show that $\\partial \\mathrm { M S E } / \\partial \\tilde { \\bf w } _ { i } = 0$ , since otherwise we can find an infinitesimal perturbation of $( \\mathbf { W } , \\mathbf { z } )$ which decreases the MSE, contradicting the assumption that $( \\mathbf { W } , \\mathbf { z } )$ is a local minimum. For each row $i$ , we divide into two cases: ", + "bbox": [ + 174, + 208, + 825, + 252 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "First, we consider the case $z _ { i } \\neq 0$ . In this case, any infinitesimal perturbation $\\mathbf { q } _ { i }$ in $\\tilde { \\mathbf { w } } _ { \\mathbf { i } }$ can be produced by an infinitesimal perturbation in $\\mathbf { w } _ { i }$ $\\mathbf { \\widetilde { w } } _ { i } + \\mathbf { q } _ { i } = ( \\mathbf { w } _ { i } + \\mathbf { q } _ { i } / z _ { i } ) z _ { i }$ . Therefore, unless the gradient $\\partial \\mathrm { { M S E } } / \\partial \\tilde { \\bf { w } } _ { i }$ is equal to zero, we can choose an infinitesimal perturbation $\\mathbf { q } _ { i }$ in the opposite direction to this gradient, which will decrease the MSE. ", + "bbox": [ + 173, + 257, + 825, + 314 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Second, we consider the case $z _ { i } ~ = ~ 0$ . In this case, the MSE is not affected by changes made exclusively to $\\mathbf { w } _ { i }$ . Therefore, all $\\mathbf { w } _ { i }$ derivatives of the MSE are equal to zero $( \\partial ^ { \\bar { k } } \\mathrm { M S E } / \\partial ^ { k } \\mathbf { w } _ { i }$ , to any order $k$ ) . Also, since we are at a differentiable local minimum, $\\partial \\mathrm { M S E } / \\partial z _ { i } = 0$ . Thus, using a Taylor expansion, if we perturb $\\left( \\mathbf { w } _ { i } , z _ { i } \\right)$ by $( \\hat { \\mathbf { w } } _ { i } , \\hat { z } _ { i } )$ then the MSE is perturbed by ", + "bbox": [ + 173, + 320, + 825, + 378 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/73b6d9dc4e8bace7bb108ccc01fd884a9a40029b8f814c56447e2b1bb5026b32.jpg", + "text": "$$\n\\hat { z } _ { i } \\hat { \\mathbf { w } } _ { i } ^ { \\top } \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\frac { \\partial } { \\partial z _ { i } } \\mathrm { M S E } + O ( \\hat { z } _ { i } ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 398, + 383, + 598, + 416 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Therefore, unless $\\partial ^ { 2 } \\mathrm { M S E } / \\left( \\partial \\mathbf { w } _ { i } \\partial z _ { i } \\right) = 0$ we can choose $\\hat { \\mathbf { w } } _ { i }$ and a sufficiently small $\\hat { z } _ { i }$ such that the MSE is decreased. Lastly, using the chain rule ", + "bbox": [ + 174, + 422, + 825, + 450 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/7d71967f3b5e10b684deb34f92ad19e3784bffb8435cfb91b4371b6167b7ec2d.jpg", + "text": "$$\n\\frac { \\partial } { \\partial z _ { i } } \\frac { \\partial } { \\partial \\mathbf { w } _ { i } } \\mathrm { M S E } = \\frac { \\partial } { \\partial z _ { i } } \\left[ z _ { i } \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\mathrm { M S E } \\right] = \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\mathrm { M S E } .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 455, + 669, + 491 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Thus, $\\partial \\mathrm { M S E } / \\partial \\tilde { \\bf w } _ { i } = 0$ . This implies that w˜ is also a $\\mathrm { D L M } ^ { 5 }$ of eq. (7.2), which entails ", + "bbox": [ + 171, + 503, + 740, + 521 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/f5f4459e41a48b3ff6b81ddac936f4dfd80546b6aa1626a5ef534239ab7d74e9.jpg", + "text": "$$\n0 = - \\frac { N } { 2 } \\frac { \\partial } { \\partial \\tilde { \\bf w } _ { i } } \\mathrm { M S E } = { \\bf G } \\left( { \\bf y } - { \\bf G } ^ { \\top } \\tilde { \\bf w } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 525, + 632, + 559 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Since $\\mathbf { G } = \\mathbf { A } \\circ \\mathbf { X }$ and $\\mathbf { e } = \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } }$ this proves eq. (7.1). Now, for any two solutions $\\tilde { \\mathbf { w } } _ { 1 }$ and $\\tilde { \\mathbf { w } } _ { 2 }$ of eq. (7.3), we have ", + "bbox": [ + 173, + 564, + 823, + 594 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/bc2aed7f2e7d864aae15f079bebe04835bc820a833fedd0f842f036cf2500a89.jpg", + "text": "$$\n0 = \\mathbf { G } \\left( \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } \\right) - \\mathbf { G } \\left( \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } \\right) = \\mathbf { G } \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 598, + 709, + 619 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Multiplying by $\\big ( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\big ) ^ { \\top }$ from the left we obtain ", + "bbox": [ + 173, + 626, + 514, + 643 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/2911e0b23e361adb36317d03e00496ae34fd047df2e7089b09dd5ad53e75c87a.jpg", + "text": "$$\n\\left\\| \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) \\right\\| ^ { 2 } = 0 \\Rightarrow \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 648, + 663, + 672 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Therefore, the MNN output and the residual error $\\mathbf { e }$ are equal for all DLMs in ${ \\mathcal { D } } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)$ . ", + "bbox": [ + 169, + 676, + 743, + 693 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "8 SUB-OPTIMAL DIFFERENTIABLE LOCAL MINIMA: PROOF OF THEOREM 6 AND ITS COROLLARY ", + "text_level": 1, + "bbox": [ + 174, + 712, + 810, + 746 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Theorem 13. (Theorem 6 restated) Given assumptions 1-4, the expected angular volume of suboptimal DLMs, with $\\mathrm { M C E } > \\epsilon > 0$ , is exponentially vanishing in $N$ as ", + "bbox": [ + 173, + 761, + 823, + 790 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c6a8ccb7d4fbd7a26598b9b10ca2720f2827e2e52de76f4a69940c947a3a78c8.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 795, + 668, + 821 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "To prove this theorem we upper bound the angular volume of $\\mathcal { L } _ { \\epsilon }$ (definition 4), i.e., differentiable regions in which there exist DLMs with $\\mathrm { M C E } > \\epsilon > 0$ . Our proof uses the first order necessary condition for DLMs from Lemma 2, $\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0$ , to find which configurations of $\\mathbf { A }$ allow for a high residual error e with $\\mathrm { M C E } > \\epsilon > 0$ . In these configurations $\\mathbf { A } \\circ \\mathbf { X }$ cannot have full rank, and therefore, as we show (Lemma 14 below), $\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)$ must have a low rank. However, $\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)$ has a low rank with exponentially low probability when $\\mathbf x \\sim \\mathcal N$ and $\\mathbf { W } \\sim { \\mathcal { N } }$ (Lemmas 15 and 16 below). Thus, we derive an upper bound on $\\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right)$ . ", + "bbox": [ + 174, + 857, + 825, + 901 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 826, + 160 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Before we begin, let us recall some notation: $[ L ] \\triangleq \\{ 1 , 2 , \\dots , L \\} , \\mathbf { M } > 0$ implies that $\\forall i , j : M _ { i j } >$ 0, $\\mathbf { M } _ { S }$ is the matrix composed of the columns of M that are in the index set $S$ , $\\left\\| \\mathbf { v } \\right\\| _ { 0 }$ as the $L _ { 0 }$ “norm” that counts the number of non-zero values in $\\mathbf { v }$ . First we consider the case $\\rho \\neq 0$ . Also, we denote $K _ { r } \\triangleq \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]$ . ", + "bbox": [ + 173, + 167, + 826, + 227 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "First we consider the case $\\rho \\neq 0$ . ", + "bbox": [ + 174, + 232, + 390, + 247 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "From definition 3 of the angular volume ", + "bbox": [ + 174, + 253, + 439, + 270 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/a5512ce672566576d61aae699006856a63d849992703e5b9a63acd0ed1a077bd.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathfrak { r } _ { \\mathbf { x } \\times \\mathcal { N } } \\mathcal { V } ( \\mathcal { L } , ( \\mathbf { X } , \\mathbf { y } ) ) = \\mathfrak { h } _ { ( \\mathbf { x } , \\mathbf { y } ) \\sim \\mathcal { N } _ { \\mathbf { x } \\times \\mathbf { y } } } \\times \\mathbf { w } \\times ( \\mathbf { \\overline { { B } } } \\otimes \\mathcal { L } , ( \\mathbf { x } , \\mathbf { x } , \\mathbf { y } ) ) } & { } \\\\ { \\overset { ( 1 ) } { \\leq } \\mathbf { R } _ { ( \\mathbf { x } , \\mathbf { y } ) \\sim \\mathcal { D } _ { \\mathbf { x } \\times \\mathbf { y } } } \\times \\mathbf { w } \\times \\times ( \\mathbf { a } \\in \\{ \\rho _ { 1 } \\} ) ^ { d _ { 1 } \\times \\mathbb { N } } , \\ \\mathbf { W } \\in \\mathcal { D } _ { \\mathbf { A } } ( \\mathbf { X } ) , \\mathbf { v } \\in \\mathbb { R } ^ { N } \\ ; \\ ( \\mathbf { A } \\circ \\mathbf { X } ) \\times \\mathbf { v } \\times \\mathbf { w } } & { } \\\\ { \\overset { ( 2 ) } { \\leq } \\mathbf { R } _ { \\mathbf { x } \\times \\mathcal { N } , \\mathbf { w } \\times \\mathbf { N } } \\times ( \\mathbf { A } \\in \\{ \\rho _ { 1 } \\} ) ^ { d _ { 1 } \\times \\mathbb { N } } , \\ \\mathbf { W } \\in \\mathcal { D } _ { \\mathbf { A } } ( \\mathbf { X } ) , \\mathbf { v } \\in \\mathbb { R } ^ { N } \\ ; \\ ( \\mathbf { A } \\circ \\mathbf { X } ) \\times \\mathbf { v } = 0 , } & { } \\\\ { \\overset { ( 4 ) } { \\leq } \\mathbf { R } _ { \\mathbf { x } \\times \\mathcal { N } , \\mathbf { w } \\times \\mathbf { N } } ( \\mathbf { S } \\subset [ \\mathbf { N } ] \\ \\cdot \\{ | \\mathbf { S } | \\ \\ \\geq \\operatorname* { m a x } [ N _ { c , \\mathrm { t r a l i k } } ( \\alpha ( \\langle \\mathbf { W } \\mathbf { X } ) ) d _ { 0 } + 1 ] \\} } & { } \\\\ - \\mathbf { B } _ { \\mathbf { x } \\times \\mathbf { y } } [ \\mathbb { W } _ { \\mathbf { w } \\times \\mathbf { x } } ( \\mathcal { S } \\subset [ \\mathbf { N } ] \\ \\cdot \\{ | \\mathbf { S } | \\ \\geq \\operatorname* { m a x } [ \\mathcal { N } _ { c , \\mathrm { r a l i k } } ( \\alpha ( \\mathbf { W } \\mathbf { X } ) ) d _ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 286, + 839, + 518 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 535, + 215, + 549 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. If we are at DLM a in $\\mathcal { D } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)$ , then Lemma 2 implies $\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0$ . Also, if $e ^ { ( n ) } = 0$ on some sample, we necessarily classify it correctly, and therefore $\\mathrm { M C E } \\leq \\| \\mathbf { e } \\| _ { 0 } / N$ . Since $\\mathrm { M C E } > \\epsilon$ in $\\mathcal { L } _ { \\epsilon }$ this implies that $N \\bar { \\epsilon } < \\| \\mathbf { e } \\| _ { 0 }$ . Thus, this inequality holds for ${ \\bf v } = { \\bf e }$ . ", + "bbox": [ + 212, + 566, + 825, + 611 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2. We apply assumption 1, that $\\mathbf { X } \\sim { \\mathcal { N } }$ . ", + "bbox": [ + 212, + 626, + 480, + 642 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. Assumption 4 implies $d _ { 0 } d _ { 1 } { \\dot { > } } N \\log ^ { 4 } N \\geq N$ . Thus, we can apply the following Lemma, proven in appendix section 12.1: ", + "bbox": [ + 212, + 659, + 825, + 689 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma 14. Let $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ , $\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }$ , $S \\subset [ N ]$ and $d _ { 0 } d _ { 1 } \\geq N$ . Then, simultaneously for every possible A and $S$ such that ", + "bbox": [ + 230, + 698, + 825, + 731 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/9b6ba8097ff70acf75b1e4aad86305a5e9c2b30cee691a85f1618c8d704fe10d.jpg", + "text": "$$\n\\left. S \\right. \\leq \\mathrm { r a n k } \\left( { \\bf A } _ { S } \\right) d _ { 0 } ,\n$$", + "text_format": "latex", + "bbox": [ + 455, + 750, + 598, + 767 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "we have that, $\\mathbf { X }$ -a.e., $\\mathbf { \\Delta } \\nexists \\mathbf { v } \\in \\mathbb { R } ^ { N }$ such that $v _ { n } \\neq 0 \\forall n \\in S$ and $( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0$ . ", + "bbox": [ + 230, + 784, + 746, + 801 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. Recall that $K _ { r } \\triangleq \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]$ . We use the union bound over all possible ranks $r \\geq 1$ we ignore the $r = 0$ case since for $\\rho \\neq 0$ (see eq. (2.1)) there is zero probability that rank $( a ( \\mathbf { W } \\mathbf { X } _ { S } ) ) = 0$ for some non-empty $S$ . For each rank $r \\geq 1$ , it is required that $| S | > K _ { r } = \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]$ , so $| S | = K _ { r }$ is a relaxation of the original condition, and thus its probability is not lower. ", + "bbox": [ + 212, + 820, + 825, + 892 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. We again use the union bound over all possible subsets $S$ of size $K _ { r }$ . ", + "bbox": [ + 209, + 909, + 684, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Thus, from eq. (8.1), we have ", + "bbox": [ + 173, + 103, + 370, + 118 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/714be8079b3dc2117ce2a44181decc63dd50b6d59b105c131bcb1d880015d3dd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\mathbf { x } \\times \\mathbf { x } ^ { \\beta } } \\nu ( \\xi , \\alpha ( \\mathbf { x } , \\mathbf { x } ) ) } \\\\ & = \\frac { \\nu ( \\mathbf { x } ) } { 2 } \\sum _ { i = 1 } ^ { N } \\sum _ { \\substack { s \\geq 0 , i \\geq 1 } } \\sum _ { \\substack { [ \\xi \\in \\mathcal { S } _ { \\mathbf { x } } \\times \\mathcal { N } _ { i } ] ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } \\times \\{ i \\} ) = ( \\xi \\cup \\mathbf { X } _ { \\mathbf { x } } ) ] = ( \\xi \\mid \\mathbf { X } _ { \\mathbf { x } } ) } } \\\\ & { \\leq \\frac { \\nu ( \\mathbf { x } ) } { 2 } \\sum _ { i = 1 } ^ { N } \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) } [ \\xi \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) ] } ( \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) \\{ ( \\xi \\mathbf { X } _ { \\mathbf { x } } ) \\} = \\{ \\alpha \\} } ] } \\\\ & \\overset { ( a ) } { \\geq } \\sum _ { i = 1 } ^ { N } ( \\frac { \\nu } { L } ) \\sum _ { \\substack { ( \\xi ) \\leq ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ) = ( \\xi ) } } \\\\ & \\overset { ( a ) } { \\leq } \\sum _ { i = 1 } ^ { N } ( \\frac { \\nu } { L } ) \\sum _ \\substack { ( \\xi ) \\leq ( \\xi ) ( \\xi ) \\leq ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) \\leq ( \\xi ) \\mathrm { h } } \\{ \\alpha \\} \\exp \\{ ( \\xi \\} \\sum _ \\mathbf { x } \\in \\{ \\xi \\} \\} \\xi ) ( \\xi ) ( \\xi ) ( \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 122, + 743, + 411 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "1. Since we take the expectation over $\\mathbf { X }$ , the location of $S$ does not affect the probability. Therefore, we can set without loss of generality $S = \\left[ K _ { r } \\right]$ . ", + "bbox": [ + 214, + 416, + 825, + 446 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2. Note that $r \\leq N / d _ { 0 } \\dot { < } \\operatorname* { m i n } \\left[ d _ { 0 } , d _ { 1 } \\right]$ from assumptions 3 and 4. Thus, with $k = K _ { r } \\ge d _ { 0 }$ , we apply the following Lemma, proven in appendix section 12.2: ", + "bbox": [ + 212, + 449, + 823, + 478 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma 15. Let $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }$ be a random matrix with independent and identically distributed columns, and $\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ an independent standard random Gaussian matrix. Then, in the limit m $\\mathrm { i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r$ , ", + "bbox": [ + 228, + 481, + 825, + 525 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1afe02259f7d479d0be27534469a0955224cee70cfd0bc061e9c2014f5cf4e70.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( a \\left( \\mathbf { W } \\mathbf { X } \\right) \\right) = r \\right) \\dot { \\leq } 2 ^ { k + r d _ { 0 } \\left( \\log d _ { 1 } + \\log k \\right) + r ^ { 2 } } \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 529, + 763, + 550 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "3. Note that $K _ { r } \\geq N \\epsilon \\dot { = } N > 2 d _ { 1 }$ , and $\\operatorname* { m i n } \\left[ K _ { r } , d _ { 0 } , d _ { 1 } \\right] \\dot { > } d _ { 0 } \\underline { { d _ { 1 } } } / K _ { r } \\dot { > } 1$ from assumptions 2 and 4. Thus, we apply the following Lemma (with $\\mathbf { C } = \\mathbf { X } ^ { \\top }$ , $\\mathbf { B } = \\mathbf { W } ^ { \\top }$ , $M = d _ { 0 }$ , $L = d _ { 1 }$ and $N = K _ { r } / 2$ ), proven in appendix section 12.3: ", + "bbox": [ + 215, + 556, + 825, + 601 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma 16. Let $\\mathbf { C } \\in \\mathbb { R } ^ { N \\times M }$ and $\\mathbf { B } \\in \\mathbb { R } ^ { M \\times L }$ be two independent standard random Gaussian matrices. Without loss of generality, assume $N \\geq L ,$ , and denote $\\alpha \\triangleq M L / N$ . Then, in the regime $M \\leq N$ and in the limit min $[ N , M , L ] \\dot { > } \\alpha \\dot { > } 1$ , we have ", + "bbox": [ + 228, + 602, + 825, + 648 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/44fede03f3bb993bbce2b480d94bb9545c17da4fe315581887002fc34564bf6d.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 406, + 652, + 647, + 679 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "4. We use $r d _ { 0 } \\leq N , \\left( \\begin{array} { c } { { N } } \\\\ { { K _ { r } } } \\end{array} \\right) \\leq 2 ^ { N } , K _ { r } \\leq N ,$ , and $d _ { 1 } \\dot { < } N$ (from assumption 4) and $r ^ { 2 } \\leq$ $N ^ { 2 } / d _ { 0 } ^ { 2 } { \\dot { < } } N$ (from assumption (3)) to simplify the combintaorial expressions. ", + "bbox": [ + 212, + 688, + 825, + 734 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "5. First, note that $r = 1$ is the maximal term in the sum, so we can neglect the other, exponentially smaller, terms. Second, from assumption 3 we have $d _ { 0 } { \\dot { \\le } } N$ , so ", + "bbox": [ + 209, + 737, + 826, + 767 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/80d1b85478f1bf71c4868633b88c85b366ae645e89dd57b5fd47fd444dc364e0.jpg", + "text": "$$\n\\operatorname* { l i m } _ { N \\to \\infty } 0 . 2 3 \\operatorname* { m a x } \\left[ \\epsilon , d _ { 0 } \\left( N \\right) / N \\right] ^ { 3 / 4 } = 0 . 2 3 \\operatorname* { m a x } \\left[ \\epsilon , \\operatorname* { l i m } _ { N \\to \\infty } d _ { 0 } \\left( N \\right) / N \\right] ^ { 3 / 4 } = \\gamma _ { \\epsilon } .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 770, + 785, + 801 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Third, from assumption 4 we have $N \\log ^ { 4 } N { \\dot { < } } d _ { 0 } d _ { 1 }$ , so the $2 ^ { N \\log N }$ term is negligible. ", + "bbox": [ + 232, + 806, + 795, + 824 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Thus, ", + "bbox": [ + 173, + 834, + 212, + 848 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/44540abf479f600cdb5f005c7773d0fd64ab85113d457f17797f804f8388a75e.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 845, + 669, + 873 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "which proves the Theorem for the case $\\rho \\neq 0$ . ", + "bbox": [ + 176, + 873, + 475, + 888 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Next, we consider the case $\\rho = 0$ . In this case, we need to change transition (4) in eq. (8.1), so the sum starts from $r = 0$ , since now we can have rank $( a ( \\mathbf { W } \\mathbf { X } _ { S } ) \\bar { ) } = 0$ . Following exactly the same ", + "bbox": [ + 174, + 895, + 823, + 925 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "logic (except the modification to the sum), we only need to modify transition (5)in eq. (8.2) – since now the maximal term in the sum is at $r = 0$ . This entails $\\gamma _ { \\epsilon } = 0 . 2 3 \\epsilon ^ { 3 / 4 }$ . ", + "bbox": [ + 171, + 103, + 825, + 135 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Corollary 17. (Corollary 7 restated) Given assumptions 1-4, for any $\\delta > 0$ (possibly a vanishing function of $N _ { . }$ ), we have, with probability $1 - \\delta$ , that the angular volume of sub-optimal DLMs, with $\\mathrm { M C E } > \\epsilon > 0$ , is exponentially vanishing in $N$ as ", + "bbox": [ + 173, + 157, + 826, + 202 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/1ec95b09718b5c62f1740c9605819e83bf645fd3ed4f09bdf5076451cd3778dd.jpg", + "text": "$$\n{ \\mathcal { V } } \\left( { \\mathcal { L } } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 348, + 204, + 648, + 234 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. Since $\\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\geq 0$ we can use Markov’s Theorem $( \\mathrm { F a c t } 1 1 ) \\forall \\eta > 0$ : ", + "bbox": [ + 173, + 247, + 710, + 263 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2dc91f38ea1fb3c23cbd1245a42972f4f8edc0e8870805621a60c5e4b1976bd0.jpg", + "text": "$$\n\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) < \\eta \\right) > 1 - \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\eta }\n$$", + "text_format": "latex", + "bbox": [ + 310, + 267, + 686, + 301 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "denoting $\\begin{array} { r } { \\eta = \\frac { 1 } { \\delta } \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\end{array}$ , and using Theorem (6) we prove the corollary. ", + "bbox": [ + 176, + 306, + 723, + 323 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/36d6b97111f482b5888329e9e8c4b873fb425980acee6b653550d879f9ba423e.jpg", + "text": "$$\n\\begin{array} { r l } & { 1 - \\delta < \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( \\displaystyle \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) < \\frac { 1 } { \\delta } \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\right) } \\\\ & { \\quad \\quad \\quad < \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 327, + 714, + 398 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where we note that replacing a regular inequality $<$ with inequality in the leading order $\\dot { \\leq }$ only removes constraints, and therefore increases the probability. □ ", + "bbox": [ + 171, + 402, + 825, + 433 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "9 CONSTRUCTION OF GLOBAL MINIMA: PROOF OF THEOREM 8: ", + "text_level": 1, + "bbox": [ + 173, + 452, + 720, + 468 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Recall the LReLU non-linearity ", + "bbox": [ + 174, + 482, + 385, + 497 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/3a9836c54a3588da2a9b52a87454af7cf7fc0020bf26f65f4a5504f2551c1d03.jpg", + "text": "$$\nf \\left( x \\right) \\triangleq { \\left\\{ \\begin{array} { l l } { \\rho x } & { , { \\mathrm { i f ~ } } x < 0 } \\\\ { x } & { , { \\mathrm { i f ~ } } x \\geq 0 } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 413, + 496, + 584, + 531 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "in eq. (2.1), where $\\rho \\neq 1$ ", + "bbox": [ + 173, + 534, + 343, + 549 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Theorem 18. (Theorem 8 restated) For any $\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }$ and $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ almost everywhere we find matrices $\\mathbf { W ^ { * } } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }$ and $\\mathbf { z } ^ { * } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } }$ , such that $\\mathbf { y } ~ = ~ f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }$ , where $d _ { 1 } ^ { * } \\ \\triangleq$ $4 \\left\\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\right\\rceil$ and $\\forall i , n : \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0$ . Therefore, every MNN with $d _ { 1 } \\ \\geq \\ d _ { 1 } ^ { * }$ has a DLM which achieves zero error $\\mathbf { e } = 0$ . ", + "bbox": [ + 173, + 550, + 825, + 613 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We prove the existence of a solution $( \\mathbf { W } ^ { * } , \\mathbf { z } ^ { * } )$ , by explicitly constructing it. This construction is a variant of (Baum, 1988, Theorem 1), except we use LReLU without bias and MSE – instead of threshold units with bias and MCE. First, we note that for any $\\epsilon _ { 1 } > \\epsilon _ { 2 } > 0$ , the following trapezoid function can be written as a scaled sum of four LReLU: ", + "bbox": [ + 174, + 622, + 825, + 679 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/a211f0f15b159948d9a8919c4ce08f0aa24f7a3aaa0f9c176ebfeede943d3075.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\tau \\left( x \\right) \\triangleq \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { , ~ i f ~ } | x | > \\epsilon _ { 1 } } \\\\ { 1 } & { \\mathrm { , ~ i f ~ } | x | \\leq \\epsilon _ { 2 } } \\\\ { \\frac { \\epsilon _ { 1 } - | x | } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } } & { \\mathrm { , ~ i f ~ } \\epsilon _ { 2 } < | x | \\leq \\epsilon _ { 1 } } \\\\ { = \\displaystyle \\frac { 1 } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } \\frac { 1 } { 1 - \\rho } \\left[ f \\left( x + \\epsilon _ { 1 } \\right) - f \\left( x + \\epsilon _ { 2 } \\right) - f \\left( x - \\epsilon _ { 2 } \\right) + f \\left( x - \\epsilon _ { 1 } \\right) \\right] . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 681, + 753, + 773 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Next, we examine the set of data points which are classified to 1: $S ^ { + } \\triangleq \\left\\{ n \\in [ N ] | y ^ { ( n ) } = 1 \\right\\}$ . Without loss of generality, assume $\\vert S ^ { + } \\vert \\le \\frac { N } { 2 }$ . We partition $S ^ { + }$ to ", + "bbox": [ + 168, + 776, + 825, + 809 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/298eb14ca98d1ab7bf9d82caf0c9900c1010dcd88684e1a7a497acf1788b749c.jpg", + "text": "$$\nK = \\left\\lceil { \\frac { | S ^ { + } | } { d _ { 0 } - 1 } } \\right\\rceil \\leq \\left\\lceil { \\frac { N } { 2 \\left( d _ { 0 } - 1 \\right) } } \\right\\rceil\n$$", + "text_format": "latex", + "bbox": [ + 387, + 814, + 609, + 849 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "subsets $\\left\\{ { \\cal { S } } _ { i } ^ { + } \\right\\} _ { i = 1 } ^ { K }$ , each with no more than $d _ { 0 } - 1$ samples. For almost any dataset we can find K $K$ i=1hyperplanes passing through the origin, with normals such that each hyperplane contains all $d _ { 0 } - 1$ points in subset ${ S } _ { i } ^ { + }$ , i.e., ", + "bbox": [ + 173, + 854, + 825, + 907 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/238355f0a586b842f9f44bbbc43072e2fb15c002b2e06ae364765f711adafc1e.jpg", + "text": "$$\n\\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { X } _ { S _ { i } ^ { + } } = 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 449, + 905, + 547, + 928 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "but no other point, so $\\forall n \\notin S _ { i } ^ { + } : \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0$ , ", + "bbox": [ + 171, + 102, + 488, + 119 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "If $\\epsilon _ { 1 } , \\epsilon _ { 2 }$ in eq. (9.1) are sufficiently small $( \\forall n \\notin S _ { i } ^ { + } : \\left| \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| > \\epsilon _ { 1 } )$ then we have ", + "bbox": [ + 174, + 125, + 725, + 143 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/39c0d71a88b161c651edff22966bbc63e28e949c7befc8a6f217f5ba7b69bcdd.jpg", + "text": "$$\n\\tau \\left( \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathrm { , ~ i f ~ } n \\in S _ { i } ^ { + } } \\\\ { 0 } & { \\mathrm { , ~ e l s e } } \\end{array} \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 150, + 614, + 185 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Then we have ", + "bbox": [ + 173, + 190, + 267, + 205 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/43dbb6a9c734ac8055f7747e130567295986012c8b93c54d8b8b5d133f50b499.jpg", + "text": "$$\n\\sum _ { i = 1 } ^ { K } \\tau \\left( \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) = \\left\\{ \\mathbf { 1 } \\quad , \\mathrm { i f } n \\in \\mathcal { S } ^ { + } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 372, + 200, + 624, + 243 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "which gives the correct classification on all the data points. Thus, from eq. (9.1), we can construct a MNN with ", + "bbox": [ + 173, + 244, + 823, + 272 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/4a1636f1b72d5f09df421358844bfa49ae0f29a037c105cc437a509de7b17342.jpg", + "text": "$$\nd _ { 1 } ^ { * } = 4 K\n$$", + "text_format": "latex", + "bbox": [ + 465, + 271, + 532, + 289 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "hidden neurons which achieves zero error. This is straightforward to do if we have a bias in each neuron. To construct this MNN even without bias, we first find a vector $\\hat { \\mathbf { w } } _ { i }$ such that ", + "bbox": [ + 173, + 290, + 823, + 320 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d6a39518b58bafbea6f9cab74cef8a5a57426c65877e0453d35ddd5fcc80fe23.jpg", + "text": "$$\n\\hat { \\mathbf { w } } _ { i } ^ { \\top } \\left[ { \\bf X } _ { S _ { i } ^ { + } } , \\tilde { \\mathbf { w } } _ { i } \\right] = \\left[ 1 , \\ldots , 1 , 1 , 0 \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 323, + 612, + 351 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Note that this is possible since $\\left[ { \\bf X } _ { S _ { i } ^ { + } } , \\tilde { { \\bf w } } _ { i } \\right]$ has full rank $\\mathbf { X }$ -a.e. (the matrix $\\mathbf { X } _ { S _ { i } ^ { + } } \\in \\mathbb { R } ^ { d _ { 0 } \\times d _ { 0 } - 1 }$ has, $\\mathbf { X }$ -a.e., one zero left eigenvector, which is $\\tilde { \\mathbf { w } } _ { i }$ , according to eq. (9.2)). Additionally, we can set ", + "bbox": [ + 173, + 356, + 833, + 393 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/e26911de7f5643e2fff69dd4a6c1e834106f6828abbecff66e2455b77b0eba5a.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\tilde { \\mathbf { w } } _ { i } \\| = \\| \\hat { \\mathbf { w } } _ { i } \\| ~ , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 447, + 398, + 552, + 416 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "since changing the scale of $\\mathbf { w } _ { i }$ would not affect the validity of eq. (9.2). Then, we denote ", + "bbox": [ + 173, + 421, + 756, + 436 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/fb72c4f1cd99152d3d4743f76989306f68dc90ad884ff9d6e9cc9703458a7844.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\mathbf w } _ { i } ^ { ( 1 ) } \\triangleq \\tilde { \\mathbf { w } } _ { i } + \\epsilon _ { 1 } \\hat { \\mathbf { w } } _ { i } ; { \\mathbf w } _ { i } ^ { ( 2 ) } \\triangleq \\tilde { \\mathbf { w } } _ { i } + \\epsilon _ { 2 } \\hat { \\mathbf { w } } _ { i } } \\\\ & { { \\mathbf w } _ { i } ^ { ( 3 ) } \\triangleq \\tilde { \\mathbf { w } } _ { i } - \\epsilon _ { 2 } \\hat { \\mathbf { w } } _ { i } ; { \\mathbf w } _ { i } ^ { ( 4 ) } \\triangleq \\tilde { \\mathbf { w } } _ { i } - \\epsilon _ { 1 } \\hat { \\mathbf { w } } _ { i } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 359, + 440, + 638, + 486 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Note, from eqs. (9.2) and (9.4) that this choice satisfies ", + "bbox": [ + 173, + 488, + 537, + 505 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/686a7cd3c7ec2e1e87d5c46ba3744f4040b00fbe0448a55af5107de2835d23d5.jpg", + "text": "$$\n\\forall n \\in S _ { i } ^ { + } : \\mathbf { w } _ { i } ^ { ( j ) \\top } \\mathbf { x } ^ { ( n ) } = \\left\\{ \\begin{array} { l l } { \\epsilon _ { 1 } } & { , \\mathrm { i f } j = 1 } \\\\ { \\epsilon _ { 2 } } & { , \\mathrm { i f } j = 2 } \\\\ { - \\epsilon _ { 2 } } & { , \\mathrm { i f } j = 3 } \\\\ { - \\epsilon _ { 1 } } & { , \\mathrm { i f } j = 4 } \\end{array} \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 511, + 651, + 579 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Also, to ensure that $\\forall n \\notin S _ { i } ^ { + }$ the sign of $\\mathbf { w } _ { i } ^ { ( j ) ^ { \\top } } \\mathbf { x } ^ { ( n ) }$ does not change for different $j$ , for some $\\beta , \\gamma < 1$ we define ", + "bbox": [ + 174, + 587, + 826, + 617 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/77d7088f44a181a7bf4a78e445507146df6d836e960f6a84758d3c5480dcd350.jpg", + "text": "$$\n\\epsilon _ { 1 } = \\beta \\frac { \\operatorname* { m i n } _ { n \\not \\in { \\cal S } _ { i } ^ { + } } \\left| \\tilde { \\mathbf { w } } _ { i } ^ { \\top } { \\mathbf { x } } ^ { ( n ) } \\right| } { \\operatorname* { m a x } _ { n \\not \\in { \\cal S } _ { i } ^ { + } } \\left| \\hat { \\mathbf { w } } _ { i } ^ { \\top } { \\mathbf { x } } ^ { ( n ) } \\right| } , \\epsilon _ { 2 } = \\gamma \\epsilon _ { 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 367, + 613, + 629, + 656 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where with probability $\\begin{array} { r } { 1 , \\operatorname* { m i n } _ { n \\not \\in \\mathcal { S } _ { i } ^ { + } } \\left| \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| > 0 } \\end{array}$ and $\\mathrm { m a x } _ { n \\notin { \\cal S } _ { i } ^ { + } } \\left| \\hat { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| > 0$ . Defining ", + "bbox": [ + 171, + 659, + 764, + 680 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/bbbe1c0f2f4bcb486181e4c90035f9f8508a185ede55327c0496643895955582.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { W } _ { i } \\triangleq \\left[ \\mathbf { w } _ { i } ^ { ( 1 ) } , \\mathbf { w } _ { i } ^ { ( 2 ) } , \\mathbf { w } _ { i } ^ { ( 3 ) } , \\mathbf { w } _ { i } ^ { ( 4 ) } \\right] ^ { \\top } \\in \\mathbb { R } ^ { 4 K \\times d _ { 0 } } } \\\\ & { \\mathbf { z } _ { i } \\triangleq \\left[ 1 , - 1 , - 1 , 1 \\right] ^ { \\top } \\in \\mathbb { R } ^ { 4 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 685, + 647, + 738 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "and combining all the above facts, we have ", + "bbox": [ + 173, + 741, + 457, + 756 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d755a9fdd395387d76ca4382021cbbc0e401fd48dec2b9018647aa0832a68d6a.jpg", + "text": "$$\n\\begin{array} { l } { f \\left( \\mathbf { W } _ { i } \\mathbf { x } ^ { ( n ) } \\right) ^ { \\top } \\mathbf { z } _ { i } } \\\\ { \\ = \\displaystyle \\frac { 1 } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } \\frac { 1 } { 1 - \\rho } \\left[ f \\left( \\mathbf { w } _ { i } ^ { ( 1 ) \\top } \\mathbf { x } ^ { ( n ) } \\right) - f \\left( \\mathbf { w } _ { i } ^ { ( 2 ) \\top } \\mathbf { x } ^ { ( n ) } \\right) - f \\left( \\mathbf { w } _ { i } ^ { ( 3 ) \\top } \\mathbf { x } ^ { ( n ) } \\right) + f \\left( \\mathbf { w } _ { i } ^ { ( 3 ) \\top } \\mathbf { x } ^ { ( n ) } \\right) \\right] } \\\\ { \\ = \\displaystyle \\frac { 1 } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } \\frac { 1 } { 1 - \\rho } \\left[ f \\left( \\widetilde { \\mathbf { w } } _ { i } ^ { ( 1 ) } \\mathbf { x } ^ { ( n ) } + \\epsilon _ { 1 } \\widehat { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) - f \\left( \\widetilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } + \\epsilon _ { 2 } \\widehat { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) \\right. } \\\\ { \\ - \\left. f \\left( \\widetilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } - \\epsilon _ { 2 } \\widehat { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) + f \\left( \\widetilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } - \\epsilon _ { 1 } \\widehat { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) \\right] } \\\\ { \\ = \\displaystyle \\frac { 1 } { \\left\\{ 0 \\right. \\ } , \\ \\mathrm { i f } n \\in \\mathcal { S } _ { i } ^ { + } } . \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 761, + 802, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Thus, for ", + "bbox": [ + 173, + 103, + 236, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8948c052d416b99ce37bf214ddf04bf73f55c18c95026c6f80adc6406d8f24c3.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle { \\bf W } ^ { * } = \\left[ { \\bf W } _ { 1 } ^ { \\top } , \\ldots , { \\bf W } _ { K } ^ { \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { 4 \\times d _ { 0 } } } } \\\\ { { \\displaystyle { \\bf z } ^ { * } = \\frac { 1 } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } \\frac { 1 } { 1 - \\rho } \\cdot \\left[ { \\bf z } _ { 1 } , \\ldots , { \\bf z } _ { K } \\right] \\in \\mathbb { R } ^ { 4 K } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 123, + 643, + 183 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "we obtain a MNN that implements ", + "bbox": [ + 173, + 186, + 401, + 202 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/cba5051c74129ca5ceccf2b4e282245cc350b1ce31f2a0e096f57efed8a0e0eb.jpg", + "text": "$$\nf \\left( \\mathbf { W } ^ { * } \\mathbf { x } ^ { ( n ) } \\right) ^ { \\top } \\mathbf { z } ^ { * } = { \\left\\{ \\begin{array} { l l } { 1 } & { , { \\mathrm { i f } } \\ n \\in S ^ { + } } \\\\ { 0 } & { , { \\mathrm { e l s e } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 370, + 209, + 625, + 246 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "and thus achieves zero error. Clearly, from this construction, if $\\mathbf { w } _ { i }$ is a row of $\\mathbf { W } ^ { * }$ , then $\\forall n \\in S _ { i } ^ { + } , \\forall i :$ $| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } | \\geq \\epsilon _ { 2 }$ , and with probability $1 \\forall n \\notin S _ { i } ^ { + } , \\forall i : \\left| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| > 0$ , so this construction does not touch any non-differentiable region of the MSE. \u0004 ", + "bbox": [ + 173, + 255, + 826, + 301 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "10 GLOBAL MINIMA: PROOF OF THEOREM 9 ", + "text_level": 1, + "bbox": [ + 174, + 320, + 566, + 338 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Theorem 19. (Theorem 9 restated). Given assumptions 1-3, we set $\\delta \\dot { = } \\sqrt { { \\textstyle \\frac { 8 } { \\pi } } } d _ { 0 } ^ { - 1 / 2 } \\ +$ $2 d _ { 0 } ^ { 1 / 2 } \\sqrt { { \\log d _ { 0 } } } / { N }$ and $d _ { 1 } ^ { * } = 2 N / d _ { 0 }$ , or if assumption $^ { 5 }$ holds, we set $d _ { 1 } ^ { * }$ as in this assumption. Then, with probability $1 - \\delta$ , the angular volume of global minima is lower bounded as, ", + "bbox": [ + 173, + 351, + 828, + 407 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d4feb8dffc8a9e1fcee5f934e9a750e147c09afd2ac75e09bede36b743b4ac2f.jpg", + "text": "$$\n\\mathscr { V } \\left( \\mathscr { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { > } \\exp \\left( - d _ { 1 } ^ { \\ast } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 412, + 687, + 431 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section we lower bound the angular volume of $\\mathcal { G }$ (definition 4), i.e., differentiable regions in which there exist DLMs with $\\mathrm { { M C E } = 0 }$ . We lower bound $\\mathcal { V } \\left( \\mathcal { G } \\right)$ using the angular volume corresponding to the differentiable region containing a single global minimum. ", + "bbox": [ + 173, + 445, + 826, + 489 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "From assumption 4, we have $d _ { 0 } d _ { 1 } \\dot { > } N$ , so we can apply Theorem 8 and say that the labels are generated using a $( \\mathbf { X } , \\mathbf { y } )$ -dependent MNN: $\\mathbf { y } ~ = ~ f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }$ with target weights $\\begin{array} { r l } { \\mathbf { W } ^ { * } } & { { } = } \\end{array}$ $\\left[ { \\bf w } _ { 1 } ^ { * \\top } , \\ldots , { \\bf w } _ { d _ { 1 } ^ { * } } ^ { * \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }$ and $\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } }$ . If, in addition, assumption 5 holds then we can assume $\\mathbf { W } ^ { * }$ and $\\mathbf { z } ^ { \\ast }$ are independent from $( \\mathbf { X } , \\mathbf { y } )$ . In both cases, the following differentiable region ", + "bbox": [ + 173, + 494, + 825, + 568 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/396df204c6dbeabe1d7d8b292ea954d4be1899341b3edd1520d420754b6f14f1.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } } | \\forall i \\le d _ { 1 } ^ { * } : \\ \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } \\right) = \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { * ^ { \\top } } \\mathbf { X } \\right) \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 575, + 743, + 595 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "also contains a differentiable global minimum (just set $\\mathbf { w } _ { i } = \\mathbf { w } _ { i } ^ { * }$ , $z _ { i } = z _ { i } ^ { * } \\forall i \\leq d _ { 1 } ^ { * }$ , and $z _ { i } = 0$ $\\forall i > d _ { 1 } ^ { * } ,$ ), and therefore $\\forall \\mathbf { X } , \\mathbf { y }$ and their corresponding $\\mathbf { W } ^ { * }$ , we have ", + "bbox": [ + 173, + 602, + 825, + 631 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/045a31095a5ab73d446c31cf5b5781057deb5585c87c32d7bbdd3bb458554b0c.jpg", + "text": "$$\n\\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\supset \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 419, + 637, + 578, + 656 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Also, we will make use of the following definition. ", + "bbox": [ + 173, + 664, + 508, + 679 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition 20. Let $\\mathbf { X }$ have an angular margin $\\alpha$ from $\\mathbf { W } ^ { * }$ if all datapoints (columns in $\\mathbf { X }$ ) are at an angle of at least $\\alpha$ from all the weight hyperplanes (rows of $\\mathbf { W } ^ { * }$ ) , i.e., $\\mathbf { X }$ is in the set ", + "bbox": [ + 173, + 683, + 825, + 713 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/ad940362f0ac2a63b6765a0667ffa0faebc3ddf014ee3fbb9c76cf460a0d6cf4.jpg", + "text": "$$\n\\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left. \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\forall i , n : \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\mathrm { s i n } \\alpha \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 289, + 718, + 709, + 762 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Using the definitions in eqs. (10.3) and (10.1), we prove the Theorem using the following three Lemmas. ", + "bbox": [ + 173, + 775, + 825, + 805 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "First, In appendix section 13.2 we prove ", + "bbox": [ + 174, + 810, + 439, + 825 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma 21. For any $\\alpha$ , $i f \\mathbf { W } ^ { * }$ is independent from W then, in the limit $N \\to \\infty$ , $\\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)$ with l $\\mathrm { { o g } s i n } \\alpha \\dot { > } d _ { 0 } ^ { - 1 } \\log d _ { 0 }$ ", + "bbox": [ + 174, + 829, + 823, + 861 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/cd09fc5a1737d4dfe7704c0adbb9fc4c50aa5b54c92f4fcbf4cdab6c572c2045.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { V } \\left( \\tilde { \\mathcal { G } } \\right) = \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\geq } \\exp \\left( d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 868, + 700, + 896 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Second, in appendix section 13.3 we prove ", + "bbox": [ + 174, + 909, + 455, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma 22. Let $\\mathbf { W } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }$ a fixed matrix independent of $\\mathbf { X }$ . Then, in the limit $N \\infty$ with $d _ { 1 } ^ { * } { \\dot { \\leq } } d _ { 0 } { \\dot { \\leq } } N$ , the probability of not having an angular margin $\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)$ (eq. (10.3)) is upper bounded by ", + "bbox": [ + 173, + 102, + 826, + 145 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d5457133b26b2af46099b687e0136ea8dd1b7806d8276b51705f7cd20c1e348f.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 }\n$$", + "text_format": "latex", + "bbox": [ + 388, + 143, + 607, + 179 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lastly, in appendix section 13.4 we prove ", + "bbox": [ + 176, + 189, + 447, + 204 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma 23. Let $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ be a standard random Gaussian matrix of datapoints. Then we can find, with probability $I$ , $( \\mathbf { X } , \\mathbf { y } )$ -dependent matrices $\\mathbf { W } ^ { * }$ and $\\mathbf { z } ^ { \\ast }$ as in Theorem 8 (where $d _ { 1 } ^ { * }$ , $4 \\ : \\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\rceil $ . Moreover, in the limit $N \\to \\infty$ , where $N / d _ { 0 } { \\dot { \\le } } d _ { 0 } { \\dot { \\le } } N$ , for any y, we can bound the probability of not having an angular margin (eq. (10.3)) with $\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)$ by ", + "bbox": [ + 173, + 207, + 826, + 268 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d6aea7d1ff018291e63c0f48c89869863a15ecab0467eb0d30553d0cbdfe3a62.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N }\n$$", + "text_format": "latex", + "bbox": [ + 333, + 273, + 665, + 310 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Recall that $\\forall \\mathbf { X } , \\mathbf { y }$ and their corresponding $\\mathbf { W } ^ { * }$ , we have $\\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\subset \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right)$ (eq. (10.2)). Thus, combining Lemmas 21 with $\\sin \\alpha = 1 / \\left( \\bar { d } _ { 1 } ^ { * } d _ { 0 } N \\right)$ together with either Lemma 22 or 23, we prove the first (left) inequality of Theorem 9: ", + "bbox": [ + 173, + 325, + 826, + 371 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ab32765f735b173f9ee86eda78742cb481fbd4dc0e053ef310f3000c814e25a9.jpg", + "text": "$$\n{ \\mathcal { V } } \\left( { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) { \\dot { \\geq } } \\exp \\left( - d _ { 1 } ^ { * } d _ { 0 } \\log N \\right)\n$$", + "text_format": "latex", + "bbox": [ + 382, + 387, + 614, + 406 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Next, if $d _ { 1 } ^ { * } = 2 N / d _ { 0 }$ or $d _ { 1 } ^ { * } { \\dot { < } } N / d _ { 0 }$ (is assumption 5 holds), we obtain the second (right) inequality ", + "bbox": [ + 173, + 409, + 816, + 425 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/c55f578000c4433de41349c9e548a1e7dd839da142cdf75226dce81b424c4488.jpg", + "text": "$$\n\\exp \\left( - d _ { 1 } ^ { \\ast } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 431, + 633, + 450 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "11 VOLUME RATIO OF GLOBAL AND LOCAL MINIMA: PROOF OF THEOREM 10 ", + "bbox": [ + 171, + 492, + 825, + 508 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem 24. (Theorem 10 restated) Given assumptions 1-3, we set $\\delta \\doteq \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } +$ $2 d _ { 0 } ^ { 1 / 2 } \\sqrt { { \\log d _ { 0 } } } / { N }$ . Then, with probability $1 - \\delta$ , the angular volume of sub-optimal DLMs, with $\\mathrm { M C E } > \\epsilon > 0 ,$ , is exponentially vanishing in $N ,$ , in comparison to the angular volume of global minima with $\\mathrm { M C E } = 0$ ", + "bbox": [ + 173, + 521, + 828, + 590 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/36040bc717ea29143e129efa4ee3e4f4f0e6c4936b233a4011a7aaa839ef899d.jpg", + "text": "$$\n\\frac { \\mathscr { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathscr { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N \\log N \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 595, + 718, + 631 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To prove this theorem we first calculate the expectation of the angular volume ratio given the $\\mathbf { X }$ -event that the bound in Theorem 9 holds (given assumptions 1-3), i.e., $\\mathcal { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\geq } \\dot { \\exp } \\left( - 2 N \\log N \\right)$ . Denoting this even $ { \\varepsilon } ^ { 6 }$ as $\\mathcal { M }$ , we find: ", + "bbox": [ + 174, + 643, + 826, + 685 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/b31cb03a028f2e5b199a72dbc8b7525e5a526aa82c1330973c2eca3888b63a92.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } | { \\mathcal { M } } \\right] \\overset { ( 1 ) } { \\leq } \\frac { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\big | { \\mathcal { M } } \\right] } { \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 2 ) } { \\leq } } \\\\ & { \\frac { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\right] } { \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( { \\mathcal { M } } \\right) \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 3 ) } { \\leq } \\frac { \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } { \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( { \\mathcal { M } } \\right) \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 4 ) } { \\leq } } \\\\ & { \\quad \\quad \\frac { \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } { \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 5 ) } { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 694, + 720, + 821 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 827, + 217, + 840 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "1. We apply Theorem 9. \n2. We use the following fact ", + "bbox": [ + 210, + 852, + 405, + 887 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Fact 25. For any variable $X \\geq 0$ and event $\\mathcal { A }$ (where $\\bar { A }$ is its complement) ", + "bbox": [ + 232, + 102, + 730, + 119 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/883661c6a334e5fdb44fe9e8595b6615d7d8d470455e1c5c6e3ffcaf20278de4.jpg", + "text": "$$\n\\mathbb { E } \\left[ X \\right] = \\mathbb { E } \\left[ X | { \\cal { A } } \\right] \\mathbb { P } \\left( { \\cal { A } } \\right) + \\mathbb { E } \\left[ X | { \\bar { \\cal { A } } } \\right] ( 1 - \\mathbb { P } \\left( { \\cal { A } } \\right) ) \\geq \\mathbb { E } \\left[ X | { \\cal { A } } \\right] \\mathbb { P } \\left( { \\cal { A } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 307, + 123, + 746, + 143 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "3. We apply Theorem 6. ", + "bbox": [ + 212, + 151, + 375, + 166 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "4. We apply Theorem 9. ", + "bbox": [ + 212, + 170, + 375, + 185 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "5. We use assumption 4, which implies $\\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\dot { > } 2 N \\log N .$ ", + "bbox": [ + 209, + 189, + 674, + 208 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For simplicity, in the reminder of the proof we denote ", + "bbox": [ + 171, + 218, + 529, + 233 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/1d0887721e78acce5607b525a4b567e015b3451e8e8c92eca4dc5c567f1ac83c.jpg", + "text": "$$\nR \\left( \\mathbf { X } \\right) \\triangleq \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } .\n$$", + "text_format": "latex", + "bbox": [ + 415, + 239, + 581, + 273 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "From Markov inequality (Fact 11), since $R \\left( { \\bf X } \\right) \\geq 0$ , we have $\\forall \\eta \\left( N \\right) > 0$ : ", + "bbox": [ + 174, + 280, + 665, + 296 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/c6168072c13db79989e917fe2037d2d71918323a17814e36ba59a0ea35f295a7.jpg", + "text": "$$\n\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\geq \\eta \\left( N \\right) \\vert \\mathcal { M } \\right] \\leq \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] } { \\eta \\left( N \\right) }\n$$", + "text_format": "latex", + "bbox": [ + 326, + 301, + 669, + 337 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "On the other hand, from fact 25, we have ", + "bbox": [ + 174, + 342, + 444, + 357 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/3c64c08f9d4128005c15ac8f4a4b72111bbd0dcc4e4c8905d594204996080c5c.jpg", + "text": "$$\n\\begin{array} { r } { 1 - \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\vert \\mathcal { M } \\right] \\ge 1 - \\frac { \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] } { \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 362, + 718, + 396 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Combining Eqs. (11.2)-(11.3) we obtain ", + "bbox": [ + 173, + 401, + 441, + 417 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/6935d7f495a50a7bfe5f0c399d742e00a6aeed8fd5a66fee9db324f082b631d5.jpg", + "text": "$$\n\\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] } { \\eta \\left( N \\right) } \\geq 1 - \\frac { \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] } { \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) } ,\n$$", + "text_format": "latex", + "bbox": [ + 321, + 422, + 674, + 458 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "and so ", + "bbox": [ + 173, + 464, + 220, + 478 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/1df218dc6a30435f3ba3020e939e8a2a2e39a69d85a35bad38ef7191c288080f.jpg", + "text": "$$\n\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) - \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\middle | \\mathcal { M } \\right] } { \\eta \\left( N \\right) } \\leq \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 482, + 751, + 516 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We choose ", + "bbox": [ + 173, + 522, + 246, + 536 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/c4037cf0ca606d8d7e5abeb3654a18534a91fdee6ef24880989ea7b15aa10d41.jpg", + "text": "$$\n\\eta \\left( { \\cal N } \\right) = N \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( { \\cal M } \\right) \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] \\doteq \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 258, + 541, + 736, + 568 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "so that ", + "bbox": [ + 173, + 574, + 220, + 588 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/19c15e4885625997a3766e3b177b1110c78b2058312390172cc9e462e99f0eba.jpg", + "text": "$$\n\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) - \\frac { 1 } { N } \\le \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\dot { \\le } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 582, + 722, + 613 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Then, from Theorem 9 we have ", + "bbox": [ + 173, + 614, + 382, + 630 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/b691e9b99b15197ab7d70e68bfc6d003ef9ae79903069ffb8279980d1c9005a6.jpg", + "text": "$$\n1 - \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 635, + 656, + 671 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "so we obtain the first (left) inequality in the Theorem (10) ", + "bbox": [ + 173, + 676, + 553, + 693 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/05bd8a33be4b4cd8eb1097bc374d96fd669c2339592f3e9b49bcee9b2fdd168f.jpg", + "text": "$$\n\\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } \\dot { \\geq 1 } - \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 205, + 708, + 790, + 746 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lastly, we note that assumption 4 implies $\\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\dot { > } N \\log N$ , which proves the second (right) inequality of the theorem. ", + "bbox": [ + 173, + 750, + 826, + 781 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Part II ", + "text_level": 1, + "bbox": [ + 174, + 98, + 246, + 119 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proofs of technical results ", + "text_level": 1, + "bbox": [ + 174, + 128, + 549, + 157 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this part we prove the technical results used in part I. ", + "bbox": [ + 173, + 178, + 537, + 194 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "12 UPPER BOUNDING THE ANGULAR VOLUME OF SUB-OPTIMAL DIFFERENTIABLE LOCAL MINIMA: PROOFS OF LEMMAS USED IN SECTION 8 ", + "bbox": [ + 176, + 213, + 826, + 263 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "12.1 PROOF OF LEMMA 14 ", + "text_level": 1, + "bbox": [ + 176, + 279, + 375, + 294 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this section we will prove Lemma 14 in subsection 12.3.3. Recall the following definition Definition 26. Let ", + "bbox": [ + 171, + 304, + 779, + 338 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/331ff2f543779ae800d15425c08fef97cf3f06a106c32ad743dcd3f82ed2c481.jpg", + "text": "$$\n{ \\bf A } = [ { \\pmb a } _ { 1 } , \\ldots , { \\pmb a } _ { N } ] \\ ; \\ { \\bf X } = [ { \\pmb x } _ { 1 } , \\ldots , { \\pmb x } _ { N } ] ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 344, + 632, + 362 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ and $\\mathbf { A } \\in \\mathbb { R } ^ { d _ { 1 } \\times N }$ . The Khatari-Rao product between the two matrices is defined as ", + "bbox": [ + 166, + 367, + 825, + 397 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/6b03bf222e0a853578cc407ae3fdaef4f1c40d92cfe079d8042e9c75e965e61c.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbf { A } \\circ \\mathbf { X } } & { \\triangleq } & { \\left[ \\pmb { a } _ { 1 } \\otimes \\mathbf { x } _ { 1 } , \\pmb { a } _ { 2 } \\otimes \\mathbf { x } _ { 2 } , . . . \\pmb { a } _ { N } \\otimes \\mathbf { x } _ { N } \\right] } \\\\ & { = } & { \\left( \\begin{array} { c c c } { a _ { 1 1 } \\mathbf { x } _ { 1 } } & { a _ { 1 2 } \\mathbf { x } _ { 2 } } & { . . . } \\\\ { a _ { 2 1 } \\mathbf { x } _ { 1 } } & { a _ { 2 2 } \\mathbf { x } _ { 2 } } & { \\ddots } \\\\ { \\vdots } & { \\ddots } & { \\ddots } \\end{array} \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 400, + 651, + 486 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma 27. (Lemma 14 restated) Let $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ , $\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }$ , $S \\subset [ N ]$ and $d _ { 0 } d _ { 1 } \\geq N$ . Then, simultaneously for every possible A and $S$ such that ", + "bbox": [ + 169, + 491, + 826, + 522 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/1e38cae94396787c732a3790bc0b443713c44fea948bca616575253ccb971463.jpg", + "text": "$$\n\\left. S \\right. \\leq \\mathrm { r a n k } \\left( { \\bf A } _ { S } \\right) d _ { 0 } ,\n$$", + "text_format": "latex", + "bbox": [ + 426, + 527, + 568, + 545 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "we have that, X-a.e., $\\mathbf { \\Delta } \\nexists \\mathbf { v } \\in \\mathbb { R } ^ { N }$ such that $v _ { n } \\neq 0 \\forall n \\in S$ and $( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0$ ", + "bbox": [ + 173, + 549, + 686, + 565 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. We examine specific $\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }$ and $S \\subset [ N ]$ , and such that $| S | \\le d _ { S } d _ { 0 }$ , where we defined $d _ { S } \\triangleq \\operatorname { r a n k } \\left( \\mathbf { A } _ { S } \\right)$ . We assume that $d _ { S } \\geq 1$ , since otherwise the proof is trivial. Also, we assume by contradiction that $\\exists \\mathbf { v } \\in \\mathbb { R } ^ { N }$ such that $v _ { i } \\neq 0 \\forall i \\in S$ and $( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0$ . Without loss of generality, assume that $S = \\{ 1 , 2 , . . . , | S | \\}$ and that $\\mathbf { } a _ { 1 } , \\mathbf { } a _ { 2 } , . . . , \\mathbf { } a _ { d _ { S } }$ are linearly independent. Then ", + "bbox": [ + 173, + 580, + 823, + 643 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/43fc4ea2aafe35eaa4064faf92f08d348fce5085b2a8785c524cefd329e42d22.jpg", + "text": "$$\n( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = \\sum _ { n = 1 } ^ { | S | } v _ { n } a _ { k , n } \\mathbf { x } _ { n } = 0\n$$", + "text_format": "latex", + "bbox": [ + 392, + 647, + 609, + 693 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "for every $1 \\leq k \\leq d _ { 1 }$ . From the definition of $S$ we must have $v _ { n } \\neq 0$ for every $1 \\leq n \\leq | S |$ . Since $\\mathbf { } a _ { 1 } , \\mathbf { } a _ { 2 } , . . . , \\mathbf { } a _ { d _ { S } }$ are linearly independent, the rows of $\\mathbf { A } _ { d _ { S } } = [ \\pmb { a } _ { 1 } , \\pmb { a } _ { 2 } , . . . , \\pmb { a } _ { d _ { S } } ]$ span a $d _ { S }$ -dimensional space. Therefore, it is possible to find a matrix $\\mathbf { R }$ such that $\\mathbf { R } \\mathbf { A } _ { d _ { S } } = [ \\mathbf { I } _ { d _ { S } \\times d _ { S } } , 0 _ { d _ { S } \\times ( d _ { 1 } - d _ { S } ) } ] ^ { \\top }$ , where $0 _ { i \\times j }$ is the all zeros matrix with $i$ columns and $j$ rows. Consider now $\\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }$ , i.e., the matrix composed of the columns of $\\mathbf { A } \\circ \\mathbf { X }$ in $S$ . Applying $\\mathbf { R } ^ { \\prime } = \\mathbf { R } \\otimes \\mathbf { I } _ { d _ { 0 } }$ to $\\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }$ , turns (12.2) into $d _ { 0 } d _ { S }$ equations in the variables $v _ { 1 } , . . . , v _ { | S | }$ , of the form ", + "bbox": [ + 173, + 696, + 825, + 784 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/4a246a95c624c7b3cb13895d8563162746724ae3124e0d90a5fb0058d85abc2d.jpg", + "text": "$$\nv _ { k } \\mathbf { x } _ { k } + \\sum _ { n = d _ { S } + 1 } ^ { | S | } v _ { n } { \\tilde { a } } _ { k , n } \\mathbf { x } _ { n } = 0\n$$", + "text_format": "latex", + "bbox": [ + 397, + 789, + 601, + 835 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "for every $1 \\leq k \\leq d _ { S }$ . We prove by induction that for every $1 \\leq d \\leq d _ { S }$ , the first $d _ { 0 } d$ equations are linearly independent, except for a set of matrices $\\mathbf { X }$ of measure 0. This will immediately imply $| S | > d _ { S } \\dot { d } _ { 0 }$ , or else eq. 12.2 cannot be true for $\\mathbf { v } \\neq 0$ . which will contradict our assumption, as required. The induction can be viewed as carrying out Gaussian elimination of the system of equations described by (12.3), where in each elimination step we characterize the set of matrices $\\mathbf { X }$ that for which that step is impossible, and show it has measure 0. ", + "bbox": [ + 173, + 839, + 826, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For $d = 1$ , the first $d _ { 0 }$ equations read $\\begin{array} { r } { v _ { 1 } \\mathbf { x } _ { 1 } + \\sum _ { n = d _ { S } + 1 } ^ { | S | } v _ { n } \\tilde { a } _ { 1 , n } \\mathbf { x } _ { n } = 0 } \\end{array}$ , and since $v _ { 1 } \\neq 0$ , we must have $\\mathbf { x } _ { 1 } \\in \\mathrm { S p a n } \\left\\{ \\widetilde { a } _ { 1 , d _ { S } + 1 } \\mathbf { x } _ { d _ { S } + 1 } , . . . , \\widetilde { a } _ { 1 , | S | } \\mathbf { x } _ { | S | } \\right\\}$ . However, except for a set of measure 0 with respect to $\\mathbf { x } _ { 1 }$ (a linear subspace of $\\mathbb { R } ^ { d _ { 0 } }$ with dimension less than $d _ { 0 }$ ), this can only happen if dim Span $\\left\\{ \\tilde { a } _ { 1 , d _ { S } + 1 } \\mathbf { x } _ { d _ { S } + 1 } , . . . , \\tilde { a } _ { 1 , | S | } \\mathbf { x } _ { | S | } \\right\\} = d _ { 0 }$ , which implies $| S | \\ge d _ { S } - 1 + d _ { 0 } > d _ { 0 }$ and also that the first $d _ { 0 }$ rows are linearly independent (since there are $d _ { 0 }$ independent columns). ", + "bbox": [ + 173, + 101, + 825, + 181 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For a general $d$ , we begin by performing Gaussian elimination on the first $( d - 1 ) d _ { 0 }$ equations, resulting in a new set of $r _ { d }$ equations, such that every new equation contains one variable that appears in no other new equation. Let $C$ be the set of the indices (equivalently, columns) of these variables $r _ { d }$ variables. From (12.3) it is clear none of the variables $v _ { d } , v _ { d + 1 } , . . . , v _ { d _ { S } }$ appear in the first $( d - 1 ) d _ { 0 }$ equations, and therefore $C \\subseteq S ^ { \\prime } = S \\setminus \\{ d , d + 1 , . . . , d _ { S } \\}$ . By our induction assumptions, except for a set of measure 0, the first $( d - 1 ) d _ { 0 }$ are independent, which means that $| C | = r _ { d } = ( d - 1 ) d _ { 0 }$ . We now extend the Gaussian elimination to the next $d _ { 0 }$ equations, and eliminate all the variables in $C$ from them. The result of the elimination can be written down as, ", + "bbox": [ + 173, + 185, + 825, + 299 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/9bf56616592389d15d94c6c71d1c7f2b4aaee3be0edce76efacda4cc129fcfaf.jpg", + "text": "$$\nv _ { d } \\mathbf { x } _ { d } + \\sum _ { n \\in S ^ { \\prime } \\backslash C } v _ { n } \\left( \\tilde { a } _ { d , n } \\mathbf { I } _ { d _ { 0 } } - \\mathbf { Y } \\right) \\mathbf { x } _ { n } = { 0 } ,\n$$", + "text_format": "latex", + "bbox": [ + 357, + 304, + 638, + 340 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $\\mathbf { Y }$ is a square matrix of size $d _ { 0 }$ whose coefficients depend only on $\\{ \\widetilde { a } _ { k , n } \\} _ { n \\in C , d > k \\geq 1 }$ and on $\\{ \\mathbf { x } _ { n } \\} _ { n \\in C }$ , and in particular do not depend on $\\mathbf { x } _ { d }$ and $\\left\\{ \\mathbf { x } _ { n } \\right\\} _ { n \\in S ^ { \\prime } \\backslash C }$ . ", + "bbox": [ + 173, + 347, + 825, + 378 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Now set $\\tilde { \\mathbf { x } } _ { n } \\ = \\ ( \\tilde { a } _ { d , n } \\mathbf { I } _ { d _ { 0 } } - \\mathbf { Y } ) \\mathbf { x } _ { n }$ for $n \\in S ^ { \\prime } \\setminus C$ . As in the case of $d \\ : = \\ : 1$ , since $v _ { d } \\neq 0$ , $\\mathbf { x } _ { d } \\in \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C }$ . Therefore, for all values of $\\mathbf { x } _ { d } \\in \\mathbb { R } ^ { d _ { 0 } }$ but a set of measure zero (linear subspace of with dimension less than $d _ { 0 }$ ), we must have di $\\begin{array} { r } { \\operatorname* { m } \\mathrm { { S p a n } } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 } } \\end{array}$ . From the independence of $\\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C }$ on $\\mathbf { x } _ { d }$ it follows that $\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }$ holds a.e. with respect to the Lebesgue measure over $\\mathbf { x }$ . ", + "bbox": [ + 173, + 383, + 825, + 460 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Whenever $\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }$ we must have $| S ^ { \\prime } C | \\geq d _ { 0 }$ and therefore ", + "bbox": [ + 173, + 465, + 710, + 483 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/32a1ca6c153e5d5a9bf82a5fbbbd89574b4e1ccf22e1d6307718430cd1deda53.jpg", + "text": "$$\n| S | > | S ^ { \\prime } | = | C | + | S ^ { \\prime } \\setminus C | \\geq \\left( d - 1 \\right) d _ { 0 } + d _ { 0 } = d _ { 0 } d .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 489, + 683, + 507 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Moreover, $\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }$ implies that the $d _ { 0 }$ equations $\\begin{array} { r } { v _ { d } \\mathbf { x } _ { d } + \\sum _ { n \\in S ^ { \\prime } \\backslash C } v _ { n } \\tilde { \\mathbf { x } } _ { n } = 0 } \\end{array}$ are independent. Thus, we may perform another step of Gaussian elimination on these $d _ { 0 }$ equations, forming $d _ { 0 }$ new equations each with a variable unique to it. Denoting by $C ^ { \\prime }$ the set of these $d _ { 0 }$ variables, it is seen from (12.4) that $C ^ { \\prime } \\subseteq ( S ^ { \\prime } \\cup \\{ d \\} ) \\setminus C$ and in particular $C ^ { \\prime }$ is disjoint from $C$ . Thus, considering the first $( d - 1 ) d _ { 0 }$ equations together with the new $d _ { 0 }$ equations, we see that there is a set $C \\cup C ^ { \\prime }$ of $d _ { 0 } d$ variables, such that each variable in $C \\cup C ^ { \\prime }$ appears only in one of the $d _ { 0 } d$ equations, and each of the $d _ { 0 } d$ contains only a single variable in $C \\cup C ^ { \\prime }$ . This means that the first $d _ { 0 } d$ must be linearly independent for all values of $\\mathbf { X }$ except for a set of Lebesgue measure zero, completing the induction. ", + "bbox": [ + 173, + 512, + 826, + 642 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Thus, we have proven, that for some $\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }$ and $S \\subset [ N ]$ such that $| S | \\leq \\mathrm { r a n k } \\left( \\mathbf { A } _ { S } \\right) d _ { 0 }$ the event ", + "bbox": [ + 173, + 647, + 823, + 679 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/6fa3c6c034467d0cd6c07a2f70417a14cdc7c19c6ebeed96882c6294c4ee8a4f.jpg", + "text": "$$\n\\mathcal E \\left( \\mathbf A , S \\right) = \\left\\{ \\mathbf X \\in \\mathbb R ^ { d _ { 0 } \\times N } | \\exists \\mathbf v \\in \\mathbb R ^ { N } : \\left( \\mathbf A \\circ \\mathbf X \\right) \\mathbf v = 0 \\mathrm { a n d } v _ { n } \\neq 0 , \\forall n \\in S \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 683, + 751, + 710 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "has zero measure. The event discussed in the theorem is a union of these events: ", + "bbox": [ + 169, + 715, + 700, + 731 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/da3a7d613ef7ea45872d466c0bdc770b822f73ec22a66fe81e1ed814d59c888c.jpg", + "text": "$$\n\\mathcal { E } _ { 0 } \\triangleq \\bigcup _ { \\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N } } \\left[ \\bigcup _ { S \\subset [ N ] : | S | \\leq \\operatorname { r a n k } ( \\mathbf { A } _ { S } ) d _ { 0 } } \\mathcal { E } \\left( \\mathbf { A } , S \\right) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 734, + 674, + 785 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "and it also has zero measure, since it is a finite union of zero measure events. ", + "bbox": [ + 173, + 790, + 676, + 806 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For completeness we note the following corollary, which is not necessary for a our main results. ", + "bbox": [ + 174, + 820, + 797, + 837 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Corollary 28. If $N \\leq d _ { 1 } d _ { 0 }$ , then rank $( \\mathbf { A } \\circ \\mathbf { X } ) = N$ , X-a.e., if and only $i f ,$ ", + "bbox": [ + 174, + 839, + 676, + 856 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/9e7d11cb95875c223dbb401985bafeb64514cfd3273a6db22551d7b6cbef4ca5.jpg", + "text": "$$\n\\forall S \\subseteq [ N ] : | S | \\leq \\mathrm { r a n k } \\left( \\mathbf { A } _ { S } \\right) d _ { 0 } .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 862, + 609, + 880 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. We define $d _ { S } \\triangleq \\operatorname { r a n k } \\left( \\mathbf { A } _ { S } \\right)$ and $\\mathbf { A } \\circ \\mathbf { X }$ . The necessity of the condition $| S | \\le d _ { 0 } d _ { S }$ holds for every $\\mathbf { X }$ , as can be seen from the following counting argument. Since the matrix $\\mathbf { A } _ { S }$ has rank $d _ { S }$ , ", + "bbox": [ + 173, + 893, + 826, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "there exists an invertible row transformation matrix $\\mathbf { R }$ , such that $\\mathbf { R A } _ { S }$ has only $d _ { S }$ non-zero rows. Consider now $\\mathbf { G } _ { S } = \\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }$ , i.e., the matrix composed of the columns of $\\mathbf { G }$ in $S$ . We have ", + "bbox": [ + 169, + 102, + 825, + 133 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/924a84543fe5a6ef518a3f828b345ceae59266a440f2c1d625f44d3b28908187.jpg", + "text": "$$\n\\mathbf { G } _ { S } ^ { \\prime } = ( \\mathbf { R } \\mathbf { A } _ { S } ) \\circ \\mathbf { X } _ { S } = \\mathbf { R } ^ { \\prime } \\left( \\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S } \\right) = \\mathbf { R } ^ { \\prime } \\mathbf { G } _ { S } ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 147, + 663, + 165 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\mathbf { R } ^ { \\prime } = \\mathbf { R } \\otimes \\mathbf { I } _ { d _ { 0 } }$ is also an invertible row transformation matrix, which applies $\\mathbf { R }$ separately on the $d _ { 0 }$ sub-matrices of $\\mathbf { G } _ { S }$ that are constructed by taking one every $d _ { 0 }$ rows. Since $\\mathbf { G } _ { S } ^ { \\prime }$ has at most $d _ { 0 } d _ { S }$ non-zero rows, the rank of $\\mathbf { G } _ { S }$ cannot exceed $d _ { 0 } d _ { S }$ . Therefore, if $| S | > d _ { 0 } d _ { S }$ , $\\mathbf { G } _ { S }$ will not have full column rank, and hence neither will $\\mathbf { G }$ . To demonstrate sufficiency a.e., suppose $\\mathbf { G }$ does not have full column rank. Let $S$ be the minimum set of columns of $\\mathbf { G }$ which are linearly dependent. Since the columns of $\\mathbf { G } _ { S }$ are assumed linearly dependent there exists $\\mathbf { v } \\in \\mathbb { R } ^ { | S | }$ such $\\| \\mathbf { v } \\| _ { 0 } = | S |$ and $\\mathbf { G } _ { S } \\mathbf { v } = 0$ . Using Lemma 28 we complete the proof. □ ", + "bbox": [ + 173, + 179, + 826, + 280 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "12.2 PROOF OF LEMMA 15 ", + "text_level": 1, + "bbox": [ + 176, + 305, + 375, + 320 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In this section we will prove Lemma 15 in subsection 12.3.3. This proof relies on two rather basic results, which we first prove in subsections 12.2.1 and 12.2.2. ", + "bbox": [ + 173, + 335, + 823, + 364 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "12.2.1 NUMBER OF DICHOTOMIES INDUCED BY A HYPERPLANE ", + "text_level": 1, + "bbox": [ + 176, + 388, + 629, + 405 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Fact 29. A hyperplane ${ \\mathbf w } \\in d _ { 0 }$ can separate a given set of points $\\mathbf { X } = \\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ into several different dichotomies, i.e., different results for $\\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } \\right)$ . The number of dichotomies is upper bounded as follows: ", + "bbox": [ + 174, + 415, + 826, + 463 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/276611149f850fb17a545865714e1e2821ff4642a53a796c1de7f19e11080b61.jpg", + "text": "$$\n\\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { N } } \\mathscr { T } \\left( \\exists \\mathbf { w } : \\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } \\right) = \\mathbf { h } ^ { \\top } \\right) \\leq 2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( \\begin{array} { l } { N - 1 } \\\\ { k } \\end{array} \\right) \\leq 2 N ^ { d _ { 0 } } .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 477, + 738, + 525 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. See (Cover, 1965, Theorem 1) for a proof of the left inequality as equality (the Schläfli Theorem) in the case that the columns of $\\mathbf { X }$ are in “general position” (which holds $\\mathbf { X }$ -a.e, see definition in (Cover, 1965)) . If $\\mathbf { X }$ is not in general position then this result becomes an upper bound, since some dichotomies might not be possible. ", + "bbox": [ + 173, + 565, + 826, + 623 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Next, we prove the right inequality. For $N = 1$ and $N = 2$ the inequality trivially holds. For $N \\geq 3$ , we have ", + "bbox": [ + 171, + 628, + 826, + 659 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/2a0c240d5f62a5eac2e8d4314d8e8be31fcb5e7c6b97181552e8d6ceb4619032.jpg", + "text": "$$\n2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( \\begin{array} { c } { { N - 1 } } \\\\ { { k } } \\end{array} \\right) \\stackrel { \\left( 1 \\right) } { \\leq } 2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( N - 1 \\right) ^ { k } \\stackrel { \\left( 2 \\right) } { \\leq } 2 \\frac { \\left( N - 1 \\right) ^ { d _ { 0 } } - 1 } { N - 2 } \\leq 2 N ^ { d _ { 0 } } .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 666, + 732, + 712 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where in (1) we used the bound $\\left( \\begin{array} { c } { { N } } \\\\ { { k } } \\end{array} \\right) \\le N ^ { k }$ , in (2) we used the sum of a geometric series. ", + "bbox": [ + 173, + 727, + 794, + 761 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "12.2.2 A BASIC PROBABILISTIC BOUND ", + "text_level": 1, + "bbox": [ + 178, + 801, + 462, + 816 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma 30. Let $\\mathbf { H } = \\left[ \\mathbf { h } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { h } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\in \\{ - 1 , 1 \\} ^ { d _ { 1 } \\times k }$ be a deterministic binary matrix, ${ \\textbf { W } } =$ $\\left[ \\mathbf { w } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { w } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ be an independent standard random Gaussian matrix, and $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }$ be a random matrix with independent and identically distributed columns. ", + "bbox": [ + 173, + 825, + 825, + 880 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/eca24cd528c0ee1bac7acc6fa2f0a83813f1e2a9ff3bcd5b862de87b6cea02d1.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\operatorname { s i g n } \\left( { \\mathbf { W } } { \\mathbf { X } } \\right) = { \\mathbf { H } } \\right) \\leq \\left( \\begin{array} { c } { k } \\\\ { \\lfloor k / 2 \\rfloor } \\end{array} \\right) \\mathbb { P } \\left( { \\mathbf { W } } { \\mathbf { X } } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 303, + 893, + 692, + 929 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. By direct calculation ", + "bbox": [ + 173, + 103, + 361, + 118 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/d1fdd0cb76eea7fd5e7e1cd6e223b98a7ba70b9e98cb7a40eec7b5bb8f8a5526.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\widehat { \\mathbf { \\xi } } \\displaystyle \\geq ( \\mathrm { s i g n } ( \\mathbf { W X } ) = \\mathbf { H } ) = } & { \\displaystyle \\mathbb { E } [ \\mathbb { P } ( \\mathrm { s i g n } ( \\mathbf { W X } ) = \\mathbf { H } \\vert \\mathbf { X } ) ] \\stackrel { ( 1 ) } { = } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d } \\mathbb { P } ( \\mathrm { s i g n } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } ) = \\mathbf { h } _ { i } ^ { \\top } \\vert \\mathbf { X } ) ] } \\\\ & { \\stackrel { ( 2 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { \\delta ( \\mathbf { h } _ { i } ) } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 3 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S _ { i } } > 0 \\vert \\mathbf { X } ) ] } \\\\ & { \\stackrel { ( 4 ) } { = } \\mathbb { E } [ \\mathbb { P } ( \\mathbf { W X } _ { S _ { s } } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 5 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\sum _ { S \\geq [ R _ { 1 } ] : S \\mid = \\lfloor k / 2 \\rfloor } } \\\\ & { = \\displaystyle \\sum _ { S \\leq [ k ] : S = \\lfloor k / 2 \\rfloor } \\mathbb { E } [ \\mathbb { P } ( \\mathbf { W X } _ { S } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 6 ) } { = } ( \\begin{array} { l } { k } \\\\ { \\lfloor k / 2 \\rfloor } \\end{array} ) \\mathbb { P } ( \\mathbf { W } _ { \\mathbf { X } _ { \\lfloor k / 2 \\rfloor } } > 0 ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 126, + 828, + 308 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 313, + 215, + 327 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "1. We used the independence of the $\\mathbf { w } _ { i }$ . ", + "bbox": [ + 212, + 343, + 477, + 358 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "2. We define $\\hat { S } _ { \\pm } \\left( \\mathbf { h } \\right) \\ \\triangleq \\ \\left\\{ S \\subset \\left[ k \\right] : \\pm \\mathbf { h } _ { S } ^ { \\top } > 0 \\right\\}$ as the sets in which $\\mathbf { h }$ is always positive/negative, and $\\hat { S } \\left( \\mathbf { h } \\right)$ as the maximal set between these two. Note that $\\mathbf { w } _ { i }$ has a standard normal distribution which is symmetric to sign flips, so $\\forall S : \\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } > 0 | \\mathbf { X } \\right) =$ $\\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } < 0 | \\mathbf { X } \\right)$ . ", + "bbox": [ + 212, + 367, + 826, + 435 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "3. Note that $\\left| \\hat { S } \\left( \\mathbf { h } \\right) \\right| \\ge \\lfloor k / 2 \\rfloor$ . Therefore, we define S∗ = argmax $\\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } > 0 | \\mathbf { X } \\right)$ . $S C [ k ] : \\bar { | } S | = \\lfloor k / 2 \\rfloor$ ", + "bbox": [ + 210, + 443, + 820, + 474 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "4. We used the independence of the $\\mathbf { w } _ { i }$ ", + "bbox": [ + 214, + 483, + 475, + 500 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "5. The maximum is a single term in the following sum of non-negative terms. ", + "bbox": [ + 210, + 507, + 722, + 523 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "6. Taking the expectation over $\\mathbf { X }$ , since the columns of $\\mathbf { X }$ are independent and identically distributed, the location of $S$ does not affect the probability. Therefore, we can set without loss of generality $S = [ \\lfloor k / 2 \\rfloor ]$ . ", + "bbox": [ + 212, + 531, + 825, + 575 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "12.2.3 MAIN PROOF: BOUND ON THE NUMBER OF CONFIGURATIONS FOR A BINARY MATRIX WITH CERTAIN RANK ", + "text_level": 1, + "bbox": [ + 176, + 622, + 825, + 651 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Recall the function $a \\left( \\cdot \\right)$ from eq. (2.1): ", + "bbox": [ + 174, + 660, + 433, + 675 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/9b2221403addba9e787fbde7841108c759c1753d7275c9d8c998715393cc168d.jpg", + "text": "$$\na \\left( u \\right) \\triangleq \\left\\{ \\begin{array} { l l } { 1 } & { , \\mathrm { i f } , u > 0 } \\\\ { \\rho } & { , \\mathrm { i f } u < 0 } \\end{array} \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 410, + 684, + 588, + 719 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where $\\rho \\neq 1$ ", + "bbox": [ + 173, + 727, + 261, + 741 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma 31. (Lemma 15 restated). Let $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }$ be a random matrix with independent and identically distributed columns, and $\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ an independent standard random Gaussian matrix. Then, in the limit $\\operatorname* { m i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r$ , ", + "bbox": [ + 173, + 746, + 826, + 790 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/364630bd489ca246d1fca1bcf304f112c24d0e6abb1816a7e761cdeb8aa75aaf.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( a \\left( \\mathbf { W } \\mathbf { X } \\right) \\right) = r \\right) \\dot { \\leq } 2 ^ { k + r d _ { 0 } ( \\log d _ { 1 } + \\log k ) + r ^ { 2 } } \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 796, + 732, + 818 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. We denote $\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right) \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times k }$ . For any such A for which rank $( \\mathbf { A } ) = r$ , we have a collection of $r$ rows that span the remaining rows. There are $\\left( \\begin{array} { c } { d _ { 1 } } \\\\ { r } \\end{array} \\right)$ possible locations for these $r$ spanning rows. In these rows there exist a collection of $r$ columns that span the remaining columns. There are $\\left( \\begin{array} { l } { k } \\\\ { r } \\end{array} \\right)$ possible locations for these $r$ spanning columns. At the intersection of the spanning rows and columns, there exist a full rank sub-matrix D. We denote $\\tilde { \\mathbf { A } }$ as the matrix A which rows and columns are permuted so that $\\mathbf { D }$ is the lower right block ", + "bbox": [ + 173, + 833, + 826, + 929 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 823, + 132 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/dc73d2077d1f22b894cfbb102d1a4719c160a8901a19011337502ed5990b9d21.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { \\mathbf { A } } \\triangleq \\left( \\begin{array} { l l } { \\mathbf { Z } } & { \\mathbf { B } } \\\\ { \\mathbf { C } } & { \\mathbf { D } } \\end{array} \\right) = a \\left( \\begin{array} { l l } { \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } } & { \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } } \\\\ { \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } } & { \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } } \\end{array} \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 133, + 656, + 169 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where $\\mathbf { D }$ is an invertible $r \\times r$ matrix, and we divided $\\mathbf { X }$ and $\\mathbf { W }$ to the corresponding block matrices ", + "bbox": [ + 176, + 170, + 820, + 185 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/72609808dad38b145048f6a2d6e51932cbf1ba6c152e6ac4035c28ea1c2eea3e.jpg", + "text": "$$\n\\mathbf { W } \\triangleq \\left[ \\mathbf { W } _ { 1 } ^ { \\top } , \\mathbf { W } _ { 2 } ^ { \\top } \\right] ^ { \\top } , \\mathbf { X } \\triangleq \\left[ \\mathbf { X } _ { 1 } , \\mathbf { X } _ { 2 } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 188, + 624, + 210 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "with $\\mathbf { W } _ { 2 } \\in \\mathbb { R } ^ { r \\times d _ { 0 } }$ rows and $\\mathbf { X } _ { 2 } \\in \\mathbb { R } ^ { d _ { 0 } \\times r }$ . ", + "bbox": [ + 174, + 213, + 452, + 229 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Since $\\mathrm { r a n k } \\left( \\tilde { \\mathbf { A } } \\right) = r$ , the first $d _ { 1 } - r$ rows are contained in the span of the last $r$ rows. Therefore, there exists a matrix $\\mathbf { Q }$ such that $\\mathbf { Q C } = \\mathbf { Z }$ and $\\mathbf { Q D } = \\mathbf { B }$ . Since $\\mathbf { D }$ is invertible, this implies that ${ \\bf Q } = { \\bf B } { \\bf D } ^ { - 1 }$ and therefore ", + "bbox": [ + 174, + 234, + 826, + 285 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/89a36d6c33a2e98460f268c5959543252c2c44cf6910a0af2d26b6570119ee38.jpg", + "text": "$$\n{ \\bf Z } = { \\bf B } { \\bf D } ^ { - 1 } { \\bf C } ,\n$$", + "text_format": "latex", + "bbox": [ + 446, + 282, + 549, + 301 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "i.e., $\\mathbf { B } , \\mathbf { C }$ and $\\mathbf { D }$ uniquely determine $\\mathbf { Z }$ . ", + "bbox": [ + 174, + 301, + 434, + 316 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Using the union bound over all possible permutations from $\\mathbf { A }$ to $\\tilde { \\mathbf { A } }$ , and eq. (12.9), we have ", + "bbox": [ + 169, + 323, + 776, + 340 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/1994e9876af76493b779887bf45ea39c31ca38d72b680c0b30eb06e691b0feba.jpg", + "text": "$$\n\\begin{array} { r l r } & { \\mathbb { P } ( \\mathrm { r a n k } ( \\mathbf { A } ) = r ) } & { ( 1 2 . 1 0 ) } \\\\ & { \\leq ( \\begin{array} { l } { d _ { 1 } } \\\\ { r } \\end{array} ) ( \\begin{array} { l } { k } \\\\ { r } \\end{array} ) \\mathbb { P } ( \\mathrm { r a n k } ( \\bar { \\mathbf { A } } ) = r ) } \\\\ & { \\leq ( \\begin{array} { l } { d _ { 1 } } \\\\ { r } \\end{array} ) ( \\begin{array} { l } { k } \\\\ { r } \\end{array} ) \\mathbb { P } ( \\mathbf { Z } = \\mathbf { B } \\mathbf { D } ^ { - 1 } \\mathbf { C } ) } \\\\ & { = ( \\begin{array} { l } { d _ { 1 } } \\\\ { r } \\end{array} ) ( \\begin{array} { l } { k } \\\\ { r } \\end{array} ) \\mathbb { P } ( a ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } ) [ a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } ) = a ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) ) } \\\\ & = ( \\begin{array} { l } { d _ { 1 } } \\\\ { r } \\end{array} ) ( \\begin{array} { l } { k } \\\\ { r } \\end{array} ) \\underbrace { \\mathrm { H } ( a ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } ) [ a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } ) = a ( \\mathbf { H } ) | \\mathrm { s i g n } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) = \\mathbf { H } ) \\mathbb { P } ( \\mathrm { s i g n } ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 173, + 361, + 839, + 532 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Using Lemma 30, we have ", + "bbox": [ + 174, + 532, + 352, + 547 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/5584bccf19c9f652988422c828c088213c81997f0a6b151e7d528a280d792a09.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { s i g n } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } \\right) = \\mathbf { H } \\right) \\leq \\left( \\begin{array} { c } { k - r } \\\\ { \\lfloor ( k - r ) / 2 \\rfloor } \\end{array} \\right) \\mathbb { P } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { \\lfloor \\lfloor ( k - r ) / 2 \\rfloor \\rfloor } > 0 \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 549, + 712, + 584 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "an upper bound which does not depend on $\\mathbf { H }$ . So all that remains is to compute the sum: ", + "bbox": [ + 169, + 585, + 753, + 602 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/477da32748c11537e670090346b6d445a657fca658df9a11db947d0865201be1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\qquad \\displaystyle \\sum _ { \\mathbf { H } \\in \\{ - 1 , 1 \\} ^ { ( d _ { 1 } - r ) \\times ( k - r ) } } \\mathbb { P } ( a ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } ) [ a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } ) = a ( \\mathbf { H } ) | \\mathrm { s i g n } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) = \\mathbf { H } ) } \\\\ { = } & { \\qquad \\displaystyle \\sum _ { \\mathbf { H } \\in \\{ - 1 , 1 \\} ^ { ( d _ { 1 } - r ) \\times ( k - r ) } } \\mathbb { E } [ \\mathbb { P } ( a ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } ) [ a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } ) = a ( \\mathbf { H } ) | \\mathbf { W } _ { 1 } , \\mathbf { X } _ { 1 } ) | \\mathrm { s i g n } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) = \\mathbf { H } } \\\\ & { \\mathrm { H } _ { 1 } ^ { 1 } [ \\displaystyle \\sum _ { \\mathbf { H } \\in \\{ - 1 , 1 \\} ^ { ( d _ { 1 } - r ) \\times ( k - r ) } } \\mathbb { Z } ( \\exists ( \\mathbf { W } _ { 2 } , \\mathbf { X } _ { 2 } ) : a ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } ) [ a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } ) ] ^ { - 1 } a ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } ) = a ( \\mathbf { H } ) ) ] \\mathrm { s i g n } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) = \\mathbf { H } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 602, + 839, + 734 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/acfffcb8eb77f7cf0101097a160552ea99c139a5830b12a26c53e8295c70692e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\overset { 2 } { \\leq } \\mathbb { R } [ 2 ^ { r ^ { 2 } } [ \\displaystyle \\sum _ { \\mathbf { H } \\in \\{ - 1 , 1 \\} ^ { ( d _ { \\mathbf { X } } _ { 2 } : \\mathrm { ~ s i g n } } } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } ) = \\mathbf { H } ) ] [ \\displaystyle \\sum _ { \\mathbf { H } \\in \\{ - 1 , 1 \\} ^ { r \\times ( k - r ) } } \\mathbb { Z } ( \\exists \\mathbf { W } _ { 2 } : \\mathrm { s i g n } ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } ) = \\mathbf { H } ) ] ] | \\mathrm { s i g n } } \\\\ & { \\leq \\mathbb { R } [ 2 ^ { r ^ { 2 } } [ \\displaystyle \\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { ( d _ { \\mathbf { X } } : \\mathrm { s i g n } } } ( \\mathbf { W } _ { 1 } \\mathbf { x } ) = \\mathbf { h } ) ] ^ { r } [ \\displaystyle \\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { ( k - r ) } } \\mathbb { Z } ( \\exists \\mathbf { w } : \\mathrm { s i g n } ( \\mathbf { w } ^ { \\mathsf { T } } \\mathbf { X } _ { 1 } ) = \\mathbf { h } ^ { \\mathsf { T } } ) ] ^ { r } ] | \\mathrm { s i g n } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) } \\\\ & { \\overset { 3 } { \\leq } \\mathbb { R } [ 2 ^ { r ^ { 2 } } 2 ^ { r d _ { 0 } \\log ( d _ { 1 } - r ) + r _ { 2 } r d _ { 0 } \\log ( k - r ) + r } ] | \\mathrm { s i g n } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) = \\mathbf { H } ] } \\\\ & { = 2 ^ { r d _ { 0 } [ \\log ( d _ { 1 } - r ) + \\log ( k - r ) ] + r ^ { 2 } + 2 r } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 750, + 839, + 911 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 910, + 217, + 922 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "1. Given $( \\mathbf { W } _ { 1 } , \\mathbf { X } _ { 1 } )$ , and eq. (12.8), the indicator function in eq. (12.12) is equal to zero only if $\\begin{array} { r } { \\mathbb { P } \\left( a \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } \\right) \\left[ a \\left( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } \\right) \\right] ^ { - 1 } a \\left( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } \\right) = \\mathbf { A } \\mathbf { | } \\mathbf { W } _ { 1 } , \\mathbf { X } _ { 1 } \\right) = 0 } \\end{array}$ , and one otherwise. ", + "bbox": [ + 210, + 102, + 825, + 142 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "2. This sum counts the number of values of $\\mathbf { H }$ consistent with $\\mathbf { W } _ { 1 }$ and $\\mathbf { X } _ { 1 }$ . Conditioned on $( \\mathbf { W } _ { 1 } , \\mathbf { X } _ { 1 } ) , \\mathbf { D } = \\left[ a \\left( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 2 } \\right) \\right] ^ { - 1 } , \\mathbf { B } = a \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } \\right)$ and $\\mathbf { C } = a \\left( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } \\right)$ can have multiple values, depending on $\\mathbf { W } _ { 2 }$ and $\\mathbf { X } _ { 2 }$ . Also, any single value for $( { \\bf D } , { \\bf B } , { \\bf C } )$ results in a single value of $\\mathbf { H }$ . Therefore, the number of possible values of $\\mathbf { H }$ in eq. (12.12) is upper bounded by the product of the number of possible values of $\\mathbf { D }$ , $\\mathbf { B }$ and $\\mathbf { C }$ , which is product in the following equation. ", + "bbox": [ + 212, + 150, + 825, + 237 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "3. The function $\\begin{array} { r } { \\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { ( k - r ) } } \\mathscr { T } \\left( \\exists \\mathbf { w } : \\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } _ { 1 } \\right) = \\mathbf { h } ^ { \\top } \\right) } \\end{array}$ \u0001 counts the number of dichotomies that can be induced by the linear classifier w on $\\mathbf { X } _ { 1 }$ . Using eq. (12.7) we can bound this number by $2 \\left( k - r \\right) ^ { d _ { 0 } }$ . Similarly, the other sum can be bounded by $2 \\left( d _ { 1 } - r \\right) ^ { r }$ . ", + "bbox": [ + 210, + 244, + 826, + 294 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Combining eqs. (12.10), (12.11) and (12.13) we obtain ", + "bbox": [ + 173, + 309, + 537, + 327 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/6468b0a86acb106249f015f9b911214c7457c2064589c4896bb71f31ee587b31.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( \\mathbf { A } \\right) = r \\right) \\leq } \\\\ & { \\left( \\begin{array} { l } { d _ { 1 } } \\\\ { r } \\end{array} \\right) \\left( \\begin{array} { l } { k } \\\\ { r } \\end{array} \\right) \\left( \\begin{array} { c } { k - r } \\\\ { \\lfloor ( k - r ) / 2 \\rfloor } \\end{array} \\right) 2 ^ { r d _ { 0 } \\left[ \\log ( d _ { 1 } - r ) + \\log ( k - r ) \\right] + r ^ { 2 } + 2 r } \\mathbb { P } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { [ \\lfloor ( k - r ) / 2 \\rfloor ] } > 0 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 332, + 807, + 388 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Next, we take the log. To upper bound $\\left( \\begin{array} { c } { { N } } \\\\ { { k } } \\end{array} \\right)$ , for small $k$ we use $\\left( \\begin{array} { c } { { N } } \\\\ { { k } } \\end{array} \\right) \\le N ^ { k }$ , while for $k = N / 2$ , we use $\\left( \\begin{array} { c } { { N } } \\\\ { { N / 2 } } \\end{array} \\right) \\le 2 ^ { N }$ . Thus, we obtain ", + "bbox": [ + 173, + 393, + 825, + 459 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/4d194bd1eb0e846077da5483deda01a9c7113979aeb679496db42d74ef0750c2.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\log \\mathbb { P } \\left( \\mathrm { r a n k } \\left( \\mathbf { A } \\right) = r \\right) \\leq \\left( r d _ { 0 } \\left( \\log \\left( d _ { 1 } - r \\right) + \\log \\left( k - r \\right) \\right) + r ^ { 2 } + 2 r \\right) \\log 2 \\qquad ( 1 2 \\triangle ( r \\mathrm { ~ o ~ t ~ } r ) } \\\\ & { } & { + r \\log d _ { 1 } + r \\log k + ( k - r ) \\log 2 + \\log \\mathbb { P } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { [ \\vert ( k - r ) / 2 ] ] } > 0 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 465, + 799, + 507 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Recalling that $\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { ( d _ { 1 } - r ) \\times d _ { 0 } }$ while $\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ , we obtain from Jensen’s inequality ", + "bbox": [ + 171, + 513, + 766, + 531 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/cdb37740def3c8c632c0376cf0cdc0ca2adf070ab1bb586699c73e9475ff0ea9.jpg", + "text": "$$\n\\begin{array} { r } { \\log \\mathbb { P } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { [ \\lfloor ( k - r ) / 2 \\rfloor ] } > 0 \\right) \\le \\frac { \\left\\lfloor \\left( k - r \\right) / 2 \\right\\rfloor \\left\\lfloor d _ { 1 } - r \\right\\rfloor } { \\left\\lfloor k / 2 \\right\\rfloor \\left\\lfloor d _ { 1 } \\right\\rfloor } \\log \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 209, + 537, + 741, + 573 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Taking the limit $\\mathrm { n i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r$ on eqs. (12.14) and (12.15) we obtain ", + "bbox": [ + 174, + 579, + 651, + 597 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/9df6a774635dc67111cc4d5f93d21467589611f3de1be67eb153f36831e3dc06.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( \\mathbf { A } \\right) = r \\right) \\overset { * } { \\leq } 2 ^ { k + r d _ { 0 } ( \\log d _ { 1 } + \\log k ) + t ^ { 2 } } \\mathbb { P } \\left( { \\mathbf { W } } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 603, + 709, + 626 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "12.3 PROOF OF LEMMA 16 ", + "text_level": 1, + "bbox": [ + 176, + 664, + 375, + 679 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In this section we will prove Lemma 16 in subsection 12.3.3. This proof relies on more elementary results, which we first prove in subsections 12.3.1 and 12.3.2. ", + "bbox": [ + 171, + 690, + 825, + 720 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "12.3.1 ORTHANT PROBABILITY OF A RANDOM GAUSSIAN VECTOR ", + "text_level": 1, + "bbox": [ + 176, + 734, + 648, + 751 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Recall that $\\phi \\left( x \\right)$ and $\\Phi \\left( x \\right)$ are, respectively, the probability density function and cumulative distribution function for a scalar standard normal random variable. ", + "bbox": [ + 174, + 760, + 826, + 789 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition 32. We define the following functions $\\forall x \\geq 0$ ", + "bbox": [ + 173, + 794, + 553, + 809 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/00187b0bc5bea52b78790b55c395390a4a0f4057e218595d545c4df856c180f6.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { g \\left( x \\right) \\triangleq \\frac { x \\Phi \\left( x \\right) } { \\phi \\left( x \\right) } , } } \\\\ { \\displaystyle { \\psi \\left( x \\right) \\triangleq \\frac { \\left( g ^ { - 1 } \\left( x \\right) \\right) ^ { 2 } } { 2 x } - \\log \\left( \\Phi \\left( g ^ { - 1 } \\left( x \\right) \\right) \\right) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 359, + 815, + 637, + 888 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where the inverse function $g ^ { - 1 } ( x ) : [ 0 , \\infty ) [ 0 , \\infty )$ is well defined since $g \\left( x \\right)$ monotonically increase from 0 to $\\infty$ , for $x \\geq 0$ . ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Lemma 33. Let $\\mathbf { z } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } )$ be a random Gaussian vector in $\\mathbb { R } ^ { K }$ , with a covariance matrix $\\Sigma _ { i j } = \\left( 1 - \\theta K ^ { - 1 } \\right) \\delta _ { m n } + \\theta K ^ { - 1 }$ where $K \\gg \\theta > 0$ . Then, recalling $\\psi \\left( \\theta \\right)$ in eq. (12.17), we have ", + "bbox": [ + 168, + 102, + 825, + 133 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/10ceb2c2a891fbb08c5041cf502e3c6eb5698cf6b62ea6b7ded0f3b27ebbdd2b.jpg", + "text": "$$\n\\begin{array} { r } { \\log \\mathbb { P } \\left( \\forall i : z _ { i } > 0 \\right) \\le - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 344, + 140, + 651, + 159 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. Note that we can write $\\mathbf { z } = \\mathbf { u } + \\eta$ , where $\\mathbf { u } \\sim { \\mathcal { N } } \\left( 0 , \\left( 1 - \\theta K ^ { - 1 } \\right) \\mathbf { I } _ { K } \\right)$ , and $\\eta \\sim \\mathcal { N } \\left( 0 , \\theta K ^ { - 1 } \\right)$ . Using this notation, we have ", + "bbox": [ + 174, + 172, + 825, + 203 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/bc262ef4ffb1ddf52f55b8a1b9f1574b6779391626101eeb51b688c7a14ed759.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { P } \\left( \\overline { { \\psi } } : z _ { i } > 0 \\right) } \\\\ & { = \\int _ { - \\infty } ^ { \\infty } d \\eta \\left[ \\displaystyle \\prod _ { i = 1 } ^ { K } \\int _ { - \\infty } ^ { \\infty } d u _ { i } \\mathcal { T } \\left( \\sqrt { 1 - \\theta K ^ { - 1 } } u _ { i } + \\sqrt { \\theta K ^ { - 1 } } \\eta > 0 \\right) \\phi \\left( u _ { i } \\right) \\right] \\phi \\left( \\eta \\right) } \\\\ & { = \\int _ { - \\infty } ^ { \\infty } d \\eta \\left[ \\Phi \\left( \\sqrt { \\frac { \\theta K ^ { - 1 } } { 1 - \\theta K ^ { - 1 } } } \\eta \\right) \\right] ^ { K } \\phi \\left( \\eta \\right) } \\\\ & { \\stackrel { ( 1 ) } { = } \\sqrt { \\displaystyle \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\int _ { - \\infty } ^ { \\infty } d \\xi \\left[ \\Phi \\left( \\xi \\right) \\right] ^ { K } \\exp \\left( - \\frac { \\left( K - \\theta \\right) \\xi ^ { 2 } } { 2 \\theta } \\right) } \\\\ & { = \\sqrt { \\displaystyle \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\int _ { - \\infty } ^ { \\infty } d \\xi \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) \\exp \\left[ K \\left( \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } \\right) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 208, + 753, + 404 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where in (1) we changed the variable of integration to $\\xi = \\sqrt { \\theta / \\left( K - \\theta \\right) } \\eta$ . We denote, for a fixed $\\theta$ ", + "bbox": [ + 173, + 409, + 823, + 428 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/29af02d48d220a9e54bd1ba2d06556480d0167d851fe2d29a2a4847fb58909a4.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle q \\left( \\xi \\right) \\triangleq \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } } } \\\\ { { \\displaystyle h \\left( \\xi \\right) \\triangleq \\sqrt { \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 383, + 431, + 612, + 508 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "and $\\xi _ { 0 }$ as its global maximum. Since $q$ is twice differentiable, we can use Laplace’s method (e.g., (Butler, 2007)) to simplify eq. (12.18) ", + "bbox": [ + 171, + 511, + 826, + 541 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/d1670a986406af48e77385d91d0789092f6aa30650f01a29338d73edc1472a78.jpg", + "text": "$$\n\\log \\int _ { - \\infty } ^ { \\infty } h \\left( \\xi \\right) \\exp \\left( K q \\left( \\xi \\right) \\right) d \\xi = K q \\left( \\xi _ { 0 } \\right) + O \\left( \\log K \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 556, + 687, + 592 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "To find $\\xi _ { 0 }$ , we differentiate $q \\left( \\xi \\right)$ and equate to zero to obtain ", + "bbox": [ + 173, + 593, + 571, + 609 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/9e51c2e3473f8f5377e7a1a2151beb2c490589d7b94d46e2cd725ad034164c58.jpg", + "text": "$$\nq ^ { \\prime } \\left( \\xi \\right) = \\frac { \\phi \\left( \\xi \\right) } { \\Phi \\left( \\xi \\right) } - \\frac { 1 } { \\theta } \\xi = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 613, + 586, + 648 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "which implies (recall eq. (12.16)) ", + "bbox": [ + 173, + 652, + 397, + 667 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/6fc19580f5b10a49f862708606138eab92076653d13c268e9e60b5a8fd1f3b6b.jpg", + "text": "$$\ng \\left( \\xi \\right) \\triangleq \\frac { \\xi \\Phi \\left( \\xi \\right) } { \\phi \\left( \\xi \\right) } = \\theta .\n$$", + "text_format": "latex", + "bbox": [ + 428, + 665, + 570, + 699 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "This is a monotonically increasing function from 0 to $\\infty$ in the range $\\xi \\ge 0$ . Its inverse function can also be defined in that range $g ^ { - 1 } \\check { ( \\theta ) } : [ 0 , \\infty ] [ 0 , \\infty ]$ . This implies that this equation has only one solution, $\\xi _ { 0 } = g ^ { - 1 } \\left( \\theta \\right)$ . Since $\\begin{array} { r } { \\operatorname* { l i m } _ { \\xi \\to \\infty } q \\left( \\xi \\right) = - \\infty } \\end{array}$ , this $\\xi _ { 0 }$ is indeed the global maximum of $q \\left( \\xi \\right)$ Substituting this solution into $q \\left( \\xi \\right)$ , we get (recall eq. (12.17)) ", + "bbox": [ + 173, + 700, + 825, + 758 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/7cc89123c84adf644e67d0d3cde30d6ff1aa1f3403854f31f967bb1d5bb1fba8.jpg", + "text": "$$\n\\forall \\theta > 0 : q \\left( \\xi _ { 0 } \\right) = - \\psi \\left( \\theta \\right) = q \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) = \\log \\left( \\Phi \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) \\right) - \\frac { \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) ^ { 2 } } { 2 \\theta } .\n$$", + "text_format": "latex", + "bbox": [ + 218, + 763, + 732, + 799 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Using eq. (12.18), (12.21) and (12.24) we obtain: ", + "bbox": [ + 176, + 803, + 496, + 819 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/a2165f3020b1cf0998b8aee13777679a8199cf2ab84a919b3469948b6de7b1f7.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\log \\mathbb { P } \\left( \\forall i : z _ { i } > 0 \\right) } \\\\ { = \\log \\left[ \\int _ { - \\infty } ^ { \\infty } d \\xi \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) \\exp \\left[ K \\left( \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } \\right) \\right] \\right] + O \\left( \\log K \\right) } \\\\ { = - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 823, + 732, + 897 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Next, we generalize the previous Lemma to a general covariance matrix. ", + "bbox": [ + 173, + 103, + 648, + 119 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Corollary 34. Let $\\mathbf { u } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } )$ be a random Gaussian vector in $\\mathbb { R } ^ { K }$ for which $\\forall n : \\Sigma _ { n n } = 1$ , and $\\theta \\geq K \\operatorname* { m a x } _ { n , m : n \\neq m } \\Sigma _ { n m } > 0$ . Then, again, for large $K$ ", + "bbox": [ + 173, + 122, + 826, + 154 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/a119d0f58c9512b8d36a8810717f0e3d2c506d21b1da6aed63aae28952e7aaf0.jpg", + "text": "$$\n\\begin{array} { r } { \\log \\mathbb { P } \\left( \\forall i : u _ { i } > 0 \\right) \\le - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 161, + 653, + 179 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. We define $\\tilde { \\mathbf { u } } \\sim \\mathcal { N } \\left( 0 , \\tilde { \\Sigma } \\right)$ , with $\\tilde { \\Sigma } _ { m n } = \\left( 1 - \\theta K ^ { - 1 } \\right) \\delta _ { m n } + \\theta K ^ { - 1 }$ . Note that $\\forall n : \\Sigma _ { n n } =$ $\\tilde { \\Sigma } _ { n n } = 1$ and $\\forall m \\neq n$ : $\\Sigma _ { m n } \\leq \\tilde { \\Sigma } _ { m n }$ . Therefore, from Slepian’s Lemma (Slepian, 1962, Lemma 1), ", + "bbox": [ + 173, + 200, + 828, + 242 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/0244d23353d526224bb96c5b6f9940c2bce067ce9d0120fd1efe08d9c5d7349f.jpg", + "text": "$$\n\\mathbb { P } \\left( \\forall n : \\tilde { u } _ { n } > 0 \\right) \\ge \\mathbb { P } \\left( \\forall n : u _ { n } > 0 \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 250, + 622, + 267 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Using Lemma 33 on $\\tilde { \\mathbf { u } }$ completes the proof. ", + "bbox": [ + 174, + 275, + 462, + 291 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "12.3.2 MUTUAL COHERENCE BOUNDS ", + "text_level": 1, + "bbox": [ + 176, + 311, + 452, + 327 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Definition 35. We define the mutual coherence of the columns of a matrix $\\mathbf { A } = [ \\pmb { a } _ { 1 } , \\cdots , \\pmb { a } _ { N } ] \\in$ $\\mathbb { R } ^ { M \\times N }$ as the maximal angle between different columns ", + "bbox": [ + 171, + 337, + 825, + 366 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/d8d6a5d0a413b29e82574bebdcf6c510e09a916bc28476ba9bd2d480f925dbce.jpg", + "text": "$$\n\\gamma \\left( { \\bf A } \\right) \\triangleq \\operatorname* { m a x } _ { i , j : i \\neq j } \\frac { \\left| { \\pmb a } _ { i } ^ { \\top } { \\pmb a } _ { j } \\right| } { \\left\\| { \\pmb a } _ { i } \\right\\| \\left\\| { \\pmb a } _ { j } \\right\\| } .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 375, + 591, + 412 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Note that $\\gamma \\left( \\mathbf { A } \\right) \\leq 1$ and from (Welch, 1974), for $\\begin{array} { r } { N \\ge M , \\gamma \\left( { \\mathbf { A } } \\right) \\ge \\sqrt { \\frac { N - M } { M \\left( N - 1 \\right) } } . } \\end{array}$ ", + "bbox": [ + 174, + 420, + 702, + 446 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma 36. Let $\\mathbf { A } = [ \\pmb { a } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { a } _ { N } ] \\in \\mathbb { R } ^ { M \\times N }$ be a standard random Gaussian matrix, and $\\gamma \\left( \\mathbf { A } \\right)$ is the mutual coherence of it columns (see definition 35). Then ", + "bbox": [ + 171, + 450, + 826, + 481 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/ebe652bcc739fb427bf31744a0d6cc8a108c8f24088cbb865c41e57e96c9784f.jpg", + "text": "$$\n\\mathbb { P } \\left( \\gamma \\left( { \\bf A } \\right) > \\epsilon \\right) \\le 2 N ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 488, + 629, + 525 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. In this case, we have from (Chen & Peng, 2016, Appendix 1): ", + "bbox": [ + 173, + 542, + 629, + 559 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/e97351e71bd87ca29c21480bb77c0941e1dc3b1328e38237128d0d32bed82b15.jpg", + "text": "$$\n\\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) \\leq N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M a ^ { 2 } \\epsilon ^ { 2 } } { 4 \\left( 1 + \\epsilon / 2 \\right) } \\right) + \\exp \\left( - \\frac { M } { 4 } \\left( 1 - a \\right) ^ { 2 } \\right) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 233, + 566, + 761, + 603 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "for any $a \\in ( 0 , 1 )$ . Setting $a = 1 - \\epsilon / 2$ ", + "bbox": [ + 173, + 611, + 434, + 627 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/05961512a51b5398c54fa887dd4c3a2a461e2c3a377ae3c33158f1093cff5c0e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) \\le N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M \\left( 1 - \\epsilon / 2 \\right) ^ { 2 } \\epsilon ^ { 2 } } { 4 \\left( 1 + \\epsilon / 2 \\right) } \\right) + \\exp \\left( - \\frac { M } { 1 6 } \\epsilon ^ { 2 } \\right) \\right] } \\\\ & { \\qquad \\overset { \\left( 1 \\right) } { \\le } N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) + \\exp \\left( - \\frac { M } { 1 6 } \\epsilon ^ { 2 } \\right) \\right] } \\\\ & { \\qquad \\le 2 N ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 635, + 754, + 751 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where in (1) we can assume that $\\epsilon \\leq 1$ , since for $\\epsilon \\geq 1$ , we have $\\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) = 0$ (recall $\\gamma ( \\mathbf { A } ) \\leq 1 )$ . □ ", + "bbox": [ + 173, + 757, + 821, + 787 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma 37. Let $\\mathbf { B } = [ \\mathbf { b } _ { 1 } , \\cdots , \\mathbf { b } _ { L } ] \\in \\mathbb { R } ^ { M \\times L }$ be a standard random Gaussian matrix and mutual coherence $\\gamma$ as in definition 35. Then, $\\forall \\epsilon > 0$ and $\\forall K \\in [ L ]$ : ", + "bbox": [ + 171, + 799, + 825, + 832 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/257d17ca7cf63447593b7d70d782498e0bc8d1a68e7f426a55fb94cf19500879.jpg", + "text": "$$\n\\mathbb { P } \\left( \\operatorname* { m i n } _ { S \\subset [ N ] : | S | = K } \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) \\leq \\exp \\left[ \\left( 2 \\log \\left( 2 K \\right) - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 248, + 838, + 746, + 875 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. We upper bound this probability by partitioning the set of column vectors into $\\lfloor L / K \\rfloor$ subsets $S _ { i }$ of size $| \\bar { S _ { i } } \\bar { | } = K$ and require that in each subset the mutual coherence is lower bounded by $\\epsilon$ . ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Since the columns are independent, we have ", + "bbox": [ + 176, + 103, + 465, + 118 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/5987161ba4f7c4213350967d3f294afc11ad2bc19967b72147387417bf676375.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } \\left( \\underset { S \\subset [ N ] : | S | = K } { \\operatorname* { m i n } } \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\le \\displaystyle \\prod _ { i = 1 } ^ { \\lfloor L / K \\rfloor } \\mathbb { P } \\left( \\forall S = \\{ 1 + \\left( i - 1 \\right) K , 2 + \\left( 1 - i \\right) K , \\ldots , i K \\} : \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\stackrel { \\mathrm { ( 1 ) } } { \\le } \\displaystyle \\prod _ { i = 1 } ^ { L / K - 1 } 2 K ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) } \\\\ & { \\le \\exp \\left[ \\left( 2 \\log \\left( 2 K \\right) - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 122, + 740, + 286 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "where in (1) we used the bound from Lemma 36. ", + "bbox": [ + 173, + 289, + 496, + 304 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "12.3.3 MAIN PROOF: ORTHANT PROBABILITY OF A PRODUCT GAUSSIAN MATRICES ", + "bbox": [ + 173, + 318, + 766, + 333 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lemma 38. (Lemma 16 restated). Let $\\mathbf { C } = \\left[ \\mathbf { c } _ { 1 } , \\cdots , \\mathbf { c } _ { N } \\right] ^ { \\top } \\in \\mathbb { R } ^ { N \\times M }$ and $\\mathbf { B } \\in \\mathbb { R } ^ { M \\times L }$ be two independent random Gaussian matrices. Without loss of generality, assume $N \\geq L$ , and denote $\\alpha \\triangleq M L / N$ . Then, in the regime $M \\leq N$ and in the limit $\\operatorname* { m i n } \\left[ N , M , L \\right] \\dot { > } \\alpha \\dot { > } 1$ , we have ", + "bbox": [ + 173, + 340, + 825, + 387 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/7c37001828c2a3958147878c11f5978c25551509807f52bbf9da0ca0389ea6e0.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 377, + 391, + 619, + 420 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. For some $\\theta > 0$ , and subset $S$ such that $| S | = K < L$ , we have ", + "bbox": [ + 173, + 431, + 637, + 448 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/12acb15a2ef1b3a40722bb6ac8ae01e3cd3271d3f78fb61d6a26cd8a44bf1e10.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) } \\\\ & { \\le \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) + \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\le \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) + \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { = \\mathbb { E } \\left[ \\left[ \\mathbb { P } \\left( \\mathbf { c } _ { 1 } ^ { \\top } \\mathbf { B } _ { S } > 0 | \\mathbf { B } _ { S } , \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) \\right] ^ { N } | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right] + \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 452, + 790, + 535 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "where in the last equality we used the fact that the rows of $\\mathbf { C }$ are independent and identically distributed. ", + "bbox": [ + 174, + 536, + 821, + 565 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We choose a specific subset ", + "bbox": [ + 174, + 571, + 356, + 587 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/df12677b88160108afdc8e94ec88315b9144847c1649b51df5b8199e56a80116.jpg", + "text": "$$\nS ^ { * } = \\mathrm { a r g m i n } _ { S C [ L ] : | S | = K } \\gamma \\left( \\mathbf { B } _ { S } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 388, + 585, + 607, + 604 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "to minimize the second term and then upper bound it using Lemma 37 with $\\theta = K \\epsilon$ ; additionally, we apply Corollary 34 on the first term with the components of the vector $\\mathbf { u }$ being ", + "bbox": [ + 173, + 604, + 823, + 633 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/953aa31d6ac88ff24202a9ce16cb564fe6d0144fb387847d45c32cdc141ba888.jpg", + "text": "$$\nu _ { i } = \\left( \\mathbf { B } _ { S } ^ { \\top } \\mathbf { c } _ { 1 } \\right) _ { i } / \\sqrt { \\left( \\mathbf { B } _ { S } ^ { \\top } \\mathbf { B } _ { S } \\right) _ { i i } } \\in \\mathbb { R } ^ { K } ,\n$$", + "text_format": "latex", + "bbox": [ + 372, + 637, + 622, + 665 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "which is a Gaussian random vector with mean zero and covariance $\\pmb { \\Sigma }$ for which $\\forall i : \\Sigma _ { i i } = 1$ and $\\forall i \\neq j : \\Sigma _ { i j } \\leq \\epsilon = \\theta K ^ { - 1 }$ . Thus, we obtain ", + "bbox": [ + 173, + 667, + 825, + 698 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/d062c72c492847d0c1f18addb2f4a2d933e3e9844ef45fb61e6335aa987dcd0a.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { C } \\mathbf { B } > 0 \\right) \\leq \\exp \\left( - N K \\psi \\left( \\theta \\right) + O \\left( N \\log K \\right) \\right) + \\exp \\left[ \\left( \\log \\left( 2 K \\right) ^ { 2 } - \\frac { M \\theta ^ { 2 } } { 2 4 K ^ { 2 } } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 702, + 812, + 738 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "where we recall $\\psi \\left( \\theta \\right)$ is defined in eq. (12.17). ", + "bbox": [ + 174, + 756, + 482, + 772 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Next, we wish to select good values for $\\theta$ and $K$ , which minimize this bound for large $( M , N , L , K )$ Thus, keeping only the first order terms in each exponent (assuming $L \\gg K \\gg 1$ ), we aim to minimize the function as much as possible ", + "bbox": [ + 176, + 777, + 825, + 820 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/1ae5ea48a896c2784e5111959286ea52a674e9fafec0fb357d81653b8ec418e5.jpg", + "text": "$$\nf \\left( K , \\theta \\right) \\triangleq \\exp \\left( - N K \\psi \\left( \\theta \\right) \\right) + \\exp \\left( - \\frac { M \\theta ^ { 2 } L } { 2 4 K ^ { 3 } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 825, + 668, + 861 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Note that the first term is decreasing in $K$ , while the second term increases. Therefore, for any $\\theta$ the minimum of this function in $K$ would be approximately achieved when both terms are equal, i.e., ", + "bbox": [ + 173, + 863, + 823, + 892 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/93b5891fbb61d899fcfe10b9b0a7a95590e0ebf987546c57c795aa2314f88afd.jpg", + "text": "$$\nN K \\psi \\left( \\theta \\right) = \\frac { M \\theta ^ { 2 } L } { 2 4 K ^ { 3 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 426, + 896, + 570, + 929 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "so we choose ", + "bbox": [ + 173, + 104, + 264, + 117 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/388b52d07a6c88ce91624ea496107aebd945676fda8d69e93cce3af934817e11.jpg", + "text": "$$\n{ \\cal K } \\left( \\theta \\right) = \\left( { \\frac { \\theta ^ { 2 } M L } { 2 4 \\psi \\left( \\theta \\right) N } } \\right) ^ { 1 / 4 } .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 113, + 594, + 152 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Substituting $K \\left( \\theta \\right)$ into $f \\left( K , \\theta \\right)$ yields ", + "bbox": [ + 174, + 154, + 429, + 170 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/2dcba8492ca11e9de048b7def965ec4068a0ca897e5872cdaef94ca5ee040bdf.jpg", + "text": "$$\nf \\left( K \\left( \\theta \\right) , \\theta \\right) = 2 \\exp \\left( - N \\left[ \\frac { \\psi ^ { 3 } \\left( \\theta \\right) \\theta ^ { 2 } M L } { 2 4 N } \\right] ^ { 1 / 4 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 175, + 668, + 218 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "To minimize this function in $\\theta$ , we need to maximize the function $\\psi ^ { 3 } \\left( \\theta \\right) \\theta ^ { 2 }$ (which has a single maximum). Doing this numerically gives us ", + "bbox": [ + 173, + 224, + 825, + 253 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/f3863d4a882a35485e8d0e72f3151d723414518fe330ed3e35d38cdf90f45cea.jpg", + "text": "$$\n\\theta _ { * } \\approx 2 3 . 2 5 ; \\psi \\left( \\theta _ { * } \\right) \\approx 0 . 1 0 6 2 ; \\psi ^ { 3 } \\left( \\theta _ { * } \\right) \\theta _ { * } ^ { 2 } \\approx 0 . 6 4 7 8 .\n$$", + "text_format": "latex", + "bbox": [ + 321, + 258, + 674, + 277 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Substituting eqs. (12.27) and (12.28) into eq. (12.25), we obtain ", + "bbox": [ + 173, + 282, + 596, + 297 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/e1c6049da1a1a73d233eee0483fecccbd48ff299ffe851950ae2fc0f58648b6e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } \\left( \\mathbf { C } \\mathbf { B } > 0 \\right) } \\\\ & { \\leq \\exp \\left( - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + O \\left( N \\log K \\right) \\right) } \\\\ & { + \\exp \\left[ - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + 2 L \\frac { \\log K } { K } + \\frac { M \\theta ^ { 2 } } { 2 4 K ^ { 2 } } - \\log \\left( 2 K ^ { 2 } \\right) \\right] } \\\\ & { \\leq \\exp \\left( - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + O \\left( N \\log \\left( \\frac { M L } { N } \\right) \\right) \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 303, + 714, + 452 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where in the last line we used $N \\geq L , N \\geq M$ and $\\operatorname* { m i n } \\left[ N , M , L \\right] \\dot { > } \\alpha \\dot { > } 1$ . Taking the log, and denoting $\\alpha \\triangleq M L / N$ , we thus obtain ", + "bbox": [ + 174, + 455, + 823, + 488 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/bbb40ca39754ca609ff4f7962feb6b2b034e3d6237b19b321a1e99e003c0d7ba.jpg", + "text": "$$\n\\begin{array} { r } { \\log \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\le - 0 . 4 N \\alpha ^ { 1 / 4 } + O \\left( N \\log \\alpha \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 493, + 656, + 513 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Therefore, in the limit that $N \\to \\infty$ and $\\alpha ( N ) \\infty$ , with $\\alpha \\left( N \\right) \\dot { < } N$ , we have ", + "bbox": [ + 173, + 518, + 697, + 535 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/7370e6868104035fed18a36a01a9dd5aa15dfa920fb0233402f7ed6fd9df2bd5.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 377, + 540, + 619, + 568 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "13 LOWER BOUNDING THE ANGULAR VOLUME OF GLOBAL MINIMA: PROOF OF LEMMAS USED IN SECTION 10 ", + "text_level": 1, + "bbox": [ + 173, + 608, + 821, + 642 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "13.1 ANGLES BETWEEN RANDOM GAUSSIAN VECTORS", + "text_level": 1, + "bbox": [ + 173, + 656, + 570, + 671 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "To prove the results in the next appendix sections, we will rely on the following basic Lemma. ", + "bbox": [ + 169, + 681, + 789, + 698 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Lemma 39. For any vector y and $\\mathbf { x } \\sim \\mathcal { N } \\left( 0 , \\mathbf { I } _ { d _ { 0 } } \\right)$ , we have ", + "bbox": [ + 174, + 700, + 562, + 717 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/fa42f9918e706f749d456c684981ab6c710b282b72ef98328705a2dbab9e7753.jpg", + "text": "$$\n\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| > \\cos \\left( \\epsilon \\right) \\right) \\geq \\frac { 2 \\sin \\left( \\epsilon \\right) ^ { d _ { 0 } - 1 } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) }\n$$", + "text_format": "latex", + "bbox": [ + 326, + 722, + 671, + 762 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/3a77ddb5663511a0017617697064e848face4733888d795d7da832e0b5771c7e.jpg", + "text": "$$\n\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| < u \\right) \\leq \\frac { 2 u } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 775, + 625, + 814 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where we recall that $B \\left( x , y \\right)$ is the beta function. ", + "bbox": [ + 173, + 819, + 500, + 835 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. Since $\\mathcal { N } ( 0 , \\mathbf { I } _ { d _ { 0 } } )$ is spherically symmetric, we can set $\\mathbf { y } = \\left[ 1 , 0 \\ldots , 0 \\right] ^ { \\top }$ , without loss of generality. Therefore, ", + "bbox": [ + 173, + 851, + 826, + 882 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/64282d6dabc67496b878a59318c54049889bb14ef87d747dd604972e9df39a48.jpg", + "text": "$$\n\\left| { \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } } \\right| ^ { 2 } = { \\frac { x _ { 1 } ^ { 2 } } { x _ { 1 } ^ { 2 } + \\sum _ { i = 2 } ^ { d _ { 0 } } x _ { i } ^ { 2 } } } \\sim { \\mathcal { B } } \\left( { \\frac { 1 } { 2 } } , { \\frac { d _ { 0 } - 1 } { 2 } } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 328, + 887, + 666, + 929 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "the Beta distribution, since $x _ { 1 } ^ { 2 } \\sim \\chi ^ { 2 } ( 1 )$ and ${ \\textstyle \\sum _ { i = 2 } ^ { d _ { 0 } } x _ { i } ^ { 2 } \\sim \\chi ^ { 2 } \\left( d _ { 0 } - 1 \\right) }$ are independent chi-square random variables. ", + "bbox": [ + 173, + 101, + 823, + 133 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Suppose $Z \\sim B \\left( \\alpha , \\beta \\right)$ , $\\alpha \\in ( 0 , 1 )$ , and $\\beta > 1$ ", + "bbox": [ + 174, + 138, + 480, + 155 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/d06592edc9440c6ba590cbc749012fc3a053f4fa597de77e312fdb3d3bbed508.jpg", + "text": "$$\n\\mathbb { P } \\left( Z > u \\right) = \\frac { \\int _ { u } ^ { 1 } x ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } \\geq \\frac { \\int _ { u } ^ { 1 } 1 ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { \\int _ { 0 } ^ { 1 - u } x ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { \\left( 1 - u \\right) ^ { \\beta } } { \\beta B \\left( \\alpha , \\beta \\right) } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 159, + 820, + 198 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Therefore, for $\\epsilon > 0$ , ", + "bbox": [ + 173, + 202, + 313, + 217 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/20496d6d65b4764c5c1c6d5bec084da0f2f7db53e73b010b9280277409c481b1.jpg", + "text": "$$\n\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\mathsf { T } } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| ^ { 2 } > \\cos ^ { 2 } \\left( \\epsilon \\right) \\right) \\geq \\frac { 2 \\left( 1 - \\cos ^ { 2 } \\left( \\epsilon \\right) \\right) ^ { \\frac { d _ { 0 } - 1 } { 2 } } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } = \\frac { 2 \\sin \\left( \\epsilon \\right) ^ { d _ { 0 } - 1 } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,\n$$", + "text_format": "latex", + "bbox": [ + 230, + 220, + 764, + 267 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "which proves eq. (13.1). ", + "bbox": [ + 173, + 272, + 334, + 286 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Similarly, for $\\alpha \\in ( 0 , 1 )$ and $\\beta > 1$ ", + "bbox": [ + 174, + 292, + 406, + 308 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/75e2fbaa029bdf83f8ed686cb88965f0aea91e71632a33ca1bed79b49d38d636.jpg", + "text": "$$\n\\mathbb { P } \\left( Z < u \\right) = \\frac { \\int _ { 0 } ^ { u } x ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } \\leq \\frac { \\int _ { 0 } ^ { u } x ^ { \\alpha - 1 } 1 ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { u ^ { \\alpha } } { \\alpha B \\left( \\alpha , \\beta \\right) } .\n$$", + "text_format": "latex", + "bbox": [ + 253, + 314, + 743, + 352 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Therefore, for $\\epsilon > 0$ ", + "bbox": [ + 173, + 356, + 312, + 371 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/5a8694a2a3c464cde6edf409c99fed078880ca3c851011d3fbc1927b83339f9c.jpg", + "text": "$$\n\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| ^ { 2 } < u ^ { 2 } \\right) \\leq \\frac { 2 u } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,\n$$", + "text_format": "latex", + "bbox": [ + 361, + 367, + 633, + 410 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "which proves eq. (13.2). ", + "bbox": [ + 174, + 410, + 334, + 426 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "13.2 PROOF OF LEMMA 21: ", + "text_level": 1, + "bbox": [ + 176, + 441, + 380, + 457 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Given three matrices: datapoints, $\\begin{array} { r c l c l } { \\mathbf { X } } & { = } & { \\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] } & { \\in } & { \\mathbb { R } ^ { d _ { 0 } \\times N } } \\end{array}$ , weights $\\begin{array} { r l } { \\mathbf { W } } & { { } = } \\end{array}$ $\\left[ { \\bf w } _ { 1 } ^ { \\top } , \\ldots , { \\bf w } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\ \\in \\ \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }$ , and target weights $\\mathbf W ^ { * } ~ = ~ \\left[ \\mathbf w _ { 1 } ^ { * \\top } , \\ldots , \\mathbf w _ { d _ { 1 } ^ { * } } ^ { * \\top } \\right] ^ { \\top } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }$ , with $d _ { 1 } ^ { * } \\leq d _ { 1 }$ ,we recall the following definitions: ", + "bbox": [ + 173, + 465, + 826, + 525 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/86ff27d291dd647a82a9fd6936fdc712e5655c3a789fad525b8a068390284dcc.jpg", + "text": "$$\n\\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\forall i , n : \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\mathrm { s i n } \\alpha \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 540, + 702, + 583 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 585, + 202, + 598 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/c2df6c638b0f87029d4c5fbbbbee6d66561c05765861aa28872135ff68aac91e.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } } | \\forall i \\le d _ { 1 } ^ { * } : \\ \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } \\right) = \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { * } { } ^ { \\top } \\mathbf { X } \\right) \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 616, + 745, + 636 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Using these definitions, in this section we prove the following Lemma. ", + "bbox": [ + 173, + 637, + 638, + 652 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Lemma 40. (Lemma 21 restated). For any $\\alpha$ , if $\\mathbf { W } ^ { * }$ is independent from W then, in the limit $N \\to \\infty$ , $\\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)$ with $\\log \\sin \\alpha { \\dot { > } } { \\dot { d } _ { 0 } } ^ { - 1 } \\log { d _ { 0 } }$ ", + "bbox": [ + 173, + 655, + 823, + 685 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/7d30cfb94c68e8b4db2b7cc08b8fb46c3d19ca0c3651149c9f8564d792c7a8c2.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\geq } \\exp \\left( d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 690, + 671, + 717 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. To lower bound $\\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)$ , we define the event that all weight hyperplanes (with normals $\\mathbf { w } _ { i }$ ) have an angle of at least $\\alpha$ from the corresponding target hyperplanes (with normals $\\mathbf { w } _ { i } ^ { * }$ ). ", + "bbox": [ + 173, + 731, + 825, + 784 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/0eb1d3f2b46033b5524c009f51e1591b0ad4831fbbba3fa78b51c5ba9ff22647.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { \\mathcal { G } } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) = \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } } \\big | \\left| \\frac { \\mathbf { W } _ { i } ^ { \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { w } _ { i } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| < \\cos \\left( \\alpha \\right) \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 789, + 687, + 825 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "In order that $\\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) \\neq \\mathrm { s i g n } \\left( \\mathbf { w } _ { 1 } ^ { \\ast \\top } \\mathbf { x } ^ { ( n ) } \\right)$ , $\\mathbf { w } _ { i }$ must be rotated in respect to $\\mathbf { w } _ { i } ^ { * }$ by an angle greater then the angular margin $\\alpha$ , which is the minimal the angle between $\\mathbf { x } ^ { ( n ) }$ and the solution hyperplanes (with normals $\\mathbf { w } _ { i } ^ { * }$ ). Therefore, we have that, given $\\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)$ , ", + "bbox": [ + 173, + 829, + 825, + 877 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/501f94c985d7910b8e133970d075709aae0442c05b8ac0edb2aff54e130851d1.jpg", + "text": "$$\n\\forall \\alpha : \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\tilde { \\mathcal { G } } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\subset \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 882, + 614, + 928 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "And so, $\\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)$ : ", + "bbox": [ + 174, + 102, + 352, + 118 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/05bd1c472319bf537d033f55fdc94fa7a901bf13a0cec6a2464976079636f8f1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\overset { ( 1 ) } { \\geq } \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\displaystyle \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\tilde { \\mathcal { G } } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) } \\\\ & { \\overset { ( 2 ) } { = } \\displaystyle \\prod _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\overset { ( 3 ) } { \\geq } \\left[ \\frac { 2 \\sin \\left( \\alpha \\right) ^ { d _ { 0 } - 1 } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } \\right] ^ { d _ { 1 } ^ { * } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 126, + 740, + 224 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where in (1) we used eq. (13.5), in (2) we used the independence of $\\{ { \\bf w } _ { i } \\} _ { i = 1 } ^ { d _ { 1 } ^ { * } }$ and in (3) we used eq. (13.1) from Lemma 39. Lastly, to simplify this equation we use the asymptotic expansion of the beta function $B \\left( { \\textstyle \\frac { 1 } { 2 } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)$ for large $x$ : ", + "bbox": [ + 174, + 232, + 825, + 280 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/ddcbd3ee9ecae6f0edcd1cd3a6bb3d07c8c96725d5d383c6e4ca63bdb1cf72df.jpg", + "text": "$$\n\\begin{array} { r } { \\log \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\geq d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha + O \\left( d _ { 1 } ^ { * } \\log d _ { 0 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 286, + 720, + 314 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "We obtain the Lemma in the limit $N \\to \\infty$ when $\\log \\sin \\alpha { \\dot { > } } d _ { 0 } ^ { - 1 } \\log d _ { 0 }$ . ", + "bbox": [ + 174, + 321, + 642, + 339 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "13.3 PROOF OF LEMMA 22: ", + "text_level": 1, + "bbox": [ + 176, + 354, + 380, + 369 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Lemma 41. (Lemma 22 restated). Let $\\mathbf { W } ^ { * } = \\left[ \\mathbf { w } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { w } _ { d _ { 1 } ^ { * } } ^ { \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }$ a fixed matrix independent of $\\mathbf { X }$ . Then, in the limit $N \\infty$ with $d _ { 1 } ^ { * } { \\dot { \\leq } } d _ { 0 } { \\dot { \\leq } } N _ { \\mathrm { ~ } }$ , the probability of not having an angular margin $\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)$ (eq. (13.3)) is upper bounded by ", + "bbox": [ + 173, + 378, + 826, + 436 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/ce627acc7cb618f45f7c25e679dc79964d455d7f1ec26ec0b22ea5b8a6e17d35.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 }\n$$", + "text_format": "latex", + "bbox": [ + 388, + 444, + 607, + 478 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proof. We define ", + "bbox": [ + 173, + 493, + 290, + 508 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/b4b3b1e2187066ea7a7e0b408ffdccfc5a6ed26dcc5480cff6b86dcf5ce352ad.jpg", + "text": "$$\n\\mathcal M _ { n , i } ^ { \\alpha } ( \\mathbf W ^ { * } ) \\triangleq \\{ \\mathbf X \\in \\mathbb R ^ { d _ { 0 } \\times N } | | \\frac { \\mathbf x ^ { ( n ) \\top } \\mathbf w _ { i } ^ { * } } { \\| \\mathbf x ^ { ( n ) } \\| \\| \\mathbf w _ { i } ^ { * } \\| } | > \\sin ( \\alpha ) \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 299, + 515, + 697, + 558 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "and $\\begin{array} { r } { \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathcal { M } _ { n , i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) } \\end{array}$ . Since $\\begin{array} { r } { \\mathcal { M } \\left( \\mathbf { W } ^ { * } \\right) = \\bigcap _ { n = 1 } ^ { N } \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) } \\end{array}$ , we have ", + "bbox": [ + 173, + 565, + 723, + 587 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/a95c3628c61d64327a15beb88d00affb7ef970a02e5e923e78cce885f719853f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\stackrel { ( 1 ) } { = } \\prod _ { n = 1 } ^ { N } \\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) = \\prod _ { n = 1 } ^ { N } \\left[ 1 - \\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\right] } \\\\ & { \\displaystyle \\overset { ( 2 ) } { \\geq } \\prod _ { n = 1 } ^ { N } \\left[ 1 - \\sum _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { n , i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\right] \\stackrel { ( 3 ) } { \\geq } \\left[ 1 - d _ { 1 } ^ { * } \\frac { 2 \\sin \\left( \\alpha \\right) } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } \\right] ^ { N } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 593, + 799, + 689 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where in (1) we used the independence of \bx(n)\tNn= , in (2) we use the union bound, and in (3) we use eq. (13.2) from Lemma 39. Taking the log and we using the asymptotic expansion of the beta function $B \\left( { \\textstyle \\frac { 1 } { 2 } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)$ for large $x$ , we get ", + "bbox": [ + 174, + 694, + 825, + 744 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/c57dcc64b5df4df02b49cddb13a272ddc1a9c7ec5fe73de5f7b50aeb52d23dcb.jpg", + "text": "$$\n\\begin{array} { r l } & { \\log \\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\geq N \\log \\left[ 1 - \\sqrt { \\frac { 2 } { \\pi } d _ { 0 } } d _ { 1 } ^ { * } \\sin \\alpha + O \\left( d _ { 1 } ^ { * } d _ { 0 } ^ { - 1 / 2 } \\sin \\alpha \\right) \\right] } \\\\ & { \\qquad = - \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + O \\left( d _ { 0 } ^ { - 3 / 2 } / N + d _ { 0 } ^ { - 1 } N ^ { - 2 } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 751, + 754, + 832 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where in the last line we recalled $\\sin \\alpha = 1 / N$ . Recalling that $d _ { 1 } ^ { * } { \\dot { \\le } } d _ { 0 } { \\dot { \\le } } N$ , we find ", + "bbox": [ + 173, + 837, + 718, + 854 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/617122fe9337cd88a938e13238bae4a6eb639f9d238c839bf08a758137bf8de3.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\geq } 1 - \\exp \\left( - \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } \\right) \\geq \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 }\n$$", + "text_format": "latex", + "bbox": [ + 295, + 861, + 700, + 904 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "13.4 PROOF OF LEMMA 23: ", + "text_level": 1, + "bbox": [ + 176, + 103, + 379, + 118 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Lemma 42. (Lemma 23 restated). Let $\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }$ be a standard random Gaussian matrix of datapoints. Then we can find, with probability $^ { l }$ , $( \\mathbf { X } , \\mathbf { y } )$ -dependent matrices $\\mathbf { W } ^ { * }$ and $\\mathbf { z } ^ { \\ast }$ as in Theorem 8 (where $d _ { 1 } ^ { * } \\triangleq 4 \\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\rceil ,$ ). Moreover, in the limit $N \\to \\infty$ , where $N / d _ { 0 } { \\dot { \\le } } d _ { 0 } { \\dot { \\le } } N$ , for any y, we can bound the probability of not having an angular margin (eq. (13.3)) with $\\sin \\alpha =$ $1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)$ by ", + "bbox": [ + 171, + 127, + 826, + 202 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/8c1c1d6ab39ccbf0f1bc0c12b5fce8e37c5e5c850185b00e41b95515e72261b7.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N }\n$$", + "text_format": "latex", + "bbox": [ + 333, + 203, + 665, + 239 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof. In this proof we heavily rely on the notation and results from the proof of in appendix section 9. Without loss of generality we assume $S _ { 1 } ^ { + } = [ d _ { 0 } - 1 ]$ . Unfortunately, we can’t use Lemma $4 1 -$ this proof is significantly more complicated since the constructed solution $\\mathbf { W } ^ { * }$ depends on $\\mathbf { X }$ (we keep this dependence implicit, for brevity). Similarly to the proof of Lemma 41, we define, ", + "bbox": [ + 173, + 251, + 828, + 308 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/b2bf5350db9f0790fefdcb282b55831a2d3290dccabd08afd033eecad5e46087.jpg", + "text": "$$\n\\mathcal { M } _ { i , n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\sin \\left( \\alpha \\right) \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 303, + 309, + 694, + 351 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "and $\\begin{array} { r } { \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\bigcap _ { n = 1 } ^ { N } \\mathcal { M } _ { i , n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) , \\operatorname { s o } \\mathcal { M } \\left( \\mathbf { W } ^ { * } \\right) = \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) . \\operatorname { W e } } \\end{array}$ have ", + "bbox": [ + 176, + 353, + 704, + 373 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/d4e572688c4f9aea0b80e92f29acda5e5ab599bcc03bd64644cd654d071d4346.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) = 1 - \\mathbb { P } \\left( \\mathbf { X } \\not \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\overset { ( 1 ) } { \\geq } 1 - \\sum _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } \\left( \\mathbf { X } \\not \\in \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 184, + 376, + 712, + 416 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "nd in (2) we used the fact that, from symmetr. Next, we examine the minimal angular margin in $\\forall i :$ $\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) = \\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { 1 } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right)$ $\\mathcal { M } _ { 1 , n } ^ { \\alpha }$ separately for $\\forall n < d _ { 0 }$ and $\\forall n \\geq d _ { 0 }$ . Recalling the construction of in appendix section 9, we have, for $\\forall n < d _ { 0 }$ : ", + "bbox": [ + 176, + 443, + 825, + 500 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/3881bc59d5a86d1d6c7d2a758c5349ba56cd50dfd0d5e0cf7288e5a1abb2eb19.jpg", + "text": "$$\n\\operatorname* { m i n } _ { i , n < d _ { 0 } } \\left. \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right. = \\operatorname* { m i n } _ { n < d _ { 0 } , \\pm } \\frac { \\left| \\left( \\tilde { \\mathbf { w } } _ { 1 } \\pm \\epsilon _ { 2 } \\hat { \\mathbf { w } } _ { 1 } \\right) ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| } { \\left\\| \\tilde { \\mathbf { w } } _ { 1 } \\pm \\epsilon _ { 2 } \\hat { \\mathbf { w } } _ { 1 } \\right\\| \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| }\n$$", + "text_format": "latex", + "bbox": [ + 295, + 500, + 658, + 546 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/5056707ff7a08660998b56004e9f915bf9581f7e51275dca79a75fc13bdea6f5.jpg", + "text": "$$\n\\begin{array} { r } { \\underline { { \\underline { { ( 1 ) } } } } _ { n < d _ { 0 } , \\pm } \\frac { \\epsilon _ { 2 } } { \\left\\| \\tilde { \\mathbf { w } } _ { 1 } \\pm \\epsilon _ { 2 } \\hat { \\mathbf { w } } _ { 1 } \\right\\| \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } \\overset { ( 2 ) } { = } \\frac { \\gamma \\epsilon _ { 1 } / \\sqrt { 1 + \\gamma ^ { 2 } \\epsilon _ { 1 } ^ { 2 } } } { \\left\\| \\hat { \\mathbf { w } } _ { 1 } \\right\\| \\operatorname* { m a x } _ { n < d _ { 0 } } \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 549, + 702, + 587 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "where in (1) we used $\\forall n < d _ { 0 }$ : $\\mathbf { x } ^ { ( n ) \\top } \\hat { \\mathbf { w } } _ { 1 } = 1$ and $\\mathbf { x } ^ { ( n ) \\top } \\tilde { \\mathbf { w } } _ { 1 } = 0$ , from the construction of $\\tilde { \\mathbf { w } } _ { 1 }$ and $\\hat { \\mathbf { w } } _ { 1 }$ (eqs. (9.2), (9.5), and (9.4)), and in (2) we used the fact that $\\hat { \\mathbf { w } } _ { 1 } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } = 0$ from eq. (9.4) together with $\\| \\tilde { \\mathbf { w } } _ { 1 } \\| = \\| \\hat { \\mathbf { w } } _ { 1 } \\|$ from eq. (9.5), and $\\epsilon _ { 2 } = \\gamma \\epsilon _ { 1 }$ from eq. (9.7). ", + "bbox": [ + 174, + 588, + 821, + 632 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "For $\\forall n \\geq d _ { 0 }$ : ", + "bbox": [ + 173, + 637, + 267, + 652 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/0f39f3d2ed3bfc7abfe649aea372d693f09534d82139ea0d976bec123253dabe.jpg", + "text": "$$\n\\operatorname* { m i n } _ { i , n \\geq d _ { 0 } } \\left. \\frac { \\mathbf { x } ^ { \\left( n \\right) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left. \\mathbf { x } ^ { \\left( n \\right) } \\right. \\left. \\mathbf { w } _ { i } ^ { * } \\right. } \\right. = \\operatorname* { m i n } _ { n \\geq d _ { 0 } , \\pm } \\frac { \\left. \\left( \\tilde { \\mathbf { w } } _ { 1 } \\pm \\epsilon _ { 1 } \\hat { \\mathbf { w } } _ { 1 } \\right) ^ { \\top } \\mathbf { x } ^ { \\left( n \\right) } \\right. } { \\left. \\tilde { \\mathbf { w } } _ { 1 } \\pm \\epsilon _ { 1 } \\hat { \\mathbf { w } } _ { 1 } \\right. \\left. \\mathbf { x } ^ { \\left( n \\right) } \\right. } \\geq \\frac { \\left( 1 - \\gamma \\beta \\right) \\epsilon _ { 1 } } { \\gamma \\beta \\sqrt { 1 + \\epsilon _ { 1 } ^ { 2 } } } \\operatorname* { m i n } _ { n \\geq d _ { 0 } } \\frac { \\left. \\tilde { \\mathbf { w } } _ { 1 } ^ { \\top } \\mathbf { x } ^ { \\left( n \\right) } \\right. } { \\left. \\tilde { \\mathbf { w } } _ { 1 } \\right. \\left. \\mathbf { x } ^ { \\left( n \\right) } \\right. } ,\n$$", + "text_format": "latex", + "bbox": [ + 194, + 652, + 800, + 700 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "where we used the fact that $\\forall n \\geq d _ { 0 } : \\epsilon _ { 2 } \\left| \\hat { \\mathbf { w } } _ { 1 } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| \\leq \\gamma \\beta \\left| \\tilde { \\mathbf { w } } _ { 1 } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right|$ , from eq. (9.7), and also that $\\hat { \\mathbf { w } } _ { 1 } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } = 0$ from eq. (9.4). ", + "bbox": [ + 173, + 708, + 825, + 742 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We substitute eqs. (13.8) and (13.9) into $\\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } _ { 1 } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right)$ : ", + "bbox": [ + 173, + 748, + 578, + 765 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "$\\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } _ { 1 } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right)$ ", + "bbox": [ + 173, + 765, + 307, + 781 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/5830fba2aab9a37b5537d6a8273f981cb7ccf5a2e0d97b661d3d46a683fe0ad1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\geq \\mathbb { P } \\left( \\frac { \\gamma \\epsilon _ { 1 } / \\sqrt { 1 + \\gamma ^ { 2 } \\epsilon _ { 1 } ^ { 2 } } } { \\left\\| \\hat { \\mathbf { w } } _ { 1 } \\right\\| \\operatorname* { m a x } _ { n < d _ { 0 } } \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } > \\sin \\alpha , \\frac { \\left( 1 - \\gamma \\beta \\right) \\epsilon _ { 1 } } { \\gamma \\beta \\sqrt { 1 + \\epsilon _ { 1 } ^ { 2 } } } \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\frac { \\left| \\hat { \\mathbf { w } } _ { 1 } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| } { \\left\\| \\hat { \\mathbf { w } } _ { 1 } \\right\\| \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } > \\sin \\alpha \\right) } \\\\ & { \\overset { ( 1 ) } { \\geq } \\mathbb { P } \\left( \\frac { \\gamma \\kappa } { \\left\\| \\hat { \\mathbf { w } } _ { 1 } \\right\\| \\operatorname* { m a x } _ { n < d _ { 0 } } \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } > \\sin \\alpha , \\frac { \\left( 1 - \\gamma \\beta \\right) } { \\gamma \\beta } \\kappa \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\frac { x _ { 1 } ^ { ( n ) } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } > \\sin \\alpha , \\frac { \\epsilon _ { 1 } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { 2 } } } > \\kappa \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 782, + 779, + 869 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/bb3aa970614c86b8023944363b8ac89fbbbf196174560d0e3f4d5474c6393db2.jpg", + "text": "$$\n\\overset { \\mathtt { ( 2 ) } } { \\geq } \\mathbb { P } \\left( \\frac { \\gamma \\kappa } { \\eta \\sin \\alpha } > \\left. \\hat { \\mathbf { w } } _ { 1 } \\right. , \\eta > \\operatorname* { m a x } _ { n < d _ { 0 } } \\left. \\mathbf { x } ^ { ( n ) } \\right. \\right) \\mathbb { P } \\left( \\frac { ( 1 - \\gamma \\beta ) } { \\gamma \\beta } \\kappa \\operatorname* { m i n } _ { n \\geq d _ { 0 } } \\frac { x _ { 1 } ^ { ( n ) } } { \\left. \\mathbf { x } ^ { ( n ) } \\right. } > \\sin \\alpha , \\frac { \\epsilon _ { 1 } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { 2 } } } > \\kappa \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 886, + 833, + 926 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "where in (1) we rotate the axes so that $\\hat { \\mathbf { w } } _ { 1 } \\propto [ 1 , 0 , 0 \\ldots , 0 ]$ axes $\\tilde { \\mathbf { w } } _ { 1 } \\propto [ 0 , 1 , 0 , 0 \\ldots , 0 ] - \\mathrm { t h i s }$ is possible due to the spherical symmetry of $\\mathbf { x } ^ { ( n ) }$ , and the fact that $\\hat { \\mathbf { w } } _ { 1 }$ and $\\tilde { \\mathbf { w } } _ { 1 }$ are functions of $\\mathbf { x } ^ { ( n ) }$ for $n < d _ { 0 }$ (from eqs. (9.4) and (9.2)), and as such, they are independent from $\\mathbf { x } ^ { ( n ) }$ for $n \\geq d _ { 0 }$ , in (2) we use that fact that $\\left. \\hat { \\mathbf { w } } _ { 1 } \\right.$ and $\\operatorname* { m a x } _ { n < d _ { 0 } } \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\|$ are functions of $\\mathbf { x } ^ { ( n ) }$ for $n < d _ { 0 }$ , and as such, they are independent from $\\mathbf { x } ^ { ( n ) }$ for $n \\geq d _ { 0 }$ . Thus, ", + "bbox": [ + 174, + 102, + 826, + 183 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/0402b0b308e57e9839ca6aeb7277844e949615723079fba9c549efc0929b83be.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } ^ { \\nu } ( \\mathbf { X } \\times M _ { 1 } ^ { \\nu } ( \\mathbf { W } ^ { \\nu } ) ) } \\\\ & { \\geq \\left( 1 - \\mathbb { P } \\left( \\frac { \\mathcal { X } } { \\eta \\sin \\alpha } \\leq \\left. \\mathbf { w } _ { 1 } \\right. \\textnormal { \\texttt { O } } \\eta \\leq \\operatorname* { m a x } \\left. \\mathbf { s } ^ { \\nu } ( \\mathbf { u } ) \\right. \\right) \\right) } \\\\ & { \\cdot \\left( 1 - \\mathbb { P } \\left( \\frac { \\left\\{ 1 - \\eta \\beta \\right\\} } { \\gamma \\beta } \\right) \\operatorname* { m i n } \\frac { x _ { 1 } ^ { \\nu \\alpha } } { x _ { 2 } ^ { \\nu \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. } \\leq \\sin \\alpha \\alpha \\frac { \\mathbb { E } _ { 1 } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { \\alpha } } } \\leq \\kappa \\right) \\right) } \\\\ & { \\overset { ( a ) } { \\geq } \\left( 1 - \\mathbb { P } \\left( \\frac { \\mathcal { X } } { \\eta \\sin \\alpha } \\leq \\left. \\mathbf { s } \\right. _ { 1 } \\right) - \\mathbb { P } \\left( \\eta \\leq \\operatorname* { m a x } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. \\right) \\right) } \\\\ & { \\cdot \\left( 1 - \\mathbb { P } \\left( \\frac { \\left\\{ 1 - \\eta \\beta \\right\\} } { \\gamma \\beta } \\operatorname* { m i n } \\frac { x _ { 1 } ^ { \\nu \\alpha } } { x _ { 2 } ^ { \\nu \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. } \\leq \\sin \\alpha \\right) - \\mathbb { E } \\left( \\frac { \\mathcal { X } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { \\alpha } } } \\leq \\kappa \\right) \\right) } \\\\ & { = \\left( \\mathbb { P } \\left( \\eta > \\frac { \\mathcal { X } } { \\eta \\sin \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. \\right) - \\mathbb { P } \\left( \\frac { \\mathcal { Y } } { \\eta \\sin \\alpha } \\leq \\left. \\mathbf { s } \\right. \\right) \\right) } \\\\ & \\cdot \\left( \\mathbb { E } \\left( \\frac { \\left\\{ 1 - \\gamma \\beta \\right\\} } { \\gamma \\beta } \\right) \\operatorname* { m i n } \\frac { x _ { 1 } ^ { \\nu \\alpha } } { x _ { 2 } ^ { \\nu \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. } \\leq \\sin \\alpha \\right) - \\mathbb { P } \\left( \\frac { \\mathcal { Z } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { \\alpha } } } \\leq \\kappa \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 189, + 738, + 445 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "where in (1) we use the union bound on both probability terms. ", + "bbox": [ + 173, + 449, + 589, + 464 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "All that remains is to calculate each remaining probability term in eq. (13.11). First, we have ", + "bbox": [ + 171, + 469, + 782, + 486 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/53f80c8221ecaa81c7d8e78bb40a6e6c17cf17cb44996a632af8395b2203b250.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } \\left( \\frac { \\epsilon _ { 1 } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { 2 } } } \\leq \\kappa \\right) = 1 - \\mathbb { P } \\left( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } < \\epsilon _ { 1 } \\right) } \\\\ & { \\stackrel { \\left( 1 \\right) } { = } 1 - \\mathbb { P } \\left( \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\left| \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { \\left( n \\right) } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\stackrel { \\left( 2 \\right) } { = } 1 - \\mathbb { P } \\left( \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\left| \\frac { x _ { 2 } ^ { \\left( n \\right) } } { x _ { 1 } ^ { \\left( n \\right) } } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) } \\\\ & { \\stackrel { \\left( 3 \\right) } { = } 1 - \\left[ \\mathbb { P } \\left( \\left| \\frac { x _ { 2 } ^ { \\left( 1 \\right) } } { x _ { 1 } ^ { \\left( 1 \\right) } } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\right] ^ { N - d _ { 0 } - 1 } \\stackrel { \\left( 4 \\right) } { \\leq } 1 - \\left[ 1 - \\frac { 2 } { \\pi } \\arctan \\left( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\right] ^ { N } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 492, + 795, + 625 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "where in (1) we used eq. (9.7), in (2) we recall that in eq. (13.10) we rotated the axes so that $\\hat { \\mathbf { w } } _ { 1 } \\propto [ 1 , 0 , 0 \\ldots , 0 ]$ axes $\\tilde { \\mathbf { w } } _ { 1 } \\propto [ 0 , 1 , 0 , 0 \\ldots , 0 ]$ , in (3) we used the independence of different $\\mathbf { x } ^ { ( n ) }$ , and in (4) we used the fact that the ratio of two independent Gaussian variables is distributed according to the symmetric Cauchy distribution, which has the cumulative distribution function $\\mathbb { P } \\left( X > \\breve { x } \\right) = \\textstyle \\frac { 1 } { 2 } - \\breve { \\frac { } { \\pi } }$ arctan $( x )$ , and therefore $\\begin{array} { r } { \\mathbb { P } \\left( | X | > x \\right) = 1 - \\frac { 2 } { \\pi } } \\end{array}$ arctan $( x )$ . ", + "bbox": [ + 173, + 645, + 826, + 719 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Second, we use eq. (13.2) ", + "bbox": [ + 174, + 723, + 346, + 738 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/5122825d4bee63b55f03956b0fef0b3e608f80c86603b81f633760e076c1000a.jpg", + "text": "$$\n\\mathbb { P } \\left( \\operatorname* { m i n } _ { n \\geq d _ { 0 } } \\frac { x _ { 1 } ^ { ( n ) } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } > \\frac { \\gamma \\beta \\sin \\alpha } { \\left( 1 - \\gamma \\beta \\right) \\kappa } \\right) > \\left[ 1 - \\frac { 2 \\gamma \\beta \\sin \\alpha } { \\left( 1 - \\gamma \\beta \\right) \\kappa B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } \\right] ^ { N } .\n$$", + "text_format": "latex", + "bbox": [ + 238, + 755, + 710, + 801 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Third, $\\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| ^ { 2 }$ is distributed according to the chi-square distribution of order $d _ { 0 }$ , so for $\\eta ^ { 2 } > d _ { 0 }$ ", + "bbox": [ + 171, + 808, + 802, + 824 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/5168ef1936f5083237cf3d70399358245ae1e25d9de608f64ce7c7fabd05d789.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| ^ { 2 } \\geq \\eta ^ { 2 } \\right) \\leq \\left( \\eta ^ { 2 } \\exp \\left( 1 - \\eta ^ { 2 } / d _ { 0 } \\right) / d _ { 0 } \\right) ^ { d _ { 0 } / 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 832, + 678, + 867 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 873, + 245, + 887 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/3bf3d3bd0fd926e4c1128b0334680dbc6eb5c6f8ec200c33c01576f2c9c75a17.jpg", + "text": "$$\n\\mathbb { P } \\left( \\operatorname* { m a x } _ { n < d _ { 0 } } \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| ^ { 2 } < \\eta ^ { 2 } \\right) > \\left[ 1 - \\left( \\eta ^ { 2 } \\exp \\left( 1 - \\eta ^ { 2 } / d _ { 0 } \\right) / d _ { 0 } \\right) ^ { d _ { 0 } / 2 } \\right] ^ { d _ { 0 } - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 240, + 893, + 710, + 929 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lastly, we bound $\\| \\tilde { \\mathbf { w } } _ { 1 } \\| = \\| \\hat { \\mathbf { w } } _ { 1 } \\|$ (from eq. (9.5)). From eq. (9.4), we have ", + "bbox": [ + 173, + 102, + 661, + 119 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/a8286c1daec6860228ebc724a5faf9f9d0c90cb6b9f266a34e50b65fdbc77fd7.jpg", + "text": "$$\n\\hat { \\mathbf { w } } _ { 1 } ^ { \\top } \\mathbf { X } _ { [ d _ { 0 } - 1 ] } = \\left[ 1 , \\ldots , 1 , 1 \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 403, + 127, + 593, + 148 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "where $\\mathbf { X } _ { [ d _ { 0 } - 1 ] }$ has a singular value decomposition ", + "bbox": [ + 171, + 156, + 506, + 172 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/563a0bb850f1f08c9d2d2921d4df61040d0bd77004e33479a9bebef3d382294d.jpg", + "text": "$$\n\\mathbf { X } _ { [ d _ { 0 } - 1 ] } = \\sum _ { i = 1 } ^ { d _ { 0 } } \\sigma _ { i } \\mathbf { u } _ { i } \\mathbf { v } _ { i } ^ { \\top } ,\n$$", + "text_format": "latex", + "bbox": [ + 416, + 183, + 580, + 227 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "with $\\sigma _ { i }$ being the singular values, and $\\mathbf { u } _ { i }$ and $\\mathbf { v } _ { i }$ being the singular vectors. The singular values are ordered from smallest to largest, and $\\sigma _ { 1 } = 0$ with $\\mathbf { u } _ { 1 } = \\tilde { \\mathbf { w } } _ { 1 }$ , from eq. (9.2). With probability 1, the other $d _ { 0 } - 1$ singular value are non-zero: they are the square roots of the eigenvalues of the random matrix $\\mathbf { \\bar { X } } _ { [ d _ { 0 } - 1 ] } ^ { \\top } \\mathbf { \\bar { X } } _ { [ d _ { 0 } - 1 ] } \\in \\mathbb { R } ^ { d _ { 0 } - 1 \\times d _ { 0 } - 1 }$ . Taking the squared norm of eq. (13.15), we have ", + "bbox": [ + 173, + 236, + 826, + 295 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/7ea05ace27fdc487100b7fc1a50eb48c7ec98bb4302ae5a2fd707cdb34115314.jpg", + "text": "$$\nd _ { 0 } - 1 = \\hat { \\mathbf { w } } _ { 1 } ^ { \\top } \\mathbf { X } _ { [ d _ { 0 } - 1 ] } \\mathbf { X } _ { [ d _ { 0 } - 1 ] } ^ { \\top } \\hat { \\mathbf { w } } _ { 1 } = \\sum _ { i = 1 } ^ { d _ { 0 } } { \\sigma _ { i } ^ { 2 } } \\left( \\mathbf { u } _ { i } ^ { \\top } \\hat { \\mathbf { w } } _ { 1 } \\right) ^ { 2 } \\geq \\sigma _ { 2 } ^ { 2 } \\left. \\hat { \\mathbf { w } } _ { 1 } \\right. ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 274, + 305, + 722, + 349 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "where the last inequality stems from the fact that $\\mathbf { u } _ { 1 } ^ { \\top } \\hat { \\mathbf { w } } _ { 1 } = \\tilde { \\mathbf { w } } _ { 1 } ^ { \\top } \\hat { \\mathbf { w } } _ { 1 } = 0$ (from eq. (9.4)), so the minimal possible value is attained when $\\mathbf { u } _ { 2 } ^ { \\top } \\hat { \\mathbf { w } } _ { 1 } = \\| \\hat { \\mathbf { w } } _ { 1 } \\|$ . The minimal nonzero singular value, $\\sigma _ { 2 }$ , can be bounded using the following result from (Rudelson & Vershynin, 2010, eq. (3.2)) ", + "bbox": [ + 173, + 358, + 826, + 404 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/2f49f2eabf49c014f680f894969c76cc1c09bdde7d99a470dd3c1e018db796f5.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\underset { { \\mathbf { r } } \\in \\mathbb { R } ^ { d _ { 0 } } } { \\operatorname* { m i n } } \\left. \\mathbf { X } _ { [ d _ { 0 } ] } \\mathbf { r } \\right. \\leq \\eta d _ { 0 } ^ { - 1 / 2 } \\right) \\leq \\eta . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 375, + 411, + 620, + 446 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Since ", + "bbox": [ + 173, + 455, + 214, + 469 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/24ed8789565e1be82b261dd6cbefb172d100528f605cd6334cde6c9718bf2eb9.jpg", + "text": "$$\n\\sigma _ { 2 } = \\operatorname* { m i n } _ { \\mathbf { r } \\in \\mathbb { R } ^ { d _ { 0 } - 1 } } \\left\\| \\mathbf { X } _ { [ d _ { 0 } - 1 ] } \\mathbf { r } \\right\\| \\geq \\operatorname* { m i n } _ { \\mathbf { r } \\in \\mathbb { R } ^ { d _ { 0 } } } \\left\\| \\mathbf { X } _ { [ d _ { 0 } ] } \\mathbf { r } \\right\\|\n$$", + "text_format": "latex", + "bbox": [ + 351, + 492, + 647, + 518 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "we have, ", + "bbox": [ + 173, + 526, + 233, + 541 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/7ccf5f14f7eabe627e6880bcd8142c70777153042977c1b68213be9c59766fd0.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left( \\sigma _ { 2 } < \\eta d _ { 0 } ^ { - 1 / 2 } \\right) \\le \\eta . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 542, + 578, + 569 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Combining this with eq. (13.16) we get ", + "bbox": [ + 173, + 575, + 434, + 592 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/ce4c72e703f6277d2ce79bf2ed1eeed7854138445f13c8d49d913e567a3b8d01.jpg", + "text": "$$\n\\mathbb { P } \\left( \\frac { \\beta \\kappa } { \\eta \\sin \\alpha } < \\| \\mathbf { w } _ { 1 } \\| \\right) \\le \\frac { \\eta d _ { 0 } } { \\beta \\kappa } \\sin \\alpha .