| 1: Inputs: input measures μ,V; cost function c; batch size p; learning rate y. Discrete case: μ = ∑i aiδxi and u is a finite vector: u(xi) @f ui (similarly for v and v) |
| 2: 3: | Continuous case: μ is a continuous measure and u is a neural network (similarly for V and u) |
| Vindicates the gradient w.r.t. the parameters |
| 4: | while not converged do |
| 5: | sample a batch (x1,..,xp) from μ |
| 6: 7: | sample a batch (y1,.., yp) from v update u ←u+γ∑ij Vu(xi)+OuFe(u(xi),v(yj))Vu(xi) |
| 8: | updateu ← U+γ∑ij Vv(yj)+OUFe(u(xi),v(yj))Vv(yj) 9: ：end while |
+
+general functions. Genevay et al. (2016) used the same stochastic dual maximization approach to compute the regularized OT objective in the continuous-continuous setting. The difference lies in their pamaterization of the dual variables as kernel expansions, while we decide to use deep neural networks. Using a neural network for parameterizing a continuous dual variable was done also by Arjovsky et al. (2017). The same discussion also stands for the second dual variable $v$ . Our stochastic gradient algorithm is detailed in Alg. 1.
+
+Convergence rates and computational cost comparison. We first discuss convergence rates in the discrete-discrete setting (i.e. both measures are discrete), where the problem is convex, while parameterization of dual variables as neural networks in the semi-discrete or continuous-continuous settings make the problem non-convex. Because the dual (6) is not strongly convex, full-gradient descent converges at a rate of $\mathcal { O } ( 1 / k )$ , where $k$ is the iteration number. SGD with a decreasing step size converges at the inferior rate of $\mathcal { O } ( 1 / \sqrt { k } )$ (Nemirovski et al., 2009), but with a $\mathcal { O } ( 1 )$ cost per iteration. The two rates can be interpolated when using mini-batches, at the cost of $\mathcal { O } ( p ^ { 2 } )$ per iteration, where $p$ is the mini-batch size. In contrast, Genevay et al. (2016) considered a semi-dual objective of the form $\mathbb { E } _ { X \sim \mu } \left[ u ( X ) + G _ { \varepsilon } ( u ( X ) ) \right]$ , with a cost per iteration which is now ${ \mathcal { O } } ( n )$ due to the computation of the gradient of √ $G _ { \varepsilon }$ . Because that objective is not strongly convex either, SGD converges at the same $O ( 1 / \sqrt { k } )$ rate, up to problem-specific constants. As noted by Genevay et al. (2016), this rate can be improved to $\mathcal { O } ( 1 / k )$ while maintaining the same iteration cost, by using stochastic average gradient (SAG) method (Schmidt et al., 2017). However, SAG requires to store past stochastic gradients, which can be problematic in a large-scale setting.
+
+In the semi-discrete setting (i.e. one measure is discrete and the other is continuous), SGD on the semi-dual objective proposed by Genevay et al. (2016) also converges at a rate of $\mathcal { O } ( 1 / \sqrt { k } )$ , whereas we only know that Alg. 1 converges to a stationary point in this non-convex case.
+
+In the continuous-continuous setting (i.e. both measures are continuous), Genevay et al. (2016) proposed to represent the dual variables as kernel expansions. A disadvantage of their approach, however, is the $\mathcal { O } ( k ^ { 2 } )$ cost per iteration. In contrast, our approach represents dual variables as neural networks. While non-convex, our approach preserves a $\mathcal { O } ( p ^ { 2 } )$ cost per iteration. This parameterization with neural networks was also used by Arjovsky et al. (2017) who maximized the 1-Wasserstein dual-objective function $\mathbb { E } _ { ( X , Y ) \sim \mu \times \nu } \left[ u ( X ) - u ( Y ) \right]$ . Their algorithm is hence very similar to ours, with the same complexity $\mathcal { O } ( p ^ { 2 } )$ per iteration. The main difference is that they had to constrain $u$ to be a Lipschitz function and hence relied of weight clipping in-between gradient updates. The proposed algorithm is capable of computing the regularized OT objective and optimal plans between empirical measures supported on arbitrary large numbers of samples. In statistical machine learning, one aims at estimating the underlying continuous distribution from which empirical observations have been sampled. In the context of optimal transport, one would like to approximate the true (non-regularized) optimal plan between the underlying measures. The next section states theoretical guarantees regarding this problem.
+
+# 3.2 CONVERGENCE OF REGULARIZED OT PLANS
+
+Consider discrete probability measures $\begin{array} { r } { \mu _ { n } = \sum _ { i = 1 } ^ { n } a _ { i } \delta _ { x _ { i } } \in P ( \mathcal { X } ) } \end{array}$ and $\textstyle \nu _ { n } = \sum _ { j = 1 } ^ { n } b _ { j } \delta _ { y _ { j } } \in P ( \mathcal { Y } )$ Analysis of entropy-regularized linear programs (Cominetti $\&$ San Mart´ın, 1994) shows that the solution $\boldsymbol { \pi } _ { n } ^ { \varepsilon }$ of the entropy-regularized problem (3) converges exponentially fast to a solution $\pi _ { n }$ of the non-regularized OT problem (2). Also, a result about stability of optimal transport (Villani, 2008)[Theorem 5.20] states that, if $\mu _ { n } \mu$ and $\nu _ { n } \to \nu$ weakly, then a sequence $\left( \pi _ { n } \right)$ of optimal transport plans between $\mu _ { n }$ and $\nu _ { n }$ converges weakly to a solution $\pi$ of the OT problem between $\mu$ and $\nu$ . We can thus write,
+
+$$
+\operatorname* { l i m } _ { n \to \infty } \operatorname* { l i m } _ { \varepsilon \to 0 } \pi _ { n } ^ { \varepsilon } = \pi .
+$$
+
+A more refined result consists in establishing the weak convergence of $\pi _ { n } ^ { \varepsilon }$ to $\pi$ when $( n , \varepsilon )$ jointly converge to $( \infty , 0 )$ . This is the result of the following theorem which states a stability property of entropy-regularized plans (proof in the Appendix).
+
+$\mu \in P ( \mathcal { X } )$ $\nu \in P ( \mathcal { V } )$ where iscrete $\mathcal { X }$ and babi $\mathcal { V }$ are complete metric spaces. Let measures which converge weakly $\textstyle \mu _ { n } = \sum _ { i = 1 } ^ { n } a _ { i } \delta _ { x _ { i } }$ $\begin{array} { r } { \nu _ { n } = \sum _ { j = 1 } ^ { n } b _ { j } \delta _ { y _ { j } } } \end{array}$ to $\mu$ and $\nu$ respectively, and let $\left( \varepsilon _ { n } \right)$ a sequence of non-negative real numbers converging to 0 sufficiently fast. Assume the cost $c$ is continuous on $\mathcal { X } \times \mathcal { V }$ and finite. Let $\pi _ { n } ^ { \varepsilon _ { n } }$ the solution of the entropy-regularized OT problem (3) between $\mu _ { n }$ and $\nu _ { n }$ . Then, up to extraction of a subsequence, $( \pi _ { n } ^ { \varepsilon _ { n } } )$ converges weakly to the solution $\pi$ of the OT problem (2) between $\mu$ and $\nu$ ,
+
+$$
+\pi _ { n } ^ { \varepsilon _ { n } } \to \pi w e a k l y .
+$$
+
+Keeping the analogy with statistical machine learning, this result is an analog to the universal consistency property of a learning method. In most applications, we consider empirical measures and $n$ is fixed, so that regularization, besides enabling dual stochastic approach, may also help learn the optimal plan between the underlying continuous measures.
+
+So far, we have derived an algorithm for computing the regularized OT objective and regularized optimal plans regardless of $\mu$ and $\nu$ being discrete or continuous. The OT objective has been used successfully as a loss in machine learning (Montavon et al., 2015; Frogner et al., 2015; Rolet et al., 2016; 2018; Arjovsky et al., 2017; Courty et al., 2017a), whereas the use of optimal plans has straightforward applications in logistics, as well as economy (Kantorovich, 1942; Carlier, 2012) or computer graphics (Bonneel et al., 2011). In numerous applications however, we often need mappings rather than joint distributions. This is all the more motivated since Brenier (1991) proved that when the source measure is continuous, the optimal transport plan is actually induced by a map. Assuming that available data samples are sampled from some underlying continuous distributions, finding the Monge map between these continuous measures rather than a discrete optimal plan between discrete measures is essential in machine learning applications. Hence in the next section, we investigate how to recover an optimal map, i.e. find an approximate solution to the Monge problem (1), from regularized optimal plans.
+
+# 4 OPTIMAL MAPPING ESTIMATIONS
+
+A map can be obtained from a solution to the OT problem (2) or regularized OT problem (3) through the computation of its barycentric projection. Indeed, a solution $\pi$ of Problem (2) or (3) between a source measure $\mu$ and a target measure $\nu$ is, identifying the plan $\pi$ with its density w.r.t. a reference measure, a function $\pi : ( \bar { x , y } ) \in \mathcal { X } \times \mathcal { Y } \mapsto \mathbb { R } ^ { + }$ which can be seen as a weighted one-to-many map, i.e. $\pi$ sends $x$ to each location $y \in \operatorname { S u p p } ( \nu )$ where $\pi ( x , y ) > 0$ . A map can then be obtained by simply averaging over these $y$ according to the weights $\pi ( x , y )$ .
+
+Definition 1. (Barycentric projection) Let $\pi$ be a solution of the OT problem (2) or regularized OT problem (3). The barycentric projection $\bar { \pi }$ w.r.t. a convex cost $d : \mathcal { V } \times \mathcal { V } \mathbb { R } ^ { + }$ is defined as,
+
+$$
+\bar { \pi } ( x ) = \arg \operatorname* { m i n } _ { z } \mathbb { E } _ { Y \sim \pi ( \cdot | x ) } \left[ d ( z , Y ) \right] .
+$$
+
+In the special case $d ( x , y ) = \| x - y \| _ { 2 } ^ { 2 }$ , Eq. (11) has the close form solution $\bar { \pi } ( x ) = \operatorname { \mathbb { E } } _ { Y \sim \pi ( \cdot | x ) } \left[ Y \right] .$ , which is equal to $\begin{array} { r } { \bar { \boldsymbol { \pi } } = \frac { \boldsymbol { \pi } \mathbf { y } ^ { t } } { a } } \end{array}$ in a discrete setting with $\mathbf { y } = ( y _ { 1 } , \dotsb , y _ { n } )$ and $a$ the weights of $\mu$ . Moreover, for the specific squared Euclidean cost $c ( x , y ) = \| x - y \| _ { 2 } ^ { 2 }$ , the barycentric projection $\bar { \pi }$ is an optimal map (Ambrosio et al., 2006)[Theorem 12.4.4], i.e. $\bar { \pi }$ is a solution to the Monge problem (1) between the source measure $\mu$ and the target measure $\bar { \pi } \# \mu$ . Hence the barycentric projection w.r.t. the squared Euclidean cost is often used as a simple way to recover optimal maps from optimal transport plans (Reich, 2013; Wang et al., 2013; Ferradans et al., 2014; Seguy & Cuturi, 2015).
+
+| lgorithm 2 Optimal map learning with SGD |
| Inputs: input measures μ, V; cost function c; dual optimal variables u and v; map fe parame- |
| terized as a deep NN; batch size n; learning rate y. while not converged do |
| sample a batch (x1,·· ,xn) from μ |
| sample a batch (y1,:.,yn) from v |
| updateθ←0-γ∑ijHe(xi,yj)Vθd(yj,fe(xi)) end while |
+
+Formula (11) provides a pointwise value of the barycentric projection. When $\mu$ is discrete, this means that we only have mapping estimations for a finite number of points. In order to define a map which is defined everywhere, we parameterize the barycentric projection as a deep neural network. We show in the next paragraph how to efficiently learn its parameters.
+
+Optimal map learning An estimation $f$ of the barycentric projection of a regularized plan $\pi ^ { \varepsilon }$ which generalizes outside the support of $\mu$ can be obtained by learning a deep neural network which minimizes the following objective w.r.t. the parameters $\theta$ ,
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { X \sim \mu } \left[ \mathbb { E } _ { Y \sim \pi ^ { \varepsilon } ( \cdot | X ) } \left[ d ( Y , f _ { \theta } ( X ) ) \right] \right] = \mathbb { E } _ { ( X , Y ) \sim \pi ^ { \varepsilon } } \left[ d ( Y , f _ { \theta } ( X ) ) \right] } \\ & { \qquad = \mathbb { E } _ { ( X , Y ) \sim \mu \times \nu } \left[ d ( Y , f _ { \theta } ( X ) ) H _ { \varepsilon } ( X , Y ) \right] . } \end{array}
+$$
+
+When $d ( x , y ) = \| x - y \| ^ { 2 }$ , the last term in Eq. (12) is simply a weighted sum of squared errors, with possibly an infinite number of terms whenever $\mu$ or $\nu$ are continuous. We propose to minimize the objective (12) by stochastic gradient descent, which provides the simple Algorithm 2. The OT problem being symmetric, we can also compute the opposite barycentric projection $g$ w.r.t. a cost $\bar { d } : \mathcal { X } \times \mathcal { X } \overset { } { \to } \bar { \mathbb { R } } ^ { + }$ by minimizing $\mathbb { E } _ { ( X , Y ) \sim \mu \times \nu } \left[ d ( g ( \bar { Y } ) , X ) H _ { \varepsilon } ( \bar { X } , Y ) \right]$ .
+
+However, unless the plan $\pi$ is induced by a map, the averaging process results in having the image of the source measure by $\bar { \pi }$ only approximately equal to the target measure $\nu$ . Still, when the size of discrete measure is large and the regularization is small, we show in the next paragraph that 1) the barycentric projection of a regularized OT plan is close to the Monge map between the underlying continuous measures (Theorem 2) and 2) the image of the source measure by this barycentric projection should be close to the target measure $\nu$ (Corollary 1).
+
+Theoretical guarantees As stated earlier, when $\mathcal { X } = \mathcal { V }$ and $c ( x , y ) = \| x - y \| _ { 2 } ^ { 2 }$ , Brenier (1991) proved that when the source measure $\mu$ is continuous, there exists a solution to the Monge problem (1). This result was generalized to more general cost functions, see (Villani, 2008)[Corollary 9.3] for details. In that case, the plan $\pi$ between $\mu$ and $\nu$ is written as $( \operatorname { i d } , f ) \# \mu$ where $f$ is the Monge map. Now considering discrete measures $\mu _ { n }$ and $\nu _ { n }$ which converge to $\mu$ (continuous) and $\nu$ respectively, we have proved in Theorem 1 that $\boldsymbol { \pi } _ { n } ^ { \varepsilon }$ converges weakly to $\pi = ( \operatorname { i d } , f ) \# \mu$ when $( n , \varepsilon ) \to ( \infty , 0 )$ . The next theorem, proved in the Appendix, shows that the barycentric projection $\bar { \pi } _ { n } ^ { \varepsilon }$ also converges weakly to the true Monge map between $\mu$ and $\nu$ , justifying our approach.
+
+Theorem 2. Let $\mu$ be a continuous probability measure on $\mathbb { R } ^ { d }$ , and $\nu$ an arbitrary probability and measure on $\textstyle \nu _ { n } = { \frac { 1 } { n } } \sum _ { j = 1 } ^ { n } \delta _ { y _ { j } }$ $\mathbb { R } ^ { d }$ and c a cost function satisfying (Villani, 2008)[Corollary 9.3]. Let converging weakly to $\mu$ and $\nu$ n i=1 respectively. Assume that the OT solution $\begin{array} { r } { \mu _ { n } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \delta _ { x _ { i } } } \end{array}$ $\pi _ { n }$ of Problem (2) between $\mu _ { n }$ and $\nu _ { n }$ is unique for all $n$ . Let $\left( \varepsilon _ { n } \right)$ a sequence of non-negative real numbers converging sufficiently fast to 0 and $\bar { \pi } _ { n } ^ { \varepsilon _ { n } }$ the barycentric projection w.r.t. the convex cost $d = c$ of the solution $\pi _ { n } ^ { \varepsilon _ { n } }$ of the entropy-regularized OT problem (3). Then, up to extraction of $a$ subsequence,
+
+$$
+( i d , \bar { \pi } _ { n } ^ { \varepsilon _ { n } } ) \# \mu _ { n } ( i d , f ) \# \mu \ w e a k l y ,
+$$
+
+where $f$ is the solution of the Monge problem (1) between $\mu$ and $\nu$ .
+
+This theorem shows that our estimated barycentric projection is close to an optimal map between the underlying continuous measures for $n$ big and $\varepsilon$ small. The following corollary confirms the intuition that the image of the source measure by this map converges to the underlying target measure.
+
+Corollary 1. With the same assumptions as above, ${ \bar { \pi } } _ { n } ^ { \varepsilon _ { n } } \# \mu _ { n } \to \nu$ weakly.
+
+In terms of random variables, the last equation states that if $X _ { n } \sim \mu _ { n }$ and $Y \sim \nu$ , then $\bar { \pi } _ { n } ^ { \varepsilon _ { n } } \left( X _ { n } \right)$ converges in distribution to $Y$ .
+
+
+Figure 2: Convergence plots of the the Stochastic Dual Algorithm 1 against a stochastic semi-dual implementation (adapted from (Genevay et al., 2016): we use SGD instead of SAG), for several entropy-regularization values. Learning rates are $\{ 5 . , 2 0 . , 2 0 . \}$ and batch sizes $\{ 1 0 2 4 , 5 0 0 , 1 0 0 \}$ respectively and are taken the same for the dual and semi-dual methods.
+
+These theoretical results show that our estimated Monge map can thus be used to perform domain adaptation by mapping a source dataset to a target dataset, as well as perform generative modeling by mapping a continuous measure to a target discrete dataset. We demontrate this in the following section.
+
+# 5 NUMERICAL EXPERIMENTS
+
+# 5.1 DUAL VS SEMI-DUAL SPEED COMPARISONS
+
+We start by evaluating the training time of our dual stochastic algorithm 1 against a stochastic semidual approach similar to (Genevay et al., 2016). In the semi-dual approach, one of the dual variable is eliminated and is computed in close form. However, this computation has ${ \mathcal { O } } ( n )$ complexity where $n$ is the size of the target measure $\nu$ . We compute the regularized OT objective with both methods on a spectral transfer problem, which is related to the color transfer problem (Reinhard et al., 2001; Pitie et al., 2007), but where images are multispectral, ´ i.e. they share a finer sampling of the light wavelength. We take two $5 0 0 \times 5 0 0$ images from the CAVE dataset (Yasuma et al., 2010) that have 31 spectral bands. As such, the optimal transport problem is computed on two empirical distributions of 250000 samples in $\mathbb { R } ^ { 3 1 }$ on which we consider the squared Euclidean ground cost $c$ . The timing evolution of train losses are reported in Figure 2 for three different regularization values $\varepsilon = \{ 0 . 0 \bar { 2 } 5 , 0 . 1 , 1 . \}$ . In the three cases, one can observe that convergence of our proposed dual algorithm is much faster.
+
+# 5.2 LARGE SCALE DOMAIN ADAPTATION
+
+We apply here our computation framework on an unsupervised domain adaptation (DA) task, for which optimal transport has shown to perform well on small scale datasets (Courty et al., 2017b; Perrot et al., 2016; Courty et al., 2014). This restriction is mainly due to the fact that those works only consider the primal formulation of the OT problem. Our goal here is not to compete with the state-of-the-art methods in domain adaptation but to assess that our formulation allows to scale optimal transport based domain adaptation (OTDA) to large datasets. OTDA is illustrated in Fig. 3 and follows two steps: 1) learn an optimal map between the source and target distribution, 2) map the source samples and train a classifier on them in the target domain. Our formulation also allows to use any differentiable ground cost $c$ while (Courty et al., 2017b) was limited to the squared Euclidean distance.
+
+Datasets We consider the three cross-domain digit image datasets MNIST (Lecun et al., 1998), USPS, and SVHN (Netzer et al., 2011), which have 10 classes each. For the adaptation between MNIST and USPS, we use 60000 samples in the MNIST domain and 9298 samples in USPS domain. MNIST images are resized to the same resolution as USPS ones $( 1 6 \times 1 6 )$ ). For the adaptation between SVHN and MNIST, we use 73212 samples in the SVHN domain and 60000 samples in the MNIST domain. MNIST images are zero-padded to reach the same resolution as SVHN $( 3 2 \times 3 2 )$ and extended to three channels to match SVHN image sizes. The labels in the target domain are withheld during the adaptation. In the experiment, we consider the adaptation in three directions: M $\mathrm { I N I S T } \to \mathrm { U S P S }$ , $\mathrm { U S P S } \to \mathrm { M N I S T }$ , and $\mathrm { S V H N } \to \mathrm { M N I S T } .$
+
+
+Figure 3: Illustration of the OT Domain Adaptation method adapted from (Courty et al., 2017b). Source samples are mapped to the target set through the barycentric projection $\bar { \pi } ^ { \varepsilon }$ . A classifier is then learned on the mapped source samples.
+
+Table 1: Results (accuracy in $\%$ ) on domain adaptation among MNIST, USPS and SVHN datasets with entropy $( R _ { e } )$ and L2 $( R _ { L ^ { 2 } } )$ regularizations. Source only refers to 1-NN classification between source and target samples without adaptation.
+
+ | 1: Inputs: input measures μ,V; cost function c; batch size p; learning rate y. Discrete case: μ = ∑i aiδxi and u is a finite vector: u(xi) @f ui (similarly for v and v) |
| 2: 3: | Continuous case: μ is a continuous measure and u is a neural network (similarly for V and u) |
| Vindicates the gradient w.r.t. the parameters |
| 4: | while not converged do |
| 5: | sample a batch (x1,..,xp) from μ |
| 6: 7: | sample a batch (y1,.., yp) from v update u ←u+γ∑ij Vu(xi)+OuFe(u(xi),v(yj))Vu(xi) |
| 8: | updateu ← U+γ∑ij Vv(yj)+OUFe(u(xi),v(yj))Vv(yj) 9: ：end while |
",
+ "bbox": [
+ 179,
+ 122,
+ 815,
+ 272
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "general functions. Genevay et al. (2016) used the same stochastic dual maximization approach to compute the regularized OT objective in the continuous-continuous setting. The difference lies in their pamaterization of the dual variables as kernel expansions, while we decide to use deep neural networks. Using a neural network for parameterizing a continuous dual variable was done also by Arjovsky et al. (2017). The same discussion also stands for the second dual variable $v$ . Our stochastic gradient algorithm is detailed in Alg. 1. ",
+ "bbox": [
+ 174,
+ 301,
+ 825,
+ 386
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Convergence rates and computational cost comparison. We first discuss convergence rates in the discrete-discrete setting (i.e. both measures are discrete), where the problem is convex, while parameterization of dual variables as neural networks in the semi-discrete or continuous-continuous settings make the problem non-convex. Because the dual (6) is not strongly convex, full-gradient descent converges at a rate of $\\mathcal { O } ( 1 / k )$ , where $k$ is the iteration number. SGD with a decreasing step size converges at the inferior rate of $\\mathcal { O } ( 1 / \\sqrt { k } )$ (Nemirovski et al., 2009), but with a $\\mathcal { O } ( 1 )$ cost per iteration. The two rates can be interpolated when using mini-batches, at the cost of $\\mathcal { O } ( p ^ { 2 } )$ per iteration, where $p$ is the mini-batch size. In contrast, Genevay et al. (2016) considered a semi-dual objective of the form $\\mathbb { E } _ { X \\sim \\mu } \\left[ u ( X ) + G _ { \\varepsilon } ( u ( X ) ) \\right]$ , with a cost per iteration which is now ${ \\mathcal { O } } ( n )$ due to the computation of the gradient of √ $G _ { \\varepsilon }$ . Because that objective is not strongly convex either, SGD converges at the same $O ( 1 / \\sqrt { k } )$ rate, up to problem-specific constants. As noted by Genevay et al. (2016), this rate can be improved to $\\mathcal { O } ( 1 / k )$ while maintaining the same iteration cost, by using stochastic average gradient (SAG) method (Schmidt et al., 2017). However, SAG requires to store past stochastic gradients, which can be problematic in a large-scale setting. ",
+ "bbox": [
+ 173,
+ 392,
+ 825,
+ 590
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "In the semi-discrete setting (i.e. one measure is discrete and the other is continuous), SGD on the semi-dual objective proposed by Genevay et al. (2016) also converges at a rate of $\\mathcal { O } ( 1 / \\sqrt { k } )$ , whereas we only know that Alg. 1 converges to a stationary point in this non-convex case. ",
+ "bbox": [
+ 174,
+ 598,
+ 825,
+ 642
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "In the continuous-continuous setting (i.e. both measures are continuous), Genevay et al. (2016) proposed to represent the dual variables as kernel expansions. A disadvantage of their approach, however, is the $\\mathcal { O } ( k ^ { 2 } )$ cost per iteration. In contrast, our approach represents dual variables as neural networks. While non-convex, our approach preserves a $\\mathcal { O } ( p ^ { 2 } )$ cost per iteration. This parameterization with neural networks was also used by Arjovsky et al. (2017) who maximized the 1-Wasserstein dual-objective function $\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ u ( X ) - u ( Y ) \\right]$ . Their algorithm is hence very similar to ours, with the same complexity $\\mathcal { O } ( p ^ { 2 } )$ per iteration. The main difference is that they had to constrain $u$ to be a Lipschitz function and hence relied of weight clipping in-between gradient updates. The proposed algorithm is capable of computing the regularized OT objective and optimal plans between empirical measures supported on arbitrary large numbers of samples. In statistical machine learning, one aims at estimating the underlying continuous distribution from which empirical observations have been sampled. In the context of optimal transport, one would like to approximate the true (non-regularized) optimal plan between the underlying measures. The next section states theoretical guarantees regarding this problem. ",
+ "bbox": [
+ 173,
+ 648,
+ 825,
+ 845
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "3.2 CONVERGENCE OF REGULARIZED OT PLANS ",
+ "text_level": 1,
+ "bbox": [
+ 178,
+ 866,
+ 524,
+ 881
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Consider discrete probability measures $\\begin{array} { r } { \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } } \\in P ( \\mathcal { X } ) } \\end{array}$ and $\\textstyle \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } \\in P ( \\mathcal { Y } )$ Analysis of entropy-regularized linear programs (Cominetti $\\&$ San Mart´ın, 1994) shows that the solution $\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }$ of the entropy-regularized problem (3) converges exponentially fast to a solution $\\pi _ { n }$ of the non-regularized OT problem (2). Also, a result about stability of optimal transport (Villani, 2008)[Theorem 5.20] states that, if $\\mu _ { n } \\mu$ and $\\nu _ { n } \\to \\nu$ weakly, then a sequence $\\left( \\pi _ { n } \\right)$ of optimal transport plans between $\\mu _ { n }$ and $\\nu _ { n }$ converges weakly to a solution $\\pi$ of the OT problem between $\\mu$ and $\\nu$ . We can thus write, ",
+ "bbox": [
+ 174,
+ 892,
+ 823,
+ 924
+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "",
+ "bbox": [
+ 173,
+ 103,
+ 825,
+ 172
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "equation",
+ "img_path": "images/308b54b39912b32160eacaa43673af27a8b487017ac8b8801bb48ffa7bb8a6b8.jpg",
+ "text": "$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname* { l i m } _ { \\varepsilon \\to 0 } \\pi _ { n } ^ { \\varepsilon } = \\pi .\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 437,
+ 171,
+ 562,
+ 194
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "A more refined result consists in establishing the weak convergence of $\\pi _ { n } ^ { \\varepsilon }$ to $\\pi$ when $( n , \\varepsilon )$ jointly converge to $( \\infty , 0 )$ . This is the result of the following theorem which states a stability property of entropy-regularized plans (proof in the Appendix). ",
+ "bbox": [
+ 173,
+ 196,
+ 825,
+ 239
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "$\\mu \\in P ( \\mathcal { X } )$ $\\nu \\in P ( \\mathcal { V } )$ where iscrete $\\mathcal { X }$ and babi $\\mathcal { V }$ are complete metric spaces. Let measures which converge weakly $\\textstyle \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } }$ $\\begin{array} { r } { \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } } \\end{array}$ to $\\mu$ and $\\nu$ respectively, and let $\\left( \\varepsilon _ { n } \\right)$ a sequence of non-negative real numbers converging to 0 sufficiently fast. Assume the cost $c$ is continuous on $\\mathcal { X } \\times \\mathcal { V }$ and finite. Let $\\pi _ { n } ^ { \\varepsilon _ { n } }$ the solution of the entropy-regularized OT problem (3) between $\\mu _ { n }$ and $\\nu _ { n }$ . Then, up to extraction of a subsequence, $( \\pi _ { n } ^ { \\varepsilon _ { n } } )$ converges weakly to the solution $\\pi$ of the OT problem (2) between $\\mu$ and $\\nu$ , ",
+ "bbox": [
+ 173,
+ 242,
+ 825,
+ 329
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "equation",
+ "img_path": "images/3b50b27e75934696e360f025b59c2af984d64f5b6f393603e784bcc9428691e7.jpg",
+ "text": "$$\n\\pi _ { n } ^ { \\varepsilon _ { n } } \\to \\pi w e a k l y .\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 433,
+ 334,
+ 565,
+ 352
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "Keeping the analogy with statistical machine learning, this result is an analog to the universal consistency property of a learning method. In most applications, we consider empirical measures and $n$ is fixed, so that regularization, besides enabling dual stochastic approach, may also help learn the optimal plan between the underlying continuous measures. ",
+ "bbox": [
+ 174,
+ 363,
+ 825,
+ 420
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "So far, we have derived an algorithm for computing the regularized OT objective and regularized optimal plans regardless of $\\mu$ and $\\nu$ being discrete or continuous. The OT objective has been used successfully as a loss in machine learning (Montavon et al., 2015; Frogner et al., 2015; Rolet et al., 2016; 2018; Arjovsky et al., 2017; Courty et al., 2017a), whereas the use of optimal plans has straightforward applications in logistics, as well as economy (Kantorovich, 1942; Carlier, 2012) or computer graphics (Bonneel et al., 2011). In numerous applications however, we often need mappings rather than joint distributions. This is all the more motivated since Brenier (1991) proved that when the source measure is continuous, the optimal transport plan is actually induced by a map. Assuming that available data samples are sampled from some underlying continuous distributions, finding the Monge map between these continuous measures rather than a discrete optimal plan between discrete measures is essential in machine learning applications. Hence in the next section, we investigate how to recover an optimal map, i.e. find an approximate solution to the Monge problem (1), from regularized optimal plans. ",
+ "bbox": [
+ 173,
+ 426,
+ 825,
+ 608
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "4 OPTIMAL MAPPING ESTIMATIONS ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 627,
+ 490,
+ 643
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "A map can be obtained from a solution to the OT problem (2) or regularized OT problem (3) through the computation of its barycentric projection. Indeed, a solution $\\pi$ of Problem (2) or (3) between a source measure $\\mu$ and a target measure $\\nu$ is, identifying the plan $\\pi$ with its density w.r.t. a reference measure, a function $\\pi : ( \\bar { x , y } ) \\in \\mathcal { X } \\times \\mathcal { Y } \\mapsto \\mathbb { R } ^ { + }$ which can be seen as a weighted one-to-many map, i.e. $\\pi$ sends $x$ to each location $y \\in \\operatorname { S u p p } ( \\nu )$ where $\\pi ( x , y ) > 0$ . A map can then be obtained by simply averaging over these $y$ according to the weights $\\pi ( x , y )$ . ",
+ "bbox": [
+ 173,
+ 659,
+ 825,
+ 743
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "Definition 1. (Barycentric projection) Let $\\pi$ be a solution of the OT problem (2) or regularized OT problem (3). The barycentric projection $\\bar { \\pi }$ w.r.t. a convex cost $d : \\mathcal { V } \\times \\mathcal { V } \\mathbb { R } ^ { + }$ is defined as, ",
+ "bbox": [
+ 171,
+ 746,
+ 825,
+ 775
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "equation",
+ "img_path": "images/2ce2cd6aae8f1baa1888c0bbdb06cd7789f25c4559036dd75702d04fef61613c.jpg",
+ "text": "$$\n\\bar { \\pi } ( x ) = \\arg \\operatorname* { m i n } _ { z } \\mathbb { E } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ d ( z , Y ) \\right] .\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 374,
+ 780,
+ 622,
+ 804
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "In the special case $d ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }$ , Eq. (11) has the close form solution $\\bar { \\pi } ( x ) = \\operatorname { \\mathbb { E } } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ Y \\right] .$ , which is equal to $\\begin{array} { r } { \\bar { \\boldsymbol { \\pi } } = \\frac { \\boldsymbol { \\pi } \\mathbf { y } ^ { t } } { a } } \\end{array}$ in a discrete setting with $\\mathbf { y } = ( y _ { 1 } , \\dotsb , y _ { n } )$ and $a$ the weights of $\\mu$ . Moreover, for the specific squared Euclidean cost $c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }$ , the barycentric projection $\\bar { \\pi }$ is an optimal map (Ambrosio et al., 2006)[Theorem 12.4.4], i.e. $\\bar { \\pi }$ is a solution to the Monge problem (1) between the source measure $\\mu$ and the target measure $\\bar { \\pi } \\# \\mu$ . Hence the barycentric projection w.r.t. the squared Euclidean cost is often used as a simple way to recover optimal maps from optimal transport plans (Reich, 2013; Wang et al., 2013; Ferradans et al., 2014; Seguy & Cuturi, 2015). ",
+ "bbox": [
+ 173,
+ 816,
+ 825,
+ 924
+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "table",
+ "img_path": "images/005b83c7e1e2dd1b696892e55336da0baf1cf57ed3f985bcc641333679575402.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| lgorithm 2 Optimal map learning with SGD |
| Inputs: input measures μ, V; cost function c; dual optimal variables u and v; map fe parame- |
| terized as a deep NN; batch size n; learning rate y. while not converged do |
| sample a batch (x1,·· ,xn) from μ |
| sample a batch (y1,:.,yn) from v |
| updateθ←0-γ∑ijHe(xi,yj)Vθd(yj,fe(xi)) end while |
",
+ "bbox": [
+ 174,
+ 99,
+ 825,
+ 224
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "Formula (11) provides a pointwise value of the barycentric projection. When $\\mu$ is discrete, this means that we only have mapping estimations for a finite number of points. In order to define a map which is defined everywhere, we parameterize the barycentric projection as a deep neural network. We show in the next paragraph how to efficiently learn its parameters. ",
+ "bbox": [
+ 173,
+ 247,
+ 825,
+ 304
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "Optimal map learning An estimation $f$ of the barycentric projection of a regularized plan $\\pi ^ { \\varepsilon }$ which generalizes outside the support of $\\mu$ can be obtained by learning a deep neural network which minimizes the following objective w.r.t. the parameters $\\theta$ , ",
+ "bbox": [
+ 173,
+ 310,
+ 825,
+ 353
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "equation",
+ "img_path": "images/99112b75efa82526f4065b76eee6fd113d983719cb533a205bff100a18f6c5f3.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { X \\sim \\mu } \\left[ \\mathbb { E } _ { Y \\sim \\pi ^ { \\varepsilon } ( \\cdot | X ) } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] \\right] = \\mathbb { E } _ { ( X , Y ) \\sim \\pi ^ { \\varepsilon } } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( Y , f _ { \\theta } ( X ) ) H _ { \\varepsilon } ( X , Y ) \\right] . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 246,
+ 356,
+ 751,
+ 397
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "When $d ( x , y ) = \\| x - y \\| ^ { 2 }$ , the last term in Eq. (12) is simply a weighted sum of squared errors, with possibly an infinite number of terms whenever $\\mu$ or $\\nu$ are continuous. We propose to minimize the objective (12) by stochastic gradient descent, which provides the simple Algorithm 2. The OT problem being symmetric, we can also compute the opposite barycentric projection $g$ w.r.t. a cost $\\bar { d } : \\mathcal { X } \\times \\mathcal { X } \\overset { } { \\to } \\bar { \\mathbb { R } } ^ { + }$ by minimizing $\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( g ( \\bar { Y } ) , X ) H _ { \\varepsilon } ( \\bar { X } , Y ) \\right]$ . ",
+ "bbox": [
+ 173,
+ 398,
+ 825,
+ 472
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "However, unless the plan $\\pi$ is induced by a map, the averaging process results in having the image of the source measure by $\\bar { \\pi }$ only approximately equal to the target measure $\\nu$ . Still, when the size of discrete measure is large and the regularization is small, we show in the next paragraph that 1) the barycentric projection of a regularized OT plan is close to the Monge map between the underlying continuous measures (Theorem 2) and 2) the image of the source measure by this barycentric projection should be close to the target measure $\\nu$ (Corollary 1). ",
+ "bbox": [
+ 173,
+ 476,
+ 825,
+ 560
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "Theoretical guarantees As stated earlier, when $\\mathcal { X } = \\mathcal { V }$ and $c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }$ , Brenier (1991) proved that when the source measure $\\mu$ is continuous, there exists a solution to the Monge problem (1). This result was generalized to more general cost functions, see (Villani, 2008)[Corollary 9.3] for details. In that case, the plan $\\pi$ between $\\mu$ and $\\nu$ is written as $( \\operatorname { i d } , f ) \\# \\mu$ where $f$ is the Monge map. Now considering discrete measures $\\mu _ { n }$ and $\\nu _ { n }$ which converge to $\\mu$ (continuous) and $\\nu$ respectively, we have proved in Theorem 1 that $\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }$ converges weakly to $\\pi = ( \\operatorname { i d } , f ) \\# \\mu$ when $( n , \\varepsilon ) \\to ( \\infty , 0 )$ . The next theorem, proved in the Appendix, shows that the barycentric projection $\\bar { \\pi } _ { n } ^ { \\varepsilon }$ also converges weakly to the true Monge map between $\\mu$ and $\\nu$ , justifying our approach. ",
+ "bbox": [
+ 173,
+ 565,
+ 825,
+ 679
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "Theorem 2. Let $\\mu$ be a continuous probability measure on $\\mathbb { R } ^ { d }$ , and $\\nu$ an arbitrary probability and measure on $\\textstyle \\nu _ { n } = { \\frac { 1 } { n } } \\sum _ { j = 1 } ^ { n } \\delta _ { y _ { j } }$ $\\mathbb { R } ^ { d }$ and c a cost function satisfying (Villani, 2008)[Corollary 9.3]. Let converging weakly to $\\mu$ and $\\nu$ n i=1 respectively. Assume that the OT solution $\\begin{array} { r } { \\mu _ { n } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { x _ { i } } } \\end{array}$ $\\pi _ { n }$ of Problem (2) between $\\mu _ { n }$ and $\\nu _ { n }$ is unique for all $n$ . Let $\\left( \\varepsilon _ { n } \\right)$ a sequence of non-negative real numbers converging sufficiently fast to 0 and $\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } }$ the barycentric projection w.r.t. the convex cost $d = c$ of the solution $\\pi _ { n } ^ { \\varepsilon _ { n } }$ of the entropy-regularized OT problem (3). Then, up to extraction of $a$ subsequence, ",
+ "bbox": [
+ 173,
+ 681,
+ 825,
+ 784
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "equation",
+ "img_path": "images/a23a5eb78f59b0e108b2a2aa02a64a266e228755f22459bb28143378ee9dfde7.jpg",
+ "text": "$$\n( i d , \\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } ) \\# \\mu _ { n } ( i d , f ) \\# \\mu \\ w e a k l y ,\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 372,
+ 782,
+ 625,
+ 799
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "where $f$ is the solution of the Monge problem (1) between $\\mu$ and $\\nu$ . ",
+ "bbox": [
+ 174,
+ 800,
+ 612,
+ 815
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "This theorem shows that our estimated barycentric projection is close to an optimal map between the underlying continuous measures for $n$ big and $\\varepsilon$ small. The following corollary confirms the intuition that the image of the source measure by this map converges to the underlying target measure. ",
+ "bbox": [
+ 174,
+ 824,
+ 825,
+ 867
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "Corollary 1. With the same assumptions as above, ${ \\bar { \\pi } } _ { n } ^ { \\varepsilon _ { n } } \\# \\mu _ { n } \\to \\nu$ weakly. ",
+ "bbox": [
+ 176,
+ 871,
+ 666,
+ 886
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "text",
+ "text": "In terms of random variables, the last equation states that if $X _ { n } \\sim \\mu _ { n }$ and $Y \\sim \\nu$ , then $\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } \\left( X _ { n } \\right)$ converges in distribution to $Y$ . ",
+ "bbox": [
+ 173,
+ 895,
+ 820,
+ 924
+ ],
+ "page_idx": 6
+ },
+ {
+ "type": "image",
+ "img_path": "images/108bd959a523d171261a4465782b24928b96161ec71032e31d7571b8fde56237.jpg",
+ "image_caption": [
+ "Figure 2: Convergence plots of the the Stochastic Dual Algorithm 1 against a stochastic semi-dual implementation (adapted from (Genevay et al., 2016): we use SGD instead of SAG), for several entropy-regularization values. Learning rates are $\\{ 5 . , 2 0 . , 2 0 . \\}$ and batch sizes $\\{ 1 0 2 4 , 5 0 0 , 1 0 0 \\}$ respectively and are taken the same for the dual and semi-dual methods. "
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 183,
+ 102,
+ 812,
+ 236
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "These theoretical results show that our estimated Monge map can thus be used to perform domain adaptation by mapping a source dataset to a target dataset, as well as perform generative modeling by mapping a continuous measure to a target discrete dataset. We demontrate this in the following section. ",
+ "bbox": [
+ 174,
+ 332,
+ 825,
+ 387
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "5 NUMERICAL EXPERIMENTS ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 410,
+ 436,
+ 425
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "5.1 DUAL VS SEMI-DUAL SPEED COMPARISONS ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 441,
+ 516,
+ 455
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "We start by evaluating the training time of our dual stochastic algorithm 1 against a stochastic semidual approach similar to (Genevay et al., 2016). In the semi-dual approach, one of the dual variable is eliminated and is computed in close form. However, this computation has ${ \\mathcal { O } } ( n )$ complexity where $n$ is the size of the target measure $\\nu$ . We compute the regularized OT objective with both methods on a spectral transfer problem, which is related to the color transfer problem (Reinhard et al., 2001; Pitie et al., 2007), but where images are multispectral, ´ i.e. they share a finer sampling of the light wavelength. We take two $5 0 0 \\times 5 0 0$ images from the CAVE dataset (Yasuma et al., 2010) that have 31 spectral bands. As such, the optimal transport problem is computed on two empirical distributions of 250000 samples in $\\mathbb { R } ^ { 3 1 }$ on which we consider the squared Euclidean ground cost $c$ . The timing evolution of train losses are reported in Figure 2 for three different regularization values $\\varepsilon = \\{ 0 . 0 \\bar { 2 } 5 , 0 . 1 , 1 . \\}$ . In the three cases, one can observe that convergence of our proposed dual algorithm is much faster. ",
+ "bbox": [
+ 173,
+ 468,
+ 825,
+ 635
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "5.2 LARGE SCALE DOMAIN ADAPTATION ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 654,
+ 467,
+ 667
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "We apply here our computation framework on an unsupervised domain adaptation (DA) task, for which optimal transport has shown to perform well on small scale datasets (Courty et al., 2017b; Perrot et al., 2016; Courty et al., 2014). This restriction is mainly due to the fact that those works only consider the primal formulation of the OT problem. Our goal here is not to compete with the state-of-the-art methods in domain adaptation but to assess that our formulation allows to scale optimal transport based domain adaptation (OTDA) to large datasets. OTDA is illustrated in Fig. 3 and follows two steps: 1) learn an optimal map between the source and target distribution, 2) map the source samples and train a classifier on them in the target domain. Our formulation also allows to use any differentiable ground cost $c$ while (Courty et al., 2017b) was limited to the squared Euclidean distance. ",
+ "bbox": [
+ 174,
+ 680,
+ 825,
+ 819
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Datasets We consider the three cross-domain digit image datasets MNIST (Lecun et al., 1998), USPS, and SVHN (Netzer et al., 2011), which have 10 classes each. For the adaptation between MNIST and USPS, we use 60000 samples in the MNIST domain and 9298 samples in USPS domain. MNIST images are resized to the same resolution as USPS ones $( 1 6 \\times 1 6 )$ ). For the adaptation between SVHN and MNIST, we use 73212 samples in the SVHN domain and 60000 samples in the MNIST domain. MNIST images are zero-padded to reach the same resolution as SVHN $( 3 2 \\times 3 2 )$ and extended to three channels to match SVHN image sizes. The labels in the target domain are withheld during the adaptation. In the experiment, we consider the adaptation in three directions: M $\\mathrm { I N I S T } \\to \\mathrm { U S P S }$ , $\\mathrm { U S P S } \\to \\mathrm { M N I S T }$ , and $\\mathrm { S V H N } \\to \\mathrm { M N I S T } .$ ",
+ "bbox": [
+ 174,
+ 827,
+ 825,
+ 924
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "image",
+ "img_path": "images/68a2bc0b8ddaf4a608cd6baab93562a2a306655bd64ab896bcdaaba30e300dd6.jpg",
+ "image_caption": [
+ "Figure 3: Illustration of the OT Domain Adaptation method adapted from (Courty et al., 2017b). Source samples are mapped to the target set through the barycentric projection $\\bar { \\pi } ^ { \\varepsilon }$ . A classifier is then learned on the mapped source samples. "
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 204,
+ 97,
+ 794,
+ 231
+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "table",
+ "img_path": "images/7734982315e7adbb9ae54a18cbdfb39b354e1e05535cad2b853fabaeb4ad0c0d.jpg",
+ "table_caption": [
+ "Table 1: Results (accuracy in $\\%$ ) on domain adaptation among MNIST, USPS and SVHN datasets with entropy $( R _ { e } )$ and L2 $( R _ { L ^ { 2 } } )$ regularizations. Source only refers to 1-NN classification between source and target samples without adaptation. "
+ ],
+ "table_footnote": [],
+ "table_body": " | 1: Inputs: input measures μ,V; cost function c; batch size p; learning rate y. Discrete case: μ = ∑i aiδxi and u is a finite vector: u(xi) @f ui (similarly for v and v) |
| 2: 3: | Continuous case: μ is a continuous measure and u is a neural network (similarly for V and u) |
| Vindicates the gradient w.r.t. the parameters |
| 4: | while not converged do |
| 5: | sample a batch (x1,..,xp) from μ |
| 6: 7: | sample a batch (y1,.., yp) from v update u ←u+γ∑ij Vu(xi)+OuFe(u(xi),v(yj))Vu(xi) |
| 8: | updateu ← U+γ∑ij Vv(yj)+OUFe(u(xi),v(yj))Vv(yj) 9: ：end while |
",
+ "type": "table",
+ "image_path": "b3ca3f1b302e44b245c466f2b070224821caedc6528d20966e2963670c83313f.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 2,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 110,
+ 97,
+ 499,
+ 136.66666666666666
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 110,
+ 136.66666666666666,
+ 499,
+ 176.33333333333331
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 110,
+ 176.33333333333331,
+ 499,
+ 215.99999999999997
+ ],
+ "spans": [],
+ "index": 3
+ }
+ ]
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 239,
+ 505,
+ 306
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 239,
+ 506,
+ 252
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 239,
+ 506,
+ 252
+ ],
+ "score": 1.0,
+ "content": "general functions. Genevay et al. (2016) used the same stochastic dual maximization approach to",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 250,
+ 506,
+ 262
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 250,
+ 506,
+ 262
+ ],
+ "score": 1.0,
+ "content": "compute the regularized OT objective in the continuous-continuous setting. The difference lies in",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 262,
+ 505,
+ 273
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 262,
+ 505,
+ 273
+ ],
+ "score": 1.0,
+ "content": "their pamaterization of the dual variables as kernel expansions, while we decide to use deep neural",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 284
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 284
+ ],
+ "score": 1.0,
+ "content": "networks. Using a neural network for parameterizing a continuous dual variable was done also",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 284,
+ 506,
+ 296
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 284,
+ 473,
+ 296
+ ],
+ "score": 1.0,
+ "content": "by Arjovsky et al. (2017). The same discussion also stands for the second dual variable",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 473,
+ 285,
+ 479,
+ 293
+ ],
+ "score": 0.65,
+ "content": "v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 480,
+ 284,
+ 506,
+ 296
+ ],
+ "score": 1.0,
+ "content": ". Our",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 294,
+ 308,
+ 306
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 294,
+ 308,
+ 306
+ ],
+ "score": 1.0,
+ "content": "stochastic gradient algorithm is detailed in Alg. 1.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 6.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 311,
+ 505,
+ 468
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 311,
+ 505,
+ 323
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 311,
+ 505,
+ 323
+ ],
+ "score": 1.0,
+ "content": "Convergence rates and computational cost comparison. We first discuss convergence rates in",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 322,