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 599, + 619, + 636 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Lastly, combining eqs. (13.12), (13.13), (13.14) and (13.17) into eqs. (13.7) and (13.11), we get, for $\\eta ^ { 2 } > d _ { 0 }$ , ", + "bbox": [ + 173, + 643, + 825, + 674 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/3d58370a23a3762338a47d32b9b106be588aa7edfd90ba95b179cc43e140b76f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } ( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } ( \\mathbf { W } ^ { \\star } ) ) } \\\\ & { \\geq 1 - d _ { 1 } ^ { \\ast } ( 1 - ( [ 1 - ( \\eta ^ { 2 } \\exp ( 1 - \\eta ^ { 2 } / d _ { 0 } ) / d _ { 0 } ) ^ { \\hat { d } _ { 0 } / 2 } ] ^ { d _ { 0 } - 1 } - \\frac { \\eta d _ { 0 } } { \\gamma \\kappa } \\sin \\alpha ) } \\\\ & { \\Bigg . \\Bigg . \\Bigg . } \\\\ & { \\quad \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg \\Gamma ( 1 - \\frac { 2 \\gamma \\beta \\sin \\alpha } { ( 1 - \\gamma \\beta ) \\kappa B ( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } ) } \\Bigg ] ^ { N } - [ 1 - \\frac { 2 } { \\pi } \\arctan ( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } ) ] ^ { N } \\Bigg ) \\Bigg ) } \\\\ & { \\geq 1 - d _ { 1 } ^ { \\ast } ( 1 - ( [ 1 - ( \\log d _ { 0 } \\exp ( 1 - \\log d _ { 0 } ) ) ^ { d _ { 0 } / 2 } ] ^ { d _ { 0 } - 1 } - \\frac { 2 d _ { 0 } ^ { \\hat { d } _ { 0 } ^ { \\prime } / 2 } \\sqrt { \\log { d _ { 0 } } } } { d _ { 1 } ^ { \\ast } N } ) \\Bigg . } \\\\ & { \\Bigg ( [ 1 - \\sqrt { \\frac { \\kappa } { \\pi } } \\frac { 1 } { d _ { 1 } ^ { \\ast } d _ { 0 } ^ { \\dagger } N } + \\mathcal { O } ( \\frac { 1 } { N d _ { 1 } ^ { \\ast } d _ { 0 } ^ { \\dagger } } ) ] ^ { N } - 0 . 4 5 ^ { N } \\Bigg ) \\Bigg ) ~ , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 681, + 759, + 881 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "where in the last line we take $\\beta = \\gamma = \\kappa = 1 / \\sqrt { 2 }$ , $\\eta = d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } }$ , $\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)$ . Using the asymptotic expansion of the beta function $B \\left( { \\textstyle { \\frac { 1 } { 2 } } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)$ for large $x$ , we obtain, ", + "bbox": [ + 173, + 890, + 825, + 926 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/62a23dd7085777619d5b10ec548d0f306e9a43c43ac2f00b8808073e75117e73.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\arcsin \\alpha - \\operatorname* { l i } ( \\theta C \\ln ( \\theta ) ) } \\\\ & { 1 - \\operatorname* { P } ( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } ( \\operatorname { W e V } ^ { * } ) ) } \\\\ & { \\leq d _ { 1 } ^ { \\alpha } ( 1 - ( [ 1 - \\exp ( - \\frac { d _ { 0 } } { 2 } \\operatorname* { l i m } _ { | \\phi | } ( \\frac { d _ { 0 } } { \\log ( d _ { 0 } \\phi _ { 0 } ) } ) ) ] ^ { \\theta | \\alpha - 1 } - \\frac { 2 d _ { 1 } ^ { \\alpha } | ^ { 2 } \\sqrt { \\log ( \\frac { d _ { 0 } } { \\log ( d _ { 0 } ) } } ) } { d _ { 1 } ^ { \\alpha } } ) } \\\\ & { \\cdot ( [ 1 - \\sqrt { \\frac { \\alpha } { \\sqrt { \\kappa } } \\frac { 1 } { \\sqrt { \\kappa } \\sqrt { \\kappa } \\frac { 1 } { \\log ( d _ { 0 } ^ { \\alpha } ) ^ { 2 } } } } + \\mathcal { O } ( \\frac { 1 } { \\sqrt { \\kappa } \\frac { 1 } { \\sqrt { \\kappa } \\phi _ { 0 } ^ { 2 } } \\phi _ { 0 } ^ { 2 } } ) ] ^ { \\mathbf { N } } - 2 ^ { - \\kappa } ) ) } \\\\ & { = d _ { 1 } ^ { \\alpha } ( 1 - ( 1 - \\frac { 2 d _ { 1 } ^ { \\alpha } / \\sqrt { \\kappa } \\log ( \\frac { d _ { 0 } } { \\kappa } ) } { d _ { 1 } ^ { \\alpha } \\sqrt { \\kappa } } + \\mathcal { O } ( \\frac { d _ { 0 } \\tan \\phi } { d _ { 0 } \\sqrt { \\kappa } } ( - \\frac { d _ { 0 } } { 2 } \\log ( \\frac { d _ { 0 } } { \\log ( d _ { 0 } \\phi _ { 0 } ) } ) ) ) ) } \\\\ & \\cdot ( 1 - \\sqrt \\frac { \\kappa } { \\delta } \\frac { 1 } { \\lambda _ { 0 } ^ { \\alpha } \\sqrt { \\kappa } ( \\frac { 1 } { \\log ( d _ { 0 } ^ { \\alpha } ) ^ { 2 } } } + \\mathcal { O } ( \\frac { 1 } { \\kappa \\phi _ { 0 } ^ { \\alpha } \\sqrt { \\kappa } ^ { 2 } } + \\frac { 1 } { d _ { 1 } ^ { \\alpha } \\sqrt { \\kappa } ^ { 2 } \\alpha ^ { 2 } } + \\alpha _ { 1 } ^ { \\alpha } \\mathcal { A } _ { 0 } \\exp \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 173, + 99, + 816, + 363 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Thus, taking the log, and using $\\log { ( 1 - x ) } = - x + O \\left( x ^ { 2 } \\right)$ , we obtain, forsin $\\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)$ ", + "bbox": [ + 173, + 367, + 799, + 385 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "log P (X ∈ Mα (W∗)) ", + "bbox": [ + 174, + 388, + 333, + 405 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/702bd8a3fb256df7c1fab688c836b0435e6a911052aaf4806645d318d747c021.jpg", + "text": "$$\n\\begin{array} { r l } & { \\log \\mathfrak { s } : \\mathrm { ~ } \\ll \\mathcal { N } : \\mathrm { ~ } \\ll \\mathcal { N } : \\mathrm { ~ } \\ll \\mathcal { N } } \\\\ & { \\geq \\log \\left( 1 - \\sqrt { \\frac { 8 } { \\pi } } \\frac { 1 } { d _ { 0 } ^ { 1 / 2 } } - \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } + O \\left( \\frac { 1 } { d _ { 0 } ^ { 3 / 2 } } + \\frac { d _ { 0 } ^ { 1 / 4 } } { d _ { 1 } ^ { * } N } + d _ { 1 } ^ { * } 2 ^ { - N } + d _ { 0 } \\exp \\left( - \\frac { d _ { 0 } } { 2 } \\log \\left( \\frac { d _ { 0 } } { \\log d _ { 0 } } \\right) \\right) \\right) \\right) } \\\\ & { = - \\sqrt { \\frac { 8 } { \\pi } } \\frac { 1 } { d _ { 0 } ^ { 1 / 2 } } - \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } + O \\left( \\frac { 1 } { d _ { 0 } ^ { 3 / 2 } } + \\frac { d _ { 0 } ^ { 1 / 4 } } { d _ { 1 } ^ { * } N } + d _ { 1 } ^ { * } 2 ^ { - N } + d _ { 0 } \\exp \\left( - \\frac { d _ { 0 } } { 2 } \\log \\left( \\frac { d _ { 0 } } { \\log d _ { 0 } } \\right) \\right) \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 397, + 839, + 494 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Recall that $d _ { 1 } ^ { * } \\triangleq 4 \\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\rceil \\doteq N / d _ { 0 }$ . Taking the limit $N \\to \\infty$ , $d _ { 0 } \\to \\infty$ with $d _ { 1 } ^ { * } { \\dot { \\leq } } d _ { 0 } { \\dot { \\leq } } N$ , we have ", + "bbox": [ + 173, + 497, + 826, + 527 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/b1c3fb18c2cc7620290cc47cca41907a45c63e6a9d385f4a1427f11c79885f18.jpg", + "text": "$$\n\\mathbb { P } \\left( \\mathbf { X } \\notin { \\mathcal { M } } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } 1 - \\exp \\left( - \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } - \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } \\right) \\leq \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 529, + 818, + 571 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Part III ", + "text_level": 1, + "bbox": [ + 174, + 621, + 254, + 640 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Numerical Experiments - implementation details ", + "text_level": 1, + "bbox": [ + 174, + 651, + 776, + 705 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Code and trained models for CIFAR and ImageNet results is available here https://github. com/MNNsMinima/Paper. In MNIST, CIFAR and ImageNet we performed binary classification on between the original odd and even class numbers. In we performed this binary classification between digits $0 - 4$ and $5 - 9$ . Weights were initialized to be uniform with mean zero and variance $2 / d$ , where $d$ is fan-in (here the width of the previous neuron layer), as suggested in (He et al., 2015). In each epoch we randomly permuted the dataset and used the Adam (Kingma & Ba, 2014) optimization method (a variant of SGD) with $\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 , \\varepsilon = 1 0 ^ { - 8 }$ . Different learning rates and mini-batch sizes were selected for each dataset and architecture. In CIFAR10 and ImageNet we used a learning-rate of $\\alpha = 1 0 ^ { - 3 }$ and a mini-batch size of 1024; also, ZCA whitening of the training samples was done to remove correlations between the input dimensions, allowing faster convergence. We define $L$ as the number of weight layers. For the random dataset we use a mini-batch size of $\\left\\lfloor \\operatorname* { m i n } \\left( N / 2 , d / 2 \\right) \\right\\rfloor$ with learning rate $\\alpha = 0 . 1$ and 0.05, for $L = 2$ and 3, respectively. In the random data parameter scans the training was done for no more than 4000 epochs – we stopped if $\\mathbf { M C E } = 0$ was reached. ", + "bbox": [ + 174, + 728, + 826, + 924 + ], + "page_idx": 34 + } +] \ No newline at end of file diff --git a/parse/train/Hkfmn5n6W/Hkfmn5n6W_middle.json b/parse/train/Hkfmn5n6W/Hkfmn5n6W_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..6a463345cfd325607147c554ff3e3f0f65d6fcdb --- /dev/null +++ b/parse/train/Hkfmn5n6W/Hkfmn5n6W_middle.json @@ -0,0 +1,118505 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 100 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 506, + 100 + ], + "score": 1.0, + "content": "EXPONENTIALLY VANISHING SUB-OPTIMAL LOCAL", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 102, + 440, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 440, + 118 + ], + "score": 1.0, + "content": "MINIMA IN MULTILAYER NEURAL NETWORKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 142, + 210, + 468, + 276 + ], + "lines": [ + { + "bbox": [ + 141, + 210, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 469, + 223 + ], + "score": 1.0, + "content": "Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "score": 1.0, + "content": "et al. (2015)) suggest that local minima with high error are exponentially rare in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 470, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 470, + 246 + ], + "score": 1.0, + "content": "high dimensions. However, to prove low error guarantees for Multilayer Neural", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 469, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 255 + ], + "score": 1.0, + "content": "Networks (MNNs), previous works so far required either a heavily modified MNN", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "score": 1.0, + "content": "model or training method, strong assumptions on the labels (e.g., “near” linear", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 425, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 373, + 279 + ], + "score": 1.0, + "content": "separability), or an unrealistically wide hidden layer with", + "type": "text" + }, + { + "bbox": [ + 373, + 265, + 399, + 277 + ], + "score": 0.88, + "content": "\\Omega \\left( N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 265, + 425, + 279 + ], + "score": 1.0, + "content": "units.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 143, + 278, + 469, + 374 + ], + "lines": [ + { + "bbox": [ + 141, + 277, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 470, + 290 + ], + "score": 1.0, + "content": "Results: We examine a MNN with one hidden layer of piecewise linear units, a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "score": 1.0, + "content": "single output, and a quadratic loss. We prove that, with high probability in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 175, + 312 + ], + "score": 1.0, + "content": "limit of", + "type": "text" + }, + { + "bbox": [ + 176, + 300, + 212, + 309 + ], + "score": 0.91, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 299, + 469, + 312 + ], + "score": 1.0, + "content": "datapoints, the volume of differentiable regions of the empiric", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "score": 1.0, + "content": "loss containing sub-optimal differentiable local minima is exponentially vanishing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "score": 1.0, + "content": "in comparison with the same volume of global minima, given standard normal", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 332, + 469, + 352 + ], + "spans": [ + { + "bbox": [ + 140, + 332, + 218, + 352 + ], + "score": 1.0, + "content": "input of dimension", + "type": "text" + }, + { + "bbox": [ + 218, + 332, + 280, + 352 + ], + "score": 0.95, + "content": "d _ { 0 } = \\tilde { \\Omega } \\left( \\sqrt { N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 332, + 405, + 352 + ], + "score": 1.0, + "content": ", and a more realistic number of", + "type": "text" + }, + { + "bbox": [ + 406, + 334, + 469, + 349 + ], + "score": 0.93, + "content": "d _ { 1 } = \\tilde { \\Omega } \\left( N / d _ { 0 } \\right)", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 350, + 469, + 364 + ], + "spans": [ + { + "bbox": [ + 141, + 350, + 424, + 364 + ], + "score": 1.0, + "content": "hidden units. We demonstrate our results numerically: for example,", + "type": "text" + }, + { + "bbox": [ + 425, + 351, + 439, + 362 + ], + "score": 0.86, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 350, + 469, + 364 + ], + "score": 1.0, + "content": "binary", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 362, + 451, + 374 + ], + "spans": [ + { + "bbox": [ + 142, + 363, + 336, + 374 + ], + "score": 1.0, + "content": "classification training error on CIFAR with only", + "type": "text" + }, + { + "bbox": [ + 336, + 362, + 384, + 374 + ], + "score": 0.92, + "content": "N / d _ { 0 } \\approx 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 363, + 451, + 374 + ], + "score": 1.0, + "content": "hidden neurons.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "Motivation. Multilayer Neural Networks (MNNs), trained with simple variants of stochastic gra-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "dient descent (SGD), have achieved state-of-the-art performances in many areas of machine learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "(LeCun et al., 2015). However, theoretical explanations seem to lag far behind this empirical success", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "(though many hardness results exist, e.g., (Síma, 2002; Shamir, 2016)). For example, as a common", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "rule-of-the-thumb, a MNN should have at least as many parameters as training samples. However, it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "is unclear why such over-parameterized MNNs often exhibit remarkably small generalization error", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "(i.e., difference between “training error” and “test error”), even without explicit regularization (Zhang", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 164, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 164, + 507 + ], + "score": 1.0, + "content": "et al., 2017a).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "Moreover, it has long been a mystery why MNNs often achieve low training error (Dauphin et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "2014). SGD is only guaranteed to converge to critical points in which the gradient of the expected", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "loss is zero (Bottou, 1998), and, specifically, to local minima (Pemantle, 1990) (this is true also for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "regular gradient descent (Lee et al., 2016)). Since loss functions parameterized by MNN weights are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "non-convex, it is unclear why does SGD often work well – rather than converging to sub-optimal local", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "minima with high training error, which are known to exist (Fukumizu & Amari, 2000; Swirszcz et al.,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "2016). Understanding this behavior is especially relevant in important cases where SGD does get", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "score": 1.0, + "content": "stuck (He et al., 2016) – where training error may be a bottleneck in further improving performance.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "Ideally, we would like to quantify the probability to converge to a local minimum as a function", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "of the error at this minimum, where the probability is taken with the respect to the randomness", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "of the initialization of the weights, the data and SGD. Specifically, we would like to know, under", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 636, + 507, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 507, + 652 + ], + "score": 1.0, + "content": "which conditions this probability is very small if the error is high, as was observed empirically (e.g.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "(Dauphin et al., 2014; Goodfellow et al., 2015)). However, this seems to be a daunting task for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "realistic MNNs, since it requires a characterization of the sizes and distributions of the basins of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 671, + 230, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 230, + 682 + ], + "score": 1.0, + "content": "attraction for all local minima.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Previous works (Dauphin et al., 2014; Choromanska et al., 2015), based on statistical physics", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "analogies, suggested a simpler property of MNNs: that with high probability, local minima with", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "high error diminish exponentially with the number of parameters. Though proving such a geometric", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 734 + ], + "score": 1.0, + "content": "property with realistic assumptions would not guarantee convergence to global minima, it appears to", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 100 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 506, + 100 + ], + "score": 1.0, + "content": "EXPONENTIALLY VANISHING SUB-OPTIMAL LOCAL", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 102, + 440, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 440, + 118 + ], + "score": 1.0, + "content": "MINIMA IN MULTILAYER NEURAL NETWORKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 111, + 135, + 245, + 158 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 142, + 210, + 468, + 276 + ], + "lines": [ + { + "bbox": [ + 141, + 210, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 469, + 223 + ], + "score": 1.0, + "content": "Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "score": 1.0, + "content": "et al. (2015)) suggest that local minima with high error are exponentially rare in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 470, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 470, + 246 + ], + "score": 1.0, + "content": "high dimensions. However, to prove low error guarantees for Multilayer Neural", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 469, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 255 + ], + "score": 1.0, + "content": "Networks (MNNs), previous works so far required either a heavily modified MNN", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "score": 1.0, + "content": "model or training method, strong assumptions on the labels (e.g., “near” linear", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 425, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 373, + 279 + ], + "score": 1.0, + "content": "separability), or an unrealistically wide hidden layer with", + "type": "text" + }, + { + "bbox": [ + 373, + 265, + 399, + 277 + ], + "score": 0.88, + "content": "\\Omega \\left( N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 265, + 425, + 279 + ], + "score": 1.0, + "content": "units.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 141, + 210, + 470, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 278, + 469, + 374 + ], + "lines": [ + { + "bbox": [ + 141, + 277, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 470, + 290 + ], + "score": 1.0, + "content": "Results: We examine a MNN with one hidden layer of piecewise linear units, a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "score": 1.0, + "content": "single output, and a quadratic loss. We prove that, with high probability in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 175, + 312 + ], + "score": 1.0, + "content": "limit of", + "type": "text" + }, + { + "bbox": [ + 176, + 300, + 212, + 309 + ], + "score": 0.91, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 299, + 469, + 312 + ], + "score": 1.0, + "content": "datapoints, the volume of differentiable regions of the empiric", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "score": 1.0, + "content": "loss containing sub-optimal differentiable local minima is exponentially vanishing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 333 + ], + "score": 1.0, + "content": "in comparison with the same volume of global minima, given standard normal", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 332, + 469, + 352 + ], + "spans": [ + { + "bbox": [ + 140, + 332, + 218, + 352 + ], + "score": 1.0, + "content": "input of dimension", + "type": "text" + }, + { + "bbox": [ + 218, + 332, + 280, + 352 + ], + "score": 0.95, + "content": "d _ { 0 } = \\tilde { \\Omega } \\left( \\sqrt { N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 332, + 405, + 352 + ], + "score": 1.0, + "content": ", and a more realistic number of", + "type": "text" + }, + { + "bbox": [ + 406, + 334, + 469, + 349 + ], + "score": 0.93, + "content": "d _ { 1 } = \\tilde { \\Omega } \\left( N / d _ { 0 } \\right)", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 350, + 469, + 364 + ], + "spans": [ + { + "bbox": [ + 141, + 350, + 424, + 364 + ], + "score": 1.0, + "content": "hidden units. We demonstrate our results numerically: for example,", + "type": "text" + }, + { + "bbox": [ + 425, + 351, + 439, + 362 + ], + "score": 0.86, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 350, + 469, + 364 + ], + "score": 1.0, + "content": "binary", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 362, + 451, + 374 + ], + "spans": [ + { + "bbox": [ + 142, + 363, + 336, + 374 + ], + "score": 1.0, + "content": "classification training error on CIFAR with only", + "type": "text" + }, + { + "bbox": [ + 336, + 362, + 384, + 374 + ], + "score": 0.92, + "content": "N / d _ { 0 } \\approx 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 363, + 451, + 374 + ], + "score": 1.0, + "content": "hidden neurons.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 140, + 277, + 470, + 374 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "Motivation. Multilayer Neural Networks (MNNs), trained with simple variants of stochastic gra-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "dient descent (SGD), have achieved state-of-the-art performances in many areas of machine learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "(LeCun et al., 2015). However, theoretical explanations seem to lag far behind this empirical success", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "(though many hardness results exist, e.g., (Síma, 2002; Shamir, 2016)). For example, as a common", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "rule-of-the-thumb, a MNN should have at least as many parameters as training samples. However, it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "is unclear why such over-parameterized MNNs often exhibit remarkably small generalization error", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "(i.e., difference between “training error” and “test error”), even without explicit regularization (Zhang", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 164, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 164, + 507 + ], + "score": 1.0, + "content": "et al., 2017a).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 417, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "Moreover, it has long been a mystery why MNNs often achieve low training error (Dauphin et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "2014). SGD is only guaranteed to converge to critical points in which the gradient of the expected", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "loss is zero (Bottou, 1998), and, specifically, to local minima (Pemantle, 1990) (this is true also for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "regular gradient descent (Lee et al., 2016)). Since loss functions parameterized by MNN weights are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "non-convex, it is unclear why does SGD often work well – rather than converging to sub-optimal local", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "minima with high training error, which are known to exist (Fukumizu & Amari, 2000; Swirszcz et al.,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "2016). Understanding this behavior is especially relevant in important cases where SGD does get", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "score": 1.0, + "content": "stuck (He et al., 2016) – where training error may be a bottleneck in further improving performance.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 510, + 507, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "Ideally, we would like to quantify the probability to converge to a local minimum as a function", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "of the error at this minimum, where the probability is taken with the respect to the randomness", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "of the initialization of the weights, the data and SGD. Specifically, we would like to know, under", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 636, + 507, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 507, + 652 + ], + "score": 1.0, + "content": "which conditions this probability is very small if the error is high, as was observed empirically (e.g.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "(Dauphin et al., 2014; Goodfellow et al., 2015)). However, this seems to be a daunting task for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "realistic MNNs, since it requires a characterization of the sizes and distributions of the basins of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 671, + 230, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 230, + 682 + ], + "score": 1.0, + "content": "attraction for all local minima.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 604, + 507, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Previous works (Dauphin et al., 2014; Choromanska et al., 2015), based on statistical physics", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "analogies, suggested a simpler property of MNNs: that with high probability, local minima with", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "high error diminish exponentially with the number of parameters. Though proving such a geometric", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 734 + ], + "score": 1.0, + "content": "property with realistic assumptions would not guarantee convergence to global minima, it appears to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "be a necessary first step in this direction (see discussion on section 6). It was therefore pointed out", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "as an open problem at the Conference of Learning Theory (COLT) 2015. However, one has to be", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 434, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 434, + 118 + ], + "score": 1.0, + "content": "careful and use realistic MMN architectures, or this problem becomes “too easy”.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 688, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "be a necessary first step in this direction (see discussion on section 6). It was therefore pointed out", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "as an open problem at the Conference of Learning Theory (COLT) 2015. However, one has to be", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 434, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 434, + 118 + ], + "score": 1.0, + "content": "careful and use realistic MMN architectures, or this problem becomes “too easy”.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "For example, one can easily achieve zero training error (Nilsson, 1965; Baum, 1988) – if the MNN’s", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "last hidden layer has more neurons than training samples. Such extremely wide MNNs are easy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "to optimize (Yu, 1992; Huang et al., 2006; Livni et al., 2014; Haeffele & Vidal, 2015; Shen, 2016;", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "Nguyen & Hein, 2017). In this case, the hidden layer becomes linearly separable in classification", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "tasks, with high probability over the random initialization of the weights. Thus, by training the last", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "layer we get to a global minimum (zero training error). However, such extremely wide layers are", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "not very useful, since they result in a huge number of weights, and serious overfitting issues. Also,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 507, + 210 + ], + "score": 1.0, + "content": "training only the last layer seems to take little advantage of the inherently non-linear nature of MNNs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 507, + 228 + ], + "score": 1.0, + "content": "Therefore, in this paper we are interested to understand the properties of local and global minima,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "but at a more practical number of parameters – and when at least two weight layers are trained. For", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "example, Alexnet (Krizhevsky, 2014) is trained using about 1.2 million ImageNet examples, and has", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "about 60 million parameters – 16 million of these in the two last weight layers. Suppose we now", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "train the last two weight layers in such an over-parameterized MNN. When do the sub-optimal local", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 397, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 397, + 282 + ], + "score": 1.0, + "content": "minima become exponentially rare in comparison to the global minima?", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "Main contributions. We focus on MNNs with a single hidden layer and piecewise linear units,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "optimized using the Mean Square Error (MSE) in a supervised binary classification task (Section", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 163, + 328 + ], + "score": 1.0, + "content": "2). We define", + "type": "text" + }, + { + "bbox": [ + 164, + 316, + 174, + 325 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 315, + 316, + 328 + ], + "score": 1.0, + "content": "as the number of training samples,", + "type": "text" + }, + { + "bbox": [ + 317, + 316, + 326, + 326 + ], + "score": 0.85, + "content": "d _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 315, + 404, + 328 + ], + "score": 1.0, + "content": "as the width of the", + "type": "text" + }, + { + "bbox": [ + 405, + 316, + 409, + 325 + ], + "score": 0.51, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "-th activation layer, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 323, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 107, + 327, + 158, + 340 + ], + "score": 0.92, + "content": "g \\left( x \\right) \\dot { < } h \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 323, + 396, + 347 + ], + "score": 1.0, + "content": "as an asymptotic inequality in the leading order (formally:", + "type": "text" + }, + { + "bbox": [ + 396, + 326, + 485, + 343 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { x \\infty } \\frac { \\log g ( x ) } { \\log h ( x ) } < 1 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 323, + 506, + 347 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "examine Differentiable Local Minima (DLMs) of the MSE: sub-optimal DLMs where at least a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 153, + 365 + ], + "score": 1.0, + "content": "fraction of", + "type": "text" + }, + { + "bbox": [ + 153, + 353, + 179, + 363 + ], + "score": 0.9, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "of the training samples are classified incorrectly, and global minima where all", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 235, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 235, + 376 + ], + "score": 1.0, + "content": "samples are classified correctly.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 503, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "Our main result, Theorem 10, states that, with high probability, the total volume of the differentiable", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "regions of the MSE containing sub-optimal DLMs is exponentially vanishing in comparison to the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 280, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 280, + 414 + ], + "score": 1.0, + "content": "same volume of global minima, given that:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 109, + 416, + 489, + 429 + ], + "lines": [ + { + "bbox": [ + 108, + 416, + 490, + 430 + ], + "spans": [ + { + "bbox": [ + 108, + 416, + 490, + 430 + ], + "score": 1.0, + "content": "Assumption 1. The datapoints (MNN inputs) are sampled from a standard normal distribution.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 109, + 431, + 484, + 444 + ], + "lines": [ + { + "bbox": [ + 107, + 430, + 484, + 445 + ], + "spans": [ + { + "bbox": [ + 107, + 430, + 172, + 445 + ], + "score": 1.0, + "content": "Assumption 2.", + "type": "text" + }, + { + "bbox": [ + 172, + 432, + 208, + 442 + ], + "score": 0.82, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 430, + 211, + 445 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 212, + 432, + 241, + 444 + ], + "score": 0.87, + "content": "d _ { 0 } \\left( N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 430, + 259, + 445 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 432, + 289, + 444 + ], + "score": 0.92, + "content": "d _ { 1 } \\left( N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 430, + 345, + 445 + ], + "score": 1.0, + "content": "increase with", + "type": "text" + }, + { + "bbox": [ + 346, + 432, + 356, + 442 + ], + "score": 0.76, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 430, + 384, + 445 + ], + "score": 1.0, + "content": ", while", + "type": "text" + }, + { + "bbox": [ + 384, + 432, + 424, + 444 + ], + "score": 0.92, + "content": "\\epsilon \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 430, + 484, + 445 + ], + "score": 1.0, + "content": "is a constant1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 353, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 351, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 294, + 461 + ], + "score": 1.0, + "content": "Assumption 3. The input dimension scales as", + "type": "text" + }, + { + "bbox": [ + 294, + 446, + 347, + 460 + ], + "score": 0.92, + "content": "\\sqrt { N } \\dot { < } d _ { 0 } \\dot { \\le } N", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 446, + 351, + 461 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 108, + 463, + 304, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 305, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 305, + 477 + ], + "score": 1.0, + "content": "Assumption 4. The hidden layer width scales as", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 478, + 347, + 507 + ], + "lines": [ + { + "bbox": [ + 264, + 478, + 347, + 507 + ], + "spans": [ + { + "bbox": [ + 264, + 478, + 347, + 507 + ], + "score": 0.95, + "content": "\\frac { N \\log ^ { 4 } N } { d _ { 0 } } \\dot { < } d _ { 1 } \\dot { < } N .", + "type": "interline_equation", + "image_path": "309480aa6f3b645542d7d9fa450a1d45c09539fa5867092e47f66f400a8912c2.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 264, + 478, + 347, + 492.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 264, + 492.5, + 347, + 507.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "Importantly, we use a standard, unmodified, MNN model, and make no assumptions on the target", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 434, + 535 + ], + "score": 1.0, + "content": "function. Moreover, as the number of parameters in the MNN is approximately", + "type": "text" + }, + { + "bbox": [ + 434, + 522, + 455, + 533 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 521, + 505, + 535 + ], + "score": 1.0, + "content": ", we require", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 312, + 547 + ], + "score": 1.0, + "content": "only “asymptotically mild” over-parameterization:", + "type": "text" + }, + { + "bbox": [ + 312, + 532, + 379, + 545 + ], + "score": 0.93, + "content": "d _ { 0 } d _ { 1 } { \\dot { > } } N \\log ^ { 4 } N", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "from eq. (1.1). For example, if", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 545, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 546, + 141, + 557 + ], + "score": 0.9, + "content": "d _ { 0 } \\propto N", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 545, + 212, + 559 + ], + "score": 1.0, + "content": ", we only require", + "type": "text" + }, + { + "bbox": [ + 212, + 545, + 260, + 557 + ], + "score": 0.92, + "content": "d _ { 1 } \\dot { > } \\log ^ { 4 } N", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 545, + 507, + 559 + ], + "score": 1.0, + "content": "neurons. This improves over previously known results (Yu,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016; Nguyen & Hein, 2017) – which require an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 239, + 582 + ], + "score": 1.0, + "content": "extremely wide hidden layer with", + "type": "text" + }, + { + "bbox": [ + 239, + 568, + 273, + 579 + ], + "score": 0.92, + "content": "d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 567, + 345, + 582 + ], + "score": 1.0, + "content": "neurons (and thus", + "type": "text" + }, + { + "bbox": [ + 345, + 568, + 365, + 579 + ], + "score": 0.89, + "content": "N d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "parameters) to remove sub-optimal", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 579, + 249, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 249, + 591 + ], + "score": 1.0, + "content": "local minima with high probability.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "In section 5 we validate our results numerically. We show that indeed the training error becomes low", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 282, + 619 + ], + "score": 1.0, + "content": "when the number of parameters is close to", + "type": "text" + }, + { + "bbox": [ + 283, + 607, + 293, + 617 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 606, + 505, + 619 + ], + "score": 1.0, + "content": ". For example, with binary classification on CIFAR", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 344, + 631 + ], + "score": 1.0, + "content": "and ImageNet, with only 16 and 105 hidden neurons (about", + "type": "text" + }, + { + "bbox": [ + 344, + 618, + 369, + 629 + ], + "score": 0.91, + "content": "N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "), respectively, we obtain less then", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 129, + 639 + ], + "score": 0.86, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "training error. Additionally, we find that convergence to non-differentiable critical points does", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 640, + 232, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 232, + 653 + ], + "score": 1.0, + "content": "not appear to be very common.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 656, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "Lastly, in section 6 we discuss our results might be extended, such as how to apply them to “mildly”", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 667, + 240, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 240, + 680 + ], + "score": 1.0, + "content": "non-differentiable critical points.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + }, + { + "type": "text", + "bbox": [ + 108, + 691, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "Plausibility of assumptions. Assumption 1 is common in this type of analysis (Andoni et al., 2014;", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 700, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 714 + ], + "score": 1.0, + "content": "Choromanska et al., 2015; Xie et al., 2016; Tian, 2017; Brutzkus & Globerson, 2017). At first it may", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 721, + 394, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 395, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 284, + 734 + ], + "score": 1.0, + "content": "1For brevity we will usually keep implicit the", + "type": "text" + }, + { + "bbox": [ + 285, + 723, + 293, + 730 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 720, + 355, + 734 + ], + "score": 1.0, + "content": "dependencies of", + "type": "text" + }, + { + "bbox": [ + 356, + 723, + 365, + 731 + ], + "score": 0.87, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 720, + 381, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 381, + 722, + 391, + 731 + ], + "score": 0.85, + "content": "d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 720, + 395, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "For example, one can easily achieve zero training error (Nilsson, 1965; Baum, 1988) – if the MNN’s", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "last hidden layer has more neurons than training samples. Such extremely wide MNNs are easy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "to optimize (Yu, 1992; Huang et al., 2006; Livni et al., 2014; Haeffele & Vidal, 2015; Shen, 2016;", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "Nguyen & Hein, 2017). In this case, the hidden layer becomes linearly separable in classification", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "tasks, with high probability over the random initialization of the weights. Thus, by training the last", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "layer we get to a global minimum (zero training error). However, such extremely wide layers are", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "not very useful, since they result in a huge number of weights, and serious overfitting issues. Also,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 507, + 210 + ], + "score": 1.0, + "content": "training only the last layer seems to take little advantage of the inherently non-linear nature of MNNs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 122, + 507, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 507, + 228 + ], + "score": 1.0, + "content": "Therefore, in this paper we are interested to understand the properties of local and global minima,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "but at a more practical number of parameters – and when at least two weight layers are trained. For", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "example, Alexnet (Krizhevsky, 2014) is trained using about 1.2 million ImageNet examples, and has", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "about 60 million parameters – 16 million of these in the two last weight layers. Suppose we now", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "train the last two weight layers in such an over-parameterized MNN. When do the sub-optimal local", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 397, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 397, + 282 + ], + "score": 1.0, + "content": "minima become exponentially rare in comparison to the global minima?", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 213, + 507, + 282 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "Main contributions. We focus on MNNs with a single hidden layer and piecewise linear units,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "optimized using the Mean Square Error (MSE) in a supervised binary classification task (Section", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 163, + 328 + ], + "score": 1.0, + "content": "2). We define", + "type": "text" + }, + { + "bbox": [ + 164, + 316, + 174, + 325 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 315, + 316, + 328 + ], + "score": 1.0, + "content": "as the number of training samples,", + "type": "text" + }, + { + "bbox": [ + 317, + 316, + 326, + 326 + ], + "score": 0.85, + "content": "d _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 315, + 404, + 328 + ], + "score": 1.0, + "content": "as the width of the", + "type": "text" + }, + { + "bbox": [ + 405, + 316, + 409, + 325 + ], + "score": 0.51, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "-th activation layer, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 323, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 107, + 327, + 158, + 340 + ], + "score": 0.92, + "content": "g \\left( x \\right) \\dot { < } h \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 323, + 396, + 347 + ], + "score": 1.0, + "content": "as an asymptotic inequality in the leading order (formally:", + "type": "text" + }, + { + "bbox": [ + 396, + 326, + 485, + 343 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { x \\infty } \\frac { \\log g ( x ) } { \\log h ( x ) } < 1 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 323, + 506, + 347 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "examine Differentiable Local Minima (DLMs) of the MSE: sub-optimal DLMs where at least a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 153, + 365 + ], + "score": 1.0, + "content": "fraction of", + "type": "text" + }, + { + "bbox": [ + 153, + 353, + 179, + 363 + ], + "score": 0.9, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "of the training samples are classified incorrectly, and global minima where all", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 235, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 235, + 376 + ], + "score": 1.0, + "content": "samples are classified correctly.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 293, + 506, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 503, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "Our main result, Theorem 10, states that, with high probability, the total volume of the differentiable", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "regions of the MSE containing sub-optimal DLMs is exponentially vanishing in comparison to the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 280, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 280, + 414 + ], + "score": 1.0, + "content": "same volume of global minima, given that:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 380, + 505, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 416, + 489, + 429 + ], + "lines": [ + { + "bbox": [ + 108, + 416, + 490, + 430 + ], + "spans": [ + { + "bbox": [ + 108, + 416, + 490, + 430 + ], + "score": 1.0, + "content": "Assumption 1. The datapoints (MNN inputs) are sampled from a standard normal distribution.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 108, + 416, + 490, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 431, + 484, + 444 + ], + "lines": [ + { + "bbox": [ + 107, + 430, + 484, + 445 + ], + "spans": [ + { + "bbox": [ + 107, + 430, + 172, + 445 + ], + "score": 1.0, + "content": "Assumption 2.", + "type": "text" + }, + { + "bbox": [ + 172, + 432, + 208, + 442 + ], + "score": 0.82, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 430, + 211, + 445 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 212, + 432, + 241, + 444 + ], + "score": 0.87, + "content": "d _ { 0 } \\left( N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 430, + 259, + 445 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 432, + 289, + 444 + ], + "score": 0.92, + "content": "d _ { 1 } \\left( N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 430, + 345, + 445 + ], + "score": 1.0, + "content": "increase with", + "type": "text" + }, + { + "bbox": [ + 346, + 432, + 356, + 442 + ], + "score": 0.76, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 430, + 384, + 445 + ], + "score": 1.0, + "content": ", while", + "type": "text" + }, + { + "bbox": [ + 384, + 432, + 424, + 444 + ], + "score": 0.92, + "content": "\\epsilon \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 430, + 484, + 445 + ], + "score": 1.0, + "content": "is a constant1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 107, + 430, + 484, + 445 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 353, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 351, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 294, + 461 + ], + "score": 1.0, + "content": "Assumption 3. The input dimension scales as", + "type": "text" + }, + { + "bbox": [ + 294, + 446, + 347, + 460 + ], + "score": 0.92, + "content": "\\sqrt { N } \\dot { < } d _ { 0 } \\dot { \\le } N", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 446, + 351, + 461 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 446, + 351, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 463, + 304, + 474 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 305, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 305, + 477 + ], + "score": 1.0, + "content": "Assumption 4. The hidden layer width scales as", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 462, + 305, + 477 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 478, + 347, + 507 + ], + "lines": [ + { + "bbox": [ + 264, + 478, + 347, + 507 + ], + "spans": [ + { + "bbox": [ + 264, + 478, + 347, + 507 + ], + "score": 0.95, + "content": "\\frac { N \\log ^ { 4 } N } { d _ { 0 } } \\dot { < } d _ { 1 } \\dot { < } N .", + "type": "interline_equation", + "image_path": "309480aa6f3b645542d7d9fa450a1d45c09539fa5867092e47f66f400a8912c2.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 264, + 478, + 347, + 492.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 264, + 492.5, + 347, + 507.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "Importantly, we use a standard, unmodified, MNN model, and make no assumptions on the target", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 434, + 535 + ], + "score": 1.0, + "content": "function. Moreover, as the number of parameters in the MNN is approximately", + "type": "text" + }, + { + "bbox": [ + 434, + 522, + 455, + 533 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 521, + 505, + 535 + ], + "score": 1.0, + "content": ", we require", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 312, + 547 + ], + "score": 1.0, + "content": "only “asymptotically mild” over-parameterization:", + "type": "text" + }, + { + "bbox": [ + 312, + 532, + 379, + 545 + ], + "score": 0.93, + "content": "d _ { 0 } d _ { 1 } { \\dot { > } } N \\log ^ { 4 } N", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "from eq. (1.1). For example, if", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 545, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 546, + 141, + 557 + ], + "score": 0.9, + "content": "d _ { 0 } \\propto N", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 545, + 212, + 559 + ], + "score": 1.0, + "content": ", we only require", + "type": "text" + }, + { + "bbox": [ + 212, + 545, + 260, + 557 + ], + "score": 0.92, + "content": "d _ { 1 } \\dot { > } \\log ^ { 4 } N", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 545, + 507, + 559 + ], + "score": 1.0, + "content": "neurons. This improves over previously known results (Yu,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016; Nguyen & Hein, 2017) – which require an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 239, + 582 + ], + "score": 1.0, + "content": "extremely wide hidden layer with", + "type": "text" + }, + { + "bbox": [ + 239, + 568, + 273, + 579 + ], + "score": 0.92, + "content": "d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 567, + 345, + 582 + ], + "score": 1.0, + "content": "neurons (and thus", + "type": "text" + }, + { + "bbox": [ + 345, + 568, + 365, + 579 + ], + "score": 0.89, + "content": "N d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "parameters) to remove sub-optimal", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 579, + 249, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 249, + 591 + ], + "score": 1.0, + "content": "local minima with high probability.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 510, + 507, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "In section 5 we validate our results numerically. We show that indeed the training error becomes low", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 282, + 619 + ], + "score": 1.0, + "content": "when the number of parameters is close to", + "type": "text" + }, + { + "bbox": [ + 283, + 607, + 293, + 617 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 606, + 505, + 619 + ], + "score": 1.0, + "content": ". For example, with binary classification on CIFAR", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 344, + 631 + ], + "score": 1.0, + "content": "and ImageNet, with only 16 and 105 hidden neurons (about", + "type": "text" + }, + { + "bbox": [ + 344, + 618, + 369, + 629 + ], + "score": 0.91, + "content": "N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "), respectively, we obtain less then", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 129, + 639 + ], + "score": 0.86, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "training error. Additionally, we find that convergence to non-differentiable critical points does", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 640, + 232, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 232, + 653 + ], + "score": 1.0, + "content": "not appear to be very common.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 595, + 505, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 656, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "Lastly, in section 6 we discuss our results might be extended, such as how to apply them to “mildly”", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 667, + 240, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 240, + 680 + ], + "score": 1.0, + "content": "non-differentiable critical points.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 655, + 507, + 680 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 691, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "Plausibility of assumptions. Assumption 1 is common in this type of analysis (Andoni et al., 2014;", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 700, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 714 + ], + "score": 1.0, + "content": "Choromanska et al., 2015; Xie et al., 2016; Tian, 2017; Brutzkus & Globerson, 2017). At first it may", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "appear rather unrealistic, especially since the inputs are correlated in typical datasets. However, this", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "no-correlation part of the assumption may seem more justified if we recall that datasets are many", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "times whitened before being used as inputs. Alternatively, if, as in our motivating question, we", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "consider the input to the our simple MNN to be the output of the previous layers of a deep MNN", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "score": 1.0, + "content": "with fixed random weights, this also tends to de-correlate inputs (Poole et al., 2016, Figure 3). The", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "remaining part of assumption 1, that the distribution is normal, is indeed strong, but might be relaxed", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 353, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 353, + 161 + ], + "score": 1.0, + "content": "in the future, e.g. using central limit theorem type arguments.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 47.5, + "bbox_fs": [ + 106, + 690, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "appear rather unrealistic, especially since the inputs are correlated in typical datasets. However, this", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "no-correlation part of the assumption may seem more justified if we recall that datasets are many", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "times whitened before being used as inputs. Alternatively, if, as in our motivating question, we", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "consider the input to the our simple MNN to be the output of the previous layers of a deep MNN", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "score": 1.0, + "content": "with fixed random weights, this also tends to de-correlate inputs (Poole et al., 2016, Figure 3). The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "remaining part of assumption 1, that the distribution is normal, is indeed strong, but might be relaxed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 353, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 353, + 161 + ], + "score": 1.0, + "content": "in the future, e.g. using central limit theorem type arguments.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "In assumption 2 we use this asymptotic limit to simplify our proofs and final results. Multiplicative", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 234, + 190 + ], + "score": 1.0, + "content": "constants and finite (yet large)", + "type": "text" + }, + { + "bbox": [ + 234, + 177, + 244, + 186 + ], + "score": 0.76, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "results can be found by inspection of the proofs. We assume a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 425, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 164, + 201 + ], + "score": 1.0, + "content": "constant error", + "type": "text" + }, + { + "bbox": [ + 164, + 190, + 170, + 197 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 187, + 267, + 201 + ], + "score": 1.0, + "content": "since typically the limit", + "type": "text" + }, + { + "bbox": [ + 267, + 187, + 293, + 198 + ], + "score": 0.9, + "content": "\\epsilon 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 187, + 425, + 201 + ], + "score": 1.0, + "content": "is avoided to prevent overfitting.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 271, + 217 + ], + "score": 1.0, + "content": "In assumption 3, for simplicity we have", + "type": "text" + }, + { + "bbox": [ + 272, + 203, + 299, + 215 + ], + "score": 0.91, + "content": "d _ { 0 } { \\dot { \\le } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 204, + 373, + 217 + ], + "score": 1.0, + "content": ", since in the case", + "type": "text" + }, + { + "bbox": [ + 373, + 204, + 408, + 215 + ], + "score": 0.91, + "content": "d _ { 0 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "the input is generically", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "linearly separable, and sub-optimal local minima are not a problem (Gori & Tesi, 1992; Safran &", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 507, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 263, + 241 + ], + "score": 1.0, + "content": "Shamir, 2016). Additionally, we have", + "type": "text" + }, + { + "bbox": [ + 264, + 226, + 299, + 239 + ], + "score": 0.92, + "content": "\\sqrt { N } \\dot { < } d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 226, + 507, + 241 + ], + "score": 1.0, + "content": ", which seems very reasonable, since for example,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 238, + 417, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 238, + 185, + 250 + ], + "score": 0.7, + "content": "d _ { 0 } / N \\approx 0 . 0 1 6 , 0 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 238, + 417, + 252 + ], + "score": 1.0, + "content": "61 and 0.055 MNIST, CIFAR and ImageNet, respectively.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 270, + 267 + ], + "score": 1.0, + "content": "In assumption 4, for simplicity we have", + "type": "text" + }, + { + "bbox": [ + 271, + 255, + 298, + 266 + ], + "score": 0.91, + "content": "d _ { 1 } \\dot { < } N", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 255, + 424, + 267 + ], + "score": 1.0, + "content": ", since, as mentioned earlier, if", + "type": "text" + }, + { + "bbox": [ + 425, + 255, + 459, + 266 + ], + "score": 0.91, + "content": "d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "the hidden", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "layer is linearly separable with high probability, which removes sub-optimal local minima. The other", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 133, + 291 + ], + "score": 1.0, + "content": "bound", + "type": "text" + }, + { + "bbox": [ + 134, + 277, + 201, + 290 + ], + "score": 0.92, + "content": "N \\log ^ { 4 } N { \\dot { < } } d _ { 0 } d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "is our main innovation – a large over-parameterization which is nevertheless", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 311, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 311, + 302 + ], + "score": 1.0, + "content": "asymptotically mild and improves previous results.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 506, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 331 + ], + "score": 1.0, + "content": "Previous work. So far, general low (training or test) error guarantees for MNNs could not be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "found – unless the underlying model (MNN) or learning method (SGD or its variants) have been", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "significantly modified. For example, (Dauphin et al., 2014) made an analogy with high-dimensional", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 349, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 506, + 364 + ], + "score": 1.0, + "content": "random Gaussian functions, local minima with high error are exponentially rare in high dimensions;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 360, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 507, + 375 + ], + "score": 1.0, + "content": "(Choromanska et al., 2015; Kawaguchi, 2016) replaced the units (activation functions) with indepen-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "dent random variables; (Pennington & Bahri, 2017) replaces the weights and error residuals with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "independent random variables; (Baldi, 1989; Saxe et al., 2014; Hardt & Ma, 2017; Lu & Kawaguchi,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 507, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 507, + 408 + ], + "score": 1.0, + "content": "2017; Zhou & Feng, 2017) used linear units; (Zhang et al., 2017b) used unconventional units (e.g.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 405, + 507, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 275, + 418 + ], + "score": 1.0, + "content": "polynomials) and very large hidden layers", + "type": "text" + }, + { + "bbox": [ + 276, + 405, + 339, + 417 + ], + "score": 0.84, + "content": "( d _ { 1 } = \\mathrm { p o l y } \\left( d _ { 0 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 405, + 379, + 418 + ], + "score": 1.0, + "content": ", typically", + "type": "text" + }, + { + "bbox": [ + 379, + 405, + 403, + 416 + ], + "score": 0.86, + "content": "\\gg N", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 405, + 507, + 418 + ], + "score": 1.0, + "content": "); (Brutzkus & Globerson,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 493, + 429 + ], + "score": 1.0, + "content": "2017; Du et al., 2017; Shalev-Shwartz et al., 2017) used a modified convnet model with less then", + "type": "text" + }, + { + "bbox": [ + 493, + 416, + 504, + 427 + ], + "score": 0.86, + "content": "d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 439 + ], + "score": 1.0, + "content": "parameters (therefore, not a universal approximator (Cybenko, 1989; Hornik, 1991)); (Tian, 2017;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "Soltanolkotabi et al., 2017; Li & Yuan, 2017) assume the weights are initialized very close to those of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "the teacher generating the labels; and (Janzamin et al., 2015; Zhong et al., 2017) use a non-standard", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "tensor method during training. Such approaches fall short of explaining the widespread success of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 471, + 290, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 290, + 483 + ], + "score": 1.0, + "content": "standard MNN models and training practices.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 506, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "Other works placed strong assumptions on the target functions. For example, to prove convergence of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "the training error near the global minimum, (Gori & Tesi, 1992) assumed linearly separable datasets,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "while (Safran & Shamir, 2016) assumed strong clustering of the targets (“near” linear-separability).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 521, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 253, + 533 + ], + "score": 1.0, + "content": "Also, (Andoni et al., 2014) showed a", + "type": "text" + }, + { + "bbox": [ + 253, + 523, + 259, + 532 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 521, + 506, + 533 + ], + "score": 1.0, + "content": "-degree polynomial is learnable by a MNN, if the hidden layer", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 102, + 530, + 510, + 552 + ], + "spans": [ + { + "bbox": [ + 102, + 530, + 158, + 552 + ], + "score": 1.0, + "content": "is very large", + "type": "text" + }, + { + "bbox": [ + 159, + 532, + 219, + 551 + ], + "score": 0.94, + "content": "( d _ { 1 } = \\Omega \\left( d _ { 0 } ^ { 6 p } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 530, + 259, + 552 + ], + "score": 1.0, + "content": ", typically", + "type": "text" + }, + { + "bbox": [ + 259, + 535, + 284, + 546 + ], + "score": 0.85, + "content": "\\gg N", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 530, + 510, + 552 + ], + "score": 1.0, + "content": ") so learning the last weight layer is sufficient. However,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "these are not the typical regimes in which MNNs are required or used. In contrast, we make no", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "assumption on the target function. Other closely related results (Soudry & Carmon, 2016; Xie et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "2016) also used unrealistic assumptions, are discussed in section 6, in regards to the details of our", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 583, + 160, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 160, + 593 + ], + "score": 1.0, + "content": "main results.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 108, + 599, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "Therefore, in contrast to previous works, the assumptions in this paper are applicable in some", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 609, + 507, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 507, + 624 + ], + "score": 1.0, + "content": "situations (e.g., Gaussian input) where a MNN trained using SGD might be used and be useful (e.g.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 622, + 287, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 287, + 633 + ], + "score": 1.0, + "content": "have a lower test error then a linear classier).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 108, + 653, + 288, + 665 + ], + "lines": [ + { + "bbox": [ + 104, + 651, + 290, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 290, + 667 + ], + "score": 1.0, + "content": "2 PRELIMINARIES AND NOTATION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "Model. We examine a Multilayer Neural Network (MNN) with a single hidden layer and a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 377, + 705 + ], + "score": 1.0, + "content": "scalar output. The MNN is trained on a finite training set of", + "type": "text" + }, + { + "bbox": [ + 377, + 693, + 387, + 703 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 691, + 479, + 705 + ], + "score": 1.0, + "content": "datapoints (features)", + "type": "text" + }, + { + "bbox": [ + 480, + 691, + 505, + 703 + ], + "score": 0.53, + "content": "\\textbf { X } \\triangleq", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 703, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 705, + 221, + 720 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] \\ \\in \\ \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 703, + 323, + 721 + ], + "score": 1.0, + "content": "with their target labels", + "type": "text" + }, + { + "bbox": [ + 323, + 704, + 480, + 720 + ], + "score": 0.9, + "content": "\\mathbf { y } \\ \\triangleq \\ \\left[ y ^ { ( 1 ) } , \\ldots , y ^ { ( N ) } \\right] ^ { \\intercal } \\ \\in \\ \\left\\{ 0 , 1 \\right\\} ^ { N } -", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 703, + 506, + 721 + ], + "score": 1.0, + "content": "each", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 718, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 189, + 734 + ], + "score": 1.0, + "content": "datapoint-label pair", + "type": "text" + }, + { + "bbox": [ + 189, + 719, + 237, + 734 + ], + "score": 0.91, + "content": "\\left( \\mathbf { x } ^ { ( n ) } , y ^ { ( n ) } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 718, + 461, + 734 + ], + "score": 1.0, + "content": "is independently sampled from some joint distribution", + "type": "text" + }, + { + "bbox": [ + 461, + 721, + 484, + 733 + ], + "score": 0.9, + "content": "\\mathbb { P } _ { X , Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 718, + 505, + 734 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 83, + 506, + 161 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "In assumption 2 we use this asymptotic limit to simplify our proofs and final results. Multiplicative", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 234, + 190 + ], + "score": 1.0, + "content": "constants and finite (yet large)", + "type": "text" + }, + { + "bbox": [ + 234, + 177, + 244, + 186 + ], + "score": 0.76, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "results can be found by inspection of the proofs. We assume a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 425, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 164, + 201 + ], + "score": 1.0, + "content": "constant error", + "type": "text" + }, + { + "bbox": [ + 164, + 190, + 170, + 197 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 187, + 267, + 201 + ], + "score": 1.0, + "content": "since typically the limit", + "type": "text" + }, + { + "bbox": [ + 267, + 187, + 293, + 198 + ], + "score": 0.9, + "content": "\\epsilon 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 187, + 425, + 201 + ], + "score": 1.0, + "content": "is avoided to prevent overfitting.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 164, + 506, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 271, + 217 + ], + "score": 1.0, + "content": "In assumption 3, for simplicity we have", + "type": "text" + }, + { + "bbox": [ + 272, + 203, + 299, + 215 + ], + "score": 0.91, + "content": "d _ { 0 } { \\dot { \\le } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 204, + 373, + 217 + ], + "score": 1.0, + "content": ", since in the case", + "type": "text" + }, + { + "bbox": [ + 373, + 204, + 408, + 215 + ], + "score": 0.91, + "content": "d _ { 0 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "the input is generically", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "linearly separable, and sub-optimal local minima are not a problem (Gori & Tesi, 1992; Safran &", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 507, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 263, + 241 + ], + "score": 1.0, + "content": "Shamir, 2016). Additionally, we have", + "type": "text" + }, + { + "bbox": [ + 264, + 226, + 299, + 239 + ], + "score": 0.92, + "content": "\\sqrt { N } \\dot { < } d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 226, + 507, + 241 + ], + "score": 1.0, + "content": ", which seems very reasonable, since for example,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 238, + 417, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 238, + 185, + 250 + ], + "score": 0.7, + "content": "d _ { 0 } / N \\approx 0 . 0 1 6 , 0 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 238, + 417, + 252 + ], + "score": 1.0, + "content": "61 and 0.055 MNIST, CIFAR and ImageNet, respectively.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 203, + 507, + 252 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 270, + 267 + ], + "score": 1.0, + "content": "In assumption 4, for simplicity we have", + "type": "text" + }, + { + "bbox": [ + 271, + 255, + 298, + 266 + ], + "score": 0.91, + "content": "d _ { 1 } \\dot { < } N", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 255, + 424, + 267 + ], + "score": 1.0, + "content": ", since, as mentioned earlier, if", + "type": "text" + }, + { + "bbox": [ + 425, + 255, + 459, + 266 + ], + "score": 0.91, + "content": "d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "the hidden", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "layer is linearly separable with high probability, which removes sub-optimal local minima. The other", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 133, + 291 + ], + "score": 1.0, + "content": "bound", + "type": "text" + }, + { + "bbox": [ + 134, + 277, + 201, + 290 + ], + "score": 0.92, + "content": "N \\log ^ { 4 } N { \\dot { < } } d _ { 0 } d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "is our main innovation – a large over-parameterization which is nevertheless", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 311, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 311, + 302 + ], + "score": 1.0, + "content": "asymptotically mild and improves previous results.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 255, + 506, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 506, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 331 + ], + "score": 1.0, + "content": "Previous work. So far, general low (training or test) error guarantees for MNNs could not be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "found – unless the underlying model (MNN) or learning method (SGD or its variants) have been", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "significantly modified. For example, (Dauphin et al., 2014) made an analogy with high-dimensional", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 349, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 506, + 364 + ], + "score": 1.0, + "content": "random Gaussian functions, local minima with high error are exponentially rare in high dimensions;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 360, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 507, + 375 + ], + "score": 1.0, + "content": "(Choromanska et al., 2015; Kawaguchi, 2016) replaced the units (activation functions) with indepen-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "dent random variables; (Pennington & Bahri, 2017) replaces the weights and error residuals with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "independent random variables; (Baldi, 1989; Saxe et al., 2014; Hardt & Ma, 2017; Lu & Kawaguchi,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 507, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 507, + 408 + ], + "score": 1.0, + "content": "2017; Zhou & Feng, 2017) used linear units; (Zhang et al., 2017b) used unconventional units (e.g.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 405, + 507, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 275, + 418 + ], + "score": 1.0, + "content": "polynomials) and very large hidden layers", + "type": "text" + }, + { + "bbox": [ + 276, + 405, + 339, + 417 + ], + "score": 0.84, + "content": "( d _ { 1 } = \\mathrm { p o l y } \\left( d _ { 0 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 405, + 379, + 418 + ], + "score": 1.0, + "content": ", typically", + "type": "text" + }, + { + "bbox": [ + 379, + 405, + 403, + 416 + ], + "score": 0.86, + "content": "\\gg N", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 405, + 507, + 418 + ], + "score": 1.0, + "content": "); (Brutzkus & Globerson,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 493, + 429 + ], + "score": 1.0, + "content": "2017; Du et al., 2017; Shalev-Shwartz et al., 2017) used a modified convnet model with less then", + "type": "text" + }, + { + "bbox": [ + 493, + 416, + 504, + 427 + ], + "score": 0.86, + "content": "d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 439 + ], + "score": 1.0, + "content": "parameters (therefore, not a universal approximator (Cybenko, 1989; Hornik, 1991)); (Tian, 2017;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "Soltanolkotabi et al., 2017; Li & Yuan, 2017) assume the weights are initialized very close to those of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "the teacher generating the labels; and (Janzamin et al., 2015; Zhong et al., 2017) use a non-standard", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "tensor method during training. Such approaches fall short of explaining the widespread success of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 471, + 290, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 290, + 483 + ], + "score": 1.0, + "content": "standard MNN models and training practices.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 316, + 507, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 506, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "Other works placed strong assumptions on the target functions. For example, to prove convergence of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "the training error near the global minimum, (Gori & Tesi, 1992) assumed linearly separable datasets,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "while (Safran & Shamir, 2016) assumed strong clustering of the targets (“near” linear-separability).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 521, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 253, + 533 + ], + "score": 1.0, + "content": "Also, (Andoni et al., 2014) showed a", + "type": "text" + }, + { + "bbox": [ + 253, + 523, + 259, + 532 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 521, + 506, + 533 + ], + "score": 1.0, + "content": "-degree polynomial is learnable by a MNN, if the hidden layer", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 102, + 530, + 510, + 552 + ], + "spans": [ + { + "bbox": [ + 102, + 530, + 158, + 552 + ], + "score": 1.0, + "content": "is very large", + "type": "text" + }, + { + "bbox": [ + 159, + 532, + 219, + 551 + ], + "score": 0.94, + "content": "( d _ { 1 } = \\Omega \\left( d _ { 0 } ^ { 6 p } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 530, + 259, + 552 + ], + "score": 1.0, + "content": ", typically", + "type": "text" + }, + { + "bbox": [ + 259, + 535, + 284, + 546 + ], + "score": 0.85, + "content": "\\gg N", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 530, + 510, + 552 + ], + "score": 1.0, + "content": ") so learning the last weight layer is sufficient. However,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "these are not the typical regimes in which MNNs are required or used. In contrast, we make no", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "assumption on the target function. Other closely related results (Soudry & Carmon, 2016; Xie et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "2016) also used unrealistic assumptions, are discussed in section 6, in regards to the details of our", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 583, + 160, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 160, + 593 + ], + "score": 1.0, + "content": "main results.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 102, + 486, + 510, + 593 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 599, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "Therefore, in contrast to previous works, the assumptions in this paper are applicable in some", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 609, + 507, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 507, + 624 + ], + "score": 1.0, + "content": "situations (e.g., Gaussian input) where a MNN trained using SGD might be used and be useful (e.g.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 622, + 287, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 287, + 633 + ], + "score": 1.0, + "content": "have a lower test error then a linear classier).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 598, + 507, + 633 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 653, + 288, + 665 + ], + "lines": [ + { + "bbox": [ + 104, + 651, + 290, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 290, + 667 + ], + "score": 1.0, + "content": "2 PRELIMINARIES AND NOTATION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "Model. 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Thus, the output of the MNN on the entire dataset", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 177, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 177, + 191 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 187, + 347, + 203 + ], + "lines": [ + { + "bbox": [ + 264, + 187, + 347, + 203 + ], + "spans": [ + { + "bbox": [ + 264, + 187, + 347, + 203 + ], + "score": 0.92, + "content": "{ f } \\left( { { \\mathbf { W } } { \\mathbf { X } } } \\right) ^ { \\top } { \\mathbf { z } } \\in \\mathbb { R } ^ { N } .", + "type": "interline_equation", + "image_path": "31d1af998fb84aae4abcba4d1bf0020e8eebacdbc76bd0f09db83f51d01ccacb.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 264, + 187, + 347, + 203 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 342, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 341, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 341, + 220 + ], + "score": 1.0, + "content": "We use the mean square error (MSE) loss for optimization", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 222, + 399, + 246 + ], + "lines": [ + { + "bbox": [ + 210, + 222, + 399, + 246 + ], + "spans": [ + { + "bbox": [ + 210, + 222, + 399, + 246 + ], + "score": 0.93, + "content": "\\operatorname { M S E } \\triangleq \\frac { 1 } { N } \\left. \\mathbf { e } \\right. ^ { 2 } \\mathrm { w i t h } \\mathbf { e } \\triangleq \\mathbf { y } - f \\left( \\mathbf { W } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ,", + "type": "interline_equation", + "image_path": "e30e527d3a5961bf2ad196071dc9d2f7110c46f6ce76c43d60553cb325fef718.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 210, + 222, + 399, + 246 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 133, + 264 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 252, + 147, + 264 + ], + "score": 0.88, + "content": "\\lVert \\cdot \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "is the standard euclidean norm. Also, we measure the empiric performance as the fraction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 379, + 275 + ], + "score": 1.0, + "content": "of samples that are classified correctly using a decision threshold at", + "type": "text" + }, + { + "bbox": [ + 380, + 264, + 412, + 274 + ], + "score": 0.9, + "content": "y = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 263, + 505, + 275 + ], + "score": 1.0, + "content": ", and denote this as the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 272, + 217, + 286 + ], + "score": 1.0, + "content": "mean classification error, or", + "type": "text" + }, + { + "bbox": [ + 217, + 273, + 244, + 284 + ], + "score": 0.82, + "content": "\\mathbf { M C E } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 272, + 337, + 286 + ], + "score": 1.0, + "content": ". Note that the variables", + "type": "text" + }, + { + "bbox": [ + 338, + 276, + 344, + 284 + ], + "score": 0.4, + "content": "\\mathbf { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 272, + 506, + 286 + ], + "score": 1.0, + "content": ", MSE, MCE and other related variables", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 259, + 299 + ], + "score": 1.0, + "content": "(e.g., their derivatives) all depend on", + "type": "text" + }, + { + "bbox": [ + 259, + 285, + 306, + 296 + ], + "score": 0.91, + "content": "\\mathbf { W } , \\mathbf { z } , \\mathbf { X } , \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 284, + 325, + 299 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 325, + 287, + 332, + 297 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 284, + 506, + 299 + ], + "score": 1.0, + "content": ", but we keep this dependency implicit, to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 296, + 221, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 221, + 308 + ], + "score": 1.0, + "content": "avoid cumbersome notation.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 316, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 420, + 336 + ], + "score": 1.0, + "content": "Additional Notation. We define g (x) 0 } \\\\ { \\rho } & { \\mathrm { , ~ i f ~ } u < 0 } \\end{array} \\right. , } \\end{array}", + "type": "interline_equation", + "image_path": "7d14c788deba198f0ff60c39cd0408b4df4d1bb3ecf62504fd6993a55ad18adc.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 122, + 402, + 136.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 207, + 136.0, + 402, + 150.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 505, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 144, + 169 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 144, + 156, + 169, + 168 + ], + "score": 0.91, + "content": "\\rho \\neq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 155, + 372, + 169 + ], + "score": 1.0, + "content": "(so the MNN is non-linear) , where both functions", + "type": "text" + }, + { + "bbox": [ + 372, + 156, + 379, + 167 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 155, + 397, + 169 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 397, + 158, + 404, + 166 + ], + "score": 0.75, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 155, + 505, + 169 + ], + "score": 1.0, + "content": "operate component-wise", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 166, + 507, + 181 + ], + "spans": [ + { + "bbox": [ + 104, + 166, + 206, + 181 + ], + "score": 1.0, + "content": "(e.g., for any matrix M:", + "type": "text" + }, + { + "bbox": [ + 206, + 167, + 296, + 181 + ], + "score": 0.92, + "content": "\\left( f \\left( \\mathbf { M } \\right) \\right) _ { i j } = f \\left( M _ { i j } \\right) )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 166, + 507, + 181 + ], + "score": 1.0, + "content": ". Thus, the output of the MNN on the entire dataset", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 177, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 177, + 191 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 155, + 507, + 191 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 187, + 347, + 203 + ], + "lines": [ + { + "bbox": [ + 264, + 187, + 347, + 203 + ], + "spans": [ + { + "bbox": [ + 264, + 187, + 347, + 203 + ], + "score": 0.92, + "content": "{ f } \\left( { { \\mathbf { W } } { \\mathbf { X } } } \\right) ^ { \\top } { \\mathbf { z } } \\in \\mathbb { R } ^ { N } .", + "type": "interline_equation", + "image_path": "31d1af998fb84aae4abcba4d1bf0020e8eebacdbc76bd0f09db83f51d01ccacb.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 264, + 187, + 347, + 203 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 342, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 341, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 341, + 220 + ], + "score": 1.0, + "content": "We use the mean square error (MSE) loss for optimization", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 204, + 341, + 220 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 222, + 399, + 246 + ], + "lines": [ + { + "bbox": [ + 210, + 222, + 399, + 246 + ], + "spans": [ + { + "bbox": [ + 210, + 222, + 399, + 246 + ], + "score": 0.93, + "content": "\\operatorname { M S E } \\triangleq \\frac { 1 } { N } \\left. \\mathbf { e } \\right. ^ { 2 } \\mathrm { w i t h } \\mathbf { e } \\triangleq \\mathbf { y } - f \\left( \\mathbf { W } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ,", + "type": "interline_equation", + "image_path": "e30e527d3a5961bf2ad196071dc9d2f7110c46f6ce76c43d60553cb325fef718.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 210, + 222, + 399, + 246 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 133, + 264 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 252, + 147, + 264 + ], + "score": 0.88, + "content": "\\lVert \\cdot \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "is the standard euclidean norm. Also, we measure the empiric performance as the fraction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 379, + 275 + ], + "score": 1.0, + "content": "of samples that are classified correctly using a decision threshold at", + "type": "text" + }, + { + "bbox": [ + 380, + 264, + 412, + 274 + ], + "score": 0.9, + "content": "y = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 263, + 505, + 275 + ], + "score": 1.0, + "content": ", and denote this as the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 272, + 217, + 286 + ], + "score": 1.0, + "content": "mean classification error, or", + "type": "text" + }, + { + "bbox": [ + 217, + 273, + 244, + 284 + ], + "score": 0.82, + "content": "\\mathbf { M C E } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 272, + 337, + 286 + ], + "score": 1.0, + "content": ". Note that the variables", + "type": "text" + }, + { + "bbox": [ + 338, + 276, + 344, + 284 + ], + "score": 0.4, + "content": "\\mathbf { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 272, + 506, + 286 + ], + "score": 1.0, + "content": ", MSE, MCE and other related variables", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 259, + 299 + ], + "score": 1.0, + "content": "(e.g., their derivatives) all depend on", + "type": "text" + }, + { + "bbox": [ + 259, + 285, + 306, + 296 + ], + "score": 0.91, + "content": "\\mathbf { W } , \\mathbf { z } , \\mathbf { X } , \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 284, + 325, + 299 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 325, + 287, + 332, + 297 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 284, + 506, + 299 + ], + "score": 1.0, + "content": ", but we keep this dependency implicit, to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 296, + 221, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 221, + 308 + ], + "score": 1.0, + "content": "avoid cumbersome notation.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 252, + 506, + 308 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 316, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 420, + 336 + ], + "score": 1.0, + "content": "Additional Notation. We define g (x) \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 395, + 505, + 408 + ], + "score": 1.0, + "content": ". We would like to compare the angular volume", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 407, + 482, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 482, + 419 + ], + "score": 1.0, + "content": "(definition 3) corresponding to both types of DLMs. Thus, we make the following definitions.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 208, + 434 + ], + "score": 1.0, + "content": "Definition 4. We define3", + "type": "text" + }, + { + "bbox": [ + 209, + 420, + 264, + 432 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\epsilon } \\subset \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 418, + 506, + 434 + ], + "score": 1.0, + "content": "as the union of differentiable regions containing sub-optimal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 157, + 446 + ], + "score": 1.0, + "content": "DLMs with", + "type": "text" + }, + { + "bbox": [ + 157, + 433, + 198, + 443 + ], + "score": 0.86, + "content": "\\mathrm { M C E } > \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 429, + 222, + 446 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 222, + 432, + 275, + 443 + ], + "score": 0.91, + "content": "\\mathcal { G } \\subset \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 429, + 506, + 446 + ], + "score": 1.0, + "content": "as the union of differentiable regions containing global", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 444, + 206, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 159, + 455 + ], + "score": 1.0, + "content": "minima with", + "type": "text" + }, + { + "bbox": [ + 159, + 444, + 202, + 454 + ], + "score": 0.87, + "content": "\\mathrm { M C E } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 444, + 206, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 104, + 454, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 454, + 263, + 473 + ], + "score": 1.0, + "content": "Definition 5. We define the constant", + "type": "text" + }, + { + "bbox": [ + 263, + 461, + 274, + 470 + ], + "score": 0.83, + "content": "\\gamma _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 454, + 288, + 473 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 288, + 456, + 473, + 471 + ], + "score": 0.58, + "content": "\\begin{array} { r } { \\gamma _ { \\epsilon } \\triangleq 0 . 2 3 \\operatorname* { m a x } \\left[ \\operatorname* { l i m } _ { N \\to \\infty } \\left( d _ { 0 } \\left( N \\right) / N \\right) , \\epsilon \\right] ^ { 3 / 4 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 454, + 484, + 473 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 485, + 458, + 506, + 471 + ], + "score": 0.8, + "content": "\\rho \\neq", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 470, + 249, + 484 + ], + "spans": [ + { + "bbox": [ + 107, + 471, + 132, + 483 + ], + "score": 0.9, + "content": "\\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 470, + 152, + 484 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 153, + 470, + 210, + 483 + ], + "score": 0.91, + "content": "\\gamma _ { \\epsilon } \\triangleq 0 . 2 3 \\epsilon ^ { 3 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 470, + 220, + 484 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 221, + 471, + 245, + 483 + ], + "score": 0.9, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 470, + 249, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "In this section, we use assumptions 1-4 (stated in section 1) to bound the angular volume of the region", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 118, + 514 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 504, + 311, + 516 + ], + "score": 1.0, + "content": "encapsulating all sub-optimal DLMs, the region", + "type": "text" + }, + { + "bbox": [ + 311, + 504, + 318, + 514 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 504, + 505, + 516 + ], + "score": 1.0, + "content": ", encapsulating all global minima, and the ratio", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 515, + 177, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 177, + 526 + ], + "score": 1.0, + "content": "between the two.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 105, + 537, + 504, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 552 + ], + "score": 1.0, + "content": "Angular volume of sub-optimal DLMs. First, in appendix section 8 we prove the following upper", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 192, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 192, + 561 + ], + "score": 1.0, + "content": "bound in expectation", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 504, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 578 + ], + "score": 1.0, + "content": "Theorem 6. Given assumptions 1-4, the expected angular volume of sub-optimal DLMs, with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 575, + 310, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 167, + 587 + ], + "score": 0.86, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 575, + 287, + 588 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 287, + 576, + 297, + 586 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 575, + 310, + 588 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 592, + 410, + 614 + ], + "lines": [ + { + "bbox": [ + 200, + 592, + 410, + 614 + ], + "spans": [ + { + "bbox": [ + 200, + 592, + 410, + 614 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "f9fa76de6d9e0120508d973d5a1ef18a3e2f783e5e614e6039ab64a4e03c9b56.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 200, + 592, + 410, + 614 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 375, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 376, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 376, + 641 + ], + "score": 1.0, + "content": "and, using Markov inequality, its immediate probabilistic corollary", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 506, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 242, + 654 + ], + "score": 1.0, + "content": "Corollary 7. Given assumptions", + "type": "text" + }, + { + "bbox": [ + 242, + 641, + 248, + 650 + ], + "score": 0.35, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 639, + 291, + 654 + ], + "score": 1.0, + "content": "-4, for any", + "type": "text" + }, + { + "bbox": [ + 292, + 641, + 316, + 651 + ], + "score": 0.88, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 639, + 452, + 654 + ], + "score": 1.0, + "content": "(possibly a vanishing function of", + "type": "text" + }, + { + "bbox": [ + 452, + 641, + 462, + 651 + ], + "score": 0.74, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 639, + 506, + 654 + ], + "score": 1.0, + "content": "), we have,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 175, + 664 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 176, + 651, + 200, + 662 + ], + "score": 0.83, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 651, + 424, + 664 + ], + "score": 1.0, + "content": ", that the angular volume of sub-optimal DLMs, with", + "type": "text" + }, + { + "bbox": [ + 424, + 652, + 491, + 662 + ], + "score": 0.82, + "content": "\\mathrm { M C E } > \\epsilon > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 651, + 506, + 664 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 237, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 214, + 675 + ], + "score": 1.0, + "content": "exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 214, + 663, + 224, + 672 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 662, + 237, + 675 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 679, + 396, + 703 + ], + "lines": [ + { + "bbox": [ + 213, + 679, + 396, + 703 + ], + "spans": [ + { + "bbox": [ + 213, + 679, + 396, + 703 + ], + "score": 0.92, + "content": "{ \\mathcal { V } } \\left( { \\mathcal { L } } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - 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Clearly, for all", + "type": "text" + }, + { + "bbox": [ + 447, + 82, + 505, + 95 + ], + "score": 0.86, + "content": "\\mathbf { W } \\in { \\mathcal { D } } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "the MSE is differentiable, so any local minimum can be non-differentiable only if it is not in any", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "differentiable region. 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At all DLMs in", + "type": "text" + }, + { + "bbox": [ + 216, + 131, + 250, + 143 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 129, + 449, + 144 + ], + "score": 1.0, + "content": "the residual error e is identical, and furthermore", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 129, + 449, + 144 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 272, + 148, + 339, + 162 + ], + "lines": [ + { + "bbox": [ + 272, + 148, + 339, + 162 + ], + "spans": [ + { + "bbox": [ + 272, + 148, + 339, + 162 + ], + "score": 0.91, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0 .", + "type": "interline_equation", + "image_path": "a19b2b07c5c5407e540ec1e9105b73491fbd3ab0c176b2c68e2e243a8564f55f.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 272, + 148, + 339, + 162 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 432, + 185 + ], + "score": 1.0, + "content": "The proof is directly derived from the first order necessary condition of DLMs", + "type": "text" + }, + { + "bbox": [ + 432, + 173, + 484, + 184 + ], + "score": 0.85, + "content": "\\boldsymbol { \\nabla } \\mathrm { M S E } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 173, + 505, + 185 + ], + "score": 1.0, + "content": ") and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 360, + 196 + ], + "score": 1.0, + "content": "their stability. Note that Lemma 2 constrains the residual error", + "type": "text" + }, + { + "bbox": [ + 360, + 186, + 367, + 194 + ], + "score": 0.39, + "content": "\\mathbf { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "in the over-parameterized regime:", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 107, + 195, + 150, + 207 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 195, + 275, + 208 + ], + "score": 1.0, + "content": ". In this case eq. (3.2) implies", + "type": "text" + }, + { + "bbox": [ + 276, + 196, + 302, + 205 + ], + "score": 0.9, + "content": "\\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 195, + 337, + 208 + ], + "score": 1.0, + "content": ", if rank", + "type": "text" + }, + { + "bbox": [ + 338, + 195, + 396, + 207 + ], + "score": 0.91, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) = N", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 195, + 506, + 208 + ], + "score": 1.0, + "content": ". Therefore, we must have", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 128, + 218 + ], + "score": 1.0, + "content": "rank", + "type": "text" + }, + { + "bbox": [ + 128, + 207, + 186, + 218 + ], + "score": 0.86, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) < N", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 207, + 505, + 218 + ], + "score": 1.0, + "content": "for sub-optimal DLMs to exist. Later, we use similar rank-based constraints to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "bound the volume of differentiable regions which contain DLMs with high error. Next, we define this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 177, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 177, + 241 + ], + "score": 1.0, + "content": "volume formally.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 173, + 506, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 365, + 266 + ], + "score": 1.0, + "content": "Angular Volume. From its definition (eq. (3.1)) each region", + "type": "text" + }, + { + "bbox": [ + 365, + 252, + 399, + 264 + ], + "score": 0.74, + "content": "\\mathcal { D } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 251, + 506, + 266 + ], + "score": 1.0, + "content": "has an infinite volume in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 137, + 273 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 261, + 506, + 277 + ], + "score": 1.0, + "content": ": if we multiply a row of W by a positive scalar, we remain in the same region. Only by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 191, + 287 + ], + "score": 1.0, + "content": "rotating the rows of", + "type": "text" + }, + { + "bbox": [ + 191, + 275, + 205, + 285 + ], + "score": 0.28, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "can we move between regions. We measure this “angular volume” of a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 283, + 507, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 354, + 300 + ], + "score": 1.0, + "content": "region in a probabilistic way: we randomly sample the rows of", + "type": "text" + }, + { + "bbox": [ + 354, + 285, + 367, + 295 + ], + "score": 0.27, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 283, + 507, + 300 + ], + "score": 1.0, + "content": "from an isotropic distribution, e.g.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 183, + 310 + ], + "score": 1.0, + "content": "standard Gaussian:", + "type": "text" + }, + { + "bbox": [ + 183, + 297, + 219, + 307 + ], + "score": 0.87, + "content": "\\mathbf { W } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 295, + 369, + 310 + ], + "score": 1.0, + "content": ", and measure the probability to fall in", + "type": "text" + }, + { + "bbox": [ + 370, + 296, + 403, + 308 + ], + "score": 0.78, + "content": "\\mathcal { D } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 295, + 506, + 310 + ], + "score": 1.0, + "content": ", arriving to the following", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 251, + 507, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 396, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 397, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 224, + 325 + ], + "score": 1.0, + "content": "Definition 3. For any region", + "type": "text" + }, + { + "bbox": [ + 224, + 310, + 276, + 322 + ], + "score": 0.92, + "content": "\\mathcal { R } \\subset \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 309, + 375, + 325 + ], + "score": 1.0, + "content": ". The angular volume of", + "type": "text" + }, + { + "bbox": [ + 376, + 312, + 385, + 321 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 309, + 397, + 325 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 309, + 397, + 325 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 329, + 365, + 344 + ], + "lines": [ + { + "bbox": [ + 246, + 329, + 365, + 344 + ], + "spans": [ + { + "bbox": [ + 246, + 329, + 365, + 344 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { V } \\left( \\mathcal { R } \\right) \\triangleq \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\mathcal { R } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "58553a6b85c8c93d4a21cb9508f3208dbfcef404f9052335dc75c383e8a2582f.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 246, + 329, + 365, + 344 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 359, + 206, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 207, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 207, + 374 + ], + "score": 1.0, + "content": "4 MAIN RESULTS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 294, + 397 + ], + "score": 1.0, + "content": "Some of the DLMs are global minima, in which", + "type": "text" + }, + { + "bbox": [ + 294, + 385, + 320, + 395 + ], + "score": 0.89, + "content": "{ \\bf e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 384, + 350, + 397 + ], + "score": 1.0, + "content": "and so,", + "type": "text" + }, + { + "bbox": [ + 350, + 384, + 427, + 395 + ], + "score": 0.9, + "content": "\\mathrm { M C E } = \\mathrm { M S E } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 384, + 505, + 397 + ], + "score": 1.0, + "content": ", while other DLMs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 260, + 408 + ], + "score": 1.0, + "content": "are sub-optimal local minima in which", + "type": "text" + }, + { + "bbox": [ + 261, + 396, + 317, + 406 + ], + "score": 0.9, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 395, + 505, + 408 + ], + "score": 1.0, + "content": ". We would like to compare the angular volume", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 407, + 482, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 482, + 419 + ], + "score": 1.0, + "content": "(definition 3) corresponding to both types of DLMs. Thus, we make the following definitions.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 384, + 505, + 419 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 208, + 434 + ], + "score": 1.0, + "content": "Definition 4. We define3", + "type": "text" + }, + { + "bbox": [ + 209, + 420, + 264, + 432 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\epsilon } \\subset \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 418, + 506, + 434 + ], + "score": 1.0, + "content": "as the union of differentiable regions containing sub-optimal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 157, + 446 + ], + "score": 1.0, + "content": "DLMs with", + "type": "text" + }, + { + "bbox": [ + 157, + 433, + 198, + 443 + ], + "score": 0.86, + "content": "\\mathrm { M C E } > \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 429, + 222, + 446 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 222, + 432, + 275, + 443 + ], + "score": 0.91, + "content": "\\mathcal { G } \\subset \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 429, + 506, + 446 + ], + "score": 1.0, + "content": "as the union of differentiable regions containing global", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 444, + 206, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 159, + 455 + ], + "score": 1.0, + "content": "minima with", + "type": "text" + }, + { + "bbox": [ + 159, + 444, + 202, + 454 + ], + "score": 0.87, + "content": "\\mathrm { M C E } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 444, + 206, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 418, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 104, + 454, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 454, + 263, + 473 + ], + "score": 1.0, + "content": "Definition 5. We define the constant", + "type": "text" + }, + { + "bbox": [ + 263, + 461, + 274, + 470 + ], + "score": 0.83, + "content": "\\gamma _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 454, + 288, + 473 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 288, + 456, + 473, + 471 + ], + "score": 0.58, + "content": "\\begin{array} { r } { \\gamma _ { \\epsilon } \\triangleq 0 . 2 3 \\operatorname* { m a x } \\left[ \\operatorname* { l i m } _ { N \\to \\infty } \\left( d _ { 0 } \\left( N \\right) / N \\right) , \\epsilon \\right] ^ { 3 / 4 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 454, + 484, + 473 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 485, + 458, + 506, + 471 + ], + "score": 0.8, + "content": "\\rho \\neq", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 470, + 249, + 484 + ], + "spans": [ + { + "bbox": [ + 107, + 471, + 132, + 483 + ], + "score": 0.9, + "content": "\\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 470, + 152, + 484 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 153, + 470, + 210, + 483 + ], + "score": 0.91, + "content": "\\gamma _ { \\epsilon } \\triangleq 0 . 2 3 \\epsilon ^ { 3 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 470, + 220, + 484 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 221, + 471, + 245, + 483 + ], + "score": 0.9, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 470, + 249, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 454, + 506, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "In this section, we use assumptions 1-4 (stated in section 1) to bound the angular volume of the region", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 118, + 514 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 504, + 311, + 516 + ], + "score": 1.0, + "content": "encapsulating all sub-optimal DLMs, the region", + "type": "text" + }, + { + "bbox": [ + 311, + 504, + 318, + 514 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 504, + 505, + 516 + ], + "score": 1.0, + "content": ", encapsulating all global minima, and the ratio", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 515, + 177, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 177, + 526 + ], + "score": 1.0, + "content": "between the two.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 491, + 505, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 537, + 504, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 552 + ], + "score": 1.0, + "content": "Angular volume of sub-optimal DLMs. First, in appendix section 8 we prove the following upper", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 192, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 192, + 561 + ], + "score": 1.0, + "content": "bound in expectation", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 536, + 505, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 504, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 578 + ], + "score": 1.0, + "content": "Theorem 6. Given assumptions 1-4, the expected angular volume of sub-optimal DLMs, with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 575, + 310, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 167, + 587 + ], + "score": 0.86, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 575, + 287, + 588 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 287, + 576, + 297, + 586 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 575, + 310, + 588 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 563, + 506, + 588 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 592, + 410, + 614 + ], + "lines": [ + { + "bbox": [ + 200, + 592, + 410, + 614 + ], + "spans": [ + { + "bbox": [ + 200, + 592, + 410, + 614 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "f9fa76de6d9e0120508d973d5a1ef18a3e2f783e5e614e6039ab64a4e03c9b56.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 200, + 592, + 410, + 614 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 375, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 376, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 376, + 641 + ], + "score": 1.0, + "content": "and, using Markov inequality, its immediate probabilistic corollary", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 622, + 376, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 506, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 242, + 654 + ], + "score": 1.0, + "content": "Corollary 7. Given assumptions", + "type": "text" + }, + { + "bbox": [ + 242, + 641, + 248, + 650 + ], + "score": 0.35, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 639, + 291, + 654 + ], + "score": 1.0, + "content": "-4, for any", + "type": "text" + }, + { + "bbox": [ + 292, + 641, + 316, + 651 + ], + "score": 0.88, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 639, + 452, + 654 + ], + "score": 1.0, + "content": "(possibly a vanishing function of", + "type": "text" + }, + { + "bbox": [ + 452, + 641, + 462, + 651 + ], + "score": 0.74, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 639, + 506, + 654 + ], + "score": 1.0, + "content": "), we have,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 175, + 664 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 176, + 651, + 200, + 662 + ], + "score": 0.83, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 651, + 424, + 664 + ], + "score": 1.0, + "content": ", that the angular volume of sub-optimal DLMs, with", + "type": "text" + }, + { + "bbox": [ + 424, + 652, + 491, + 662 + ], + "score": 0.82, + "content": "\\mathrm { M C E } > \\epsilon > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 651, + 506, + 664 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 237, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 214, + 675 + ], + "score": 1.0, + "content": "exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 214, + 663, + 224, + 672 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 662, + 237, + 675 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 639, + 506, + 675 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 679, + 396, + 703 + ], + "lines": [ + { + "bbox": [ + 213, + 679, + 396, + 703 + ], + "spans": [ + { + "bbox": [ + 213, + 679, + 396, + 703 + ], + "score": 0.92, + "content": "{ \\mathcal { V } } \\left( { \\mathcal { L } } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)", + "type": "interline_equation", + "image_path": "c6785e691f46a9da4a76dd01010670a877cd784df1e265a9f23c23c259459f39.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 213, + 679, + 396, + 703 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 418, + 95 + ], + "score": 1.0, + "content": "Proof idea of Theorem 6: we first show that in differentiable regions with", + "type": "text" + }, + { + "bbox": [ + 419, + 83, + 485, + 93 + ], + "score": 0.9, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 83, + 505, + 95 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 200, + 106 + ], + "score": 1.0, + "content": "condition in Lemma 2,", + "type": "text" + }, + { + "bbox": [ + 201, + 93, + 261, + 105 + ], + "score": 0.84, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 94, + 315, + 106 + ], + "score": 1.0, + "content": ", implies that", + "type": "text" + }, + { + "bbox": [ + 315, + 93, + 373, + 105 + ], + "score": 0.42, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "must have a low rank. Then, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 173, + 118 + ], + "score": 1.0, + "content": "show that, when", + "type": "text" + }, + { + "bbox": [ + 173, + 105, + 206, + 115 + ], + "score": 0.9, + "content": "\\mathbf { X } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 103, + 225, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 105, + 261, + 115 + ], + "score": 0.83, + "content": "\\mathbf { W } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 103, + 309, + 118 + ], + "score": 1.0, + "content": ", the matrix", + "type": "text" + }, + { + "bbox": [ + 309, + 105, + 368, + 117 + ], + "score": 0.76, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "has a low rank with exponentially", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 349, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 349, + 127 + ], + "score": 1.0, + "content": "low probability. Combining both facts, we obtain the bound.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "Existence of global minima. Next, to compare the volume of sub-optimal DLMs with that of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "global minima, in appendix section 9 we show first that, generically, global minima do exist (using a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 308, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 308, + 173 + ], + "score": 1.0, + "content": "variant of the proof of (Baum, 1988, Theorem 1)):", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 104, + 170, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 104, + 170, + 194, + 187 + ], + "score": 1.0, + "content": "Theorem 8. For any", + "type": "text" + }, + { + "bbox": [ + 194, + 173, + 249, + 186 + ], + "score": 0.93, + "content": "\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 170, + 268, + 187 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 269, + 173, + 321, + 184 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 170, + 476, + 187 + ], + "score": 1.0, + "content": "almost everywhere4 we find matrices", + "type": "text" + }, + { + "bbox": [ + 476, + 174, + 505, + 185 + ], + "score": 0.86, + "content": "\\mathbf { W } ^ { * } \\in", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 137, + 197 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 184, + 158, + 200 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 186, + 198, + 198 + ], + "score": 0.91, + "content": "\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 184, + 244, + 200 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 244, + 185, + 327, + 199 + ], + "score": 0.91, + "content": "\\mathbf { y } = f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 184, + 361, + 200 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 361, + 186, + 457, + 200 + ], + "score": 0.89, + "content": "d _ { 1 } ^ { * } \\triangleq 4 \\lceil N / ( 2 d _ { 0 } - 2 ) \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 184, + 477, + 200 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 187, + 501, + 199 + ], + "score": 0.64, + "content": "\\forall i , n", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 184, + 507, + 200 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 490, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 159, + 212 + ], + "score": 0.93, + "content": "\\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 198, + 273, + 213 + ], + "score": 1.0, + "content": ". Therefore, every MNN with", + "type": "text" + }, + { + "bbox": [ + 273, + 200, + 307, + 212 + ], + "score": 0.91, + "content": "d _ { 1 } \\geq d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 198, + 461, + 213 + ], + "score": 1.0, + "content": "has a DLM which achieves zero error", + "type": "text" + }, + { + "bbox": [ + 461, + 200, + 486, + 210 + ], + "score": 0.89, + "content": "\\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 198, + 490, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 104, + 218, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 506, + 233 + ], + "score": 1.0, + "content": "Recently (Zhang et al., 2017a, Theorem 1) similarly proved that a 2-layer MNN with approximately", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 122, + 241 + ], + "score": 0.76, + "content": "2 N", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 230, + 373, + 244 + ], + "score": 1.0, + "content": "parameters can achieve zero error. However, that proof required", + "type": "text" + }, + { + "bbox": [ + 373, + 231, + 384, + 240 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "neurons (similarly to (Nilsson,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 254 + ], + "score": 1.0, + "content": "1965; Baum, 1988; Yu, 1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016)), while Theorem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 279, + 265 + ], + "score": 1.0, + "content": "8 here requires much less: approximately", + "type": "text" + }, + { + "bbox": [ + 279, + 252, + 332, + 264 + ], + "score": 0.93, + "content": "d _ { 1 } ^ { * } \\approx 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 252, + 506, + 265 + ], + "score": 1.0, + "content": ". Also, (Hardt & Ma, 2017, Theorem 3.2)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 259, + 275 + ], + "score": 1.0, + "content": "showed a deep residual network with", + "type": "text" + }, + { + "bbox": [ + 259, + 264, + 295, + 275 + ], + "score": 0.89, + "content": "N \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 264, + 505, + 275 + ], + "score": 1.0, + "content": "parameters can achieve zero error. In contrast, here", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 322, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 257, + 287 + ], + "score": 1.0, + "content": "we require just one hidden layer with", + "type": "text" + }, + { + "bbox": [ + 257, + 275, + 272, + 284 + ], + "score": 0.78, + "content": "2 N", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 274, + 322, + 287 + ], + "score": 1.0, + "content": "parameters.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "Note the construction in Theorem 8 here achieves zero training error by overfitting to the data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 302, + 504, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 504, + 314 + ], + "score": 1.0, + "content": "realization, so it is not expected to be a “good” solution in terms of generalization. To get good", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 392, + 326 + ], + "score": 1.0, + "content": "generalization, one needs to add additional assumptions on the data (", + "type": "text" + }, + { + "bbox": [ + 392, + 313, + 402, + 323 + ], + "score": 0.48, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "and y). Such a possible", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "(common yet insufficient for MNNs) assumption is that the problem is “realizable”, i.e., there exist a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 334, + 451, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 451, + 348 + ], + "score": 1.0, + "content": "small “solution MNN”, which achieves low error. For example, in the zero error case:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 350, + 504, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 399, + 363 + ], + "score": 1.0, + "content": "Assumption 5. (Optional) The labels are generated by some teacher", + "type": "text" + }, + { + "bbox": [ + 399, + 348, + 483, + 363 + ], + "score": 0.9, + "content": "\\mathbf { y } = f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 348, + 505, + 363 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 449, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 172, + 377 + ], + "score": 1.0, + "content": "weight matrices", + "type": "text" + }, + { + "bbox": [ + 172, + 362, + 232, + 373 + ], + "score": 0.92, + "content": "\\mathbf { W } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 360, + 251, + 377 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 251, + 362, + 290, + 373 + ], + "score": 0.9, + "content": "\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 360, + 353, + 377 + ], + "score": 1.0, + "content": "independent of", + "type": "text" + }, + { + "bbox": [ + 353, + 363, + 363, + 373 + ], + "score": 0.45, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 360, + 403, + 377 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 404, + 363, + 445, + 375 + ], + "score": 0.92, + "content": "d _ { 1 } ^ { * } { \\dot { < } } N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 360, + 449, + 377 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 394 + ], + "score": 1.0, + "content": "This assumption is not required for our main result (Theorem 10) – it is merely helpful in improving", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 390, + 255, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 227, + 404 + ], + "score": 1.0, + "content": "the following lower bound on", + "type": "text" + }, + { + "bbox": [ + 227, + 391, + 251, + 403 + ], + "score": 0.92, + "content": "\\mathcal { V } \\left( \\mathcal { G } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 390, + 255, + 404 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 397, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 397, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 397, + 428 + ], + "score": 1.0, + "content": "Angular volume of global minima. We prove in appendix section 10:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 102, + 426, + 509, + 448 + ], + "spans": [ + { + "bbox": [ + 102, + 426, + 281, + 448 + ], + "score": 1.0, + "content": "Theorem 9. Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 281, + 427, + 417, + 447 + ], + "score": 0.93, + "content": "\\delta { \\doteq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } / N", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 426, + 435, + 448 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 432, + 488, + 444 + ], + "score": 0.92, + "content": "d _ { 1 } ^ { * } = 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 426, + 509, + 448 + ], + "score": 1.0, + "content": ", or", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 445, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 227, + 460 + ], + "score": 1.0, + "content": "if assumption 5 holds, we set", + "type": "text" + }, + { + "bbox": [ + 228, + 446, + 239, + 458 + ], + "score": 0.88, + "content": "d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 445, + 427, + 460 + ], + "score": 1.0, + "content": "as in this assumption. Then, with probability", + "type": "text" + }, + { + "bbox": [ + 427, + 446, + 451, + 457 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 445, + 506, + 460 + ], + "score": 1.0, + "content": ", the angular", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 457, + 293, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 293, + 470 + ], + "score": 1.0, + "content": "volume of global minima is lower bounded as,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 470, + 421, + 484 + ], + "lines": [ + { + "bbox": [ + 189, + 470, + 421, + 484 + ], + "spans": [ + { + "bbox": [ + 189, + 470, + 421, + 484 + ], + "score": 0.86, + "content": "\\mathscr { V } \\left( \\mathscr { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { > } \\exp \\left( - d _ { 1 } ^ { * } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .", + "type": "interline_equation", + "image_path": "b1bb19380c59ff35fb86408a28e30cca957030b3e32dc7c2ba7e2b505f44b117.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 189, + 470, + 421, + 484 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 247, + 505 + ], + "score": 1.0, + "content": "Proof idea: First, we lower bound", + "type": "text" + }, + { + "bbox": [ + 248, + 492, + 271, + 504 + ], + "score": 0.92, + "content": "\\mathcal { V } \\left( \\mathcal { G } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "with the angular volume of a single differentiable region", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 205, + 516 + ], + "score": 1.0, + "content": "of one global minimum", + "type": "text" + }, + { + "bbox": [ + 205, + 504, + 244, + 515 + ], + "score": 0.86, + "content": "( \\mathbf { W } ^ { * } , \\mathbf { z } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "– either from Theorem 8, or from assumption 5. Then we show", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 303, + 527 + ], + "score": 1.0, + "content": "that this angular volume is lower bounded when", + "type": "text" + }, + { + "bbox": [ + 303, + 514, + 339, + 525 + ], + "score": 0.88, + "content": "\\mathbf { W } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 514, + 505, + 527 + ], + "score": 1.0, + "content": ", given a certain angular margin between", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 524, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 175, + 539 + ], + "score": 1.0, + "content": "the datapoints in", + "type": "text" + }, + { + "bbox": [ + 175, + 526, + 185, + 535 + ], + "score": 0.52, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 524, + 250, + 539 + ], + "score": 1.0, + "content": "and the rows of", + "type": "text" + }, + { + "bbox": [ + 250, + 525, + 268, + 536 + ], + "score": 0.85, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 524, + 505, + 539 + ], + "score": 1.0, + "content": ". We then calculate the probability of obtaining this margin", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 536, + 371, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 130, + 548 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 536, + 163, + 547 + ], + "score": 0.89, + "content": "\\mathbf { X } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 536, + 371, + 548 + ], + "score": 1.0, + "content": ". Combining both results, we obtain the final bound.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 105, + 559, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Main result: angular volume ratio. Finally, combining Theorems 6 and 9 it is straightforward to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 381, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 381, + 582 + ], + "score": 1.0, + "content": "prove our main result in this paper, as we do in appendix section 11:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 509, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 300, + 605 + ], + "score": 1.0, + "content": "Theorem 10. Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 300, + 583, + 449, + 604 + ], + "score": 0.92, + "content": "\\delta \\doteq \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } / N", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 580, + 509, + 605 + ], + "score": 1.0, + "content": ". Then, with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 153, + 615 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 154, + 602, + 177, + 613 + ], + "score": 0.67, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 601, + 375, + 615 + ], + "score": 1.0, + "content": ", the angular volume of sub-optimal DLMs, with", + "type": "text" + }, + { + "bbox": [ + 375, + 603, + 435, + 613 + ], + "score": 0.83, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 601, + 505, + 615 + ], + "score": 1.0, + "content": ", is exponentially", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 613, + 452, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 157, + 626 + ], + "score": 1.0, + "content": "vanishing in", + "type": "text" + }, + { + "bbox": [ + 158, + 614, + 166, + 623 + ], + "score": 0.69, + "content": "N _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 613, + 410, + 626 + ], + "score": 1.0, + "content": ", in comparison to the angular volume of global minima with", + "type": "text" + }, + { + "bbox": [ + 410, + 614, + 452, + 624 + ], + "score": 0.85, + "content": "\\mathrm { M C E } = 0", + "type": "inline_equation" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 627, + 439, + 654 + ], + "lines": [ + { + "bbox": [ + 171, + 627, + 439, + 654 + ], + "spans": [ + { + "bbox": [ + 171, + 627, + 439, + 654 + ], + "score": 0.91, + "content": "\\frac { \\mathscr { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathscr { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N \\log N \\right) .", + "type": "interline_equation", + "image_path": "3af6fc16ee09e676267d0a9d7849e66e063aa542e2cd863187ff96b954d366e5.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 171, + 627, + 439, + 654 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 665, + 267, + 679 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 268, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 268, + 680 + ], + "score": 1.0, + "content": "5 NUMERICAL EXPERIMENTS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "Theorem 10 implies that, with “asymptotically mild” over-parameterization (i.e. in which #parameters", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 701, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 141, + 714 + ], + "score": 0.91, + "content": "\\mathrm { = } \\tilde { \\Omega } \\left( N \\right) )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 701, + 505, + 716 + ], + "score": 1.0, + "content": ", differentiable regions in weight space containing sub-optimal DLMs (with high MCE) are", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 112, + 721, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 208, + 734 + ], + "score": 1.0, + "content": "4i.e., the set of entries of", + "type": "text" + }, + { + "bbox": [ + 209, + 723, + 217, + 730 + ], + "score": 0.34, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 719, + 505, + 734 + ], + "score": 1.0, + "content": ", for which the following statement does not hold, has zero measure (Lebesgue).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 418, + 95 + ], + "score": 1.0, + "content": "Proof idea of Theorem 6: we first show that in differentiable regions with", + "type": "text" + }, + { + "bbox": [ + 419, + 83, + 485, + 93 + ], + "score": 0.9, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 83, + 505, + 95 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 200, + 106 + ], + "score": 1.0, + "content": "condition in Lemma 2,", + "type": "text" + }, + { + "bbox": [ + 201, + 93, + 261, + 105 + ], + "score": 0.84, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 94, + 315, + 106 + ], + "score": 1.0, + "content": ", implies that", + "type": "text" + }, + { + "bbox": [ + 315, + 93, + 373, + 105 + ], + "score": 0.42, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "must have a low rank. Then, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 173, + 118 + ], + "score": 1.0, + "content": "show that, when", + "type": "text" + }, + { + "bbox": [ + 173, + 105, + 206, + 115 + ], + "score": 0.9, + "content": "\\mathbf { X } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 103, + 225, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 105, + 261, + 115 + ], + "score": 0.83, + "content": "\\mathbf { W } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 103, + 309, + 118 + ], + "score": 1.0, + "content": ", the matrix", + "type": "text" + }, + { + "bbox": [ + 309, + 105, + 368, + 117 + ], + "score": 0.76, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "has a low rank with exponentially", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 349, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 349, + 127 + ], + "score": 1.0, + "content": "low probability. Combining both facts, we obtain the bound.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 83, + 505, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "Existence of global minima. Next, to compare the volume of sub-optimal DLMs with that of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "global minima, in appendix section 9 we show first that, generically, global minima do exist (using a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 308, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 308, + 173 + ], + "score": 1.0, + "content": "variant of the proof of (Baum, 1988, Theorem 1)):", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 138, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 104, + 170, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 104, + 170, + 194, + 187 + ], + "score": 1.0, + "content": "Theorem 8. For any", + "type": "text" + }, + { + "bbox": [ + 194, + 173, + 249, + 186 + ], + "score": 0.93, + "content": "\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 170, + 268, + 187 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 269, + 173, + 321, + 184 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 170, + 476, + 187 + ], + "score": 1.0, + "content": "almost everywhere4 we find matrices", + "type": "text" + }, + { + "bbox": [ + 476, + 174, + 505, + 185 + ], + "score": 0.86, + "content": "\\mathbf { W } ^ { * } \\in", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 137, + 197 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 184, + 158, + 200 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 186, + 198, + 198 + ], + "score": 0.91, + "content": "\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 184, + 244, + 200 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 244, + 185, + 327, + 199 + ], + "score": 0.91, + "content": "\\mathbf { y } = f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 184, + 361, + 200 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 361, + 186, + 457, + 200 + ], + "score": 0.89, + "content": "d _ { 1 } ^ { * } \\triangleq 4 \\lceil N / ( 2 d _ { 0 } - 2 ) \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 184, + 477, + 200 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 187, + 501, + 199 + ], + "score": 0.64, + "content": "\\forall i , n", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 184, + 507, + 200 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 490, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 159, + 212 + ], + "score": 0.93, + "content": "\\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 198, + 273, + 213 + ], + "score": 1.0, + "content": ". Therefore, every MNN with", + "type": "text" + }, + { + "bbox": [ + 273, + 200, + 307, + 212 + ], + "score": 0.91, + "content": "d _ { 1 } \\geq d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 198, + 461, + 213 + ], + "score": 1.0, + "content": "has a DLM which achieves zero error", + "type": "text" + }, + { + "bbox": [ + 461, + 200, + 486, + 210 + ], + "score": 0.89, + "content": "\\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 198, + 490, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 170, + 507, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 104, + 218, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 506, + 233 + ], + "score": 1.0, + "content": "Recently (Zhang et al., 2017a, Theorem 1) similarly proved that a 2-layer MNN with approximately", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 122, + 241 + ], + "score": 0.76, + "content": "2 N", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 230, + 373, + 244 + ], + "score": 1.0, + "content": "parameters can achieve zero error. However, that proof required", + "type": "text" + }, + { + "bbox": [ + 373, + 231, + 384, + 240 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "neurons (similarly to (Nilsson,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 254 + ], + "score": 1.0, + "content": "1965; Baum, 1988; Yu, 1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016)), while Theorem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 279, + 265 + ], + "score": 1.0, + "content": "8 here requires much less: approximately", + "type": "text" + }, + { + "bbox": [ + 279, + 252, + 332, + 264 + ], + "score": 0.93, + "content": "d _ { 1 } ^ { * } \\approx 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 252, + 506, + 265 + ], + "score": 1.0, + "content": ". Also, (Hardt & Ma, 2017, Theorem 3.2)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 259, + 275 + ], + "score": 1.0, + "content": "showed a deep residual network with", + "type": "text" + }, + { + "bbox": [ + 259, + 264, + 295, + 275 + ], + "score": 0.89, + "content": "N \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 264, + 505, + 275 + ], + "score": 1.0, + "content": "parameters can achieve zero error. In contrast, here", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 322, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 257, + 287 + ], + "score": 1.0, + "content": "we require just one hidden layer with", + "type": "text" + }, + { + "bbox": [ + 257, + 275, + 272, + 284 + ], + "score": 0.78, + "content": "2 N", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 274, + 322, + 287 + ], + "score": 1.0, + "content": "parameters.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 218, + 506, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "Note the construction in Theorem 8 here achieves zero training error by overfitting to the data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 302, + 504, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 504, + 314 + ], + "score": 1.0, + "content": "realization, so it is not expected to be a “good” solution in terms of generalization. To get good", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 392, + 326 + ], + "score": 1.0, + "content": "generalization, one needs to add additional assumptions on the data (", + "type": "text" + }, + { + "bbox": [ + 392, + 313, + 402, + 323 + ], + "score": 0.48, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "and y). Such a possible", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "(common yet insufficient for MNNs) assumption is that the problem is “realizable”, i.e., there exist a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 334, + 451, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 451, + 348 + ], + "score": 1.0, + "content": "small “solution MNN”, which achieves low error. For example, in the zero error case:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 291, + 506, + 348 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 350, + 504, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 399, + 363 + ], + "score": 1.0, + "content": "Assumption 5. (Optional) The labels are generated by some teacher", + "type": "text" + }, + { + "bbox": [ + 399, + 348, + 483, + 363 + ], + "score": 0.9, + "content": "\\mathbf { y } = f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 348, + 505, + 363 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 449, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 172, + 377 + ], + "score": 1.0, + "content": "weight matrices", + "type": "text" + }, + { + "bbox": [ + 172, + 362, + 232, + 373 + ], + "score": 0.92, + "content": "\\mathbf { W } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 360, + 251, + 377 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 251, + 362, + 290, + 373 + ], + "score": 0.9, + "content": "\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 360, + 353, + 377 + ], + "score": 1.0, + "content": "independent of", + "type": "text" + }, + { + "bbox": [ + 353, + 363, + 363, + 373 + ], + "score": 0.45, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 360, + 403, + 377 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 404, + 363, + 445, + 375 + ], + "score": 0.92, + "content": "d _ { 1 } ^ { * } { \\dot { < } } N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 360, + 449, + 377 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 348, + 505, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 394 + ], + "score": 1.0, + "content": "This assumption is not required for our main result (Theorem 10) – it is merely helpful in improving", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 390, + 255, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 227, + 404 + ], + "score": 1.0, + "content": "the following lower bound on", + "type": "text" + }, + { + "bbox": [ + 227, + 391, + 251, + 403 + ], + "score": 0.92, + "content": "\\mathcal { V } \\left( \\mathcal { G } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 390, + 255, + 404 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 378, + 505, + 404 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 397, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 397, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 397, + 428 + ], + "score": 1.0, + "content": "Angular volume of global minima. We prove in appendix section 10:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 413, + 397, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 102, + 426, + 509, + 448 + ], + "spans": [ + { + "bbox": [ + 102, + 426, + 281, + 448 + ], + "score": 1.0, + "content": "Theorem 9. Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 281, + 427, + 417, + 447 + ], + "score": 0.93, + "content": "\\delta { \\doteq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } / N", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 426, + 435, + 448 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 432, + 488, + 444 + ], + "score": 0.92, + "content": "d _ { 1 } ^ { * } = 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 426, + 509, + 448 + ], + "score": 1.0, + "content": ", or", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 445, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 227, + 460 + ], + "score": 1.0, + "content": "if assumption 5 holds, we set", + "type": "text" + }, + { + "bbox": [ + 228, + 446, + 239, + 458 + ], + "score": 0.88, + "content": "d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 445, + 427, + 460 + ], + "score": 1.0, + "content": "as in this assumption. Then, with probability", + "type": "text" + }, + { + "bbox": [ + 427, + 446, + 451, + 457 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 445, + 506, + 460 + ], + "score": 1.0, + "content": ", the angular", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 457, + 293, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 293, + 470 + ], + "score": 1.0, + "content": "volume of global minima is lower bounded as,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 102, + 426, + 509, + 470 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 470, + 421, + 484 + ], + "lines": [ + { + "bbox": [ + 189, + 470, + 421, + 484 + ], + "spans": [ + { + "bbox": [ + 189, + 470, + 421, + 484 + ], + "score": 0.86, + "content": "\\mathscr { V } \\left( \\mathscr { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { > } \\exp \\left( - d _ { 1 } ^ { * } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .", + "type": "interline_equation", + "image_path": "b1bb19380c59ff35fb86408a28e30cca957030b3e32dc7c2ba7e2b505f44b117.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 189, + 470, + 421, + 484 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 247, + 505 + ], + "score": 1.0, + "content": "Proof idea: First, we lower bound", + "type": "text" + }, + { + "bbox": [ + 248, + 492, + 271, + 504 + ], + "score": 0.92, + "content": "\\mathcal { V } \\left( \\mathcal { G } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "with the angular volume of a single differentiable region", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 205, + 516 + ], + "score": 1.0, + "content": "of one global minimum", + "type": "text" + }, + { + "bbox": [ + 205, + 504, + 244, + 515 + ], + "score": 0.86, + "content": "( \\mathbf { W } ^ { * } , \\mathbf { z } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "– either from Theorem 8, or from assumption 5. Then we show", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 303, + 527 + ], + "score": 1.0, + "content": "that this angular volume is lower bounded when", + "type": "text" + }, + { + "bbox": [ + 303, + 514, + 339, + 525 + ], + "score": 0.88, + "content": "\\mathbf { W } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 514, + 505, + 527 + ], + "score": 1.0, + "content": ", given a certain angular margin between", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 524, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 175, + 539 + ], + "score": 1.0, + "content": "the datapoints in", + "type": "text" + }, + { + "bbox": [ + 175, + 526, + 185, + 535 + ], + "score": 0.52, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 524, + 250, + 539 + ], + "score": 1.0, + "content": "and the rows of", + "type": "text" + }, + { + "bbox": [ + 250, + 525, + 268, + 536 + ], + "score": 0.85, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 524, + 505, + 539 + ], + "score": 1.0, + "content": ". We then calculate the probability of obtaining this margin", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 536, + 371, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 130, + 548 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 536, + 163, + 547 + ], + "score": 0.89, + "content": "\\mathbf { X } \\sim { \\mathcal { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 536, + 371, + 548 + ], + "score": 1.0, + "content": ". Combining both results, we obtain the final bound.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 492, + 505, + 548 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 559, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Main result: angular volume ratio. Finally, combining Theorems 6 and 9 it is straightforward to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 381, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 381, + 582 + ], + "score": 1.0, + "content": "prove our main result in this paper, as we do in appendix section 11:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 559, + 505, + 582 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 509, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 300, + 605 + ], + "score": 1.0, + "content": "Theorem 10. Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 300, + 583, + 449, + 604 + ], + "score": 0.92, + "content": "\\delta \\doteq \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } / N", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 580, + 509, + 605 + ], + "score": 1.0, + "content": ". Then, with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 153, + 615 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 154, + 602, + 177, + 613 + ], + "score": 0.67, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 601, + 375, + 615 + ], + "score": 1.0, + "content": ", the angular volume of sub-optimal DLMs, with", + "type": "text" + }, + { + "bbox": [ + 375, + 603, + 435, + 613 + ], + "score": 0.83, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 601, + 505, + 615 + ], + "score": 1.0, + "content": ", is exponentially", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 613, + 452, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 157, + 626 + ], + "score": 1.0, + "content": "vanishing in", + "type": "text" + }, + { + "bbox": [ + 158, + 614, + 166, + 623 + ], + "score": 0.69, + "content": "N _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 613, + 410, + 626 + ], + "score": 1.0, + "content": ", in comparison to the angular volume of global minima with", + "type": "text" + }, + { + "bbox": [ + 410, + 614, + 452, + 624 + ], + "score": 0.85, + "content": "\\mathrm { M C E } = 0", + "type": "inline_equation" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 580, + 509, + 626 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 627, + 439, + 654 + ], + "lines": [ + { + "bbox": [ + 171, + 627, + 439, + 654 + ], + "spans": [ + { + "bbox": [ + 171, + 627, + 439, + 654 + ], + "score": 0.91, + "content": "\\frac { \\mathscr { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathscr { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N \\log N \\right) .", + "type": "interline_equation", + "image_path": "3af6fc16ee09e676267d0a9d7849e66e063aa542e2cd863187ff96b954d366e5.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 171, + 627, + 439, + 654 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 665, + 267, + 679 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 268, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 268, + 680 + ], + "score": 1.0, + "content": "5 NUMERICAL EXPERIMENTS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "Theorem 10 implies that, with “asymptotically mild” over-parameterization (i.e. in which #parameters", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 701, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 141, + 714 + ], + "score": 0.91, + "content": "\\mathrm { = } \\tilde { \\Omega } \\left( N \\right) )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 701, + 505, + 716 + ], + "score": 1.0, + "content": ", differentiable regions in weight space containing sub-optimal DLMs (with high MCE) are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "exponentially small in comparison with the same regions for global minima. Since these results", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 183, + 407 + ], + "score": 1.0, + "content": "are asymptotic in", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 183, + 395, + 225, + 405 + ], + "score": 0.9, + "content": "N \\infty", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 225, + 395, + 506, + 407 + ], + "score": 1.0, + "content": ", in this section we examine it numerically for a finite number of", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 507, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 507, + 419 + ], + "score": 1.0, + "content": "samples and parameters. 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Since these results", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 183, + 407 + ], + "score": 1.0, + "content": "are asymptotic in", + "type": "text" + }, + { + "bbox": [ + 183, + 395, + 225, + 405 + ], + "score": 0.9, + "content": "N \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 395, + 506, + 407 + ], + "score": 1.0, + "content": ", in this section we examine it numerically for a finite number of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 507, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 507, + 419 + ], + "score": 1.0, + "content": "samples and parameters. We perform experiments on random data, MNIST, CIFAR10 and ImageNet-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 416, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 358, + 430 + ], + "score": 1.0, + "content": "ILSVRC2012. 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Additional implementation details are given in appendix part III.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "First, on the small synthetic Gaussian random data (matching our assumptions) we perform a scan", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "on various networks and dataset sizes. With either one or two hidden layers (Figure 5.1) , the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 439, + 491 + ], + "score": 1.0, + "content": "error goes to zero when the number of non-redundant parameters (approximately", + "type": "text" + }, + { + "bbox": [ + 440, + 478, + 461, + 489 + ], + "score": 0.88, + "content": "d _ { 0 } d _ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 478, + 506, + 491 + ], + "score": 1.0, + "content": ") is greater", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "than the number of samples, as suggested by our asymptotic results. Second, on the non-syntehtic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 469, + 512 + ], + "score": 1.0, + "content": "datasets, MNIST, CIFAR and ImageNet (In ImageNet we downsampled the images to size", + "type": "text" + }, + { + "bbox": [ + 469, + 500, + 503, + 510 + ], + "score": 0.91, + "content": "6 4 \\times 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 500, + 506, + 512 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "to allow input whitening) we only perform a simulation with a single 1-hidden layer MNN for which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 165, + 534 + ], + "score": 1.0, + "content": "#parameters", + "type": "text" + }, + { + "bbox": [ + 165, + 522, + 187, + 532 + ], + "score": 0.86, + "content": "\\approx N", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 522, + 506, + 534 + ], + "score": 1.0, + "content": ", and again find (Table 1) that the final error is zero (for MNIST and CIFAR) or", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 194, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 194, + 545 + ], + "score": 1.0, + "content": "very low (ImageNet).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "Lastly, in Figure 5.2 we find that, on the Gaussian dataset, the inputs to the hidden neurons converge", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "to a distinctly non-zero value. This indicates we converged to differentiable critical points – since non-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 570, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 507, + 584 + ], + "score": 1.0, + "content": "differentiable critical points must have zero neural inputs. Note that occasionally, during optimization,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "we could find some neural inputs with very low values near numerical precision level, so convergence", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "to non-differentiable minima may be possible. However, as explained in the next section, as long as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 604, + 502, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 502, + 617 + ], + "score": 1.0, + "content": "the number of neural inputs equal to zero are not too large, our bounds also hold for these minima.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 107, + 632, + 190, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 192, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 192, + 647 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "In this paper we examine Differentiable Local Minima (DLMs) of the empiric loss of Multilayer", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Neural Networks (MNNs) with one hidden layer, scalar output, and LReLU nonlinearities (section 2).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "We prove (Theorem 10) that with high probability the angular volume (definition 3) of sub-optimal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "DLMs is exponentially vanishing in comparison to the angular volume of global minima (definition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 700, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 714 + ], + "score": 1.0, + "content": "4), under assumptions 1-4. 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In", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 351, + 368, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 368, + 366 + ], + "score": 1.0, + "content": "ImageNet we downsampled the images to allow input whitening.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 450 + ], + "lines": [], + "index": 15.5, + "bbox_fs": [ + 105, + 384, + 507, + 452 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "First, on the small synthetic Gaussian random data (matching our assumptions) we perform a scan", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "on various networks and dataset sizes. With either one or two hidden layers (Figure 5.1) , the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 439, + 491 + ], + "score": 1.0, + "content": "error goes to zero when the number of non-redundant parameters (approximately", + "type": "text" + }, + { + "bbox": [ + 440, + 478, + 461, + 489 + ], + "score": 0.88, + "content": "d _ { 0 } d _ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 478, + 506, + 491 + ], + "score": 1.0, + "content": ") is greater", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "than the number of samples, as suggested by our asymptotic results. Second, on the non-syntehtic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 469, + 512 + ], + "score": 1.0, + "content": "datasets, MNIST, CIFAR and ImageNet (In ImageNet we downsampled the images to size", + "type": "text" + }, + { + "bbox": [ + 469, + 500, + 503, + 510 + ], + "score": 0.91, + "content": "6 4 \\times 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 500, + 506, + 512 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "to allow input whitening) we only perform a simulation with a single 1-hidden layer MNN for which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 165, + 534 + ], + "score": 1.0, + "content": "#parameters", + "type": "text" + }, + { + "bbox": [ + 165, + 522, + 187, + 532 + ], + "score": 0.86, + "content": "\\approx N", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 522, + 506, + 534 + ], + "score": 1.0, + "content": ", and again find (Table 1) that the final error is zero (for MNIST and CIFAR) or", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 194, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 194, + 545 + ], + "score": 1.0, + "content": "very low (ImageNet).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 455, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "Lastly, in Figure 5.2 we find that, on the Gaussian dataset, the inputs to the hidden neurons converge", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "to a distinctly non-zero value. This indicates we converged to differentiable critical points – since non-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 570, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 507, + 584 + ], + "score": 1.0, + "content": "differentiable critical points must have zero neural inputs. Note that occasionally, during optimization,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "we could find some neural inputs with very low values near numerical precision level, so convergence", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "to non-differentiable minima may be possible. However, as explained in the next section, as long as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 604, + 502, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 502, + 617 + ], + "score": 1.0, + "content": "the number of neural inputs equal to zero are not too large, our bounds also hold for these minima.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 549, + 507, + 617 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 632, + 190, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 192, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 192, + 647 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "In this paper we examine Differentiable Local Minima (DLMs) of the empiric loss of Multilayer", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Neural Networks (MNNs) with one hidden layer, scalar output, and LReLU nonlinearities (section 2).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "We prove (Theorem 10) that with high probability the angular volume (definition 3) of sub-optimal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "DLMs is exponentially vanishing in comparison to the angular volume of global minima (definition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 700, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 714 + ], + "score": 1.0, + "content": "4), under assumptions 1-4. 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We trained MNNs", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 307, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 307, + 226 + ], + "score": 1.0, + "content": "with one hidden layer on the Gaussian dataset from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 307, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 236, + 237 + ], + "score": 1.0, + "content": "Figure 5.1, with various widths", + "type": "text" + }, + { + "bbox": [ + 236, + 226, + 289, + 237 + ], + "score": 0.93, + "content": "d = d _ { 0 } = d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 226, + 307, + 237 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 236, + 307, + 250 + ], + "spans": [ + { + "bbox": [ + 107, + 236, + 159, + 250 + ], + "score": 0.93, + "content": "N = \\lfloor d ^ { 2 } / 5 \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 236, + 307, + 249 + ], + "score": 1.0, + "content": "for 1000 epochs, then decreased the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 307, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 307, + 260 + ], + "score": 1.0, + "content": "learning rate exponentially for another 1000 epo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 307, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 281, + 270 + ], + "score": 1.0, + "content": "chs. This was repeated 30 times. For all", + "type": "text" + }, + { + "bbox": [ + 281, + 259, + 288, + 268 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 258, + 307, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 307, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 307, + 281 + ], + "score": 1.0, + "content": "repeats, we see that (left) the final absolute value", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 280, + 307, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 215, + 296 + ], + "score": 1.0, + "content": "of the minimal neural input", + "type": "text" + }, + { + "bbox": [ + 216, + 280, + 307, + 295 + ], + "score": 0.85, + "content": "( i . e . , \\operatorname* { m i n } _ { i , n } \\left| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| )", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 291, + 309, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 167, + 307 + ], + "score": 1.0, + "content": "in the range of", + "type": "text" + }, + { + "bbox": [ + 168, + 293, + 216, + 305 + ], + "score": 0.91, + "content": "1 0 ^ { - 3 } - 1 0 ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 291, + 309, + 307 + ], + "score": 1.0, + "content": ", which is much larger", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 305, + 307, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 127, + 316 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 128, + 306, + 154, + 316 + ], + "score": 0.74, + "content": "( \\mathrm { r } i g h t )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 305, + 268, + 316 + ], + "score": 1.0, + "content": "the final MSE error for all", + "type": "text" + }, + { + "bbox": [ + 268, + 306, + 275, + 315 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 305, + 307, + 316 + ], + "score": 1.0, + "content": "and all", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 256, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 194, + 328 + ], + "score": 1.0, + "content": "repeats – in the range", + "type": "text" + }, + { + "bbox": [ + 194, + 315, + 253, + 327 + ], + "score": 0.91, + "content": "1 0 ^ { - 3 1 } - 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 315, + 256, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12.5 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 315, + 82, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 314, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 314, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Convergence of SGD to DLMs. These re-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 314, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 314, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "sults suggest a mechanism through which low", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 313, + 105, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 313, + 105, + 506, + 115 + ], + "score": 1.0, + "content": "training error is obtained in such MNNs. Ho-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 313, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 313, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "wever, they do not guarantee it. One issue is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 313, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 313, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "that sub-optimal DLMs may have exponentially", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 313, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 313, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "large basins of attraction. We see two possi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 313, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 313, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "ble paths that might address this issue in future", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 314, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 314, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "work, using additional assumptions on y. One", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 313, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 313, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "approach is to show that, with high probability,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 313, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 313, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "no sub optimal DLM falls within the vanishingly", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 313, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 313, + 192, + 506, + 204 + ], + "score": 1.0, + "content": "small differentiable regions we bounded in The-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 313, + 203, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 313, + 203, + 506, + 214 + ], + "score": 1.0, + "content": "orem 6. Another approach would be to bound", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 313, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 313, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "the size of these basins of attraction, by showing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 313, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 313, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "that sufficiently large of number of differentia-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 313, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 313, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "ble regions near the DLM are also vanishingly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 313, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 313, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "small (other methods might also help here (Free-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 314, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 314, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "man & Bruna, 2016)). Another issue is that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 313, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 313, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "SGD might get stuck near differentiable saddle", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 313, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 313, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "points, if their Hessian does not have strictly", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 313, + 291, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 313, + 291, + 506, + 302 + ], + "score": 1.0, + "content": "negative eigenvalues (i.e., the strict saddle pro-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 313, + 302, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 313, + 302, + 506, + 314 + ], + "score": 1.0, + "content": "perty (Sun et al., 2015)). It should be straig-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 314, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 314, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "htforward to show that such points also have", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 313, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 313, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "exponentially vanishing angular volume, similar", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 313, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 313, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "to sub-optimal DLMs. Lastly, SGD might also", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 108, + 346, + 381, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 383, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 383, + 359 + ], + "score": 1.0, + "content": "converge to non-differentiable critical points, which we discuss next.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "Non-differentiable critical points. The proof of Theorem 6 stems from a first order necessary", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 202, + 395 + ], + "score": 1.0, + "content": "condition (Lemma 2):", + "type": "text" + }, + { + "bbox": [ + 202, + 383, + 264, + 395 + ], + "score": 0.92, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { \\bar { X } } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 383, + 505, + 395 + ], + "score": 1.0, + "content": ", which is true for any DLM. However, non-differentiable", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "score": 1.0, + "content": "critical points, in which some neural inputs are exactly zero, may also exist (though, numerically, they", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "don’t seem very common – see Figure 5.2). In this case, to derive a similar bound, we can replace the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 168, + 428 + ], + "score": 1.0, + "content": "condition with", + "type": "text" + }, + { + "bbox": [ + 168, + 416, + 238, + 428 + ], + "score": 0.9, + "content": "\\mathbf { P } \\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 416, + 269, + 428 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 269, + 416, + 279, + 426 + ], + "score": 0.65, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "is a projection matrix to the subspace orthogonal to the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "non-differentiable directions. As long as there are not too many zero neural inputs, we should be able", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 351, + 451 + ], + "score": 1.0, + "content": "to obtain similar results. For example, if only a constant ratio", + "type": "text" + }, + { + "bbox": [ + 352, + 440, + 358, + 448 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "of the neural inputs are zero, we can", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 166, + 461 + ], + "score": 1.0, + "content": "simply choose", + "type": "text" + }, + { + "bbox": [ + 167, + 449, + 176, + 459 + ], + "score": 0.58, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 449, + 266, + 461 + ], + "score": 1.0, + "content": "to remove all rows of", + "type": "text" + }, + { + "bbox": [ + 266, + 449, + 300, + 460 + ], + "score": 0.88, + "content": "( \\mathbf { A } \\circ \\mathbf { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "corresponding to those neurons, and proceed with", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 258, + 473 + ], + "score": 1.0, + "content": "exactly the same proof as before, with", + "type": "text" + }, + { + "bbox": [ + 258, + 460, + 269, + 471 + ], + "score": 0.89, + "content": "d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 459, + 326, + 473 + ], + "score": 1.0, + "content": "replaced with", + "type": "text" + }, + { + "bbox": [ + 326, + 460, + 368, + 471 + ], + "score": 0.92, + "content": "( 1 - r ) d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 459, + 505, + 473 + ], + "score": 1.0, + "content": ". It remains a theoretical challenge", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "to find reasonable assumptions under which the number of non-differentiable directions (i.e., zero", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 481, + 273, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 273, + 494 + ], + "score": 1.0, + "content": "neural inputs) does not become too large.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 50 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 403, + 520 + ], + "score": 1.0, + "content": "Related results. Two works have also derived related results using the", + "type": "text" + }, + { + "bbox": [ + 403, + 508, + 463, + 520 + ], + "score": 0.92, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "condition", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 519, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 494, + 530 + ], + "score": 1.0, + "content": "from Lemma 2. In (Soudry & Carmon, 2016), it was noticed that an infinitesimal perturbation of", + "type": "text" + }, + { + "bbox": [ + 495, + 519, + 504, + 529 + ], + "score": 0.29, + "content": "\\mathbf { A }", + "type": "inline_equation" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 176, + 542 + ], + "score": 1.0, + "content": "makes the matrix", + "type": "text" + }, + { + "bbox": [ + 176, + 530, + 203, + 540 + ], + "score": 0.83, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "full rank with probability 1 (Allman et al., 2009, Lemma 13) – which entails", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 123, + 553 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 123, + 541, + 149, + 551 + ], + "score": 0.9, + "content": "\\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "at all DLMs. Though a simple and intuitive approach, such an infinitesimal perturbation is", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "problematic: from continuity, it cannot change the original MSE at sub-optimal DLMs – unless the", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "score": 1.0, + "content": "weights go to infinity, or the DLM becomes non-differentiable – which are both undesirable results.", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 439, + 586 + ], + "score": 1.0, + "content": "An extension of this analysis was also done to constrain e using the singular values of", + "type": "text" + }, + { + "bbox": [ + 439, + 574, + 462, + 584 + ], + "score": 0.7, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "(Xie et al.,", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 583, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 598 + ], + "score": 1.0, + "content": "2016), deriving bounds that are easier to combine with generalization bounds. Though a promising", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "approach, the size of the sub-optimal regions (where the error is high) does not vanish exponentially", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "in the derived bounds. More importantly, these bounds require assumptions on the activation kernel", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 145, + 630 + ], + "score": 1.0, + "content": "spectrum", + "type": "text" + }, + { + "bbox": [ + 145, + 619, + 159, + 629 + ], + "score": 0.86, + "content": "\\gamma _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 618, + 505, + 630 + ], + "score": 1.0, + "content": ", which do not appear to hold in practice (e.g., (Xie et al., 2016, Theorems 1,3) require", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 628, + 477, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 150, + 640 + ], + "score": 0.91, + "content": "m \\gamma _ { m } \\gg 1", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 628, + 294, + 641 + ], + "score": 1.0, + "content": "to hold with high probability, while", + "type": "text" + }, + { + "bbox": [ + 295, + 628, + 351, + 640 + ], + "score": 0.92, + "content": "m \\gamma _ { m } < \\bar { 1 0 } ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 628, + 477, + 641 + ], + "score": 1.0, + "content": "in (Xie et al., 2016, Figure 1)).", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 61.5 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 504, + 667 + ], + "score": 1.0, + "content": "Modifications and extensions. There are many relatively simple extensions of these results: the", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "Gaussian assumption could be relaxed to other near-isotropic distributions (e.g., sparse-land model,", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "score": 1.0, + "content": "(Elad, 2010, Section 9.2)) and other convex loss functions are possible instead of the quadratic loss.", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "More challenging directions are extending our results to MNNs with multi-output and multiple hidden", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "layers, or combining our training error results with novel generalization bounds which might be better", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "suited for MNNs (e.g., (Feng et al., 2016; Sokolic et al., 2016; Dziugaite & Roy, 2017)) than previous", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 105, + 721, + 242, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 242, + 733 + ], + "score": 1.0, + "content": "approaches (Zhang et al., 2017a).", + "type": "text" + } + ], + "index": 74 + } + ], + "index": 71 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 83, + 306, + 160 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 83, + 306, + 160 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 83, + 306, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 306, + 160 + ], + "score": 0.969, + "type": "image", + "image_path": "e75b9ef834dd7be439435fa2e13f28f31f7a9f670be1cdff33d9185308bb07d8.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 106, + 83, + 306, + 95.83333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 95.83333333333333, + 306, + 108.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 108.66666666666666, + 306, + 121.49999999999999 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 106, + 121.49999999999999, + 306, + 134.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 106, + 134.33333333333331, + 306, + 147.16666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 106, + 147.16666666666666, + 306, + 160.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 171, + 306, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 170, + 307, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 307, + 183 + ], + "score": 1.0, + "content": "Figure 5.2: Gaussian data: convergence of the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 307, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 307, + 194 + ], + "score": 1.0, + "content": "MSE to differentiable critical points, as indica-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 307, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 307, + 205 + ], + "score": 1.0, + "content": "ted by the convergence of the neural inputs to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 307, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 307, + 215 + ], + "score": 1.0, + "content": "distinctly non-zero values. We trained MNNs", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 307, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 307, + 226 + ], + "score": 1.0, + "content": "with one hidden layer on the Gaussian dataset from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 307, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 236, + 237 + ], + "score": 1.0, + "content": "Figure 5.1, with various widths", + "type": "text" + }, + { + "bbox": [ + 236, + 226, + 289, + 237 + ], + "score": 0.93, + "content": "d = d _ { 0 } = d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 226, + 307, + 237 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 236, + 307, + 250 + ], + "spans": [ + { + "bbox": [ + 107, + 236, + 159, + 250 + ], + "score": 0.93, + "content": "N = \\lfloor d ^ { 2 } / 5 \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 236, + 307, + 249 + ], + "score": 1.0, + "content": "for 1000 epochs, then decreased the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 307, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 307, + 260 + ], + "score": 1.0, + "content": "learning rate exponentially for another 1000 epo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 307, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 281, + 270 + ], + "score": 1.0, + "content": "chs. This was repeated 30 times. For all", + "type": "text" + }, + { + "bbox": [ + 281, + 259, + 288, + 268 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 258, + 307, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 307, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 307, + 281 + ], + "score": 1.0, + "content": "repeats, we see that (left) the final absolute value", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 280, + 307, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 215, + 296 + ], + "score": 1.0, + "content": "of the minimal neural input", + "type": "text" + }, + { + "bbox": [ + 216, + 280, + 307, + 295 + ], + "score": 0.85, + "content": "( i . e . , \\operatorname* { m i n } _ { i , n } \\left| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| )", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 291, + 309, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 167, + 307 + ], + "score": 1.0, + "content": "in the range of", + "type": "text" + }, + { + "bbox": [ + 168, + 293, + 216, + 305 + ], + "score": 0.91, + "content": "1 0 ^ { - 3 } - 1 0 ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 291, + 309, + 307 + ], + "score": 1.0, + "content": ", which is much larger", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 305, + 307, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 127, + 316 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 128, + 306, + 154, + 316 + ], + "score": 0.74, + "content": "( \\mathrm { r } i g h t )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 305, + 268, + 316 + ], + "score": 1.0, + "content": "the final MSE error for all", + "type": "text" + }, + { + "bbox": [ + 268, + 306, + 275, + 315 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 305, + 307, + 316 + ], + "score": 1.0, + "content": "and all", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 256, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 194, + 328 + ], + "score": 1.0, + "content": "repeats – in the range", + "type": "text" + }, + { + "bbox": [ + 194, + 315, + 253, + 327 + ], + "score": 0.91, + "content": "1 0 ^ { - 3 1 } - 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 315, + 256, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12.5 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 315, + 82, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 314, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 314, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Convergence of SGD to DLMs. These re-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 314, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 314, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "sults suggest a mechanism through which low", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 313, + 105, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 313, + 105, + 506, + 115 + ], + "score": 1.0, + "content": "training error is obtained in such MNNs. Ho-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 313, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 313, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "wever, they do not guarantee it. One issue is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 313, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 313, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "that sub-optimal DLMs may have exponentially", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 313, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 313, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "large basins of attraction. We see two possi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 313, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 313, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "ble paths that might address this issue in future", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 314, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 314, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "work, using additional assumptions on y. One", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 313, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 313, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "approach is to show that, with high probability,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 313, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 313, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "no sub optimal DLM falls within the vanishingly", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 313, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 313, + 192, + 506, + 204 + ], + "score": 1.0, + "content": "small differentiable regions we bounded in The-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 313, + 203, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 313, + 203, + 506, + 214 + ], + "score": 1.0, + "content": "orem 6. Another approach would be to bound", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 313, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 313, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "the size of these basins of attraction, by showing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 313, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 313, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "that sufficiently large of number of differentia-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 313, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 313, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "ble regions near the DLM are also vanishingly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 313, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 313, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "small (other methods might also help here (Free-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 314, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 314, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "man & Bruna, 2016)). Another issue is that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 313, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 313, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "SGD might get stuck near differentiable saddle", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 313, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 313, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "points, if their Hessian does not have strictly", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 313, + 291, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 313, + 291, + 506, + 302 + ], + "score": 1.0, + "content": "negative eigenvalues (i.e., the strict saddle pro-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 313, + 302, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 313, + 302, + 506, + 314 + ], + "score": 1.0, + "content": "perty (Sun et al., 2015)). It should be straig-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 314, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 314, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "htforward to show that such points also have", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 313, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 313, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "exponentially vanishing angular volume, similar", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 313, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 313, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "to sub-optimal DLMs. Lastly, SGD might also", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 345, + 383, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 383, + 359 + ], + "score": 1.0, + "content": "converge to non-differentiable critical points, which we discuss next.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 31.5, + "bbox_fs": [ + 313, + 82, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 346, + 381, + 357 + ], + "lines": [], + "index": 44, + "bbox_fs": [ + 105, + 345, + 383, + 359 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "Non-differentiable critical points. The proof of Theorem 6 stems from a first order necessary", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 202, + 395 + ], + "score": 1.0, + "content": "condition (Lemma 2):", + "type": "text" + }, + { + "bbox": [ + 202, + 383, + 264, + 395 + ], + "score": 0.92, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { \\bar { X } } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 383, + 505, + 395 + ], + "score": 1.0, + "content": ", which is true for any DLM. However, non-differentiable", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "score": 1.0, + "content": "critical points, in which some neural inputs are exactly zero, may also exist (though, numerically, they", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "don’t seem very common – see Figure 5.2). In this case, to derive a similar bound, we can replace the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 168, + 428 + ], + "score": 1.0, + "content": "condition with", + "type": "text" + }, + { + "bbox": [ + 168, + 416, + 238, + 428 + ], + "score": 0.9, + "content": "\\mathbf { P } \\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 416, + 269, + 428 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 269, + 416, + 279, + 426 + ], + "score": 0.65, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "is a projection matrix to the subspace orthogonal to the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "non-differentiable directions. As long as there are not too many zero neural inputs, we should be able", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 351, + 451 + ], + "score": 1.0, + "content": "to obtain similar results. For example, if only a constant ratio", + "type": "text" + }, + { + "bbox": [ + 352, + 440, + 358, + 448 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "of the neural inputs are zero, we can", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 166, + 461 + ], + "score": 1.0, + "content": "simply choose", + "type": "text" + }, + { + "bbox": [ + 167, + 449, + 176, + 459 + ], + "score": 0.58, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 449, + 266, + 461 + ], + "score": 1.0, + "content": "to remove all rows of", + "type": "text" + }, + { + "bbox": [ + 266, + 449, + 300, + 460 + ], + "score": 0.88, + "content": "( \\mathbf { A } \\circ \\mathbf { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "corresponding to those neurons, and proceed with", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 258, + 473 + ], + "score": 1.0, + "content": "exactly the same proof as before, with", + "type": "text" + }, + { + "bbox": [ + 258, + 460, + 269, + 471 + ], + "score": 0.89, + "content": "d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 459, + 326, + 473 + ], + "score": 1.0, + "content": "replaced with", + "type": "text" + }, + { + "bbox": [ + 326, + 460, + 368, + 471 + ], + "score": 0.92, + "content": "( 1 - r ) d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 459, + 505, + 473 + ], + "score": 1.0, + "content": ". It remains a theoretical challenge", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "to find reasonable assumptions under which the number of non-differentiable directions (i.e., zero", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 481, + 273, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 273, + 494 + ], + "score": 1.0, + "content": "neural inputs) does not become too large.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 371, + 506, + 494 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 403, + 520 + ], + "score": 1.0, + "content": "Related results. Two works have also derived related results using the", + "type": "text" + }, + { + "bbox": [ + 403, + 508, + 463, + 520 + ], + "score": 0.92, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "condition", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 519, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 494, + 530 + ], + "score": 1.0, + "content": "from Lemma 2. In (Soudry & Carmon, 2016), it was noticed that an infinitesimal perturbation of", + "type": "text" + }, + { + "bbox": [ + 495, + 519, + 504, + 529 + ], + "score": 0.29, + "content": "\\mathbf { A }", + "type": "inline_equation" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 176, + 542 + ], + "score": 1.0, + "content": "makes the matrix", + "type": "text" + }, + { + "bbox": [ + 176, + 530, + 203, + 540 + ], + "score": 0.83, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "full rank with probability 1 (Allman et al., 2009, Lemma 13) – which entails", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 123, + 553 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 123, + 541, + 149, + 551 + ], + "score": 0.9, + "content": "\\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "at all DLMs. Though a simple and intuitive approach, such an infinitesimal perturbation is", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "problematic: from continuity, it cannot change the original MSE at sub-optimal DLMs – unless the", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "score": 1.0, + "content": "weights go to infinity, or the DLM becomes non-differentiable – which are both undesirable results.", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 439, + 586 + ], + "score": 1.0, + "content": "An extension of this analysis was also done to constrain e using the singular values of", + "type": "text" + }, + { + "bbox": [ + 439, + 574, + 462, + 584 + ], + "score": 0.7, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "(Xie et al.,", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 583, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 598 + ], + "score": 1.0, + "content": "2016), deriving bounds that are easier to combine with generalization bounds. Though a promising", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "approach, the size of the sub-optimal regions (where the error is high) does not vanish exponentially", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "in the derived bounds. More importantly, these bounds require assumptions on the activation kernel", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 145, + 630 + ], + "score": 1.0, + "content": "spectrum", + "type": "text" + }, + { + "bbox": [ + 145, + 619, + 159, + 629 + ], + "score": 0.86, + "content": "\\gamma _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 618, + 505, + 630 + ], + "score": 1.0, + "content": ", which do not appear to hold in practice (e.g., (Xie et al., 2016, Theorems 1,3) require", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 628, + 477, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 150, + 640 + ], + "score": 0.91, + "content": "m \\gamma _ { m } \\gg 1", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 628, + 294, + 641 + ], + "score": 1.0, + "content": "to hold with high probability, while", + "type": "text" + }, + { + "bbox": [ + 295, + 628, + 351, + 640 + ], + "score": 0.92, + "content": "m \\gamma _ { m } < \\bar { 1 0 } ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 628, + 477, + 641 + ], + "score": 1.0, + "content": "in (Xie et al., 2016, Figure 1)).", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 61.5, + "bbox_fs": [ + 105, + 508, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 504, + 667 + ], + "score": 1.0, + "content": "Modifications and extensions. There are many relatively simple extensions of these results: the", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "Gaussian assumption could be relaxed to other near-isotropic distributions (e.g., sparse-land model,", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "score": 1.0, + "content": "(Elad, 2010, Section 9.2)) and other convex loss functions are possible instead of the quadratic loss.", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "More challenging directions are extending our results to MNNs with multi-output and multiple hidden", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "layers, or combining our training error results with novel generalization bounds which might be better", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "suited for MNNs (e.g., (Feng et al., 2016; Sokolic et al., 2016; Dziugaite & Roy, 2017)) than previous", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 105, + 721, + 242, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 242, + 733 + ], + "score": 1.0, + "content": "approaches (Zhang et al., 2017a).", + "type": "text" + } + ], + "index": 74 + } + ], + "index": 71, + "bbox_fs": [ + 105, + 655, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 104, + 175, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 176, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 176, + 117 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 129, + 507, + 733 + ], + "lines": [ + { + "bbox": [ + 115, + 131, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 115, + 131, + 506, + 144 + ], + "score": 1.0, + "content": "models with many observed variables. 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(Markov Inequality) For any random variable", + "type": "text" + }, + { + "bbox": [ + 337, + 499, + 365, + 510 + ], + "score": 0.9, + "content": "X \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 498, + 403, + 512 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 404, + 499, + 434, + 510 + ], + "score": 0.89, + "content": "\\forall \\eta > 0", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 498, + 434, + 512 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 515, + 346, + 542 + ], + "lines": [ + { + "bbox": [ + 264, + 515, + 346, + 542 + ], + "spans": [ + { + "bbox": [ + 264, + 515, + 346, + 542 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( X \\geq \\eta \\right) \\leq \\frac { \\mathbb { E } X } { \\eta } .", + "type": "interline_equation", + "image_path": "ea33d643859be498473f709832490660f7942fb65f2fb63c65a2e5aecca6cdef.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 264, + 515, + 346, + 542 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 563, + 145, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 146, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 146, + 580 + ], + "score": 1.0, + "content": "Part I", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 106, + 587, + 334, + 609 + ], + "lines": [ + { + "bbox": [ + 104, + 582, + 336, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 336, + 612 + ], + "score": 1.0, + "content": "Proofs of the main results", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 368, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 368, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 368, + 642 + ], + "score": 1.0, + "content": "7 FIRST ORDER CONDITION: PROOF OF LEMMA 2", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 627, + 368, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 651, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 314, + 665 + ], + "score": 1.0, + "content": "Lemma 12. (Lemma 2 restated) At all DLMs in", + "type": "text" + }, + { + "bbox": [ + 315, + 653, + 349, + 665 + ], + "score": 0.88, + "content": "{ \\mathcal { D } } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 652, + 506, + 665 + ], + "score": 1.0, + "content": "the residual error e is identical, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 663, + 158, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 158, + 677 + ], + "score": 1.0, + "content": "furthermore", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 652, + 506, + 677 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 272, + 675, + 340, + 688 + ], + "lines": [ + { + "bbox": [ + 272, + 675, + 340, + 688 + ], + "spans": [ + { + "bbox": [ + 272, + 675, + 340, + 688 + ], + "score": 0.88, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0 .", + "type": "interline_equation", + "image_path": "4a2db5c58e310b6fb8cb380966205883c913f5184f7326ec966df11ca7b27d1e.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 272, + 675, + 340, + 688 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 701, + 505, + 735 + ], + "lines": [ + { + "bbox": [ + 104, + 699, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 153, + 719 + ], + "score": 1.0, + "content": "Proof. 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Then, we can re-write the MSE (eq. 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For each row", + "type": "text" + }, + { + "bbox": [ + 205, + 190, + 210, + 198 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 188, + 316, + 200 + ], + "score": 1.0, + "content": ", we divide into two cases:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 220, + 217 + ], + "score": 1.0, + "content": "First, we consider the case", + "type": "text" + }, + { + "bbox": [ + 221, + 205, + 251, + 217 + ], + "score": 0.91, + "content": "z _ { i } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 205, + 436, + 217 + ], + "score": 1.0, + "content": ". In this case, any infinitesimal perturbation", + "type": "text" + }, + { + "bbox": [ + 437, + 207, + 447, + 217 + ], + "score": 0.86, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 205, + 460, + 217 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 461, + 206, + 474, + 216 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { w } } _ { \\mathbf { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 286, + 230 + ], + "score": 1.0, + "content": "produced by an infinitesimal perturbation in", + "type": "text" + }, + { + "bbox": [ + 286, + 217, + 299, + 227 + ], + "score": 0.66, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 216, + 414, + 228 + ], + "score": 0.87, + "content": "\\mathbf { \\widetilde { w } } _ { i } + \\mathbf { q } _ { i } = ( \\mathbf { w } _ { i } + \\mathbf { q } _ { i } / z _ { i } ) z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 216, + 505, + 230 + ], + "score": 1.0, + "content": ". Therefore, unless the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 141, + 240 + ], + "score": 1.0, + "content": "gradient", + "type": "text" + }, + { + "bbox": [ + 141, + 227, + 192, + 239 + ], + "score": 0.71, + "content": "\\partial \\mathrm { { M S E } } / \\partial \\tilde { \\bf { w } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 226, + 432, + 240 + ], + "score": 1.0, + "content": "is equal to zero, we can choose an infinitesimal perturbation", + "type": "text" + }, + { + "bbox": [ + 433, + 228, + 443, + 239 + ], + "score": 0.86, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "in the opposite", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 238, + 330, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 330, + 250 + ], + "score": 1.0, + "content": "direction to this gradient, which will decrease the MSE.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 233, + 267 + ], + "score": 1.0, + "content": "Second, we consider the case", + "type": "text" + }, + { + "bbox": [ + 233, + 256, + 265, + 266 + ], + "score": 0.91, + "content": "z _ { i } ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 255, + 505, + 267 + ], + "score": 1.0, + "content": ". In this case, the MSE is not affected by changes made", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 165, + 278 + ], + "score": 1.0, + "content": "exclusively to", + "type": "text" + }, + { + "bbox": [ + 166, + 267, + 178, + 277 + ], + "score": 0.87, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 265, + 243, + 278 + ], + "score": 1.0, + "content": ". Therefore, all", + "type": "text" + }, + { + "bbox": [ + 243, + 268, + 256, + 277 + ], + "score": 0.88, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 265, + 428, + 278 + ], + "score": 1.0, + "content": "derivatives of the MSE are equal to zero", + "type": "text" + }, + { + "bbox": [ + 428, + 266, + 490, + 277 + ], + "score": 0.91, + "content": "( \\partial ^ { \\bar { k } } \\mathrm { M S E } / \\partial ^ { k } \\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 265, + 506, + 278 + ], + "score": 1.0, + "content": ", to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 147, + 290 + ], + "score": 1.0, + "content": "any order", + "type": "text" + }, + { + "bbox": [ + 147, + 277, + 154, + 287 + ], + "score": 0.71, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 276, + 379, + 290 + ], + "score": 1.0, + "content": ") . Also, since we are at a differentiable local minimum,", + "type": "text" + }, + { + "bbox": [ + 379, + 277, + 445, + 289 + ], + "score": 0.9, + "content": "\\partial \\mathrm { M S E } / \\partial z _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 276, + 506, + 290 + ], + "score": 1.0, + "content": ". Thus, using a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 434, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 234, + 301 + ], + "score": 1.0, + "content": "Taylor expansion, if we perturb", + "type": "text" + }, + { + "bbox": [ + 234, + 288, + 266, + 300 + ], + "score": 0.95, + "content": "\\left( \\mathbf { w } _ { i } , z _ { i } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 288, + 280, + 301 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 281, + 288, + 312, + 300 + ], + "score": 0.89, + "content": "( \\hat { \\mathbf { w } } _ { i } , \\hat { z } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 288, + 434, + 301 + ], + "score": 1.0, + "content": "then the MSE is perturbed by", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 304, + 366, + 330 + ], + "lines": [ + { + "bbox": [ + 244, + 304, + 366, + 330 + ], + "spans": [ + { + "bbox": [ + 244, + 304, + 366, + 330 + ], + "score": 0.94, + "content": "\\hat { z } _ { i } \\hat { \\mathbf { w } } _ { i } ^ { \\top } \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\frac { \\partial } { \\partial z _ { i } } \\mathrm { M S E } + O ( \\hat { z } _ { i } ^ { 2 } )", + "type": "interline_equation", + "image_path": "73b6d9dc4e8bace7bb108ccc01fd884a9a40029b8f814c56447e2b1bb5026b32.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 244, + 304, + 366, + 330 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 177, + 347 + ], + "score": 1.0, + "content": "Therefore, unless", + "type": "text" + }, + { + "bbox": [ + 177, + 335, + 274, + 347 + ], + "score": 0.87, + "content": "\\partial ^ { 2 } \\mathrm { M S E } / \\left( \\partial \\mathbf { w } _ { i } \\partial z _ { i } \\right) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 335, + 334, + 347 + ], + "score": 1.0, + "content": "we can choose", + "type": "text" + }, + { + "bbox": [ + 335, + 336, + 347, + 346 + ], + "score": 0.89, + "content": "\\hat { \\mathbf { w } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 335, + 442, + 347 + ], + "score": 1.0, + "content": "and a sufficiently small", + "type": "text" + }, + { + "bbox": [ + 443, + 336, + 451, + 346 + ], + "score": 0.88, + "content": "\\hat { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "such that the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 345, + 293, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 293, + 358 + ], + "score": 1.0, + "content": "MSE is decreased. Lastly, using the chain rule", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 361, + 410, + 389 + ], + "lines": [ + { + "bbox": [ + 200, + 361, + 410, + 389 + ], + "spans": [ + { + "bbox": [ + 200, + 361, + 410, + 389 + ], + "score": 0.91, + "content": "\\frac { \\partial } { \\partial z _ { i } } \\frac { \\partial } { \\partial \\mathbf { w } _ { i } } \\mathrm { M S E } = \\frac { \\partial } { \\partial z _ { i } } \\left[ z _ { i } \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\mathrm { M S E } \\right] = \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\mathrm { M S E } .", + "type": "interline_equation", + "image_path": "7d71967f3b5e10b684deb34f92ad19e3784bffb8435cfb91b4371b6167b7ec2d.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 361, + 410, + 375.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 200, + 375.0, + 410, + 389.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 399, + 453, + 413 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 453, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 131, + 414 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + }, + { + "bbox": [ + 131, + 401, + 200, + 412 + ], + "score": 0.89, + "content": "\\partial \\mathrm { M S E } / \\partial \\tilde { \\bf w } _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 399, + 319, + 414 + ], + "score": 1.0, + "content": ". This implies that w˜ is also a", + "type": "text" + }, + { + "bbox": [ + 320, + 400, + 347, + 411 + ], + "score": 0.61, + "content": "\\mathrm { D L M } ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 399, + 453, + 414 + ], + "score": 1.0, + "content": "of eq. (7.2), which entails", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 416, + 387, + 443 + ], + "lines": [ + { + "bbox": [ + 223, + 416, + 387, + 443 + ], + "spans": [ + { + "bbox": [ + 223, + 416, + 387, + 443 + ], + "score": 0.94, + "content": "0 = - \\frac { N } { 2 } \\frac { \\partial } { \\partial \\tilde { \\bf w } _ { i } } \\mathrm { M S E } = { \\bf G } \\left( { \\bf y } - { \\bf G } ^ { \\top } \\tilde { \\bf w } \\right) .", + "type": "interline_equation", + "image_path": "f5f4459e41a48b3ff6b81ddac936f4dfd80546b6aa1626a5ef534239ab7d74e9.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 416, + 387, + 429.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 223, + 429.5, + 387, + 443.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 504, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 131, + 462 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 448, + 181, + 459 + ], + "score": 0.9, + "content": "\\mathbf { G } = \\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 446, + 199, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 447, + 262, + 460 + ], + "score": 0.9, + "content": "\\mathbf { e } = \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 446, + 457, + 462 + ], + "score": 1.0, + "content": "this proves eq. (7.1). Now, for any two solutions", + "type": "text" + }, + { + "bbox": [ + 458, + 449, + 471, + 459 + ], + "score": 0.9, + "content": "\\tilde { \\mathbf { w } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 446, + 489, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 449, + 504, + 459 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { w } } _ { 2 }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 459, + 192, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 192, + 472 + ], + "score": 1.0, + "content": "of eq. (7.3), we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 474, + 434, + 491 + ], + "lines": [ + { + "bbox": [ + 176, + 474, + 434, + 491 + ], + "spans": [ + { + "bbox": [ + 176, + 474, + 434, + 491 + ], + "score": 0.92, + "content": "0 = \\mathbf { G } \\left( \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } \\right) - \\mathbf { G } \\left( \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } \\right) = \\mathbf { G } \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) .", + "type": "interline_equation", + "image_path": "bc2aed7f2e7d864aae15f079bebe04835bc820a833fedd0f842f036cf2500a89.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 176, + 474, + 434, + 491 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 315, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 316, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 169, + 512 + ], + "score": 1.0, + "content": "Multiplying by", + "type": "text" + }, + { + "bbox": [ + 170, + 496, + 221, + 510 + ], + "score": 0.93, + "content": "\\big ( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\big ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 495, + 316, + 512 + ], + "score": 1.0, + "content": "from the left we obtain", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 514, + 406, + 533 + ], + "lines": [ + { + "bbox": [ + 205, + 514, + 406, + 533 + ], + "spans": [ + { + "bbox": [ + 205, + 514, + 406, + 533 + ], + "score": 0.91, + "content": "\\left\\| \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) \\right\\| ^ { 2 } = 0 \\Rightarrow \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) = 0 .", + "type": "interline_equation", + "image_path": "2911e0b23e361adb36317d03e00496ae34fd047df2e7089b09dd5ad53e75c87a.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 205, + 514, + 406, + 533 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 536, + 455, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 456, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 306, + 550 + ], + "score": 1.0, + "content": "Therefore, the MNN output and the residual error", + "type": "text" + }, + { + "bbox": [ + 306, + 540, + 313, + 547 + ], + "score": 0.5, + "content": "\\mathbf { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 536, + 418, + 550 + ], + "score": 1.0, + "content": "are equal for all DLMs in", + "type": "text" + }, + { + "bbox": [ + 418, + 537, + 452, + 549 + ], + "score": 0.88, + "content": "{ \\mathcal { D } } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 536, + 456, + 550 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 496, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 495, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 117, + 577 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 119, + 564, + 495, + 578 + ], + "score": 1.0, + "content": "SUB-OPTIMAL DIFFERENTIABLE LOCAL MINIMA: PROOF OF THEOREM 6", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 124, + 579, + 234, + 591 + ], + "spans": [ + { + "bbox": [ + 124, + 579, + 234, + 591 + ], + "score": 1.0, + "content": "AND ITS COROLLARY", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 504, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "Theorem 13. (Theorem 6 restated) Given assumptions 1-4, the expected angular volume of sub-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 614, + 392, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 189, + 627 + ], + "score": 1.0, + "content": "optimal DLMs, with", + "type": "text" + }, + { + "bbox": [ + 190, + 615, + 249, + 625 + ], + "score": 0.86, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 614, + 369, + 627 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 369, + 615, + 380, + 624 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 614, + 392, + 627 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 630, + 409, + 651 + ], + "lines": [ + { + "bbox": [ + 200, + 630, + 409, + 651 + ], + "spans": [ + { + "bbox": [ + 200, + 630, + 409, + 651 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "c6a8ccb7d4fbd7a26598b9b10ca2720f2827e2e52de76f4a69940c947a3a78c8.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 200, + 630, + 409, + 651 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 655, + 483, + 672 + ], + "lines": [], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 129, + 655, + 483, + 672 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 679, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 358, + 693 + ], + "score": 1.0, + "content": "To prove this theorem we upper bound the angular volume of", + "type": "text" + }, + { + "bbox": [ + 359, + 680, + 370, + 691 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "(definition 4), i.e., differentiable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 273, + 704 + ], + "score": 1.0, + "content": "regions in which there exist DLMs with", + "type": "text" + }, + { + "bbox": [ + 273, + 691, + 336, + 701 + ], + "score": 0.89, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 690, + 506, + 704 + ], + "score": 1.0, + "content": ". Our proof uses the first order necessary", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 702, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 259, + 714 + ], + "score": 1.0, + "content": "condition for DLMs from Lemma 2,", + "type": "text" + }, + { + "bbox": [ + 259, + 702, + 321, + 714 + ], + "score": 0.9, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 702, + 454, + 714 + ], + "score": 1.0, + "content": ", to find which configurations of", + "type": "text" + }, + { + "bbox": [ + 455, + 702, + 464, + 712 + ], + "score": 0.42, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 702, + 505, + 714 + ], + "score": 1.0, + "content": "allow for", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 117, + 721, + 450, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 451, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 419, + 734 + ], + "score": 1.0, + "content": "5Note that the converse argument is not true – a DLM in w˜ might not be a DLM in", + "type": "text" + }, + { + "bbox": [ + 419, + 721, + 446, + 732 + ], + "score": 0.91, + "content": "( \\mathbf { W } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 719, + 451, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 537, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 495, + 538, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 495, + 538, + 505, + 549 + ], + "score": 0.996, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 434, + 95 + ], + "score": 1.0, + "content": "diagonal, and vec (M) is vector obtained by stacking the columns of the matrix", + "type": "text" + }, + { + "bbox": [ + 434, + 83, + 447, + 93 + ], + "score": 0.39, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "on top of one", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 323, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 323, + 106 + ], + "score": 1.0, + "content": "another. Then, we can re-write the MSE (eq. (2.3)) as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 109, + 385, + 133 + ], + "lines": [ + { + "bbox": [ + 224, + 109, + 385, + 133 + ], + "spans": [ + { + "bbox": [ + 224, + 109, + 385, + 133 + ], + "score": 0.94, + "content": "\\mathrm { M S E } = { \\frac { 1 } { N } } \\left\\| \\mathbf { y } - \\mathbf { G } ^ { \\top } { \\tilde { \\mathbf { w } } } \\right\\| ^ { 2 } = { \\frac { 1 } { N } } \\left\\| \\mathbf { e } \\right\\| ^ { 2 } ,", + "type": "interline_equation", + "image_path": "2030c62eb28304531a776e0e08483ac21c2f2e8d5d3f66eaf371a620dfdcbb7a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 224, + 109, + 385, + 133 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 138, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 132, + 151 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 137, + 158, + 149 + ], + "score": 0.9, + "content": "\\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 137, + 295, + 151 + ], + "score": 1.0, + "content": "is the output of the MNN. Now, if", + "type": "text" + }, + { + "bbox": [ + 296, + 138, + 325, + 150 + ], + "score": 0.91, + "content": "( \\mathbf { W } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "is a DLM of the MSE in eq. (2.3), then there", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 376, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 243, + 162 + ], + "score": 1.0, + "content": "is no infinitesimal perturbation of", + "type": "text" + }, + { + "bbox": [ + 243, + 150, + 272, + 162 + ], + "score": 0.91, + "content": "( \\mathbf { W } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 149, + 376, + 162 + ], + "score": 1.0, + "content": "which reduces this MSE.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 137, + 505, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 181, + 178 + ], + "score": 1.0, + "content": "Next, for each row", + "type": "text" + }, + { + "bbox": [ + 181, + 168, + 186, + 176 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 167, + 259, + 178 + ], + "score": 1.0, + "content": ", we will show that", + "type": "text" + }, + { + "bbox": [ + 260, + 166, + 329, + 178 + ], + "score": 0.71, + "content": "\\partial \\mathrm { M S E } / \\partial \\tilde { \\bf w } _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 167, + 505, + 178 + ], + "score": 1.0, + "content": ", since otherwise we can find an infinitesimal", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 167, + 190 + ], + "score": 1.0, + "content": "perturbation of", + "type": "text" + }, + { + "bbox": [ + 168, + 178, + 197, + 189 + ], + "score": 0.9, + "content": "( \\mathbf { W } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 177, + 437, + 190 + ], + "score": 1.0, + "content": "which decreases the MSE, contradicting the assumption that", + "type": "text" + }, + { + "bbox": [ + 437, + 178, + 466, + 190 + ], + "score": 0.92, + "content": "( \\mathbf { W } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "is a local", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 188, + 316, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 205, + 200 + ], + "score": 1.0, + "content": "minimum. For each row", + "type": "text" + }, + { + "bbox": [ + 205, + 190, + 210, + 198 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 188, + 316, + 200 + ], + "score": 1.0, + "content": ", we divide into two cases:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 166, + 505, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 220, + 217 + ], + "score": 1.0, + "content": "First, we consider the case", + "type": "text" + }, + { + "bbox": [ + 221, + 205, + 251, + 217 + ], + "score": 0.91, + "content": "z _ { i } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 205, + 436, + 217 + ], + "score": 1.0, + "content": ". In this case, any infinitesimal perturbation", + "type": "text" + }, + { + "bbox": [ + 437, + 207, + 447, + 217 + ], + "score": 0.86, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 205, + 460, + 217 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 461, + 206, + 474, + 216 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { w } } _ { \\mathbf { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 286, + 230 + ], + "score": 1.0, + "content": "produced by an infinitesimal perturbation in", + "type": "text" + }, + { + "bbox": [ + 286, + 217, + 299, + 227 + ], + "score": 0.66, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 216, + 414, + 228 + ], + "score": 0.87, + "content": "\\mathbf { \\widetilde { w } } _ { i } + \\mathbf { q } _ { i } = ( \\mathbf { w } _ { i } + \\mathbf { q } _ { i } / z _ { i } ) z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 216, + 505, + 230 + ], + "score": 1.0, + "content": ". Therefore, unless the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 141, + 240 + ], + "score": 1.0, + "content": "gradient", + "type": "text" + }, + { + "bbox": [ + 141, + 227, + 192, + 239 + ], + "score": 0.71, + "content": "\\partial \\mathrm { { M S E } } / \\partial \\tilde { \\bf { w } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 226, + 432, + 240 + ], + "score": 1.0, + "content": "is equal to zero, we can choose an infinitesimal perturbation", + "type": "text" + }, + { + "bbox": [ + 433, + 228, + 443, + 239 + ], + "score": 0.86, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "in the opposite", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 238, + 330, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 330, + 250 + ], + "score": 1.0, + "content": "direction to this gradient, which will decrease the MSE.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 205, + 505, + 250 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 233, + 267 + ], + "score": 1.0, + "content": "Second, we consider the case", + "type": "text" + }, + { + "bbox": [ + 233, + 256, + 265, + 266 + ], + "score": 0.91, + "content": "z _ { i } ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 255, + 505, + 267 + ], + "score": 1.0, + "content": ". In this case, the MSE is not affected by changes made", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 265, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 165, + 278 + ], + "score": 1.0, + "content": "exclusively to", + "type": "text" + }, + { + "bbox": [ + 166, + 267, + 178, + 277 + ], + "score": 0.87, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 265, + 243, + 278 + ], + "score": 1.0, + "content": ". Therefore, all", + "type": "text" + }, + { + "bbox": [ + 243, + 268, + 256, + 277 + ], + "score": 0.88, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 265, + 428, + 278 + ], + "score": 1.0, + "content": "derivatives of the MSE are equal to zero", + "type": "text" + }, + { + "bbox": [ + 428, + 266, + 490, + 277 + ], + "score": 0.91, + "content": "( \\partial ^ { \\bar { k } } \\mathrm { M S E } / \\partial ^ { k } \\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 265, + 506, + 278 + ], + "score": 1.0, + "content": ", to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 147, + 290 + ], + "score": 1.0, + "content": "any order", + "type": "text" + }, + { + "bbox": [ + 147, + 277, + 154, + 287 + ], + "score": 0.71, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 276, + 379, + 290 + ], + "score": 1.0, + "content": ") . Also, since we are at a differentiable local minimum,", + "type": "text" + }, + { + "bbox": [ + 379, + 277, + 445, + 289 + ], + "score": 0.9, + "content": "\\partial \\mathrm { M S E } / \\partial z _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 276, + 506, + 290 + ], + "score": 1.0, + "content": ". Thus, using a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 434, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 234, + 301 + ], + "score": 1.0, + "content": "Taylor expansion, if we perturb", + "type": "text" + }, + { + "bbox": [ + 234, + 288, + 266, + 300 + ], + "score": 0.95, + "content": "\\left( \\mathbf { w } _ { i } , z _ { i } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 288, + 280, + 301 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 281, + 288, + 312, + 300 + ], + "score": 0.89, + "content": "( \\hat { \\mathbf { w } } _ { i } , \\hat { z } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 288, + 434, + 301 + ], + "score": 1.0, + "content": "then the MSE is perturbed by", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 255, + 506, + 301 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 304, + 366, + 330 + ], + "lines": [ + { + "bbox": [ + 244, + 304, + 366, + 330 + ], + "spans": [ + { + "bbox": [ + 244, + 304, + 366, + 330 + ], + "score": 0.94, + "content": "\\hat { z } _ { i } \\hat { \\mathbf { w } } _ { i } ^ { \\top } \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\frac { \\partial } { \\partial z _ { i } } \\mathrm { M S E } + O ( \\hat { z } _ { i } ^ { 2 } )", + "type": "interline_equation", + "image_path": "73b6d9dc4e8bace7bb108ccc01fd884a9a40029b8f814c56447e2b1bb5026b32.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 244, + 304, + 366, + 330 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 177, + 347 + ], + "score": 1.0, + "content": "Therefore, unless", + "type": "text" + }, + { + "bbox": [ + 177, + 335, + 274, + 347 + ], + "score": 0.87, + "content": "\\partial ^ { 2 } \\mathrm { M S E } / \\left( \\partial \\mathbf { w } _ { i } \\partial z _ { i } \\right) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 335, + 334, + 347 + ], + "score": 1.0, + "content": "we can choose", + "type": "text" + }, + { + "bbox": [ + 335, + 336, + 347, + 346 + ], + "score": 0.89, + "content": "\\hat { \\mathbf { w } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 335, + 442, + 347 + ], + "score": 1.0, + "content": "and a sufficiently small", + "type": "text" + }, + { + "bbox": [ + 443, + 336, + 451, + 346 + ], + "score": 0.88, + "content": "\\hat { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "such that the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 345, + 293, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 293, + 358 + ], + "score": 1.0, + "content": "MSE is decreased. Lastly, using the chain rule", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 335, + 505, + 358 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 361, + 410, + 389 + ], + "lines": [ + { + "bbox": [ + 200, + 361, + 410, + 389 + ], + "spans": [ + { + "bbox": [ + 200, + 361, + 410, + 389 + ], + "score": 0.91, + "content": "\\frac { \\partial } { \\partial z _ { i } } \\frac { \\partial } { \\partial \\mathbf { w } _ { i } } \\mathrm { M S E } = \\frac { \\partial } { \\partial z _ { i } } \\left[ z _ { i } \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\mathrm { M S E } \\right] = \\frac { \\partial } { \\partial \\tilde { \\mathbf { w } } _ { i } } \\mathrm { M S E } .", + "type": "interline_equation", + "image_path": "7d71967f3b5e10b684deb34f92ad19e3784bffb8435cfb91b4371b6167b7ec2d.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 361, + 410, + 375.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 200, + 375.0, + 410, + 389.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 399, + 453, + 413 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 453, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 131, + 414 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + }, + { + "bbox": [ + 131, + 401, + 200, + 412 + ], + "score": 0.89, + "content": "\\partial \\mathrm { M S E } / \\partial \\tilde { \\bf w } _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 399, + 319, + 414 + ], + "score": 1.0, + "content": ". This implies that w˜ is also a", + "type": "text" + }, + { + "bbox": [ + 320, + 400, + 347, + 411 + ], + "score": 0.61, + "content": "\\mathrm { D L M } ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 399, + 453, + 414 + ], + "score": 1.0, + "content": "of eq. (7.2), which entails", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 399, + 453, + 414 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 416, + 387, + 443 + ], + "lines": [ + { + "bbox": [ + 223, + 416, + 387, + 443 + ], + "spans": [ + { + "bbox": [ + 223, + 416, + 387, + 443 + ], + "score": 0.94, + "content": "0 = - \\frac { N } { 2 } \\frac { \\partial } { \\partial \\tilde { \\bf w } _ { i } } \\mathrm { M S E } = { \\bf G } \\left( { \\bf y } - { \\bf G } ^ { \\top } \\tilde { \\bf w } \\right) .", + "type": "interline_equation", + "image_path": "f5f4459e41a48b3ff6b81ddac936f4dfd80546b6aa1626a5ef534239ab7d74e9.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 416, + 387, + 429.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 223, + 429.5, + 387, + 443.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 504, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 131, + 462 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 448, + 181, + 459 + ], + "score": 0.9, + "content": "\\mathbf { G } = \\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 446, + 199, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 447, + 262, + 460 + ], + "score": 0.9, + "content": "\\mathbf { e } = \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 446, + 457, + 462 + ], + "score": 1.0, + "content": "this proves eq. (7.1). Now, for any two solutions", + "type": "text" + }, + { + "bbox": [ + 458, + 449, + 471, + 459 + ], + "score": 0.9, + "content": "\\tilde { \\mathbf { w } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 446, + 489, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 490, + 449, + 504, + 459 + ], + "score": 0.87, + "content": "\\tilde { \\mathbf { w } } _ { 2 }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 459, + 192, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 192, + 472 + ], + "score": 1.0, + "content": "of eq. (7.3), we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 446, + 504, + 472 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 474, + 434, + 491 + ], + "lines": [ + { + "bbox": [ + 176, + 474, + 434, + 491 + ], + "spans": [ + { + "bbox": [ + 176, + 474, + 434, + 491 + ], + "score": 0.92, + "content": "0 = \\mathbf { G } \\left( \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } \\right) - \\mathbf { G } \\left( \\mathbf { y } - \\mathbf { G } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } \\right) = \\mathbf { G } \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) .", + "type": "interline_equation", + "image_path": "bc2aed7f2e7d864aae15f079bebe04835bc820a833fedd0f842f036cf2500a89.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 176, + 474, + 434, + 491 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 315, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 316, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 169, + 512 + ], + "score": 1.0, + "content": "Multiplying by", + "type": "text" + }, + { + "bbox": [ + 170, + 496, + 221, + 510 + ], + "score": 0.93, + "content": "\\big ( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\big ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 495, + 316, + 512 + ], + "score": 1.0, + "content": "from the left we obtain", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 495, + 316, + 512 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 514, + 406, + 533 + ], + "lines": [ + { + "bbox": [ + 205, + 514, + 406, + 533 + ], + "spans": [ + { + "bbox": [ + 205, + 514, + 406, + 533 + ], + "score": 0.91, + "content": "\\left\\| \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) \\right\\| ^ { 2 } = 0 \\Rightarrow \\mathbf { G } ^ { \\top } \\left( \\tilde { \\mathbf { w } } _ { 2 } - \\tilde { \\mathbf { w } } _ { 1 } \\right) = 0 .", + "type": "interline_equation", + "image_path": "2911e0b23e361adb36317d03e00496ae34fd047df2e7089b09dd5ad53e75c87a.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 205, + 514, + 406, + 533 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 536, + 455, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 456, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 306, + 550 + ], + "score": 1.0, + "content": "Therefore, the MNN output and the residual error", + "type": "text" + }, + { + "bbox": [ + 306, + 540, + 313, + 547 + ], + "score": 0.5, + "content": "\\mathbf { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 536, + 418, + 550 + ], + "score": 1.0, + "content": "are equal for all DLMs in", + "type": "text" + }, + { + "bbox": [ + 418, + 537, + 452, + 549 + ], + "score": 0.88, + "content": "{ \\mathcal { D } } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 536, + 456, + 550 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 536, + 456, + 550 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 496, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 495, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 117, + 577 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 119, + 564, + 495, + 578 + ], + "score": 1.0, + "content": "SUB-OPTIMAL DIFFERENTIABLE LOCAL MINIMA: PROOF OF THEOREM 6", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 124, + 579, + 234, + 591 + ], + "spans": [ + { + "bbox": [ + 124, + 579, + 234, + 591 + ], + "score": 1.0, + "content": "AND ITS COROLLARY", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 504, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "Theorem 13. (Theorem 6 restated) Given assumptions 1-4, the expected angular volume of sub-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 614, + 392, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 189, + 627 + ], + "score": 1.0, + "content": "optimal DLMs, with", + "type": "text" + }, + { + "bbox": [ + 190, + 615, + 249, + 625 + ], + "score": 0.86, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 614, + 369, + 627 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 369, + 615, + 380, + 624 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 614, + 392, + 627 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 603, + 505, + 627 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 630, + 409, + 651 + ], + "lines": [ + { + "bbox": [ + 200, + 630, + 409, + 651 + ], + "spans": [ + { + "bbox": [ + 200, + 630, + 409, + 651 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "c6a8ccb7d4fbd7a26598b9b10ca2720f2827e2e52de76f4a69940c947a3a78c8.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 200, + 630, + 409, + 651 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 655, + 483, + 672 + ], + "lines": [], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 129, + 655, + 483, + 672 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 679, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 358, + 693 + ], + "score": 1.0, + "content": "To prove this theorem we upper bound the angular volume of", + "type": "text" + }, + { + "bbox": [ + 359, + 680, + 370, + 691 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "(definition 4), i.e., differentiable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 273, + 704 + ], + "score": 1.0, + "content": "regions in which there exist DLMs with", + "type": "text" + }, + { + "bbox": [ + 273, + 691, + 336, + 701 + ], + "score": 0.89, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 690, + 506, + 704 + ], + "score": 1.0, + "content": ". Our proof uses the first order necessary", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 702, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 259, + 714 + ], + "score": 1.0, + "content": "condition for DLMs from Lemma 2,", + "type": "text" + }, + { + "bbox": [ + 259, + 702, + 321, + 714 + ], + "score": 0.9, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 702, + 454, + 714 + ], + "score": 1.0, + "content": ", to find which configurations of", + "type": "text" + }, + { + "bbox": [ + 455, + 702, + 464, + 712 + ], + "score": 0.42, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 702, + 505, + 714 + ], + "score": 1.0, + "content": "allow for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 221, + 95 + ], + "score": 1.0, + "content": "a high residual error e with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 221, + 83, + 284, + 93 + ], + "score": 0.89, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 284, + 82, + 385, + 95 + ], + "score": 1.0, + "content": ". In these configurations", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 385, + 83, + 414, + 93 + ], + "score": 0.83, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 414, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "cannot have full rank,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 304, + 106 + ], + "score": 1.0, + "content": "and therefore, as we show (Lemma 14 below),", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 304, + 93, + 365, + 106 + ], + "score": 0.88, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 365, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "must have a low rank. However,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 168, + 116 + ], + "score": 0.82, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 168, + 104, + 407, + 117 + ], + "score": 1.0, + "content": "has a low rank with exponentially low probability when", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 407, + 105, + 443, + 114 + ], + "score": 0.89, + "content": "\\mathbf x \\sim \\mathcal N", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 444, + 104, + 464, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 464, + 104, + 504, + 115 + ], + "score": 0.87, + "content": "\\mathbf { W } \\sim { \\mathcal { N } }", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 452, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 364, + 128 + ], + "score": 1.0, + "content": "(Lemmas 15 and 16 below). Thus, we derive an upper bound on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 364, + 115, + 448, + 127 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right)", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 448, + 115, + 452, + 128 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 679, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 221, + 95 + ], + "score": 1.0, + "content": "a high residual error e with", + "type": "text" + }, + { + "bbox": [ + 221, + 83, + 284, + 93 + ], + "score": 0.89, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 82, + 385, + 95 + ], + "score": 1.0, + "content": ". In these configurations", + "type": "text" + }, + { + "bbox": [ + 385, + 83, + 414, + 93 + ], + "score": 0.83, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "cannot have full rank,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 304, + 106 + ], + "score": 1.0, + "content": "and therefore, as we show (Lemma 14 below),", + "type": "text" + }, + { + "bbox": [ + 304, + 93, + 365, + 106 + ], + "score": 0.88, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "must have a low rank. However,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 168, + 116 + ], + "score": 0.82, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 104, + 407, + 117 + ], + "score": 1.0, + "content": "has a low rank with exponentially low probability when", + "type": "text" + }, + { + "bbox": [ + 407, + 105, + 443, + 114 + ], + "score": 0.89, + "content": "\\mathbf x \\sim \\mathcal N", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 104, + 464, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 464, + 104, + 504, + 115 + ], + "score": 0.87, + "content": "\\mathbf { W } \\sim { \\mathcal { N } }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 452, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 364, + 128 + ], + "score": 1.0, + "content": "(Lemmas 15 and 16 below). Thus, we derive an upper bound on", + "type": "text" + }, + { + "bbox": [ + 364, + 115, + 448, + 127 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 115, + 452, + 128 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 506, + 180 + ], + "lines": [ + { + "bbox": [ + 104, + 131, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 286, + 147 + ], + "score": 1.0, + "content": "Before we begin, let us recall some notation:", + "type": "text" + }, + { + "bbox": [ + 286, + 133, + 397, + 146 + ], + "score": 0.88, + "content": "[ L ] \\triangleq \\{ 1 , 2 , \\dots , L \\} , \\mathbf { M } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 131, + 447, + 147 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 448, + 134, + 505, + 146 + ], + "score": 0.88, + "content": "\\forall i , j : M _ { i j } >", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 115, + 158 + ], + "score": 1.0, + "content": "0,", + "type": "text" + }, + { + "bbox": [ + 116, + 145, + 134, + 156 + ], + "score": 0.75, + "content": "\\mathbf { M } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 144, + 403, + 158 + ], + "score": 1.0, + "content": "is the matrix composed of the columns of M that are in the index set", + "type": "text" + }, + { + "bbox": [ + 403, + 145, + 411, + 155 + ], + "score": 0.7, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 144, + 414, + 158 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 415, + 145, + 436, + 157 + ], + "score": 0.89, + "content": "\\left\\| \\mathbf { v } \\right\\| _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 144, + 461, + 158 + ], + "score": 1.0, + "content": "as the", + "type": "text" + }, + { + "bbox": [ + 461, + 146, + 474, + 156 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 144, + 506, + 158 + ], + "score": 1.0, + "content": "“norm”", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 289, + 168 + ], + "score": 1.0, + "content": "that counts the number of non-zero values in", + "type": "text" + }, + { + "bbox": [ + 289, + 158, + 297, + 166 + ], + "score": 0.46, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 155, + 408, + 168 + ], + "score": 1.0, + "content": ". First we consider the case", + "type": "text" + }, + { + "bbox": [ + 408, + 157, + 433, + 168 + ], + "score": 0.9, + "content": "\\rho \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 155, + 505, + 168 + ], + "score": 1.0, + "content": ". Also, we denote", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 166, + 199, + 181 + ], + "spans": [ + { + "bbox": [ + 107, + 167, + 194, + 180 + ], + "score": 0.89, + "content": "K _ { r } \\triangleq \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 166, + 199, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 239, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 240, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 212, + 198 + ], + "score": 1.0, + "content": "First we consider the case", + "type": "text" + }, + { + "bbox": [ + 212, + 185, + 236, + 196 + ], + "score": 0.93, + "content": "\\rho \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 183, + 240, + 198 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 269, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 269, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 269, + 215 + ], + "score": 1.0, + "content": "From definition 3 of the angular volume", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 227, + 514, + 411 + ], + "lines": [ + { + "bbox": [ + 111, + 227, + 514, + 411 + ], + "spans": [ + { + "bbox": [ + 111, + 227, + 514, + 411 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\mathfrak { r } _ { \\mathbf { x } \\times \\mathcal { N } } \\mathcal { V } ( \\mathcal { L } , ( \\mathbf { X } , \\mathbf { y } ) ) = \\mathfrak { h } _ { ( \\mathbf { x } , \\mathbf { y } ) \\sim \\mathcal { N } _ { \\mathbf { x } \\times \\mathbf { y } } } \\times \\mathbf { w } \\times ( \\mathbf { \\overline { { B } } } \\otimes \\mathcal { L } , ( \\mathbf { x } , \\mathbf { x } , \\mathbf { y } ) ) } & { } \\\\ { \\overset { ( 1 ) } { \\leq } \\mathbf { R } _ { ( \\mathbf { x } , \\mathbf { y } ) \\sim \\mathcal { D } _ { \\mathbf { x } \\times \\mathbf { y } } } \\times \\mathbf { w } \\times \\times ( \\mathbf { a } \\in \\{ \\rho _ { 1 } \\} ) ^ { d _ { 1 } \\times \\mathbb { N } } , \\ \\mathbf { W } \\in \\mathcal { D } _ { \\mathbf { A } } ( \\mathbf { X } ) , \\mathbf { v } \\in \\mathbb { R } ^ { N } \\ ; \\ ( \\mathbf { A } \\circ \\mathbf { X } ) \\times \\mathbf { v } \\times \\mathbf { w } } & { } \\\\ { \\overset { ( 2 ) } { \\leq } \\mathbf { R } _ { \\mathbf { x } \\times \\mathcal { N } , \\mathbf { w } \\times \\mathbf { N } } \\times ( \\mathbf { A } \\in \\{ \\rho _ { 1 } \\} ) ^ { d _ { 1 } \\times \\mathbb { N } } , \\ \\mathbf { W } \\in \\mathcal { D } _ { \\mathbf { A } } ( \\mathbf { X } ) , \\mathbf { v } \\in \\mathbb { R } ^ { N } \\ ; \\ ( \\mathbf { A } \\circ \\mathbf { X } ) \\times \\mathbf { v } = 0 , } & { } \\\\ { \\overset { ( 4 ) } { \\leq } \\mathbf { R } _ { \\mathbf { x } \\times \\mathcal { N } , \\mathbf { w } \\times \\mathbf { N } } ( \\mathbf { S } \\subset [ \\mathbf { N } ] \\ \\cdot \\{ | \\mathbf { S } | \\ \\ \\geq \\operatorname* { m a x } [ N _ { c , \\mathrm { t r a l i k } } ( \\alpha ( \\langle \\mathbf { W } \\mathbf { X } ) ) d _ { 0 } + 1 ] \\} } & { } \\\\ - \\mathbf { B } _ { \\mathbf { x } \\times \\mathbf { y } } [ \\mathbb { W } _ { \\mathbf { w } \\times \\mathbf { x } } ( \\mathcal { S } \\subset [ \\mathbf { N } ] \\ \\cdot \\{ | \\mathbf { S } | \\ \\geq \\operatorname* { m a x } [ \\mathcal { N } _ { c , \\mathrm { r a l i k } } ( \\alpha ( \\mathbf { W } \\mathbf { X } ) ) d _ \\end{array}", + "type": "interline_equation", + "image_path": "a5512ce672566576d61aae699006856a63d849992703e5b9a63acd0ed1a077bd.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 111, + 227, + 514, + 288.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 288.3333333333333, + 514, + 349.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 349.66666666666663, + 514, + 410.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 132, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 134, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 134, + 436 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 130, + 449, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 128, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 128, + 448, + 230, + 462 + ], + "score": 1.0, + "content": "1. 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Also, if", + "type": "text" + }, + { + "bbox": [ + 456, + 448, + 492, + 460 + ], + "score": 0.91, + "content": "e ^ { ( n ) } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 141, + 460, + 402, + 474 + ], + "score": 1.0, + "content": "some sample, we necessarily classify it correctly, and therefore", + "type": "text" + }, + { + "bbox": [ + 403, + 461, + 475, + 473 + ], + "score": 0.91, + "content": "\\mathrm { M C E } \\leq \\| \\mathbf { e } \\| _ { 0 } / N", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 460, + 505, + 474 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 470, + 482, + 486 + ], + "spans": [ + { + "bbox": [ + 142, + 472, + 185, + 482 + ], + "score": 0.82, + "content": "\\mathrm { M C E } > \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 470, + 195, + 486 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 195, + 472, + 208, + 483 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 470, + 275, + 486 + ], + "score": 1.0, + "content": "this implies that", + "type": "text" + }, + { + "bbox": [ + 275, + 471, + 322, + 484 + ], + "score": 0.92, + "content": "N \\bar { \\epsilon } < \\| \\mathbf { e } \\| _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 470, + 452, + 486 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 210, + 555, + 261, + 566 + ], + "score": 0.87, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 553, + 264, + 569 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 265, + 553, + 333, + 568 + ], + "score": 0.87, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 553, + 336, + 569 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 337, + 554, + 372, + 568 + ], + "score": 0.82, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 553, + 391, + 569 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 391, + 556, + 434, + 567 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 553, + 508, + 569 + ], + "score": 1.0, + "content": ". 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Recall that", + "type": "text" + }, + { + "bbox": [ + 189, + 651, + 275, + 663 + ], + "score": 0.91, + "content": "K _ { r } \\triangleq \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 651, + 477, + 664 + ], + "score": 1.0, + "content": ". We use the union bound over all possible ranks", + "type": "text" + }, + { + "bbox": [ + 477, + 652, + 502, + 663 + ], + "score": 0.9, + "content": "r \\geq 1", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 141, + 663, + 202, + 676 + ], + "score": 1.0, + "content": "we ignore the", + "type": "text" + }, + { + "bbox": [ + 202, + 664, + 230, + 673 + ], + "score": 0.88, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 663, + 291, + 676 + ], + "score": 1.0, + "content": "case since for", + "type": "text" + }, + { + "bbox": [ + 292, + 663, + 320, + 675 + ], + "score": 0.9, + "content": "\\rho \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "(see eq. (2.1)) there is zero probability that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 673, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 141, + 673, + 164, + 686 + ], + "score": 1.0, + "content": "rank", + "type": "text" + }, + { + "bbox": [ + 165, + 673, + 235, + 685 + ], + "score": 0.87, + "content": "( a ( \\mathbf { W } \\mathbf { X } _ { S } ) ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 673, + 324, + 686 + ], + "score": 1.0, + "content": "for some non-empty", + "type": "text" + }, + { + "bbox": [ + 324, + 674, + 332, + 684 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 673, + 399, + 686 + ], + "score": 1.0, + "content": ". For each rank", + "type": "text" + }, + { + "bbox": [ + 399, + 674, + 426, + 685 + ], + "score": 0.9, + "content": "r \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 673, + 506, + 686 + ], + "score": 1.0, + "content": ", it is required that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 143, + 684, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 143, + 685, + 253, + 696 + ], + "score": 0.89, + "content": "| S | > K _ { r } = \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 684, + 268, + 697 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 268, + 685, + 307, + 696 + ], + "score": 0.93, + "content": "| S | = K _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 684, + 506, + 697 + ], + "score": 1.0, + "content": "is a relaxation of the original condition, and thus", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 695, + 251, + 707 + ], + "spans": [ + { + "bbox": [ + 141, + 695, + 251, + 707 + ], + "score": 1.0, + "content": "its probability is not lower.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 128, + 720, + 419, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 719, + 419, + 734 + ], + "spans": [ + { + "bbox": [ + 129, + 719, + 363, + 734 + ], + "score": 1.0, + "content": "5. We again use the union bound over all possible subsets", + "type": "text" + }, + { + "bbox": [ + 364, + 721, + 372, + 730 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 719, + 402, + 734 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 402, + 721, + 416, + 732 + ], + "score": 0.88, + "content": "K _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 719, + 419, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 506, + 180 + ], + "lines": [ + { + "bbox": [ + 104, + 131, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 286, + 147 + ], + "score": 1.0, + "content": "Before we begin, let us recall some notation:", + "type": "text" + }, + { + "bbox": [ + 286, + 133, + 397, + 146 + ], + "score": 0.88, + "content": "[ L ] \\triangleq \\{ 1 , 2 , \\dots , L \\} , \\mathbf { M } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 131, + 447, + 147 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 448, + 134, + 505, + 146 + ], + "score": 0.88, + "content": "\\forall i , j : M _ { i j } >", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 115, + 158 + ], + "score": 1.0, + "content": "0,", + "type": "text" + }, + { + "bbox": [ + 116, + 145, + 134, + 156 + ], + "score": 0.75, + "content": "\\mathbf { M } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 144, + 403, + 158 + ], + "score": 1.0, + "content": "is the matrix composed of the columns of M that are in the index set", + "type": "text" + }, + { + "bbox": [ + 403, + 145, + 411, + 155 + ], + "score": 0.7, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 144, + 414, + 158 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 415, + 145, + 436, + 157 + ], + "score": 0.89, + "content": "\\left\\| \\mathbf { v } \\right\\| _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 144, + 461, + 158 + ], + "score": 1.0, + "content": "as the", + "type": "text" + }, + { + "bbox": [ + 461, + 146, + 474, + 156 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 144, + 506, + 158 + ], + "score": 1.0, + "content": "“norm”", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 289, + 168 + ], + "score": 1.0, + "content": "that counts the number of non-zero values in", + "type": "text" + }, + { + "bbox": [ + 289, + 158, + 297, + 166 + ], + "score": 0.46, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 155, + 408, + 168 + ], + "score": 1.0, + "content": ". First we consider the case", + "type": "text" + }, + { + "bbox": [ + 408, + 157, + 433, + 168 + ], + "score": 0.9, + "content": "\\rho \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 155, + 505, + 168 + ], + "score": 1.0, + "content": ". Also, we denote", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 166, + 199, + 181 + ], + "spans": [ + { + "bbox": [ + 107, + 167, + 194, + 180 + ], + "score": 0.89, + "content": "K _ { r } \\triangleq \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 166, + 199, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 131, + 506, + 181 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 239, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 240, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 212, + 198 + ], + "score": 1.0, + "content": "First we consider the case", + "type": "text" + }, + { + "bbox": [ + 212, + 185, + 236, + 196 + ], + "score": 0.93, + "content": "\\rho \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 183, + 240, + 198 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 183, + 240, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 269, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 269, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 269, + 215 + ], + "score": 1.0, + "content": "From definition 3 of the angular volume", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 200, + 269, + 215 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 227, + 514, + 411 + ], + "lines": [ + { + "bbox": [ + 111, + 227, + 514, + 411 + ], + "spans": [ + { + "bbox": [ + 111, + 227, + 514, + 411 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\mathfrak { r } _ { \\mathbf { x } \\times \\mathcal { N } } \\mathcal { V } ( \\mathcal { L } , ( \\mathbf { X } , \\mathbf { y } ) ) = \\mathfrak { h } _ { ( \\mathbf { x } , \\mathbf { y } ) \\sim \\mathcal { N } _ { \\mathbf { x } \\times \\mathbf { y } } } \\times \\mathbf { w } \\times ( \\mathbf { \\overline { { B } } } \\otimes \\mathcal { L } , ( \\mathbf { x } , \\mathbf { x } , \\mathbf { y } ) ) } & { } \\\\ { \\overset { ( 1 ) } { \\leq } \\mathbf { R } _ { ( \\mathbf { x } , \\mathbf { y } ) \\sim \\mathcal { D } _ { \\mathbf { x } \\times \\mathbf { y } } } \\times \\mathbf { w } \\times \\times ( \\mathbf { a } \\in \\{ \\rho _ { 1 } \\} ) ^ { d _ { 1 } \\times \\mathbb { N } } , \\ \\mathbf { W } \\in \\mathcal { D } _ { \\mathbf { A } } ( \\mathbf { X } ) , \\mathbf { v } \\in \\mathbb { R } ^ { N } \\ ; \\ ( \\mathbf { A } \\circ \\mathbf { X } ) \\times \\mathbf { v } \\times \\mathbf { w } } & { } \\\\ { \\overset { ( 2 ) } { \\leq } \\mathbf { R } _ { \\mathbf { x } \\times \\mathcal { N } , \\mathbf { w } \\times \\mathbf { N } } \\times ( \\mathbf { A } \\in \\{ \\rho _ { 1 } \\} ) ^ { d _ { 1 } \\times \\mathbb { N } } , \\ \\mathbf { W } \\in \\mathcal { D } _ { \\mathbf { A } } ( \\mathbf { X } ) , \\mathbf { v } \\in \\mathbb { R } ^ { N } \\ ; \\ ( \\mathbf { A } \\circ \\mathbf { X } ) \\times \\mathbf { v } = 0 , } & { } \\\\ { \\overset { ( 4 ) } { \\leq } \\mathbf { R } _ { \\mathbf { x } \\times \\mathcal { N } , \\mathbf { w } \\times \\mathbf { N } } ( \\mathbf { S } \\subset [ \\mathbf { N } ] \\ \\cdot \\{ | \\mathbf { S } | \\ \\ \\geq \\operatorname* { m a x } [ N _ { c , \\mathrm { t r a l i k } } ( \\alpha ( \\langle \\mathbf { W } \\mathbf { X } ) ) d _ { 0 } + 1 ] \\} } & { } \\\\ - \\mathbf { B } _ { \\mathbf { x } \\times \\mathbf { y } } [ \\mathbb { W } _ { \\mathbf { w } \\times \\mathbf { x } } ( \\mathcal { S } \\subset [ \\mathbf { N } ] \\ \\cdot \\{ | \\mathbf { S } | \\ \\geq \\operatorname* { m a x } [ \\mathcal { N } _ { c , \\mathrm { r a l i k } } ( \\alpha ( \\mathbf { W } \\mathbf { X } ) ) d _ \\end{array}", + "type": "interline_equation", + "image_path": "a5512ce672566576d61aae699006856a63d849992703e5b9a63acd0ed1a077bd.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 111, + 227, + 514, + 288.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 288.3333333333333, + 514, + 349.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 349.66666666666663, + 514, + 410.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 132, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 134, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 134, + 436 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 422, + 134, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 449, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 128, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 128, + 448, + 230, + 462 + ], + "score": 1.0, + "content": "1. If we are at DLM a in", + "type": "text" + }, + { + "bbox": [ + 230, + 450, + 264, + 461 + ], + "score": 0.85, + "content": "\\mathcal { D } _ { \\mathbf { A } } \\left( \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 448, + 359, + 462 + ], + "score": 1.0, + "content": ", then Lemma 2 implies", + "type": "text" + }, + { + "bbox": [ + 360, + 450, + 420, + 461 + ], + "score": 0.91, + "content": "\\left( \\mathbf { A } \\circ \\mathbf { X } \\right) \\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 448, + 456, + 462 + ], + "score": 1.0, + "content": ". 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Since", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 470, + 482, + 486 + ], + "spans": [ + { + "bbox": [ + 142, + 472, + 185, + 482 + ], + "score": 0.82, + "content": "\\mathrm { M C E } > \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 470, + 195, + 486 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 195, + 472, + 208, + 483 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 470, + 275, + 486 + ], + "score": 1.0, + "content": "this implies that", + "type": "text" + }, + { + "bbox": [ + 275, + 471, + 322, + 484 + ], + "score": 0.92, + "content": "N \\bar { \\epsilon } < \\| \\mathbf { e } \\| _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 470, + 452, + 486 + ], + "score": 1.0, + "content": ". Thus, this inequality holds for", + "type": "text" + }, + { + "bbox": [ + 452, + 474, + 478, + 482 + ], + "score": 0.87, + "content": "{ \\bf v } = { \\bf e }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 470, + 482, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 128, + 448, + 506, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 496, + 294, + 509 + ], + "lines": [ + { + "bbox": [ + 128, + 496, + 294, + 510 + ], + "spans": [ + { + "bbox": [ + 128, + 496, + 258, + 510 + ], + "score": 1.0, + "content": "2. 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Thus, we can apply the following Lemma,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 534, + 275, + 546 + ], + "spans": [ + { + "bbox": [ + 141, + 534, + 275, + 546 + ], + "score": 1.0, + "content": "proven in appendix section 12.1:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 128, + 520, + 507, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 553, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 139, + 553, + 508, + 569 + ], + "spans": [ + { + "bbox": [ + 139, + 553, + 210, + 569 + ], + "score": 1.0, + "content": "Lemma 14. Let", + "type": "text" + }, + { + "bbox": [ + 210, + 555, + 261, + 566 + ], + "score": 0.87, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 553, + 264, + 569 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 265, + 553, + 333, + 568 + ], + "score": 0.87, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 553, + 336, + 569 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 337, + 554, + 372, + 568 + ], + "score": 0.82, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 553, + 391, + 569 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 391, + 556, + 434, + 567 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 553, + 508, + 569 + ], + "score": 1.0, + "content": ". Then, simultane-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 567, + 316, + 579 + ], + "spans": [ + { + "bbox": [ + 141, + 567, + 268, + 579 + ], + "score": 1.0, + "content": "ously for every possible A and", + "type": "text" + }, + { + "bbox": [ + 268, + 568, + 276, + 577 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 567, + 316, + 579 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 139, + 553, + 508, + 579 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 279, + 594, + 366, + 608 + ], + "lines": [ + { + "bbox": [ + 279, + 594, + 366, + 608 + ], + "spans": [ + { + "bbox": [ + 279, + 594, + 366, + 608 + ], + "score": 0.83, + "content": "\\left. S \\right. \\leq \\mathrm { r a n k } \\left( { \\bf A } _ { S } \\right) d _ { 0 } ,", + "type": "interline_equation", + "image_path": "9b6ba8097ff70acf75b1e4aad86305a5e9c2b30cee691a85f1618c8d704fe10d.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 279, + 594, + 366, + 608 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 621, + 457, + 635 + ], + "lines": [ + { + "bbox": [ + 141, + 620, + 459, + 637 + ], + "spans": [ + { + "bbox": [ + 141, + 620, + 198, + 637 + ], + "score": 1.0, + "content": "we have that,", + "type": "text" + }, + { + "bbox": [ + 198, + 623, + 208, + 633 + ], + "score": 0.29, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 620, + 229, + 637 + ], + "score": 1.0, + "content": "-a.e.,", + "type": "text" + }, + { + "bbox": [ + 229, + 622, + 268, + 634 + ], + "score": 0.91, + "content": "\\mathbf { \\Delta } \\nexists \\mathbf { v } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 620, + 309, + 637 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 309, + 623, + 371, + 635 + ], + "score": 0.91, + "content": "v _ { n } \\neq 0 \\forall n \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 620, + 390, + 637 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 391, + 623, + 453, + 635 + ], + "score": 0.89, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 620, + 459, + 637 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 141, + 620, + 459, + 637 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 650, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 128, + 651, + 502, + 664 + ], + "spans": [ + { + "bbox": [ + 128, + 651, + 189, + 664 + ], + "score": 1.0, + "content": "4. Recall that", + "type": "text" + }, + { + "bbox": [ + 189, + 651, + 275, + 663 + ], + "score": 0.91, + "content": "K _ { r } \\triangleq \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 651, + 477, + 664 + ], + "score": 1.0, + "content": ". We use the union bound over all possible ranks", + "type": "text" + }, + { + "bbox": [ + 477, + 652, + 502, + 663 + ], + "score": 0.9, + "content": "r \\geq 1", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 141, + 663, + 202, + 676 + ], + "score": 1.0, + "content": "we ignore the", + "type": "text" + }, + { + "bbox": [ + 202, + 664, + 230, + 673 + ], + "score": 0.88, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 663, + 291, + 676 + ], + "score": 1.0, + "content": "case since for", + "type": "text" + }, + { + "bbox": [ + 292, + 663, + 320, + 675 + ], + "score": 0.9, + "content": "\\rho \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "(see eq. (2.1)) there is zero probability that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 673, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 141, + 673, + 164, + 686 + ], + "score": 1.0, + "content": "rank", + "type": "text" + }, + { + "bbox": [ + 165, + 673, + 235, + 685 + ], + "score": 0.87, + "content": "( a ( \\mathbf { W } \\mathbf { X } _ { S } ) ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 673, + 324, + 686 + ], + "score": 1.0, + "content": "for some non-empty", + "type": "text" + }, + { + "bbox": [ + 324, + 674, + 332, + 684 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 673, + 399, + 686 + ], + "score": 1.0, + "content": ". For each rank", + "type": "text" + }, + { + "bbox": [ + 399, + 674, + 426, + 685 + ], + "score": 0.9, + "content": "r \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 673, + 506, + 686 + ], + "score": 1.0, + "content": ", it is required that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 143, + 684, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 143, + 685, + 253, + 696 + ], + "score": 0.89, + "content": "| S | > K _ { r } = \\operatorname* { m a x } \\left[ N \\epsilon , r d _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 684, + 268, + 697 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 268, + 685, + 307, + 696 + ], + "score": 0.93, + "content": "| S | = K _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 684, + 506, + 697 + ], + "score": 1.0, + "content": "is a relaxation of the original condition, and thus", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 695, + 251, + 707 + ], + "spans": [ + { + "bbox": [ + 141, + 695, + 251, + 707 + ], + "score": 1.0, + "content": "its probability is not lower.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 128, + 651, + 506, + 707 + ] + }, + { + "type": "text", + "bbox": [ + 128, + 720, + 419, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 719, + 419, + 734 + ], + "spans": [ + { + "bbox": [ + 129, + 719, + 363, + 734 + ], + "score": 1.0, + "content": "5. We again use the union bound over all possible subsets", + "type": "text" + }, + { + "bbox": [ + 364, + 721, + 372, + 730 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 719, + 402, + 734 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 402, + 721, + 416, + 732 + ], + "score": 0.88, + "content": "K _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 719, + 419, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 129, + 719, + 419, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 227, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 227, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 227, + 95 + ], + "score": 1.0, + "content": "Thus, from eq. (8.1), we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 97, + 455, + 326 + ], + "lines": [ + { + "bbox": [ + 152, + 97, + 455, + 326 + ], + "spans": [ + { + "bbox": [ + 152, + 97, + 455, + 326 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\mathbf { x } \\times \\mathbf { x } ^ { \\beta } } \\nu ( \\xi , \\alpha ( \\mathbf { x } , \\mathbf { x } ) ) } \\\\ & = \\frac { \\nu ( \\mathbf { x } ) } { 2 } \\sum _ { i = 1 } ^ { N } \\sum _ { \\substack { s \\geq 0 , i \\geq 1 } } \\sum _ { \\substack { [ \\xi \\in \\mathcal { S } _ { \\mathbf { x } } \\times \\mathcal { N } _ { i } ] ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } \\times \\{ i \\} ) = ( \\xi \\cup \\mathbf { X } _ { \\mathbf { x } } ) ] = ( \\xi \\mid \\mathbf { X } _ { \\mathbf { x } } ) } } \\\\ & { \\leq \\frac { \\nu ( \\mathbf { x } ) } { 2 } \\sum _ { i = 1 } ^ { N } \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) } [ \\xi \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) ] } ( \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) \\{ ( \\xi \\mathbf { X } _ { \\mathbf { x } } ) \\} = \\{ \\alpha \\} } ] } \\\\ & \\overset { ( a ) } { \\geq } \\sum _ { i = 1 } ^ { N } ( \\frac { \\nu } { L } ) \\sum _ { \\substack { ( \\xi ) \\leq ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ) = ( \\xi ) } } \\\\ & \\overset { ( a ) } { \\leq } \\sum _ { i = 1 } ^ { N } ( \\frac { \\nu } { L } ) \\sum _ \\substack { ( \\xi ) \\leq ( \\xi ) ( \\xi ) \\leq ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) \\leq ( \\xi ) \\mathrm { h } } \\{ \\alpha \\} \\exp \\{ ( \\xi \\} \\sum _ \\mathbf { x } \\in \\{ \\xi \\} \\} \\xi ) ( \\xi ) ( \\xi ) ( \\end{array}", + "type": "interline_equation", + "image_path": "714be8079b3dc2117ce2a44181decc63dd50b6d59b105c131bcb1d880015d3dd.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 152, + 97, + 455, + 111.3125 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 152, + 111.3125, + 455, + 125.625 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 152, + 125.625, + 455, + 139.9375 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 152, + 139.9375, + 455, + 154.25 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 152, + 154.25, + 455, + 168.5625 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 152, + 168.5625, + 455, + 182.875 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 152, + 182.875, + 455, + 197.1875 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 152, + 197.1875, + 455, + 211.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 152, + 211.5, + 455, + 225.8125 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 152, + 225.8125, + 455, + 240.125 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 152, + 240.125, + 455, + 254.4375 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 152, + 254.4375, + 455, + 268.75 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 152, + 268.75, + 455, + 283.0625 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 152, + 283.0625, + 455, + 297.375 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 152, + 297.375, + 455, + 311.6875 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 152, + 311.6875, + 455, + 326.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 131, + 330, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 129, + 329, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 129, + 329, + 290, + 344 + ], + "score": 1.0, + "content": "1. Since we take the expectation over", + "type": "text" + }, + { + "bbox": [ + 290, + 331, + 300, + 341 + ], + "score": 0.72, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 329, + 368, + 344 + ], + "score": 1.0, + "content": ", the location of", + "type": "text" + }, + { + "bbox": [ + 368, + 331, + 376, + 341 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 329, + 507, + 344 + ], + "score": 1.0, + "content": "does not affect the probability.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 340, + 378, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 334, + 355 + ], + "score": 1.0, + "content": "Therefore, we can set without loss of generality", + "type": "text" + }, + { + "bbox": [ + 335, + 342, + 374, + 354 + ], + "score": 0.93, + "content": "S = \\left[ K _ { r } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 340, + 378, + 355 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 130, + 356, + 504, + 379 + ], + "lines": [ + { + "bbox": [ + 128, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 128, + 355, + 181, + 369 + ], + "score": 1.0, + "content": "2. Note that", + "type": "text" + }, + { + "bbox": [ + 181, + 356, + 280, + 369 + ], + "score": 0.91, + "content": "r \\leq N / d _ { 0 } \\dot { < } \\operatorname* { m i n } \\left[ d _ { 0 } , d _ { 1 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 355, + 432, + 369 + ], + "score": 1.0, + "content": "from assumptions 3 and 4. Thus, with", + "type": "text" + }, + { + "bbox": [ + 432, + 357, + 487, + 368 + ], + "score": 0.92, + "content": "k = K _ { r } \\ge d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 355, + 505, + 369 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 367, + 390, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 390, + 380 + ], + "score": 1.0, + "content": "apply the following Lemma, proven in appendix section 12.2:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 140, + 381, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 141, + 379, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 210, + 394 + ], + "score": 1.0, + "content": "Lemma 15. Let", + "type": "text" + }, + { + "bbox": [ + 210, + 380, + 258, + 392 + ], + "score": 0.9, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 379, + 507, + 394 + ], + "score": 1.0, + "content": "be a random matrix with independent and identically distribu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 140, + 389, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 140, + 389, + 212, + 406 + ], + "score": 1.0, + "content": "ted columns, and", + "type": "text" + }, + { + "bbox": [ + 212, + 392, + 266, + 403 + ], + "score": 0.89, + "content": "\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 389, + 507, + 406 + ], + "score": 1.0, + "content": "an independent standard random Gaussian matrix. Then, in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 403, + 254, + 416 + ], + "spans": [ + { + "bbox": [ + 142, + 403, + 187, + 416 + ], + "score": 1.0, + "content": "the limit m", + "type": "text" + }, + { + "bbox": [ + 188, + 403, + 250, + 416 + ], + "score": 0.82, + "content": "\\mathrm { i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 403, + 254, + 416 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 419, + 467, + 436 + ], + "lines": [ + { + "bbox": [ + 180, + 419, + 467, + 436 + ], + "spans": [ + { + "bbox": [ + 180, + 419, + 467, + 436 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( a \\left( \\mathbf { W } \\mathbf { X } \\right) \\right) = r \\right) \\dot { \\leq } 2 ^ { k + r d _ { 0 } \\left( \\log d _ { 1 } + \\log k \\right) + r ^ { 2 } } \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "1afe02259f7d479d0be27534469a0955224cee70cfd0bc061e9c2014f5cf4e70.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 180, + 419, + 467, + 436 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 132, + 441, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 129, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 129, + 441, + 183, + 455 + ], + "score": 1.0, + "content": "3. Note that", + "type": "text" + }, + { + "bbox": [ + 183, + 442, + 272, + 454 + ], + "score": 0.92, + "content": "K _ { r } \\geq N \\epsilon \\dot { = } N > 2 d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 441, + 294, + 455 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 294, + 442, + 420, + 454 + ], + "score": 0.88, + "content": "\\operatorname* { m i n } \\left[ K _ { r } , d _ { 0 } , d _ { 1 } \\right] \\dot { > } d _ { 0 } \\underline { { d _ { 1 } } } / K _ { r } \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "from assumptions 2", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 452, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 349, + 466 + ], + "score": 1.0, + "content": "and 4. Thus, we apply the following Lemma (with", + "type": "text" + }, + { + "bbox": [ + 349, + 454, + 386, + 464 + ], + "score": 0.79, + "content": "\\mathbf { C } = \\mathbf { X } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 452, + 389, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 390, + 453, + 430, + 464 + ], + "score": 0.84, + "content": "\\mathbf { B } = \\mathbf { W } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 452, + 434, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 435, + 453, + 469, + 464 + ], + "score": 0.87, + "content": "M = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 452, + 473, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 474, + 453, + 504, + 464 + ], + "score": 0.88, + "content": "L = d _ { 1 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 464, + 346, + 477 + ], + "spans": [ + { + "bbox": [ + 142, + 464, + 159, + 477 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 464, + 206, + 477 + ], + "score": 0.91, + "content": "N = K _ { r } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 464, + 346, + 477 + ], + "score": 1.0, + "content": "), proven in appendix section 12.3:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 140, + 477, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 141, + 475, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 213, + 491 + ], + "score": 1.0, + "content": "Lemma 16. Let", + "type": "text" + }, + { + "bbox": [ + 214, + 477, + 269, + 489 + ], + "score": 0.91, + "content": "\\mathbf { C } \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 475, + 290, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 290, + 477, + 344, + 489 + ], + "score": 0.9, + "content": "\\mathbf { B } \\in \\mathbb { R } ^ { M \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 475, + 506, + 491 + ], + "score": 1.0, + "content": "be two independent standard random", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 142, + 491, + 366, + 503 + ], + "score": 1.0, + "content": "Gaussian matrices. Without loss of generality, assume", + "type": "text" + }, + { + "bbox": [ + 367, + 491, + 399, + 501 + ], + "score": 0.88, + "content": "N \\geq L ,", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 491, + 450, + 503 + ], + "score": 1.0, + "content": ", and denote", + "type": "text" + }, + { + "bbox": [ + 450, + 489, + 503, + 502 + ], + "score": 0.92, + "content": "\\alpha \\triangleq M L / N", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 491, + 506, + 503 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 501, + 447, + 515 + ], + "spans": [ + { + "bbox": [ + 141, + 501, + 221, + 515 + ], + "score": 1.0, + "content": "Then, in the regime", + "type": "text" + }, + { + "bbox": [ + 221, + 502, + 255, + 513 + ], + "score": 0.91, + "content": "M \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 501, + 338, + 515 + ], + "score": 1.0, + "content": "and in the limit min", + "type": "text" + }, + { + "bbox": [ + 339, + 502, + 407, + 514 + ], + "score": 0.62, + "content": "[ N , M , L ] \\dot { > } \\alpha \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 501, + 447, + 515 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 517, + 396, + 538 + ], + "lines": [ + { + "bbox": [ + 249, + 517, + 396, + 538 + ], + "spans": [ + { + "bbox": [ + 249, + 517, + 396, + 538 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "44fede03f3bb993bbce2b480d94bb9545c17da4fe315581887002fc34564bf6d.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 249, + 517, + 396, + 538 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 130, + 545, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 127, + 543, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 127, + 543, + 175, + 569 + ], + "score": 1.0, + "content": "4. We use", + "type": "text" + }, + { + "bbox": [ + 176, + 544, + 326, + 570 + ], + "score": 0.89, + "content": "r d _ { 0 } \\leq N , \\left( \\begin{array} { c } { { N } } \\\\ { { K _ { r } } } \\end{array} \\right) \\leq 2 ^ { N } , K _ { r } \\leq N ,", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 546, + 348, + 567 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 348, + 551, + 375, + 562 + ], + "score": 0.9, + "content": "d _ { 1 } \\dot { < } N", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 546, + 482, + 567 + ], + "score": 1.0, + "content": "(from assumption 4) and", + "type": "text" + }, + { + "bbox": [ + 482, + 550, + 505, + 563 + ], + "score": 0.87, + "content": "r ^ { 2 } \\leq", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 567, + 451, + 583 + ], + "spans": [ + { + "bbox": [ + 142, + 569, + 189, + 582 + ], + "score": 0.92, + "content": "N ^ { 2 } / d _ { 0 } ^ { 2 } { \\dot { < } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 567, + 451, + 583 + ], + "score": 1.0, + "content": "(from assumption (3)) to simplify the combintaorial expressions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 128, + 584, + 506, + 608 + ], + "lines": [ + { + "bbox": [ + 129, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 129, + 584, + 201, + 597 + ], + "score": 1.0, + "content": "5. 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Following exactly the same", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 227, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 227, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 227, + 95 + ], + "score": 1.0, + "content": "Thus, from eq. (8.1), we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 227, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 97, + 455, + 326 + ], + "lines": [ + { + "bbox": [ + 152, + 97, + 455, + 326 + ], + "spans": [ + { + "bbox": [ + 152, + 97, + 455, + 326 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\mathbf { x } \\times \\mathbf { x } ^ { \\beta } } \\nu ( \\xi , \\alpha ( \\mathbf { x } , \\mathbf { x } ) ) } \\\\ & = \\frac { \\nu ( \\mathbf { x } ) } { 2 } \\sum _ { i = 1 } ^ { N } \\sum _ { \\substack { s \\geq 0 , i \\geq 1 } } \\sum _ { \\substack { [ \\xi \\in \\mathcal { S } _ { \\mathbf { x } } \\times \\mathcal { N } _ { i } ] ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } \\times \\{ i \\} ) = ( \\xi \\cup \\mathbf { X } _ { \\mathbf { x } } ) ] = ( \\xi \\mid \\mathbf { X } _ { \\mathbf { x } } ) } } \\\\ & { \\leq \\frac { \\nu ( \\mathbf { x } ) } { 2 } \\sum _ { i = 1 } ^ { N } \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) } [ \\xi \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) ] } ( \\operatorname* { s u p } _ { ( \\xi \\in \\mathcal { S } _ { \\mathbf { x } } ) \\{ ( \\xi \\mathbf { X } _ { \\mathbf { x } } ) \\} = \\{ \\alpha \\} } ] } \\\\ & \\overset { ( a ) } { \\geq } \\sum _ { i = 1 } ^ { N } ( \\frac { \\nu } { L } ) \\sum _ { \\substack { ( \\xi ) \\leq ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ) = ( \\xi ) } } \\\\ & \\overset { ( a ) } { \\leq } \\sum _ { i = 1 } ^ { N } ( \\frac { \\nu } { L } ) \\sum _ \\substack { ( \\xi ) \\leq ( \\xi ) ( \\xi ) \\leq ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) ( \\xi ) \\leq ( \\xi ) \\mathrm { h } } \\{ \\alpha \\} \\exp \\{ ( \\xi \\} \\sum _ \\mathbf { x } \\in \\{ \\xi \\} \\} \\xi ) ( \\xi ) ( \\xi ) ( \\end{array}", + "type": "interline_equation", + "image_path": "714be8079b3dc2117ce2a44181decc63dd50b6d59b105c131bcb1d880015d3dd.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 152, + 97, + 455, + 111.3125 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 152, + 111.3125, + 455, + 125.625 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 152, + 125.625, + 455, + 139.9375 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 152, + 139.9375, + 455, + 154.25 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 152, + 154.25, + 455, + 168.5625 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 152, + 168.5625, + 455, + 182.875 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 152, + 182.875, + 455, + 197.1875 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 152, + 197.1875, + 455, + 211.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 152, + 211.5, + 455, + 225.8125 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 152, + 225.8125, + 455, + 240.125 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 152, + 240.125, + 455, + 254.4375 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 152, + 254.4375, + 455, + 268.75 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 152, + 268.75, + 455, + 283.0625 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 152, + 283.0625, + 455, + 297.375 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 152, + 297.375, + 455, + 311.6875 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 152, + 311.6875, + 455, + 326.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 131, + 330, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 129, + 329, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 129, + 329, + 290, + 344 + ], + "score": 1.0, + "content": "1. Since we take the expectation over", + "type": "text" + }, + { + "bbox": [ + 290, + 331, + 300, + 341 + ], + "score": 0.72, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 329, + 368, + 344 + ], + "score": 1.0, + "content": ", the location of", + "type": "text" + }, + { + "bbox": [ + 368, + 331, + 376, + 341 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 329, + 507, + 344 + ], + "score": 1.0, + "content": "does not affect the probability.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 340, + 378, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 334, + 355 + ], + "score": 1.0, + "content": "Therefore, we can set without loss of generality", + "type": "text" + }, + { + "bbox": [ + 335, + 342, + 374, + 354 + ], + "score": 0.93, + "content": "S = \\left[ K _ { r } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 340, + 378, + 355 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 129, + 329, + 507, + 355 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 356, + 504, + 379 + ], + "lines": [ + { + "bbox": [ + 128, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 128, + 355, + 181, + 369 + ], + "score": 1.0, + "content": "2. Note that", + "type": "text" + }, + { + "bbox": [ + 181, + 356, + 280, + 369 + ], + "score": 0.91, + "content": "r \\leq N / d _ { 0 } \\dot { < } \\operatorname* { m i n } \\left[ d _ { 0 } , d _ { 1 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 355, + 432, + 369 + ], + "score": 1.0, + "content": "from assumptions 3 and 4. Thus, with", + "type": "text" + }, + { + "bbox": [ + 432, + 357, + 487, + 368 + ], + "score": 0.92, + "content": "k = K _ { r } \\ge d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 355, + 505, + 369 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 367, + 390, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 390, + 380 + ], + "score": 1.0, + "content": "apply the following Lemma, proven in appendix section 12.2:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 128, + 355, + 505, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 140, + 381, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 141, + 379, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 210, + 394 + ], + "score": 1.0, + "content": "Lemma 15. Let", + "type": "text" + }, + { + "bbox": [ + 210, + 380, + 258, + 392 + ], + "score": 0.9, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 379, + 507, + 394 + ], + "score": 1.0, + "content": "be a random matrix with independent and identically distribu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 140, + 389, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 140, + 389, + 212, + 406 + ], + "score": 1.0, + "content": "ted columns, and", + "type": "text" + }, + { + "bbox": [ + 212, + 392, + 266, + 403 + ], + "score": 0.89, + "content": "\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 389, + 507, + 406 + ], + "score": 1.0, + "content": "an independent standard random Gaussian matrix. Then, in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 403, + 254, + 416 + ], + "spans": [ + { + "bbox": [ + 142, + 403, + 187, + 416 + ], + "score": 1.0, + "content": "the limit m", + "type": "text" + }, + { + "bbox": [ + 188, + 403, + 250, + 416 + ], + "score": 0.82, + "content": "\\mathrm { i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 403, + 254, + 416 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 140, + 379, + 507, + 416 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 419, + 467, + 436 + ], + "lines": [ + { + "bbox": [ + 180, + 419, + 467, + 436 + ], + "spans": [ + { + "bbox": [ + 180, + 419, + 467, + 436 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( a \\left( \\mathbf { W } \\mathbf { X } \\right) \\right) = r \\right) \\dot { \\leq } 2 ^ { k + r d _ { 0 } \\left( \\log d _ { 1 } + \\log k \\right) + r ^ { 2 } } \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "1afe02259f7d479d0be27534469a0955224cee70cfd0bc061e9c2014f5cf4e70.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 180, + 419, + 467, + 436 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 132, + 441, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 129, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 129, + 441, + 183, + 455 + ], + "score": 1.0, + "content": "3. Note that", + "type": "text" + }, + { + "bbox": [ + 183, + 442, + 272, + 454 + ], + "score": 0.92, + "content": "K _ { r } \\geq N \\epsilon \\dot { = } N > 2 d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 441, + 294, + 455 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 294, + 442, + 420, + 454 + ], + "score": 0.88, + "content": "\\operatorname* { m i n } \\left[ K _ { r } , d _ { 0 } , d _ { 1 } \\right] \\dot { > } d _ { 0 } \\underline { { d _ { 1 } } } / K _ { r } \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "from assumptions 2", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 452, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 349, + 466 + ], + "score": 1.0, + "content": "and 4. Thus, we apply the following Lemma (with", + "type": "text" + }, + { + "bbox": [ + 349, + 454, + 386, + 464 + ], + "score": 0.79, + "content": "\\mathbf { C } = \\mathbf { X } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 452, + 389, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 390, + 453, + 430, + 464 + ], + "score": 0.84, + "content": "\\mathbf { B } = \\mathbf { W } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 452, + 434, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 435, + 453, + 469, + 464 + ], + "score": 0.87, + "content": "M = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 452, + 473, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 474, + 453, + 504, + 464 + ], + "score": 0.88, + "content": "L = d _ { 1 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 464, + 346, + 477 + ], + "spans": [ + { + "bbox": [ + 142, + 464, + 159, + 477 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 464, + 206, + 477 + ], + "score": 0.91, + "content": "N = K _ { r } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 464, + 346, + 477 + ], + "score": 1.0, + "content": "), proven in appendix section 12.3:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 129, + 441, + 505, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 140, + 477, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 141, + 475, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 213, + 491 + ], + "score": 1.0, + "content": "Lemma 16. Let", + "type": "text" + }, + { + "bbox": [ + 214, + 477, + 269, + 489 + ], + "score": 0.91, + "content": "\\mathbf { C } \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 475, + 290, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 290, + 477, + 344, + 489 + ], + "score": 0.9, + "content": "\\mathbf { B } \\in \\mathbb { R } ^ { M \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 475, + 506, + 491 + ], + "score": 1.0, + "content": "be two independent standard random", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 142, + 491, + 366, + 503 + ], + "score": 1.0, + "content": "Gaussian matrices. Without loss of generality, assume", + "type": "text" + }, + { + "bbox": [ + 367, + 491, + 399, + 501 + ], + "score": 0.88, + "content": "N \\geq L ,", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 491, + 450, + 503 + ], + "score": 1.0, + "content": ", and denote", + "type": "text" + }, + { + "bbox": [ + 450, + 489, + 503, + 502 + ], + "score": 0.92, + "content": "\\alpha \\triangleq M L / N", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 491, + 506, + 503 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 501, + 447, + 515 + ], + "spans": [ + { + "bbox": [ + 141, + 501, + 221, + 515 + ], + "score": 1.0, + "content": "Then, in the regime", + "type": "text" + }, + { + "bbox": [ + 221, + 502, + 255, + 513 + ], + "score": 0.91, + "content": "M \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 501, + 338, + 515 + ], + "score": 1.0, + "content": "and in the limit min", + "type": "text" + }, + { + "bbox": [ + 339, + 502, + 407, + 514 + ], + "score": 0.62, + "content": "[ N , M , L ] \\dot { > } \\alpha \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 501, + 447, + 515 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 141, + 475, + 506, + 515 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 517, + 396, + 538 + ], + "lines": [ + { + "bbox": [ + 249, + 517, + 396, + 538 + ], + "spans": [ + { + "bbox": [ + 249, + 517, + 396, + 538 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "44fede03f3bb993bbce2b480d94bb9545c17da4fe315581887002fc34564bf6d.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 249, + 517, + 396, + 538 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 130, + 545, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 127, + 543, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 127, + 543, + 175, + 569 + ], + "score": 1.0, + "content": "4. We use", + "type": "text" + }, + { + "bbox": [ + 176, + 544, + 326, + 570 + ], + "score": 0.89, + "content": "r d _ { 0 } \\leq N , \\left( \\begin{array} { c } { { N } } \\\\ { { K _ { r } } } \\end{array} \\right) \\leq 2 ^ { N } , K _ { r } \\leq N ,", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 546, + 348, + 567 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 348, + 551, + 375, + 562 + ], + "score": 0.9, + "content": "d _ { 1 } \\dot { < } N", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 546, + 482, + 567 + ], + "score": 1.0, + "content": "(from assumption 4) and", + "type": "text" + }, + { + "bbox": [ + 482, + 550, + 505, + 563 + ], + "score": 0.87, + "content": "r ^ { 2 } \\leq", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 567, + 451, + 583 + ], + "spans": [ + { + "bbox": [ + 142, + 569, + 189, + 582 + ], + "score": 0.92, + "content": "N ^ { 2 } / d _ { 0 } ^ { 2 } { \\dot { < } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 567, + 451, + 583 + ], + "score": 1.0, + "content": "(from assumption (3)) to simplify the combintaorial expressions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 127, + 543, + 505, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 128, + 584, + 506, + 608 + ], + "lines": [ + { + "bbox": [ + 129, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 129, + 584, + 201, + 597 + ], + "score": 1.0, + "content": "5. First, note that", + "type": "text" + }, + { + "bbox": [ + 201, + 585, + 226, + 594 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "is the maximal term in the sum, so we can neglect the other, exponenti-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 142, + 595, + 411, + 608 + ], + "spans": [ + { + "bbox": [ + 142, + 595, + 368, + 608 + ], + "score": 1.0, + "content": "ally smaller, terms. Second, from assumption 3 we have", + "type": "text" + }, + { + "bbox": [ + 368, + 595, + 396, + 607 + ], + "score": 0.91, + "content": "d _ { 0 } { \\dot { \\le } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 595, + 411, + 608 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 129, + 584, + 505, + 608 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 610, + 481, + 635 + ], + "lines": [ + { + "bbox": [ + 165, + 610, + 481, + 635 + ], + "spans": [ + { + "bbox": [ + 165, + 610, + 481, + 635 + ], + "score": 0.91, + "content": "\\operatorname* { l i m } _ { N \\to \\infty } 0 . 2 3 \\operatorname* { m a x } \\left[ \\epsilon , d _ { 0 } \\left( N \\right) / N \\right] ^ { 3 / 4 } = 0 . 2 3 \\operatorname* { m a x } \\left[ \\epsilon , \\operatorname* { l i m } _ { N \\to \\infty } d _ { 0 } \\left( N \\right) / N \\right] ^ { 3 / 4 } = \\gamma _ { \\epsilon } .", + "type": "interline_equation", + "image_path": "80d1b85478f1bf71c4868633b88c85b366ae645e89dd57b5fd47fd444dc364e0.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 165, + 610, + 481, + 635 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 639, + 487, + 653 + ], + "lines": [ + { + "bbox": [ + 140, + 637, + 489, + 656 + ], + "spans": [ + { + "bbox": [ + 140, + 637, + 281, + 656 + ], + "score": 1.0, + "content": "Third, from assumption 4 we have", + "type": "text" + }, + { + "bbox": [ + 281, + 639, + 348, + 652 + ], + "score": 0.92, + "content": "N \\log ^ { 4 } N { \\dot { < } } d _ { 0 } d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 637, + 378, + 656 + ], + "score": 1.0, + "content": ", so the", + "type": "text" + }, + { + "bbox": [ + 379, + 640, + 412, + 651 + ], + "score": 0.9, + "content": "2 ^ { N \\log N }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 637, + 489, + 656 + ], + "score": 1.0, + "content": "term is negligible.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 140, + 637, + 489, + 656 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 661, + 130, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 133, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 133, + 676 + ], + "score": 1.0, + "content": "Thus,", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 659, + 133, + 676 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 670, + 410, + 692 + ], + "lines": [ + { + "bbox": [ + 200, + 670, + 410, + 692 + ], + "spans": [ + { + "bbox": [ + 200, + 670, + 410, + 692 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "44540abf479f600cdb5f005c7773d0fd64ab85113d457f17797f804f8388a75e.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 200, + 670, + 410, + 692 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 692, + 291, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 291, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 263, + 706 + ], + "score": 1.0, + "content": "which proves the Theorem for the case", + "type": "text" + }, + { + "bbox": [ + 264, + 693, + 288, + 704 + ], + "score": 0.9, + "content": "\\rho \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 691, + 291, + 706 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 691, + 291, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 216, + 722 + ], + "score": 1.0, + "content": "Next, we consider the case", + "type": "text" + }, + { + "bbox": [ + 217, + 710, + 241, + 721 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ". In this case, we need to change transition (4) in eq. (8.1), so the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 172, + 734 + ], + "score": 1.0, + "content": "sum starts from", + "type": "text" + }, + { + "bbox": [ + 172, + 721, + 197, + 731 + ], + "score": 0.9, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 720, + 318, + 734 + ], + "score": 1.0, + "content": ", since now we can have rank", + "type": "text" + }, + { + "bbox": [ + 319, + 720, + 386, + 732 + ], + "score": 0.88, + "content": "( a ( \\mathbf { W } \\mathbf { X } _ { S } ) \\bar { ) } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 720, + 505, + 734 + ], + "score": 1.0, + "content": ". Following exactly the same", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 107 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "logic (except the modification to the sum), we only need to modify transition (5)in eq. (8.2) – since", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 400, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 261, + 106 + ], + "score": 1.0, + "content": "now the maximal term in the sum is at", + "type": "text" + }, + { + "bbox": [ + 262, + 95, + 286, + 105 + ], + "score": 0.89, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 94, + 338, + 106 + ], + "score": 1.0, + "content": ". This entails", + "type": "text" + }, + { + "bbox": [ + 339, + 93, + 396, + 106 + ], + "score": 0.94, + "content": "\\gamma _ { \\epsilon } = 0 . 2 3 \\epsilon ^ { 3 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 94, + 400, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 125, + 506, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 391, + 140 + ], + "score": 1.0, + "content": "Corollary 17. (Corollary 7 restated) Given assumptions 1-4, for any", + "type": "text" + }, + { + "bbox": [ + 391, + 126, + 416, + 137 + ], + "score": 0.89, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 123, + 506, + 140 + ], + "score": 1.0, + "content": "(possibly a vanishing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 151, + 150 + ], + "score": 1.0, + "content": "function of", + "type": "text" + }, + { + "bbox": [ + 152, + 137, + 162, + 147 + ], + "score": 0.72, + "content": "N _ { . }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 136, + 271, + 150 + ], + "score": 1.0, + "content": "), we have, with probability", + "type": "text" + }, + { + "bbox": [ + 272, + 137, + 295, + 147 + ], + "score": 0.85, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 136, + 505, + 150 + ], + "score": 1.0, + "content": ", that the angular volume of sub-optimal DLMs, with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 310, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 166, + 159 + ], + "score": 0.89, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 148, + 287, + 160 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 287, + 149, + 297, + 158 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 148, + 310, + 160 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 162, + 397, + 186 + ], + "lines": [ + { + "bbox": [ + 213, + 162, + 397, + 186 + ], + "spans": [ + { + "bbox": [ + 213, + 162, + 397, + 186 + ], + "score": 0.93, + "content": "{ \\mathcal { V } } \\left( { \\mathcal { L } } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)", + "type": "interline_equation", + "image_path": "1ec95b09718b5c62f1740c9605819e83bf645fd3ed4f09bdf5076451cd3778dd.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 213, + 162, + 397, + 186 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 435, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 435, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 160, + 210 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 161, + 197, + 236, + 209 + ], + "score": 0.93, + "content": "\\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 196, + 363, + 210 + ], + "score": 1.0, + "content": "we can use Markov’s Theorem", + "type": "text" + }, + { + "bbox": [ + 363, + 198, + 432, + 209 + ], + "score": 0.37, + "content": "( \\mathrm { F a c t } 1 1 ) \\forall \\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 196, + 435, + 210 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 212, + 420, + 239 + ], + "lines": [ + { + "bbox": [ + 190, + 212, + 420, + 239 + ], + "spans": [ + { + "bbox": [ + 190, + 212, + 420, + 239 + ], + "score": 0.9, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) < \\eta \\right) > 1 - \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\eta }", + "type": "interline_equation", + "image_path": "2dc91f38ea1fb3c23cbd1245a42972f4f8edc0e8870805621a60c5e4b1976bd0.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 190, + 212, + 420, + 239 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 243, + 443, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 445, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 144, + 258 + ], + "score": 1.0, + "content": "denoting", + "type": "text" + }, + { + "bbox": [ + 144, + 243, + 253, + 257 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\eta = \\frac { 1 } { \\delta } \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 241, + 445, + 258 + ], + "score": 1.0, + "content": ", and using Theorem (6) we prove the corollary.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 259, + 437, + 316 + ], + "lines": [ + { + "bbox": [ + 173, + 259, + 437, + 316 + ], + "spans": [ + { + "bbox": [ + 173, + 259, + 437, + 316 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { 1 - \\delta < \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( \\displaystyle \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) < \\frac { 1 } { \\delta } \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\right) } \\\\ & { \\quad \\quad \\quad < \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "36d6b97111f482b5888329e9e8c4b873fb425980acee6b653550d879f9ba423e.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 173, + 259, + 437, + 278.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 173, + 278.0, + 437, + 297.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 173, + 297.0, + 437, + 316.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 319, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 295, + 332 + ], + "score": 1.0, + "content": "where we note that replacing a regular inequality", + "type": "text" + }, + { + "bbox": [ + 295, + 321, + 304, + 330 + ], + "score": 0.8, + "content": "<", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 319, + 441, + 332 + ], + "score": 1.0, + "content": "with inequality in the leading order", + "type": "text" + }, + { + "bbox": [ + 441, + 319, + 450, + 331 + ], + "score": 0.71, + "content": "\\dot { \\leq }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "only removes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 331, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 311, + 344 + ], + "score": 1.0, + "content": "constraints, and therefore increases the probability.", + "type": "text" + }, + { + "bbox": [ + 495, + 333, + 504, + 340 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 106, + 358, + 441, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 442, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 442, + 372 + ], + "score": 1.0, + "content": "9 CONSTRUCTION OF GLOBAL MINIMA: PROOF OF THEOREM 8:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 236, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 235, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 235, + 397 + ], + "score": 1.0, + "content": "Recall the LReLU non-linearity", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 393, + 358, + 421 + ], + "lines": [ + { + "bbox": [ + 253, + 393, + 358, + 421 + ], + "spans": [ + { + "bbox": [ + 253, + 393, + 358, + 421 + ], + "score": 0.94, + "content": "f \\left( x \\right) \\triangleq { \\left\\{ \\begin{array} { l l } { \\rho x } & { , { \\mathrm { i f ~ } } x < 0 } \\\\ { x } & { , { \\mathrm { i f ~ } } x \\geq 0 } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "3a9836c54a3588da2a9b52a87454af7cf7fc0020bf26f65f4a5504f2551c1d03.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 253, + 393, + 358, + 421 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 210, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 207, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 182, + 437 + ], + "score": 1.0, + "content": "in eq. (2.1), where", + "type": "text" + }, + { + "bbox": [ + 183, + 423, + 207, + 435 + ], + "score": 0.91, + "content": "\\rho \\neq 1", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 294, + 450 + ], + "score": 1.0, + "content": "Theorem 18. (Theorem 8 restated) For any", + "type": "text" + }, + { + "bbox": [ + 294, + 436, + 350, + 449 + ], + "score": 0.93, + "content": "\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 434, + 370, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 436, + 424, + 447 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 434, + 506, + 450 + ], + "score": 1.0, + "content": "almost everywhere", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 447, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 180, + 464 + ], + "score": 1.0, + "content": "we find matrices", + "type": "text" + }, + { + "bbox": [ + 180, + 449, + 245, + 461 + ], + "score": 0.9, + "content": "\\mathbf { W ^ { * } } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 447, + 267, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 268, + 449, + 312, + 461 + ], + "score": 0.91, + "content": "\\mathbf { z } ^ { * } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 447, + 360, + 464 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 360, + 449, + 445, + 462 + ], + "score": 0.91, + "content": "\\mathbf { y } ~ = ~ f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 447, + 479, + 464 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 479, + 449, + 505, + 462 + ], + "score": 0.78, + "content": "d _ { 1 } ^ { * } \\ \\triangleq", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 460, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 177, + 475 + ], + "score": 0.9, + "content": "4 \\left\\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\right\\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 460, + 198, + 476 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 461, + 290, + 475 + ], + "score": 0.92, + "content": "\\forall i , n : \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 460, + 413, + 476 + ], + "score": 1.0, + "content": ". Therefore, every MNN with", + "type": "text" + }, + { + "bbox": [ + 413, + 463, + 452, + 475 + ], + "score": 0.93, + "content": "d _ { 1 } \\ \\geq \\ d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 460, + 506, + 476 + ], + "score": 1.0, + "content": "has a DLM", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 474, + 240, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 211, + 485 + ], + "score": 1.0, + "content": "which achieves zero error", + "type": "text" + }, + { + "bbox": [ + 211, + 475, + 236, + 484 + ], + "score": 0.87, + "content": "\\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 474, + 240, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 259, + 506 + ], + "score": 1.0, + "content": "We prove the existence of a solution", + "type": "text" + }, + { + "bbox": [ + 259, + 494, + 296, + 505 + ], + "score": 0.9, + "content": "( \\mathbf { W } ^ { * } , \\mathbf { z } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 494, + 506, + 506 + ], + "score": 1.0, + "content": ", by explicitly constructing it. This construction is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 516 + ], + "score": 1.0, + "content": "a variant of (Baum, 1988, Theorem 1), except we use LReLU without bias and MSE – instead of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 356, + 529 + ], + "score": 1.0, + "content": "threshold units with bias and MCE. First, we note that for any", + "type": "text" + }, + { + "bbox": [ + 356, + 516, + 406, + 527 + ], + "score": 0.91, + "content": "\\epsilon _ { 1 } > \\epsilon _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 515, + 505, + 529 + ], + "score": 1.0, + "content": ", the following trapezoid", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 526, + 331, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 331, + 538 + ], + "score": 1.0, + "content": "function can be written as a scaled sum of four LReLU:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 540, + 461, + 613 + ], + "lines": [ + { + "bbox": [ + 149, + 540, + 461, + 613 + ], + "spans": [ + { + "bbox": [ + 149, + 540, + 461, + 613 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\tau \\left( x \\right) \\triangleq \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { , ~ i f ~ } | x | > \\epsilon _ { 1 } } \\\\ { 1 } & { \\mathrm { , ~ i f ~ } | x | \\leq \\epsilon _ { 2 } } \\\\ { \\frac { \\epsilon _ { 1 } - | x | } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } } & { \\mathrm { , ~ i f ~ } \\epsilon _ { 2 } < | x | \\leq \\epsilon _ { 1 } } \\\\ { = \\displaystyle \\frac { 1 } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } \\frac { 1 } { 1 - \\rho } \\left[ f \\left( x + \\epsilon _ { 1 } \\right) - f \\left( x + \\epsilon _ { 2 } \\right) - f \\left( x - \\epsilon _ { 2 } \\right) + f \\left( x - \\epsilon _ { 1 } \\right) \\right] . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "a211f0f15b159948d9a8919c4ce08f0aa24f7a3aaa0f9c176ebfeede943d3075.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 149, + 540, + 461, + 564.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 149, + 564.3333333333334, + 461, + 588.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 149, + 588.6666666666667, + 461, + 613.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 615, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 382, + 630 + ], + "score": 1.0, + "content": "Next, we examine the set of data points which are classified to 1:", + "type": "text" + }, + { + "bbox": [ + 382, + 614, + 500, + 629 + ], + "score": 0.81, + "content": "S ^ { + } \\triangleq \\left\\{ n \\in [ N ] | y ^ { ( n ) } = 1 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 613, + 506, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 627, + 372, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 245, + 644 + ], + "score": 1.0, + "content": "Without loss of generality, assume", + "type": "text" + }, + { + "bbox": [ + 246, + 628, + 288, + 642 + ], + "score": 0.94, + "content": "\\vert S ^ { + } \\vert \\le \\frac { N } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 627, + 344, + 644 + ], + "score": 1.0, + "content": ". We partition", + "type": "text" + }, + { + "bbox": [ + 344, + 629, + 359, + 639 + ], + "score": 0.89, + "content": "S ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 627, + 372, + 644 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 645, + 373, + 673 + ], + "lines": [ + { + "bbox": [ + 237, + 645, + 373, + 673 + ], + "spans": [ + { + "bbox": [ + 237, + 645, + 373, + 673 + ], + "score": 0.94, + "content": "K = \\left\\lceil { \\frac { | S ^ { + } | } { d _ { 0 } - 1 } } \\right\\rceil \\leq \\left\\lceil { \\frac { N } { 2 \\left( d _ { 0 } - 1 \\right) } } \\right\\rceil", + "type": "interline_equation", + "image_path": "298eb14ca98d1ab7bf9d82caf0c9900c1010dcd88684e1a7a497acf1788b749c.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 237, + 645, + 373, + 673 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 505, + 719 + ], + "lines": [ + { + "bbox": [ + 101, + 670, + 504, + 704 + ], + "spans": [ + { + "bbox": [ + 101, + 670, + 138, + 704 + ], + "score": 1.0, + "content": "subsets", + "type": "text" + }, + { + "bbox": [ + 139, + 677, + 178, + 694 + ], + "score": 0.93, + "content": "\\left\\{ { \\cal { S } } _ { i } ^ { + } \\right\\} _ { i = 1 } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 670, + 280, + 704 + ], + "score": 1.0, + "content": ", each with no more than", + "type": "text" + }, + { + "bbox": [ + 281, + 680, + 309, + 691 + ], + "score": 0.9, + "content": "d _ { 0 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 670, + 493, + 704 + ], + "score": 1.0, + "content": "samples. For almost any dataset we can find K", + "type": "text" + }, + { + "bbox": [ + 493, + 680, + 504, + 690 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 102, + 687, + 509, + 713 + ], + "spans": [ + { + "bbox": [ + 102, + 687, + 317, + 713 + ], + "score": 1.0, + "content": "i=1hyperplanes passing through the origin, with normals", + "type": "text" + }, + { + "bbox": [ + 353, + 687, + 509, + 713 + ], + "score": 1.0, + "content": "such that each hyperplane contains all", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 706, + 236, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 707, + 135, + 719 + ], + "score": 0.89, + "content": "d _ { 0 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 706, + 200, + 721 + ], + "score": 1.0, + "content": "points in subset", + "type": "text" + }, + { + "bbox": [ + 200, + 707, + 214, + 720 + ], + "score": 0.91, + "content": "{ S } _ { i } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 706, + 236, + 721 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 275, + 717, + 335, + 735 + ], + "lines": [ + { + "bbox": [ + 275, + 717, + 335, + 735 + ], + "spans": [ + { + "bbox": [ + 275, + 717, + 335, + 735 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { X } _ { S _ { i } ^ { + } } = 0 ,", + "type": "interline_equation", + "image_path": "238355f0a586b842f9f44bbbc43072e2fb15c002b2e06ae364765f711adafc1e.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 275, + 717, + 335, + 735 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 111, + 504, + 122 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 107 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "logic (except the modification to the sum), we only need to modify transition (5)in eq. (8.2) – since", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 400, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 261, + 106 + ], + "score": 1.0, + "content": "now the maximal term in the sum is at", + "type": "text" + }, + { + "bbox": [ + 262, + 95, + 286, + 105 + ], + "score": 0.89, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 94, + 338, + 106 + ], + "score": 1.0, + "content": ". This entails", + "type": "text" + }, + { + "bbox": [ + 339, + 93, + 396, + 106 + ], + "score": 0.94, + "content": "\\gamma _ { \\epsilon } = 0 . 2 3 \\epsilon ^ { 3 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 94, + 400, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 81, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 125, + 506, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 391, + 140 + ], + "score": 1.0, + "content": "Corollary 17. (Corollary 7 restated) Given assumptions 1-4, for any", + "type": "text" + }, + { + "bbox": [ + 391, + 126, + 416, + 137 + ], + "score": 0.89, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 123, + 506, + 140 + ], + "score": 1.0, + "content": "(possibly a vanishing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 151, + 150 + ], + "score": 1.0, + "content": "function of", + "type": "text" + }, + { + "bbox": [ + 152, + 137, + 162, + 147 + ], + "score": 0.72, + "content": "N _ { . }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 136, + 271, + 150 + ], + "score": 1.0, + "content": "), we have, with probability", + "type": "text" + }, + { + "bbox": [ + 272, + 137, + 295, + 147 + ], + "score": 0.85, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 136, + 505, + 150 + ], + "score": 1.0, + "content": ", that the angular volume of sub-optimal DLMs, with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 310, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 166, + 159 + ], + "score": 0.89, + "content": "\\mathrm { M C E } > \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 148, + 287, + 160 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 287, + 149, + 297, + 158 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 148, + 310, + 160 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 123, + 506, + 160 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 162, + 397, + 186 + ], + "lines": [ + { + "bbox": [ + 213, + 162, + 397, + 186 + ], + "spans": [ + { + "bbox": [ + 213, + 162, + 397, + 186 + ], + "score": 0.93, + "content": "{ \\mathcal { V } } \\left( { \\mathcal { L } } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)", + "type": "interline_equation", + "image_path": "1ec95b09718b5c62f1740c9605819e83bf645fd3ed4f09bdf5076451cd3778dd.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 213, + 162, + 397, + 186 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 435, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 435, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 160, + 210 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 161, + 197, + 236, + 209 + ], + "score": 0.93, + "content": "\\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 196, + 363, + 210 + ], + "score": 1.0, + "content": "we can use Markov’s Theorem", + "type": "text" + }, + { + "bbox": [ + 363, + 198, + 432, + 209 + ], + "score": 0.37, + "content": "( \\mathrm { F a c t } 1 1 ) \\forall \\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 196, + 435, + 210 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 196, + 435, + 210 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 212, + 420, + 239 + ], + "lines": [ + { + "bbox": [ + 190, + 212, + 420, + 239 + ], + "spans": [ + { + "bbox": [ + 190, + 212, + 420, + 239 + ], + "score": 0.9, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) < \\eta \\right) > 1 - \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\eta }", + "type": "interline_equation", + "image_path": "2dc91f38ea1fb3c23cbd1245a42972f4f8edc0e8870805621a60c5e4b1976bd0.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 190, + 212, + 420, + 239 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 243, + 443, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 445, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 144, + 258 + ], + "score": 1.0, + "content": "denoting", + "type": "text" + }, + { + "bbox": [ + 144, + 243, + 253, + 257 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\eta = \\frac { 1 } { \\delta } \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 241, + 445, + 258 + ], + "score": 1.0, + "content": ", and using Theorem (6) we prove the corollary.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 241, + 445, + 258 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 259, + 437, + 316 + ], + "lines": [ + { + "bbox": [ + 173, + 259, + 437, + 316 + ], + "spans": [ + { + "bbox": [ + 173, + 259, + 437, + 316 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { 1 - \\delta < \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( \\displaystyle \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) < \\frac { 1 } { \\delta } \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\right) } \\\\ & { \\quad \\quad \\quad < \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\dot { \\leq } \\frac { 1 } { \\delta } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "36d6b97111f482b5888329e9e8c4b873fb425980acee6b653550d879f9ba423e.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 173, + 259, + 437, + 278.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 173, + 278.0, + 437, + 297.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 173, + 297.0, + 437, + 316.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 319, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 295, + 332 + ], + "score": 1.0, + "content": "where we note that replacing a regular inequality", + "type": "text" + }, + { + "bbox": [ + 295, + 321, + 304, + 330 + ], + "score": 0.8, + "content": "<", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 319, + 441, + 332 + ], + "score": 1.0, + "content": "with inequality in the leading order", + "type": "text" + }, + { + "bbox": [ + 441, + 319, + 450, + 331 + ], + "score": 0.71, + "content": "\\dot { \\leq }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "only removes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 331, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 311, + 344 + ], + "score": 1.0, + "content": "constraints, and therefore increases the probability.", + "type": "text" + }, + { + "bbox": [ + 495, + 333, + 504, + 340 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 319, + 505, + 344 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 358, + 441, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 442, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 442, + 372 + ], + "score": 1.0, + "content": "9 CONSTRUCTION OF GLOBAL MINIMA: PROOF OF THEOREM 8:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 236, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 235, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 235, + 397 + ], + "score": 1.0, + "content": "Recall the LReLU non-linearity", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 381, + 235, + 397 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 393, + 358, + 421 + ], + "lines": [ + { + "bbox": [ + 253, + 393, + 358, + 421 + ], + "spans": [ + { + "bbox": [ + 253, + 393, + 358, + 421 + ], + "score": 0.94, + "content": "f \\left( x \\right) \\triangleq { \\left\\{ \\begin{array} { l l } { \\rho x } & { , { \\mathrm { i f ~ } } x < 0 } \\\\ { x } & { , { \\mathrm { i f ~ } } x \\geq 0 } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "3a9836c54a3588da2a9b52a87454af7cf7fc0020bf26f65f4a5504f2551c1d03.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 253, + 393, + 358, + 421 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 210, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 207, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 182, + 437 + ], + "score": 1.0, + "content": "in eq. (2.1), where", + "type": "text" + }, + { + "bbox": [ + 183, + 423, + 207, + 435 + ], + "score": 0.91, + "content": "\\rho \\neq 1", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 421, + 207, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 294, + 450 + ], + "score": 1.0, + "content": "Theorem 18. (Theorem 8 restated) For any", + "type": "text" + }, + { + "bbox": [ + 294, + 436, + 350, + 449 + ], + "score": 0.93, + "content": "\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 434, + 370, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 436, + 424, + 447 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 434, + 506, + 450 + ], + "score": 1.0, + "content": "almost everywhere", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 447, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 180, + 464 + ], + "score": 1.0, + "content": "we find matrices", + "type": "text" + }, + { + "bbox": [ + 180, + 449, + 245, + 461 + ], + "score": 0.9, + "content": "\\mathbf { W ^ { * } } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 447, + 267, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 268, + 449, + 312, + 461 + ], + "score": 0.91, + "content": "\\mathbf { z } ^ { * } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 447, + 360, + 464 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 360, + 449, + 445, + 462 + ], + "score": 0.91, + "content": "\\mathbf { y } ~ = ~ f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 447, + 479, + 464 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 479, + 449, + 505, + 462 + ], + "score": 0.78, + "content": "d _ { 1 } ^ { * } \\ \\triangleq", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 460, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 177, + 475 + ], + "score": 0.9, + "content": "4 \\left\\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\right\\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 460, + 198, + 476 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 461, + 290, + 475 + ], + "score": 0.92, + "content": "\\forall i , n : \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 460, + 413, + 476 + ], + "score": 1.0, + "content": ". Therefore, every MNN with", + "type": "text" + }, + { + "bbox": [ + 413, + 463, + 452, + 475 + ], + "score": 0.93, + "content": "d _ { 1 } \\ \\geq \\ d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 460, + 506, + 476 + ], + "score": 1.0, + "content": "has a DLM", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 474, + 240, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 211, + 485 + ], + "score": 1.0, + "content": "which achieves zero error", + "type": "text" + }, + { + "bbox": [ + 211, + 475, + 236, + 484 + ], + "score": 0.87, + "content": "\\mathbf { e } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 474, + 240, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 434, + 506, + 485 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 259, + 506 + ], + "score": 1.0, + "content": "We prove the existence of a solution", + "type": "text" + }, + { + "bbox": [ + 259, + 494, + 296, + 505 + ], + "score": 0.9, + "content": "( \\mathbf { W } ^ { * } , \\mathbf { z } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 494, + 506, + 506 + ], + "score": 1.0, + "content": ", by explicitly constructing it. This construction is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 516 + ], + "score": 1.0, + "content": "a variant of (Baum, 1988, Theorem 1), except we use LReLU without bias and MSE – instead of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 356, + 529 + ], + "score": 1.0, + "content": "threshold units with bias and MCE. First, we note that for any", + "type": "text" + }, + { + "bbox": [ + 356, + 516, + 406, + 527 + ], + "score": 0.91, + "content": "\\epsilon _ { 1 } > \\epsilon _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 515, + 505, + 529 + ], + "score": 1.0, + "content": ", the following trapezoid", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 526, + 331, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 331, + 538 + ], + "score": 1.0, + "content": "function can be written as a scaled sum of four LReLU:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 494, + 506, + 538 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 540, + 461, + 613 + ], + "lines": [ + { + "bbox": [ + 149, + 540, + 461, + 613 + ], + "spans": [ + { + "bbox": [ + 149, + 540, + 461, + 613 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\tau \\left( x \\right) \\triangleq \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { , ~ i f ~ } | x | > \\epsilon _ { 1 } } \\\\ { 1 } & { \\mathrm { , ~ i f ~ } | x | \\leq \\epsilon _ { 2 } } \\\\ { \\frac { \\epsilon _ { 1 } - | x | } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } } & { \\mathrm { , ~ i f ~ } \\epsilon _ { 2 } < | x | \\leq \\epsilon _ { 1 } } \\\\ { = \\displaystyle \\frac { 1 } { \\epsilon _ { 1 } - \\epsilon _ { 2 } } \\frac { 1 } { 1 - \\rho } \\left[ f \\left( x + \\epsilon _ { 1 } \\right) - f \\left( x + \\epsilon _ { 2 } \\right) - f \\left( x - \\epsilon _ { 2 } \\right) + f \\left( x - \\epsilon _ { 1 } \\right) \\right] . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "a211f0f15b159948d9a8919c4ce08f0aa24f7a3aaa0f9c176ebfeede943d3075.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 149, + 540, + 461, + 564.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 149, + 564.3333333333334, + 461, + 588.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 149, + 588.6666666666667, + 461, + 613.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 615, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 382, + 630 + ], + "score": 1.0, + "content": "Next, we examine the set of data points which are classified to 1:", + "type": "text" + }, + { + "bbox": [ + 382, + 614, + 500, + 629 + ], + "score": 0.81, + "content": "S ^ { + } \\triangleq \\left\\{ n \\in [ N ] | y ^ { ( n ) } = 1 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 613, + 506, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 627, + 372, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 245, + 644 + ], + "score": 1.0, + "content": "Without loss of generality, assume", + "type": "text" + }, + { + "bbox": [ + 246, + 628, + 288, + 642 + ], + "score": 0.94, + "content": "\\vert S ^ { + } \\vert \\le \\frac { N } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 627, + 344, + 644 + ], + "score": 1.0, + "content": ". We partition", + "type": "text" + }, + { + "bbox": [ + 344, + 629, + 359, + 639 + ], + "score": 0.89, + "content": "S ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 627, + 372, + 644 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 613, + 506, + 644 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 645, + 373, + 673 + ], + "lines": [ + { + "bbox": [ + 237, + 645, + 373, + 673 + ], + "spans": [ + { + "bbox": [ + 237, + 645, + 373, + 673 + ], + "score": 0.94, + "content": "K = \\left\\lceil { \\frac { | S ^ { + } | } { d _ { 0 } - 1 } } \\right\\rceil \\leq \\left\\lceil { \\frac { N } { 2 \\left( d _ { 0 } - 1 \\right) } } \\right\\rceil", + "type": "interline_equation", + "image_path": "298eb14ca98d1ab7bf9d82caf0c9900c1010dcd88684e1a7a497acf1788b749c.