+ 505,
+ 334
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 322,
+ 505,
+ 334
+ ],
+ "score": 1.0,
+ "content": "the discrete-discrete setting (i.e. both measures are discrete), where the problem is convex, while",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 333,
+ 506,
+ 346
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 333,
+ 506,
+ 346
+ ],
+ "score": 1.0,
+ "content": "parameterization of dual variables as neural networks in the semi-discrete or continuous-continuous",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 356
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 356
+ ],
+ "score": 1.0,
+ "content": "settings make the problem non-convex. Because the dual (6) is not strongly convex, full-gradient",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 353,
+ 506,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 353,
+ 232,
+ 369
+ ],
+ "score": 1.0,
+ "content": "descent converges at a rate of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 355,
+ 264,
+ 367
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / k )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 353,
+ 297,
+ 369
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 355,
+ 304,
+ 365
+ ],
+ "score": 0.8,
+ "content": "k",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 305,
+ 353,
+ 506,
+ 369
+ ],
+ "score": 1.0,
+ "content": "is the iteration number. SGD with a decreasing",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 366,
+ 506,
+ 381
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 367,
+ 272,
+ 381
+ ],
+ "score": 1.0,
+ "content": "step size converges at the inferior rate of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 272,
+ 366,
+ 312,
+ 380
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / \\sqrt { k } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 313,
+ 367,
+ 463,
+ 381
+ ],
+ "score": 1.0,
+ "content": "(Nemirovski et al., 2009), but with a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 463,
+ 367,
+ 485,
+ 379
+ ],
+ "score": 0.84,
+ "content": "\\mathcal { O } ( 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 485,
+ 367,
+ 506,
+ 381
+ ],
+ "score": 1.0,
+ "content": "cost",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 104,
+ 377,
+ 506,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 377,
+ 461,
+ 393
+ ],
+ "score": 1.0,
+ "content": "per iteration. The two rates can be interpolated when using mini-batches, at the cost of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 462,
+ 379,
+ 488,
+ 390
+ ],
+ "score": 0.91,
+ "content": "\\mathcal { O } ( p ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 488,
+ 377,
+ 506,
+ 393
+ ],
+ "score": 1.0,
+ "content": "per",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 389,
+ 506,
+ 402
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 389,
+ 172,
+ 402
+ ],
+ "score": 1.0,
+ "content": "iteration, where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 391,
+ 179,
+ 401
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 179,
+ 389,
+ 506,
+ 402
+ ],
+ "score": 1.0,
+ "content": "is the mini-batch size. In contrast, Genevay et al. (2016) considered a semi-dual",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 399,
+ 506,
+ 415
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 194,
+ 415
+ ],
+ "score": 1.0,
+ "content": "objective of the form",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 401,
+ 305,
+ 413
+ ],
+ "score": 0.92,
+ "content": "\\mathbb { E } _ { X \\sim \\mu } \\left[ u ( X ) + G _ { \\varepsilon } ( u ( X ) ) \\right]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 305,
+ 399,
+ 463,
+ 415
+ ],
+ "score": 1.0,
+ "content": ", with a cost per iteration which is now",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 401,
+ 487,
+ 412
+ ],
+ "score": 0.91,
+ "content": "{ \\mathcal { O } } ( n )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 399,
+ 506,
+ 415
+ ],
+ "score": 1.0,
+ "content": "due",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 412,
+ 505,
+ 425
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 412,
+ 255,
+ 425
+ ],
+ "score": 1.0,
+ "content": "to the computation of the gradient of √",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 412,
+ 268,
+ 423
+ ],
+ "score": 0.87,
+ "content": "G _ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 269,
+ 412,
+ 505,
+ 425
+ ],
+ "score": 1.0,
+ "content": ". Because that objective is not strongly convex either, SGD",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 423,
+ 505,
+ 438
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 423,
+ 196,
+ 438
+ ],
+ "score": 1.0,
+ "content": "converges at the same",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 423,
+ 237,
+ 436
+ ],
+ "score": 0.93,
+ "content": "O ( 1 / \\sqrt { k } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 423,
+ 505,
+ 438
+ ],
+ "score": 1.0,
+ "content": "rate, up to problem-specific constants. As noted by Genevay et al.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 433,
+ 506,
+ 449
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 433,
+ 258,
+ 449
+ ],
+ "score": 1.0,
+ "content": "(2016), this rate can be improved to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 435,
+ 290,
+ 447
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / k )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 433,
+ 506,
+ 449
+ ],
+ "score": 1.0,
+ "content": "while maintaining the same iteration cost, by using",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "score": 1.0,
+ "content": "stochastic average gradient (SAG) method (Schmidt et al., 2017). However, SAG requires to store",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 457,
+ 408,
+ 471
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 457,
+ 408,
+ 471
+ ],
+ "score": 1.0,
+ "content": "past stochastic gradients, which can be problematic in a large-scale setting.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 16.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 474,
+ 505,
+ 509
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 473,
+ 505,
+ 486
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 473,
+ 505,
+ 486
+ ],
+ "score": 1.0,
+ "content": "In the semi-discrete setting (i.e. one measure is discrete and the other is continuous), SGD on the",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 485,
+ 505,
+ 499
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 426,
+ 499
+ ],
+ "score": 1.0,
+ "content": "semi-dual objective proposed by Genevay et al. (2016) also converges at a rate of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 485,
+ 466,
+ 498
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / \\sqrt { k } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 486,
+ 505,
+ 499
+ ],
+ "score": 1.0,
+ "content": ", whereas",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 498,
+ 433,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 498,
+ 433,
+ 509
+ ],
+ "score": 1.0,
+ "content": "we only know that Alg. 1 converges to a stationary point in this non-convex case.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 670
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 505,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 505,
+ 527
+ ],
+ "score": 1.0,
+ "content": "In the continuous-continuous setting (i.e. both measures are continuous), Genevay et al. (2016)",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 525,
+ 505,
+ 538
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 525,
+ 505,
+ 538
+ ],
+ "score": 1.0,
+ "content": "proposed to represent the dual variables as kernel expansions. A disadvantage of their approach,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 535,
+ 506,
+ 549
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 535,
+ 173,
+ 549
+ ],
+ "score": 1.0,
+ "content": "however, is the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 173,
+ 536,
+ 200,
+ 548
+ ],
+ "score": 0.92,
+ "content": "\\mathcal { O } ( k ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 535,
+ 506,
+ 549
+ ],
+ "score": 1.0,
+ "content": "cost per iteration. In contrast, our approach represents dual variables as",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 547,
+ 505,
+ 560
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 547,
+ 364,
+ 560
+ ],
+ "score": 1.0,
+ "content": "neural networks. While non-convex, our approach preserves a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 364,
+ 547,
+ 390,
+ 559
+ ],
+ "score": 0.92,
+ "content": "\\mathcal { O } ( p ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 390,
+ 547,
+ 505,
+ 560
+ ],
+ "score": 1.0,
+ "content": "cost per iteration. This pa-",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 558,
+ 506,
+ 571
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 558,
+ 506,
+ 571
+ ],
+ "score": 1.0,
+ "content": "rameterization with neural networks was also used by Arjovsky et al. (2017) who maximized the",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 568,
+ 505,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 261,
+ 583
+ ],
+ "score": 1.0,
+ "content": "1-Wasserstein dual-objective function",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 262,
+ 569,
+ 376,
+ 582
+ ],
+ "score": 0.89,
+ "content": "\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ u ( X ) - u ( Y ) \\right]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 568,
+ 505,
+ 583
+ ],
+ "score": 1.0,
+ "content": ". Their algorithm is hence very",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 104,
+ 580,
+ 505,
+ 596
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 580,
+ 282,
+ 596
+ ],
+ "score": 1.0,
+ "content": "similar to ours, with the same complexity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 283,
+ 582,
+ 309,
+ 594
+ ],
+ "score": 0.91,
+ "content": "\\mathcal { O } ( p ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 580,
+ 505,
+ 596
+ ],
+ "score": 1.0,
+ "content": "per iteration. The main difference is that they",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 591,
+ 505,
+ 606
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 591,
+ 174,
+ 606
+ ],
+ "score": 1.0,
+ "content": "had to constrain",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 595,
+ 182,
+ 603
+ ],
+ "score": 0.69,
+ "content": "u",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 182,
+ 591,
+ 505,
+ 606
+ ],
+ "score": 1.0,
+ "content": "to be a Lipschitz function and hence relied of weight clipping in-between gra-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 603,
+ 506,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 603,
+ 506,
+ 617
+ ],
+ "score": 1.0,
+ "content": "dient updates. The proposed algorithm is capable of computing the regularized OT objective and",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 628
+ ],
+ "score": 1.0,
+ "content": "optimal plans between empirical measures supported on arbitrary large numbers of samples. In sta-",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 625,
+ 505,
+ 637
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 625,
+ 505,
+ 637
+ ],
+ "score": 1.0,
+ "content": "tistical machine learning, one aims at estimating the underlying continuous distribution from which",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 649
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 649
+ ],
+ "score": 1.0,
+ "content": "empirical observations have been sampled. In the context of optimal transport, one would like to",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 647,
+ 506,
+ 660
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 647,
+ 506,
+ 660
+ ],
+ "score": 1.0,
+ "content": "approximate the true (non-regularized) optimal plan between the underlying measures. The next",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 659,
+ 346,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 659,
+ 346,
+ 671
+ ],
+ "score": 1.0,
+ "content": "section states theoretical guarantees regarding this problem.",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 33.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 109,
+ 686,
+ 321,
+ 698
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 685,
+ 323,
+ 699
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 685,
+ 323,
+ 699
+ ],
+ "score": 1.0,
+ "content": "3.2 CONVERGENCE OF REGULARIZED OT PLANS",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ }
+ ],
+ "index": 41
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 707,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 102,
+ 702,
+ 501,
+ 727
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 702,
+ 265,
+ 727
+ ],
+ "score": 1.0,
+ "content": "Consider discrete probability measures",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 265,
+ 708,
+ 375,
+ 721
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } } \\in P ( \\mathcal { X } ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 702,
+ 393,
+ 727
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 393,
+ 708,
+ 501,
+ 722
+ ],
+ "score": 0.93,
+ "content": "\\textstyle \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } \\in P ( \\mathcal { Y } )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 719,
+ 506,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 719,
+ 353,
+ 732
+ ],
+ "score": 1.0,
+ "content": "Analysis of entropy-regularized linear programs (Cominetti",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 353,
+ 721,
+ 363,
+ 731
+ ],
+ "score": 0.25,
+ "content": "\\&",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 363,
+ 719,
+ 506,
+ 732
+ ],
+ "score": 1.0,
+ "content": "San Mart´ın, 1994) shows that the",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 42.5
+ }
+ ],
+ "page_idx": 4,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 108,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2018",
+ "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": "title",
+ "bbox": [
+ 108,
+ 82,
+ 271,
+ 94
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 272,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 81,
+ 272,
+ 96
+ ],
+ "score": 1.0,
+ "content": "Algorithm 1 Stochastic OT computation",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 110,
+ 97,
+ 499,
+ 216
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 110,
+ 97,
+ 499,
+ 216
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 110,
+ 97,
+ 499,
+ 216
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 110,
+ 97,
+ 499,
+ 216
+ ],
+ "score": 0.509,
+ "html": " | 1: Inputs: input measures μ,V; cost function c; batch size p; learning rate y. Discrete case: μ = ∑i aiδxi and u is a finite vector: u(xi) @f ui (similarly for v and v) |
| 2: 3: | Continuous case: μ is a continuous measure and u is a neural network (similarly for V and u) |
| Vindicates the gradient w.r.t. the parameters |
| 4: | while not converged do |
| 5: | sample a batch (x1,..,xp) from μ |
| 6: 7: | sample a batch (y1,.., yp) from v update u ←u+γ∑ij Vu(xi)+OuFe(u(xi),v(yj))Vu(xi) |
| 8: | updateu ← U+γ∑ij Vv(yj)+OUFe(u(xi),v(yj))Vv(yj) 9: ：end while |
",
+ "type": "table",
+ "image_path": "b3ca3f1b302e44b245c466f2b070224821caedc6528d20966e2963670c83313f.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 2,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 110,
+ 97,
+ 499,
+ 136.66666666666666
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 110,
+ 136.66666666666666,
+ 499,
+ 176.33333333333331
+ ],
+ "spans": [],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 110,
+ 176.33333333333331,
+ 499,
+ 215.99999999999997
+ ],
+ "spans": [],
+ "index": 3
+ }
+ ]
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 239,
+ 505,
+ 306
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 239,
+ 506,
+ 252
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 239,
+ 506,
+ 252
+ ],
+ "score": 1.0,
+ "content": "general functions. Genevay et al. (2016) used the same stochastic dual maximization approach to",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 105,
+ 250,
+ 506,
+ 262
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 250,
+ 506,
+ 262
+ ],
+ "score": 1.0,
+ "content": "compute the regularized OT objective in the continuous-continuous setting. The difference lies in",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 262,
+ 505,
+ 273
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 262,
+ 505,
+ 273
+ ],
+ "score": 1.0,
+ "content": "their pamaterization of the dual variables as kernel expansions, while we decide to use deep neural",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 284
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 284
+ ],
+ "score": 1.0,
+ "content": "networks. Using a neural network for parameterizing a continuous dual variable was done also",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 284,
+ 506,
+ 296
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 284,
+ 473,
+ 296
+ ],
+ "score": 1.0,
+ "content": "by Arjovsky et al. (2017). The same discussion also stands for the second dual variable",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 473,
+ 285,
+ 479,
+ 293
+ ],
+ "score": 0.65,
+ "content": "v",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 480,
+ 284,
+ 506,
+ 296
+ ],
+ "score": 1.0,
+ "content": ". Our",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 294,
+ 308,
+ 306
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 294,
+ 308,
+ 306
+ ],
+ "score": 1.0,
+ "content": "stochastic gradient algorithm is detailed in Alg. 1.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 6.5,
+ "bbox_fs": [
+ 105,
+ 239,
+ 506,
+ 306
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 311,
+ 505,
+ 468
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 311,
+ 505,
+ 323
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 311,
+ 505,
+ 323
+ ],
+ "score": 1.0,
+ "content": "Convergence rates and computational cost comparison. We first discuss convergence rates in",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 106,
+ 322,
+ 505,
+ 334
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 322,
+ 505,
+ 334
+ ],
+ "score": 1.0,
+ "content": "the discrete-discrete setting (i.e. both measures are discrete), where the problem is convex, while",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 333,
+ 506,
+ 346
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 333,
+ 506,
+ 346
+ ],
+ "score": 1.0,
+ "content": "parameterization of dual variables as neural networks in the semi-discrete or continuous-continuous",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 356
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 356
+ ],
+ "score": 1.0,
+ "content": "settings make the problem non-convex. Because the dual (6) is not strongly convex, full-gradient",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 105,
+ 353,
+ 506,
+ 369
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 353,
+ 232,
+ 369
+ ],
+ "score": 1.0,
+ "content": "descent converges at a rate of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 232,
+ 355,
+ 264,
+ 367
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / k )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 353,
+ 297,
+ 369
+ ],
+ "score": 1.0,
+ "content": ", where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 297,
+ 355,
+ 304,
+ 365
+ ],
+ "score": 0.8,
+ "content": "k",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 305,
+ 353,
+ 506,
+ 369
+ ],
+ "score": 1.0,
+ "content": "is the iteration number. SGD with a decreasing",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 105,
+ 366,
+ 506,
+ 381
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 367,
+ 272,
+ 381
+ ],
+ "score": 1.0,
+ "content": "step size converges at the inferior rate of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 272,
+ 366,
+ 312,
+ 380
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / \\sqrt { k } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 313,
+ 367,
+ 463,
+ 381
+ ],
+ "score": 1.0,
+ "content": "(Nemirovski et al., 2009), but with a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 463,
+ 367,
+ 485,
+ 379
+ ],
+ "score": 0.84,
+ "content": "\\mathcal { O } ( 1 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 485,
+ 367,
+ 506,
+ 381
+ ],
+ "score": 1.0,
+ "content": "cost",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 104,
+ 377,
+ 506,
+ 393
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 377,
+ 461,
+ 393
+ ],
+ "score": 1.0,
+ "content": "per iteration. The two rates can be interpolated when using mini-batches, at the cost of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 462,
+ 379,
+ 488,
+ 390
+ ],
+ "score": 0.91,
+ "content": "\\mathcal { O } ( p ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 488,
+ 377,
+ 506,
+ 393
+ ],
+ "score": 1.0,
+ "content": "per",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 389,
+ 506,
+ 402
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 389,
+ 172,
+ 402
+ ],
+ "score": 1.0,
+ "content": "iteration, where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 172,
+ 391,
+ 179,
+ 401
+ ],
+ "score": 0.8,
+ "content": "p",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 179,
+ 389,
+ 506,
+ 402
+ ],
+ "score": 1.0,
+ "content": "is the mini-batch size. In contrast, Genevay et al. (2016) considered a semi-dual",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 399,
+ 506,
+ 415
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 194,
+ 415
+ ],
+ "score": 1.0,
+ "content": "objective of the form",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 194,
+ 401,
+ 305,
+ 413
+ ],
+ "score": 0.92,
+ "content": "\\mathbb { E } _ { X \\sim \\mu } \\left[ u ( X ) + G _ { \\varepsilon } ( u ( X ) ) \\right]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 305,
+ 399,
+ 463,
+ 415
+ ],
+ "score": 1.0,
+ "content": ", with a cost per iteration which is now",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 464,
+ 401,
+ 487,
+ 412
+ ],
+ "score": 0.91,
+ "content": "{ \\mathcal { O } } ( n )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 487,
+ 399,
+ 506,
+ 415
+ ],
+ "score": 1.0,
+ "content": "due",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 412,
+ 505,
+ 425
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 412,
+ 255,
+ 425
+ ],
+ "score": 1.0,
+ "content": "to the computation of the gradient of √",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 412,
+ 268,
+ 423
+ ],
+ "score": 0.87,
+ "content": "G _ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 269,
+ 412,
+ 505,
+ 425
+ ],
+ "score": 1.0,
+ "content": ". Because that objective is not strongly convex either, SGD",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 423,
+ 505,
+ 438
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 423,
+ 196,
+ 438
+ ],
+ "score": 1.0,
+ "content": "converges at the same",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 423,
+ 237,
+ 436
+ ],
+ "score": 0.93,
+ "content": "O ( 1 / \\sqrt { k } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 423,
+ 505,
+ 438
+ ],
+ "score": 1.0,
+ "content": "rate, up to problem-specific constants. As noted by Genevay et al.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 433,
+ 506,
+ 449
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 433,
+ 258,
+ 449
+ ],
+ "score": 1.0,
+ "content": "(2016), this rate can be improved to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 258,
+ 435,
+ 290,
+ 447
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / k )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 433,
+ 506,
+ 449
+ ],
+ "score": 1.0,
+ "content": "while maintaining the same iteration cost, by using",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 446,
+ 505,
+ 459
+ ],
+ "score": 1.0,
+ "content": "stochastic average gradient (SAG) method (Schmidt et al., 2017). However, SAG requires to store",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 457,
+ 408,
+ 471
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 457,
+ 408,
+ 471
+ ],
+ "score": 1.0,
+ "content": "past stochastic gradients, which can be problematic in a large-scale setting.",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 16.5,
+ "bbox_fs": [
+ 104,
+ 311,
+ 506,
+ 471
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 474,
+ 505,
+ 509
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 473,
+ 505,
+ 486
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 473,
+ 505,
+ 486
+ ],
+ "score": 1.0,
+ "content": "In the semi-discrete setting (i.e. one measure is discrete and the other is continuous), SGD on the",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 485,
+ 505,
+ 499
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 486,
+ 426,
+ 499
+ ],
+ "score": 1.0,
+ "content": "semi-dual objective proposed by Genevay et al. (2016) also converges at a rate of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 485,
+ 466,
+ 498
+ ],
+ "score": 0.93,
+ "content": "\\mathcal { O } ( 1 / \\sqrt { k } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 466,
+ 486,
+ 505,
+ 499
+ ],
+ "score": 1.0,
+ "content": ", whereas",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 498,
+ 433,
+ 509
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 498,
+ 433,
+ 509
+ ],
+ "score": 1.0,
+ "content": "we only know that Alg. 1 converges to a stationary point in this non-convex case.",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ }
+ ],
+ "index": 25,
+ "bbox_fs": [
+ 105,
+ 473,
+ 505,
+ 509
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 514,
+ 505,
+ 670
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 505,
+ 527
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 514,
+ 505,
+ 527
+ ],
+ "score": 1.0,
+ "content": "In the continuous-continuous setting (i.e. both measures are continuous), Genevay et al. (2016)",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 525,
+ 505,
+ 538
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 525,
+ 505,
+ 538
+ ],
+ "score": 1.0,
+ "content": "proposed to represent the dual variables as kernel expansions. A disadvantage of their approach,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 535,
+ 506,
+ 549
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 535,
+ 173,
+ 549
+ ],
+ "score": 1.0,
+ "content": "however, is the",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 173,
+ 536,
+ 200,
+ 548
+ ],
+ "score": 0.92,
+ "content": "\\mathcal { O } ( k ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 535,
+ 506,
+ 549
+ ],
+ "score": 1.0,
+ "content": "cost per iteration. In contrast, our approach represents dual variables as",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 547,
+ 505,
+ 560
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 547,
+ 364,
+ 560
+ ],
+ "score": 1.0,
+ "content": "neural networks. While non-convex, our approach preserves a",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 364,
+ 547,
+ 390,
+ 559
+ ],
+ "score": 0.92,
+ "content": "\\mathcal { O } ( p ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 390,
+ 547,
+ 505,
+ 560
+ ],
+ "score": 1.0,
+ "content": "cost per iteration. This pa-",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 558,
+ 506,
+ 571
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 558,
+ 506,
+ 571
+ ],
+ "score": 1.0,
+ "content": "rameterization with neural networks was also used by Arjovsky et al. (2017) who maximized the",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 105,
+ 568,
+ 505,
+ 583
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 568,
+ 261,
+ 583
+ ],
+ "score": 1.0,
+ "content": "1-Wasserstein dual-objective function",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 262,
+ 569,
+ 376,
+ 582
+ ],
+ "score": 0.89,
+ "content": "\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ u ( X ) - u ( Y ) \\right]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 568,
+ 505,
+ 583
+ ],
+ "score": 1.0,
+ "content": ". Their algorithm is hence very",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 104,
+ 580,
+ 505,
+ 596
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 580,
+ 282,
+ 596
+ ],
+ "score": 1.0,
+ "content": "similar to ours, with the same complexity",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 283,
+ 582,
+ 309,
+ 594
+ ],
+ "score": 0.91,
+ "content": "\\mathcal { O } ( p ^ { 2 } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 309,
+ 580,
+ 505,
+ 596
+ ],
+ "score": 1.0,
+ "content": "per iteration. The main difference is that they",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 591,
+ 505,
+ 606
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 591,
+ 174,
+ 606
+ ],
+ "score": 1.0,
+ "content": "had to constrain",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 174,
+ 595,
+ 182,
+ 603
+ ],
+ "score": 0.69,
+ "content": "u",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 182,
+ 591,
+ 505,
+ 606
+ ],
+ "score": 1.0,
+ "content": "to be a Lipschitz function and hence relied of weight clipping in-between gra-",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 603,
+ 506,
+ 617
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 603,
+ 506,
+ 617
+ ],
+ "score": 1.0,
+ "content": "dient updates. The proposed algorithm is capable of computing the regularized OT objective and",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 615,
+ 505,
+ 628
+ ],
+ "score": 1.0,
+ "content": "optimal plans between empirical measures supported on arbitrary large numbers of samples. In sta-",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 625,
+ 505,
+ 637
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 625,
+ 505,
+ 637
+ ],
+ "score": 1.0,
+ "content": "tistical machine learning, one aims at estimating the underlying continuous distribution from which",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 649
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 506,
+ 649
+ ],
+ "score": 1.0,
+ "content": "empirical observations have been sampled. In the context of optimal transport, one would like to",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 647,
+ 506,
+ 660
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 647,
+ 506,
+ 660
+ ],
+ "score": 1.0,
+ "content": "approximate the true (non-regularized) optimal plan between the underlying measures. The next",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 106,
+ 659,
+ 346,
+ 671
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 659,
+ 346,
+ 671
+ ],
+ "score": 1.0,
+ "content": "section states theoretical guarantees regarding this problem.",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ }
+ ],
+ "index": 33.5,
+ "bbox_fs": [
+ 104,
+ 514,
+ 506,
+ 671
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 109,
+ 686,
+ 321,
+ 698
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 685,
+ 323,
+ 699
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 685,
+ 323,
+ 699
+ ],
+ "score": 1.0,
+ "content": "3.2 CONVERGENCE OF REGULARIZED OT PLANS",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ }
+ ],
+ "index": 41
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 707,
+ 504,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 102,
+ 702,
+ 501,
+ 727
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 102,
+ 702,
+ 265,
+ 727
+ ],
+ "score": 1.0,
+ "content": "Consider discrete probability measures",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 265,
+ 708,
+ 375,
+ 721
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } } \\in P ( \\mathcal { X } ) } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 702,
+ 393,
+ 727
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 393,
+ 708,
+ 501,
+ 722
+ ],
+ "score": 0.93,
+ "content": "\\textstyle \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } \\in P ( \\mathcal { Y } )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 105,
+ 719,
+ 506,
+ 732
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 719,
+ 353,
+ 732
+ ],
+ "score": 1.0,
+ "content": "Analysis of entropy-regularized linear programs (Cominetti",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 353,
+ 721,
+ 363,
+ 731
+ ],
+ "score": 0.25,
+ "content": "\\&",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 363,
+ 719,
+ 506,
+ 732
+ ],
+ "score": 1.0,
+ "content": "San Mart´ın, 1994) shows that the",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 105,
+ 81,
+ 504,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 142,
+ 96
+ ],
+ "score": 1.0,
+ "content": "solution",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 142,
+ 83,
+ 154,
+ 95
+ ],
+ "score": 0.89,
+ "content": "\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 155,
+ 81,
+ 491,
+ 96
+ ],
+ "score": 1.0,
+ "content": "of the entropy-regularized problem (3) converges exponentially fast to a solution",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 491,
+ 85,
+ 504,
+ 94
+ ],
+ "score": 0.83,
+ "content": "\\pi _ { n }",
+ "type": "inline_equation",
+ "cross_page": true
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 107
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 107
+ ],
+ "score": 1.0,
+ "content": "of the non-regularized OT problem (2). Also, a result about stability of optimal transport (Villani,",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 104,
+ 505,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 104,
+ 249,
+ 117
+ ],
+ "score": 1.0,
+ "content": "2008)[Theorem 5.20] states that, if",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 250,
+ 106,
+ 285,
+ 116
+ ],
+ "score": 0.91,
+ "content": "\\mu _ { n } \\mu",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 286,
+ 104,
+ 303,
+ 117
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 304,
+ 106,
+ 338,
+ 115
+ ],
+ "score": 0.9,
+ "content": "\\nu _ { n } \\to \\nu",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 338,
+ 104,
+ 440,
+ 117
+ ],
+ "score": 1.0,
+ "content": "weakly, then a sequence",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 440,
+ 105,
+ 459,
+ 116
+ ],
+ "score": 0.85,
+ "content": "\\left( \\pi _ { n } \\right)",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 459,
+ 104,
+ 505,
+ 117
+ ],
+ "score": 1.0,
+ "content": "of optimal",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 114,
+ 504,
+ 130
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 114,
+ 204,
+ 130
+ ],
+ "score": 1.0,
+ "content": "transport plans between",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 204,
+ 117,
+ 217,
+ 127
+ ],
+ "score": 0.87,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 217,
+ 114,
+ 235,
+ 130
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 236,
+ 117,
+ 247,
+ 127
+ ],
+ "score": 0.88,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 247,
+ 114,
+ 374,
+ 130
+ ],
+ "score": 1.0,
+ "content": "converges weakly to a solution",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 375,
+ 117,
+ 382,
+ 125
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 383,
+ 114,
+ 496,
+ 130
+ ],
+ "score": 1.0,
+ "content": "of the OT problem between",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 496,
+ 117,
+ 504,
+ 127
+ ],
+ "score": 0.78,
+ "content": "\\mu",
+ "type": "inline_equation",
+ "cross_page": true
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 125,
+ 212,
+ 140
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 125,
+ 123,
+ 140
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 123,
+ 128,
+ 130,
+ 136
+ ],
+ "score": 0.7,
+ "content": "\\nu",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 131,
+ 125,
+ 212,
+ 140
+ ],
+ "score": 1.0,
+ "content": ". We can thus write,",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 42.5,
+ "bbox_fs": [
+ 102,
+ 702,
+ 506,
+ 732
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 137
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 504,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 81,
+ 142,
+ 96
+ ],
+ "score": 1.0,
+ "content": "solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 142,
+ 83,
+ 154,
+ 95
+ ],
+ "score": 0.89,
+ "content": "\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 155,
+ 81,
+ 491,
+ 96
+ ],
+ "score": 1.0,
+ "content": "of the entropy-regularized problem (3) converges exponentially fast to a solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 491,
+ 85,
+ 504,
+ 94
+ ],
+ "score": 0.83,
+ "content": "\\pi _ { n }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 107
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 93,
+ 505,
+ 107
+ ],
+ "score": 1.0,
+ "content": "of the non-regularized OT problem (2). Also, a result about stability of optimal transport (Villani,",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 104,
+ 505,
+ 117
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 104,
+ 249,
+ 117
+ ],
+ "score": 1.0,
+ "content": "2008)[Theorem 5.20] states that, if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 250,
+ 106,
+ 285,
+ 116
+ ],
+ "score": 0.91,
+ "content": "\\mu _ { n } \\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 104,
+ 303,
+ 117
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 304,
+ 106,
+ 338,
+ 115
+ ],
+ "score": 0.9,
+ "content": "\\nu _ { n } \\to \\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 338,
+ 104,
+ 440,
+ 117
+ ],
+ "score": 1.0,
+ "content": "weakly, then a sequence",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 440,
+ 105,
+ 459,
+ 116
+ ],
+ "score": 0.85,
+ "content": "\\left( \\pi _ { n } \\right)",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 459,
+ 104,
+ 505,
+ 117
+ ],
+ "score": 1.0,
+ "content": "of optimal",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 114,
+ 504,
+ 130
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 114,
+ 204,
+ 130
+ ],
+ "score": 1.0,
+ "content": "transport plans between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 204,
+ 117,
+ 217,
+ 127
+ ],
+ "score": 0.87,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 217,
+ 114,
+ 235,
+ 130
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 236,
+ 117,
+ 247,
+ 127
+ ],
+ "score": 0.88,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 247,
+ 114,
+ 374,
+ 130
+ ],
+ "score": 1.0,
+ "content": "converges weakly to a solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 375,
+ 117,
+ 382,
+ 125
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 383,
+ 114,
+ 496,
+ 130
+ ],
+ "score": 1.0,
+ "content": "of the OT problem between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 117,
+ 504,
+ 127
+ ],
+ "score": 0.78,
+ "content": "\\mu",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 125,
+ 212,
+ 140
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 125,
+ 123,
+ 140
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 128,
+ 130,
+ 136
+ ],
+ "score": 0.7,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 131,
+ 125,
+ 212,
+ 140
+ ],
+ "score": 1.0,
+ "content": ". We can thus write,",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "score": 0.93,
+ "content": "\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname* { l i m } _ { \\varepsilon \\to 0 } \\pi _ { n } ^ { \\varepsilon } = \\pi .",
+ "type": "interline_equation",
+ "image_path": "308b54b39912b32160eacaa43673af27a8b487017ac8b8801bb48ffa7bb8a6b8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 156,
+ 505,
+ 190
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 156,
+ 504,
+ 168
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 156,
+ 393,
+ 168
+ ],
+ "score": 1.0,
+ "content": "A more refined result consists in establishing the weak convergence of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 393,
+ 157,
+ 405,
+ 168
+ ],
+ "score": 0.9,
+ "content": "\\pi _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 156,
+ 417,
+ 168
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 417,
+ 158,
+ 425,
+ 166
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 425,
+ 156,
+ 451,
+ 168
+ ],
+ "score": 1.0,
+ "content": "when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 156,
+ 475,
+ 168
+ ],
+ "score": 0.92,
+ "content": "( n , \\varepsilon )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 475,
+ 156,
+ 504,
+ 168
+ ],
+ "score": 1.0,
+ "content": "jointly",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 167,
+ 506,
+ 180
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 167,
+ 156,
+ 180
+ ],
+ "score": 1.0,
+ "content": "converge to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 167,
+ 184,
+ 179
+ ],
+ "score": 0.92,
+ "content": "( \\infty , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 184,
+ 167,
+ 506,
+ 180
+ ],
+ "score": 1.0,
+ "content": ". This is the result of the following theorem which states a stability property of",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 178,
+ 310,
+ 191
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 178,
+ 310,
+ 191
+ ],
+ "score": 1.0,
+ "content": "entropy-regularized plans (proof in the Appendix).",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 192,
+ 505,
+ 261
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 178,
+ 191,
+ 509,
+ 223
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 178,
+ 192,
+ 227,
+ 204
+ ],
+ "score": 0.91,
+ "content": "\\mu \\in P ( \\mathcal { X } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 192,
+ 295,
+ 204
+ ],
+ "score": 0.91,
+ "content": "\\nu \\in P ( \\mathcal { V } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 191,
+ 325,
+ 223
+ ],
+ "score": 1.0,
+ "content": "where iscrete",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 193,
+ 335,
+ 203
+ ],
+ "score": 0.8,
+ "content": "\\mathcal { X }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 336,
+ 191,
+ 356,
+ 223
+ ],
+ "score": 1.0,
+ "content": "and babi",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 357,
+ 193,
+ 366,
+ 204
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 366,
+ 191,
+ 509,
+ 223
+ ],
+ "score": 1.0,
+ "content": "are complete metric spaces. Let measures which converge weakly",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 107,
+ 203,
+ 273,
+ 217
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 203,
+ 180,
+ 217
+ ],
+ "score": 0.92,
+ "content": "\\textstyle \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 203,
+ 273,
+ 217
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } } \\end{array}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 216,
+ 506,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 216,
+ 118,
+ 230
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 218,
+ 126,
+ 228
+ ],
+ "score": 0.68,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 126,
+ 216,
+ 146,
+ 230
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 218,
+ 154,
+ 226
+ ],
+ "score": 0.66,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 155,
+ 216,
+ 243,
+ 230
+ ],
+ "score": 1.0,
+ "content": "respectively, and let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 243,
+ 217,
+ 261,
+ 228
+ ],
+ "score": 0.83,
+ "content": "\\left( \\varepsilon _ { n } \\right)",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 216,
+ 506,
+ 230
+ ],
+ "score": 1.0,
+ "content": "a sequence of non-negative real numbers converging to 0",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 226,
+ 505,
+ 241
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 226,
+ 241,
+ 241
+ ],
+ "score": 1.0,
+ "content": "sufficiently fast. Assume the cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 242,
+ 230,
+ 248,
+ 237
+ ],
+ "score": 0.52,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 226,
+ 318,
+ 241
+ ],
+ "score": 1.0,
+ "content": "is continuous on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 319,
+ 227,
+ 349,
+ 238
+ ],
+ "score": 0.91,
+ "content": "\\mathcal { X } \\times \\mathcal { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 349,
+ 226,
+ 411,
+ 241
+ ],
+ "score": 1.0,
+ "content": "and finite. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 228,
+ 427,
+ 239
+ ],
+ "score": 0.89,
+ "content": "\\pi _ { n } ^ { \\varepsilon _ { n } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 427,
+ 226,
+ 505,
+ 241
+ ],
+ "score": 1.0,
+ "content": "the solution of the",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 237,
+ 505,
+ 252
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 237,
+ 289,
+ 252
+ ],
+ "score": 1.0,
+ "content": "entropy-regularized OT problem (3) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 289,
+ 240,
+ 302,
+ 250
+ ],
+ "score": 0.86,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 237,
+ 321,
+ 252
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 240,
+ 333,
+ 249
+ ],
+ "score": 0.86,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 333,
+ 237,
+ 505,
+ 252
+ ],
+ "score": 1.0,
+ "content": ". Then, up to extraction of a subsequence,",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 107,
+ 248,
+ 433,
+ 264
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 249,
+ 130,
+ 261
+ ],
+ "score": 0.9,
+ "content": "( \\pi _ { n } ^ { \\varepsilon _ { n } } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 131,
+ 248,
+ 263,
+ 264
+ ],
+ "score": 1.0,
+ "content": "converges weakly to the solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 252,
+ 271,
+ 259
+ ],
+ "score": 0.62,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 248,
+ 396,
+ 264
+ ],
+ "score": 1.0,
+ "content": "of the OT problem (2) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 251,
+ 404,
+ 261
+ ],
+ "score": 0.5,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 404,
+ 248,
+ 422,
+ 264
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 251,
+ 429,
+ 259
+ ],
+ "score": 0.6,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 429,
+ 248,
+ 433,
+ 264
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 11.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "score": 0.89,
+ "content": "\\pi _ { n } ^ { \\varepsilon _ { n } } \\to \\pi w e a k l y .",
+ "type": "interline_equation",
+ "image_path": "3b50b27e75934696e360f025b59c2af984d64f5b6f393603e784bcc9428691e7.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 15,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "spans": [],
+ "index": 15
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 288,
+ 505,
+ 333
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 301
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 301
+ ],
+ "score": 1.0,
+ "content": "Keeping the analogy with statistical machine learning, this result is an analog to the universal con-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 312
+ ],
+ "score": 1.0,
+ "content": "sistency property of a learning method. In most applications, we consider empirical measures and",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 311,
+ 504,
+ 323
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 313,
+ 114,
+ 321
+ ],
+ "score": 0.7,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 114,
+ 311,
+ 504,
+ 323
+ ],
+ "score": 1.0,
+ "content": "is fixed, so that regularization, besides enabling dual stochastic approach, may also help learn the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 322,
+ 343,
+ 334
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 322,
+ 343,
+ 334
+ ],
+ "score": 1.0,
+ "content": "optimal plan between the underlying continuous measures.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 338,
+ 505,
+ 482
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 352
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 352
+ ],
+ "score": 1.0,
+ "content": "So far, we have derived an algorithm for computing the regularized OT objective and regularized",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 349,
+ 505,
+ 362
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 349,
+ 218,
+ 362
+ ],
+ "score": 1.0,
+ "content": "optimal plans regardless of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 218,
+ 351,
+ 226,
+ 361
+ ],
+ "score": 0.82,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 349,
+ 244,
+ 362
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 244,
+ 351,
+ 251,
+ 360
+ ],
+ "score": 0.73,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 252,
+ 349,
+ 505,
+ 362
+ ],
+ "score": 1.0,
+ "content": "being discrete or continuous. The OT objective has been used",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 361,
+ 505,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 361,
+ 505,
+ 374
+ ],
+ "score": 1.0,
+ "content": "successfully as a loss in machine learning (Montavon et al., 2015; Frogner et al., 2015; Rolet et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 371,
+ 505,
+ 384
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 371,
+ 505,
+ 384
+ ],
+ "score": 1.0,
+ "content": "2016; 2018; Arjovsky et al., 2017; Courty et al., 2017a), whereas the use of optimal plans has",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 382,
+ 506,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 382,
+ 506,
+ 396
+ ],
+ "score": 1.0,
+ "content": "straightforward applications in logistics, as well as economy (Kantorovich, 1942; Carlier, 2012) or",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 393,
+ 505,
+ 408
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 393,
+ 505,
+ 408
+ ],
+ "score": 1.0,
+ "content": "computer graphics (Bonneel et al., 2011). In numerous applications however, we often need map-",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 404,
+ 506,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 404,
+ 506,
+ 417
+ ],
+ "score": 1.0,
+ "content": "pings rather than joint distributions. This is all the more motivated since Brenier (1991) proved that",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 414,
+ 505,
+ 430
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 414,
+ 505,
+ 430
+ ],
+ "score": 1.0,
+ "content": "when the source measure is continuous, the optimal transport plan is actually induced by a map.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "score": 1.0,
+ "content": "Assuming that available data samples are sampled from some underlying continuous distributions,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 437,
+ 504,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 437,
+ 504,
+ 450
+ ],
+ "score": 1.0,
+ "content": "finding the Monge map between these continuous measures rather than a discrete optimal plan be-",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "score": 1.0,
+ "content": "tween discrete measures is essential in machine learning applications. Hence in the next section, we",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 459,
+ 505,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 459,
+ 505,
+ 472
+ ],
+ "score": 1.0,
+ "content": "investigate how to recover an optimal map, i.e. find an approximate solution to the Monge problem",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 470,
+ 250,
+ 483
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 470,
+ 250,
+ 483
+ ],
+ "score": 1.0,
+ "content": "(1), from regularized optimal plans.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 26
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 497,
+ 300,
+ 510
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 496,
+ 302,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 496,
+ 302,
+ 512
+ ],
+ "score": 1.0,
+ "content": "4 OPTIMAL MAPPING ESTIMATIONS",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 522,
+ 505,
+ 589
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 523,
+ 504,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 523,
+ 504,
+ 534
+ ],
+ "score": 1.0,
+ "content": "A map can be obtained from a solution to the OT problem (2) or regularized OT problem (3) through",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 533,
+ 506,
+ 546
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 533,
+ 366,
+ 546
+ ],
+ "score": 1.0,
+ "content": "the computation of its barycentric projection. Indeed, a solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 535,
+ 374,
+ 543
+ ],
+ "score": 0.78,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 533,
+ 506,
+ 546
+ ],
+ "score": 1.0,
+ "content": "of Problem (2) or (3) between a",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 544,
+ 506,
+ 557
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 544,
+ 170,
+ 557
+ ],
+ "score": 1.0,
+ "content": "source measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 171,
+ 546,
+ 178,
+ 556
+ ],
+ "score": 0.81,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 179,
+ 544,
+ 263,
+ 557
+ ],
+ "score": 1.0,
+ "content": "and a target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 546,
+ 271,
+ 554
+ ],
+ "score": 0.79,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 544,
+ 363,
+ 557
+ ],
+ "score": 1.0,
+ "content": "is, identifying the plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 364,
+ 546,
+ 371,
+ 554
+ ],
+ "score": 0.76,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 544,
+ 506,
+ 557
+ ],
+ "score": 1.0,
+ "content": "with its density w.r.t. a reference",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 554,
+ 505,
+ 569
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 554,
+ 186,
+ 569
+ ],
+ "score": 1.0,
+ "content": "measure, a function",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 555,
+ 298,
+ 567
+ ],
+ "score": 0.91,
+ "content": "\\pi : ( \\bar { x , y } ) \\in \\mathcal { X } \\times \\mathcal { Y } \\mapsto \\mathbb { R } ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 554,
+ 505,
+ 569
+ ],
+ "score": 1.0,
+ "content": "which can be seen as a weighted one-to-many map,",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 566,
+ 505,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 566,
+ 123,
+ 579
+ ],
+ "score": 1.0,
+ "content": "i.e.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 568,
+ 131,
+ 576
+ ],
+ "score": 0.7,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 132,
+ 566,
+ 158,
+ 579
+ ],
+ "score": 1.0,
+ "content": "sends",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 158,
+ 568,
+ 165,
+ 576
+ ],
+ "score": 0.73,
+ "content": "x",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 165,
+ 566,
+ 234,
+ 579
+ ],
+ "score": 1.0,
+ "content": "to each location",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 567,
+ 288,
+ 578
+ ],
+ "score": 0.86,
+ "content": "y \\in \\operatorname { S u p p } ( \\nu )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 289,
+ 566,
+ 318,
+ 579
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 318,
+ 566,
+ 369,
+ 578
+ ],
+ "score": 0.91,
+ "content": "\\pi ( x , y ) > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 369,
+ 566,
+ 505,
+ 579
+ ],
+ "score": 1.0,
+ "content": ". A map can then be obtained by",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 106,
+ 577,
+ 362,
+ 590
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 577,
+ 220,
+ 590
+ ],
+ "score": 1.0,
+ "content": "simply averaging over these",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 220,
+ 579,
+ 227,
+ 589
+ ],
+ "score": 0.79,
+ "content": "y",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 227,
+ 577,
+ 328,
+ 590
+ ],
+ "score": 1.0,
+ "content": "according to the weights",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 329,
+ 578,
+ 358,
+ 589
+ ],
+ "score": 0.92,
+ "content": "\\pi ( x , y )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 358,
+ 577,
+ 362,
+ 590
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 36.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 591,
+ 505,
+ 614
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 590,
+ 506,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 590,
+ 278,
+ 605
+ ],
+ "score": 1.0,
+ "content": "Definition 1. (Barycentric projection) Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 594,
+ 286,
+ 601
+ ],
+ "score": 0.33,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 590,
+ 506,
+ 605
+ ],
+ "score": 1.0,
+ "content": "be a solution of the OT problem (2) or regularized OT",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 104,
+ 602,
+ 486,
+ 616
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 602,
+ 267,
+ 616
+ ],
+ "score": 1.0,
+ "content": "problem (3). The barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 604,
+ 275,
+ 613
+ ],
+ "score": 0.64,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 276,
+ 602,
+ 354,
+ 616
+ ],
+ "score": 1.0,
+ "content": "w.r.t. a convex cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 603,
+ 430,
+ 614
+ ],
+ "score": 0.9,
+ "content": "d : \\mathcal { V } \\times \\mathcal { V } \\mathbb { R } ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 430,
+ 602,
+ 486,
+ 616
+ ],
+ "score": 1.0,
+ "content": "is defined as,",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ }
+ ],
+ "index": 40.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\pi } ( x ) = \\arg \\operatorname* { m i n } _ { z } \\mathbb { E } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ d ( z , Y ) \\right] .",
+ "type": "interline_equation",
+ "image_path": "2ce2cd6aae8f1baa1888c0bbdb06cd7789f25c4559036dd75702d04fef61613c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 42,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "spans": [],
+ "index": 42
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 647,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 647,
+ 505,
+ 663
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 647,
+ 182,
+ 663
+ ],
+ "score": 1.0,
+ "content": "In the special case",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 182,
+ 649,
+ 261,
+ 661
+ ],
+ "score": 0.95,
+ "content": "d ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 261,
+ 647,
+ 411,
+ 663
+ ],
+ "score": 1.0,
+ "content": ", Eq. (11) has the close form solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 412,
+ 649,
+ 502,
+ 662
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\pi } ( x ) = \\operatorname { \\mathbb { E } } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ Y \\right] .",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 647,
+ 505,
+ 663
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 104,
+ 662,
+ 505,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 662,
+ 180,
+ 678
+ ],
+ "score": 1.0,
+ "content": "which is equal to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 662,
+ 221,
+ 677
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\bar { \\boldsymbol { \\pi } } = \\frac { \\boldsymbol { \\pi } \\mathbf { y } ^ { t } } { a } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 221,
+ 662,
+ 327,
+ 678
+ ],
+ "score": 1.0,
+ "content": "in a discrete setting with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 327,
+ 664,
+ 403,
+ 676
+ ],
+ "score": 0.91,
+ "content": "\\mathbf { y } = ( y _ { 1 } , \\dotsb , y _ { n } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 403,
+ 662,
+ 423,
+ 678
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 667,
+ 430,
+ 675
+ ],
+ "score": 0.76,
+ "content": "a",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 430,
+ 662,
+ 494,
+ 678
+ ],
+ "score": 1.0,
+ "content": "the weights of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 667,
+ 501,
+ 676
+ ],
+ "score": 0.78,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 662,
+ 505,
+ 678
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 677,
+ 506,
+ 690
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 305,
+ 690
+ ],
+ "score": 1.0,
+ "content": "Moreover, for the specific squared Euclidean cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 305,
+ 677,
+ 380,
+ 689
+ ],
+ "score": 0.92,
+ "content": "c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 380,
+ 677,
+ 487,
+ 690
+ ],
+ "score": 1.0,
+ "content": ", the barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 488,
+ 678,
+ 495,
+ 687
+ ],
+ "score": 0.8,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 677,
+ 506,
+ 690
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 106,
+ 687,
+ 505,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 687,
+ 354,
+ 701
+ ],
+ "score": 1.0,
+ "content": "an optimal map (Ambrosio et al., 2006)[Theorem 12.4.4], i.e.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 355,
+ 689,
+ 362,
+ 698
+ ],
+ "score": 0.75,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 687,
+ 505,
+ 701
+ ],
+ "score": 1.0,
+ "content": "is a solution to the Monge problem",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 712
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 238,
+ 712
+ ],
+ "score": 1.0,
+ "content": "(1) between the source measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 238,
+ 701,
+ 246,
+ 710
+ ],
+ "score": 0.82,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 699,
+ 342,
+ 712
+ ],
+ "score": 1.0,
+ "content": "and the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 343,
+ 699,
+ 364,
+ 711
+ ],
+ "score": 0.81,
+ "content": "\\bar { \\pi } \\# \\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 364,
+ 699,
+ 505,
+ 712
+ ],
+ "score": 1.0,
+ "content": ". Hence the barycentric projection",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "w.r.t. the squared Euclidean cost is often used as a simple way to recover optimal maps from optimal",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 106,
+ 720,
+ 487,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 720,
+ 487,
+ 734
+ ],
+ "score": 1.0,
+ "content": "transport plans (Reich, 2013; Wang et al., 2013; Ferradans et al., 2014; Seguy & Cuturi, 2015).",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ }
+ ],
+ "index": 46
+ }
+ ],
+ "page_idx": 5,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2018",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 310,
+ 761
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 310,
+ 761
+ ],
+ "score": 1.0,
+ "content": "6",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 82,
+ 505,
+ 137
+ ],
+ "lines": [],
+ "index": 2,
+ "bbox_fs": [
+ 105,
+ 81,
+ 505,
+ 140
+ ],
+ "lines_deleted": true
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "score": 0.93,
+ "content": "\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname* { l i m } _ { \\varepsilon \\to 0 } \\pi _ { n } ^ { \\varepsilon } = \\pi .",
+ "type": "interline_equation",
+ "image_path": "308b54b39912b32160eacaa43673af27a8b487017ac8b8801bb48ffa7bb8a6b8.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 268,
+ 136,
+ 344,
+ 154
+ ],
+ "spans": [],
+ "index": 5
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 156,
+ 505,
+ 190
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 156,
+ 504,
+ 168
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 156,
+ 393,
+ 168
+ ],
+ "score": 1.0,
+ "content": "A more refined result consists in establishing the weak convergence of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 393,
+ 157,
+ 405,
+ 168
+ ],
+ "score": 0.9,
+ "content": "\\pi _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 406,
+ 156,
+ 417,
+ 168
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 417,
+ 158,
+ 425,
+ 166
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 425,
+ 156,
+ 451,
+ 168
+ ],
+ "score": 1.0,
+ "content": "when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 451,
+ 156,
+ 475,
+ 168
+ ],
+ "score": 0.92,
+ "content": "( n , \\varepsilon )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 475,
+ 156,
+ 504,
+ 168
+ ],
+ "score": 1.0,
+ "content": "jointly",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 105,
+ 167,
+ 506,
+ 180
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 167,
+ 156,
+ 180
+ ],
+ "score": 1.0,
+ "content": "converge to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 156,
+ 167,
+ 184,
+ 179
+ ],
+ "score": 0.92,
+ "content": "( \\infty , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 184,
+ 167,
+ 506,
+ 180
+ ],
+ "score": 1.0,
+ "content": ". This is the result of the following theorem which states a stability property of",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 178,
+ 310,
+ 191
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 178,
+ 310,
+ 191
+ ],
+ "score": 1.0,
+ "content": "entropy-regularized plans (proof in the Appendix).",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7,
+ "bbox_fs": [
+ 105,
+ 156,
+ 506,
+ 191
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 192,
+ 505,
+ 261
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 178,
+ 191,
+ 509,
+ 223
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 178,
+ 192,
+ 227,
+ 204
+ ],
+ "score": 0.91,
+ "content": "\\mu \\in P ( \\mathcal { X } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 192,
+ 295,
+ 204
+ ],
+ "score": 0.91,
+ "content": "\\nu \\in P ( \\mathcal { V } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 191,
+ 325,
+ 223
+ ],
+ "score": 1.0,
+ "content": "where iscrete",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 326,
+ 193,
+ 335,
+ 203
+ ],
+ "score": 0.8,
+ "content": "\\mathcal { X }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 336,
+ 191,
+ 356,
+ 223
+ ],
+ "score": 1.0,
+ "content": "and babi",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 357,
+ 193,
+ 366,
+ 204
+ ],
+ "score": 0.81,
+ "content": "\\mathcal { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 366,
+ 191,
+ 509,
+ 223
+ ],
+ "score": 1.0,
+ "content": "are complete metric spaces. Let measures which converge weakly",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 107,
+ 203,
+ 273,
+ 217
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 203,
+ 180,
+ 217
+ ],
+ "score": 0.92,
+ "content": "\\textstyle \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 200,
+ 203,
+ 273,
+ 217
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r } { \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } } \\end{array}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 216,
+ 506,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 216,
+ 118,
+ 230
+ ],
+ "score": 1.0,
+ "content": "to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 118,
+ 218,
+ 126,
+ 228
+ ],
+ "score": 0.68,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 126,
+ 216,
+ 146,
+ 230
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 147,
+ 218,
+ 154,
+ 226
+ ],
+ "score": 0.66,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 155,
+ 216,
+ 243,
+ 230
+ ],
+ "score": 1.0,
+ "content": "respectively, and let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 243,
+ 217,
+ 261,
+ 228
+ ],
+ "score": 0.83,
+ "content": "\\left( \\varepsilon _ { n } \\right)",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 216,
+ 506,
+ 230
+ ],
+ "score": 1.0,
+ "content": "a sequence of non-negative real numbers converging to 0",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 226,
+ 505,
+ 241
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 226,
+ 241,
+ 241
+ ],
+ "score": 1.0,
+ "content": "sufficiently fast. Assume the cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 242,
+ 230,
+ 248,
+ 237
+ ],
+ "score": 0.52,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 248,
+ 226,
+ 318,
+ 241
+ ],
+ "score": 1.0,
+ "content": "is continuous on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 319,
+ 227,
+ 349,
+ 238
+ ],
+ "score": 0.91,
+ "content": "\\mathcal { X } \\times \\mathcal { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 349,
+ 226,
+ 411,
+ 241
+ ],
+ "score": 1.0,
+ "content": "and finite. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 228,
+ 427,
+ 239
+ ],
+ "score": 0.89,
+ "content": "\\pi _ { n } ^ { \\varepsilon _ { n } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 427,
+ 226,
+ 505,
+ 241
+ ],
+ "score": 1.0,
+ "content": "the solution of the",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 237,
+ 505,
+ 252
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 237,
+ 289,
+ 252
+ ],
+ "score": 1.0,
+ "content": "entropy-regularized OT problem (3) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 289,
+ 240,
+ 302,
+ 250
+ ],
+ "score": 0.86,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 302,
+ 237,
+ 321,
+ 252
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 240,
+ 333,
+ 249
+ ],
+ "score": 0.86,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 333,
+ 237,
+ 505,
+ 252
+ ],
+ "score": 1.0,
+ "content": ". Then, up to extraction of a subsequence,",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 107,
+ 248,
+ 433,
+ 264
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 249,
+ 130,
+ 261
+ ],
+ "score": 0.9,
+ "content": "( \\pi _ { n } ^ { \\varepsilon _ { n } } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 131,
+ 248,
+ 263,
+ 264
+ ],
+ "score": 1.0,
+ "content": "converges weakly to the solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 252,
+ 271,
+ 259
+ ],
+ "score": 0.62,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 248,
+ 396,
+ 264
+ ],
+ "score": 1.0,
+ "content": "of the OT problem (2) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 397,
+ 251,
+ 404,
+ 261
+ ],
+ "score": 0.5,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 404,
+ 248,
+ 422,
+ 264
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 251,
+ 429,
+ 259
+ ],
+ "score": 0.6,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 429,
+ 248,
+ 433,
+ 264
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ }
+ ],
+ "index": 11.5,
+ "bbox_fs": [
+ 105,
+ 191,
+ 509,
+ 264
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "score": 0.89,
+ "content": "\\pi _ { n } ^ { \\varepsilon _ { n } } \\to \\pi w e a k l y .",
+ "type": "interline_equation",
+ "image_path": "3b50b27e75934696e360f025b59c2af984d64f5b6f393603e784bcc9428691e7.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 15,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 265,
+ 265,
+ 346,
+ 279
+ ],
+ "spans": [],
+ "index": 15
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 288,
+ 505,
+ 333
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 301
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 288,
+ 505,
+ 301
+ ],
+ "score": 1.0,
+ "content": "Keeping the analogy with statistical machine learning, this result is an analog to the universal con-",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 300,
+ 505,
+ 312
+ ],
+ "score": 1.0,
+ "content": "sistency property of a learning method. In most applications, we consider empirical measures and",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 106,
+ 311,
+ 504,
+ 323
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 313,
+ 114,
+ 321
+ ],
+ "score": 0.7,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 114,
+ 311,
+ 504,
+ 323
+ ],
+ "score": 1.0,
+ "content": "is fixed, so that regularization, besides enabling dual stochastic approach, may also help learn the",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 106,
+ 322,
+ 343,
+ 334
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 322,
+ 343,
+ 334
+ ],
+ "score": 1.0,
+ "content": "optimal plan between the underlying continuous measures.",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ }
+ ],
+ "index": 17.5,
+ "bbox_fs": [
+ 105,
+ 288,
+ 505,
+ 334
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 338,
+ 505,
+ 482
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 352
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 352
+ ],
+ "score": 1.0,
+ "content": "So far, we have derived an algorithm for computing the regularized OT objective and regularized",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 106,
+ 349,
+ 505,
+ 362
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 349,
+ 218,
+ 362
+ ],
+ "score": 1.0,
+ "content": "optimal plans regardless of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 218,
+ 351,
+ 226,
+ 361
+ ],
+ "score": 0.82,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 226,
+ 349,
+ 244,
+ 362
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 244,
+ 351,
+ 251,
+ 360
+ ],
+ "score": 0.73,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 252,
+ 349,
+ 505,
+ 362
+ ],
+ "score": 1.0,
+ "content": "being discrete or continuous. The OT objective has been used",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 361,
+ 505,
+ 374
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 361,
+ 505,
+ 374
+ ],
+ "score": 1.0,
+ "content": "successfully as a loss in machine learning (Montavon et al., 2015; Frogner et al., 2015; Rolet et al.,",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 371,
+ 505,
+ 384
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 371,
+ 505,
+ 384
+ ],
+ "score": 1.0,
+ "content": "2016; 2018; Arjovsky et al., 2017; Courty et al., 2017a), whereas the use of optimal plans has",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 382,
+ 506,
+ 396
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 382,
+ 506,
+ 396
+ ],
+ "score": 1.0,
+ "content": "straightforward applications in logistics, as well as economy (Kantorovich, 1942; Carlier, 2012) or",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 393,
+ 505,
+ 408
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 393,
+ 505,
+ 408
+ ],
+ "score": 1.0,
+ "content": "computer graphics (Bonneel et al., 2011). In numerous applications however, we often need map-",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 404,
+ 506,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 404,
+ 506,
+ 417
+ ],
+ "score": 1.0,
+ "content": "pings rather than joint distributions. This is all the more motivated since Brenier (1991) proved that",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 105,
+ 414,
+ 505,
+ 430
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 414,
+ 505,
+ 430
+ ],
+ "score": 1.0,
+ "content": "when the source measure is continuous, the optimal transport plan is actually induced by a map.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "score": 1.0,
+ "content": "Assuming that available data samples are sampled from some underlying continuous distributions,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 437,
+ 504,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 437,
+ 504,
+ 450
+ ],
+ "score": 1.0,
+ "content": "finding the Monge map between these continuous measures rather than a discrete optimal plan be-",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "score": 1.0,
+ "content": "tween discrete measures is essential in machine learning applications. Hence in the next section, we",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 459,
+ 505,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 459,
+ 505,
+ 472
+ ],
+ "score": 1.0,
+ "content": "investigate how to recover an optimal map, i.e. find an approximate solution to the Monge problem",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 106,
+ 470,
+ 250,
+ 483
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 470,
+ 250,
+ 483
+ ],
+ "score": 1.0,
+ "content": "(1), from regularized optimal plans.",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ }
+ ],
+ "index": 26,
+ "bbox_fs": [
+ 105,
+ 339,
+ 506,
+ 483
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 497,
+ 300,
+ 510
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 496,
+ 302,
+ 512
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 496,
+ 302,
+ 512
+ ],
+ "score": 1.0,
+ "content": "4 OPTIMAL MAPPING ESTIMATIONS",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 33
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 522,
+ 505,
+ 589
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 523,
+ 504,
+ 534
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 523,
+ 504,
+ 534
+ ],
+ "score": 1.0,
+ "content": "A map can be obtained from a solution to the OT problem (2) or regularized OT problem (3) through",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 533,
+ 506,
+ 546
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 533,
+ 366,
+ 546
+ ],
+ "score": 1.0,
+ "content": "the computation of its barycentric projection. Indeed, a solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 535,
+ 374,
+ 543
+ ],
+ "score": 0.78,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 374,
+ 533,
+ 506,
+ 546
+ ],
+ "score": 1.0,
+ "content": "of Problem (2) or (3) between a",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 105,
+ 544,
+ 506,
+ 557
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 544,
+ 170,
+ 557
+ ],
+ "score": 1.0,
+ "content": "source measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 171,
+ 546,
+ 178,
+ 556
+ ],
+ "score": 0.81,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 179,
+ 544,
+ 263,
+ 557
+ ],
+ "score": 1.0,
+ "content": "and a target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 264,
+ 546,
+ 271,
+ 554
+ ],
+ "score": 0.79,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 271,
+ 544,
+ 363,
+ 557
+ ],
+ "score": 1.0,
+ "content": "is, identifying the plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 364,
+ 546,
+ 371,
+ 554
+ ],
+ "score": 0.76,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 544,
+ 506,
+ 557
+ ],
+ "score": 1.0,
+ "content": "with its density w.r.t. a reference",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 554,
+ 505,
+ 569
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 554,
+ 186,
+ 569
+ ],
+ "score": 1.0,
+ "content": "measure, a function",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 187,
+ 555,
+ 298,
+ 567
+ ],
+ "score": 0.91,
+ "content": "\\pi : ( \\bar { x , y } ) \\in \\mathcal { X } \\times \\mathcal { Y } \\mapsto \\mathbb { R } ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 554,
+ 505,
+ 569
+ ],
+ "score": 1.0,
+ "content": "which can be seen as a weighted one-to-many map,",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 566,
+ 505,
+ 579
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 566,
+ 123,
+ 579
+ ],
+ "score": 1.0,
+ "content": "i.e.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 123,
+ 568,
+ 131,
+ 576
+ ],
+ "score": 0.7,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 132,
+ 566,
+ 158,
+ 579
+ ],
+ "score": 1.0,
+ "content": "sends",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 158,
+ 568,
+ 165,
+ 576
+ ],
+ "score": 0.73,
+ "content": "x",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 165,
+ 566,
+ 234,
+ 579
+ ],
+ "score": 1.0,
+ "content": "to each location",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 234,
+ 567,
+ 288,
+ 578
+ ],
+ "score": 0.86,
+ "content": "y \\in \\operatorname { S u p p } ( \\nu )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 289,
+ 566,
+ 318,
+ 579
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 318,
+ 566,
+ 369,
+ 578
+ ],
+ "score": 0.91,
+ "content": "\\pi ( x , y ) > 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 369,
+ 566,
+ 505,
+ 579
+ ],
+ "score": 1.0,
+ "content": ". A map can then be obtained by",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 106,
+ 577,
+ 362,
+ 590
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 577,
+ 220,
+ 590
+ ],
+ "score": 1.0,
+ "content": "simply averaging over these",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 220,
+ 579,
+ 227,
+ 589
+ ],
+ "score": 0.79,
+ "content": "y",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 227,
+ 577,
+ 328,
+ 590
+ ],
+ "score": 1.0,
+ "content": "according to the weights",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 329,
+ 578,
+ 358,
+ 589
+ ],
+ "score": 0.92,
+ "content": "\\pi ( x , y )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 358,
+ 577,
+ 362,
+ 590
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 36.5,
+ "bbox_fs": [
+ 105,
+ 523,
+ 506,
+ 590
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 105,
+ 591,
+ 505,
+ 614
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 590,
+ 506,
+ 605
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 590,
+ 278,
+ 605
+ ],
+ "score": 1.0,
+ "content": "Definition 1. (Barycentric projection) Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 279,
+ 594,
+ 286,
+ 601
+ ],
+ "score": 0.33,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 286,
+ 590,
+ 506,
+ 605
+ ],
+ "score": 1.0,
+ "content": "be a solution of the OT problem (2) or regularized OT",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 104,
+ 602,
+ 486,
+ 616
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 602,
+ 267,
+ 616
+ ],
+ "score": 1.0,
+ "content": "problem (3). The barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 268,
+ 604,
+ 275,
+ 613
+ ],
+ "score": 0.64,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 276,
+ 602,
+ 354,
+ 616
+ ],
+ "score": 1.0,
+ "content": "w.r.t. a convex cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 354,
+ 603,
+ 430,
+ 614
+ ],
+ "score": 0.9,
+ "content": "d : \\mathcal { V } \\times \\mathcal { V } \\mathbb { R } ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 430,
+ 602,
+ 486,
+ 616
+ ],
+ "score": 1.0,
+ "content": "is defined as,",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ }
+ ],
+ "index": 40.5,
+ "bbox_fs": [
+ 104,
+ 590,
+ 506,
+ 616
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\pi } ( x ) = \\arg \\operatorname* { m i n } _ { z } \\mathbb { E } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ d ( z , Y ) \\right] .",
+ "type": "interline_equation",
+ "image_path": "2ce2cd6aae8f1baa1888c0bbdb06cd7789f25c4559036dd75702d04fef61613c.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 42,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 229,
+ 618,
+ 381,
+ 637
+ ],
+ "spans": [],
+ "index": 42
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 647,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 647,
+ 505,
+ 663
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 647,
+ 182,
+ 663
+ ],
+ "score": 1.0,
+ "content": "In the special case",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 182,
+ 649,
+ 261,
+ 661
+ ],
+ "score": 0.95,
+ "content": "d ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 261,
+ 647,
+ 411,
+ 663
+ ],
+ "score": 1.0,
+ "content": ", Eq. (11) has the close form solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 412,
+ 649,
+ 502,
+ 662
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\pi } ( x ) = \\operatorname { \\mathbb { E } } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ Y \\right] .",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 647,
+ 505,
+ 663
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ },
+ {
+ "bbox": [
+ 104,
+ 662,
+ 505,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 662,
+ 180,
+ 678
+ ],
+ "score": 1.0,
+ "content": "which is equal to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 181,
+ 662,
+ 221,
+ 677
+ ],
+ "score": 0.93,
+ "content": "\\begin{array} { r } { \\bar { \\boldsymbol { \\pi } } = \\frac { \\boldsymbol { \\pi } \\mathbf { y } ^ { t } } { a } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 221,
+ 662,
+ 327,
+ 678
+ ],
+ "score": 1.0,
+ "content": "in a discrete setting with",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 327,
+ 664,
+ 403,
+ 676
+ ],
+ "score": 0.91,
+ "content": "\\mathbf { y } = ( y _ { 1 } , \\dotsb , y _ { n } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 403,
+ 662,
+ 423,
+ 678
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 423,
+ 667,
+ 430,
+ 675
+ ],
+ "score": 0.76,
+ "content": "a",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 430,
+ 662,
+ 494,
+ 678
+ ],
+ "score": 1.0,
+ "content": "the weights of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 494,
+ 667,
+ 501,
+ 676
+ ],
+ "score": 0.78,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 662,
+ 505,
+ 678
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 105,
+ 677,
+ 506,
+ 690
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 677,
+ 305,
+ 690
+ ],
+ "score": 1.0,
+ "content": "Moreover, for the specific squared Euclidean cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 305,
+ 677,
+ 380,
+ 689
+ ],
+ "score": 0.92,
+ "content": "c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 380,
+ 677,
+ 487,
+ 690
+ ],
+ "score": 1.0,
+ "content": ", the barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 488,
+ 678,
+ 495,
+ 687
+ ],
+ "score": 0.8,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 495,
+ 677,
+ 506,
+ 690
+ ],
+ "score": 1.0,
+ "content": "is",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ },
+ {
+ "bbox": [
+ 106,
+ 687,
+ 505,
+ 701
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 687,
+ 354,
+ 701
+ ],
+ "score": 1.0,
+ "content": "an optimal map (Ambrosio et al., 2006)[Theorem 12.4.4], i.e.",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 355,
+ 689,
+ 362,
+ 698
+ ],
+ "score": 0.75,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 362,
+ 687,
+ 505,
+ 701
+ ],
+ "score": 1.0,
+ "content": "is a solution to the Monge problem",
+ "type": "text"
+ }
+ ],
+ "index": 46
+ },
+ {
+ "bbox": [
+ 106,
+ 699,
+ 505,
+ 712
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 699,
+ 238,
+ 712
+ ],
+ "score": 1.0,
+ "content": "(1) between the source measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 238,
+ 701,
+ 246,
+ 710
+ ],
+ "score": 0.82,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 246,
+ 699,
+ 342,
+ 712
+ ],
+ "score": 1.0,
+ "content": "and the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 343,
+ 699,
+ 364,
+ 711
+ ],
+ "score": 0.81,
+ "content": "\\bar { \\pi } \\# \\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 364,
+ 699,
+ 505,
+ 712
+ ],
+ "score": 1.0,
+ "content": ". Hence the barycentric projection",
+ "type": "text"
+ }
+ ],
+ "index": 47
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 505,
+ 722
+ ],
+ "score": 1.0,
+ "content": "w.r.t. the squared Euclidean cost is often used as a simple way to recover optimal maps from optimal",
+ "type": "text"
+ }
+ ],
+ "index": 48
+ },
+ {
+ "bbox": [
+ 106,
+ 720,
+ 487,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 720,
+ 487,
+ 734
+ ],
+ "score": 1.0,
+ "content": "transport plans (Reich, 2013; Wang et al., 2013; Ferradans et al., 2014; Seguy & Cuturi, 2015).",
+ "type": "text"
+ }
+ ],
+ "index": 49
+ }
+ ],
+ "index": 46,
+ "bbox_fs": [
+ 104,
+ 647,
+ 506,
+ 734
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 107,
+ 79,
+ 505,
+ 178
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 107,
+ 79,
+ 505,
+ 178
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 113,
+ 79,
+ 505,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 113,
+ 79,
+ 505,
+ 178
+ ],
+ "score": 0.442,
+ "html": "| lgorithm 2 Optimal map learning with SGD |
| Inputs: input measures μ, V; cost function c; dual optimal variables u and v; map fe parame- |
| terized as a deep NN; batch size n; learning rate y. while not converged do |
| sample a batch (x1,·· ,xn) from μ |
| sample a batch (y1,:.,yn) from v |
| updateθ←0-γ∑ijHe(xi,yj)Vθd(yj,fe(xi)) end while |
",
+ "type": "table",
+ "image_path": "005b83c7e1e2dd1b696892e55336da0baf1cf57ed3f985bcc641333679575402.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 107,
+ 79,
+ 505,
+ 112.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 107,
+ 112.0,
+ 505,
+ 145.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 145.0,
+ 505,
+ 178.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 241
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 196,
+ 505,
+ 209
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 196,
+ 430,
+ 209
+ ],
+ "score": 1.0,
+ "content": "Formula (11) provides a pointwise value of the barycentric projection. When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 430,
+ 199,
+ 438,
+ 209
+ ],
+ "score": 0.8,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 439,
+ 196,
+ 505,
+ 209
+ ],
+ "score": 1.0,
+ "content": "is discrete, this",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 222
+ ],
+ "score": 1.0,
+ "content": "means that we only have mapping estimations for a finite number of points. In order to define a map",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 219,
+ 505,
+ 232
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 219,
+ 505,
+ 232
+ ],
+ "score": 1.0,
+ "content": "which is defined everywhere, we parameterize the barycentric projection as a deep neural network.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 230,
+ 388,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 230,
+ 388,
+ 243
+ ],
+ "score": 1.0,
+ "content": "We show in the next paragraph how to efficiently learn its parameters.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 246,
+ 505,
+ 280
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 246,
+ 504,
+ 259
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 246,
+ 272,
+ 259
+ ],
+ "score": 1.0,
+ "content": "Optimal map learning An estimation",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 272,
+ 247,
+ 280,
+ 258
+ ],
+ "score": 0.86,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 246,
+ 492,
+ 259
+ ],
+ "score": 1.0,
+ "content": "of the barycentric projection of a regularized plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 492,
+ 247,
+ 504,
+ 257
+ ],
+ "score": 0.83,
+ "content": "\\pi ^ { \\varepsilon }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 258,
+ 505,
+ 270
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 258,
+ 268,
+ 270
+ ],
+ "score": 1.0,
+ "content": "which generalizes outside the support of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 259,
+ 276,
+ 270
+ ],
+ "score": 0.85,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 258,
+ 505,
+ 270
+ ],
+ "score": 1.0,
+ "content": "can be obtained by learning a deep neural network which",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 269,
+ 339,
+ 281
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 269,
+ 328,
+ 281
+ ],
+ "score": 1.0,
+ "content": "minimizes the following objective w.r.t. the parameters",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 329,
+ 269,
+ 334,
+ 279
+ ],
+ "score": 0.8,
+ "content": "\\theta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 269,
+ 339,
+ 281
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 8
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 315
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 315
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 315
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r l } & { \\mathbb { E } _ { X \\sim \\mu } \\left[ \\mathbb { E } _ { Y \\sim \\pi ^ { \\varepsilon } ( \\cdot | X ) } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] \\right] = \\mathbb { E } _ { ( X , Y ) \\sim \\pi ^ { \\varepsilon } } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( Y , f _ { \\theta } ( X ) ) H _ { \\varepsilon } ( X , Y ) \\right] . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "99112b75efa82526f4065b76eee6fd113d983719cb533a205bff100a18f6c5f3.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 11,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 293.0
+ ],
+ "spans": [],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 151,
+ 293.0,
+ 460,
+ 304.0
+ ],
+ "spans": [],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 151,
+ 304.0,
+ 460,
+ 315.0
+ ],
+ "spans": [],
+ "index": 12
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 316,
+ 505,
+ 374
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 316,
+ 506,
+ 330
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 316,
+ 133,
+ 330
+ ],
+ "score": 1.0,
+ "content": "When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 316,
+ 215,
+ 329
+ ],
+ "score": 0.92,
+ "content": "d ( x , y ) = \\| x - y \\| ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 215,
+ 316,
+ 506,
+ 330
+ ],
+ "score": 1.0,
+ "content": ", the last term in Eq. (12) is simply a weighted sum of squared errors,",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 327,
+ 505,
+ 340
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 327,
+ 314,
+ 340
+ ],
+ "score": 1.0,
+ "content": "with possibly an infinite number of terms whenever",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 314,
+ 330,
+ 321,
+ 339
+ ],
+ "score": 0.83,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 327,
+ 333,
+ 340
+ ],
+ "score": 1.0,
+ "content": "or",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 330,
+ 340,
+ 338
+ ],
+ "score": 0.75,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 341,
+ 327,
+ 505,
+ 340
+ ],
+ "score": 1.0,
+ "content": "are continuous. We propose to minimize",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 351
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 351
+ ],
+ "score": 1.0,
+ "content": "the objective (12) by stochastic gradient descent, which provides the simple Algorithm 2. The OT",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 350,
+ 506,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 350,
+ 446,
+ 363
+ ],
+ "score": 1.0,
+ "content": "problem being symmetric, we can also compute the opposite barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 447,
+ 352,
+ 453,
+ 362
+ ],
+ "score": 0.79,
+ "content": "g",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 454,
+ 350,
+ 506,
+ 363
+ ],
+ "score": 1.0,
+ "content": "w.r.t. a cost",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 107,
+ 358,
+ 399,
+ 376
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 361,
+ 185,
+ 371
+ ],
+ "score": 0.9,
+ "content": "\\bar { d } : \\mathcal { X } \\times \\mathcal { X } \\overset { } { \\to } \\bar { \\mathbb { R } } ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 358,
+ 247,
+ 376
+ ],
+ "score": 1.0,
+ "content": "by minimizing",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 247,
+ 361,
+ 394,
+ 374
+ ],
+ "score": 0.91,
+ "content": "\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( g ( \\bar { Y } ) , X ) H _ { \\varepsilon } ( \\bar { X } , Y ) \\right]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 358,
+ 399,
+ 376
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 15
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 377,
+ 505,
+ 444
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 378,
+ 505,
+ 390
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 378,
+ 209,
+ 390
+ ],
+ "score": 1.0,
+ "content": "However, unless the plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 210,
+ 380,
+ 217,
+ 388
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 378,
+ 505,
+ 390
+ ],
+ "score": 1.0,
+ "content": "is induced by a map, the averaging process results in having the image",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 389,
+ 506,
+ 401
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 389,
+ 215,
+ 401
+ ],
+ "score": 1.0,
+ "content": "of the source measure by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 390,
+ 223,
+ 399
+ ],
+ "score": 0.78,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 389,
+ 425,
+ 401
+ ],
+ "score": 1.0,
+ "content": "only approximately equal to the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 391,
+ 433,
+ 399
+ ],
+ "score": 0.71,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 433,
+ 389,
+ 506,
+ 401
+ ],
+ "score": 1.0,
+ "content": ". Still, when the",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "score": 1.0,
+ "content": "size of discrete measure is large and the regularization is small, we show in the next paragraph",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 411,
+ 505,
+ 424
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 411,
+ 505,
+ 424
+ ],
+ "score": 1.0,
+ "content": "that 1) the barycentric projection of a regularized OT plan is close to the Monge map between",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 434
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 434
+ ],
+ "score": 1.0,
+ "content": "the underlying continuous measures (Theorem 2) and 2) the image of the source measure by this",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 432,
+ 411,
+ 445
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 346,
+ 445
+ ],
+ "score": 1.0,
+ "content": "barycentric projection should be close to the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 346,
+ 434,
+ 353,
+ 442
+ ],
+ "score": 0.72,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 432,
+ 411,
+ 445
+ ],
+ "score": 1.0,
+ "content": "(Corollary 1).",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 20.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 448,
+ 505,
+ 538
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 449,
+ 505,
+ 462
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 449,
+ 306,
+ 462
+ ],
+ "score": 1.0,
+ "content": "Theoretical guarantees As stated earlier, when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 307,
+ 450,
+ 339,
+ 460
+ ],
+ "score": 0.92,
+ "content": "\\mathcal { X } = \\mathcal { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 449,
+ 358,
+ 462
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 359,
+ 449,
+ 438,
+ 461
+ ],
+ "score": 0.93,
+ "content": "c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 439,
+ 449,
+ 505,
+ 462
+ ],
+ "score": 1.0,
+ "content": ", Brenier (1991)",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 461,
+ 505,
+ 473
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 461,
+ 257,
+ 473
+ ],
+ "score": 1.0,
+ "content": "proved that when the source measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 462,
+ 265,
+ 472
+ ],
+ "score": 0.82,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 461,
+ 505,
+ 473
+ ],
+ "score": 1.0,
+ "content": "is continuous, there exists a solution to the Monge problem",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 471,
+ 505,
+ 484
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 471,
+ 505,
+ 484
+ ],
+ "score": 1.0,
+ "content": "(1). This result was generalized to more general cost functions, see (Villani, 2008)[Corollary 9.3] for",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 104,
+ 481,
+ 505,
+ 496
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 481,
+ 222,
+ 496
+ ],
+ "score": 1.0,
+ "content": "details. In that case, the plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 222,
+ 484,
+ 230,
+ 493
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 231,
+ 481,
+ 266,
+ 496
+ ],
+ "score": 1.0,
+ "content": "between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 267,
+ 484,
+ 274,
+ 494
+ ],
+ "score": 0.8,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 481,
+ 292,
+ 496
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 292,
+ 484,
+ 299,
+ 492
+ ],
+ "score": 0.75,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 481,
+ 351,
+ 496
+ ],
+ "score": 1.0,
+ "content": "is written as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 483,
+ 392,
+ 494
+ ],
+ "score": 0.73,
+ "content": "( \\operatorname { i d } , f ) \\# \\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 481,
+ 420,
+ 496
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 420,
+ 483,
+ 427,
+ 494
+ ],
+ "score": 0.86,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 481,
+ 505,
+ 496
+ ],
+ "score": 1.0,
+ "content": "is the Monge map.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 493,
+ 505,
+ 506
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 493,
+ 249,
+ 506
+ ],
+ "score": 1.0,
+ "content": "Now considering discrete measures",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 249,
+ 495,
+ 262,
+ 505
+ ],
+ "score": 0.87,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 493,
+ 280,
+ 506
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 495,
+ 291,
+ 504
+ ],
+ "score": 0.87,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 292,
+ 493,
+ 367,
+ 506
+ ],
+ "score": 1.0,
+ "content": "which converge to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 367,
+ 496,
+ 375,
+ 505
+ ],
+ "score": 0.85,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 493,
+ 445,
+ 506
+ ],
+ "score": 1.0,
+ "content": "(continuous) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 445,
+ 495,
+ 452,
+ 503
+ ],
+ "score": 0.78,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 452,
+ 493,
+ 505,
+ 506
+ ],
+ "score": 1.0,
+ "content": "respectively,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 504,
+ 505,
+ 518
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 504,
+ 247,
+ 518
+ ],
+ "score": 1.0,
+ "content": "we have proved in Theorem 1 that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 247,
+ 505,
+ 259,
+ 516
+ ],