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 237, + 645, + 373, + 673 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 505, + 719 + ], + "lines": [ + { + "bbox": [ + 101, + 670, + 504, + 704 + ], + "spans": [ + { + "bbox": [ + 101, + 670, + 138, + 704 + ], + "score": 1.0, + "content": "subsets", + "type": "text" + }, + { + "bbox": [ + 139, + 677, + 178, + 694 + ], + "score": 0.93, + "content": "\\left\\{ { \\cal { S } } _ { i } ^ { + } \\right\\} _ { i = 1 } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 670, + 280, + 704 + ], + "score": 1.0, + "content": ", each with no more than", + "type": "text" + }, + { + "bbox": [ + 281, + 680, + 309, + 691 + ], + "score": 0.9, + "content": "d _ { 0 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 670, + 493, + 704 + ], + "score": 1.0, + "content": "samples. For almost any dataset we can find K", + "type": "text" + }, + { + "bbox": [ + 493, + 680, + 504, + 690 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 102, + 687, + 509, + 713 + ], + "spans": [ + { + "bbox": [ + 102, + 687, + 317, + 713 + ], + "score": 1.0, + "content": "i=1hyperplanes passing through the origin, with normals", + "type": "text" + }, + { + "bbox": [ + 353, + 687, + 509, + 713 + ], + "score": 1.0, + "content": "such that each hyperplane contains all", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 706, + 236, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 707, + 135, + 719 + ], + "score": 0.89, + "content": "d _ { 0 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 706, + 200, + 721 + ], + "score": 1.0, + "content": "points in subset", + "type": "text" + }, + { + "bbox": [ + 200, + 707, + 214, + 720 + ], + "score": 0.91, + "content": "{ S } _ { i } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 706, + 236, + 721 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 101, + 670, + 509, + 721 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 275, + 717, + 335, + 735 + ], + "lines": [ + { + "bbox": [ + 275, + 717, + 335, + 735 + ], + "spans": [ + { + "bbox": [ + 275, + 717, + 335, + 735 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { X } _ { S _ { i } ^ { + } } = 0 ,", + "type": "interline_equation", + "image_path": "238355f0a586b842f9f44bbbc43072e2fb15c002b2e06ae364765f711adafc1e.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 275, + 717, + 335, + 735 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 299, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 300, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 193, + 96 + ], + "score": 1.0, + "content": "but no other point, so", + "type": "text" + }, + { + "bbox": [ + 193, + 81, + 294, + 95 + ], + "score": 0.84, + "content": "\\forall n \\notin S _ { i } ^ { + } : \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 80, + 300, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 444, + 114 + ], + "lines": [ + { + "bbox": [ + 104, + 98, + 446, + 116 + ], + "spans": [ + { + "bbox": [ + 104, + 98, + 115, + 116 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 102, + 138, + 113 + ], + "score": 0.89, + "content": "\\epsilon _ { 1 } , \\epsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 98, + 272, + 116 + ], + "score": 1.0, + "content": "in eq. (9.1) are sufficiently small", + "type": "text" + }, + { + "bbox": [ + 273, + 100, + 388, + 114 + ], + "score": 0.91, + "content": "( \\forall n \\notin S _ { i } ^ { + } : \\left| \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| > \\epsilon _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 98, + 446, + 116 + ], + "score": 1.0, + "content": "then we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 119, + 376, + 147 + ], + "lines": [ + { + "bbox": [ + 234, + 119, + 376, + 147 + ], + "spans": [ + { + "bbox": [ + 234, + 119, + 376, + 147 + ], + "score": 0.93, + "content": "\\tau \\left( \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathrm { , ~ i f ~ } n \\in S _ { i } ^ { + } } \\\\ { 0 } & { \\mathrm { , ~ e l s e } } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "39c0d71a88b161c651edff22966bbc63e28e949c7befc8a6f217f5ba7b69bcdd.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 234, + 119, + 376, + 147 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 151, + 164, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 164, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 164, + 164 + ], + "score": 1.0, + "content": "Then we have", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 159, + 382, + 193 + ], + "lines": [ + { + "bbox": [ + 228, + 159, + 382, + 193 + ], + "spans": [ + { + "bbox": [ + 228, + 159, + 382, + 193 + ], + "score": 0.93, + "content": "\\sum _ { i = 1 } ^ { K } \\tau \\left( \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) = \\left\\{ \\mathbf { 1 } \\quad , \\mathrm { i f } n \\in \\mathcal { S } ^ { + } \\right.", + "type": "interline_equation", + "image_path": "43dbb6a9c734ac8055f7747e130567295986012c8b93c54d8b8b5d133f50b499.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 159, + 382, + 176.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 228, + 176.0, + 382, + 193.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 194, + 504, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 208 + ], + "score": 1.0, + "content": "which gives the correct classification on all the data points. Thus, from eq. 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This is straightforward to do if we have a bias in each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 447, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 393, + 254 + ], + "score": 1.0, + "content": "neuron. 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Defining", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 543, + 396, + 585 + ], + "lines": [ + { + "bbox": [ + 214, + 543, + 396, + 585 + ], + "spans": [ + { + "bbox": [ + 214, + 543, + 396, + 585 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbf { W } _ { i } \\triangleq \\left[ \\mathbf { w } _ { i } ^ { ( 1 ) } , \\mathbf { w } _ { i } ^ { ( 2 ) } , \\mathbf { w } _ { i } ^ { ( 3 ) } , \\mathbf { w } _ { i } ^ { ( 4 ) } \\right] ^ { \\top } \\in \\mathbb { R } ^ { 4 K \\times d _ { 0 } } } \\\\ & { \\mathbf { z } _ { i } \\triangleq \\left[ 1 , - 1 , - 1 , 1 \\right] ^ { \\top } \\in \\mathbb { R } ^ { 4 } } \\end{array}", + "type": "interline_equation", + "image_path": "bbbe1c0f2f4bcb486181e4c90035f9f8508a185ede55327c0496643895955582.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 543, + 396, + 564.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 214, + 564.0, + 396, + 585.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 280, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 281, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 281, + 599 + ], + "score": 1.0, + "content": "and combining all the above facts, we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 603, + 491, + 732 + ], + "lines": [ + { + "bbox": [ + 120, + 603, + 491, + 732 + ], + "spans": [ + { + "bbox": [ + 120, + 603, + 491, + 732 + ], + "score": 0.93, + "content": "\\begin{array} { l } { f \\left( \\mathbf { W } _ { i } \\mathbf { x } ^ { ( n ) } \\right) ^ { \\top } \\mathbf { z } _ { i } } \\\\ { \\ = \\displaystyle \\frac { 1 } { \\epsilon _ { 1 } - 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Clearly, from this construction, if", + "type": "text" + }, + { + "bbox": [ + 352, + 204, + 365, + 214 + ], + "score": 0.87, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 201, + 409, + 215 + ], + "score": 1.0, + "content": "is a row of", + "type": "text" + }, + { + "bbox": [ + 410, + 203, + 426, + 213 + ], + "score": 0.87, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 201, + 449, + 215 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 450, + 202, + 504, + 215 + ], + "score": 0.88, + "content": "\\forall n \\in S _ { i } ^ { + } , \\forall i :", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 212, + 507, + 230 + ], + "spans": [ + { + "bbox": [ + 107, + 214, + 169, + 228 + ], + "score": 0.93, + "content": "| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } | \\geq \\epsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 212, + 258, + 230 + ], + "score": 1.0, + "content": ", and with probability", + "type": "text" + }, + { + "bbox": [ + 258, + 214, + 383, + 228 + ], + "score": 0.91, + "content": "1 \\forall n \\notin S _ { i } ^ { + } , \\forall i : \\left| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 212, + 507, + 230 + ], + "score": 1.0, + "content": ", so this construction does not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 301, + 239 + ], + "score": 1.0, + "content": "touch any non-differentiable region of the MSE.", + "type": "text" + }, + { + "bbox": [ + 495, + 227, + 505, + 236 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 254, + 347, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 348, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 348, + 271 + ], + "score": 1.0, + "content": "10 GLOBAL MINIMA: PROOF OF THEOREM 9", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 507, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 507, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 437, + 299 + ], + "score": 1.0, + "content": "Theorem 19. (Theorem 9 restated). Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 437, + 277, + 507, + 299 + ], + "score": 0.88, + "content": "\\delta \\dot { = } \\sqrt { { \\textstyle \\frac { 8 } { \\pi } } } d _ { 0 } ^ { - 1 / 2 } \\ +", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 295, + 510, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 176, + 312 + ], + "score": 0.92, + "content": "2 d _ { 0 } ^ { 1 / 2 } \\sqrt { { \\log d _ { 0 } } } / { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 295, + 197, + 317 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 198, + 300, + 253, + 312 + ], + "score": 0.92, + "content": "d _ { 1 } ^ { * } = 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 295, + 333, + 317 + ], + "score": 1.0, + "content": ", or if assumption", + "type": "text" + }, + { + "bbox": [ + 333, + 301, + 340, + 310 + ], + "score": 0.39, + "content": "^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 295, + 400, + 317 + ], + "score": 1.0, + "content": "holds, we set", + "type": "text" + }, + { + "bbox": [ + 400, + 300, + 411, + 312 + ], + "score": 0.89, + "content": "d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 295, + 510, + 317 + ], + "score": 1.0, + "content": "as in this assumption.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 311, + 460, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 197, + 324 + ], + "score": 1.0, + "content": "Then, with probability", + "type": "text" + }, + { + "bbox": [ + 197, + 312, + 220, + 322 + ], + "score": 0.85, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 311, + 460, + 324 + ], + "score": 1.0, + "content": ", the angular volume of global minima is lower bounded as,", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 327, + 421, + 342 + ], + "lines": [ + { + "bbox": [ + 189, + 327, + 421, + 342 + ], + "spans": [ + { + "bbox": [ + 189, + 327, + 421, + 342 + ], + "score": 0.86, + "content": "\\mathscr { V } \\left( \\mathscr { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { > } \\exp \\left( - d _ { 1 } ^ { \\ast } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .", + "type": "interline_equation", + "image_path": "d4feb8dffc8a9e1fcee5f934e9a750e147c09afd2ac75e09bede36b743b4ac2f.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 189, + 327, + 421, + 342 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 506, + 388 + ], + "lines": [ + { + "bbox": [ + 104, + 352, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 329, + 367 + ], + "score": 1.0, + "content": "In this section we lower bound the angular volume of", + "type": "text" + }, + { + "bbox": [ + 329, + 354, + 337, + 365 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 352, + 505, + 367 + ], + "score": 1.0, + "content": "(definition 4), i.e., differentiable regions", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 244, + 378 + ], + "score": 1.0, + "content": "in which there exist DLMs with", + "type": "text" + }, + { + "bbox": [ + 245, + 365, + 291, + 376 + ], + "score": 0.9, + "content": "\\mathrm { { M C E } = 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 364, + 370, + 378 + ], + "score": 1.0, + "content": ". We lower bound", + "type": "text" + }, + { + "bbox": [ + 370, + 365, + 394, + 377 + ], + "score": 0.92, + "content": "\\mathcal { V } \\left( \\mathcal { G } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "using the angular volume", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 376, + 424, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 424, + 389 + ], + "score": 1.0, + "content": "corresponding to the differentiable region containing a single global minimum.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 230, + 405 + ], + "score": 1.0, + "content": "From assumption 4, we have", + "type": "text" + }, + { + "bbox": [ + 231, + 392, + 268, + 404 + ], + "score": 0.92, + "content": "d _ { 0 } d _ { 1 } \\dot { > } N", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 392, + 505, + 405 + ], + "score": 1.0, + "content": ", so we can apply Theorem 8 and say that the labels are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 404, + 504, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 186, + 418 + ], + "score": 1.0, + "content": "generated using a", + "type": "text" + }, + { + "bbox": [ + 186, + 405, + 213, + 417 + ], + "score": 0.89, + "content": "( \\mathbf { X } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 405, + 296, + 418 + ], + "score": 1.0, + "content": "-dependent MNN:", + "type": "text" + }, + { + "bbox": [ + 297, + 404, + 383, + 418 + ], + "score": 0.92, + "content": "\\mathbf { y } ~ = ~ f \\left( \\mathbf { W } ^ { * } \\mathbf { X } \\right) ^ { \\top } \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 405, + 471, + 418 + ], + "score": 1.0, + "content": "with target weights", + "type": "text" + }, + { + "bbox": [ + 472, + 405, + 504, + 417 + ], + "score": 0.84, + "content": "\\begin{array} { r l } { \\mathbf { W } ^ { * } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 417, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 107, + 417, + 231, + 439 + ], + "score": 0.93, + "content": "\\left[ { \\bf w } _ { 1 } ^ { * \\top } , \\ldots , { \\bf w } _ { d _ { 1 } ^ { * } } ^ { * \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 420, + 252, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 422, + 294, + 434 + ], + "score": 0.91, + "content": "\\mathbf { z } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 420, + 506, + 437 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 184, + 542, + 194, + 552 + ], + "score": 0.62, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 541, + 291, + 554 + ], + "score": 1.0, + "content": "have an angular margin", + "type": "text" + }, + { + "bbox": [ + 292, + 544, + 300, + 552 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 541, + 322, + 554 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 322, + 542, + 340, + 552 + ], + "score": 0.8, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 541, + 455, + 554 + ], + "score": 1.0, + "content": "if all datapoints (columns in", + "type": "text" + }, + { + "bbox": [ + 455, + 543, + 465, + 552 + ], + "score": 0.71, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 541, + 505, + 554 + ], + "score": 1.0, + "content": ") are at an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 450, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 172, + 565 + ], + "score": 1.0, + "content": "angle of at least", + "type": "text" + }, + { + "bbox": [ + 172, + 555, + 180, + 563 + ], + "score": 0.81, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 552, + 345, + 565 + ], + "score": 1.0, + "content": "from all the weight hyperplanes (rows of", + "type": "text" + }, + { + "bbox": [ + 346, + 553, + 363, + 563 + ], + "score": 0.85, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 552, + 390, + 565 + ], + "score": 1.0, + "content": ") , i.e.,", + "type": "text" + }, + { + "bbox": [ + 390, + 553, + 400, + 563 + ], + "score": 0.6, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 552, + 450, + 565 + ], + "score": 1.0, + "content": "is in the set", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 569, + 434, + 604 + ], + "lines": [ + { + "bbox": [ + 177, + 569, + 434, + 604 + ], + "spans": [ + { + "bbox": [ + 177, + 569, + 434, + 604 + ], + "score": 0.95, + "content": "\\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left. \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\forall i , n : \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\mathrm { s i n } \\alpha \\right. .", + "type": "interline_equation", + "image_path": "ad940362f0ac2a63b6765a0667ffa0faebc3ddf014ee3fbb9c76cf460a0d6cf4.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 177, + 569, + 434, + 580.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 177, + 580.6666666666666, + 434, + 592.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 177, + 592.3333333333333, + 434, + 603.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "score": 1.0, + "content": "Using the definitions in eqs. (10.3) and (10.1), we prove the Theorem using the following three", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 624, + 147, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 147, + 639 + ], + "score": 1.0, + "content": "Lemmas.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 269, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 269, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 269, + 655 + ], + "score": 1.0, + "content": "First, In appendix section 13.2 we prove", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 192, + 671 + ], + "score": 1.0, + "content": "Lemma 21. For any", + "type": "text" + }, + { + "bbox": [ + 192, + 660, + 200, + 668 + ], + "score": 0.64, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 655, + 203, + 671 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 203, + 658, + 230, + 669 + ], + "score": 0.35, + "content": "i f \\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 655, + 394, + 671 + ], + "score": 1.0, + "content": "is independent from W then, in the limit", + "type": "text" + }, + { + "bbox": [ + 394, + 658, + 429, + 668 + ], + "score": 0.66, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 655, + 433, + 671 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 434, + 658, + 505, + 670 + ], + "score": 0.75, + "content": "\\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 668, + 211, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 131, + 684 + ], + "score": 1.0, + "content": "with l", + "type": "text" + }, + { + "bbox": [ + 131, + 669, + 211, + 682 + ], + "score": 0.69, + "content": "\\mathrm { { o g } s i n } \\alpha \\dot { > } d _ { 0 } ^ { - 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Clearly, from this construction, if", + "type": "text" + }, + { + "bbox": [ + 352, + 204, + 365, + 214 + ], + "score": 0.87, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 201, + 409, + 215 + ], + "score": 1.0, + "content": "is a row of", + "type": "text" + }, + { + "bbox": [ + 410, + 203, + 426, + 213 + ], + "score": 0.87, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 201, + 449, + 215 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 450, + 202, + 504, + 215 + ], + "score": 0.88, + "content": "\\forall n \\in S _ { i } ^ { + } , \\forall i :", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 212, + 507, + 230 + ], + "spans": [ + { + "bbox": [ + 107, + 214, + 169, + 228 + ], + "score": 0.93, + "content": "| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } | \\geq \\epsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 212, + 258, + 230 + ], + "score": 1.0, + "content": ", and with probability", + "type": "text" + }, + { + "bbox": [ + 258, + 214, + 383, + 228 + ], + "score": 0.91, + "content": "1 \\forall n \\notin S _ { i } ^ { + } , \\forall i : \\left| \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right| > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 212, + 507, + 230 + ], + "score": 1.0, + "content": ", so this construction does not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 301, + 239 + ], + "score": 1.0, + "content": "touch any non-differentiable region of the MSE.", + "type": "text" + }, + { + "bbox": [ + 495, + 227, + 505, + 236 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 201, + 507, + 239 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 254, + 347, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 348, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 348, + 271 + ], + "score": 1.0, + "content": "10 GLOBAL MINIMA: PROOF OF THEOREM 9", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 507, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 507, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 437, + 299 + ], + "score": 1.0, + "content": "Theorem 19. (Theorem 9 restated). Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 437, + 277, + 507, + 299 + ], + "score": 0.88, + "content": "\\delta \\dot { = } \\sqrt { { \\textstyle \\frac { 8 } { \\pi } } } d _ { 0 } ^ { - 1 / 2 } \\ +", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 295, + 510, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 176, + 312 + ], + "score": 0.92, + "content": "2 d _ { 0 } ^ { 1 / 2 } \\sqrt { { \\log d _ { 0 } } } / { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 295, + 197, + 317 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 198, + 300, + 253, + 312 + ], + "score": 0.92, + "content": "d _ { 1 } ^ { * } = 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 295, + 333, + 317 + ], + "score": 1.0, + "content": ", or if assumption", + "type": "text" + }, + { + "bbox": [ + 333, + 301, + 340, + 310 + ], + "score": 0.39, + "content": "^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 295, + 400, + 317 + ], + "score": 1.0, + "content": "holds, we set", + "type": "text" + }, + { + "bbox": [ + 400, + 300, + 411, + 312 + ], + "score": 0.89, + "content": "d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 295, + 510, + 317 + ], + "score": 1.0, + "content": "as in this assumption.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 311, + 460, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 197, + 324 + ], + "score": 1.0, + "content": "Then, with probability", + "type": "text" + }, + { + "bbox": [ + 197, + 312, + 220, + 322 + ], + "score": 0.85, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 311, + 460, + 324 + ], + "score": 1.0, + "content": ", the angular volume of global minima is lower bounded as,", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 277, + 510, + 324 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 327, + 421, + 342 + ], + "lines": [ + { + "bbox": [ + 189, + 327, + 421, + 342 + ], + "spans": [ + { + "bbox": [ + 189, + 327, + 421, + 342 + ], + "score": 0.86, + "content": "\\mathscr { V } \\left( \\mathscr { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { > } \\exp \\left( - d _ { 1 } ^ { \\ast } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .", + "type": "interline_equation", + "image_path": "d4feb8dffc8a9e1fcee5f934e9a750e147c09afd2ac75e09bede36b743b4ac2f.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 189, + 327, + 421, + 342 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 506, + 388 + ], + "lines": [ + { + "bbox": [ + 104, + 352, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 329, + 367 + ], + "score": 1.0, + "content": "In this section we lower bound the angular volume of", + "type": "text" + }, + { + "bbox": [ + 329, + 354, + 337, + 365 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 352, + 505, + 367 + ], + "score": 1.0, + "content": "(definition 4), i.e., differentiable regions", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 244, + 378 + ], + "score": 1.0, + "content": "in which there exist DLMs with", + "type": "text" + }, + { + "bbox": [ + 245, + 365, + 291, + 376 + ], + "score": 0.9, + "content": "\\mathrm { { M C E } = 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 364, + 370, + 378 + ], + "score": 1.0, + "content": ". 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In both cases, the following differentiable region", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 392, + 506, + 451 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 456, + 455, + 472 + ], + "lines": [ + { + "bbox": [ + 155, + 456, + 455, + 472 + ], + "spans": [ + { + "bbox": [ + 155, + 456, + 455, + 472 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } } | \\forall i \\le d _ { 1 } ^ { * } : \\ \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } \\right) = \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { * ^ { \\top } } \\mathbf { X } \\right) \\right\\} , } \\end{array}", + "type": "interline_equation", + "image_path": "396df204c6dbeabe1d7d8b292ea954d4be1899341b3edd1520d420754b6f14f1.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 155, + 456, + 455, + 472 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 477, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 334, + 491 + ], + "score": 1.0, + "content": "also contains a differentiable global minimum (just set", + "type": "text" + }, + { + "bbox": [ + 335, + 478, + 375, + 489 + ], + "score": 0.89, + "content": "\\mathbf { w } _ { i } = \\mathbf { w } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 475, + 380, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 381, + 477, + 452, + 489 + ], + "score": 0.92, + "content": "z _ { i } = z _ { i } ^ { * } \\forall i \\leq d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 475, + 474, + 491 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 474, + 478, + 504, + 488 + ], + "score": 0.91, + "content": "z _ { i } = 0", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 488, + 384, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 141, + 500 + ], + "score": 0.88, + "content": "\\forall i > d _ { 1 } ^ { * } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 488, + 201, + 501 + ], + "score": 1.0, + "content": "), and therefore", + "type": "text" + }, + { + "bbox": [ + 202, + 488, + 228, + 499 + ], + "score": 0.6, + "content": "\\forall \\mathbf { X } , \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 488, + 326, + 501 + ], + "score": 1.0, + "content": "and their corresponding", + "type": "text" + }, + { + "bbox": [ + 327, + 489, + 344, + 498 + ], + "score": 0.86, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 488, + 384, + 501 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 475, + 504, + 501 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 505, + 354, + 520 + ], + "lines": [ + { + "bbox": [ + 257, + 505, + 354, + 520 + ], + "spans": [ + { + "bbox": [ + 257, + 505, + 354, + 520 + ], + "score": 0.92, + "content": "\\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\supset \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right)", + "type": "interline_equation", + "image_path": "045a31095a5ab73d446c31cf5b5781057deb5585c87c32d7bbdd3bb458554b0c.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 257, + 505, + 354, + 520 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 526, + 311, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 311, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 311, + 540 + ], + "score": 1.0, + "content": "Also, we will make use of the following definition.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 525, + 311, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 184, + 554 + ], + "score": 1.0, + "content": "Definition 20. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 542, + 194, + 552 + ], + "score": 0.62, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 541, + 291, + 554 + ], + "score": 1.0, + "content": "have an angular margin", + "type": "text" + }, + { + "bbox": [ + 292, + 544, + 300, + 552 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 541, + 322, + 554 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 322, + 542, + 340, + 552 + ], + "score": 0.8, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 541, + 455, + 554 + ], + "score": 1.0, + "content": "if all datapoints (columns in", + "type": "text" + }, + { + "bbox": [ + 455, + 543, + 465, + 552 + ], + "score": 0.71, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 541, + 505, + 554 + ], + "score": 1.0, + "content": ") are at an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 450, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 172, + 565 + ], + "score": 1.0, + "content": "angle of at least", + "type": "text" + }, + { + "bbox": [ + 172, + 555, + 180, + 563 + ], + "score": 0.81, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 552, + 345, + 565 + ], + "score": 1.0, + "content": "from all the weight hyperplanes (rows of", + "type": "text" + }, + { + "bbox": [ + 346, + 553, + 363, + 563 + ], + "score": 0.85, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 552, + 390, + 565 + ], + "score": 1.0, + "content": ") , i.e.,", + "type": "text" + }, + { + "bbox": [ + 390, + 553, + 400, + 563 + ], + "score": 0.6, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 552, + 450, + 565 + ], + "score": 1.0, + "content": "is in the set", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 541, + 505, + 565 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 569, + 434, + 604 + ], + "lines": [ + { + "bbox": [ + 177, + 569, + 434, + 604 + ], + "spans": [ + { + "bbox": [ + 177, + 569, + 434, + 604 + ], + "score": 0.95, + "content": "\\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left. \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\forall i , n : \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\mathrm { s i n } \\alpha \\right. .", + "type": "interline_equation", + "image_path": "ad940362f0ac2a63b6765a0667ffa0faebc3ddf014ee3fbb9c76cf460a0d6cf4.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 177, + 569, + 434, + 580.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 177, + 580.6666666666666, + 434, + 592.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 177, + 592.3333333333333, + 434, + 603.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "score": 1.0, + "content": "Using the definitions in eqs. 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For any", + "type": "text" + }, + { + "bbox": [ + 192, + 660, + 200, + 668 + ], + "score": 0.64, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 655, + 203, + 671 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 203, + 658, + 230, + 669 + ], + "score": 0.35, + "content": "i f \\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 655, + 394, + 671 + ], + "score": 1.0, + "content": "is independent from W then, in the limit", + "type": "text" + }, + { + "bbox": [ + 394, + 658, + 429, + 668 + ], + "score": 0.66, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 655, + 433, + 671 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 434, + 658, + 505, + 670 + ], + "score": 0.75, + "content": "\\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 668, + 211, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 131, + 684 + ], + "score": 1.0, + "content": "with l", + "type": "text" + }, + { + "bbox": [ + 131, + 669, + 211, + 682 + ], + "score": 0.69, + "content": "\\mathrm { { o g } s i n } \\alpha \\dot { > } d _ { 0 } ^ { - 1 } \\log d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 655, + 505, + 684 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 688, + 429, + 710 + ], + "lines": [ + { + "bbox": [ + 180, + 688, + 429, + 710 + ], + "spans": [ + { + "bbox": [ + 180, + 688, + 429, + 710 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { V } \\left( \\tilde { \\mathcal { G } } \\right) = \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\geq } \\exp \\left( d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "cd09fc5a1737d4dfe7704c0adbb9fc4c50aa5b54c92f4fcbf4cdab6c572c2045.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 180, + 688, + 429, + 710 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 279, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 280, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 280, + 734 + ], + "score": 1.0, + "content": "Second, in appendix section 13.3 we prove", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 720, + 280, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 175, + 95 + ], + "score": 1.0, + "content": "Lemma 22. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 81, + 236, + 93 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 80, + 360, + 95 + ], + "score": 1.0, + "content": "a fixed matrix independent of", + "type": "text" + }, + { + "bbox": [ + 360, + 83, + 370, + 93 + ], + "score": 0.39, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 80, + 447, + 95 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 178, + 164, + 233, + 176 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 163, + 506, + 179 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix of datapoints. Then we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 212, + 190 + ], + "score": 1.0, + "content": "can find, with probability", + "type": "text" + }, + { + "bbox": [ + 212, + 178, + 217, + 187 + ], + "score": 0.49, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 177, + 222, + 190 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 222, + 177, + 250, + 190 + ], + "score": 0.89, + "content": "( \\mathbf { X } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 177, + 334, + 190 + ], + "score": 1.0, + "content": "-dependent matrices", + "type": "text" + }, + { + "bbox": [ + 334, + 177, + 352, + 188 + ], + "score": 0.82, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 177, + 371, + 190 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 178, + 382, + 188 + ], + "score": 0.84, + "content": "\\mathbf { z } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 177, + 482, + 190 + ], + "score": 1.0, + "content": "as in Theorem 8 (where", + "type": "text" + }, + { + "bbox": [ + 482, + 177, + 493, + 190 + ], + "score": 0.73, + "content": "d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 177, + 505, + 190 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 107, + 189, + 179, + 201 + ], + "score": 0.87, + "content": "4 \\ : \\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\rceil ", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 189, + 275, + 203 + ], + "score": 1.0, + "content": ". Moreover, in the limit", + "type": "text" + }, + { + "bbox": [ + 275, + 190, + 311, + 200 + ], + "score": 0.89, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 189, + 342, + 203 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 342, + 189, + 400, + 201 + ], + "score": 0.9, + "content": "N / d _ { 0 } { \\dot { \\le } } d _ { 0 } { \\dot { \\le } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 189, + 506, + 203 + ], + "score": 1.0, + "content": ", for any y, we can bound", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 464, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 370, + 214 + ], + "score": 1.0, + "content": "the probability of not having an angular margin (eq. (10.3)) with", + "type": "text" + }, + { + "bbox": [ + 370, + 201, + 450, + 213 + ], + "score": 0.87, + "content": "\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 200, + 464, + 214 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 217, + 407, + 246 + ], + "lines": [ + { + "bbox": [ + 204, + 217, + 407, + 246 + ], + "spans": [ + { + "bbox": [ + 204, + 217, + 407, + 246 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N }", + "type": "interline_equation", + "image_path": "d6aea7d1ff018291e63c0f48c89869863a15ecab0467eb0d30553d0cbdfe3a62.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 217, + 407, + 231.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 204, + 231.5, + 407, + 246.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 506, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 151, + 272 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + }, + { + "bbox": [ + 152, + 260, + 178, + 271 + ], + "score": 0.61, + "content": "\\forall \\mathbf { X } , \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 258, + 276, + 272 + ], + "score": 1.0, + "content": "and their corresponding", + "type": "text" + }, + { + "bbox": [ + 277, + 260, + 294, + 270 + ], + "score": 0.86, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 258, + 333, + 272 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 334, + 258, + 428, + 271 + ], + "score": 0.94, + "content": "\\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\subset \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "(eq. (10.2)). Thus,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 219, + 282 + ], + "score": 1.0, + "content": "combining Lemmas 21 with", + "type": "text" + }, + { + "bbox": [ + 219, + 270, + 302, + 282 + ], + "score": 0.91, + "content": "\\sin \\alpha = 1 / \\left( \\bar { d } _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "together with either Lemma 22 or 23, we prove the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 281, + 249, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 249, + 294 + ], + "score": 1.0, + "content": "first (left) inequality of Theorem 9:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 307, + 376, + 322 + ], + "lines": [ + { + "bbox": [ + 234, + 307, + 376, + 322 + ], + "spans": [ + { + "bbox": [ + 234, + 307, + 376, + 322 + ], + "score": 0.9, + "content": "{ \\mathcal { V } } \\left( { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) { \\dot { \\geq } } \\exp \\left( - d _ { 1 } ^ { * } d _ { 0 } \\log N \\right)", + "type": "interline_equation", + "image_path": "ab32765f735b173f9ee86eda78742cb481fbd4dc0e053ef310f3000c814e25a9.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 234, + 307, + 376, + 322 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 500, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 500, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 139, + 340 + ], + "score": 1.0, + "content": "Next, if", + "type": "text" + }, + { + "bbox": [ + 140, + 325, + 191, + 337 + ], + "score": 0.93, + "content": "d _ { 1 } ^ { * } = 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 322, + 203, + 340 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 204, + 325, + 245, + 337 + ], + "score": 0.93, + "content": "d _ { 1 } ^ { * } { \\dot { < } } N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 322, + 500, + 340 + ], + "score": 1.0, + "content": "(is assumption 5 holds), we obtain the second (right) inequality", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 342, + 388, + 357 + ], + "lines": [ + { + "bbox": [ + 222, + 342, + 388, + 357 + ], + "spans": [ + { + "bbox": [ + 222, + 342, + 388, + 357 + ], + "score": 0.91, + "content": "\\exp \\left( - d _ { 1 } ^ { \\ast } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .", + "type": "interline_equation", + "image_path": "c55f578000c4433de41349c9e548a1e7dd839da142cdf75226dce81b424c4488.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 222, + 342, + 388, + 357 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 390, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 107, + 390, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 107, + 390, + 504, + 405 + ], + "score": 1.0, + "content": "11 VOLUME RATIO OF GLOBAL AND LOCAL MINIMA: PROOF OF THEOREM 10", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 507, + 468 + ], + "lines": [ + { + "bbox": [ + 102, + 413, + 507, + 435 + ], + "spans": [ + { + "bbox": [ + 102, + 413, + 420, + 435 + ], + "score": 1.0, + "content": "Theorem 24. (Theorem 10 restated) Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 420, + 413, + 507, + 434 + ], + "score": 0.92, + "content": "\\delta \\doteq \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } +", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 431, + 507, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 176, + 447 + ], + "score": 0.9, + "content": "2 d _ { 0 } ^ { 1 / 2 } \\sqrt { { \\log d _ { 0 } } } / { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 431, + 277, + 450 + ], + "score": 1.0, + "content": ". Then, with probability", + "type": "text" + }, + { + "bbox": [ + 278, + 435, + 302, + 446 + ], + "score": 0.76, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 431, + 507, + 450 + ], + "score": 1.0, + "content": ", the angular volume of sub-optimal DLMs, with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 444, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 172, + 457 + ], + "score": 0.85, + "content": "\\mathrm { M C E } > \\epsilon > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 444, + 296, + 460 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 297, + 447, + 306, + 456 + ], + "score": 0.61, + "content": "N ,", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 444, + 506, + 460 + ], + "score": 1.0, + "content": ", in comparison to the angular volume of global", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 457, + 201, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 158, + 468 + ], + "score": 1.0, + "content": "minima with", + "type": "text" + }, + { + "bbox": [ + 158, + 458, + 201, + 468 + ], + "score": 0.84, + "content": "\\mathrm { M C E } = 0", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 472, + 440, + 500 + ], + "lines": [ + { + "bbox": [ + 171, + 472, + 440, + 500 + ], + "spans": [ + { + "bbox": [ + 171, + 472, + 440, + 500 + ], + "score": 0.91, + "content": "\\frac { \\mathscr { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathscr { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N \\log N \\right) .", + "type": "interline_equation", + "image_path": "36040bc717ea29143e129efa4ee3e4f4f0e6c4936b233a4011a7aaa839ef899d.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 171, + 472, + 440, + 500 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 506, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 470, + 523 + ], + "score": 1.0, + "content": "To prove this theorem we first calculate the expectation of the angular volume ratio given the", + "type": "text" + }, + { + "bbox": [ + 470, + 511, + 479, + 520 + ], + "score": 0.71, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "-event", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 520, + 507, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 367, + 534 + ], + "score": 1.0, + "content": "that the bound in Theorem 9 holds (given assumptions 1-3), i.e.,", + "type": "text" + }, + { + "bbox": [ + 368, + 521, + 503, + 533 + ], + "score": 0.92, + "content": "\\mathcal { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\geq } \\dot { \\exp } \\left( - 2 N \\log N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 520, + 507, + 534 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 532, + 253, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 183, + 544 + ], + "score": 1.0, + "content": "Denoting this even", + "type": "text" + }, + { + "bbox": [ + 184, + 532, + 189, + 542 + ], + "score": 0.26, + "content": " { \\varepsilon } ^ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 532, + 201, + 544 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 201, + 532, + 214, + 542 + ], + "score": 0.83, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 532, + 253, + 544 + ], + "score": 1.0, + "content": ", we find:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 550, + 441, + 651 + ], + "lines": [ + { + "bbox": [ + 169, + 550, + 441, + 651 + ], + "spans": [ + { + "bbox": [ + 169, + 550, + 441, + 651 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } | { \\mathcal { M } } \\right] \\overset { ( 1 ) } { \\leq } \\frac { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\big | { \\mathcal { M } } \\right] } { \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 2 ) } { \\leq } } \\\\ & { \\frac { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\right] } { \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( { \\mathcal { M } } \\right) \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 3 ) } { \\leq } \\frac { \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } { \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( { \\mathcal { M } } \\right) \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 4 ) } { \\leq } } \\\\ & { \\quad \\quad \\frac { \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } { \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 5 ) } { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "b31cb03a028f2e5b199a72dbc8b7525e5a526aa82c1330973c2eca3888b63a92.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 169, + 550, + 441, + 583.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 169, + 583.6666666666666, + 441, + 617.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 169, + 617.3333333333333, + 441, + 650.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 133, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 135, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 135, + 667 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 129, + 675, + 248, + 703 + ], + "lines": [ + { + "bbox": [ + 130, + 676, + 231, + 688 + ], + "spans": [ + { + "bbox": [ + 130, + 676, + 231, + 688 + ], + "score": 1.0, + "content": "1. We apply Theorem 9.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 129, + 691, + 247, + 703 + ], + "spans": [ + { + "bbox": [ + 129, + 691, + 247, + 703 + ], + "score": 1.0, + "content": "2. We use the following fact", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 104, + 711, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 258, + 725 + ], + "score": 1.0, + "content": "6This event was previously denoted as", + "type": "text" + }, + { + "bbox": [ + 258, + 711, + 319, + 722 + ], + "score": 0.84, + "content": "\\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 708, + 506, + 725 + ], + "score": 1.0, + "content": "in the proof of Theorem 9, but this is not important", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 720, + 267, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 267, + 733 + ], + "score": 1.0, + "content": "for this proof, so we simplified the notation.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 363, + 504, + 373 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 175, + 95 + ], + "score": 1.0, + "content": "Lemma 22. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 81, + 236, + 93 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { * } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 80, + 360, + 95 + ], + "score": 1.0, + "content": "a fixed matrix independent of", + "type": "text" + }, + { + "bbox": [ + 360, + 83, + 370, + 93 + ], + "score": 0.39, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 80, + 447, + 95 + ], + "score": 1.0, + "content": ". Then, in the limit", + "type": "text" + }, + { + "bbox": [ + 447, + 83, + 484, + 92 + ], + "score": 0.86, + "content": "N \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 80, + 505, + 95 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 151, + 105 + ], + "score": 0.91, + "content": "d _ { 1 } ^ { * } { \\dot { \\leq } } d _ { 0 } { \\dot { \\leq } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 93, + 357, + 106 + ], + "score": 1.0, + "content": ", the probability of not having an angular margin", + "type": "text" + }, + { + "bbox": [ + 358, + 93, + 442, + 106 + ], + "score": 0.91, + "content": "\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "(eq. 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Let", + "type": "text" + }, + { + "bbox": [ + 178, + 164, + 233, + 176 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 163, + 506, + 179 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix of datapoints. Then we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 212, + 190 + ], + "score": 1.0, + "content": "can find, with probability", + "type": "text" + }, + { + "bbox": [ + 212, + 178, + 217, + 187 + ], + "score": 0.49, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 177, + 222, + 190 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 222, + 177, + 250, + 190 + ], + "score": 0.89, + "content": "( \\mathbf { X } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 177, + 334, + 190 + ], + "score": 1.0, + "content": "-dependent matrices", + "type": "text" + }, + { + "bbox": [ + 334, + 177, + 352, + 188 + ], + "score": 0.82, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 177, + 371, + 190 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 178, + 382, + 188 + ], + "score": 0.84, + "content": "\\mathbf { z } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 177, + 482, + 190 + ], + "score": 1.0, + "content": "as in Theorem 8 (where", + "type": "text" + }, + { + "bbox": [ + 482, + 177, + 493, + 190 + ], + "score": 0.73, + "content": "d _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 177, + 505, + 190 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 107, + 189, + 179, + 201 + ], + "score": 0.87, + "content": "4 \\ : \\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\rceil ", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 189, + 275, + 203 + ], + "score": 1.0, + "content": ". 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(10.2)). Thus,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 219, + 282 + ], + "score": 1.0, + "content": "combining Lemmas 21 with", + "type": "text" + }, + { + "bbox": [ + 219, + 270, + 302, + 282 + ], + "score": 0.91, + "content": "\\sin \\alpha = 1 / \\left( \\bar { d } _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "together with either Lemma 22 or 23, we prove the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 281, + 249, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 249, + 294 + ], + "score": 1.0, + "content": "first (left) inequality of Theorem 9:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 258, + 506, + 294 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 307, + 376, + 322 + ], + "lines": [ + { + "bbox": [ + 234, + 307, + 376, + 322 + ], + "spans": [ + { + "bbox": [ + 234, + 307, + 376, + 322 + ], + "score": 0.9, + "content": "{ \\mathcal { V } } \\left( { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) { \\dot { \\geq } } \\exp \\left( - d _ { 1 } ^ { * } d _ { 0 } \\log N \\right)", + "type": "interline_equation", + "image_path": "ab32765f735b173f9ee86eda78742cb481fbd4dc0e053ef310f3000c814e25a9.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 234, + 307, + 376, + 322 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 500, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 500, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 139, + 340 + ], + "score": 1.0, + "content": "Next, if", + "type": "text" + }, + { + "bbox": [ + 140, + 325, + 191, + 337 + ], + "score": 0.93, + "content": "d _ { 1 } ^ { * } = 2 N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 322, + 203, + 340 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 204, + 325, + 245, + 337 + ], + "score": 0.93, + "content": "d _ { 1 } ^ { * } { \\dot { < } } N / d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 322, + 500, + 340 + ], + "score": 1.0, + "content": "(is assumption 5 holds), we obtain the second (right) inequality", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 322, + 500, + 340 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 342, + 388, + 357 + ], + "lines": [ + { + "bbox": [ + 222, + 342, + 388, + 357 + ], + "spans": [ + { + "bbox": [ + 222, + 342, + 388, + 357 + ], + "score": 0.91, + "content": "\\exp \\left( - d _ { 1 } ^ { \\ast } d _ { 0 } \\log N \\right) \\dot { \\geq } \\exp \\left( - 2 N \\log N \\right) .", + "type": "interline_equation", + "image_path": "c55f578000c4433de41349c9e548a1e7dd839da142cdf75226dce81b424c4488.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 222, + 342, + 388, + 357 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 390, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 107, + 390, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 107, + 390, + 504, + 405 + ], + "score": 1.0, + "content": "11 VOLUME RATIO OF GLOBAL AND LOCAL MINIMA: PROOF OF THEOREM 10", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 107, + 390, + 504, + 405 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 507, + 468 + ], + "lines": [ + { + "bbox": [ + 102, + 413, + 507, + 435 + ], + "spans": [ + { + "bbox": [ + 102, + 413, + 420, + 435 + ], + "score": 1.0, + "content": "Theorem 24. (Theorem 10 restated) Given assumptions 1-3, we set", + "type": "text" + }, + { + "bbox": [ + 420, + 413, + 507, + 434 + ], + "score": 0.92, + "content": "\\delta \\doteq \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } +", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 431, + 507, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 176, + 447 + ], + "score": 0.9, + "content": "2 d _ { 0 } ^ { 1 / 2 } \\sqrt { { \\log d _ { 0 } } } / { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 431, + 277, + 450 + ], + "score": 1.0, + "content": ". Then, with probability", + "type": "text" + }, + { + "bbox": [ + 278, + 435, + 302, + 446 + ], + "score": 0.76, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 431, + 507, + 450 + ], + "score": 1.0, + "content": ", the angular volume of sub-optimal DLMs, with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 444, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 172, + 457 + ], + "score": 0.85, + "content": "\\mathrm { M C E } > \\epsilon > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 444, + 296, + 460 + ], + "score": 1.0, + "content": ", is exponentially vanishing in", + "type": "text" + }, + { + "bbox": [ + 297, + 447, + 306, + 456 + ], + "score": 0.61, + "content": "N ,", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 444, + 506, + 460 + ], + "score": 1.0, + "content": ", in comparison to the angular volume of global", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 457, + 201, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 158, + 468 + ], + "score": 1.0, + "content": "minima with", + "type": "text" + }, + { + "bbox": [ + 158, + 458, + 201, + 468 + ], + "score": 0.84, + "content": "\\mathrm { M C E } = 0", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 102, + 413, + 507, + 468 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 472, + 440, + 500 + ], + "lines": [ + { + "bbox": [ + 171, + 472, + 440, + 500 + ], + "spans": [ + { + "bbox": [ + 171, + 472, + 440, + 500 + ], + "score": 0.91, + "content": "\\frac { \\mathscr { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathscr { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N \\log N \\right) .", + "type": "interline_equation", + "image_path": "36040bc717ea29143e129efa4ee3e4f4f0e6c4936b233a4011a7aaa839ef899d.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 171, + 472, + 440, + 500 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 506, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 470, + 523 + ], + "score": 1.0, + "content": "To prove this theorem we first calculate the expectation of the angular volume ratio given the", + "type": "text" + }, + { + "bbox": [ + 470, + 511, + 479, + 520 + ], + "score": 0.71, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "-event", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 520, + 507, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 367, + 534 + ], + "score": 1.0, + "content": "that the bound in Theorem 9 holds (given assumptions 1-3), i.e.,", + "type": "text" + }, + { + "bbox": [ + 368, + 521, + 503, + 533 + ], + "score": 0.92, + "content": "\\mathcal { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) \\dot { \\geq } \\dot { \\exp } \\left( - 2 N \\log N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 520, + 507, + 534 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 532, + 253, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 183, + 544 + ], + "score": 1.0, + "content": "Denoting this even", + "type": "text" + }, + { + "bbox": [ + 184, + 532, + 189, + 542 + ], + "score": 0.26, + "content": " { \\varepsilon } ^ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 532, + 201, + 544 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 201, + 532, + 214, + 542 + ], + "score": 0.83, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 532, + 253, + 544 + ], + "score": 1.0, + "content": ", we find:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 510, + 507, + 544 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 550, + 441, + 651 + ], + "lines": [ + { + "bbox": [ + 169, + 550, + 441, + 651 + ], + "spans": [ + { + "bbox": [ + 169, + 550, + 441, + 651 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } | { \\mathcal { M } } \\right] \\overset { ( 1 ) } { \\leq } \\frac { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\big | { \\mathcal { M } } \\right] } { \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 2 ) } { \\leq } } \\\\ & { \\frac { \\mathbb { E } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) \\right] } { \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( { \\mathcal { M } } \\right) \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 3 ) } { \\leq } \\frac { \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } { \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left( { \\mathcal { M } } \\right) \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 4 ) } { \\leq } } \\\\ & { \\quad \\quad \\frac { \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } { \\exp \\left( - 2 N \\log N \\right) } \\overset { ( 5 ) } { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "b31cb03a028f2e5b199a72dbc8b7525e5a526aa82c1330973c2eca3888b63a92.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 169, + 550, + 441, + 583.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 169, + 583.6666666666666, + 441, + 617.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 169, + 617.3333333333333, + 441, + 650.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 133, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 135, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 135, + 667 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 653, + 135, + 667 + ] + }, + { + "type": "index", + "bbox": [ + 129, + 675, + 248, + 703 + ], + "lines": [ + { + "bbox": [ + 130, + 676, + 231, + 688 + ], + "spans": [ + { + "bbox": [ + 130, + 676, + 231, + 688 + ], + "score": 1.0, + "content": "1. We apply Theorem 9.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 129, + 691, + 247, + 703 + ], + "spans": [ + { + "bbox": [ + 129, + 691, + 247, + 703 + ], + "score": 1.0, + "content": "2. We use the following fact", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + } + ], + "index": 30.5, + "bbox_fs": [ + 129, + 676, + 247, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 142, + 81, + 447, + 95 + ], + "lines": [ + { + "bbox": [ + 140, + 79, + 447, + 97 + ], + "spans": [ + { + "bbox": [ + 140, + 79, + 249, + 97 + ], + "score": 1.0, + "content": "Fact 25. For any variable", + "type": "text" + }, + { + "bbox": [ + 250, + 83, + 279, + 93 + ], + "score": 0.9, + "content": "X \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 79, + 320, + 97 + ], + "score": 1.0, + "content": "and event", + "type": "text" + }, + { + "bbox": [ + 321, + 83, + 330, + 93 + ], + "score": 0.78, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 79, + 361, + 97 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 361, + 82, + 370, + 93 + ], + "score": 0.82, + "content": "\\bar { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 79, + 447, + 97 + ], + "score": 1.0, + "content": "is its complement)", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 98, + 457, + 114 + ], + "lines": [ + { + "bbox": [ + 188, + 98, + 457, + 114 + ], + "spans": [ + { + "bbox": [ + 188, + 98, + 457, + 114 + ], + "score": 0.88, + "content": "\\mathbb { E } \\left[ X \\right] = \\mathbb { E } \\left[ X | { \\cal { A } } \\right] \\mathbb { P } \\left( { \\cal { A } } \\right) + \\mathbb { E } \\left[ X | { \\bar { \\cal { A } } } \\right] ( 1 - \\mathbb { P } \\left( { \\cal { A } } \\right) ) \\geq \\mathbb { E } \\left[ X | { \\cal { A } } \\right] \\mathbb { P } \\left( { \\cal { A } } \\right)", + "type": "interline_equation", + "image_path": "883661c6a334e5fdb44fe9e8595b6615d7d8d470455e1c5c6e3ffcaf20278de4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 188, + 98, + 457, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 130, + 120, + 230, + 132 + ], + "lines": [ + { + "bbox": [ + 129, + 119, + 231, + 134 + ], + "spans": [ + { + "bbox": [ + 129, + 119, + 231, + 134 + ], + "score": 1.0, + "content": "3. We apply Theorem 6.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 130, + 135, + 230, + 147 + ], + "lines": [ + { + "bbox": [ + 129, + 134, + 231, + 149 + ], + "spans": [ + { + "bbox": [ + 129, + 134, + 231, + 149 + ], + "score": 1.0, + "content": "4. We apply Theorem 9.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 128, + 150, + 413, + 165 + ], + "lines": [ + { + "bbox": [ + 127, + 147, + 411, + 168 + ], + "spans": [ + { + "bbox": [ + 127, + 147, + 290, + 168 + ], + "score": 1.0, + "content": "5. We use assumption 4, which implies", + "type": "text" + }, + { + "bbox": [ + 290, + 150, + 411, + 165 + ], + "score": 0.92, + "content": "\\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\dot { > } 2 N \\log N .", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 173, + 324, + 185 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 323, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 323, + 187 + ], + "score": 1.0, + "content": "For simplicity, in the reminder of the proof we denote", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 190, + 356, + 217 + ], + "lines": [ + { + "bbox": [ + 254, + 190, + 356, + 217 + ], + "spans": [ + { + "bbox": [ + 254, + 190, + 356, + 217 + ], + "score": 0.94, + "content": "R \\left( \\mathbf { X } \\right) \\triangleq \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } .", + "type": "interline_equation", + "image_path": "1d0887721e78acce5607b525a4b567e015b3451e8e8c92eca4dc5c567f1ac83c.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 254, + 190, + 356, + 217 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 407, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 408, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 271, + 236 + ], + "score": 1.0, + "content": "From Markov inequality (Fact 11), since", + "type": "text" + }, + { + "bbox": [ + 271, + 223, + 316, + 235 + ], + "score": 0.93, + "content": "R \\left( { \\bf X } \\right) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 222, + 354, + 236 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 355, + 223, + 403, + 235 + ], + "score": 0.91, + "content": "\\forall \\eta \\left( N \\right) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 222, + 408, + 236 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 239, + 410, + 267 + ], + "lines": [ + { + "bbox": [ + 200, + 239, + 410, + 267 + ], + "spans": [ + { + "bbox": [ + 200, + 239, + 410, + 267 + ], + "score": 0.91, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\geq \\eta \\left( N \\right) \\vert \\mathcal { M } \\right] \\leq \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] } { \\eta \\left( N \\right) }", + "type": "interline_equation", + "image_path": "c6168072c13db79989e917fe2037d2d71918323a17814e36ba59a0ea35f295a7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 200, + 239, + 410, + 267 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 272, + 283 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 272, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 272, + 284 + ], + "score": 1.0, + "content": "On the other hand, from fact 25, we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 287, + 440, + 314 + ], + "lines": [ + { + "bbox": [ + 171, + 287, + 440, + 314 + ], + "spans": [ + { + "bbox": [ + 171, + 287, + 440, + 314 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 1 - \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\vert \\mathcal { M } \\right] \\ge 1 - \\frac { \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] } { \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) } . } \\end{array}", + "type": "interline_equation", + "image_path": "3c64c08f9d4128005c15ac8f4a4b72111bbd0dcc4e4c8905d594204996080c5c.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 171, + 287, + 440, + 314 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 270, + 331 + ], + "lines": [ + { + "bbox": [ + 107, + 318, + 270, + 332 + ], + "spans": [ + { + "bbox": [ + 107, + 318, + 270, + 332 + ], + "score": 1.0, + "content": "Combining Eqs. (11.2)-(11.3) we obtain", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 335, + 413, + 363 + ], + "lines": [ + { + "bbox": [ + 197, + 335, + 413, + 363 + ], + "spans": [ + { + "bbox": [ + 197, + 335, + 413, + 363 + ], + "score": 0.91, + "content": "\\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] } { \\eta \\left( N \\right) } \\geq 1 - \\frac { \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] } { \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) } ,", + "type": "interline_equation", + "image_path": "6935d7f495a50a7bfe5f0c399d742e00a6aeed8fd5a66fee9db324f082b631d5.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 197, + 335, + 413, + 363 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 135, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 135, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 135, + 379 + ], + "score": 1.0, + "content": "and so", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 382, + 460, + 409 + ], + "lines": [ + { + "bbox": [ + 150, + 382, + 460, + 409 + ], + "spans": [ + { + "bbox": [ + 150, + 382, + 460, + 409 + ], + "score": 0.94, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) - \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\middle | \\mathcal { M } \\right] } { \\eta \\left( N \\right) } \\leq \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] .", + "type": "interline_equation", + "image_path": "1df218dc6a30435f3ba3020e939e8a2a2e39a69d85a35bad38ef7191c288080f.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 150, + 382, + 460, + 409 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 151, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 153, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 153, + 426 + ], + "score": 1.0, + "content": "We choose", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 429, + 451, + 450 + ], + "lines": [ + { + "bbox": [ + 158, + 429, + 451, + 450 + ], + "spans": [ + { + "bbox": [ + 158, + 429, + 451, + 450 + ], + "score": 0.88, + "content": "\\eta \\left( { \\cal N } \\right) = N \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( { \\cal M } \\right) \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] \\doteq \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)", + "type": "interline_equation", + "image_path": "c4037cf0ca606d8d7e5abeb3654a18534a91fdee6ef24880989ea7b15aa10d41.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 158, + 429, + 451, + 450 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 135, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 136, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 136, + 466 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 461, + 442, + 486 + ], + "lines": [ + { + "bbox": [ + 168, + 461, + 442, + 486 + ], + "spans": [ + { + "bbox": [ + 168, + 461, + 442, + 486 + ], + "score": 0.9, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) - \\frac { 1 } { N } \\le \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\dot { \\le } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right] .", + "type": "interline_equation", + "image_path": "19c15e4885625997a3766e3b177b1110c78b2058312390172cc9e462e99f0eba.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 168, + 461, + 442, + 486 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 234, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 235, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 235, + 500 + ], + "score": 1.0, + "content": "Then, from Theorem 9 we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 503, + 402, + 532 + ], + "lines": [ + { + "bbox": [ + 209, + 503, + 402, + 532 + ], + "spans": [ + { + "bbox": [ + 209, + 503, + 402, + 532 + ], + "score": 0.94, + "content": "1 - \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } .", + "type": "interline_equation", + "image_path": "b691e9b99b15197ab7d70e68bfc6d003ef9ae79903069ffb8279980d1c9005a6.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 503, + 402, + 517.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 209, + 517.5, + 402, + 532.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 339, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 339, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 339, + 550 + ], + "score": 1.0, + "content": "so we obtain the first (left) inequality in the Theorem (10)", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 561, + 484, + 591 + ], + "lines": [ + { + "bbox": [ + 126, + 561, + 484, + 591 + ], + "spans": [ + { + "bbox": [ + 126, + 561, + 484, + 591 + ], + "score": 0.92, + "content": "\\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } \\dot { \\geq 1 } - \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right] .", + "type": "interline_equation", + "image_path": "05bd8a33be4b4cd8eb1097bc374d96fd669c2339592f3e9b49bcee9b2fdd168f.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 126, + 561, + 484, + 571.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 126, + 571.0, + 484, + 581.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 126, + 581.0, + 484, + 591.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 506, + 619 + ], + "lines": [ + { + "bbox": [ + 104, + 592, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 281, + 611 + ], + "score": 1.0, + "content": "Lastly, we note that assumption 4 implies", + "type": "text" + }, + { + "bbox": [ + 281, + 594, + 396, + 608 + ], + "score": 0.93, + "content": "\\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\dot { > } N \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 592, + 506, + 611 + ], + "score": 1.0, + "content": ", which proves the second", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 607, + 239, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 239, + 620 + ], + "score": 1.0, + "content": "(right) inequality of the theorem.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 624, + 504, + 635 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 142, + 81, + 447, + 95 + ], + "lines": [ + { + "bbox": [ + 140, + 79, + 447, + 97 + ], + "spans": [ + { + "bbox": [ + 140, + 79, + 249, + 97 + ], + "score": 1.0, + "content": "Fact 25. For any variable", + "type": "text" + }, + { + "bbox": [ + 250, + 83, + 279, + 93 + ], + "score": 0.9, + "content": "X \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 79, + 320, + 97 + ], + "score": 1.0, + "content": "and event", + "type": "text" + }, + { + "bbox": [ + 321, + 83, + 330, + 93 + ], + "score": 0.78, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 79, + 361, + 97 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 361, + 82, + 370, + 93 + ], + "score": 0.82, + "content": "\\bar { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 79, + 447, + 97 + ], + "score": 1.0, + "content": "is its complement)", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 140, + 79, + 447, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 98, + 457, + 114 + ], + "lines": [ + { + "bbox": [ + 188, + 98, + 457, + 114 + ], + "spans": [ + { + "bbox": [ + 188, + 98, + 457, + 114 + ], + "score": 0.88, + "content": "\\mathbb { E } \\left[ X \\right] = \\mathbb { E } \\left[ X | { \\cal { A } } \\right] \\mathbb { P } \\left( { \\cal { A } } \\right) + \\mathbb { E } \\left[ X | { \\bar { \\cal { A } } } \\right] ( 1 - \\mathbb { P } \\left( { \\cal { A } } \\right) ) \\geq \\mathbb { E } \\left[ X | { \\cal { A } } \\right] \\mathbb { P } \\left( { \\cal { A } } \\right)", + "type": "interline_equation", + "image_path": "883661c6a334e5fdb44fe9e8595b6615d7d8d470455e1c5c6e3ffcaf20278de4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 188, + 98, + 457, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 130, + 120, + 230, + 132 + ], + "lines": [ + { + "bbox": [ + 129, + 119, + 231, + 134 + ], + "spans": [ + { + "bbox": [ + 129, + 119, + 231, + 134 + ], + "score": 1.0, + "content": "3. We apply Theorem 6.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 129, + 119, + 231, + 134 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 135, + 230, + 147 + ], + "lines": [ + { + "bbox": [ + 129, + 134, + 231, + 149 + ], + "spans": [ + { + "bbox": [ + 129, + 134, + 231, + 149 + ], + "score": 1.0, + "content": "4. We apply Theorem 9.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 129, + 134, + 231, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 128, + 150, + 413, + 165 + ], + "lines": [ + { + "bbox": [ + 127, + 147, + 411, + 168 + ], + "spans": [ + { + "bbox": [ + 127, + 147, + 290, + 168 + ], + "score": 1.0, + "content": "5. We use assumption 4, which implies", + "type": "text" + }, + { + "bbox": [ + 290, + 150, + 411, + 165 + ], + "score": 0.92, + "content": "\\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\dot { > } 2 N \\log N .", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 127, + 147, + 411, + 168 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 173, + 324, + 185 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 323, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 323, + 187 + ], + "score": 1.0, + "content": "For simplicity, in the reminder of the proof we denote", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 172, + 323, + 187 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 190, + 356, + 217 + ], + "lines": [ + { + "bbox": [ + 254, + 190, + 356, + 217 + ], + "spans": [ + { + "bbox": [ + 254, + 190, + 356, + 217 + ], + "score": 0.94, + "content": "R \\left( \\mathbf { X } \\right) \\triangleq \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( \\mathbf { X } , \\mathbf { y } \\right) \\right) } .", + "type": "interline_equation", + "image_path": "1d0887721e78acce5607b525a4b567e015b3451e8e8c92eca4dc5c567f1ac83c.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 254, + 190, + 356, + 217 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 407, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 408, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 271, + 236 + ], + "score": 1.0, + "content": "From Markov inequality (Fact 11), since", + "type": "text" + }, + { + "bbox": [ + 271, + 223, + 316, + 235 + ], + "score": 0.93, + "content": "R \\left( { \\bf X } \\right) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 222, + 354, + 236 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 355, + 223, + 403, + 235 + ], + "score": 0.91, + "content": "\\forall \\eta \\left( N \\right) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 222, + 408, + 236 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 222, + 408, + 236 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 239, + 410, + 267 + ], + "lines": [ + { + "bbox": [ + 200, + 239, + 410, + 267 + ], + "spans": [ + { + "bbox": [ + 200, + 239, + 410, + 267 + ], + "score": 0.91, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\geq \\eta \\left( N \\right) \\vert \\mathcal { M } \\right] \\leq \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] } { \\eta \\left( N \\right) }", + "type": "interline_equation", + "image_path": "c6168072c13db79989e917fe2037d2d71918323a17814e36ba59a0ea35f295a7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 200, + 239, + 410, + 267 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 272, + 283 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 272, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 272, + 284 + ], + "score": 1.0, + "content": "On the other hand, from fact 25, we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 270, + 272, + 284 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 287, + 440, + 314 + ], + "lines": [ + { + "bbox": [ + 171, + 287, + 440, + 314 + ], + "spans": [ + { + "bbox": [ + 171, + 287, + 440, + 314 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 1 - \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\vert \\mathcal { M } \\right] \\ge 1 - \\frac { \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] } { \\operatorname { \\mathbb { P } } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) } . } \\end{array}", + "type": "interline_equation", + "image_path": "3c64c08f9d4128005c15ac8f4a4b72111bbd0dcc4e4c8905d594204996080c5c.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 171, + 287, + 440, + 314 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 270, + 331 + ], + "lines": [ + { + "bbox": [ + 107, + 318, + 270, + 332 + ], + "spans": [ + { + "bbox": [ + 107, + 318, + 270, + 332 + ], + "score": 1.0, + "content": "Combining Eqs. (11.2)-(11.3) we obtain", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 107, + 318, + 270, + 332 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 335, + 413, + 363 + ], + "lines": [ + { + "bbox": [ + 197, + 335, + 413, + 363 + ], + "spans": [ + { + "bbox": [ + 197, + 335, + 413, + 363 + ], + "score": 0.91, + "content": "\\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] } { \\eta \\left( N \\right) } \\geq 1 - \\frac { \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] } { \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) } ,", + "type": "interline_equation", + "image_path": "6935d7f495a50a7bfe5f0c399d742e00a6aeed8fd5a66fee9db324f082b631d5.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 197, + 335, + 413, + 363 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 135, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 135, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 135, + 379 + ], + "score": 1.0, + "content": "and so", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 367, + 135, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 382, + 460, + 409 + ], + "lines": [ + { + "bbox": [ + 150, + 382, + 460, + 409 + ], + "spans": [ + { + "bbox": [ + 150, + 382, + 460, + 409 + ], + "score": 0.94, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) - \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) \\frac { \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\middle | \\mathcal { M } \\right] } { \\eta \\left( N \\right) } \\leq \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) < \\eta \\left( N \\right) \\right] .", + "type": "interline_equation", + "image_path": "1df218dc6a30435f3ba3020e939e8a2a2e39a69d85a35bad38ef7191c288080f.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 150, + 382, + 460, + 409 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 151, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 153, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 153, + 426 + ], + "score": 1.0, + "content": "We choose", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 412, + 153, + 426 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 429, + 451, + 450 + ], + "lines": [ + { + "bbox": [ + 158, + 429, + 451, + 450 + ], + "spans": [ + { + "bbox": [ + 158, + 429, + 451, + 450 + ], + "score": 0.88, + "content": "\\eta \\left( { \\cal N } \\right) = N \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( { \\cal M } \\right) \\mathbb { E } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\vert \\mathcal { M } \\right] \\doteq \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right)", + "type": "interline_equation", + "image_path": "c4037cf0ca606d8d7e5abeb3654a18534a91fdee6ef24880989ea7b15aa10d41.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 158, + 429, + 451, + 450 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 135, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 136, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 136, + 466 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 453, + 136, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 461, + 442, + 486 + ], + "lines": [ + { + "bbox": [ + 168, + 461, + 442, + 486 + ], + "spans": [ + { + "bbox": [ + 168, + 461, + 442, + 486 + ], + "score": 0.9, + "content": "\\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) - \\frac { 1 } { N } \\le \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left[ R \\left( \\mathbf { X } \\right) \\dot { \\le } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right] .", + "type": "interline_equation", + "image_path": "19c15e4885625997a3766e3b177b1110c78b2058312390172cc9e462e99f0eba.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 168, + 461, + 442, + 486 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 234, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 235, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 235, + 500 + ], + "score": 1.0, + "content": "Then, from Theorem 9 we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 486, + 235, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 503, + 402, + 532 + ], + "lines": [ + { + "bbox": [ + 209, + 503, + 402, + 532 + ], + "spans": [ + { + "bbox": [ + 209, + 503, + 402, + 532 + ], + "score": 0.94, + "content": "1 - \\mathbb { P } _ { \\mathbf { X } \\sim \\mathcal { N } } \\left( \\mathcal { M } \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } .", + "type": "interline_equation", + "image_path": "b691e9b99b15197ab7d70e68bfc6d003ef9ae79903069ffb8279980d1c9005a6.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 503, + 402, + 517.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 209, + 517.5, + 402, + 532.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 339, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 339, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 339, + 550 + ], + "score": 1.0, + "content": "so we obtain the first (left) inequality in the Theorem (10)", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 536, + 339, + 550 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 561, + 484, + 591 + ], + "lines": [ + { + "bbox": [ + 126, + 561, + 484, + 591 + ], + "spans": [ + { + "bbox": [ + 126, + 561, + 484, + 591 + ], + "score": 0.92, + "content": "\\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N } \\dot { \\geq 1 } - \\mathbb { P } _ { { \\mathbf { X } } \\sim { \\mathcal { N } } } \\left[ \\frac { \\mathcal { V } \\left( \\mathcal { L } _ { \\epsilon } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } { \\mathcal { V } \\left( \\mathcal { G } \\left( { \\mathbf { X } } , { \\mathbf { y } } \\right) \\right) } \\dot { \\leq } \\exp \\left( - \\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\right) \\right] .", + "type": "interline_equation", + "image_path": "05bd8a33be4b4cd8eb1097bc374d96fd669c2339592f3e9b49bcee9b2fdd168f.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 126, + 561, + 484, + 571.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 126, + 571.0, + 484, + 581.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 126, + 581.0, + 484, + 591.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 506, + 619 + ], + "lines": [ + { + "bbox": [ + 104, + 592, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 281, + 611 + ], + "score": 1.0, + "content": "Lastly, we note that assumption 4 implies", + "type": "text" + }, + { + "bbox": [ + 281, + 594, + 396, + 608 + ], + "score": 0.93, + "content": "\\gamma _ { \\epsilon } N ^ { 3 / 4 } \\left[ d _ { 1 } d _ { 0 } \\right] ^ { 1 / 4 } \\dot { > } N \\log N", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 592, + 506, + 611 + ], + "score": 1.0, + "content": ", which proves the second", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 607, + 239, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 239, + 620 + ], + "score": 1.0, + "content": "(right) inequality of the theorem.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 592, + 506, + 620 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 151, + 95 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 153, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 153, + 96 + ], + "score": 1.0, + "content": "Part II", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 102, + 336, + 125 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 337, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 337, + 126 + ], + "score": 1.0, + "content": "Proofs of technical results", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 329, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 330, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 330, + 155 + ], + "score": 1.0, + "content": "In this part we prove the technical results used in part I.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 108, + 169, + 506, + 209 + ], + "lines": [ + { + "bbox": [ + 107, + 169, + 443, + 183 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 126, + 182 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 130, + 169, + 443, + 183 + ], + "score": 1.0, + "content": "UPPER BOUNDING THE ANGULAR VOLUME OF SUB-OPTIMAL", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 131, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 131, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "DIFFERENTIABLE LOCAL MINIMA: PROOFS OF LEMMAS USED IN SECTION", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 131, + 196, + 141, + 210 + ], + "spans": [ + { + "bbox": [ + 131, + 196, + 141, + 210 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 221, + 230, + 233 + ], + "lines": [ + { + "bbox": [ + 106, + 220, + 231, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 231, + 235 + ], + "score": 1.0, + "content": "12.1 PROOF OF LEMMA 14", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 105, + 241, + 477, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 476, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 476, + 255 + ], + "score": 1.0, + "content": "In this section we will prove Lemma 14 in subsection 12.3.3. Recall the following definition", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 256, + 185, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 185, + 269 + ], + "score": 1.0, + "content": "Definition 26. Let", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 273, + 387, + 287 + ], + "lines": [ + { + "bbox": [ + 223, + 273, + 387, + 287 + ], + "spans": [ + { + "bbox": [ + 223, + 273, + 387, + 287 + ], + "score": 0.9, + "content": "{ \\bf A } = [ { \\pmb a } _ { 1 } , \\ldots , { \\pmb a } _ { N } ] \\ ; \\ { \\bf X } = [ { \\pmb x } _ { 1 } , \\ldots , { \\pmb x } _ { N } ] ,", + "type": "interline_equation", + "image_path": "331ff2f543779ae800d15425c08fef97cf3f06a106c32ad743dcd3f82ed2c481.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 223, + 273, + 387, + 287 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 102, + 291, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 133, + 305 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 291, + 184, + 303 + ], + "score": 0.92, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 289, + 202, + 305 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 291, + 253, + 303 + ], + "score": 0.92, + "content": "\\mathbf { A } \\in \\mathbb { R } ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 289, + 506, + 305 + ], + "score": 1.0, + "content": ". The Khatari-Rao product between the two matrices is defined", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 304, + 118, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 118, + 316 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 317, + 399, + 385 + ], + "lines": [ + { + "bbox": [ + 212, + 317, + 399, + 385 + ], + "spans": [ + { + "bbox": [ + 212, + 317, + 399, + 385 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\mathbf { A } \\circ \\mathbf { X } } & { \\triangleq } & { \\left[ \\pmb { a } _ { 1 } \\otimes \\mathbf { x } _ { 1 } , \\pmb { a } _ { 2 } \\otimes \\mathbf { x } _ { 2 } , . . . \\pmb { a } _ { N } \\otimes \\mathbf { x } _ { N } \\right] } \\\\ & { = } & { \\left( \\begin{array} { c c c } { a _ { 1 1 } \\mathbf { x } _ { 1 } } & { a _ { 1 2 } \\mathbf { x } _ { 2 } } & { . . . } \\\\ { a _ { 2 1 } \\mathbf { x } _ { 1 } } & { a _ { 2 2 } \\mathbf { x } _ { 2 } } & { \\ddots } \\\\ { \\vdots } & { \\ddots } & { \\ddots } \\end{array} \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "6b03bf222e0a853578cc407ae3fdaef4f1c40d92cfe079d8042e9c75e965e61c.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 317, + 399, + 334.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 212, + 334.0, + 399, + 351.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 212, + 351.0, + 399, + 368.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 212, + 368.0, + 399, + 385.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 389, + 506, + 414 + ], + "lines": [ + { + "bbox": [ + 104, + 385, + 508, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 255, + 406 + ], + "score": 1.0, + "content": "Lemma 27. (Lemma 14 restated) Let", + "type": "text" + }, + { + "bbox": [ + 255, + 390, + 305, + 402 + ], + "score": 0.89, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 385, + 309, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 309, + 389, + 376, + 403 + ], + "score": 0.86, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 385, + 381, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 381, + 390, + 417, + 403 + ], + "score": 0.83, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 385, + 435, + 406 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 435, + 392, + 478, + 403 + ], + "score": 0.91, + "content": "d _ { 0 } d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 385, + 508, + 406 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 402, + 319, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 271, + 415 + ], + "score": 1.0, + "content": "simultaneously for every possible A and", + "type": "text" + }, + { + "bbox": [ + 271, + 403, + 279, + 412 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 402, + 319, + 415 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 418, + 348, + 432 + ], + "lines": [ + { + "bbox": [ + 261, + 418, + 348, + 432 + ], + "spans": [ + { + "bbox": [ + 261, + 418, + 348, + 432 + ], + "score": 0.9, + "content": "\\left. S \\right. \\leq \\mathrm { r a n k } \\left( { \\bf A } _ { S } \\right) d _ { 0 } ,", + "type": "interline_equation", + "image_path": "1e38cae94396787c732a3790bc0b443713c44fea948bca616575253ccb971463.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 261, + 418, + 348, + 432 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 420, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 417, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 193, + 450 + ], + "score": 1.0, + "content": "we have that, X-a.e.,", + "type": "text" + }, + { + "bbox": [ + 193, + 435, + 233, + 448 + ], + "score": 0.75, + "content": "\\mathbf { \\Delta } \\nexists \\mathbf { v } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 434, + 273, + 450 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 273, + 436, + 335, + 448 + ], + "score": 0.9, + "content": "v _ { n } \\neq 0 \\forall n \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 434, + 355, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 355, + 436, + 417, + 448 + ], + "score": 0.74, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 103, + 460, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 103, + 460, + 223, + 477 + ], + "score": 1.0, + "content": "Proof. We examine specific", + "type": "text" + }, + { + "bbox": [ + 224, + 460, + 293, + 475 + ], + "score": 0.94, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 460, + 312, + 477 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 462, + 349, + 475 + ], + "score": 0.92, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 460, + 410, + 477 + ], + "score": 1.0, + "content": ", and such that", + "type": "text" + }, + { + "bbox": [ + 411, + 462, + 459, + 475 + ], + "score": 0.92, + "content": "| S | \\le d _ { S } d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 460, + 506, + 477 + ], + "score": 1.0, + "content": ", where we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 473, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 139, + 489 + ], + "score": 1.0, + "content": "defined", + "type": "text" + }, + { + "bbox": [ + 140, + 474, + 210, + 487 + ], + "score": 0.78, + "content": "d _ { S } \\triangleq \\operatorname { r a n k } \\left( \\mathbf { A } _ { S } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 473, + 282, + 489 + ], + "score": 1.0, + "content": ". We assume that", + "type": "text" + }, + { + "bbox": [ + 282, + 475, + 315, + 486 + ], + "score": 0.91, + "content": "d _ { S } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 473, + 506, + 489 + ], + "score": 1.0, + "content": ", since otherwise the proof is trivial. Also, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 226, + 499 + ], + "score": 1.0, + "content": "assume by contradiction that", + "type": "text" + }, + { + "bbox": [ + 226, + 486, + 265, + 496 + ], + "score": 0.91, + "content": "\\exists \\mathbf { v } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 485, + 306, + 499 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 306, + 486, + 364, + 497 + ], + "score": 0.91, + "content": "v _ { i } \\neq 0 \\forall i \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 485, + 383, + 499 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 383, + 486, + 445, + 498 + ], + "score": 0.9, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 485, + 506, + 499 + ], + "score": 1.0, + "content": ". Without loss", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 209, + 510 + ], + "score": 1.0, + "content": "of generality, assume that", + "type": "text" + }, + { + "bbox": [ + 210, + 497, + 284, + 509 + ], + "score": 0.93, + "content": "S = \\{ 1 , 2 , . . . , | S | \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 497, + 320, + 510 + ], + "score": 1.0, + "content": "and that", + "type": "text" + }, + { + "bbox": [ + 320, + 499, + 380, + 509 + ], + "score": 0.91, + "content": "\\mathbf { } a _ { 1 } , \\mathbf { } a _ { 2 } , . . . , \\mathbf { } a _ { d _ { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "are linearly independent. Then", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 513, + 373, + 549 + ], + "lines": [ + { + "bbox": [ + 240, + 513, + 373, + 549 + ], + "spans": [ + { + "bbox": [ + 240, + 513, + 373, + 549 + ], + "score": 0.95, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = \\sum _ { n = 1 } ^ { | S | } v _ { n } a _ { k , n } \\mathbf { x } _ { n } = 0", + "type": "interline_equation", + "image_path": "43fc4ea2aafe35eaa4064faf92f08d348fce5085b2a8785c524cefd329e42d22.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 513, + 373, + 531.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 240, + 531.0, + 373, + 549.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 145, + 566 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 145, + 553, + 193, + 564 + ], + "score": 0.95, + "content": "1 \\leq k \\leq d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 551, + 288, + 566 + ], + "score": 1.0, + "content": ". From the definition of", + "type": "text" + }, + { + "bbox": [ + 289, + 553, + 297, + 563 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 551, + 355, + 566 + ], + "score": 1.0, + "content": "we must have", + "type": "text" + }, + { + "bbox": [ + 355, + 553, + 385, + 564 + ], + "score": 0.9, + "content": "v _ { n } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 551, + 425, + 566 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 425, + 553, + 475, + 565 + ], + "score": 0.92, + "content": "1 \\leq n \\leq | S |", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 551, + 506, + 566 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 107, + 566, + 166, + 576 + ], + "score": 0.92, + "content": "\\mathbf { } a _ { 1 } , \\mathbf { } a _ { 2 } , . . . , \\mathbf { } a _ { d _ { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 563, + 315, + 577 + ], + "score": 1.0, + "content": "are linearly independent, the rows of", + "type": "text" + }, + { + "bbox": [ + 315, + 564, + 412, + 576 + ], + "score": 0.86, + "content": "\\mathbf { A } _ { d _ { S } } = [ \\pmb { a } _ { 1 } , \\pmb { a } _ { 2 } , . . . , \\pmb { a } _ { d _ { S } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 563, + 440, + 577 + ], + "score": 1.0, + "content": "span a", + "type": "text" + }, + { + "bbox": [ + 441, + 565, + 452, + 575 + ], + "score": 0.85, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 563, + 505, + 577 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 573, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 289, + 589 + ], + "score": 1.0, + "content": "space. Therefore, it is possible to find a matrix", + "type": "text" + }, + { + "bbox": [ + 289, + 576, + 299, + 585 + ], + "score": 0.69, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 573, + 338, + 589 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 338, + 576, + 475, + 587 + ], + "score": 0.84, + "content": "\\mathbf { R } \\mathbf { A } _ { d _ { S } } = [ \\mathbf { I } _ { d _ { S } \\times d _ { S } } , 0 _ { d _ { S } \\times ( d _ { 1 } - d _ { S } ) } ] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 573, + 506, + 589 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 126, + 599 + ], + "score": 0.91, + "content": "0 _ { i \\times j }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 586, + 241, + 599 + ], + "score": 1.0, + "content": "is the all zeros matrix with", + "type": "text" + }, + { + "bbox": [ + 241, + 587, + 246, + 596 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 586, + 303, + 599 + ], + "score": 1.0, + "content": "columns and", + "type": "text" + }, + { + "bbox": [ + 303, + 587, + 309, + 598 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 586, + 398, + 599 + ], + "score": 1.0, + "content": "rows. Consider now", + "type": "text" + }, + { + "bbox": [ + 398, + 587, + 438, + 597 + ], + "score": 0.84, + "content": "\\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 586, + 505, + 599 + ], + "score": 1.0, + "content": ", i.e., the matrix", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 225, + 609 + ], + "score": 1.0, + "content": "composed of the columns of", + "type": "text" + }, + { + "bbox": [ + 225, + 597, + 253, + 607 + ], + "score": 0.88, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 596, + 266, + 609 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 266, + 597, + 273, + 607 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 596, + 320, + 609 + ], + "score": 1.0, + "content": ". Applying", + "type": "text" + }, + { + "bbox": [ + 320, + 597, + 380, + 609 + ], + "score": 0.91, + "content": "\\mathbf { R } ^ { \\prime } = \\mathbf { R } \\otimes \\mathbf { I } _ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 596, + 392, + 609 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 393, + 597, + 432, + 608 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 596, + 505, + 609 + ], + "score": 1.0, + "content": ", turns (12.2) into", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 608, + 328, + 621 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 128, + 619 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 608, + 233, + 621 + ], + "score": 1.0, + "content": "equations in the variables", + "type": "text" + }, + { + "bbox": [ + 234, + 610, + 276, + 621 + ], + "score": 0.91, + "content": "v _ { 1 } , . . . , v _ { | S | }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 608, + 328, + 621 + ], + "score": 1.0, + "content": ", of the form", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 625, + 368, + 662 + ], + "lines": [ + { + "bbox": [ + 243, + 625, + 368, + 662 + ], + "spans": [ + { + "bbox": [ + 243, + 625, + 368, + 662 + ], + "score": 0.94, + "content": "v _ { k } \\mathbf { x } _ { k } + \\sum _ { n = d _ { S } + 1 } ^ { | S | } v _ { n } { \\tilde { a } } _ { k , n } \\mathbf { x } _ { n } = 0", + "type": "interline_equation", + "image_path": "4a246a95c624c7b3cb13895d8563162746724ae3124e0d90a5fb0058d85abc2d.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 243, + 625, + 368, + 643.5 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 243, + 643.5, + 368, + 662.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 146, + 678 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 146, + 666, + 198, + 677 + ], + "score": 0.91, + "content": "1 \\leq k \\leq d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 665, + 356, + 678 + ], + "score": 1.0, + "content": ". 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This will immediately imply", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 157, + 700 + ], + "score": 0.92, + "content": "| S | > d _ { S } \\dot { d } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 687, + 310, + 700 + ], + "score": 1.0, + "content": ", or else eq. 12.2 cannot be true for", + "type": "text" + }, + { + "bbox": [ + 311, + 688, + 339, + 699 + ], + "score": 0.89, + "content": "\\mathbf { v } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ". which will contradict our assumption,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "as required. The induction can be viewed as carrying out Gaussian elimination of the system of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 494, + 722 + ], + "score": 1.0, + "content": "equations described by (12.3), where in each elimination step we characterize the set of matrices", + "type": "text" + }, + { + "bbox": [ + 495, + 710, + 504, + 720 + ], + "score": 0.49, + "content": "\\mathbf { X }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 721, + 368, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 368, + 732 + ], + "score": 1.0, + "content": "that for which that step is impossible, and show it has measure 0.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 151, + 95 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 153, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 153, + 96 + ], + "score": 1.0, + "content": "Part II", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 102, + 336, + 125 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 337, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 337, + 126 + ], + "score": 1.0, + "content": "Proofs of technical results", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 329, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 330, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 330, + 155 + ], + "score": 1.0, + "content": "In this part we prove the technical results used in part I.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 141, + 330, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 169, + 506, + 209 + ], + "lines": [ + { + "bbox": [ + 107, + 169, + 443, + 183 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 126, + 182 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 130, + 169, + 443, + 183 + ], + "score": 1.0, + "content": "UPPER BOUNDING THE ANGULAR VOLUME OF SUB-OPTIMAL", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 131, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 131, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "DIFFERENTIABLE LOCAL MINIMA: PROOFS OF LEMMAS USED IN SECTION", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 131, + 196, + 141, + 210 + ], + "spans": [ + { + "bbox": [ + 131, + 196, + 141, + 210 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 107, + 169, + 505, + 210 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 221, + 230, + 233 + ], + "lines": [ + { + "bbox": [ + 106, + 220, + 231, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 231, + 235 + ], + "score": 1.0, + "content": "12.1 PROOF OF LEMMA 14", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 105, + 241, + 477, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 476, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 476, + 255 + ], + "score": 1.0, + "content": "In this section we will prove Lemma 14 in subsection 12.3.3. 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(Lemma 14 restated) Let", + "type": "text" + }, + { + "bbox": [ + 255, + 390, + 305, + 402 + ], + "score": 0.89, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 385, + 309, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 309, + 389, + 376, + 403 + ], + "score": 0.86, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 385, + 381, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 381, + 390, + 417, + 403 + ], + "score": 0.83, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 385, + 435, + 406 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 435, + 392, + 478, + 403 + ], + "score": 0.91, + "content": "d _ { 0 } d _ { 1 } \\geq N", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 385, + 508, + 406 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 402, + 319, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 271, + 415 + ], + "score": 1.0, + "content": "simultaneously for every possible A and", + "type": "text" + }, + { + "bbox": [ + 271, + 403, + 279, + 412 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 402, + 319, + 415 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 385, + 508, + 415 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 418, + 348, + 432 + ], + "lines": [ + { + "bbox": [ + 261, + 418, + 348, + 432 + ], + "spans": [ + { + "bbox": [ + 261, + 418, + 348, + 432 + ], + "score": 0.9, + "content": "\\left. S \\right. \\leq \\mathrm { r a n k } \\left( { \\bf A } _ { S } \\right) d _ { 0 } ,", + "type": "interline_equation", + "image_path": "1e38cae94396787c732a3790bc0b443713c44fea948bca616575253ccb971463.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 261, + 418, + 348, + 432 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 420, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 417, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 193, + 450 + ], + "score": 1.0, + "content": "we have that, X-a.e.,", + "type": "text" + }, + { + "bbox": [ + 193, + 435, + 233, + 448 + ], + "score": 0.75, + "content": "\\mathbf { \\Delta } \\nexists \\mathbf { v } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 434, + 273, + 450 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 273, + 436, + 335, + 448 + ], + "score": 0.9, + "content": "v _ { n } \\neq 0 \\forall n \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 434, + 355, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 355, + 436, + 417, + 448 + ], + "score": 0.74, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 434, + 417, + 450 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 103, + 460, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 103, + 460, + 223, + 477 + ], + "score": 1.0, + "content": "Proof. We examine specific", + "type": "text" + }, + { + "bbox": [ + 224, + 460, + 293, + 475 + ], + "score": 0.94, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 460, + 312, + 477 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 462, + 349, + 475 + ], + "score": 0.92, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 460, + 410, + 477 + ], + "score": 1.0, + "content": ", and such that", + "type": "text" + }, + { + "bbox": [ + 411, + 462, + 459, + 475 + ], + "score": 0.92, + "content": "| S | \\le d _ { S } d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 460, + 506, + 477 + ], + "score": 1.0, + "content": ", where we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 473, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 139, + 489 + ], + "score": 1.0, + "content": "defined", + "type": "text" + }, + { + "bbox": [ + 140, + 474, + 210, + 487 + ], + "score": 0.78, + "content": "d _ { S } \\triangleq \\operatorname { r a n k } \\left( \\mathbf { A } _ { S } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 473, + 282, + 489 + ], + "score": 1.0, + "content": ". 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Also, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 226, + 499 + ], + "score": 1.0, + "content": "assume by contradiction that", + "type": "text" + }, + { + "bbox": [ + 226, + 486, + 265, + 496 + ], + "score": 0.91, + "content": "\\exists \\mathbf { v } \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 485, + 306, + 499 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 306, + 486, + 364, + 497 + ], + "score": 0.91, + "content": "v _ { i } \\neq 0 \\forall i \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 485, + 383, + 499 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 383, + 486, + 445, + 498 + ], + "score": 0.9, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) \\mathbf { v } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 485, + 506, + 499 + ], + "score": 1.0, + "content": ". Without loss", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 209, + 510 + ], + "score": 1.0, + "content": "of generality, assume that", + "type": "text" + }, + { + "bbox": [ + 210, + 497, + 284, + 509 + ], + "score": 0.93, + "content": "S = \\{ 1 , 2 , . . . , | S | \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 497, + 320, + 510 + ], + "score": 1.0, + "content": "and that", + "type": "text" + }, + { + "bbox": [ + 320, + 499, + 380, + 509 + ], + "score": 0.91, + "content": "\\mathbf { } a _ { 1 } , \\mathbf { } a _ { 2 } , . . . , \\mathbf { } a _ { d _ { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "are linearly independent. 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From the definition of", + "type": "text" + }, + { + "bbox": [ + 289, + 553, + 297, + 563 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 551, + 355, + 566 + ], + "score": 1.0, + "content": "we must have", + "type": "text" + }, + { + "bbox": [ + 355, + 553, + 385, + 564 + ], + "score": 0.9, + "content": "v _ { n } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 551, + 425, + 566 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 425, + 553, + 475, + 565 + ], + "score": 0.92, + "content": "1 \\leq n \\leq | S |", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 551, + 506, + 566 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 107, + 566, + 166, + 576 + ], + "score": 0.92, + "content": "\\mathbf { } a _ { 1 } , \\mathbf { } a _ { 2 } , . . . , \\mathbf { } a _ { d _ { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 563, + 315, + 577 + ], + "score": 1.0, + "content": "are linearly independent, the rows of", + "type": "text" + }, + { + "bbox": [ + 315, + 564, + 412, + 576 + ], + "score": 0.86, + "content": "\\mathbf { A } _ { d _ { S } } = [ \\pmb { a } _ { 1 } , \\pmb { a } _ { 2 } , . . . , \\pmb { a } _ { d _ { S } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 563, + 440, + 577 + ], + "score": 1.0, + "content": "span a", + "type": "text" + }, + { + "bbox": [ + 441, + 565, + 452, + 575 + ], + "score": 0.85, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 563, + 505, + 577 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 573, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 289, + 589 + ], + "score": 1.0, + "content": "space. Therefore, it is possible to find a matrix", + "type": "text" + }, + { + "bbox": [ + 289, + 576, + 299, + 585 + ], + "score": 0.69, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 573, + 338, + 589 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 338, + 576, + 475, + 587 + ], + "score": 0.84, + "content": "\\mathbf { R } \\mathbf { A } _ { d _ { S } } = [ \\mathbf { I } _ { d _ { S } \\times d _ { S } } , 0 _ { d _ { S } \\times ( d _ { 1 } - d _ { S } ) } ] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 573, + 506, + 589 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 126, + 599 + ], + "score": 0.91, + "content": "0 _ { i \\times j }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 586, + 241, + 599 + ], + "score": 1.0, + "content": "is the all zeros matrix with", + "type": "text" + }, + { + "bbox": [ + 241, + 587, + 246, + 596 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 586, + 303, + 599 + ], + "score": 1.0, + "content": "columns and", + "type": "text" + }, + { + "bbox": [ + 303, + 587, + 309, + 598 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 586, + 398, + 599 + ], + "score": 1.0, + "content": "rows. Consider now", + "type": "text" + }, + { + "bbox": [ + 398, + 587, + 438, + 597 + ], + "score": 0.84, + "content": "\\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 586, + 505, + 599 + ], + "score": 1.0, + "content": ", i.e., the matrix", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 225, + 609 + ], + "score": 1.0, + "content": "composed of the columns of", + "type": "text" + }, + { + "bbox": [ + 225, + 597, + 253, + 607 + ], + "score": 0.88, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 596, + 266, + 609 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 266, + 597, + 273, + 607 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 596, + 320, + 609 + ], + "score": 1.0, + "content": ". Applying", + "type": "text" + }, + { + "bbox": [ + 320, + 597, + 380, + 609 + ], + "score": 0.91, + "content": "\\mathbf { R } ^ { \\prime } = \\mathbf { R } \\otimes \\mathbf { I } _ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 596, + 392, + 609 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 393, + 597, + 432, + 608 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 596, + 505, + 609 + ], + "score": 1.0, + "content": ", turns (12.2) into", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 608, + 328, + 621 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 128, + 619 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 608, + 233, + 621 + ], + "score": 1.0, + "content": "equations in the variables", + "type": "text" + }, + { + "bbox": [ + 234, + 610, + 276, + 621 + ], + "score": 0.91, + "content": "v _ { 1 } , . . . , v _ { | S | }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 608, + 328, + 621 + ], + "score": 1.0, + "content": ", of the form", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 551, + 506, + 621 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 625, + 368, + 662 + ], + "lines": [ + { + "bbox": [ + 243, + 625, + 368, + 662 + ], + "spans": [ + { + "bbox": [ + 243, + 625, + 368, + 662 + ], + "score": 0.94, + "content": "v _ { k } \\mathbf { x } _ { k } + \\sum _ { n = d _ { S } + 1 } ^ { | S | } v _ { n } { \\tilde { a } } _ { k , n } \\mathbf { x } _ { n } = 0", + "type": "interline_equation", + "image_path": "4a246a95c624c7b3cb13895d8563162746724ae3124e0d90a5fb0058d85abc2d.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 243, + 625, + 368, + 643.5 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 243, + 643.5, + 368, + 662.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 146, + 678 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 146, + 666, + 198, + 677 + ], + "score": 0.91, + "content": "1 \\leq k \\leq d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 665, + 356, + 678 + ], + "score": 1.0, + "content": ". 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This will immediately imply", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 157, + 700 + ], + "score": 0.92, + "content": "| S | > d _ { S } \\dot { d } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 687, + 310, + 700 + ], + "score": 1.0, + "content": ", or else eq. 12.2 cannot be true for", + "type": "text" + }, + { + "bbox": [ + 311, + 688, + 339, + 699 + ], + "score": 0.89, + "content": "\\mathbf { v } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ". which will contradict our assumption,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "as required. 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Let", + "type": "text" + }, + { + "bbox": [ + 225, + 171, + 234, + 180 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 169, + 493, + 184 + ], + "score": 1.0, + "content": "be the set of the indices (equivalently, columns) of these variables", + "type": "text" + }, + { + "bbox": [ + 493, + 173, + 504, + 181 + ], + "score": 0.8, + "content": "r _ { d }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 504, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 322, + 195 + ], + "score": 1.0, + "content": "variables. From (12.3) it is clear none of the variables", + "type": "text" + }, + { + "bbox": [ + 322, + 183, + 388, + 193 + ], + "score": 0.86, + "content": "v _ { d } , v _ { d + 1 } , . . . , v _ { d _ { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 180, + 462, + 195 + ], + "score": 1.0, + "content": "appear in the first", + "type": "text" + }, + { + "bbox": [ + 462, + 181, + 504, + 193 + ], + "score": 0.91, + "content": "( d - 1 ) d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 204, + 205 + ], + "score": 1.0, + "content": "equations, and therefore", + "type": "text" + }, + { + "bbox": [ + 204, + 192, + 335, + 204 + ], + "score": 0.91, + "content": "C \\subseteq S ^ { \\prime } = S \\setminus \\{ d , d + 1 , . . . , d _ { S } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 192, + 506, + 205 + ], + "score": 1.0, + "content": ". By our induction assumptions, except for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 202, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 104, + 202, + 221, + 216 + ], + "score": 1.0, + "content": "a set of measure 0, the first", + "type": "text" + }, + { + "bbox": [ + 221, + 204, + 263, + 215 + ], + "score": 0.91, + "content": "( d - 1 ) d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 202, + 409, + 216 + ], + "score": 1.0, + "content": "are independent, which means that", + "type": "text" + }, + { + "bbox": [ + 410, + 203, + 503, + 215 + ], + "score": 0.93, + "content": "| C | = r _ { d } = ( d - 1 ) d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 202, + 506, + 216 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 504, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 311, + 226 + ], + "score": 1.0, + "content": "We now extend the Gaussian elimination to the next", + "type": "text" + }, + { + "bbox": [ + 311, + 214, + 322, + 225 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 213, + 495, + 226 + ], + "score": 1.0, + "content": "equations, and eliminate all the variables in", + "type": "text" + }, + { + "bbox": [ + 495, + 215, + 504, + 224 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 224, + 366, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 366, + 237 + ], + "score": 1.0, + "content": "from them. The result of the elimination can be written down as,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 241, + 391, + 270 + ], + "lines": [ + { + "bbox": [ + 219, + 241, + 391, + 270 + ], + "spans": [ + { + "bbox": [ + 219, + 241, + 391, + 270 + ], + "score": 0.93, + "content": "v _ { d } \\mathbf { x } _ { d } + \\sum _ { n \\in S ^ { \\prime } \\backslash C } v _ { n } \\left( \\tilde { a } _ { d , n } \\mathbf { I } _ { d _ { 0 } } - \\mathbf { Y } \\right) \\mathbf { x } _ { n } = { 0 } ,", + "type": "interline_equation", + "image_path": "9bf56616592389d15d94c6c71d1c7f2b4aaee3be0edce76efacda4cc129fcfaf.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 219, + 241, + 391, + 270 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 133, + 289 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 276, + 143, + 285 + ], + "score": 0.75, + "content": "\\mathbf { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 273, + 248, + 289 + ], + "score": 1.0, + "content": "is a square matrix of size", + "type": "text" + }, + { + "bbox": [ + 248, + 276, + 258, + 286 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 273, + 402, + 289 + ], + "score": 1.0, + "content": "whose coefficients depend only on", + "type": "text" + }, + { + "bbox": [ + 402, + 275, + 474, + 288 + ], + "score": 0.92, + "content": "\\{ \\widetilde { a } _ { k , n } \\} _ { n \\in C , d > k \\geq 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 273, + 505, + 289 + ], + "score": 1.0, + "content": "and on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 282, + 379, + 304 + ], + "spans": [ + { + "bbox": [ + 107, + 286, + 146, + 299 + ], + "score": 0.92, + "content": "\\{ \\mathbf { x } _ { n } \\} _ { n \\in C }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 282, + 291, + 304 + ], + "score": 1.0, + "content": ", and in particular do not depend on", + "type": "text" + }, + { + "bbox": [ + 291, + 288, + 304, + 298 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 282, + 322, + 304 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 286, + 373, + 301 + ], + "score": 0.93, + "content": "\\left\\{ \\mathbf { x } _ { n } \\right\\} _ { n \\in S ^ { \\prime } \\backslash C }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 282, + 379, + 304 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 145, + 319 + ], + "score": 1.0, + "content": "Now set", + "type": "text" + }, + { + "bbox": [ + 145, + 306, + 249, + 318 + ], + "score": 0.91, + "content": "\\tilde { \\mathbf { x } } _ { n } \\ = \\ ( \\tilde { a } _ { d , n } \\mathbf { I } _ { d _ { 0 } } - \\mathbf { Y } ) \\mathbf { x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 304, + 268, + 319 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 268, + 305, + 321, + 318 + ], + "score": 0.93, + "content": "n \\in S ^ { \\prime } \\setminus C", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 304, + 407, + 319 + ], + "score": 1.0, + "content": ". As in the case of", + "type": "text" + }, + { + "bbox": [ + 408, + 306, + 437, + 316 + ], + "score": 0.89, + "content": "d \\ : = \\ : 1", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 304, + 468, + 319 + ], + "score": 1.0, + "content": ", since", + "type": "text" + }, + { + "bbox": [ + 468, + 306, + 502, + 317 + ], + "score": 0.91, + "content": "v _ { d } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 304, + 506, + 319 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 204, + 331 + ], + "score": 0.91, + "content": "\\mathbf { x } _ { d } \\in \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 317, + 325, + 330 + ], + "score": 1.0, + "content": ". Therefore, for all values of", + "type": "text" + }, + { + "bbox": [ + 325, + 317, + 367, + 329 + ], + "score": 0.92, + "content": "\\mathbf { x } _ { d } \\in \\mathbb { R } ^ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "but a set of measure zero (linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 328, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 263, + 344 + ], + "score": 1.0, + "content": "subspace of with dimension less than", + "type": "text" + }, + { + "bbox": [ + 263, + 330, + 274, + 341 + ], + "score": 0.84, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 328, + 351, + 344 + ], + "score": 1.0, + "content": "), we must have di", + "type": "text" + }, + { + "bbox": [ + 351, + 330, + 459, + 343 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\operatorname* { m } \\mathrm { { S p a n } } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 328, + 506, + 344 + ], + "score": 1.0, + "content": ". From the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 341, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 178, + 357 + ], + "score": 1.0, + "content": "independence of", + "type": "text" + }, + { + "bbox": [ + 178, + 342, + 229, + 355 + ], + "score": 0.92, + "content": "\\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 341, + 244, + 357 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 245, + 344, + 257, + 353 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 341, + 320, + 357 + ], + "score": 1.0, + "content": "it follows that", + "type": "text" + }, + { + "bbox": [ + 320, + 342, + 437, + 355 + ], + "score": 0.85, + "content": "\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 341, + 506, + 357 + ], + "score": 1.0, + "content": "holds a.e. with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 269, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 258, + 365 + ], + "score": 1.0, + "content": "respect to the Lebesgue measure over", + "type": "text" + }, + { + "bbox": [ + 258, + 355, + 266, + 363 + ], + "score": 0.58, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 353, + 269, + 365 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 435, + 383 + ], + "lines": [ + { + "bbox": [ + 104, + 367, + 436, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 150, + 384 + ], + "score": 1.0, + "content": "Whenever", + "type": "text" + }, + { + "bbox": [ + 150, + 370, + 263, + 383 + ], + "score": 0.77, + "content": "\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 367, + 322, + 384 + ], + "score": 1.0, + "content": "we must have", + "type": "text" + }, + { + "bbox": [ + 322, + 369, + 378, + 382 + ], + "score": 0.93, + "content": "| S ^ { \\prime } C | \\geq d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 367, + 436, + 384 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 388, + 418, + 402 + ], + "lines": [ + { + "bbox": [ + 195, + 388, + 418, + 402 + ], + "spans": [ + { + "bbox": [ + 195, + 388, + 418, + 402 + ], + "score": 0.88, + "content": "| S | > | S ^ { \\prime } | = | C | + | S ^ { \\prime } \\setminus C | \\geq \\left( d - 1 \\right) d _ { 0 } + d _ { 0 } = d _ { 0 } d .", + "type": "interline_equation", + "image_path": "32a1ca6c153e5d5a9bf82a5fbbbd89574b4e1ccf22e1d6307718430cd1deda53.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 195, + 388, + 418, + 402 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 104, + 405, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 151, + 424 + ], + "score": 1.0, + "content": "Moreover,", + "type": "text" + }, + { + "bbox": [ + 151, + 407, + 267, + 420 + ], + "score": 0.88, + "content": "\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 405, + 333, + 424 + ], + "score": 1.0, + "content": "implies that the", + "type": "text" + }, + { + "bbox": [ + 334, + 407, + 345, + 419 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 405, + 388, + 424 + ], + "score": 1.0, + "content": "equations", + "type": "text" + }, + { + "bbox": [ + 388, + 407, + 505, + 421 + ], + "score": 0.91, + "content": "\\begin{array} { r } { v _ { d } \\mathbf { x } _ { d } + \\sum _ { n \\in S ^ { \\prime } \\backslash C } v _ { n } \\tilde { \\mathbf { x } } _ { n } = 0 } \\end{array}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 420, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 451, + 433 + ], + "score": 1.0, + "content": "are independent. Thus, we may perform another step of Gaussian elimination on these", + "type": "text" + }, + { + "bbox": [ + 451, + 421, + 462, + 431 + ], + "score": 0.86, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 420, + 507, + 433 + ], + "score": 1.0, + "content": "equations,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 142, + 444 + ], + "score": 1.0, + "content": "forming", + "type": "text" + }, + { + "bbox": [ + 142, + 431, + 153, + 442 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 430, + 414, + 444 + ], + "score": 1.0, + "content": "new equations each with a variable unique to it. 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This means that the first", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 484, + 507, + 499 + ], + "spans": [ + { + "bbox": [ + 107, + 486, + 123, + 497 + ], + "score": 0.91, + "content": "d _ { 0 } d", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 484, + 315, + 499 + ], + "score": 1.0, + "content": "must be linearly independent for all values of", + "type": "text" + }, + { + "bbox": [ + 315, + 486, + 325, + 496 + ], + "score": 0.66, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 484, + 507, + 499 + ], + "score": 1.0, + "content": "except for a set of Lebesgue measure zero,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 498, + 210, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 210, + 509 + ], + "score": 1.0, + "content": "completing the induction.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 102, + 509, + 504, + 531 + ], + "spans": [ + { + "bbox": [ + 102, + 509, + 258, + 531 + ], + "score": 1.0, + "content": "Thus, we have proven, that for some", + "type": "text" + }, + { + "bbox": [ + 258, + 514, + 326, + 528 + ], + "score": 0.92, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 509, + 344, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 515, + 381, + 528 + ], + "score": 0.91, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 509, + 422, + 531 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 423, + 515, + 504, + 528 + ], + "score": 0.85, + "content": "| S | \\leq \\mathrm { r a n k } \\left( \\mathbf { A } _ { S } \\right) d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 525, + 146, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 146, + 539 + ], + "score": 1.0, + "content": "the event", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 541, + 460, + 563 + ], + "lines": [ + { + "bbox": [ + 151, + 541, + 460, + 563 + ], + "spans": [ + { + "bbox": [ + 151, + 541, + 460, + 563 + ], + "score": 0.9, + "content": "\\mathcal E \\left( \\mathbf A , S \\right) = \\left\\{ \\mathbf X \\in \\mathbb R ^ { d _ { 0 } \\times N } | \\exists \\mathbf v \\in \\mathbb R ^ { N } : \\left( \\mathbf A \\circ \\mathbf X \\right) \\mathbf v = 0 \\mathrm { a n d } v _ { n } \\neq 0 , \\forall n \\in S \\right\\}", + "type": "interline_equation", + "image_path": "6fa3c6c034467d0cd6c07a2f70417a14cdc7c19c6ebeed96882c6294c4ee8a4f.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 151, + 541, + 460, + 563 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 567, + 429, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 429, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 429, + 580 + ], + "score": 1.0, + "content": "has zero measure. The event discussed in the theorem is a union of these events:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 582, + 413, + 622 + ], + "lines": [ + { + "bbox": [ + 196, + 582, + 413, + 622 + ], + "spans": [ + { + "bbox": [ + 196, + 582, + 413, + 622 + ], + "score": 0.93, + "content": "\\mathcal { E } _ { 0 } \\triangleq \\bigcup _ { \\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N } } \\left[ \\bigcup _ { S \\subset [ N ] : | S | \\leq \\operatorname { r a n k } ( \\mathbf { A } _ { S } ) d _ { 0 } } \\mathcal { E } \\left( \\mathbf { A } , S \\right) \\right] ,", + "type": "interline_equation", + "image_path": "da3a7d613ef7ea45872d466c0bdc770b822f73ec22a66fe81e1ed814d59c888c.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 582, + 413, + 602.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 196, + 602.0, + 413, + 622.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 414, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 414, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 414, + 639 + ], + "score": 1.0, + "content": "and it also has zero measure, since it is a finite union of zero measure events.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 488, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 490, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 490, + 664 + ], + "score": 1.0, + "content": "For completeness we note the following corollary, which is not necessary for a our main results.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 414, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 410, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 176, + 681 + ], + "score": 1.0, + "content": "Corollary 28. If", + "type": "text" + }, + { + "bbox": [ + 177, + 666, + 219, + 677 + ], + "score": 0.91, + "content": "N \\leq d _ { 1 } d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 664, + 264, + 681 + ], + "score": 1.0, + "content": ", then rank", + "type": "text" + }, + { + "bbox": [ + 265, + 666, + 322, + 678 + ], + "score": 0.9, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) = N", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 664, + 402, + 681 + ], + "score": 1.0, + "content": ", X-a.e., if and only", + "type": "text" + }, + { + "bbox": [ + 402, + 667, + 410, + 678 + ], + "score": 0.63, + "content": "i f ,", + "type": "inline_equation" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 683, + 373, + 697 + ], + "lines": [ + { + "bbox": [ + 236, + 683, + 373, + 697 + ], + "spans": [ + { + "bbox": [ + 236, + 683, + 373, + 697 + ], + "score": 0.9, + "content": "\\forall S \\subseteq [ N ] : | S | \\leq \\mathrm { r a n k } \\left( \\mathbf { A } _ { S } \\right) d _ { 0 } .", + "type": "interline_equation", + "image_path": "9e7d11cb95875c223dbb401985bafeb64514cfd3273a6db22551d7b6cbef4ca5.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 236, + 683, + 373, + 697 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 708, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 178, + 723 + ], + "score": 1.0, + "content": "Proof. We define", + "type": "text" + }, + { + "bbox": [ + 178, + 709, + 247, + 721 + ], + "score": 0.89, + "content": "d _ { S } \\triangleq \\operatorname { r a n k } \\left( \\mathbf { A } _ { S } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 708, + 265, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 710, + 293, + 720 + ], + "score": 0.84, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 708, + 419, + 723 + ], + "score": 1.0, + "content": ". 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As in the case of", + "type": "text" + }, + { + "bbox": [ + 408, + 306, + 437, + 316 + ], + "score": 0.89, + "content": "d \\ : = \\ : 1", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 304, + 468, + 319 + ], + "score": 1.0, + "content": ", since", + "type": "text" + }, + { + "bbox": [ + 468, + 306, + 502, + 317 + ], + "score": 0.91, + "content": "v _ { d } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 304, + 506, + 319 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 204, + 331 + ], + "score": 0.91, + "content": "\\mathbf { x } _ { d } \\in \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 317, + 325, + 330 + ], + "score": 1.0, + "content": ". Therefore, for all values of", + "type": "text" + }, + { + "bbox": [ + 325, + 317, + 367, + 329 + ], + "score": 0.92, + "content": "\\mathbf { x } _ { d } \\in \\mathbb { R } ^ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "but a set of measure zero (linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 328, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 263, + 344 + ], + "score": 1.0, + "content": "subspace of with dimension less than", + "type": "text" + }, + { + "bbox": [ + 263, + 330, + 274, + 341 + ], + "score": 0.84, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 328, + 351, + 344 + ], + "score": 1.0, + "content": "), we must have di", + "type": "text" + }, + { + "bbox": [ + 351, + 330, + 459, + 343 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\operatorname* { m } \\mathrm { { S p a n } } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 328, + 506, + 344 + ], + "score": 1.0, + "content": ". From the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 341, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 178, + 357 + ], + "score": 1.0, + "content": "independence of", + "type": "text" + }, + { + "bbox": [ + 178, + 342, + 229, + 355 + ], + "score": 0.92, + "content": "\\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 341, + 244, + 357 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 245, + 344, + 257, + 353 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 341, + 320, + 357 + ], + "score": 1.0, + "content": "it follows that", + "type": "text" + }, + { + "bbox": [ + 320, + 342, + 437, + 355 + ], + "score": 0.85, + "content": "\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 341, + 506, + 357 + ], + "score": 1.0, + "content": "holds a.e. with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 269, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 258, + 365 + ], + "score": 1.0, + "content": "respect to the Lebesgue measure over", + "type": "text" + }, + { + "bbox": [ + 258, + 355, + 266, + 363 + ], + "score": 0.58, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 353, + 269, + 365 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 304, + 506, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 435, + 383 + ], + "lines": [ + { + "bbox": [ + 104, + 367, + 436, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 150, + 384 + ], + "score": 1.0, + "content": "Whenever", + "type": "text" + }, + { + "bbox": [ + 150, + 370, + 263, + 383 + ], + "score": 0.77, + "content": "\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 367, + 322, + 384 + ], + "score": 1.0, + "content": "we must have", + "type": "text" + }, + { + "bbox": [ + 322, + 369, + 378, + 382 + ], + "score": 0.93, + "content": "| S ^ { \\prime } C | \\geq d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 367, + 436, + 384 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 367, + 436, + 384 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 388, + 418, + 402 + ], + "lines": [ + { + "bbox": [ + 195, + 388, + 418, + 402 + ], + "spans": [ + { + "bbox": [ + 195, + 388, + 418, + 402 + ], + "score": 0.88, + "content": "| S | > | S ^ { \\prime } | = | C | + | S ^ { \\prime } \\setminus C | \\geq \\left( d - 1 \\right) d _ { 0 } + d _ { 0 } = d _ { 0 } d .", + "type": "interline_equation", + "image_path": "32a1ca6c153e5d5a9bf82a5fbbbd89574b4e1ccf22e1d6307718430cd1deda53.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 195, + 388, + 418, + 402 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 104, + 405, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 151, + 424 + ], + "score": 1.0, + "content": "Moreover,", + "type": "text" + }, + { + "bbox": [ + 151, + 407, + 267, + 420 + ], + "score": 0.88, + "content": "\\mathrm { d i m } \\mathrm { S p a n } \\{ \\tilde { \\mathbf { x } } _ { n } \\} _ { n \\in S ^ { \\prime } \\backslash C } = d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 405, + 333, + 424 + ], + "score": 1.0, + "content": "implies that the", + "type": "text" + }, + { + "bbox": [ + 334, + 407, + 345, + 419 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 405, + 388, + 424 + ], + "score": 1.0, + "content": "equations", + "type": "text" + }, + { + "bbox": [ + 388, + 407, + 505, + 421 + ], + "score": 0.91, + "content": "\\begin{array} { r } { v _ { d } \\mathbf { x } _ { d } + \\sum _ { n \\in S ^ { \\prime } \\backslash C } v _ { n } \\tilde { \\mathbf { x } } _ { n } = 0 } \\end{array}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 420, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 451, + 433 + ], + "score": 1.0, + "content": "are independent. Thus, we may perform another step of Gaussian elimination on these", + "type": "text" + }, + { + "bbox": [ + 451, + 421, + 462, + 431 + ], + "score": 0.86, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 420, + 507, + 433 + ], + "score": 1.0, + "content": "equations,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 142, + 444 + ], + "score": 1.0, + "content": "forming", + "type": "text" + }, + { + "bbox": [ + 142, + 431, + 153, + 442 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 430, + 414, + 444 + ], + "score": 1.0, + "content": "new equations each with a variable unique to it. Denoting by", + "type": "text" + }, + { + "bbox": [ + 414, + 432, + 425, + 441 + ], + "score": 0.87, + "content": "C ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 430, + 493, + 444 + ], + "score": 1.0, + "content": "the set of these", + "type": "text" + }, + { + "bbox": [ + 493, + 432, + 504, + 442 + ], + "score": 0.86, + "content": "d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 255, + 455 + ], + "score": 1.0, + "content": "variables, it is seen from (12.4) that", + "type": "text" + }, + { + "bbox": [ + 256, + 442, + 343, + 454 + ], + "score": 0.93, + "content": "C ^ { \\prime } \\subseteq ( S ^ { \\prime } \\cup \\{ d \\} ) \\setminus C", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 441, + 415, + 455 + ], + "score": 1.0, + "content": "and in particular", + "type": "text" + }, + { + "bbox": [ + 415, + 442, + 426, + 452 + ], + "score": 0.85, + "content": "C ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 441, + 494, + 455 + ], + "score": 1.0, + "content": "is disjoint from", + "type": "text" + }, + { + "bbox": [ + 495, + 444, + 503, + 452 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 441, + 506, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 212, + 466 + ], + "score": 1.0, + "content": "Thus, considering the first", + "type": "text" + }, + { + "bbox": [ + 212, + 453, + 253, + 465 + ], + "score": 0.92, + "content": "( d - 1 ) d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 452, + 382, + 466 + ], + "score": 1.0, + "content": "equations together with the new", + "type": "text" + }, + { + "bbox": [ + 383, + 453, + 393, + 464 + ], + "score": 0.87, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "equations, we see that there", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 462, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 137, + 477 + ], + "score": 1.0, + "content": "is a set", + "type": "text" + }, + { + "bbox": [ + 137, + 464, + 168, + 474 + ], + "score": 0.9, + "content": "C \\cup C ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 462, + 181, + 477 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 181, + 464, + 197, + 475 + ], + "score": 0.9, + "content": "d _ { 0 } d", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 462, + 346, + 477 + ], + "score": 1.0, + "content": "variables, such that each variable in", + "type": "text" + }, + { + "bbox": [ + 347, + 464, + 377, + 474 + ], + "score": 0.9, + "content": "C \\cup C ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 462, + 487, + 477 + ], + "score": 1.0, + "content": "appears only in one of the", + "type": "text" + }, + { + "bbox": [ + 488, + 464, + 504, + 475 + ], + "score": 0.88, + "content": "d _ { 0 } d", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 215, + 487 + ], + "score": 1.0, + "content": "equations, and each of the", + "type": "text" + }, + { + "bbox": [ + 215, + 475, + 232, + 486 + ], + "score": 0.89, + "content": "d _ { 0 } d", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 475, + 369, + 487 + ], + "score": 1.0, + "content": "contains only a single variable in", + "type": "text" + }, + { + "bbox": [ + 369, + 475, + 400, + 485 + ], + "score": 0.91, + "content": "C \\cup C ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 475, + 506, + 487 + ], + "score": 1.0, + "content": ". This means that the first", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 484, + 507, + 499 + ], + "spans": [ + { + "bbox": [ + 107, + 486, + 123, + 497 + ], + "score": 0.91, + "content": "d _ { 0 } d", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 484, + 315, + 499 + ], + "score": 1.0, + "content": "must be linearly independent for all values of", + "type": "text" + }, + { + "bbox": [ + 315, + 486, + 325, + 496 + ], + "score": 0.66, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 484, + 507, + 499 + ], + "score": 1.0, + "content": "except for a set of Lebesgue measure zero,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 498, + 210, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 210, + 509 + ], + "score": 1.0, + "content": "completing the induction.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 405, + 507, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 102, + 509, + 504, + 531 + ], + "spans": [ + { + "bbox": [ + 102, + 509, + 258, + 531 + ], + "score": 1.0, + "content": "Thus, we have proven, that for some", + "type": "text" + }, + { + "bbox": [ + 258, + 514, + 326, + 528 + ], + "score": 0.92, + "content": "\\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 509, + 344, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 515, + 381, + 528 + ], + "score": 0.91, + "content": "S \\subset [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 509, + 422, + 531 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 423, + 515, + 504, + 528 + ], + "score": 0.85, + "content": "| S | \\leq \\mathrm { r a n k } \\left( \\mathbf { A } _ { S } \\right) d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 525, + 146, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 146, + 539 + ], + "score": 1.0, + "content": "the event", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 102, + 509, + 504, + 539 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 541, + 460, + 563 + ], + "lines": [ + { + "bbox": [ + 151, + 541, + 460, + 563 + ], + "spans": [ + { + "bbox": [ + 151, + 541, + 460, + 563 + ], + "score": 0.9, + "content": "\\mathcal E \\left( \\mathbf A , S \\right) = \\left\\{ \\mathbf X \\in \\mathbb R ^ { d _ { 0 } \\times N } | \\exists \\mathbf v \\in \\mathbb R ^ { N } : \\left( \\mathbf A \\circ \\mathbf X \\right) \\mathbf v = 0 \\mathrm { a n d } v _ { n } \\neq 0 , \\forall n \\in S \\right\\}", + "type": "interline_equation", + "image_path": "6fa3c6c034467d0cd6c07a2f70417a14cdc7c19c6ebeed96882c6294c4ee8a4f.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 151, + 541, + 460, + 563 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 567, + 429, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 429, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 429, + 580 + ], + "score": 1.0, + "content": "has zero measure. The event discussed in the theorem is a union of these events:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 566, + 429, + 580 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 582, + 413, + 622 + ], + "lines": [ + { + "bbox": [ + 196, + 582, + 413, + 622 + ], + "spans": [ + { + "bbox": [ + 196, + 582, + 413, + 622 + ], + "score": 0.93, + "content": "\\mathcal { E } _ { 0 } \\triangleq \\bigcup _ { \\mathbf { A } \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times N } } \\left[ \\bigcup _ { S \\subset [ N ] : | S | \\leq \\operatorname { r a n k } ( \\mathbf { A } _ { S } ) d _ { 0 } } \\mathcal { E } \\left( \\mathbf { A } , S \\right) \\right] ,", + "type": "interline_equation", + "image_path": "da3a7d613ef7ea45872d466c0bdc770b822f73ec22a66fe81e1ed814d59c888c.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 582, + 413, + 602.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 196, + 602.0, + 413, + 622.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 414, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 414, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 414, + 639 + ], + "score": 1.0, + "content": "and it also has zero measure, since it is a finite union of zero measure events.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 626, + 414, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 488, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 490, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 490, + 664 + ], + "score": 1.0, + "content": "For completeness we note the following corollary, which is not necessary for a our main results.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 106, + 650, + 490, + 664 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 414, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 410, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 176, + 681 + ], + "score": 1.0, + "content": "Corollary 28. If", + "type": "text" + }, + { + "bbox": [ + 177, + 666, + 219, + 677 + ], + "score": 0.91, + "content": "N \\leq d _ { 1 } d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 664, + 264, + 681 + ], + "score": 1.0, + "content": ", then rank", + "type": "text" + }, + { + "bbox": [ + 265, + 666, + 322, + 678 + ], + "score": 0.9, + "content": "( \\mathbf { A } \\circ \\mathbf { X } ) = N", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 664, + 402, + 681 + ], + "score": 1.0, + "content": ", X-a.e., if and only", + "type": "text" + }, + { + "bbox": [ + 402, + 667, + 410, + 678 + ], + "score": 0.63, + "content": "i f ,", + "type": "inline_equation" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 106, + 664, + 410, + 681 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 683, + 373, + 697 + ], + "lines": [ + { + "bbox": [ + 236, + 683, + 373, + 697 + ], + "spans": [ + { + "bbox": [ + 236, + 683, + 373, + 697 + ], + "score": 0.9, + "content": "\\forall S \\subseteq [ N ] : | S | \\leq \\mathrm { r a n k } \\left( \\mathbf { A } _ { S } \\right) d _ { 0 } .", + "type": "interline_equation", + "image_path": "9e7d11cb95875c223dbb401985bafeb64514cfd3273a6db22551d7b6cbef4ca5.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 236, + 683, + 373, + 697 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 708, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 178, + 723 + ], + "score": 1.0, + "content": "Proof. We define", + "type": "text" + }, + { + "bbox": [ + 178, + 709, + 247, + 721 + ], + "score": 0.89, + "content": "d _ { S } \\triangleq \\operatorname { r a n k } \\left( \\mathbf { A } _ { S } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 708, + 265, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 710, + 293, + 720 + ], + "score": 0.84, + "content": "\\mathbf { A } \\circ \\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 708, + 419, + 723 + ], + "score": 1.0, + "content": ". The necessity of the condition", + "type": "text" + }, + { + "bbox": [ + 419, + 709, + 466, + 721 + ], + "score": 0.92, + "content": "| S | \\le d _ { 0 } d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 719, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 131, + 734 + ], + "score": 1.0, + "content": "every", + "type": "text" + }, + { + "bbox": [ + 132, + 721, + 141, + 730 + ], + "score": 0.56, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 719, + 436, + 734 + ], + "score": 1.0, + "content": ", as can be seen from the following counting argument. Since the matrix", + "type": "text" + }, + { + "bbox": [ + 437, + 722, + 452, + 732 + ], + "score": 0.87, + "content": "\\mathbf { A } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 719, + 490, + 734 + ], + "score": 1.0, + "content": "has rank", + "type": "text" + }, + { + "bbox": [ + 490, + 721, + 502, + 732 + ], + "score": 0.88, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 719, + 507, + 734 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 708, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 316, + 95 + ], + "score": 1.0, + "content": "there exists an invertible row transformation matrix", + "type": "text" + }, + { + "bbox": [ + 316, + 83, + 326, + 93 + ], + "score": 0.55, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 83, + 368, + 95 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 369, + 83, + 393, + 93 + ], + "score": 0.88, + "content": "\\mathbf { R A } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 83, + 430, + 95 + ], + "score": 1.0, + "content": "has only", + "type": "text" + }, + { + "bbox": [ + 430, + 83, + 443, + 93 + ], + "score": 0.88, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "non-zero rows.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 483, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 164, + 106 + ], + "score": 1.0, + "content": "Consider now", + "type": "text" + }, + { + "bbox": [ + 165, + 93, + 232, + 105 + ], + "score": 0.92, + "content": "\\mathbf { G } _ { S } = \\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 93, + 411, + 106 + ], + "score": 1.0, + "content": ", i.e., the matrix composed of the columns of", + "type": "text" + }, + { + "bbox": [ + 412, + 94, + 421, + 104 + ], + "score": 0.72, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 93, + 433, + 106 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 433, + 95, + 441, + 104 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 93, + 483, + 106 + ], + "score": 1.0, + "content": ". We have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 117, + 406, + 131 + ], + "lines": [ + { + "bbox": [ + 204, + 117, + 406, + 131 + ], + "spans": [ + { + "bbox": [ + 204, + 117, + 406, + 131 + ], + "score": 0.89, + "content": "\\mathbf { G } _ { S } ^ { \\prime } = ( \\mathbf { R } \\mathbf { A } _ { S } ) \\circ \\mathbf { X } _ { S } = \\mathbf { R } ^ { \\prime } \\left( \\mathbf { A } _ { S } \\circ \\mathbf { X } _ { S } \\right) = \\mathbf { R } ^ { \\prime } \\mathbf { G } _ { S } ,", + "type": "interline_equation", + "image_path": "924a84543fe5a6ef518a3f828b345ceae59266a440f2c1d625f44d3b28908187.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 204, + 117, + 406, + 131 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 506, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 133, + 156 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 143, + 192, + 155 + ], + "score": 0.92, + "content": "\\mathbf { R } ^ { \\prime } = \\mathbf { R } \\otimes \\mathbf { I } _ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 143, + 439, + 156 + ], + "score": 1.0, + "content": "is also an invertible row transformation matrix, which applies", + "type": "text" + }, + { + "bbox": [ + 439, + 144, + 449, + 153 + ], + "score": 0.64, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "separately on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 121, + 168 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 155, + 132, + 165 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 154, + 198, + 168 + ], + "score": 1.0, + "content": "sub-matrices of", + "type": "text" + }, + { + "bbox": [ + 198, + 155, + 214, + 165 + ], + "score": 0.9, + "content": "\\mathbf { G } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 154, + 378, + 168 + ], + "score": 1.0, + "content": "that are constructed by taking one every", + "type": "text" + }, + { + "bbox": [ + 378, + 155, + 389, + 165 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 154, + 440, + 168 + ], + "score": 1.0, + "content": "rows. Since", + "type": "text" + }, + { + "bbox": [ + 440, + 154, + 456, + 166 + ], + "score": 0.89, + "content": "\\mathbf { G } _ { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "has at most", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 107, + 166, + 128, + 177 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 164, + 241, + 178 + ], + "score": 1.0, + "content": "non-zero rows, the rank of", + "type": "text" + }, + { + "bbox": [ + 241, + 165, + 257, + 177 + ], + "score": 0.9, + "content": "\\mathbf { G } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 164, + 319, + 178 + ], + "score": 1.0, + "content": "cannot exceed", + "type": "text" + }, + { + "bbox": [ + 319, + 165, + 341, + 177 + ], + "score": 0.9, + "content": "d _ { 0 } d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 164, + 401, + 178 + ], + "score": 1.0, + "content": ". Therefore, if", + "type": "text" + }, + { + "bbox": [ + 401, + 165, + 449, + 177 + ], + "score": 0.89, + "content": "| S | > d _ { 0 } d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 164, + 453, + 178 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 453, + 166, + 469, + 177 + ], + "score": 0.85, + "content": "\\mathbf { G } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "will not", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 292, + 189 + ], + "score": 1.0, + "content": "have full column rank, and hence neither will", + "type": "text" + }, + { + "bbox": [ + 292, + 176, + 302, + 186 + ], + "score": 0.37, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 175, + 472, + 189 + ], + "score": 1.0, + "content": ". To demonstrate sufficiency a.e., suppose", + "type": "text" + }, + { + "bbox": [ + 473, + 177, + 483, + 186 + ], + "score": 0.65, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "does", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 229, + 199 + ], + "score": 1.0, + "content": "not have full column rank. Let", + "type": "text" + }, + { + "bbox": [ + 229, + 187, + 237, + 197 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 187, + 376, + 199 + ], + "score": 1.0, + "content": "be the minimum set of columns of", + "type": "text" + }, + { + "bbox": [ + 376, + 187, + 387, + 197 + ], + "score": 0.53, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "which are linearly dependent.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 191, + 212 + ], + "score": 1.0, + "content": "Since the columns of", + "type": "text" + }, + { + "bbox": [ + 192, + 199, + 207, + 210 + ], + "score": 0.93, + "content": "\\mathbf { G } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 196, + 380, + 212 + ], + "score": 1.0, + "content": "are assumed linearly dependent there exists", + "type": "text" + }, + { + "bbox": [ + 381, + 198, + 418, + 209 + ], + "score": 0.92, + "content": "\\mathbf { v } \\in \\mathbb { R } ^ { | S | }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 196, + 440, + 212 + ], + "score": 1.0, + "content": "such", + "type": "text" + }, + { + "bbox": [ + 440, + 199, + 487, + 212 + ], + "score": 0.92, + "content": "\\| \\mathbf { v } \\| _ { 0 } = | S |", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 196, + 506, + 212 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 147, + 221 + ], + "score": 0.91, + "content": "\\mathbf { G } _ { S } \\mathbf { v } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 210, + 317, + 222 + ], + "score": 1.0, + "content": ". Using Lemma 28 we complete the proof.", + "type": "text" + }, + { + "bbox": [ + 494, + 210, + 505, + 221 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 242, + 230, + 254 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 230, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 230, + 255 + ], + "score": 1.0, + "content": "12.2 PROOF OF LEMMA 15", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 266, + 504, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "In this section we will prove Lemma 15 in subsection 12.3.3. This proof relies on two rather basic", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 277, + 354, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 354, + 288 + ], + "score": 1.0, + "content": "results, which we first prove in subsections 12.2.1 and 12.2.2.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 308, + 385, + 321 + ], + "lines": [ + { + "bbox": [ + 107, + 309, + 386, + 321 + ], + "spans": [ + { + "bbox": [ + 107, + 309, + 386, + 321 + ], + "score": 1.0, + "content": "12.2.1 NUMBER OF DICHOTOMIES INDUCED BY A HYPERPLANE", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 104, + 329, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 201, + 344 + ], + "score": 1.0, + "content": "Fact 29. A hyperplane", + "type": "text" + }, + { + "bbox": [ + 202, + 332, + 233, + 343 + ], + "score": 0.9, + "content": "{ \\mathbf w } \\in d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 329, + 372, + 344 + ], + "score": 1.0, + "content": "can separate a given set of points", + "type": "text" + }, + { + "bbox": [ + 373, + 330, + 504, + 344 + ], + "score": 0.93, + "content": "\\mathbf { X } = \\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 338, + 357 + ], + "score": 1.0, + "content": "into several different dichotomies, i.e., different results for", + "type": "text" + }, + { + "bbox": [ + 338, + 343, + 390, + 357 + ], + "score": 0.78, + "content": "\\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 343, + 505, + 357 + ], + "score": 1.0, + "content": ". The number of dichotomies", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 355, + 224, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 224, + 368 + ], + "score": 1.0, + "content": "is upper bounded as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 378, + 452, + 416 + ], + "lines": [ + { + "bbox": [ + 156, + 378, + 452, + 416 + ], + "spans": [ + { + "bbox": [ + 156, + 378, + 452, + 416 + ], + "score": 0.92, + "content": "\\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { N } } \\mathscr { T } \\left( \\exists \\mathbf { w } : \\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } \\right) = \\mathbf { h } ^ { \\top } \\right) \\leq 2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( \\begin{array} { l } { N - 1 } \\\\ { k } \\end{array} \\right) \\leq 2 N ^ { d _ { 0 } } .", + "type": "interline_equation", + "image_path": "276611149f850fb17a545865714e1e2821ff4642a53a796c1de7f19e11080b61.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 156, + 378, + 452, + 390.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 156, + 390.6666666666667, + 452, + 403.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 156, + 403.33333333333337, + 452, + 416.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 506, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "Proof. See (Cover, 1965, Theorem 1) for a proof of the left inequality as equality (the Schläfli", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 283, + 473 + ], + "score": 1.0, + "content": "Theorem) in the case that the columns of", + "type": "text" + }, + { + "bbox": [ + 284, + 461, + 294, + 471 + ], + "score": 0.56, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 460, + 459, + 473 + ], + "score": 1.0, + "content": "are in “general position” (which holds", + "type": "text" + }, + { + "bbox": [ + 460, + 461, + 469, + 471 + ], + "score": 0.58, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "-a.e, see", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 232, + 484 + ], + "score": 1.0, + "content": "definition in (Cover, 1965)) . If", + "type": "text" + }, + { + "bbox": [ + 232, + 472, + 242, + 482 + ], + "score": 0.44, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "is not in general position then this result becomes an upper bound,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 294, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 294, + 495 + ], + "score": 1.0, + "content": "since some dichotomies might not be possible.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 105, + 498, + 506, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 498, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 264, + 513 + ], + "score": 1.0, + "content": "Next, we prove the right inequality. For", + "type": "text" + }, + { + "bbox": [ + 265, + 500, + 294, + 510 + ], + "score": 0.89, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 498, + 311, + 513 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 500, + 340, + 510 + ], + "score": 0.89, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 498, + 474, + 513 + ], + "score": 1.0, + "content": "the inequality trivially holds. For", + "type": "text" + }, + { + "bbox": [ + 474, + 500, + 502, + 510 + ], + "score": 0.89, + "content": "N \\geq 3", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 498, + 507, + 513 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 509, + 143, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 143, + 524 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 528, + 448, + 564 + ], + "lines": [ + { + "bbox": [ + 162, + 528, + 448, + 564 + ], + "spans": [ + { + "bbox": [ + 162, + 528, + 448, + 564 + ], + "score": 0.95, + "content": "2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( \\begin{array} { c } { { N - 1 } } \\\\ { { k } } \\end{array} \\right) \\stackrel { \\left( 1 \\right) } { \\leq } 2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( N - 1 \\right) ^ { k } \\stackrel { \\left( 2 \\right) } { \\leq } 2 \\frac { \\left( N - 1 \\right) ^ { d _ { 0 } } - 1 } { N - 2 } \\leq 2 N ^ { d _ { 0 } } .", + "type": "interline_equation", + "image_path": "2a0c240d5f62a5eac2e8d4314d8e8be31fcb5e7c6b97181552e8d6ceb4619032.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 162, + 528, + 448, + 540.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 162, + 540.0, + 448, + 552.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 162, + 552.0, + 448, + 564.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 486, + 603 + ], + "lines": [ + { + "bbox": [ + 102, + 576, + 488, + 603 + ], + "spans": [ + { + "bbox": [ + 102, + 576, + 236, + 601 + ], + "score": 1.0, + "content": "where in (1) we used the bound", + "type": "text" + }, + { + "bbox": [ + 236, + 576, + 298, + 603 + ], + "score": 0.92, + "content": "\\left( \\begin{array} { c } { { N } } \\\\ { { k } } \\end{array} \\right) \\le N ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 578, + 488, + 599 + ], + "score": 1.0, + "content": ", in (2) we used the sum of a geometric series.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 109, + 635, + 283, + 647 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 284, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 284, + 648 + ], + "score": 1.0, + "content": "12.2.2 A BASIC PROBABILISTIC BOUND", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 177, + 672 + ], + "score": 1.0, + "content": "Lemma 30. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 654, + 336, + 672 + ], + "score": 0.92, + "content": "\\mathbf { H } = \\left[ \\mathbf { h } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { h } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\in \\{ - 1 , 1 \\} ^ { d _ { 1 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 654, + 478, + 672 + ], + "score": 1.0, + "content": "be a deterministic binary matrix,", + "type": "text" + }, + { + "bbox": [ + 478, + 658, + 505, + 670 + ], + "score": 0.85, + "content": "{ \\textbf { W } } =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 670, + 504, + 687 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 219, + 687 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { w } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { w } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 670, + 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Using Lemma 28 we complete the proof.", + "type": "text" + }, + { + "bbox": [ + 494, + 210, + 505, + 221 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 143, + 506, + 222 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 242, + 230, + 254 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 230, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 230, + 255 + ], + "score": 1.0, + "content": "12.2 PROOF OF LEMMA 15", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 266, + 504, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "In this section we will prove Lemma 15 in subsection 12.3.3. This proof relies on two rather basic", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 277, + 354, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 354, + 288 + ], + "score": 1.0, + "content": "results, which we first prove in subsections 12.2.1 and 12.2.2.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 266, + 505, + 288 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 308, + 385, + 321 + ], + "lines": [ + { + "bbox": [ + 107, + 309, + 386, + 321 + ], + "spans": [ + { + "bbox": [ + 107, + 309, + 386, + 321 + ], + "score": 1.0, + "content": "12.2.1 NUMBER OF DICHOTOMIES INDUCED BY A HYPERPLANE", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 104, + 329, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 201, + 344 + ], + "score": 1.0, + "content": "Fact 29. A hyperplane", + "type": "text" + }, + { + "bbox": [ + 202, + 332, + 233, + 343 + ], + "score": 0.9, + "content": "{ \\mathbf w } \\in d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 329, + 372, + 344 + ], + "score": 1.0, + "content": "can separate a given set of points", + "type": "text" + }, + { + "bbox": [ + 373, + 330, + 504, + 344 + ], + "score": 0.93, + "content": "\\mathbf { X } = \\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 338, + 357 + ], + "score": 1.0, + "content": "into several different dichotomies, i.e., different results for", + "type": "text" + }, + { + "bbox": [ + 338, + 343, + 390, + 357 + ], + "score": 0.78, + "content": "\\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 343, + 505, + 357 + ], + "score": 1.0, + "content": ". The number of dichotomies", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 355, + 224, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 224, + 368 + ], + "score": 1.0, + "content": "is upper bounded as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 329, + 505, + 368 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 378, + 452, + 416 + ], + "lines": [ + { + "bbox": [ + 156, + 378, + 452, + 416 + ], + "spans": [ + { + "bbox": [ + 156, + 378, + 452, + 416 + ], + "score": 0.92, + "content": "\\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { N } } \\mathscr { T } \\left( \\exists \\mathbf { w } : \\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } \\right) = \\mathbf { h } ^ { \\top } \\right) \\leq 2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( \\begin{array} { l } { N - 1 } \\\\ { k } \\end{array} \\right) \\leq 2 N ^ { d _ { 0 } } .", + "type": "interline_equation", + "image_path": "276611149f850fb17a545865714e1e2821ff4642a53a796c1de7f19e11080b61.