+ "score": 0.89,
+ "content": "\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 260,
+ 504,
+ 345,
+ 518
+ ],
+ "score": 1.0,
+ "content": "converges weakly to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 346,
+ 505,
+ 408,
+ 516
+ ],
+ "score": 0.92,
+ "content": "\\pi = ( \\operatorname { i d } , f ) \\# \\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 504,
+ 434,
+ 518
+ ],
+ "score": 1.0,
+ "content": "when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 504,
+ 501,
+ 516
+ ],
+ "score": 0.92,
+ "content": "( n , \\varepsilon ) \\to ( \\infty , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 504,
+ 505,
+ 518
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 104,
+ 514,
+ 506,
+ 530
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 514,
+ 430,
+ 530
+ ],
+ "score": 1.0,
+ "content": "The next theorem, proved in the Appendix, shows that the barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 516,
+ 443,
+ 527
+ ],
+ "score": 0.89,
+ "content": "\\bar { \\pi } _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 443,
+ 514,
+ 506,
+ 530
+ ],
+ "score": 1.0,
+ "content": "also converges",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 527,
+ 400,
+ 539
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 527,
+ 266,
+ 539
+ ],
+ "score": 1.0,
+ "content": "weakly to the true Monge map between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 267,
+ 528,
+ 274,
+ 538
+ ],
+ "score": 0.81,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 527,
+ 292,
+ 539
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 292,
+ 528,
+ 299,
+ 536
+ ],
+ "score": 0.74,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 527,
+ 400,
+ 539
+ ],
+ "score": 1.0,
+ "content": ", justifying our approach.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 27.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 540,
+ 505,
+ 621
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 506,
+ 553
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 178,
+ 553
+ ],
+ "score": 1.0,
+ "content": "Theorem 2. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 542,
+ 187,
+ 552
+ ],
+ "score": 0.73,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 539,
+ 357,
+ 553
+ ],
+ "score": 1.0,
+ "content": "be a continuous probability measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 358,
+ 539,
+ 370,
+ 550
+ ],
+ "score": 0.86,
+ "content": "\\mathbb { R } ^ { d }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 539,
+ 395,
+ 553
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 395,
+ 542,
+ 402,
+ 550
+ ],
+ "score": 0.71,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 539,
+ 506,
+ 553
+ ],
+ "score": 1.0,
+ "content": "an arbitrary probability",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 100,
+ 549,
+ 504,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 100,
+ 556,
+ 124,
+ 582
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 104,
+ 549,
+ 154,
+ 567
+ ],
+ "score": 1.0,
+ "content": "measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 563,
+ 198,
+ 578
+ ],
+ "score": 0.93,
+ "content": "\\textstyle \\nu _ { n } = { \\frac { 1 } { n } } \\sum _ { j = 1 } ^ { n } \\delta _ { y _ { j } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 154,
+ 551,
+ 167,
+ 561
+ ],
+ "score": 0.88,
+ "content": "\\mathbb { R } ^ { d }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 549,
+ 431,
+ 567
+ ],
+ "score": 1.0,
+ "content": "and c a cost function satisfying (Villani, 2008)[Corollary 9.3]. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 556,
+ 287,
+ 582
+ ],
+ "score": 1.0,
+ "content": "converging weakly to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 287,
+ 566,
+ 295,
+ 576
+ ],
+ "score": 0.57,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 556,
+ 313,
+ 582
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 314,
+ 566,
+ 321,
+ 574
+ ],
+ "score": 0.52,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 556,
+ 491,
+ 582
+ ],
+ "score": 1.0,
+ "content": "n i=1 respectively. Assume that the OT solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 551,
+ 504,
+ 564
+ ],
+ "score": 0.88,
+ "content": "\\begin{array} { r } { \\mu _ { n } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { x _ { i } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 492,
+ 566,
+ 504,
+ 575
+ ],
+ "score": 0.81,
+ "content": "\\pi _ { n }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 577,
+ 506,
+ 590
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 577,
+ 205,
+ 590
+ ],
+ "score": 1.0,
+ "content": "of Problem (2) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 206,
+ 579,
+ 218,
+ 588
+ ],
+ "score": 0.83,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 577,
+ 238,
+ 590
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 578,
+ 250,
+ 588
+ ],
+ "score": 0.86,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 250,
+ 577,
+ 321,
+ 590
+ ],
+ "score": 1.0,
+ "content": "is unique for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 579,
+ 329,
+ 587
+ ],
+ "score": 0.72,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 329,
+ 577,
+ 352,
+ 590
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 577,
+ 370,
+ 588
+ ],
+ "score": 0.83,
+ "content": "\\left( \\varepsilon _ { n } \\right)",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 577,
+ 506,
+ 590
+ ],
+ "score": 1.0,
+ "content": "a sequence of non-negative real",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 587,
+ 506,
+ 600
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 587,
+ 291,
+ 600
+ ],
+ "score": 1.0,
+ "content": "numbers converging sufficiently fast to 0 and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 588,
+ 307,
+ 600
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 307,
+ 587,
+ 506,
+ 600
+ ],
+ "score": 1.0,
+ "content": "the barycentric projection w.r.t. the convex cost",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 107,
+ 597,
+ 504,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 599,
+ 133,
+ 609
+ ],
+ "score": 0.89,
+ "content": "d = c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 134,
+ 597,
+ 196,
+ 614
+ ],
+ "score": 1.0,
+ "content": "of the solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 599,
+ 212,
+ 611
+ ],
+ "score": 0.9,
+ "content": "\\pi _ { n } ^ { \\varepsilon _ { n } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 213,
+ 597,
+ 497,
+ 614
+ ],
+ "score": 1.0,
+ "content": "of the entropy-regularized OT problem (3). Then, up to extraction of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 498,
+ 600,
+ 504,
+ 609
+ ],
+ "score": 0.54,
+ "content": "a",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 609,
+ 162,
+ 623
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 162,
+ 623
+ ],
+ "score": 1.0,
+ "content": "subsequence,",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 34.5
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "score": 0.9,
+ "content": "( i d , \\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } ) \\# \\mu _ { n } ( i d , f ) \\# \\mu \\ w e a k l y ,",
+ "type": "interline_equation",
+ "image_path": "a23a5eb78f59b0e108b2a2aa02a64a266e228755f22459bb28143378ee9dfde7.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 38,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "spans": [],
+ "index": 38
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 634,
+ 375,
+ 646
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 633,
+ 375,
+ 647
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 633,
+ 133,
+ 647
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 635,
+ 140,
+ 646
+ ],
+ "score": 0.84,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 141,
+ 633,
+ 339,
+ 647
+ ],
+ "score": 1.0,
+ "content": "is the solution of the Monge problem (1) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 339,
+ 636,
+ 347,
+ 646
+ ],
+ "score": 0.73,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 347,
+ 633,
+ 365,
+ 647
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 636,
+ 372,
+ 644
+ ],
+ "score": 0.64,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 633,
+ 375,
+ 647
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 39
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 653,
+ 505,
+ 687
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 653,
+ 505,
+ 667
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 653,
+ 505,
+ 667
+ ],
+ "score": 1.0,
+ "content": "This theorem shows that our estimated barycentric projection is close to an optimal map between the",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 106,
+ 665,
+ 505,
+ 677
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 665,
+ 250,
+ 677
+ ],
+ "score": 1.0,
+ "content": "underlying continuous measures for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 250,
+ 667,
+ 257,
+ 675
+ ],
+ "score": 0.78,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 665,
+ 288,
+ 677
+ ],
+ "score": 1.0,
+ "content": "big and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 289,
+ 667,
+ 295,
+ 675
+ ],
+ "score": 0.78,
+ "content": "\\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 665,
+ 505,
+ 677
+ ],
+ "score": 1.0,
+ "content": "small. The following corollary confirms the intuition",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 675,
+ 480,
+ 690
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 675,
+ 480,
+ 690
+ ],
+ "score": 1.0,
+ "content": "that the image of the source measure by this map converges to the underlying target measure.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 690,
+ 408,
+ 702
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 688,
+ 409,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 688,
+ 313,
+ 704
+ ],
+ "score": 1.0,
+ "content": "Corollary 1. With the same assumptions as above,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 690,
+ 371,
+ 702
+ ],
+ "score": 0.87,
+ "content": "{ \\bar { \\pi } } _ { n } ^ { \\varepsilon _ { n } } \\# \\mu _ { n } \\to \\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 688,
+ 409,
+ 704
+ ],
+ "score": 1.0,
+ "content": "weakly.",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 43
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 709,
+ 502,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 707,
+ 504,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 707,
+ 351,
+ 723
+ ],
+ "score": 1.0,
+ "content": "In terms of random variables, the last equation states that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 710,
+ 392,
+ 721
+ ],
+ "score": 0.92,
+ "content": "X _ { n } \\sim \\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 707,
+ 411,
+ 723
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 710,
+ 441,
+ 720
+ ],
+ "score": 0.89,
+ "content": "Y \\sim \\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 707,
+ 465,
+ 723
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 466,
+ 709,
+ 504,
+ 722
+ ],
+ "score": 0.91,
+ "content": "\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } \\left( X _ { n } \\right)",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 106,
+ 721,
+ 230,
+ 731
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 721,
+ 217,
+ 731
+ ],
+ "score": 1.0,
+ "content": "converges in distribution to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 218,
+ 721,
+ 226,
+ 730
+ ],
+ "score": 0.81,
+ "content": "Y",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 227,
+ 721,
+ 230,
+ 731
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ }
+ ],
+ "index": 44.5
+ }
+ ],
+ "page_idx": 6,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 2018",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 302,
+ 751,
+ 309,
+ 759
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 762
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 302,
+ 750,
+ 309,
+ 762
+ ],
+ "score": 1.0,
+ "content": "7",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "table",
+ "bbox": [
+ 107,
+ 79,
+ 505,
+ 178
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 107,
+ 79,
+ 505,
+ 178
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 113,
+ 79,
+ 505,
+ 178
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 113,
+ 79,
+ 505,
+ 178
+ ],
+ "score": 0.442,
+ "html": "| lgorithm 2 Optimal map learning with SGD |
| Inputs: input measures μ, V; cost function c; dual optimal variables u and v; map fe parame- |
| terized as a deep NN; batch size n; learning rate y. while not converged do |
| sample a batch (x1,·· ,xn) from μ |
| sample a batch (y1,:.,yn) from v |
| updateθ←0-γ∑ijHe(xi,yj)Vθd(yj,fe(xi)) end while |
",
+ "type": "table",
+ "image_path": "005b83c7e1e2dd1b696892e55336da0baf1cf57ed3f985bcc641333679575402.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 107,
+ 79,
+ 505,
+ 112.0
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 107,
+ 112.0,
+ 505,
+ 145.0
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 107,
+ 145.0,
+ 505,
+ 178.0
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ }
+ ],
+ "index": 1
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 241
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 196,
+ 505,
+ 209
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 196,
+ 430,
+ 209
+ ],
+ "score": 1.0,
+ "content": "Formula (11) provides a pointwise value of the barycentric projection. When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 430,
+ 199,
+ 438,
+ 209
+ ],
+ "score": 0.8,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 439,
+ 196,
+ 505,
+ 209
+ ],
+ "score": 1.0,
+ "content": "is discrete, this",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 222
+ ],
+ "score": 1.0,
+ "content": "means that we only have mapping estimations for a finite number of points. In order to define a map",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 219,
+ 505,
+ 232
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 219,
+ 505,
+ 232
+ ],
+ "score": 1.0,
+ "content": "which is defined everywhere, we parameterize the barycentric projection as a deep neural network.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 106,
+ 230,
+ 388,
+ 243
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 230,
+ 388,
+ 243
+ ],
+ "score": 1.0,
+ "content": "We show in the next paragraph how to efficiently learn its parameters.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4.5,
+ "bbox_fs": [
+ 105,
+ 196,
+ 505,
+ 243
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 246,
+ 505,
+ 280
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 246,
+ 504,
+ 259
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 246,
+ 272,
+ 259
+ ],
+ "score": 1.0,
+ "content": "Optimal map learning An estimation",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 272,
+ 247,
+ 280,
+ 258
+ ],
+ "score": 0.86,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 280,
+ 246,
+ 492,
+ 259
+ ],
+ "score": 1.0,
+ "content": "of the barycentric projection of a regularized plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 492,
+ 247,
+ 504,
+ 257
+ ],
+ "score": 0.83,
+ "content": "\\pi ^ { \\varepsilon }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 258,
+ 505,
+ 270
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 258,
+ 268,
+ 270
+ ],
+ "score": 1.0,
+ "content": "which generalizes outside the support of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 269,
+ 259,
+ 276,
+ 270
+ ],
+ "score": 0.85,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 277,
+ 258,
+ 505,
+ 270
+ ],
+ "score": 1.0,
+ "content": "can be obtained by learning a deep neural network which",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 269,
+ 339,
+ 281
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 269,
+ 328,
+ 281
+ ],
+ "score": 1.0,
+ "content": "minimizes the following objective w.r.t. the parameters",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 329,
+ 269,
+ 334,
+ 279
+ ],
+ "score": 0.8,
+ "content": "\\theta",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 335,
+ 269,
+ 339,
+ 281
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 8,
+ "bbox_fs": [
+ 106,
+ 246,
+ 505,
+ 281
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 315
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 315
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 315
+ ],
+ "score": 0.92,
+ "content": "\\begin{array} { r l } & { \\mathbb { E } _ { X \\sim \\mu } \\left[ \\mathbb { E } _ { Y \\sim \\pi ^ { \\varepsilon } ( \\cdot | X ) } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] \\right] = \\mathbb { E } _ { ( X , Y ) \\sim \\pi ^ { \\varepsilon } } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( Y , f _ { \\theta } ( X ) ) H _ { \\varepsilon } ( X , Y ) \\right] . } \\end{array}",
+ "type": "interline_equation",
+ "image_path": "99112b75efa82526f4065b76eee6fd113d983719cb533a205bff100a18f6c5f3.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 11,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 151,
+ 282,
+ 460,
+ 293.0
+ ],
+ "spans": [],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 151,
+ 293.0,
+ 460,
+ 304.0
+ ],
+ "spans": [],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 151,
+ 304.0,
+ 460,
+ 315.0
+ ],
+ "spans": [],
+ "index": 12
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 316,
+ 505,
+ 374
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 316,
+ 506,
+ 330
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 316,
+ 133,
+ 330
+ ],
+ "score": 1.0,
+ "content": "When",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 316,
+ 215,
+ 329
+ ],
+ "score": 0.92,
+ "content": "d ( x , y ) = \\| x - y \\| ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 215,
+ 316,
+ 506,
+ 330
+ ],
+ "score": 1.0,
+ "content": ", the last term in Eq. (12) is simply a weighted sum of squared errors,",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 327,
+ 505,
+ 340
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 327,
+ 314,
+ 340
+ ],
+ "score": 1.0,
+ "content": "with possibly an infinite number of terms whenever",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 314,
+ 330,
+ 321,
+ 339
+ ],
+ "score": 0.83,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 322,
+ 327,
+ 333,
+ 340
+ ],
+ "score": 1.0,
+ "content": "or",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 333,
+ 330,
+ 340,
+ 338
+ ],
+ "score": 0.75,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 341,
+ 327,
+ 505,
+ 340
+ ],
+ "score": 1.0,
+ "content": "are continuous. We propose to minimize",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 351
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 339,
+ 505,
+ 351
+ ],
+ "score": 1.0,
+ "content": "the objective (12) by stochastic gradient descent, which provides the simple Algorithm 2. The OT",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 350,
+ 506,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 350,
+ 446,
+ 363
+ ],
+ "score": 1.0,
+ "content": "problem being symmetric, we can also compute the opposite barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 447,
+ 352,
+ 453,
+ 362
+ ],
+ "score": 0.79,
+ "content": "g",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 454,
+ 350,
+ 506,
+ 363
+ ],
+ "score": 1.0,
+ "content": "w.r.t. a cost",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 107,
+ 358,
+ 399,
+ 376
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 361,
+ 185,
+ 371
+ ],
+ "score": 0.9,
+ "content": "\\bar { d } : \\mathcal { X } \\times \\mathcal { X } \\overset { } { \\to } \\bar { \\mathbb { R } } ^ { + }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 185,
+ 358,
+ 247,
+ 376
+ ],
+ "score": 1.0,
+ "content": "by minimizing",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 247,
+ 361,
+ 394,
+ 374
+ ],
+ "score": 0.91,
+ "content": "\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( g ( \\bar { Y } ) , X ) H _ { \\varepsilon } ( \\bar { X } , Y ) \\right]",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 394,
+ 358,
+ 399,
+ 376
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 15,
+ "bbox_fs": [
+ 105,
+ 316,
+ 506,
+ 376
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 377,
+ 505,
+ 444
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 378,
+ 505,
+ 390
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 378,
+ 209,
+ 390
+ ],
+ "score": 1.0,
+ "content": "However, unless the plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 210,
+ 380,
+ 217,
+ 388
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 378,
+ 505,
+ 390
+ ],
+ "score": 1.0,
+ "content": "is induced by a map, the averaging process results in having the image",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 389,
+ 506,
+ 401
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 389,
+ 215,
+ 401
+ ],
+ "score": 1.0,
+ "content": "of the source measure by",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 216,
+ 390,
+ 223,
+ 399
+ ],
+ "score": 0.78,
+ "content": "\\bar { \\pi }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 224,
+ 389,
+ 425,
+ 401
+ ],
+ "score": 1.0,
+ "content": "only approximately equal to the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 391,
+ 433,
+ 399
+ ],
+ "score": 0.71,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 433,
+ 389,
+ 506,
+ 401
+ ],
+ "score": 1.0,
+ "content": ". Still, when the",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 399,
+ 505,
+ 412
+ ],
+ "score": 1.0,
+ "content": "size of discrete measure is large and the regularization is small, we show in the next paragraph",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 411,
+ 505,
+ 424
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 411,
+ 505,
+ 424
+ ],
+ "score": 1.0,
+ "content": "that 1) the barycentric projection of a regularized OT plan is close to the Monge map between",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 434
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 421,
+ 505,
+ 434
+ ],
+ "score": 1.0,
+ "content": "the underlying continuous measures (Theorem 2) and 2) the image of the source measure by this",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 105,
+ 432,
+ 411,
+ 445
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 432,
+ 346,
+ 445
+ ],
+ "score": 1.0,
+ "content": "barycentric projection should be close to the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 346,
+ 434,
+ 353,
+ 442
+ ],
+ "score": 0.72,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 353,
+ 432,
+ 411,
+ 445
+ ],
+ "score": 1.0,
+ "content": "(Corollary 1).",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ }
+ ],
+ "index": 20.5,
+ "bbox_fs": [
+ 105,
+ 378,
+ 506,
+ 445
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 448,
+ 505,
+ 538
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 449,
+ 505,
+ 462
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 449,
+ 306,
+ 462
+ ],
+ "score": 1.0,
+ "content": "Theoretical guarantees As stated earlier, when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 307,
+ 450,
+ 339,
+ 460
+ ],
+ "score": 0.92,
+ "content": "\\mathcal { X } = \\mathcal { V }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 340,
+ 449,
+ 358,
+ 462
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 359,
+ 449,
+ 438,
+ 461
+ ],
+ "score": 0.93,
+ "content": "c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 439,
+ 449,
+ 505,
+ 462
+ ],
+ "score": 1.0,
+ "content": ", Brenier (1991)",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ },
+ {
+ "bbox": [
+ 105,
+ 461,
+ 505,
+ 473
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 461,
+ 257,
+ 473
+ ],
+ "score": 1.0,
+ "content": "proved that when the source measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 257,
+ 462,
+ 265,
+ 472
+ ],
+ "score": 0.82,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 265,
+ 461,
+ 505,
+ 473
+ ],
+ "score": 1.0,
+ "content": "is continuous, there exists a solution to the Monge problem",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 105,
+ 471,
+ 505,
+ 484
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 471,
+ 505,
+ 484
+ ],
+ "score": 1.0,
+ "content": "(1). This result was generalized to more general cost functions, see (Villani, 2008)[Corollary 9.3] for",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 104,
+ 481,
+ 505,
+ 496
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 481,
+ 222,
+ 496
+ ],
+ "score": 1.0,
+ "content": "details. In that case, the plan",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 222,
+ 484,
+ 230,
+ 493
+ ],
+ "score": 0.75,
+ "content": "\\pi",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 231,
+ 481,
+ 266,
+ 496
+ ],
+ "score": 1.0,
+ "content": "between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 267,
+ 484,
+ 274,
+ 494
+ ],
+ "score": 0.8,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 481,
+ 292,
+ 496
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 292,
+ 484,
+ 299,
+ 492
+ ],
+ "score": 0.75,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 481,
+ 351,
+ 496
+ ],
+ "score": 1.0,
+ "content": "is written as",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 483,
+ 392,
+ 494
+ ],
+ "score": 0.73,
+ "content": "( \\operatorname { i d } , f ) \\# \\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 481,
+ 420,
+ 496
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 420,
+ 483,
+ 427,
+ 494
+ ],
+ "score": 0.86,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 428,
+ 481,
+ 505,
+ 496
+ ],
+ "score": 1.0,
+ "content": "is the Monge map.",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 493,
+ 505,
+ 506
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 493,
+ 249,
+ 506
+ ],
+ "score": 1.0,
+ "content": "Now considering discrete measures",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 249,
+ 495,
+ 262,
+ 505
+ ],
+ "score": 0.87,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 262,
+ 493,
+ 280,
+ 506
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 280,
+ 495,
+ 291,
+ 504
+ ],
+ "score": 0.87,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 292,
+ 493,
+ 367,
+ 506
+ ],
+ "score": 1.0,
+ "content": "which converge to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 367,
+ 496,
+ 375,
+ 505
+ ],
+ "score": 0.85,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 375,
+ 493,
+ 445,
+ 506
+ ],
+ "score": 1.0,
+ "content": "(continuous) and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 445,
+ 495,
+ 452,
+ 503
+ ],
+ "score": 0.78,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 452,
+ 493,
+ 505,
+ 506
+ ],
+ "score": 1.0,
+ "content": "respectively,",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 504,
+ 505,
+ 518
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 504,
+ 247,
+ 518
+ ],
+ "score": 1.0,
+ "content": "we have proved in Theorem 1 that",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 247,
+ 505,
+ 259,
+ 516
+ ],
+ "score": 0.89,
+ "content": "\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 260,
+ 504,
+ 345,
+ 518
+ ],
+ "score": 1.0,
+ "content": "converges weakly to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 346,
+ 505,
+ 408,
+ 516
+ ],
+ "score": 0.92,
+ "content": "\\pi = ( \\operatorname { i d } , f ) \\# \\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 408,
+ 504,
+ 434,
+ 518
+ ],
+ "score": 1.0,
+ "content": "when",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 504,
+ 501,
+ 516
+ ],
+ "score": 0.92,
+ "content": "( n , \\varepsilon ) \\to ( \\infty , 0 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 501,
+ 504,
+ 505,
+ 518
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 104,
+ 514,
+ 506,
+ 530
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 514,
+ 430,
+ 530
+ ],
+ "score": 1.0,
+ "content": "The next theorem, proved in the Appendix, shows that the barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 516,
+ 443,
+ 527
+ ],
+ "score": 0.89,
+ "content": "\\bar { \\pi } _ { n } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 443,
+ 514,
+ 506,
+ 530
+ ],
+ "score": 1.0,
+ "content": "also converges",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 106,
+ 527,
+ 400,
+ 539
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 527,
+ 266,
+ 539
+ ],
+ "score": 1.0,
+ "content": "weakly to the true Monge map between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 267,
+ 528,
+ 274,
+ 538
+ ],
+ "score": 0.81,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 275,
+ 527,
+ 292,
+ 539
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 292,
+ 528,
+ 299,
+ 536
+ ],
+ "score": 0.74,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 299,
+ 527,
+ 400,
+ 539
+ ],
+ "score": 1.0,
+ "content": ", justifying our approach.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 27.5,
+ "bbox_fs": [
+ 104,
+ 449,
+ 506,
+ 539
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 540,
+ 505,
+ 621
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 506,
+ 553
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 539,
+ 178,
+ 553
+ ],
+ "score": 1.0,
+ "content": "Theorem 2. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 179,
+ 542,
+ 187,
+ 552
+ ],
+ "score": 0.73,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 187,
+ 539,
+ 357,
+ 553
+ ],
+ "score": 1.0,
+ "content": "be a continuous probability measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 358,
+ 539,
+ 370,
+ 550
+ ],
+ "score": 0.86,
+ "content": "\\mathbb { R } ^ { d }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 539,
+ 395,
+ 553
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 395,
+ 542,
+ 402,
+ 550
+ ],
+ "score": 0.71,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 402,
+ 539,
+ 506,
+ 553
+ ],
+ "score": 1.0,
+ "content": "an arbitrary probability",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 100,
+ 549,
+ 504,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 100,
+ 556,
+ 124,
+ 582
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 104,
+ 549,
+ 154,
+ 567
+ ],
+ "score": 1.0,
+ "content": "measure on",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 124,
+ 563,
+ 198,
+ 578
+ ],
+ "score": 0.93,
+ "content": "\\textstyle \\nu _ { n } = { \\frac { 1 } { n } } \\sum _ { j = 1 } ^ { n } \\delta _ { y _ { j } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 154,
+ 551,
+ 167,
+ 561
+ ],
+ "score": 0.88,
+ "content": "\\mathbb { R } ^ { d }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 168,
+ 549,
+ 431,
+ 567
+ ],
+ "score": 1.0,
+ "content": "and c a cost function satisfying (Villani, 2008)[Corollary 9.3]. Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 198,
+ 556,
+ 287,
+ 582
+ ],
+ "score": 1.0,
+ "content": "converging weakly to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 287,
+ 566,
+ 295,
+ 576
+ ],
+ "score": 0.57,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 556,
+ 313,
+ 582
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 314,
+ 566,
+ 321,
+ 574
+ ],
+ "score": 0.52,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 556,
+ 491,
+ 582
+ ],
+ "score": 1.0,
+ "content": "n i=1 respectively. Assume that the OT solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 431,
+ 551,
+ 504,
+ 564
+ ],
+ "score": 0.88,
+ "content": "\\begin{array} { r } { \\mu _ { n } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { x _ { i } } } \\end{array}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 492,
+ 566,
+ 504,
+ 575
+ ],
+ "score": 0.81,
+ "content": "\\pi _ { n }",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 106,
+ 577,
+ 506,
+ 590
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 577,
+ 205,
+ 590
+ ],
+ "score": 1.0,
+ "content": "of Problem (2) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 206,
+ 579,
+ 218,
+ 588
+ ],
+ "score": 0.83,
+ "content": "\\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 218,
+ 577,
+ 238,
+ 590
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 239,
+ 578,
+ 250,
+ 588
+ ],
+ "score": 0.86,
+ "content": "\\nu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 250,
+ 577,
+ 321,
+ 590
+ ],
+ "score": 1.0,
+ "content": "is unique for all",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 579,
+ 329,
+ 587
+ ],
+ "score": 0.72,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 329,
+ 577,
+ 352,
+ 590
+ ],
+ "score": 1.0,
+ "content": ". Let",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 352,
+ 577,
+ 370,
+ 588
+ ],
+ "score": 0.83,
+ "content": "\\left( \\varepsilon _ { n } \\right)",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 577,
+ 506,
+ 590
+ ],
+ "score": 1.0,
+ "content": "a sequence of non-negative real",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 587,
+ 506,
+ 600
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 587,
+ 291,
+ 600
+ ],
+ "score": 1.0,
+ "content": "numbers converging sufficiently fast to 0 and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 291,
+ 588,
+ 307,
+ 600
+ ],
+ "score": 0.92,
+ "content": "\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 307,
+ 587,
+ 506,
+ 600
+ ],
+ "score": 1.0,
+ "content": "the barycentric projection w.r.t. the convex cost",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 107,
+ 597,
+ 504,
+ 614
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 107,
+ 599,
+ 133,
+ 609
+ ],
+ "score": 0.89,
+ "content": "d = c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 134,
+ 597,
+ 196,
+ 614
+ ],
+ "score": 1.0,
+ "content": "of the solution",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 197,
+ 599,
+ 212,
+ 611
+ ],
+ "score": 0.9,
+ "content": "\\pi _ { n } ^ { \\varepsilon _ { n } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 213,
+ 597,
+ 497,
+ 614
+ ],
+ "score": 1.0,
+ "content": "of the entropy-regularized OT problem (3). Then, up to extraction of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 498,
+ 600,
+ 504,
+ 609
+ ],
+ "score": 0.54,
+ "content": "a",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 609,
+ 162,
+ 623
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 609,
+ 162,
+ 623
+ ],
+ "score": 1.0,
+ "content": "subsequence,",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ }
+ ],
+ "index": 34.5,
+ "bbox_fs": [
+ 100,
+ 539,
+ 506,
+ 623
+ ]
+ },
+ {
+ "type": "interline_equation",
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "score": 0.9,
+ "content": "( i d , \\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } ) \\# \\mu _ { n } ( i d , f ) \\# \\mu \\ w e a k l y ,",
+ "type": "interline_equation",
+ "image_path": "a23a5eb78f59b0e108b2a2aa02a64a266e228755f22459bb28143378ee9dfde7.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 38,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 228,
+ 620,
+ 383,
+ 633
+ ],
+ "spans": [],
+ "index": 38
+ }
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 634,
+ 375,
+ 646
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 633,
+ 375,
+ 647
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 633,
+ 133,
+ 647
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 133,
+ 635,
+ 140,
+ 646
+ ],
+ "score": 0.84,
+ "content": "f",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 141,
+ 633,
+ 339,
+ 647
+ ],
+ "score": 1.0,
+ "content": "is the solution of the Monge problem (1) between",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 339,
+ 636,
+ 347,
+ 646
+ ],
+ "score": 0.73,
+ "content": "\\mu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 347,
+ 633,
+ 365,
+ 647
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 365,
+ 636,
+ 372,
+ 644
+ ],
+ "score": 0.64,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 372,
+ 633,
+ 375,
+ 647
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "index": 39,
+ "bbox_fs": [
+ 105,
+ 633,
+ 375,
+ 647
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 653,
+ 505,
+ 687
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 653,
+ 505,
+ 667
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 653,
+ 505,
+ 667
+ ],
+ "score": 1.0,
+ "content": "This theorem shows that our estimated barycentric projection is close to an optimal map between the",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 106,
+ 665,
+ 505,
+ 677
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 665,
+ 250,
+ 677
+ ],
+ "score": 1.0,
+ "content": "underlying continuous measures for",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 250,
+ 667,
+ 257,
+ 675
+ ],
+ "score": 0.78,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 257,
+ 665,
+ 288,
+ 677
+ ],
+ "score": 1.0,
+ "content": "big and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 289,
+ 667,
+ 295,
+ 675
+ ],
+ "score": 0.78,
+ "content": "\\varepsilon",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 295,
+ 665,
+ 505,
+ 677
+ ],
+ "score": 1.0,
+ "content": "small. The following corollary confirms the intuition",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 675,
+ 480,
+ 690
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 675,
+ 480,
+ 690
+ ],
+ "score": 1.0,
+ "content": "that the image of the source measure by this map converges to the underlying target measure.",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 41,
+ "bbox_fs": [
+ 105,
+ 653,
+ 505,
+ 690
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 108,
+ 690,
+ 408,
+ 702
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 688,
+ 409,
+ 704
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 688,
+ 313,
+ 704
+ ],
+ "score": 1.0,
+ "content": "Corollary 1. With the same assumptions as above,",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 313,
+ 690,
+ 371,
+ 702
+ ],
+ "score": 0.87,
+ "content": "{ \\bar { \\pi } } _ { n } ^ { \\varepsilon _ { n } } \\# \\mu _ { n } \\to \\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 371,
+ 688,
+ 409,
+ 704
+ ],
+ "score": 1.0,
+ "content": "weakly.",
+ "type": "text"
+ }
+ ],
+ "index": 43
+ }
+ ],
+ "index": 43,
+ "bbox_fs": [
+ 106,
+ 688,
+ 409,
+ 704
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 709,
+ 502,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 707,
+ 504,
+ 723
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 707,
+ 351,
+ 723
+ ],
+ "score": 1.0,
+ "content": "In terms of random variables, the last equation states that if",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 351,
+ 710,
+ 392,
+ 721
+ ],
+ "score": 0.92,
+ "content": "X _ { n } \\sim \\mu _ { n }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 393,
+ 707,
+ 411,
+ 723
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 411,
+ 710,
+ 441,
+ 720
+ ],
+ "score": 0.89,
+ "content": "Y \\sim \\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 442,
+ 707,
+ 465,
+ 723
+ ],
+ "score": 1.0,
+ "content": ", then",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 466,
+ 709,
+ 504,
+ 722
+ ],
+ "score": 0.91,
+ "content": "\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } \\left( X _ { n } \\right)",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 44
+ },
+ {
+ "bbox": [
+ 106,
+ 721,
+ 230,
+ 731
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 721,
+ 217,
+ 731
+ ],
+ "score": 1.0,
+ "content": "converges in distribution to",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 218,
+ 721,
+ 226,
+ 730
+ ],
+ "score": 0.81,
+ "content": "Y",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 227,
+ 721,
+ 230,
+ 731
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 45
+ }
+ ],
+ "index": 44.5,
+ "bbox_fs": [
+ 104,
+ 707,
+ 504,
+ 731
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "score": 0.97,
+ "type": "image",
+ "image_path": "108bd959a523d171261a4465782b24928b96161ec71032e31d7571b8fde56237.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 116.33333333333334
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 112,
+ 116.33333333333334,
+ 497,
+ 151.66666666666669
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 112,
+ 151.66666666666669,
+ 497,
+ 187.00000000000003
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 240
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 207
+ ],
+ "score": 1.0,
+ "content": "Figure 2: Convergence plots of the the Stochastic Dual Algorithm 1 against a stochastic semi-dual",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 219
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 219
+ ],
+ "score": 1.0,
+ "content": "implementation (adapted from (Genevay et al., 2016): we use SGD instead of SAG), for several",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 218,
+ 504,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 218,
+ 312,
+ 230
+ ],
+ "score": 1.0,
+ "content": "entropy-regularization values. Learning rates are",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 218,
+ 365,
+ 230
+ ],
+ "score": 0.85,
+ "content": "\\{ 5 . , 2 0 . , 2 0 . \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 366,
+ 218,
+ 434,
+ 230
+ ],
+ "score": 1.0,
+ "content": "and batch sizes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 218,
+ 504,
+ 230
+ ],
+ "score": 0.79,
+ "content": "\\{ 1 0 2 4 , 5 0 0 , 1 0 0 \\}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 229,
+ 396,
+ 241
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 229,
+ 396,
+ 241
+ ],
+ "score": 1.0,
+ "content": "respectively and are taken the same for the dual and semi-dual methods.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4.5
+ }
+ ],
+ "index": 2.75
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 263,
+ 505,
+ 307
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 262,
+ 505,
+ 275
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 262,
+ 505,
+ 275
+ ],
+ "score": 1.0,
+ "content": "These theoretical results show that our estimated Monge map can thus be used to perform domain",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 286
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 286
+ ],
+ "score": 1.0,
+ "content": "adaptation by mapping a source dataset to a target dataset, as well as perform generative modeling",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 285,
+ 506,
+ 298
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 285,
+ 506,
+ 298
+ ],
+ "score": 1.0,
+ "content": "by mapping a continuous measure to a target discrete dataset. We demontrate this in the following",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 296,
+ 140,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 296,
+ 140,
+ 307
+ ],
+ "score": 1.0,
+ "content": "section.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 8.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 325,
+ 267,
+ 337
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 323,
+ 268,
+ 339
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 323,
+ 268,
+ 339
+ ],
+ "score": 1.0,
+ "content": "5 NUMERICAL EXPERIMENTS",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 350,
+ 316,
+ 361
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 350,
+ 318,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 350,
+ 318,
+ 363
+ ],
+ "score": 1.0,
+ "content": "5.1 DUAL VS SEMI-DUAL SPEED COMPARISONS",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 371,
+ 505,
+ 503
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 384
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 384
+ ],
+ "score": 1.0,
+ "content": "We start by evaluating the training time of our dual stochastic algorithm 1 against a stochastic semi-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 383,
+ 505,
+ 395
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 383,
+ 505,
+ 395
+ ],
+ "score": 1.0,
+ "content": "dual approach similar to (Genevay et al., 2016). In the semi-dual approach, one of the dual variable",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 104,
+ 392,
+ 505,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 392,
+ 432,
+ 407
+ ],
+ "score": 1.0,
+ "content": "is eliminated and is computed in close form. However, this computation has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 432,
+ 393,
+ 455,
+ 405
+ ],
+ "score": 0.91,
+ "content": "{ \\mathcal { O } } ( n )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 392,
+ 505,
+ 407
+ ],
+ "score": 1.0,
+ "content": "complexity",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 405,
+ 505,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 405,
+ 134,
+ 417
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 406,
+ 142,
+ 414
+ ],
+ "score": 0.7,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 142,
+ 405,
+ 277,
+ 417
+ ],
+ "score": 1.0,
+ "content": "is the size of the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 277,
+ 406,
+ 284,
+ 414
+ ],
+ "score": 0.65,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 284,
+ 405,
+ 505,
+ 417
+ ],
+ "score": 1.0,
+ "content": ". We compute the regularized OT objective with both",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 415,
+ 506,
+ 428
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 415,
+ 506,
+ 428
+ ],
+ "score": 1.0,
+ "content": "methods on a spectral transfer problem, which is related to the color transfer problem (Reinhard",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "score": 1.0,
+ "content": "et al., 2001; Pitie et al., 2007), but where images are multispectral, ´ i.e. they share a finer sampling of",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 437,
+ 506,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 437,
+ 245,
+ 450
+ ],
+ "score": 1.0,
+ "content": "the light wavelength. We take two",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 246,
+ 437,
+ 290,
+ 448
+ ],
+ "score": 0.91,
+ "content": "5 0 0 \\times 5 0 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 437,
+ 506,
+ 450
+ ],
+ "score": 1.0,
+ "content": "images from the CAVE dataset (Yasuma et al., 2010)",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "score": 1.0,
+ "content": "that have 31 spectral bands. As such, the optimal transport problem is computed on two empirical",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 457,
+ 506,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 457,
+ 247,
+ 472
+ ],
+ "score": 1.0,
+ "content": "distributions of 250000 samples in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 247,
+ 459,
+ 264,
+ 469
+ ],
+ "score": 0.89,
+ "content": "\\mathbb { R } ^ { 3 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 457,
+ 496,
+ 472
+ ],
+ "score": 1.0,
+ "content": "on which we consider the squared Euclidean ground cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 461,
+ 502,
+ 469
+ ],
+ "score": 0.46,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 457,
+ 506,
+ 472
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 470,
+ 505,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 470,
+ 505,
+ 482
+ ],
+ "score": 1.0,
+ "content": "The timing evolution of train losses are reported in Figure 2 for three different regularization values",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 481,
+ 506,
+ 494
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 481,
+ 191,
+ 493
+ ],
+ "score": 0.9,
+ "content": "\\varepsilon = \\{ 0 . 0 \\bar { 2 } 5 , 0 . 1 , 1 . \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 191,
+ 481,
+ 506,
+ 494
+ ],
+ "score": 1.0,
+ "content": ". In the three cases, one can observe that convergence of our proposed dual",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 492,
+ 208,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 492,
+ 208,
+ 504
+ ],
+ "score": 1.0,
+ "content": "algorithm is much faster.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 18.5
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 518,
+ 286,
+ 529
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 518,
+ 288,
+ 531
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 518,
+ 288,
+ 531
+ ],
+ "score": 1.0,
+ "content": "5.2 LARGE SCALE DOMAIN ADAPTATION",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 539,
+ 505,
+ 649
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 538,
+ 506,
+ 552
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 538,
+ 506,
+ 552
+ ],
+ "score": 1.0,
+ "content": "We apply here our computation framework on an unsupervised domain adaptation (DA) task, for",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 551,
+ 505,
+ 562
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 551,
+ 505,
+ 562
+ ],
+ "score": 1.0,
+ "content": "which optimal transport has shown to perform well on small scale datasets (Courty et al., 2017b;",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 561,
+ 505,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 561,
+ 505,
+ 573
+ ],
+ "score": 1.0,
+ "content": "Perrot et al., 2016; Courty et al., 2014). This restriction is mainly due to the fact that those works",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 572,
+ 505,
+ 585
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 572,
+ 505,
+ 585
+ ],
+ "score": 1.0,
+ "content": "only consider the primal formulation of the OT problem. Our goal here is not to compete with",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 106,
+ 583,
+ 505,
+ 595
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 583,
+ 505,
+ 595
+ ],
+ "score": 1.0,
+ "content": "the state-of-the-art methods in domain adaptation but to assess that our formulation allows to scale",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "score": 1.0,
+ "content": "optimal transport based domain adaptation (OTDA) to large datasets. OTDA is illustrated in Fig. 3",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 104,
+ 603,
+ 506,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 603,
+ 506,
+ 619
+ ],
+ "score": 1.0,
+ "content": "and follows two steps: 1) learn an optimal map between the source and target distribution, 2) map",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 617,
+ 505,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 505,
+ 628
+ ],
+ "score": 1.0,
+ "content": "the source samples and train a classifier on them in the target domain. Our formulation also allows",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 261,
+ 639
+ ],
+ "score": 1.0,
+ "content": "to use any differentiable ground cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 262,
+ 629,
+ 268,
+ 637
+ ],
+ "score": 0.62,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 627,
+ 505,
+ 639
+ ],
+ "score": 1.0,
+ "content": "while (Courty et al., 2017b) was limited to the squared",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 637,
+ 186,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 186,
+ 650
+ ],
+ "score": 1.0,
+ "content": "Euclidean distance.",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 30.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 655,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 505,
+ 667
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 505,
+ 667
+ ],
+ "score": 1.0,
+ "content": "Datasets We consider the three cross-domain digit image datasets MNIST (Lecun et al., 1998),",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 665,
+ 505,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 665,
+ 505,
+ 678
+ ],
+ "score": 1.0,
+ "content": "USPS, and SVHN (Netzer et al., 2011), which have 10 classes each. For the adaptation between",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 676,
+ 505,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 676,
+ 505,
+ 689
+ ],
+ "score": 1.0,
+ "content": "MNIST and USPS, we use 60000 samples in the MNIST domain and 9298 samples in USPS domain.",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 380,
+ 700
+ ],
+ "score": 1.0,
+ "content": "MNIST images are resized to the same resolution as USPS ones",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 380,
+ 688,
+ 418,
+ 699
+ ],
+ "score": 0.86,
+ "content": "( 1 6 \\times 1 6 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 418,
+ 687,
+ 505,
+ 700
+ ],
+ "score": 1.0,
+ "content": "). For the adaptation",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 699,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 699,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "between SVHN and MNIST, we use 73212 samples in the SVHN domain and 60000 samples in the",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 503,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 465,
+ 722
+ ],
+ "score": 1.0,
+ "content": "MNIST domain. MNIST images are zero-padded to reach the same resolution as SVHN",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 465,
+ 710,
+ 503,
+ 721
+ ],
+ "score": 0.85,
+ "content": "( 3 2 \\times 3 2 )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 734
+ ],
+ "score": 1.0,
+ "content": "and extended to three channels to match SVHN image sizes. The labels in the target domain are",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ }
+ ],
+ "index": 39
+ }
+ ],
+ "page_idx": 7,
+ "page_size": [
+ 612,
+ 792
+ ],
+ "discarded_blocks": [
+ {
+ "type": "discarded",
+ "bbox": [
+ 107,
+ 27,
+ 293,
+ 37
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 26,
+ 294,
+ 38
+ ],
+ "score": 1.0,
+ "content": "Published as a conference paper at ICLR 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": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 187
+ ],
+ "score": 0.97,
+ "type": "image",
+ "image_path": "108bd959a523d171261a4465782b24928b96161ec71032e31d7571b8fde56237.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 112,
+ 81,
+ 497,
+ 116.33333333333334
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 112,
+ 116.33333333333334,
+ 497,
+ 151.66666666666669
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 112,
+ 151.66666666666669,
+ 497,
+ 187.00000000000003
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 240
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 196,
+ 505,
+ 207
+ ],
+ "score": 1.0,
+ "content": "Figure 2: Convergence plots of the the Stochastic Dual Algorithm 1 against a stochastic semi-dual",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 219
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 207,
+ 505,
+ 219
+ ],
+ "score": 1.0,
+ "content": "implementation (adapted from (Genevay et al., 2016): we use SGD instead of SAG), for several",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 218,
+ 504,
+ 230
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 218,
+ 312,
+ 230
+ ],
+ "score": 1.0,
+ "content": "entropy-regularization values. Learning rates are",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 312,
+ 218,
+ 365,
+ 230
+ ],
+ "score": 0.85,
+ "content": "\\{ 5 . , 2 0 . , 2 0 . \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 366,
+ 218,
+ 434,
+ 230
+ ],
+ "score": 1.0,
+ "content": "and batch sizes",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 434,
+ 218,
+ 504,
+ 230
+ ],
+ "score": 0.79,
+ "content": "\\{ 1 0 2 4 , 5 0 0 , 1 0 0 \\}",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 105,
+ 229,
+ 396,
+ 241
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 229,
+ 396,
+ 241
+ ],
+ "score": 1.0,
+ "content": "respectively and are taken the same for the dual and semi-dual methods.",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ }
+ ],
+ "index": 4.5
+ }
+ ],
+ "index": 2.75
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 263,
+ 505,
+ 307
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 262,
+ 505,
+ 275
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 262,
+ 505,
+ 275
+ ],
+ "score": 1.0,
+ "content": "These theoretical results show that our estimated Monge map can thus be used to perform domain",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 286
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 273,
+ 505,
+ 286
+ ],
+ "score": 1.0,
+ "content": "adaptation by mapping a source dataset to a target dataset, as well as perform generative modeling",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 105,
+ 285,
+ 506,
+ 298
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 285,
+ 506,
+ 298
+ ],
+ "score": 1.0,
+ "content": "by mapping a continuous measure to a target discrete dataset. We demontrate this in the following",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 105,
+ 296,
+ 140,
+ 307
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 296,
+ 140,
+ 307
+ ],
+ "score": 1.0,
+ "content": "section.",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 8.5,
+ "bbox_fs": [
+ 105,
+ 262,
+ 506,
+ 307
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 325,
+ 267,
+ 337
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 323,
+ 268,
+ 339
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 323,
+ 268,
+ 339
+ ],
+ "score": 1.0,
+ "content": "5 NUMERICAL EXPERIMENTS",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ }
+ ],
+ "index": 11
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 350,
+ 316,
+ 361
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 350,
+ 318,
+ 363
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 350,
+ 318,
+ 363
+ ],
+ "score": 1.0,
+ "content": "5.1 DUAL VS SEMI-DUAL SPEED COMPARISONS",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ }
+ ],
+ "index": 12
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 371,
+ 505,
+ 503
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 384
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 370,
+ 505,
+ 384
+ ],
+ "score": 1.0,
+ "content": "We start by evaluating the training time of our dual stochastic algorithm 1 against a stochastic semi-",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 383,
+ 505,
+ 395
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 383,
+ 505,
+ 395
+ ],
+ "score": 1.0,
+ "content": "dual approach similar to (Genevay et al., 2016). In the semi-dual approach, one of the dual variable",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 104,
+ 392,
+ 505,
+ 407
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 392,
+ 432,
+ 407
+ ],
+ "score": 1.0,
+ "content": "is eliminated and is computed in close form. However, this computation has",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 432,
+ 393,
+ 455,
+ 405
+ ],
+ "score": 0.91,
+ "content": "{ \\mathcal { O } } ( n )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 455,
+ 392,
+ 505,
+ 407
+ ],
+ "score": 1.0,
+ "content": "complexity",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 405,
+ 505,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 405,
+ 134,
+ 417
+ ],
+ "score": 1.0,
+ "content": "where",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 134,
+ 406,
+ 142,
+ 414
+ ],
+ "score": 0.7,
+ "content": "n",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 142,
+ 405,
+ 277,
+ 417
+ ],
+ "score": 1.0,
+ "content": "is the size of the target measure",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 277,
+ 406,
+ 284,
+ 414
+ ],
+ "score": 0.65,
+ "content": "\\nu",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 284,
+ 405,
+ 505,
+ 417
+ ],
+ "score": 1.0,
+ "content": ". We compute the regularized OT objective with both",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 415,
+ 506,
+ 428
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 415,
+ 506,
+ 428
+ ],
+ "score": 1.0,
+ "content": "methods on a spectral transfer problem, which is related to the color transfer problem (Reinhard",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 426,
+ 506,
+ 439
+ ],
+ "score": 1.0,
+ "content": "et al., 2001; Pitie et al., 2007), but where images are multispectral, ´ i.e. they share a finer sampling of",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 437,
+ 506,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 437,
+ 245,
+ 450
+ ],
+ "score": 1.0,
+ "content": "the light wavelength. We take two",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 246,
+ 437,
+ 290,
+ 448
+ ],
+ "score": 0.91,
+ "content": "5 0 0 \\times 5 0 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 290,
+ 437,
+ 506,
+ 450
+ ],
+ "score": 1.0,
+ "content": "images from the CAVE dataset (Yasuma et al., 2010)",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 448,
+ 505,
+ 461
+ ],
+ "score": 1.0,
+ "content": "that have 31 spectral bands. As such, the optimal transport problem is computed on two empirical",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 457,
+ 506,
+ 472
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 457,
+ 247,
+ 472
+ ],
+ "score": 1.0,
+ "content": "distributions of 250000 samples in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 247,
+ 459,
+ 264,
+ 469
+ ],
+ "score": 0.89,
+ "content": "\\mathbb { R } ^ { 3 1 }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 264,
+ 457,
+ 496,
+ 472
+ ],
+ "score": 1.0,
+ "content": "on which we consider the squared Euclidean ground cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 496,
+ 461,
+ 502,
+ 469
+ ],
+ "score": 0.46,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 502,
+ 457,
+ 506,
+ 472
+ ],
+ "score": 1.0,
+ "content": ".",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 470,
+ 505,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 470,
+ 505,
+ 482
+ ],
+ "score": 1.0,
+ "content": "The timing evolution of train losses are reported in Figure 2 for three different regularization values",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 106,
+ 481,
+ 506,
+ 494
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 481,
+ 191,
+ 493
+ ],
+ "score": 0.9,
+ "content": "\\varepsilon = \\{ 0 . 0 \\bar { 2 } 5 , 0 . 1 , 1 . \\}",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 191,
+ 481,
+ 506,
+ 494
+ ],
+ "score": 1.0,
+ "content": ". In the three cases, one can observe that convergence of our proposed dual",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 492,
+ 208,
+ 504
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 492,
+ 208,
+ 504
+ ],
+ "score": 1.0,
+ "content": "algorithm is much faster.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 18.5,
+ "bbox_fs": [
+ 104,
+ 370,
+ 506,
+ 504
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 108,
+ 518,
+ 286,
+ 529
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 518,
+ 288,
+ 531
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 518,
+ 288,
+ 531
+ ],
+ "score": 1.0,
+ "content": "5.2 LARGE SCALE DOMAIN ADAPTATION",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ }
+ ],
+ "index": 25
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 539,
+ 505,
+ 649
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 538,
+ 506,
+ 552
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 538,
+ 506,
+ 552
+ ],
+ "score": 1.0,
+ "content": "We apply here our computation framework on an unsupervised domain adaptation (DA) task, for",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 551,
+ 505,
+ 562
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 551,
+ 505,
+ 562
+ ],
+ "score": 1.0,
+ "content": "which optimal transport has shown to perform well on small scale datasets (Courty et al., 2017b;",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 561,
+ 505,
+ 573
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 561,
+ 505,
+ 573
+ ],
+ "score": 1.0,
+ "content": "Perrot et al., 2016; Courty et al., 2014). This restriction is mainly due to the fact that those works",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 572,
+ 505,
+ 585
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 572,
+ 505,
+ 585
+ ],
+ "score": 1.0,
+ "content": "only consider the primal formulation of the OT problem. Our goal here is not to compete with",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 106,
+ 583,
+ 505,
+ 595
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 583,
+ 505,
+ 595
+ ],
+ "score": 1.0,
+ "content": "the state-of-the-art methods in domain adaptation but to assess that our formulation allows to scale",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 594,
+ 506,
+ 606
+ ],
+ "score": 1.0,
+ "content": "optimal transport based domain adaptation (OTDA) to large datasets. OTDA is illustrated in Fig. 3",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ },
+ {
+ "bbox": [
+ 104,
+ 603,
+ 506,
+ 619
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 603,
+ 506,
+ 619
+ ],
+ "score": 1.0,
+ "content": "and follows two steps: 1) learn an optimal map between the source and target distribution, 2) map",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 105,
+ 617,
+ 505,
+ 628
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 617,
+ 505,
+ 628
+ ],
+ "score": 1.0,
+ "content": "the source samples and train a classifier on them in the target domain. Our formulation also allows",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ },
+ {
+ "bbox": [
+ 105,
+ 627,
+ 505,
+ 639
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 627,
+ 261,
+ 639
+ ],
+ "score": 1.0,
+ "content": "to use any differentiable ground cost",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 262,
+ 629,
+ 268,
+ 637
+ ],
+ "score": 0.62,
+ "content": "c",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 268,
+ 627,
+ 505,
+ 639
+ ],
+ "score": 1.0,
+ "content": "while (Courty et al., 2017b) was limited to the squared",
+ "type": "text"
+ }
+ ],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 105,
+ 637,
+ 186,
+ 650
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 637,
+ 186,
+ 650
+ ],
+ "score": 1.0,
+ "content": "Euclidean distance.",
+ "type": "text"
+ }
+ ],
+ "index": 35
+ }
+ ],
+ "index": 30.5,
+ "bbox_fs": [
+ 104,
+ 538,
+ 506,
+ 650
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 655,
+ 505,
+ 732
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 505,
+ 667
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 655,
+ 505,
+ 667
+ ],
+ "score": 1.0,
+ "content": "Datasets We consider the three cross-domain digit image datasets MNIST (Lecun et al., 1998),",
+ "type": "text"
+ }
+ ],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 105,
+ 665,
+ 505,
+ 678
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 665,
+ 505,
+ 678
+ ],
+ "score": 1.0,
+ "content": "USPS, and SVHN (Netzer et al., 2011), which have 10 classes each. For the adaptation between",
+ "type": "text"
+ }
+ ],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 105,
+ 676,
+ 505,
+ 689
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 676,
+ 505,
+ 689
+ ],
+ "score": 1.0,
+ "content": "MNIST and USPS, we use 60000 samples in the MNIST domain and 9298 samples in USPS domain.",
+ "type": "text"
+ }
+ ],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 105,
+ 687,
+ 505,
+ 700
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 687,
+ 380,
+ 700
+ ],
+ "score": 1.0,
+ "content": "MNIST images are resized to the same resolution as USPS ones",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 380,
+ 688,
+ 418,
+ 699
+ ],
+ "score": 0.86,
+ "content": "( 1 6 \\times 1 6 )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 418,
+ 687,
+ 505,
+ 700
+ ],
+ "score": 1.0,
+ "content": "). For the adaptation",
+ "type": "text"
+ }
+ ],
+ "index": 39
+ },
+ {
+ "bbox": [
+ 105,
+ 699,
+ 505,
+ 711
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 699,
+ 505,
+ 711
+ ],
+ "score": 1.0,
+ "content": "between SVHN and MNIST, we use 73212 samples in the SVHN domain and 60000 samples in the",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 709,
+ 503,
+ 722
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 709,
+ 465,
+ 722
+ ],
+ "score": 1.0,
+ "content": "MNIST domain. MNIST images are zero-padded to reach the same resolution as SVHN",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 465,
+ 710,
+ 503,
+ 721
+ ],
+ "score": 0.85,
+ "content": "( 3 2 \\times 3 2 )",
+ "type": "inline_equation"
+ }
+ ],
+ "index": 41
+ },
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 734
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 720,
+ 505,
+ 734
+ ],
+ "score": 1.0,
+ "content": "and extended to three channels to match SVHN image sizes. The labels in the target domain are",
+ "type": "text"
+ }
+ ],
+ "index": 42
+ },
+ {
+ "bbox": [
+ 106,
+ 396,
+ 504,
+ 409
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 396,
+ 504,
+ 409
+ ],
+ "score": 1.0,
+ "content": "withheld during the adaptation. In the experiment, we consider the adaptation in three directions:",
+ "type": "text",
+ "cross_page": true
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 106,
+ 407,
+ 347,
+ 418
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 407,
+ 114,
+ 418
+ ],
+ "score": 1.0,
+ "content": "M",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 114,
+ 407,
+ 177,
+ 417
+ ],
+ "score": 0.27,
+ "content": "\\mathrm { I N I S T } \\to \\mathrm { U S P S }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 177,
+ 407,
+ 181,
+ 418
+ ],
+ "score": 1.0,
+ "content": ",",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 181,
+ 407,
+ 252,
+ 418
+ ],
+ "score": 0.49,
+ "content": "\\mathrm { U S P S } \\to \\mathrm { M N I S T }",
+ "type": "inline_equation",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 252,
+ 407,
+ 272,
+ 418
+ ],
+ "score": 1.0,
+ "content": ", and",
+ "type": "text",
+ "cross_page": true
+ },
+ {
+ "bbox": [
+ 272,
+ 407,
+ 347,
+ 417
+ ],
+ "score": 0.59,
+ "content": "\\mathrm { S V H N } \\to \\mathrm { M N I S T } .",
+ "type": "inline_equation",
+ "cross_page": true
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 39,
+ "bbox_fs": [
+ 105,
+ 655,
+ 505,
+ 734
+ ]
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "image",
+ "bbox": [
+ 125,
+ 77,
+ 486,
+ 183
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 125,
+ 77,
+ 486,
+ 183
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 125,
+ 77,
+ 486,
+ 183
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 125,
+ 77,
+ 486,
+ 183
+ ],
+ "score": 0.969,
+ "type": "image",
+ "image_path": "68a2bc0b8ddaf4a608cd6baab93562a2a306655bd64ab896bcdaaba30e300dd6.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 1,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 125,
+ 77,
+ 486,
+ 112.33333333333334
+ ],
+ "spans": [],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 125,
+ 112.33333333333334,
+ 486,
+ 147.66666666666669
+ ],
+ "spans": [],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 125,
+ 147.66666666666669,
+ 486,
+ 183.00000000000003
+ ],
+ "spans": [],
+ "index": 2
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 107,
+ 191,
+ 504,
+ 226
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 191,
+ 505,
+ 205
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 191,
+ 505,
+ 205
+ ],
+ "score": 1.0,
+ "content": "Figure 3: Illustration of the OT Domain Adaptation method adapted from (Courty et al., 2017b).",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 203,
+ 505,
+ 215
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 203,
+ 429,
+ 215
+ ],
+ "score": 1.0,
+ "content": "Source samples are mapped to the target set through the barycentric projection",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 429,
+ 203,
+ 441,
+ 213
+ ],
+ "score": 0.86,
+ "content": "\\bar { \\pi } ^ { \\varepsilon }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 441,
+ 203,
+ 505,
+ 215
+ ],
+ "score": 1.0,
+ "content": ". A classifier is",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
+ 106,
+ 213,
+ 285,
+ 227
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 213,
+ 285,
+ 227
+ ],
+ "score": 1.0,
+ "content": "then learned on the mapped source samples.",
+ "type": "text"
+ }
+ ],
+ "index": 5
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 126,
+ 284,
+ 484,
+ 371
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 108,
+ 240,
+ 503,
+ 275
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 241,
+ 505,
+ 253
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 241,
+ 227,
+ 253
+ ],
+ "score": 1.0,
+ "content": "Table 1: Results (accuracy in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 227,
+ 241,
+ 236,
+ 251
+ ],
+ "score": 0.75,
+ "content": "\\%",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 241,
+ 505,
+ 253
+ ],
+ "score": 1.0,
+ "content": ") on domain adaptation among MNIST, USPS and SVHN datasets",
+ "type": "text"
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 106,
+ 252,
+ 505,
+ 264
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 252,
+ 160,
+ 264
+ ],
+ "score": 1.0,
+ "content": "with entropy",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 161,
+ 252,
+ 179,
+ 263
+ ],
+ "score": 0.79,
+ "content": "( R _ { e } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 179,
+ 252,
+ 212,
+ 264
+ ],
+ "score": 1.0,
+ "content": "and L2",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 212,
+ 252,
+ 237,
+ 263
+ ],
+ "score": 0.84,
+ "content": "( R _ { L ^ { 2 } } )",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 237,
+ 252,
+ 505,
+ 264
+ ],
+ "score": 1.0,
+ "content": "regularizations. Source only refers to 1-NN classification between",
+ "type": "text"
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 106,
+ 263,
+ 290,
+ 275
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 263,
+ 290,
+ 275
+ ],
+ "score": 1.0,
+ "content": "source and target samples without adaptation.",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ }
+ ],
+ "index": 7
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 126,
+ 284,
+ 484,
+ 371
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 126,
+ 284,
+ 484,
+ 371
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 126,
+ 284,
+ 484,
+ 371
+ ],
+ "score": 0.984,
+ "html": " | 1: Inputs: input measures μ,V; cost function c; batch size p; learning rate y. Discrete case: μ = ∑i aiδxi and u is a finite vector: u(xi) @f ui (similarly for v and v) |
| 2: 3: | Continuous case: μ is a continuous measure and u is a neural network (similarly for V and u) |
| Vindicates the gradient w.r.t. the parameters |
| 4: | while not converged do |
| 5: | sample a batch (x1,..,xp) from μ |
| 6: 7: | sample a batch (y1,.., yp) from v update u ←u+γ∑ij Vu(xi)+OuFe(u(xi),v(yj))Vu(xi) |
| 8: | updateu ← U+γ∑ij Vv(yj)+OUFe(u(xi),v(yj))Vv(yj) 9: ：end while |
"
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 306,
+ 270,
+ 1388,
+ 270,
+ 1388,
+ 600,
+ 306,
+ 600
+ ],
+ "score": 0.48
+ },
+ {
+ "category_id": 6,
+ "poly": [
+ 302,
+ 229,
+ 754,
+ 229,
+ 754,
+ 262,
+ 302,
+ 262
+ ],
+ "score": 0.122
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 646,
+ 987,
+ 735,
+ 987,
+ 735,
+ 1021,
+ 646,
+ 1021
+ ],
+ "score": 0.93,
+ "latex": "\\mathcal { O } ( 1 / k )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1094,
+ 1967,
+ 1394,
+ 1967,
+ 1394,
+ 2006,
+ 1094,
+ 2006
+ ],
+ "score": 0.93,
+ "latex": "\\textstyle \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } \\in P ( \\mathcal { Y } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 738,
+ 1967,
+ 1042,
+ 1967,
+ 1042,
+ 2004,
+ 738,
+ 2004
+ ],
+ "score": 0.93,
+ "latex": "\\begin{array} { r } { \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } } \\in P ( \\mathcal { X } ) } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 548,
+ 1175,
+ 659,
+ 1175,
+ 659,
+ 1213,
+ 548,
+ 1213
+ ],
+ "score": 0.93,
+ "latex": "O ( 1 / \\sqrt { k } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 757,
+ 1019,
+ 869,
+ 1019,
+ 869,
+ 1057,
+ 757,
+ 1057
+ ],
+ "score": 0.93,
+ "latex": "\\mathcal { O } ( 1 / \\sqrt { k } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 718,
+ 1210,
+ 807,
+ 1210,
+ 807,
+ 1243,
+ 718,
+ 1243
+ ],
+ "score": 0.93,
+ "latex": "\\mathcal { O } ( 1 / k )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1185,
+ 1348,
+ 1296,
+ 1348,
+ 1296,
+ 1386,
+ 1185,
+ 1386
+ ],
+ "score": 0.93,
+ "latex": "\\mathcal { O } ( 1 / \\sqrt { k } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1013,
+ 1520,
+ 1085,
+ 1520,
+ 1085,
+ 1554,
+ 1013,
+ 1554
+ ],
+ "score": 0.92,
+ "latex": "\\mathcal { O } ( p ^ { 2 } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 482,
+ 1489,
+ 556,
+ 1489,
+ 556,
+ 1524,
+ 482,
+ 1524
+ ],
+ "score": 0.92,
+ "latex": "\\mathcal { O } ( k ^ { 2 } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 509,
+ 309,
+ 665,
+ 309,
+ 665,
+ 346,
+ 509,
+ 346
+ ],
+ "score": 0.92,
+ "latex": "\\begin{array} { r } { \\mu = \\sum _ { i } a _ { i } \\delta _ { x _ { i } } } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 933,
+ 305,
+ 1064,
+ 305,
+ 1064,
+ 344,
+ 933,
+ 344
+ ],
+ "score": 0.92,
+ "latex": "u ( x _ { i } ) \\stackrel { \\mathrm { d e f . } } { = } u _ { i }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 540,
+ 1114,
+ 848,
+ 1114,
+ 848,
+ 1148,
+ 540,
+ 1148
+ ],
+ "score": 0.92,
+ "latex": "\\mathbb { E } _ { X \\sim \\mu } \\left[ u ( X ) + G _ { \\varepsilon } ( u ( X ) ) \\right]"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1289,
+ 1114,
+ 1353,
+ 1114,
+ 1353,
+ 1147,
+ 1289,
+ 1147
+ ],
+ "score": 0.91,
+ "latex": "{ \\mathcal { O } } ( n )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1284,
+ 1053,
+ 1356,
+ 1053,
+ 1356,
+ 1086,
+ 1284,
+ 1086
+ ],
+ "score": 0.91,
+ "latex": "\\mathcal { O } ( p ^ { 2 } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 787,
+ 1617,
+ 859,
+ 1617,
+ 859,
+ 1650,
+ 787,
+ 1650
+ ],
+ "score": 0.91,
+ "latex": "\\mathcal { O } ( p ^ { 2 } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 453,
+ 495,
+ 1060,
+ 495,
+ 1060,
+ 528,
+ 453,
+ 528
+ ],
+ "score": 0.9,
+ "latex": "\\begin{array} { r } { u u + \\gamma \\sum _ { i j } \\nabla u ( x _ { i } ) + \\partial _ { u } F _ { \\varepsilon } ( u ( x _ { i } ) , v ( y _ { j } ) ) \\nabla u ( x _ { i } ) } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 728,
+ 1582,
+ 1045,
+ 1582,
+ 1045,
+ 1617,
+ 728,
+ 1617
+ ],
+ "score": 0.89,
+ "latex": "\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ u ( X ) - u ( Y ) \\right]"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 711,
+ 1147,
+ 747,
+ 1147,
+ 747,
+ 1175,
+ 711,
+ 1175
+ ],
+ "score": 0.87,
+ "latex": "G _ { \\varepsilon }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 546,
+ 466,
+ 686,
+ 466,
+ 686,
+ 496,
+ 546,
+ 496
+ ],
+ "score": 0.86,
+ "latex": "( y _ { 1 } , \\cdots , y _ { p } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1288,
+ 1022,
+ 1348,
+ 1022,
+ 1348,
+ 1054,
+ 1288,
+ 1054
+ ],
+ "score": 0.84,
+ "latex": "\\mathcal { O } ( 1 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 455,
+ 529,
+ 1057,
+ 529,
+ 1057,
+ 567,
+ 455,
+ 567
+ ],
+ "score": 0.84,
+ "latex": "\\begin{array} { r } { \\boldsymbol { v } \\boldsymbol { v } + \\gamma \\sum _ { i j } \\nabla \\boldsymbol { v } ( y _ { j } ) + \\partial _ { \\boldsymbol { v } } F _ { \\varepsilon } ( \\boldsymbol { u } ( x _ { i } ) , \\boldsymbol { v } ( y _ { j } ) ) \\nabla \\boldsymbol { v } ( y _ { j } ) } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 545,
+ 433,
+ 690,
+ 433,
+ 690,
+ 466,
+ 545,
+ 466
+ ],
+ "score": 0.83,
+ "latex": "( x _ { 1 } , \\cdots , x _ { p } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 479,
+ 1088,
+ 498,
+ 1088,
+ 498,
+ 1115,
+ 479,
+ 1115
+ ],
+ "score": 0.8,
+ "latex": "p"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 827,
+ 988,
+ 847,
+ 988,
+ 847,
+ 1015,
+ 827,
+ 1015
+ ],
+ "score": 0.8,
+ "latex": "k"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 542,
+ 349,
+ 563,
+ 349,
+ 563,
+ 374,
+ 542,
+ 374
+ ],
+ "score": 0.8,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 716,
+ 315,
+ 736,
+ 315,
+ 736,
+ 338,
+ 716,
+ 338
+ ],
+ "score": 0.78,
+ "latex": "u"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1280,
+ 347,
+ 1299,
+ 347,
+ 1299,
+ 369,
+ 1280,
+ 369
+ ],
+ "score": 0.78,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1142,
+ 280,
+ 1161,
+ 280,
+ 1161,
+ 305,
+ 1142,
+ 305
+ ],
+ "score": 0.77,
+ "latex": "\\gamma"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 967,
+ 279,
+ 985,
+ 279,
+ 985,
+ 304,
+ 967,
+ 304
+ ],
+ "score": 0.76,
+ "latex": "p"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 886,
+ 348,
+ 905,
+ 348,
+ 905,
+ 369,
+ 886,
+ 369
+ ],
+ "score": 0.75,
+ "latex": "u"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 756,
+ 439,
+ 776,
+ 439,
+ 776,
+ 465,
+ 756,
+ 465
+ ],
+ "score": 0.74,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1222,
+ 316,
+ 1241,
+ 316,
+ 1241,
+ 338,
+ 1222,
+ 338
+ ],
+ "score": 0.73,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 486,
+ 1653,
+ 507,
+ 1653,
+ 507,
+ 1675,
+ 486,
+ 1675
+ ],
+ "score": 0.69,
+ "latex": "u"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 824,
+ 280,
+ 840,
+ 280,
+ 840,
+ 300,
+ 824,
+ 300
+ ],
+ "score": 0.69,
+ "latex": "c"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 752,
+ 470,
+ 770,
+ 470,
+ 770,
+ 491,
+ 752,
+ 491
+ ],
+ "score": 0.68,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1316,
+ 794,
+ 1333,
+ 794,
+ 1333,
+ 816,
+ 1316,
+ 816
+ ],
+ "score": 0.65,
+ "latex": "v"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1291,
+ 317,
+ 1309,
+ 317,
+ 1309,
+ 338,
+ 1291,
+ 338
+ ],
+ "score": 0.64,
+ "latex": "v"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 343,
+ 372,
+ 371,
+ 372,
+ 371,
+ 401,
+ 343,
+ 401
+ ],
+ "score": 0.61,
+ "latex": "\\nabla"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1349,
+ 349,
+ 1366,
+ 349,
+ 1366,
+ 369,
+ 1349,
+ 369
+ ],
+ "score": 0.59,
+ "latex": "v"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 613,
+ 280,
+ 632,
+ 280,
+ 632,
+ 304,
+ 613,
+ 304
+ ],
+ "score": 0.45,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 612,
+ 279,
+ 662,
+ 279,
+ 662,
+ 304,
+ 612,
+ 304
+ ],
+ "score": 0.33,
+ "latex": "\\mu , \\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 983,
+ 2005,
+ 1009,
+ 2005,
+ 1009,
+ 2031,
+ 983,
+ 2031
+ ],
+ "score": 0.25,
+ "latex": "\\&"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 73.0,
+ 817.0,
+ 73.0,
+ 817.0,
+ 108.0,
+ 296.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1905.0,
+ 899.0,
+ 1905.0,
+ 899.0,
+ 1944.0,
+ 296.0,
+ 1944.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 839.0,
+ 2085.0,
+ 861.0,
+ 2085.0,
+ 861.0,
+ 2120.0,
+ 839.0,
+ 2120.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 226.0,
+ 758.0,
+ 226.0,
+ 758.0,
+ 268.0,
+ 297.0,
+ 268.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 226.0,
+ 758.0,
+ 226.0,
+ 758.0,
+ 268.0,
+ 297.0,
+ 268.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1429.0,
+ 1404.0,
+ 1429.0,
+ 1404.0,
+ 1465.0,
+ 294.0,
+ 1465.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1461.0,
+ 1404.0,
+ 1461.0,
+ 1404.0,
+ 1497.0,
+ 294.0,
+ 1497.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1488.0,
+ 481.0,
+ 1488.0,
+ 481.0,
+ 1526.0,
+ 292.0,
+ 1526.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 557.0,
+ 1488.0,
+ 1406.0,
+ 1488.0,
+ 1406.0,
+ 1526.0,
+ 557.0,
+ 1526.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1521.0,
+ 1012.0,
+ 1521.0,
+ 1012.0,
+ 1557.0,
+ 294.0,
+ 1557.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1086.0,
+ 1521.0,
+ 1404.0,
+ 1521.0,
+ 1404.0,
+ 1557.0,
+ 1086.0,
+ 1557.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1552.0,
+ 1406.0,
+ 1552.0,
+ 1406.0,
+ 1588.0,
+ 292.0,
+ 1588.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1580.0,
+ 727.0,
+ 1580.0,
+ 727.0,
+ 1621.0,
+ 294.0,
+ 1621.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1046.0,
+ 1580.0,
+ 1404.0,
+ 1580.0,
+ 1404.0,
+ 1621.0,
+ 1046.0,
+ 1621.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1612.0,
+ 786.0,
+ 1612.0,
+ 786.0,
+ 1656.0,
+ 291.0,
+ 1656.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 860.0,
+ 1612.0,
+ 1405.0,
+ 1612.0,
+ 1405.0,
+ 1656.0,
+ 860.0,
+ 1656.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1644.0,
+ 485.0,
+ 1644.0,
+ 485.0,
+ 1685.0,
+ 292.0,
+ 1685.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 508.0,
+ 1644.0,
+ 1404.0,
+ 1644.0,
+ 1404.0,
+ 1685.0,
+ 508.0,
+ 1685.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1676.0,
+ 1406.0,
+ 1676.0,
+ 1406.0,
+ 1716.0,
+ 292.0,
+ 1716.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1710.0,
+ 1404.0,
+ 1710.0,
+ 1404.0,
+ 1745.0,
+ 295.0,
+ 1745.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1738.0,
+ 1405.0,
+ 1738.0,
+ 1405.0,
+ 1772.0,
+ 294.0,
+ 1772.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1770.0,
+ 1406.0,
+ 1770.0,
+ 1406.0,
+ 1804.0,
+ 294.0,
+ 1804.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1799.0,
+ 1407.0,
+ 1799.0,
+ 1407.0,
+ 1836.0,
+ 292.0,
+ 1836.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1833.0,
+ 962.0,
+ 1833.0,
+ 962.0,
+ 1865.0,
+ 295.0,
+ 1865.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 866.0,
+ 1403.0,
+ 866.0,
+ 1403.0,
+ 898.0,
+ 297.0,
+ 898.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 897.0,
+ 1403.0,
+ 897.0,
+ 1403.0,
+ 929.0,
+ 296.0,
+ 929.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 926.0,
+ 1406.0,
+ 926.0,
+ 1406.0,
+ 963.0,
+ 292.0,
+ 963.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 958.0,
+ 1403.0,
+ 958.0,
+ 1403.0,
+ 990.0,
+ 294.0,
+ 990.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 983.0,
+ 645.0,
+ 983.0,
+ 645.0,
+ 1026.0,
+ 292.0,
+ 1026.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 736.0,
+ 983.0,
+ 826.0,
+ 983.0,
+ 826.0,
+ 1026.0,
+ 736.0,
+ 1026.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 848.0,
+ 983.0,
+ 1406.0,
+ 983.0,
+ 1406.0,
+ 1026.0,
+ 848.0,
+ 1026.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1021.0,
+ 756.0,
+ 1021.0,
+ 756.0,
+ 1060.0,
+ 294.0,
+ 1060.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 870.0,
+ 1021.0,
+ 1287.0,
+ 1021.0,
+ 1287.0,
+ 1060.0,
+ 870.0,
+ 1060.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1349.0,
+ 1021.0,
+ 1406.0,
+ 1021.0,
+ 1406.0,
+ 1060.0,
+ 1349.0,
+ 1060.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1048.0,
+ 1283.0,
+ 1048.0,
+ 1283.0,
+ 1092.0,
+ 291.0,
+ 1092.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1357.0,
+ 1048.0,
+ 1408.0,
+ 1048.0,
+ 1408.0,
+ 1092.0,
+ 1357.0,
+ 1092.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1083.0,
+ 478.0,
+ 1083.0,
+ 478.0,
+ 1117.0,
+ 292.0,
+ 1117.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 499.0,
+ 1083.0,
+ 1406.0,
+ 1083.0,
+ 1406.0,
+ 1117.0,
+ 499.0,
+ 1117.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1111.0,
+ 539.0,
+ 1111.0,
+ 539.0,
+ 1155.0,
+ 292.0,
+ 1155.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 849.0,
+ 1111.0,
+ 1288.0,
+ 1111.0,
+ 1288.0,
+ 1155.0,
+ 849.0,
+ 1155.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1354.0,
+ 1111.0,
+ 1408.0,
+ 1111.0,
+ 1408.0,
+ 1155.0,
+ 1354.0,
+ 1155.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1145.0,
+ 710.0,
+ 1145.0,
+ 710.0,
+ 1181.0,
+ 295.0,
+ 1181.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 748.0,
+ 1145.0,
+ 1405.0,
+ 1145.0,
+ 1405.0,
+ 1181.0,
+ 748.0,
+ 1181.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1175.0,
+ 547.0,
+ 1175.0,
+ 547.0,
+ 1217.0,
+ 292.0,
+ 1217.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 660.0,
+ 1175.0,
+ 1405.0,
+ 1175.0,
+ 1405.0,
+ 1217.0,
+ 660.0,
+ 1217.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1205.0,
+ 717.0,
+ 1205.0,
+ 717.0,
+ 1249.0,
+ 292.0,
+ 1249.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 808.0,
+ 1205.0,
+ 1406.0,
+ 1205.0,
+ 1406.0,
+ 1249.0,
+ 808.0,
+ 1249.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1239.0,
+ 1405.0,
+ 1239.0,
+ 1405.0,
+ 1277.0,
+ 292.0,
+ 1277.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1270.0,
+ 1135.0,
+ 1270.0,
+ 1135.0,
+ 1309.0,
+ 292.0,
+ 1309.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 665.0,
+ 1407.0,
+ 665.0,
+ 1407.0,
+ 702.0,
+ 292.0,
+ 702.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 697.0,
+ 1406.0,
+ 697.0,
+ 1406.0,
+ 730.0,
+ 293.0,
+ 730.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 729.0,
+ 1405.0,
+ 729.0,
+ 1405.0,
+ 761.0,
+ 296.0,
+ 761.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 759.0,
+ 1405.0,
+ 759.0,
+ 1405.0,
+ 791.0,
+ 294.0,
+ 791.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 789.0,
+ 1315.0,
+ 789.0,
+ 1315.0,
+ 824.0,
+ 294.0,
+ 824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1334.0,
+ 789.0,
+ 1406.0,
+ 789.0,
+ 1406.0,
+ 824.0,
+ 1334.0,
+ 824.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 817.0,
+ 857.0,
+ 817.0,
+ 857.0,
+ 852.0,
+ 293.0,
+ 852.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1316.0,
+ 1403.0,
+ 1316.0,
+ 1403.0,
+ 1351.0,
+ 294.0,
+ 1351.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1351.0,
+ 1184.0,
+ 1351.0,
+ 1184.0,
+ 1388.0,
+ 293.0,
+ 1388.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1297.0,
+ 1351.0,
+ 1404.0,
+ 1351.0,
+ 1404.0,
+ 1388.0,
+ 1297.0,
+ 1388.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 1385.0,
+ 1204.0,
+ 1385.0,
+ 1204.0,
+ 1416.0,
+ 297.0,
+ 1416.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 285.0,
+ 1950.0,
+ 737.0,
+ 1950.0,
+ 737.0,
+ 2020.0,
+ 285.0,
+ 2020.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1043.0,
+ 1950.0,
+ 1093.0,
+ 1950.0,
+ 1093.0,
+ 2020.0,
+ 1043.0,
+ 2020.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1395.0,
+ 1950.0,
+ 1414.0,
+ 1950.0,
+ 1414.0,
+ 2020.0,
+ 1395.0,
+ 2020.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1999.0,
+ 982.0,
+ 1999.0,
+ 982.0,
+ 2036.0,
+ 294.0,
+ 2036.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1010.0,
+ 1999.0,
+ 1406.0,
+ 1999.0,
+ 1406.0,
+ 2036.0,
+ 1010.0,
+ 2036.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 307.0,
+ 269.0,
+ 611.0,
+ 269.0,
+ 611.0,
+ 309.0,
+ 307.0,
+ 309.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 663.0,
+ 269.0,
+ 823.0,
+ 269.0,
+ 823.0,
+ 309.0,
+ 663.0,
+ 309.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 841.0,
+ 269.0,
+ 966.0,
+ 269.0,
+ 966.0,
+ 309.0,
+ 841.0,
+ 309.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 986.0,
+ 269.0,
+ 1141.0,
+ 269.0,
+ 1141.0,
+ 309.0,
+ 986.0,
+ 309.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1162.0,
+ 269.0,
+ 1172.0,
+ 269.0,
+ 1172.0,
+ 309.0,
+ 1162.0,
+ 309.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 304.0,
+ 307.0,
+ 508.0,
+ 307.0,
+ 508.0,
+ 346.0,
+ 304.0,
+ 346.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 666.0,
+ 307.0,
+ 715.0,
+ 307.0,
+ 715.0,
+ 346.0,
+ 666.0,
+ 346.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 737.0,
+ 307.0,
+ 932.0,
+ 307.0,
+ 932.0,
+ 346.0,
+ 737.0,
+ 346.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1065.0,
+ 307.0,
+ 1221.0,
+ 307.0,
+ 1221.0,
+ 346.0,
+ 1065.0,
+ 346.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1242.0,
+ 307.0,
+ 1290.0,
+ 307.0,
+ 1290.0,
+ 346.0,
+ 1242.0,
+ 346.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1310.0,
+ 307.0,
+ 1325.0,
+ 307.0,
+ 1325.0,
+ 346.0,
+ 1310.0,
+ 346.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 309.0,
+ 337.0,
+ 541.0,
+ 337.0,
+ 541.0,
+ 378.0,
+ 309.0,
+ 378.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 564.0,
+ 337.0,
+ 885.0,
+ 337.0,
+ 885.0,
+ 378.0,
+ 564.0,
+ 378.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 906.0,
+ 337.0,
+ 1279.0,
+ 337.0,
+ 1279.0,
+ 378.0,
+ 906.0,
+ 378.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1300.0,
+ 337.0,
+ 1348.0,
+ 337.0,
+ 1348.0,
+ 378.0,
+ 1300.0,
+ 378.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1367.0,
+ 337.0,
+ 1381.0,
+ 337.0,
+ 1381.0,
+ 378.0,
+ 1367.0,
+ 378.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 372.0,
+ 368.0,
+ 851.0,
+ 368.0,
+ 851.0,
+ 406.0,
+ 372.0,
+ 406.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 309.0,
+ 403.0,
+ 340.0,
+ 403.0,
+ 340.0,
+ 433.0,
+ 309.0,
+ 433.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 341.0,
+ 402.0,
+ 614.0,
+ 402.0,
+ 614.0,
+ 436.0,
+ 341.0,
+ 436.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 308.0,
+ 433.0,
+ 340.0,
+ 433.0,
+ 340.0,
+ 464.0,
+ 308.0,
+ 464.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 367.0,
+ 428.0,
+ 544.0,
+ 428.0,
+ 544.0,
+ 471.0,
+ 367.0,
+ 471.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 691.0,
+ 428.0,
+ 755.0,
+ 428.0,
+ 755.0,
+ 471.0,
+ 691.0,
+ 471.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 777.0,
+ 428.0,
+ 782.0,
+ 428.0,
+ 782.0,
+ 471.0,
+ 777.0,
+ 471.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 308.0,
+ 463.0,
+ 338.0,
+ 463.0,
+ 338.0,
+ 495.0,
+ 308.0,
+ 495.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 367.0,
+ 462.0,
+ 545.0,
+ 462.0,
+ 545.0,
+ 500.0,
+ 367.0,
+ 500.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 687.0,
+ 462.0,
+ 751.0,
+ 462.0,
+ 751.0,
+ 500.0,
+ 687.0,
+ 500.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 771.0,
+ 462.0,
+ 777.0,
+ 462.0,
+ 777.0,
+ 500.0,
+ 771.0,
+ 500.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 308.0,
+ 494.0,
+ 340.0,
+ 494.0,
+ 340.0,
+ 525.0,
+ 308.0,
+ 525.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 366.0,
+ 489.0,
+ 452.0,
+ 489.0,
+ 452.0,
+ 534.0,
+ 366.0,
+ 534.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1061.0,
+ 489.0,
+ 1064.0,
+ 489.0,
+ 1064.0,
+ 534.0,
+ 1061.0,
+ 534.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 310.0,
+ 526.0,
+ 341.0,
+ 526.0,
+ 341.0,
+ 563.0,
+ 310.0,
+ 563.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 367.0,
+ 527.0,
+ 454.0,
+ 527.0,
+ 454.0,
+ 568.0,
+ 367.0,
+ 568.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 309.0,
+ 561.0,
+ 465.0,
+ 561.0,
+ 465.0,
+ 593.0,
+ 309.0,
+ 593.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 4,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 941,
+ 1405,
+ 941,
+ 1405,
+ 1339,
+ 297,
+ 1339
+ ],
+ "score": 0.985
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1450,
+ 1405,
+ 1450,
+ 1405,
+ 1638,
+ 297,
+ 1638
+ ],
+ "score": 0.982
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1799,
+ 1404,
+ 1799,
+ 1404,
+ 2036,
+ 297,
+ 2036
+ ],
+ "score": 0.981
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 534,
+ 1404,
+ 534,
+ 1404,
+ 726,
+ 297,
+ 726
+ ],
+ "score": 0.979
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 802,
+ 1403,
+ 802,
+ 1403,
+ 926,
+ 298,
+ 926
+ ],
+ "score": 0.979
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 228,
+ 1403,
+ 228,
+ 1403,
+ 382,
+ 297,
+ 382
+ ],
+ "score": 0.976
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 435,
+ 1404,
+ 435,
+ 1404,
+ 528,
+ 297,
+ 528
+ ],
+ "score": 0.974
+ },
+ {
+ "category_id": 8,
+ "poly": [
+ 748,
+ 382,
+ 953,
+ 382,
+ 953,
+ 429,
+ 748,
+ 429
+ ],
+ "score": 0.945
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 292,
+ 1644,
+ 1403,
+ 1644,
+ 1403,
+ 1707,
+ 292,
+ 1707
+ ],
+ "score": 0.94
+ },
+ {
+ "category_id": 8,
+ "poly": [
+ 737,
+ 739,
+ 958,
+ 739,
+ 958,
+ 775,
+ 737,
+ 775
+ ],
+ "score": 0.935
+ },
+ {
+ "category_id": 8,
+ "poly": [
+ 639,
+ 1718,
+ 1057,
+ 1718,
+ 1057,
+ 1771,
+ 639,
+ 1771
+ ],
+ "score": 0.927
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 299,
+ 1382,
+ 836,
+ 1382,
+ 836,
+ 1418,
+ 299,
+ 1418
+ ],
+ "score": 0.915
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 299,
+ 75,
+ 815,
+ 75,
+ 815,
+ 105,
+ 299,
+ 105
+ ],
+ "score": 0.911
+ },
+ {
+ "category_id": 9,
+ "poly": [
+ 1352,
+ 741,
+ 1399,
+ 741,
+ 1399,
+ 771,
+ 1352,
+ 771
+ ],
+ "score": 0.905
+ },
+ {
+ "category_id": 9,
+ "poly": [
+ 1352,
+ 1722,
+ 1399,
+ 1722,
+ 1399,
+ 1752,
+ 1352,
+ 1752
+ ],
+ "score": 0.886
+ },
+ {
+ "category_id": 9,
+ "poly": [
+ 1366,
+ 384,
+ 1399,
+ 384,
+ 1399,
+ 413,
+ 1366,
+ 413
+ ],
+ "score": 0.883
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 840,
+ 2088,
+ 859,
+ 2088,
+ 859,
+ 2112,
+ 840,
+ 2112
+ ],
+ "score": 0.796
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 508,
+ 1803,
+ 726,
+ 1803,
+ 726,
+ 1838,
+ 508,
+ 1838
+ ],
+ "score": 0.95,
+ "latex": "d ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 503,
+ 1840,
+ 614,
+ 1840,
+ 614,
+ 1882,
+ 503,
+ 1882
+ ],
+ "score": 0.93,
+ "latex": "\\begin{array} { r } { \\bar { \\boldsymbol { \\pi } } = \\frac { \\boldsymbol { \\pi } \\mathbf { y } ^ { t } } { a } } \\end{array}"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 746,
+ 380,
+ 957,
+ 380,
+ 957,
+ 430,
+ 746,
+ 430
+ ],
+ "score": 0.93,
+ "latex": "\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname* { l i m } _ { \\varepsilon \\to 0 } \\pi _ { n } ^ { \\varepsilon } = \\pi ."
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1145,
+ 1803,
+ 1395,
+ 1803,
+ 1395,
+ 1841,
+ 1145,
+ 1841
+ ],
+ "score": 0.92,
+ "latex": "\\bar { \\pi } ( x ) = \\operatorname { \\mathbb { E } } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ Y \\right] ."
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 558,
+ 566,
+ 760,
+ 566,
+ 760,
+ 605,
+ 558,
+ 605
+ ],
+ "score": 0.92,
+ "latex": "\\begin{array} { r } { \\nu _ { n } = \\sum _ { j = 1 } ^ { n } b _ { j } \\delta _ { y _ { j } } } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 436,
+ 466,
+ 512,
+ 466,
+ 512,
+ 499,
+ 436,
+ 499
+ ],
+ "score": 0.92,
+ "latex": "( \\infty , 0 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 849,
+ 1881,
+ 1056,
+ 1881,
+ 1056,
+ 1915,
+ 849,
+ 1915
+ ],
+ "score": 0.92,
+ "latex": "c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 638,
+ 1718,
+ 1061,
+ 1718,
+ 1061,
+ 1772,
+ 638,
+ 1772
+ ],
+ "score": 0.92,
+ "latex": "\\bar { \\pi } ( x ) = \\arg \\operatorname* { m i n } _ { z } \\mathbb { E } _ { Y \\sim \\pi ( \\cdot | x ) } \\left[ d ( z , Y ) \\right] ."
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 914,
+ 1607,
+ 996,
+ 1607,
+ 996,
+ 1638,
+ 914,
+ 1638
+ ],
+ "score": 0.92,
+ "latex": "\\pi ( x , y )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1255,
+ 436,
+ 1320,
+ 436,
+ 1320,
+ 469,
+ 1255,
+ 469
+ ],
+ "score": 0.92,
+ "latex": "( n , \\varepsilon )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 298,
+ 565,
+ 501,
+ 565,
+ 501,
+ 603,
+ 298,
+ 603
+ ],
+ "score": 0.92,
+ "latex": "\\textstyle \\mu _ { n } = \\sum _ { i = 1 } ^ { n } a _ { i } \\delta _ { x _ { i } }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 911,
+ 1845,
+ 1121,
+ 1845,
+ 1121,
+ 1880,
+ 911,
+ 1880
+ ],
+ "score": 0.91,
+ "latex": "\\mathbf { y } = ( y _ { 1 } , \\dotsb , y _ { n } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 885,
+ 1573,
+ 1026,
+ 1573,
+ 1026,
+ 1606,
+ 885,
+ 1606
+ ],
+ "score": 0.91,
+ "latex": "\\pi ( x , y ) > 0"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 520,
+ 1543,
+ 830,
+ 1543,
+ 830,
+ 1576,
+ 520,
+ 1576
+ ],
+ "score": 0.91,
+ "latex": "\\pi : ( \\bar { x , y } ) \\in \\mathcal { X } \\times \\mathcal { Y } \\mapsto \\mathbb { R } ^ { + }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 497,
+ 535,
+ 632,
+ 535,
+ 632,
+ 569,
+ 497,
+ 569
+ ],
+ "score": 0.91,
+ "latex": "\\mu \\in P ( \\mathcal { X } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 695,
+ 295,
+ 794,
+ 295,
+ 794,
+ 324,
+ 695,
+ 324
+ ],
+ "score": 0.91,
+ "latex": "\\mu _ { n } \\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 887,
+ 633,
+ 971,
+ 633,
+ 971,
+ 663,
+ 887,
+ 663
+ ],
+ "score": 0.91,
+ "latex": "\\mathcal { X } \\times \\mathcal { V }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 691,
+ 535,
+ 821,
+ 535,
+ 821,
+ 569,
+ 691,
+ 569
+ ],
+ "score": 0.91,
+ "latex": "\\nu \\in P ( \\mathcal { V } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1093,
+ 437,
+ 1127,
+ 437,
+ 1127,
+ 469,
+ 1093,
+ 469
+ ],
+ "score": 0.9,
+ "latex": "\\pi _ { n } ^ { \\varepsilon }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 845,
+ 295,
+ 940,
+ 295,
+ 940,
+ 322,
+ 845,
+ 322
+ ],
+ "score": 0.9,
+ "latex": "\\nu _ { n } \\to \\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 985,
+ 1675,
+ 1196,
+ 1675,
+ 1196,
+ 1706,
+ 985,
+ 1706
+ ],
+ "score": 0.9,
+ "latex": "d : \\mathcal { V } \\times \\mathcal { V } \\mathbb { R } ^ { + }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 299,
+ 694,
+ 363,
+ 694,
+ 363,
+ 727,
+ 299,
+ 727
+ ],
+ "score": 0.9,
+ "latex": "( \\pi _ { n } ^ { \\varepsilon _ { n } } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1143,
+ 634,
+ 1187,
+ 634,
+ 1187,
+ 666,
+ 1143,
+ 666
+ ],
+ "score": 0.89,
+ "latex": "\\pi _ { n } ^ { \\varepsilon _ { n } }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 396,
+ 232,
+ 430,
+ 232,
+ 430,
+ 264,
+ 396,
+ 264
+ ],
+ "score": 0.89,
+ "latex": "\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 738,
+ 738,
+ 962,
+ 738,
+ 962,
+ 775,
+ 738,
+ 775
+ ],
+ "score": 0.89,
+ "latex": "\\pi _ { n } ^ { \\varepsilon _ { n } } \\to \\pi w e a k l y ."