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 156, + 378, + 452, + 390.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 156, + 390.6666666666667, + 452, + 403.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 156, + 403.33333333333337, + 452, + 416.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 506, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "Proof. See (Cover, 1965, Theorem 1) for a proof of the left inequality as equality (the Schläfli", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 283, + 473 + ], + "score": 1.0, + "content": "Theorem) in the case that the columns of", + "type": "text" + }, + { + "bbox": [ + 284, + 461, + 294, + 471 + ], + "score": 0.56, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 460, + 459, + 473 + ], + "score": 1.0, + "content": "are in “general position” (which holds", + "type": "text" + }, + { + "bbox": [ + 460, + 461, + 469, + 471 + ], + "score": 0.58, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "-a.e, see", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 232, + 484 + ], + "score": 1.0, + "content": "definition in (Cover, 1965)) . If", + "type": "text" + }, + { + "bbox": [ + 232, + 472, + 242, + 482 + ], + "score": 0.44, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "is not in general position then this result becomes an upper bound,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 294, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 294, + 495 + ], + "score": 1.0, + "content": "since some dichotomies might not be possible.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 449, + 506, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 498, + 506, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 498, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 264, + 513 + ], + "score": 1.0, + "content": "Next, we prove the right inequality. For", + "type": "text" + }, + { + "bbox": [ + 265, + 500, + 294, + 510 + ], + "score": 0.89, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 498, + 311, + 513 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 500, + 340, + 510 + ], + "score": 0.89, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 498, + 474, + 513 + ], + "score": 1.0, + "content": "the inequality trivially holds. For", + "type": "text" + }, + { + "bbox": [ + 474, + 500, + 502, + 510 + ], + "score": 0.89, + "content": "N \\geq 3", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 498, + 507, + 513 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 509, + 143, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 143, + 524 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 498, + 507, + 524 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 528, + 448, + 564 + ], + "lines": [ + { + "bbox": [ + 162, + 528, + 448, + 564 + ], + "spans": [ + { + "bbox": [ + 162, + 528, + 448, + 564 + ], + "score": 0.95, + "content": "2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( \\begin{array} { c } { { N - 1 } } \\\\ { { k } } \\end{array} \\right) \\stackrel { \\left( 1 \\right) } { \\leq } 2 \\sum _ { k = 0 } ^ { d _ { 0 } - 1 } \\left( N - 1 \\right) ^ { k } \\stackrel { \\left( 2 \\right) } { \\leq } 2 \\frac { \\left( N - 1 \\right) ^ { d _ { 0 } } - 1 } { N - 2 } \\leq 2 N ^ { d _ { 0 } } .", + "type": "interline_equation", + "image_path": "2a0c240d5f62a5eac2e8d4314d8e8be31fcb5e7c6b97181552e8d6ceb4619032.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 162, + 528, + 448, + 540.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 162, + 540.0, + 448, + 552.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 162, + 552.0, + 448, + 564.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 486, + 603 + ], + "lines": [ + { + "bbox": [ + 102, + 576, + 488, + 603 + ], + "spans": [ + { + "bbox": [ + 102, + 576, + 236, + 601 + ], + "score": 1.0, + "content": "where in (1) we used the bound", + "type": "text" + }, + { + "bbox": [ + 236, + 576, + 298, + 603 + ], + "score": 0.92, + "content": "\\left( \\begin{array} { c } { { N } } \\\\ { { k } } \\end{array} \\right) \\le N ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 578, + 488, + 599 + ], + "score": 1.0, + "content": ", in (2) we used the sum of a geometric series.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 102, + 576, + 488, + 603 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 635, + 283, + 647 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 284, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 284, + 648 + ], + "score": 1.0, + "content": "12.2.2 A BASIC PROBABILISTIC BOUND", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 177, + 672 + ], + "score": 1.0, + "content": "Lemma 30. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 654, + 336, + 672 + ], + "score": 0.92, + "content": "\\mathbf { H } = \\left[ \\mathbf { h } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { h } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\in \\{ - 1 , 1 \\} ^ { d _ { 1 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 654, + 478, + 672 + ], + "score": 1.0, + "content": "be a deterministic binary matrix,", + "type": "text" + }, + { + "bbox": [ + 478, + 658, + 505, + 670 + ], + "score": 0.85, + "content": "{ \\textbf { W } } =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 670, + 504, + 687 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 219, + 687 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { w } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { w } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 670, + 455, + 687 + ], + "score": 1.0, + "content": "be an independent standard random Gaussian matrix, and", + "type": "text" + }, + { + "bbox": [ + 456, + 672, + 504, + 684 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 684, + 405, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 405, + 698 + ], + "score": 1.0, + "content": "be a random matrix with independent and identically distributed columns.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 654, + 505, + 698 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 708, + 424, + 736 + ], + "lines": [ + { + "bbox": [ + 186, + 708, + 424, + 736 + ], + "spans": [ + { + "bbox": [ + 186, + 708, + 424, + 736 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\operatorname { s i g n } \\left( { \\mathbf { W } } { \\mathbf { X } } \\right) = { \\mathbf { H } } \\right) \\leq \\left( \\begin{array} { c } { k } \\\\ { \\lfloor k / 2 \\rfloor } \\end{array} \\right) \\mathbb { P } \\left( { \\mathbf { W } } { \\mathbf { X } } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "eca24cd528c0ee1bac7acc6fa2f0a83813f1e2a9ff3bcd5b862de87b6cea02d1.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 708, + 424, + 722.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 186, + 722.0, + 424, + 736.0 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 221, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 222, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 222, + 95 + ], + "score": 1.0, + "content": "Proof. By direct calculation", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 100, + 507, + 244 + ], + "lines": [ + { + "bbox": [ + 111, + 100, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 507, + 244 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\displaystyle \\widehat { \\mathbf { \\xi } } \\displaystyle \\geq ( \\mathrm { s i g n } ( \\mathbf { W X } ) = \\mathbf { H } ) = } & { \\displaystyle \\mathbb { E } [ \\mathbb { P } ( \\mathrm { s i g n } ( \\mathbf { W X } ) = \\mathbf { H } \\vert \\mathbf { X } ) ] \\stackrel { ( 1 ) } { = } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d } \\mathbb { P } ( \\mathrm { s i g n } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } ) = \\mathbf { h } _ { i } ^ { \\top } \\vert \\mathbf { X } ) ] } \\\\ & { \\stackrel { ( 2 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { \\delta ( \\mathbf { h } _ { i } ) } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 3 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S _ { i } } > 0 \\vert \\mathbf { X } ) ] } \\\\ & { \\stackrel { ( 4 ) } { = } \\mathbb { E } [ \\mathbb { P } ( \\mathbf { W X } _ { S _ { s } } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 5 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\sum _ { S \\geq [ R _ { 1 } ] : S \\mid = \\lfloor k / 2 \\rfloor } } \\\\ & { = \\displaystyle \\sum _ { S \\leq [ k ] : S = \\lfloor k / 2 \\rfloor } \\mathbb { E } [ \\mathbb { P } ( \\mathbf { W X } _ { S } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 6 ) } { = } ( \\begin{array} { l } { k } \\\\ { \\lfloor k / 2 \\rfloor } \\end{array} ) \\mathbb { P } ( \\mathbf { W } _ { \\mathbf { X } _ { \\lfloor k / 2 \\rfloor } } > 0 ) . } \\end{array}", + "type": "interline_equation", + "image_path": "d1fdd0cb76eea7fd5e7e1cd6e223b98a7ba70b9e98cb7a40eec7b5bb8f8a5526.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 100, + 507, + 148.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 148.0, + 507, + 196.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 196.0, + 507, + 244.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 132, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 134, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 134, + 261 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 130, + 272, + 292, + 284 + ], + "lines": [ + { + "bbox": [ + 129, + 270, + 293, + 286 + ], + "spans": [ + { + "bbox": [ + 129, + 270, + 276, + 286 + ], + "score": 1.0, + "content": "1. We used the independence of the", + "type": "text" + }, + { + "bbox": [ + 276, + 274, + 289, + 283 + ], + "score": 0.87, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 270, + 293, + 286 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 130, + 291, + 506, + 345 + ], + "lines": [ + { + "bbox": [ + 128, + 290, + 507, + 306 + ], + "spans": [ + { + "bbox": [ + 128, + 290, + 190, + 306 + ], + "score": 1.0, + "content": "2. We define", + "type": "text" + }, + { + "bbox": [ + 190, + 291, + 333, + 306 + ], + "score": 0.92, + "content": "\\hat { S } _ { \\pm } \\left( \\mathbf { h } \\right) \\ \\triangleq \\ \\left\\{ S \\subset \\left[ k \\right] : \\pm \\mathbf { h } _ { S } ^ { \\top } > 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 290, + 428, + 306 + ], + "score": 1.0, + "content": "as the sets in which", + "type": "text" + }, + { + "bbox": [ + 428, + 294, + 437, + 303 + ], + "score": 0.36, + "content": "\\mathbf { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 290, + 507, + 306 + ], + "score": 1.0, + "content": "is always posi-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 217, + 318 + ], + "score": 1.0, + "content": "tive/negative, and", + "type": "text" + }, + { + "bbox": [ + 217, + 305, + 241, + 319 + ], + "score": 0.91, + "content": "\\hat { S } \\left( \\mathbf { h } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 306, + 446, + 318 + ], + "score": 1.0, + "content": "as the maximal set between these two. Note that", + "type": "text" + }, + { + "bbox": [ + 446, + 308, + 459, + 318 + ], + "score": 0.85, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 306, + 506, + 318 + ], + "score": 1.0, + "content": "has a stan-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 317, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 393, + 332 + ], + "score": 1.0, + "content": "dard normal distribution which is symmetric to sign flips, so", + "type": "text" + }, + { + "bbox": [ + 393, + 318, + 505, + 332 + ], + "score": 0.91, + "content": "\\forall S : \\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } > 0 | \\mathbf { X } \\right) =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 329, + 224, + 346 + ], + "spans": [ + { + "bbox": [ + 142, + 330, + 219, + 345 + ], + "score": 0.9, + "content": "\\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } < 0 | \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 329, + 224, + 346 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 129, + 351, + 502, + 376 + ], + "lines": [ + { + "bbox": [ + 128, + 351, + 502, + 372 + ], + "spans": [ + { + "bbox": [ + 128, + 351, + 182, + 371 + ], + "score": 1.0, + "content": "3. Note that", + "type": "text" + }, + { + "bbox": [ + 182, + 352, + 249, + 372 + ], + "score": 0.92, + "content": "\\left| \\hat { S } \\left( \\mathbf { h } \\right) \\right| \\ge \\lfloor k / 2 \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 351, + 366, + 371 + ], + "score": 1.0, + "content": ". Therefore, we define S∗ =", + "type": "text" + }, + { + "bbox": [ + 374, + 354, + 421, + 369 + ], + "score": 1.0, + "content": "argmax", + "type": "text" + }, + { + "bbox": [ + 421, + 354, + 499, + 368 + ], + "score": 0.59, + "content": "\\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } > 0 | \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 354, + 502, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 363, + 367, + 423, + 376 + ], + "spans": [ + { + "bbox": [ + 363, + 367, + 423, + 376 + ], + "score": 0.59, + "content": "S C [ k ] : \\bar { | } S | = \\lfloor k / 2 \\rfloor", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 131, + 383, + 291, + 396 + ], + "lines": [ + { + "bbox": [ + 128, + 381, + 289, + 398 + ], + "spans": [ + { + "bbox": [ + 128, + 381, + 276, + 398 + ], + "score": 1.0, + "content": "4. We used the independence of the", + "type": "text" + }, + { + "bbox": [ + 276, + 385, + 289, + 394 + ], + "score": 0.9, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 129, + 402, + 442, + 415 + ], + "lines": [ + { + "bbox": [ + 129, + 401, + 443, + 416 + ], + "spans": [ + { + "bbox": [ + 129, + 401, + 443, + 416 + ], + "score": 1.0, + "content": "5. The maximum is a single term in the following sum of non-negative terms.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 130, + 421, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 128, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 128, + 420, + 259, + 435 + ], + "score": 1.0, + "content": "6. Taking the expectation over", + "type": "text" + }, + { + "bbox": [ + 259, + 422, + 269, + 432 + ], + "score": 0.63, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 420, + 362, + 435 + ], + "score": 1.0, + "content": ", since the columns of", + "type": "text" + }, + { + "bbox": [ + 362, + 422, + 372, + 432 + ], + "score": 0.42, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "are independent and identically", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 250, + 446 + ], + "score": 1.0, + "content": "distributed, the location of", + "type": "text" + }, + { + "bbox": [ + 250, + 433, + 258, + 443 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "does not affect the probability. Therefore, we can set without", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 443, + 268, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 213, + 457 + ], + "score": 1.0, + "content": "loss of generality", + "type": "text" + }, + { + "bbox": [ + 214, + 443, + 264, + 456 + ], + "score": 0.93, + "content": "S = [ \\lfloor k / 2 \\rfloor ]", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 443, + 268, + 457 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 107, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 107, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "12.2.3 MAIN PROOF: BOUND ON THE NUMBER OF CONFIGURATIONS FOR A BINARY MATRIX", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 144, + 506, + 236, + 516 + ], + "spans": [ + { + "bbox": [ + 144, + 506, + 236, + 516 + ], + "score": 1.0, + "content": "WITH CERTAIN RANK", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 265, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 266, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 185, + 538 + ], + "score": 1.0, + "content": "Recall the function", + "type": "text" + }, + { + "bbox": [ + 185, + 523, + 203, + 536 + ], + "score": 0.89, + "content": "a \\left( \\cdot \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 522, + 266, + 538 + ], + "score": 1.0, + "content": "from eq. (2.1):", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 542, + 360, + 570 + ], + "lines": [ + { + "bbox": [ + 251, + 542, + 360, + 570 + ], + "spans": [ + { + "bbox": [ + 251, + 542, + 360, + 570 + ], + "score": 0.94, + "content": "a \\left( u \\right) \\triangleq \\left\\{ \\begin{array} { l l } { 1 } & { , \\mathrm { i f } , u > 0 } \\\\ { \\rho } & { , \\mathrm { i f } u < 0 } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "9b2221403addba9e787fbde7841108c759c1753d7275c9d8c998715393cc168d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 251, + 542, + 360, + 570 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 160, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 158, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 133, + 589 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 576, + 158, + 588 + ], + "score": 0.9, + "content": "\\rho \\neq 1", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 506, + 626 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 274, + 604 + ], + "score": 1.0, + "content": "Lemma 31. (Lemma 15 restated). Let", + "type": "text" + }, + { + "bbox": [ + 275, + 590, + 326, + 601 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 589, + 507, + 604 + ], + "score": 1.0, + "content": "be a random matrix with independent and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 599, + 507, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 249, + 615 + ], + "score": 1.0, + "content": "identically distributed columns, and", + "type": "text" + }, + { + "bbox": [ + 250, + 602, + 304, + 613 + ], + "score": 0.89, + "content": "\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 599, + 507, + 615 + ], + "score": 1.0, + "content": "an independent standard random Gaussian matrix.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 612, + 253, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 178, + 626 + ], + "score": 1.0, + "content": "Then, in the limit", + "type": "text" + }, + { + "bbox": [ + 178, + 613, + 249, + 625 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 612, + 253, + 626 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 631, + 448, + 648 + ], + "lines": [ + { + "bbox": [ + 162, + 631, + 448, + 648 + ], + "spans": [ + { + "bbox": [ + 162, + 631, + 448, + 648 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( a \\left( \\mathbf { W } \\mathbf { X } \\right) \\right) = r \\right) \\dot { \\leq } 2 ^ { k + r d _ { 0 } ( \\log d _ { 1 } + \\log k ) + r ^ { 2 } } \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "364630bd489ca246d1fca1bcf304f112c24d0e6abb1816a7e761cdeb8aa75aaf.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 162, + 631, + 448, + 648 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 660, + 506, + 736 + ], + "lines": [ + { + "bbox": [ + 104, + 658, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 180, + 678 + ], + "score": 1.0, + "content": "Proof. We denote", + "type": "text" + }, + { + "bbox": [ + 180, + 661, + 294, + 675 + ], + "score": 0.93, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right) \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 658, + 424, + 678 + ], + "score": 1.0, + "content": ". For any such A for which rank", + "type": "text" + }, + { + "bbox": [ + 425, + 663, + 460, + 675 + ], + "score": 0.91, + "content": "( \\mathbf { A } ) = r", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 658, + 507, + 678 + ], + "score": 1.0, + "content": ", we have a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 674, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 159, + 694 + ], + "score": 1.0, + "content": "collection of", + "type": "text" + }, + { + "bbox": [ + 160, + 684, + 165, + 691 + ], + "score": 0.72, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 679, + 351, + 694 + ], + "score": 1.0, + "content": "rows that span the remaining rows. There are", + "type": "text" + }, + { + "bbox": [ + 351, + 674, + 385, + 700 + ], + "score": 0.92, + "content": "\\left( \\begin{array} { c } { d _ { 1 } } \\\\ { r } \\end{array} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 680, + 497, + 695 + ], + "score": 1.0, + "content": "possible locations for these", + "type": "text" + }, + { + "bbox": [ + 498, + 684, + 504, + 691 + ], + "score": 0.67, + "content": "r", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 329, + 711 + ], + "score": 1.0, + "content": "spanning rows. In these rows there exist a collection of", + "type": "text" + }, + { + "bbox": [ + 329, + 701, + 334, + 708 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "columns that span the remaining columns.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 709, + 506, + 736 + ], + "spans": [ + { + "bbox": [ + 105, + 714, + 146, + 731 + ], + "score": 1.0, + "content": "There are", + "type": "text" + }, + { + "bbox": [ + 146, + 709, + 176, + 736 + ], + "score": 0.92, + "content": "\\left( \\begin{array} { l } { k } \\\\ { r } \\end{array} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 714, + 285, + 731 + ], + "score": 1.0, + "content": "possible locations for these", + "type": "text" + }, + { + "bbox": [ + 286, + 720, + 291, + 726 + ], + "score": 0.75, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 714, + 506, + 731 + ], + "score": 1.0, + "content": "spanning columns. At the intersection of the spanning", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 468, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 495, + 469, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 495, + 469, + 505, + 479 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 221, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 222, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 222, + 95 + ], + "score": 1.0, + "content": "Proof. By direct calculation", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 222, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 100, + 507, + 244 + ], + "lines": [ + { + "bbox": [ + 111, + 100, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 507, + 244 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\displaystyle \\widehat { \\mathbf { \\xi } } \\displaystyle \\geq ( \\mathrm { s i g n } ( \\mathbf { W X } ) = \\mathbf { H } ) = } & { \\displaystyle \\mathbb { E } [ \\mathbb { P } ( \\mathrm { s i g n } ( \\mathbf { W X } ) = \\mathbf { H } \\vert \\mathbf { X } ) ] \\stackrel { ( 1 ) } { = } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d } \\mathbb { P } ( \\mathrm { s i g n } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } ) = \\mathbf { h } _ { i } ^ { \\top } \\vert \\mathbf { X } ) ] } \\\\ & { \\stackrel { ( 2 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { \\delta ( \\mathbf { h } _ { i } ) } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 3 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\prod _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } ( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S _ { i } } > 0 \\vert \\mathbf { X } ) ] } \\\\ & { \\stackrel { ( 4 ) } { = } \\mathbb { E } [ \\mathbb { P } ( \\mathbf { W X } _ { S _ { s } } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 5 ) } { \\leq } \\mathbb { E } [ \\displaystyle \\sum _ { S \\geq [ R _ { 1 } ] : S \\mid = \\lfloor k / 2 \\rfloor } } \\\\ & { = \\displaystyle \\sum _ { S \\leq [ k ] : S = \\lfloor k / 2 \\rfloor } \\mathbb { E } [ \\mathbb { P } ( \\mathbf { W X } _ { S } > 0 \\vert \\mathbf { X } ) ] \\stackrel { ( 6 ) } { = } ( \\begin{array} { l } { k } \\\\ { \\lfloor k / 2 \\rfloor } \\end{array} ) \\mathbb { P } ( \\mathbf { W } _ { \\mathbf { X } _ { \\lfloor k / 2 \\rfloor } } > 0 ) . } \\end{array}", + "type": "interline_equation", + "image_path": "d1fdd0cb76eea7fd5e7e1cd6e223b98a7ba70b9e98cb7a40eec7b5bb8f8a5526.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 100, + 507, + 148.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 148.0, + 507, + 196.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 196.0, + 507, + 244.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 132, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 134, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 134, + 261 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 246, + 134, + 261 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 272, + 292, + 284 + ], + "lines": [ + { + "bbox": [ + 129, + 270, + 293, + 286 + ], + "spans": [ + { + "bbox": [ + 129, + 270, + 276, + 286 + ], + "score": 1.0, + "content": "1. We used the independence of the", + "type": "text" + }, + { + "bbox": [ + 276, + 274, + 289, + 283 + ], + "score": 0.87, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 270, + 293, + 286 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 129, + 270, + 293, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 291, + 506, + 345 + ], + "lines": [ + { + "bbox": [ + 128, + 290, + 507, + 306 + ], + "spans": [ + { + "bbox": [ + 128, + 290, + 190, + 306 + ], + "score": 1.0, + "content": "2. We define", + "type": "text" + }, + { + "bbox": [ + 190, + 291, + 333, + 306 + ], + "score": 0.92, + "content": "\\hat { S } _ { \\pm } \\left( \\mathbf { h } \\right) \\ \\triangleq \\ \\left\\{ S \\subset \\left[ k \\right] : \\pm \\mathbf { h } _ { S } ^ { \\top } > 0 \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 290, + 428, + 306 + ], + "score": 1.0, + "content": "as the sets in which", + "type": "text" + }, + { + "bbox": [ + 428, + 294, + 437, + 303 + ], + "score": 0.36, + "content": "\\mathbf { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 290, + 507, + 306 + ], + "score": 1.0, + "content": "is always posi-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 217, + 318 + ], + "score": 1.0, + "content": "tive/negative, and", + "type": "text" + }, + { + "bbox": [ + 217, + 305, + 241, + 319 + ], + "score": 0.91, + "content": "\\hat { S } \\left( \\mathbf { h } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 306, + 446, + 318 + ], + "score": 1.0, + "content": "as the maximal set between these two. Note that", + "type": "text" + }, + { + "bbox": [ + 446, + 308, + 459, + 318 + ], + "score": 0.85, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 306, + 506, + 318 + ], + "score": 1.0, + "content": "has a stan-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 317, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 393, + 332 + ], + "score": 1.0, + "content": "dard normal distribution which is symmetric to sign flips, so", + "type": "text" + }, + { + "bbox": [ + 393, + 318, + 505, + 332 + ], + "score": 0.91, + "content": "\\forall S : \\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } > 0 | \\mathbf { X } \\right) =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 329, + 224, + 346 + ], + "spans": [ + { + "bbox": [ + 142, + 330, + 219, + 345 + ], + "score": 0.9, + "content": "\\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } < 0 | \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 329, + 224, + 346 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 128, + 290, + 507, + 346 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 351, + 502, + 376 + ], + "lines": [ + { + "bbox": [ + 128, + 351, + 502, + 372 + ], + "spans": [ + { + "bbox": [ + 128, + 351, + 182, + 371 + ], + "score": 1.0, + "content": "3. Note that", + "type": "text" + }, + { + "bbox": [ + 182, + 352, + 249, + 372 + ], + "score": 0.92, + "content": "\\left| \\hat { S } \\left( \\mathbf { h } \\right) \\right| \\ge \\lfloor k / 2 \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 351, + 366, + 371 + ], + "score": 1.0, + "content": ". Therefore, we define S∗ =", + "type": "text" + }, + { + "bbox": [ + 374, + 354, + 421, + 369 + ], + "score": 1.0, + "content": "argmax", + "type": "text" + }, + { + "bbox": [ + 421, + 354, + 499, + 368 + ], + "score": 0.59, + "content": "\\mathbb { P } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } _ { S } > 0 | \\mathbf { X } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 354, + 502, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 363, + 367, + 423, + 376 + ], + "spans": [ + { + "bbox": [ + 363, + 367, + 423, + 376 + ], + "score": 0.59, + "content": "S C [ k ] : \\bar { | } S | = \\lfloor k / 2 \\rfloor", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 128, + 351, + 502, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 383, + 291, + 396 + ], + "lines": [ + { + "bbox": [ + 128, + 381, + 289, + 398 + ], + "spans": [ + { + "bbox": [ + 128, + 381, + 276, + 398 + ], + "score": 1.0, + "content": "4. We used the independence of the", + "type": "text" + }, + { + "bbox": [ + 276, + 385, + 289, + 394 + ], + "score": 0.9, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 128, + 381, + 289, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 402, + 442, + 415 + ], + "lines": [ + { + "bbox": [ + 129, + 401, + 443, + 416 + ], + "spans": [ + { + "bbox": [ + 129, + 401, + 443, + 416 + ], + "score": 1.0, + "content": "5. The maximum is a single term in the following sum of non-negative terms.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 129, + 401, + 443, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 421, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 128, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 128, + 420, + 259, + 435 + ], + "score": 1.0, + "content": "6. Taking the expectation over", + "type": "text" + }, + { + "bbox": [ + 259, + 422, + 269, + 432 + ], + "score": 0.63, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 420, + 362, + 435 + ], + "score": 1.0, + "content": ", since the columns of", + "type": "text" + }, + { + "bbox": [ + 362, + 422, + 372, + 432 + ], + "score": 0.42, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "are independent and identically", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 250, + 446 + ], + "score": 1.0, + "content": "distributed, the location of", + "type": "text" + }, + { + "bbox": [ + 250, + 433, + 258, + 443 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "does not affect the probability. Therefore, we can set without", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 443, + 268, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 213, + 457 + ], + "score": 1.0, + "content": "loss of generality", + "type": "text" + }, + { + "bbox": [ + 214, + 443, + 264, + 456 + ], + "score": 0.93, + "content": "S = [ \\lfloor k / 2 \\rfloor ]", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 443, + 268, + 457 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 128, + 420, + 506, + 457 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 107, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 107, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "12.2.3 MAIN PROOF: BOUND ON THE NUMBER OF CONFIGURATIONS FOR A BINARY MATRIX", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 144, + 506, + 236, + 516 + ], + "spans": [ + { + "bbox": [ + 144, + 506, + 236, + 516 + ], + "score": 1.0, + "content": "WITH CERTAIN RANK", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 265, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 266, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 185, + 538 + ], + "score": 1.0, + "content": "Recall the function", + "type": "text" + }, + { + "bbox": [ + 185, + 523, + 203, + 536 + ], + "score": 0.89, + "content": "a \\left( \\cdot \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 522, + 266, + 538 + ], + "score": 1.0, + "content": "from eq. (2.1):", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 522, + 266, + 538 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 542, + 360, + 570 + ], + "lines": [ + { + "bbox": [ + 251, + 542, + 360, + 570 + ], + "spans": [ + { + "bbox": [ + 251, + 542, + 360, + 570 + ], + "score": 0.94, + "content": "a \\left( u \\right) \\triangleq \\left\\{ \\begin{array} { l l } { 1 } & { , \\mathrm { i f } , u > 0 } \\\\ { \\rho } & { , \\mathrm { i f } u < 0 } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "9b2221403addba9e787fbde7841108c759c1753d7275c9d8c998715393cc168d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 251, + 542, + 360, + 570 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 160, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 158, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 133, + 589 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 576, + 158, + 588 + ], + "score": 0.9, + "content": "\\rho \\neq 1", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 574, + 158, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 506, + 626 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 274, + 604 + ], + "score": 1.0, + "content": "Lemma 31. (Lemma 15 restated). Let", + "type": "text" + }, + { + "bbox": [ + 275, + 590, + 326, + 601 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 589, + 507, + 604 + ], + "score": 1.0, + "content": "be a random matrix with independent and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 599, + 507, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 249, + 615 + ], + "score": 1.0, + "content": "identically distributed columns, and", + "type": "text" + }, + { + "bbox": [ + 250, + 602, + 304, + 613 + ], + "score": 0.89, + "content": "\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 599, + 507, + 615 + ], + "score": 1.0, + "content": "an independent standard random Gaussian matrix.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 612, + 253, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 178, + 626 + ], + "score": 1.0, + "content": "Then, in the limit", + "type": "text" + }, + { + "bbox": [ + 178, + 613, + 249, + 625 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 612, + 253, + 626 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 589, + 507, + 626 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 631, + 448, + 648 + ], + "lines": [ + { + "bbox": [ + 162, + 631, + 448, + 648 + ], + "spans": [ + { + "bbox": [ + 162, + 631, + 448, + 648 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( a \\left( \\mathbf { W } \\mathbf { X } \\right) \\right) = r \\right) \\dot { \\leq } 2 ^ { k + r d _ { 0 } ( \\log d _ { 1 } + \\log k ) + r ^ { 2 } } \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "364630bd489ca246d1fca1bcf304f112c24d0e6abb1816a7e761cdeb8aa75aaf.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 162, + 631, + 448, + 648 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 660, + 506, + 736 + ], + "lines": [ + { + "bbox": [ + 104, + 658, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 180, + 678 + ], + "score": 1.0, + "content": "Proof. We denote", + "type": "text" + }, + { + "bbox": [ + 180, + 661, + 294, + 675 + ], + "score": 0.93, + "content": "\\mathbf { A } = a \\left( \\mathbf { W } \\mathbf { X } \\right) \\in \\{ \\rho , 1 \\} ^ { d _ { 1 } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 658, + 424, + 678 + ], + "score": 1.0, + "content": ". For any such A for which rank", + "type": "text" + }, + { + "bbox": [ + 425, + 663, + 460, + 675 + ], + "score": 0.91, + "content": "( \\mathbf { A } ) = r", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 658, + 507, + 678 + ], + "score": 1.0, + "content": ", we have a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 674, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 159, + 694 + ], + "score": 1.0, + "content": "collection of", + "type": "text" + }, + { + "bbox": [ + 160, + 684, + 165, + 691 + ], + "score": 0.72, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 679, + 351, + 694 + ], + "score": 1.0, + "content": "rows that span the remaining rows. There are", + "type": "text" + }, + { + "bbox": [ + 351, + 674, + 385, + 700 + ], + "score": 0.92, + "content": "\\left( \\begin{array} { c } { d _ { 1 } } \\\\ { r } \\end{array} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 680, + 497, + 695 + ], + "score": 1.0, + "content": "possible locations for these", + "type": "text" + }, + { + "bbox": [ + 498, + 684, + 504, + 691 + ], + "score": 0.67, + "content": "r", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 329, + 711 + ], + "score": 1.0, + "content": "spanning rows. 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We denote", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 379, + 81, + 389, + 93 + ], + "score": 0.82, + "content": "\\tilde { \\mathbf { A } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 389, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "as the matrix A which rows", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 349, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 243, + 106 + ], + "score": 1.0, + "content": "and columns are permuted so that", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 243, + 94, + 253, + 104 + ], + "score": 0.68, + "content": "\\mathbf { D }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 254, + 93, + 349, + 106 + ], + "score": 1.0, + "content": "is the lower right block", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 658, + 507, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 379, + 96 + ], + "score": 1.0, + "content": "rows and columns, there exist a full rank sub-matrix D. We denote", + "type": "text" + }, + { + "bbox": [ + 379, + 81, + 389, + 93 + ], + "score": 0.82, + "content": "\\tilde { \\mathbf { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "as the matrix A which rows", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 349, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 243, + 106 + ], + "score": 1.0, + "content": "and columns are permuted so that", + "type": "text" + }, + { + "bbox": [ + 243, + 94, + 253, + 104 + ], + "score": 0.68, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 93, + 349, + 106 + ], + "score": 1.0, + "content": "is the lower right block", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 106, + 402, + 134 + ], + "lines": [ + { + "bbox": [ + 207, + 106, + 402, + 134 + ], + "spans": [ + { + "bbox": [ + 207, + 106, 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1 , 1 \\} ^ { ( d _ { \\mathbf { X } } _ { 2 } : \\mathrm { ~ s i g n } } } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 2 } ) = \\mathbf { H } ) ] [ \\displaystyle \\sum _ { \\mathbf { H } \\in \\{ - 1 , 1 \\} ^ { r \\times ( k - r ) } } \\mathbb { Z } ( \\exists \\mathbf { W } _ { 2 } : \\mathrm { s i g n } ( \\mathbf { W } _ { 2 } \\mathbf { X } _ { 1 } ) = \\mathbf { H } ) ] ] | \\mathrm { s i g n } } \\\\ & { \\leq \\mathbb { R } [ 2 ^ { r ^ { 2 } } [ \\displaystyle \\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { ( d _ { \\mathbf { X } } : \\mathrm { s i g n } } } ( \\mathbf { W } _ { 1 } \\mathbf { x } ) = \\mathbf { h } ) ] ^ { r } [ \\displaystyle \\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { ( k - r ) } } \\mathbb { Z } ( \\exists \\mathbf { w } : \\mathrm { s i g n } ( \\mathbf { w } ^ { \\mathsf { T } } \\mathbf { X } _ { 1 } ) = \\mathbf { h } ^ { \\mathsf { T } } ) ] ^ { r } ] | \\mathrm { s i g n } ( \\mathbf { W } _ { 1 } \\mathbf { X } _ { 1 } ) } \\\\ & { \\overset { 3 } { \\leq } \\mathbb { R } [ 2 ^ { r ^ { 2 } } 2 ^ { r d _ { 0 } \\log ( d _ { 1 } - 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(12.12) is upper bounded", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 141, + 165, + 352, + 177 + ], + "score": 1.0, + "content": "by the product of the number of possible values of", + "type": "text" + }, + { + "bbox": [ + 352, + 165, + 362, + 175 + ], + "score": 0.25, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 165, + 366, + 177 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 366, + 165, + 376, + 175 + ], + "score": 0.31, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 165, + 394, + 177 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 165, + 404, + 175 + ], + "score": 0.57, + "content": "\\mathbf { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 165, + 506, + 177 + ], + "score": 1.0, + "content": ", which is product in the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 175, + 223, + 190 + ], + "spans": [ + { + "bbox": [ + 141, + 175, + 223, + 190 + ], + "score": 1.0, + "content": "following equation.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 129, + 194, + 506, + 233 + ], + "lines": [ + { + "bbox": [ + 128, + 192, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 128, + 192, + 195, + 210 + ], + "score": 1.0, + "content": "3. The function", + "type": "text" + }, + { + "bbox": [ + 195, + 194, + 381, + 209 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { \\mathbf { h } \\in \\{ - 1 , 1 \\} ^ { ( k - r ) } } \\mathscr { T } \\left( \\exists \\mathbf { w } : \\mathrm { s i g n } \\left( \\mathbf { w } ^ { \\top } \\mathbf { X } _ { 1 } \\right) = \\mathbf { h } ^ { \\top } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 192, + 507, + 210 + ], + "score": 1.0, + "content": "\u0001 counts the number of dichoto-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 207, + 504, + 220 + ], + "spans": [ + { + "bbox": [ + 142, + 207, + 358, + 220 + ], + "score": 1.0, + "content": "mies that can be induced by the linear classifier w on", + "type": "text" + }, + { + "bbox": [ + 359, + 209, + 373, + 219 + ], + "score": 0.87, + "content": "\\mathbf { X } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 207, + 504, + 220 + ], + "score": 1.0, + "content": ". Using eq. (12.7) we can bound", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 218, + 483, + 235 + ], + "spans": [ + { + "bbox": [ + 141, + 218, + 205, + 235 + ], + "score": 1.0, + "content": "this number by", + "type": "text" + }, + { + "bbox": [ + 205, + 219, + 250, + 233 + ], + "score": 0.91, + "content": "2 \\left( k - r \\right) ^ { d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 218, + 431, + 235 + ], + "score": 1.0, + "content": ". Similarly, the other sum can be bounded by", + "type": "text" + }, + { + "bbox": [ + 431, + 220, + 477, + 233 + ], + "score": 0.92, + "content": "2 \\left( d _ { 1 } - r \\right) ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 218, + 483, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 329, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 328, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 328, + 260 + ], + "score": 1.0, + "content": "Combining eqs. (12.10), (12.11) and (12.13) we obtain", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 263, + 494, + 308 + ], + "lines": [ + { + "bbox": [ + 116, + 263, + 494, + 308 + ], + "spans": [ + { + "bbox": [ + 116, + 263, + 494, + 308 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( \\mathbf { A } \\right) = r \\right) \\leq } \\\\ & { \\left( \\begin{array} { l } { d _ { 1 } } \\\\ { r } \\end{array} \\right) \\left( \\begin{array} { l } { k } \\\\ { r } \\end{array} \\right) \\left( \\begin{array} { c } { k - r } \\\\ { \\lfloor ( k - r ) / 2 \\rfloor } \\end{array} \\right) 2 ^ { r d _ { 0 } \\left[ \\log ( d _ { 1 } - r ) + \\log ( k - r ) \\right] + r ^ { 2 } + 2 r } \\mathbb { P } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { [ \\lfloor ( k - r ) / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "6468b0a86acb106249f015f9b911214c7457c2064589c4896bb71f31ee587b31.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 116, + 263, + 494, + 278.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 116, + 278.0, + 494, + 293.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 116, + 293.0, + 494, + 308.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 103, + 309, + 511, + 342 + ], + "spans": [ + { + "bbox": [ + 103, + 313, + 274, + 337 + ], + "score": 1.0, + "content": "Next, we take the log. To upper bound", + "type": "text" + }, + { + "bbox": [ + 274, + 312, + 308, + 340 + ], + "score": 0.92, + "content": "\\left( \\begin{array} { c } { { N } } \\\\ { { k } } \\end{array} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 309, + 353, + 342 + ], + "score": 1.0, + "content": ", for small", + "type": "text" + }, + { + "bbox": [ + 354, + 321, + 361, + 330 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 309, + 395, + 342 + ], + "score": 1.0, + "content": "we use", + "type": "text" + }, + { + "bbox": [ + 395, + 313, + 460, + 340 + ], + "score": 0.93, + "content": "\\left( \\begin{array} { c } { { N } } \\\\ { { k } } \\end{array} \\right) \\le N ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 309, + 511, + 342 + ], + "score": 1.0, + "content": ", while for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 330, + 329, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 145, + 357 + ], + "score": 0.89, + "content": "k = N / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 330, + 179, + 371 + ], + "score": 1.0, + "content": ", we use", + "type": "text" + }, + { + "bbox": [ + 179, + 337, + 249, + 365 + ], + "score": 0.93, + "content": "\\left( \\begin{array} { c } { { N } } \\\\ { { N / 2 } } \\end{array} \\right) \\le 2 ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 330, + 329, + 371 + ], + "score": 1.0, + "content": ". Thus, we obtain", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 369, + 489, + 402 + ], + "lines": [ + { + "bbox": [ + 120, + 369, + 489, + 402 + ], + "spans": [ + { + "bbox": [ + 120, + 369, + 489, + 402 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } & { } & { \\log \\mathbb { P } \\left( \\mathrm { r a n k } \\left( \\mathbf { A } \\right) = r \\right) \\leq \\left( r d _ { 0 } \\left( \\log \\left( d _ { 1 } - r \\right) + \\log \\left( k - r \\right) \\right) + r ^ { 2 } + 2 r \\right) \\log 2 \\qquad ( 1 2 \\triangle ( r \\mathrm { ~ o ~ t ~ } r ) } \\\\ & { } & { + r \\log d _ { 1 } + r \\log k + ( k - r ) \\log 2 + \\log \\mathbb { P } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { [ \\vert ( k - r ) / 2 ] ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "4d194bd1eb0e846077da5483deda01a9c7113979aeb679496db42d74ef0750c2.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 120, + 369, + 489, + 380.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 120, + 380.0, + 489, + 391.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 120, + 391.0, + 489, + 402.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 407, + 469, + 421 + ], + "lines": [ + { + "bbox": [ + 104, + 405, + 468, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 165, + 424 + ], + "score": 1.0, + "content": "Recalling that", + "type": "text" + }, + { + "bbox": [ + 165, + 407, + 240, + 420 + ], + "score": 0.93, + "content": "\\mathbf { W } _ { 1 } \\in \\mathbb { R } ^ { ( d _ { 1 } - r ) \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 405, + 267, + 424 + ], + "score": 1.0, + "content": "while", + "type": "text" + }, + { + "bbox": [ + 267, + 408, + 321, + 419 + ], + "score": 0.92, + "content": "\\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 405, + 468, + 424 + ], + "score": 1.0, + "content": ", we obtain from Jensen’s inequality", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 426, + 454, + 454 + ], + "lines": [ + { + "bbox": [ + 128, + 426, + 454, + 454 + ], + "spans": [ + { + "bbox": [ + 128, + 426, + 454, + 454 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\log \\mathbb { P } \\left( \\mathbf { W } _ { 1 } \\mathbf { X } _ { [ \\lfloor ( k - r ) / 2 \\rfloor ] } > 0 \\right) \\le \\frac { \\left\\lfloor \\left( k - r \\right) / 2 \\right\\rfloor \\left\\lfloor d _ { 1 } - r \\right\\rfloor } { \\left\\lfloor k / 2 \\right\\rfloor \\left\\lfloor d _ { 1 } \\right\\rfloor } \\log \\mathbb { P } \\left( \\mathbf { W } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "cdb37740def3c8c632c0376cf0cdc0ca2adf070ab1bb586699c73e9475ff0ea9.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 128, + 426, + 454, + 454 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 399, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 399, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 177, + 474 + ], + "score": 1.0, + "content": "Taking the limit", + "type": "text" + }, + { + "bbox": [ + 177, + 459, + 245, + 472 + ], + "score": 0.85, + "content": "\\mathrm { n i n } \\left[ k , d _ { 0 } , d _ { 1 } \\right] \\dot { > } r", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 459, + 399, + 474 + ], + "score": 1.0, + "content": "on eqs. 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This proof relies on more elementary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 559, + 354, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 354, + 570 + ], + "score": 1.0, + "content": "results, which we first prove in subsections 12.3.1 and 12.3.2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 582, + 397, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 398, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 398, + 596 + ], + "score": 1.0, + "content": "12.3.1 ORTHANT PROBABILITY OF A RANDOM GAUSSIAN VECTOR", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 602, + 506, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 151, + 615 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + }, + { + "bbox": [ + 151, + 603, + 173, + 615 + ], + "score": 0.91, + "content": "\\phi \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 603, + 191, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 603, + 214, + 615 + ], + "score": 0.92, + "content": "\\Phi \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "are, respectively, the probability density function and cumulative distribu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 614, + 343, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 343, + 625 + ], + "score": 1.0, + "content": "tion function for a scalar standard normal random variable.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 339, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 337, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 306, + 642 + ], + "score": 1.0, + "content": "Definition 32. 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(12.14) and (12.15) we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 459, + 399, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 478, + 434, + 496 + ], + "lines": [ + { + "bbox": [ + 176, + 478, + 434, + 496 + ], + "spans": [ + { + "bbox": [ + 176, + 478, + 434, + 496 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathrm { r a n k } \\left( \\mathbf { A } \\right) = r \\right) \\overset { * } { \\leq } 2 ^ { k + r d _ { 0 } ( \\log d _ { 1 } + \\log k ) + t ^ { 2 } } \\mathbb { P } \\left( { \\mathbf { W } } \\mathbf { X } _ { [ \\lfloor k / 2 \\rfloor ] } > 0 \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "9df6a774635dc67111cc4d5f93d21467589611f3de1be67eb153f36831e3dc06.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 176, + 478, + 434, + 496 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 526, + 230, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 231, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 231, + 540 + ], + "score": 1.0, + "content": "12.3 PROOF OF LEMMA 16", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 105, + 547, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "score": 1.0, + "content": "In this section we will prove Lemma 16 in subsection 12.3.3. This proof relies on more elementary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 559, + 354, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 354, + 570 + ], + "score": 1.0, + "content": "results, which we first prove in subsections 12.3.1 and 12.3.2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 546, + 506, + 570 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 582, + 397, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 398, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 398, + 596 + ], + "score": 1.0, + "content": "12.3.1 ORTHANT PROBABILITY OF A RANDOM GAUSSIAN VECTOR", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 602, + 506, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 151, + 615 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + }, + { + "bbox": [ + 151, + 603, + 173, + 615 + ], + "score": 0.91, + "content": "\\phi \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 603, + 191, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 603, + 214, + 615 + ], + "score": 0.92, + "content": "\\Phi \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "are, respectively, the probability density function and cumulative distribu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 614, + 343, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 343, + 625 + ], + "score": 1.0, + "content": "tion function for a scalar standard normal random variable.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 603, + 506, + 625 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 339, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 337, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 306, + 642 + ], + "score": 1.0, + "content": "Definition 32. We define the following functions", + "type": "text" + }, + { + "bbox": [ + 306, + 630, + 337, + 640 + ], + "score": 0.9, + "content": "\\forall x \\geq 0", + "type": "inline_equation" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 628, + 337, + 642 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 646, + 390, + 704 + ], + "lines": [ + { + "bbox": [ + 220, + 646, + 390, + 704 + ], + "spans": [ + { + "bbox": [ + 220, + 646, + 390, + 704 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle { g \\left( x \\right) \\triangleq \\frac { x \\Phi \\left( x \\right) } { \\phi \\left( x \\right) } , } } \\\\ { \\displaystyle { \\psi \\left( x \\right) \\triangleq \\frac { \\left( g ^ { - 1 } \\left( x \\right) \\right) ^ { 2 } } { 2 x } - \\log \\left( \\Phi \\left( g ^ { - 1 } \\left( x \\right) \\right) \\right) , } } \\end{array}", + "type": "interline_equation", + "image_path": "00187b0bc5bea52b78790b55c395390a4a0f4057e218595d545c4df856c180f6.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 220, + 646, + 390, + 665.3333333333334 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 220, + 665.3333333333334, + 390, + 684.6666666666667 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 220, + 684.6666666666667, + 390, + 704.0000000000001 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 218, + 723 + ], + "score": 1.0, + "content": "where the inverse function", + "type": "text" + }, + { + "bbox": [ + 219, + 709, + 331, + 722 + ], + "score": 0.92, + "content": "g ^ { - 1 } ( x ) : [ 0 , \\infty ) [ 0 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 707, + 421, + 723 + ], + "score": 1.0, + "content": "is well defined since", + "type": "text" + }, + { + "bbox": [ + 421, + 709, + 442, + 722 + ], + "score": 0.94, + "content": "g \\left( x \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 707, + 505, + 723 + ], + "score": 1.0, + "content": "monotonically", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 720, + 240, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 181, + 733 + ], + "score": 1.0, + "content": "increase from 0 to", + "type": "text" + }, + { + "bbox": [ + 181, + 722, + 192, + 730 + ], + "score": 0.78, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 720, + 210, + 733 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 210, + 722, + 235, + 731 + ], + "score": 0.89, + "content": "x \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 720, + 240, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 707, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 176, + 96 + ], + "score": 1.0, + "content": "Lemma 33. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 82, + 236, + 94 + ], + "score": 0.9, + "content": "\\mathbf { z } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 80, + 376, + 96 + ], + "score": 1.0, + "content": "be a random Gaussian vector in", + "type": "text" + }, + { + "bbox": [ + 377, + 82, + 393, + 93 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 80, + 507, + 96 + ], + "score": 1.0, + "content": ", with a covariance matrix", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 241, + 108 + ], + "score": 0.88, + "content": "\\Sigma _ { i j } = \\left( 1 - \\theta K ^ { - 1 } \\right) \\delta _ { m n } + \\theta K ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 93, + 269, + 108 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 270, + 94, + 318, + 105 + ], + "score": 0.91, + "content": "K \\gg \\theta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 93, + 385, + 108 + ], + "score": 1.0, + "content": ". Then, recalling", + "type": "text" + }, + { + "bbox": [ + 386, + 94, + 408, + 106 + ], + "score": 0.89, + "content": "\\psi \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 93, + 504, + 108 + ], + "score": 1.0, + "content": "in eq. (12.17), we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 111, + 399, + 126 + ], + "lines": [ + { + "bbox": [ + 211, + 111, + 399, + 126 + ], + "spans": [ + { + "bbox": [ + 211, + 111, + 399, + 126 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\log \\mathbb { P } \\left( \\forall i : z _ { i } > 0 \\right) \\le - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "10ceb2c2a891fbb08c5041cf502e3c6eb5698cf6b62ea6b7ded0f3b27ebbdd2b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 211, + 111, + 399, + 126 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 137, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 225, + 153 + ], + "score": 1.0, + "content": "Proof. Note that we can write", + "type": "text" + }, + { + "bbox": [ + 225, + 140, + 264, + 150 + ], + "score": 0.91, + "content": "\\mathbf { z } = \\mathbf { u } + \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 135, + 293, + 153 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 294, + 137, + 408, + 152 + ], + "score": 0.91, + "content": "\\mathbf { u } \\sim { \\mathcal { N } } \\left( 0 , \\left( 1 - \\theta K ^ { - 1 } \\right) \\mathbf { I } _ { K } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 135, + 428, + 153 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 429, + 137, + 503, + 151 + ], + "score": 0.91, + "content": "\\eta \\sim \\mathcal { N } \\left( 0 , \\theta K ^ { - 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 135, + 506, + 153 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 223, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 223, + 162 + ], + "score": 1.0, + "content": "Using this notation, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 165, + 461, + 320 + ], + "lines": [ + { + "bbox": [ + 151, + 165, + 461, + 320 + ], + "spans": [ + { + "bbox": [ + 151, + 165, + 461, + 320 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\mathbb { P } \\left( \\overline { { \\psi } } : z _ { i } > 0 \\right) } \\\\ & { = \\int _ { - \\infty } ^ { \\infty } d \\eta \\left[ \\displaystyle \\prod _ { i = 1 } ^ { K } \\int _ { - \\infty } ^ { \\infty } d u _ { i } \\mathcal { T } \\left( \\sqrt { 1 - \\theta K ^ { - 1 } } u _ { i } + \\sqrt { \\theta K ^ { - 1 } } \\eta > 0 \\right) \\phi \\left( u _ { i } \\right) \\right] \\phi \\left( \\eta \\right) } \\\\ & { = \\int _ { - \\infty } ^ { \\infty } d \\eta \\left[ \\Phi \\left( \\sqrt { \\frac { \\theta K ^ { - 1 } } { 1 - \\theta K ^ { - 1 } } } \\eta \\right) \\right] ^ { K } \\phi \\left( \\eta \\right) } \\\\ & { \\stackrel { ( 1 ) } { = } \\sqrt { \\displaystyle \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\int _ { - \\infty } ^ { \\infty } d \\xi \\left[ \\Phi \\left( \\xi \\right) \\right] ^ { K } \\exp \\left( - \\frac { \\left( K - \\theta \\right) \\xi ^ { 2 } } { 2 \\theta } \\right) } \\\\ & { = \\sqrt { \\displaystyle \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\int _ { - \\infty } ^ { \\infty } d \\xi \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) \\exp \\left[ K \\left( \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "bc262ef4ffb1ddf52f55b8a1b9f1574b6779391626101eeb51b688c7a14ed759.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 151, + 165, + 461, + 216.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 151, + 216.66666666666666, + 461, + 268.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 151, + 268.3333333333333, + 461, + 320.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 502, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 322, + 339 + ], + "score": 1.0, + "content": "where in (1) we changed the variable of integration to", + "type": "text" + }, + { + "bbox": [ + 322, + 324, + 402, + 339 + ], + "score": 0.93, + "content": "\\xi = \\sqrt { \\theta / \\left( K - \\theta \\right) } \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 324, + 496, + 339 + ], + "score": 1.0, + "content": ". 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Since", + "type": "text" + }, + { + "bbox": [ + 259, + 408, + 266, + 417 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 403, + 507, + 420 + ], + "score": 1.0, + "content": "is twice differentiable, we can use Laplace’s method (e.g.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 417, + 262, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 262, + 430 + ], + "score": 1.0, + "content": "(Butler, 2007)) to simplify eq. (12.18)", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 441, + 421, + 469 + ], + "lines": [ + { + "bbox": [ + 190, + 441, + 421, + 469 + ], + "spans": [ + { + "bbox": [ + 190, + 441, + 421, + 469 + ], + "score": 0.92, + "content": "\\log \\int _ { - \\infty } ^ { \\infty } h \\left( \\xi \\right) \\exp \\left( K q \\left( \\xi \\right) \\right) d \\xi = K q \\left( \\xi _ { 0 } \\right) + O \\left( \\log K \\right) .", + "type": "interline_equation", + "image_path": "d1670a986406af48e77385d91d0789092f6aa30650f01a29338d73edc1472a78.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 190, + 441, + 421, + 469 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 350, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 350, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 137, + 484 + ], + "score": 1.0, + "content": "To find", + "type": "text" + }, + { + "bbox": [ + 137, + 471, + 147, + 482 + ], + "score": 0.88, + "content": "\\xi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 469, + 216, + 484 + ], + "score": 1.0, + "content": ", we differentiate", + "type": "text" + }, + { + "bbox": [ + 216, + 471, + 236, + 483 + ], + "score": 0.92, + "content": "q \\left( \\xi \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 469, + 350, + 484 + ], + "score": 1.0, + "content": "and equate to zero to obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 486, + 359, + 514 + ], + "lines": [ + { + "bbox": [ + 252, + 486, + 359, + 514 + ], + "spans": [ + { + "bbox": [ + 252, + 486, + 359, + 514 + ], + "score": 0.95, + "content": "q ^ { \\prime } \\left( \\xi \\right) = \\frac { \\phi \\left( \\xi \\right) } { \\Phi \\left( \\xi \\right) } - \\frac { 1 } { \\theta } \\xi = 0 .", + "type": "interline_equation", + "image_path": "9e51c2e3473f8f5377e7a1a2151beb2c490589d7b94d46e2cd725ad034164c58.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 252, + 486, + 359, + 514 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 243, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 243, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 243, + 532 + ], + "score": 1.0, + "content": "which implies (recall eq. (12.16))", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 527, + 349, + 554 + ], + "lines": [ + { + "bbox": [ + 262, + 527, + 349, + 554 + ], + "spans": [ + { + "bbox": [ + 262, + 527, + 349, + 554 + ], + "score": 0.95, + "content": "g \\left( \\xi \\right) \\triangleq \\frac { \\xi \\Phi \\left( \\xi \\right) } { \\phi \\left( \\xi \\right) } = \\theta .", + "type": "interline_equation", + "image_path": "6fc19580f5b10a49f862708606138eab92076653d13c268e9e60b5a8fd1f3b6b.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 262, + 527, + 349, + 554 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 320, + 568 + ], + "score": 1.0, + "content": "This is a monotonically increasing function from 0 to", + "type": "text" + }, + { + "bbox": [ + 320, + 558, + 331, + 566 + ], + "score": 0.82, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 555, + 381, + 568 + ], + "score": 1.0, + "content": "in the range", + "type": "text" + }, + { + "bbox": [ + 382, + 556, + 406, + 568 + ], + "score": 0.91, + "content": "\\xi \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 555, + 505, + 568 + ], + "score": 1.0, + "content": ". Its inverse function can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 221, + 580 + ], + "score": 1.0, + "content": "also be defined in that range", + "type": "text" + }, + { + "bbox": [ + 221, + 566, + 326, + 578 + ], + "score": 0.89, + "content": "g ^ { - 1 } \\check { ( \\theta ) } : [ 0 , \\infty ] [ 0 , \\infty ]", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 566, + 506, + 580 + ], + "score": 1.0, + "content": ". This implies that this equation has only one", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 577, + 503, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 144, + 591 + ], + "score": 1.0, + "content": "solution,", + "type": "text" + }, + { + "bbox": [ + 144, + 577, + 198, + 589 + ], + "score": 0.74, + "content": "\\xi _ { 0 } = g ^ { - 1 } \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 577, + 227, + 591 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 227, + 578, + 314, + 590 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { \\xi \\to \\infty } q \\left( \\xi \\right) = - \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 577, + 335, + 591 + ], + "score": 1.0, + "content": ", this", + "type": "text" + }, + { + "bbox": [ + 335, + 578, + 345, + 589 + ], + "score": 0.88, + "content": "\\xi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 577, + 483, + 591 + ], + "score": 1.0, + "content": "is indeed the global maximum of", + "type": "text" + }, + { + "bbox": [ + 483, + 578, + 503, + 590 + ], + "score": 0.91, + "content": "q \\left( \\xi \\right)", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 587, + 357, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 226, + 602 + ], + "score": 1.0, + "content": "Substituting this solution into", + "type": "text" + }, + { + "bbox": [ + 227, + 589, + 246, + 601 + ], + "score": 0.9, + "content": "q \\left( \\xi \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 587, + 357, + 602 + ], + "score": 1.0, + "content": ", we get (recall eq. (12.17))", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 605, + 448, + 633 + ], + "lines": [ + { + "bbox": [ + 134, + 605, + 448, + 633 + ], + "spans": [ + { + "bbox": [ + 134, + 605, + 448, + 633 + ], + "score": 0.89, + "content": "\\forall \\theta > 0 : q \\left( \\xi _ { 0 } \\right) = - \\psi \\left( \\theta \\right) = q \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) = \\log \\left( \\Phi \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) \\right) - \\frac { \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) ^ { 2 } } { 2 \\theta } .", + "type": "interline_equation", + "image_path": "7cc89123c84adf644e67d0d3cde30d6ff1aa1f3403854f31f967bb1d5bb1fba8.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 134, + 605, + 448, + 633 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 636, + 304, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 306, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 306, + 651 + ], + "score": 1.0, + "content": "Using eq. (12.18), (12.21) and (12.24) we obtain:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 652, + 448, + 711 + ], + "lines": [ + { + "bbox": [ + 162, + 652, + 448, + 711 + ], + "spans": [ + { + "bbox": [ + 162, + 652, + 448, + 711 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\log \\mathbb { P } \\left( \\forall i : z _ { i } > 0 \\right) } \\\\ { = \\log \\left[ \\int _ { - \\infty } ^ { \\infty } d \\xi \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) \\exp \\left[ K \\left( \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } \\right) \\right] \\right] + O \\left( \\log K \\right) } \\\\ { = - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "a2165f3020b1cf0998b8aee13777679a8199cf2ab84a919b3469948b6de7b1f7.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 162, + 652, + 448, + 671.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 162, + 671.6666666666666, + 448, + 691.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 162, + 691.3333333333333, + 448, + 710.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 721, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 722, + 504, + 731 + ], + "spans": [ + { + "bbox": [ + 496, + 722, + 504, + 731 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 103, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 176, + 96 + ], + "score": 1.0, + "content": "Lemma 33. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 82, + 236, + 94 + ], + "score": 0.9, + "content": "\\mathbf { z } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 80, + 376, + 96 + ], + "score": 1.0, + "content": "be a random Gaussian vector in", + "type": "text" + }, + { + "bbox": [ + 377, + 82, + 393, + 93 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 80, + 507, + 96 + ], + "score": 1.0, + "content": ", with a covariance matrix", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 241, + 108 + ], + "score": 0.88, + "content": "\\Sigma _ { i j } = \\left( 1 - \\theta K ^ { - 1 } \\right) \\delta _ { m n } + \\theta K ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 93, + 269, + 108 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 270, + 94, + 318, + 105 + ], + "score": 0.91, + "content": "K \\gg \\theta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 93, + 385, + 108 + ], + "score": 1.0, + "content": ". Then, recalling", + "type": "text" + }, + { + "bbox": [ + 386, + 94, + 408, + 106 + ], + "score": 0.89, + "content": "\\psi \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 93, + 504, + 108 + ], + "score": 1.0, + "content": "in eq. (12.17), we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 80, + 507, + 108 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 111, + 399, + 126 + ], + "lines": [ + { + "bbox": [ + 211, + 111, + 399, + 126 + ], + "spans": [ + { + "bbox": [ + 211, + 111, + 399, + 126 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\log \\mathbb { P } \\left( \\forall i : z _ { i } > 0 \\right) \\le - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "10ceb2c2a891fbb08c5041cf502e3c6eb5698cf6b62ea6b7ded0f3b27ebbdd2b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 211, + 111, + 399, + 126 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 137, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 225, + 153 + ], + "score": 1.0, + "content": "Proof. Note that we can write", + "type": "text" + }, + { + "bbox": [ + 225, + 140, + 264, + 150 + ], + "score": 0.91, + "content": "\\mathbf { z } = \\mathbf { u } + \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 135, + 293, + 153 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 294, + 137, + 408, + 152 + ], + "score": 0.91, + "content": "\\mathbf { u } \\sim { \\mathcal { N } } \\left( 0 , \\left( 1 - \\theta K ^ { - 1 } \\right) \\mathbf { I } _ { K } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 135, + 428, + 153 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 429, + 137, + 503, + 151 + ], + "score": 0.91, + "content": "\\eta \\sim \\mathcal { N } \\left( 0 , \\theta K ^ { - 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 135, + 506, + 153 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 223, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 223, + 162 + ], + "score": 1.0, + "content": "Using this notation, we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 135, + 506, + 162 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 165, + 461, + 320 + ], + "lines": [ + { + "bbox": [ + 151, + 165, + 461, + 320 + ], + "spans": [ + { + "bbox": [ + 151, + 165, + 461, + 320 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\mathbb { P } \\left( \\overline { { \\psi } } : z _ { i } > 0 \\right) } \\\\ & { = \\int _ { - \\infty } ^ { \\infty } d \\eta \\left[ \\displaystyle \\prod _ { i = 1 } ^ { K } \\int _ { - \\infty } ^ { \\infty } d u _ { i } \\mathcal { T } \\left( \\sqrt { 1 - \\theta K ^ { - 1 } } u _ { i } + \\sqrt { \\theta K ^ { - 1 } } \\eta > 0 \\right) \\phi \\left( u _ { i } \\right) \\right] \\phi \\left( \\eta \\right) } \\\\ & { = \\int _ { - \\infty } ^ { \\infty } d \\eta \\left[ \\Phi \\left( \\sqrt { \\frac { \\theta K ^ { - 1 } } { 1 - \\theta K ^ { - 1 } } } \\eta \\right) \\right] ^ { K } \\phi \\left( \\eta \\right) } \\\\ & { \\stackrel { ( 1 ) } { = } \\sqrt { \\displaystyle \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\int _ { - \\infty } ^ { \\infty } d \\xi \\left[ \\Phi \\left( \\xi \\right) \\right] ^ { K } \\exp \\left( - \\frac { \\left( K - \\theta \\right) \\xi ^ { 2 } } { 2 \\theta } \\right) } \\\\ & { = \\sqrt { \\displaystyle \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\int _ { - \\infty } ^ { \\infty } d \\xi \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) \\exp \\left[ K \\left( \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "bc262ef4ffb1ddf52f55b8a1b9f1574b6779391626101eeb51b688c7a14ed759.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 151, + 165, + 461, + 216.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 151, + 216.66666666666666, + 461, + 268.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 151, + 268.3333333333333, + 461, + 320.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 502, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 322, + 339 + ], + "score": 1.0, + "content": "where in (1) we changed the variable of integration to", + "type": "text" + }, + { + "bbox": [ + 322, + 324, + 402, + 339 + ], + "score": 0.93, + "content": "\\xi = \\sqrt { \\theta / \\left( K - \\theta \\right) } \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 324, + 496, + 339 + ], + "score": 1.0, + "content": ". We denote, for a fixed", + "type": "text" + }, + { + "bbox": [ + 496, + 326, + 502, + 336 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 324, + 502, + 339 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 342, + 375, + 403 + ], + "lines": [ + { + "bbox": [ + 235, + 342, + 375, + 403 + ], + "spans": [ + { + "bbox": [ + 235, + 342, + 375, + 403 + ], + "score": 0.93, + "content": "\\begin{array} { l } { { \\displaystyle q \\left( \\xi \\right) \\triangleq \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } } } \\\\ { { \\displaystyle h \\left( \\xi \\right) \\triangleq \\sqrt { \\frac { \\theta } { 2 \\pi \\left( K - \\theta \\right) } } \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "29af02d48d220a9e54bd1ba2d06556480d0167d851fe2d29a2a4847fb58909a4.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 342, + 375, + 372.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 235, + 372.5, + 375, + 403.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 405, + 506, + 429 + ], + "lines": [ + { + "bbox": [ + 104, + 403, + 507, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 123, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 406, + 134, + 417 + ], + "score": 0.88, + "content": "\\xi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 403, + 259, + 420 + ], + "score": 1.0, + "content": "as its global maximum. Since", + "type": "text" + }, + { + "bbox": [ + 259, + 408, + 266, + 417 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 403, + 507, + 420 + ], + "score": 1.0, + "content": "is twice differentiable, we can use Laplace’s method (e.g.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 417, + 262, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 262, + 430 + ], + "score": 1.0, + "content": "(Butler, 2007)) to simplify eq. (12.18)", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 403, + 507, + 430 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 441, + 421, + 469 + ], + "lines": [ + { + "bbox": [ + 190, + 441, + 421, + 469 + ], + "spans": [ + { + "bbox": [ + 190, + 441, + 421, + 469 + ], + "score": 0.92, + "content": "\\log \\int _ { - \\infty } ^ { \\infty } h \\left( \\xi \\right) \\exp \\left( K q \\left( \\xi \\right) \\right) d \\xi = K q \\left( \\xi _ { 0 } \\right) + O \\left( \\log K \\right) .", + "type": "interline_equation", + "image_path": "d1670a986406af48e77385d91d0789092f6aa30650f01a29338d73edc1472a78.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 190, + 441, + 421, + 469 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 350, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 350, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 137, + 484 + ], + "score": 1.0, + "content": "To find", + "type": "text" + }, + { + "bbox": [ + 137, + 471, + 147, + 482 + ], + "score": 0.88, + "content": "\\xi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 469, + 216, + 484 + ], + "score": 1.0, + "content": ", we differentiate", + "type": "text" + }, + { + "bbox": [ + 216, + 471, + 236, + 483 + ], + "score": 0.92, + "content": "q \\left( \\xi \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 469, + 350, + 484 + ], + "score": 1.0, + "content": "and equate to zero to obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 469, + 350, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 486, + 359, + 514 + ], + "lines": [ + { + "bbox": [ + 252, + 486, + 359, + 514 + ], + "spans": [ + { + "bbox": [ + 252, + 486, + 359, + 514 + ], + "score": 0.95, + "content": "q ^ { \\prime } \\left( \\xi \\right) = \\frac { \\phi \\left( \\xi \\right) } { \\Phi \\left( \\xi \\right) } - \\frac { 1 } { \\theta } \\xi = 0 .", + "type": "interline_equation", + "image_path": "9e51c2e3473f8f5377e7a1a2151beb2c490589d7b94d46e2cd725ad034164c58.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 252, + 486, + 359, + 514 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 243, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 243, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 243, + 532 + ], + "score": 1.0, + "content": "which implies (recall eq. (12.16))", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 516, + 243, + 532 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 527, + 349, + 554 + ], + "lines": [ + { + "bbox": [ + 262, + 527, + 349, + 554 + ], + "spans": [ + { + "bbox": [ + 262, + 527, + 349, + 554 + ], + "score": 0.95, + "content": "g \\left( \\xi \\right) \\triangleq \\frac { \\xi \\Phi \\left( \\xi \\right) } { \\phi \\left( \\xi \\right) } = \\theta .", + "type": "interline_equation", + "image_path": "6fc19580f5b10a49f862708606138eab92076653d13c268e9e60b5a8fd1f3b6b.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 262, + 527, + 349, + 554 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 320, + 568 + ], + "score": 1.0, + "content": "This is a monotonically increasing function from 0 to", + "type": "text" + }, + { + "bbox": [ + 320, + 558, + 331, + 566 + ], + "score": 0.82, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 555, + 381, + 568 + ], + "score": 1.0, + "content": "in the range", + "type": "text" + }, + { + "bbox": [ + 382, + 556, + 406, + 568 + ], + "score": 0.91, + "content": "\\xi \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 555, + 505, + 568 + ], + "score": 1.0, + "content": ". Its inverse function can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 221, + 580 + ], + "score": 1.0, + "content": "also be defined in that range", + "type": "text" + }, + { + "bbox": [ + 221, + 566, + 326, + 578 + ], + "score": 0.89, + "content": "g ^ { - 1 } \\check { ( \\theta ) } : [ 0 , \\infty ] [ 0 , \\infty ]", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 566, + 506, + 580 + ], + "score": 1.0, + "content": ". This implies that this equation has only one", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 577, + 503, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 144, + 591 + ], + "score": 1.0, + "content": "solution,", + "type": "text" + }, + { + "bbox": [ + 144, + 577, + 198, + 589 + ], + "score": 0.74, + "content": "\\xi _ { 0 } = g ^ { - 1 } \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 577, + 227, + 591 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 227, + 578, + 314, + 590 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { \\xi \\to \\infty } q \\left( \\xi \\right) = - \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 577, + 335, + 591 + ], + "score": 1.0, + "content": ", this", + "type": "text" + }, + { + "bbox": [ + 335, + 578, + 345, + 589 + ], + "score": 0.88, + "content": "\\xi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 577, + 483, + 591 + ], + "score": 1.0, + "content": "is indeed the global maximum of", + "type": "text" + }, + { + "bbox": [ + 483, + 578, + 503, + 590 + ], + "score": 0.91, + "content": "q \\left( \\xi \\right)", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 587, + 357, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 226, + 602 + ], + "score": 1.0, + "content": "Substituting this solution into", + "type": "text" + }, + { + "bbox": [ + 227, + 589, + 246, + 601 + ], + "score": 0.9, + "content": "q \\left( \\xi \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 587, + 357, + 602 + ], + "score": 1.0, + "content": ", we get (recall eq. (12.17))", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 555, + 506, + 602 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 605, + 448, + 633 + ], + "lines": [ + { + "bbox": [ + 134, + 605, + 448, + 633 + ], + "spans": [ + { + "bbox": [ + 134, + 605, + 448, + 633 + ], + "score": 0.89, + "content": "\\forall \\theta > 0 : q \\left( \\xi _ { 0 } \\right) = - \\psi \\left( \\theta \\right) = q \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) = \\log \\left( \\Phi \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) \\right) - \\frac { \\left( g ^ { - 1 } \\left( \\theta \\right) \\right) ^ { 2 } } { 2 \\theta } .", + "type": "interline_equation", + "image_path": "7cc89123c84adf644e67d0d3cde30d6ff1aa1f3403854f31f967bb1d5bb1fba8.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 134, + 605, + 448, + 633 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 636, + 304, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 306, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 306, + 651 + ], + "score": 1.0, + "content": "Using eq. (12.18), (12.21) and (12.24) we obtain:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 636, + 306, + 651 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 652, + 448, + 711 + ], + "lines": [ + { + "bbox": [ + 162, + 652, + 448, + 711 + ], + "spans": [ + { + "bbox": [ + 162, + 652, + 448, + 711 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\log \\mathbb { P } \\left( \\forall i : z _ { i } > 0 \\right) } \\\\ { = \\log \\left[ \\int _ { - \\infty } ^ { \\infty } d \\xi \\exp \\left( \\frac { \\xi ^ { 2 } } { 2 } \\right) \\exp \\left[ K \\left( \\log \\Phi \\left( \\xi \\right) - \\frac { \\xi ^ { 2 } } { 2 \\theta } \\right) \\right] \\right] + O \\left( \\log K \\right) } \\\\ { = - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "a2165f3020b1cf0998b8aee13777679a8199cf2ab84a919b3469948b6de7b1f7.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 162, + 652, + 448, + 671.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 162, + 671.6666666666666, + 448, + 691.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 162, + 691.3333333333333, + 448, + 710.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 397, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 397, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 397, + 95 + ], + "score": 1.0, + "content": "Next, we generalize the previous Lemma to a general covariance matrix.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 97, + 506, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 506, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 182, + 111 + ], + "score": 1.0, + "content": "Corollary 34. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 98, + 240, + 110 + ], + "score": 0.88, + "content": "\\mathbf { u } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 95, + 369, + 111 + ], + "score": 1.0, + "content": "be a random Gaussian vector in", + "type": "text" + }, + { + "bbox": [ + 369, + 97, + 385, + 109 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 95, + 425, + 111 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 426, + 99, + 484, + 109 + ], + "score": 0.9, + "content": "\\forall n : \\Sigma _ { n n } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 95, + 506, + 111 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 108, + 338, + 123 + ], + "spans": [ + { + "bbox": [ + 107, + 110, + 232, + 122 + ], + "score": 0.86, + "content": "\\theta \\geq K \\operatorname* { m a x } _ { n , m : n \\neq m } \\Sigma _ { n m } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 108, + 327, + 123 + ], + "score": 1.0, + "content": ". Then, again, for large", + "type": "text" + }, + { + "bbox": [ + 327, + 110, + 338, + 119 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 128, + 400, + 142 + ], + "lines": [ + { + "bbox": [ + 210, + 128, + 400, + 142 + ], + "spans": [ + { + "bbox": [ + 210, + 128, + 400, + 142 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\log \\mathbb { P } \\left( \\forall i : u _ { i } > 0 \\right) \\le - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "a119d0f58c9512b8d36a8810717f0e3d2c506d21b1da6aed63aae28952e7aaf0.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 210, + 128, + 400, + 142 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 507, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 179, + 178 + ], + "score": 1.0, + "content": "Proof. We define", + "type": "text" + }, + { + "bbox": [ + 180, + 159, + 241, + 179 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { u } } \\sim \\mathcal { N } \\left( 0 , \\tilde { \\Sigma } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 159, + 266, + 178 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 266, + 161, + 407, + 176 + ], + "score": 0.91, + "content": "\\tilde { \\Sigma } _ { m n } = \\left( 1 - \\theta K ^ { - 1 } \\right) \\delta _ { m n } + \\theta K ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 159, + 452, + 178 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 452, + 163, + 506, + 175 + ], + "score": 0.89, + "content": "\\forall n : \\Sigma _ { n n } =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 177, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 144, + 191 + ], + "score": 0.91, + "content": "\\tilde { \\Sigma } _ { n n } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 177, + 161, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 180, + 196, + 191 + ], + "score": 0.83, + "content": "\\forall m \\neq n", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 177, + 201, + 194 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 201, + 179, + 254, + 191 + ], + "score": 0.8, + "content": "\\Sigma _ { m n } \\leq \\tilde { \\Sigma } _ { m n }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 177, + 506, + 194 + ], + "score": 1.0, + "content": ". Therefore, from Slepian’s Lemma (Slepian, 1962, Lemma 1),", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 198, + 381, + 212 + ], + "lines": [ + { + "bbox": [ + 228, + 198, + 381, + 212 + ], + "spans": [ + { + "bbox": [ + 228, + 198, + 381, + 212 + ], + "score": 0.9, + "content": "\\mathbb { P } \\left( \\forall n : \\tilde { u } _ { n } > 0 \\right) \\ge \\mathbb { P } \\left( \\forall n : u _ { n } > 0 \\right) .", + "type": "interline_equation", + "image_path": "0244d23353d526224bb96c5b6f9940c2bce067ce9d0120fd1efe08d9c5d7349f.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 228, + 198, + 381, + 212 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 283, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 283, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 191, + 232 + ], + "score": 1.0, + "content": "Using Lemma 33 on", + "type": "text" + }, + { + "bbox": [ + 191, + 219, + 199, + 229 + ], + "score": 0.39, + "content": "\\tilde { \\mathbf { u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 217, + 283, + 232 + ], + "score": 1.0, + "content": "completes the proof.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 247, + 277, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 279, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 279, + 261 + ], + "score": 1.0, + "content": "12.3.2 MUTUAL COHERENCE BOUNDS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 105, + 267, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 415, + 281 + ], + "score": 1.0, + "content": "Definition 35. We define the mutual coherence of the columns of a matrix", + "type": "text" + }, + { + "bbox": [ + 415, + 267, + 505, + 280 + ], + "score": 0.91, + "content": "\\mathbf { A } = [ \\pmb { a } _ { 1 } , \\cdots , \\pmb { a } _ { N } ] \\in", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 275, + 335, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 137, + 289 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { M \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 275, + 335, + 293 + ], + "score": 1.0, + "content": "as the maximal angle between different columns", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 297, + 362, + 327 + ], + "lines": [ + { + "bbox": [ + 248, + 297, + 362, + 327 + ], + "spans": [ + { + "bbox": [ + 248, + 297, + 362, + 327 + ], + "score": 0.93, + "content": "\\gamma \\left( { \\bf A } \\right) \\triangleq \\operatorname* { m a x } _ { i , j : i \\neq j } \\frac { \\left| { \\pmb a } _ { i } ^ { \\top } { \\pmb a } _ { j } \\right| } { \\left\\| { \\pmb a } _ { i } \\right\\| \\left\\| { \\pmb a } _ { j } \\right\\| } .", + "type": "interline_equation", + "image_path": "d8d6a5d0a413b29e82574bebdcf6c510e09a916bc28476ba9bd2d480f925dbce.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 248, + 297, + 362, + 327 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 430, + 354 + ], + "lines": [ + { + "bbox": [ + 104, + 332, + 429, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 145, + 354 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 337, + 189, + 349 + ], + "score": 0.92, + "content": "\\gamma \\left( \\mathbf { A } \\right) \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 332, + 305, + 354 + ], + "score": 1.0, + "content": "and from (Welch, 1974), for", + "type": "text" + }, + { + "bbox": [ + 305, + 333, + 429, + 354 + ], + "score": 0.8, + "content": "\\begin{array} { r } { N \\ge M , \\gamma \\left( { \\mathbf { A } } \\right) \\ge \\sqrt { \\frac { N - M } { M \\left( N - 1 \\right) } } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 506, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 174, + 372 + ], + "score": 1.0, + "content": "Lemma 36. Let", + "type": "text" + }, + { + "bbox": [ + 174, + 357, + 293, + 370 + ], + "score": 0.91, + "content": "\\mathbf { A } = [ \\pmb { a } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { a } _ { N } ] \\in \\mathbb { R } ^ { M \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 355, + 470, + 372 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix, and", + "type": "text" + }, + { + "bbox": [ + 470, + 358, + 495, + 370 + ], + "score": 0.91, + "content": "\\gamma \\left( \\mathbf { A } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 355, + 506, + 372 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 370, + 348, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 348, + 381 + ], + "score": 1.0, + "content": "the mutual coherence of it columns (see definition 35). Then", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 387, + 385, + 416 + ], + "lines": [ + { + "bbox": [ + 225, + 387, + 385, + 416 + ], + "spans": [ + { + "bbox": [ + 225, + 387, + 385, + 416 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\gamma \\left( { \\bf A } \\right) > \\epsilon \\right) \\le 2 N ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) .", + "type": "interline_equation", + "image_path": "ebe652bcc739fb427bf31744a0d6cc8a108c8f24088cbb865c41e57e96c9784f.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 225, + 387, + 385, + 416 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 385, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 385, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 385, + 445 + ], + "score": 1.0, + "content": "Proof. In this case, we have from (Chen & Peng, 2016, Appendix 1):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 449, + 466, + 478 + ], + "lines": [ + { + "bbox": [ + 143, + 449, + 466, + 478 + ], + "spans": [ + { + "bbox": [ + 143, + 449, + 466, + 478 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) \\leq N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M a ^ { 2 } \\epsilon ^ { 2 } } { 4 \\left( 1 + \\epsilon / 2 \\right) } \\right) + \\exp \\left( - \\frac { M } { 4 } \\left( 1 - a \\right) ^ { 2 } \\right) \\right] ,", + "type": "interline_equation", + "image_path": "e97351e71bd87ca29c21480bb77c0941e1dc3b1328e38237128d0d32bed82b15.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 143, + 449, + 466, + 478 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 266, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 264, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 137, + 498 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 137, + 484, + 178, + 497 + ], + "score": 0.93, + "content": "a \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 483, + 213, + 498 + ], + "score": 1.0, + "content": ". Setting", + "type": "text" + }, + { + "bbox": [ + 213, + 484, + 264, + 497 + ], + "score": 0.9, + "content": "a = 1 - \\epsilon / 2", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 503, + 462, + 595 + ], + "lines": [ + { + "bbox": [ + 148, + 503, + 462, + 595 + ], + "spans": [ + { + "bbox": [ + 148, + 503, + 462, + 595 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) \\le N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M \\left( 1 - \\epsilon / 2 \\right) ^ { 2 } \\epsilon ^ { 2 } } { 4 \\left( 1 + \\epsilon / 2 \\right) } \\right) + \\exp \\left( - \\frac { M } { 1 6 } \\epsilon ^ { 2 } \\right) \\right] } \\\\ & { \\qquad \\overset { \\left( 1 \\right) } { \\le } N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) + \\exp \\left( - \\frac { M } { 1 6 } \\epsilon ^ { 2 } \\right) \\right] } \\\\ & { \\qquad \\le 2 N ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "05961512a51b5398c54fa887dd4c3a2a461e2c3a377ae3c33158f1093cff5c0e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 148, + 503, + 462, + 533.6666666666666 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 148, + 533.6666666666666, + 462, + 564.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 148, + 564.3333333333333, + 462, + 594.9999999999999 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 503, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 248, + 614 + ], + "score": 1.0, + "content": "where in (1) we can assume that", + "type": "text" + }, + { + "bbox": [ + 249, + 601, + 276, + 612 + ], + "score": 0.9, + "content": "\\epsilon \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 599, + 322, + 614 + ], + "score": 1.0, + "content": ", since for", + "type": "text" + }, + { + "bbox": [ + 322, + 601, + 350, + 612 + ], + "score": 0.9, + "content": "\\epsilon \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 599, + 393, + 614 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 393, + 600, + 474, + 613 + ], + "score": 0.91, + "content": "\\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "(recall", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 610, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 151, + 624 + ], + "score": 0.9, + "content": "\\gamma ( \\mathbf { A } ) \\leq 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 610, + 159, + 626 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 614, + 504, + 622 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 105, + 633, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 175, + 648 + ], + "score": 1.0, + "content": "Lemma 37. Let", + "type": "text" + }, + { + "bbox": [ + 175, + 634, + 292, + 647 + ], + "score": 0.92, + "content": "\\mathbf { B } = [ \\mathbf { b } _ { 1 } , \\cdots , \\mathbf { b } _ { L } ] \\in \\mathbb { R } ^ { M \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix and mutual", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 644, + 352, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 149, + 660 + ], + "score": 1.0, + "content": "coherence", + "type": "text" + }, + { + "bbox": [ + 150, + 649, + 157, + 658 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 644, + 259, + 660 + ], + "score": 1.0, + "content": "as in definition 35. Then,", + "type": "text" + }, + { + "bbox": [ + 260, + 647, + 289, + 657 + ], + "score": 0.86, + "content": "\\forall \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 644, + 307, + 660 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 646, + 348, + 658 + ], + "score": 0.92, + "content": "\\forall K \\in [ L ]", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 644, + 352, + 660 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 664, + 457, + 693 + ], + "lines": [ + { + "bbox": [ + 152, + 664, + 457, + 693 + ], + "spans": [ + { + "bbox": [ + 152, + 664, + 457, + 693 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\operatorname* { m i n } _ { S \\subset [ N ] : | S | = K } \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) \\leq \\exp \\left[ \\left( 2 \\log \\left( 2 K \\right) - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] .", + "type": "interline_equation", + "image_path": "257d17ca7cf63447593b7d70d782498e0bc8d1a68e7f426a55fb94cf19500879.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 152, + 664, + 457, + 693 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 443, + 722 + ], + "score": 1.0, + "content": "Proof. We upper bound this probability by partitioning the set of column vectors into", + "type": "text" + }, + { + "bbox": [ + 443, + 710, + 473, + 722 + ], + "score": 0.92, + "content": "\\lfloor L / K \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "subsets", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 117, + 732 + ], + "score": 0.86, + "content": "S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 720, + 149, + 733 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 149, + 720, + 190, + 732 + ], + "score": 0.93, + "content": "| \\bar { S _ { i } } \\bar { | } = K", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 720, + 497, + 733 + ], + "score": 1.0, + "content": "and require that in each subset the mutual coherence is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 498, + 723, + 503, + 730 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "page_idx": 26, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 750, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "27", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 219, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 496, + 221, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 496, + 221, + 505, + 230 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 397, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 397, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 397, + 95 + ], + "score": 1.0, + "content": "Next, we generalize the previous Lemma to a general covariance matrix.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 397, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 97, + 506, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 506, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 182, + 111 + ], + "score": 1.0, + "content": "Corollary 34. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 98, + 240, + 110 + ], + "score": 0.88, + "content": "\\mathbf { u } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 95, + 369, + 111 + ], + "score": 1.0, + "content": "be a random Gaussian vector in", + "type": "text" + }, + { + "bbox": [ + 369, + 97, + 385, + 109 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 95, + 425, + 111 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 426, + 99, + 484, + 109 + ], + "score": 0.9, + "content": "\\forall n : \\Sigma _ { n n } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 95, + 506, + 111 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 108, + 338, + 123 + ], + "spans": [ + { + "bbox": [ + 107, + 110, + 232, + 122 + ], + "score": 0.86, + "content": "\\theta \\geq K \\operatorname* { m a x } _ { n , m : n \\neq m } \\Sigma _ { n m } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 108, + 327, + 123 + ], + "score": 1.0, + "content": ". Then, again, for large", + "type": "text" + }, + { + "bbox": [ + 327, + 110, + 338, + 119 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 95, + 506, + 123 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 128, + 400, + 142 + ], + "lines": [ + { + "bbox": [ + 210, + 128, + 400, + 142 + ], + "spans": [ + { + "bbox": [ + 210, + 128, + 400, + 142 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\log \\mathbb { P } \\left( \\forall i : u _ { i } > 0 \\right) \\le - K \\psi \\left( \\theta \\right) + O \\left( \\log K \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "a119d0f58c9512b8d36a8810717f0e3d2c506d21b1da6aed63aae28952e7aaf0.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 210, + 128, + 400, + 142 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 507, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 179, + 178 + ], + "score": 1.0, + "content": "Proof. We define", + "type": "text" + }, + { + "bbox": [ + 180, + 159, + 241, + 179 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { u } } \\sim \\mathcal { N } \\left( 0 , \\tilde { \\Sigma } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 159, + 266, + 178 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 266, + 161, + 407, + 176 + ], + "score": 0.91, + "content": "\\tilde { \\Sigma } _ { m n } = \\left( 1 - \\theta K ^ { - 1 } \\right) \\delta _ { m n } + \\theta K ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 159, + 452, + 178 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 452, + 163, + 506, + 175 + ], + "score": 0.89, + "content": "\\forall n : \\Sigma _ { n n } =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 177, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 144, + 191 + ], + "score": 0.91, + "content": "\\tilde { \\Sigma } _ { n n } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 177, + 161, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 180, + 196, + 191 + ], + "score": 0.83, + "content": "\\forall m \\neq n", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 177, + 201, + 194 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 201, + 179, + 254, + 191 + ], + "score": 0.8, + "content": "\\Sigma _ { m n } \\leq \\tilde { \\Sigma } _ { m n }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 177, + 506, + 194 + ], + "score": 1.0, + "content": ". Therefore, from Slepian’s Lemma (Slepian, 1962, Lemma 1),", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 159, + 506, + 194 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 198, + 381, + 212 + ], + "lines": [ + { + "bbox": [ + 228, + 198, + 381, + 212 + ], + "spans": [ + { + "bbox": [ + 228, + 198, + 381, + 212 + ], + "score": 0.9, + "content": "\\mathbb { P } \\left( \\forall n : \\tilde { u } _ { n } > 0 \\right) \\ge \\mathbb { P } \\left( \\forall n : u _ { n } > 0 \\right) .", + "type": "interline_equation", + "image_path": "0244d23353d526224bb96c5b6f9940c2bce067ce9d0120fd1efe08d9c5d7349f.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 228, + 198, + 381, + 212 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 283, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 283, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 191, + 232 + ], + "score": 1.0, + "content": "Using Lemma 33 on", + "type": "text" + }, + { + "bbox": [ + 191, + 219, + 199, + 229 + ], + "score": 0.39, + "content": "\\tilde { \\mathbf { u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 217, + 283, + 232 + ], + "score": 1.0, + "content": "completes the proof.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 217, + 283, + 232 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 247, + 277, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 279, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 279, + 261 + ], + "score": 1.0, + "content": "12.3.2 MUTUAL COHERENCE BOUNDS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 105, + 267, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 415, + 281 + ], + "score": 1.0, + "content": "Definition 35. We define the mutual coherence of the columns of a matrix", + "type": "text" + }, + { + "bbox": [ + 415, + 267, + 505, + 280 + ], + "score": 0.91, + "content": "\\mathbf { A } = [ \\pmb { a } _ { 1 } , \\cdots , \\pmb { a } _ { N } ] \\in", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 275, + 335, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 137, + 289 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { M \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 275, + 335, + 293 + ], + "score": 1.0, + "content": "as the maximal angle between different columns", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 266, + 505, + 293 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 297, + 362, + 327 + ], + "lines": [ + { + "bbox": [ + 248, + 297, + 362, + 327 + ], + "spans": [ + { + "bbox": [ + 248, + 297, + 362, + 327 + ], + "score": 0.93, + "content": "\\gamma \\left( { \\bf A } \\right) \\triangleq \\operatorname* { m a x } _ { i , j : i \\neq j } \\frac { \\left| { \\pmb a } _ { i } ^ { \\top } { \\pmb a } _ { j } \\right| } { \\left\\| { \\pmb a } _ { i } \\right\\| \\left\\| { \\pmb a } _ { j } \\right\\| } .", + "type": "interline_equation", + "image_path": "d8d6a5d0a413b29e82574bebdcf6c510e09a916bc28476ba9bd2d480f925dbce.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 248, + 297, + 362, + 327 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 430, + 354 + ], + "lines": [ + { + "bbox": [ + 104, + 332, + 429, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 145, + 354 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 337, + 189, + 349 + ], + "score": 0.92, + "content": "\\gamma \\left( \\mathbf { A } \\right) \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 332, + 305, + 354 + ], + "score": 1.0, + "content": "and from (Welch, 1974), for", + "type": "text" + }, + { + "bbox": [ + 305, + 333, + 429, + 354 + ], + "score": 0.8, + "content": "\\begin{array} { r } { N \\ge M , \\gamma \\left( { \\mathbf { A } } \\right) \\ge \\sqrt { \\frac { N - M } { M \\left( N - 1 \\right) } } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 332, + 429, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 506, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 174, + 372 + ], + "score": 1.0, + "content": "Lemma 36. Let", + "type": "text" + }, + { + "bbox": [ + 174, + 357, + 293, + 370 + ], + "score": 0.91, + "content": "\\mathbf { A } = [ \\pmb { a } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { a } _ { N } ] \\in \\mathbb { R } ^ { M \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 355, + 470, + 372 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix, and", + "type": "text" + }, + { + "bbox": [ + 470, + 358, + 495, + 370 + ], + "score": 0.91, + "content": "\\gamma \\left( \\mathbf { A } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 355, + 506, + 372 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 370, + 348, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 348, + 381 + ], + "score": 1.0, + "content": "the mutual coherence of it columns (see definition 35). Then", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 355, + 506, + 381 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 387, + 385, + 416 + ], + "lines": [ + { + "bbox": [ + 225, + 387, + 385, + 416 + ], + "spans": [ + { + "bbox": [ + 225, + 387, + 385, + 416 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\gamma \\left( { \\bf A } \\right) > \\epsilon \\right) \\le 2 N ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) .", + "type": "interline_equation", + "image_path": "ebe652bcc739fb427bf31744a0d6cc8a108c8f24088cbb865c41e57e96c9784f.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 225, + 387, + 385, + 416 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 385, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 385, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 385, + 445 + ], + "score": 1.0, + "content": "Proof. In this case, we have from (Chen & Peng, 2016, Appendix 1):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 429, + 385, + 445 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 449, + 466, + 478 + ], + "lines": [ + { + "bbox": [ + 143, + 449, + 466, + 478 + ], + "spans": [ + { + "bbox": [ + 143, + 449, + 466, + 478 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) \\leq N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M a ^ { 2 } \\epsilon ^ { 2 } } { 4 \\left( 1 + \\epsilon / 2 \\right) } \\right) + \\exp \\left( - \\frac { M } { 4 } \\left( 1 - a \\right) ^ { 2 } \\right) \\right] ,", + "type": "interline_equation", + "image_path": "e97351e71bd87ca29c21480bb77c0941e1dc3b1328e38237128d0d32bed82b15.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 143, + 449, + 466, + 478 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 266, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 264, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 137, + 498 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 137, + 484, + 178, + 497 + ], + "score": 0.93, + "content": "a \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 483, + 213, + 498 + ], + "score": 1.0, + "content": ". Setting", + "type": "text" + }, + { + "bbox": [ + 213, + 484, + 264, + 497 + ], + "score": 0.9, + "content": "a = 1 - \\epsilon / 2", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 483, + 264, + 498 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 503, + 462, + 595 + ], + "lines": [ + { + "bbox": [ + 148, + 503, + 462, + 595 + ], + "spans": [ + { + "bbox": [ + 148, + 503, + 462, + 595 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) \\le N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M \\left( 1 - \\epsilon / 2 \\right) ^ { 2 } \\epsilon ^ { 2 } } { 4 \\left( 1 + \\epsilon / 2 \\right) } \\right) + \\exp \\left( - \\frac { M } { 1 6 } \\epsilon ^ { 2 } \\right) \\right] } \\\\ & { \\qquad \\overset { \\left( 1 \\right) } { \\le } N \\left( N - 1 \\right) \\left[ \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) + \\exp \\left( - \\frac { M } { 1 6 } \\epsilon ^ { 2 } \\right) \\right] } \\\\ & { \\qquad \\le 2 N ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "05961512a51b5398c54fa887dd4c3a2a461e2c3a377ae3c33158f1093cff5c0e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 148, + 503, + 462, + 533.6666666666666 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 148, + 533.6666666666666, + 462, + 564.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 148, + 564.3333333333333, + 462, + 594.9999999999999 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 503, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 248, + 614 + ], + "score": 1.0, + "content": "where in (1) we can assume that", + "type": "text" + }, + { + "bbox": [ + 249, + 601, + 276, + 612 + ], + "score": 0.9, + "content": "\\epsilon \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 599, + 322, + 614 + ], + "score": 1.0, + "content": ", since for", + "type": "text" + }, + { + "bbox": [ + 322, + 601, + 350, + 612 + ], + "score": 0.9, + "content": "\\epsilon \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 599, + 393, + 614 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 393, + 600, + 474, + 613 + ], + "score": 0.91, + "content": "\\mathbb { P } \\left( \\gamma \\left( \\mathbf { A } \\right) > \\epsilon \\right) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "(recall", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 610, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 151, + 624 + ], + "score": 0.9, + "content": "\\gamma ( \\mathbf { A } ) \\leq 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 610, + 159, + 626 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 614, + 504, + 622 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 599, + 505, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 633, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 175, + 648 + ], + "score": 1.0, + "content": "Lemma 37. Let", + "type": "text" + }, + { + "bbox": [ + 175, + 634, + 292, + 647 + ], + "score": 0.92, + "content": "\\mathbf { B } = [ \\mathbf { b } _ { 1 } , \\cdots , \\mathbf { b } _ { L } ] \\in \\mathbb { R } ^ { M \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix and mutual", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 644, + 352, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 149, + 660 + ], + "score": 1.0, + "content": "coherence", + "type": "text" + }, + { + "bbox": [ + 150, + 649, + 157, + 658 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 644, + 259, + 660 + ], + "score": 1.0, + "content": "as in definition 35. Then,", + "type": "text" + }, + { + "bbox": [ + 260, + 647, + 289, + 657 + ], + "score": 0.86, + "content": "\\forall \\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 644, + 307, + 660 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 308, + 646, + 348, + 658 + ], + "score": 0.92, + "content": "\\forall K \\in [ L ]", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 644, + 352, + 660 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 632, + 506, + 660 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 664, + 457, + 693 + ], + "lines": [ + { + "bbox": [ + 152, + 664, + 457, + 693 + ], + "spans": [ + { + "bbox": [ + 152, + 664, + 457, + 693 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\operatorname* { m i n } _ { S \\subset [ N ] : | S | = K } \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) \\leq \\exp \\left[ \\left( 2 \\log \\left( 2 K \\right) - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] .", + "type": "interline_equation", + "image_path": "257d17ca7cf63447593b7d70d782498e0bc8d1a68e7f426a55fb94cf19500879.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 152, + 664, + 457, + 693 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 443, + 722 + ], + "score": 1.0, + "content": "Proof. We upper bound this probability by partitioning the set of column vectors into", + "type": "text" + }, + { + "bbox": [ + 443, + 710, + 473, + 722 + ], + "score": 0.92, + "content": "\\lfloor L / K \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "subsets", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 117, + 732 + ], + "score": 0.86, + "content": "S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 720, + 149, + 733 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 149, + 720, + 190, + 732 + ], + "score": 0.93, + "content": "| \\bar { S _ { i } } \\bar { | } = K", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 720, + 497, + 733 + ], + "score": 1.0, + "content": "and require that in each subset the mutual coherence is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 498, + 723, + 503, + 730 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 709, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 285, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 285, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 285, + 96 + ], + "score": 1.0, + "content": "Since the columns are independent, we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 97, + 453, + 227 + ], + "lines": [ + { + "bbox": [ + 156, + 97, + 453, + 227 + ], + "spans": [ + { + "bbox": [ + 156, + 97, + 453, + 227 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\underset { S \\subset [ N ] : | S | = K } { \\operatorname* { m i n } } \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\le \\displaystyle \\prod _ { i = 1 } ^ { \\lfloor L / K \\rfloor } \\mathbb { P } \\left( \\forall S = \\{ 1 + \\left( i - 1 \\right) K , 2 + \\left( 1 - i \\right) K , \\ldots , i K \\} : \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\stackrel { \\mathrm { ( 1 ) } } { \\le } \\displaystyle \\prod _ { i = 1 } ^ { L / K - 1 } 2 K ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) } \\\\ & { \\le \\exp \\left[ \\left( 2 \\log \\left( 2 K \\right) - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "5987161ba4f7c4213350967d3f294afc11ad2bc19967b72147387417bf676375.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 156, + 97, + 453, + 140.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 156, + 140.33333333333334, + 453, + 183.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 156, + 183.66666666666669, + 453, + 227.00000000000003 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 304, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 304, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 304, + 242 + ], + "score": 1.0, + "content": "where in (1) we used the bound from Lemma 36.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 469, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 470, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 470, + 265 + ], + "score": 1.0, + "content": "12.3.3 MAIN PROOF: ORTHANT PROBABILITY OF A PRODUCT GAUSSIAN MATRICES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 104, + 268, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 273, + 286 + ], + "score": 1.0, + "content": "Lemma 38. (Lemma 16 restated). Let", + "type": "text" + }, + { + "bbox": [ + 273, + 270, + 401, + 284 + ], + "score": 0.92, + "content": "\\mathbf { C } = \\left[ \\mathbf { c } _ { 1 } , \\cdots , \\mathbf { c } _ { N } \\right] ^ { \\top } \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 268, + 421, + 286 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 421, + 271, + 473, + 282 + ], + "score": 0.91, + "content": "\\mathbf { B } \\in \\mathbb { R } ^ { M \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 268, + 506, + 286 + ], + "score": 1.0, + "content": "be two", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 420, + 296 + ], + "score": 1.0, + "content": "independent random Gaussian matrices. Without loss of generality, assume", + "type": "text" + }, + { + "bbox": [ + 421, + 284, + 453, + 294 + ], + "score": 0.91, + "content": "N \\geq L", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 283, + 505, + 296 + ], + "score": 1.0, + "content": ", and denote", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 294, + 468, + 308 + ], + "spans": [ + { + "bbox": [ + 107, + 295, + 158, + 308 + ], + "score": 0.91, + "content": "\\alpha \\triangleq M L / N", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 294, + 242, + 308 + ], + "score": 1.0, + "content": ". Then, in the regime", + "type": "text" + }, + { + "bbox": [ + 242, + 295, + 276, + 307 + ], + "score": 0.91, + "content": "M \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 294, + 341, + 308 + ], + "score": 1.0, + "content": "and in the limit", + "type": "text" + }, + { + "bbox": [ + 342, + 295, + 429, + 308 + ], + "score": 0.9, + "content": "\\operatorname* { m i n } \\left[ N , M , L \\right] \\dot { > } \\alpha \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 294, + 468, + 308 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 310, + 379, + 333 + ], + "lines": [ + { + "bbox": [ + 231, + 310, + 379, + 333 + ], + "spans": [ + { + "bbox": [ + 231, + 310, + 379, + 333 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7c37001828c2a3958147878c11f5978c25551509807f52bbf9da0ca0389ea6e0.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 231, + 310, + 379, + 333 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 390, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 390, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 176, + 356 + ], + "score": 1.0, + "content": "Proof. For some", + "type": "text" + }, + { + "bbox": [ + 176, + 344, + 200, + 353 + ], + "score": 0.9, + "content": "\\theta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 342, + 248, + 356 + ], + "score": 1.0, + "content": ", and subset", + "type": "text" + }, + { + "bbox": [ + 249, + 344, + 256, + 353 + ], + "score": 0.85, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 342, + 296, + 356 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 296, + 343, + 352, + 355 + ], + "score": 0.92, + "content": "| S | = K < L", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 342, + 390, + 356 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 358, + 484, + 424 + ], + "lines": [ + { + "bbox": [ + 123, + 358, + 484, + 424 + ], + "spans": [ + { + "bbox": [ + 123, + 358, + 484, + 424 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\quad \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) } \\\\ & { \\le \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) + \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\le \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) + \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { = \\mathbb { E } \\left[ \\left[ \\mathbb { P } \\left( \\mathbf { c } _ { 1 } ^ { \\top } \\mathbf { B } _ { S } > 0 | \\mathbf { B } _ { S } , \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) \\right] ^ { N } | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right] + \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "12acb15a2ef1b3a40722bb6ac8ae01e3cd3271d3f78fb61d6a26cd8a44bf1e10.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 123, + 358, + 484, + 380.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 123, + 380.0, + 484, + 402.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 123, + 402.0, + 484, + 424.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 503, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 360, + 439 + ], + "score": 1.0, + "content": "where in the last equality we used the fact that the rows of", + "type": "text" + }, + { + "bbox": [ + 360, + 426, + 370, + 435 + ], + "score": 0.82, + "content": "\\mathbf { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 424, + 505, + 439 + ], + "score": 1.0, + "content": "are independent and identically", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 436, + 155, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 155, + 448 + ], + "score": 1.0, + "content": "distributed.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 218, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 220, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 220, + 466 + ], + "score": 1.0, + "content": "We choose a specific subset", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 464, + 372, + 479 + ], + "lines": [ + { + "bbox": [ + 238, + 464, + 372, + 479 + ], + "spans": [ + { + "bbox": [ + 238, + 464, + 372, + 479 + ], + "score": 0.91, + "content": "S ^ { * } = \\mathrm { a r g m i n } _ { S C [ L ] : | S | = K } \\gamma \\left( \\mathbf { B } _ { S } \\right)", + "type": "interline_equation", + "image_path": "df12677b88160108afdc8e94ec88315b9144847c1649b51df5b8199e56a80116.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 238, + 464, + 372, + 479 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 504, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 403, + 493 + ], + "score": 1.0, + "content": "to minimize the second term and then upper bound it using Lemma 37 with", + "type": "text" + }, + { + "bbox": [ + 403, + 479, + 437, + 489 + ], + "score": 0.9, + "content": "\\theta = K \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 478, + 506, + 493 + ], + "score": 1.0, + "content": "; additionally, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 490, + 422, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 388, + 504 + ], + "score": 1.0, + "content": "apply Corollary 34 on the first term with the components of the vector", + "type": "text" + }, + { + "bbox": [ + 389, + 493, + 396, + 500 + ], + "score": 0.48, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 490, + 422, + 504 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 505, + 381, + 527 + ], + "lines": [ + { + "bbox": [ + 228, + 505, + 381, + 527 + ], + "spans": [ + { + "bbox": [ + 228, + 505, + 381, + 527 + ], + "score": 0.94, + "content": "u _ { i } = \\left( \\mathbf { B } _ { S } ^ { \\top } \\mathbf { c } _ { 1 } \\right) _ { i } / \\sqrt { \\left( \\mathbf { B } _ { S } ^ { \\top } \\mathbf { B } _ { S } \\right) _ { i i } } \\in \\mathbb { R } ^ { K } ,", + "type": "interline_equation", + "image_path": "953aa31d6ac88ff24202a9ce16cb564fe6d0144fb387847d45c32cdc141ba888.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 228, + 505, + 381, + 527 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 380, + 542 + ], + "score": 1.0, + "content": "which is a Gaussian random vector with mean zero and covariance", + "type": "text" + }, + { + "bbox": [ + 381, + 530, + 390, + 540 + ], + "score": 0.83, + "content": "\\pmb { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 529, + 433, + 542 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 434, + 530, + 487, + 541 + ], + "score": 0.91, + "content": "\\forall i : \\Sigma _ { i i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 540, + 283, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 212, + 553 + ], + "score": 0.92, + "content": "\\forall i \\neq j : \\Sigma _ { i j } \\leq \\epsilon = \\theta K ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 540, + 283, + 553 + ], + "score": 1.0, + "content": ". Thus, we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 556, + 497, + 585 + ], + "lines": [ + { + "bbox": [ + 111, + 556, + 497, + 585 + ], + "spans": [ + { + "bbox": [ + 111, + 556, + 497, + 585 + ], + "score": 0.94, + "content": "\\mathbb { P } \\left( \\mathbf { C } \\mathbf { B } > 0 \\right) \\leq \\exp \\left( - N K \\psi \\left( \\theta \\right) + O \\left( N \\log K \\right) \\right) + \\exp \\left[ \\left( \\log \\left( 2 K \\right) ^ { 2 } - \\frac { M \\theta ^ { 2 } } { 2 4 K ^ { 2 } } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] ,", + "type": "interline_equation", + "image_path": "d062c72c492847d0c1f18addb2f4a2d933e3e9844ef45fb61e6335aa987dcd0a.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 111, + 556, + 497, + 565.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 565.6666666666666, + 497, + 575.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 575.3333333333333, + 497, + 584.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 295, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 295, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 172, + 614 + ], + "score": 1.0, + "content": "where we recall", + "type": "text" + }, + { + "bbox": [ + 172, + 600, + 194, + 612 + ], + "score": 0.92, + "content": "\\psi \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 597, + 295, + 614 + ], + "score": 1.0, + "content": "is defined in eq. (12.17).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 108, + 616, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 503, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 263, + 630 + ], + "score": 1.0, + "content": "Next, we wish to select good values for", + "type": "text" + }, + { + "bbox": [ + 263, + 618, + 270, + 627 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 616, + 287, + 630 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 287, + 618, + 297, + 627 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 616, + 447, + 630 + ], + "score": 1.0, + "content": ", which minimize this bound for large", + "type": "text" + }, + { + "bbox": [ + 447, + 617, + 503, + 629 + ], + "score": 0.92, + "content": "( M , N , L , K )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 393, + 640 + ], + "score": 1.0, + "content": "Thus, keeping only the first order terms in each exponent (assuming", + "type": "text" + }, + { + "bbox": [ + 393, + 628, + 453, + 639 + ], + "score": 0.9, + "content": "L \\gg K \\gg 1", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "), we aim to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 639, + 279, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 279, + 651 + ], + "score": 1.0, + "content": "minimize the function as much as possible", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 654, + 409, + 682 + ], + "lines": [ + { + "bbox": [ + 202, + 654, + 409, + 682 + ], + "spans": [ + { + "bbox": [ + 202, + 654, + 409, + 682 + ], + "score": 0.93, + "content": "f \\left( K , \\theta \\right) \\triangleq \\exp \\left( - N K \\psi \\left( \\theta \\right) \\right) + \\exp \\left( - \\frac { M \\theta ^ { 2 } L } { 2 4 K ^ { 3 } } \\right) .", + "type": "interline_equation", + "image_path": "1ae5ea48a896c2784e5111959286ea52a674e9fafec0fb357d81653b8ec418e5.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 202, + 654, + 409, + 682 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 684, + 504, + 707 + ], + "lines": [ + { + "bbox": [ + 106, + 685, + 504, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 263, + 696 + ], + "score": 1.0, + "content": "Note that the first term is decreasing in", + "type": "text" + }, + { + "bbox": [ + 263, + 686, + 273, + 694 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 685, + 483, + 696 + ], + "score": 1.0, + "content": ", while the second term increases. Therefore, for any", + "type": "text" + }, + { + "bbox": [ + 483, + 685, + 489, + 694 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 685, + 504, + 696 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 695, + 497, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 222, + 709 + ], + "score": 1.0, + "content": "minimum of this function in", + "type": "text" + }, + { + "bbox": [ + 222, + 697, + 231, + 705 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 695, + 497, + 709 + ], + "score": 1.0, + "content": "would be approximately achieved when both terms are equal, i.e.,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 710, + 349, + 736 + ], + "lines": [ + { + "bbox": [ + 261, + 710, + 349, + 736 + ], + "spans": [ + { + "bbox": [ + 261, + 710, + 349, + 736 + ], + "score": 0.94, + "content": "N K \\psi \\left( \\theta \\right) = \\frac { M \\theta ^ { 2 } L } { 2 4 K ^ { 3 } } ,", + "type": "interline_equation", + "image_path": "93b5891fbb61d899fcfe10b9b0a7a95590e0ebf987546c57c795aa2314f88afd.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 261, + 710, + 349, + 736 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "page_idx": 27, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "28", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 229, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 496, + 231, + 504, + 239 + ], + "spans": [ + { + "bbox": [ + 496, + 231, + 504, + 239 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 285, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 285, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 285, + 96 + ], + "score": 1.0, + "content": "Since the columns are independent, we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 285, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 97, + 453, + 227 + ], + "lines": [ + { + "bbox": [ + 156, + 97, + 453, + 227 + ], + "spans": [ + { + "bbox": [ + 156, + 97, + 453, + 227 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\underset { S \\subset [ N ] : | S | = K } { \\operatorname* { m i n } } \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\le \\displaystyle \\prod _ { i = 1 } ^ { \\lfloor L / K \\rfloor } \\mathbb { P } \\left( \\forall S = \\{ 1 + \\left( i - 1 \\right) K , 2 + \\left( 1 - i \\right) K , \\ldots , i K \\} : \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\stackrel { \\mathrm { ( 1 ) } } { \\le } \\displaystyle \\prod _ { i = 1 } ^ { L / K - 1 } 2 K ^ { 2 } \\exp \\left( - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) } \\\\ & { \\le \\exp \\left[ \\left( 2 \\log \\left( 2 K \\right) - \\frac { M \\epsilon ^ { 2 } } { 2 4 } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "5987161ba4f7c4213350967d3f294afc11ad2bc19967b72147387417bf676375.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 156, + 97, + 453, + 140.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 156, + 140.33333333333334, + 453, + 183.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 156, + 183.66666666666669, + 453, + 227.00000000000003 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 304, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 304, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 304, + 242 + ], + "score": 1.0, + "content": "where in (1) we used the bound from Lemma 36.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 106, + 228, + 304, + 242 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 469, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 470, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 470, + 265 + ], + "score": 1.0, + "content": "12.3.3 MAIN PROOF: ORTHANT PROBABILITY OF A PRODUCT GAUSSIAN MATRICES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 252, + 470, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 104, + 268, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 273, + 286 + ], + "score": 1.0, + "content": "Lemma 38. (Lemma 16 restated). Let", + "type": "text" + }, + { + "bbox": [ + 273, + 270, + 401, + 284 + ], + "score": 0.92, + "content": "\\mathbf { C } = \\left[ \\mathbf { c } _ { 1 } , \\cdots , \\mathbf { c } _ { N } \\right] ^ { \\top } \\in \\mathbb { R } ^ { N \\times M }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 268, + 421, + 286 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 421, + 271, + 473, + 282 + ], + "score": 0.91, + "content": "\\mathbf { B } \\in \\mathbb { R } ^ { M \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 268, + 506, + 286 + ], + "score": 1.0, + "content": "be two", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 420, + 296 + ], + "score": 1.0, + "content": "independent random Gaussian matrices. Without loss of generality, assume", + "type": "text" + }, + { + "bbox": [ + 421, + 284, + 453, + 294 + ], + "score": 0.91, + "content": "N \\geq L", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 283, + 505, + 296 + ], + "score": 1.0, + "content": ", and denote", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 294, + 468, + 308 + ], + "spans": [ + { + "bbox": [ + 107, + 295, + 158, + 308 + ], + "score": 0.91, + "content": "\\alpha \\triangleq M L / N", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 294, + 242, + 308 + ], + "score": 1.0, + "content": ". Then, in the regime", + "type": "text" + }, + { + "bbox": [ + 242, + 295, + 276, + 307 + ], + "score": 0.91, + "content": "M \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 294, + 341, + 308 + ], + "score": 1.0, + "content": "and in the limit", + "type": "text" + }, + { + "bbox": [ + 342, + 295, + 429, + 308 + ], + "score": 0.9, + "content": "\\operatorname* { m i n } \\left[ N , M , L \\right] \\dot { > } \\alpha \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 294, + 468, + 308 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 268, + 506, + 308 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 310, + 379, + 333 + ], + "lines": [ + { + "bbox": [ + 231, + 310, + 379, + 333 + ], + "spans": [ + { + "bbox": [ + 231, + 310, + 379, + 333 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7c37001828c2a3958147878c11f5978c25551509807f52bbf9da0ca0389ea6e0.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 231, + 310, + 379, + 333 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 390, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 390, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 176, + 356 + ], + "score": 1.0, + "content": "Proof. For some", + "type": "text" + }, + { + "bbox": [ + 176, + 344, + 200, + 353 + ], + "score": 0.9, + "content": "\\theta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 342, + 248, + 356 + ], + "score": 1.0, + "content": ", and subset", + "type": "text" + }, + { + "bbox": [ + 249, + 344, + 256, + 353 + ], + "score": 0.85, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 342, + 296, + 356 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 296, + 343, + 352, + 355 + ], + "score": 0.92, + "content": "| S | = K < L", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 342, + 390, + 356 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 342, + 390, + 356 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 358, + 484, + 424 + ], + "lines": [ + { + "bbox": [ + 123, + 358, + 484, + 424 + ], + "spans": [ + { + "bbox": [ + 123, + 358, + 484, + 424 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\quad \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) } \\\\ & { \\le \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) + \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { \\le \\mathbb { P } \\left( \\mathbf { C B } _ { S } > 0 | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) + \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) } \\\\ & { = \\mathbb { E } \\left[ \\left[ \\mathbb { P } \\left( \\mathbf { c } _ { 1 } ^ { \\top } \\mathbf { B } _ { S } > 0 | \\mathbf { B } _ { S } , \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right) \\right] ^ { N } | \\gamma \\left( \\mathbf { B } _ { S } \\right) \\le \\epsilon \\right] + \\mathbb { P } \\left( \\gamma \\left( \\mathbf { B } _ { S } \\right) > \\epsilon \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "12acb15a2ef1b3a40722bb6ac8ae01e3cd3271d3f78fb61d6a26cd8a44bf1e10.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 123, + 358, + 484, + 380.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 123, + 380.0, + 484, + 402.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 123, + 402.0, + 484, + 424.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 503, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 360, + 439 + ], + "score": 1.0, + "content": "where in the last equality we used the fact that the rows of", + 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464, + 372, + 479 + ], + "lines": [ + { + "bbox": [ + 238, + 464, + 372, + 479 + ], + "spans": [ + { + "bbox": [ + 238, + 464, + 372, + 479 + ], + "score": 0.91, + "content": "S ^ { * } = \\mathrm { a r g m i n } _ { S C [ L ] : | S | = K } \\gamma \\left( \\mathbf { B } _ { S } \\right)", + "type": "interline_equation", + "image_path": "df12677b88160108afdc8e94ec88315b9144847c1649b51df5b8199e56a80116.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 238, + 464, + 372, + 479 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 504, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 403, + 493 + ], + "score": 1.0, + "content": "to minimize the second term and then upper bound it using Lemma 37 with", + "type": "text" + }, + { + "bbox": [ + 403, + 479, + 437, + 489 + ], + "score": 0.9, + "content": "\\theta = K \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 478, + 506, + 493 + ], + "score": 1.0, + "content": "; additionally, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 490, + 422, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 388, + 504 + ], + "score": 1.0, + "content": "apply Corollary 34 on the first term with the components of the vector", + "type": "text" + }, + { + "bbox": [ + 389, + 493, + 396, + 500 + ], + "score": 0.48, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 490, + 422, + 504 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 478, + 506, + 504 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 505, + 381, + 527 + ], + "lines": [ + { + "bbox": [ + 228, + 505, + 381, + 527 + ], + "spans": [ + { + "bbox": [ + 228, + 505, + 381, + 527 + ], + "score": 0.94, + "content": "u _ { i } = \\left( \\mathbf { B } _ { S } ^ { \\top } \\mathbf { c } _ { 1 } \\right) _ { i } / \\sqrt { \\left( \\mathbf { B } _ { S } ^ { \\top } \\mathbf { B } _ { S } \\right) _ { i i } } \\in \\mathbb { R } ^ { K } ,", + "type": "interline_equation", + "image_path": "953aa31d6ac88ff24202a9ce16cb564fe6d0144fb387847d45c32cdc141ba888.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 228, + 505, + 381, + 527 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 380, + 542 + ], + "score": 1.0, + "content": "which is a Gaussian random vector with mean zero and covariance", + "type": "text" + }, + { + "bbox": [ + 381, + 530, + 390, + 540 + ], + "score": 0.83, + "content": "\\pmb { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 529, + 433, + 542 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 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Thus, we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 106, + 529, + 505, + 553 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 556, + 497, + 585 + ], + "lines": [ + { + "bbox": [ + 111, + 556, + 497, + 585 + ], + "spans": [ + { + "bbox": [ + 111, + 556, + 497, + 585 + ], + "score": 0.94, + "content": "\\mathbb { P } \\left( \\mathbf { C } \\mathbf { B } > 0 \\right) \\leq \\exp \\left( - N K \\psi \\left( \\theta \\right) + O \\left( N \\log K \\right) \\right) + \\exp \\left[ \\left( \\log \\left( 2 K \\right) ^ { 2 } - \\frac { M \\theta ^ { 2 } } { 2 4 K ^ { 2 } } \\right) \\left( \\frac { L } { K } - 1 \\right) \\right] ,", + "type": "interline_equation", + "image_path": "d062c72c492847d0c1f18addb2f4a2d933e3e9844ef45fb61e6335aa987dcd0a.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 111, + 556, + 497, + 565.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 565.6666666666666, + 497, + 575.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 575.3333333333333, + 497, + 584.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 295, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 295, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 172, + 614 + ], + "score": 1.0, + "content": "where we recall", + "type": "text" + }, + { + "bbox": [ + 172, + 600, + 194, + 612 + ], + "score": 0.92, + "content": "\\psi \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 597, + 295, + 614 + ], + "score": 1.0, + "content": "is defined in eq. (12.17).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 597, + 295, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 616, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 503, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 263, + 630 + ], + "score": 1.0, + "content": "Next, we wish to select good values for", + "type": "text" + }, + { + "bbox": [ + 263, + 618, + 270, + 627 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 616, + 287, + 630 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 287, + 618, + 297, + 627 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 616, + 447, + 630 + ], + "score": 1.0, + "content": ", which minimize this bound for large", + "type": "text" + }, + { + "bbox": [ + 447, + 617, + 503, + 629 + ], + "score": 0.92, + "content": "( M , N , L , K )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 393, + 640 + ], + "score": 1.0, + "content": "Thus, keeping only the first order terms in each exponent (assuming", + "type": "text" + }, + { + "bbox": [ + 393, + 628, + 453, + 639 + ], + "score": 0.9, + "content": "L \\gg K \\gg 1", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "), we aim to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 639, + 279, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 279, + 651 + ], + "score": 1.0, + "content": "minimize the function as much as possible", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 616, + 505, + 651 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 654, + 409, + 682 + ], + "lines": [ + { + "bbox": [ + 202, + 654, + 409, + 682 + ], + "spans": [ + { + "bbox": [ + 202, + 654, + 409, + 682 + ], + "score": 0.93, + "content": "f \\left( K , \\theta \\right) \\triangleq \\exp \\left( - N K \\psi \\left( \\theta \\right) \\right) + \\exp \\left( - \\frac { M \\theta ^ { 2 } L } { 2 4 K ^ { 3 } } \\right) .", + "type": "interline_equation", + "image_path": "1ae5ea48a896c2784e5111959286ea52a674e9fafec0fb357d81653b8ec418e5.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 202, + 654, + 409, + 682 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 684, + 504, + 707 + ], + "lines": [ + { + "bbox": [ + 106, + 685, + 504, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 263, + 696 + ], + "score": 1.0, + "content": "Note that the first term is decreasing in", + "type": "text" + }, + { + "bbox": [ + 263, + 686, + 273, + 694 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 685, + 483, + 696 + ], + "score": 1.0, + "content": ", while the second term increases. Therefore, for any", + "type": "text" + }, + { + "bbox": [ + 483, + 685, + 489, + 694 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 685, + 504, + 696 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 695, + 497, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 222, + 709 + ], + "score": 1.0, + "content": "minimum of this function in", + "type": "text" + }, + { + "bbox": [ + 222, + 697, + 231, + 705 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 695, + 497, + 709 + ], + "score": 1.0, + "content": "would be approximately achieved when both terms are equal, i.e.,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 685, + 504, + 709 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 710, + 349, + 736 + ], + "lines": [ + { + "bbox": [ + 261, + 710, + 349, + 736 + ], + "spans": [ + { + "bbox": [ + 261, + 710, + 349, + 736 + ], + "score": 0.94, + "content": "N K \\psi \\left( \\theta \\right) = \\frac { M \\theta ^ { 2 } L } { 2 4 K ^ { 3 } } ,", + "type": "interline_equation", + "image_path": "93b5891fbb61d899fcfe10b9b0a7a95590e0ebf987546c57c795aa2314f88afd.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 261, + 710, + 349, + 736 + ], + "spans": [], + "index": 33 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 162, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 163, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 163, + 95 + ], + "score": 1.0, + "content": "so we choose", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 90, + 364, + 121 + ], + "lines": [ + { + "bbox": [ + 247, + 90, + 364, + 121 + ], + "spans": [ + { + "bbox": [ + 247, + 90, + 364, + 121 + ], + "score": 0.95, + "content": "{ \\cal K } \\left( \\theta \\right) = \\left( { \\frac { \\theta ^ { 2 } M L } { 2 4 \\psi \\left( \\theta \\right) N } } \\right) ^ { 1 / 4 } .", + "type": "interline_equation", + "image_path": "388b52d07a6c88ce91624ea496107aebd945676fda8d69e93cce3af934817e11.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 247, + 90, + 364, + 105.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 247, + 105.5, + 364, + 121.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 263, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 263, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 157, + 136 + ], + "score": 1.0, + "content": "Substituting", + "type": "text" + }, + { + "bbox": [ + 157, + 123, + 182, + 135 + ], + "score": 0.92, + "content": "K \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 122, + 201, + 136 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 201, + 123, + 235, + 135 + ], + "score": 0.93, + "content": "f \\left( K , \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 122, + 263, + 136 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 139, + 409, + 173 + ], + "lines": [ + { + "bbox": [ + 201, + 139, + 409, + 173 + ], + "spans": [ + { + "bbox": [ + 201, + 139, + 409, + 173 + ], + "score": 0.93, + "content": "f \\left( K \\left( \\theta \\right) , \\theta \\right) = 2 \\exp \\left( - N \\left[ \\frac { \\psi ^ { 3 } \\left( \\theta \\right) \\theta ^ { 2 } M L } { 2 4 N } \\right] ^ { 1 / 4 } \\right) .", + "type": "interline_equation", + "image_path": "2dcba8492ca11e9de048b7def965ec4068a0ca897e5872cdaef94ca5ee040bdf.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 139, + 409, + 156.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 201, + 156.0, + 409, + 173.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 227, + 191 + ], + "score": 1.0, + "content": "To minimize this function in", + "type": "text" + }, + { + "bbox": [ + 227, + 179, + 234, + 189 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 177, + 382, + 191 + ], + "score": 1.0, + "content": ", we need to maximize the function", + "type": "text" + }, + { + "bbox": [ + 383, + 178, + 420, + 191 + ], + "score": 0.93, + "content": "\\psi ^ { 3 } \\left( \\theta \\right) \\theta ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 177, + 505, + 191 + ], + "score": 1.0, + "content": "(which has a single", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 284, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 284, + 201 + ], + "score": 1.0, + "content": "maximum). Doing this numerically gives us", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 205, + 413, + 220 + ], + "lines": [ + { + "bbox": [ + 197, + 205, + 413, + 220 + ], + "spans": [ + { + "bbox": [ + 197, + 205, + 413, + 220 + ], + "score": 0.88, + "content": "\\theta _ { * } \\approx 2 3 . 2 5 ; \\psi \\left( \\theta _ { * } \\right) \\approx 0 . 1 0 6 2 ; \\psi ^ { 3 } \\left( \\theta _ { * } \\right) \\theta _ { * } ^ { 2 } \\approx 0 . 6 4 7 8 .", + "type": "interline_equation", + "image_path": "f3863d4a882a35485e8d0e72f3151d723414518fe330ed3e35d38cdf90f45cea.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 197, + 205, + 413, + 220 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 224, + 365, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 365, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 365, + 238 + ], + "score": 1.0, + "content": "Substituting eqs. (12.27) and (12.28) into eq. (12.25), we obtain", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 240, + 437, + 358 + ], + "lines": [ + { + "bbox": [ + 173, + 240, + 437, + 358 + ], + "spans": [ + { + "bbox": [ + 173, + 240, + 437, + 358 + ], + "score": 0.97, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\mathbf { C } \\mathbf { B } > 0 \\right) } \\\\ & { \\leq \\exp \\left( - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + O \\left( N \\log K \\right) \\right) } \\\\ & { + \\exp \\left[ - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + 2 L \\frac { \\log K } { K } + \\frac { M \\theta ^ { 2 } } { 2 4 K ^ { 2 } } - \\log \\left( 2 K ^ { 2 } \\right) \\right] } \\\\ & { \\leq \\exp \\left( - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + O \\left( N \\log \\left( \\frac { M L } { N } \\right) \\right) \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "e1c6049da1a1a73d233eee0483fecccbd48ff299ffe851950ae2fc0f58648b6e.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 173, + 240, + 437, + 279.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 173, + 279.3333333333333, + 437, + 318.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 173, + 318.66666666666663, + 437, + 357.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 504, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 234, + 376 + ], + "score": 1.0, + "content": "where in the last line we used", + "type": "text" + }, + { + "bbox": [ + 235, + 362, + 307, + 374 + ], + "score": 0.86, + "content": "N \\geq L , N \\geq M", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 361, + 326, + 376 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 327, + 362, + 415, + 375 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } \\left[ N , M , L \\right] \\dot { > } \\alpha \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 361, + 505, + 376 + ], + "score": 1.0, + "content": ". Taking the log, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 374, + 260, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 144, + 388 + ], + "score": 1.0, + "content": "denoting", + "type": "text" + }, + { + "bbox": [ + 144, + 374, + 196, + 387 + ], + "score": 0.93, + "content": "\\alpha \\triangleq M L / N", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 374, + 260, + 388 + ], + "score": 1.0, + "content": ", we thus obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 391, + 402, + 407 + ], + "lines": [ + { + "bbox": [ + 207, + 391, + 402, + 407 + ], + "spans": [ + { + "bbox": [ + 207, + 391, + 402, + 407 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\log \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\le - 0 . 4 N \\alpha ^ { 1 / 4 } + O \\left( N \\log \\alpha \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "bbb40ca39754ca609ff4f7962feb6b2b034e3d6237b19b321a1e99e003c0d7ba.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 207, + 391, + 402, + 407 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 427, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 428, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 214, + 425 + ], + "score": 1.0, + "content": "Therefore, in the limit that", + "type": "text" + }, + { + "bbox": [ + 214, + 412, + 250, + 422 + ], + "score": 0.91, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 410, + 268, + 425 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 268, + 412, + 320, + 424 + ], + "score": 0.92, + "content": "\\alpha ( N ) \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 410, + 343, + 425 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 344, + 412, + 388, + 424 + ], + "score": 0.93, + "content": "\\alpha \\left( N \\right) \\dot { < } N", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 410, + 428, + 425 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 428, + 379, + 450 + ], + "lines": [ + { + "bbox": [ + 231, + 428, + 379, + 450 + ], + "spans": [ + { + "bbox": [ + 231, + 428, + 379, + 450 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7370e6868104035fed18a36a01a9dd5aa15dfa920fb0233402f7ed6fd9df2bd5.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 231, + 428, + 379, + 450 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 482, + 503, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 503, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 503, + 495 + ], + "score": 1.0, + "content": "13 LOWER BOUNDING THE ANGULAR VOLUME OF GLOBAL MINIMA: PROOF", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 131, + 496, + 304, + 508 + ], + "spans": [ + { + "bbox": [ + 131, + 496, + 304, + 508 + ], + "score": 1.0, + "content": "OF LEMMAS USED IN SECTION 10", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 106, + 520, + 349, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 349, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 349, + 533 + ], + "score": 1.0, + "content": "13.1 ANGLES BETWEEN RANDOM GAUSSIAN VECTORS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 104, + 540, + 483, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 484, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 484, + 554 + ], + "score": 1.0, + "content": "To prove the results in the next appendix sections, we will rely on the following basic Lemma.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 344, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 347, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 245, + 570 + ], + "score": 1.0, + "content": "Lemma 39. For any vector y and", + "type": "text" + }, + { + "bbox": [ + 246, + 555, + 307, + 568 + ], + "score": 0.92, + "content": "\\mathbf { x } \\sim \\mathcal { N } \\left( 0 , \\mathbf { I } _ { d _ { 0 } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 554, + 347, + 570 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 572, + 411, + 604 + ], + "lines": [ + { + "bbox": [ + 200, + 572, + 411, + 604 + ], + "spans": [ + { + "bbox": [ + 200, + 572, + 411, + 604 + ], + "score": 0.87, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| > \\cos \\left( \\epsilon \\right) \\right) \\geq \\frac { 2 \\sin \\left( \\epsilon \\right) ^ { d _ { 0 } - 1 } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) }", + "type": "interline_equation", + "image_path": "fa42f9918e706f749d456c684981ab6c710b282b72ef98328705a2dbab9e7753.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 572, + 411, + 588.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 200, + 588.0, + 411, + 604.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 614, + 383, + 645 + ], + "lines": [ + { + "bbox": [ + 227, + 614, + 383, + 645 + ], + "spans": [ + { + "bbox": [ + 227, + 614, + 383, + 645 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| < u \\right) \\leq \\frac { 2 u } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,", + "type": "interline_equation", + "image_path": "3a77ddb5663511a0017617697064e848face4733888d795d7da832e0b5771c7e.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 227, + 614, + 383, + 629.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 227, + 629.5, + 383, + 645.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 649, + 306, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 306, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 190, + 662 + ], + "score": 1.0, + "content": "where we recall that", + "type": "text" + }, + { + "bbox": [ + 190, + 649, + 224, + 662 + ], + "score": 0.93, + "content": "B \\left( x , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 649, + 306, + 662 + ], + "score": 1.0, + "content": "is the beta function.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 506, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 673, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 162, + 689 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 162, + 676, + 205, + 688 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , \\mathbf { I } _ { d _ { 0 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 673, + 361, + 689 + ], + "score": 1.0, + "content": "is spherically symmetric, we can set", + "type": "text" + }, + { + "bbox": [ + 361, + 673, + 435, + 688 + ], + "score": 0.92, + "content": "\\mathbf { y } = \\left[ 1 , 0 \\ldots , 0 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 673, + 506, + 689 + ], + "score": 1.0, + "content": ", without loss of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 686, + 196, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 196, + 700 + ], + "score": 1.0, + "content": "generality. Therefore,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 703, + 408, + 736 + ], + "lines": [ + { + "bbox": [ + 201, + 703, + 408, + 736 + ], + "spans": [ + { + "bbox": [ + 201, + 703, + 408, + 736 + ], + "score": 0.93, + "content": "\\left| { \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } } \\right| ^ { 2 } = { \\frac { x _ { 1 } ^ { 2 } } { x _ { 1 } ^ { 2 } + \\sum _ { i = 2 } ^ { d _ { 0 } } x _ { i } ^ { 2 } } } \\sim { \\mathcal { B } } \\left( { \\frac { 1 } { 2 } } , { \\frac { d _ { 0 } - 1 } { 2 } } \\right) ,", + "type": "interline_equation", + "image_path": "64282d6dabc67496b878a59318c54049889bb14ef87d747dd604972e9df39a48.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 703, + 408, + 719.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 201, + 719.5, + 408, + 736.0 + ], + "spans": [], + "index": 31 + } + ] + } + ], + "page_idx": 28, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 454, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 496, + 456, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 496, + 456, + 505, + 466 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 162, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 163, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 163, + 95 + ], + "score": 1.0, + "content": "so we choose", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 163, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 90, + 364, + 121 + ], + "lines": [ + { + "bbox": [ + 247, + 90, + 364, + 121 + ], + "spans": [ + { + "bbox": [ + 247, + 90, + 364, + 121 + ], + "score": 0.95, + "content": "{ \\cal K } \\left( \\theta \\right) = \\left( { \\frac { \\theta ^ { 2 } M L } { 2 4 \\psi \\left( \\theta \\right) N } } \\right) ^ { 1 / 4 } .", + "type": "interline_equation", + "image_path": "388b52d07a6c88ce91624ea496107aebd945676fda8d69e93cce3af934817e11.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 247, + 90, + 364, + 105.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 247, + 105.5, + 364, + 121.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 263, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 263, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 157, + 136 + ], + "score": 1.0, + "content": "Substituting", + "type": "text" + }, + { + "bbox": [ + 157, + 123, + 182, + 135 + ], + "score": 0.92, + "content": "K \\left( \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 122, + 201, + 136 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 201, + 123, + 235, + 135 + ], + "score": 0.93, + "content": "f \\left( K , \\theta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 122, + 263, + 136 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 106, + 122, + 263, + 136 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 139, + 409, + 173 + ], + "lines": [ + { + "bbox": [ + 201, + 139, + 409, + 173 + ], + "spans": [ + { + "bbox": [ + 201, + 139, + 409, + 173 + ], + "score": 0.93, + "content": "f \\left( K \\left( \\theta \\right) , \\theta \\right) = 2 \\exp \\left( - N \\left[ \\frac { \\psi ^ { 3 } \\left( \\theta \\right) \\theta ^ { 2 } M L } { 2 4 N } \\right] ^ { 1 / 4 } \\right) .", + "type": "interline_equation", + "image_path": "2dcba8492ca11e9de048b7def965ec4068a0ca897e5872cdaef94ca5ee040bdf.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 139, + 409, + 156.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 201, + 156.0, + 409, + 173.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 227, + 191 + ], + "score": 1.0, + "content": "To minimize this function in", + "type": "text" + }, + { + "bbox": [ + 227, + 179, + 234, + 189 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 177, + 382, + 191 + ], + "score": 1.0, + "content": ", we need to maximize the function", + "type": "text" + }, + { + "bbox": [ + 383, + 178, + 420, + 191 + ], + "score": 0.93, + "content": "\\psi ^ { 3 } \\left( \\theta \\right) \\theta ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 177, + 505, + 191 + ], + "score": 1.0, + "content": "(which has a single", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 284, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 284, + 201 + ], + "score": 1.0, + "content": "maximum). Doing this numerically gives us", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 177, + 505, + 201 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 205, + 413, + 220 + ], + "lines": [ + { + "bbox": [ + 197, + 205, + 413, + 220 + ], + "spans": [ + { + "bbox": [ + 197, + 205, + 413, + 220 + ], + "score": 0.88, + "content": "\\theta _ { * } \\approx 2 3 . 2 5 ; \\psi \\left( \\theta _ { * } \\right) \\approx 0 . 1 0 6 2 ; \\psi ^ { 3 } \\left( \\theta _ { * } \\right) \\theta _ { * } ^ { 2 } \\approx 0 . 6 4 7 8 .", + "type": "interline_equation", + "image_path": "f3863d4a882a35485e8d0e72f3151d723414518fe330ed3e35d38cdf90f45cea.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 197, + 205, + 413, + 220 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 224, + 365, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 365, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 365, + 238 + ], + "score": 1.0, + "content": "Substituting eqs. (12.27) and (12.28) into eq. (12.25), we obtain", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 223, + 365, + 238 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 240, + 437, + 358 + ], + "lines": [ + { + "bbox": [ + 173, + 240, + 437, + 358 + ], + "spans": [ + { + "bbox": [ + 173, + 240, + 437, + 358 + ], + "score": 0.97, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\mathbf { C } \\mathbf { B } > 0 \\right) } \\\\ & { \\leq \\exp \\left( - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + O \\left( N \\log K \\right) \\right) } \\\\ & { + \\exp \\left[ - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + 2 L \\frac { \\log K } { K } + \\frac { M \\theta ^ { 2 } } { 2 4 K ^ { 2 } } - \\log \\left( 2 K ^ { 2 } \\right) \\right] } \\\\ & { \\leq \\exp \\left( - N \\left[ \\frac { M L } { 3 7 . 0 5 N } \\right] ^ { 1 / 4 } + O \\left( N \\log \\left( \\frac { M L } { N } \\right) \\right) \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "e1c6049da1a1a73d233eee0483fecccbd48ff299ffe851950ae2fc0f58648b6e.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 173, + 240, + 437, + 279.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 173, + 279.3333333333333, + 437, + 318.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 173, + 318.66666666666663, + 437, + 357.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 504, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 234, + 376 + ], + "score": 1.0, + "content": "where in the last line we used", + "type": "text" + }, + { + "bbox": [ + 235, + 362, + 307, + 374 + ], + "score": 0.86, + "content": "N \\geq L , N \\geq M", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 361, + 326, + 376 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 327, + 362, + 415, + 375 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } \\left[ N , M , L \\right] \\dot { > } \\alpha \\dot { > } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 361, + 505, + 376 + ], + "score": 1.0, + "content": ". Taking the log, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 374, + 260, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 144, + 388 + ], + "score": 1.0, + "content": "denoting", + "type": "text" + }, + { + "bbox": [ + 144, + 374, + 196, + 387 + ], + "score": 0.93, + "content": "\\alpha \\triangleq M L / N", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 374, + 260, + 388 + ], + "score": 1.0, + "content": ", we thus obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 361, + 505, + 388 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 391, + 402, + 407 + ], + "lines": [ + { + "bbox": [ + 207, + 391, + 402, + 407 + ], + "spans": [ + { + "bbox": [ + 207, + 391, + 402, + 407 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\log \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\le - 0 . 4 N \\alpha ^ { 1 / 4 } + O \\left( N \\log \\alpha \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "bbb40ca39754ca609ff4f7962feb6b2b034e3d6237b19b321a1e99e003c0d7ba.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 207, + 391, + 402, + 407 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 427, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 428, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 214, + 425 + ], + "score": 1.0, + "content": "Therefore, in the limit that", + "type": "text" + }, + { + "bbox": [ + 214, + 412, + 250, + 422 + ], + "score": 0.91, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 410, + 268, + 425 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 268, + 412, + 320, + 424 + ], + "score": 0.92, + "content": "\\alpha ( N ) \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 410, + 343, + 425 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 344, + 412, + 388, + 424 + ], + "score": 0.93, + "content": "\\alpha \\left( N \\right) \\dot { < } N", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 410, + 428, + 425 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 410, + 428, + 425 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 428, + 379, + 450 + ], + "lines": [ + { + "bbox": [ + 231, + 428, + 379, + 450 + ], + "spans": [ + { + "bbox": [ + 231, + 428, + 379, + 450 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\mathbf { C B } > 0 \\right) \\dot { \\leq } \\exp \\left( - 0 . 4 N \\alpha ^ { 1 / 4 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7370e6868104035fed18a36a01a9dd5aa15dfa920fb0233402f7ed6fd9df2bd5.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 231, + 428, + 379, + 450 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 482, + 503, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 503, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 503, + 495 + ], + "score": 1.0, + "content": "13 LOWER BOUNDING THE ANGULAR VOLUME OF GLOBAL MINIMA: PROOF", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 131, + 496, + 304, + 508 + ], + "spans": [ + { + "bbox": [ + 131, + 496, + 304, + 508 + ], + "score": 1.0, + "content": "OF LEMMAS USED IN SECTION 10", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 106, + 520, + 349, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 349, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 349, + 533 + ], + "score": 1.0, + "content": "13.1 ANGLES BETWEEN RANDOM GAUSSIAN VECTORS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 104, + 540, + 483, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 484, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 484, + 554 + ], + "score": 1.0, + "content": "To prove the results in the next appendix sections, we will rely on the following basic Lemma.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 540, + 484, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 344, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 347, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 245, + 570 + ], + "score": 1.0, + "content": "Lemma 39. For any vector y and", + "type": "text" + }, + { + "bbox": [ + 246, + 555, + 307, + 568 + ], + "score": 0.92, + "content": "\\mathbf { x } \\sim \\mathcal { N } \\left( 0 , \\mathbf { I } _ { d _ { 0 } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 554, + 347, + 570 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 554, + 347, + 570 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 572, + 411, + 604 + ], + "lines": [ + { + "bbox": [ + 200, + 572, + 411, + 604 + ], + "spans": [ + { + "bbox": [ + 200, + 572, + 411, + 604 + ], + "score": 0.87, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| > \\cos \\left( \\epsilon \\right) \\right) \\geq \\frac { 2 \\sin \\left( \\epsilon \\right) ^ { d _ { 0 } - 1 } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) }", + "type": "interline_equation", + "image_path": "fa42f9918e706f749d456c684981ab6c710b282b72ef98328705a2dbab9e7753.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 572, + 411, + 588.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 200, + 588.0, + 411, + 604.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 614, + 383, + 645 + ], + "lines": [ + { + "bbox": [ + 227, + 614, + 383, + 645 + ], + "spans": [ + { + "bbox": [ + 227, + 614, + 383, + 645 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| < u \\right) \\leq \\frac { 2 u } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,", + "type": "interline_equation", + "image_path": "3a77ddb5663511a0017617697064e848face4733888d795d7da832e0b5771c7e.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 227, + 614, + 383, + 629.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 227, + 629.5, + 383, + 645.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 649, + 306, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 306, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 190, + 662 + ], + "score": 1.0, + "content": "where we recall that", + "type": "text" + }, + { + "bbox": [ + 190, + 649, + 224, + 662 + ], + "score": 0.93, + "content": "B \\left( x , y \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 649, + 306, + 662 + ], + "score": 1.0, + "content": "is the beta function.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 649, + 306, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 506, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 673, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 162, + 689 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 162, + 676, + 205, + 688 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , \\mathbf { I } _ { d _ { 0 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 673, + 361, + 689 + ], + "score": 1.0, + "content": "is spherically symmetric, we can set", + "type": "text" + }, + { + "bbox": [ + 361, + 673, + 435, + 688 + ], + "score": 0.92, + "content": "\\mathbf { y } = \\left[ 1 , 0 \\ldots , 0 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 673, + 506, + 689 + ], + "score": 1.0, + "content": ", without loss of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 686, + 196, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 196, + 700 + ], + "score": 1.0, + "content": "generality. Therefore,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 673, + 506, + 700 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 703, + 408, + 736 + ], + "lines": [ + { + "bbox": [ + 201, + 703, + 408, + 736 + ], + "spans": [ + { + "bbox": [ + 201, + 703, + 408, + 736 + ], + "score": 0.93, + "content": "\\left| { \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } } \\right| ^ { 2 } = { \\frac { x _ { 1 } ^ { 2 } } { x _ { 1 } ^ { 2 } + \\sum _ { i = 2 } ^ { d _ { 0 } } x _ { i } ^ { 2 } } } \\sim { \\mathcal { B } } \\left( { \\frac { 1 } { 2 } } , { \\frac { d _ { 0 } - 1 } { 2 } } \\right) ,", + "type": "interline_equation", + "image_path": "64282d6dabc67496b878a59318c54049889bb14ef87d747dd604972e9df39a48.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 703, + 408, + 719.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 201, + 719.5, + 408, + 736.0 + ], + "spans": [], + "index": 31 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 102, + 76, + 508, + 100 + ], + "spans": [ + { + "bbox": [ + 102, + 76, + 219, + 100 + ], + "score": 1.0, + "content": "the Beta distribution, since", + "type": "text" + }, + { + "bbox": [ + 220, + 81, + 272, + 95 + ], + "score": 0.92, + "content": "x _ { 1 } ^ { 2 } \\sim \\chi ^ { 2 } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 76, + 291, + 100 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 80, + 390, + 95 + ], + "score": 0.92, + "content": "{ \\textstyle \\sum _ { i = 2 } ^ { d _ { 0 } } x _ { i } ^ { 2 } \\sim \\chi ^ { 2 } \\left( d _ { 0 } - 1 \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 76, + 508, + 100 + ], + "score": 1.0, + "content": "are independent chi-square", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 179, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 179, + 105 + ], + "score": 1.0, + "content": "random variables.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 294, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 109, + 290, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 109, + 143, + 125 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 143, + 110, + 198, + 123 + ], + "score": 0.84, + "content": "Z \\sim B \\left( \\alpha , \\beta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 109, + 201, + 125 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 202, + 110, + 243, + 123 + ], + "score": 0.89, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 109, + 264, + 125 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 264, + 110, + 290, + 122 + ], + "score": 0.9, + "content": "\\beta > 1", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 126, + 502, + 157 + ], + "lines": [ + { + "bbox": [ + 111, + 126, + 502, + 157 + ], + "spans": [ + { + "bbox": [ + 111, + 126, + 502, + 157 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( Z > u \\right) = \\frac { \\int _ { u } ^ { 1 } x ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } \\geq \\frac { \\int _ { u } ^ { 1 } 1 ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { \\int _ { 0 } ^ { 1 - u } x ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { \\left( 1 - u \\right) ^ { \\beta } } { \\beta B \\left( \\alpha , \\beta \\right) } .", + "type": "interline_equation", + "image_path": "d06592edc9440c6ba590cbc749012fc3a053f4fa597de77e312fdb3d3bbed508.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 126, + 502, + 136.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 136.33333333333334, + 502, + 146.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 146.66666666666669, + 502, + 157.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 192, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 192, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 164, + 173 + ], + "score": 1.0, + "content": "Therefore, for", + "type": "text" + }, + { + "bbox": [ + 165, + 161, + 188, + 171 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 159, + 192, + 173 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 175, + 468, + 212 + ], + "lines": [ + { + "bbox": [ + 141, + 175, + 468, + 212 + ], + "spans": [ + { + "bbox": [ + 141, + 175, + 468, + 212 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\mathsf { T } } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| ^ { 2 } > \\cos ^ { 2 } \\left( \\epsilon \\right) \\right) \\geq \\frac { 2 \\left( 1 - \\cos ^ { 2 } \\left( \\epsilon \\right) \\right) ^ { \\frac { d _ { 0 } - 1 } { 2 } } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } = \\frac { 2 \\sin \\left( \\epsilon \\right) ^ { d _ { 0 } - 1 } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,", + "type": "interline_equation", + "image_path": "20496d6d65b4764c5c1c6d5bec084da0f2f7db53e73b010b9280277409c481b1.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 141, + 175, + 468, + 187.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 141, + 187.33333333333334, + 468, + 199.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 141, + 199.66666666666669, + 468, + 212.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 205, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 205, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 205, + 230 + ], + "score": 1.0, + "content": "which proves eq. (13.1).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 249, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 248, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 162, + 246 + ], + "score": 1.0, + "content": "Similarly, for", + "type": "text" + }, + { + "bbox": [ + 162, + 232, + 204, + 245 + ], + "score": 0.93, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 231, + 222, + 246 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 222, + 232, + 248, + 244 + ], + "score": 0.89, + "content": "\\beta > 1", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 249, + 455, + 279 + ], + "lines": [ + { + "bbox": [ + 155, + 249, + 455, + 279 + ], + "spans": [ + { + "bbox": [ + 155, + 249, + 455, + 279 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( Z < u \\right) = \\frac { \\int _ { 0 } ^ { u } x ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } \\leq \\frac { \\int _ { 0 } ^ { u } x ^ { \\alpha - 1 } 1 ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { u ^ { \\alpha } } { \\alpha B \\left( \\alpha , \\beta \\right) } .", + "type": "interline_equation", + "image_path": "75e2fbaa029bdf83f8ed686cb88965f0aea91e71632a33ca1bed79b49d38d636.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 155, + 249, + 455, + 279 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 191, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 188, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 164, + 295 + ], + "score": 1.0, + "content": "Therefore, for", + "type": "text" + }, + { + "bbox": [ + 165, + 283, + 188, + 293 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 291, + 388, + 325 + ], + "lines": [ + { + "bbox": [ + 221, + 291, + 388, + 325 + ], + "spans": [ + { + "bbox": [ + 221, + 291, + 388, + 325 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| ^ { 2 } < u ^ { 2 } \\right) \\leq \\frac { 2 u } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,", + "type": "interline_equation", + "image_path": "5a8694a2a3c464cde6edf409c99fed078880ca3c851011d3fbc1927b83339f9c.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 221, + 291, + 388, + 308.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 221, + 308.0, + 388, + 325.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 205, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 206, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 206, + 340 + ], + "score": 1.0, + "content": "which proves eq. (13.2).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 350, + 233, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 234, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 234, + 363 + ], + "score": 1.0, + "content": "13.2 PROOF OF LEMMA 21:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 104, + 364, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 264, + 387 + ], + "score": 1.0, + "content": "Given three matrices: datapoints,", + "type": "text" + }, + { + "bbox": [ + 264, + 369, + 426, + 385 + ], + "score": 0.89, + "content": "\\begin{array} { r c l c l } { \\mathbf { X } } & { = } & { \\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] } & { \\in } & { \\mathbb { R } ^ { d _ { 0 } \\times N } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 364, + 472, + 387 + ], + "score": 1.0, + "content": ", weights", + "type": "text" + }, + { + "bbox": [ + 472, + 370, + 505, + 383 + ], + "score": 0.8, + "content": "\\begin{array} { r l } { \\mathbf { W } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 108, + 380, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 108, + 385, + 226, + 403 + ], + "score": 0.92, + "content": "\\left[ { \\bf w } _ { 1 } ^ { \\top } , \\ldots , { \\bf w } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\ \\in \\ \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 380, + 314, + 410 + ], + "score": 1.0, + "content": ", and target weights", + "type": "text" + }, + { + "bbox": [ + 314, + 384, + 478, + 406 + ], + "score": 0.9, + "content": "\\mathbf W ^ { * } ~ = ~ \\left[ \\mathbf w _ { 1 } ^ { * \\top } , \\ldots , \\mathbf w _ { d _ { 1 } ^ { * } } ^ { * \\top } \\right] ^ { \\top } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 387, + 506, + 402 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 403, + 284, + 417 + ], + "spans": [ + { + "bbox": [ + 107, + 404, + 140, + 416 + ], + "score": 0.91, + "content": "d _ { 1 } ^ { * } \\leq d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 403, + 284, + 417 + ], + "score": 1.0, + "content": ",we recall the following definitions:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 428, + 430, + 462 + ], + "lines": [ + { + "bbox": [ + 180, + 428, + 430, + 462 + ], + "spans": [ + { + "bbox": [ + 180, + 428, + 430, + 462 + ], + "score": 0.94, + "content": "\\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\forall i , n : \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\mathrm { s i n } \\alpha \\right\\}", + "type": "interline_equation", + "image_path": "86ff27d291dd647a82a9fd6936fdc712e5655c3a789fad525b8a068390284dcc.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 180, + 428, + 430, + 439.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 180, + 439.3333333333333, + 430, + 450.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 180, + 450.66666666666663, + 430, + 461.99999999999994 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 124, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 124, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 124, + 474 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 488, + 456, + 504 + ], + "lines": [ + { + "bbox": [ + 155, + 488, + 456, + 504 + ], + "spans": [ + { + "bbox": [ + 155, + 488, + 456, + 504 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } } | \\forall i \\le d _ { 1 } ^ { * } : \\ \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } \\right) = \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { * } { } ^ { \\top } \\mathbf { X } \\right) \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "c2df6c638b0f87029d4c5fbbbbee6d66561c05765861aa28872135ff68aac91e.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 155, + 488, + 456, + 504 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 391, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 390, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 390, + 518 + ], + "score": 1.0, + "content": "Using these definitions, in this section we prove the following Lemma.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 504, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 293, + 533 + ], + "score": 1.0, + "content": "Lemma 40. (Lemma 21 restated). For any", + "type": "text" + }, + { + "bbox": [ + 293, + 522, + 300, + 530 + ], + "score": 0.65, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 518, + 314, + 533 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 315, + 520, + 332, + 531 + ], + "score": 0.55, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 518, + 506, + 533 + ], + "score": 1.0, + "content": "is independent from W then, in the limit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 530, + 324, + 544 + ], + "spans": [ + { + "bbox": [ + 107, + 531, + 142, + 541 + ], + "score": 0.83, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 530, + 146, + 544 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 146, + 531, + 217, + 543 + ], + "score": 0.73, + "content": "\\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 530, + 238, + 544 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 239, + 530, + 324, + 543 + ], + "score": 0.81, + "content": "\\log \\sin \\alpha { \\dot { > } } { \\dot { d } _ { 0 } } ^ { - 1 } \\log { d _ { 0 } }", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 547, + 411, + 568 + ], + "lines": [ + { + "bbox": [ + 199, + 547, + 411, + 568 + ], + "spans": [ + { + "bbox": [ + 199, + 547, + 411, + 568 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\geq } \\exp \\left( d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7d30cfb94c68e8b4db2b7cc08b8fb46c3d19ca0c3651149c9f8564d792c7a8c2.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 199, + 547, + 411, + 568 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 104, + 579, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 203, + 597 + ], + "score": 1.0, + "content": "Proof. To lower bound", + "type": "text" + }, + { + "bbox": [ + 203, + 579, + 390, + 600 + ], + "score": 0.94, + "content": "\\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 580, + 506, + 597 + ], + "score": 1.0, + "content": ", we define the event that all", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 596, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 249, + 612 + ], + "score": 1.0, + "content": "weight hyperplanes (with normals", + "type": "text" + }, + { + "bbox": [ + 249, + 600, + 262, + 609 + ], + "score": 0.81, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 596, + 370, + 612 + ], + "score": 1.0, + "content": ") have an angle of at least", + "type": "text" + }, + { + "bbox": [ + 371, + 600, + 378, + 608 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 596, + 506, + 612 + ], + "score": 1.0, + "content": "from the corresponding target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 609, + 236, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 215, + 622 + ], + "score": 1.0, + "content": "hyperplanes (with normals", + "type": "text" + }, + { + "bbox": [ + 215, + 610, + 229, + 621 + ], + "score": 0.88, + "content": "\\mathbf { w } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 609, + 236, + 622 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 625, + 421, + 654 + ], + "lines": [ + { + "bbox": [ + 189, + 625, + 421, + 654 + ], + "spans": [ + { + "bbox": [ + 189, + 625, + 421, + 654 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { \\mathcal { G } } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) = \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } } \\big | \\left| \\frac { \\mathbf { W } _ { i } ^ { \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { w } _ { i } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| < \\cos \\left( \\alpha \\right) \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "0eb1d3f2b46033b5524c009f51e1591b0ad4831fbbba3fa78b51c5ba9ff22647.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 189, + 625, + 421, + 654 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 160, + 673 + ], + "score": 1.0, + "content": "In order that", + "type": "text" + }, + { + "bbox": [ + 161, + 657, + 301, + 672 + ], + "score": 0.87, + "content": "\\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { x } ^ { ( n ) } \\right) \\neq \\mathrm { s i g n } \\left( \\mathbf { w } _ { 1 } ^ { \\ast \\top } \\mathbf { x } ^ { ( n ) } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 657, + 305, + 673 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 306, + 659, + 318, + 671 + ], + "score": 0.8, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 657, + 439, + 673 + ], + "score": 1.0, + "content": "must be rotated in respect to", + "type": "text" + }, + { + "bbox": [ + 439, + 659, + 453, + 671 + ], + "score": 0.9, + "content": "\\mathbf { w } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 657, + 506, + 673 + ], + "score": 1.0, + "content": "by an angle", + "type": "text" + } + ], + "index": 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1 \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 76, + 508, + 100 + ], + "score": 1.0, + "content": "are independent chi-square", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 179, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 179, + 105 + ], + "score": 1.0, + "content": "random variables.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 102, + 76, + 508, + 105 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 294, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 109, + 290, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 109, + 143, + 125 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 143, + 110, + 198, + 123 + ], + "score": 0.84, + "content": "Z \\sim B \\left( \\alpha , \\beta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 109, + 201, + 125 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 202, + 110, + 243, + 123 + ], + "score": 0.89, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 109, + 264, + 125 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 264, + 110, + 290, + 122 + ], + "score": 0.9, + "content": "\\beta > 1", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 109, + 290, + 125 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 126, + 502, + 157 + ], + "lines": [ + { + "bbox": [ + 111, + 126, + 502, + 157 + ], + "spans": [ + { + "bbox": [ + 111, + 126, + 502, + 157 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( Z > u \\right) = \\frac { \\int _ { u } ^ { 1 } x ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } \\geq \\frac { \\int _ { u } ^ { 1 } 1 ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { \\int _ { 0 } ^ { 1 - u } x ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { \\left( 1 - u \\right) ^ { \\beta } } { \\beta B \\left( \\alpha , \\beta \\right) } .", + "type": "interline_equation", + "image_path": "d06592edc9440c6ba590cbc749012fc3a053f4fa597de77e312fdb3d3bbed508.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 126, + 502, + 136.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 136.33333333333334, + 502, + 146.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 146.66666666666669, + 502, + 157.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 192, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 192, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 164, + 173 + ], + "score": 1.0, + "content": "Therefore, for", + "type": "text" + }, + { + "bbox": [ + 165, + 161, + 188, + 171 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 159, + 192, + 173 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 159, + 192, + 173 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 175, + 468, + 212 + ], + "lines": [ + { + "bbox": [ + 141, + 175, + 468, + 212 + ], + "spans": [ + { + "bbox": [ + 141, + 175, + 468, + 212 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\mathsf { T } } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| ^ { 2 } > \\cos ^ { 2 } \\left( \\epsilon \\right) \\right) \\geq \\frac { 2 \\left( 1 - \\cos ^ { 2 } \\left( \\epsilon \\right) \\right) ^ { \\frac { d _ { 0 } - 1 } { 2 } } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } = \\frac { 2 \\sin \\left( \\epsilon \\right) ^ { d _ { 0 } - 1 } } { \\left( d _ { 0 } - 1 \\right) B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,", + "type": "interline_equation", + "image_path": "20496d6d65b4764c5c1c6d5bec084da0f2f7db53e73b010b9280277409c481b1.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 141, + 175, + 468, + 187.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 141, + 187.33333333333334, + 468, + 199.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 141, + 199.66666666666669, + 468, + 212.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 205, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 205, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 205, + 230 + ], + "score": 1.0, + "content": "which proves eq. (13.1).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 214, + 205, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 249, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 248, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 162, + 246 + ], + "score": 1.0, + "content": "Similarly, for", + "type": "text" + }, + { + "bbox": [ + 162, + 232, + 204, + 245 + ], + "score": 0.93, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 231, + 222, + 246 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 222, + 232, + 248, + 244 + ], + "score": 0.89, + "content": "\\beta > 1", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 231, + 248, + 246 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 249, + 455, + 279 + ], + "lines": [ + { + "bbox": [ + 155, + 249, + 455, + 279 + ], + "spans": [ + { + "bbox": [ + 155, + 249, + 455, + 279 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( Z < u \\right) = \\frac { \\int _ { 0 } ^ { u } x ^ { \\alpha - 1 } \\left( 1 - x \\right) ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } \\leq \\frac { \\int _ { 0 } ^ { u } x ^ { \\alpha - 1 } 1 ^ { \\beta - 1 } d x } { B \\left( \\alpha , \\beta \\right) } = \\frac { u ^ { \\alpha } } { \\alpha B \\left( \\alpha , \\beta \\right) } .", + "type": "interline_equation", + "image_path": "75e2fbaa029bdf83f8ed686cb88965f0aea91e71632a33ca1bed79b49d38d636.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 155, + 249, + 455, + 279 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 191, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 188, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 164, + 295 + ], + "score": 1.0, + "content": "Therefore, for", + "type": "text" + }, + { + "bbox": [ + 165, + 283, + 188, + 293 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 280, + 188, + 295 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 291, + 388, + 325 + ], + "lines": [ + { + "bbox": [ + 221, + 291, + 388, + 325 + ], + "spans": [ + { + "bbox": [ + 221, + 291, + 388, + 325 + ], + "score": 0.93, + "content": "\\mathbb { P } \\left( \\left| \\frac { \\mathbf { x } ^ { \\top } \\mathbf { y } } { \\left\\| \\mathbf { x } \\right\\| \\left\\| \\mathbf { y } \\right\\| } \\right| ^ { 2 } < u ^ { 2 } \\right) \\leq \\frac { 2 u } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } ,", + "type": "interline_equation", + "image_path": "5a8694a2a3c464cde6edf409c99fed078880ca3c851011d3fbc1927b83339f9c.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 221, + 291, + 388, + 308.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 221, + 308.0, + 388, + 325.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 205, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 206, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 206, + 340 + ], + "score": 1.0, + "content": "which proves eq. (13.2).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 324, + 206, + 340 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 350, + 233, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 234, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 234, + 363 + ], + "score": 1.0, + "content": "13.2 PROOF OF LEMMA 21:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 104, + 364, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 264, + 387 + ], + "score": 1.0, + "content": "Given three matrices: datapoints,", + "type": "text" + }, + { + "bbox": [ + 264, + 369, + 426, + 385 + ], + "score": 0.89, + "content": "\\begin{array} { r c l c l } { \\mathbf { X } } & { = } & { \\left[ \\mathbf { x } ^ { ( 1 ) } , \\ldots , \\mathbf { x } ^ { ( N ) } \\right] } & { \\in } & { \\mathbb { R } ^ { d _ { 0 } \\times N } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 364, + 472, + 387 + ], + "score": 1.0, + "content": ", weights", + "type": "text" + }, + { + "bbox": [ + 472, + 370, + 505, + 383 + ], + "score": 0.8, + "content": "\\begin{array} { r l } { \\mathbf { W } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 108, + 380, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 108, + 385, + 226, + 403 + ], + "score": 0.92, + "content": "\\left[ { \\bf w } _ { 1 } ^ { \\top } , \\ldots , { \\bf w } _ { d _ { 1 } } ^ { \\top } \\right] ^ { \\top } \\ \\in \\ \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 380, + 314, + 410 + ], + "score": 1.0, + "content": ", and target weights", + "type": "text" + }, + { + "bbox": [ + 314, + 384, + 478, + 406 + ], + "score": 0.9, + "content": "\\mathbf W ^ { * } ~ = ~ \\left[ \\mathbf w _ { 1 } ^ { * \\top } , \\ldots , \\mathbf w _ { d _ { 1 } ^ { * } } ^ { * \\top } \\right] ^ { \\top } ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 387, + 506, + 402 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 403, + 284, + 417 + ], + "spans": [ + { + "bbox": [ + 107, + 404, + 140, + 416 + ], + "score": 0.91, + "content": "d _ { 1 } ^ { * } \\leq d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 403, + 284, + 417 + ], + "score": 1.0, + "content": ",we recall the following definitions:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 364, + 506, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 428, + 430, + 462 + ], + "lines": [ + { + "bbox": [ + 180, + 428, + 430, + 462 + ], + "spans": [ + { + "bbox": [ + 180, + 428, + 430, + 462 + ], + "score": 0.94, + "content": "\\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\forall i , n : \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\mathrm { s i n } \\alpha \\right\\}", + "type": "interline_equation", + "image_path": "86ff27d291dd647a82a9fd6936fdc712e5655c3a789fad525b8a068390284dcc.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 180, + 428, + 430, + 439.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 180, + 439.3333333333333, + 430, + 450.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 180, + 450.66666666666663, + 430, + 461.99999999999994 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 124, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 124, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 124, + 474 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 463, + 124, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 488, + 456, + 504 + ], + "lines": [ + { + "bbox": [ + 155, + 488, + 456, + 504 + ], + "spans": [ + { + "bbox": [ + 155, + 488, + 456, + 504 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { W } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { 0 } } | \\forall i \\le d _ { 1 } ^ { * } : \\ \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { \\top } \\mathbf { X } \\right) = \\mathrm { s i g n } \\left( \\mathbf { w } _ { i } ^ { * } { } ^ { \\top } \\mathbf { X } \\right) \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": 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(Lemma 21 restated). For any", + "type": "text" + }, + { + "bbox": [ + 293, + 522, + 300, + 530 + ], + "score": 0.65, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 518, + 314, + 533 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 315, + 520, + 332, + 531 + ], + "score": 0.55, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 518, + 506, + 533 + ], + "score": 1.0, + "content": "is independent from W then, in the limit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 530, + 324, + 544 + ], + "spans": [ + { + "bbox": [ + 107, + 531, + 142, + 541 + ], + "score": 0.83, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 530, + 146, + 544 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 146, + 531, + 217, + 543 + ], + "score": 0.73, + "content": "\\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 530, + 238, + 544 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 239, + 530, + 324, + 543 + ], + "score": 0.81, + "content": "\\log \\sin \\alpha { \\dot { > } } { \\dot { d } _ { 0 } } ^ { - 1 } \\log { d _ { 0 } }", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 518, + 506, + 544 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 547, + 411, + 568 + ], + "lines": [ + { + "bbox": [ + 199, + 547, + 411, + 568 + ], + "spans": [ + { + "bbox": [ + 199, + 547, + 411, + 568 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\geq } \\exp \\left( d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "7d30cfb94c68e8b4db2b7cc08b8fb46c3d19ca0c3651149c9f8564d792c7a8c2.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 199, + 547, + 411, + 568 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 104, + 579, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 203, + 597 + ], + "score": 1.0, + "content": "Proof. To lower bound", + "type": "text" + }, + { + "bbox": [ + 203, + 579, + 390, + 600 + ], + "score": 0.94, + "content": "\\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\forall \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 580, + 506, + 597 + ], + "score": 1.0, + "content": ", we define the event that all", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 596, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 249, + 612 + ], + "score": 1.0, + "content": "weight hyperplanes (with normals", + "type": "text" + }, + { + "bbox": [ + 249, + 600, + 262, + 609 + ], + "score": 0.81, + "content": "\\mathbf { w } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 596, + 370, + 612 + ], + "score": 1.0, + "content": ") have an angle of at least", + "type": "text" + }, + { + "bbox": [ + 371, + 600, + 378, + 608 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 596, + 506, + 612 + ], + "score": 1.0, + "content": "from the corresponding target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 609, + 236, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 215, + 622 + ], + "score": 1.0, + "content": "hyperplanes (with normals", + "type": "text" + }, + { + "bbox": [ + 215, + 610, + 229, + 621 + ], + "score": 0.88, + "content": "\\mathbf { w } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 609, + 236, + 622 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 579, + 506, + 622 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 625, + 421, + 654 + ], + "lines": [ + { + "bbox": [ + 189, + 625, + 421, + 654 + ], 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angle", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 669, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 238, + 686 + ], + "score": 1.0, + "content": "greater then the angular margin", + "type": "text" + }, + { + "bbox": [ + 238, + 675, + 245, + 682 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 669, + 416, + 686 + ], + "score": 1.0, + "content": ", which is the minimal the angle between", + "type": "text" + }, + { + "bbox": [ + 416, + 671, + 435, + 682 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 669, + 506, + 686 + ], + "score": 1.0, + "content": "and the solution", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 682, + 430, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 215, + 696 + ], + "score": 1.0, + "content": "hyperplanes (with normals", + "type": "text" + }, + { + "bbox": [ 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(13.5), in (2) we used the independence of", + "type": "text" + }, + { + "bbox": [ + 379, + 182, + 415, + 198 + ], + "score": 0.94, + "content": "\\{ { \\bf w } _ { i } \\} _ { i = 1 } ^ { d _ { 1 } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 179, + 506, + 203 + ], + "score": 1.0, + "content": "and in (3) we used eq.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "(13.1) from Lemma 39. Lastly, to simplify this equation we use the asymptotic expansion of the beta", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 205, + 325, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 142, + 224 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 208, + 273, + 222 + ], + "score": 0.92, + "content": "B \\left( { \\textstyle \\frac { 1 } { 2 } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 205, + 311, + 224 + ], + "score": 1.0, + "content": "for large", + "type": "text" + }, + { + "bbox": [ + 312, + 211, + 318, + 219 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 205, + 325, + 224 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 227, + 441, + 249 + ], + "lines": [ + { + "bbox": [ + 169, + 227, + 441, + 249 + ], + "spans": [ + { + "bbox": [ + 169, + 227, + 441, + 249 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\log \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\geq d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha + O \\left( d _ { 1 } ^ { * } \\log d _ { 0 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "ddcbd3ee9ecae6f0edcd1cd3a6bb3d07c8c96725d5d383c6e4ca63bdb1cf72df.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 169, + 227, + 441, + 249 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 393, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 393, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 243, + 271 + ], + "score": 1.0, + "content": "We obtain the Lemma in the limit", + "type": "text" + }, + { + "bbox": [ + 243, + 257, + 279, + 267 + ], + "score": 0.9, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 253, + 304, + 271 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 304, + 255, + 389, + 268 + ], + "score": 0.86, + "content": "\\log \\sin \\alpha { \\dot { > } } d _ { 0 } ^ { - 1 } \\log d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 253, + 393, + 271 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 281, + 233, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 234, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 234, + 295 + ], + "score": 1.0, + "content": "13.3 PROOF OF LEMMA 22:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 506, + 346 + ], + "lines": [ + { + "bbox": [ + 101, + 299, + 508, + 325 + ], + "spans": [ + { + "bbox": [ + 101, + 299, + 268, + 325 + ], + "score": 1.0, + "content": "Lemma 41. (Lemma 22 restated). Let", + "type": "text" + }, + { + "bbox": [ + 268, + 300, + 412, + 323 + ], + "score": 0.92, + "content": "\\mathbf { W } ^ { * } = \\left[ \\mathbf { w } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { w } _ { d _ { 1 } ^ { * } } ^ { \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 302, + 508, + 322 + ], + "score": 1.0, + "content": "a fixed matrix indepen-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 320, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 138, + 336 + ], + "score": 1.0, + "content": "dent of", + "type": "text" + }, + { + "bbox": [ + 138, + 323, + 148, + 333 + ], + "score": 0.42, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 320, + 226, + 336 + ], + "score": 1.0, + "content": ". Then, in the limit", + "type": "text" + }, + { + "bbox": [ + 227, + 323, + 264, + 333 + ], + "score": 0.89, + "content": "N \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 320, + 286, + 336 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 287, + 323, + 332, + 335 + ], + "score": 0.86, + "content": "d _ { 1 } ^ { * } { \\dot { \\leq } } d _ { 0 } { \\dot { \\leq } } N _ { \\mathrm { ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 320, + 506, + 336 + ], + "score": 1.0, + "content": ", the probability of not having an angular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 332, + 354, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 139, + 348 + ], + "score": 1.0, + "content": "margin", + "type": "text" + }, + { + "bbox": [ + 139, + 334, + 220, + 346 + ], + "score": 0.87, + "content": "\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 332, + 354, + 348 + ], + "score": 1.0, + "content": "(eq. (13.3)) is upper bounded by", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 352, + 372, + 379 + ], + "lines": [ + { + "bbox": [ + 238, + 352, + 372, + 379 + ], + "spans": [ + { + "bbox": [ + 238, + 352, + 372, + 379 + ], + "score": 0.94, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 }", + "type": "interline_equation", + "image_path": "ce627acc7cb618f45f7c25e679dc79964d455d7f1ec26ec0b22ea5b8a6e17d35.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 238, + 352, + 372, + 379 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 178, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 179, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 179, + 404 + ], + "score": 1.0, + "content": "Proof. We define", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 408, + 427, + 442 + ], + "lines": [ + { + "bbox": [ + 183, + 408, + 427, + 442 + ], + "spans": [ + { + "bbox": [ + 183, + 408, + 427, + 442 + ], + "score": 0.94, + "content": "\\mathcal M _ { n , i } ^ { \\alpha } ( \\mathbf W ^ { * } ) \\triangleq \\{ \\mathbf X \\in \\mathbb R ^ { d _ { 0 } \\times N } | | \\frac { \\mathbf x ^ { ( n ) \\top } \\mathbf w _ { i } ^ { * } } { \\| \\mathbf x ^ { ( n ) } \\| \\| \\mathbf w _ { i } ^ { * } \\| } | > \\sin ( \\alpha ) \\} ,", + "type": "interline_equation", + "image_path": "b4b3b1e2187066ea7a7e0b408ffdccfc5a6ed26dcc5480cff6b86dcf5ce352ad.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 408, + 427, + 425.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 183, + 425.0, + 427, + 442.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 443, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 444, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 124, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 447, + 253, + 465 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathcal { M } _ { n , i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 447, + 282, + 465 + ], + "score": 1.0, + "content": ". 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Taking the log and we using the asymptotic expansion of the beta", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 573, + 353, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 142, + 592 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 576, + 273, + 591 + ], + "score": 0.94, + "content": "B \\left( { \\textstyle \\frac { 1 } { 2 } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 573, + 311, + 592 + ], + "score": 1.0, + "content": "for large", + "type": "text" + }, + { + "bbox": [ + 312, + 579, + 318, + 587 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 573, + 353, + 592 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 595, + 462, + 659 + ], + "lines": [ + { + "bbox": [ + 147, + 595, + 462, + 659 + ], + "spans": [ + { + "bbox": [ + 147, + 595, + 462, + 659 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\log \\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\geq N \\log \\left[ 1 - \\sqrt { \\frac { 2 } { \\pi } d _ { 0 } } d _ { 1 } ^ { * } \\sin \\alpha + O \\left( d _ { 1 } ^ { * } d _ { 0 } ^ { - 1 / 2 } \\sin \\alpha \\right) \\right] } \\\\ & { \\qquad = - \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + O \\left( d _ { 0 } ^ { - 3 / 2 } / N + d _ { 0 } ^ { - 1 } N ^ { - 2 } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "c57dcc64b5df4df02b49cddb13a272ddc1a9c7ec5fe73de5f7b50aeb52d23dcb.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 147, + 595, + 462, + 616.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 147, + 616.3333333333334, + 462, + 637.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 147, + 637.6666666666667, + 462, + 659.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 440, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 440, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 240, + 677 + ], + "score": 1.0, + "content": "where in the last line we recalled", + "type": "text" + }, + { + "bbox": [ + 240, + 664, + 293, + 677 + ], + "score": 0.92, + "content": "\\sin \\alpha = 1 / N", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 663, + 356, + 677 + ], + "score": 1.0, + "content": ". 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Lastly, to simplify this equation we use the asymptotic expansion of the beta", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 205, + 325, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 142, + 224 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 208, + 273, + 222 + ], + "score": 0.92, + "content": "B \\left( { \\textstyle \\frac { 1 } { 2 } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 205, + 311, + 224 + ], + "score": 1.0, + "content": "for large", + "type": "text" + }, + { + "bbox": [ + 312, + 211, + 318, + 219 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 205, + 325, + 224 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 103, + 179, + 506, + 224 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 227, + 441, + 249 + ], + "lines": [ + { + "bbox": [ + 169, + 227, + 441, + 249 + ], + "spans": [ + { + "bbox": [ + 169, + 227, + 441, + 249 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\log \\mathbb { P } _ { \\mathbf { W } \\sim \\mathcal { N } } \\left( \\mathbf { W } \\in \\tilde { \\mathcal { G } } \\left( \\mathbf { X } , \\mathbf { W } ^ { * } \\right) \\right) \\geq d _ { 0 } d _ { 1 } ^ { * } \\log \\sin \\alpha + O \\left( d _ { 1 } ^ { * } \\log d _ { 0 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "ddcbd3ee9ecae6f0edcd1cd3a6bb3d07c8c96725d5d383c6e4ca63bdb1cf72df.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 169, + 227, + 441, + 249 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 393, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 393, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 243, + 271 + ], + "score": 1.0, + "content": "We obtain the Lemma in the limit", + "type": "text" + }, + { + "bbox": [ + 243, + 257, + 279, + 267 + ], + "score": 0.9, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 253, + 304, + 271 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 304, + 255, + 389, + 268 + ], + "score": 0.86, + "content": "\\log \\sin \\alpha { \\dot { > } } d _ { 0 } ^ { - 1 } \\log d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 253, + 393, + 271 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 253, + 393, + 271 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 281, + 233, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 234, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 234, + 295 + ], + "score": 1.0, + "content": "13.3 PROOF OF LEMMA 22:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 506, + 346 + ], + "lines": [ + { + "bbox": [ + 101, + 299, + 508, + 325 + ], + "spans": [ + { + "bbox": [ + 101, + 299, + 268, + 325 + ], + "score": 1.0, + "content": "Lemma 41. (Lemma 22 restated). Let", + "type": "text" + }, + { + "bbox": [ + 268, + 300, + 412, + 323 + ], + "score": 0.92, + "content": "\\mathbf { W } ^ { * } = \\left[ \\mathbf { w } _ { 1 } ^ { \\top } , \\ldots , \\mathbf { w } _ { d _ { 1 } ^ { * } } ^ { \\top } \\right] ^ { \\top } \\in \\mathbb { R } ^ { d _ { 1 } ^ { * } \\times d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 302, + 508, + 322 + ], + "score": 1.0, + "content": "a fixed matrix indepen-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 320, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 138, + 336 + ], + "score": 1.0, + "content": "dent of", + "type": "text" + }, + { + "bbox": [ + 138, + 323, + 148, + 333 + ], + "score": 0.42, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 320, + 226, + 336 + ], + "score": 1.0, + "content": ". Then, in the limit", + "type": "text" + }, + { + "bbox": [ + 227, + 323, + 264, + 333 + ], + "score": 0.89, + "content": "N \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 320, + 286, + 336 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 287, + 323, + 332, + 335 + ], + "score": 0.86, + "content": "d _ { 1 } ^ { * } { \\dot { \\leq } } d _ { 0 } { \\dot { \\leq } } N _ { \\mathrm { ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 320, + 506, + 336 + ], + "score": 1.0, + "content": ", the probability of not having an angular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 332, + 354, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 139, + 348 + ], + "score": 1.0, + "content": "margin", + "type": "text" + }, + { + "bbox": [ + 139, + 334, + 220, + 346 + ], + "score": 0.87, + "content": "\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 332, + 354, + 348 + ], + "score": 1.0, + "content": "(eq. (13.3)) is upper bounded by", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 101, + 299, + 508, + 348 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 352, + 372, + 379 + ], + "lines": [ + { + "bbox": [ + 238, + 352, + 372, + 379 + ], + "spans": [ + { + "bbox": [ + 238, + 352, + 372, + 379 + ], + "score": 0.94, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 }", + "type": "interline_equation", + "image_path": "ce627acc7cb618f45f7c25e679dc79964d455d7f1ec26ec0b22ea5b8a6e17d35.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 238, + 352, + 372, + 379 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 178, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 179, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 179, + 404 + ], + "score": 1.0, + "content": "Proof. We define", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 390, + 179, + 404 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 408, + 427, + 442 + ], + "lines": [ + { + "bbox": [ + 183, + 408, + 427, + 442 + ], + "spans": [ + { + "bbox": [ + 183, + 408, + 427, + 442 + ], + "score": 0.94, + "content": "\\mathcal M _ { n , i } ^ { \\alpha } ( \\mathbf W ^ { * } ) \\triangleq \\{ \\mathbf X \\in \\mathbb R ^ { d _ { 0 } \\times N } | | \\frac { \\mathbf x ^ { ( n ) \\top } \\mathbf w _ { i } ^ { * } } { \\| \\mathbf x ^ { ( n ) } \\| \\| \\mathbf w _ { i } ^ { * } \\| } | > \\sin ( \\alpha ) \\} ,", + "type": "interline_equation", + "image_path": "b4b3b1e2187066ea7a7e0b408ffdccfc5a6ed26dcc5480cff6b86dcf5ce352ad.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 408, + 427, + 425.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 183, + 425.0, + 427, + 442.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 443, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 444, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 124, + 465 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 447, + 253, + 465 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathcal { M } _ { n , i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 447, + 282, + 465 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 283, + 448, + 404, + 464 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { M } \\left( \\mathbf { W } ^ { * } \\right) = \\bigcap _ { n = 1 } ^ { N } \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 447, + 444, + 465 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 447, + 444, + 465 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 470, + 489, + 546 + ], + "lines": [ + { + "bbox": [ + 120, + 470, + 489, + 546 + ], + "spans": [ + { + "bbox": [ + 120, + 470, + 489, + 546 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\displaystyle \\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\stackrel { ( 1 ) } { = } \\prod _ { n = 1 } ^ { N } \\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) = \\prod _ { n = 1 } ^ { N } \\left[ 1 - \\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\right] } \\\\ & { \\displaystyle \\overset { ( 2 ) } { \\geq } \\prod _ { n = 1 } ^ { N } \\left[ 1 - \\sum _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { n , i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\right] \\stackrel { ( 3 ) } { \\geq } \\left[ 1 - d _ { 1 } ^ { * } \\frac { 2 \\sin \\left( \\alpha \\right) } { B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } \\right] ^ { N } , } \\end{array}", + "type": "interline_equation", + "image_path": "a95c3628c61d64327a15beb88d00affb7ef970a02e5e923e78cce885f719853f.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 120, + 470, + 489, + 495.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 120, + 495.3333333333333, + 489, + 520.6666666666666 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 120, + 520.6666666666666, + 489, + 546.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 102, + 547, + 507, + 570 + ], + "spans": [ + { + "bbox": [ + 102, + 547, + 318, + 570 + ], + "score": 1.0, + "content": "where in (1) we used the independence of \bx(n)\tNn=", + "type": "text" + }, + { + "bbox": [ + 322, + 551, + 507, + 569 + ], + "score": 1.0, + "content": ", in (2) we use the union bound, and in (3) we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 563, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 579 + ], + "score": 1.0, + "content": "use eq. (13.2) from Lemma 39. Taking the log and we using the asymptotic expansion of the beta", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 573, + 353, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 142, + 592 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 576, + 273, + 591 + ], + "score": 0.94, + "content": "B \\left( { \\textstyle \\frac { 1 } { 2 } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 573, + 311, + 592 + ], + "score": 1.0, + "content": "for large", + "type": "text" + }, + { + "bbox": [ + 312, + 579, + 318, + 587 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 573, + 353, + 592 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 102, + 547, + 507, + 592 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 595, + 462, + 659 + ], + "lines": [ + { + "bbox": [ + 147, + 595, + 462, + 659 + ], + "spans": [ + { + "bbox": [ + 147, + 595, + 462, + 659 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\log \\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\geq N \\log \\left[ 1 - \\sqrt { \\frac { 2 } { \\pi } d _ { 0 } } d _ { 1 } ^ { * } \\sin \\alpha + O \\left( d _ { 1 } ^ { * } d _ { 0 } ^ { - 1 / 2 } \\sin \\alpha \\right) \\right] } \\\\ & { \\qquad = - \\sqrt { \\frac { 2 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + O \\left( d _ { 0 } ^ { - 3 / 2 } / N + d _ { 0 } ^ { - 1 } N ^ { - 2 } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "c57dcc64b5df4df02b49cddb13a272ddc1a9c7ec5fe73de5f7b50aeb52d23dcb.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 147, + 595, + 462, + 616.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 147, + 616.3333333333334, + 462, + 637.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 147, + 637.6666666666667, + 462, + 659.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 440, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 440, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 240, + 677 + ], + "score": 1.0, + "content": "where in the last line we recalled", + "type": "text" + }, + { + "bbox": [ + 240, + 664, + 293, + 677 + ], + "score": 0.92, + "content": "\\sin \\alpha = 1 / N", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 663, + 356, + 677 + ], + "score": 1.0, + "content": ". 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(Lemma 23 restated). Let", + "type": "text" + }, + { + "bbox": [ + 274, + 102, + 327, + 113 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 100, + 507, + 116 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 305, + 127 + ], + "score": 1.0, + "content": "datapoints. Then we can find, with probability", + "type": "text" + }, + { + "bbox": [ + 305, + 115, + 311, + 124 + ], + "score": 0.56, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 114, + 316, + 127 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 317, + 114, + 344, + 126 + ], + "score": 0.92, + "content": "( \\mathbf { X } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 114, + 430, + 127 + ], + "score": 1.0, + "content": "-dependent matrices", + "type": "text" + }, + { + "bbox": [ + 430, + 114, + 448, + 124 + ], + "score": 0.83, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 114, + 469, + 127 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 469, + 115, + 480, + 124 + ], + "score": 0.85, + "content": "\\mathbf { z } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "as in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 183, + 139 + ], + "score": 1.0, + "content": "Theorem 8 (where", + "type": "text" + }, + { + "bbox": [ + 183, + 126, + 279, + 139 + ], + "score": 0.9, + "content": "d _ { 1 } ^ { * } \\triangleq 4 \\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\rceil ,", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 126, + 377, + 139 + ], + "score": 1.0, + "content": "). Moreover, in the limit", + "type": "text" + }, + { + "bbox": [ + 377, + 127, + 413, + 137 + ], + "score": 0.9, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 126, + 444, + 139 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 445, + 126, + 503, + 138 + ], + "score": 0.9, + "content": "N / d _ { 0 } { \\dot { \\le } } d _ { 0 } { \\dot { \\le } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 126, + 506, + 139 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 471, + 150 + ], + "score": 1.0, + "content": "for any y, we can bound the probability of not having an angular margin (eq. (13.3)) with", + "type": "text" + }, + { + "bbox": [ + 471, + 139, + 506, + 149 + ], + "score": 0.65, + "content": "\\sin \\alpha =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 146, + 169, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 155, + 160 + ], + "score": 0.92, + "content": "1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 146, + 169, + 162 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 161, + 407, + 190 + ], + "lines": [ + { + "bbox": [ + 204, + 161, + 407, + 190 + ], + "spans": [ + { + "bbox": [ + 204, + 161, + 407, + 190 + ], + "score": 0.91, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N }", + "type": "interline_equation", + "image_path": "8c1c1d6ab39ccbf0f1bc0c12b5fce8e37c5e5c850185b00e41b95515e72261b7.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 161, + 407, + 175.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 204, + 175.5, + 407, + 190.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 507, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "Proof. In this proof we heavily rely on the notation and results from the proof of in appendix section", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 272, + 223 + ], + "score": 1.0, + "content": "9. Without loss of generality we assume", + "type": "text" + }, + { + "bbox": [ + 273, + 210, + 333, + 223 + ], + "score": 0.93, + "content": "S _ { 1 } ^ { + } = [ d _ { 0 } - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 210, + 486, + 223 + ], + "score": 1.0, + "content": ". Unfortunately, we can’t use Lemma", + "type": "text" + }, + { + "bbox": [ + 486, + 211, + 506, + 221 + ], + "score": 0.59, + "content": "4 1 -", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 408, + 235 + ], + "score": 1.0, + "content": "this proof is significantly more complicated since the constructed solution", + "type": "text" + }, + { + "bbox": [ + 409, + 222, + 426, + 232 + ], + "score": 0.83, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 221, + 476, + 235 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 476, + 222, + 486, + 232 + ], + "score": 0.7, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "(we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 472, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 472, + 245 + ], + "score": 1.0, + "content": "keep this dependence implicit, for brevity). Similarly to the proof of Lemma 41, we define,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 245, + 425, + 278 + ], + "lines": [ + { + "bbox": [ + 186, + 245, + 425, + 278 + ], + "spans": [ + { + "bbox": [ + 186, + 245, + 425, + 278 + ], + "score": 0.87, + "content": "\\mathcal { M } _ { i , n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\sin \\left( \\alpha \\right) \\right\\}", + "type": "interline_equation", + "image_path": "b2bf5350db9f0790fefdcb282b55831a2d3290dccabd08afd033eecad5e46087.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 245, + 425, + 261.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 186, + 261.5, + 425, + 278.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 280, + 431, + 296 + ], + "lines": [ + { + "bbox": [ + 104, + 278, + 433, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 124, + 297 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 281, + 408, + 295 + ], + "score": 0.35, + "content": "\\begin{array} { r } { \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\bigcap _ { n = 1 } ^ { N } \\mathcal { M } _ { i , n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) , \\operatorname { s o } \\mathcal { M } \\left( \\mathbf { W } ^ { * } \\right) = \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) . \\operatorname { W e } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 278, + 433, + 297 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 298, + 436, + 330 + ], + "lines": [ + { + "bbox": [ + 113, + 298, + 436, + 330 + ], + "spans": [ + { + "bbox": [ + 113, + 298, + 436, + 330 + ], + "score": 0.63, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) = 1 - \\mathbb { P } \\left( \\mathbf { X } \\not \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\overset { ( 1 ) } { \\geq } 1 - \\sum _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } \\left( \\mathbf { X } \\not \\in \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right)", + "type": "interline_equation", + "image_path": "d4e572688c4f9aea0b80e92f29acda5e5ab599bcc03bd64644cd654d071d4346.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 113, + 298, + 436, + 308.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 113, + 308.6666666666667, + 436, + 319.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 113, + 319.33333333333337, + 436, + 330.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 351, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 280, + 351, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 280, + 351, + 478, + 380 + ], + "score": 1.0, + "content": "nd in (2) we used the fact that, from symmetr. Next, we examine the minimal angular margin in", + "type": "text" + }, + { + "bbox": [ + 487, + 352, + 506, + 362 + ], + "score": 0.73, + "content": "\\forall i :", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 362, + 502, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 280, + 374 + ], + "score": 0.89, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) = \\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } _ { 1 } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 363, + 502, + 375 + ], + "score": 0.9, + "content": "\\mathcal { M } _ { 1 , n } ^ { \\alpha }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 164, + 386 + ], + "score": 1.0, + "content": "separately for", + "type": "text" + }, + { + "bbox": [ + 165, + 374, + 201, + 384 + ], + "score": 0.88, + "content": "\\forall n < d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 373, + 220, + 386 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 220, + 374, + 257, + 384 + ], + "score": 0.9, + "content": "\\forall n \\geq d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 373, + 383, + 386 + ], + "score": 1.0, + "content": ". 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(Lemma 23 restated). Let", + "type": "text" + }, + { + "bbox": [ + 274, + 102, + 327, + 113 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 100, + 507, + 116 + ], + "score": 1.0, + "content": "be a standard random Gaussian matrix of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 305, + 127 + ], + "score": 1.0, + "content": "datapoints. Then we can find, with probability", + "type": "text" + }, + { + "bbox": [ + 305, + 115, + 311, + 124 + ], + "score": 0.56, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 114, + 316, + 127 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 317, + 114, + 344, + 126 + ], + "score": 0.92, + "content": "( \\mathbf { X } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 114, + 430, + 127 + ], + "score": 1.0, + "content": "-dependent matrices", + "type": "text" + }, + { + "bbox": [ + 430, + 114, + 448, + 124 + ], + "score": 0.83, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 114, + 469, + 127 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 469, + 115, + 480, + 124 + ], + "score": 0.85, + "content": "\\mathbf { z } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "as in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 183, + 139 + ], + "score": 1.0, + "content": "Theorem 8 (where", + "type": "text" + }, + { + "bbox": [ + 183, + 126, + 279, + 139 + ], + "score": 0.9, + "content": "d _ { 1 } ^ { * } \\triangleq 4 \\lceil N / \\left( 2 d _ { 0 } - 2 \\right) \\rceil ,", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 126, + 377, + 139 + ], + "score": 1.0, + "content": "). Moreover, in the limit", + "type": "text" + }, + { + "bbox": [ + 377, + 127, + 413, + 137 + ], + "score": 0.9, + "content": "N \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 126, + 444, + 139 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 445, + 126, + 503, + 138 + ], + "score": 0.9, + "content": "N / d _ { 0 } { \\dot { \\le } } d _ { 0 } { \\dot { \\le } } N", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 126, + 506, + 139 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 471, + 150 + ], + "score": 1.0, + "content": "for any y, we can bound the probability of not having an angular margin (eq. (13.3)) with", + "type": "text" + }, + { + "bbox": [ + 471, + 139, + 506, + 149 + ], + "score": 0.65, + "content": "\\sin \\alpha =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 146, + 169, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 155, + 160 + ], + "score": 0.92, + "content": "1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 146, + 169, + 162 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 100, + 507, + 162 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 161, + 407, + 190 + ], + "lines": [ + { + "bbox": [ + 204, + 161, + 407, + 190 + ], + "spans": [ + { + "bbox": [ + 204, + 161, + 407, + 190 + ], + "score": 0.91, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\notin \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\dot { \\leq } \\sqrt { \\frac { 8 } { \\pi } } d _ { 0 } ^ { - 1 / 2 } + \\frac { 2 d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } } } { N }", + "type": "interline_equation", + "image_path": "8c1c1d6ab39ccbf0f1bc0c12b5fce8e37c5e5c850185b00e41b95515e72261b7.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 161, + 407, + 175.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 204, + 175.5, + 407, + 190.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 507, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "Proof. In this proof we heavily rely on the notation and results from the proof of in appendix section", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 272, + 223 + ], + "score": 1.0, + "content": "9. Without loss of generality we assume", + "type": "text" + }, + { + "bbox": [ + 273, + 210, + 333, + 223 + ], + "score": 0.93, + "content": "S _ { 1 } ^ { + } = [ d _ { 0 } - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 210, + 486, + 223 + ], + "score": 1.0, + "content": ". Unfortunately, we can’t use Lemma", + "type": "text" + }, + { + "bbox": [ + 486, + 211, + 506, + 221 + ], + "score": 0.59, + "content": "4 1 -", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 408, + 235 + ], + "score": 1.0, + "content": "this proof is significantly more complicated since the constructed solution", + "type": "text" + }, + { + "bbox": [ + 409, + 222, + 426, + 232 + ], + "score": 0.83, + "content": "\\mathbf { W } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 221, + 476, + 235 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 476, + 222, + 486, + 232 + ], + "score": 0.7, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "(we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 472, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 472, + 245 + ], + "score": 1.0, + "content": "keep this dependence implicit, for brevity). Similarly to the proof of Lemma 41, we define,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 200, + 506, + 245 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 245, + 425, + 278 + ], + "lines": [ + { + "bbox": [ + 186, + 245, + 425, + 278 + ], + "spans": [ + { + "bbox": [ + 186, + 245, + 425, + 278 + ], + "score": 0.87, + "content": "\\mathcal { M } _ { i , n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\left\\{ \\mathbf { X } \\in \\mathbb { R } ^ { d _ { 0 } \\times N } | \\left| \\frac { \\mathbf { x } ^ { ( n ) \\top } \\mathbf { w } _ { i } ^ { * } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| \\left\\| \\mathbf { w } _ { i } ^ { * } \\right\\| } \\right| > \\sin \\left( \\alpha \\right) \\right\\}", + "type": "interline_equation", + "image_path": "b2bf5350db9f0790fefdcb282b55831a2d3290dccabd08afd033eecad5e46087.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 245, + 425, + 261.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 186, + 261.5, + 425, + 278.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 280, + 431, + 296 + ], + "lines": [ + { + "bbox": [ + 104, + 278, + 433, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 124, + 297 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 281, + 408, + 295 + ], + "score": 0.35, + "content": "\\begin{array} { r } { \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\triangleq \\bigcap _ { n = 1 } ^ { N } \\mathcal { M } _ { i , n } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) , \\operatorname { s o } \\mathcal { M } \\left( \\mathbf { W } ^ { * } \\right) = \\bigcap _ { i = 1 } ^ { d _ { 1 } ^ { * } } \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) . \\operatorname { W e } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 278, + 433, + 297 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 278, + 433, + 297 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 298, + 436, + 330 + ], + "lines": [ + { + "bbox": [ + 113, + 298, + 436, + 330 + ], + "spans": [ + { + "bbox": [ + 113, + 298, + 436, + 330 + ], + "score": 0.63, + "content": "\\mathbb { P } \\left( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) = 1 - \\mathbb { P } \\left( \\mathbf { X } \\not \\in \\mathcal { M } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right) \\overset { ( 1 ) } { \\geq } 1 - \\sum _ { i = 1 } ^ { d _ { 1 } } \\mathbb { P } \\left( \\mathbf { X } \\not \\in \\mathcal { M } _ { i } ^ { \\alpha } \\left( \\mathbf { W } ^ { * } \\right) \\right)", + "type": "interline_equation", + "image_path": "d4e572688c4f9aea0b80e92f29acda5e5ab599bcc03bd64644cd654d071d4346.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 113, + 298, + 436, + 308.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 113, + 308.6666666666667, + 436, + 319.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 113, + 319.33333333333337, + 436, + 330.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 351, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 280, + 351, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 280, + 351, + 478, + 380 + ], + "score": 1.0, + "content": "nd in (2) we used the fact that, from symmetr. 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(9.2), (9.5), and (9.4)), and in (2) we used the fact that", + "type": "text" + }, + { + "bbox": [ + 363, + 477, + 410, + 490 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { w } } _ { 1 } ^ { \\top } \\tilde { \\mathbf { w } } _ { 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "from eq. (9.4) together", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 365, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 127, + 502 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 489, + 186, + 501 + ], + "score": 0.93, + "content": "\\| \\tilde { \\mathbf { w } } _ { 1 } \\| = \\| \\hat { \\mathbf { w } } _ { 1 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 488, + 265, + 502 + ], + "score": 1.0, + "content": "from eq. 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(13.11). First, we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 390, + 487, + 495 + ], + "lines": [ + { + "bbox": [ + 125, + 390, + 487, + 495 + ], + "spans": [ + { + "bbox": [ + 125, + 390, + 487, + 495 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\frac { \\epsilon _ { 1 } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { 2 } } } \\leq \\kappa \\right) = 1 - \\mathbb { P } \\left( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } < \\epsilon _ { 1 } \\right) } \\\\ & { \\stackrel { \\left( 1 \\right) } { = } 1 - \\mathbb { P } \\left( \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\left| \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { \\left( n \\right) } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\stackrel { \\left( 2 \\right) } { = } 1 - \\mathbb { P } \\left( \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\left| \\frac { x _ { 2 } ^ { \\left( n \\right) } } { x _ { 1 } ^ { \\left( n \\right) } } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) } \\\\ & { \\stackrel { \\left( 3 \\right) } { = } 1 - \\left[ \\mathbb { P } \\left( \\left| \\frac { x _ { 2 } ^ { \\left( 1 \\right) } } { x _ { 1 } ^ { \\left( 1 \\right) } } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\right] ^ { N - d _ { 0 } - 1 } \\stackrel { \\left( 4 \\right) } { \\leq } 1 - \\left[ 1 - \\frac { 2 } { \\pi } \\arctan \\left( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\right] ^ { N } , } \\end{array}", + "type": "interline_equation", + "image_path": "53f80c8221ecaa81c7d8e78bb40a6e6c17cf17cb44996a632af8395b2203b250.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 125, + 390, + 487, + 425.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 125, + 425.0, + 487, + 460.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 125, + 460.0, + 487, + 495.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 506, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "where in (1) we used eq. (9.7), in (2) we recall that in eq. (13.10) we rotated the axes so that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 523, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 191, + 534 + ], + "score": 0.9, + "content": "\\hat { \\mathbf { w } } _ { 1 } \\propto [ 1 , 0 , 0 \\ldots , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 523, + 215, + 534 + ], + "score": 1.0, + "content": "axes", + "type": "text" + }, + { + "bbox": [ + 216, + 523, + 309, + 534 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { w } } _ { 1 } \\propto [ 0 , 1 , 0 , 0 \\ldots , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 523, + 506, + 534 + ], + "score": 1.0, + "content": ", in (3) we used the independence of different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 532, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 124, + 546 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 532, + 506, + 549 + ], + "score": 1.0, + "content": ", and in (4) we used the fact that the ratio of two independent Gaussian variables is distributed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "according to the symmetric Cauchy distribution, which has the cumulative distribution function", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 554, + 433, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 190, + 570 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( X > \\breve { x } \\right) = \\textstyle \\frac { 1 } { 2 } - \\breve { \\frac { } { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 554, + 220, + 572 + ], + "score": 1.0, + "content": "arctan", + "type": "text" + }, + { + "bbox": [ + 221, + 557, + 235, + 569 + ], + "score": 0.82, + "content": "( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 554, + 294, + 572 + ], + "score": 1.0, + "content": ", and therefore", + "type": "text" + }, + { + "bbox": [ + 294, + 556, + 381, + 570 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( | X | > x \\right) = 1 - \\frac { 2 } { \\pi } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 554, + 412, + 572 + ], + "score": 1.0, + "content": "arctan", + "type": "text" + }, + { + "bbox": [ + 412, + 557, + 426, + 569 + ], + "score": 0.79, + "content": "( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 554, + 433, + 572 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 212, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 213, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 213, + 587 + ], + "score": 1.0, + "content": "Second, we use eq. 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\\left( \\eta ^ { 2 } \\exp \\left( 1 - \\eta ^ { 2 } / d _ { 0 } \\right) / d _ { 0 } \\right) ^ { d _ { 0 } / 2 } \\right] ^ { d _ { 0 } - 1 } .", + "type": "interline_equation", + "image_path": "3bf3d3bd0fd926e4c1128b0334680dbc6eb5c6f8ec200c33c01576f2c9c75a17.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 147, + 708, + 435, + 736 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "page_idx": 32, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "33", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 506, + 145 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 269, + 95 + ], + "score": 1.0, + "content": "where in (1) we rotate the axes so that", + "type": "text" + }, + { + "bbox": [ + 270, + 82, + 352, + 95 + ], + "score": 0.9, + "content": "\\hat { \\mathbf { w } } _ { 1 } \\propto [ 1 , 0 , 0 \\ldots , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 82, + 375, + 95 + ], + "score": 1.0, + "content": "axes", + "type": "text" + }, + { + "bbox": [ + 375, + 82, + 493, + 95 + ], + "score": 0.85, + "content": "\\tilde { \\mathbf { w } } _ { 1 } \\propto [ 0 , 1 , 0 , 0 \\ldots , 0 ] - 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(9.4) and (9.2)), and as such, they are independent from", + "type": "text" + }, + { + "bbox": [ + 412, + 106, + 430, + 117 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 105, + 446, + 120 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 446, + 107, + 477, + 118 + ], + "score": 0.91, + "content": "n \\geq d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 105, + 506, + 120 + ], + "score": 1.0, + "content": ", in (2)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 117, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 104, + 117, + 187, + 133 + ], + "score": 1.0, + "content": "we use that fact that", + "type": "text" + }, + { + "bbox": [ + 188, + 119, + 211, + 131 + ], + "score": 0.92, + "content": "\\left. \\hat { \\mathbf { w } } _ { 1 } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 117, + 230, + 133 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 118, + 299, + 132 + ], + "score": 0.93, + "content": "\\operatorname* { m a x } _ { n < d _ { 0 } } \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\|", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 117, + 365, + 133 + ], + "score": 1.0, + "content": "are functions of", + "type": "text" + }, + { + "bbox": [ + 365, + 118, + 384, + 129 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 117, + 399, + 133 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 399, + 119, + 429, + 130 + ], + "score": 0.92, + "content": "n < d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 117, + 506, + 133 + ], + "score": 1.0, + "content": ", and as such, they", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 130, + 288, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 194, + 146 + ], + "score": 1.0, + "content": "are independent from", + "type": "text" + }, + { + "bbox": [ + 194, + 132, + 213, + 142 + ], + "score": 0.9, + "content": "\\mathbf { x } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 130, + 228, + 146 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 228, + 133, + 258, + 144 + ], + "score": 0.9, + "content": "n \\geq d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 130, + 288, + 146 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 82, + 507, + 146 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 150, + 452, + 353 + ], + "lines": [ + { + "bbox": [ + 159, + 150, + 452, + 353 + ], + "spans": [ + { + "bbox": [ + 159, + 150, + 452, + 353 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbb { P } ^ { \\nu } ( \\mathbf { X } \\times M _ { 1 } ^ { \\nu } ( \\mathbf { W } ^ { \\nu } ) ) } \\\\ & { \\geq \\left( 1 - \\mathbb { P } \\left( \\frac { \\mathcal { X } } { \\eta \\sin \\alpha } \\leq \\left. \\mathbf { w } _ { 1 } \\right. \\textnormal { \\texttt { O } } \\eta \\leq \\operatorname* { m a x } \\left. \\mathbf { s } ^ { \\nu } ( \\mathbf { u } ) \\right. \\right) \\right) } \\\\ & { \\cdot \\left( 1 - \\mathbb { P } \\left( \\frac { \\left\\{ 1 - \\eta \\beta \\right\\} } { \\gamma \\beta } \\right) \\operatorname* { m i n } \\frac { x _ { 1 } ^ { \\nu \\alpha } } { x _ { 2 } ^ { \\nu \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. } \\leq \\sin \\alpha \\alpha \\frac { \\mathbb { E } _ { 1 } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { \\alpha } } } \\leq \\kappa \\right) \\right) } \\\\ & { \\overset { ( a ) } { \\geq } \\left( 1 - \\mathbb { P } \\left( \\frac { \\mathcal { X } } { \\eta \\sin \\alpha } \\leq \\left. \\mathbf { s } \\right. _ { 1 } \\right) - \\mathbb { P } \\left( \\eta \\leq \\operatorname* { m a x } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. \\right) \\right) } \\\\ & { \\cdot \\left( 1 - \\mathbb { P } \\left( \\frac { \\left\\{ 1 - \\eta \\beta \\right\\} } { \\gamma \\beta } \\operatorname* { m i n } \\frac { x _ { 1 } ^ { \\nu \\alpha } } { x _ { 2 } ^ { \\nu \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. } \\leq \\sin \\alpha \\right) - \\mathbb { E } \\left( \\frac { \\mathcal { X } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { \\alpha } } } \\leq \\kappa \\right) \\right) } \\\\ & { = \\left( \\mathbb { P } \\left( \\eta > \\frac { \\mathcal { X } } { \\eta \\sin \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. \\right) - \\mathbb { P } \\left( \\frac { \\mathcal { Y } } { \\eta \\sin \\alpha } \\leq \\left. \\mathbf { s } \\right. \\right) \\right) } \\\\ & \\cdot \\left( \\mathbb { E } \\left( \\frac { \\left\\{ 1 - \\gamma \\beta \\right\\} } { \\gamma \\beta } \\right) \\operatorname* { m i n } \\frac { x _ { 1 } ^ { \\nu \\alpha } } { x _ { 2 } ^ { \\nu \\alpha } \\left. \\mathbf { s } ^ { \\nu \\alpha } \\right. } \\leq \\sin \\alpha \\right) - \\mathbb { P } \\left( \\frac { \\mathcal { Z } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { \\alpha } } } \\leq \\kappa \\end{array}", + "type": "interline_equation", + "image_path": "0402b0b308e57e9839ca6aeb7277844e949615723079fba9c549efc0929b83be.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 159, + 150, + 452, + 165.6153846153846 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 159, + 165.6153846153846, + 452, + 181.23076923076923 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 159, + 181.23076923076923, + 452, + 196.84615384615384 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 159, + 196.84615384615384, + 452, + 212.46153846153845 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 159, + 212.46153846153845, + 452, + 228.07692307692307 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 159, + 228.07692307692307, + 452, + 243.69230769230768 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 159, + 243.69230769230768, + 452, + 259.3076923076923 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 159, + 259.3076923076923, + 452, + 274.92307692307696 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 159, + 274.92307692307696, + 452, + 290.5384615384616 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 159, + 290.5384615384616, + 452, + 306.15384615384625 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 159, + 306.15384615384625, + 452, + 321.7692307692309 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 159, + 321.7692307692309, + 452, + 337.38461538461553 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 159, + 337.38461538461553, + 452, + 353.00000000000017 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 361, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 361, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 361, + 370 + ], + "score": 1.0, + "content": "where in (1) we use the union bound on both probability terms.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 354, + 361, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 372, + 479, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 479, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 479, + 386 + ], + "score": 1.0, + "content": "All that remains is to calculate each remaining probability term in eq. (13.11). First, we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 372, + 479, + 386 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 390, + 487, + 495 + ], + "lines": [ + { + "bbox": [ + 125, + 390, + 487, + 495 + ], + "spans": [ + { + "bbox": [ + 125, + 390, + 487, + 495 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left( \\frac { \\epsilon _ { 1 } } { \\sqrt { 1 + \\epsilon _ { 1 } ^ { 2 } } } \\leq \\kappa \\right) = 1 - \\mathbb { P } \\left( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } < \\epsilon _ { 1 } \\right) } \\\\ & { \\stackrel { \\left( 1 \\right) } { = } 1 - \\mathbb { P } \\left( \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\left| \\tilde { \\mathbf { w } } _ { i } ^ { \\top } \\mathbf { x } ^ { \\left( n \\right) } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\stackrel { \\left( 2 \\right) } { = } 1 - \\mathbb { P } \\left( \\underset { n \\geq d _ { 0 } } { \\operatorname* { m i n } } \\left| \\frac { x _ { 2 } ^ { \\left( n \\right) } } { x _ { 1 } ^ { \\left( n \\right) } } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) } \\\\ & { \\stackrel { \\left( 3 \\right) } { = } 1 - \\left[ \\mathbb { P } \\left( \\left| \\frac { x _ { 2 } ^ { \\left( 1 \\right) } } { x _ { 1 } ^ { \\left( 1 \\right) } } \\right| > \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\right] ^ { N - d _ { 0 } - 1 } \\stackrel { \\left( 4 \\right) } { \\leq } 1 - \\left[ 1 - \\frac { 2 } { \\pi } \\arctan \\left( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } \\right) \\right] ^ { N } , } \\end{array}", + "type": "interline_equation", + "image_path": "53f80c8221ecaa81c7d8e78bb40a6e6c17cf17cb44996a632af8395b2203b250.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 125, + 390, + 487, + 425.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 125, + 425.0, + 487, + 460.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 125, + 460.0, + 487, + 495.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 506, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "where in (1) we used eq. (9.7), in (2) we recall that in eq. (13.10) we rotated the axes so that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 523, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 191, + 534 + ], + "score": 0.9, + "content": "\\hat { \\mathbf { w } } _ { 1 } \\propto [ 1 , 0 , 0 \\ldots , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 523, + 215, + 534 + ], + "score": 1.0, + "content": "axes", + "type": "text" + }, + { + "bbox": [ + 216, + 523, + 309, + 534 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { w } } _ { 1 } \\propto [ 0 , 1 , 0 , 0 \\ldots , 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 523, + 506, + 534 + ], + "score": 1.0, + "content": ", in (3) we used the independence of different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 532, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 124, + 546 + ], + "score": 0.87, + "content": "\\mathbf { x } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 532, + 506, + 549 + ], + "score": 1.0, + "content": ", and in (4) we used the fact that the ratio of two independent Gaussian variables is distributed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "according to the symmetric Cauchy distribution, which has the cumulative distribution function", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 554, + 433, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 190, + 570 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( X > \\breve { x } \\right) = \\textstyle \\frac { 1 } { 2 } - \\breve { \\frac { } { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 554, + 220, + 572 + ], + "score": 1.0, + "content": "arctan", + "type": "text" + }, + { + "bbox": [ + 221, + 557, + 235, + 569 + ], + "score": 0.82, + "content": "( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 554, + 294, + 572 + ], + "score": 1.0, + "content": ", and therefore", + "type": "text" + }, + { + "bbox": [ + 294, + 556, + 381, + 570 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( | X | > x \\right) = 1 - \\frac { 2 } { \\pi } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 554, + 412, + 572 + ], + "score": 1.0, + "content": "arctan", + "type": "text" + }, + { + "bbox": [ + 412, + 557, + 426, + 569 + ], + "score": 0.79, + "content": "( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 554, + 433, + 572 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 511, + 506, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 212, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 213, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 213, + 587 + ], + "score": 1.0, + "content": "Second, we use eq. (13.2)", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 571, + 213, + 587 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 598, + 435, + 635 + ], + "lines": [ + { + "bbox": [ + 146, + 598, + 435, + 635 + ], + "spans": [ + { + "bbox": [ + 146, + 598, + 435, + 635 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( \\operatorname* { m i n } _ { n \\geq d _ { 0 } } \\frac { x _ { 1 } ^ { ( n ) } } { \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| } > \\frac { \\gamma \\beta \\sin \\alpha } { \\left( 1 - \\gamma \\beta \\right) \\kappa } \\right) > \\left[ 1 - \\frac { 2 \\gamma \\beta \\sin \\alpha } { \\left( 1 - \\gamma \\beta \\right) \\kappa B \\left( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } \\right) } \\right] ^ { N } .", + "type": "interline_equation", + "image_path": "5122825d4bee63b55f03956b0fef0b3e608f80c86603b81f633760e076c1000a.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 146, + 598, + 435, + 610.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 146, + 610.3333333333334, + 435, + 622.6666666666667 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 146, + 622.6666666666667, + 435, + 635.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 640, + 491, + 653 + ], + "lines": [ + { + "bbox": [ + 104, + 637, + 488, + 656 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 131, + 656 + ], + "score": 1.0, + "content": "Third,", + "type": "text" + }, + { + "bbox": [ + 132, + 637, + 165, + 654 + ], + "score": 0.93, + "content": "\\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 637, + 414, + 656 + ], + "score": 1.0, + "content": "is distributed according to the chi-square distribution of order", + "type": "text" + }, + { + "bbox": [ + 414, + 641, + 424, + 652 + ], + "score": 0.88, + "content": "d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 637, + 454, + 656 + ], + "score": 1.0, + "content": ", so for", + "type": "text" + }, + { + "bbox": [ + 454, + 640, + 488, + 653 + ], + "score": 0.92, + "content": "\\eta ^ { 2 } > d _ { 0 }", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 637, + 488, + 656 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 659, + 415, + 687 + ], + "lines": [ + { + "bbox": [ + 195, + 659, + 415, + 687 + ], + "spans": [ + { + "bbox": [ + 195, + 659, + 415, + 687 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| ^ { 2 } \\geq \\eta ^ { 2 } \\right) \\leq \\left( \\eta ^ { 2 } \\exp \\left( 1 - \\eta ^ { 2 } / d _ { 0 } \\right) / d _ { 0 } \\right) ^ { d _ { 0 } / 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "5168ef1936f5083237cf3d70399358245ae1e25d9de608f64ce7c7fabd05d789.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 195, + 659, + 415, + 687 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 692, + 150, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 151, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 151, + 705 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 690, + 151, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 708, + 435, + 736 + ], + "lines": [ + { + "bbox": [ + 147, + 708, + 435, + 736 + ], + "spans": [ + { + "bbox": [ + 147, + 708, + 435, + 736 + ], + "score": 0.92, + "content": "\\mathbb { P } \\left( \\operatorname* { m a x } _ { n < d _ { 0 } } \\left\\| \\mathbf { x } ^ { ( n ) } \\right\\| ^ { 2 } < \\eta ^ { 2 } \\right) > \\left[ 1 - \\left( \\eta ^ { 2 } \\exp \\left( 1 - \\eta ^ { 2 } / d _ { 0 } \\right) / d _ { 0 } \\right) ^ { d _ { 0 } / 2 } \\right] ^ { d _ { 0 } - 1 } .", + "type": "interline_equation", + "image_path": "3bf3d3bd0fd926e4c1128b0334680dbc6eb5c6f8ec200c33c01576f2c9c75a17.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 147, + 708, + 435, + 736 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 405, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 405, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 177, + 96 + ], + "score": 1.0, + "content": "Lastly, we bound", + "type": "text" + }, + { + "bbox": [ + 178, + 82, + 237, + 95 + ], + "score": 0.94, + "content": "\\| \\tilde { \\mathbf { w } } _ { 1 } \\| = \\| \\hat { \\mathbf { w } } _ { 1 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 81, + 405, + 96 + ], + "score": 1.0, + "content": "(from eq. (9.5)). From eq. 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The singular values are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 256, + 210 + ], + "score": 1.0, + "content": "ordered from smallest to largest, and", + "type": "text" + }, + { + "bbox": [ + 257, + 199, + 286, + 209 + ], + "score": 0.91, + "content": "\\sigma _ { 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 198, + 307, + 210 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 308, + 199, + 345, + 209 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { 1 } = \\tilde { \\mathbf { w } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 198, + 505, + 210 + ], + "score": 1.0, + "content": ", from eq. (9.2). With probability 1, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 129, + 222 + ], + "score": 1.0, + "content": "other", + "type": "text" + }, + { + "bbox": [ + 129, + 209, + 158, + 220 + ], + "score": 0.88, + "content": "d _ { 0 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "singular value are non-zero: they are the square roots of the eigenvalues of the random", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 217, + 467, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 135, + 235 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 136, + 219, + 262, + 235 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { X } } _ { [ d _ { 0 } - 1 ] } ^ { \\top } \\mathbf { \\bar { X } } _ { [ d _ { 0 } - 1 ] } \\in \\mathbb { R } ^ { d _ { 0 } - 1 \\times d _ { 0 } - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 217, + 467, + 235 + ], + "score": 1.0, + "content": ". Taking the squared norm of eq. 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(9.4)), so the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 295, + 508, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 271, + 311 + ], + "score": 1.0, + "content": "minimal possible value is attained when", + "type": "text" + }, + { + "bbox": [ + 272, + 296, + 334, + 309 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { 2 } ^ { \\top } \\hat { \\mathbf { w } } _ { 1 } = \\| \\hat { \\mathbf { w } } _ { 1 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 295, + 491, + 311 + ], + "score": 1.0, + "content": ". The minimal nonzero singular value,", + "type": "text" + }, + { + "bbox": [ + 491, + 298, + 502, + 308 + ], + "score": 0.79, + "content": "\\sigma _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 295, + 508, + 311 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 308, + 461, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 461, + 320 + ], + "score": 1.0, + "content": "can be bounded using the following result from (Rudelson & Vershynin, 2010, eq. (3.2))", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 326, + 380, + 354 + ], + "lines": [ + { + "bbox": [ + 230, + 326, + 380, + 354 + ], + "spans": [ + { + "bbox": [ + 230, + 326, + 380, + 354 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\underset { { \\mathbf { r } } \\in \\mathbb { R } ^ { d _ { 0 } } } { \\operatorname* { m i n } } \\left. \\mathbf { X } _ { [ d _ { 0 } ] } \\mathbf { r } \\right. \\leq \\eta d _ { 0 } ^ { - 1 / 2 } \\right) \\leq \\eta . } \\end{array}", + "type": "interline_equation", + "image_path": "2f49f2eabf49c014f680f894969c76cc1c09bdde7d99a470dd3c1e018db796f5.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 230, + 326, + 380, + 354 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 131, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 132, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 132, + 374 + ], + "score": 1.0, + "content": "Since", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 390, + 396, + 411 + ], + "lines": [ + { + "bbox": [ + 215, + 390, + 396, + 411 + ], + "spans": [ + { + "bbox": [ + 215, + 390, + 396, + 411 + ], + "score": 0.91, + "content": "\\sigma _ { 2 } = \\operatorname* { m i n } _ { \\mathbf { r } \\in \\mathbb { R } ^ { d _ { 0 } - 1 } } \\left\\| \\mathbf { X } _ { [ d _ { 0 } - 1 ] } \\mathbf { r } \\right\\| \\geq \\operatorname* { m i n } _ { \\mathbf { r } \\in \\mathbb { R } ^ { d _ { 0 } } } \\left\\| \\mathbf { X } _ { [ d _ { 0 } ] } \\mathbf { r } \\right\\|", + "type": "interline_equation", + "image_path": "24ed8789565e1be82b261dd6cbefb172d100528f605cd6334cde6c9718bf2eb9.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 215, + 390, + 396, + 411 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 143, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 416, + 145, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 145, + 430 + ], + "score": 1.0, + "content": "we have,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 430, + 354, + 451 + ], + "lines": [ + { + "bbox": [ + 257, + 430, + 354, + 451 + ], + "spans": [ + { + "bbox": [ + 257, + 430, + 354, + 451 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { P } \\left( \\sigma _ { 2 } < \\eta d _ { 0 } ^ { - 1 / 2 } \\right) \\le \\eta . } \\end{array}", + "type": "interline_equation", + "image_path": "7ccf5f14f7eabe627e6880bcd8142c70777153042977c1b68213be9c59766fd0.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 257, + 430, + 354, + 451 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 266, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 267, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 267, + 471 + ], + "score": 1.0, + "content": "Combining this with eq. (13.16) we get", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 475, + 379, + 504 + ], + "lines": [ + { + "bbox": [ + 232, + 475, + 379, + 504 + ], + "spans": [ + { + "bbox": [ + 232, + 475, + 379, + 504 + ], + "score": 0.94, + "content": "\\mathbb { P } \\left( \\frac { \\beta \\kappa } { \\eta \\sin \\alpha } < \\| \\mathbf { w } _ { 1 } \\| \\right) \\le \\frac { \\eta d _ { 0 } } { \\beta \\kappa } \\sin \\alpha .", + "type": "interline_equation", + "image_path": "ce4c72e703f6277d2ce79bf2ed1eeed7854138445f13c8d49d913e567a3b8d01.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 232, + 475, + 379, + 504 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 506, + 524 + ], + "score": 1.0, + "content": "Lastly, combining eqs. (13.12), (13.13), (13.14) and (13.17) into eqs. (13.7) and (13.11), we get, for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 518, + 146, + 536 + ], + "spans": [ + { + "bbox": [ + 107, + 521, + 140, + 533 + ], + "score": 0.91, + "content": "\\eta ^ { 2 } > d _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 518, + 146, + 536 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 540, + 465, + 698 + ], + "lines": [ + { + "bbox": [ + 146, + 540, + 465, + 698 + ], + "spans": [ + { + "bbox": [ + 146, + 540, + 465, + 698 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbb { P } ( \\mathbf { X } \\in \\mathcal { M } ^ { \\alpha } ( \\mathbf { W } ^ { \\star } ) ) } \\\\ & { \\geq 1 - d _ { 1 } ^ { \\ast } ( 1 - ( [ 1 - ( \\eta ^ { 2 } \\exp ( 1 - \\eta ^ { 2 } / d _ { 0 } ) / d _ { 0 } ) ^ { \\hat { d } _ { 0 } / 2 } ] ^ { d _ { 0 } - 1 } - \\frac { \\eta d _ { 0 } } { \\gamma \\kappa } \\sin \\alpha ) } \\\\ & { \\Bigg . \\Bigg . \\Bigg . } \\\\ & { \\quad \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg . \\Bigg \\Gamma ( 1 - \\frac { 2 \\gamma \\beta \\sin \\alpha } { ( 1 - \\gamma \\beta ) \\kappa B ( \\frac { 1 } { 2 } , \\frac { d _ { 0 } - 1 } { 2 } ) } \\Bigg ] ^ { N } - [ 1 - \\frac { 2 } { \\pi } \\arctan ( \\frac { \\kappa } { \\sqrt { 1 - \\kappa ^ { 2 } } } \\frac { 1 } { \\beta } ) ] ^ { N } \\Bigg ) \\Bigg ) } \\\\ & { \\geq 1 - d _ { 1 } ^ { \\ast } ( 1 - ( [ 1 - ( \\log d _ { 0 } \\exp ( 1 - \\log d _ { 0 } ) ) ^ { d _ { 0 } / 2 } ] ^ { d _ { 0 } - 1 } - \\frac { 2 d _ { 0 } ^ { \\hat { d } _ { 0 } ^ { \\prime } / 2 } \\sqrt { \\log { d _ { 0 } } } } { d _ { 1 } ^ { \\ast } N } ) \\Bigg . } \\\\ & { \\Bigg ( [ 1 - \\sqrt { \\frac { \\kappa } { \\pi } } \\frac { 1 } { d _ { 1 } ^ { \\ast } d _ { 0 } ^ { \\dagger } N } + \\mathcal { O } ( \\frac { 1 } { N d _ { 1 } ^ { \\ast } d _ { 0 } ^ { \\dagger } } ) ] ^ { N } - 0 . 4 5 ^ { N } \\Bigg ) \\Bigg ) ~ , } \\end{array}", + "type": "interline_equation", + "image_path": "3d58370a23a3762338a47d32b9b106be588aa7edfd90ba95b179cc43e140b76f.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 146, + 540, + 465, + 592.6666666666666 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 146, + 592.6666666666666, + 465, + 645.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 146, + 645.3333333333333, + 465, + 697.9999999999999 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 705, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 104, + 702, + 507, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 702, + 228, + 724 + ], + "score": 1.0, + "content": "where in the last line we take", + "type": "text" + }, + { + "bbox": [ + 228, + 706, + 311, + 720 + ], + "score": 0.86, + "content": "\\beta = \\gamma = \\kappa = 1 / \\sqrt { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 702, + 315, + 724 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 705, + 386, + 720 + ], + "score": 0.85, + "content": "\\eta = d _ { 0 } ^ { 1 / 2 } \\sqrt { \\log d _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 702, + 390, + 724 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 390, + 707, + 473, + 720 + ], + "score": 0.9, + "content": "\\sin \\alpha = 1 / \\left( d _ { 1 } ^ { * } d _ { 0 } N \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 702, + 507, + 724 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 717, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 286, + 735 + ], + "score": 1.0, + "content": "the asymptotic expansion of the beta function", + "type": "text" + }, + { + "bbox": [ + 286, + 720, + 416, + 734 + ], + "score": 0.91, + "content": "B \\left( { \\textstyle { \\frac { 1 } { 2 } } } , x \\right) = { \\sqrt { \\pi / x } } + O \\left( x ^ { - 3 / 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 717, + 452, + 735 + ], + "score": 1.0, + "content": "for large", + "type": "text" + }, + { + "bbox": [ + 453, + 723, + 460, + 730 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 717, + 507, + 735 + ], + "score": 1.0, + "content": ", we obtain,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 405, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 405, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 177, + 96 + ], + "score": 1.0, + "content": "Lastly, we bound", + "type": "text" + }, + { + "bbox": [ + 178, + 82, + 237, + 95 + ], + "score": 0.94, + "content": "\\| \\tilde { \\mathbf { w } } _ { 1 } \\| = \\| \\hat { \\mathbf { w } } _ { 1 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 81, + 405, + 96 + ], + "score": 1.0, + "content": "(from eq. (9.5)). From eq. 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(13.16) we get", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 455, + 267, + 471 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 475, + 379, + 504 + ], + "lines": [ + { + "bbox": [ + 232, + 475, + 379, + 504 + ], + "spans": [ + { + "bbox": [ + 232, + 475, + 379, + 504 + ], + "score": 0.94, + "content": "\\mathbb { P } \\left( \\frac { \\beta \\kappa } { \\eta \\sin \\alpha } < \\| \\mathbf { w } _ { 1 } \\| \\right) \\le \\frac { \\eta d _ { 0 } } { \\beta \\kappa } \\sin \\alpha .", + "type": "interline_equation", + "image_path": "ce4c72e703f6277d2ce79bf2ed1eeed7854138445f13c8d49d913e567a3b8d01.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 232, + 475, + 379, + 504 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 506, + 524 + ], + "score": 1.0, + "content": "Lastly, combining eqs. 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CIFAR0%3072165·1040.983
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diff --git a/parse/train/S1vyujVye/S1vyujVye.md b/parse/train/S1vyujVye/S1vyujVye.md new file mode 100644 index 0000000000000000000000000000000000000000..13c21e0e9a64fffa2bea8feb62bb98ce4785a24a --- /dev/null +++ b/parse/train/S1vyujVye/S1vyujVye.md @@ -0,0 +1,273 @@ +# 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 + +
ModelSTL-10 test accuracy
Zero-bias Convnets - Paine et al. (2014)70.2%
Triplet network -Hoffer & Ailon (2015)70.7%
Exemplar Convnets - Dosovitskiy et al. (2014)72.8%
Target Coding - Yang et al. (2015)73.15%
Stacked what-where AE - Zhao et al. (2015)74.33%
Spatial contrasting initialization (this work)81.34% ± 0.1
The same model without initialization72.6%±0.1
+ +• 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 + +
ModelCifar10 (400 per class) test accuracy
Convolutional K-means Network - Coates & Ng (2012)70.7%
View-Invariant K-means -Hui (2013)72.6%
DCGAN - Radford et al. (2015)73.8%
Exemplar Convnets - Dosovitskiy et al. (2014)76.6%
Ladder networks - Rasmus et al. (2015)79.6%
Conv-CatGan Springenberg (2016)80.42% (± 0.58)
ImprovedGan Salimans et al. (2016)81.37% (± 2.32)
Spatial contrasting initialization (this work)79.2%(±0.3)
The same model without initialization72.4%(±0.1)
+ +Table 3: results on MNIST dataset + +
ModelMNIST test error
Stacked what-where AE - Zhao et al. (2015)0.71%
Triplet network - Hoffer & Ailon (2015)0.56%
Jarrett et al. (2009)0.53%
Ladder networks - Rasmus et al. (2015)0.36%
DropConnect - Wan et al. (2013)0.21%
Spatial contrasting initialization (this work)0.34%± 0.02
The same model without initialization0.63%± 0.02
+ +# 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 + +Vassileios Balntas, Edward Johns, Lilian Tang, and Krystian Mikolajczyk. Pn-net: Conjoined triple deep network for learning local image descriptors. arXiv preprint arXiv:1601.05030, 2016. + +Jane Bromley, James W Bentz, Leon Bottou, Isabelle Guyon, Yann LeCun, Cliff Moore, Eduard ´ Sackinger, and Roopak Shah. Signature verification using a siamese time delay neural network.¨ International Journal of Pattern Recognition and Artificial Intelligence, 7(04):669–688, 1993. + +Sumit Chopra, Raia Hadsell, and Yann LeCun. Learning a similarity metric discriminatively, with application to face verification. In Computer Vision and Pattern Recognition, 2005. CVPR 2005. 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Nature, 518(7540):529–533, 2015. + +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. + +Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015. + +Antti Rasmus, Mathias Berglund, Mikko Honkala, Harri Valpola, and Tapani Raiko. Semisupervised learning with ladder networks. In Advances in Neural Information Processing Systems, pp. 3532–3540, 2015. + +Ali Razavian, Hossein Azizpour, Josephine Sullivan, and Stefan Carlsson. Cnn features off-theshelf: an astounding baseline for recognition. 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URL https://arxiv.org/abs/1511.06390. + +Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–9, 2015. + +Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning, pp. 1096–1103. ACM, 2008. + +Li Wan, Matthew Zeiler, Sixin Zhang, Yann L Cun, and Rob Fergus. Regularization of neural networks using dropconnect. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 1058–1066, 2013. + +Shuo Yang, Ping Luo, Chen Change Loy, Kenneth W Shum, and Xiaoou Tang. Deep representation learning with target coding. 2015. + +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) + + +
Model
STL10CIFAR-10MNIST
Input: 96×96 RGBInput:32×32RGBInput:28 × 28 monochrome
5 ×5 conv. 64 BN ReLU3 × 3 conv. 96 BN LeakyReLU5 ×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
1 ×1 conv.160 BN ReLU3 ×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
1 ×1 conv. 96 BN ReLU
3 × 3 max-pooling, stride 2
5 × 5 conv. 192 BN ReLU
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
Spatial contrasting criterion
3×3 conv.256 ReLU3 × 3 max-pooling,stride 2dropout, p = 0.53 × 3 conv.128 ReLUdropout, p = 0.5fully-connected 103×3 conv.192BNLeakyReLU1 ×1 conv.192 BNLeakyReLU1 ×1 conv.10 BNLeakyReLUglobal average pooling3×3 conv.128 BN ReLU1 × 1 conv.10 BN ReLUglobal average pooling
+ +10-way softmax + +![](images/dbd928541e70043deb704a91118b985563d8e3c93a5cbdcb28b93b6bd8b098bc.jpg) +Figure 2: First layer convolutional filters after spatial-contrasting training \ No newline at end of file diff --git a/parse/train/S1vyujVye/S1vyujVye_content_list.json b/parse/train/S1vyujVye/S1vyujVye_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..3ad1d839caf1dcd0c3ad8f2317b4997df41ee86f --- /dev/null +++ b/parse/train/S1vyujVye/S1vyujVye_content_list.json @@ -0,0 +1,1429 @@ +[ + { + "type": "text", + "text": "DEEP UNSUPERVISED LEARNING THROUGH SPATIAL CONTRASTING ", + "text_level": 1, + "bbox": [ + 174, + 101, + 823, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "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 ", + "bbox": [ + 184, + 170, + 450, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 545, + 170, + 813, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 184, + 247, + 450, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 339, + 544, + 354 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 233, + 371, + 764, + 511 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 536, + 334, + 553 + ], + "page_idx": 0 + }, + { + "type": "text", + "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). ", + "bbox": [ + 174, + 568, + 825, + 651 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 178, + 651, + 820, + 694 + ], + "page_idx": 0 + }, + { + "type": "text", + "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.. ", + "bbox": [ + 174, + 700, + 823, + 797 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 805, + 823, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PREVIOUS WORKS ", + "text_level": 1, + "bbox": [ + 176, + 155, + 357, + 172 + ], + "page_idx": 1 + }, + { + "type": "text", + "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. ", + "bbox": [ + 176, + 189, + 823, + 232 + ], + "page_idx": 1 + }, + { + "type": "text", + "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: ", + "bbox": [ + 173, + 251, + 825, + 335 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/5bd4f185929c9804dd1e8cad87809e51fb2d7db4b5572fb153eb7d7ef8d273db.jpg", + "text": "$$\n\\hat { x } = F ( G ( x ; w _ { 1 } ) ; w _ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 424, + 345, + 573, + 363 + ], + "page_idx": 1 + }, + { + "type": "text", + "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) ", + "bbox": [ + 173, + 372, + 825, + 443 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/7ab54084fb1f2b1bb0e333109ca419d603e264ec6c57d96ec21422cf2e2b0272.jpg", + "text": "$$\nm i n \\lVert x - { \\hat { x } } \\rVert ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 450, + 452, + 547, + 470 + ], + "page_idx": 1 + }, + { + "type": "text", + "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: ", + "bbox": [ + 173, + 497, + 825, + 679 + ], + "page_idx": 1 + }, + { + "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", + "bbox": [ + 264, + 689, + 732, + 708 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 173, + 736, + 825, + 820 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "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. ", + "bbox": [ + 174, + 103, + 825, + 228 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 246, + 825, + 343 + ], + "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. ", + "bbox": [ + 174, + 359, + 825, + 498 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 PROBLEMS WITH CURRENT APPROACHES", + "text_level": 1, + "bbox": [ + 176, + 517, + 503, + 530 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 542, + 823, + 570 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 173, + 578, + 825, + 717 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 LEARNING BY COMPARISONS ", + "text_level": 1, + "bbox": [ + 178, + 738, + 450, + 755 + ], + "page_idx": 2 + }, + { + "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 ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 3 + }, + { + "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": [ + 259, + 170, + 738, + 188 + ], + "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: ", + "bbox": [ + 174, + 196, + 825, + 267 + ], + "page_idx": 3 + }, + { + "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": [ + 315, + 276, + 678, + 310 + ], + "page_idx": 3 + }, + { + "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 ", + "bbox": [ + 174, + 320, + 825, + 377 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 OUR CONTRIBUTION: SPATIAL CONTRASTING ", + "text_level": 1, + "bbox": [ + 174, + 400, + 593, + 417 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 173, + 434, + 825, + 574 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 173, + 580, + 825, + 678 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 FORMULATION ", + "text_level": 1, + "bbox": [ + 174, + 699, + 318, + 713 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 726, + 825, + 785 + ], + "page_idx": 3 + }, + { + "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", + "bbox": [ + 174, + 786, + 825, + 830 + ], + "page_idx": 3 + }, + { + "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 ", + "bbox": [ + 173, + 830, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 103, + 825, + 133 + ], + "page_idx": 4 + }, + { + "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": [ + 318, + 148, + 679, + 189 + ], + "page_idx": 4 + }, + { + "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": [ + 173, + 198, + 825, + 253 + ], + "page_idx": 4 + }, + { + "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", + "bbox": [ + 338, + 261, + 658, + 291 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/f412cdc401c9a392a006634319c43ab46c8b2578a7cd2274d9536d24baa26d3b.jpg", + "image_caption": [ + "Figure 1: Spatial contrasting depiction. " + ], + "image_footnote": [], + "bbox": [ + 367, + 310, + 625, + 647 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 699, + 279, + 713 + ], + "page_idx": 4 + }, + { + "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: ", + "bbox": [ + 173, + 724, + 825, + 810 + ], + "page_idx": 4 + }, + { + "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": [ + 236, + 815, + 761, + 862 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 202 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1 Calculation the spatial contrasting loss ", + "bbox": [ + 178, + 218, + 514, + 232 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Require: $X = \\{ x \\} _ { i = 1 } ^ { N }$ # Training on batches of images ", + "bbox": [ + 174, + 236, + 544, + 252 + ], + "page_idx": 5 + }, + { + "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": [ + 186, + 263, + 553, + 310 + ], + "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", + "bbox": [ + 191, + 469, + 374, + 494 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "# Spatial contrasting loss is the mean of distance ratios return $\\textstyle { \\frac { 1 } { N } } \\sum _ { i = 1 } ^ { N } d _ { i }$ g ", + "bbox": [ + 187, + 506, + 549, + 539 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 573, + 326, + 589 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 606, + 825, + 675 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 683, + 825, + 821 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The datasets we used are: ", + "bbox": [ + 176, + 821, + 341, + 835 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 215, + 847, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/f70e680c4da009ba6c944376177710a0a64eecc57507eb76161f1f476fa15955.jpg", + "table_caption": [ + "Table 1: State of the art results on STL-10 dataset " + ], + "table_footnote": [], + "table_body": "
ModelSTL-10 test accuracy
Zero-bias Convnets - Paine et al. (2014)70.2%
Triplet network -Hoffer & Ailon (2015)70.7%
Exemplar Convnets - Dosovitskiy et al. (2014)72.8%
Target Coding - Yang et al. (2015)73.15%
Stacked what-where AE - Zhao et al. (2015)74.33%
Spatial contrasting initialization (this work)81.34% ± 0.1
The same model without initialization72.6%±0.1
", + "bbox": [ + 251, + 126, + 746, + 257 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 230, + 284, + 823, + 311 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 217, + 318, + 823, + 373 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 178, + 386, + 825, + 428 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 RESULTS ON STL10 ", + "text_level": 1, + "bbox": [ + 176, + 445, + 356, + 460 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 173, + 473, + 825, + 638 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 RESULTS ON CIFAR10 ", + "text_level": 1, + "bbox": [ + 176, + 657, + 369, + 671 + ], + "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. ", + "bbox": [ + 174, + 684, + 825, + 739 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 176, + 739, + 825, + 809 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 RESULTS ON MNIST ", + "text_level": 1, + "bbox": [ + 176, + 827, + 361, + 842 + ], + "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). ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 6 + }, + { + "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": "
ModelCifar10 (400 per class) test accuracy
Convolutional K-means Network - Coates & Ng (2012)70.7%
View-Invariant K-means -Hui (2013)72.6%
DCGAN - Radford et al. (2015)73.8%
Exemplar Convnets - Dosovitskiy et al. (2014)76.6%
Ladder networks - Rasmus et al. (2015)79.6%
Conv-CatGan Springenberg (2016)80.42% (± 0.58)
ImprovedGan Salimans et al. (2016)81.37% (± 2.32)
Spatial contrasting initialization (this work)79.2%(±0.3)
The same model without initialization72.4%(±0.1)
", + "bbox": [ + 171, + 127, + 828, + 287 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e1e70e8fd494548a2eb23c22fe6d4cb4355a7dd6bbd7414322563dce6d766cd4.jpg", + "table_caption": [ + "Table 3: results on MNIST dataset " + ], + "table_footnote": [], + "table_body": "
ModelMNIST test error
Stacked what-where AE - Zhao et al. (2015)0.71%
Triplet network - Hoffer & Ailon (2015)0.56%
Jarrett et al. (2009)0.53%
Ladder networks - Rasmus et al. (2015)0.36%
DropConnect - Wan et al. (2013)0.21%
Spatial contrasting initialization (this work)0.34%± 0.02
The same model without initialization0.63%± 0.02
", + "bbox": [ + 271, + 325, + 725, + 457 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 483, + 825, + 554 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSIONS AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 575, + 498, + 590 + ], + "page_idx": 7 + }, + { + "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. ", + "bbox": [ + 174, + 607, + 825, + 690 + ], + "page_idx": 7 + }, + { + "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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(2015) ", + "" + ], + "table_footnote": [], + "table_body": "
Model
STL10CIFAR-10MNIST
Input: 96×96 RGBInput:32×32RGBInput:28 × 28 monochrome
5 ×5 conv. 64 BN ReLU3 × 3 conv. 96 BN LeakyReLU5 ×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
1 ×1 conv.160 BN ReLU3 ×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
1 ×1 conv. 96 BN ReLU
3 × 3 max-pooling, stride 2
5 × 5 conv. 192 BN ReLU
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
Spatial contrasting criterion
3×3 conv.256 ReLU3 × 3 max-pooling,stride 2dropout, p = 0.53 × 3 conv.128 ReLUdropout, p = 0.5fully-connected 103×3 conv.192BNLeakyReLU1 ×1 conv.192 BNLeakyReLU1 ×1 conv.10 BNLeakyReLUglobal average pooling3×3 conv.128 BN ReLU1 × 1 conv.10 BN ReLUglobal average pooling
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These attempts require different architectures and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 349, + 470, + 362 + ], + "spans": [ + { + "bbox": [ + 141, + 349, + 470, + 362 + ], + "score": 1.0, + "content": "training methods. In this work we present a novel approach for unsupervised", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 360, + 469, + 372 + ], + "spans": [ + { + "bbox": [ + 142, + 360, + 469, + 372 + ], + "score": 1.0, + "content": "training of Convolutional networks that is based on contrasting between spatial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 371, + 470, + 384 + ], + "spans": [ + { + "bbox": [ + 141, + 371, + 470, + 384 + ], + "score": 1.0, + "content": "regions within images. This criterion can be employed within conventional neu-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 382, + 469, + 394 + ], + "spans": [ + { + "bbox": [ + 141, + 382, + 469, + 394 + ], + "score": 1.0, + "content": "ral networks and optimized using standard techniques such as SGD and back-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 394, + 363, + 406 + ], + "spans": [ + { + "bbox": [ + 141, + 394, + 363, + 406 + ], + "score": 1.0, + "content": "propagation, thus complementing supervised methods.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 141, + 294, + 470, + 406 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 205, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 208, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 208, + 441 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "For the past few years convolutional networks (ConvNets, CNNs) LeCun et al. (1998) have proven", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "themselves as a successful model for vision related tasks Krizhevsky et al. (2012) Mnih et al. (2015)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "Pinheiro et al. (2015) Razavian et al. (2014). A convolutional network is composed of multiple", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "convolutional and pooling layers, followed by a fully-connected affine transformations. 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Autoencoders are NNs which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "aim to transform inputs into outputs with the least possible amount of distortion. 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VAE", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 368, + 506 + ], + "score": 1.0, + "content": "assumes that the data is generated by a directed graphical model", + "type": "text" + }, + { + "bbox": [ + 368, + 494, + 396, + 506 + ], + "score": 0.91, + "content": "p ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "and require the encoder to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 201, + 518 + ], + "score": 1.0, + "content": "learn an approximation", + "type": "text" + }, + { + "bbox": [ + 202, + 505, + 238, + 517 + ], + "score": 0.93, + "content": "q _ { w _ { 1 } } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 505, + 350, + 518 + ], + "score": 1.0, + "content": "to the posterior distribution", + "type": "text" + }, + { + "bbox": [ + 351, + 505, + 388, + 517 + ], + "score": 0.92, + "content": "p _ { w _ { 2 } } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 505, + 416, + 518 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 416, + 506, + 429, + 516 + ], + "score": 0.86, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 505, + 447, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 447, + 506, + 460, + 516 + ], + "score": 0.88, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "denote the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "parameters of the encoder and decoder. The objective of the variational autoencoder in that case has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 527, + 186, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 186, + 539 + ], + "score": 1.0, + "content": "the following form:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 394, + 506, + 539 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 546, + 448, + 561 + ], + "lines": [ + { + "bbox": [ + 162, + 546, + 448, + 561 + ], + "spans": [ + { + "bbox": [ + 162, + 546, + 448, + 561 + ], + "score": 0.87, + "content": "\\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 )", + "type": "interline_equation", + "image_path": "b80a08937376cb8f40f66af938aa10f2a90d8332197f760430b1611dab07074e.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 162, + 546, + 448, + 561 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 596 + ], + "score": 1.0, + "content": "Recently, a stacked set of denoising autoencoders architectures showed promising results in both", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "semi-supervised and unsupervised tasks. A stacked what-where autoencoder by Zhao et al. (2015)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "computes a set of complementary variables that enable reconstruction whenever a layer implements", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 629 + ], + "score": 1.0, + "content": "a many-to-one mapping. Ladder networks by Rasmus et al. (2015) - use lateral connections and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "layer-wise cost functions to allow the higher levels of an autoencoder to focus on invariant abstract", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 639, + 143, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 143, + 651 + ], + "score": 1.0, + "content": "features.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 585, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "Exemplar Networks: The unsupervised method introduced byDosovitskiy et al. (2014) takes a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "different approach to this task and trains the network to discriminate between a set of pseudo-classes.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "Each pseudo-class is formed by applying multiple transformations to a randomly sampled image", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "patch. The number of pseudo-classes can be as big as the size of the input samples. This criterion", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "ensures that different input samples would be distinguished while providing robustness to the applied", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 482, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 482, + 732 + ], + "score": 1.0, + "content": "transformations. In this work we will explore an alternative method with a similar motivation.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 664, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Context prediction Another method for unsupervised learning by context was introduced by Do-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "ersch et al. (2015). This method uses an auxiliary criterion of predicting the location of an image", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "patch given another from the same image. This is done by classification to 1 of 9 possible locations.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "Although the work of Doersch et al. (2015) and ours both use patches from an image to perform", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "unsupervised learning, the methods are quite different. Whereas the former used a classification", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "criterion over the spatial location of each patch within a single image, our work is concerned with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "comparing patches from several images to each other. We claim that this encourages discriminability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "between images (which we feel to be important aspect of feature learning), and was not an explicit", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 199, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 199, + 183 + ], + "score": 1.0, + "content": "goal in previous work.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 195, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 195, + 504, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 504, + 206 + ], + "score": 1.0, + "content": "Adversarial Generative Models: This a recently introduced model that can be used in an unsu-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "pervised fashion Goodfellow et al. 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By propagating the gradient through the paired networks, the model learns to generate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "samples that are distributed similarly to the source data. As shown by Radford et al. 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In these methods, roughly speaking, the poste-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 172, + 320 + ], + "score": 1.0, + "content": "rior probability", + "type": "text" + }, + { + "bbox": [ + 172, + 307, + 203, + 319 + ], + "score": 0.93, + "content": "P ( t | y _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 307, + 312, + 320 + ], + "score": 1.0, + "content": "of the ground-truth target", + "type": "text" + }, + { + "bbox": [ + 313, + 308, + 319, + 317 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "given the model output on an input sampled", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 213, + 331 + ], + "score": 1.0, + "content": "from the true distribution", + "type": "text" + }, + { + "bbox": [ + 214, + 318, + 261, + 330 + ], + "score": 0.93, + "content": "y _ { t } = F ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 317, + 412, + 331 + ], + "score": 1.0, + "content": "is maximized, while the probability", + "type": "text" + }, + { + "bbox": [ + 413, + 318, + 446, + 330 + ], + "score": 0.93, + "content": "P ( t | y _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "given a noise", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 162, + 342 + ], + "score": 1.0, + "content": "measurement", + "type": "text" + }, + { + "bbox": [ + 163, + 329, + 206, + 341 + ], + "score": 0.95, + "content": "y = F ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "is minimized. This was successfully used in a language domain to learn", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "unsupervised representation of words. The most noteworthy case is the word2vec model introduced", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 350, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 364 + ], + "score": 1.0, + "content": "by Mikolov et al. (2013). When using this setting in language applications, a natural contrasting", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "noise is a smooth approximation of the Unigram distribution. A suitable contrasting distribution is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "less obvious when data points are sampled from a high dimensional continuous space, such as the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 198, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 198, + 396 + ], + "score": 1.0, + "content": "case of image patches.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 410, + 308, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 309, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 309, + 422 + ], + "score": 1.0, + "content": "2.1 PROBLEMS WITH CURRENT APPROACHES", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 504, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "Only recently the potential of ConvNets in an unsupervised environment began to bear fruit, still we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 237, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 237, + 453 + ], + "score": 1.0, + "content": "believe it is not fully uncovered.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "The majority of unsupervised optimization criteria currently used are based on variations of recon-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "struction losses. One limitation of this fact is that a pixel level reconstruction is non-compliant with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "the idea of a discriminative objective, which is expected to be agnostic to low level information in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 503 + ], + "score": 1.0, + "content": "input. In addition, it is evident that MSE is not best suited as a measurement to compare images, for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 505, + 516 + ], + "score": 1.0, + "content": "example, viewing the possibly large square-error between an image and a single pixel shifted copy", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "of it. Another problem with recent approaches such as Rasmus et al. (2015); Zeiler et al. (2010)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 523, + 506, + 538 + ], + "score": 1.0, + "content": "is their need to extensively modify the original convolutional network model. This leads to a gap", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "between unsupervised method and the state-of-the-art, supervised, models for classification - which", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "can hurt future attempt to reconcile them in a unified framework, as well as efficiently leverage", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 309, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 309, + 570 + ], + "score": 1.0, + "content": "unlabeled data with otherwise supervised regimes.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 109, + 585, + 276, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 279, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 279, + 600 + ], + "score": 1.0, + "content": "3 LEARNING BY COMPARISONS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "The most common way to train NN is by defining a loss function between the target values and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "the network output. Learning by comparison approaches the supervised task from a different angle.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "The main idea is to use distance comparisons between samples to learn useful representations. For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 375, + 656 + ], + "score": 1.0, + "content": "example, we consider relative and qualitative examples of the form", + "type": "text" + }, + { + "bbox": [ + 375, + 644, + 389, + 655 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 644, + 435, + 656 + ], + "score": 1.0, + "content": "is closer to", + "type": "text" + }, + { + "bbox": [ + 436, + 644, + 450, + 655 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 644, + 470, + 656 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 470, + 644, + 484, + 655 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "is to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 120, + 666 + ], + "score": 0.87, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 654, + 505, + 668 + ], + "score": 1.0, + "content": ". Using a comparative measure with neural network to learn embedding space was introduced in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the “Siamese network” framework by Bromley et al. (1993) and later used in the works of Chopra", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 676, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 505, + 691 + ], + "score": 1.0, + "content": "et al. (2005). One use for this methods is when the number of classes is too large or expected to vary", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "over time, as in the case of face verification, where a face contained in an image has to compared", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "against another image of a face. This problem was recently tackled by Schroff et al. (2015) for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "training a convolutional network model on triplets of examples. 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(2015). This method uses an auxiliary criterion of predicting the location of an image", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "patch given another from the same image. This is done by classification to 1 of 9 possible locations.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "Although the work of Doersch et al. (2015) and ours both use patches from an image to perform", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "unsupervised learning, the methods are quite different. 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By propagating the gradient through the paired networks, the model learns to generate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "samples that are distributed similarly to the source data. As shown by Radford et al. 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A suitable contrasting distribution is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "less obvious when data points are sampled from a high dimensional continuous space, such as the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 198, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 198, + 396 + ], + "score": 1.0, + "content": "case of image patches.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 284, + 506, + 396 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 410, + 308, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 309, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 309, + 422 + ], + "score": 1.0, + "content": "2.1 PROBLEMS WITH CURRENT APPROACHES", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 504, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "Only recently the potential of ConvNets in an unsupervised environment began to bear fruit, still we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 237, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 237, + 453 + ], + "score": 1.0, + "content": "believe it is not fully uncovered.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 430, + 505, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "The majority of unsupervised optimization criteria currently used are based on variations of recon-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "struction losses. 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Learning by comparison approaches the supervised task from a different angle.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "The main idea is to use distance comparisons between samples to learn useful representations. For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 375, + 656 + ], + "score": 1.0, + "content": "example, we consider relative and qualitative examples of the form", + "type": "text" + }, + { + "bbox": [ + 375, + 644, + 389, + 655 + ], + "score": 0.88, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 644, + 435, + 656 + ], + "score": 1.0, + "content": "is closer to", + "type": "text" + }, + { + "bbox": [ + 436, + 644, + 450, + 655 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 644, + 470, + 656 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 470, + 644, + 484, + 655 + ], + "score": 0.89, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "is to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 120, + 666 + ], + "score": 0.87, + "content": "X _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 654, + 505, + 668 + ], + "score": 1.0, + "content": ". Using a comparative measure with neural network to learn embedding space was introduced in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the “Siamese network” framework by Bromley et al. (1993) and later used in the works of Chopra", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 676, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 505, + 691 + ], + "score": 1.0, + "content": "et al. (2005). One use for this methods is when the number of classes is too large or expected to vary", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "over time, as in the case of face verification, where a face contained in an image has to compared", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "against another image of a face. This problem was recently tackled by Schroff et al. (2015) for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "training a convolutional network model on triplets of examples. 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More precisely, the loss function used in this case was defined as", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 45, + "bbox_fs": [ + 104, + 610, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 363, + 95 + ], + "score": 1.0, + "content": "of the face of the same person) together with a negative example", + "type": "text" + }, + { + "bbox": [ + 363, + 84, + 377, + 94 + ], + "score": 0.86, + "content": "x _ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", containing a face of a different", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "person. The training objective was on the embedded distance of the input faces, where the distance", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 474, + 117 + ], + "score": 1.0, + "content": "between the anchor and positive example is adjusted to be smaller by at least some constant", + "type": "text" + }, + { + "bbox": [ + 474, + 106, + 482, + 114 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 456, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 456, + 127 + ], + "score": 1.0, + "content": "the negative distance. More precisely, the loss function used in this case was defined as", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 135, + 452, + 149 + ], + "lines": [ + { + "bbox": [ + 159, + 135, + 452, + 149 + ], + "spans": [ + { + "bbox": [ + 159, + 135, + 452, + 149 + ], + "score": 0.88, + "content": "L ( x , x _ { + } , x _ { - } ) = \\operatorname* { m a x } \\left\\{ \\| F ( x ) - F ( x _ { + } ) \\| _ { 2 } - \\| F ( x ) - F ( x _ { - } ) \\| _ { 2 } + \\alpha , 0 \\right\\}", + "type": "interline_equation", + "image_path": "484d212cac854adcb5189435d1af769a6d494575cce095b409510330c3aaa3c2.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 159, + 135, + 452, + 149 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 156, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 133, + 169 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 157, + 156, + 169 + ], + "score": 0.92, + "content": "F ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 157, + 435, + 169 + ], + "score": 1.0, + "content": "is the embedding (the output of a convolutional neural network), and", + "type": "text" + }, + { + "bbox": [ + 435, + 159, + 443, + 167 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "is a predefined", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "margin constant. Another similar model used by Hoffer & Ailon (2015) with triplets comparisons for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 192 + ], + "score": 1.0, + "content": "classification, where examples from the same class were trained to have a lower embedded distance", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 190, + 504, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 504, + 202 + ], + "score": 1.0, + "content": "than that of two images from distinct classes. This work introduced a concept of a distance ratio", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 291, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 291, + 213 + ], + "score": 1.0, + "content": "loss, where the defined measure amounted to:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 219, + 415, + 246 + ], + "lines": [ + { + "bbox": [ + 193, + 219, + 415, + 246 + ], + "spans": [ + { + "bbox": [ + 193, + 219, + 415, + 246 + ], + "score": 0.94, + "content": "L ( x , x _ { + } , x _ { - } ) = \\frac { e ^ { - \\| F ( x ) - F ( x _ { + } ) \\| _ { 2 } } } { e ^ { - \\| F ( x ) - F ( x _ { + } ) \\| _ { 2 } } + e ^ { - \\| F ( x ) - F ( x _ { - } ) \\| _ { 2 } } }", + "type": "interline_equation", + "image_path": "4fb638bda29994a614957769f108d611ae9a93ad6b9a11972de0d8d2dfad9e99.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 193, + 219, + 415, + 232.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 193, + 232.5, + 415, + 246.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "This loss has a flavor of a probability of a biased coin flip. By ‘pushing’ this probability to zero,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "we express the objective that pairs of samples coming from distinct classes should be less similar to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 504, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 504, + 288 + ], + "score": 1.0, + "content": "each other, compared to pairs of samples coming from the same class. It was shown empirical by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 287, + 490, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 490, + 300 + ], + "score": 1.0, + "content": "Balntas et al. (2016) to provide better feature embeddings than the margin based distance loss 1", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 317, + 363, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 364, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 364, + 333 + ], + "score": 1.0, + "content": "4 OUR CONTRIBUTION: SPATIAL CONTRASTING", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "One implicit assumption in convolutional networks, is that features are gradually learned hierar-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "chically, each level in the hierarchy corresponding to a layer in the network. Each spatial location", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "within a layer corresponds to a region in the original image. It is empirically observed that deeper", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "layers tend to contain more ‘abstract’ information from the image. Intuitively, features describing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "different regions within the same image are likely to be semantically similar (e.g. different parts", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "of an animal), and indeed the corresponding deep representations tend to be similar. Conversely,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "regions from two probably unrelated images (say, two images chosen at random) tend to be far from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "each other in the deep representation. This logic is commonly used in modern deep networks such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "as Szegedy et al. (2015) Lin et al. (2013) He et al. (2015), where a global average pooling is used to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 370, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 370, + 456 + ], + "score": 1.0, + "content": "aggregate spatial features in the final layer used for classification.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 474 + ], + "score": 1.0, + "content": "Our suggestion is that this property, often observed as a side effect of supervised applications, can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "be used as a desired objective when learning deep representations in an unsupervised task. Later, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "score": 1.0, + "content": "resulting representation can be used, as typically done, as a starting point or a supervised learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "task. We call this idea which we formalize below Spatial contrasting. The spatial contrasting crite-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "rion is similar to noise contrasting estimation Gutmann & Hyvarinen (2010) Mnih & Kavukcuoglu ¨", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "(2013), in trying to train a model by maximizing the expected probability on desired inputs, while", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 527, + 320, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 320, + 539 + ], + "score": 1.0, + "content": "minimizing it on contrasting sampled measurements.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 554, + 195, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 197, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 197, + 567 + ], + "score": 1.0, + "content": "4.1 FORMULATION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 575, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 341, + 589 + ], + "score": 1.0, + "content": "We will concern ourselves with samples of images patches", + "type": "text" + }, + { + "bbox": [ + 341, + 575, + 361, + 586 + ], + "score": 0.91, + "content": "\\tilde { x } ^ { ( m ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 574, + 446, + 589 + ], + "score": 1.0, + "content": "taken from an image", + "type": "text" + }, + { + "bbox": [ + 446, + 578, + 453, + 586 + ], + "score": 0.67, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 574, + 505, + 589 + ], + "score": 1.0, + "content": ". 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By ‘pushing’ this probability to zero,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "we express the objective that pairs of samples coming from distinct classes should be less similar to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 504, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 504, + 288 + ], + "score": 1.0, + "content": "each other, compared to pairs of samples coming from the same class. It was shown empirical by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 287, + 490, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 490, + 300 + ], + "score": 1.0, + "content": "Balntas et al. (2016) to provide better feature embeddings than the margin based distance loss 1", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 253, + 505, + 300 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 317, + 363, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 364, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 364, + 333 + ], + "score": 1.0, + "content": "4 OUR CONTRIBUTION: SPATIAL CONTRASTING", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "One implicit assumption in convolutional networks, is that features are gradually learned hierar-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "chically, each level in the hierarchy corresponding to a layer in the network. Each spatial location", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "within a layer corresponds to a region in the original image. It is empirically observed that deeper", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "layers tend to contain more ‘abstract’ information from the image. Intuitively, features describing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "different regions within the same image are likely to be semantically similar (e.g. different parts", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "of an animal), and indeed the corresponding deep representations tend to be similar. Conversely,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "regions from two probably unrelated images (say, two images chosen at random) tend to be far from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "each other in the deep representation. This logic is commonly used in modern deep networks such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "as Szegedy et al. (2015) Lin et al. (2013) He et al. (2015), where a global average pooling is used to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 370, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 370, + 456 + ], + "score": 1.0, + "content": "aggregate spatial features in the final layer used for classification.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 344, + 506, + 456 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 474 + ], + "score": 1.0, + "content": "Our suggestion is that this property, often observed as a side effect of supervised applications, can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "be used as a desired objective when learning deep representations in an unsupervised task. Later, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "score": 1.0, + "content": "resulting representation can be used, as typically done, as a starting point or a supervised learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "task. We call this idea which we formalize below Spatial contrasting. The spatial contrasting crite-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "rion is similar to noise contrasting estimation Gutmann & Hyvarinen (2010) Mnih & Kavukcuoglu ¨", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "(2013), in trying to train a model by maximizing the expected probability on desired inputs, while", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 527, + 320, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 320, + 539 + ], + "score": 1.0, + "content": "minimizing it on contrasting sampled measurements.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 460, + 505, + 539 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 554, + 195, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 197, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 197, + 567 + ], + "score": 1.0, + "content": "4.1 FORMULATION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 575, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 341, + 589 + ], + "score": 1.0, + "content": "We will concern ourselves with samples of images patches", + "type": "text" + }, + { + "bbox": [ + 341, + 575, + 361, + 586 + ], + "score": 0.91, + "content": "\\tilde { x } ^ { ( m ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 574, + 446, + 589 + ], + "score": 1.0, + "content": "taken from an image", + "type": "text" + }, + { + "bbox": [ + 446, + 578, + 453, + 586 + ], + "score": 0.67, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 574, + 505, + 589 + ], + "score": 1.0, + "content": ". Our convo-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 586, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 586, + 250, + 602 + ], + "score": 1.0, + "content": "lutional network model, denoted by", + "type": "text" + }, + { + "bbox": [ + 250, + 588, + 272, + 600 + ], + "score": 0.92, + "content": "F ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 586, + 372, + 602 + ], + "score": 1.0, + "content": ", extracts spatial features", + "type": "text" + }, + { + "bbox": [ + 372, + 588, + 379, + 600 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 586, + 409, + 602 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 409, + 587, + 478, + 600 + ], + "score": 0.93, + "content": "f ^ { ( m ) } = F ( \\tilde { x } ^ { ( m ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 586, + 506, + 602 + ], + "score": 1.0, + "content": "for an", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 597, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 158, + 615 + ], + "score": 1.0, + "content": "image patch", + "type": "text" + }, + { + "bbox": [ + 158, + 599, + 178, + 611 + ], + "score": 0.9, + "content": "\\tilde { x } ^ { ( m ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 597, + 264, + 615 + ], + "score": 1.0, + "content": ". We will also define", + "type": "text" + }, + { + "bbox": [ + 265, + 600, + 303, + 613 + ], + "score": 0.93, + "content": "P ( f _ { 1 } | f _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 597, + 443, + 615 + ], + "score": 1.0, + "content": "as the probability for two features", + "type": "text" + }, + { + "bbox": [ + 444, + 601, + 468, + 612 + ], + "score": 0.91, + "content": "f _ { 1 } , f _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 597, + 506, + 615 + ], + "score": 1.0, + "content": "to occur", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 612, + 219, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 219, + 624 + ], + "score": 1.0, + "content": "together in the same image.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 574, + 506, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "We wish to optimize our model such that for two features representing patches taken from the same", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 99, + 627, + 509, + 657 + ], + "spans": [ + { + "bbox": [ + 99, + 628, + 161, + 657 + ], + "score": 1.0, + "content": "image x˜(1)i , x˜", + "type": "text" + }, + { + "bbox": [ + 134, + 633, + 195, + 648 + ], + "score": 0.93, + "content": "\\tilde { x } _ { i } ^ { ( 1 ) } , \\bar { x _ { i } ^ { ( 2 ) } } \\in x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 627, + 240, + 654 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 240, + 633, + 304, + 648 + ], + "score": 0.94, + "content": "f _ { i } ^ { ( 1 ) } = F ( \\tilde { x } _ { i } ^ { ( 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 627, + 324, + 654 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 633, + 389, + 648 + ], + "score": 0.9, + "content": "f _ { i } ^ { ( 2 ) } \\stackrel { - } { = } F ( \\tilde { x } _ { i } ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 627, + 393, + 654 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 633, + 446, + 648 + ], + "score": 0.91, + "content": "P ( f _ { i } ^ { ( 1 ) } | f _ { i } ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 627, + 509, + 654 + ], + "score": 1.0, + "content": "will be maxi-", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 99, + 622, + 509, + 657 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "This means that features from a patch taken from a specific image can effectively predict, under our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "model, features extracted from other patches in the same image. Conversely, we want our model", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 679, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 159, + 693 + ], + "score": 1.0, + "content": "to minimize", + "type": "text" + }, + { + "bbox": [ + 159, + 680, + 195, + 692 + ], + "score": 0.94, + "content": "P ( f _ { i } | f _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 679, + 213, + 693 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 213, + 680, + 226, + 691 + ], + "score": 0.88, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 679, + 505, + 693 + ], + "score": 1.0, + "content": "being two patches taken from distinct images. 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This formulation is portrayed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 181 + ], + "score": 1.0, + "content": "in figure 4.1. Since we sample our contrasting sample from the same underlying distribution, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "can evaluate this loss considering the image patch as both patch compared (anchor) and contrast", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 190, + 412, + 203 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 412, + 203 + ], + "score": 1.0, + "content": "symmetrically. 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This formulation is portrayed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 181 + ], + "score": 1.0, + "content": "in figure 4.1. Since we sample our contrasting sample from the same underlying distribution, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "can evaluate this loss considering the image patch as both patch compared (anchor) and contrast", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 190, + 412, + 203 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 412, + 203 + ], + "score": 1.0, + "content": "symmetrically. 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Each image", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "serves as both anchor and positive patches, for which the corresponding features should be closer,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 408, + 620 + ], + "score": 1.0, + "content": "as well as contrasting samples for other images in that batch. For a batch of", + "type": "text" + }, + { + "bbox": [ + 409, + 609, + 419, + 618 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "images, two samples", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 242, + 631 + ], + "score": 1.0, + "content": "from each image are taken, and", + "type": "text" + }, + { + "bbox": [ + 243, + 618, + 257, + 628 + ], + "score": 0.88, + "content": "N ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "different distance comparisons are made. The final loss is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 630, + 362, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 362, + 642 + ], + "score": 1.0, + "content": "defined as the average distance ratio for all images in the batch:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 573, + 506, + 642 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 646, + 466, + 683 + ], + "lines": [ + { + "bbox": [ + 145, + 646, + 466, + 683 + ], + "spans": [ + { + "bbox": [ + 145, + 646, + 466, + 683 + ], + "score": 0.94, + "content": "\\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 } } }", + "type": "interline_equation", + "image_path": "8b403568fa24822998826ddcf40e5c6cf8dbd89796e01189a57ab00cf49c6a19.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 145, + 646, + 466, + 658.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 145, + 658.3333333333334, + 466, + 670.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 145, + 670.6666666666667, + 466, + 683.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "Since the criterion is differentiable with respect to its inputs, it is fully compliant with standard", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "methods for training convolutional network and specifically using backpropagation and gradient", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "descent. Furthermore, SC can be applied to any layer in the network hierarchy. In fact, SC can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "be used at multiple layers within the same convolutional network. The spatial properties of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 354, + 96 + ], + "score": 1.0, + "content": "features means that we can sample directly from feature space", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 354, + 81, + 393, + 94 + ], + "score": 0.93, + "content": "\\tilde { f } ^ { ( m ) } \\in f", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 394, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "instead of from the original", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "image. Therefore SC has a simple implementation which doesn’t require substation amount of", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "computation. The complete algorithm for batch training is described in Algorithm (1). Similar to", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the batch normalization (BN) layer Ioffe & Szegedy (2015), a recent usage for batch statistics in", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 504, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 504, + 138 + ], + "score": 1.0, + "content": "neural networks, SC also uses the batch statistics. While BN normalize the input based on the batch", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "statistics, SC sample from it. This can be viewed as a simple sampling from the space of possible", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 255, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 255, + 162 + ], + "score": 1.0, + "content": "features describing a patch of image.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 686, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 354, + 96 + ], + "score": 1.0, + "content": "features means that we can sample directly from feature space", + "type": "text" + }, + { + "bbox": [ + 354, + 81, + 393, + 94 + ], + "score": 0.93, + "content": "\\tilde { f } ^ { ( m ) } \\in f", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "instead of from the original", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "image. 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We used mild data augmentations, such as small translations and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 639, + 192, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 192, + 653 + ], + "score": 1.0, + "content": "horizontal mirroring.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 540, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 651, + 209, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 211, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 211, + 663 + ], + "score": 1.0, + "content": "The datasets we used are:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 650, + 211, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 671, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 133, + 672, + 359, + 684 + ], + "score": 1.0, + "content": "• STL10 (Coates et al. (2011)). This dataset consists of", + "type": "text" + }, + { + "bbox": [ + 360, + 672, + 428, + 683 + ], + "score": 0.82, + "content": "1 0 0 , 0 0 0 9 6 \\times 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "colored, unlabeled", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 141, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "images, together with another set of 5, 000 labeled training images and 8, 000 test images .", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 694, + 323, + 707 + ], + "spans": [ + { + "bbox": [ + 142, + 694, + 323, + 707 + ], + "score": 1.0, + "content": "The label space consists of 10 object classes.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 136, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 136, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "• Cifar10 (Krizhevsky & Hinton (2009)). The well known CIFAR-10 is an image classifi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "cation benchmark dataset containing 50, 000 training images and 10, 000 test images. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 223, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 191, + 239 + ], + "score": 1.0, + "content": "image sizes", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 191, + 225, + 225, + 236 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 226, + 223, + 505, + 239 + ], + "score": 1.0, + "content": "pixels, with color. The classes are airplanes, automobiles, birds, cats,", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 142, + 236, + 313, + 249 + ], + "spans": [ + { + "bbox": [ + 142, + 236, + 313, + 249 + ], + "score": 1.0, + "content": "deer, dogs, frogs, horses, ships and trucks.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 34, + "bbox_fs": [ + 133, + 672, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 154, + 100, + 457, + 204 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 204, + 90, + 406, + 100 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 204, + 88, + 407, + 101 + ], + "spans": [ + { + "bbox": [ + 204, + 88, + 407, + 101 + ], + "score": 1.0, + "content": "Table 1: State of the art results on STL-10 dataset", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 154, + 100, + 457, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 100, + 457, + 204 + ], + "spans": [ + { + "bbox": [ + 154, + 100, + 457, + 204 + ], + "score": 0.984, + "html": "
ModelSTL-10 test accuracy
Zero-bias Convnets - Paine et al. (2014)70.2%
Triplet network -Hoffer & Ailon (2015)70.7%
Exemplar Convnets - Dosovitskiy et al. (2014)72.8%
Target Coding - Yang et al. (2015)73.15%
Stacked what-where AE - Zhao et al. (2015)74.33%
Spatial contrasting initialization (this work)81.34% ± 0.1
The same model without initialization72.6%±0.1