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 656,
+ 326,
+ 687,
+ 326,
+ 687,
+ 353,
+ 656,
+ 353
+ ],
+ "score": 0.88,
+ "latex": "\\nu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 569,
+ 326,
+ 604,
+ 326,
+ 604,
+ 354,
+ 569,
+ 354
+ ],
+ "score": 0.87,
+ "latex": "\\mu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 805,
+ 669,
+ 839,
+ 669,
+ 839,
+ 696,
+ 805,
+ 696
+ ],
+ "score": 0.86,
+ "latex": "\\mu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 652,
+ 1575,
+ 802,
+ 1575,
+ 802,
+ 1607,
+ 652,
+ 1607
+ ],
+ "score": 0.86,
+ "latex": "y \\in \\operatorname { S u p p } ( \\nu )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 894,
+ 668,
+ 925,
+ 668,
+ 925,
+ 694,
+ 894,
+ 694
+ ],
+ "score": 0.86,
+ "latex": "\\nu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1224,
+ 292,
+ 1276,
+ 292,
+ 1276,
+ 324,
+ 1224,
+ 324
+ ],
+ "score": 0.85,
+ "latex": "\\left( \\pi _ { n } \\right)"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 676,
+ 605,
+ 727,
+ 605,
+ 727,
+ 635,
+ 676,
+ 635
+ ],
+ "score": 0.83,
+ "latex": "\\left( \\varepsilon _ { n } \\right)"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1366,
+ 237,
+ 1400,
+ 237,
+ 1400,
+ 262,
+ 1366,
+ 262
+ ],
+ "score": 0.83,
+ "latex": "\\pi _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 663,
+ 1948,
+ 685,
+ 1948,
+ 685,
+ 1974,
+ 663,
+ 1974
+ ],
+ "score": 0.82,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 607,
+ 977,
+ 628,
+ 977,
+ 628,
+ 1004,
+ 607,
+ 1004
+ ],
+ "score": 0.82,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 953,
+ 1944,
+ 1012,
+ 1944,
+ 1012,
+ 1975,
+ 953,
+ 1975
+ ],
+ "score": 0.81,
+ "latex": "\\bar { \\pi } \\# \\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 475,
+ 1518,
+ 497,
+ 1518,
+ 497,
+ 1545,
+ 475,
+ 1545
+ ],
+ "score": 0.81,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 992,
+ 538,
+ 1018,
+ 538,
+ 1018,
+ 567,
+ 992,
+ 567
+ ],
+ "score": 0.81,
+ "latex": "\\mathcal { V }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 906,
+ 538,
+ 933,
+ 538,
+ 933,
+ 564,
+ 906,
+ 564
+ ],
+ "score": 0.8,
+ "latex": "\\mathcal { X }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1356,
+ 1884,
+ 1376,
+ 1884,
+ 1376,
+ 1910,
+ 1356,
+ 1910
+ ],
+ "score": 0.8,
+ "latex": "\\bar { \\pi }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 613,
+ 1610,
+ 632,
+ 1610,
+ 632,
+ 1637,
+ 613,
+ 1637
+ ],
+ "score": 0.79,
+ "latex": "y"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 734,
+ 1519,
+ 753,
+ 1519,
+ 753,
+ 1541,
+ 734,
+ 1541
+ ],
+ "score": 0.79,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1374,
+ 1853,
+ 1394,
+ 1853,
+ 1394,
+ 1879,
+ 1374,
+ 1879
+ ],
+ "score": 0.78,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1380,
+ 327,
+ 1401,
+ 327,
+ 1401,
+ 353,
+ 1380,
+ 353
+ ],
+ "score": 0.78,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1018,
+ 1488,
+ 1039,
+ 1488,
+ 1039,
+ 1510,
+ 1018,
+ 1510
+ ],
+ "score": 0.78,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1012,
+ 1519,
+ 1033,
+ 1519,
+ 1033,
+ 1541,
+ 1012,
+ 1541
+ ],
+ "score": 0.76,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1177,
+ 1853,
+ 1196,
+ 1853,
+ 1196,
+ 1875,
+ 1177,
+ 1875
+ ],
+ "score": 0.76,
+ "latex": "a"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 987,
+ 1916,
+ 1006,
+ 1916,
+ 1006,
+ 1939,
+ 987,
+ 1939
+ ],
+ "score": 0.75,
+ "latex": "\\bar { \\pi }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1042,
+ 327,
+ 1063,
+ 327,
+ 1063,
+ 349,
+ 1042,
+ 349
+ ],
+ "score": 0.75,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1161,
+ 441,
+ 1182,
+ 441,
+ 1182,
+ 463,
+ 1161,
+ 463
+ ],
+ "score": 0.75,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 440,
+ 1579,
+ 460,
+ 1579,
+ 460,
+ 1601,
+ 440,
+ 1601
+ ],
+ "score": 0.73,
+ "latex": "x"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 680,
+ 977,
+ 699,
+ 977,
+ 699,
+ 1001,
+ 680,
+ 1001
+ ],
+ "score": 0.73,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 344,
+ 357,
+ 363,
+ 357,
+ 363,
+ 380,
+ 344,
+ 380
+ ],
+ "score": 0.7,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 297,
+ 871,
+ 317,
+ 871,
+ 317,
+ 892,
+ 297,
+ 892
+ ],
+ "score": 0.7,
+ "latex": "n"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 344,
+ 1579,
+ 366,
+ 1579,
+ 366,
+ 1601,
+ 344,
+ 1601
+ ],
+ "score": 0.7,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 329,
+ 608,
+ 350,
+ 608,
+ 350,
+ 634,
+ 329,
+ 634
+ ],
+ "score": 0.68,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 409,
+ 608,
+ 430,
+ 608,
+ 430,
+ 630,
+ 409,
+ 630
+ ],
+ "score": 0.66,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 745,
+ 1679,
+ 766,
+ 1679,
+ 766,
+ 1703,
+ 745,
+ 1703
+ ],
+ "score": 0.64,
+ "latex": "\\bar { \\pi }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 734,
+ 700,
+ 754,
+ 700,
+ 754,
+ 721,
+ 734,
+ 721
+ ],
+ "score": 0.62,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1175,
+ 699,
+ 1192,
+ 699,
+ 1192,
+ 721,
+ 1175,
+ 721
+ ],
+ "score": 0.6,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 673,
+ 640,
+ 689,
+ 640,
+ 689,
+ 660,
+ 673,
+ 660
+ ],
+ "score": 0.52,
+ "latex": "c"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1103,
+ 699,
+ 1123,
+ 699,
+ 1123,
+ 726,
+ 1103,
+ 726
+ ],
+ "score": 0.5,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 775,
+ 1650,
+ 795,
+ 1650,
+ 795,
+ 1672,
+ 775,
+ 1672
+ ],
+ "score": 0.33,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1378.0,
+ 840.0,
+ 1378.0,
+ 840.0,
+ 1423.0,
+ 292.0,
+ 1423.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 73.0,
+ 818.0,
+ 73.0,
+ 818.0,
+ 108.0,
+ 295.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 840.0,
+ 2086.0,
+ 862.0,
+ 2086.0,
+ 862.0,
+ 2116.0,
+ 840.0,
+ 2116.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 942.0,
+ 1405.0,
+ 942.0,
+ 1405.0,
+ 978.0,
+ 295.0,
+ 978.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 971.0,
+ 606.0,
+ 971.0,
+ 606.0,
+ 1008.0,
+ 295.0,
+ 1008.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 629.0,
+ 971.0,
+ 679.0,
+ 971.0,
+ 679.0,
+ 1008.0,
+ 629.0,
+ 1008.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 700.0,
+ 971.0,
+ 1405.0,
+ 971.0,
+ 1405.0,
+ 1008.0,
+ 700.0,
+ 1008.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1003.0,
+ 1405.0,
+ 1003.0,
+ 1405.0,
+ 1039.0,
+ 294.0,
+ 1039.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1032.0,
+ 1405.0,
+ 1032.0,
+ 1405.0,
+ 1069.0,
+ 295.0,
+ 1069.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1062.0,
+ 1407.0,
+ 1062.0,
+ 1407.0,
+ 1102.0,
+ 294.0,
+ 1102.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1092.0,
+ 1405.0,
+ 1092.0,
+ 1405.0,
+ 1134.0,
+ 292.0,
+ 1134.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1123.0,
+ 1406.0,
+ 1123.0,
+ 1406.0,
+ 1161.0,
+ 292.0,
+ 1161.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1150.0,
+ 1405.0,
+ 1150.0,
+ 1405.0,
+ 1196.0,
+ 292.0,
+ 1196.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1185.0,
+ 1406.0,
+ 1185.0,
+ 1406.0,
+ 1222.0,
+ 294.0,
+ 1222.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1215.0,
+ 1402.0,
+ 1215.0,
+ 1402.0,
+ 1250.0,
+ 294.0,
+ 1250.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1245.0,
+ 1405.0,
+ 1245.0,
+ 1405.0,
+ 1283.0,
+ 292.0,
+ 1283.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1276.0,
+ 1405.0,
+ 1276.0,
+ 1405.0,
+ 1312.0,
+ 294.0,
+ 1312.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1307.0,
+ 697.0,
+ 1307.0,
+ 697.0,
+ 1344.0,
+ 295.0,
+ 1344.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1453.0,
+ 1402.0,
+ 1453.0,
+ 1402.0,
+ 1485.0,
+ 296.0,
+ 1485.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1481.0,
+ 1017.0,
+ 1481.0,
+ 1017.0,
+ 1518.0,
+ 294.0,
+ 1518.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1040.0,
+ 1481.0,
+ 1407.0,
+ 1481.0,
+ 1407.0,
+ 1518.0,
+ 1040.0,
+ 1518.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1512.0,
+ 474.0,
+ 1512.0,
+ 474.0,
+ 1549.0,
+ 292.0,
+ 1549.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 498.0,
+ 1512.0,
+ 733.0,
+ 1512.0,
+ 733.0,
+ 1549.0,
+ 498.0,
+ 1549.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 754.0,
+ 1512.0,
+ 1011.0,
+ 1512.0,
+ 1011.0,
+ 1549.0,
+ 754.0,
+ 1549.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1034.0,
+ 1512.0,
+ 1406.0,
+ 1512.0,
+ 1406.0,
+ 1549.0,
+ 1034.0,
+ 1549.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1539.0,
+ 519.0,
+ 1539.0,
+ 519.0,
+ 1583.0,
+ 292.0,
+ 1583.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 831.0,
+ 1539.0,
+ 1405.0,
+ 1539.0,
+ 1405.0,
+ 1583.0,
+ 831.0,
+ 1583.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1574.0,
+ 343.0,
+ 1574.0,
+ 343.0,
+ 1610.0,
+ 294.0,
+ 1610.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 367.0,
+ 1574.0,
+ 439.0,
+ 1574.0,
+ 439.0,
+ 1610.0,
+ 367.0,
+ 1610.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 461.0,
+ 1574.0,
+ 651.0,
+ 1574.0,
+ 651.0,
+ 1610.0,
+ 461.0,
+ 1610.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 803.0,
+ 1574.0,
+ 884.0,
+ 1574.0,
+ 884.0,
+ 1610.0,
+ 803.0,
+ 1610.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 1574.0,
+ 1405.0,
+ 1574.0,
+ 1405.0,
+ 1610.0,
+ 1027.0,
+ 1610.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1604.0,
+ 612.0,
+ 1604.0,
+ 612.0,
+ 1640.0,
+ 295.0,
+ 1640.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 633.0,
+ 1604.0,
+ 913.0,
+ 1604.0,
+ 913.0,
+ 1640.0,
+ 633.0,
+ 1640.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 997.0,
+ 1604.0,
+ 1007.0,
+ 1604.0,
+ 1007.0,
+ 1640.0,
+ 997.0,
+ 1640.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1799.0,
+ 507.0,
+ 1799.0,
+ 507.0,
+ 1844.0,
+ 292.0,
+ 1844.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 727.0,
+ 1799.0,
+ 1144.0,
+ 1799.0,
+ 1144.0,
+ 1844.0,
+ 727.0,
+ 1844.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1396.0,
+ 1799.0,
+ 1405.0,
+ 1799.0,
+ 1405.0,
+ 1844.0,
+ 1396.0,
+ 1844.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1840.0,
+ 502.0,
+ 1840.0,
+ 502.0,
+ 1886.0,
+ 291.0,
+ 1886.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 615.0,
+ 1840.0,
+ 910.0,
+ 1840.0,
+ 910.0,
+ 1886.0,
+ 615.0,
+ 1886.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1122.0,
+ 1840.0,
+ 1176.0,
+ 1840.0,
+ 1176.0,
+ 1886.0,
+ 1122.0,
+ 1886.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1197.0,
+ 1840.0,
+ 1373.0,
+ 1840.0,
+ 1373.0,
+ 1886.0,
+ 1197.0,
+ 1886.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1395.0,
+ 1840.0,
+ 1405.0,
+ 1840.0,
+ 1405.0,
+ 1886.0,
+ 1395.0,
+ 1886.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1881.0,
+ 848.0,
+ 1881.0,
+ 848.0,
+ 1917.0,
+ 294.0,
+ 1917.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1057.0,
+ 1881.0,
+ 1355.0,
+ 1881.0,
+ 1355.0,
+ 1917.0,
+ 1057.0,
+ 1917.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1377.0,
+ 1881.0,
+ 1406.0,
+ 1881.0,
+ 1406.0,
+ 1917.0,
+ 1377.0,
+ 1917.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1911.0,
+ 986.0,
+ 1911.0,
+ 986.0,
+ 1948.0,
+ 295.0,
+ 1948.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1007.0,
+ 1911.0,
+ 1404.0,
+ 1911.0,
+ 1404.0,
+ 1948.0,
+ 1007.0,
+ 1948.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1943.0,
+ 662.0,
+ 1943.0,
+ 662.0,
+ 1979.0,
+ 295.0,
+ 1979.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 686.0,
+ 1943.0,
+ 952.0,
+ 1943.0,
+ 952.0,
+ 1979.0,
+ 686.0,
+ 1979.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1013.0,
+ 1943.0,
+ 1405.0,
+ 1943.0,
+ 1405.0,
+ 1979.0,
+ 1013.0,
+ 1979.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1971.0,
+ 1405.0,
+ 1971.0,
+ 1405.0,
+ 2007.0,
+ 294.0,
+ 2007.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 2002.0,
+ 1353.0,
+ 2002.0,
+ 1353.0,
+ 2039.0,
+ 295.0,
+ 2039.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 283.0,
+ 531.0,
+ 297.0,
+ 531.0,
+ 297.0,
+ 620.0,
+ 283.0,
+ 620.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 822.0,
+ 531.0,
+ 905.0,
+ 531.0,
+ 905.0,
+ 620.0,
+ 822.0,
+ 620.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 934.0,
+ 531.0,
+ 991.0,
+ 531.0,
+ 991.0,
+ 620.0,
+ 934.0,
+ 620.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1019.0,
+ 531.0,
+ 1415.0,
+ 531.0,
+ 1415.0,
+ 620.0,
+ 1019.0,
+ 620.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 600.0,
+ 328.0,
+ 600.0,
+ 328.0,
+ 640.0,
+ 294.0,
+ 640.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 351.0,
+ 600.0,
+ 408.0,
+ 600.0,
+ 408.0,
+ 640.0,
+ 351.0,
+ 640.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 431.0,
+ 600.0,
+ 675.0,
+ 600.0,
+ 675.0,
+ 640.0,
+ 431.0,
+ 640.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 728.0,
+ 600.0,
+ 1407.0,
+ 600.0,
+ 1407.0,
+ 640.0,
+ 728.0,
+ 640.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 630.0,
+ 672.0,
+ 630.0,
+ 672.0,
+ 670.0,
+ 292.0,
+ 670.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 690.0,
+ 630.0,
+ 886.0,
+ 630.0,
+ 886.0,
+ 670.0,
+ 690.0,
+ 670.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 972.0,
+ 630.0,
+ 1142.0,
+ 630.0,
+ 1142.0,
+ 670.0,
+ 972.0,
+ 670.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1188.0,
+ 630.0,
+ 1405.0,
+ 630.0,
+ 1405.0,
+ 670.0,
+ 1188.0,
+ 670.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 660.0,
+ 804.0,
+ 660.0,
+ 804.0,
+ 700.0,
+ 292.0,
+ 700.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 840.0,
+ 660.0,
+ 893.0,
+ 660.0,
+ 893.0,
+ 700.0,
+ 840.0,
+ 700.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 926.0,
+ 660.0,
+ 1404.0,
+ 660.0,
+ 1404.0,
+ 700.0,
+ 926.0,
+ 700.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 690.0,
+ 298.0,
+ 690.0,
+ 298.0,
+ 734.0,
+ 294.0,
+ 734.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 364.0,
+ 690.0,
+ 733.0,
+ 690.0,
+ 733.0,
+ 734.0,
+ 364.0,
+ 734.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 755.0,
+ 690.0,
+ 1102.0,
+ 690.0,
+ 1102.0,
+ 734.0,
+ 755.0,
+ 734.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1124.0,
+ 690.0,
+ 1174.0,
+ 690.0,
+ 1174.0,
+ 734.0,
+ 1124.0,
+ 734.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1193.0,
+ 690.0,
+ 1205.0,
+ 690.0,
+ 1205.0,
+ 734.0,
+ 1193.0,
+ 734.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 802.0,
+ 1404.0,
+ 802.0,
+ 1404.0,
+ 838.0,
+ 294.0,
+ 838.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 836.0,
+ 1404.0,
+ 836.0,
+ 1404.0,
+ 869.0,
+ 294.0,
+ 869.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 866.0,
+ 296.0,
+ 866.0,
+ 296.0,
+ 899.0,
+ 293.0,
+ 899.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 318.0,
+ 866.0,
+ 1402.0,
+ 866.0,
+ 1402.0,
+ 899.0,
+ 318.0,
+ 899.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 896.0,
+ 955.0,
+ 896.0,
+ 955.0,
+ 928.0,
+ 296.0,
+ 928.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 227.0,
+ 395.0,
+ 227.0,
+ 395.0,
+ 268.0,
+ 294.0,
+ 268.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 431.0,
+ 227.0,
+ 1365.0,
+ 227.0,
+ 1365.0,
+ 268.0,
+ 431.0,
+ 268.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1401.0,
+ 227.0,
+ 1405.0,
+ 227.0,
+ 1405.0,
+ 268.0,
+ 1401.0,
+ 268.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 259.0,
+ 1405.0,
+ 259.0,
+ 1405.0,
+ 298.0,
+ 292.0,
+ 298.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 290.0,
+ 694.0,
+ 290.0,
+ 694.0,
+ 327.0,
+ 295.0,
+ 327.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 795.0,
+ 290.0,
+ 844.0,
+ 290.0,
+ 844.0,
+ 327.0,
+ 795.0,
+ 327.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 941.0,
+ 290.0,
+ 1223.0,
+ 290.0,
+ 1223.0,
+ 327.0,
+ 941.0,
+ 327.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1277.0,
+ 290.0,
+ 1405.0,
+ 290.0,
+ 1405.0,
+ 327.0,
+ 1277.0,
+ 327.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 317.0,
+ 568.0,
+ 317.0,
+ 568.0,
+ 362.0,
+ 292.0,
+ 362.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 605.0,
+ 317.0,
+ 655.0,
+ 317.0,
+ 655.0,
+ 362.0,
+ 605.0,
+ 362.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 688.0,
+ 317.0,
+ 1041.0,
+ 317.0,
+ 1041.0,
+ 362.0,
+ 688.0,
+ 362.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1064.0,
+ 317.0,
+ 1379.0,
+ 317.0,
+ 1379.0,
+ 362.0,
+ 1064.0,
+ 362.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1402.0,
+ 317.0,
+ 1408.0,
+ 317.0,
+ 1408.0,
+ 362.0,
+ 1402.0,
+ 362.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 348.0,
+ 343.0,
+ 348.0,
+ 343.0,
+ 390.0,
+ 294.0,
+ 390.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 364.0,
+ 348.0,
+ 589.0,
+ 348.0,
+ 589.0,
+ 390.0,
+ 364.0,
+ 390.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 436.0,
+ 1092.0,
+ 436.0,
+ 1092.0,
+ 469.0,
+ 295.0,
+ 469.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1128.0,
+ 436.0,
+ 1160.0,
+ 436.0,
+ 1160.0,
+ 469.0,
+ 1128.0,
+ 469.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1183.0,
+ 436.0,
+ 1254.0,
+ 436.0,
+ 1254.0,
+ 469.0,
+ 1183.0,
+ 469.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1321.0,
+ 436.0,
+ 1401.0,
+ 436.0,
+ 1401.0,
+ 469.0,
+ 1321.0,
+ 469.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 465.0,
+ 435.0,
+ 465.0,
+ 435.0,
+ 502.0,
+ 294.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 513.0,
+ 465.0,
+ 1407.0,
+ 465.0,
+ 1407.0,
+ 502.0,
+ 513.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 496.0,
+ 863.0,
+ 496.0,
+ 863.0,
+ 532.0,
+ 296.0,
+ 532.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1640.0,
+ 774.0,
+ 1640.0,
+ 774.0,
+ 1681.0,
+ 292.0,
+ 1681.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 796.0,
+ 1640.0,
+ 1406.0,
+ 1640.0,
+ 1406.0,
+ 1681.0,
+ 796.0,
+ 1681.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1673.0,
+ 744.0,
+ 1673.0,
+ 744.0,
+ 1712.0,
+ 291.0,
+ 1712.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 767.0,
+ 1673.0,
+ 984.0,
+ 1673.0,
+ 984.0,
+ 1712.0,
+ 767.0,
+ 1712.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1197.0,
+ 1673.0,
+ 1350.0,
+ 1673.0,
+ 1350.0,
+ 1712.0,
+ 1197.0,
+ 1712.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 5,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 296,
+ 1247,
+ 1405,
+ 1247,
+ 1405,
+ 1497,
+ 296,
+ 1497
+ ],
+ "score": 0.981
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 296,
+ 1500,
+ 1404,
+ 1500,
+ 1404,
+ 1725,
+ 296,
+ 1725
+ ],
+ "score": 0.98
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 296,
+ 1049,
+ 1404,
+ 1049,
+ 1404,
+ 1235,
+ 296,
+ 1235
+ ],
+ "score": 0.977
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 547,
+ 1405,
+ 547,
+ 1405,
+ 672,
+ 297,
+ 672
+ ],
+ "score": 0.975
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 296,
+ 879,
+ 1404,
+ 879,
+ 1404,
+ 1039,
+ 296,
+ 1039
+ ],
+ "score": 0.975
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 685,
+ 1403,
+ 685,
+ 1403,
+ 779,
+ 297,
+ 779
+ ],
+ "score": 0.973
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1816,
+ 1405,
+ 1816,
+ 1405,
+ 1911,
+ 298,
+ 1911
+ ],
+ "score": 0.971
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 296,
+ 1971,
+ 1397,
+ 1971,
+ 1397,
+ 2034,
+ 296,
+ 2034
+ ],
+ "score": 0.943
+ },
+ {
+ "category_id": 8,
+ "poly": [
+ 421,
+ 787,
+ 1278,
+ 787,
+ 1278,
+ 872,
+ 421,
+ 872
+ ],
+ "score": 0.942
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 300,
+ 1917,
+ 1136,
+ 1917,
+ 1136,
+ 1951,
+ 300,
+ 1951
+ ],
+ "score": 0.92
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 299,
+ 75,
+ 815,
+ 75,
+ 815,
+ 105,
+ 299,
+ 105
+ ],
+ "score": 0.904
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1762,
+ 1043,
+ 1762,
+ 1043,
+ 1795,
+ 298,
+ 1795
+ ],
+ "score": 0.899
+ },
+ {
+ "category_id": 9,
+ "poly": [
+ 1352,
+ 1726,
+ 1399,
+ 1726,
+ 1399,
+ 1756,
+ 1352,
+ 1756
+ ],
+ "score": 0.886
+ },
+ {
+ "category_id": 9,
+ "poly": [
+ 1352,
+ 813,
+ 1399,
+ 813,
+ 1399,
+ 844,
+ 1352,
+ 844
+ ],
+ "score": 0.883
+ },
+ {
+ "category_id": 8,
+ "poly": [
+ 637,
+ 1724,
+ 1064,
+ 1724,
+ 1064,
+ 1760,
+ 637,
+ 1760
+ ],
+ "score": 0.865
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 841,
+ 2087,
+ 858,
+ 2087,
+ 858,
+ 2111,
+ 841,
+ 2111
+ ],
+ "score": 0.706
+ },
+ {
+ "category_id": 6,
+ "poly": [
+ 298,
+ 229,
+ 814,
+ 229,
+ 814,
+ 262,
+ 298,
+ 262
+ ],
+ "score": 0.649
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 841,
+ 2087,
+ 859,
+ 2087,
+ 859,
+ 2111,
+ 841,
+ 2111
+ ],
+ "score": 0.18
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 998,
+ 1248,
+ 1219,
+ 1248,
+ 1219,
+ 1283,
+ 998,
+ 1283
+ ],
+ "score": 0.93,
+ "latex": "c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 347,
+ 1564,
+ 550,
+ 1564,
+ 550,
+ 1607,
+ 347,
+ 1607
+ ],
+ "score": 0.93,
+ "latex": "\\textstyle \\nu _ { n } = { \\frac { 1 } { n } } \\sum _ { j = 1 } ^ { n } \\delta _ { y _ { j } }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 373,
+ 880,
+ 598,
+ 880,
+ 598,
+ 915,
+ 373,
+ 915
+ ],
+ "score": 0.92,
+ "latex": "d ( x , y ) = \\| x - y \\| ^ { 2 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1207,
+ 1402,
+ 1393,
+ 1402,
+ 1393,
+ 1436,
+ 1207,
+ 1436
+ ],
+ "score": 0.92,
+ "latex": "( n , \\varepsilon ) \\to ( \\infty , 0 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 811,
+ 1636,
+ 854,
+ 1636,
+ 854,
+ 1667,
+ 811,
+ 1667
+ ],
+ "score": 0.92,
+ "latex": "\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 976,
+ 1974,
+ 1091,
+ 1974,
+ 1091,
+ 2005,
+ 976,
+ 2005
+ ],
+ "score": 0.92,
+ "latex": "X _ { n } \\sim \\mu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 853,
+ 1250,
+ 944,
+ 1250,
+ 944,
+ 1280,
+ 853,
+ 1280
+ ],
+ "score": 0.92,
+ "latex": "\\mathcal { X } = \\mathcal { V }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 962,
+ 1403,
+ 1134,
+ 1403,
+ 1134,
+ 1436,
+ 962,
+ 1436
+ ],
+ "score": 0.92,
+ "latex": "\\pi = ( \\operatorname { i d } , f ) \\# \\mu"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 420,
+ 786,
+ 1280,
+ 786,
+ 1280,
+ 876,
+ 420,
+ 876
+ ],
+ "score": 0.92,
+ "latex": "\\begin{array} { r l } & { \\mathbb { E } _ { X \\sim \\mu } \\left[ \\mathbb { E } _ { Y \\sim \\pi ^ { \\varepsilon } ( \\cdot | X ) } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] \\right] = \\mathbb { E } _ { ( X , Y ) \\sim \\pi ^ { \\varepsilon } } \\left[ d ( Y , f _ { \\theta } ( X ) ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( Y , f _ { \\theta } ( X ) ) H _ { \\varepsilon } ( X , Y ) \\right] . } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1295,
+ 1972,
+ 1400,
+ 1972,
+ 1400,
+ 2007,
+ 1295,
+ 2007
+ ],
+ "score": 0.91,
+ "latex": "\\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } \\left( X _ { n } \\right)"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 688,
+ 1003,
+ 1096,
+ 1003,
+ 1096,
+ 1040,
+ 688,
+ 1040
+ ],
+ "score": 0.91,
+ "latex": "\\mathbb { E } _ { ( X , Y ) \\sim \\mu \\times \\nu } \\left[ d ( g ( \\bar { Y } ) , X ) H _ { \\varepsilon } ( \\bar { X } , Y ) \\right]"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 635,
+ 1723,
+ 1064,
+ 1723,
+ 1064,
+ 1760,
+ 635,
+ 1760
+ ],
+ "score": 0.9,
+ "latex": "( i d , \\bar { \\pi } _ { n } ^ { \\varepsilon _ { n } } ) \\# \\mu _ { n } ( i d , f ) \\# \\mu \\ w e a k l y ,"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 548,
+ 1666,
+ 591,
+ 1666,
+ 591,
+ 1698,
+ 548,
+ 1698
+ ],
+ "score": 0.9,
+ "latex": "\\pi _ { n } ^ { \\varepsilon _ { n } }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 298,
+ 1003,
+ 514,
+ 1003,
+ 514,
+ 1033,
+ 298,
+ 1033
+ ],
+ "score": 0.9,
+ "latex": "\\bar { d } : \\mathcal { X } \\times \\mathcal { X } \\overset { } { \\to } \\bar { \\mathbb { R } } ^ { + }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1198,
+ 1435,
+ 1231,
+ 1435,
+ 1231,
+ 1465,
+ 1198,
+ 1465
+ ],
+ "score": 0.89,
+ "latex": "\\bar { \\pi } _ { n } ^ { \\varepsilon }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 298,
+ 1665,
+ 372,
+ 1665,
+ 372,
+ 1693,
+ 298,
+ 1693
+ ],
+ "score": 0.89,
+ "latex": "d = c"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 688,
+ 1405,
+ 722,
+ 1405,
+ 722,
+ 1436,
+ 688,
+ 1436
+ ],
+ "score": 0.89,
+ "latex": "\\boldsymbol { \\pi } _ { n } ^ { \\varepsilon }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 453,
+ 426,
+ 934,
+ 426,
+ 934,
+ 464,
+ 453,
+ 464
+ ],
+ "score": 0.89,
+ "latex": "\\begin{array} { r } { \\theta \\theta - \\gamma \\sum _ { i j } H _ { \\varepsilon } ( x _ { i } , y _ { j } ) \\nabla _ { \\theta } d ( y _ { j } , f _ { \\theta } ( x _ { i } ) ) } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1144,
+ 1973,
+ 1227,
+ 1973,
+ 1227,
+ 2001,
+ 1144,
+ 2001
+ ],
+ "score": 0.89,
+ "latex": "Y \\sim \\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 430,
+ 1531,
+ 466,
+ 1531,
+ 466,
+ 1560,
+ 430,
+ 1560
+ ],
+ "score": 0.88,
+ "latex": "\\mathbb { R } ^ { d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1199,
+ 1531,
+ 1400,
+ 1531,
+ 1400,
+ 1568,
+ 1199,
+ 1568
+ ],
+ "score": 0.88,
+ "latex": "\\begin{array} { r } { \\mu _ { n } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\delta _ { x _ { i } } } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 779,
+ 1376,
+ 811,
+ 1376,
+ 811,
+ 1402,
+ 779,
+ 1402
+ ],
+ "score": 0.87,
+ "latex": "\\nu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 694,
+ 1377,
+ 729,
+ 1377,
+ 729,
+ 1403,
+ 694,
+ 1403
+ ],
+ "score": 0.87,
+ "latex": "\\mu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 871,
+ 1918,
+ 1031,
+ 1918,
+ 1031,
+ 1951,
+ 871,
+ 1951
+ ],
+ "score": 0.87,
+ "latex": "{ \\bar { \\pi } } _ { n } ^ { \\varepsilon _ { n } } \\# \\mu _ { n } \\to \\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 995,
+ 1499,
+ 1030,
+ 1499,
+ 1030,
+ 1530,
+ 995,
+ 1530
+ ],
+ "score": 0.86,
+ "latex": "\\mathbb { R } ^ { d }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 664,
+ 1607,
+ 696,
+ 1607,
+ 696,
+ 1634,
+ 664,
+ 1634
+ ],
+ "score": 0.86,
+ "latex": "\\nu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 758,
+ 688,
+ 779,
+ 688,
+ 779,
+ 719,
+ 758,
+ 719
+ ],
+ "score": 0.86,
+ "latex": "f"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 545,
+ 398,
+ 688,
+ 398,
+ 688,
+ 426,
+ 545,
+ 426
+ ],
+ "score": 0.86,
+ "latex": "( y _ { 1 } , \\cdots , y _ { n } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1168,
+ 1343,
+ 1188,
+ 1343,
+ 1188,
+ 1373,
+ 1168,
+ 1373
+ ],
+ "score": 0.86,
+ "latex": "f"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 545,
+ 364,
+ 692,
+ 364,
+ 692,
+ 396,
+ 545,
+ 396
+ ],
+ "score": 0.85,
+ "latex": "( x _ { 1 } , \\cdots , x _ { n } )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 748,
+ 722,
+ 769,
+ 722,
+ 769,
+ 750,
+ 748,
+ 750
+ ],
+ "score": 0.85,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1022,
+ 1378,
+ 1042,
+ 1378,
+ 1042,
+ 1403,
+ 1022,
+ 1403
+ ],
+ "score": 0.85,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 371,
+ 1764,
+ 391,
+ 1764,
+ 391,
+ 1795,
+ 371,
+ 1795
+ ],
+ "score": 0.84,
+ "latex": "f"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 573,
+ 1609,
+ 607,
+ 1609,
+ 607,
+ 1635,
+ 573,
+ 1635
+ ],
+ "score": 0.83,
+ "latex": "\\mu _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1369,
+ 688,
+ 1401,
+ 688,
+ 1401,
+ 715,
+ 1369,
+ 715
+ ],
+ "score": 0.83,
+ "latex": "\\pi ^ { \\varepsilon }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 980,
+ 1603,
+ 1030,
+ 1603,
+ 1030,
+ 1636,
+ 980,
+ 1636
+ ],
+ "score": 0.83,
+ "latex": "\\left( \\varepsilon _ { n } \\right)"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 874,
+ 919,
+ 894,
+ 919,
+ 894,
+ 944,
+ 874,
+ 944
+ ],
+ "score": 0.83,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 716,
+ 1285,
+ 737,
+ 1285,
+ 737,
+ 1313,
+ 716,
+ 1313
+ ],
+ "score": 0.82,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 883,
+ 310,
+ 903,
+ 310,
+ 903,
+ 336,
+ 883,
+ 336
+ ],
+ "score": 0.82,
+ "latex": "\\gamma"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 742,
+ 1469,
+ 763,
+ 1469,
+ 763,
+ 1496,
+ 742,
+ 1496
+ ],
+ "score": 0.81,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 606,
+ 2004,
+ 630,
+ 2004,
+ 630,
+ 2030,
+ 606,
+ 2030
+ ],
+ "score": 0.81,
+ "latex": "Y"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1367,
+ 1574,
+ 1400,
+ 1574,
+ 1400,
+ 1599,
+ 1367,
+ 1599
+ ],
+ "score": 0.81,
+ "latex": "\\pi _ { n }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 914,
+ 749,
+ 930,
+ 749,
+ 930,
+ 775,
+ 914,
+ 775
+ ],
+ "score": 0.8,
+ "latex": "\\theta"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 742,
+ 1347,
+ 763,
+ 1347,
+ 763,
+ 1374,
+ 742,
+ 1374
+ ],
+ "score": 0.8,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1197,
+ 553,
+ 1219,
+ 553,
+ 1219,
+ 581,
+ 1197,
+ 581
+ ],
+ "score": 0.8,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1273,
+ 274,
+ 1302,
+ 274,
+ 1302,
+ 305,
+ 1273,
+ 305
+ ],
+ "score": 0.8,
+ "latex": "f _ { \\theta }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1242,
+ 979,
+ 1261,
+ 979,
+ 1261,
+ 1006,
+ 1242,
+ 1006
+ ],
+ "score": 0.79,
+ "latex": "g"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 600,
+ 1084,
+ 622,
+ 1084,
+ 622,
+ 1110,
+ 600,
+ 1110
+ ],
+ "score": 0.78,
+ "latex": "\\bar { \\pi }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1238,
+ 1377,
+ 1257,
+ 1377,
+ 1257,
+ 1399,
+ 1238,
+ 1399
+ ],
+ "score": 0.78,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 803,
+ 1854,
+ 820,
+ 1854,
+ 820,
+ 1876,
+ 803,
+ 1876
+ ],
+ "score": 0.78,
+ "latex": "\\varepsilon"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 696,
+ 1854,
+ 715,
+ 1854,
+ 715,
+ 1876,
+ 696,
+ 1876
+ ],
+ "score": 0.78,
+ "latex": "n"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 706,
+ 309,
+ 726,
+ 309,
+ 726,
+ 331,
+ 706,
+ 331
+ ],
+ "score": 0.77,
+ "latex": "n"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 584,
+ 1056,
+ 605,
+ 1056,
+ 605,
+ 1078,
+ 584,
+ 1078
+ ],
+ "score": 0.75,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 619,
+ 1346,
+ 641,
+ 1346,
+ 641,
+ 1370,
+ 619,
+ 1370
+ ],
+ "score": 0.75,
+ "latex": "\\pi"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 927,
+ 919,
+ 947,
+ 919,
+ 947,
+ 940,
+ 927,
+ 940
+ ],
+ "score": 0.75,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 813,
+ 1346,
+ 832,
+ 1346,
+ 832,
+ 1369,
+ 813,
+ 1369
+ ],
+ "score": 0.75,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1116,
+ 279,
+ 1137,
+ 279,
+ 1137,
+ 300,
+ 1116,
+ 300
+ ],
+ "score": 0.74,
+ "latex": "u"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 813,
+ 1469,
+ 831,
+ 1469,
+ 831,
+ 1491,
+ 813,
+ 1491
+ ],
+ "score": 0.74,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 498,
+ 1507,
+ 520,
+ 1507,
+ 520,
+ 1534,
+ 498,
+ 1534
+ ],
+ "score": 0.73,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 758,
+ 370,
+ 778,
+ 370,
+ 778,
+ 396,
+ 758,
+ 396
+ ],
+ "score": 0.73,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 944,
+ 1769,
+ 964,
+ 1769,
+ 964,
+ 1795,
+ 944,
+ 1795
+ ],
+ "score": 0.73,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 977,
+ 1342,
+ 1091,
+ 1342,
+ 1091,
+ 1374,
+ 977,
+ 1374
+ ],
+ "score": 0.73,
+ "latex": "( \\operatorname { i d } , f ) \\# \\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 894,
+ 1609,
+ 914,
+ 1609,
+ 914,
+ 1631,
+ 894,
+ 1631
+ ],
+ "score": 0.72,
+ "latex": "n"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 963,
+ 1208,
+ 982,
+ 1208,
+ 982,
+ 1230,
+ 963,
+ 1230
+ ],
+ "score": 0.72,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1184,
+ 1088,
+ 1203,
+ 1088,
+ 1203,
+ 1109,
+ 1184,
+ 1109
+ ],
+ "score": 0.71,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1099,
+ 1507,
+ 1118,
+ 1507,
+ 1118,
+ 1529,
+ 1099,
+ 1529
+ ],
+ "score": 0.71,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1187,
+ 279,
+ 1206,
+ 279,
+ 1206,
+ 300,
+ 1187,
+ 300
+ ],
+ "score": 0.68,
+ "latex": "v"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 835,
+ 280,
+ 851,
+ 280,
+ 851,
+ 300,
+ 835,
+ 300
+ ],
+ "score": 0.65,
+ "latex": "c"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1016,
+ 1768,
+ 1034,
+ 1768,
+ 1034,
+ 1790,
+ 1016,
+ 1790
+ ],
+ "score": 0.64,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 754,
+ 401,
+ 773,
+ 401,
+ 773,
+ 422,
+ 754,
+ 422
+ ],
+ "score": 0.62,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 615,
+ 280,
+ 636,
+ 280,
+ 636,
+ 304,
+ 615,
+ 304
+ ],
+ "score": 0.58,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 799,
+ 1573,
+ 820,
+ 1573,
+ 820,
+ 1600,
+ 799,
+ 1600
+ ],
+ "score": 0.57,
+ "latex": "\\mu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1384,
+ 1669,
+ 1402,
+ 1669,
+ 1402,
+ 1692,
+ 1384,
+ 1692
+ ],
+ "score": 0.54,
+ "latex": "a"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 873,
+ 1574,
+ 892,
+ 1574,
+ 892,
+ 1595,
+ 873,
+ 1595
+ ],
+ "score": 0.52,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 615,
+ 278,
+ 672,
+ 278,
+ 672,
+ 305,
+ 615,
+ 305
+ ],
+ "score": 0.39,
+ "latex": "\\mu , \\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 652,
+ 280,
+ 671,
+ 280,
+ 671,
+ 302,
+ 652,
+ 302
+ ],
+ "score": 0.29,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 5,
+ "poly": [
+ 314,
+ 222,
+ 1403,
+ 222,
+ 1403,
+ 496,
+ 314,
+ 496
+ ],
+ "score": 0.442,
+ "html": "| lgorithm 2 Optimal map learning with SGD |
| Inputs: input measures μ, V; cost function c; dual optimal variables u and v; map fe parame- |
| terized as a deep NN; batch size n; learning rate y. while not converged do |
| sample a batch (x1,·· ,xn) from μ |
| sample a batch (y1,:.,yn) from v |
| updateθ←0-γ∑ijHe(xi,yj)Vθd(yj,fe(xi)) end while |
"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 73.0,
+ 818.0,
+ 73.0,
+ 818.0,
+ 108.0,
+ 295.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 839.0,
+ 2086.0,
+ 860.0,
+ 2086.0,
+ 860.0,
+ 2117.0,
+ 839.0,
+ 2117.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 226.0,
+ 816.0,
+ 226.0,
+ 816.0,
+ 268.0,
+ 296.0,
+ 268.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 839.0,
+ 2086.0,
+ 860.0,
+ 2086.0,
+ 860.0,
+ 2118.0,
+ 839.0,
+ 2118.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1248.0,
+ 852.0,
+ 1248.0,
+ 852.0,
+ 1286.0,
+ 295.0,
+ 1286.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 945.0,
+ 1248.0,
+ 997.0,
+ 1248.0,
+ 997.0,
+ 1286.0,
+ 945.0,
+ 1286.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1220.0,
+ 1248.0,
+ 1405.0,
+ 1248.0,
+ 1405.0,
+ 1286.0,
+ 1220.0,
+ 1286.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1281.0,
+ 715.0,
+ 1281.0,
+ 715.0,
+ 1315.0,
+ 294.0,
+ 1315.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 738.0,
+ 1281.0,
+ 1405.0,
+ 1281.0,
+ 1405.0,
+ 1315.0,
+ 738.0,
+ 1315.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1311.0,
+ 1403.0,
+ 1311.0,
+ 1403.0,
+ 1345.0,
+ 294.0,
+ 1345.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1337.0,
+ 618.0,
+ 1337.0,
+ 618.0,
+ 1379.0,
+ 291.0,
+ 1379.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 642.0,
+ 1337.0,
+ 741.0,
+ 1337.0,
+ 741.0,
+ 1379.0,
+ 642.0,
+ 1379.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 764.0,
+ 1337.0,
+ 812.0,
+ 1337.0,
+ 812.0,
+ 1379.0,
+ 764.0,
+ 1379.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 833.0,
+ 1337.0,
+ 976.0,
+ 1337.0,
+ 976.0,
+ 1379.0,
+ 833.0,
+ 1379.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1092.0,
+ 1337.0,
+ 1167.0,
+ 1337.0,
+ 1167.0,
+ 1379.0,
+ 1092.0,
+ 1379.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1189.0,
+ 1337.0,
+ 1405.0,
+ 1337.0,
+ 1405.0,
+ 1379.0,
+ 1189.0,
+ 1379.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1371.0,
+ 693.0,
+ 1371.0,
+ 693.0,
+ 1408.0,
+ 293.0,
+ 1408.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 730.0,
+ 1371.0,
+ 778.0,
+ 1371.0,
+ 778.0,
+ 1408.0,
+ 730.0,
+ 1408.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 812.0,
+ 1371.0,
+ 1021.0,
+ 1371.0,
+ 1021.0,
+ 1408.0,
+ 812.0,
+ 1408.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1043.0,
+ 1371.0,
+ 1237.0,
+ 1371.0,
+ 1237.0,
+ 1408.0,
+ 1043.0,
+ 1408.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1258.0,
+ 1371.0,
+ 1405.0,
+ 1371.0,
+ 1405.0,
+ 1408.0,
+ 1258.0,
+ 1408.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1401.0,
+ 687.0,
+ 1401.0,
+ 687.0,
+ 1439.0,
+ 293.0,
+ 1439.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 723.0,
+ 1401.0,
+ 961.0,
+ 1401.0,
+ 961.0,
+ 1439.0,
+ 723.0,
+ 1439.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1135.0,
+ 1401.0,
+ 1206.0,
+ 1401.0,
+ 1206.0,
+ 1439.0,
+ 1135.0,
+ 1439.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1394.0,
+ 1401.0,
+ 1405.0,
+ 1401.0,
+ 1405.0,
+ 1439.0,
+ 1394.0,
+ 1439.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1429.0,
+ 1197.0,
+ 1429.0,
+ 1197.0,
+ 1473.0,
+ 291.0,
+ 1473.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1232.0,
+ 1429.0,
+ 1407.0,
+ 1429.0,
+ 1407.0,
+ 1473.0,
+ 1232.0,
+ 1473.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1464.0,
+ 741.0,
+ 1464.0,
+ 741.0,
+ 1498.0,
+ 296.0,
+ 1498.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 764.0,
+ 1464.0,
+ 812.0,
+ 1464.0,
+ 812.0,
+ 1498.0,
+ 764.0,
+ 1498.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 832.0,
+ 1464.0,
+ 1112.0,
+ 1464.0,
+ 1112.0,
+ 1498.0,
+ 832.0,
+ 1498.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1498.0,
+ 497.0,
+ 1498.0,
+ 497.0,
+ 1538.0,
+ 295.0,
+ 1538.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 521.0,
+ 1498.0,
+ 994.0,
+ 1498.0,
+ 994.0,
+ 1538.0,
+ 521.0,
+ 1538.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1031.0,
+ 1498.0,
+ 1098.0,
+ 1498.0,
+ 1098.0,
+ 1538.0,
+ 1031.0,
+ 1538.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1119.0,
+ 1498.0,
+ 1406.0,
+ 1498.0,
+ 1406.0,
+ 1538.0,
+ 1119.0,
+ 1538.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 289.0,
+ 1526.0,
+ 429.0,
+ 1526.0,
+ 429.0,
+ 1577.0,
+ 289.0,
+ 1577.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 467.0,
+ 1526.0,
+ 1198.0,
+ 1526.0,
+ 1198.0,
+ 1577.0,
+ 467.0,
+ 1577.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1401.0,
+ 1526.0,
+ 1407.0,
+ 1526.0,
+ 1407.0,
+ 1577.0,
+ 1401.0,
+ 1577.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 280.0,
+ 1545.0,
+ 346.0,
+ 1545.0,
+ 346.0,
+ 1617.0,
+ 280.0,
+ 1617.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 551.0,
+ 1545.0,
+ 798.0,
+ 1545.0,
+ 798.0,
+ 1617.0,
+ 551.0,
+ 1617.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 821.0,
+ 1545.0,
+ 872.0,
+ 1545.0,
+ 872.0,
+ 1617.0,
+ 821.0,
+ 1617.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 893.0,
+ 1545.0,
+ 1366.0,
+ 1545.0,
+ 1366.0,
+ 1617.0,
+ 893.0,
+ 1617.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1401.0,
+ 1545.0,
+ 1417.0,
+ 1545.0,
+ 1417.0,
+ 1617.0,
+ 1401.0,
+ 1617.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1604.0,
+ 572.0,
+ 1604.0,
+ 572.0,
+ 1639.0,
+ 295.0,
+ 1639.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 608.0,
+ 1604.0,
+ 663.0,
+ 1604.0,
+ 663.0,
+ 1639.0,
+ 608.0,
+ 1639.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 697.0,
+ 1604.0,
+ 893.0,
+ 1604.0,
+ 893.0,
+ 1639.0,
+ 697.0,
+ 1639.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 915.0,
+ 1604.0,
+ 979.0,
+ 1604.0,
+ 979.0,
+ 1639.0,
+ 915.0,
+ 1639.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1031.0,
+ 1604.0,
+ 1406.0,
+ 1604.0,
+ 1406.0,
+ 1639.0,
+ 1031.0,
+ 1639.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1633.0,
+ 810.0,
+ 1633.0,
+ 810.0,
+ 1668.0,
+ 294.0,
+ 1668.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 855.0,
+ 1633.0,
+ 1406.0,
+ 1633.0,
+ 1406.0,
+ 1668.0,
+ 855.0,
+ 1668.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 290.0,
+ 1659.0,
+ 297.0,
+ 1659.0,
+ 297.0,
+ 1706.0,
+ 290.0,
+ 1706.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 373.0,
+ 1659.0,
+ 547.0,
+ 1659.0,
+ 547.0,
+ 1706.0,
+ 373.0,
+ 1706.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 592.0,
+ 1659.0,
+ 1383.0,
+ 1659.0,
+ 1383.0,
+ 1706.0,
+ 592.0,
+ 1706.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1403.0,
+ 1659.0,
+ 1410.0,
+ 1659.0,
+ 1410.0,
+ 1706.0,
+ 1403.0,
+ 1706.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1692.0,
+ 451.0,
+ 1692.0,
+ 451.0,
+ 1732.0,
+ 293.0,
+ 1732.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1050.0,
+ 583.0,
+ 1050.0,
+ 583.0,
+ 1086.0,
+ 294.0,
+ 1086.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 606.0,
+ 1050.0,
+ 1405.0,
+ 1050.0,
+ 1405.0,
+ 1086.0,
+ 606.0,
+ 1086.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1081.0,
+ 599.0,
+ 1081.0,
+ 599.0,
+ 1116.0,
+ 294.0,
+ 1116.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 623.0,
+ 1081.0,
+ 1183.0,
+ 1081.0,
+ 1183.0,
+ 1116.0,
+ 623.0,
+ 1116.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1204.0,
+ 1081.0,
+ 1406.0,
+ 1081.0,
+ 1406.0,
+ 1116.0,
+ 1204.0,
+ 1116.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1110.0,
+ 1404.0,
+ 1110.0,
+ 1404.0,
+ 1147.0,
+ 293.0,
+ 1147.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1142.0,
+ 1405.0,
+ 1142.0,
+ 1405.0,
+ 1178.0,
+ 294.0,
+ 1178.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1170.0,
+ 1405.0,
+ 1170.0,
+ 1405.0,
+ 1207.0,
+ 293.0,
+ 1207.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1202.0,
+ 962.0,
+ 1202.0,
+ 962.0,
+ 1238.0,
+ 294.0,
+ 1238.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 983.0,
+ 1202.0,
+ 1144.0,
+ 1202.0,
+ 1144.0,
+ 1238.0,
+ 983.0,
+ 1238.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 547.0,
+ 1196.0,
+ 547.0,
+ 1196.0,
+ 583.0,
+ 292.0,
+ 583.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1220.0,
+ 547.0,
+ 1405.0,
+ 547.0,
+ 1405.0,
+ 583.0,
+ 1220.0,
+ 583.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 577.0,
+ 1405.0,
+ 577.0,
+ 1405.0,
+ 617.0,
+ 292.0,
+ 617.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 609.0,
+ 1405.0,
+ 609.0,
+ 1405.0,
+ 646.0,
+ 295.0,
+ 646.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 639.0,
+ 1078.0,
+ 639.0,
+ 1078.0,
+ 675.0,
+ 295.0,
+ 675.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 878.0,