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The classes are airplanes, automobiles, birds, cats,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 142, + 236, + 313, + 249 + ], + "spans": [ + { + "bbox": [ + 142, + 236, + 313, + 249 + ], + "score": 1.0, + "content": "deer, dogs, frogs, horses, ships and trucks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 133, + 252, + 504, + 296 + ], + "lines": [ + { + "bbox": [ + 132, + 252, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 132, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "• MNIST (LeCun et al. (1998)). The MNIST database of handwritten digits is one of the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "most studied dataset benchmark for image classification. The dataset contains 60,000 ex-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 143, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 143, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "amples of handwritten digits from 0 to 9 for training and 10,000 additional examples for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 284, + 369, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 243, + 298 + ], + "score": 1.0, + "content": "testing. Each sample is a", + "type": "text" + }, + { + "bbox": [ + 244, + 285, + 275, + 295 + ], + "score": 0.82, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 284, + 369, + 298 + ], + "score": 1.0, + "content": "pixel gray level image.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 109, + 306, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 505, + 319 + ], + "score": 1.0, + "content": "All experiments were conducted using the Torch7 framework by Collobert et al. (2011).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "Code reproducing these results will by available at https://github.com/eladhoffer/", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 327, + 220, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 220, + 341 + ], + "score": 1.0, + "content": "SpatialContrasting.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 353, + 218, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 219, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 219, + 367 + ], + "score": 1.0, + "content": "5.1 RESULTS ON STL10", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "Since STL10 dataset is comprised of mostly unlabeled data, it is most suitable to highlight the ben-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "efits of the spatial contrasting criterion. The initial training was unsupervised, as described earlier,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "using the entire set of 105, 000 samples (union of the original unlabeled set and labeled training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "score": 1.0, + "content": "set). The representation outputted by the training, was used to initialize supervised training on the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 417, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 505, + 433 + ], + "score": 1.0, + "content": "5, 000 labeled images. Evaluation was done on a separate test set of 8, 000 samples. Comparing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 337, + 443 + ], + "score": 1.0, + "content": "with state of the art results, we see an improvement of", + "type": "text" + }, + { + "bbox": [ + 338, + 430, + 353, + 440 + ], + "score": 0.84, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "in test accuracy over the best model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 324, + 452 + ], + "score": 1.0, + "content": "by Zhao et al. (2015), setting the SC as best model at", + "type": "text" + }, + { + "bbox": [ + 324, + 441, + 351, + 451 + ], + "score": 0.88, + "content": "8 1 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "test classification accuracy (see Table", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "(1)). We note that the results of Dosovitskiy et al. (2014) are achieved with no fine-tuning over", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "score": 1.0, + "content": "labeled examples, which may be unfair to this work. We also compare with the same network, but", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 371, + 485 + ], + "score": 1.0, + "content": "without SC initialization, which achieves a lower classification of", + "type": "text" + }, + { + "bbox": [ + 371, + 473, + 399, + 484 + ], + "score": 0.88, + "content": "7 2 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 473, + 505, + 485 + ], + "score": 1.0, + "content": ". This is an indication that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "indeed SC managed to leverage unlabeled examples to provide a better initialization point for the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 496, + 182, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 182, + 507 + ], + "score": 1.0, + "content": "supervised model.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 521, + 226, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 227, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 227, + 534 + ], + "score": 1.0, + "content": "5.2 RESULTS ON CIFAR10", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 332, + 554 + ], + "score": 1.0, + "content": "For Cifar10 dataset, we use the same setting as Coates &", + "type": "text" + }, + { + "bbox": [ + 332, + 542, + 375, + 554 + ], + "score": 0.34, + "content": "\\mathrm { N g } \\left( 2 0 1 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "and Hui (2013) to test a model’s", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "ability to learn from unlabeled images. Here, only 4, 000 samples out of 50, 000 are used with their", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "label annotation, and the rest of the samples can be used only in an unsupervised manner. The final", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 576, + 327, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 327, + 587 + ], + "score": 1.0, + "content": "test accuracy is measured on the entire 10, 000 test set.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 108, + 586, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "In our experiments, we trained our model using SC criterion on the entire dataset, and then used", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "only 400 labeled samples per class (for a total of 4000) in a supervised regime over the initialized", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "network. The results are compared with previous efforts in Table (2). Using the SC criterion allowed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 186, + 631 + ], + "score": 1.0, + "content": "an improvement of", + "type": "text" + }, + { + "bbox": [ + 187, + 619, + 209, + 630 + ], + "score": 0.87, + "content": "6 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 619, + 474, + 631 + ], + "score": 1.0, + "content": "over a non-initialized model, and achieved a final test accuracy of", + "type": "text" + }, + { + "bbox": [ + 474, + 619, + 501, + 630 + ], + "score": 0.85, + "content": "7 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 619, + 505, + 631 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 630, + 362, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 362, + 642 + ], + "score": 1.0, + "content": "This is a competitive result with current state-of-the-art models.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 221, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 223, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 223, + 669 + ], + "score": 1.0, + "content": "5.3 RESULTS ON MNIST", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "The MNIST dataset is very different in nature from the Cifar10 and STL10 datasets, we experi-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "mented earlier. The biggest difference, relevant to this work, is that spatial regions sampled from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "MNIST images usually provide very little, or no information. Thus, SC is much less suited for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "MNIST dataset, and was conjured to have little benefit. We still, however, experimented with ini-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "tializing a model with SC criterion and continuing with a fully-supervised regime over all labeled", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 154, + 100, + 457, + 204 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 204, + 90, + 406, + 100 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 204, + 88, + 407, + 101 + ], + "spans": [ + { + "bbox": [ + 204, + 88, + 407, + 101 + ], + "score": 1.0, + "content": "Table 1: State of the art results on STL-10 dataset", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 154, + 100, + 457, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 100, + 457, + 204 + ], + "spans": [ + { + "bbox": [ + 154, + 100, + 457, + 204 + ], + "score": 0.984, + "html": "
ModelSTL-10 test accuracy
Zero-bias Convnets - Paine et al. (2014)70.2%
Triplet network -Hoffer & Ailon (2015)70.7%
Exemplar Convnets - Dosovitskiy et al. (2014)72.8%
Target Coding - Yang et al. (2015)73.15%
Stacked what-where AE - Zhao et al. (2015)74.33%
Spatial contrasting initialization (this work)81.34% ± 0.1
The same model without initialization72.6%±0.1
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The MNIST database of handwritten digits is one of the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "most studied dataset benchmark for image classification. The dataset contains 60,000 ex-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 143, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 143, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "amples of handwritten digits from 0 to 9 for training and 10,000 additional examples for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 284, + 369, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 243, + 298 + ], + "score": 1.0, + "content": "testing. Each sample is a", + "type": "text" + }, + { + "bbox": [ + 244, + 285, + 275, + 295 + ], + "score": 0.82, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 284, + 369, + 298 + ], + "score": 1.0, + "content": "pixel gray level image.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 132, + 252, + 505, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 306, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 505, + 319 + ], + "score": 1.0, + "content": "All experiments were conducted using the Torch7 framework by Collobert et al. (2011).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "Code reproducing these results will by available at https://github.com/eladhoffer/", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 327, + 220, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 220, + 341 + ], + "score": 1.0, + "content": "SpatialContrasting.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 305, + 505, + 341 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 353, + 218, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 219, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 219, + 367 + ], + "score": 1.0, + "content": "5.1 RESULTS ON STL10", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "Since STL10 dataset is comprised of mostly unlabeled data, it is most suitable to highlight the ben-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "efits of the spatial contrasting criterion. The initial training was unsupervised, as described earlier,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "using the entire set of 105, 000 samples (union of the original unlabeled set and labeled training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "score": 1.0, + "content": "set). The representation outputted by the training, was used to initialize supervised training on the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 417, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 505, + 433 + ], + "score": 1.0, + "content": "5, 000 labeled images. Evaluation was done on a separate test set of 8, 000 samples. Comparing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 337, + 443 + ], + "score": 1.0, + "content": "with state of the art results, we see an improvement of", + "type": "text" + }, + { + "bbox": [ + 338, + 430, + 353, + 440 + ], + "score": 0.84, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "in test accuracy over the best model", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 324, + 452 + ], + "score": 1.0, + "content": "by Zhao et al. (2015), setting the SC as best model at", + "type": "text" + }, + { + "bbox": [ + 324, + 441, + 351, + 451 + ], + "score": 0.88, + "content": "8 1 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "test classification accuracy (see Table", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "(1)). We note that the results of Dosovitskiy et al. (2014) are achieved with no fine-tuning over", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "score": 1.0, + "content": "labeled examples, which may be unfair to this work. We also compare with the same network, but", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 371, + 485 + ], + "score": 1.0, + "content": "without SC initialization, which achieves a lower classification of", + "type": "text" + }, + { + "bbox": [ + 371, + 473, + 399, + 484 + ], + "score": 0.88, + "content": "7 2 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 473, + 505, + 485 + ], + "score": 1.0, + "content": ". This is an indication that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "indeed SC managed to leverage unlabeled examples to provide a better initialization point for the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 496, + 182, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 182, + 507 + ], + "score": 1.0, + "content": "supervised model.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 375, + 506, + 507 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 521, + 226, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 227, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 227, + 534 + ], + "score": 1.0, + "content": "5.2 RESULTS ON CIFAR10", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 332, + 554 + ], + "score": 1.0, + "content": "For Cifar10 dataset, we use the same setting as Coates &", + "type": "text" + }, + { + "bbox": [ + 332, + 542, + 375, + 554 + ], + "score": 0.34, + "content": "\\mathrm { N g } \\left( 2 0 1 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "and Hui (2013) to test a model’s", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "ability to learn from unlabeled images. Here, only 4, 000 samples out of 50, 000 are used with their", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "label annotation, and the rest of the samples can be used only in an unsupervised manner. The final", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 576, + 327, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 327, + 587 + ], + "score": 1.0, + "content": "test accuracy is measured on the entire 10, 000 test set.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 542, + 506, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 586, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "In our experiments, we trained our model using SC criterion on the entire dataset, and then used", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "only 400 labeled samples per class (for a total of 4000) in a supervised regime over the initialized", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "network. The results are compared with previous efforts in Table (2). Using the SC criterion allowed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 186, + 631 + ], + "score": 1.0, + "content": "an improvement of", + "type": "text" + }, + { + "bbox": [ + 187, + 619, + 209, + 630 + ], + "score": 0.87, + "content": "6 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 619, + 474, + 631 + ], + "score": 1.0, + "content": "over a non-initialized model, and achieved a final test accuracy of", + "type": "text" + }, + { + "bbox": [ + 474, + 619, + 501, + 630 + ], + "score": 0.85, + "content": "7 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 619, + 505, + 631 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 630, + 362, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 362, + 642 + ], + "score": 1.0, + "content": "This is a competitive result with current state-of-the-art models.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 586, + 505, + 642 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 221, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 223, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 223, + 669 + ], + "score": 1.0, + "content": "5.3 RESULTS ON MNIST", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "The MNIST dataset is very different in nature from the Cifar10 and STL10 datasets, we experi-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "mented earlier. The biggest difference, relevant to this work, is that spatial regions sampled from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "MNIST images usually provide very little, or no information. Thus, SC is much less suited for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "MNIST dataset, and was conjured to have little benefit. We still, however, experimented with ini-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "tializing a model with SC criterion and continuing with a fully-supervised regime over all labeled", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 383, + 504, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 504, + 395 + ], + "score": 1.0, + "content": "examples. We found again that this provided benefit over training the same network without pre-", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 254, + 407 + ], + "score": 1.0, + "content": "initialization, improving results from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 255, + 394, + 282, + 405 + ], + "score": 0.89, + "content": "{ \\bar { 0 } } . 6 3 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 282, + 394, + 292, + 407 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 293, + 394, + 320, + 405 + ], + "score": 0.9, + "content": "0 . 3 4 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 321, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "error on test set. As mentioned previously, the", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "effective compared patches of MNIST covered almost the entire image area. This can be attributed", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "to the fact that MNIST requires global features to differentiate between digits. The results, compared", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 426, + 302, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 302, + 441 + ], + "score": 1.0, + "content": "with previous attempts are included in Table (3).", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 676, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 101, + 507, + 228 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 140, + 89, + 470, + 100 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 87, + 470, + 102 + ], + "spans": [ + { + "bbox": [ + 140, + 87, + 470, + 102 + ], + "score": 1.0, + "content": "Table 2: State of the art results on Cifar10 dataset with only 4000 labeled samples", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 105, + 101, + 507, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 101, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 507, + 228 + ], + "score": 0.982, + "html": "
ModelCifar10 (400 per class) test accuracy
Convolutional K-means Network - Coates & Ng (2012)70.7%
View-Invariant K-means -Hui (2013)72.6%
DCGAN - Radford et al. (2015)73.8%
Exemplar Convnets - Dosovitskiy et al. (2014)76.6%
Ladder networks - Rasmus et al. (2015)79.6%
Conv-CatGan Springenberg (2016)80.42% (± 0.58)
ImprovedGan Salimans et al. (2016)81.37% (± 2.32)
Spatial contrasting initialization (this work)79.2%(±0.3)
The same model without initialization72.4%(±0.1)
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ModelMNIST test error
Stacked what-where AE - Zhao et al. (2015)0.71%
Triplet network - Hoffer & Ailon (2015)0.56%
Jarrett et al. (2009)0.53%
Ladder networks - Rasmus et al. (2015)0.36%
DropConnect - Wan et al. (2013)0.21%
Spatial contrasting initialization (this work)0.34%± 0.02
The same model without initialization0.63%± 0.02
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We found again that this provided benefit over training the same network without pre-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 254, + 407 + ], + "score": 1.0, + "content": "initialization, improving results from", + "type": "text" + }, + { + "bbox": [ + 255, + 394, + 282, + 405 + ], + "score": 0.89, + "content": "{ \\bar { 0 } } . 6 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 394, + 292, + 407 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 293, + 394, + 320, + 405 + ], + "score": 0.9, + "content": "0 . 3 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "error on test set. As mentioned previously, the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "effective compared patches of MNIST covered almost the entire image area. This can be attributed", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "to the fact that MNIST requires global features to differentiate between digits. The results, compared", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 426, + 302, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 302, + 441 + ], + "score": 1.0, + "content": "with previous attempts are included in Table (3).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 456, + 305, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 306, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 306, + 470 + ], + "score": 1.0, + "content": "6 CONCLUSIONS AND FUTURE WORK", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "score": 1.0, + "content": "In this work we presented spatial contrasting - a novel unsupervised criterion for training convo-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "lutional networks on unlabeled data. 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ModelCifar10 (400 per class) test accuracy
Convolutional K-means Network - Coates & Ng (2012)70.7%
View-Invariant K-means -Hui (2013)72.6%
DCGAN - Radford et al. (2015)73.8%
Exemplar Convnets - Dosovitskiy et al. (2014)76.6%
Ladder networks - Rasmus et al. (2015)79.6%
Conv-CatGan Springenberg (2016)80.42% (± 0.58)
ImprovedGan Salimans et al. (2016)81.37% (± 2.32)
Spatial contrasting initialization (this work)79.2%(±0.3)
The same model without initialization72.4%(±0.1)
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ModelMNIST test error
Stacked what-where AE - Zhao et al. (2015)0.71%
Triplet network - Hoffer & Ailon (2015)0.56%
Jarrett et al. (2009)0.53%
Ladder networks - Rasmus et al. (2015)0.36%
DropConnect - Wan et al. (2013)0.21%
Spatial contrasting initialization (this work)0.34%± 0.02
The same model without initialization0.63%± 0.02
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Model
STL10CIFAR-10MNIST
Input: 96×96 RGBInput:32×32RGBInput:28 × 28 monochrome
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1 ×1 conv.160 BN ReLU3 ×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
1 ×1 conv. 96 BN ReLU
3 × 3 max-pooling, stride 2
5 × 5 conv. 192 BN ReLU
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
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3×3 conv.256 ReLU3 × 3 max-pooling,stride 2dropout, p = 0.53 × 3 conv.128 ReLUdropout, p = 0.5fully-connected 103×3 conv.192BNLeakyReLU1 ×1 conv.192 BNLeakyReLU1 ×1 conv.10 BNLeakyReLUglobal average pooling3×3 conv.128 BN ReLU1 × 1 conv.10 BN ReLUglobal average pooling
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Model
STL10CIFAR-10MNIST
Input: 96×96 RGBInput:32×32RGBInput:28 × 28 monochrome
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1 ×1 conv.160 BN ReLU3 ×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
1 ×1 conv. 96 BN ReLU
3 × 3 max-pooling, stride 2
5 × 5 conv. 192 BN ReLU
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
Spatial contrasting criterion
3×3 conv.256 ReLU3 × 3 max-pooling,stride 2dropout, p = 0.53 × 3 conv.128 ReLUdropout, p = 0.5fully-connected 103×3 conv.192BNLeakyReLU1 ×1 conv.192 BNLeakyReLU1 ×1 conv.10 BNLeakyReLUglobal average pooling3×3 conv.128 BN ReLU1 × 1 conv.10 BN ReLUglobal average pooling
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ModelSTL-10 test accuracy
Zero-bias Convnets - Paine et al. (2014)70.2%
Triplet network -Hoffer & Ailon (2015)70.7%
Exemplar Convnets - Dosovitskiy et al. (2014)72.8%
Target Coding - Yang et al. (2015)73.15%
Stacked what-where AE - Zhao et al. (2015)74.33%
Spatial contrasting initialization (this work)81.34% ± 0.1
The same model without initialization72.6%±0.1
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ModelCifar10 (400 per class) test accuracy
Convolutional K-means Network - Coates & Ng (2012)70.7%
View-Invariant K-means -Hui (2013)72.6%
DCGAN - Radford et al. (2015)73.8%
Exemplar Convnets - Dosovitskiy et al. (2014)76.6%
Ladder networks - Rasmus et al. (2015)79.6%
Conv-CatGan Springenberg (2016)80.42% (± 0.58)
ImprovedGan Salimans et al. (2016)81.37% (± 2.32)
Spatial contrasting initialization (this work)79.2%(±0.3)
The same model without initialization72.4%(±0.1)
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ModelMNIST test error
Stacked what-where AE - Zhao et al. (2015)0.71%
Triplet network - Hoffer & Ailon (2015)0.56%
Jarrett et al. (2009)0.53%
Ladder networks - Rasmus et al. (2015)0.36%
DropConnect - Wan et al. (2013)0.21%
Spatial contrasting initialization (this work)0.34%± 0.02
The same model without initialization0.63%± 0.02
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Model
STL10CIFAR-10MNIST
Input: 96×96 RGBInput:32×32RGBInput:28 × 28 monochrome
5 ×5 conv. 64 BN ReLU3 × 3 conv. 96 BN LeakyReLU5 ×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
1 ×1 conv.160 BN ReLU3 ×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
1 ×1 conv. 96 BN ReLU
3 × 3 max-pooling, stride 2
5 × 5 conv. 192 BN ReLU
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
Spatial contrasting criterion
3×3 conv.256 ReLU3 × 3 max-pooling,stride 2dropout, p = 0.53 × 3 conv.128 ReLUdropout, p = 0.5fully-connected 103×3 conv.192BNLeakyReLU1 ×1 conv.192 BNLeakyReLU1 ×1 conv.10 BNLeakyReLUglobal average pooling3×3 conv.128 BN ReLU1 × 1 conv.10 BN ReLUglobal average pooling
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100644 index 0000000000000000000000000000000000000000..370c11a8d2c6ccae83d8c909880104fbd4508a36 --- /dev/null +++ b/parse/train/S1x1IkHtPr/S1x1IkHtPr.md @@ -0,0 +1,346 @@ +# 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. + +
Median MMDMean Ranking
RDKITDL4CHEMGRAPHDGRDKITDL4CHEMGRAPHDG
p(dk|9)0.55 (0.01)1.11 (0.01)0.38 (0.02)1.71 (0.03)2.74 (0.02)1.51 (0.03)
p(dk,di/9)0.53 (0.01)1.09 ( (0.01)0.34 (0.01)1.66 (0.02)2.92 (0.01)1.43 ( (0.02)
p({d}9）0.60 (0.01)1.07 (0.03)0.44 (0.05)1.58 (0.05)2.90 (0.05)1.45 (0.02)
+ +![](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. + +
RDKITGRAPHDG
Eelec42.7 (4.3)58.0 (21.0)
€HOMO0.08 (0.04)0.10 (0.05)
ELUMO0.15 (0.03)0.09 (0.05)
从0.29 (0.05)0.33 (0.09)
+ +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. 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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. + +
FeatureData TypeDimension
atomic numberinteger1
chiral tagone-hot (R, S,and N/A)3
+ +Table 4: Edge features. + +
FeatureData TypeDimension
kindone-hot (indicating whether e is in Ebond,Eangle,or Edihedral)3
stereo chemistryone-hot (E,Z,Any,None, and N/A)5
typeinteger (single, double, triple or N/A)1
is aromaticbinary1
is conjugatedbinary1
is in ring of sizeone-hot (3,4,...,9) and N/A8
+ +# 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 \ No newline at end of file diff --git a/parse/train/S1x1IkHtPr/S1x1IkHtPr_content_list.json b/parse/train/S1x1IkHtPr/S1x1IkHtPr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2fa8de38e73178e697c240e218316d91be5d2193 --- /dev/null +++ b/parse/train/S1x1IkHtPr/S1x1IkHtPr_content_list.json @@ -0,0 +1,1808 @@ +[ + { + "type": "text", + "text": "A GENERATIVE MODEL FOR MOLECULAR DISTANCE GEOMETRY ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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": [ + 233, + 267, + 764, + 420 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 446, + 336, + 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": [ + 173, + 479, + 825, + 688 + ], + "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": [ + 174, + 695, + 825, + 862 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This work includes the following key contributions: ", + "bbox": [ + 176, + 869, + 513, + 883 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• We introduce a novel probabilistic model for learning conformational distributions of molecules with graph neural networks. ", + "bbox": [ + 215, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "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": [], + "bbox": [ + 173, + 99, + 813, + 198 + ], + "page_idx": 1 + }, + { + "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": [ + 214, + 308, + 825, + 462 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 487, + 282, + 503 + ], + "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": [ + 173, + 518, + 825, + 631 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 EXTENDED MOLECULAR GRAPHS AND DISTANCE GEOMETRY ", + "text_level": 1, + "bbox": [ + 174, + 647, + 645, + 662 + ], + "page_idx": 1 + }, + { + "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": [ + 174, + 674, + 825, + 762 + ], + "page_idx": 1 + }, + { + "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": [ + 173, + 768, + 825, + 924 + ], + "page_idx": 1 + }, + { + "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, + 821, + 228 + ], + "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": [ + 174, + 353, + 825, + 424 + ], + "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": [ + 176, + 429, + 823, + 477 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 GENERATIVE MODEL ", + "text_level": 1, + "bbox": [ + 176, + 497, + 362, + 512 + ], + "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, + 580 + ], + "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, + 825, + 657 + ], + "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, + 804 + ], + "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, + 810, + 825, + 854 + ], + "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. " + ], + "image_footnote": [], + "bbox": [ + 196, + 99, + 825, + 327 + ], + "page_idx": 3 + }, + { + "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": [ + 173, + 396, + 825, + 487 + ], + "page_idx": 3 + }, + { + "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": [ + 245, + 497, + 751, + 568 + ], + "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": [ + 171, + 578, + 826, + 683 + ], + "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": [ + 245, + 694, + 751, + 765 + ], + "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": [ + 173, + 775, + 826, + 823 + ], + "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, + 854 + ], + "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": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 CONFORMATION GENERATION THROUGH EUCLIDEAN DISTANCE GEOMETRY", + "text_level": 1, + "bbox": [ + 174, + 103, + 746, + 118 + ], + "page_idx": 4 + }, + { + "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": [ + 174, + 128, + 823, + 174 + ], + "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": [ + 173, + 180, + 825, + 327 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 CALCULATION OF MOLECULAR PROPERTIES", + "text_level": 1, + "bbox": [ + 178, + 342, + 524, + 356 + ], + "page_idx": 4 + }, + { + "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": [ + 173, + 367, + 825, + 525 + ], + "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, + 676, + 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": [ + 174, + 577, + 823, + 604 + ], + "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, + 637, + 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": [ + 174, + 648, + 825, + 680 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 176, + 699, + 354, + 715 + ], + "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": [ + 173, + 729, + 825, + 800 + ], + "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, + 808, + 821, + 835 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "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, + 825, + 277 + ], + "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, + 285, + 825, + 354 + ], + "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": [ + 174, + 361, + 825, + 445 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We argue that our model has several advantages over the approaches reviewed above: ", + "bbox": [ + 174, + 452, + 730, + 467 + ], + "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": [ + 214, + 482, + 825, + 578 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 THE CONF17 BENCHMARK ", + "text_level": 1, + "bbox": [ + 176, + 607, + 431, + 623 + ], + "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": [ + 174, + 643, + 825, + 755 + ], + "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": [ + 174, + 762, + 825, + 845 + ], + "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": [ + 205, + 98, + 784, + 377 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 465, + 326, + 482 + ], + "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": [ + 174, + 501, + 825, + 585 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 DISTRIBUTIONS OVER DISTANCES ", + "text_level": 1, + "bbox": [ + 176, + 609, + 454, + 625 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 173, + 638, + 825, + 819 + ], + "page_idx": 6 + }, + { + "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 $p ( d _ { i } , d _ { j } | \\mathcal { G } )$ . ", + "bbox": [ + 174, + 827, + 825, + 924 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 173, + 101, + 825, + 185 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e83dbdf3c3547b9f3252fec33f50c007fb77c3c8439737f6fe4f14904c384495.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Median MMDMean Ranking
RDKITDL4CHEMGRAPHDGRDKITDL4CHEMGRAPHDG
p(dk|9)0.55 (0.01)1.11 (0.01)0.38 (0.02)1.71 (0.03)2.74 (0.02)1.51 (0.03)
p(dk,di/9)0.53 (0.01)1.09 ( (0.01)0.34 (0.01)1.66 (0.02)2.92 (0.01)1.43 ( (0.02)
p({d}9）0.60 (0.01)1.07 (0.03)0.44 (0.05)1.58 (0.05)2.90 (0.05)1.45 (0.02)
", + "bbox": [ + 174, + 195, + 820, + 272 + ], + "page_idx": 7 + }, + { + "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": [], + "bbox": [ + 223, + 289, + 777, + 505 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 GENERATION OF CONFORMATIONS ", + "text_level": 1, + "bbox": [ + 178, + 603, + 457, + 617 + ], + "page_idx": 7 + }, + { + "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. ", + "bbox": [ + 173, + 630, + 825, + 782 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 CALCULATION OF MOLECULAR PROPERTIES", + "text_level": 1, + "bbox": [ + 174, + 801, + 524, + 815 + ], + "page_idx": 7 + }, + { + "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 ", + "bbox": [ + 174, + 828, + 825, + 883 + ], + "page_idx": 7 + }, + { + "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": [ + 179, + 102, + 823, + 297 + ], + "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. ", + "bbox": [ + 173, + 353, + 825, + 410 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/28fdddd3902b37a3a8fedd86b4e37bef3145c3d79b8a57f2f80e6e8539ef3e2e.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
RDKITGRAPHDG
Eelec42.7 (4.3)58.0 (21.0)
€HOMO0.08 (0.04)0.10 (0.05)
ELUMO0.15 (0.03)0.09 (0.05)
从0.29 (0.05)0.33 (0.09)
", + "bbox": [ + 370, + 420, + 625, + 501 + ], + "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. ", + "bbox": [ + 174, + 529, + 825, + 599 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 LIMITATIONS ", + "text_level": 1, + "bbox": [ + 174, + 622, + 316, + 638 + ], + "page_idx": 8 + }, + { + "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": [ + 174, + 656, + 825, + 713 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 737, + 328, + 752 + ], + "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. ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 287, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "F. H. Allen. The Cambridge Structural Database: A quarter of a million crystal structures and rising. Acta Crystallogr., Sect. B: Struct. Sci, 58(3):380–388, 2002. doi: 10.1107/S0108768102003890. 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", + "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": "
FeatureData TypeDimension
atomic numberinteger1
chiral tagone-hot (R, S,and N/A)3
", + "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": "
FeatureData TypeDimension
kindone-hot (indicating whether e is in Ebond,Eangle,or Edihedral)3
stereo chemistryone-hot (E,Z,Any,None, and N/A)5
typeinteger (single, double, triple or N/A)1
is aromaticbinary1
is conjugatedbinary1
is in ring of sizeone-hot (3,4,...,9) and N/A8
", + "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, + 825, + 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, + 252, + 800, + 733 + ], + "page_idx": 21 + } +] \ No newline at end of file diff --git a/parse/train/S1x1IkHtPr/S1x1IkHtPr_middle.json b/parse/train/S1x1IkHtPr/S1x1IkHtPr_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..b0001a6babf185f02788804ac21c5614438723ac --- /dev/null +++ b/parse/train/S1x1IkHtPr/S1x1IkHtPr_middle.json @@ -0,0 +1,41154 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "score": 1.0, + "content": "A GENERATIVE MODEL FOR MOLECULAR DISTANCE", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 98, + 194, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 98, + 194, + 117 + ], + "score": 1.0, + "content": "GEOMETRY", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 212, + 468, + 333 + ], + "lines": [ + { + "bbox": [ + 142, + 212, + 470, + 225 + ], + "spans": [ + { + "bbox": [ + 142, + 212, + 470, + 225 + ], + "score": 1.0, + "content": "Computing equilibrium states for many-body systems, such as molecules, is a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 222, + 469, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 469, + 236 + ], + "score": 1.0, + "content": "long-standing challenge. 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These methods operate using graph representations of molecules in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "score": 1.0, + "content": "which nodes and edges represent atoms and bonds, respectively. A representation that is closer to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "the physical system is one in which a molecule is described by its geometry or conformation. 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Under a wide", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 253, + 525 + ], + "score": 1.0, + "content": "range of conditions, the probability", + "type": "text" + }, + { + "bbox": [ + 253, + 512, + 273, + 524 + ], + "score": 0.9, + "content": "p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 513, + 350, + 525 + ], + "score": 1.0, + "content": "of a conformation", + "type": "text" + }, + { + "bbox": [ + 351, + 514, + 358, + 522 + ], + "score": 0.31, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 513, + 505, + 525 + ], + "score": 1.0, + "content": ", is governed by the Boltzmann dis-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 522, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 232, + 537 + ], + "score": 1.0, + "content": "tribution and is proportional to", + "type": "text" + }, + { + "bbox": [ + 232, + 523, + 312, + 535 + ], + "score": 0.91, + "content": "\\exp \\{ - E ( \\mathbf { x } ) / k _ { B } T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 522, + 343, + 537 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 343, + 523, + 385, + 535 + ], + "score": 0.93, + "content": "E ( \\mathbf { x } ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 522, + 506, + 537 + ], + "score": 1.0, + "content": "is the conformation’s energy,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 534, + 332, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 120, + 545 + ], + "score": 0.88, + "content": "k _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 535, + 246, + 546 + ], + "score": 1.0, + "content": "is the Boltzmann constant, and", + "type": "text" + }, + { + "bbox": [ + 246, + 535, + 255, + 544 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 535, + 332, + 546 + ], + "score": 1.0, + "content": "is the temperature.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 392, + 565 + ], + "score": 1.0, + "content": "To compute a molecular property for a molecule, one must sample from", + "type": "text" + }, + { + "bbox": [ + 393, + 551, + 412, + 563 + ], + "score": 0.91, + "content": "p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 550, + 506, + 565 + ], + "score": 1.0, + "content": ". The main approach is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "to start with one conformation and make small changes to it over time, e.g., by using Markov chain", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "Monte Carlo (MCMC) or molecular dynamics (MD). 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In the absence of methods for generating statistically in-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 233, + 469, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 247 + ], + "score": 1.0, + "content": "dependent samples, great computational effort is invested in simulating these sys-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 469, + 257 + ], + "score": 1.0, + "content": "tems using, for example, Markov chain Monte Carlo. We present a probabilistic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "score": 1.0, + "content": "model that generates such samples for molecules from their graph representations.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 267, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 470, + 279 + ], + "score": 1.0, + "content": "Our model learns a low-dimensional manifold that preserves the geometry of local", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "score": 1.0, + "content": "atomic neighborhoods through a principled learning representation that is based", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "on Euclidean distance geometry. In a new benchmark for molecular conformation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 311 + ], + "score": 1.0, + "content": "generation, we show experimentally that our generative model achieves state-of-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 311, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 142, + 311, + 469, + 322 + ], + "score": 1.0, + "content": "the-art accuracy. 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These methods operate using graph representations of molecules in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "score": 1.0, + "content": "which nodes and edges represent atoms and bonds, respectively. A representation that is closer to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "the physical system is one in which a molecule is described by its geometry or conformation. 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Under a wide", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 253, + 525 + ], + "score": 1.0, + "content": "range of conditions, the probability", + "type": "text" + }, + { + "bbox": [ + 253, + 512, + 273, + 524 + ], + "score": 0.9, + "content": "p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 513, + 350, + 525 + ], + "score": 1.0, + "content": "of a conformation", + "type": "text" + }, + { + "bbox": [ + 351, + 514, + 358, + 522 + ], + "score": 0.31, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 513, + 505, + 525 + ], + "score": 1.0, + "content": ", is governed by the Boltzmann dis-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 522, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 232, + 537 + ], + "score": 1.0, + "content": "tribution and is proportional to", + "type": "text" + }, + { + "bbox": [ + 232, + 523, + 312, + 535 + ], + "score": 0.91, + "content": "\\exp \\{ - E ( \\mathbf { x } ) / k _ { B } T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 522, + 343, + 537 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 343, + 523, + 385, + 535 + ], + "score": 0.93, + "content": "E ( \\mathbf { x } ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 522, + 506, + 537 + ], + "score": 1.0, + "content": "is the conformation’s energy,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 534, + 332, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 120, + 545 + ], + "score": 0.88, + "content": "k _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 535, + 246, + 546 + ], + "score": 1.0, + "content": "is the Boltzmann constant, and", + "type": "text" + }, + { + "bbox": [ + 246, + 535, + 255, + 544 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 535, + 332, + 546 + ], + "score": 1.0, + "content": "is the temperature.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 24, + "bbox_fs": [ + 103, + 379, + 507, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 392, + 565 + ], + "score": 1.0, + "content": "To compute a molecular property for a molecule, one must sample from", + "type": "text" + }, + { + "bbox": [ + 393, + 551, + 412, + 563 + ], + "score": 0.91, + "content": "p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 550, + 506, + 565 + ], + "score": 1.0, + "content": ". The main approach is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "to start with one conformation and make small changes to it over time, e.g., by using Markov chain", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "Monte Carlo (MCMC) or molecular dynamics (MD). These methods can be used to accurately", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "sample equilibrium states of molecules, but they become computationally expensive for larger ones", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "(Shim & MacKerell, 2011; Ballard et al., 2015; De Vivo et al., 2016). Other heuristic approaches ex-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 604, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 505, + 619 + ], + "score": 1.0, + "content": "ist in which distances between atoms are set to fixed idealized values (Havel, 2002; Blaney & Dixon,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "score": 1.0, + "content": "2007). Several methods based on statistical learning have also recently been developed to tackle the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "issue of conformation generation. However, they are mainly geared towards studying proteins and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 652 + ], + "score": 1.0, + "content": "their folding dynamics (AlQuraishi, 2019). Some of these models are not targeting a distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "over conformations but the most stable folded configuration (Evans et al., 2018; Ingraham et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 661, + 504, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 504, + 672 + ], + "score": 1.0, + "content": "2019), while others are not transferable between different molecules (Lemke & Peter, 2019; Noe´", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 672, + 159, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 159, + 684 + ], + "score": 1.0, + "content": "et al., 2019).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 550, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 314, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 316, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 316, + 702 + ], + "score": 1.0, + "content": "This work includes the following key contributions:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 687, + 316, + 702 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 133, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "• We introduce a novel probabilistic model for learning conformational distributions of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 720, + 299, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 720, + 299, + 733 + ], + "score": 1.0, + "content": "molecules with graph neural networks.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 133, + 709, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 79, + 498, + 157 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 79, + 498, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 498, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 498, + 157 + ], + "score": 0.941, + "type": "image", + "image_path": "434d845d7c8ccab7aba98bd6f5f40a3ddc2c0a355bf36f4ceacd18ef6cbb7a0b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 79, + 498, + 105.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 105.0, + 498, + 131.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 131.0, + 498, + 157.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 166, + 505, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "Figure 1: Standard graph representation of a molecule (left) with a set of possible conformations", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 107, + 178, + 127, + 190 + ], + "score": 0.91, + "content": "\\{ { \\bf { x } } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "(right). 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A CVAE first encodes", + "type": "text" + }, + { + "bbox": [ + 357, + 428, + 365, + 438 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "together with d into a latent space", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 436, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 151, + 449 + ], + "score": 0.91, + "content": "\\mathbf { z } ~ \\in ~ \\mathbb { R } ^ { k N _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 436, + 185, + 452 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 185, + 438, + 222, + 449 + ], + "score": 0.91, + "content": "k \\in \\mathbb { N } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 436, + 298, + 452 + ], + "score": 1.0, + "content": ", with an encoder", + "type": "text" + }, + { + "bbox": [ + 298, + 438, + 341, + 451 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 436, + 460, + 452 + ], + "score": 1.0, + "content": ". Subsequently, the decoder", + "type": "text" + }, + { + "bbox": [ + 460, + 438, + 504, + 450 + ], + "score": 0.93, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 448, + 426, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 141, + 461 + ], + "score": 1.0, + "content": "decodes", + "type": "text" + }, + { + "bbox": [ + 141, + 451, + 148, + 459 + ], + "score": 0.53, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 448, + 426, + 461 + ], + "score": 1.0, + "content": "back into a set of distances. A graphical model is shown in Fig. 2 C).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 240, + 478 + ], + "score": 1.0, + "content": "A conformation has, in general,", + "type": "text" + }, + { + "bbox": [ + 241, + 466, + 279, + 477 + ], + "score": 0.89, + "content": "3 N _ { v } - 6", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "spatial degrees of freedom (dofs): one dof per spacial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 475, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 491 + ], + "score": 1.0, + "content": "dimension per atom minus three translational and three rotational dofs. 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In addition, the latent space should be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 157, + 511 + ], + "score": 1.0, + "content": "smaller than", + "type": "text" + }, + { + "bbox": [ + 157, + 499, + 176, + 510 + ], + "score": 0.9, + "content": "3 N _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "as it is the role of the encoder to project the conformation into a lower-dimensional", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 510, + 317, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 209, + 523 + ], + "score": 1.0, + "content": "space. 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The MPNN is a well-studied technique that achieves", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "state-of-the-art performance in representation learning for molecules (Kipf & Welling, 2017; Duve-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "naud et al., 2015; Kearnes et al., 2016; Schutt et al., 2017b; Gilmer et al., 2017; Kusner et al., 2017; ¨", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 624, + 205, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 205, + 637 + ], + "score": 1.0, + "content": "Bradshaw et al., 2019a).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "In the following, we describe the details of the model.2 In Fig. 3, an illustration of the model is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 195, + 666 + ], + "score": 1.0, + "content": "shown. 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A CVAE first encodes", + "type": "text" + }, + { + "bbox": [ + 357, + 428, + 365, + 438 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "together with d into a latent space", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 436, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 151, + 449 + ], + "score": 0.91, + "content": "\\mathbf { z } ~ \\in ~ \\mathbb { R } ^ { k N _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 436, + 185, + 452 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 185, + 438, + 222, + 449 + ], + "score": 0.91, + "content": "k \\in \\mathbb { N } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 436, + 298, + 452 + ], + "score": 1.0, + "content": ", with an encoder", + "type": "text" + }, + { + "bbox": [ + 298, + 438, + 341, + 451 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 436, + 460, + 452 + ], + "score": 1.0, + "content": ". Subsequently, the decoder", + "type": "text" + }, + { + "bbox": [ + 460, + 438, + 504, + 450 + ], + "score": 0.93, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 448, + 426, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 141, + 461 + ], + "score": 1.0, + "content": "decodes", + "type": "text" + }, + { + "bbox": [ + 141, + 451, + 148, + 459 + ], + "score": 0.53, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 448, + 426, + 461 + ], + "score": 1.0, + "content": "back into a set of distances. A graphical model is shown in Fig. 2 C).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 416, + 505, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 240, + 478 + ], + "score": 1.0, + "content": "A conformation has, in general,", + "type": "text" + }, + { + "bbox": [ + 241, + 466, + 279, + 477 + ], + "score": 0.89, + "content": "3 N _ { v } - 6", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "spatial degrees of freedom (dofs): one dof per spacial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 475, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 491 + ], + "score": 1.0, + "content": "dimension per atom minus three translational and three rotational dofs. 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In addition, the latent space should be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 157, + 511 + ], + "score": 1.0, + "content": "smaller than", + "type": "text" + }, + { + "bbox": [ + 157, + 499, + 176, + 510 + ], + "score": 0.9, + "content": "3 N _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "as it is the role of the encoder to project the conformation into a lower-dimensional", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 510, + 317, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 209, + 523 + ], + "score": 1.0, + "content": "space. As a result, we set", + "type": "text" + }, + { + "bbox": [ + 209, + 510, + 235, + 520 + ], + "score": 0.91, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 510, + 317, + 523 + ], + "score": 1.0, + "content": "to avoid overfitting.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 466, + 505, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 526, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 132, + 539 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 132, + 527, + 176, + 539 + ], + "score": 0.92, + "content": "q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 527, + 196, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 527, + 240, + 538 + ], + "score": 0.93, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 527, + 506, + 539 + ], + "score": 1.0, + "content": "are Gaussian distributions, the mean and variance of which are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "modeled by two artificial neural networks. At the center of this model are message-passing neural", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "networks (MPNNs) (Gilmer et al., 2017) with multi-head attention (Velickovi ˇ c et al., 2018). In short, ´", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "an MPNN is a convolutional neural network that allows end-to-end learning of prediction pipelines", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "whose inputs are graphs of arbitrary size and shape. In a convolution, neighboring nodes exchange", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "so-called messages between neighbors to update their attributes. Edges update their attributes with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "the features of the nodes they are connecting. 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Since we cannot draw", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 163, + 337 + ], + "score": 1.0, + "content": "samples from", + "type": "text" + }, + { + "bbox": [ + 164, + 325, + 192, + 336 + ], + "score": 0.93, + "content": "p ( \\mathbf { x } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "directly, we employ an IS integration scheme (Bishop, 2009) with our CVAE", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "as the proposal distribution. 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Since the EDG algorithm is map-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 190, + 381 + ], + "score": 1.0, + "content": "ping the distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 369, + 233, + 380 + ], + "score": 0.93, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 367, + 307, + 381 + ], + "score": 1.0, + "content": "to a point mass in", + "type": "text" + }, + { + "bbox": [ + 307, + 368, + 330, + 379 + ], + "score": 0.9, + "content": " { \\mathbb { R } } ^ { 3 N _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 367, + 506, + 381 + ], + "score": 1.0, + "content": ", the MC estimate for the resulting distribu-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 378, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 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\\mathbf { x } - \\mathbf { x } _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 388, + 507, + 410 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 404, + 376, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 263, + 417 + ], + "score": 1.0, + "content": "The IS estimator for the expectation of", + "type": "text" + }, + { + "bbox": [ + 264, + 405, + 273, + 414 + ], + "score": 0.82, + "content": "\\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 404, + 301, + 417 + ], + "score": 1.0, + "content": "w. r. t.", + "type": "text" + }, + { + "bbox": [ + 301, + 405, + 330, + 416 + ], + "score": 0.91, + "content": "\\tilde { p } ( \\mathbf { x } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 404, + 376, + 417 + ], + "score": 1.0, + "content": "then reads", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": 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When dividing two delta functions wee arbitrarily large finite value.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 554, + 217, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 218, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 218, + 569 + ], + "score": 1.0, + "content": "3 RELATED WORKS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "The standard approach for generating molecular conformations is to start with one, and make small", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "changes to it over time, e.g., by using MCMC or MD. These methods are considered the gold", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "standard for sampling equilibrium states, but they are computationally expensive, especially if the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "molecule is large and the Hamiltonian is based on quantum-mechanical principles (Shim & MacK-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 622, + 323, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 323, + 636 + ], + "score": 1.0, + "content": "erell, 2011; Ballard et al., 2015; De Vivo et al., 2016).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 503, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 652 + ], + "score": 1.0, + "content": "A much faster but more approximate approach for conformation generation is EDG (Havel, 2002;", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "Blaney & Dixon, 2007; Lagorce et al., 2009; Riniker & Landrum, 2015). Lower and upper distance", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 669, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 667, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 118, + 667, + 505, + 683 + ], + "score": 1.0, + "content": "3Even though quantum-chemical methods require the input to be in Cartesian coordinates, calculated prop-", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 680, + 357, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 357, + 691 + ], + "score": 1.0, + "content": "erties, such as the energy, are invariant under translation and rotation.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 689, + 416, + 703 + ], + "score": 1.0, + "content": "4There are additional constraints due to chirality. However, since they are given by", + "type": "text" + }, + { + "bbox": [ + 416, + 691, + 423, + 700 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "and are fixed, they are", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 701, + 210, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 210, + 712 + ], + "score": 1.0, + "content": "not modeled by our method.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "5Often there exist multiple solutions for the same set of bounds. As the bounds are generally tight, the", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 452, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 452, + 732 + ], + "score": 1.0, + "content": "solutions are very similar. Therefore, we only generate one set of coordinates per set of bounds.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 457, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 460, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 460, + 95 + ], + "score": 1.0, + "content": "2.3 CONFORMATION GENERATION THROUGH EUCLIDEAN DISTANCE GEOMETRY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 504, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 506, + 116 + ], + "score": 1.0, + "content": "To compute molecular properties, quantum-chemical methods need to be employed which require", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "the input, i.e., the molecule, to be in Cartesian coordinates.3 Therefore, we use an EDG algorithm", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 102, + 120, + 428, + 144 + ], + "spans": [ + { + "bbox": [ + 102, + 120, + 231, + 144 + ], + "score": 1.0, + "content": "to translate the set of distances", + "type": "text" + }, + { + "bbox": [ + 231, + 125, + 267, + 139 + ], + "score": 0.93, + "content": "\\{ d _ { k } \\} _ { k = 1 } ^ { N _ { e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 120, + 388, + 144 + ], + "score": 1.0, + "content": "to a set of atomic coordinates", + "type": "text" + }, + { + "bbox": [ + 388, + 125, + 420, + 139 + ], + "score": 0.92, + "content": "\\{ \\mathbf { r } _ { i } \\} _ { i = 1 } ^ { N _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 120, + 428, + 144 + ], + "score": 1.0, + "content": ". 4", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 102, + 101, + 506, + 144 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "EDG is the mathematical basis for a geometric theory of molecular conformation. In the field of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "machine learning, Weinberger & Saul (2006) used it for learning image manifolds, Tenenbaum", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "et al. (2000) for image understanding and handwriting recognition, Jain & Saul (2004) for speech", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "and music, and Demaine et al. (2009) for music and musical rhythms. An EDG description of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "a molecular system consists of a list of lower and upper bounds on the distances between pairs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 197, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 146, + 216 + ], + "score": 1.0, + "content": "of atoms", + "type": "text" + }, + { + "bbox": [ + 146, + 198, + 231, + 211 + ], + "score": 0.91, + "content": "\\{ ( d _ { k , \\operatorname* { m i n } } , d _ { k , \\operatorname* { m a x } } ) \\} _ { k = 1 } ^ { N _ { e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 197, + 264, + 216 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + }, + { + "bbox": [ + 265, + 199, + 308, + 212 + ], + "score": 0.92, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 197, + 506, + 216 + ], + "score": 1.0, + "content": "is used to model these bounds, namely, we set", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 208, + 507, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 164, + 226 + ], + "score": 1.0, + "content": "the bounds to", + "type": "text" + }, + { + "bbox": [ + 165, + 211, + 272, + 223 + ], + "score": 0.9, + "content": "\\{ ( \\mu _ { d _ { k } } - \\sigma _ { d _ { k } } , \\mu _ { d _ { k } } + \\sigma _ { d _ { k } } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 208, + 304, + 226 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 304, + 213, + 320, + 222 + ], + "score": 0.88, + "content": "\\mu _ { d _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 208, + 339, + 226 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 212, + 355, + 222 + ], + "score": 0.88, + "content": "\\sigma _ { d _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 208, + 507, + 226 + ], + "score": 1.0, + "content": "are the mean and standard deviation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 220, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 179, + 235 + ], + "score": 1.0, + "content": "for each distance", + "type": "text" + }, + { + "bbox": [ + 179, + 222, + 191, + 232 + ], + "score": 0.86, + "content": "d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 220, + 506, + 235 + ], + "score": 1.0, + "content": "given by the CVAE. Then, an EDG algorithm determines a set of Cartesian", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 102, + 228, + 508, + 252 + ], + "spans": [ + { + "bbox": [ + 102, + 228, + 155, + 252 + ], + "score": 1.0, + "content": "coordinates", + "type": "text" + }, + { + "bbox": [ + 155, + 232, + 187, + 246 + ], + "score": 0.92, + "content": "\\{ \\mathbf { r } _ { i } \\} _ { i = 1 } ^ { N _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 228, + 508, + 252 + ], + "score": 1.0, + "content": "so that these bounds are fulfilled (see Appendix A.2 for details).5 Together with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 242, + 411, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 242, + 257, + 263 + ], + "score": 1.0, + "content": "the corresponding chemical elements", + "type": "text" + }, + { + "bbox": [ + 258, + 245, + 289, + 259 + ], + "score": 0.93, + "content": "\\{ \\epsilon _ { i } \\} _ { i = 1 } ^ { N _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 242, + 397, + 263 + ], + "score": 1.0, + "content": ", we obtain a conformation", + "type": "text" + }, + { + "bbox": [ + 397, + 249, + 405, + 257 + ], + "score": 0.5, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 242, + 411, + 263 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5, + "bbox_fs": [ + 102, + 143, + 508, + 263 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 271, + 321, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 322, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 322, + 284 + ], + "score": 1.0, + "content": "2.4 CALCULATION OF MOLECULAR PROPERTIES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 300, + 305 + ], + "score": 1.0, + "content": "We can get an MC estimate of the expectation", + "type": "text" + }, + { + "bbox": [ + 301, + 292, + 328, + 304 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mathcal { G } } [ \\mathcal { O } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 290, + 387, + 305 + ], + "score": 1.0, + "content": "of a property", + "type": "text" + }, + { + "bbox": [ + 388, + 293, + 397, + 302 + ], + "score": 0.79, + "content": "\\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 290, + 505, + 305 + ], + "score": 1.0, + "content": "(e.g., the dipole moment)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 302, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 230, + 317 + ], + "score": 1.0, + "content": "for a molecule represented by", + "type": "text" + }, + { + "bbox": [ + 230, + 303, + 238, + 314 + ], + "score": 0.81, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 302, + 386, + 317 + ], + "score": 1.0, + "content": "by drawing conformational samples", + "type": "text" + }, + { + "bbox": [ + 387, + 303, + 441, + 315 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { i } \\sim p ( \\mathbf { x } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 302, + 505, + 317 + ], + "score": 1.0, + "content": "and computing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 107, + 314, + 153, + 326 + ], + "score": 0.92, + "content": "\\mathcal { O } ( \\mathbf { x } _ { i } ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "with a quantum-chemical method (e.g., density functional theory). Since we cannot draw", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 163, + 337 + ], + "score": 1.0, + "content": "samples from", + "type": "text" + }, + { + "bbox": [ + 164, + 325, + 192, + 336 + ], + "score": 0.93, + "content": "p ( \\mathbf { x } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "directly, we employ an IS integration scheme (Bishop, 2009) with our CVAE", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "as the proposal distribution. We assume that we can readily evaluate the unnormalized probability of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 169, + 360 + ], + "score": 1.0, + "content": "a conformation", + "type": "text" + }, + { + "bbox": [ + 169, + 347, + 290, + 359 + ], + "score": 0.93, + "content": "\\tilde { p } ( \\mathbf { x } | \\mathcal { G } ) = \\exp \\{ - E ( \\mathbf { x } ) / k _ { B } T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 347, + 320, + 360 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 320, + 348, + 328, + 357 + ], + "score": 0.62, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "must be a conformation of the molecule and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 104, + 356, + 150, + 371 + ], + "score": 1.0, + "content": "the energy", + "type": "text" + }, + { + "bbox": [ + 151, + 358, + 173, + 370 + ], + "score": 0.91, + "content": "E ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 356, + 505, + 371 + ], + "score": 1.0, + "content": "is determined with a quantum-chemical method. Since the EDG algorithm is map-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 190, + 381 + ], + "score": 1.0, + "content": "ping the distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 369, + 233, + 380 + ], + "score": 0.93, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 367, + 307, + 381 + ], + "score": 1.0, + "content": "to a point mass in", + "type": "text" + }, + { + "bbox": [ + 307, + 368, + 330, + 379 + ], + "score": 0.9, + "content": " { \\mathbb { R } } ^ { 3 N _ { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 367, + 506, + 381 + ], + "score": 1.0, + "content": ", the MC estimate for the resulting distribu-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 378, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 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(A) Structural formulae of a random selection of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 319, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 319, + 286, + 336 + ], + "score": 1.0, + "content": "molecules. 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We generated 3000 conformations", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 453, + 315, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 315, + 465 + ], + "score": 1.0, + "content": "with each method for molecular graphs in a test set.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 483, + 278, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 280, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 280, + 497 + ], + "score": 1.0, + "content": "5.1 DISTRIBUTIONS OVER DISTANCES", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "We assessed the accuracy of the distance distributions of RDKIT, DL4CHEM, and GRAPHDG by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "score": 1.0, + "content": "calculating the maximum mean discrepancy (MMD) (Gretton et al., 2012) to the ground-truth dis-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "tribution. We compute the MMD using a Gaussian kernel, where we set the standard deviation to be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 264, + 552 + ], + "score": 1.0, + "content": "the median distance between distances", + "type": "text" + }, + { + "bbox": [ + 264, + 540, + 272, + 550 + ], + "score": 0.26, + "content": "\\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "in the aggregate sample. 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RDKIT is slightly worse than", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "GRAPHDG while DL4CHEM seems to struggle with the complexity of the molecules and the small", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 639, + 253, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 253, + 650 + ], + "score": 1.0, + "content": "number of graphs in the training set.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 457, + 667 + ], + "score": 1.0, + "content": "In Fig. 5, we showcase the accuracy of our model by plotting the marginal distributions", + "type": "text" + }, + { + "bbox": [ + 458, + 655, + 489, + 667 + ], + "score": 0.93, + "content": "p ( d _ { i } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "distances between C and O atoms given a molecular graph from a test set. 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We trained GRAPHDG and DL4CHEM on three different training and test splits", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "of the CONF17 benchmark using Adam (Kingma & Ba, 2014). We generated 3000 conformations", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 453, + 315, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 315, + 465 + ], + "score": 1.0, + "content": "with each method for molecular graphs in a test set.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 397, + 506, + 465 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 483, + 278, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 280, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 280, + 497 + ], + "score": 1.0, + "content": "5.1 DISTRIBUTIONS OVER DISTANCES", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "We assessed the accuracy of the distance distributions of RDKIT, DL4CHEM, and GRAPHDG by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "score": 1.0, + "content": "calculating the maximum mean discrepancy (MMD) (Gretton et al., 2012) to the ground-truth dis-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "tribution. We compute the MMD using a Gaussian kernel, where we set the standard deviation to be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 264, + 552 + ], + "score": 1.0, + "content": "the median distance between distances", + "type": "text" + }, + { + "bbox": [ + 264, + 540, + 272, + 550 + ], + "score": 0.26, + "content": "\\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "in the aggregate sample. For this, we determined the dis-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "tances in the conformations from the ground-truth and those generated by RDKIT and DL4CHEM.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 241, + 574 + ], + "score": 1.0, + "content": "For each train-test split and each", + "type": "text" + }, + { + "bbox": [ + 241, + 562, + 249, + 572 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "in a test set, we compute the MMD of the joint distribution of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 208, + 584 + ], + "score": 1.0, + "content": "distances between C and", + "type": "text" + }, + { + "bbox": [ + 208, + 573, + 217, + 582 + ], + "score": 0.43, + "content": "\\mathrm { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "atoms (H atoms are usually ignored), the MMDs of pair-wise distances", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 582, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 152, + 596 + ], + "score": 0.93, + "content": "p ( d _ { i } , d _ { j } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 582, + 409, + 597 + ], + "score": 1.0, + "content": ", and the MMDs between the marginals of individual distances", + "type": "text" + }, + { + "bbox": [ + 410, + 583, + 441, + 595 + ], + "score": 0.93, + "content": "p ( d _ { i } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 582, + 506, + 597 + ], + "score": 1.0, + "content": ". We aggregate", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "the results of three train-test splits, and, finally, compute the median MMDs and average rankings.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "The results are summarized in Table 1. It can be seen that the samples from GRAPHDG are signifi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "cantly closer to the ground-truth distribution than the other methods. RDKIT is slightly worse than", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "GRAPHDG while DL4CHEM seems to struggle with the complexity of the molecules and the small", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 639, + 253, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 253, + 650 + ], + "score": 1.0, + "content": "number of graphs in the training set.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 505, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 457, + 667 + ], + "score": 1.0, + "content": "In Fig. 5, we showcase the accuracy of our model by plotting the marginal distributions", + "type": "text" + }, + { + "bbox": [ + 458, + 655, + 489, + 667 + ], + "score": 0.93, + "content": "p ( d _ { i } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "distances between C and O atoms given a molecular graph from a test set. It can be seen that RDKIT", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "consistently underestimates the marginal variances. This is because this method aims to predict the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "most stable conformation, i.e., the distribution’s mode. In contrast, DL4CHEM often fails to predict", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the correct mean. For this molecule, GRAPHDG is the most accurate, predicting the right mean and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "variance in most cases. Additional figures can be found in the Appendix A.4, where we also show", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 296, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 247, + 734 + ], + "score": 1.0, + "content": "plots for the marginal distributions", + "type": "text" + }, + { + "bbox": [ + 247, + 721, + 292, + 733 + ], + "score": 0.93, + "content": "p ( d _ { i } , d _ { j } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 720, + 296, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 654, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 1: Assessment of the accuracy of the distributions over conformations generated by three", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "score": 1.0, + "content": "models compared to the ground-truth. We compare the distributions with respect to the marginals", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 101, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 107, + 102, + 140, + 114 + ], + "score": 0.88, + "content": "p ( d _ { k } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 101, + 144, + 115 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 144, + 102, + 190, + 114 + ], + "score": 0.89, + "content": "p ( d _ { k } , d _ { l } | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 101, + 398, + 115 + ], + "score": 1.0, + "content": ", and the distribution over all edges between C and", + "type": "text" + }, + { + "bbox": [ + 398, + 102, + 408, + 113 + ], + "score": 0.26, + "content": "\\mathrm { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 101, + 436, + 115 + ], + "score": 1.0, + "content": "atoms", + "type": "text" + }, + { + "bbox": [ + 436, + 102, + 479, + 114 + ], + "score": 0.93, + "content": "p ( \\{ d _ { k } \\} | \\mathcal { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 101, + 506, + 115 + ], + "score": 1.0, + "content": ". Two", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "score": 1.0, + "content": "different metrics are used: median MMD between ground-truth conformations and generated ones,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 124, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 506, + 136 + ], + "score": 1.0, + "content": "and mean ranking (1 to 3) based on the MMD. Reported are the results for molecular graphs in a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 135, + 396, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 396, + 147 + ], + "score": 1.0, + "content": "test set from three train-test splits. Standard errors are given in brackets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 107, + 155, + 502, + 216 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 155, + 502, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 155, + 502, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 502, + 216 + ], + "score": 0.966, + "html": "
Median MMDMean Ranking
RDKITDL4CHEMGRAPHDGRDKITDL4CHEMGRAPHDG
p(dk|9)0.55 (0.01)1.11 (0.01)0.38 (0.02)1.71 (0.03)2.74 (0.02)1.51 (0.03)
p(dk,di/9)0.53 (0.01)1.09 ( (0.01)0.34 (0.01)1.66 (0.02)2.92 (0.01)1.43 ( (0.02)
p({d}9）0.60 (0.01)1.07 (0.03)0.44 (0.05)1.58 (0.05)2.90 (0.05)1.45 (0.02)
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The atoms connected by each edge", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 118, + 444 + ], + "score": 0.87, + "content": "d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 432, + 239, + 446 + ], + "score": 1.0, + "content": "are indicated in each subplot", + "type": "text" + }, + { + "bbox": [ + 239, + 434, + 271, + 444 + ], + "score": 0.91, + "content": "\\left( \\boldsymbol { s } _ { k } \\mathrm { - } \\boldsymbol { r } _ { k } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 432, + 506, + 446 + ], + "score": 1.0, + "content": ". In the 3D structure of the molecule, carbon and oxygen", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 443, + 415, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 287, + 457 + ], + "score": 1.0, + "content": "atoms are colored gray and red, respectively.", + "type": "text" + }, + { + "bbox": [ + 287, + 444, + 296, + 454 + ], + "score": 0.26, + "content": "_ \\mathrm { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 443, + 415, + 457 + ], + "score": 1.0, + "content": "atoms are omitted for clarity.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + } + ], + "index": 11.75 + }, + { + "type": "title", + "bbox": [ + 109, + 478, + 280, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 282, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 282, + 491 + ], + "score": 1.0, + "content": "5.2 GENERATION OF CONFORMATIONS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "We passed the distances from our generative model to an EDG algorithm to obtain conformations.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 123, + 523 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 510, + 150, + 521 + ], + "score": 0.86, + "content": "9 9 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 509, + 383, + 523 + ], + "score": 1.0, + "content": "of the sets of distances, all triangle inequalities held. For", + "type": "text" + }, + { + "bbox": [ + 383, + 510, + 403, + 521 + ], + "score": 0.85, + "content": "94 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "of the molecular graphs,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "the algorithm succeeded which is 8 pp higher than the success rate we observed for RDKIT. For", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 532, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 544 + ], + "score": 1.0, + "content": "each molecular graph in a test set, we generated 50 conformations with each method. This took", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 506, + 556 + ], + "score": 1.0, + "content": "DL4CHEM, RDKIT, and GRAPHDG on average around hundreds of milliseconds per molecule.8", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "In contrast, a single conformation in the ISO17 dataset takes around a minute to compute. In Fig. 6,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "an overlay of these conformations of six molecules generated by the different methods is shown.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "It can be seen that RDKIT’s conformations show too little variance, while DL4CHEM’s structures", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 587, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 504, + 599 + ], + "score": 1.0, + "content": "are mostly invalid, which is due in part to its failure to predict the correct interatomic angles. Our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "method slightly overestimates the structural variance (see, for example, Fig. 6, top row, second", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 609, + 415, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 415, + 621 + ], + "score": 1.0, + "content": "column), but produces conformations that are the closest to the ground-truth.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 635, + 321, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 322, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 322, + 647 + ], + "score": 1.0, + "content": "5.3 CALCULATION OF MOLECULAR PROPERTIES", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 466, + 668 + ], + "score": 1.0, + "content": "We estimate expected molecular properties for molecular graphs from the test set with", + "type": "text" + }, + { + "bbox": [ + 466, + 656, + 504, + 666 + ], + "score": 0.88, + "content": "N = 5 0", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 478, + 680 + ], + "score": 1.0, + "content": "conformational samples each. 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Standard errors are given in brackets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 79, + 506, + 147 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 155, + 502, + 216 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 155, + 502, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 155, + 502, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 502, + 216 + ], + "score": 0.966, + "html": "
Median MMDMean Ranking
RDKITDL4CHEMGRAPHDGRDKITDL4CHEMGRAPHDG
p(dk|9)0.55 (0.01)1.11 (0.01)0.38 (0.02)1.71 (0.03)2.74 (0.02)1.51 (0.03)
p(dk,di/9)0.53 (0.01)1.09 ( (0.01)0.34 (0.01)1.66 (0.02)2.92 (0.01)1.43 ( (0.02)
p({d}9）0.60 (0.01)1.07 (0.03)0.44 (0.05)1.58 (0.05)2.90 (0.05)1.45 (0.02)
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This took", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 506, + 556 + ], + "score": 1.0, + "content": "DL4CHEM, RDKIT, and GRAPHDG on average around hundreds of milliseconds per molecule.8", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "In contrast, a single conformation in the ISO17 dataset takes around a minute to compute. In Fig. 6,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "an overlay of these conformations of six molecules generated by the different methods is shown.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "It can be seen that RDKIT’s conformations show too little variance, while DL4CHEM’s structures", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 587, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 504, + 599 + ], + "score": 1.0, + "content": "are mostly invalid, which is due in part to its failure to predict the correct interatomic angles. 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However, both methods are still highly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 164, + 442 + ], + "score": 1.0, + "content": "inaccurate for", + "type": "text" + }, + { + "bbox": [ + 165, + 430, + 185, + 441 + ], + "score": 0.9, + "content": "E _ { \\mathrm { e l e c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 430, + 335, + 442 + ], + "score": 1.0, + "content": "(in practice, an accuracy of less than", + "type": "text" + }, + { + "bbox": [ + 335, + 430, + 372, + 441 + ], + "score": 0.64, + "content": "5 \\ \\mathrm { k J / m o l }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "is required). Close inspection of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "the conformations shows that, even though GRAPHDG predicts the most accurate distances overall,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "the variances of certain strongly constrained distances (e.g., triple bonds) are overestimated so that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 462, + 323, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 323, + 477 + ], + "score": 1.0, + "content": "the energies of the conformations increase drastically.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 493, + 194, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 195, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 195, + 509 + ], + "score": 1.0, + "content": "6 LIMITATIONS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "The first limitation of this work is that the CVAE can sample (with low probability) invalid sets of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "distances for which there exists no 3D structure. 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Future work will address these points.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 201, + 596 + ], + "lines": [ + { + "bbox": [ + 104, + 582, + 203, + 600 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 203, + 600 + ], + "score": 1.0, + "content": "7 CONCLUSIONS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "We presented GRAPHDG, a transferable, generative model that allows sampling from a distribution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "over molecular conformations. 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However, both methods are still highly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 164, + 442 + ], + "score": 1.0, + "content": "inaccurate for", + "type": "text" + }, + { + "bbox": [ + 165, + 430, + 185, + 441 + ], + "score": 0.9, + "content": "E _ { \\mathrm { e l e c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 430, + 335, + 442 + ], + "score": 1.0, + "content": "(in practice, an accuracy of less than", + "type": "text" + }, + { + "bbox": [ + 335, + 430, + 372, + 441 + ], + "score": 0.64, + "content": "5 \\ \\mathrm { k J / m o l }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "is required). Close inspection of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "the conformations shows that, even though GRAPHDG predicts the most accurate distances overall,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "the variances of certain strongly constrained distances (e.g., triple bonds) are overestimated so that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 462, + 323, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 323, + 477 + ], + "score": 1.0, + "content": "the energies of the conformations increase drastically.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 419, + 506, + 477 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 493, + 194, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 195, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 195, + 509 + ], + "score": 1.0, + "content": "6 LIMITATIONS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "The first limitation of this work is that the CVAE can sample (with low probability) invalid sets of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "distances for which there exists no 3D structure. Second, the CONF17 benchmark covers only a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "small portion of chemical space. Finally, a large set of auxiliary edges would be required to capture", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 553, + 430, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 430, + 566 + ], + "score": 1.0, + "content": "long-range correlations (e.g., in proteins). Future work will address these points.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 519, + 506, + 566 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 201, + 596 + ], + "lines": [ + { + "bbox": [ + 104, + 582, + 203, + 600 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 203, + 600 + ], + "score": 1.0, + "content": "7 CONCLUSIONS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "We presented GRAPHDG, a transferable, generative model that allows sampling from a distribution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "over molecular conformations. We developed a principled learning representation of conforma-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "tions that is based on distances between atoms. Then, we proposed a challenging benchmark for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "comparing molecular conformation generators. With this benchmark, we show experimentally that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "conformations generated by GRAPHDG are closer to the ground-truth than those generated by other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "methods. Finally, we employ our model as a proposal distribution in an IS integration scheme to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "estimate molecular properties. While orbital energies and the dipole moments were predicted well,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "a larger and more diverse dataset will be necessary for meaningful estimates of electronic energies.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "Further, methods have to be devised to estimate how many conformations need to be generated", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "score": 1.0, + "content": "to ensure all important conformations have been sampled. 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Auxiliary", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "edges between all second neighbors were added. This can lead to a slight over-specification of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "system’s geometry, however, this did not pose a problem in our experiments. In addition, auxiliary", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 228 + ], + "score": 1.0, + "content": "edges between third neighbors were added to fix dihedral angles. 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Median MMDMean Ranking
RDKITDL4CHEMGRAPHDGRDKITDL4CHEMGRAPHDG
p(dk|9)0.55 (0.01)1.11 (0.01)0.38 (0.02)1.71 (0.03)2.74 (0.02)1.51 (0.03)
p(dk,di/9)0.53 (0.01)1.09 ( (0.01)0.34 (0.01)1.66 (0.02)2.92 (0.01)1.43 ( (0.02)
p({d}9）0.60 (0.01)1.07 (0.03)0.44 (0.05)1.58 (0.05)2.90 (0.05)1.45 (0.02)
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We evaluate a diverse array of classifiers pre-trained on ImageNet", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 253, + 470, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 253, + 470, + 266 + ], + "score": 1.0, + "content": "and show a median classification accuracy drop of 16 and 10 percent on our two", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 265, + 470, + 276 + ], + "spans": [ + { + "bbox": [ + 142, + 265, + 470, + 276 + ], + "score": 1.0, + "content": "datasets. Additionally, we evaluate three detection models and show that natural", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 275, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 288 + ], + "score": 1.0, + "content": "perturbations induce both classification as well as localization errors, leading", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 287, + 469, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 298 + ], + "score": 1.0, + "content": "to a median drop in detection mAP of 14 points. Our analysis demonstrates", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "score": 1.0, + "content": "that perturbations occurring naturally in videos pose a substantial and realistic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "challenge to deploying convolutional neural networks in environments that require", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 318, + 310, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 310, + 331 + ], + "score": 1.0, + "content": "both reliable and low-latency predictions.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 349, + 206, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 208, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 208, + 364 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "Convolutional neural networks (CNNs) still exhibit many troubling failure modes. At one extreme,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 116, + 398 + ], + "score": 0.87, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "-adversarial examples cause large drops in accuracy for state-of-the-art models while relying only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "on visually imperceptible changes to the input image (Goodfellow et al., 2014; Biggio and Roli,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 406, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 422 + ], + "score": 1.0, + "content": "2018). However, this failure mode usually does not pose a problem outside a fully adversarial context", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 468, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 208, + 432 + ], + "score": 1.0, + "content": "because carefully crafted", + "type": "text" + }, + { + "bbox": [ + 209, + 419, + 218, + 432 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 419, + 468, + 432 + ], + "score": 1.0, + "content": "-perturbations are unlikely to occur naturally in the real world.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "To study more realistic failure modes, researchers have investigated benign image perturbations such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "as rotations & translations, colorspace changes, and various image corruptions (Fawzi and Frossard,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 507, + 473 + ], + "score": 1.0, + "content": "2015; Engstrom et al., 2017; Fawzi and Frossard, 2015; Hendrycks and Dietterich, 2019). However,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "it is still unclear whether these perturbations reflect the robustness challenges arising in real data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 373, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 373, + 494 + ], + "score": 1.0, + "content": "since the perturbations also rely on synthetic image modifications.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "Recent work has therefore turned to videos as a source of naturally occurring perturbations of images", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "(Zheng et al., 2016; Azulay and Weiss, 2018; Gu et al., 2019). In contrast to other failure modes, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "perturbed images are taken from existing image data without further modifications that make the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "task more difficult. As a result, robustness to such perturbations directly corresponds to performance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 543, + 217, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 217, + 554 + ], + "score": 1.0, + "content": "improvements on real data.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "However, it is currently unclear to what extent such video perturbations pose a significant robustness", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "challenge. Azulay and Weiss (2018) and Zheng et al. (2016) only provide anecdotal evidence from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "a small number of videos. Gu et al. (2019) go beyond individual videos and utilize a large video", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "dataset (Real et al., 2017) in order to measure the effect of video perturbations more quantitatively. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 314, + 617 + ], + "score": 1.0, + "content": "their evaluation, the best image classifiers lose about", + "type": "text" + }, + { + "bbox": [ + 315, + 604, + 329, + 615 + ], + "score": 0.85, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "accuracy for video frames up to 0.3 seconds", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "away. However, the authors did not employ humans to review the frames in their videos. Hence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "the accuracy drop could also be caused by significant changes in the video frames (e.g., due to fast", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 257, + 649 + ], + "score": 1.0, + "content": "camera or object motion). Since the", + "type": "text" + }, + { + "bbox": [ + 257, + 637, + 273, + 648 + ], + "score": 0.85, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "accuracy drop is small to begin with, it remains unclear", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 443, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 443, + 662 + ], + "score": 1.0, + "content": "whether video perturbations are a robustness challenge for current image classifiers.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We address these issues by conducting a thorough evaluation of robustness to natural perturbations", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "arising in videos. As a cornerstone of our investigation, we introduce two test sets for evaluating", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "model robustness: ImageNet-Vid-Robust and YTBB-Robust, carefully curated from the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "ImageNet-Vid and Youtube-BB datasets, respectively (Russakovsky et al., 2015; Real et al., 2017).", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "All images in the two datasets were screened by a set of expert labelers to ensure high annotation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "quality and minimize selection biases that arise when filtering a dataset with CNNs. To the best of", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 96 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 506, + 98 + ], + "score": 1.0, + "content": "DO IMAGE CLASSIFIERS GENERALIZE ACROSS TIME?", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 115, + 244, + 137 + ], + "lines": [ + { + "bbox": [ + 113, + 116, + 201, + 127 + ], + "spans": [ + { + "bbox": [ + 113, + 116, + 201, + 127 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 112, + 126, + 245, + 138 + ], + "spans": [ + { + "bbox": [ + 112, + 126, + 245, + 138 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 112, + 116, + 245, + 138 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 166, + 333, + 179 + ], + "lines": [ + { + "bbox": [ + 276, + 165, + 336, + 181 + ], + "spans": [ + { + "bbox": [ + 276, + 165, + 336, + 181 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 143, + 187, + 468, + 330 + ], + "lines": [ + { + "bbox": [ + 142, + 188, + 469, + 200 + ], + "spans": [ + { + "bbox": [ + 142, + 188, + 469, + 200 + ], + "score": 1.0, + "content": "We study the robustness of image classifiers to temporal perturbations derived", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 198, + 469, + 211 + ], + "spans": [ + { + "bbox": [ + 141, + 198, + 469, + 211 + ], + "score": 1.0, + "content": "from videos. As part of this study, we construct ImageNet-Vid-Robust", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 210, + 470, + 222 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 470, + 222 + ], + "score": 1.0, + "content": "and YTBB-Robust, containing a total 57,897 images grouped into 3,139 sets", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 221, + 469, + 232 + ], + "spans": [ + { + "bbox": [ + 142, + 221, + 469, + 232 + ], + "score": 1.0, + "content": "of perceptually similar images. Our datasets were derived from ImageNet-Vid", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 231, + 470, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 470, + 244 + ], + "score": 1.0, + "content": "and Youtube-BB respectively and thoroughly re-annotated by human experts for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 243, + 470, + 254 + ], + "spans": [ + { + "bbox": [ + 142, + 243, + 470, + 254 + ], + "score": 1.0, + "content": "image similarity. We evaluate a diverse array of classifiers pre-trained on ImageNet", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 253, + 470, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 253, + 470, + 266 + ], + "score": 1.0, + "content": "and show a median classification accuracy drop of 16 and 10 percent on our two", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 265, + 470, + 276 + ], + "spans": [ + { + "bbox": [ + 142, + 265, + 470, + 276 + ], + "score": 1.0, + "content": "datasets. Additionally, we evaluate three detection models and show that natural", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 275, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 288 + ], + "score": 1.0, + "content": "perturbations induce both classification as well as localization errors, leading", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 287, + 469, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 298 + ], + "score": 1.0, + "content": "to a median drop in detection mAP of 14 points. Our analysis demonstrates", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 469, + 309 + ], + "score": 1.0, + "content": "that perturbations occurring naturally in videos pose a substantial and realistic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "challenge to deploying convolutional neural networks in environments that require", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 318, + 310, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 310, + 331 + ], + "score": 1.0, + "content": "both reliable and low-latency predictions.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10, + "bbox_fs": [ + 141, + 188, + 470, + 331 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 349, + 206, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 208, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 208, + 364 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "Convolutional neural networks (CNNs) still exhibit many troubling failure modes. At one extreme,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 116, + 398 + ], + "score": 0.87, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "-adversarial examples cause large drops in accuracy for state-of-the-art models while relying only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "on visually imperceptible changes to the input image (Goodfellow et al., 2014; Biggio and Roli,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 406, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 422 + ], + "score": 1.0, + "content": "2018). However, this failure mode usually does not pose a problem outside a fully adversarial context", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 468, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 208, + 432 + ], + "score": 1.0, + "content": "because carefully crafted", + "type": "text" + }, + { + "bbox": [ + 209, + 419, + 218, + 432 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 419, + 468, + 432 + ], + "score": 1.0, + "content": "-perturbations are unlikely to occur naturally in the real world.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 375, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "To study more realistic failure modes, researchers have investigated benign image perturbations such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "as rotations & translations, colorspace changes, and various image corruptions (Fawzi and Frossard,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 507, + 473 + ], + "score": 1.0, + "content": "2015; Engstrom et al., 2017; Fawzi and Frossard, 2015; Hendrycks and Dietterich, 2019). However,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "it is still unclear whether these perturbations reflect the robustness challenges arising in real data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 373, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 373, + 494 + ], + "score": 1.0, + "content": "since the perturbations also rely on synthetic image modifications.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 437, + 507, + 494 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "Recent work has therefore turned to videos as a source of naturally occurring perturbations of images", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "(Zheng et al., 2016; Azulay and Weiss, 2018; Gu et al., 2019). In contrast to other failure modes, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "perturbed images are taken from existing image data without further modifications that make the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "task more difficult. As a result, robustness to such perturbations directly corresponds to performance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 543, + 217, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 217, + 554 + ], + "score": 1.0, + "content": "improvements on real data.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 498, + 506, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "However, it is currently unclear to what extent such video perturbations pose a significant robustness", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "challenge. Azulay and Weiss (2018) and Zheng et al. (2016) only provide anecdotal evidence from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "a small number of videos. Gu et al. (2019) go beyond individual videos and utilize a large video", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "dataset (Real et al., 2017) in order to measure the effect of video perturbations more quantitatively. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 314, + 617 + ], + "score": 1.0, + "content": "their evaluation, the best image classifiers lose about", + "type": "text" + }, + { + "bbox": [ + 315, + 604, + 329, + 615 + ], + "score": 0.85, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "accuracy for video frames up to 0.3 seconds", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "away. However, the authors did not employ humans to review the frames in their videos. Hence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "the accuracy drop could also be caused by significant changes in the video frames (e.g., due to fast", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 257, + 649 + ], + "score": 1.0, + "content": "camera or object motion). Since the", + "type": "text" + }, + { + "bbox": [ + 257, + 637, + 273, + 648 + ], + "score": 0.85, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "accuracy drop is small to begin with, it remains unclear", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 443, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 443, + 662 + ], + "score": 1.0, + "content": "whether video perturbations are a robustness challenge for current image classifiers.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 561, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We address these issues by conducting a thorough evaluation of robustness to natural perturbations", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "arising in videos. As a cornerstone of our investigation, we introduce two test sets for evaluating", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "model robustness: ImageNet-Vid-Robust and YTBB-Robust, carefully curated from the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "ImageNet-Vid and Youtube-BB datasets, respectively (Russakovsky et al., 2015; Real et al., 2017).", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "All images in the two datasets were screened by a set of expert labelers to ensure high annotation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "quality and minimize selection biases that arise when filtering a dataset with CNNs. To the best of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "our knowledge these are the first datasets of their kind, containing tens of thousands of images that", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "are human reviewed and grouped into thousands of perceptually similar sets. In total, our datasets", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 474, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 474, + 117 + ], + "score": 1.0, + "content": "contain 3,139 sets of temporally adjacent and visually similar images (57,897 images total).", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 665, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "our knowledge these are the first datasets of their kind, containing tens of thousands of images that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "are human reviewed and grouped into thousands of perceptually similar sets. In total, our datasets", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 474, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 474, + 117 + ], + "score": 1.0, + "content": "contain 3,139 sets of temporally adjacent and visually similar images (57,897 images total).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 507, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 507, + 134 + ], + "score": 1.0, + "content": "We then utilize these datasets to measure the accuracy of current CNNs to small, naturally oc-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "curring perturbations. Our testbed contains over 45 different models, varying both architecture", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "and training methodology (adversarial training, data augmentation, etc.). To better understand the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "drop in accuracy due to natural perturbations, we also introduce a robustness metric that is more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "stringent than those employed in prior work. Under this metric, we find that natural perturba-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 486, + 188 + ], + "score": 1.0, + "content": "tions from ImageNet-Vid-Robust and YTBB-Robust induce a median accuracy drop of", + "type": "text" + }, + { + "bbox": [ + 486, + 176, + 505, + 187 + ], + "score": 0.86, + "content": "16 \\%", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 124, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 187, + 145, + 198 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "respectively for classification tasks and a median 14 point drop in mAP for detection", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 470, + 211 + ], + "score": 1.0, + "content": "tasks.1 Even for the best-performing classification models, we observe an accuracy drop of", + "type": "text" + }, + { + "bbox": [ + 471, + 198, + 490, + 208 + ], + "score": 0.86, + "content": "14 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 340, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 239, + 222 + ], + "score": 1.0, + "content": "ImageNet-Vid-Robust and", + "type": "text" + }, + { + "bbox": [ + 239, + 209, + 254, + 219 + ], + "score": 0.85, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 209, + 340, + 222 + ], + "score": 1.0, + "content": "for YTBB-Robust.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "Our results show that robustness to natural perturbations in videos is indeed a significant challenge", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "for current CNNs. As these models are increasingly deployed in safety-critical environments that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "require both high accuracy and low latency (e.g., autonomous vehicles), ensuring reliable predictions", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 376, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 376, + 271 + ], + "score": 1.0, + "content": "on every frame of a video is an important direction for future work.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "image", + "bbox": [ + 108, + 281, + 504, + 385 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 281, + 504, + 385 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 281, + 504, + 385 + ], + "spans": [ + { + "bbox": [ + 108, + 281, + 504, + 385 + ], + "score": 0.964, + "type": "image", + "image_path": "3f6b30513a9f6483b745608a87d39877b703a1ece944de817966f2872e542011.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 108, + 281, + 504, + 315.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 108, + 315.6666666666667, + 504, + 350.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 108, + 350.33333333333337, + 504, + 385.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 393, + 506, + 426 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "Figure 1: Three examples of natural perturbations from nearby video frames and resulting classifier", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "confidences from a ResNet-152 model fine-tuned on ImageNet-Vid. While the images appear almost", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 414, + 404, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 404, + 428 + ], + "score": 1.0, + "content": "identical to the human eye, the classifier confidence changes substantially.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 366, + 442 + ], + "lines": [ + { + "bbox": [ + 104, + 428, + 367, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 367, + 444 + ], + "score": 1.0, + "content": "2 CONSTRUCTING A TEST SET FOR ROBUSTNESS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 504, + 466 + ], + "score": 1.0, + "content": "ImageNet-Vid-Robust and YTBB-Robust are sourced from videos in the ImageNet-Vid and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "Youtube-BB datasets (Russakovsky et al., 2015; Real et al., 2017). All object classes in ImageNet-Vid", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "and Youtube-BB are from the WordNet hierarchy (Miller, 1995) and direct ancestors of ILSVRC-2012", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 488, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 506, + 500 + ], + "score": 1.0, + "content": "classes. Using the WordNet hierarchy, we construct a canonical mapping from ILSVRC-2012 classes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "to ImageNet-Vid and Youtube-BB classes, which allows us to evaluate off-the-shelf ILSVRC-2012", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "models on ImageNet-Vid-Robust and YTBB-Robust. We provide more background on the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 231, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 231, + 533 + ], + "score": 1.0, + "content": "source datsets in Appendix A.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 108, + 546, + 405, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 407, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 407, + 558 + ], + "score": 1.0, + "content": "2.1 CONSTRUCTING IM A G ENE T-VI D-RO B U S T AND YTBB-RO B U S T", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Next, we describe how we extracted sets of naturally perturbed frames from ImageNet-Vid and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "Youtube-BB to create ImageNet-Vid-Robust and YTBB-Robust. A straightforward approach", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "would be to select a set of anchor frames and use temporally adjacent frames in the video with the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "assumption that such frames contain only small perturbations from the anchor. However, as Fig. 2", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 609, + 493, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 493, + 623 + ], + "score": 1.0, + "content": "illustrates, this assumption is frequently violated, especially due to fast camera or object motion.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "Instead, we first collect preliminary datasets of natural perturbations following the same approach,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "and then manually review each of the frame sets. For each video, we randomly sample an anchor", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 168, + 660 + ], + "score": 1.0, + "content": "frame and take", + "type": "text" + }, + { + "bbox": [ + 168, + 649, + 199, + 659 + ], + "score": 0.9, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 647, + 506, + 660 + ], + "score": 1.0, + "content": "frames before and after the anchor frame as candidate perturbation images.2", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 657, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 673 + ], + "score": 1.0, + "content": "This results in two datasets containing one anchor frame each from 3,139 videos, with approximately", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 669, + 288, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 288, + 682 + ], + "score": 1.0, + "content": "20 candidate perturbation per anchor frame.3", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 689, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 118, + 687, + 505, + 702 + ], + "score": 1.0, + "content": "1We only evaluated detection on ImageNet-Vid-Robust as bounding-box annotations in Youtube-BB", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 700, + 372, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 372, + 711 + ], + "score": 1.0, + "content": "were only at 1 frame-per-second and not dense enough for our evaluation.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 708, + 424, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 424, + 723 + ], + "score": 1.0, + "content": "2For YTBB-Robust we use a subset of the anchor frames used by Gu et al. (2019).", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 719, + 443, + 733 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 443, + 733 + ], + "score": 1.0, + "content": "3Anchor frames near the start or end of the video may have less than 20 candidate frames.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 507, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 507, + 134 + ], + "score": 1.0, + "content": "We then utilize these datasets to measure the accuracy of current CNNs to small, naturally oc-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "curring perturbations. Our testbed contains over 45 different models, varying both architecture", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "and training methodology (adversarial training, data augmentation, etc.). To better understand the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "drop in accuracy due to natural perturbations, we also introduce a robustness metric that is more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "stringent than those employed in prior work. Under this metric, we find that natural perturba-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 486, + 188 + ], + "score": 1.0, + "content": "tions from ImageNet-Vid-Robust and YTBB-Robust induce a median accuracy drop of", + "type": "text" + }, + { + "bbox": [ + 486, + 176, + 505, + 187 + ], + "score": 0.86, + "content": "16 \\%", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 124, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 187, + 145, + 198 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "respectively for classification tasks and a median 14 point drop in mAP for detection", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 470, + 211 + ], + "score": 1.0, + "content": "tasks.1 Even for the best-performing classification models, we observe an accuracy drop of", + "type": "text" + }, + { + "bbox": [ + 471, + 198, + 490, + 208 + ], + "score": 0.86, + "content": "14 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 340, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 239, + 222 + ], + "score": 1.0, + "content": "ImageNet-Vid-Robust and", + "type": "text" + }, + { + "bbox": [ + 239, + 209, + 254, + 219 + ], + "score": 0.85, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 209, + 340, + 222 + ], + "score": 1.0, + "content": "for YTBB-Robust.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 120, + 507, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "Our results show that robustness to natural perturbations in videos is indeed a significant challenge", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "for current CNNs. As these models are increasingly deployed in safety-critical environments that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "require both high accuracy and low latency (e.g., autonomous vehicles), ensuring reliable predictions", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 376, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 376, + 271 + ], + "score": 1.0, + "content": "on every frame of a video is an important direction for future work.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 225, + 506, + 271 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 281, + 504, + 385 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 281, + 504, + 385 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 281, + 504, + 385 + ], + "spans": [ + { + "bbox": [ + 108, + 281, + 504, + 385 + ], + "score": 0.964, + "type": "image", + "image_path": "3f6b30513a9f6483b745608a87d39877b703a1ece944de817966f2872e542011.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 108, + 281, + 504, + 315.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 108, + 315.6666666666667, + 504, + 350.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 108, + 350.33333333333337, + 504, + 385.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 393, + 506, + 426 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "Figure 1: Three examples of natural perturbations from nearby video frames and resulting classifier", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "confidences from a ResNet-152 model fine-tuned on ImageNet-Vid. While the images appear almost", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 414, + 404, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 404, + 428 + ], + "score": 1.0, + "content": "identical to the human eye, the classifier confidence changes substantially.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 366, + 442 + ], + "lines": [ + { + "bbox": [ + 104, + 428, + 367, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 367, + 444 + ], + "score": 1.0, + "content": "2 CONSTRUCTING A TEST SET FOR ROBUSTNESS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 504, + 466 + ], + "score": 1.0, + "content": "ImageNet-Vid-Robust and YTBB-Robust are sourced from videos in the ImageNet-Vid and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "Youtube-BB datasets (Russakovsky et al., 2015; Real et al., 2017). All object classes in ImageNet-Vid", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "and Youtube-BB are from the WordNet hierarchy (Miller, 1995) and direct ancestors of ILSVRC-2012", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 488, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 506, + 500 + ], + "score": 1.0, + "content": "classes. Using the WordNet hierarchy, we construct a canonical mapping from ILSVRC-2012 classes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "to ImageNet-Vid and Youtube-BB classes, which allows us to evaluate off-the-shelf ILSVRC-2012", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "models on ImageNet-Vid-Robust and YTBB-Robust. We provide more background on the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 520, + 231, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 231, + 533 + ], + "score": 1.0, + "content": "source datsets in Appendix A.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 455, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 546, + 405, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 407, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 407, + 558 + ], + "score": 1.0, + "content": "2.1 CONSTRUCTING IM A G ENE T-VI D-RO B U S T AND YTBB-RO B U S T", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 544, + 407, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Next, we describe how we extracted sets of naturally perturbed frames from ImageNet-Vid and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "Youtube-BB to create ImageNet-Vid-Robust and YTBB-Robust. A straightforward approach", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "would be to select a set of anchor frames and use temporally adjacent frames in the video with the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "assumption that such frames contain only small perturbations from the anchor. However, as Fig. 2", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 609, + 493, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 493, + 623 + ], + "score": 1.0, + "content": "illustrates, this assumption is frequently violated, especially due to fast camera or object motion.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 565, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "Instead, we first collect preliminary datasets of natural perturbations following the same approach,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "and then manually review each of the frame sets. 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We show three randomly sampled", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "frame pairs where the nearby frame was marked as “dissimilar” to the anchor frame during human", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 183, + 283, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 283, + 195 + ], + "score": 1.0, + "content": "review and then discarded from our dataset.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 120, + 206, + 491, + 279 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 120, + 206, + 491, + 279 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 206, + 491, + 279 + ], + "spans": [ + { + "bbox": [ + 120, + 206, + 491, + 279 + ], + "score": 0.975, + "html": "
ImageNet-Vid-RobustYTBB-Robust
Anchor framesReviewed1,3142,467
Accepted1,109 (84%)2,030 (82%)
Labels updated1834 (41%)
Frame pairsReviewed26,02945,631
Accepted21,070 (80.9%)36,827 (80.7%)
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ImageNet-Vid-RobustYTBB-Robust
Anchor framesReviewed1,3142,467
Accepted1,109 (84%)2,030 (82%)
Labels updated1834 (41%)
Frame pairsReviewed26,02945,631
Accepted21,070 (80.9%)36,827 (80.7%)
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We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "asked reviewers to mark each dissimilar frame with one of these transformations, or “other”, and to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "mark a pair of images as dissimilar if a distinctive feature of the object is only visible in one of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 508, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 519 + ], + "score": 1.0, + "content": "two frames (such as the face of a dog). If an annotator was unsure about the correct label, she could", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "mark the pair as “unsure”. Second, we present only a single pair of frames at a time to reviewers", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "because presenting videos or groups of frames could cause them to miss large changes due to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 541, + 307, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 307, + 553 + ], + "score": 1.0, + "content": "phenomenon of change blindness (Pashler, 1988).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 431, + 507, + 553 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "Verification. In the previous stage, all annotators were given identical labeling instructions and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "score": 1.0, + "content": "individually reviewed a total of 71,660 images pairs. To increase consistency in annotation, annotators", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 581, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 592 + ], + "score": 1.0, + "content": "jointly reviewed all frames marked as dissimilar, incorrectly labeled, or “unsure”. A frame was only", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 475, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 475, + 603 + ], + "score": 1.0, + "content": "considered similar to its anchor if a strict majority of the annotators marked the pair as such.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 557, + 506, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 608, + 504, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "After the reviewing was complete, we discarded all anchor frames and candidate perturbations that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 619, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 631 + ], + "score": 1.0, + "content": "annotators marked as dissimilar or incorrectly labeled. The final datasets contain a combined total of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 631, + 355, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 355, + 641 + ], + "score": 1.0, + "content": "3,139 anchor frames with a median of 20 similar frames each.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 607, + 506, + 641 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 268, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 270, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 270, + 668 + ], + "score": 1.0, + "content": "2.2 THE P M-K EVALUATION METRIC", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 688 + ], + "score": 1.0, + "content": "Given the datasets introduced above, we propose a metric to measure a model’s robustness to natural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 229, + 700 + ], + "score": 1.0, + "content": "perturbations. In particular, let", + "type": "text" + }, + { + "bbox": [ + 230, + 688, + 298, + 700 + ], + "score": 0.92, + "content": "A = \\{ a _ { 1 } , . . . , a _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "be the set of valid anchor frames in our dataset. Let", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 176, + 711 + ], + "score": 0.92, + "content": "Y = \\{ y _ { 1 } , . . . , y _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 698, + 268, + 711 + ], + "score": 1.0, + "content": "be the set of labels for", + "type": "text" + }, + { + "bbox": [ + 268, + 700, + 276, + 708 + ], + "score": 0.81, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 698, + 309, + 711 + ], + "score": 1.0, + "content": ". We let", + "type": "text" + }, + { + "bbox": [ + 309, + 699, + 339, + 711 + ], + "score": 0.91, + "content": "\\textstyle { \\mathcal { N } } _ { k } ( a _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "be the set of frames marked as similar to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 163, + 722 + ], + "score": 1.0, + "content": "anchor frame", + "type": "text" + }, + { + "bbox": [ + 163, + 712, + 172, + 721 + ], + "score": 0.82, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 710, + 236, + 722 + ], + "score": 1.0, + "content": ". In our setting,", + "type": "text" + }, + { + "bbox": [ + 236, + 710, + 250, + 721 + ], + "score": 0.88, + "content": "\\mathcal { N } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 710, + 321, + 722 + ], + "score": 1.0, + "content": "is a subset of the", + "type": "text" + }, + { + "bbox": [ + 322, + 711, + 333, + 720 + ], + "score": 0.85, + "content": "2 k", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 710, + 497, + 722 + ], + "score": 1.0, + "content": "temporally adjacent frames (plus/minus", + "type": "text" + }, + { + "bbox": [ + 497, + 710, + 505, + 720 + ], + "score": 0.27, + "content": "\\mathbf { k }", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 719, + 208, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 208, + 732 + ], + "score": 1.0, + "content": "frames from the anchor).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 676, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 102, + 74, + 508, + 101 + ], + "spans": [ + { + "bbox": [ + 102, + 74, + 320, + 101 + ], + "score": 1.0, + "content": "Classification. Classification accuracy is defined as", + "type": "text" + }, + { + "bbox": [ + 320, + 80, + 475, + 96 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathrm { a c c } _ { \\mathrm { o r i g } } = 1 - \\frac { 1 } { N } \\sum _ { i = 0 } ^ { N } \\mathcal { L } _ { 0 / 1 } ( f ( a _ { i } ) , y _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 74, + 508, + 101 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 434, + 108 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 127, + 107 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { 0 / 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 92, + 317, + 108 + ], + "score": 1.0, + "content": "is the standard 0-1 loss function. We define the", + "type": "text" + }, + { + "bbox": [ + 318, + 95, + 343, + 105 + ], + "score": 0.81, + "content": "\\mathrm { p m - k }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 92, + 434, + 108 + ], + "score": 1.0, + "content": "analog of accuracy as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 114, + 401, + 149 + ], + "lines": [ + { + "bbox": [ + 209, + 114, + 401, + 149 + ], + "spans": [ + { + "bbox": [ + 209, + 114, + 401, + 149 + ], + "score": 0.94, + "content": "\\mathrm { a c c } _ { \\mathrm { p m k } } = 1 - \\frac { 1 } { N } \\sum _ { i = 0 } ^ { N } \\operatorname* { m a x } _ { b \\in \\mathcal { N } _ { k } ( a _ { i } ) } \\mathcal { L } _ { 0 / 1 } ( f ( b ) , y _ { i } ) ,", + "type": "interline_equation", + "image_path": "71701e6adb581e0f883f201bedca69f386a5fac163fdf05b2c7fd2e3e8fc5c13.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 114, + 401, + 131.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 209, + 131.5, + 401, + 149.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 490, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 492, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 347, + 172 + ], + "score": 1.0, + "content": "which corresponds to picking the worst frame from each set", + "type": "text" + }, + { + "bbox": [ + 347, + 156, + 378, + 168 + ], + "score": 0.93, + "content": "\\textstyle { \\mathcal { N } } _ { k } ( a _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 153, + 492, + 172 + ], + "score": 1.0, + "content": "before computing accuracy.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 184, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "Detection. The standard metric for detection is mean average precision (mAP) of the predictions", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 449, + 209 + ], + "score": 1.0, + "content": "at a fixed intersection-over-union (IoU) threshold Lin et al. (2014). We define the", + "type": "text" + }, + { + "bbox": [ + 449, + 197, + 475, + 208 + ], + "score": 0.86, + "content": "\\mathrm { p m - k }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "metric", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 220 + ], + "score": 1.0, + "content": "analogous to that for classification: We replace each anchor frame with the nearest frame that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "minimizes the average precision (AP, averaged over recall thresholds) of the predictions, and compute", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 228, + 344, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 132, + 241 + ], + "score": 0.77, + "content": "\\mathrm { p m - k }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 228, + 344, + 242 + ], + "score": 1.0, + "content": "as the mAP on these worst-case neighboring frames.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 262, + 204, + 274 + ], + "lines": [ + { + "bbox": [ + 104, + 260, + 205, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 205, + 277 + ], + "score": 1.0, + "content": "3 MAIN RESULTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "image", + "bbox": [ + 145, + 293, + 465, + 445 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 293, + 465, + 445 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 293, + 465, + 445 + ], + "spans": [ + { + "bbox": [ + 145, + 293, + 465, + 445 + ], + "score": 0.972, + "type": "image", + "image_path": "ce35b09ba50d55d64192d79ef52cb386fa1ca0e6d3e8f2da696a1913a871875d.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 145, + 293, + 465, + 343.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 145, + 343.6666666666667, + 465, + 394.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 145, + 394.33333333333337, + 465, + 445.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 454, + 505, + 498 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "Figure 3: Model accuracy on original vs. perturbed images. Each data point corresponds to one", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 240, + 478 + ], + "score": 1.0, + "content": "model in our testbed (shown with", + "type": "text" + }, + { + "bbox": [ + 241, + 465, + 260, + 475 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "Clopper-Pearson confidence intervals). Each perturbed frame", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "was taken from a ten frame neighborhood of the original frame (approximately 0.3 seconds). All", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 487, + 443, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 443, + 500 + ], + "score": 1.0, + "content": "frames were reviewed by humans to confirm visual similarity to the original frames.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + } + ], + "index": 13.75 + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 504, + 527 + ], + "score": 1.0, + "content": "We evaluate a testbed of 45 classification and three detection models on ImageNet-Vid-Robust", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "and YTBB-Robust. We first discuss the various types of classification models evaluated with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 122, + 550 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 123, + 538, + 148, + 549 + ], + "score": 0.72, + "content": "\\mathrm { p m - k }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "classification metric. Second, we evaluate the performance of detection models on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "ImageNet-Vid-Robust using use the bounding box annotations inherited from ImageNet-Vid", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 177, + 571 + ], + "score": 1.0, + "content": "using a variant of", + "type": "text" + }, + { + "bbox": [ + 178, + 560, + 203, + 570 + ], + "score": 0.73, + "content": "\\mathrm { p m - k }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "for detection. We then analyze the errors made on the detection adversarial", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 570, + 412, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 412, + 583 + ], + "score": 1.0, + "content": "examples to isolate the effects of localization errors vs. classification errors.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 600, + 203, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 204, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 204, + 613 + ], + "score": 1.0, + "content": "3.1 CLASSIFICATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 266, + 637 + ], + "score": 1.0, + "content": "The classification robustness metric is", + "type": "text" + }, + { + "bbox": [ + 267, + 626, + 294, + 636 + ], + "score": 0.86, + "content": "\\mathrm { a c c } _ { \\mathrm { p m k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "defined in Equation (1). For frames with multiple", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "labels, we count a prediction as correct if the model predicts any of the correct classes for a frame. 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Each data point corresponds to one", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 240, + 478 + ], + "score": 1.0, + "content": "model in our testbed (shown with", + "type": "text" + }, + { + "bbox": [ + 241, + 465, + 260, + 475 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "Clopper-Pearson confidence intervals). Each perturbed frame", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "was taken from a ten frame neighborhood of the original frame (approximately 0.3 seconds). 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We then analyze the errors made on the detection adversarial", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 570, + 412, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 412, + 583 + ], + "score": 1.0, + "content": "examples to isolate the effects of localization errors vs. classification errors.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 515, + 505, + 583 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 600, + 203, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 204, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 204, + 613 + ], + "score": 1.0, + "content": "3.1 CLASSIFICATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 266, + 637 + ], + "score": 1.0, + "content": "The classification robustness metric is", + "type": "text" + }, + { + "bbox": [ + 267, + 626, + 294, + 636 + ], + "score": 0.86, + "content": "\\mathrm { a c c } _ { \\mathrm { p m k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "defined in Equation (1). For frames with multiple", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "labels, we count a prediction as correct if the model predicts any of the correct classes for a frame. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 257, + 660 + ], + "score": 1.0, + "content": "Figure 3, we plot the benign accuracy,", + "type": "text" + }, + { + "bbox": [ + 258, + 648, + 284, + 658 + ], + "score": 0.85, + "content": "\\operatorname { a c c } _ { \\mathrm { o r i g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 646, + 394, + 660 + ], + "score": 1.0, + "content": ", versus the robust accuracy,", + "type": "text" + }, + { + "bbox": [ + 394, + 648, + 421, + 658 + ], + "score": 0.83, + "content": "\\mathrm { a c c } _ { \\mathrm { p m k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 646, + 506, + 660 + ], + "score": 1.0, + "content": ", for all classification", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 656, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 341, + 671 + ], + "score": 1.0, + "content": "models in our test bed and find that the relationship between", + "type": "text" + }, + { + "bbox": [ + 341, + 659, + 367, + 670 + ], + "score": 0.87, + "content": "\\operatorname { a c c } _ { \\mathrm { o r i g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 656, + 384, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 659, + 412, + 669 + ], + "score": 0.87, + "content": "\\mathrm { a c c } _ { \\mathrm { p m k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 656, + 506, + 671 + ], + "score": 1.0, + "content": "is approximately linear.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "This relationship indicates that improvements in the benign accuracy do result in improvements in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 679, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 506, + 692 + ], + "score": 1.0, + "content": "the worst-case accuracy, but do not suffice to resolve the accuracy drop due to natural perturbations.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 623, + 506, + 692 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Our test bed consists of five model types with increasing levels of supervision. We present results for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "representative models from each model type in Table 2 and defer the full classification results table to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 167, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 167, + 733 + ], + "score": 1.0, + "content": "Appendix B.2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 135, + 111, + 476, + 286 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "Table 2: Accuracies of five different model types and the best performing model. The model", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 501, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 501, + 102 + ], + "score": 1.0, + "content": "architecture is ResNet-50 unless noted otherwise. ‘FT’ is ‘fine-tuning.’ See Section 3.1 for details.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 135, + 111, + 476, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 111, + 476, + 286 + ], + "spans": [ + { + "bbox": [ + 135, + 111, + 476, + 286 + ], + "score": 0.981, + "html": "
Model Type Accuracy Accuracy △PerturbedOriginal
ImageNet-Vid-Robust
Trained on ILSVRC 67.5 [64.7, 70.3] 52.5 [49.5, 55.5] 15.0
+ Noise Augmentation 68.8 [66.0, 71.5] 53.2 [50.2, 56.2] 15.6
+lrobustness (ResNext-101) 54.3 [51.3, 57.2] 40.8 [39.0, 43.7] 12.4
+ FT on ImageNet- Vid 80.8 [78.3, 83.1] 65.7 [62.9, 68.5] 15.1
+ FT on ImageNet-Vid (ResNet-152) 84.8 [82.5, 86.8]70.2 [67.4, 72.8] 14.6
+ FT on ImageNet-Vid-Det 77.6 [75.1, 80.0] 65.4 [62.5, 68.1] 12.3
YTBB-Robust
Trained on ILSVRC 57.0 [54.9, 59.2]43.8 [41.7, 46.0] 13.2
+Noise Augmentation 62.3 [60.2, 64.4]45.7 [43.5, 47.9]16.6
+ l robustness (ResNext-101) 53.6 [51.4, 55.8]43.2 [41.0, 45.3]10.4
+ FT on Youtube-BB 91.4 [90.1, 92.6]82.0 [80.3, 83.7]9.4
+ FT on Youtube-BB (ResNet-152) 92.9 [91.6, 93.9]84.7 [83.0, 86.2]8.2
", + "type": "table", + "image_path": "93504e29d42d2d6cda7c44c90142ee15e467a45dfec98b321b9089ed326f2725.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 135, + 111, + 476, + 169.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 135, + 169.33333333333334, + 476, + 227.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 135, + 227.66666666666669, + 476, + 286.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "ILSVRC Trained The WordNet hierarchy enables us to repurpose models trained for the 1,000", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "class ILSVRC dataset on ImageNet-Vid-Robust and YTBB-Robust (see Appendix A.1).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "We evaluate a wide array of ILSVRC-2012 models (available from Cadene) against our natural", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "perturbations. Since these datasets present a substantial distribution shift from the original ILSVRC-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 346, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 297, + 362 + ], + "score": 1.0, + "content": "2012 validation, we expect the benign accuracy", + "type": "text" + }, + { + "bbox": [ + 297, + 349, + 324, + 360 + ], + "score": 0.85, + "content": "\\operatorname { a c c } _ { \\mathrm { o r i g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 346, + 506, + 362 + ], + "score": 1.0, + "content": "to be lower than the comparable accuracy on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "the ILSVRC-2012 validation set. However, our main interest here is in the difference between the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 243, + 384 + ], + "score": 1.0, + "content": "original and perturbed accuracies", + "type": "text" + }, + { + "bbox": [ + 243, + 371, + 303, + 382 + ], + "score": 0.86, + "content": "\\operatorname { a c c } _ { \\mathrm { o r i g } } - \\operatorname { a c c } _ { \\mathrm { p m k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 369, + 506, + 384 + ], + "score": 1.0, + "content": ". A small drop in accuracy would indicate that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 381, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 506, + 393 + ], + "score": 1.0, + "content": "model is robust to small changes that occur naturally in videos. Instead, we find significant drops of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 391, + 462, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 133, + 402 + ], + "score": 0.85, + "content": "1 5 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 391, + 151, + 405 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 151, + 392, + 178, + 402 + ], + "score": 0.85, + "content": "1 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 391, + 462, + 405 + ], + "score": 1.0, + "content": "in accuracy on our two datasets, indicating sensitivity to such changes.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 424 + ], + "score": 1.0, + "content": "Noise augmentation One hypothesis for the accuracy drop from original to perturbed accuracy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "is that subtle artifacts and corruptions introduced by video compression schemes could degrade", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 450, + 445 + ], + "score": 1.0, + "content": "performance when evaluating on these corrupted frames. The worst-case nature of the", + "type": "text" + }, + { + "bbox": [ + 451, + 433, + 476, + 444 + ], + "score": 0.76, + "content": "\\mathrm { p m - k }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "metric", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 444, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 455 + ], + "score": 1.0, + "content": "could then be focusing on these corrupted frames. One model for these corruptions are the perturba-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "tions introduced in Hendrycks and Dietterich (2019). To test this hypothesis, we evaluate models", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "augmented with a subset of the perturbations (exactly one of: Gaussian noise, Gaussian blur, shot", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "noise, contrast change, impulse noise, or JPEG compression). We found that these augmentation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "score": 1.0, + "content": "schemes did not improve robustness against our perturbations substantially, and still result in accuracy", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 498, + 292, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 138, + 510 + ], + "score": 1.0, + "content": "drop of", + "type": "text" + }, + { + "bbox": [ + 138, + 498, + 165, + 509 + ], + "score": 0.84, + "content": "1 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 498, + 183, + 510 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 498, + 210, + 509 + ], + "score": 0.85, + "content": "1 6 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 498, + 292, + 510 + ], + "score": 1.0, + "content": "on the two datasets.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 120, + 528 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "robustness. We evaluate the model from Xie et al. (2018), which currently performs best", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 137, + 541 + ], + "score": 1.0, + "content": "against", + "type": "text" + }, + { + "bbox": [ + 137, + 528, + 151, + 539 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "attacks on ImageNet. We find that this model has a smaller accuracy drop than the two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "aforementioned model types on both datasets. However, we note that the robust model achieves", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "significantly lower original and perturbed accuracy than either of the two model types above, and the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 561, + 427, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 214, + 574 + ], + "score": 1.0, + "content": "robustness gain is modest", + "type": "text" + }, + { + "bbox": [ + 214, + 561, + 229, + 572 + ], + "score": 0.81, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 561, + 427, + 574 + ], + "score": 1.0, + "content": "compared to models of similar benign accuracy).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "Fine-tuning on video frames. To adapt to the new class vocabulary and the video domain, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "fine-tune several network architectures on the ImageNet-Vid and Youtube-BB training sets. For", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "Youtube-BB, we train on the anchor frames used for training in Gu et al. (2019), and for ImageNet-Vid", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 492, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 492, + 624 + ], + "score": 1.0, + "content": "we use all frames in the training set. We provide hyperparameters for all models in Appendix K.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 108, + 625, + 506, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "score": 1.0, + "content": "The resulting models significantly improve in accuracy over their ILSVRC pre-trained counterparts", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 129, + 649 + ], + "score": 1.0, + "content": "(e.g.,", + "type": "text" + }, + { + "bbox": [ + 130, + 636, + 150, + 647 + ], + "score": 0.86, + "content": "13 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 635, + 298, + 649 + ], + "score": 1.0, + "content": "on ImageNet-Vid-Robust and", + "type": "text" + }, + { + "bbox": [ + 298, + 636, + 318, + 646 + ], + "score": 0.87, + "content": "34 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "on YTBB-Robust for ResNet-50). This im-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 647, + 507, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 507, + 660 + ], + "score": 1.0, + "content": "provement in accuracy results in a modest improvement in the accuracy drop for YTBB-Robust,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 657, + 507, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 327, + 671 + ], + "score": 1.0, + "content": "but a finetuned ResNet-50 still suffers from a significant", + "type": "text" + }, + { + "bbox": [ + 327, + 658, + 349, + 668 + ], + "score": 0.87, + "content": "9 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 657, + 507, + 671 + ], + "score": 1.0, + "content": "drop. On ImageNet-Vid-Robust,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 668, + 384, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 316, + 681 + ], + "score": 1.0, + "content": "there is almost no change in the accuracy drop from", + "type": "text" + }, + { + "bbox": [ + 316, + 669, + 343, + 680 + ], + "score": 0.86, + "content": "1 5 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 668, + 354, + 681 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 354, + 669, + 381, + 680 + ], + "score": 0.87, + "content": "1 5 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 668, + 384, + 681 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Fine-tuning for detection on video frames. We further analyze whether additional supervision", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "in the form of bounding box annotations improves robustness. To this end, we train the Faster", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "R-CNN detection model Ren et al. (2015) with a ResNet-50 backbone on ImageNet-Vid. Following", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "standard practice, the detection backbone is pre-trained on ILSVRC-2012. To evaluate this detector", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 135, + 111, + 476, + 286 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "Table 2: Accuracies of five different model types and the best performing model. The model", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 501, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 501, + 102 + ], + "score": 1.0, + "content": "architecture is ResNet-50 unless noted otherwise. ‘FT’ is ‘fine-tuning.’ See Section 3.1 for details.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 135, + 111, + 476, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 111, + 476, + 286 + ], + "spans": [ + { + "bbox": [ + 135, + 111, + 476, + 286 + ], + "score": 0.981, + "html": "
Model Type Accuracy Accuracy △PerturbedOriginal
ImageNet-Vid-Robust
Trained on ILSVRC 67.5 [64.7, 70.3] 52.5 [49.5, 55.5] 15.0
+ Noise Augmentation 68.8 [66.0, 71.5] 53.2 [50.2, 56.2] 15.6
+lrobustness (ResNext-101) 54.3 [51.3, 57.2] 40.8 [39.0, 43.7] 12.4
+ FT on ImageNet- Vid 80.8 [78.3, 83.1] 65.7 [62.9, 68.5] 15.1
+ FT on ImageNet-Vid (ResNet-152) 84.8 [82.5, 86.8]70.2 [67.4, 72.8] 14.6
+ FT on ImageNet-Vid-Det 77.6 [75.1, 80.0] 65.4 [62.5, 68.1] 12.3
YTBB-Robust
Trained on ILSVRC 57.0 [54.9, 59.2]43.8 [41.7, 46.0] 13.2
+Noise Augmentation 62.3 [60.2, 64.4]45.7 [43.5, 47.9]16.6
+ l robustness (ResNext-101) 53.6 [51.4, 55.8]43.2 [41.0, 45.3]10.4
+ FT on Youtube-BB 91.4 [90.1, 92.6]82.0 [80.3, 83.7]9.4
+ FT on Youtube-BB (ResNet-152) 92.9 [91.6, 93.9]84.7 [83.0, 86.2]8.2