+ 372.0,
+ 878.0,
+ 372.0,
+ 918.0,
+ 294.0,
+ 918.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 599.0,
+ 878.0,
+ 1406.0,
+ 878.0,
+ 1406.0,
+ 918.0,
+ 599.0,
+ 918.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 910.0,
+ 873.0,
+ 910.0,
+ 873.0,
+ 947.0,
+ 295.0,
+ 947.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 895.0,
+ 910.0,
+ 926.0,
+ 910.0,
+ 926.0,
+ 947.0,
+ 895.0,
+ 947.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 948.0,
+ 910.0,
+ 1405.0,
+ 910.0,
+ 1405.0,
+ 947.0,
+ 948.0,
+ 947.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 943.0,
+ 1404.0,
+ 943.0,
+ 1404.0,
+ 977.0,
+ 296.0,
+ 977.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 973.0,
+ 1241.0,
+ 973.0,
+ 1241.0,
+ 1010.0,
+ 295.0,
+ 1010.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1262.0,
+ 973.0,
+ 1406.0,
+ 973.0,
+ 1406.0,
+ 1010.0,
+ 1262.0,
+ 1010.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 290.0,
+ 997.0,
+ 297.0,
+ 997.0,
+ 297.0,
+ 1046.0,
+ 290.0,
+ 1046.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 515.0,
+ 997.0,
+ 687.0,
+ 997.0,
+ 687.0,
+ 1046.0,
+ 515.0,
+ 1046.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1097.0,
+ 997.0,
+ 1111.0,
+ 997.0,
+ 1111.0,
+ 1046.0,
+ 1097.0,
+ 1046.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 685.0,
+ 757.0,
+ 685.0,
+ 757.0,
+ 722.0,
+ 296.0,
+ 722.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 780.0,
+ 685.0,
+ 1368.0,
+ 685.0,
+ 1368.0,
+ 722.0,
+ 780.0,
+ 722.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 717.0,
+ 747.0,
+ 717.0,
+ 747.0,
+ 751.0,
+ 296.0,
+ 751.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 770.0,
+ 717.0,
+ 1404.0,
+ 717.0,
+ 1404.0,
+ 751.0,
+ 770.0,
+ 751.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 748.0,
+ 913.0,
+ 748.0,
+ 913.0,
+ 782.0,
+ 295.0,
+ 782.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 931.0,
+ 748.0,
+ 942.0,
+ 748.0,
+ 942.0,
+ 782.0,
+ 931.0,
+ 782.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1816.0,
+ 1405.0,
+ 1816.0,
+ 1405.0,
+ 1854.0,
+ 295.0,
+ 1854.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1848.0,
+ 695.0,
+ 1848.0,
+ 695.0,
+ 1883.0,
+ 295.0,
+ 1883.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 716.0,
+ 1848.0,
+ 802.0,
+ 1848.0,
+ 802.0,
+ 1883.0,
+ 716.0,
+ 1883.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 821.0,
+ 1848.0,
+ 1405.0,
+ 1848.0,
+ 1405.0,
+ 1883.0,
+ 821.0,
+ 1883.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1876.0,
+ 1334.0,
+ 1876.0,
+ 1334.0,
+ 1917.0,
+ 292.0,
+ 1917.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1964.0,
+ 975.0,
+ 1964.0,
+ 975.0,
+ 2011.0,
+ 291.0,
+ 2011.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1092.0,
+ 1964.0,
+ 1143.0,
+ 1964.0,
+ 1143.0,
+ 2011.0,
+ 1092.0,
+ 2011.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1228.0,
+ 1964.0,
+ 1294.0,
+ 1964.0,
+ 1294.0,
+ 2011.0,
+ 1228.0,
+ 2011.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 2003.0,
+ 605.0,
+ 2003.0,
+ 605.0,
+ 2033.0,
+ 295.0,
+ 2033.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 631.0,
+ 2003.0,
+ 639.0,
+ 2003.0,
+ 639.0,
+ 2033.0,
+ 631.0,
+ 2033.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1912.0,
+ 870.0,
+ 1912.0,
+ 870.0,
+ 1956.0,
+ 295.0,
+ 1956.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1032.0,
+ 1912.0,
+ 1138.0,
+ 1912.0,
+ 1138.0,
+ 1956.0,
+ 1032.0,
+ 1956.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1759.0,
+ 370.0,
+ 1759.0,
+ 370.0,
+ 1799.0,
+ 294.0,
+ 1799.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 392.0,
+ 1759.0,
+ 943.0,
+ 1759.0,
+ 943.0,
+ 1799.0,
+ 392.0,
+ 1799.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 965.0,
+ 1759.0,
+ 1015.0,
+ 1759.0,
+ 1015.0,
+ 1799.0,
+ 965.0,
+ 1799.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1035.0,
+ 1759.0,
+ 1044.0,
+ 1759.0,
+ 1044.0,
+ 1799.0,
+ 1035.0,
+ 1799.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 6,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1499,
+ 1404,
+ 1499,
+ 1404,
+ 1803,
+ 298,
+ 1803
+ ],
+ "score": 0.983
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1033,
+ 1404,
+ 1033,
+ 1404,
+ 1398,
+ 297,
+ 1398
+ ],
+ "score": 0.983
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1820,
+ 1403,
+ 1820,
+ 1403,
+ 2034,
+ 298,
+ 2034
+ ],
+ "score": 0.978
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 731,
+ 1403,
+ 731,
+ 1403,
+ 854,
+ 299,
+ 854
+ ],
+ "score": 0.974
+ },
+ {
+ "category_id": 3,
+ "poly": [
+ 313,
+ 225,
+ 1383,
+ 225,
+ 1383,
+ 520,
+ 313,
+ 520
+ ],
+ "score": 0.97
+ },
+ {
+ "category_id": 4,
+ "poly": [
+ 296,
+ 545,
+ 1403,
+ 545,
+ 1403,
+ 668,
+ 296,
+ 668
+ ],
+ "score": 0.966
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 302,
+ 903,
+ 742,
+ 903,
+ 742,
+ 938,
+ 302,
+ 938
+ ],
+ "score": 0.896
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 302,
+ 1441,
+ 796,
+ 1441,
+ 796,
+ 1470,
+ 302,
+ 1470
+ ],
+ "score": 0.89
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 299,
+ 76,
+ 814,
+ 76,
+ 814,
+ 104,
+ 299,
+ 104
+ ],
+ "score": 0.888
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 302,
+ 974,
+ 878,
+ 974,
+ 878,
+ 1005,
+ 302,
+ 1005
+ ],
+ "score": 0.848
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 841,
+ 2089,
+ 858,
+ 2089,
+ 858,
+ 2111,
+ 841,
+ 2111
+ ],
+ "score": 0.785
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1202,
+ 1093,
+ 1265,
+ 1093,
+ 1265,
+ 1127,
+ 1202,
+ 1127
+ ],
+ "score": 0.91,
+ "latex": "{ \\mathcal { O } } ( n )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 684,
+ 1216,
+ 806,
+ 1216,
+ 806,
+ 1245,
+ 684,
+ 1245
+ ],
+ "score": 0.91,
+ "latex": "5 0 0 \\times 5 0 0"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 297,
+ 1337,
+ 532,
+ 1337,
+ 532,
+ 1371,
+ 297,
+ 1371
+ ],
+ "score": 0.9,
+ "latex": "\\varepsilon = \\{ 0 . 0 \\bar { 2 } 5 , 0 . 1 , 1 . \\}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 688,
+ 1275,
+ 734,
+ 1275,
+ 734,
+ 1304,
+ 688,
+ 1304
+ ],
+ "score": 0.89,
+ "latex": "\\mathbb { R } ^ { 3 1 }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1058,
+ 1913,
+ 1162,
+ 1913,
+ 1162,
+ 1942,
+ 1058,
+ 1942
+ ],
+ "score": 0.86,
+ "latex": "( 1 6 \\times 1 6 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1293,
+ 1973,
+ 1399,
+ 1973,
+ 1399,
+ 2003,
+ 1293,
+ 2003
+ ],
+ "score": 0.85,
+ "latex": "( 3 2 \\times 3 2 )"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 868,
+ 606,
+ 1016,
+ 606,
+ 1016,
+ 640,
+ 868,
+ 640
+ ],
+ "score": 0.85,
+ "latex": "\\{ 5 . , 2 0 . , 2 0 . \\}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1208,
+ 606,
+ 1401,
+ 606,
+ 1401,
+ 641,
+ 1208,
+ 641
+ ],
+ "score": 0.79,
+ "latex": "\\{ 1 0 2 4 , 5 0 0 , 1 0 0 \\}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 374,
+ 1130,
+ 395,
+ 1130,
+ 395,
+ 1152,
+ 374,
+ 1152
+ ],
+ "score": 0.7,
+ "latex": "n"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 771,
+ 1130,
+ 790,
+ 1130,
+ 790,
+ 1152,
+ 771,
+ 1152
+ ],
+ "score": 0.65,
+ "latex": "\\nu"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 728,
+ 1749,
+ 745,
+ 1749,
+ 745,
+ 1770,
+ 728,
+ 1770
+ ],
+ "score": 0.62,
+ "latex": "c"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1379,
+ 1282,
+ 1395,
+ 1282,
+ 1395,
+ 1304,
+ 1379,
+ 1304
+ ],
+ "score": 0.46,
+ "latex": "c"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 448.0,
+ 227.0,
+ 588.0,
+ 227.0,
+ 588.0,
+ 254.0,
+ 448.0,
+ 254.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1201.0,
+ 230.0,
+ 1263.0,
+ 230.0,
+ 1263.0,
+ 255.0,
+ 1201.0,
+ 255.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 312.0,
+ 249.0,
+ 385.0,
+ 249.0,
+ 385.0,
+ 400.0,
+ 312.0,
+ 400.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 686.0,
+ 252.0,
+ 737.0,
+ 252.0,
+ 737.0,
+ 307.0,
+ 686.0,
+ 307.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 266.0,
+ 1093.0,
+ 266.0,
+ 1093.0,
+ 295.0,
+ 1027.0,
+ 295.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1149.0,
+ 259.0,
+ 1200.0,
+ 259.0,
+ 1200.0,
+ 279.0,
+ 1149.0,
+ 279.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 327.0,
+ 307.0,
+ 379.0,
+ 307.0,
+ 379.0,
+ 332.0,
+ 327.0,
+ 332.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 687.0,
+ 302.0,
+ 736.0,
+ 302.0,
+ 736.0,
+ 333.0,
+ 687.0,
+ 333.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 293.0,
+ 1094.0,
+ 293.0,
+ 1094.0,
+ 322.0,
+ 1027.0,
+ 322.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 331.0,
+ 337.0,
+ 379.0,
+ 337.0,
+ 379.0,
+ 363.0,
+ 331.0,
+ 363.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 687.0,
+ 331.0,
+ 736.0,
+ 331.0,
+ 736.0,
+ 356.0,
+ 687.0,
+ 356.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 321.0,
+ 1094.0,
+ 321.0,
+ 1094.0,
+ 349.0,
+ 1027.0,
+ 349.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 330.0,
+ 364.0,
+ 380.0,
+ 364.0,
+ 380.0,
+ 393.0,
+ 330.0,
+ 393.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 687.0,
+ 354.0,
+ 737.0,
+ 354.0,
+ 737.0,
+ 384.0,
+ 687.0,
+ 384.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 350.0,
+ 1093.0,
+ 350.0,
+ 1093.0,
+ 376.0,
+ 1027.0,
+ 376.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 312.0,
+ 384.0,
+ 381.0,
+ 384.0,
+ 381.0,
+ 467.0,
+ 312.0,
+ 467.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 554.0,
+ 411.0,
+ 654.0,
+ 411.0,
+ 654.0,
+ 465.0,
+ 554.0,
+ 465.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 686.0,
+ 381.0,
+ 737.0,
+ 381.0,
+ 737.0,
+ 461.0,
+ 686.0,
+ 461.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 910.0,
+ 411.0,
+ 1011.0,
+ 411.0,
+ 1011.0,
+ 465.0,
+ 910.0,
+ 465.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 376.0,
+ 1094.0,
+ 376.0,
+ 1094.0,
+ 459.0,
+ 1027.0,
+ 459.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1267.0,
+ 411.0,
+ 1368.0,
+ 411.0,
+ 1368.0,
+ 465.0,
+ 1267.0,
+ 465.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 330.0,
+ 451.0,
+ 381.0,
+ 451.0,
+ 381.0,
+ 481.0,
+ 330.0,
+ 481.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 687.0,
+ 456.0,
+ 743.0,
+ 456.0,
+ 743.0,
+ 486.0,
+ 687.0,
+ 486.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 458.0,
+ 1094.0,
+ 458.0,
+ 1094.0,
+ 486.0,
+ 1027.0,
+ 486.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 380.0,
+ 477.0,
+ 401.0,
+ 477.0,
+ 401.0,
+ 500.0,
+ 380.0,
+ 500.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 409.0,
+ 475.0,
+ 666.0,
+ 475.0,
+ 666.0,
+ 502.0,
+ 409.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 736.0,
+ 477.0,
+ 757.0,
+ 477.0,
+ 757.0,
+ 498.0,
+ 736.0,
+ 498.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 765.0,
+ 475.0,
+ 1025.0,
+ 475.0,
+ 1025.0,
+ 502.0,
+ 765.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1093.0,
+ 477.0,
+ 1116.0,
+ 477.0,
+ 1116.0,
+ 501.0,
+ 1093.0,
+ 501.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1122.0,
+ 475.0,
+ 1381.0,
+ 475.0,
+ 1381.0,
+ 502.0,
+ 1122.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 452.0,
+ 495.0,
+ 586.0,
+ 495.0,
+ 586.0,
+ 522.0,
+ 452.0,
+ 522.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 809.0,
+ 495.0,
+ 943.0,
+ 495.0,
+ 943.0,
+ 522.0,
+ 809.0,
+ 522.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1167.0,
+ 495.0,
+ 1301.0,
+ 495.0,
+ 1301.0,
+ 522.0,
+ 1167.0,
+ 522.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 826.25,
+ 225.0,
+ 924.25,
+ 225.0,
+ 924.25,
+ 258.0,
+ 826.25,
+ 258.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 545.0,
+ 1404.0,
+ 545.0,
+ 1404.0,
+ 577.0,
+ 295.0,
+ 577.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 575.0,
+ 1405.0,
+ 575.0,
+ 1405.0,
+ 611.0,
+ 294.0,
+ 611.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 607.0,
+ 867.0,
+ 607.0,
+ 867.0,
+ 641.0,
+ 295.0,
+ 641.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1017.0,
+ 607.0,
+ 1207.0,
+ 607.0,
+ 1207.0,
+ 641.0,
+ 1017.0,
+ 641.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 637.0,
+ 1101.0,
+ 637.0,
+ 1101.0,
+ 670.0,
+ 294.0,
+ 670.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 898.0,
+ 747.0,
+ 898.0,
+ 747.0,
+ 944.0,
+ 292.0,
+ 944.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1439.0,
+ 802.0,
+ 1439.0,
+ 802.0,
+ 1475.0,
+ 295.0,
+ 1475.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 73.0,
+ 818.0,
+ 73.0,
+ 818.0,
+ 108.0,
+ 296.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 973.0,
+ 885.0,
+ 973.0,
+ 885.0,
+ 1009.0,
+ 295.0,
+ 1009.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 839.0,
+ 2086.0,
+ 860.0,
+ 2086.0,
+ 860.0,
+ 2116.0,
+ 839.0,
+ 2116.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1497.0,
+ 1406.0,
+ 1497.0,
+ 1406.0,
+ 1534.0,
+ 295.0,
+ 1534.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1531.0,
+ 1404.0,
+ 1531.0,
+ 1404.0,
+ 1563.0,
+ 296.0,
+ 1563.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1561.0,
+ 1404.0,
+ 1561.0,
+ 1404.0,
+ 1593.0,
+ 294.0,
+ 1593.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1590.0,
+ 1404.0,
+ 1590.0,
+ 1404.0,
+ 1626.0,
+ 294.0,
+ 1626.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1622.0,
+ 1405.0,
+ 1622.0,
+ 1405.0,
+ 1654.0,
+ 296.0,
+ 1654.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1652.0,
+ 1406.0,
+ 1652.0,
+ 1406.0,
+ 1686.0,
+ 294.0,
+ 1686.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1676.0,
+ 1408.0,
+ 1676.0,
+ 1408.0,
+ 1720.0,
+ 291.0,
+ 1720.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1714.0,
+ 1405.0,
+ 1714.0,
+ 1405.0,
+ 1746.0,
+ 294.0,
+ 1746.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1742.0,
+ 727.0,
+ 1742.0,
+ 727.0,
+ 1777.0,
+ 293.0,
+ 1777.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 746.0,
+ 1742.0,
+ 1405.0,
+ 1742.0,
+ 1405.0,
+ 1777.0,
+ 746.0,
+ 1777.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1772.0,
+ 518.0,
+ 1772.0,
+ 518.0,
+ 1806.0,
+ 293.0,
+ 1806.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1030.0,
+ 1405.0,
+ 1030.0,
+ 1405.0,
+ 1069.0,
+ 294.0,
+ 1069.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1064.0,
+ 1405.0,
+ 1064.0,
+ 1405.0,
+ 1098.0,
+ 295.0,
+ 1098.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1091.0,
+ 1201.0,
+ 1091.0,
+ 1201.0,
+ 1131.0,
+ 291.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1266.0,
+ 1091.0,
+ 1404.0,
+ 1091.0,
+ 1404.0,
+ 1131.0,
+ 1266.0,
+ 1131.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1125.0,
+ 373.0,
+ 1125.0,
+ 373.0,
+ 1159.0,
+ 295.0,
+ 1159.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 396.0,
+ 1125.0,
+ 770.0,
+ 1125.0,
+ 770.0,
+ 1159.0,
+ 396.0,
+ 1159.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 791.0,
+ 1125.0,
+ 1405.0,
+ 1125.0,
+ 1405.0,
+ 1159.0,
+ 791.0,
+ 1159.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1155.0,
+ 1406.0,
+ 1155.0,
+ 1406.0,
+ 1189.0,
+ 295.0,
+ 1189.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1184.0,
+ 1407.0,
+ 1184.0,
+ 1407.0,
+ 1222.0,
+ 294.0,
+ 1222.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1214.0,
+ 683.0,
+ 1214.0,
+ 683.0,
+ 1252.0,
+ 294.0,
+ 1252.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 807.0,
+ 1214.0,
+ 1406.0,
+ 1214.0,
+ 1406.0,
+ 1252.0,
+ 807.0,
+ 1252.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1245.0,
+ 1405.0,
+ 1245.0,
+ 1405.0,
+ 1281.0,
+ 292.0,
+ 1281.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1272.0,
+ 687.0,
+ 1272.0,
+ 687.0,
+ 1312.0,
+ 292.0,
+ 1312.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 735.0,
+ 1272.0,
+ 1378.0,
+ 1272.0,
+ 1378.0,
+ 1312.0,
+ 735.0,
+ 1312.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1396.0,
+ 1272.0,
+ 1406.0,
+ 1272.0,
+ 1406.0,
+ 1312.0,
+ 1396.0,
+ 1312.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1308.0,
+ 1404.0,
+ 1308.0,
+ 1404.0,
+ 1341.0,
+ 295.0,
+ 1341.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1337.0,
+ 296.0,
+ 1337.0,
+ 296.0,
+ 1374.0,
+ 291.0,
+ 1374.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 533.0,
+ 1337.0,
+ 1406.0,
+ 1337.0,
+ 1406.0,
+ 1374.0,
+ 533.0,
+ 1374.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1368.0,
+ 580.0,
+ 1368.0,
+ 580.0,
+ 1402.0,
+ 293.0,
+ 1402.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1820.0,
+ 1404.0,
+ 1820.0,
+ 1404.0,
+ 1855.0,
+ 294.0,
+ 1855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1848.0,
+ 1405.0,
+ 1848.0,
+ 1405.0,
+ 1885.0,
+ 293.0,
+ 1885.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1879.0,
+ 1404.0,
+ 1879.0,
+ 1404.0,
+ 1916.0,
+ 292.0,
+ 1916.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1911.0,
+ 1057.0,
+ 1911.0,
+ 1057.0,
+ 1947.0,
+ 292.0,
+ 1947.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1163.0,
+ 1911.0,
+ 1405.0,
+ 1911.0,
+ 1405.0,
+ 1947.0,
+ 1163.0,
+ 1947.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1942.0,
+ 1405.0,
+ 1942.0,
+ 1405.0,
+ 1976.0,
+ 293.0,
+ 1976.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1972.0,
+ 1292.0,
+ 1972.0,
+ 1292.0,
+ 2007.0,
+ 294.0,
+ 2007.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1400.0,
+ 1972.0,
+ 1404.0,
+ 1972.0,
+ 1404.0,
+ 2007.0,
+ 1400.0,
+ 2007.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 2001.0,
+ 1405.0,
+ 2001.0,
+ 1405.0,
+ 2039.0,
+ 293.0,
+ 2039.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 730.0,
+ 1405.0,
+ 730.0,
+ 1405.0,
+ 765.0,
+ 294.0,
+ 765.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 761.0,
+ 1404.0,
+ 761.0,
+ 1404.0,
+ 797.0,
+ 294.0,
+ 797.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 792.0,
+ 1407.0,
+ 792.0,
+ 1407.0,
+ 829.0,
+ 294.0,
+ 829.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 824.0,
+ 391.0,
+ 824.0,
+ 391.0,
+ 855.0,
+ 293.0,
+ 855.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 7,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 5,
+ "poly": [
+ 352,
+ 790,
+ 1347,
+ 790,
+ 1347,
+ 1033,
+ 352,
+ 1033
+ ],
+ "score": 0.984,
+ "html": "| Algorithm1Exploration through the Substochastic Successor Representation (ESSR) |
| n(s,s')←0 ∀s,s'∈S | |
| t(s,a,s')←1 | ∀s,s'∈S,∀a∈A |
| r(s,a)←0 | ∀s∈S,∀a∈A |
| P(s,a)←1/|Sl | ∀s∈S,∀a∈A |
| P(s,s')←0 | As,s'∈S |
| T←random over A |
| while episode is not over do Observe s E S, take action a E A selected according to π(s),and observe a reward R and a |
| next state s' ∈ S n(s,s')← n(s,s')+1 |
| t(s,a,s')←t(s,a,s')+1 |
| n(s)←∑x',bt(s,b,x') |
| n(s,a)←∑x,t(s,a,x') |
| r(s,a,s') ← (t(s,a,s)-2)xf(s,a,s')+R |
| t(s,a,s')-1 for each state x' ∈ Sdo |
| P(s,a,x'）← t(s,a,x') |
| n(s,a) n(s,x') |
| P(s,x')← n(s)+1 |
| end for √←(I-γP)-1 |
| rint←-e |
| T ← POLICYITERATION(P,𝑟+ βrint) |
| end while |
+
+# DESCRIPTION OF RIVERSWIM AND SIXARMS
+
+The two domains we used as testbed to evaluate the proposed model-based algorithm with the exploration bonus generated by the substochastic successor representation are shown in Figure 2. These domains are the same used by Strehl & Littman (2008). For SixArms, the agent starts in state 0. For RiverSwim, the agent starts in either state 1 or 2 with equal probability.
+
+
+Figure 2: Domains used as testbed in the tabular case. The tuples in each transition should be read as haction id, probability, rewardi. See text for details.
+
+# EVALUATION THE IMPACT OF THE AUXILIARY TASK IN ESSR
+
+The algorithm we introduced in the paper, ESSR, relies on a network that estimates the state-action value function, the successor representation, and the next observation to be seen given the agent’s current observation and action. While the results depicted in Table 2 allow us to clearly see the benefit of using an exploration bonus derived from the successor representation, they do not inform us about the impact of the auxiliary task in the results. The experiments in this section aim at addressing this issue. We focus on Montezumas Revenge because it is the game where the problem of exploration is maximized, with most algorithms not being able to do anything without an exploration bonus.
+
+The first question we asked was whether the auxiliary task was necessary in our algorithm. We evaluated this by dropping the reconstruction module from the network to test whether the initial random noise generated by the successor representation is enough to drive representation learning. It is not. When dropping the auxiliary task, the average performance of this baseline over 4 seeds in MONTEZUMA’S REVENGE after 100 million frames was 100.0 points $\textstyle \cdot \sigma ^ { 2 } = 2 0 0 . 0$ ; min: 0.0, max: 400.0). As comparison, our algorithm obtains 1778.6 points $\cdot \sigma ^ { 2 } = 9 0 3 . 6$ , min: 400.0, max: 2500.0). These results suggest that auxiliary tasks seem to be necessary for our method to perform well.
+
+We also evaluated whether the auxiliary task was sufficient to generate the results we observed. To do so we dropped the SR module and set $\beta = 0 . 0$ to evaluate whether our exploration bonus was actually improving the agent’s performance or whether the auxiliary task was doing it. The exploration bonus seems to be essential in our algorithm. When dropping the exploration bonus and the successor representation module, the average performance of this baseline over 4 seeds in MONTEZUMA’S REVENGE after 100 million frames was 398.5 points $\sigma ^ { 2 } = 2 3 0 . 1$ ; min: 0.0, max: 400.0). Again, clearly, the auxiliary task is not a sufficient condition for the performance we report.
+
+The reported results use the same parameters as those reported in the main paper. Learning curves for each individual run are depicted in Figure 3.
+
+
+Figure 3: Evaluation of the sufficiency and necessity of the auxiliary task in $\mathrm { D Q N } _ { e } ^ { \mathrm { M M C } } { + } \mathrm { S R }$ . The learning curves are smoothed with a running average computed using a window of size 100.
+
+ADDITIONAL RESULTS FOR $\mathrm { D Q N } _ { e } ^ { \mathrm { M M C } } + \mathrm { S R }$ AND $\mathrm { D Q N } _ { e } ^ { \mathrm { M M C } }$ IN THE ATARI 2600 GAMES
+
+As recommended by Machado et al. (2018a), we report the performance of $\mathrm { D Q N } _ { e } ^ { \mathrm { M M C } } + \mathrm { S R }$ and $\mathrm { D Q N } _ { e } ^ { \mathrm { M M C } }$ after different amounts of experience (10, 50, and 100 million frames) in Tables 3 and 4.
+
+Finally, Figure 4 depicts the learning curves obtained with the evaluated algorithms in each game. Lighter lines represent individual runs while the solid lines encode the average over the multiple runs.
+Table 3: Results obtained with $\mathrm { D Q N } _ { e } ^ { \mathrm { M M C } } + \mathrm { S R }$ after different amounts of experience.
+
+| Algorithm1Exploration through the Substochastic Successor Representation (ESSR) |
| n(s,s')←0 ∀s,s'∈S | |
| t(s,a,s')←1 | ∀s,s'∈S,∀a∈A |
| r(s,a)←0 | ∀s∈S,∀a∈A |
| P(s,a)←1/|Sl | ∀s∈S,∀a∈A |
| P(s,s')←0 | As,s'∈S |
| T←random over A |
| while episode is not over do Observe s E S, take action a E A selected according to π(s),and observe a reward R and a |
| next state s' ∈ S n(s,s')← n(s,s')+1 |
| t(s,a,s')←t(s,a,s')+1 |
| n(s)←∑x',bt(s,b,x') |
| n(s,a)←∑x,t(s,a,x') |
| r(s,a,s') ← (t(s,a,s)-2)xf(s,a,s')+R |
| t(s,a,s')-1 for each state x' ∈ Sdo |
| P(s,a,x'）← t(s,a,x') |
| n(s,a) n(s,x') |
| P(s,x')← n(s)+1 |
| end for √←(I-γP)-1 |
| rint←-e |
| T ← POLICYITERATION(P,𝑟+ βrint) |
| end while |
",
+ "bbox": [
+ 181,
+ 402,
+ 826,
+ 828
+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "DESCRIPTION OF RIVERSWIM AND SIXARMS ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 103,
+ 490,
+ 117
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "The two domains we used as testbed to evaluate the proposed model-based algorithm with the exploration bonus generated by the substochastic successor representation are shown in Figure 2. These domains are the same used by Strehl & Littman (2008). For SixArms, the agent starts in state 0. For RiverSwim, the agent starts in either state 1 or 2 with equal probability. ",
+ "bbox": [
+ 174,
+ 130,
+ 826,
+ 185
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "image",
+ "img_path": "images/5668527eec930a068c0c61f8e7ce73a9f8d40ed779bcb71ff47bdc04db2eb83b.jpg",
+ "image_caption": [
+ "Figure 2: Domains used as testbed in the tabular case. The tuples in each transition should be read as haction id, probability, rewardi. See text for details. "
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 176,
+ 196,
+ 820,
+ 315
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "EVALUATION THE IMPACT OF THE AUXILIARY TASK IN ESSR ",
+ "text_level": 1,
+ "bbox": [
+ 181,
+ 380,
+ 591,
+ 393
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "The algorithm we introduced in the paper, ESSR, relies on a network that estimates the state-action value function, the successor representation, and the next observation to be seen given the agent’s current observation and action. While the results depicted in Table 2 allow us to clearly see the benefit of using an exploration bonus derived from the successor representation, they do not inform us about the impact of the auxiliary task in the results. The experiments in this section aim at addressing this issue. We focus on Montezumas Revenge because it is the game where the problem of exploration is maximized, with most algorithms not being able to do anything without an exploration bonus. ",
+ "bbox": [
+ 173,
+ 405,
+ 825,
+ 502
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "The first question we asked was whether the auxiliary task was necessary in our algorithm. We evaluated this by dropping the reconstruction module from the network to test whether the initial random noise generated by the successor representation is enough to drive representation learning. It is not. When dropping the auxiliary task, the average performance of this baseline over 4 seeds in MONTEZUMA’S REVENGE after 100 million frames was 100.0 points $\\textstyle \\cdot \\sigma ^ { 2 } = 2 0 0 . 0$ ; min: 0.0, max: 400.0). As comparison, our algorithm obtains 1778.6 points $\\cdot \\sigma ^ { 2 } = 9 0 3 . 6$ , min: 400.0, max: 2500.0). These results suggest that auxiliary tasks seem to be necessary for our method to perform well. ",
+ "bbox": [
+ 173,
+ 510,
+ 826,
+ 608
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "We also evaluated whether the auxiliary task was sufficient to generate the results we observed. To do so we dropped the SR module and set $\\beta = 0 . 0$ to evaluate whether our exploration bonus was actually improving the agent’s performance or whether the auxiliary task was doing it. The exploration bonus seems to be essential in our algorithm. When dropping the exploration bonus and the successor representation module, the average performance of this baseline over 4 seeds in MONTEZUMA’S REVENGE after 100 million frames was 398.5 points $\\sigma ^ { 2 } = 2 3 0 . 1$ ; min: 0.0, max: 400.0). Again, clearly, the auxiliary task is not a sufficient condition for the performance we report. ",
+ "bbox": [
+ 174,
+ 614,
+ 825,
+ 712
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "The reported results use the same parameters as those reported in the main paper. Learning curves for each individual run are depicted in Figure 3. ",
+ "bbox": [
+ 173,
+ 718,
+ 823,
+ 747
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "image",
+ "img_path": "images/d6d739ad291411fdbea9cc046ca7a4a0c3c72cd8cbb1947ce71e832e5aae22ea.jpg",
+ "image_caption": [
+ "Figure 3: Evaluation of the sufficiency and necessity of the auxiliary task in $\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } { + } \\mathrm { S R }$ . The learning curves are smoothed with a running average computed using a window of size 100. "
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 295,
+ 765,
+ 692,
+ 876
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "ADDITIONAL RESULTS FOR $\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } + \\mathrm { S R }$ AND $\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } }$ IN THE ATARI 2600 GAMES ",
+ "bbox": [
+ 169,
+ 102,
+ 759,
+ 119
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "As recommended by Machado et al. (2018a), we report the performance of $\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } + \\mathrm { S R }$ and $\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } }$ after different amounts of experience (10, 50, and 100 million frames) in Tables 3 and 4. ",
+ "bbox": [
+ 173,
+ 128,
+ 823,
+ 159
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "table",
+ "img_path": "images/5618f78349bcdb9597ea629f0f175ee3dd97eba228e5a3c7f4d763ed84195aee.jpg",
+ "table_caption": [
+ "Finally, Figure 4 depicts the learning curves obtained with the evaluated algorithms in each game. Lighter lines represent individual runs while the solid lines encode the average over the multiple runs. ",
+ "Table 3: Results obtained with $\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } + \\mathrm { S R }$ after different amounts of experience. "
+ ],
+ "table_footnote": [],
+ "table_body": "| Algorithm1Exploration through the Substochastic Successor Representation (ESSR) |
| n(s,s')←0 ∀s,s'∈S | |
| t(s,a,s')←1 | ∀s,s'∈S,∀a∈A |
| r(s,a)←0 | ∀s∈S,∀a∈A |
| P(s,a)←1/|Sl | ∀s∈S,∀a∈A |
| P(s,s')←0 | As,s'∈S |
| T←random over A |
| while episode is not over do Observe s E S, take action a E A selected according to π(s),and observe a reward R and a |
| next state s' ∈ S n(s,s')← n(s,s')+1 |
| t(s,a,s')←t(s,a,s')+1 |
| n(s)←∑x',bt(s,b,x') |
| n(s,a)←∑x,t(s,a,x') |
| r(s,a,s') ← (t(s,a,s)-2)xf(s,a,s')+R |
| t(s,a,s')-1 for each state x' ∈ Sdo |
| P(s,a,x'）← t(s,a,x') |
| n(s,a) n(s,x') |
| P(s,x')← n(s)+1 |
| end for √←(I-γP)-1 |
| rint←-e |
| T ← POLICYITERATION(P,𝑟+ βrint) |
| end while |
",
+ "type": "table",
+ "image_path": "fb5781582a0947e1de03549818a77980bd64bf8e1179bb182a987e4ccba94093.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 15,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 111,
+ 319,
+ 506,
+ 431.3333333333333
+ ],
+ "spans": [],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 111,
+ 431.3333333333333,
+ 506,
+ 543.6666666666666
+ ],
+ "spans": [],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 111,
+ 543.6666666666666,
+ 506,
+ 656.0
+ ],
+ "spans": [],
+ "index": 16
+ }
+ ]
+ }
+ ],
+ "index": 15
+ }
+ ],
+ "page_idx": 10,
+ "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 2019",
+ "type": "text"
+ }
+ ]
+ }
+ ]
+ },
+ {
+ "type": "discarded",
+ "bbox": [
+ 300,
+ 751,
+ 310,
+ 760
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 765
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 299,
+ 750,
+ 312,
+ 765
+ ],
+ "score": 1.0,
+ "content": "",
+ "type": "text",
+ "height": 15,
+ "width": 13
+ }
+ ]
+ }
+ ]
+ }
+ ],
+ "para_blocks": [
+ {
+ "type": "title",
+ "bbox": [
+ 119,
+ 78,
+ 493,
+ 111
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 233,
+ 79,
+ 378,
+ 96
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 233,
+ 79,
+ 378,
+ 96
+ ],
+ "score": 1.0,
+ "content": "Supplemental Material",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ },
+ {
+ "bbox": [
+ 118,
+ 93,
+ 493,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 118,
+ 93,
+ 493,
+ 115
+ ],
+ "score": 1.0,
+ "content": "Count-Based Exploration with the Successor Representation",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ }
+ ],
+ "index": 0.5
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 120,
+ 505,
+ 144
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 506,
+ 133
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 120,
+ 506,
+ 133
+ ],
+ "score": 1.0,
+ "content": "This supplementary material contains details omitted from the main text due to space constraints.",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 132,
+ 224,
+ 144
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 132,
+ 224,
+ 144
+ ],
+ "score": 1.0,
+ "content": "The list of contents is below:",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ }
+ ],
+ "index": 2.5,
+ "bbox_fs": [
+ 105,
+ 120,
+ 506,
+ 144
+ ]
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 132,
+ 153,
+ 505,
+ 232
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 132,
+ 154,
+ 409,
+ 165
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 132,
+ 154,
+ 409,
+ 165
+ ],
+ "score": 1.0,
+ "content": "• Pseudo-code of the model-based algorithm discussed in Section 3;",
+ "type": "text"
+ }
+ ],
+ "index": 4,
+ "is_list_start_line": true,
+ "is_list_end_line": true
+ },
+ {
+ "bbox": [
+ 131,
+ 168,
+ 495,
+ 181
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 131,
+ 168,
+ 495,
+ 181
+ ],
+ "score": 1.0,
+ "content": "• Description of RiverSwim and SixArms, the tabular domains we used in our evaluation;",
+ "type": "text"
+ }
+ ],
+ "index": 5,
+ "is_list_start_line": true
+ },
+ {
+ "bbox": [
+ 131,
+ 181,
+ 506,
+ 196
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 131,
+ 181,
+ 219,
+ 196
+ ],
+ "score": 1.0,
+ "content": "• Learning curves of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 220,
+ 183,
+ 246,
+ 194
+ ],
+ "score": 0.89,
+ "content": "\\mathrm { D Q N } _ { e }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 247,
+ 181,
+ 264,
+ 196
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 265,
+ 182,
+ 321,
+ 195
+ ],
+ "score": 0.89,
+ "content": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } + \\mathrm { S R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 321,
+ 181,
+ 506,
+ 196
+ ],
+ "score": 1.0,
+ "content": "and their performance after different amounts",
+ "type": "text"
+ }
+ ],
+ "index": 6,
+ "is_list_start_line": true
+ },
+ {
+ "bbox": [
+ 142,
+ 194,
+ 378,
+ 207
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 142,
+ 194,
+ 378,
+ 207
+ ],
+ "score": 1.0,
+ "content": "of experience in the Atari 2600 games used for evaluation;",
+ "type": "text"
+ }
+ ],
+ "index": 7,
+ "is_list_end_line": true
+ },
+ {
+ "bbox": [
+ 137,
+ 208,
+ 505,
+ 222
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 137,
+ 208,
+ 505,
+ 222
+ ],
+ "score": 1.0,
+ "content": "Results of additional experiments designed to evaluate the role of the auxiliary task in the",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 141,
+ 220,
+ 299,
+ 232
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 141,
+ 220,
+ 299,
+ 232
+ ],
+ "score": 1.0,
+ "content": "results reported in the paper for ESSR.",
+ "type": "text"
+ }
+ ],
+ "index": 9,
+ "is_list_end_line": true
+ }
+ ],
+ "index": 6.5,
+ "bbox_fs": [
+ 131,
+ 154,
+ 506,
+ 232
+ ]
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 255,
+ 441,
+ 267
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 256,
+ 442,
+ 267
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 256,
+ 442,
+ 267
+ ],
+ "score": 1.0,
+ "content": "EXPLORATION THROUGH THE SUBSTOCHASTIC SUCCESSOR REPRESENTATION",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 276,
+ 506,
+ 309
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 275,
+ 506,
+ 289
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 275,
+ 506,
+ 289
+ ],
+ "score": 1.0,
+ "content": "In the main paper we described our algorithm as a standard model-based algorithm where the agent",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 105,
+ 287,
+ 506,
+ 299
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 287,
+ 506,
+ 299
+ ],
+ "score": 1.0,
+ "content": "updates its transition probability model and reward model through Equation 2 and its SSR estimate as",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 298,
+ 507,
+ 311
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 298,
+ 507,
+ 311
+ ],
+ "score": 1.0,
+ "content": "in Definition 3.1. The pseudo-code with details about the implementation is presented in Algorithm 1.",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ }
+ ],
+ "index": 12,
+ "bbox_fs": [
+ 105,
+ 275,
+ 507,
+ 311
+ ]
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 111,
+ 319,
+ 506,
+ 656
+ ],
+ "blocks": [
+ {
+ "type": "table_body",
+ "bbox": [
+ 111,
+ 319,
+ 506,
+ 656
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 111,
+ 319,
+ 506,
+ 656
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 111,
+ 319,
+ 506,
+ 656
+ ],
+ "score": 0.711,
+ "html": "| Algorithm1Exploration through the Substochastic Successor Representation (ESSR) |
| n(s,s')←0 ∀s,s'∈S | |
| t(s,a,s')←1 | ∀s,s'∈S,∀a∈A |
| r(s,a)←0 | ∀s∈S,∀a∈A |
| P(s,a)←1/|Sl | ∀s∈S,∀a∈A |
| P(s,s')←0 | As,s'∈S |
| T←random over A |
| while episode is not over do Observe s E S, take action a E A selected according to π(s),and observe a reward R and a |
| next state s' ∈ S n(s,s')← n(s,s')+1 |
| t(s,a,s')←t(s,a,s')+1 |
| n(s)←∑x',bt(s,b,x') |
| n(s,a)←∑x,t(s,a,x') |
| r(s,a,s') ← (t(s,a,s)-2)xf(s,a,s')+R |
| t(s,a,s')-1 for each state x' ∈ Sdo |
| P(s,a,x'）← t(s,a,x') |
| n(s,a) n(s,x') |
| P(s,x')← n(s)+1 |
| end for √←(I-γP)-1 |
| rint←-e |
| T ← POLICYITERATION(P,𝑟+ βrint) |
| end while |
",
+ "type": "table",
+ "image_path": "fb5781582a0947e1de03549818a77980bd64bf8e1179bb182a987e4ccba94093.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 15,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 111,
+ 319,
+ 506,
+ 431.3333333333333
+ ],
+ "spans": [],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 111,
+ 431.3333333333333,
+ 506,
+ 543.6666666666666
+ ],
+ "spans": [],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 111,
+ 543.6666666666666,
+ 506,
+ 656.0
+ ],
+ "spans": [],
+ "index": 16
+ }
+ ]
+ }
+ ],
+ "index": 15
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "title",
+ "bbox": [
+ 107,
+ 82,
+ 300,
+ 93
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 301,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 301,
+ 95
+ ],
+ "score": 1.0,
+ "content": "DESCRIPTION OF RIVERSWIM AND SIXARMS",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 103,
+ 506,
+ 147
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 102,
+ 505,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 102,
+ 505,
+ 115
+ ],
+ "score": 1.0,
+ "content": "The two domains we used as testbed to evaluate the proposed model-based algorithm with the",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 114,
+ 507,
+ 126
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 114,
+ 507,
+ 126
+ ],
+ "score": 1.0,
+ "content": "exploration bonus generated by the substochastic successor representation are shown in Figure 2.",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 124,
+ 507,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 124,
+ 507,
+ 137
+ ],
+ "score": 1.0,
+ "content": "These domains are the same used by Strehl & Littman (2008). For SixArms, the agent starts in state 0.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 135,
+ 409,
+ 149
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 135,
+ 409,
+ 149
+ ],
+ "score": 1.0,
+ "content": "For RiverSwim, the agent starts in either state 1 or 2 with equal probability.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "score": 0.962,
+ "type": "image",
+ "image_path": "5668527eec930a068c0c61f8e7ce73a9f8d40ed779bcb71ff47bdc04db2eb83b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 6,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 187.33333333333334
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 108,
+ 187.33333333333334,
+ 502,
+ 218.66666666666669
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 108,
+ 218.66666666666669,
+ 502,
+ 250.00000000000003
+ ],
+ "spans": [],
+ "index": 7
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 259,
+ 505,
+ 281
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 258,
+ 505,
+ 271
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 258,
+ 505,
+ 271
+ ],
+ "score": 1.0,
+ "content": "Figure 2: Domains used as testbed in the tabular case. The tuples in each transition should be read as",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 270,
+ 314,
+ 282
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 314,
+ 282
+ ],
+ "score": 1.0,
+ "content": "haction id, probability, rewardi. See text for details.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 8.5
+ }
+ ],
+ "index": 7.25
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 111,
+ 301,
+ 362,
+ 312
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 109,
+ 301,
+ 364,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 109,
+ 301,
+ 364,
+ 312
+ ],
+ "score": 1.0,
+ "content": "EVALUATION THE IMPACT OF THE AUXILIARY TASK IN ESSR",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 321,
+ 505,
+ 398
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 505,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 505,
+ 333
+ ],
+ "score": 1.0,
+ "content": "The algorithm we introduced in the paper, ESSR, relies on a network that estimates the state-action",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 333,
+ 505,
+ 344
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 333,
+ 505,
+ 344
+ ],
+ "score": 1.0,
+ "content": "value function, the successor representation, and the next observation to be seen given the agent’s",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 354
+ ],
+ "score": 1.0,
+ "content": "current observation and action. While the results depicted in Table 2 allow us to clearly see the benefit",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 355,
+ 505,
+ 366
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 355,
+ 505,
+ 366
+ ],
+ "score": 1.0,
+ "content": "of using an exploration bonus derived from the successor representation, they do not inform us about",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 366,
+ 505,
+ 377
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 366,
+ 505,
+ 377
+ ],
+ "score": 1.0,
+ "content": "the impact of the auxiliary task in the results. The experiments in this section aim at addressing this",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 376,
+ 505,
+ 389
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 376,
+ 505,
+ 389
+ ],
+ "score": 1.0,