", + "type": "table", + "image_path": "93504e29d42d2d6cda7c44c90142ee15e467a45dfec98b321b9089ed326f2725.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 135, + 111, + 476, + 169.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 135, + 169.33333333333334, + 476, + 227.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 135, + 227.66666666666669, + 476, + 286.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 106, + 303, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "ILSVRC Trained The WordNet hierarchy enables us to repurpose models trained for the 1,000", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "class ILSVRC dataset on ImageNet-Vid-Robust and YTBB-Robust (see Appendix A.1).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "We evaluate a wide array of ILSVRC-2012 models (available from Cadene) against our natural", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "perturbations. Since these datasets present a substantial distribution shift from the original ILSVRC-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 346, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 297, + 362 + ], + "score": 1.0, + "content": "2012 validation, we expect the benign accuracy", + "type": "text" + }, + { + "bbox": [ + 297, + 349, + 324, + 360 + ], + "score": 0.85, + "content": "\\operatorname { a c c } _ { \\mathrm { o r i g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 346, + 506, + 362 + ], + "score": 1.0, + "content": "to be lower than the comparable accuracy on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "the ILSVRC-2012 validation set. However, our main interest here is in the difference between the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 243, + 384 + ], + "score": 1.0, + "content": "original and perturbed accuracies", + "type": "text" + }, + { + "bbox": [ + 243, + 371, + 303, + 382 + ], + "score": 0.86, + "content": "\\operatorname { a c c } _ { \\mathrm { o r i g } } - \\operatorname { a c c } _ { \\mathrm { p m k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 369, + 506, + 384 + ], + "score": 1.0, + "content": ". A small drop in accuracy would indicate that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 381, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 506, + 393 + ], + "score": 1.0, + "content": "model is robust to small changes that occur naturally in videos. Instead, we find significant drops of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 391, + 462, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 133, + 402 + ], + "score": 0.85, + "content": "1 5 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 391, + 151, + 405 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 151, + 392, + 178, + 402 + ], + "score": 0.85, + "content": "1 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 391, + 462, + 405 + ], + "score": 1.0, + "content": "in accuracy on our two datasets, indicating sensitivity to such changes.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 303, + 506, + 405 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 424 + ], + "score": 1.0, + "content": "Noise augmentation One hypothesis for the accuracy drop from original to perturbed accuracy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "is that subtle artifacts and corruptions introduced by video compression schemes could degrade", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 450, + 445 + ], + "score": 1.0, + "content": "performance when evaluating on these corrupted frames. The worst-case nature of the", + "type": "text" + }, + { + "bbox": [ + 451, + 433, + 476, + 444 + ], + "score": 0.76, + "content": "\\mathrm { p m - k }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "metric", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 444, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 455 + ], + "score": 1.0, + "content": "could then be focusing on these corrupted frames. One model for these corruptions are the perturba-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "tions introduced in Hendrycks and Dietterich (2019). To test this hypothesis, we evaluate models", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "augmented with a subset of the perturbations (exactly one of: Gaussian noise, Gaussian blur, shot", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "noise, contrast change, impulse noise, or JPEG compression). We found that these augmentation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "score": 1.0, + "content": "schemes did not improve robustness against our perturbations substantially, and still result in accuracy", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 498, + 292, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 138, + 510 + ], + "score": 1.0, + "content": "drop of", + "type": "text" + }, + { + "bbox": [ + 138, + 498, + 165, + 509 + ], + "score": 0.84, + "content": "1 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 498, + 183, + 510 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 498, + 210, + 509 + ], + "score": 0.85, + "content": "1 6 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 498, + 292, + 510 + ], + "score": 1.0, + "content": "on the two datasets.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 409, + 506, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 120, + 528 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "robustness. We evaluate the model from Xie et al. (2018), which currently performs best", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 137, + 541 + ], + "score": 1.0, + "content": "against", + "type": "text" + }, + { + "bbox": [ + 137, + 528, + 151, + 539 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "attacks on ImageNet. We find that this model has a smaller accuracy drop than the two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "aforementioned model types on both datasets. However, we note that the robust model achieves", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "significantly lower original and perturbed accuracy than either of the two model types above, and the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 561, + 427, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 214, + 574 + ], + "score": 1.0, + "content": "robustness gain is modest", + "type": "text" + }, + { + "bbox": [ + 214, + 561, + 229, + 572 + ], + "score": 0.81, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 561, + 427, + 574 + ], + "score": 1.0, + "content": "compared to models of similar benign accuracy).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 517, + 506, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "Fine-tuning on video frames. To adapt to the new class vocabulary and the video domain, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "fine-tune several network architectures on the ImageNet-Vid and Youtube-BB training sets. For", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "Youtube-BB, we train on the anchor frames used for training in Gu et al. (2019), and for ImageNet-Vid", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 492, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 492, + 624 + ], + "score": 1.0, + "content": "we use all frames in the training set. We provide hyperparameters for all models in Appendix K.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 579, + 506, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 625, + 506, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "score": 1.0, + "content": "The resulting models significantly improve in accuracy over their ILSVRC pre-trained counterparts", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 129, + 649 + ], + "score": 1.0, + "content": "(e.g.,", + "type": "text" + }, + { + "bbox": [ + 130, + 636, + 150, + 647 + ], + "score": 0.86, + "content": "13 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 635, + 298, + 649 + ], + "score": 1.0, + "content": "on ImageNet-Vid-Robust and", + "type": "text" + }, + { + "bbox": [ + 298, + 636, + 318, + 646 + ], + "score": 0.87, + "content": "34 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "on YTBB-Robust for ResNet-50). This im-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 647, + 507, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 507, + 660 + ], + "score": 1.0, + "content": "provement in accuracy results in a modest improvement in the accuracy drop for YTBB-Robust,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 657, + 507, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 327, + 671 + ], + "score": 1.0, + "content": "but a finetuned ResNet-50 still suffers from a significant", + "type": "text" + }, + { + "bbox": [ + 327, + 658, + 349, + 668 + ], + "score": 0.87, + "content": "9 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 657, + 507, + 671 + ], + "score": 1.0, + "content": "drop. On ImageNet-Vid-Robust,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 668, + 384, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 316, + 681 + ], + "score": 1.0, + "content": "there is almost no change in the accuracy drop from", + "type": "text" + }, + { + "bbox": [ + 316, + 669, + 343, + 680 + ], + "score": 0.86, + "content": "1 5 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 668, + 354, + 681 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 354, + 669, + 381, + 680 + ], + "score": 0.87, + "content": "1 5 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 668, + 384, + 681 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 624, + 507, + 681 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Fine-tuning for detection on video frames. We further analyze whether additional supervision", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "in the form of bounding box annotations improves robustness. To this end, we train the Faster", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "R-CNN detection model Ren et al. (2015) with a ResNet-50 backbone on ImageNet-Vid. Following", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "standard practice, the detection backbone is pre-trained on ILSVRC-2012. To evaluate this detector", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "for classification, we assign the class with the most confident bounding box as label to the image. We", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 476, + 107 + ], + "score": 1.0, + "content": "find that this transformation reduces accuracy compared to the model trained for classification", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 477, + 94, + 505, + 105 + ], + "score": 0.82, + "content": "( 7 7 . 6 \\%", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 120, + 117 + ], + "score": 1.0, + "content": "vs.", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 120, + 105, + 147, + 115 + ], + "score": 0.84, + "content": "8 0 . 8 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 147, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "). While there is a slight reduction in the accuracy drop caused by natural perturbations, the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 322, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 322, + 127 + ], + "score": 1.0, + "content": "reduction is well within the error bars for this test set.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 688, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "for classification, we assign the class with the most confident bounding box as label to the image. We", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 476, + 107 + ], + "score": 1.0, + "content": "find that this transformation reduces accuracy compared to the model trained for classification", + "type": "text" + }, + { + "bbox": [ + 477, + 94, + 505, + 105 + ], + "score": 0.82, + "content": "( 7 7 . 6 \\%", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 120, + 117 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 120, + 105, + 147, + 115 + ], + "score": 0.84, + "content": "8 0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "). While there is a slight reduction in the accuracy drop caused by natural perturbations, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 322, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 322, + 127 + ], + "score": 1.0, + "content": "reduction is well within the error bars for this test set.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 137, + 182, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 184, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 184, + 150 + ], + "score": 1.0, + "content": "3.2 DETECTION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "We further study the impact of natural perturbations on object detection. Specifically, we report", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "score": 1.0, + "content": "results for two related tasks: object localization and detection. Object detection is the standard", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "score": 1.0, + "content": "computer vision task of correctly classifying an object and finding the coordinates of a tight bounding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "score": 1.0, + "content": "box containing the object. “Object localization”, meanwhile, refers to only the subtask of finding the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 369, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 369, + 210 + ], + "score": 1.0, + "content": "bounding box, without attempting to correctly classify the object.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "We present our results on ImageNet-Vid-Robust, which contains dense bounding box labels", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "unlike Youtube-BB, which only labels boxes at 1 frame per second. We use the popular Faster", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "R-CNN Ren et al. (2015) and R-FCN Dai et al. (2016); Xiao and Jae Lee (2018) architectures for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "object detection and localization and report results in Table 3. For the R-FCN architecture, we use the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 225, + 265 + ], + "score": 1.0, + "content": "model from Xiao and Jae Lee", + "type": "text" + }, + { + "bbox": [ + 225, + 253, + 255, + 265 + ], + "score": 0.55, + "content": "( 2 0 1 8 ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 253, + 506, + 265 + ], + "score": 1.0, + "content": ". We first note the significant drop in mAP of 12 – 15 points for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 265, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 506, + 276 + ], + "score": 1.0, + "content": "object detection due to perturbed frames for both the Faster R-CNN and R-FCN architectures. Next,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "we show that localization is indeed easier than detection, as the mAP is higher for localization than", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "for detection (e.g., 76.6 vs 62.8 for Faster R-CNN with a ResNet-50 backbone). Perhaps surprisingly,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "however, switching to the localization task does not improve the drop between original and perturbed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "frames, indicating that natural perturbations induce both classification and localization errors. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 301, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 301, + 332 + ], + "score": 1.0, + "content": "show examples of detection failures in Figure 4.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "image", + "bbox": [ + 107, + 343, + 504, + 520 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 343, + 504, + 520 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 343, + 504, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 343, + 504, + 520 + ], + "score": 0.976, + "type": "image", + "image_path": "66cd8c2b8564417377f00f08cfec935c33d0a2cce3d3f68b46ab2790725206ef.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 107, + 343, + 504, + 402.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 107, + 402.0, + 504, + 461.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 107, + 461.0, + 504, + 520.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 529, + 505, + 595 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Figure 4: Naturally perturbed examples for detection. Red boxes indicate false positives; green boxes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "indicate true positives; white boxes are ground truth. Classification errors are common failures, such", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "as the fox on the left, which is classified correctly in the anchor frame, and misclassified as a sheep in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "score": 1.0, + "content": "a nearby frame. However, detection models also have localization errors, where the object of interest", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "is not correctly localized in addition to being misclassified, such as the airplane (middle) and the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 584, + 461, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 461, + 596 + ], + "score": 1.0, + "content": "motorcycle (right). All visualizations show predictions with confidence greater than 0.5.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + } + ], + "index": 24.25 + }, + { + "type": "title", + "bbox": [ + 108, + 619, + 257, + 631 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 258, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 258, + 632 + ], + "score": 1.0, + "content": "3.3 IMPACT OF DATASET REVIEW", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "We analyze the impact of our human review, described in Section 2.1, on the classifiers in our test", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 647, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 506, + 659 + ], + "score": 1.0, + "content": "bed. First, we compare the original and perturbed accuracies of a representative classifier (ResNet-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "score": 1.0, + "content": "152 finetuned) with and without review in Table 4. Our review improves the original accuracy by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 667, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 130, + 679 + ], + "score": 0.85, + "content": "3- 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 667, + 505, + 683 + ], + "score": 1.0, + "content": "by throwing away mislabeled or blurry anchor frames, and improves perturbed accuracy by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 130, + 690 + ], + "score": 0.82, + "content": "5- 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 680, + 469, + 692 + ], + "score": 1.0, + "content": "by discarding pairs of dissimilar frames. Our review reduces the accuracy drop by", + "type": "text" + }, + { + "bbox": [ + 469, + 680, + 491, + 690 + ], + "score": 0.85, + "content": "1 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 117, + 698, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 698, + 506, + 714 + ], + "score": 1.0, + "content": "4This model was originally trained on the 2015 subset of ImageNet-Vid. We evaluated this model on the", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "2015 validation set because the method requires access to pre-computed bounding box proposals which are", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 722, + 295, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 295, + 732 + ], + "score": 1.0, + "content": "available only for the 2015 subset of ImageNet-Vid.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 505, + 127 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 137, + 182, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 184, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 184, + 150 + ], + "score": 1.0, + "content": "3.2 DETECTION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "We further study the impact of natural perturbations on object detection. Specifically, we report", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "score": 1.0, + "content": "results for two related tasks: object localization and detection. Object detection is the standard", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 189 + ], + "score": 1.0, + "content": "computer vision task of correctly classifying an object and finding the coordinates of a tight bounding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "score": 1.0, + "content": "box containing the object. “Object localization”, meanwhile, refers to only the subtask of finding the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 369, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 369, + 210 + ], + "score": 1.0, + "content": "bounding box, without attempting to correctly classify the object.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 154, + 506, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "We present our results on ImageNet-Vid-Robust, which contains dense bounding box labels", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "unlike Youtube-BB, which only labels boxes at 1 frame per second. We use the popular Faster", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "R-CNN Ren et al. (2015) and R-FCN Dai et al. (2016); Xiao and Jae Lee (2018) architectures for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "object detection and localization and report results in Table 3. For the R-FCN architecture, we use the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 225, + 265 + ], + "score": 1.0, + "content": "model from Xiao and Jae Lee", + "type": "text" + }, + { + "bbox": [ + 225, + 253, + 255, + 265 + ], + "score": 0.55, + "content": "( 2 0 1 8 ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 253, + 506, + 265 + ], + "score": 1.0, + "content": ". We first note the significant drop in mAP of 12 – 15 points for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 265, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 506, + 276 + ], + "score": 1.0, + "content": "object detection due to perturbed frames for both the Faster R-CNN and R-FCN architectures. Next,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "we show that localization is indeed easier than detection, as the mAP is higher for localization than", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "for detection (e.g., 76.6 vs 62.8 for Faster R-CNN with a ResNet-50 backbone). Perhaps surprisingly,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "however, switching to the localization task does not improve the drop between original and perturbed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "frames, indicating that natural perturbations induce both classification and localization errors. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 301, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 301, + 332 + ], + "score": 1.0, + "content": "show examples of detection failures in Figure 4.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 209, + 506, + 332 + ] + }, + { + "type": "image", + "bbox": [ + 107, + 343, + 504, + 520 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 343, + 504, + 520 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 343, + 504, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 343, + 504, + 520 + ], + "score": 0.976, + "type": "image", + "image_path": "66cd8c2b8564417377f00f08cfec935c33d0a2cce3d3f68b46ab2790725206ef.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 107, + 343, + 504, + 402.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 107, + 402.0, + 504, + 461.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 107, + 461.0, + 504, + 520.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 529, + 505, + 595 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Figure 4: Naturally perturbed examples for detection. Red boxes indicate false positives; green boxes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "indicate true positives; white boxes are ground truth. Classification errors are common failures, such", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "as the fox on the left, which is classified correctly in the anchor frame, and misclassified as a sheep in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "score": 1.0, + "content": "a nearby frame. 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Both detection and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 506, + 104 + ], + "score": 1.0, + "content": "localization suffer from significant drops in mAP due to the perturbations. (*Model trained on", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 490, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 490, + 114 + ], + "score": 1.0, + "content": "ILSVRC Det and VID 2015 datasets, and evaluated on the 2015 subset of ILSVRC-VID 2017.)", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 111, + 122, + 497, + 220 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 122, + 497, + 220 + ], + "spans": [ + { + "bbox": [ + 111, + 122, + 497, + 220 + ], + "score": 0.951, + "html": "
TaskModelmAP OriginalmAP PerturbedmAP △
DetectionFRCNN,ResNet562.848.814.0
FRCNN,ResNet 10163.150.612.5
R-FCN, ResNet 101 Xiao and Jae Lee (2018)*79.4*63.7*15.7*
LocalizationFRCNN,ResNet5076.664.212.4
FRCNN, ResNet 10177.866.311.5
R-FCN, ResNet 101*80.9*70.3*10.6*
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Accuracy
ReviewedOriginalPerturbedDrop
ImageNet-Vid-Robust80.364.116.2
X84.870.214.4
YTBB-Robust88.178.110.0
X92.984.78.9
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TaskModelmAP OriginalmAP PerturbedmAP △
DetectionFRCNN,ResNet562.848.814.0
FRCNN,ResNet 10163.150.612.5
R-FCN, ResNet 101 Xiao and Jae Lee (2018)*79.4*63.7*15.7*
LocalizationFRCNN,ResNet5076.664.212.4
FRCNN, ResNet 10177.866.311.5
R-FCN, ResNet 101*80.9*70.3*10.6*
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Accuracy
ReviewedOriginalPerturbedDrop
ImageNet-Vid-Robust80.364.116.2
X84.870.214.4
YTBB-Robust88.178.110.0
X92.984.78.9
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While various forms of adversarial examples have been studied, the major-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 209, + 690 + ], + "score": 1.0, + "content": "ity of research focuses on", + "type": "text" + }, + { + "bbox": [ + 210, + 677, + 220, + 689 + ], + "score": 0.87, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 676, + 507, + 690 + ], + "score": 1.0, + "content": "robustness Goodfellow et al. (2014); Biggio and Roli (2018). However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "it is unclear whether adversarial examples pose a problem for classifier robustness outside of a truly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 432, + 712 + ], + "score": 1.0, + "content": "worst case context. It is an open question whether perfect robustness against a", + "type": "text" + }, + { + "bbox": [ + 433, + 699, + 443, + 711 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "adversary will", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "induce robustness to realistic image distortions such as those studied in this paper. Recent work has", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 383, + 733 + ], + "score": 1.0, + "content": "proposed more realistic image modifications such as small rotations", + "type": "text" + }, + { + "bbox": [ + 383, + 721, + 392, + 730 + ], + "score": 0.45, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "translations Engstrom et al.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 665, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "(2017); Azulay and Weiss (2018); Fawzi and Frossard (2015); Kanbak et al. (2017), hue and color", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "changes Hosseini and Poovendran (2018), image stylization Geirhos et al. (2018a) and synthetic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "image corruptions such as Gaussian blur and JPEG compression Hendrycks and Dietterich (2019);", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 443, + 129 + ], + "score": 1.0, + "content": "Geirhos et al. (2018b). Even though the above examples are more realistic than the", + "type": "text" + }, + { + "bbox": [ + 444, + 115, + 454, + 128 + ], + "score": 0.89, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "model, they", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "still synthetically modify the input images to generate perturbed versions. In contrast, our work", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 468, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 468, + 150 + ], + "score": 1.0, + "content": "performs no synthetic modification and instead uses images that naturally occur in videos.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 159, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Utilizing videos to study robustness. In work concurrent to ours, Gu et al. (2019) exploit the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "temporal structure in videos to study robustness. However, their experiments suggest a substantially", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "smaller drop in classification accuracy. The primary reason for this is a less stringent metric used", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "in Gu et al. (2019). By contrast, our “pm-k” metric is inspired by the “worst-of-k” metric used in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "score": 1.0, + "content": "prior work Engstrom et al. (2017), highlighting the sensitivity of models to natural perturbations.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "In Appendix E we study the differences between the two metrics in more detail. Furthermore, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 224, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 237 + ], + "score": 1.0, + "content": "lack of human review and the high label error-rate we discovered in Youtube-BB(Table 1) presents a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 338, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 338, + 248 + ], + "score": 1.0, + "content": "troubling confounding factor that we resolve in our work.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "Distribution shift. Small, benign changes in the test distribution are often referred to as distribution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "shift. Recht et al. (2019) explore this phenomenon by constructing new test sets for CIFAR-10 and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 279, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 292 + ], + "score": 1.0, + "content": "ImageNet and observe performance drops for a large suite of models on the newly constructed test", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "sets. Similar to our Figure 3, the relationship between original and new test set accuracy is also", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "approximately linear. However, the images in their test set bear little visual similarity to images in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "original test set, while all of our failure cases in ImageNet-Vid-Robust and YTBB-Robust are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "on perceptually similar images. In a similar vein of study, Torralba et al. (2011) studies distribution", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 333, + 483, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 483, + 347 + ], + "score": 1.0, + "content": "shift across different computer vision data sets such as Caltech-101, PASCAL, and ImageNet.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 507, + 370 + ], + "score": 1.0, + "content": "Computer vision. A common issue when applying image based models to videos is flickering,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "where object detectors spuriously produce false-positives or false-negatives in isolated frames or", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "groups of frames. Jin et al. (2018) explicitly identify such failures and use a technique reminiscent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "of adversarially robust training to improve image-based models. A similar line of work focuses on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "score": 1.0, + "content": "improving object detection in videos as objects become occluded or move quickly Kang et al. (2017);", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "Feichtenhofer et al. (2017); Zhu et al. (2017); Xiao and Jae Lee (2018). The focus in this line of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "work has generally been on improving object detection when objects transform in a way that makes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "score": 1.0, + "content": "recognition difficult from a single frame, such as fast motion or occlusion. In this work, we document", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "score": 1.0, + "content": "a broader set of failure cases for image-based classifiers and detectors and show that failures occur", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 455, + 336, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 336, + 467 + ], + "score": 1.0, + "content": "when the neighboring frames are imperceptibly different.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 195, + 492 + ], + "lines": [ + { + "bbox": [ + 104, + 478, + 197, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 197, + 496 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "Our study quantifies the sensitivity of image classifiers to naturally occuring temporal perturbations.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "We show that these perturbations can cause significant drops in accuracy for a wide range of models", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "for both classification and detection. Our work on analyzing this failure mode opens multiple avenues", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 534, + 185, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 185, + 546 + ], + "score": 1.0, + "content": "for future research:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "score": 1.0, + "content": "Building more robust models. Our ImageNet-Vid-Robust and YTBB-Robust datasets", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "provide a standard measure for robustness that can be applied to any classification or detection", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "model. In Table 2, we evaluated several commonly used models and found that all of them suffer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "from substantial accuracy drops due to natural perturbations. In particular, we found that model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 439, + 612 + ], + "score": 1.0, + "content": "improvements with respect to artificial perturbations (such as image corruptions or", + "type": "text" + }, + { + "bbox": [ + 440, + 600, + 453, + 611 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "adversaries)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "induce at best modest improvements in robustness. We hope that our standardized datasets and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 498, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 498, + 635 + ], + "score": 1.0, + "content": "evaluation metric will enable future work to quantify improvements in natural robustness directly.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Further natural perturbations. Videos provide a straightforward method for collecting natural", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "perturbations of images, admitting the study of realistic forms of robustness for machine learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "methods. Other methods for generating these natural perturbations are likely to provide additional", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "insights into model robustness. As an example, photo sharing websites contain a large number of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 507, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 507, + 701 + ], + "score": 1.0, + "content": "near-duplicate images: pairs of images of the same scene captured at different times, viewpoints,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "or from a different camera Recht et al. (2019). More generally, devising similar, domain-specific", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "strategies to collect, verify, and measure robustness to natural perturbations in domains such as natural", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 441, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 441, + 733 + ], + "score": 1.0, + "content": "language processing or speech recognition is a promising direction for future work.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "(2017); Azulay and Weiss (2018); Fawzi and Frossard (2015); Kanbak et al. (2017), hue and color", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "changes Hosseini and Poovendran (2018), image stylization Geirhos et al. (2018a) and synthetic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "image corruptions such as Gaussian blur and JPEG compression Hendrycks and Dietterich (2019);", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 443, + 129 + ], + "score": 1.0, + "content": "Geirhos et al. (2018b). Even though the above examples are more realistic than the", + "type": "text" + }, + { + "bbox": [ + 444, + 115, + 454, + 128 + ], + "score": 0.89, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "model, they", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "still synthetically modify the input images to generate perturbed versions. In contrast, our work", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 468, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 468, + 150 + ], + "score": 1.0, + "content": "performs no synthetic modification and instead uses images that naturally occur in videos.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 159, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Utilizing videos to study robustness. In work concurrent to ours, Gu et al. (2019) exploit the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "temporal structure in videos to study robustness. However, their experiments suggest a substantially", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "smaller drop in classification accuracy. The primary reason for this is a less stringent metric used", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "in Gu et al. (2019). By contrast, our “pm-k” metric is inspired by the “worst-of-k” metric used in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 507, + 215 + ], + "score": 1.0, + "content": "prior work Engstrom et al. (2017), highlighting the sensitivity of models to natural perturbations.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "In Appendix E we study the differences between the two metrics in more detail. Furthermore, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 224, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 237 + ], + "score": 1.0, + "content": "lack of human review and the high label error-rate we discovered in Youtube-BB(Table 1) presents a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 338, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 338, + 248 + ], + "score": 1.0, + "content": "troubling confounding factor that we resolve in our work.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 158, + 507, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "Distribution shift. Small, benign changes in the test distribution are often referred to as distribution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "shift. Recht et al. (2019) explore this phenomenon by constructing new test sets for CIFAR-10 and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 279, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 292 + ], + "score": 1.0, + "content": "ImageNet and observe performance drops for a large suite of models on the newly constructed test", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "sets. Similar to our Figure 3, the relationship between original and new test set accuracy is also", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "approximately linear. However, the images in their test set bear little visual similarity to images in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "original test set, while all of our failure cases in ImageNet-Vid-Robust and YTBB-Robust are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "on perceptually similar images. In a similar vein of study, Torralba et al. (2011) studies distribution", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 333, + 483, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 483, + 347 + ], + "score": 1.0, + "content": "shift across different computer vision data sets such as Caltech-101, PASCAL, and ImageNet.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 258, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 507, + 370 + ], + "score": 1.0, + "content": "Computer vision. A common issue when applying image based models to videos is flickering,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "where object detectors spuriously produce false-positives or false-negatives in isolated frames or", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "groups of frames. Jin et al. (2018) explicitly identify such failures and use a technique reminiscent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "of adversarially robust training to improve image-based models. A similar line of work focuses on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "score": 1.0, + "content": "improving object detection in videos as objects become occluded or move quickly Kang et al. (2017);", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "Feichtenhofer et al. (2017); Zhu et al. (2017); Xiao and Jae Lee (2018). The focus in this line of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "work has generally been on improving object detection when objects transform in a way that makes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 444 + ], + "score": 1.0, + "content": "recognition difficult from a single frame, such as fast motion or occlusion. In this work, we document", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "score": 1.0, + "content": "a broader set of failure cases for image-based classifiers and detectors and show that failures occur", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 455, + 336, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 336, + 467 + ], + "score": 1.0, + "content": "when the neighboring frames are imperceptibly different.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 354, + 507, + 467 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 195, + 492 + ], + "lines": [ + { + "bbox": [ + 104, + 478, + 197, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 197, + 496 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "Our study quantifies the sensitivity of image classifiers to naturally occuring temporal perturbations.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "We show that these perturbations can cause significant drops in accuracy for a wide range of models", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "for both classification and detection. Our work on analyzing this failure mode opens multiple avenues", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 534, + 185, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 185, + 546 + ], + "score": 1.0, + "content": "for future research:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 501, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "score": 1.0, + "content": "Building more robust models. Our ImageNet-Vid-Robust and YTBB-Robust datasets", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "provide a standard measure for robustness that can be applied to any classification or detection", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "model. In Table 2, we evaluated several commonly used models and found that all of them suffer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "from substantial accuracy drops due to natural perturbations. In particular, we found that model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 439, + 612 + ], + "score": 1.0, + "content": "improvements with respect to artificial perturbations (such as image corruptions or", + "type": "text" + }, + { + "bbox": [ + 440, + 600, + 453, + 611 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "adversaries)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "induce at best modest improvements in robustness. We hope that our standardized datasets and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 498, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 498, + 635 + ], + "score": 1.0, + "content": "evaluation metric will enable future work to quantify improvements in natural robustness directly.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 557, + 505, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Further natural perturbations. Videos provide a straightforward method for collecting natural", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "perturbations of images, admitting the study of realistic forms of robustness for machine learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "methods. Other methods for generating these natural perturbations are likely to provide additional", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "insights into model robustness. As an example, photo sharing websites contain a large number of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 507, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 507, + 701 + ], + "score": 1.0, + "content": "near-duplicate images: pairs of images of the same scene captured at different times, viewpoints,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "or from a different camera Recht et al. (2019). 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ModelAccuracy OriginalAccuracy Perturbed△
resnet152_finetuned84.8 [82.5, 86.8]70.2 [67.4, 72.8]14.6
resnet50_finetuned80.8 [78.3, 83.1]65.7 [62.9, 68.5]15.1
vgg16bn_finetuned78.0 [75.4, 80.4]61.0 [58.1, 63.9]17.0
nasnetalarge_imagenet_pretrained77.6 [75.1, 80.1]62.1 [59.2, 65.0]15.5
resnet50_detection77.6 [75.1, 80.1]65.0 [62.1, 67.8]12.6
inceptionresnetv2_imagenet_pretrained75.7 [73.1, 78.2]58.7 [55.7, 61.6]17.0
dpn107_imagenet_pretrained75.6 [72.9, 78.1]59.1 [56.1, 62.0]16.5
inceptionv4_imagenet_pretrained75.3 [72.6, 77.8]59.0 [56.0, 61.9]16.3
dpn92_imagenet_pretrained74.4 [71.7, 76.9]56.8 [53.8, 59.7]17.6
dpn131_imagenet_pretrained74.0 [71.3, 76.6]59.9 [56.9, 62.8]14.1
dpn68b_imagenet_pretrained73.7 [71.0, 76.2]54.0 [51.0, 57.0]19.7
resnext101_32x4d_imagenet_pretrained73.3 [70.6, 75.9]57.2 [54.2, 60.1]16.1
resnext101_64x4d_imagenet_pretrained72.9 [70.1, 75.5]56.6 [53.7, 59.6]16.3
resnet152_imagenet_pretrained72.8 [70.0, 75.4]57.0 [54.0, 59.9]15.8
resnet1O1_imagenet_pretrained71.5 [68.7, 74.1]53.7 [50.8, 56.7]17.8
fbresnet152_imagenet_pretrained71.5 [68.7, 74.1]54.5 [51.5, 57.4]17.0
densenet161_imagenet_pretrained71.4 [68.7, 74.1]55.1 [52.1, 58.1]16.3
densenet169_imagenet_pretrained70.2 [67.5, 72.9]53.1 [50.1, 56.1]17.1
densenet2O1_imagenet_pretrained70.2 [67.5, 72.9]53.4 [50.4, 56.4]16.8
dpn68_imagenet_pretrained69.4 [66.6, 72.1]53.3 [50.3, 56.3]16.1
bninception_imagenet_pretrained69.0 [66.2, 71.7]49.0 [46.0, 51.9]20.0
densenet121_imagenet_pretrained69.0 [66.2, 71.7]50.9 [47.9, 53.8]18.1
nasnetamobile_imagenet_pretrained68.8 [66.0, 71.5]48.4 [45.4, 51.4]20.4
resnet50_augment_ jpeg_compression68.8 [66.0, 71.5]53.2 [50.2, 56.2]15.6
resnet34_imagenet_pretrained68.0 [65.2, 70.7]48.0 [45.0, 51.0]20.0
resnet50_augment impulse_noise67.7 [64.9, 70.5]50.2 [47.2, 53.2]17.5
resnet50_augment_gaussian_blur67.7 [64.9, 70.5]52.5 [49.5, 55.5]15.2
resnet5O_imagenet_pretrained67.5 [64.7, 70.3]52.5 [49.5, 55.5]15.0
resnet50_augment gaussian_noise67.4 [64.5, 70.1]50.6 [47.6, 53.6]16.8
resnet50_augment shot_noise66.5 [63.6, 69.2]51.1 [48.1, 54.1]15.4
vgg16_bn_imagenet_pretrained66.4 [63.5, 69.1]47.4 [44.5, 50.4]19.0
resnet50_augment_ _defocus_blur66.3 [63.4, 69.1]47.6 [44.6, 50.6]18.7
vgg19_bn_imagenet_pretrained65.6 [62.7, 68.4]46.6 [43.6, 49.6]19.0
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(2016)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 506, + 141 + ], + "score": 1.0, + "content": "as well as trackers Bertinetto et al. (2016). We chose to work with the 2017 ImageNet-Vid dataset", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 506, + 152 + ], + "score": 1.0, + "content": "because it is a superset of the 2015 dataset. In total, the 2017 ImageNet-Vid dataset consists of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "1,181,113 training frames from 4,000 videos and 512,360 validation frames from 1,314 videos. The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "score": 1.0, + "content": "videos have frame rates ranging from 9 to 59 frames per second (fps), with a median fps of 29. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "videos range from 0.44 to 96 seconds in duration with a median duration of 12 seconds. Each frame", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 198 + ], + "score": 1.0, + "content": "is annotated with labels indicating the presence or absence of 30 object classes and corresponding", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "bounding boxes for any label present in the frame. The 30 classes are ancestors of 293 of the 1,000", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 200, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 200, + 217 + ], + "score": 1.0, + "content": "ILSVRC-2012 classes.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 118, + 506, + 217 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 226, + 195, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 224, + 196, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 196, + 238 + ], + "score": 1.0, + "content": "A.2 YOUTUBE-BB", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "The 2017 Youtube-BB is a a large scale dataset with 8,146,143 annotated training frames 253,569", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "unique videos and with 1,013,246 validation frames from 31,829 videos. The video segments are", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "approximately 19 seconds long on average. Each frame is annotated with exactly one label indicating", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "the presence of 22 object classes, all of which are ancestors of 229 out of the ILSVRC-2012 classes.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 106, + 242, + 506, + 287 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 298, + 363, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 364, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 364, + 313 + ], + "score": 1.0, + "content": "B FULL ORIGINAL VS PERTURBED ACCURACIES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 319, + 247, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 248, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 248, + 330 + ], + "score": 1.0, + "content": "B.1 IM A G ENE T-VI D-RO B U S T", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 318, + 248, + 330 + ] + }, + { + "type": "table", + "bbox": [ + 129, + 338, + 482, + 732 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 129, + 338, + 482, + 732 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 129, + 338, + 482, + 732 + ], + "spans": [ + { + "bbox": [ + 129, + 338, + 482, + 732 + ], + "score": 0.984, + "html": "
ModelAccuracy OriginalAccuracy Perturbed△
resnet152_finetuned84.8 [82.5, 86.8]70.2 [67.4, 72.8]14.6
resnet50_finetuned80.8 [78.3, 83.1]65.7 [62.9, 68.5]15.1
vgg16bn_finetuned78.0 [75.4, 80.4]61.0 [58.1, 63.9]17.0
nasnetalarge_imagenet_pretrained77.6 [75.1, 80.1]62.1 [59.2, 65.0]15.5
resnet50_detection77.6 [75.1, 80.1]65.0 [62.1, 67.8]12.6
inceptionresnetv2_imagenet_pretrained75.7 [73.1, 78.2]58.7 [55.7, 61.6]17.0
dpn107_imagenet_pretrained75.6 [72.9, 78.1]59.1 [56.1, 62.0]16.5
inceptionv4_imagenet_pretrained75.3 [72.6, 77.8]59.0 [56.0, 61.9]16.3
dpn92_imagenet_pretrained74.4 [71.7, 76.9]56.8 [53.8, 59.7]17.6
dpn131_imagenet_pretrained74.0 [71.3, 76.6]59.9 [56.9, 62.8]14.1
dpn68b_imagenet_pretrained73.7 [71.0, 76.2]54.0 [51.0, 57.0]19.7
resnext101_32x4d_imagenet_pretrained73.3 [70.6, 75.9]57.2 [54.2, 60.1]16.1
resnext101_64x4d_imagenet_pretrained72.9 [70.1, 75.5]56.6 [53.7, 59.6]16.3
resnet152_imagenet_pretrained72.8 [70.0, 75.4]57.0 [54.0, 59.9]15.8
resnet1O1_imagenet_pretrained71.5 [68.7, 74.1]53.7 [50.8, 56.7]17.8
fbresnet152_imagenet_pretrained71.5 [68.7, 74.1]54.5 [51.5, 57.4]17.0
densenet161_imagenet_pretrained71.4 [68.7, 74.1]55.1 [52.1, 58.1]16.3
densenet169_imagenet_pretrained70.2 [67.5, 72.9]53.1 [50.1, 56.1]17.1
densenet2O1_imagenet_pretrained70.2 [67.5, 72.9]53.4 [50.4, 56.4]16.8
dpn68_imagenet_pretrained69.4 [66.6, 72.1]53.3 [50.3, 56.3]16.1
bninception_imagenet_pretrained69.0 [66.2, 71.7]49.0 [46.0, 51.9]20.0
densenet121_imagenet_pretrained69.0 [66.2, 71.7]50.9 [47.9, 53.8]18.1
nasnetamobile_imagenet_pretrained68.8 [66.0, 71.5]48.4 [45.4, 51.4]20.4
resnet50_augment_ jpeg_compression68.8 [66.0, 71.5]53.2 [50.2, 56.2]15.6
resnet34_imagenet_pretrained68.0 [65.2, 70.7]48.0 [45.0, 51.0]20.0
resnet50_augment impulse_noise67.7 [64.9, 70.5]50.2 [47.2, 53.2]17.5
resnet50_augment_gaussian_blur67.7 [64.9, 70.5]52.5 [49.5, 55.5]15.2
resnet5O_imagenet_pretrained67.5 [64.7, 70.3]52.5 [49.5, 55.5]15.0
resnet50_augment gaussian_noise67.4 [64.5, 70.1]50.6 [47.6, 53.6]16.8
resnet50_augment shot_noise66.5 [63.6, 69.2]51.1 [48.1, 54.1]15.4
vgg16_bn_imagenet_pretrained66.4 [63.5, 69.1]47.4 [44.5, 50.4]19.0
resnet50_augment_ _defocus_blur66.3 [63.4, 69.1]47.6 [44.6, 50.6]18.7
vgg19_bn_imagenet_pretrained65.6 [62.7, 68.4]46.6 [43.6, 49.6]19.0
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vgg19_imagenet_pretrained63.2 [60.3, 66.1]45.4 [42.4, 48.3]17.8
resnet18_imagenet_pretrained61.9 [59.0, 64.8]41.5 [38.6, 44.4]20.4
vgg13_bn_imagenet_pretrained61.9 [59.0, 64.8]43.3 [40.3, 46.3]18.6
vgg16_imagenet_pretrained61.4 [58.5, 64.3]43.1 [40.2, 46.1]18.3
vgg11_bn_imagenet_pretrained60.9 [57.9, 63.8]43.2 [40.3, 46.2]17.7
vgg13_imagenet_pretrained59.6 [56.6, 62.5]41.1 [38.2, 44.1]18.5
vgg11_imagenet_pretrained57.3 [54.4, 60.3]41.3 [38.4, 44.3]16.0
alexnet_finetuned57.3 [54.3, 60.2]43.6 [40.7, 46.6]13.7
ResNeXtDenoiseAll-101_robust_pgd54.3 [51.3, 57.2]40.8 [37.8, 43.7]13.5
squeezenet1_1_imagenet_pretrained49.8 [46.8, 52.8]31.7 [28.9, 34.5]18.1
alexnet_imagenet_pretrained49.4 [46.4, 52.4]32.0 [29.3, 34.8]17.4
resnet50_augment contrast_change38.3 [35.5, 41.3]23.3 [20.8, 25.9]15.0
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ModelAccuracy OriginalAccuracy Perturbed△
resnet152_finetuned92.9 [91.2, 94.3]84.7 [82.4, 86.8]8.2
resnet50_finetuned91.4 [89.6, 93.0]82.0 [79.6, 84.2]9.4
inceptionresnetv2_finetuned91.3 [89.5, 92.9]79.0 [76.4, 81.3]12.3
vgg19_finetuned90.5 [88.6, 92.2]79.1 [76.5, 81.4]11.4
vgg16_finetuned89.1 [87.1, 90.8]78.0 [75.4, 80.4]11.1
inceptionv4_finetuned88.5 [86.5, 90.3]76.3 [73.6, 78.7]12.2
resnet18_finetuned88.0 [85.9, 89.8]76.2 [73.6, 78.7]11.8
alexnet_finetuned80.6 [78.2, 82.9]64.4 [61.5, 67.3]16.2
pnasnet5large_imagenet_pretrained65.2 [62.3, 68.0]51.0 [48.0, 54.0]14.2
nasnetalarge_imagenet_pretrained64.9 [62.0, 67.7]51.4 [48.4, 54.4]13.5
inceptionresnetv2_imagenet_pretrained64.5 [61.6, 67.4] 50.4 [47.5, 53.4]14.1
dpn98_imagenet_pretrained64.1 [61.2, 66.9]49.0 [46.0, 52.0]15.1
dpn107_imagenet_pretrained64.1 [61.2, 66.9]50.1 [47.2, 53.1]14.0
dpn131_imagenet_pretrained64.0 [61.1, 66.8]49.9 [46.9, 52.9]14.1
inceptionv4_imagenet_pretrained63.6 [60.7, 66.4]48.8 [45.8, 51.8]14.8
Xception_imagenet_pretrained63.2 [60.2, 66.0]47.6 [44.6, 50.6]15.6
dpn92_imagenet_pretrained62.3 [59.3, 65.1]47.7 [44.8, 50.7]14.6
resnet50_augment_jpeg_compressioon62.3 [59.4, 65.2]45.7 [42.8, 48.7]16.6
polynet_imagenet_pretrained61.4 [58.4, 64.3]47.3 [44.4, 50.3]14.1
nasnetamobile_imagenet_pretrained61.4 [58.4, 64.3]43.0 [40.1, 46.0]18.4
resnet50_augment__shot_noise61.3 [58.3, 64.2]46.4 [43.4, 49.3]14.9
dpn68_imagenet_pretrained61.2 [58.3, 64.1]44.2 [41.2, 47.2]17.0
fbresnet152_imagenet_pretrained61.1 [58.1, 64.0]45.9 [42.9, 48.8]15.2
resnet152_imagenet_pretrained60.8 [57.8, 63.7]46.5 [43.5, 49.5]14.3
resnet101_imagenet_pretrained60.8 [57.8, 63.7]45.2 [42.2, 48.2]15.6
senet154_imagenet_pretrained60.7 [57.7, 63.6]47.2 [44.3, 50.2]13.5
resnet50_augment__impulse_noise60.6 [57.7, 63.5]45.5 [42.6, 48.5]15.1
se_resnet101_imagenet_pretrained60.5 [57.6, 63.4]45.6 [42.6, 48.6]14.9
bninception_imagenet_pretrained60.4 [57.4, 63.3]41.8 [38.9, 44.7]18.6
densenetl61_imagenet_pretrained60.2 [57.3, 63.1]46.4 [43.4, 49.4]13.8
resnet50_augment_gaussian_noise60.2 [57.3, 63.1]45.7 [42.8, 48.7]14.5
se_resnext50_32x4d_imagenet_pretrained59.9 [56.9, 62.8]45.7 [42.7, 48.6]14.2
dpn68b_imagenet_pretrained59.7 [56.7, 62.6]45.9 [42.9, 48.8]13.8
inceptionv3_imagenet_pretrained59.6 [56.6, 62.5]43.8 [40.8, 46.8]15.8
densenet121_imagenet_pretrained59.5 [56.5, 62.4]43.1 [40.1, 46.0]16.4
se_resnext101_32x4d_imagenet_pretrained59.2 [56.3, 62.1]45.2 [42.3, 48.2]14.0
densenet2O1_imagenet_pretrained59.2 [56.2, 62.1]44.8 [41.8, 47.8]14.4
densenet169_imagenet_pretrained59.2 [56.2, 62.1]44.6 [41.7, 47.6]14.6
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vgg19_imagenet_pretrained63.2 [60.3, 66.1]45.4 [42.4, 48.3]17.8
resnet18_imagenet_pretrained61.9 [59.0, 64.8]41.5 [38.6, 44.4]20.4
vgg13_bn_imagenet_pretrained61.9 [59.0, 64.8]43.3 [40.3, 46.3]18.6
vgg16_imagenet_pretrained61.4 [58.5, 64.3]43.1 [40.2, 46.1]18.3
vgg11_bn_imagenet_pretrained60.9 [57.9, 63.8]43.2 [40.3, 46.2]17.7
vgg13_imagenet_pretrained59.6 [56.6, 62.5]41.1 [38.2, 44.1]18.5
vgg11_imagenet_pretrained57.3 [54.4, 60.3]41.3 [38.4, 44.3]16.0
alexnet_finetuned57.3 [54.3, 60.2]43.6 [40.7, 46.6]13.7
ResNeXtDenoiseAll-101_robust_pgd54.3 [51.3, 57.2]40.8 [37.8, 43.7]13.5
squeezenet1_1_imagenet_pretrained49.8 [46.8, 52.8]31.7 [28.9, 34.5]18.1
alexnet_imagenet_pretrained49.4 [46.4, 52.4]32.0 [29.3, 34.8]17.4
resnet50_augment contrast_change38.3 [35.5, 41.3]23.3 [20.8, 25.9]15.0
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ModelAccuracy OriginalAccuracy Perturbed△
resnet152_finetuned92.9 [91.2, 94.3]84.7 [82.4, 86.8]8.2
resnet50_finetuned91.4 [89.6, 93.0]82.0 [79.6, 84.2]9.4
inceptionresnetv2_finetuned91.3 [89.5, 92.9]79.0 [76.4, 81.3]12.3
vgg19_finetuned90.5 [88.6, 92.2]79.1 [76.5, 81.4]11.4
vgg16_finetuned89.1 [87.1, 90.8]78.0 [75.4, 80.4]11.1
inceptionv4_finetuned88.5 [86.5, 90.3]76.3 [73.6, 78.7]12.2
resnet18_finetuned88.0 [85.9, 89.8]76.2 [73.6, 78.7]11.8
alexnet_finetuned80.6 [78.2, 82.9]64.4 [61.5, 67.3]16.2
pnasnet5large_imagenet_pretrained65.2 [62.3, 68.0]51.0 [48.0, 54.0]14.2
nasnetalarge_imagenet_pretrained64.9 [62.0, 67.7]51.4 [48.4, 54.4]13.5
inceptionresnetv2_imagenet_pretrained64.5 [61.6, 67.4] 50.4 [47.5, 53.4]14.1
dpn98_imagenet_pretrained64.1 [61.2, 66.9]49.0 [46.0, 52.0]15.1
dpn107_imagenet_pretrained64.1 [61.2, 66.9]50.1 [47.2, 53.1]14.0
dpn131_imagenet_pretrained64.0 [61.1, 66.8]49.9 [46.9, 52.9]14.1
inceptionv4_imagenet_pretrained63.6 [60.7, 66.4]48.8 [45.8, 51.8]14.8
Xception_imagenet_pretrained63.2 [60.2, 66.0]47.6 [44.6, 50.6]15.6
dpn92_imagenet_pretrained62.3 [59.3, 65.1]47.7 [44.8, 50.7]14.6
resnet50_augment_jpeg_compressioon62.3 [59.4, 65.2]45.7 [42.8, 48.7]16.6
polynet_imagenet_pretrained61.4 [58.4, 64.3]47.3 [44.4, 50.3]14.1
nasnetamobile_imagenet_pretrained61.4 [58.4, 64.3]43.0 [40.1, 46.0]18.4
resnet50_augment__shot_noise61.3 [58.3, 64.2]46.4 [43.4, 49.3]14.9
dpn68_imagenet_pretrained61.2 [58.3, 64.1]44.2 [41.2, 47.2]17.0
fbresnet152_imagenet_pretrained61.1 [58.1, 64.0]45.9 [42.9, 48.8]15.2
resnet152_imagenet_pretrained60.8 [57.8, 63.7]46.5 [43.5, 49.5]14.3
resnet101_imagenet_pretrained60.8 [57.8, 63.7]45.2 [42.2, 48.2]15.6
senet154_imagenet_pretrained60.7 [57.7, 63.6]47.2 [44.3, 50.2]13.5
resnet50_augment__impulse_noise60.6 [57.7, 63.5]45.5 [42.6, 48.5]15.1
se_resnet101_imagenet_pretrained60.5 [57.6, 63.4]45.6 [42.6, 48.6]14.9
bninception_imagenet_pretrained60.4 [57.4, 63.3]41.8 [38.9, 44.7]18.6
densenetl61_imagenet_pretrained60.2 [57.3, 63.1]46.4 [43.4, 49.4]13.8
resnet50_augment_gaussian_noise60.2 [57.3, 63.1]45.7 [42.8, 48.7]14.5
se_resnext50_32x4d_imagenet_pretrained59.9 [56.9, 62.8]45.7 [42.7, 48.6]14.2
dpn68b_imagenet_pretrained59.7 [56.7, 62.6]45.9 [42.9, 48.8]13.8
inceptionv3_imagenet_pretrained59.6 [56.6, 62.5]43.8 [40.8, 46.8]15.8
densenet121_imagenet_pretrained59.5 [56.5, 62.4]43.1 [40.1, 46.0]16.4
se_resnext101_32x4d_imagenet_pretrained59.2 [56.3, 62.1]45.2 [42.3, 48.2]14.0
densenet2O1_imagenet_pretrained59.2 [56.2, 62.1]44.8 [41.8, 47.8]14.4
densenet169_imagenet_pretrained59.2 [56.2, 62.1]44.6 [41.7, 47.6]14.6
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resnet50_augment__brightness_change58.9 [56.0, 61.8]42.6 [39.6, 45.5]16.3
se_resnet50_imagenet_pretrained58.8 [55.9, 61.7]44.1 [41.1, 47.1]14.7
se_resnet152_imagenet_pretrained58.8 [55.9, 61.7]44.8 [41.9, 47.8]14.0
cafferesnetlO1_imagenet_pretrained58.2 [55.2, 61.1]44.3 [41.3, 47.3]13.9
resnet50_augment__regular58.0 [55.1, 61.0]42.9 [39.9, 45.8]15.1
resnet34_imagenet_pretrained57.9 [55.0, 60.9]42.8 [39.8, 45.7]15.1
vgg19_imagenet_pretrained57.5 [54.6, 60.5]40.1 [37.2, 43.1]17.4
resnet50_augment t_gaussian_blur57.5 [54.5, 60.4]41.8 [38.9, 44.7]15.7
vgg16_bn_imagenet_pretrained57.2 [54.2, 60.1]39.6 [36.7, 42.6]17.6
resnet5O_imagenet_pretrained57.0 [54.1, 60.0]43.8 [40.9, 46.8]13.2
vgg19_bn_imagenet_pretrained56.8 [53.9, 59.8]40.6 [37.7, 43.5]16.2
vgg16_imagenet_pretrained55.4 [52.4, 58.4]40.1 [37.2, 43.1]15.3
vgg13_bn_imagenet_pretrained54.8 [51.8, 57.7]38.6 [35.7, 41.6]16.2
vgg11_bn_imagenet_pretrained54.8 [51.8, 57.7]38.8 [35.9, 41.8]16.0
vgg11_imagenet_pretrained54.7 [51.7, 57.6]38.4 [35.5, 41.3]16.3
resnetl8_imagenet_pretrained54.4 [51.4, 57.4]38.1 [35.2, 41.0]16.3
vgg13_imagenet_pretrained54.2 [51.3, 57.2]37.7 [34.9, 40.7]16.5
ResNeXtDenoiseAll-101_robust_pgd53.6 [50.7, 56.6]43.2 [40.2, 46.1]10.4
squeezenet1_O_imagenet_pretrained51.1 [48.1, 54.1]33.1 [30.3, 36.0]18.0
squeezenetl_1_imagenet_pretrained48.6 [45.6, 51.6]31.3 [28.6, 34.2]17.3
resnet50_augment__defocus_blur48.4 [45.4, 51.4]29.1 [26.4, 31.8]19.3
alexnet_imagenet_pretrained45.3 [42.4, 48.3]30.5 [27.8, 33.3]14.8
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In Figure 6 we study this static set of perturbations", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "across all models and see a substantial (but smaller) drop in accuracy for both models. The static", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "set of perturbations were chosen by choosing the neighbor frame that the largest number of models", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 451, + 193, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 193, + 464 + ], + "score": 1.0, + "content": "classified incorrectly.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "image", + "bbox": [ + 145, + 475, + 465, + 622 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 475, + 465, + 622 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 475, + 465, + 622 + ], + "spans": [ + { + "bbox": [ + 145, + 475, + 465, + 622 + ], + "score": 0.972, + "type": "image", + "image_path": "f98513794c5a4b0f8364b14f6fc979d78acf7145cef44bd78021361128348e1b.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 145, + 475, + 465, + 524.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 145, + 524.0, + 465, + 573.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 145, + 573.0, + 465, + 622.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 631, + 504, + 664 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "Figure 6: Model accuracy on original vs. perturbed images for a static set of perturbed frames across", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "all models. 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resnet50_augment__brightness_change58.9 [56.0, 61.8]42.6 [39.6, 45.5]16.3
se_resnet50_imagenet_pretrained58.8 [55.9, 61.7]44.1 [41.1, 47.1]14.7
se_resnet152_imagenet_pretrained58.8 [55.9, 61.7]44.8 [41.9, 47.8]14.0
cafferesnetlO1_imagenet_pretrained58.2 [55.2, 61.1]44.3 [41.3, 47.3]13.9
resnet50_augment__regular58.0 [55.1, 61.0]42.9 [39.9, 45.8]15.1
resnet34_imagenet_pretrained57.9 [55.0, 60.9]42.8 [39.8, 45.7]15.1
vgg19_imagenet_pretrained57.5 [54.6, 60.5]40.1 [37.2, 43.1]17.4
resnet50_augment t_gaussian_blur57.5 [54.5, 60.4]41.8 [38.9, 44.7]15.7
vgg16_bn_imagenet_pretrained57.2 [54.2, 60.1]39.6 [36.7, 42.6]17.6
resnet5O_imagenet_pretrained57.0 [54.1, 60.0]43.8 [40.9, 46.8]13.2
vgg19_bn_imagenet_pretrained56.8 [53.9, 59.8]40.6 [37.7, 43.5]16.2
vgg16_imagenet_pretrained55.4 [52.4, 58.4]40.1 [37.2, 43.1]15.3
vgg13_bn_imagenet_pretrained54.8 [51.8, 57.7]38.6 [35.7, 41.6]16.2
vgg11_bn_imagenet_pretrained54.8 [51.8, 57.7]38.8 [35.9, 41.8]16.0
vgg11_imagenet_pretrained54.7 [51.7, 57.6]38.4 [35.5, 41.3]16.3
resnetl8_imagenet_pretrained54.4 [51.4, 57.4]38.1 [35.2, 41.0]16.3
vgg13_imagenet_pretrained54.2 [51.3, 57.2]37.7 [34.9, 40.7]16.5
ResNeXtDenoiseAll-101_robust_pgd53.6 [50.7, 56.6]43.2 [40.2, 46.1]10.4
squeezenet1_O_imagenet_pretrained51.1 [48.1, 54.1]33.1 [30.3, 36.0]18.0
squeezenetl_1_imagenet_pretrained48.6 [45.6, 51.6]31.3 [28.6, 34.2]17.3
resnet50_augment__defocus_blur48.4 [45.4, 51.4]29.1 [26.4, 31.8]19.3
alexnet_imagenet_pretrained45.3 [42.4, 48.3]30.5 [27.8, 33.3]14.8
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While", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "there are modest differences in accuracy due to compression, this analysis suggests that the sensitivity", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 626, + 501, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 501, + 637 + ], + "score": 1.0, + "content": "of models is not significantly due to the differences in quality of frames due to video compression.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 593, + 506, + 637 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 655, + 200, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 201, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 201, + 670 + ], + "score": 1.0, + "content": "I FPS ANALYSIS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 680, + 244, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 679, + 245, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 245, + 692 + ], + "score": 1.0, + "content": "I.1 IM A G ENE T-VI D-RO B U S T", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "To analyze the impact of frame-rate on accuracy, we show results on subsets of videos with fixed fps", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 709, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 225, + 722 + ], + "score": 1.0, + "content": "(25, 29, and 30, which cover", + "type": "text" + }, + { + "bbox": [ + 225, + 710, + 245, + 720 + ], + "score": 0.86, + "content": "89 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 709, + 507, + 722 + ], + "score": 1.0, + "content": "of the dataset) using a fine-tuned ResNet-152 model in Table 8.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 721, + 479, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 479, + 732 + ], + "score": 1.0, + "content": "The accuracy drop is similar across the subsets, and similar to the drop for the whole dataset.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 698, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 171, + 80, + 440, + 128 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 171, + 80, + 440, + 128 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 171, + 80, + 440, + 128 + ], + "spans": [ + { + "bbox": [ + 171, + 80, + 440, + 128 + ], + "score": 0.979, + "html": "
FPSAcc. Orig.Acc.PerturbedDrop# Videos
2587.3 [83.0, 90.9]73.3[67.8, 78.3]14.0292
2987.7 [84.0, 90.8]74.9[70.3, 79.2]12.8383
3078.3 [73.3, 82.7]61.7 [56.0, 67.1]16.6313
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ModelAcc. Orig.Acc. PerturbedDrop
ILSVRC-3061.044.915.1
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We searched for learning rates between", + "type": "text" + }, + { + "bbox": [ + 308, + 374, + 330, + 385 + ], + "score": 0.9, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 372, + 347, + 387 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 374, + 370, + 385 + ], + "score": 0.93, + "content": "1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 372, + 431, + 387 + ], + "score": 1.0, + "content": "for all models.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "We additionally detail hyperparameters for detection models in Table 11. Detection experiments", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "were conducted with PyTorch version 1.0.1 on a machine with 4 Titan X GPUs, using the Mask", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "R-CNN benchmark repositoryMassa and Girshick (2018). We used the default learning rate provided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 504, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 504, + 432 + ], + "score": 1.0, + "content": "in Massa and Girshick (2018). 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ModelBase Learning RateLearning Rate ScheduleBatch SizeEpochs
resnet15210-4ReduceLR On Plateau3210
resnet5010-4ReduceLR On Plateau3210
alexnet10-5Reduce LR On Plateau3210
vgg1610-5Reduce LR On Plateau3210
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ModelBase Learning RateLearning Rate ScheduleBatch SizeIterations
F-RCNN ResNet-5010-2Step 20k，30k840k
F-RCNN ResNet-10110-2Step 20k，30k840k
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FPSAcc. Orig.Acc.PerturbedDrop# Videos
2587.3 [83.0, 90.9]73.3[67.8, 78.3]14.0292
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ModelAcc. Orig.Acc. PerturbedDrop
ILSVRC-3061.044.915.1
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ModelBase Learning RateLearning Rate ScheduleBatch SizeEpochs
resnet15210-4ReduceLR On Plateau3210
resnet5010-4ReduceLR On Plateau3210
alexnet10-5Reduce LR On Plateau3210
vgg1610-5Reduce LR On Plateau3210
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ModelBase Learning RateLearning Rate ScheduleBatch SizeIterations
F-RCNN ResNet-5010-2Step 20k，30k840k
F-RCNN ResNet-10110-2Step 20k，30k840k
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Model Type Accuracy Accuracy △PerturbedOriginal
ImageNet-Vid-Robust
Trained on ILSVRC 67.5 [64.7, 70.3] 52.5 [49.5, 55.5] 15.0
+ Noise Augmentation 68.8 [66.0, 71.5] 53.2 [50.2, 56.2] 15.6
+lrobustness (ResNext-101) 54.3 [51.3, 57.2] 40.8 [39.0, 43.7] 12.4
+ FT on ImageNet- Vid 80.8 [78.3, 83.1] 65.7 [62.9, 68.5] 15.1
+ FT on ImageNet-Vid (ResNet-152) 84.8 [82.5, 86.8]70.2 [67.4, 72.8] 14.6
+ FT on ImageNet-Vid-Det 77.6 [75.1, 80.0] 65.4 [62.5, 68.1] 12.3
YTBB-Robust
Trained on ILSVRC 57.0 [54.9, 59.2]43.8 [41.7, 46.0] 13.2
+Noise Augmentation 62.3 [60.2, 64.4]45.7 [43.5, 47.9]16.6
+ l robustness (ResNext-101) 53.6 [51.4, 55.8]43.2 [41.0, 45.3]10.4
+ FT on Youtube-BB 91.4 [90.1, 92.6]82.0 [80.3, 83.7]9.4
+ FT on Youtube-BB (ResNet-152) 92.9 [91.6, 93.9]84.7 [83.0, 86.2]8.2
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Accuracy
ReviewedOriginalPerturbedDrop
ImageNet-Vid-Robust80.364.116.2
X84.870.214.4
YTBB-Robust88.178.110.0
X92.984.78.9
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TaskModelmAP OriginalmAP PerturbedmAP △
DetectionFRCNN,ResNet562.848.814.0
FRCNN,ResNet 10163.150.612.5
R-FCN, ResNet 101 Xiao and Jae Lee (2018)*79.4*63.7*15.7*
LocalizationFRCNN,ResNet5076.664.212.4
FRCNN, ResNet 10177.866.311.5
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ModelAccuracy OriginalAccuracy Perturbed△
resnet152_finetuned84.8 [82.5, 86.8]70.2 [67.4, 72.8]14.6
resnet50_finetuned80.8 [78.3, 83.1]65.7 [62.9, 68.5]15.1
vgg16bn_finetuned78.0 [75.4, 80.4]61.0 [58.1, 63.9]17.0
nasnetalarge_imagenet_pretrained77.6 [75.1, 80.1]62.1 [59.2, 65.0]15.5
resnet50_detection77.6 [75.1, 80.1]65.0 [62.1, 67.8]12.6
inceptionresnetv2_imagenet_pretrained75.7 [73.1, 78.2]58.7 [55.7, 61.6]17.0
dpn107_imagenet_pretrained75.6 [72.9, 78.1]59.1 [56.1, 62.0]16.5
inceptionv4_imagenet_pretrained75.3 [72.6, 77.8]59.0 [56.0, 61.9]16.3
dpn92_imagenet_pretrained74.4 [71.7, 76.9]56.8 [53.8, 59.7]17.6
dpn131_imagenet_pretrained74.0 [71.3, 76.6]59.9 [56.9, 62.8]14.1
dpn68b_imagenet_pretrained73.7 [71.0, 76.2]54.0 [51.0, 57.0]19.7
resnext101_32x4d_imagenet_pretrained73.3 [70.6, 75.9]57.2 [54.2, 60.1]16.1
resnext101_64x4d_imagenet_pretrained72.9 [70.1, 75.5]56.6 [53.7, 59.6]16.3
resnet152_imagenet_pretrained72.8 [70.0, 75.4]57.0 [54.0, 59.9]15.8
resnet1O1_imagenet_pretrained71.5 [68.7, 74.1]53.7 [50.8, 56.7]17.8
fbresnet152_imagenet_pretrained71.5 [68.7, 74.1]54.5 [51.5, 57.4]17.0
densenet161_imagenet_pretrained71.4 [68.7, 74.1]55.1 [52.1, 58.1]16.3
densenet169_imagenet_pretrained70.2 [67.5, 72.9]53.1 [50.1, 56.1]17.1
densenet2O1_imagenet_pretrained70.2 [67.5, 72.9]53.4 [50.4, 56.4]16.8
dpn68_imagenet_pretrained69.4 [66.6, 72.1]53.3 [50.3, 56.3]16.1
bninception_imagenet_pretrained69.0 [66.2, 71.7]49.0 [46.0, 51.9]20.0
densenet121_imagenet_pretrained69.0 [66.2, 71.7]50.9 [47.9, 53.8]18.1
nasnetamobile_imagenet_pretrained68.8 [66.0, 71.5]48.4 [45.4, 51.4]20.4
resnet50_augment_ jpeg_compression68.8 [66.0, 71.5]53.2 [50.2, 56.2]15.6
resnet34_imagenet_pretrained68.0 [65.2, 70.7]48.0 [45.0, 51.0]20.0
resnet50_augment impulse_noise67.7 [64.9, 70.5]50.2 [47.2, 53.2]17.5
resnet50_augment_gaussian_blur67.7 [64.9, 70.5]52.5 [49.5, 55.5]15.2
resnet5O_imagenet_pretrained67.5 [64.7, 70.3]52.5 [49.5, 55.5]15.0
resnet50_augment gaussian_noise67.4 [64.5, 70.1]50.6 [47.6, 53.6]16.8
resnet50_augment shot_noise66.5 [63.6, 69.2]51.1 [48.1, 54.1]15.4
vgg16_bn_imagenet_pretrained66.4 [63.5, 69.1]47.4 [44.5, 50.4]19.0
resnet50_augment_ _defocus_blur66.3 [63.4, 69.1]47.6 [44.6, 50.6]18.7
vgg19_bn_imagenet_pretrained65.6 [62.7, 68.4]46.6 [43.6, 49.6]19.0
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ModelAccuracy OriginalAccuracy Perturbed△
resnet152_finetuned92.9 [91.2, 94.3]84.7 [82.4, 86.8]8.2
resnet50_finetuned91.4 [89.6, 93.0]82.0 [79.6, 84.2]9.4
inceptionresnetv2_finetuned91.3 [89.5, 92.9]79.0 [76.4, 81.3]12.3
vgg19_finetuned90.5 [88.6, 92.2]79.1 [76.5, 81.4]11.4
vgg16_finetuned89.1 [87.1, 90.8]78.0 [75.4, 80.4]11.1
inceptionv4_finetuned88.5 [86.5, 90.3]76.3 [73.6, 78.7]12.2
resnet18_finetuned88.0 [85.9, 89.8]76.2 [73.6, 78.7]11.8
alexnet_finetuned80.6 [78.2, 82.9]64.4 [61.5, 67.3]16.2
pnasnet5large_imagenet_pretrained65.2 [62.3, 68.0]51.0 [48.0, 54.0]14.2
nasnetalarge_imagenet_pretrained64.9 [62.0, 67.7]51.4 [48.4, 54.4]13.5
inceptionresnetv2_imagenet_pretrained64.5 [61.6, 67.4] 50.4 [47.5, 53.4]14.1
dpn98_imagenet_pretrained64.1 [61.2, 66.9]49.0 [46.0, 52.0]15.1
dpn107_imagenet_pretrained64.1 [61.2, 66.9]50.1 [47.2, 53.1]14.0
dpn131_imagenet_pretrained64.0 [61.1, 66.8]49.9 [46.9, 52.9]14.1
inceptionv4_imagenet_pretrained63.6 [60.7, 66.4]48.8 [45.8, 51.8]14.8
Xception_imagenet_pretrained63.2 [60.2, 66.0]47.6 [44.6, 50.6]15.6
dpn92_imagenet_pretrained62.3 [59.3, 65.1]47.7 [44.8, 50.7]14.6
resnet50_augment_jpeg_compressioon62.3 [59.4, 65.2]45.7 [42.8, 48.7]16.6
polynet_imagenet_pretrained61.4 [58.4, 64.3]47.3 [44.4, 50.3]14.1
nasnetamobile_imagenet_pretrained61.4 [58.4, 64.3]43.0 [40.1, 46.0]18.4
resnet50_augment__shot_noise61.3 [58.3, 64.2]46.4 [43.4, 49.3]14.9
dpn68_imagenet_pretrained61.2 [58.3, 64.1]44.2 [41.2, 47.2]17.0
fbresnet152_imagenet_pretrained61.1 [58.1, 64.0]45.9 [42.9, 48.8]15.2
resnet152_imagenet_pretrained60.8 [57.8, 63.7]46.5 [43.5, 49.5]14.3
resnet101_imagenet_pretrained60.8 [57.8, 63.7]45.2 [42.2, 48.2]15.6
senet154_imagenet_pretrained60.7 [57.7, 63.6]47.2 [44.3, 50.2]13.5
resnet50_augment__impulse_noise60.6 [57.7, 63.5]45.5 [42.6, 48.5]15.1
se_resnet101_imagenet_pretrained60.5 [57.6, 63.4]45.6 [42.6, 48.6]14.9
bninception_imagenet_pretrained60.4 [57.4, 63.3]41.8 [38.9, 44.7]18.6
densenetl61_imagenet_pretrained60.2 [57.3, 63.1]46.4 [43.4, 49.4]13.8
resnet50_augment_gaussian_noise60.2 [57.3, 63.1]45.7 [42.8, 48.7]14.5
se_resnext50_32x4d_imagenet_pretrained59.9 [56.9, 62.8]45.7 [42.7, 48.6]14.2
dpn68b_imagenet_pretrained59.7 [56.7, 62.6]45.9 [42.9, 48.8]13.8
inceptionv3_imagenet_pretrained59.6 [56.6, 62.5]43.8 [40.8, 46.8]15.8
densenet121_imagenet_pretrained59.5 [56.5, 62.4]43.1 [40.1, 46.0]16.4
se_resnext101_32x4d_imagenet_pretrained59.2 [56.3, 62.1]45.2 [42.3, 48.2]14.0
densenet2O1_imagenet_pretrained59.2 [56.2, 62.1]44.8 [41.8, 47.8]14.4
densenet169_imagenet_pretrained59.2 [56.2, 62.1]44.6 [41.7, 47.6]14.6
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vgg19_imagenet_pretrained63.2 [60.3, 66.1]45.4 [42.4, 48.3]17.8
resnet18_imagenet_pretrained61.9 [59.0, 64.8]41.5 [38.6, 44.4]20.4
vgg13_bn_imagenet_pretrained61.9 [59.0, 64.8]43.3 [40.3, 46.3]18.6
vgg16_imagenet_pretrained61.4 [58.5, 64.3]43.1 [40.2, 46.1]18.3
vgg11_bn_imagenet_pretrained60.9 [57.9, 63.8]43.2 [40.3, 46.2]17.7
vgg13_imagenet_pretrained59.6 [56.6, 62.5]41.1 [38.2, 44.1]18.5
vgg11_imagenet_pretrained57.3 [54.4, 60.3]41.3 [38.4, 44.3]16.0
alexnet_finetuned57.3 [54.3, 60.2]43.6 [40.7, 46.6]13.7
ResNeXtDenoiseAll-101_robust_pgd54.3 [51.3, 57.2]40.8 [37.8, 43.7]13.5
squeezenet1_1_imagenet_pretrained49.8 [46.8, 52.8]31.7 [28.9, 34.5]18.1
alexnet_imagenet_pretrained49.4 [46.4, 52.4]32.0 [29.3, 34.8]17.4
resnet50_augment contrast_change38.3 [35.5, 41.3]23.3 [20.8, 25.9]15.0
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resnet50_augment__brightness_change58.9 [56.0, 61.8]42.6 [39.6, 45.5]16.3
se_resnet50_imagenet_pretrained58.8 [55.9, 61.7]44.1 [41.1, 47.1]14.7
se_resnet152_imagenet_pretrained58.8 [55.9, 61.7]44.8 [41.9, 47.8]14.0
cafferesnetlO1_imagenet_pretrained58.2 [55.2, 61.1]44.3 [41.3, 47.3]13.9
resnet50_augment__regular58.0 [55.1, 61.0]42.9 [39.9, 45.8]15.1
resnet34_imagenet_pretrained57.9 [55.0, 60.9]42.8 [39.8, 45.7]15.1
vgg19_imagenet_pretrained57.5 [54.6, 60.5]40.1 [37.2, 43.1]17.4
resnet50_augment t_gaussian_blur57.5 [54.5, 60.4]41.8 [38.9, 44.7]15.7
vgg16_bn_imagenet_pretrained57.2 [54.2, 60.1]39.6 [36.7, 42.6]17.6
resnet5O_imagenet_pretrained57.0 [54.1, 60.0]43.8 [40.9, 46.8]13.2
vgg19_bn_imagenet_pretrained56.8 [53.9, 59.8]40.6 [37.7, 43.5]16.2
vgg16_imagenet_pretrained55.4 [52.4, 58.4]40.1 [37.2, 43.1]15.3
vgg13_bn_imagenet_pretrained54.8 [51.8, 57.7]38.6 [35.7, 41.6]16.2
vgg11_bn_imagenet_pretrained54.8 [51.8, 57.7]38.8 [35.9, 41.8]16.0
vgg11_imagenet_pretrained54.7 [51.7, 57.6]38.4 [35.5, 41.3]16.3
resnetl8_imagenet_pretrained54.4 [51.4, 57.4]38.1 [35.2, 41.0]16.3
vgg13_imagenet_pretrained54.2 [51.3, 57.2]37.7 [34.9, 40.7]16.5
ResNeXtDenoiseAll-101_robust_pgd53.6 [50.7, 56.6]43.2 [40.2, 46.1]10.4
squeezenet1_O_imagenet_pretrained51.1 [48.1, 54.1]33.1 [30.3, 36.0]18.0
squeezenetl_1_imagenet_pretrained48.6 [45.6, 51.6]31.3 [28.6, 34.2]17.3
resnet50_augment__defocus_blur48.4 [45.4, 51.4]29.1 [26.4, 31.8]19.3
alexnet_imagenet_pretrained45.3 [42.4, 48.3]30.5 [27.8, 33.3]14.8
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ModelBase Learning RateLearning Rate ScheduleBatch SizeEpochs
resnet15210-4ReduceLR On Plateau3210
resnet5010-4ReduceLR On Plateau3210
alexnet10-5Reduce LR On Plateau3210
vgg1610-5Reduce LR On Plateau3210
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FPSAcc. Orig.Acc.PerturbedDrop# Videos
2587.3 [83.0, 90.9]73.3[67.8, 78.3]14.0292
2987.7 [84.0, 90.8]74.9[70.3, 79.2]12.8383
3078.3 [73.3, 82.7]61.7 [56.0, 67.1]16.6313
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ModelBase Learning RateLearning Rate ScheduleBatch SizeIterations
F-RCNN ResNet-5010-2Step 20k，30k840k
F-RCNN ResNet-10110-2Step 20k，30k840k
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ModelAcc. Orig.Acc. PerturbedDrop
ILSVRC-3061.044.915.1
ILSVRC-30 + FT77.859.917.9
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+ 1405.0, + 803.0, + 294.0, + 803.0 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 16, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/rklEj2EFvB/rklEj2EFvB.md b/parse/train/rklEj2EFvB/rklEj2EFvB.md new file mode 100644 index 0000000000000000000000000000000000000000..579bc95d00b4b09cf064dfd2df07761a79e5d434 --- /dev/null +++ b/parse/train/rklEj2EFvB/rklEj2EFvB.md @@ -0,0 +1,788 @@ +# ESTIMATING GRADIENTS FOR DISCRETE RANDOM VARIABLES BY SAMPLING WITHOUT REPLACEMENT + +Herke van Hoof University of Amsterdam h.c.vanhoof@uva.nl + +Wouter Kool +University of Amsterdam ORTEC +w.w.m.kool@uva.nl Max Welling +University of Amsterdam CIFAR +m.welling@uva.nl + +# ABSTRACT + +We derive an unbiased estimator for expectations over discrete random variables based on sampling without replacement, which reduces variance as it avoids duplicate samples. We show that our estimator can be derived as the RaoBlackwellization of three different estimators. Combining our estimator with REINFORCE, we obtain a policy gradient estimator and we reduce its variance using a built-in control variate which is obtained without additional model evaluations. The resulting estimator is closely related to other gradient estimators. Experiments with a toy problem, a categorical Variational Auto-Encoder and a structured prediction problem show that our estimator is the only estimator that is consistently among the best estimators in both high and low entropy settings. + +# 1 INTRODUCTION + +Put replacement in your basement! We derive the unordered set estimator1: an unbiased (gradient) estimator for expectations over discrete random variables based on (unordered sets of) samples without replacement. In particular, we consider the problem of estimating (the gradient of) the expectation of $f ( { \pmb x } )$ where $_ { \textbf { \em x } }$ has a discrete distribution $p$ over the domain $D$ , i.e. + +$$ +\mathbb { E } _ { { \pmb x } \sim p ( { \pmb x } ) } [ f ( { \pmb x } ) ] = \sum _ { { \pmb x } \in D } p ( { \pmb x } ) f ( { \pmb x } ) . +$$ + +This expectation comes up in reinforcement learning, discrete latent variable modelling (e.g. for compression), structured prediction (e.g. for translation), hard attention and many other tasks that use models with discrete operations in their computational graphs (see e.g. Jang et al. (2016)). In general, $_ { \textbf { \em x } }$ has structure (such as a sequence), but we can treat it as a ‘flat’ distribution, omitting the bold notation, so $x$ has a categorical distribution over $D$ given by $p ( x ) , x \in D$ . Typically, the distribution has parameters $\pmb { \theta }$ , which are learnt through gradient descent. This requires estimating the gradient $\nabla _ { \pmb { \theta } } \bar { \mathbb { E } } _ { \boldsymbol { x } \sim p _ { \pmb { \theta } } ( \boldsymbol { x } ) } [ f ( \boldsymbol { x } ) ]$ , using a set of samples $S$ . A gradient estimate $e ( S )$ is unbiased if + +$$ +\mathbb { E } _ { S } [ e ( S ) ] = \nabla _ { \pmb { \theta } } \mathbb { E } _ { \pmb { x } \sim p _ { \pmb { \theta } } ( \pmb { x } ) } [ f ( \pmb { x } ) ] . +$$ + +The samples $S$ can be sampled independently or using alternatives such as stratified sampling which reduce variance to increase the speed of learning. In this paper, we derive an unbiased gradient estimator that reduces variance by avoiding duplicate samples, i.e. by sampling $S$ without replacement. This is challenging as samples without replacement are dependent and have marginal distributions that are different from $p ( x )$ . We further reduce the variance by deriving a built-in control variate, which maintains the unbiasedness and does not require additional samples. + +Related work. Many algorithms for estimating gradients for discrete distributions have been proposed. A general and widely used estimator is REINFORCE (Williams, 1992). Biased gradients based on a continuous relaxations of the discrete distribution (known as Gumbel-Softmax or Concrete) were jointly introduced by Jang et al. (2016) and Maddison et al. (2016). These can be combined with the straight through estimator (Bengio et al., 2013) if the model requires discrete samples or be used to construct control variates for REINFORCE, as in REBAR (Tucker et al., 2017) or + +RELAX (Grathwohl et al., 2018). Many other methods use control variates and other techniques to reduce the variance of REINFORCE (Paisley et al., 2012; Ranganath et al., 2014; Gregor et al., 2014; Mnih & Gregor, 2014; Gu et al., 2016; Mnih & Rezende, 2016). + +Some works rely on explicit summation of the expectation, either for the marginal distribution (Titsias & Lazaro-Gredilla, 2015) or globally summing some categories while sampling from the re- ´ mainder (Liang et al., 2018; Liu et al., 2019). Other approaches use a finite difference approximation to the gradient (Lorberbom et al., 2018; 2019). Yin et al. (2019) introduced ARSM, which uses multiple model evaluations where the number adapts automatically to the uncertainty. + +In the structured prediction setting, there are many algorithms for optimizing a quantity under a sequence of discrete decisions, using (weak) supervision, multiple samples (or deterministic model evaluations), or a combination both (Ranzato et al., 2016; Shen et al., 2016; He et al., 2016; Norouzi et al., 2016; Bahdanau et al., 2017; Edunov et al., 2018; Leblond et al., 2018; Negrinho et al., 2018). Most of these algorithms are biased and rely on pretraining using maximum likelihood or gradually transitioning from supervised to reinforcement learning. Using Gumbel-Softmax based approaches in a sequential setting is difficult as the bias accumulates because of mixing errors (Gu et al., 2018). + +# 2 PRELIMINARIES + +Throughout this paper, we will denote with $B ^ { k }$ an ordered sample without replacement of size $k$ and with $S ^ { k }$ an unordered sample (of size $k$ ) from the categorical distribution $p$ . + +Restricted distribution. When sampling without replacement, we remove the set $C \subset D$ already sampled from the domain and we denote with $p ^ { D \setminus C }$ the distribution restricted to the domain $D \backslash C$ : + +$$ +p ^ { D \setminus C } ( x ) = \frac { p ( x ) } { 1 - \sum _ { c \in C } p ( c ) } , \quad x \in D \setminus C . +$$ + +Ordered sample without replacement $B ^ { k }$ . Let $B ^ { k } = ( b _ { 1 } , . . . , b _ { k } ) , b _ { i } \in D$ be an ordered sample without replacement, which is generated from the distribution $p$ as follows: first, sample $b _ { 1 } \sim p$ , then sample $b _ { 2 } \sim p ^ { D \setminus \{ b _ { 1 } \} }$ , $b _ { 3 } ^ { ^ { - } } \sim p ^ { D \setminus \{ b _ { 1 } , b _ { 2 } \} }$ , etc. i.e. elements are sampled one by one without replacement. Using this procedure, $B ^ { k }$ can be seen as a (partial) ranking according to the PlackettLuce model (Plackett, 1975; Luce, 1959) and the probability of obtaining the vector $B ^ { k }$ is + +$$ +p ( B ^ { k } ) = \prod _ { i = 1 } ^ { k } p ^ { D \setminus B ^ { i - 1 } } ( b _ { i } ) = \prod _ { i = 1 } ^ { k } \frac { p ( b _ { i } ) } { 1 - \sum _ { j < i } p ( b _ { j } ) } . +$$ + +We can also restrict $B ^ { k }$ to the domain $D \backslash C$ , which means that $b _ { i } \notin C$ for $i = 1 , . . . , k$ : + +$$ +p ^ { D \setminus C } ( B ^ { k } ) = \prod _ { i = 1 } ^ { k } { \frac { p ^ { D \setminus C } ( b _ { i } ) } { 1 - \sum _ { j < i } p ^ { D \setminus C } ( b _ { j } ) } } = \prod _ { i = 1 } ^ { k } { \frac { p ( b _ { i } ) } { 1 - \sum _ { c \in C } p ( c ) - \sum _ { j < i } p ( b _ { j } ) } } . +$$ + +Unordered sample without replacement. Let $S ^ { k } \subseteq D$ be an unordered sample without replacement from the distribution $p$ , which can be generated simply by generating an ordered sample and discarding the order. We denote elements in the sample with $s \in \breve { S } ^ { k }$ (so without index) and we write $B ( S ^ { k } )$ as the set of all $k !$ ! permutations (orderings) $\bar { B ^ { k } }$ that correspond to (could have generated) $S ^ { k }$ . It follows that the probability for sampling $S ^ { k }$ is given by: + +$$ +p ( S ^ { k } ) = \sum _ { B ^ { k } \in B ( S ^ { k } ) } p ( B ^ { k } ) = \sum _ { B ^ { k } \in B ( S ^ { k } ) } \prod _ { i = 1 } ^ { k } \frac { p ( b _ { i } ) } { 1 - \sum _ { j < i } p ( b _ { j } ) } = \left( \prod _ { s \in S ^ { k } } p ( s ) \right) \cdot \sum _ { B ^ { k } \in B ( S ^ { k } ) } \prod _ { i = 1 } ^ { k } \frac { 1 } { 1 - \sum _ { j < i } p ( b _ { j } ) } . +$$ + +The last step follows since $B ^ { k } \in B ( S ^ { k } )$ is an ordering of $S ^ { k }$ , such that $\begin{array} { r } { \prod _ { i = 1 } ^ { k } p ( b _ { i } ) = \prod _ { s \in S } p ( s ) } \end{array}$ . +Naive computation of $p ( S ^ { k } )$ is $O ( k ! )$ , but in Appendix $\mathbf { B }$ we show how to compute it efficiently. + +When sampling from the distribution restricted to $D \backslash C$ , we sample $S ^ { k } \subseteq D \setminus C$ with probability: + +$$ +p ^ { D \setminus C } ( S ^ { k } ) = \left( \prod _ { s \in S ^ { k } } p ( s ) \right) \cdot \sum _ { B ^ { k } \in \mathcal { B } ( S ^ { k } ) } \prod _ { i = 1 } ^ { k } \frac { 1 } { 1 - \sum _ { c \in C } p ( c ) - \sum _ { j < i } p ( b _ { j } ) } . +$$ + +The Gumbel-Top- $k$ trick. As an alternative to sequential sampling, we can also sample $B ^ { k }$ and $S ^ { k }$ by taking the top $k$ of Gumbel variables (Yellott, 1977; Vieira, 2014; Kim et al., 2016). Following notation from Kool et al. $( 2 0 1 9 \mathrm { c } )$ , we define the perturbed log-probability $g _ { \phi _ { i } } = \phi _ { i } + g _ { i }$ , where $\phi _ { i } = \log p ( i )$ and $g _ { i } \sim \mathrm { G u m b e l } ( 0 )$ . Then let $b _ { 1 } = \arg \operatorname* { m a x } _ { i \in D } g _ { \phi _ { i } }$ , $b _ { 2 } = \arg \operatorname* { m a x } _ { i \in D \backslash \{ b _ { 1 } \} } g _ { \phi _ { i } }$ , etc., so $B ^ { k }$ is the top $k$ of the perturbed log-probabilities in decreasing order. The probability of obtaining $B _ { k }$ using this procedure is given by equation 4, so this provides an alternative sampling method which is effectively a (non-differentiable) reparameterization of sampling without replacement. For a differentiable reparameterization, see Grover et al. (2019). + +It follows that taking the top $k$ perturbed log-probabilities without order, we obtain the unordered sample set $S ^ { k }$ . This way of sampling underlies the efficient computation of $p ( S ^ { k } )$ in Appendix B. + +# 3 METHODOLOGY + +In this section, we derive the unordered set policy gradient estimator: a low-variance, unbiased estimator of $\nabla _ { \pmb { \theta } } \mathbb { E } _ { p _ { \pmb { \theta } } ( \pmb { x } ) } [ f ( \pmb { x } ) ]$ based on an unordered sample without replacement $S ^ { k }$ . First, we derive the generic (non-gradient) estimator for $\mathbb { E } [ f ( x ) ]$ as the Rao-Blackwellized version of a single sample Monte Carlo estimator (and two other estimators!). Then we combine this estimator with REINFORCE (Williams, 1992) and we show how to reduce its variance using a built-in baseline. + +# 3.1 RAO-BLACKWELLIZATION OF THE SINGLE SAMPLE ESTIMATOR + +A very crude but simple estimator for $\mathbb { E } [ f ( x ) ]$ based on the ordered sample $B ^ { k }$ is to only use the first element $b _ { 1 }$ , which by definition is a sample from the distribution $p$ . We define this estimator as the single sample estimator, which is unbiased, since + +$$ +\begin{array} { r } { \mathbb { E } _ { B ^ { k } \sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \mathbb { E } _ { b _ { 1 } \sim p ( b _ { 1 } ) } [ f ( b _ { 1 } ) ] = \mathbb { E } _ { x \sim p ( x ) } [ f ( x ) ] . } \end{array} +$$ + +Discarding all but one sample, the single sample estimator is inefficient, but we can use RaoBlackwellization (Casella $\&$ Robert, 1996) to signficantly improve it. To this end, we consider the distribution $B ^ { k } | S ^ { k }$ , which is, knowing the unordered sample $S ^ { k }$ , the conditional distribution over ordered samples $B ^ { k } \in B ( S ^ { k } )$ that could have generated $S ^ { \hat { k } }$ .2 Using $B ^ { k } | S ^ { k }$ , we rewrite $\mathbb { E } [ f ( b _ { 1 } ) ]$ as + +$$ +\begin{array} { r } { \mathbb { E } _ { B ^ { k } \sim p ( B ^ { k } ) } [ f ( b _ { 1 } ) ] = \mathbb { E } _ { S ^ { k } \sim p ( S ^ { k } ) } \left[ \mathbb { E } _ { B ^ { k } \sim p ( B ^ { k } \mid S ^ { k } ) } \left[ f ( b _ { 1 } ) \right] \right] = \mathbb { E } _ { S ^ { k } \sim p ( S ^ { k } ) } \left[ \mathbb { E } _ { b _ { 1 } \sim p ( b _ { 1 } \mid S ^ { k } ) } \left[ f ( b _ { 1 } ) \right] \right] . } \end{array} +$$ + +The Rao-Blackwellized version of the single sample estimator computes the inner conditional expectation exactly. Since $B ^ { k }$ is an ordering of $S ^ { k }$ , we have $b _ { 1 } \in S ^ { k }$ and we can compute this as + +$$ +\mathbb { E } _ { b _ { 1 } \sim p ( b _ { 1 } | S ^ { k } ) } \left[ f ( b _ { 1 } ) \right] = \sum _ { s \in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s ) +$$ + +where, in a slight abuse of notation, $P ( b _ { 1 } = s | S ^ { k } )$ is the probability that the first sampled element $b _ { 1 }$ takes the value $s$ , given that the complete set of $k$ samples is $S ^ { k }$ . Using Bayes’ Theorem we find + +$$ +P ( b _ { 1 } = s | S ^ { k } ) = \frac { p ( S ^ { k } | b _ { 1 } = s ) P ( b _ { 1 } = s ) } { p ( S ^ { k } ) } = \frac { p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) p ( s ) } { p ( S ^ { k } ) } . +$$ + +The step $p ( S ^ { k } | b _ { 1 } = s ) = p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} )$ comes from analyzing sequential sampling without replacement: given that the first element sampled is $s$ , the remaining elements have a distribution restricted to $D \backslash \{ s \}$ , so sampling $S ^ { k }$ (including $s$ ) given the first element $s$ is equivalent to sampling the remainder $S ^ { k } \setminus \{ s \}$ from the restricted distribution, which has probability $p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} )$ (see equation 7). + +The unordered set estimator. For notational convenience, we introduce the leave-one-out ratio. +Definition 1. The leave-one-out ratio of $s$ w.r.t. the set $S$ is given by $\begin{array} { r } { R ( S ^ { k } , s ) = \frac { p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) } { p ( S ^ { k } ) } } \end{array}$ . + +Rewriting equation 10 as $P ( b _ { 1 } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )$ shows that the probability of sampling $s$ first, given $S ^ { k }$ , is simply the unconditional probability multiplied by the leave-one-out ratio. We now define the unordered set estimator as the Rao-Blackwellized version of the single-sample estimator. + +Theorem 1. The unordered set estimator, given by + +$$ +e ^ { U S } ( S ^ { k } ) = \sum _ { s \in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) +$$ + +is the Rao-Blackwellized version of the (unbiased!) single sample estimator. + +Proof. Using $P ( b _ { 1 } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s )$ in equation 9 we have + +$$ +\mathbb { E } _ { b _ { 1 } \sim p ( b _ { 1 } | S ^ { k } ) } \left[ f ( b _ { 1 } ) \right] = \sum _ { s \in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) f ( s ) = \sum _ { s \in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) . +$$ + +The implication of this theorem is that the unordered set estimator, in explicit form given by equation 11, is an unbiased estimator of $\mathbb { E } [ f ( x ) ]$ since it is the Rao-Blackwellized version of the unbiased single sample estimator. Also, as expected by taking multiple samples, it has variance equal or lower than the single sample estimator by the Rao-Blackwell Theorem (Lehmann & Scheffe, 1950). ´ + +# 3.2 RAO-BLACKWELLIZATION OF OTHER ESTIMATORS + +The unordered set estimator is also the result of Rao-Blackwellizing two other unbiased estimators: the stochastic sum-and-sample estimator and the importance-weighted estimator. + +The sum-and-sample estimator. We define as sum-and-sample estimator any estimator that relies on the identity that for any $C \subset D$ + +$$ +\mathbb { E } _ { x \sim p ( x ) } [ f ( x ) ] = \mathbb { E } _ { x \sim p ^ { D \setminus C } ( x ) } \left[ \sum _ { c \in C } p ( c ) f ( c ) + \left( 1 - \sum _ { c \in C } p ( c ) \right) f ( x ) \right] . +$$ + +For the derivation, see Appendix C.1 or Liang et al. (2018); Liu et al. (2019). In general, a sum-andsample estimator with a budget of $k > 1$ evaluations sums expectation terms for a set of categories $C$ (s.t. $| C | < k )$ explicitly (e.g. selected by their value $f$ (Liang et al., 2018) or probability $p$ (Liu et al., 2019)), and uses $k - | C |$ (down-weighted) samples from $D \backslash C$ to estimate the remaining terms. As is noted by Liu et al. (2019), selecting $C$ such that $\frac { 1 - \sum _ { c \in C } p ( c ) } { k - | C | }$ is minimized guarantees to reduce variance compared to a standard minibatch of $k$ samples (which is equivalent to setting $C = \varnothing$ ). See also Fearnhead $\&$ Clifford (2003) for a discussion on selecting $C$ optimally. The ability to optimize $C$ depends on whether $p ( c )$ can be computed efficiently a-priori (before sampling). This is difficult in high-dimensional settings, e.g. sequence models which compute the probability incrementally while ancestral sampling. An alternative is to select $C$ stochastically (as equation 13 holds for any $C$ ), and we choose $C = B ^ { k - 1 }$ to define the stochastic sum-and-sample estimator: + +$$ +e ^ { { \mathrm { S S A S } } } ( B ^ { k } ) = \sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) f ( b _ { j } ) + \left( 1 - \sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) \right) f ( b _ { k } ) . +$$ + +For simplicity, we consider the version that sums $k - 1$ terms here, but the following results also hold for a version that sums $k - m$ terms and uses $m$ samples (without replacement) (see Appendix C.3). Sampling without replacement, it holds that bk|Bk−1 ∼ pD\Bk−1, so the unbiasedness follows from equation 13 by separating the expectation over $B ^ { k }$ into expectations over $B ^ { k - 1 }$ and $b _ { k } | B ^ { k - 1 }$ : + +$$ +\begin{array} { r } { \mathbb { E } _ { B ^ { k - 1 } \sim p ( B ^ { k - 1 } ) } \left[ \mathbb { E } _ { b _ { k } \sim p ( b _ { k } | B ^ { k - 1 } ) } \left[ e ^ { S S A S } ( B ^ { k } ) \right] \right] = \mathbb { E } _ { B ^ { k - 1 } \sim p ( B ^ { k - 1 } ) } \left[ \mathbb { E } [ f ( x ) ] \right] = \mathbb { E } [ f ( x ) ] . } \end{array} +$$ + +In general, a sum-and-sample estimator reduces variance if the probability mass is concentrated on the summed categories. As typically high probability categories are sampled first, the stochastic sum-and-sample estimator sums high probability categories, similar to the estimator by Liu et al. (2019) which we refer to as the deterministic sum-and-sample estimator. As we show in Appendix C.2, Rao-Blackwellizing the stochastic sum-and-sample estimator also results in the unordered set estimator. This even holds for a version that uses $m$ samples and $k - m$ summed terms (see Appendix C.3), which means that the unordered set estimator has equal or lower variance than the optimal (in terms of $m$ ) stochastic sum-and-sample estimator, but conveniently does not need to choose $m$ . + +The importance-weighted estimator. The importance-weighted estimator (Vieira, 2017) is + +$$ +e ^ { \mathrm { I W } } ( S ^ { k } , \kappa ) = \sum _ { s \in S ^ { k } } { \frac { p ( s ) } { q ( s , \kappa ) } } f ( s ) . +$$ + +This estimator is based on the idea of priority sampling (Duffield et al., 2007). It does not use the order of the sample, but assumes sampling using the Gumbel-Top- $k$ trick and requires access to $\kappa$ , the $( k + 1 )$ -th largest perturbed log-probability, which can be seen as the ‘threshold’ since $g _ { \phi _ { s } } ~ > ~ \kappa ~ \forall s ~ \in ~ S ^ { k }$ . $\bar { q ( s , a ) } = P ( g _ { \phi _ { s } } > a )$ can be interpreted as the inclusion probability of $s \in S ^ { k }$ (assuming a fixed threshold $a$ instead of a fixed sample size $k$ ). For details and a proof of unbiasedness, see Vieira (2017) or Kool et al. (2019c). As the estimator has high variance, Kool et al. (2019c) resort to normalizing the importance weights, resulting in biased estimates. Instead, we use Rao-Blackwellization to eliminate stochasticity by $\kappa$ . Again, the result is the unordered set estimator (see Appendix D.1), which thus has equal or lower variance. + +# 3.3 THE UNORDERED SET POLICY GRADIENT ESTIMATOR + +Writing $p _ { \pmb { \theta } }$ to indicate the dependency on the model parameters $\pmb { \theta }$ , we can combine the unordered set estimator with REINFORCE (Williams, 1992) to obtain the unordered set policy gradient estimator. + +Corollary 1. The unordered set policy gradient estimator, given by + +$$ +e ^ { U S P G } ( S ^ { k } ) = \sum _ { s \in S ^ { k } } p _ { \theta } ( s ) R ( S ^ { k } , s ) \nabla _ { \theta } \log p _ { \theta } ( s ) f ( s ) = \sum _ { s \in S ^ { k } } \nabla _ { \theta } p _ { \theta } ( s ) R ( S ^ { k } , s ) f ( s ) , +$$ + +is an unbiased estimate of the policy gradient. + +Proof. Using REINFORCE (Williams, 1992) combined with the unordered set estimator we find: + +$$ +\nabla _ { \theta } \mathbb { E } _ { p _ { \theta } ( x ) } [ f ( x ) ] = \mathbb { E } _ { p _ { \theta } ( x ) } [ \nabla _ { \theta } \log p _ { \theta } ( x ) f ( x ) ] = \mathbb { E } _ { S ^ { k } \sim p _ { \theta } ( S ^ { k } ) } \left[ \sum _ { s \in S ^ { k } } p _ { \theta } ( s ) R ( S ^ { k } , s ) \nabla _ { \theta } \log p _ { \theta } ( s ) f ( s ) \right] . +$$ + +Variance reduction using a built-in control variate. The variance of REINFORCE can be reduced by subtracting a baseline from $f$ . When taking multiple samples (with replacement), a simple and effective baseline is to take the mean of other (independent!) samples (Mnih & Rezende, 2016). Sampling without replacement, we can use the same idea to construct a baseline based on the other samples, but we have to correct for the fact that the samples are not independent. + +Theorem 2. The unordered set policy gradient estimator with baseline, given by + +$$ +e ^ { U S P G B L } ( S ^ { k } ) = \sum _ { s \in S ^ { k } } \nabla _ { \theta } p _ { \theta } ( s ) R ( S ^ { k } , s ) \left( f ( s ) - \sum _ { s ^ { \prime } \in S ^ { k } } p _ { \theta } ( s ^ { \prime } ) R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) f ( s ^ { \prime } ) \right) , +$$ + +where + +$$ +R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) = { \frac { p _ { \theta } ^ { D \setminus \{ s , s ^ { \prime } \} } ( S ^ { k } \setminus \{ s , s ^ { \prime } \} ) } { p _ { \theta } ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) } } +$$ + +is the second order leave-one-out ratio, is an unbiased estimate of the policy gradient. + +Proof. See Appendix E.1. + +This theorem shows how to include a built-in baseline based on dependent samples (without replacement), without introducing bias. By having a built-in baseline, the value $f ( s )$ for sample $s$ is compared against an estimate of its expectation $\mathbb { E } [ f ( s ) ]$ , based on the other samples. The difference is an estimate of the advantage (Sutton & Barto, 2018), which is positive if the sample $s$ is ‘better’ than average, causing $p _ { \pmb { \theta } } ( s )$ to be increased (reinforced) through the sign of the gradient, and vice versa. By sampling without replacement, the unordered set estimator forces the estimator to compare different alternatives, and reinforces the best among them. + +Including the pathwise derivative. So far, we have only considered the scenario where $f$ does not depend on $\pmb \theta$ . If $f$ does depend on $\pmb \theta$ , for example in a VAE (Kingma & Welling, 2014; Rezende et al., 2014), then we use the notation $f _ { \theta }$ and we can write the gradient (Schulman et al., 2015) as + +$$ +\nabla _ { \pmb { \theta } } \mathbb { E } _ { p _ { \pmb { \theta } } ( \pmb { x } ) } [ f _ { \pmb { \theta } } ( \pmb { x } ) ] = \mathbb { E } _ { p _ { \pmb { \theta } } ( \pmb { x } ) } [ \nabla _ { \pmb { \theta } } \log p _ { \pmb { \theta } } ( \pmb { x } ) f _ { \pmb { \theta } } ( \pmb { x } ) + \nabla _ { \pmb { \theta } } f _ { \pmb { \theta } } ( \pmb { x } ) ] . +$$ + +The additional second (‘pathwise’) term can be estimated (using the same samples) with the standard unordered set estimator. This results in the full unordered set policy gradient estimator: + +$$ +\begin{array} { l } { { \displaystyle e ^ { \mathrm { F U S P G } } ( S ^ { k } ) = \sum _ { s \in S ^ { k } } \nabla _ { \theta } p _ { \theta } ( s ) R ( S ^ { k } , s ) f _ { \theta } ( s ) + \sum _ { s \in S ^ { k } } p _ { \theta } ( s ) R ( S ^ { k } , s ) \nabla _ { \theta } f _ { \theta } ( s ) } } \\ { { \displaystyle \quad = \sum _ { s \in S ^ { k } } R ( S ^ { k } , s ) \nabla _ { \theta } \left( p _ { \theta } ( s ) f _ { \theta } ( s ) \right) } } \end{array} +$$ + +Equation 20 is straightforward to implement using an automatic differentiation library. We can also include the baseline (as in equation 17) but we must make sure to call STOP GRADIENT (DETACH in PyTorch) on the baseline (but not on $f _ { \pmb \theta } ( s ) ! )$ . Importantly, we should never track gradients through the leave-one-out ratio $R ( S ^ { k } , s )$ which means it can be efficiently computed in pure inference mode. + +Scope & limitations. We can use the unordered set estimator for any discrete distribution from which we can sample without replacement, by treating it as a univariate categorical distribution over its domain. This includes sequence models, from which we can sample using Stochastic Beam Search (Kool et al., 2019c), as well as multivariate categorical distributions which can also be treated as sequence models (see Section 4.2). In the presence of continuous variables or a stochastic function $f$ , we may separate this stochasticity from the stochasticity over the discrete distribution, as in Lorberbom et al. (2019). The computation of the leave-one-out ratios adds some overhead, although they can be computed efficiently, even for large $k$ (see Appendix B). For a moderately sized model, the costs of model evaluation and backpropagation dominate the cost of computing the estimator. + +# 3.4 RELATION TO OTHER MULTI-SAMPLE ESTIMATORS + +Relation to Murthy’s estimator. We found out that the ‘vanilla’ unordered set estimator (equation 11) is actually a special case of the estimator by Murthy (1957), known in statistics literature for estimation of a population total $\begin{array} { r } { \Theta = \sum _ { i \in D } y _ { i } } \end{array}$ . Using $y _ { i } = p ( i ) f ( i )$ , we have $\Theta = \mathbb { E } [ f ( i ) ]$ , so Murthy’s estimator can be used to estimate expectations (see equation 11). Murthy derives the estimator by ‘unordering’ a convex combination of Raj (1956) estimators, which, using $y _ { i } = p ( i ) f ( i )$ , are stochastic sum-and-sample estimators in our analogy. + +Murthy (1957) also provides an unbiased estimator of the variance, which may be interesting for future applications. Since Murthy’s estimator can be used with arbitrary sampling distribution, it is straightforward to derive importance-sampling versions of our estimators. In particular, we can sample $S$ without replacement using $q ( x ) > 0 , x \in D$ , and use equations 11, 16, 17 and 20, as long as we compute the leave-one-out ratio $R ( S ^ { k } , s )$ using $q$ . + +While part of our derivation coincides with Murthy (1957), we are not aware of previous work using this estimator to estimate expectations. Additionally, we discuss practical computation of $p ( S )$ (Appendix B), we show the relation to the importance-weighted estimator, and we provide the extension to estimating policy gradients, especially including a built-in baseline without adding bias. + +Relation to the empirical risk estimator. The empirical risk loss (Edunov et al., 2018) estimates the expectation in equation 1 by summing only a subset $S$ of the domain, using normalized probabilities $\begin{array} { r } { \hat { p } _ { \pmb { \theta } } ( s ) = \frac { \bar { p } _ { \pmb { \theta } } ( s ) } { \sum _ { s ^ { \prime } \in S } p _ { \pmb { \theta } } ( s ) } } \end{array}$ . Using this loss, the (biased) estimate of the gradient is given by + +$$ +e ^ { \mathrm { R I S K } } ( S ^ { k } ) = \sum _ { s \in S ^ { k } } \nabla _ { \pmb { \theta } } \left( \frac { p _ { \pmb { \theta } } ( s ) } { \sum _ { s ^ { \prime } \in S ^ { k } } p _ { \pmb { \theta } } ( s ^ { \prime } ) } \right) f ( s ) . +$$ + +The risk estimator is similar to the unordered set policy gradient estimator, with two important differences: 1) the individual terms are normalized by the total probability mass rather than the leave-one-out ratio and 2) the gradient w.r.t. the normalization factor is taken into account. As a result, samples ‘compete’ for probability mass and only the best can be reinforced. This has the same effect as using a built-in baseline, which we prove in the following theorem. + +Theorem 3. By taking the gradient w.r.t. the normalization factor into account, the risk estimator has a built-in baseline, which means it can be written as + +$$ +e ^ { R I S K } ( S ^ { k } ) = \sum _ { s \in S ^ { k } } \nabla _ { \theta } p _ { \theta } ( s ) \frac { 1 } { \sum _ { s ^ { \prime \prime } \in S ^ { k } } p _ { \theta } ( s ^ { \prime \prime } ) } \left( f ( s ) - \sum _ { s ^ { \prime } \in S ^ { k } } p _ { \theta } ( s ^ { \prime } ) \frac { 1 } { \sum _ { s ^ { \prime \prime } \in S ^ { k } } p _ { \theta } ( s ^ { \prime \prime } ) } f ( s ^ { \prime } ) \right) . +$$ + +Proof. See Appendix F.1 + +This theorem highlights the similarity between the biased risk estimator and our unbiased estimator (equation 17), and suggests that their only difference is the weighting of terms. Unfortunately, the implementation by Edunov et al. (2018) has more sources of bias (e.g. length normalization), which are not compatible with our estimator. However, we believe that our analysis helps analyze the bias of the risk estimator and is a step towards developing unbiased estimators for structured prediction. + +Relation to VIMCO. VIMCO (Mnih & Rezende, 2016) is an estimator that uses $k$ samples (with replacement) to optimize an objective of the form log $\textstyle { \frac { 1 } { k } } \sum _ { i } f ( x _ { i } )$ , which is a multi-sample stochastic lower bound in the context of variational inference. VIMCO reduces the variance by using a local baseline for each of the $k$ samples, based on the other $k - 1$ samples. While we do not have a log term, as our goal is to optimize general $\mathbb { E } [ f ( x ) ]$ , we adopt the idea of forming a baseline based on the other samples, and we define REINFORCE with replacement (with built-in baseline) as the estimator that computes the gradient estimate using samples with replacement $X ^ { k } = ( x _ { 1 } , . . . , x _ { k } )$ as + +$$ +e ^ { \mathrm { R F W R } } ( X ^ { k } ) = \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \nabla _ { \theta } \log p _ { \theta } ( x _ { i } ) \left( f ( x _ { i } ) - \frac { 1 } { k - 1 } \sum _ { j \ne i } f ( x _ { j } ) \right) . +$$ + +This estimator is unbiased, as $\mathbb { E } _ { x _ { i } , x _ { j } } [ \nabla _ { \pmb { \theta } } \log p _ { \pmb { \theta } } ( x _ { i } ) f ( x _ { j } ) ] = 0$ for $i \neq j$ (see also Kool et al. (2019b)). We think of the unordered set estimator as the without-replacement version of this estimator, which weights terms by $p _ { \pmb { \theta } } ( s ) R ( S ^ { k } , s )$ instead of $\frac { 1 } { k }$ . This puts more weight on higher probability elements to compensate for sampling without replacement. If probabilities are small and (close to) uniform, there are (almost) no duplicate samples and the weights will be close to $\frac { 1 } { k }$ , so the gradient estimate of the with- and without-replacement versions are similar. + +Relation to ARSM. ARSM (Yin et al., 2019) also uses multiple evaluations (‘pseudo-samples’) of $p _ { \theta }$ and $f$ . This can be seen as similar to sampling without replacement, and the estimator also has a built-in control variate. Compared to ARSM, our estimator allows direct control over the computational cost (through the sample size $k$ ) and has wider applicability, for example it also applies to multivariate categorical variables with different numbers of categories per dimension. + +Relation to stratified/systematic sampling. Our estimator aims to reduce variance by changing the sampling distribution for multiple samples by sampling without replacement. There are alternatives, such as using stratified or systematic sampling (see, e.g. Douc & Cappe (2005)). Both partition ´ the domain $D$ into $k$ strata and take a single sample from each stratum, where systematic sampling uses common random numbers for each stratum. In applications involving high-dimensional or structured domains, it is unclear how to partition the domain and how to sample from each partition. Additionally, as samples are not independent, it is non-trivial to include a built-in baseline, which we find is a key component that makes our estimator perform well. + +![](images/1827e94fdb564b2088e10a208547d35fffe208d74d3041a8d194d3868ada9f35.jpg) +Figure 1: Bernoulli gradient variance (on log scale) as a function of the number of model evaluations (including baseline evaluations, so the sum-and-sample estimators with sampled baselines use twice as many evaluations). Note that for some estimators, the variance is 0 (log variance $- \infty )$ for $k = 8$ . + +# 4 EXPERIMENTS + +# 4.1 BERNOULLI TOY EXPERIMENT + +We use the code by Liu et al. (2019) to reproduce their Bernoulli toy experiment. Given a vector $\mathbf { p } =$ (0.6, 0.51, 0.48) the goal is to minimize the loss $\begin{array} { r } { \mathcal { L } ( \eta ) \ : = \ : \mathbb { E } _ { x _ { 1 } , x _ { 2 } , x _ { 3 } \sim \mathrm { B e r n } ( \sigma ( \eta ) ) } \left[ \sum _ { i = 1 } ^ { 3 } ( x _ { i } - p _ { i } ) ^ { 2 } \right] } \end{array}$ . Here $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are i.i.d. from the Bernoulli $( \sigma ( \eta ) )$ distribution, parameterized by a scalar $\eta \in \mathbb { R }$ , where $\sigma ( \eta ) = ( 1 + \exp ( - \eta ) ) ^ { - 1 }$ is the sigmoid function. We compare different estimators, with and without baseline (either ‘built-in’ or using additional samples, referred to as REINFORCE+ in Liu et al. (2019)). We report the (log-)variance of the scalar gradient $\frac { \partial \mathcal { L } } { \partial \eta }$ as a function of the number of model evaluations, which is twice as high when using a sampled baseline (for each term). + +As can be seen in Figure 1, the unordered set estimator is the only estimator that has consistently the lowest (or comparable) variance in both the high $\langle \eta = 0 \rangle$ ) and low entropy $\hphantom { - } \eta = - 4 )$ regimes and for different number of samples/model evaluations. This suggests that it combines the advantages of the other estimators. We also ran the actual optimization experiment, where with as few as $k = 3$ samples the trajectory was indistinguishable from using the exact gradient (see Liu et al. (2019)). + +# 4.2 CATEGORICAL VARIATIONAL AUTO-ENCODER + +We use the code from Yin et al. (2019) to train a categorical Variational Auto-Encoder (VAE) with 20 dimensional latent space, with 10 categories per dimension (details in Appendix G.1). To use our estimator, we treat this as a single factorized distribution with $1 0 ^ { 2 0 }$ categories from which we can sample without replacement using Stochastic Beam Search (Kool et al., 2019c), sequentially sampling each dimension as if it were a sequence model. We also perform experiments with $1 0 ^ { \dot { 2 } }$ latent space, which provides a lower entropy setting, to highlight the advantage of our estimator. + +Measuring the variance. In Table 1, we report the variance of different gradient estimators with $k = 4$ samples, evaluated on a trained model. The unordered set estimator has the lowest variance in both the small and large domain (low and high entropy) setting, being on-par with the best of the (stochastic3) sum-and-sample estimator and REINFORCE with replacement4. This confirms the toy experiment, suggesting that the unordered set estimator provides the best of both estimators. In Appendix G.2 we repeat the same experiment at different stages of training, with similar results. + +![](images/38fff474333f745080d8627c5fb4566e88f353059b7a66955110f7f26afb9b13.jpg) +Figure 2: VAE smoothed training curves (-ELBO) of two independent runs when training with different estimators with $k = 1$ , 4 or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value. + +ELBO optimization. We use different estimators to optimize the ELBO (details in Appendix G.1). Additionally to the baselines by Yin et al. (2019) we compare against REINFORCE with replacement and the stochastic sum-and-sample estimator. In Figure 2 we observe that our estimator performs on par with REINFORCE with replacement (and built-in baseline, equation 23) and outperforms other estimators in at least one of the settings. There are a lot of other factors, e.g. exploration that may explain why we do not get a strictly better result despite the lower variance. We note some overfitting (see validation curves in Appendix G.2), but since our goal is to show improved optimization, and to keep results directly comparable to Yin et al. (2019), we consider regularization a separate issue outside the scope of this paper. These results are using MNIST binarized by a threshold of 0.5. In Appendix G.2 we report results using the standard binarized MNIST dataset from Salakhutdinov & Murray (2008). + +# 4.3 STRUCTURED PREDICTION FOR THE TRAVELLING SALESMAN PROBLEM + +To show the wide applicability of our estimator, we consider the structured prediction task of predicting routes (sequences) for the Travelling Salesman Problem (TSP) (Vinyals et al., 2015; Bello et al., 2016; Kool et al., 2019a). We use the code by Kool et al. (2019a)5 to reproduce their TSP experiment with 20 nodes. For details, see Appendix H. + +We implement REINFORCE with replacement (and built-in baseline) as well as the stochastic sumand-sample estimator and our estimator, using Stochastic Beam Search (Kool et al., 2019c) for sampling. Also, we include results using the biased normalized importance-weighted policy gradient estimator with built-in baseline (derived in Kool et al. (2019b), see Appendix D.2). Additionally, we compare against REINFORCE with greedy rollout baseline (Rennie et al., 2017) used by Kool et al. (2019c) and a batch-average baseline. For reference, we also include the biased risk estimator, either ‘sampling’ using stochastic or deterministic beam search (as in Edunov et al. (2018)). + +In Figure 3a, we compare training progress (measured on the validation set) as a function of the number of training steps, where we divide the batch size by $k$ to keep the total number of samples equal. Our estimator outperforms REINFORCE with replacement, the stochastic sum-and-sample estimator and the strong greedy rollout baseline (which uses additional baseline model evaluations) and performs on-par with the biased risk estimator. In Figure 3b, we plot the same results against the number of instances, which shows that, compared to the single sample estimators, we can train with less data and less computational cost (as we only need to run the encoder once for each instance). + +![](images/3cf7aa8c67e64f59fc3b77ff931ff72bff898a29559db5fa6f7eddaa573687f1.jpg) +Figure 3: TSP validation set optimality gap measured during training. Raw results are light, smoothed results are darker (2 random seeds). We compare our estimator against different unbiased and biased (dotted) multi-sample estimators and against single-sample REINFORCE, with batch-average or greedy rollout baseline. + +# 5 DISCUSSION + +We introduced the unordered set estimator, a low-variance, unbiased gradient estimator based on sampling without replacement, which can be used as an alternative to the popular biased GumbelSoftmax estimator (Jang et al., 2016; Maddison et al., 2016). Our estimator is the result of RaoBlackwellizing three existing estimators, which guarantees equal or lower variance, and is closely related to a number of other estimators. It has wide applicability, is parameter free (except for the sample size $k$ ) and has competitive performance to the best of alternatives in both high and low entropy regimes. + +In our experiments, we found that REINFORCE with replacement, with multiple samples and a built-in baseline as inspired by VIMCO (Mnih & Rezende, 2016), is a simple yet strong estimator which has performance similar to our estimator in the high entropy setting. We are not aware of any recent work on gradient estimators for discrete distributions that has considered this estimator as baseline, while it may be often preferred given its simplicity. In future work, we want to investigate if we can apply our estimator to estimate gradients ‘locally’ (Titsias & Lazaro-Gredilla, 2015), as ´ locally we have a smaller domain and expect more duplicate samples. + +# ACKNOWLEDGMENTS + +This research was funded by ORTEC. We would like to thank anonymous reviewers for their feedback that helped improve the paper. + +# REFERENCES + +Dzmitry Bahdanau, Philemon Brakel, Kelvin Xu, Anirudh Goyal, Ryan Lowe, Joelle Pineau, Aaron Courville, and Yoshua Bengio. An actor-critic algorithm for sequence prediction. In International Conference on Learning Representations, 2017. + +Irwan Bello, Hieu Pham, Quoc V Le, Mohammad Norouzi, and Samy Bengio. Neural combinatorial optimization with reinforcement learning. arXiv preprint arXiv:1611.09940, 2016. + +Yoshua Bengio, Nicholas Leonard, and Aaron Courville. 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(2014): + +$$ +\begin{array} { l } { { e _ { \phi } ( g ) = \exp ( - g + \phi ) } } \\ { { F _ { \phi } ( g ) = \exp ( - \exp ( - g + \phi ) ) } } \\ { { f _ { \phi } ( g ) = e _ { \phi } ( g ) F _ { \phi } ( g ) . } } \end{array} +$$ + +This means that $F _ { \phi } ( g )$ is the CDF and $f _ { \phi } ( g )$ the PDF of the Gumbel $\left( \phi \right)$ distribution. Additionally we will use the identities by Maddison et al. (2014): + +$$ +\begin{array} { c } { { F _ { \phi } ( g ) F _ { \gamma } ( g ) = F _ { \log ( \exp ( \phi ) + \exp ( \gamma ) ) } ( g ) } } \\ { { \displaystyle \int _ { g = a } ^ { b } e _ { \gamma } ( g ) F _ { \phi } ( g ) \partial g = ( F _ { \phi } ( b ) - F _ { \phi } ( a ) ) \frac { \exp ( \gamma ) } { \exp ( \phi ) } . } } \end{array} +$$ + +Also, we will use the following notation, definitions and identities (see Kool et al. (2019c)): + +$$ +\begin{array} { r l } & { \displaystyle \phi _ { i } = \log p ( i ) } \\ & { \displaystyle \phi _ { S } = \log \displaystyle \sum _ { i \in S } p ( i ) = \log \sum _ { i \in S } \exp \phi _ { i } } \\ & { \displaystyle \phi _ { D \setminus S } = \log \sum _ { i \in D \setminus S } p ( i ) = \log \left( 1 - \sum _ { i \in S } p ( i ) \right) = \log ( 1 - \exp ( \phi _ { S } ) ) } \\ & { \displaystyle { G _ { \phi _ { i } } \sim \mathrm { G u m b e l } ( \phi _ { i } ) } } \\ & { \displaystyle { G _ { \phi _ { S } } = \sum _ { i \in S } G _ { \phi _ { i } } \sim \mathrm { G u m b e l } ( \phi _ { S } ) } } \end{array} +$$ + +For a proof of equation 30, see Maddison et al. (2014). + +$$ +p ( S ^ { k } ) , p ^ { D \setminus C } ( S \setminus C ) \ \mathrm { A N D } \ R ( S ^ { k } , s ) +$$ + +We can sample the set $S ^ { k }$ from the Plackett-Luce distribution using the Gumbel-Top- $k$ trick by drawing Gumbel variables $G _ { \phi _ { i } } \sim \mathrm { G u m b e l } ( \phi _ { i } )$ for each element and returning the indices of the $k$ largest Gumbels. If we ignore the ordering, this means we will obtain the set $S ^ { k }$ if $\mathrm { m i n } _ { i \in S ^ { k } } G _ { \phi _ { i } } >$ $\operatorname* { m a x } _ { i \in D \setminus S ^ { k } } G _ { \phi _ { i } }$ . Omitting the superscript $k$ for clarity, we can use the Gumbel-Max trick, i.e. that $G _ { \phi _ { D \setminus S } } = \operatorname* { m a x } _ { i \notin S } G _ { \phi _ { i } } \sim \mathrm { G u m b e l } ( \phi _ { D \setminus S } )$ (equation 30) and marginalize over $G _ { \phi _ { D \setminus S } }$ : + +$$ +\begin{array} { r l } { \displaystyle p ( S ) = P ( \underset { i \in S } { \operatorname* { m i n } } G _ { \phi _ { i } \setminus } \ G _ { \phi _ { D \setminus S } } ) } \\ { \displaystyle } & { = P ( G _ { \phi _ { i } } > G _ { \phi _ { D \setminus S } } , i \in S ) } \\ { \displaystyle } & { = \int _ { g _ { \phi _ { D \setminus S } } = - \infty } ^ { \infty } f _ { \phi _ { D \setminus S } } ( g _ { \phi _ { D \setminus S } } ) P ( G _ { \phi _ { i } } > g _ { \phi _ { D \setminus S } } , i \in S ) \partial g _ { \phi _ { D \setminus S } } } \\ { \displaystyle } & { = \int _ { g _ { \phi _ { D \setminus S } } = - \infty } ^ { \infty } f _ { \phi _ { D \setminus S } } ( g _ { \phi _ { D \setminus S } } ) \prod _ { i \in S } \left( 1 - F _ { \phi _ { \star } } ( g _ { \phi _ { D \setminus S } } ) \right) \partial g _ { \phi _ { D \setminus S } } } \\ { \displaystyle } & { = \int _ { u = 0 } ^ { 1 } \prod _ { i \in S } \left( 1 - F _ { \phi _ { i } } \left( F _ { \phi _ { D \setminus S } } ^ { - 1 } ( u ) \right) \right) \partial u } \end{array} +$$ + +Here we have used a change of variables $u = F _ { \phi _ { D \setminus S } } ( g _ { \phi _ { D \setminus S } } )$ . This expression can be efficiently numerically integrated (although another change of variables may be required for numerical stability depending on the values of $\phi$ ). + +Exact computation in $O ( 2 ^ { k } )$ . The integral in equation 31 can be computed exactly using the identity + +$$ +\prod _ { i \in S } ( a _ { i } - b _ { i } ) = \sum _ { C \subseteq S } ( - 1 ) ^ { | C | } \prod _ { i \in C } b _ { i } \prod _ { i \in S \setminus C } a _ { i } +$$ + +which gives + +$$ +\begin{array} { r l } { { \mu ( S ) = \int _ { s _ { \phi ( S ) - } - \infty } ^ { \infty } \longrightarrow \int _ { \partial \sigma _ { \phi ( S ) } } \bigl ( g _ { \phi ( S ) , s } \bigr ) \prod ( 1 - F _ { \phi _ { \phi } } ( g _ { \phi ( S ) , s } ) ) \partial g _ { \phi _ { \phi ( S ) , s } } } } \\ & { = \sum _ { c \in S } ^ { \infty } ( - 1 ) ^ { C } \big | \int _ { s _ { \phi ( S ) - } - \infty } ^ { \infty } \int _ { \Phi _ { \phi ( S ) - } ( \partial g _ { \phi ( S ) - } ) } \prod \int _ { s _ { \phi } } F _ { \phi _ { \phi } } ( g _ { \phi ( S ) - s } ) \prod \big | \partial g _ { \phi _ { \phi ( S ) - } s } } \\ & { = \sum _ { c \in S } ^ { \infty } ( - 1 ) ^ { C } \big | \int _ { s _ { \phi ( S ) - } - \infty } ^ { \infty } e _ { \phi _ { \phi ( S ) - } ( \partial g _ { \phi ( S ) - } ) } F _ { \phi _ { \phi _ { \phi ( S ) - } s } } ( g _ { \phi _ { S } , s } ) F _ { \phi _ { \phi ( S ) - } } ( g _ { \phi _ { S } , s } ) \big | \mathcal { P } _ { \phi _ { \phi ( S ) - } } \big | \mathcal { P } _ { \phi _ { \phi ( S ) - } } \big | \mathcal { P } _ { \phi _ { \phi ( S ) - } } } \\ & { = \sum _ { c \in S } ^ { \infty } ( - 1 ) ^ { C } \big | \int _ { s _ { \phi ( S ) - } - \infty } ^ { \infty } e _ { \phi _ { \phi ( S ) - } ( \partial g _ { \phi ( S ) - } ) } F _ { \phi _ { \phi ( S ) - } , \phi _ { \phi ( S ) - } } ( g _ { \phi _ { S } , s } ) \big | \partial g _ { \phi _ { \phi ( S ) - } } } \\ & { = \sum _ { c \in S } ^ { \infty } ( - 1 ) ^ { C } ( 1 - 0 ) \frac { \exp ( \phi _ { \phi ( S ) - } ) } { \exp ( \phi _ { \phi ( S ) - } ) } } \\ & { = \sum _ { c \in S } ^ { \infty } ( - 1 ) ^ { C } ( 1 - 0 ) \exp ( \phi _ { \phi ( S ) - } ) } \\ & = \sum _ { c \in S } ^ { \infty } ( - 1 ) ^ { C } ( 1 - \sum \end{array} +$$ + +Computation of $p ^ { D \setminus C } ( S \setminus C )$ . When using the Gumbel-Top- $k$ trick over the restricted domain $D \backslash C$ , we do not need to renormalize the log-probabilities $\phi _ { s } , s \in D \setminus C$ since the Gumbel-Top- $k$ trick applies to unnormalized log-probabilities. Also, assuming $C \subseteq S ^ { k }$ , it holds that $\left( D \backslash C \right) \backslash ( \mathbf { \bar { S } } )$ $C ) = D \backslash S$ . This means that we can compute $p ^ { D \setminus C } ( S \setminus C )$ similar to equation 31: + +$$ +\begin{array} { r l } { \displaystyle p ^ { D \setminus C } ( S \setminus C ) = P ( \underset { i \in S \backslash C } { \mathrm { m i n } } G _ { \phi _ { i } } > G _ { \phi _ { ( D \setminus C ) \setminus ( S \setminus C ) } } ) } & { } \\ { = P ( \underset { i \in S \backslash C } { \mathrm { m i n } } G _ { \phi _ { i } } > G _ { \phi _ { D \setminus S } } ) } & { } \\ { = \displaystyle \int _ { g _ { \phi _ { D \setminus S } } = - \infty } ^ { \infty } f _ { \phi _ { D \setminus S } } \big ( g _ { \phi _ { D \setminus S } } \big ) \prod _ { i \in S \backslash C } \left( 1 - F _ { \phi _ { i } } \big ( g _ { \phi _ { D \setminus S } } \big ) \right) \partial g _ { \phi _ { D \setminus S } } . } \end{array} +$$ + +Computation of $R ( S ^ { k } , s )$ . Note that, using equation 10, it holds that + +$$ +\sum _ { s \in S ^ { k } } { \frac { p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) p ( s ) } { p ( S ^ { k } ) } } = \sum _ { s \in S ^ { k } } P ( b _ { 1 } = s | S ^ { k } ) = 1 +$$ + +from which it follows that + +$$ +p ( S ^ { k } ) = \sum _ { s \in S ^ { k } } p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) p ( s ) +$$ + +such that + +$$ +R ( S ^ { k } , s ) = { \frac { p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) } { p ( S ^ { k } ) } } = { \frac { p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) } { \sum _ { s ^ { \prime } \in S ^ { k } } p ^ { D \setminus \{ s ^ { \prime } \} } ( S ^ { k } \setminus \{ s ^ { \prime } \} ) p ( s ^ { \prime } ) } } . +$$ + +This means that, to compute the leave-one-out ratio for all $s ~ \in ~ S ^ { k }$ , we only need to compute $p ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} )$ for $s \in S ^ { k }$ . When using the numerical integration or summation in $O ( 2 ^ { k } )$ , we can reuse computation, whereas using the naive method, the cost is $O ( k \cdot ( k - 1 ) ! ) = O ( k ! )$ , making the total computational cost comparable to computing just $p ( S ^ { k } )$ , and the same holds when computing the ‘second-order’ leave one out ratios for the built-in baseline (equation 17). + +Details of numerical integration. For computation of the leave-one-out ratio (equation 35) for large $k$ we can use the numerical integration, where we need to compute equation 34 with $C = \{ s \}$ . For this purpose, we rewrite the integral as + +$$ +\begin{array} { l } { \displaystyle p ^ { D \setminus C } ( S \setminus C ) = \int _ { \phi _ { \partial \setminus \mathcal { D } } \setminus \phi = - \phi } ^ { \phi _ { \partial \setminus \mathcal { D } } } f _ { \phi _ { D \setminus \mathcal { D } } } ( g _ { \phi \phi _ { D \setminus \mathcal { D } } } , s ) \prod _ { i \in S \setminus \mathcal { C } } \left( 1 - F _ { \phi _ { i } } ( g _ { \phi _ { D \setminus i } } , s ) \right) \partial g _ { \phi _ { D \setminus i } } } \\ { \displaystyle \qquad = \int _ { u = 0 } ^ { 1 } \prod _ { i \in S \setminus \mathcal { C } } \Big ( 1 - F _ { \phi _ { i } } \Big ( F _ { \phi _ { D \setminus i } } ^ { - 1 } \Big ( u \Big ) \Big ) \Big ) \partial u } \\ { \displaystyle \qquad = \int _ { u = 0 } ^ { 1 } \prod _ { i \in S \setminus \mathcal { C } } \Big ( 1 - u ^ { \alpha \setminus \phi _ { i } ( \phi _ { i } - \phi _ { D \setminus i } ) s } \Big ) \partial u } \\ { \displaystyle \qquad = \exp ( b ) \cdot \int _ { v = 0 } ^ { 1 } v ^ { \alpha \setminus \phi _ { i } ( b ) - 1 } \prod _ { i \in S \setminus \mathcal { C } } \Big ( 1 - v ^ { \alpha \setminus \Phi _ { i } ( \phi _ { i } - \phi _ { D \setminus i } - \phi _ { i } ) } \Big ) \partial v } \\ { \displaystyle \qquad = \exp ( a + \phi _ { D \setminus S } ) \cdot \int _ { v = 0 } ^ { 1 } v ^ { \alpha \setminus \Phi _ { i } ( \phi _ { i } + \phi _ { D \setminus i } - s ) - 1 } \prod _ { i \in S \setminus \mathcal { C } } \Big ( 1 - v ^ { \alpha \setminus \Phi _ { i } ( \phi _ { i } + s ) } \Big ) \partial v } \\ { \displaystyle \qquad = \exp ( a + \phi _ { D \setminus \mathcal { D } } ) \cdot \int _ { v = 0 } ^ { 1 } v ^ { \alpha \setminus \Phi _ { i } ( \phi _ { i } + \phi _ { D \setminus i } - s ) - 1 } v \in \mathrm { S } ^ { \setminus \Phi _ { i } } } \end{array} +$$ + +Here we have used change of variables $v = u ^ { e x p ( - b ) }$ and $a = b - \phi _ { D \backslash S }$ . This form allows to compute the integrands efficiently, as + +$$ +\prod _ { i \in S \setminus C } \left( 1 - v ^ { \exp ( \phi _ { i } + a ) } \right) = \frac { \prod _ { i \in S } \left( 1 - v ^ { \exp ( \phi _ { i } + a ) } \right) } { \prod _ { i \in C } \left( 1 - v ^ { \exp ( \phi _ { i } + a ) } \right) } +$$ + +where the numerator only needs to computed once, and, since $C = \{ s \}$ when computing equation 35, the denominator only consists of a single term. + +The choice of $a$ may depend on the setting, but we found that $a \ : = \ : 5$ is a good default option which leads to an integral that is generally smooth and can be accurately approximated using the trapezoid rule. We compute the integrands in logarithmic space and sum the terms using the stable LOGSUMEXP trick. In our code we provide an implementation which also computes all second-order leave-one-out ratios efficiently. + +# C THE SUM-AND-SAMPLE ESTIMATOR + +# C.1 UNBIASEDNESS OF THE SUM-AND-SAMPLE ESTIMATOR + +We show that the sum-and-sample estimator is unbiased for any set $C \subset D$ (see also Liang et al. (2018); Liu et al. (2019)): + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \mathrm { s e } \sim p ^ { N } \times ( \pi ) } \left[ \underset { s \in \mathcal { C } } { \sum } p ( c ) f ( c ) + \left( 1 - \underset { s \in \mathcal { C } } { \sum } p ( c ) \right) f ( s ) \right] } \\ & { = \underset { s \in \mathcal { C } } { \sum } p ( c ) f ( e ) + \left( 1 - \underset { s \in \mathcal { C } } { \sum } p ( c ) \right) \mathbb { E } _ { \alpha \sim p ^ { N } \times ( \pi ) } [ f ( x ) ] } \\ & { = \underset { s \in \mathcal { C } } { \sum } p ( c ) f ( e ) + \left( 1 - \underset { s \in \mathcal { C } } { \sum } p ( c ) \right) \underset { s \in \mathcal { C } } { \sum } \frac { p ( x ) } { 1 - \sum _ { s \in \mathcal { C } } p ( c ) } f ( x ) } \\ & { = \underset { s \in \mathcal { C } } { \sum } p ( c ) f ( e ) + \underset { s \in \mathcal { D } } { \sum } p ( x ) f ( x ) } \\ & { = \underset { s \in \mathcal { C } } { \sum } p ( c ) f ( x ) + \underset { s \in \mathcal { D } \backslash \mathcal { C } } { \sum } } \\ & { = \underset { s \in \mathcal { D } } { \sum } p ( x ) f ( x ) } \\ & { = \underset { c \in \mathcal { C } \sim ( \mathcal { C } ) } { \sum } [ f ( x ) ] } \end{array} +$$ + +In this section we give the proof that Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered set estimator. + +Theorem 4. Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered set estimator, i.e. + +$$ +\mathbb { E } _ { B ^ { k } \sim p ( B ^ { k } | S ^ { k } ) } \left[ \sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) f ( b _ { j } ) + \left( 1 - \sum _ { j = 1 } ^ { k - 1 } p ( b _ { j } ) \right) f ( b _ { k } ) \right] = \sum _ { s \in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) . +$$ + +Proof. To give the proof, we first prove three Lemmas. + +Lemma 1. + +$$ +P ( b _ { k } = s | S ^ { k } ) = { \frac { p ( S ^ { k } \setminus \{ s \} ) } { p ( S ^ { k } ) } } { \frac { p ( s ) } { 1 - \sum _ { s ^ { \prime } \in S ^ { k } \setminus \{ s \} } p ( s ^ { \prime } ) } } +$$ + +Proof. Similar to the derivation of $P ( b _ { 1 } = s | S ^ { k } )$ (equation 10 in the main paper), we can write: + +$$ +\begin{array} { r l } & { P ( b _ { k } = s | S ^ { k } ) = \displaystyle \frac { P ( S ^ { k } \cap b _ { k } = s ) } { p ( S ^ { k } ) } } \\ & { \qquad = \displaystyle \frac { p ( S ^ { k } \setminus \{ s \} ) p ^ { D \setminus ( S ^ { k } \setminus \{ s \} ) } ( s ) } { p ( S ^ { k } ) } } \\ & { \qquad = \displaystyle \frac { p ( S ^ { k } \setminus \{ s \} ) } { p ( S ^ { k } ) } \displaystyle \frac { p ( s ) } { 1 - \sum _ { s ^ { \prime } \in S ^ { k } \setminus \{ s \} } p ( s ^ { \prime } ) } . } \end{array} +$$ + +The step from the first to the second row comes from analyzing the event $S ^ { k } \cap b _ { k } = s$ using sequential sampling: to sample $S ^ { k }$ (including $s$ ) with $s$ being the $k$ -th element means that we should first sample $S ^ { k } \setminus \{ s \bar { \} }$ (in any order), and then sample $s$ from the distribution restricted to $D \setminus \left( S ^ { k } \setminus \left\{ s \right\} \right)$ . □ + +Lemma 2. + +$$ +p ( S ) + p ( S \setminus \{ s \} ) { \frac { 1 - \sum _ { s ^ { \prime } \in S } p ( s ^ { \prime } ) } { 1 - \sum _ { s ^ { \prime } \in S \setminus \{ s \} } p ( s ^ { \prime } ) } } = p ^ { D \setminus \{ s \} } ( S \setminus \{ s \} ) +$$ + +Dividing equation 33 by $\begin{array} { r } { 1 - \sum _ { s ^ { \prime } \in S } p ( s ^ { \prime } ) } \end{array}$ on both sides, we obtain + +Proof. + +$$ +\begin{array} { r l } & { \quad _ { 1 } - \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } ( \mathsf { E } ^ { ( k ) } , \mathsf { \Lambda } _ { k } ^ { 2 } ) } \\ & { = \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } ( \mathsf { E } ^ { ( k ) } , \mathsf { \Lambda } _ { k } ^ { 2 } ) \le \epsilon _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } } \\ & { \quad _ { 1 } - \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } \Big ( \mathsf { E } ^ { ( k ) } \frac { \epsilon _ { k \in \mathcal { N } _ { k } } } { 1 - \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } ( \mathsf { E } ^ { ( k ) } , \mathsf { \Lambda } _ { k } ^ { 2 } ) } + \mathsf { E } \eta ^ { \mathsf { C P S } _ { k } } \Big ) + \mathsf { E } ^ { ( k ) } \frac { \epsilon _ { k \in \mathcal { N } _ { k } } } { 1 - \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } ( \mathsf { E } ^ { ( k ) } , \mathsf { \Lambda } _ { k } ^ { 2 } ) } } \\ & = \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } ( \mathsf { E } ^ { ( k ) } , \mathsf { \Lambda } _ { k } ^ { 2 } ) \le \epsilon _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } \le \epsilon _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } \le \epsilon _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } - \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } \frac { \mathsf { E } ^ { ( k ) } } { 1 - \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } ( \mathsf { E } ^ { ( k ) } , \mathsf { \Lambda } _ { k } ^ { 2 } ) } \\ & \quad _ { 1 } - \sum _ { k \in \mathcal { N } _ { k } } ^ { \mathsf { C P S } _ { k } } ( \mathsf { E } ^ { ( k ) } \frac { \epsilon _ { k \in \mathcal { N } _ { k } } } 1 - \sum _ k \ \end{array} +$$ + +Multiplying by $\begin{array} { r } { 1 - \sum _ { s ^ { \prime } \in S } p ( s ^ { \prime } ) } \end{array}$ and rearranging terms proves Lemma 2. + +Lemma 3. + +$$ +p ( s ) + \left( 1 - \sum _ { s ^ { \prime } \in S ^ { k } } p ( s ^ { \prime } ) \right) P ( b _ { k } = s | S ^ { k } ) = p ( s ) R ( S ^ { k } , s ) +$$ + +Proof. First using Lemma 1 and then Lemma 2 we find + +$$ +\begin{array} { r l } & { \quad p ( s ) + \left( 1 - \displaystyle \sum _ { s ^ { \prime } \in S ^ { k } } p ( s ^ { \prime } ) \right) P ( b _ { k } = s | S ^ { k } ) } \\ & { = p ( s ) + \left( 1 - \displaystyle \sum _ { s ^ { \prime } \in S ^ { k } } p ( s ^ { \prime } ) \right) \frac { p ( S ^ { k } \setminus \{ S \} ) } { p ( S ^ { k } ) } \frac { p ( s ) } { 1 - \sum _ { s ^ { \prime } \in S ^ { k } \setminus \{ S \} } p ( s ^ { \prime } ) } } \\ & { = \frac { p ( S ) } { p ( S ^ { k } ) } \left( p ( S ^ { k } ) + \displaystyle \frac { 1 - \sum _ { s ^ { \prime } \in S ^ { k } \setminus \{ P ( s ^ { \prime } ) \} } p ( s ^ { \prime } ) } { 1 - \sum _ { s ^ { \prime } \in S ^ { k } \setminus \{ S \} } p ( s ^ { \prime } ) } p ( S ^ { k } \setminus \{ s \} ) \right) } \\ & { = \frac { p ( s ) } { p ( S ^ { k } ) } p ^ { D \setminus \{ S \} } ( S ^ { k } \setminus \{ s \} ) } \\ & { \quad - p ( s ) R ( S ^ { k } , s ) . } \end{array} +$$ + +Now we can complete the proof of Theorem 4 by adding $p ( b _ { k } ) f ( b _ { k } ) - p ( b _ { k } ) f ( b _ { k } ) = 0$ to the estimator, moving the terms independent of $B ^ { k }$ outside the expectation and using Lemma 3: + +$$ +\begin{array} { r l } & \quad \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } \\ & \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } \\ & \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } \\ & { \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \infty } [ ( 1 - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } ) f ^ { ( i i i i ) } } ] f ^ { ( i i i i ) } } \\ & \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } \\ & { \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } [ f ^ { ( i i ) } + \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } ] f ^ { ( i i i i ) } } \\ & { \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } ( f ^ { ( i i i ) } + \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } ) f ^ { ( i i i i ) } } \\ & { \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } ( f ^ { ( i i i ) } + \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } ) f ^ { ( i i i ) } } \\ & \quad \quad - \sum _ { \varrho ^ { \prime } \to \varrho ^ { \prime } \to \overline { { \wp } } } ( f ^ { ( i i ) } + \sum _ \end{array} +$$ + +# C.3 THE STOCHASTIC SUM-AND-SAMPLE ESTIMATOR WITH MULTIPLE SAMPLES + +As was discussed in Liu et al. (2019), one can trade off the number of summed terms and number of sampled terms to maximize the achieved variance reduction. As a generalization of Theorem 4 (the stochastic sum-and-sample estimator with $k - 1$ summed terms), we introduce here the stochastic sum-and-sample estimator that sums $k - m$ terms and samples $m > 1$ terms without replacement. To estimate the sampled term, we use the unordered set estimator on the $m$ samples without replacement, on the domain restricted to $D \setminus B ^ { k - m }$ . In general, we denote the unordered set estimator restricted to the domain $D \backslash C$ by + +$$ +e ^ { { \bf U S } , D \setminus C } ( S ^ { k } ) = \sum _ { s \in S ^ { k } \setminus C } p ( s ) R ^ { D \setminus C } ( S ^ { k } , s ) f ( s ) +$$ + +where $R ^ { D \setminus C } ( S ^ { k } , s )$ is the leave-one-out ratio restricted to the domain $D \backslash C$ , similar to the second order leave-one-out ratio in equation 18: + +$$ +R ^ { D \setminus C } ( S ^ { k } , s ) = { \frac { p _ { \theta } ^ { ( D \setminus C ) \setminus \{ s \} } ( ( S ^ { k } \setminus C ) \setminus \{ s \} ) } { p _ { \theta } ^ { D \setminus C } ( S ^ { k } \setminus C ) } } . +$$ + +While we can also constrain $S ^ { k } \subseteq ( D \backslash C )$ , this definition is consistent with equation 18 and allows simplified notation. + +Theorem 5. Rao-Blackwellizing the stochastic sum-and-sample estimator with $m \ > \ 1$ samples results in the unordered set estimator, i.e. + +$$ +\ ? \ ? \complement _ { B ^ { k } \sim p ( B ^ { k } \mid S ^ { k } ) } \left[ \sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) f ( b _ { j } ) + \left( 1 - \sum _ { j = 1 } ^ { k - m } p ( b _ { j } ) \right) e ^ { U S , D \setminus B ^ { k - m } } ( S ^ { k } ) \right] = \sum _ { s \in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) . +$$ + +Proof. Recall that for the unordered set estimator, it holds that + +$$ +e ^ { \mathrm { U S } } ( S ^ { k } ) = \mathbb { E } _ { b _ { 1 } \sim p ( b _ { 1 } | S ^ { k } ) } \left[ f ( b _ { 1 } ) \right] = \mathbb { E } _ { x \sim p ( x ) } \left[ f ( x ) \middle | x \in S ^ { k } \right] +$$ + +which for the restricted equivalent (with restricted distribution $p ^ { D \setminus C }$ ) translates into + +$$ +e ^ { \mathrm { U S } , D \setminus C } ( S ^ { k } ) = \mathbb { E } _ { x \sim p ^ { D \setminus C } ( x ) } \left[ f ( x ) \big | x \in S ^ { k } \right] = \mathbb { E } _ { x \sim p ( x ) } \left[ f ( x ) \big | x \in S ^ { k } , x \not \in C \right] . +$$ + +Now we consider the distribution $b _ { k - m + 1 } | S ^ { k } , B ^ { k - m }$ : the distribution of the first element sampled (without replacement) after sampling $B ^ { k - m }$ , given (conditionally on the event) that the set of $k$ samples is $S ^ { k }$ , so we have $b _ { k - m + 1 } \stackrel { - } { \in } S ^ { k }$ and $\mathsf { \bar { b } } _ { k - m + 1 } \notin B ^ { k - m }$ . This means that its conditional expectation of $f ( b _ { k - m + 1 } )$ is the restricted unordered set estimator for $C = B ^ { k - m }$ since + +$$ +\begin{array} { r l } { { e ^ { \mathrm { U S } , D \setminus B ^ { k - m } } ( S ^ { k } ) = \mathbb { E } _ { x \sim p ( x ) } [ f ( x ) \vert x \in S ^ { k } , x \not \in B ^ { k - m } ] } \quad } & { } \\ & { = \mathbb { E } _ { b _ { k - m + 1 } \sim p ( b _ { k - m + 1 } \mid S ^ { k } , B ^ { k - m } ) } [ f ( b _ { k - m + 1 } ) ] . } \end{array} +$$ + +Observing that the definition (equation 42) of the stochastic sum-and-sample estimator does not depend on the actual order of the $m$ samples, and using equation 45, we can reduce the multisample estimator to the stochastic sum-and-sample estimator with $k ^ { \prime } = k - m + 1$ , such that the result follows from equation 36. + +$$ +\begin{array} { c } { { \displaystyle \begin{array} { l } { { { \cal L } _ { \mathrm { e x c } , \mathrm { t r i g h } , \mathrm { \scriptsize ~ 1 . . } } \sum _ { j = 0 } ^ { N } \phi _ { j } \Big | Z _ { j } ^ { ( k ) } \phi _ { j } \Big | + ( \displaystyle { 1 - \sum _ { j = 0 } ^ { N } \phi _ { j } } ) Z _ { j } ^ { ( k ) } z ^ { ( k ) } z ^ { ( k ) } - ( \beta _ { j } - \gamma ) \delta _ { j } ^ { \prime } \Big | } } } \\ { { \displaystyle - \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \mathrm { \cdots ~ 1 } } } \end{array} \} } \\ \displaystyle - \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } ( \displaystyle { 1 - \sum _ { j = 0 } ^ { N } \phi _ { j } } ) \Bigg | \displaystyle { 1 - \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { k = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { l = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { l = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { l = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { j = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } \sum _ { l = 0 } ^ { N } \mathrm { \scriptsize ~ 1 . } } \\ \displaystyle - \sum _ j = \end{array} +$$ + +# D THE IMPORTANCE-WEIGHTED ESTIMATOR + +# D.1 RAO-BLACKWELLIZATION OF THE IMPORTANCE-WEIGHTED ESTIMATOR + +In this section we give the proof that Rao-Blackwellizing the importance-weighted estimator results in the unordered set estimator. + +Theorem 6. Rao-Blackwellizing the importance-weighted estimator results in the unordered set estimator, i.e.: + +$$ +\mathbb { E } _ { \kappa \sim p ( \kappa | S ^ { k } ) } \left[ \sum _ { s \in S ^ { k } } \frac { p ( s ) } { 1 - F _ { \phi _ { s } } ( \kappa ) } f ( s ) \right] = \sum _ { s \in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) . +$$ + +Here we have slightly rewritten the definition of the importance-weighted estimator, using that $q ( s , a ) = P ( g _ { \phi _ { s } } ^ { \mathrm { ~ ~ } } > \stackrel { \cdot } { a } ) = 1 - F _ { \phi _ { s } } ( a )$ , where $F _ { \phi _ { s } }$ is the CDF of the Gumbel distribution (see Appendix A). + +Proof. We first prove the following Lemma: + +Lemma 4. + +$$ +\mathbb { E } _ { \kappa \sim p ( \kappa | S ^ { k } ) } \left[ \frac { 1 } { 1 - F _ { \phi _ { s } } ( \kappa ) } \right] = R ( S ^ { k } , s ) +$$ + +Proof. Conditioning on $S ^ { k }$ , we know that the elements in $S ^ { k }$ have the $k$ largest perturbed logprobabilities, so $\kappa$ , the $( k + 1 )$ -th largest perturbed log-probability is the largest perturbed logprobability in $D \backslash S ^ { k }$ , and satisfies $\begin{array} { r } { \kappa = \operatorname* { m a x } _ { s \in D \backslash S ^ { k } } g _ { \phi _ { s } } = g _ { \phi _ { D \backslash S ^ { k } } } \sim \mathrm { G u m b e l } ( \phi _ { D \backslash S ^ { k } } ) } \end{array}$ . Computing $p ( \kappa | S ^ { k } )$ using Bayes’ Theorem, we have + +$$ +p ( \kappa | S ^ { k } ) = \frac { p ( S ^ { k } | \kappa ) p ( \kappa ) } { p ( S ^ { k } ) } = \frac { \prod _ { s \in S ^ { k } } ( 1 - F _ { \phi _ { s } } ( \kappa ) ) f _ { \phi _ { D \setminus S ^ { k } } } ( \kappa ) } { p ( S ^ { k } ) } +$$ + +which allows us to compute (using equation 34 with $C = \{ s \}$ and $g _ { \phi _ { D \setminus S } } = \kappa$ + +$$ +\begin{array} { r l } & { \mathbb { E } _ { n \sim p ( n | S ^ { k } ) } \left[ \frac { 1 } { 1 - F _ { \phi _ { n } } ( \kappa ) } \right] } \\ & { = \int _ { n = - \infty } ^ { \infty } p ( \kappa | S ^ { k } ) \frac { 1 } { 1 - F _ { \phi _ { n } } ( \kappa ) } \partial \kappa } \\ & { = \int _ { \kappa = - \infty } ^ { \infty } \frac { \prod _ { s \in S ^ { k } } ( 1 - F _ { \phi _ { s } } ( \kappa ) ) f _ { \phi _ { n ; s } } ( \kappa ) } { p ( S ^ { k } ) } \frac { 1 } { 1 - F _ { \phi _ { n } } ( \kappa ) } \partial \kappa } \\ & { = \frac { 1 } { p ( S ^ { k } ) } \int _ { \kappa = - \infty } ^ { \infty } \underset { s \in S ^ { k } \backslash \{ s \} } { \prod } \ ( 1 - F _ { \phi _ { s } } ( \kappa ) ) f _ { \phi _ { n ; s } \mu \kappa } ( \kappa ) \partial \kappa } \\ & { = \frac { 1 } { p ( S ^ { k } ) } p ^ { D \cup S } ( S \setminus \{ s \} ) } \\ & { = R ( S ^ { k } , s ) , } \end{array} +$$ + +Using Lemma 4 we find + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \kappa \sim p ( \kappa | S ^ { k } ) } \left[ \displaystyle \sum _ { s \in S ^ { k } } \frac { p ( s ) } { 1 - F _ { \phi _ { s } } ( \kappa ) } f ( s ) \right] } \\ & { = \displaystyle \sum _ { s \in S ^ { k } } p ( s ) \mathbb { E } _ { \kappa \sim p ( \kappa | S ^ { k } ) } \left[ \frac { 1 } { 1 - F _ { \phi _ { s } } ( \kappa ) } \right] f ( s ) } \\ & { = \displaystyle \sum _ { s \in S ^ { k } } p ( s ) R ( S ^ { k } , s ) f ( s ) . } \end{array} +$$ + +# D.2 THE IMPORTANCE-WEIGHTED POLICY GRADIENT ESTIMATOR WITH BUILT-IN BASELINE + +For self-containment we include this section, which is adapted from our unpublished workshop paper (Kool et al., 2019b). The importance-weighted policy gradient estimator combines REINFORCE (Williams, 1992) with the importance-weighted estimator (Duffield et al., 2007; Vieira, 2017) in equation 15 which results in an unbiased estimator of the policy gradient $\nabla _ { \pmb { \theta } } \mathbb { E } _ { p _ { \pmb { \theta } } ( \pmb { x } ) } [ f _ { \pmb { \theta } } ( \pmb { x } ) ]$ : + +$$ +e ^ { \mathrm { I W P G } } ( S ^ { k } , \kappa ) = \sum _ { s \in S ^ { k } } \frac { p _ { \theta } ( s ) } { q _ { \theta , \kappa } ( s ) } \nabla _ { \theta } \log p _ { \theta } ( s ) f ( s ) = \sum _ { s \in S ^ { k } } \frac { \nabla _ { \theta } p _ { \theta } ( s ) } { q _ { \theta , \kappa } ( s ) } f ( s ) +$$ + +Recall that $\kappa$ is the $( k + 1 )$ -th largest perturbed log-probability (see Section 3.2). We compute a lower variance but biased variant by normalizing the importance weights using the normalization W (Sk) = Ps∈Sk pθ(s)qθ,κ(s) . + +As we show in Kool et al. (2019b), we can include a ‘baseline’ B(Sk) = Ps∈Sk pθ(s)qθ,κ(s) f and correct for the bias (since it depends on the complete sample $S ^ { k }$ ) by weighting individual terms of the estimator by $\begin{array} { r } { 1 - p _ { \pmb { \theta } } ( s ) + \frac { p _ { \pmb { \theta } } ( s ) } { q _ { \pmb { \theta } , \kappa } ( s ) } } \end{array}$ : + +$$ +e ^ { \mathrm { I W P G B L } } ( S ^ { k } , \kappa ) = \sum _ { s \in S ^ { k } } { \frac { \nabla _ { \theta } p _ { \theta } ( s ) } { q _ { \theta , \kappa } ( s ) } } \left( f ( s ) \left( 1 - p _ { \theta } ( s ) + { \frac { p _ { \theta } ( s ) } { q _ { \theta , \kappa } ( s ) } } \right) - B ( S ^ { k } ) \right) +$$ + +For the normalized version, we use the normalization W (Sk) = Ps∈Sk q for the baseline, and $\begin{array} { r } { W _ { i } ( S ^ { k } ) = W ( S ^ { k } ) - \frac { p _ { \theta } ( s ) } { q _ { \theta , \kappa } ( s ) } + p _ { \theta } ( s ) } \end{array}$ to normalize the individual terms: + +$$ +\nabla _ { \pmb \theta } \mathbb { E } _ { \pmb \theta \sim p _ { \pmb \theta } ( \pmb y ) } \left[ f ( \pmb y ) \right] \approx \sum _ { s \in S ^ { k } } \frac { 1 } { W _ { i } ( S ^ { k } ) } \cdot \frac { \nabla _ { \pmb \theta } p _ { \pmb \theta } ( s ) } { q _ { \pmb \theta , \kappa } ( s ) } \left( f ( s ) - \frac { B ( S ^ { k } ) } { W ( S ^ { k } ) } \right) +$$ + +It seems odd to normalize the terms in the outer sum by $\frac { 1 } { W _ { i } ( S ^ { k } ) }$ instead of 1W (Sk) , but equation 52 can be rewritten into a form similar to equation 17, i.e. with a different baseline for each sample, but this form is more convenient for implementation (Kool et al., 2019b). + +# E THE UNORDERED SET POLICY GRADIENT ESTIMATOR + +E.1 PROOF OF UNBIASEDNESS OF THE UNORDERED SET POLICY GRADIENT ESTIMATOR WITH BASELINE + +To prove the unbiasedness of result we need to prove that the control variate has expectation 0: Lemma 5. + +$$ +\mathbb { E } _ { S ^ { k } \sim p _ { \theta } ( S ^ { k } ) } \left[ \sum _ { s \in S ^ { k } } \nabla _ { \theta } p _ { \theta } ( s ) R ( S ^ { k } , s ) \sum _ { s ^ { \prime } \in S ^ { k } } p _ { \theta } ( s ^ { \prime } ) R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) f ( s ^ { \prime } ) \right] = 0 . +$$ + +Proof. Similar to equation 10, we apply Bayes’ Theorem conditionally on $b _ { 1 } ~ = ~ s$ to derive for $s ^ { \prime } \neq s$ + +$$ +\begin{array} { c } { P ( b _ { 2 } = s ^ { \prime } | S ^ { k } , b _ { 1 } = s ) = \displaystyle { \frac { P ( S ^ { k } | b _ { 2 } = s ^ { \prime } , b _ { 1 } = s ) P ( b _ { 2 } = s ^ { \prime } | b _ { 1 } = s ^ { \prime } ) } { P ( S ^ { k } | b _ { 1 } = s ) } } } \\ { = \displaystyle { \frac { p _ { \theta } ^ { D \setminus \{ s , s ^ { \prime } \} } ( S ^ { k } \setminus \{ s , s ^ { \prime } \} ) p _ { \theta } ^ { D \setminus \{ s \} } ( s ^ { \prime } ) } { p _ { \theta } ^ { D \setminus \{ s \} } ( S ^ { k } \setminus \{ s \} ) } } } \\ { { = \displaystyle { \frac { p _ { \theta } ( s ^ { \prime } ) } { 1 - p _ { \theta } ( s ) } R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) } . } } \end{array} +$$ + +For $s ^ { \prime } = s$ we have $R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) = 1$ by definition, so using equation 54 we can show that + +$$ +\begin{array} { r l } & { \displaystyle \sum _ { s ^ { \prime } \in S ^ { k } } p _ { \theta } ( s ^ { \prime } ) R ^ { D N \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) f ( s ^ { \prime } ) } \\ & { = p _ { \theta } ( s ) f ( s ) + \displaystyle \sum _ { s ^ { \prime } \in S ^ { k \setminus \{ s \} } } p _ { \theta } ( s ^ { \prime } ) R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) f ( s ^ { \prime } ) } \\ & { = p _ { \theta } ( s ) f ( s ) + ( 1 - p _ { \theta } ( s ) ) \displaystyle \sum _ { s ^ { \prime } \in S ^ { k \setminus \{ s \} } \setminus \{ s \} } \frac { p _ { \theta } ( s ^ { \prime } ) } { 1 - p _ { \theta } ( s ) } R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) f ( s ^ { \prime } ) } \\ & { = p _ { \theta } ( s ) f ( s ) + ( 1 - p _ { \theta } ( s ) ) \displaystyle \sum _ { s ^ { \prime } \in S ^ { k \setminus \{ s \} } \setminus \{ s \} } P ( b _ { 2 } = s ^ { \prime } | S ^ { k } , b _ { 1 } = s ) f ( s ^ { \prime } ) } \\ & { = p _ { \theta } ( s ) f ( s ) + ( 1 - p _ { \theta } ( s ) ) \| g _ { s ^ { \prime } \in S ^ { k \setminus \{ s \} } \setminus \{ s \} } } \\ & { = p _ { \theta } ( s ) f ( s ) + ( 1 - p _ { \theta } ( s ) ) \| g _ { s ^ { \prime } \circ p _ { \theta } ( b _ { 2 } \mid S ^ { k } , b _ { 1 } = s ) } [ f ( b _ { 2 } ) ] } \\ & { = \mathbb { E } _ { \delta \to p _ { \theta } ( b _ { 2 } \mid S ^ { k } , b _ { 1 } = s ) } [ p _ { \theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ] . } \end{array} +$$ + +Now we can show that the control variate is actually the result of Rao-Blackwellization: + +$$ +\begin{array} { r l } & { \mathbb { E } _ { \theta ^ { \star } \sim \Theta ( \theta ^ { ( k ) } ) } [ \displaystyle \sum _ { \ell \in \mathbb { N } ^ { k } } \nabla \theta ^ { ( k ) } ( \theta ^ { k ) } H ^ { ( S ^ { k } , \ell ) } \cdot \sum _ { \ell \in \mathbb { N } ^ { k } } \mathcal { P } \varphi ( \theta ^ { ( k ) } H ^ { ( S ^ { k } , \ell ) } ( \theta ^ { k } , s ^ { \prime } ) f ( s ^ { \prime } ) ] } \\ & { = \mathbb { E } _ { \theta ^ { \star } \sim \Theta ( \theta ^ { ( k ) } ) } [ \displaystyle \sum _ { \ell \in \mathbb { N } ^ { k } } \mathcal { P } ( \theta ^ { ( k ) } ) H ( \theta ^ { ( k ) } , s ^ { \prime } ) \nabla \theta ( \theta , \theta ^ { \prime } ) \wedge \sum _ { \ell \in \mathbb { N } ^ { k } } \mathcal { P } ( \theta ^ { ( k ) } H ^ { ( S ^ { k } , \ell ) } ( \theta ^ { k } ) \cdot \xi ^ { \prime } ) f ( s ^ { \prime } ) ] } \\ & { = \mathbb { E } _ { \theta ^ { \star } \sim \Theta ( \theta ^ { ( k ) } ) } [ \displaystyle \sum _ { \ell \in \mathbb { N } ^ { k } } \mathcal { P } ( \theta ^ { ( k ) } - s | S ^ { k } | \nabla \theta ) \cdot \Big ( \displaystyle \sum _ { \ell \in \mathbb { N } ^ { k } } \mathcal { P } ( \theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \ell ) } ( \theta ^ { k } ) \Big ) ] } \\ & { = \mathbb { E } _ { \theta ^ { \star } \sim \Theta ( \theta ^ { ( k ) } ) } [ \displaystyle \sum _ { \ell \in \mathbb { N } ^ { k } } \mathcal { P } ( \hat { b } _ { 1 } - s | S ^ { k } | \nabla \theta ) \cdot \Big ( \displaystyle \sum _ { \ell \in \mathbb { N } ^ { k } } \mathcal { P } ( \theta ^ { ( k ) } ) H ^ { ( S ^ { k } , \ell ) } ( \theta ^ { ( k ) } , \theta ^ { \prime } ) f ( s ^ { \prime } ) ] } \\ & = \mathbb { E } _ { \theta ^ { \star } \sim \Theta ( \theta ^ { ( k ) } ) } [ \displaystyle \operatorname* { m i n } _ { \theta ^ { \star } \sim \Theta ( \theta ^ { ( k ) } ) } [ \nabla \theta ^ { ( k ) } \theta ^ { ( k ) } H ^ { ( S ^ { k } , \ell ) } ] \xi ^ { \prime } s ^ \end{array} +$$ + +This expression depends only on $b _ { 1 }$ and $b _ { 2 }$ and we recognize the stochastic sum-and-sample estimator for $k = 2$ used as ‘baseline’. As a special case of equation 13 for $C = \{ b _ { 1 } \}$ , we have + +$$ +\mathbb { E } _ { b _ { 2 } \sim p _ { \theta } ( b _ { 2 } \mid b _ { 1 } ) } \left[ \left( p _ { \theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) \right) \right] = \mathbb { E } _ { i \sim p _ { \theta } ( i ) } \left[ f ( i ) \right] . +$$ + +Using this, and the fact that $\begin{array} { r } { \mathbb { E } _ { b _ { 1 } \sim p _ { \theta } ( b _ { 1 } ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( b _ { 1 } ) \right] = \nabla _ { \theta } \mathbb { E } _ { b _ { 1 } \sim p _ { \theta } ( b _ { 1 } ) } \left[ 1 \right] = \nabla _ { \theta } 1 = 0 } \end{array}$ we find + +$$ +\begin{array} { r l } & { \mathbb { E } _ { S ^ { k } \sim p _ { \theta } ( S ^ { k } ) } \left[ \displaystyle \sum _ { s \in S ^ { k } } \nabla _ { \theta } p _ { \theta } ( s ) R ( S ^ { k } , s ) \displaystyle \sum _ { s ^ { \prime } \in S ^ { k } } p _ { \theta } ( s ^ { \prime } ) R ^ { D \setminus \{ s \} } ( S ^ { k } , s ^ { \prime } ) f ( s ^ { \prime } ) \right] } \\ & { = \mathbb { E } _ { B ^ { k } \sim p _ { \theta } ( B ^ { k } ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( b _ { 1 } ) \left( p _ { \theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) \right) \right] } \\ & { = \mathbb { E } _ { b _ { 1 } \sim p _ { \theta } ( b _ { 1 } ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( b _ { 1 } ) \mathbb { E } _ { b _ { 2 } \sim p _ { \theta } ( b _ { 2 } | b _ { 1 } ) } \left[ ( p _ { \theta } ( b _ { 1 } ) f ( b _ { 1 } ) + ( 1 - p _ { \theta } ( b _ { 1 } ) ) f ( b _ { 2 } ) ) \right] \right] } \\ & { = \mathbb { E } _ { b _ { 1 } \sim p _ { \theta } ( b _ { 1 } ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( b _ { 1 } ) \mathbb { E } _ { x \sim p _ { \theta } ( x ) } \left[ f ( x ) \right] \right] } \\ & { = \mathbb { E } _ { b _ { 1 } \sim p _ { \theta } ( b _ { 1 } ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( b _ { 1 } ) \right] \mathbb { E } _ { x \sim p _ { \theta } ( x ) } \left[ f ( x ) \right] } \\ & { = 0 \cdot \mathbb { E } _ { x \sim p _ { \theta } ( x ) } \left[ f ( x ) \right] } \\ & { = 0 } \end{array} +$$ + +# F THE RISK ESTIMATOR + +# F.1 PROOF OF BUILT-IN BASELINE + +We show that the RISK estimator, taking gradients through the normalization factor actually has a built-in baseline. We first use the log-derivative trick to rewrite the gradient of the ratio as the ratio times the logarithm of the gradient, and then swap the summation variables in the double sum that arises: + +$$ +\begin{array} { r l } \varepsilon ^ { \mathrm { s c } } > 5 1 - \sum _ { j \in \mathcal { K } } \frac { \partial } { \partial x _ { j } } \varepsilon ( \frac \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \cdot \partial \end{array} +$$ + +# G CATEGORICAL VARIATIONAL AUTO-ENCODER + +# G.1 EXPERIMENTAL DETAILS + +We use the code6 by Yin et al. (2019) to reproduce their categorical VAE experiment, of which we include details here for self-containment. The dataset is MNIST, statically binarized by thresholding at 0.5 (although we include results using the standard binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray (2011) in Section G.2). The latent representation $_ { z }$ is $K = 2 0$ dimensional with $C = 1 0$ categories per dimension with a uniform prior $p ( z _ { k } = c ) = 1 / C , k = 1 , . . . , K$ . The encoder is parameterized by $\phi$ as $\begin{array} { r } { q _ { \phi } ( z | \pmb { x } ) = \prod _ { k } q _ { \phi } ( z _ { k } | \pmb { \bar { x } } ) } \end{array}$ and has two fully connected hidden layers with 512 and 256 hidden nodes respectively, with LeakyReLU $\mathit { \Phi } _ { \mathrm { ( \alpha ) } } = 0 . 1$ ) activations. The decoder, parameterized by $\pmb \theta$ , is given by $\begin{array} { r } { \bar { p _ { \pmb \theta } } ( \pmb x | z ) \dot { = } \prod _ { i } p _ { \pmb \theta } ( x _ { i } | \pmb z ) } \end{array}$ , where $x _ { i } \in \{ 0 , 1 \}$ are the pixel values, and has fully connected hidden layers with 256 and 512 nodes and LeakyReLU activation. + +ELBO optimization. The evidence lower bound (ELBO) that we optimize is given by + +$$ +\begin{array} { r } { \mathcal { L } ( \phi , \pmb { \theta } ) = \mathbb { E } _ { z \sim q _ { \phi } ( z \vert x ) } \left[ \ln p _ { \pmb { \theta } } ( \pmb { x } \vert z ) + \ln p ( z ) - \ln q _ { \phi } ( z \vert x ) \right] } \\ { = \mathbb { E } _ { z \sim q _ { \phi } ( z \vert x ) } \left[ \ln p _ { \pmb { \theta } } ( \pmb { x } \vert z ) \right] - K L ( q _ { \phi } ( z \vert x ) \vert \vert p ( z ) ) . } \end{array} +$$ + +For the decoder parameters $\pmb \theta$ , since $q _ { \phi } ( z | \boldsymbol { x } )$ does not depend on $\pmb \theta$ , it follows that + +$$ +\nabla _ { \pmb \theta } \mathcal L ( \phi , \pmb \theta ) = \mathbb E _ { z \sim q _ { \phi } ( z | x ) } \left[ \nabla _ { \pmb \theta } \ln p _ { \pmb \theta } ( \pmb x | z ) \right] . +$$ + +or the encoder parameters $\phi$ , we can write $\nabla _ { \phi } \mathcal { L } ( \phi , \theta )$ using equation 57 and equation 19 as + +$$ +\nabla _ { \phi } \mathcal { L } ( \phi , \pmb { \theta } ) = \mathbb { E } _ { z \sim q _ { \phi } ( z | \pmb { x } ) } \left[ \nabla _ { \phi } \ln q _ { \phi } ( z | \pmb { x } ) \ln p _ { \theta } ( \pmb { x } | z ) \right] - \nabla _ { \phi } K L ( q _ { \phi } ( z | \pmb { x } ) | | p ( z ) ) . +$$ + +This assumes we can compute the $\mathrm { K L }$ divergence analytically. Alternatively, we can use a sample estimate for the KL divergence, and use equation 56 with equation 19 to obtain + +$$ +\begin{array} { l } { \nabla _ { \phi } \mathcal { L } ( \phi , \theta ) = \mathbb { E } _ { z \sim q _ { \phi } ( z | x ) } \left[ \nabla _ { \phi } \ln q _ { \phi } ( z | x ) ( \ln p _ { \theta } ( x | z ) + \ln p ( z ) - \ln q _ { \phi } ( z | x ) ) + \nabla _ { \phi } \ln q _ { \phi } ( z | x ) \right] , } \\ { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ { = \mathbb { E } _ { z \sim q _ { \phi } ( z | x ) } \left[ \nabla _ { \phi } \ln q _ { \phi } ( z | x ) ( \ln p _ { \theta } ( x | z ) - \ln q _ { \phi } ( z | x ) ) \right] . } \end{array} +$$ + +Here we have left out the term $\begin{array} { r } { \mathbb { E } _ { z \sim q _ { \phi } ( z | \pmb { x } ) } \left[ \nabla _ { \phi } \ln q _ { \phi } ( z | \pmb { x } ) \right] = 0 } \end{array}$ , similar to Roeder et al. (2017), and, assuming a uniform (i.e. constant) prior $\ln p ( z )$ , the term ${ { \mathbb E } } _ { z \sim q _ { \phi } ( z | x ) } \left[ \nabla _ { \phi } \ln q _ { \phi } ( z | x ) \ln p ( z ) \right] = 0$ . With a built-in baseline, this second term cancels out automatically, even if it is implemented. Despite the similarity of the equation 56 and equation 57, their gradient estimates (equation 60 and equation 59) are structurally dissimilar and care should be taken to implement the REINFORCE estimator (or related estimators such as ARSM and the unordered set estimator) correctly using automatic differentiation software. Using Gumbel-Softmax and RELAX, we take gradients ‘directly’ through the objective in equation 57. + +We optimize the ELBO using the analytic KL for 1000 epochs using the Adam (Kingma & Ba, 2015) optimizer. We use a learning rate of $\mathrm { i 0 ^ { - 3 } }$ for all estimators except Gumbel-Softmax and RELAX, which use a learning rate of $\bar { 1 } 0 ^ { - 4 }$ as we found they diverged with a higher learning rate. For ARSM, as an exception we use the sample KL, and a learning rate of $3 \cdot 1 0 ^ { - \overline { { 4 } } }$ , as suggested by the authors. All reported ELBO values are computed using the analytic KL. Our code is publicly available7. + +# G.2 ADDITIONAL RESULTS + +Gradient variance during training. We also evaluate gradient variance of different estimators during different stages of training. We measure the variance of different estimators with $k = 4$ samples during training with REINFORCE with replacement, such that all estimators are computed for the same model parameters. The results during training, given in Figure 4, are similar to the results for the trained model in Table 1, except for at the beginning of training, although the rankings of different estimator are mostly the same. + +Negative ELBO on validation set. Figure 5 shows the -ELBO evaluated during training on the validation set. For the large latent space, we see validation error quickly increase (after reaching a minimum) which is likely because of overfitting (due to improved optimization), a phenomenon observed before (Tucker et al., 2017; Grathwohl et al., 2018). Note that before the overfitting starts, both REINFORCE without replacement and the unordered set estimator achieve a validation error similar to the other estimators, such that in a practical setting, one can use early stopping. + +Results using standard binarized MNIST dataset. Instead of using the MNIST dataset binarized by thresholding values at 0.5 (as in the code and paper by Yin et al. (2019)) we also experiment with the standard (fixed) binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray (2011), for which we plot train and validation curves for two runs on the small and large domain in Figure 6. This gives more realistic (higher) -ELBO scores, although we still observe the effect of overfitting. As this is a bit more unstable setting, one of the runs using REINFORCE with replacement diverged, but in general the relative performance of estimators is similar to using the dataset with 0.5 threshold. + +![](images/3cf64085ef97333430dfc6ae343dba9cc3bb81a28a8d5d46e4e908f59c58c425.jpg) +Training log variance (10 2 latent space), $k = 4$ samples +Figure 4: Gradient log variance of different unbiased estimators with $k = 4$ samples, estimated every 100 (out of 1000) epochs while training using REINFORCE with replacement. Each estimator is computed 1000 times with different latent samples for a fixed minibatch (the first 100 records of training data). We report (the logarithm of) the sum of the variances per parameter (trace of the covariance matrix). Some lines coincide, so we sort the legend by the last measurement and report its value. + +![](images/9db0e1232d97d8b19cc6e8b5122295f21e9ef07cda2dbb6e7624da7dadc43d3b.jpg) +Figure 5: Smoothed validation -ELBO curves during training of two independent runs when with different estimators with $k = 1$ , 4 or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value. + +![](images/d1fca44c1c3c18bdebf0fcdc5ea9a18b01b73cd2389f61f27b68f5e6d89e5bae.jpg) +Figure 6: Smoothed training and validation -ELBO curves during training on the standard binarized MNIST dataset (Salakhutdinov & Murray, 2008; Larochelle & Murray, 2011) of two independent runs when with different estimators with $k = 1$ , 4 or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value. + +# H TRAVELLING SALESMAN PROBLEM + +The Travelling Salesman Problem (TSP) is a discrete optimization problem that consists of finding the order in which to visit a set of locations, given as $x , y$ coordinates, to minimize the total length of the tour, starting and ending at the same location. As a tour can be considered a sequence of locations, this problem can be set up as a sequence modelling problem, that can be either addressed using supervised (Vinyals et al., 2015) or reinforcement learning (Bello et al., 2016; Kool et al., 2019a). + +Kool et al. (2019a) introduced the Attention Model, which is an encoder-decoder model which considers a TSP instances as a fully connected graph. The encoder computes embeddings for all nodes (locations) and the decoder produces a tour, which is sequence of nodes, selecting one note at the time using an attention mechanism, and uses this autoregressively as input to select the next node. In Kool et al. (2019a), this model is trained using REINFORCE, with a greedy rollout used as baseline to reduce variance. + +We use the code by Kool et al. (2019a) to train the exact same Attention Model (for details we refer to Kool et al. (2019a)), and minimize the expected length of a tour predicted by the model, using different gradient estimators. We did not do any hyperparameter optimization and used the exact same training details, using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of $1 0 ^ { - 4 }$ (no decay) for 100 epochs for all estimators. For the baselines, we used the same batch size of 512, but for estimators that use $k = 4$ samples, we used a batch size of $\frac { 5 1 2 } { 4 } = 1 2 8$ to compensate for the additional samples (this makes multi-sample methods actually faster since the encoder still needs to be evaluated only once). \ No newline at end of file diff --git a/parse/train/rklEj2EFvB/rklEj2EFvB_model.json b/parse/train/rklEj2EFvB/rklEj2EFvB_model.json new file mode 100644 index 0000000000000000000000000000000000000000..44ab31762fe128fadd316ffe6ceda57aae2a25c5 --- /dev/null +++ b/parse/train/rklEj2EFvB/rklEj2EFvB_model.json @@ -0,0 +1,36407 @@ +[ + { + "layout_dets": [ + { + "category_id": 1, + "poly": [ + 297, + 1293, + 1404, + 1293, + 1404, + 1510, + 297, + 1510 + ], + "score": 0.982 + }, + { + "category_id": 1, + "poly": [ + 398, + 650, + 1302, + 650, + 1302, + 957, + 398, + 957 + ], + "score": 0.981 + }, + { + "category_id": 1, + "poly": [ + 297, + 1579, + 1404, + 1579, + 1404, + 1764, + 297, + 1764 + 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\\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } ) f ^ { ( i i i i ) } } ] f ^ { ( i i i i ) } } \\\\ & \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } \\\\ & { \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } [ f ^ { ( i i ) } + \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ] f ^ { ( i i i i ) } } \\\\ & { \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ( f ^ { ( i i i ) } + \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ) f ^ { ( i i i i ) } } \\\\ & { \\quad - \\sum _ { \\varrho ^ { \\prime } \\to \\varrho ^ { \\prime } \\to \\overline { { \\wp } } } ( f ^ { ( i 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} \\in S ^ { k \\setminus \\{ s \\} } } p _ { \\theta } ( s ^ { \\prime } ) R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } \\frac { p _ { \\theta } ( s ^ { \\prime } ) } { 1 - p _ { \\theta } ( s ) } R ^ { D \\setminus \\{ s \\} } ( S ^ { k } , s ^ { \\prime } ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\displaystyle \\sum _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } P ( b _ { 2 } = s ^ { \\prime } | S ^ { k } , b _ { 1 } = s ) f ( s ^ { \\prime } ) } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ { s ^ { \\prime } \\in S ^ { k \\setminus \\{ s \\} } \\setminus \\{ s \\} } } \\\\ & { = p _ { \\theta } ( s ) f ( s ) + ( 1 - p _ { \\theta } ( s ) ) \\| g _ 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