+ "content": "issue. We focus on Montezumas Revenge because it is the game where the problem of exploration is",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 388,
+ 483,
+ 399
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 388,
+ 483,
+ 399
+ ],
+ "score": 1.0,
+ "content": "maximized, with most algorithms not being able to do anything without an exploration bonus.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 14
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 404,
+ 506,
+ 482
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 403,
+ 507,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 403,
+ 507,
+ 417
+ ],
+ "score": 1.0,
+ "content": "The first question we asked was whether the auxiliary task was necessary in our algorithm. We evalu-",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 415,
+ 505,
+ 427
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 415,
+ 505,
+ 427
+ ],
+ "score": 1.0,
+ "content": "ated this by dropping the reconstruction module from the network to test whether the initial random",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 426,
+ 506,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 426,
+ 506,
+ 439
+ ],
+ "score": 1.0,
+ "content": "noise generated by the successor representation is enough to drive representation learning. It is not.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 435,
+ 507,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 435,
+ 507,
+ 450
+ ],
+ "score": 1.0,
+ "content": "When dropping the auxiliary task, the average performance of this baseline over 4 seeds in MON-",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 447,
+ 506,
+ 460
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 448,
+ 366,
+ 460
+ ],
+ "score": 1.0,
+ "content": "TEZUMA’S REVENGE after 100 million frames was 100.0 points",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 447,
+ 411,
+ 459
+ ],
+ "score": 0.88,
+ "content": "\\textstyle \\cdot \\sigma ^ { 2 } = 2 0 0 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 412,
+ 448,
+ 506,
+ 460
+ ],
+ "score": 1.0,
+ "content": "; min: 0.0, max: 400.0).",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 104,
+ 458,
+ 506,
+ 471
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 458,
+ 320,
+ 471
+ ],
+ "score": 1.0,
+ "content": "As comparison, our algorithm obtains 1778.6 points",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 459,
+ 366,
+ 469
+ ],
+ "score": 0.89,
+ "content": "\\cdot \\sigma ^ { 2 } = 9 0 3 . 6",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 366,
+ 458,
+ 506,
+ 471
+ ],
+ "score": 1.0,
+ "content": ", min: 400.0, max: 2500.0). These",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 470,
+ 460,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 460,
+ 482
+ ],
+ "score": 1.0,
+ "content": "results suggest that auxiliary tasks seem to be necessary for our method to perform well.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 21
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 487,
+ 505,
+ 564
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 487,
+ 505,
+ 499
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 487,
+ 505,
+ 499
+ ],
+ "score": 1.0,
+ "content": "We also evaluated whether the auxiliary task was sufficient to generate the results we observed. To do",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 498,
+ 505,
+ 510
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 498,
+ 255,
+ 510
+ ],
+ "score": 1.0,
+ "content": "so we dropped the SR module and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 498,
+ 288,
+ 509
+ ],
+ "score": 0.92,
+ "content": "\\beta = 0 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 289,
+ 498,
+ 505,
+ 510
+ ],
+ "score": 1.0,
+ "content": "to evaluate whether our exploration bonus was actually",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 521
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 521
+ ],
+ "score": 1.0,
+ "content": "improving the agent’s performance or whether the auxiliary task was doing it. The exploration bonus",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 520,
+ 506,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 520,
+ 506,
+ 532
+ ],
+ "score": 1.0,
+ "content": "seems to be essential in our algorithm. When dropping the exploration bonus and the successor",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 531,
+ 505,
+ 542
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 531,
+ 505,
+ 542
+ ],
+ "score": 1.0,
+ "content": "representation module, the average performance of this baseline over 4 seeds in MONTEZUMA’S",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 541,
+ 506,
+ 554
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 541,
+ 329,
+ 554
+ ],
+ "score": 1.0,
+ "content": "REVENGE after 100 million frames was 398.5 points",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 330,
+ 541,
+ 376,
+ 552
+ ],
+ "score": 0.88,
+ "content": "\\sigma ^ { 2 } = 2 3 0 . 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 541,
+ 506,
+ 554
+ ],
+ "score": 1.0,
+ "content": "; min: 0.0, max: 400.0). Again,",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 551,
+ 444,
+ 565
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 551,
+ 444,
+ 565
+ ],
+ "score": 1.0,
+ "content": "clearly, the auxiliary task is not a sufficient condition for the performance we report.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 28
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 569,
+ 504,
+ 592
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 569,
+ 505,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 569,
+ 505,
+ 582
+ ],
+ "score": 1.0,
+ "content": "The reported results use the same parameters as those reported in the main paper. Learning curves for",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 106,
+ 580,
+ 284,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 580,
+ 284,
+ 592
+ ],
+ "score": 1.0,
+ "content": "each individual run are depicted in Figure 3.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 32.5
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "score": 0.96,
+ "type": "image",
+ "image_path": "d6d739ad291411fdbea9cc046ca7a4a0c3c72cd8cbb1947ce71e832e5aae22ea.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 36.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 620.6666666666666
+ ],
+ "spans": [],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 181,
+ 620.6666666666666,
+ 424,
+ 635.3333333333333
+ ],
+ "spans": [],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 181,
+ 635.3333333333333,
+ 424,
+ 649.9999999999999
+ ],
+ "spans": [],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 181,
+ 649.9999999999999,
+ 424,
+ 664.6666666666665
+ ],
+ "spans": [],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 181,
+ 664.6666666666665,
+ 424,
+ 679.3333333333331
+ ],
+ "spans": [],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 181,
+ 679.3333333333331,
+ 424,
+ 693.9999999999998
+ ],
+ "spans": [],
+ "index": 39
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 703,
+ 505,
+ 726
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 701,
+ 506,
+ 718
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 701,
+ 425,
+ 718
+ ],
+ "score": 1.0,
+ "content": "Figure 3: Evaluation of the sufficiency and necessity of the auxiliary task in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 703,
+ 482,
+ 715
+ ],
+ "score": 0.9,
+ "content": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } { + } \\mathrm { S R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 701,
+ 506,
+ 718
+ ],
+ "score": 1.0,
+ "content": ". The",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 714,
+ 475,
+ 727
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 714,
+ 475,
+ 727
+ ],
+ "score": 1.0,
+ "content": "learning curves are smoothed with a running average computed using a window of size 100.",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ }
+ ],
+ "index": 40.5
+ }
+ ],
+ "index": 38.5
+ }
+ ],
+ "page_idx": 11,
+ "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": [
+ 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": "title",
+ "bbox": [
+ 107,
+ 82,
+ 300,
+ 93
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 301,
+ 95
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 82,
+ 301,
+ 95
+ ],
+ "score": 1.0,
+ "content": "DESCRIPTION OF RIVERSWIM AND SIXARMS",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 103,
+ 506,
+ 147
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 102,
+ 505,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 102,
+ 505,
+ 115
+ ],
+ "score": 1.0,
+ "content": "The two domains we used as testbed to evaluate the proposed model-based algorithm with the",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 114,
+ 507,
+ 126
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 114,
+ 507,
+ 126
+ ],
+ "score": 1.0,
+ "content": "exploration bonus generated by the substochastic successor representation are shown in Figure 2.",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ },
+ {
+ "bbox": [
+ 105,
+ 124,
+ 507,
+ 137
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 124,
+ 507,
+ 137
+ ],
+ "score": 1.0,
+ "content": "These domains are the same used by Strehl & Littman (2008). For SixArms, the agent starts in state 0.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 105,
+ 135,
+ 409,
+ 149
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 135,
+ 409,
+ 149
+ ],
+ "score": 1.0,
+ "content": "For RiverSwim, the agent starts in either state 1 or 2 with equal probability.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 2.5,
+ "bbox_fs": [
+ 105,
+ 102,
+ 507,
+ 149
+ ]
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 250
+ ],
+ "score": 0.962,
+ "type": "image",
+ "image_path": "5668527eec930a068c0c61f8e7ce73a9f8d40ed779bcb71ff47bdc04db2eb83b.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 6,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 108,
+ 156,
+ 502,
+ 187.33333333333334
+ ],
+ "spans": [],
+ "index": 5
+ },
+ {
+ "bbox": [
+ 108,
+ 187.33333333333334,
+ 502,
+ 218.66666666666669
+ ],
+ "spans": [],
+ "index": 6
+ },
+ {
+ "bbox": [
+ 108,
+ 218.66666666666669,
+ 502,
+ 250.00000000000003
+ ],
+ "spans": [],
+ "index": 7
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 259,
+ 505,
+ 281
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 258,
+ 505,
+ 271
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 258,
+ 505,
+ 271
+ ],
+ "score": 1.0,
+ "content": "Figure 2: Domains used as testbed in the tabular case. The tuples in each transition should be read as",
+ "type": "text"
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 106,
+ 270,
+ 314,
+ 282
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 270,
+ 314,
+ 282
+ ],
+ "score": 1.0,
+ "content": "haction id, probability, rewardi. See text for details.",
+ "type": "text"
+ }
+ ],
+ "index": 9
+ }
+ ],
+ "index": 8.5
+ }
+ ],
+ "index": 7.25
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 111,
+ 301,
+ 362,
+ 312
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 109,
+ 301,
+ 364,
+ 312
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 109,
+ 301,
+ 364,
+ 312
+ ],
+ "score": 1.0,
+ "content": "EVALUATION THE IMPACT OF THE AUXILIARY TASK IN ESSR",
+ "type": "text"
+ }
+ ],
+ "index": 10
+ }
+ ],
+ "index": 10
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 321,
+ 505,
+ 398
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 505,
+ 333
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 320,
+ 505,
+ 333
+ ],
+ "score": 1.0,
+ "content": "The algorithm we introduced in the paper, ESSR, relies on a network that estimates the state-action",
+ "type": "text"
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 106,
+ 333,
+ 505,
+ 344
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 333,
+ 505,
+ 344
+ ],
+ "score": 1.0,
+ "content": "value function, the successor representation, and the next observation to be seen given the agent’s",
+ "type": "text"
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 354
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 344,
+ 505,
+ 354
+ ],
+ "score": 1.0,
+ "content": "current observation and action. While the results depicted in Table 2 allow us to clearly see the benefit",
+ "type": "text"
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 106,
+ 355,
+ 505,
+ 366
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 355,
+ 505,
+ 366
+ ],
+ "score": 1.0,
+ "content": "of using an exploration bonus derived from the successor representation, they do not inform us about",
+ "type": "text"
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 106,
+ 366,
+ 505,
+ 377
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 366,
+ 505,
+ 377
+ ],
+ "score": 1.0,
+ "content": "the impact of the auxiliary task in the results. The experiments in this section aim at addressing this",
+ "type": "text"
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 106,
+ 376,
+ 505,
+ 389
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 376,
+ 505,
+ 389
+ ],
+ "score": 1.0,
+ "content": "issue. We focus on Montezumas Revenge because it is the game where the problem of exploration is",
+ "type": "text"
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
+ 106,
+ 388,
+ 483,
+ 399
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 388,
+ 483,
+ 399
+ ],
+ "score": 1.0,
+ "content": "maximized, with most algorithms not being able to do anything without an exploration bonus.",
+ "type": "text"
+ }
+ ],
+ "index": 17
+ }
+ ],
+ "index": 14,
+ "bbox_fs": [
+ 105,
+ 320,
+ 505,
+ 399
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 404,
+ 506,
+ 482
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 403,
+ 507,
+ 417
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 403,
+ 507,
+ 417
+ ],
+ "score": 1.0,
+ "content": "The first question we asked was whether the auxiliary task was necessary in our algorithm. We evalu-",
+ "type": "text"
+ }
+ ],
+ "index": 18
+ },
+ {
+ "bbox": [
+ 105,
+ 415,
+ 505,
+ 427
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 415,
+ 505,
+ 427
+ ],
+ "score": 1.0,
+ "content": "ated this by dropping the reconstruction module from the network to test whether the initial random",
+ "type": "text"
+ }
+ ],
+ "index": 19
+ },
+ {
+ "bbox": [
+ 106,
+ 426,
+ 506,
+ 439
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 426,
+ 506,
+ 439
+ ],
+ "score": 1.0,
+ "content": "noise generated by the successor representation is enough to drive representation learning. It is not.",
+ "type": "text"
+ }
+ ],
+ "index": 20
+ },
+ {
+ "bbox": [
+ 105,
+ 435,
+ 507,
+ 450
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 435,
+ 507,
+ 450
+ ],
+ "score": 1.0,
+ "content": "When dropping the auxiliary task, the average performance of this baseline over 4 seeds in MON-",
+ "type": "text"
+ }
+ ],
+ "index": 21
+ },
+ {
+ "bbox": [
+ 106,
+ 447,
+ 506,
+ 460
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 448,
+ 366,
+ 460
+ ],
+ "score": 1.0,
+ "content": "TEZUMA’S REVENGE after 100 million frames was 100.0 points",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 366,
+ 447,
+ 411,
+ 459
+ ],
+ "score": 0.88,
+ "content": "\\textstyle \\cdot \\sigma ^ { 2 } = 2 0 0 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 412,
+ 448,
+ 506,
+ 460
+ ],
+ "score": 1.0,
+ "content": "; min: 0.0, max: 400.0).",
+ "type": "text"
+ }
+ ],
+ "index": 22
+ },
+ {
+ "bbox": [
+ 104,
+ 458,
+ 506,
+ 471
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 458,
+ 320,
+ 471
+ ],
+ "score": 1.0,
+ "content": "As comparison, our algorithm obtains 1778.6 points",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 321,
+ 459,
+ 366,
+ 469
+ ],
+ "score": 0.89,
+ "content": "\\cdot \\sigma ^ { 2 } = 9 0 3 . 6",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 366,
+ 458,
+ 506,
+ 471
+ ],
+ "score": 1.0,
+ "content": ", min: 400.0, max: 2500.0). These",
+ "type": "text"
+ }
+ ],
+ "index": 23
+ },
+ {
+ "bbox": [
+ 105,
+ 470,
+ 460,
+ 482
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 470,
+ 460,
+ 482
+ ],
+ "score": 1.0,
+ "content": "results suggest that auxiliary tasks seem to be necessary for our method to perform well.",
+ "type": "text"
+ }
+ ],
+ "index": 24
+ }
+ ],
+ "index": 21,
+ "bbox_fs": [
+ 104,
+ 403,
+ 507,
+ 482
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 107,
+ 487,
+ 505,
+ 564
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 487,
+ 505,
+ 499
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 487,
+ 505,
+ 499
+ ],
+ "score": 1.0,
+ "content": "We also evaluated whether the auxiliary task was sufficient to generate the results we observed. To do",
+ "type": "text"
+ }
+ ],
+ "index": 25
+ },
+ {
+ "bbox": [
+ 106,
+ 498,
+ 505,
+ 510
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 498,
+ 255,
+ 510
+ ],
+ "score": 1.0,
+ "content": "so we dropped the SR module and set",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 255,
+ 498,
+ 288,
+ 509
+ ],
+ "score": 0.92,
+ "content": "\\beta = 0 . 0",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 289,
+ 498,
+ 505,
+ 510
+ ],
+ "score": 1.0,
+ "content": "to evaluate whether our exploration bonus was actually",
+ "type": "text"
+ }
+ ],
+ "index": 26
+ },
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 521
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 509,
+ 505,
+ 521
+ ],
+ "score": 1.0,
+ "content": "improving the agent’s performance or whether the auxiliary task was doing it. The exploration bonus",
+ "type": "text"
+ }
+ ],
+ "index": 27
+ },
+ {
+ "bbox": [
+ 105,
+ 520,
+ 506,
+ 532
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 520,
+ 506,
+ 532
+ ],
+ "score": 1.0,
+ "content": "seems to be essential in our algorithm. When dropping the exploration bonus and the successor",
+ "type": "text"
+ }
+ ],
+ "index": 28
+ },
+ {
+ "bbox": [
+ 105,
+ 531,
+ 505,
+ 542
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 531,
+ 505,
+ 542
+ ],
+ "score": 1.0,
+ "content": "representation module, the average performance of this baseline over 4 seeds in MONTEZUMA’S",
+ "type": "text"
+ }
+ ],
+ "index": 29
+ },
+ {
+ "bbox": [
+ 105,
+ 541,
+ 506,
+ 554
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 541,
+ 329,
+ 554
+ ],
+ "score": 1.0,
+ "content": "REVENGE after 100 million frames was 398.5 points",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 330,
+ 541,
+ 376,
+ 552
+ ],
+ "score": 0.88,
+ "content": "\\sigma ^ { 2 } = 2 3 0 . 1",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 376,
+ 541,
+ 506,
+ 554
+ ],
+ "score": 1.0,
+ "content": "; min: 0.0, max: 400.0). Again,",
+ "type": "text"
+ }
+ ],
+ "index": 30
+ },
+ {
+ "bbox": [
+ 105,
+ 551,
+ 444,
+ 565
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 551,
+ 444,
+ 565
+ ],
+ "score": 1.0,
+ "content": "clearly, the auxiliary task is not a sufficient condition for the performance we report.",
+ "type": "text"
+ }
+ ],
+ "index": 31
+ }
+ ],
+ "index": 28,
+ "bbox_fs": [
+ 105,
+ 487,
+ 506,
+ 565
+ ]
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 569,
+ 504,
+ 592
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 569,
+ 505,
+ 582
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 569,
+ 505,
+ 582
+ ],
+ "score": 1.0,
+ "content": "The reported results use the same parameters as those reported in the main paper. Learning curves for",
+ "type": "text"
+ }
+ ],
+ "index": 32
+ },
+ {
+ "bbox": [
+ 106,
+ 580,
+ 284,
+ 592
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 580,
+ 284,
+ 592
+ ],
+ "score": 1.0,
+ "content": "each individual run are depicted in Figure 3.",
+ "type": "text"
+ }
+ ],
+ "index": 33
+ }
+ ],
+ "index": 32.5,
+ "bbox_fs": [
+ 106,
+ 569,
+ 505,
+ 592
+ ]
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "blocks": [
+ {
+ "type": "image_body",
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 694
+ ],
+ "score": 0.96,
+ "type": "image",
+ "image_path": "d6d739ad291411fdbea9cc046ca7a4a0c3c72cd8cbb1947ce71e832e5aae22ea.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 36.5,
+ "virtual_lines": [
+ {
+ "bbox": [
+ 181,
+ 606,
+ 424,
+ 620.6666666666666
+ ],
+ "spans": [],
+ "index": 34
+ },
+ {
+ "bbox": [
+ 181,
+ 620.6666666666666,
+ 424,
+ 635.3333333333333
+ ],
+ "spans": [],
+ "index": 35
+ },
+ {
+ "bbox": [
+ 181,
+ 635.3333333333333,
+ 424,
+ 649.9999999999999
+ ],
+ "spans": [],
+ "index": 36
+ },
+ {
+ "bbox": [
+ 181,
+ 649.9999999999999,
+ 424,
+ 664.6666666666665
+ ],
+ "spans": [],
+ "index": 37
+ },
+ {
+ "bbox": [
+ 181,
+ 664.6666666666665,
+ 424,
+ 679.3333333333331
+ ],
+ "spans": [],
+ "index": 38
+ },
+ {
+ "bbox": [
+ 181,
+ 679.3333333333331,
+ 424,
+ 693.9999999999998
+ ],
+ "spans": [],
+ "index": 39
+ }
+ ]
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 106,
+ 703,
+ 505,
+ 726
+ ],
+ "group_id": 1,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 701,
+ 506,
+ 718
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 701,
+ 425,
+ 718
+ ],
+ "score": 1.0,
+ "content": "Figure 3: Evaluation of the sufficiency and necessity of the auxiliary task in",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 426,
+ 703,
+ 482,
+ 715
+ ],
+ "score": 0.9,
+ "content": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } { + } \\mathrm { S R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 483,
+ 701,
+ 506,
+ 718
+ ],
+ "score": 1.0,
+ "content": ". The",
+ "type": "text"
+ }
+ ],
+ "index": 40
+ },
+ {
+ "bbox": [
+ 105,
+ 714,
+ 475,
+ 727
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 714,
+ 475,
+ 727
+ ],
+ "score": 1.0,
+ "content": "learning curves are smoothed with a running average computed using a window of size 100.",
+ "type": "text"
+ }
+ ],
+ "index": 41
+ }
+ ],
+ "index": 40.5
+ }
+ ],
+ "index": 38.5
+ }
+ ]
+ },
+ {
+ "preproc_blocks": [
+ {
+ "type": "text",
+ "bbox": [
+ 104,
+ 81,
+ 465,
+ 95
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 103,
+ 78,
+ 468,
+ 100
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 103,
+ 78,
+ 224,
+ 100
+ ],
+ "score": 1.0,
+ "content": "ADDITIONAL RESULTS FOR",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 224,
+ 82,
+ 284,
+ 95
+ ],
+ "score": 0.72,
+ "content": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } + \\mathrm { S R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 284,
+ 78,
+ 306,
+ 100
+ ],
+ "score": 1.0,
+ "content": "AND",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 306,
+ 82,
+ 346,
+ 95
+ ],
+ "score": 0.6,
+ "content": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 347,
+ 78,
+ 468,
+ 100
+ ],
+ "score": 1.0,
+ "content": "IN THE ATARI 2600 GAMES",
+ "type": "text"
+ }
+ ],
+ "index": 0
+ }
+ ],
+ "index": 0
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 106,
+ 102,
+ 504,
+ 126
+ ],
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 102,
+ 506,
+ 115
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 105,
+ 102,
+ 428,
+ 115
+ ],
+ "score": 1.0,
+ "content": "As recommended by Machado et al. (2018a), we report the performance of",
+ "type": "text"
+ },
+ {
+ "bbox": [
+ 429,
+ 102,
+ 486,
+ 115
+ ],
+ "score": 0.91,
+ "content": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } + \\mathrm { S R }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 486,
+ 102,
+ 506,
+ 115
+ ],
+ "score": 1.0,
+ "content": "and",
+ "type": "text"
+ }
+ ],
+ "index": 1
+ },
+ {
+ "bbox": [
+ 106,
+ 112,
+ 501,
+ 128
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 114,
+ 144,
+ 126
+ ],
+ "score": 0.83,
+ "content": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } }",
+ "type": "inline_equation"
+ },
+ {
+ "bbox": [
+ 145,
+ 112,
+ 501,
+ 128
+ ],
+ "score": 1.0,
+ "content": "after different amounts of experience (10, 50, and 100 million frames) in Tables 3 and 4.",
+ "type": "text"
+ }
+ ],
+ "index": 2
+ }
+ ],
+ "index": 1.5
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 151,
+ 162,
+ 456,
+ 241
+ ],
+ "blocks": [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 107,
+ 131,
+ 505,
+ 154
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 106,
+ 131,
+ 506,
+ 144
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 131,
+ 506,
+ 144
+ ],
+ "score": 1.0,
+ "content": "Finally, Figure 4 depicts the learning curves obtained with the evaluated algorithms in each game.",
+ "type": "text"
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
+ 106,
+ 142,
+ 506,
+ 155
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 106,
+ 142,
+ 506,
+ 155
+ ],
+ "score": 1.0,
+ "content": "Lighter lines represent individual runs while the solid lines encode the average over the multiple runs.",
+ "type": "text"
+ }
+ ],
+ "index": 4
+ }
+ ],
+ "index": 3.5
+ },
+ {
+ "type": "table_body",
+ "bbox": [
+ 151,
+ 162,
+ 456,
+ 241
+ ],
+ "group_id": 0,
+ "lines": [
+ {
+ "bbox": [
+ 151,
+ 162,
+ 456,
+ 241
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 151,
+ 162,
+ 456,
+ 241
+ ],
+ "score": 0.977,
+ "html": "| Algorithm1Exploration through the Substochastic Successor Representation (ESSR) |
| n(s,s')←0 ∀s,s'∈S | |
| t(s,a,s')←1 | ∀s,s'∈S,∀a∈A |
| r(s,a)←0 | ∀s∈S,∀a∈A |
| P(s,a)←1/|Sl | ∀s∈S,∀a∈A |
| P(s,s')←0 | As,s'∈S |
| T←random over A |
| while episode is not over do Observe s E S, take action a E A selected according to π(s),and observe a reward R and a |
| next state s' ∈ S n(s,s')← n(s,s')+1 |
| t(s,a,s')←t(s,a,s')+1 |
| n(s)←∑x',bt(s,b,x') |
| n(s,a)←∑x,t(s,a,x') |
| r(s,a,s') ← (t(s,a,s)-2)xf(s,a,s')+R |
| t(s,a,s')-1 for each state x' ∈ Sdo |
| P(s,a,x'）← t(s,a,x') |
| n(s,a) n(s,x') |
| P(s,x')← n(s)+1 |
| end for √←(I-γP)-1 |
| rint←-e |
| T ← POLICYITERATION(P,𝑟+ βrint) |
| end while |
"
+ },
+ {
+ "category_id": 8,
+ "poly": [
+ 320,
+ 936,
+ 762,
+ 936,
+ 762,
+ 1138,
+ 320,
+ 1138
+ ],
+ "score": 0.124
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 993,
+ 1211,
+ 1046,
+ 1211,
+ 1046,
+ 1244,
+ 993,
+ 1244
+ ],
+ "score": 0.92,
+ "latex": "\\pi ( s )"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 347,
+ 1276,
+ 773,
+ 1276,
+ 773,
+ 1490,
+ 347,
+ 1490
+ ],
+ "score": 0.9,
+ "latex": "\\begin{array} { r l } & { n ( s , s ^ { \\prime } ) \\gets n ( s , s ^ { \\prime } ) + 1 } \\\\ & { t ( s , a , s ^ { \\prime } ) \\gets t ( s , a , s ^ { \\prime } ) + 1 } \\\\ & { n ( s ) \\gets \\sum _ { x ^ { \\prime } , b } t ( s , b , x ^ { \\prime } ) } \\\\ & { n ( s , a ) \\gets \\sum _ { x ^ { \\prime } } t ( s , a , x ^ { \\prime } ) } \\\\ & { \\hat { r } ( s , a , s ^ { \\prime } ) \\gets \\frac { ( t ( s , a , s ^ { \\prime } ) - 2 ) \\times \\hat { r } ( s , a , s ^ { \\prime } ) + R } { t ( s , a , s ^ { \\prime } ) - 1 } } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 665,
+ 1212,
+ 740,
+ 1212,
+ 740,
+ 1239,
+ 665,
+ 1239
+ ],
+ "score": 0.9,
+ "latex": "a \\in { \\mathcal { A } }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 454,
+ 1211,
+ 522,
+ 1211,
+ 522,
+ 1239,
+ 454,
+ 1239
+ ],
+ "score": 0.89,
+ "latex": "s \\in \\mathcal { S }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 612,
+ 510,
+ 686,
+ 510,
+ 686,
+ 541,
+ 612,
+ 541
+ ],
+ "score": 0.89,
+ "latex": "\\mathrm { D Q N } _ { e }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 737,
+ 508,
+ 892,
+ 508,
+ 892,
+ 542,
+ 737,
+ 542
+ ],
+ "score": 0.89,
+ "latex": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } + \\mathrm { S R }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 465,
+ 1241,
+ 538,
+ 1241,
+ 538,
+ 1269,
+ 465,
+ 1269
+ ],
+ "score": 0.88,
+ "latex": "s ^ { \\prime } \\in \\mathcal { S }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 512,
+ 1492,
+ 588,
+ 1492,
+ 588,
+ 1522,
+ 512,
+ 1522
+ ],
+ "score": 0.87,
+ "latex": "\\boldsymbol { x } ^ { \\prime } \\in \\mathcal { S }"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 376,
+ 1525,
+ 635,
+ 1525,
+ 635,
+ 1632,
+ 376,
+ 1632
+ ],
+ "score": 0.85,
+ "latex": "\\begin{array} { r l } & { \\hat { P } ( s , a , x ^ { \\prime } ) \\gets \\frac { t ( s , a , x ^ { \\prime } ) } { n ( s , a ) } } \\\\ & { } \\\\ & { \\tilde { P } ( s , x ^ { \\prime } ) \\gets \\frac { n ( s , x ^ { \\prime } ) } { n ( s ) + 1 } } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1308,
+ 1212,
+ 1332,
+ 1212,
+ 1332,
+ 1238,
+ 1308,
+ 1238
+ ],
+ "score": 0.81,
+ "latex": "R"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 325,
+ 1145,
+ 383,
+ 1145,
+ 383,
+ 1170,
+ 325,
+ 1170
+ ],
+ "score": 0.8,
+ "latex": "\\pi "
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 325,
+ 1056,
+ 513,
+ 1056,
+ 513,
+ 1094,
+ 325,
+ 1094
+ ],
+ "score": 0.58,
+ "latex": "\\hat { P } ( s , a ) \\gets 1 / | \\mathcal { S } |"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 349,
+ 1656,
+ 790,
+ 1656,
+ 790,
+ 1786,
+ 349,
+ 1786
+ ],
+ "score": 0.49,
+ "latex": "\\begin{array} { r l } & { \\tilde { \\Psi } \\gets ( I - \\gamma \\tilde { P } ) ^ { - 1 } } \\\\ & { r _ { \\mathrm { i n t } } \\gets - \\tilde { \\Psi } \\mathbf { e } } \\\\ & { \\pi \\gets \\mathrm { P O L I C Y I T E R A T I O N } ( \\hat { P } , \\hat { r } + \\beta r _ { \\mathrm { i n t } } ) } \\end{array}"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 349,
+ 1657,
+ 560,
+ 1657,
+ 560,
+ 1740,
+ 349,
+ 1740
+ ],
+ "score": 0.42,
+ "latex": "\\begin{array} { r l } & { \\tilde { \\Psi } ( I - \\gamma \\tilde { P } ) ^ { - 1 } } \\\\ & { r _ { \\mathrm { i n t } } - \\tilde { \\Psi } \\mathbf { e } } \\end{array}"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 537,
+ 1142,
+ 564,
+ 1142,
+ 564,
+ 1169,
+ 537,
+ 1169
+ ],
+ "score": 0.41,
+ "latex": "\\mathcal A"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 325,
+ 1099,
+ 477,
+ 1099,
+ 477,
+ 1138,
+ 325,
+ 1138
+ ],
+ "score": 0.4,
+ "latex": "\\tilde { P } ( s , s ^ { \\prime } ) \\gets 0"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 352,
+ 1750,
+ 411,
+ 1750,
+ 411,
+ 1779,
+ 352,
+ 1779
+ ],
+ "score": 0.39,
+ "latex": "\\pi \\gets"
+ },
+ {
+ "category_id": 14,
+ "poly": [
+ 325,
+ 1098,
+ 477,
+ 1098,
+ 477,
+ 1138,
+ 325,
+ 1138
+ ],
+ "score": 0.25,
+ "latex": "\\tilde { P } ( s , s ^ { \\prime } ) \\gets 0"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 648.0,
+ 221.0,
+ 1051.0,
+ 221.0,
+ 1051.0,
+ 267.0,
+ 648.0,
+ 267.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 330.0,
+ 260.0,
+ 1370.0,
+ 260.0,
+ 1370.0,
+ 320.0,
+ 330.0,
+ 320.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 73.0,
+ 858.0,
+ 73.0,
+ 858.0,
+ 108.0,
+ 297.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 712.0,
+ 1228.0,
+ 712.0,
+ 1228.0,
+ 744.0,
+ 296.0,
+ 744.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 831.0,
+ 2085.0,
+ 867.0,
+ 2085.0,
+ 867.0,
+ 2125.0,
+ 831.0,
+ 2125.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 368.0,
+ 428.0,
+ 1138.0,
+ 428.0,
+ 1138.0,
+ 459.0,
+ 368.0,
+ 459.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 365.0,
+ 467.0,
+ 1376.0,
+ 467.0,
+ 1376.0,
+ 503.0,
+ 365.0,
+ 503.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 364.0,
+ 504.0,
+ 611.0,
+ 504.0,
+ 611.0,
+ 547.0,
+ 364.0,
+ 547.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 687.0,
+ 504.0,
+ 736.0,
+ 504.0,
+ 736.0,
+ 547.0,
+ 687.0,
+ 547.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 893.0,
+ 504.0,
+ 1406.0,
+ 504.0,
+ 1406.0,
+ 547.0,
+ 893.0,
+ 547.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 395.0,
+ 541.0,
+ 1052.0,
+ 541.0,
+ 1052.0,
+ 576.0,
+ 395.0,
+ 576.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 381.0,
+ 579.0,
+ 1403.0,
+ 579.0,
+ 1403.0,
+ 617.0,
+ 381.0,
+ 617.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 394.0,
+ 612.0,
+ 832.0,
+ 612.0,
+ 832.0,
+ 646.0,
+ 394.0,
+ 646.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 765.0,
+ 1407.0,
+ 765.0,
+ 1407.0,
+ 804.0,
+ 292.0,
+ 804.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 798.0,
+ 1408.0,
+ 798.0,
+ 1408.0,
+ 833.0,
+ 293.0,
+ 833.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 829.0,
+ 1410.0,
+ 829.0,
+ 1410.0,
+ 866.0,
+ 293.0,
+ 866.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 335.0,
+ 1408.0,
+ 335.0,
+ 1408.0,
+ 372.0,
+ 294.0,
+ 372.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 367.0,
+ 623.0,
+ 367.0,
+ 623.0,
+ 401.0,
+ 294.0,
+ 401.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 10,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 1353,
+ 1405,
+ 1353,
+ 1405,
+ 1568,
+ 298,
+ 1568
+ ],
+ "score": 0.983
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1123,
+ 1406,
+ 1123,
+ 1406,
+ 1339,
+ 297,
+ 1339
+ ],
+ "score": 0.981
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 892,
+ 1405,
+ 892,
+ 1405,
+ 1108,
+ 297,
+ 1108
+ ],
+ "score": 0.979
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 298,
+ 287,
+ 1407,
+ 287,
+ 1407,
+ 410,
+ 298,
+ 410
+ ],
+ "score": 0.976
+ },
+ {
+ "category_id": 3,
+ "poly": [
+ 302,
+ 436,
+ 1397,
+ 436,
+ 1397,
+ 696,
+ 302,
+ 696
+ ],
+ "score": 0.962
+ },
+ {
+ "category_id": 3,
+ "poly": [
+ 503,
+ 1684,
+ 1178,
+ 1684,
+ 1178,
+ 1928,
+ 503,
+ 1928
+ ],
+ "score": 0.96
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 297,
+ 1582,
+ 1401,
+ 1582,
+ 1401,
+ 1646,
+ 297,
+ 1646
+ ],
+ "score": 0.953
+ },
+ {
+ "category_id": 4,
+ "poly": [
+ 296,
+ 1954,
+ 1405,
+ 1954,
+ 1405,
+ 2019,
+ 296,
+ 2019
+ ],
+ "score": 0.939
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 298,
+ 75,
+ 857,
+ 75,
+ 857,
+ 105,
+ 298,
+ 105
+ ],
+ "score": 0.909
+ },
+ {
+ "category_id": 4,
+ "poly": [
+ 297,
+ 720,
+ 1403,
+ 720,
+ 1403,
+ 783,
+ 297,
+ 783
+ ],
+ "score": 0.904
+ },
+ {
+ "category_id": 2,
+ "poly": [
+ 836,
+ 2088,
+ 863,
+ 2088,
+ 863,
+ 2113,
+ 836,
+ 2113
+ ],
+ "score": 0.859
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 299,
+ 230,
+ 835,
+ 230,
+ 835,
+ 261,
+ 299,
+ 261
+ ],
+ "score": 0.788
+ },
+ {
+ "category_id": 0,
+ "poly": [
+ 309,
+ 838,
+ 1008,
+ 838,
+ 1008,
+ 867,
+ 309,
+ 867
+ ],
+ "score": 0.728
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 309,
+ 838,
+ 1008,
+ 838,
+ 1008,
+ 867,
+ 309,
+ 867
+ ],
+ "score": 0.14
+ },
+ {
+ "category_id": 1,
+ "poly": [
+ 299,
+ 230,
+ 835,
+ 230,
+ 835,
+ 261,
+ 299,
+ 261
+ ],
+ "score": 0.11
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 710,
+ 1384,
+ 802,
+ 1384,
+ 802,
+ 1415,
+ 710,
+ 1415
+ ],
+ "score": 0.92,
+ "latex": "\\beta = 0 . 0"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1184,
+ 1954,
+ 1341,
+ 1954,
+ 1341,
+ 1987,
+ 1184,
+ 1987
+ ],
+ "score": 0.9,
+ "latex": "\\mathrm { D Q N } _ { e } ^ { \\mathrm { M M C } } { + } \\mathrm { S R }"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 892,
+ 1275,
+ 1017,
+ 1275,
+ 1017,
+ 1305,
+ 892,
+ 1305
+ ],
+ "score": 0.89,
+ "latex": "\\cdot \\sigma ^ { 2 } = 9 0 3 . 6"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 1019,
+ 1244,
+ 1144,
+ 1244,
+ 1144,
+ 1275,
+ 1019,
+ 1275
+ ],
+ "score": 0.88,
+ "latex": "\\textstyle \\cdot \\sigma ^ { 2 } = 2 0 0 . 0"
+ },
+ {
+ "category_id": 13,
+ "poly": [
+ 917,
+ 1504,
+ 1045,
+ 1504,
+ 1045,
+ 1535,
+ 917,
+ 1535
+ ],
+ "score": 0.88,
+ "latex": "\\sigma ^ { 2 } = 2 3 0 . 1"
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 972.0,
+ 448.0,
+ 1058.0,
+ 448.0,
+ 1058.0,
+ 473.0,
+ 972.0,
+ 473.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1079.0,
+ 451.0,
+ 1103.0,
+ 451.0,
+ 1103.0,
+ 471.0,
+ 1079.0,
+ 471.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1108.0,
+ 446.0,
+ 1177.0,
+ 446.0,
+ 1177.0,
+ 467.0,
+ 1108.0,
+ 467.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1238.0,
+ 451.0,
+ 1258.0,
+ 451.0,
+ 1258.0,
+ 472.0,
+ 1238.0,
+ 472.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1287.0,
+ 441.0,
+ 1368.0,
+ 441.0,
+ 1368.0,
+ 480.0,
+ 1287.0,
+ 480.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 339.0,
+ 480.0,
+ 408.0,
+ 480.0,
+ 408.0,
+ 501.0,
+ 339.0,
+ 501.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 432.0,
+ 481.0,
+ 502.0,
+ 481.0,
+ 502.0,
+ 502.0,
+ 432.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 528.0,
+ 481.0,
+ 598.0,
+ 481.0,
+ 598.0,
+ 502.0,
+ 528.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 620.0,
+ 481.0,
+ 690.0,
+ 481.0,
+ 690.0,
+ 502.0,
+ 620.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 719.0,
+ 481.0,
+ 787.0,
+ 481.0,
+ 787.0,
+ 502.0,
+ 719.0,
+ 502.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 812.0,
+ 480.0,
+ 913.0,
+ 480.0,
+ 913.0,
+ 504.0,
+ 812.0,
+ 504.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1027.0,
+ 474.0,
+ 1113.0,
+ 474.0,
+ 1113.0,
+ 499.0,
+ 1027.0,
+ 499.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1218.0,
+ 471.0,
+ 1292.0,
+ 471.0,
+ 1292.0,
+ 504.0,
+ 1218.0,
+ 504.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 300.0,
+ 503.0,
+ 454.0,
+ 503.0,
+ 454.0,
+ 534.0,
+ 300.0,
+ 534.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 470.0,
+ 504.0,
+ 552.0,
+ 504.0,
+ 552.0,
+ 529.0,
+ 470.0,
+ 529.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 564.0,
+ 504.0,
+ 642.0,
+ 504.0,
+ 642.0,
+ 527.0,
+ 564.0,
+ 527.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 663.0,
+ 505.0,
+ 739.0,
+ 505.0,
+ 739.0,
+ 529.0,
+ 663.0,
+ 529.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 762.0,
+ 504.0,
+ 838.0,
+ 504.0,
+ 838.0,
+ 529.0,
+ 762.0,
+ 529.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1050.0,
+ 501.0,
+ 1115.0,
+ 501.0,
+ 1115.0,
+ 530.0,
+ 1050.0,
+ 530.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1151.0,
+ 513.0,
+ 1161.0,
+ 513.0,
+ 1161.0,
+ 522.0,
+ 1151.0,
+ 522.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 329.0,
+ 532.0,
+ 389.0,
+ 532.0,
+ 389.0,
+ 558.0,
+ 329.0,
+ 558.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 456.0,
+ 536.0,
+ 481.0,
+ 536.0,
+ 481.0,
+ 555.0,
+ 456.0,
+ 555.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 550.0,
+ 532.0,
+ 577.0,
+ 532.0,
+ 577.0,
+ 557.0,
+ 550.0,
+ 557.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 644.0,
+ 531.0,
+ 670.0,
+ 531.0,
+ 670.0,
+ 557.0,
+ 644.0,
+ 557.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 744.0,
+ 532.0,
+ 765.0,
+ 532.0,
+ 765.0,
+ 556.0,
+ 744.0,
+ 556.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 837.0,
+ 532.0,
+ 866.0,
+ 532.0,
+ 866.0,
+ 557.0,
+ 837.0,
+ 557.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 930.0,
+ 532.0,
+ 1018.0,
+ 532.0,
+ 1018.0,
+ 557.0,
+ 930.0,
+ 557.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1040.0,
+ 517.0,
+ 1143.0,
+ 517.0,
+ 1143.0,
+ 557.0,
+ 1040.0,
+ 557.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1151.0,
+ 525.0,
+ 1189.0,
+ 525.0,
+ 1189.0,
+ 571.0,
+ 1151.0,
+ 571.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1273.0,
+ 533.0,
+ 1293.0,
+ 533.0,
+ 1293.0,
+ 556.0,
+ 1273.0,
+ 556.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1301.0,
+ 527.0,
+ 1396.0,
+ 527.0,
+ 1396.0,
+ 560.0,
+ 1301.0,
+ 560.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1209.0,
+ 550.0,
+ 1272.0,
+ 550.0,
+ 1272.0,
+ 572.0,
+ 1209.0,
+ 572.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 385.0,
+ 563.0,
+ 444.0,
+ 563.0,
+ 444.0,
+ 588.0,
+ 385.0,
+ 588.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 486.0,
+ 563.0,
+ 547.0,
+ 563.0,
+ 547.0,
+ 588.0,
+ 486.0,
+ 588.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 576.0,
+ 563.0,
+ 637.0,
+ 563.0,
+ 637.0,
+ 588.0,
+ 576.0,
+ 588.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 679.0,
+ 563.0,
+ 739.0,
+ 563.0,
+ 739.0,
+ 588.0,
+ 679.0,
+ 588.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 770.0,
+ 563.0,
+ 831.0,
+ 563.0,
+ 831.0,
+ 588.0,
+ 770.0,
+ 588.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1173.0,
+ 562.0,
+ 1197.0,
+ 562.0,
+ 1197.0,
+ 581.0,
+ 1173.0,
+ 581.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1234.0,
+ 561.0,
+ 1308.0,
+ 561.0,
+ 1308.0,
+ 586.0,
+ 1234.0,
+ 586.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 379.0,
+ 581.0,
+ 448.0,
+ 581.0,
+ 448.0,
+ 603.0,
+ 379.0,
+ 603.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 481.0,
+ 580.0,
+ 551.0,
+ 580.0,
+ 551.0,
+ 605.0,
+ 481.0,
+ 605.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 572.0,
+ 581.0,
+ 640.0,
+ 581.0,
+ 640.0,
+ 602.0,
+ 572.0,
+ 602.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 673.0,
+ 580.0,
+ 742.0,
+ 580.0,
+ 742.0,
+ 605.0,
+ 673.0,
+ 605.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 765.0,
+ 580.0,
+ 835.0,
+ 580.0,
+ 835.0,
+ 605.0,
+ 765.0,
+ 605.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1219.0,
+ 588.0,
+ 1304.0,
+ 588.0,
+ 1304.0,
+ 621.0,
+ 1219.0,
+ 621.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 964.0,
+ 611.0,
+ 1110.0,
+ 611.0,
+ 1110.0,
+ 651.0,
+ 964.0,
+ 651.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1158.0,
+ 611.0,
+ 1363.0,
+ 611.0,
+ 1363.0,
+ 650.0,
+ 1158.0,
+ 650.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 531.0,
+ 666.0,
+ 685.0,
+ 666.0,
+ 685.0,
+ 692.0,
+ 531.0,
+ 692.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1097.0,
+ 666.0,
+ 1229.0,
+ 666.0,
+ 1229.0,
+ 695.0,
+ 1097.0,
+ 695.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1106.25,
+ 456.5,
+ 1237.25,
+ 456.5,
+ 1237.25,
+ 485.5,
+ 1106.25,
+ 485.5
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1170.25,
+ 511.0,
+ 1272.25,
+ 511.0,
+ 1272.25,
+ 548.5,
+ 1170.25,
+ 548.5
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1057.0,
+ 546.0,
+ 1168.0,
+ 546.0,
+ 1168.0,
+ 583.5,
+ 1057.0,
+ 583.5
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1036.75,
+ 585.0,
+ 1208.75,
+ 585.0,
+ 1208.75,
+ 625.0,
+ 1036.75,
+ 625.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 623.0,
+ 1687.0,
+ 724.0,
+ 1687.0,
+ 724.0,
+ 1707.0,
+ 623.0,
+ 1707.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 984.0,
+ 1687.0,
+ 1085.0,
+ 1687.0,
+ 1085.0,
+ 1707.0,
+ 984.0,
+ 1707.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 517.0,
+ 1709.0,
+ 545.0,
+ 1709.0,
+ 545.0,
+ 1726.0,
+ 517.0,
+ 1726.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 562.0,
+ 1704.0,
+ 681.0,
+ 1704.0,
+ 681.0,
+ 1724.0,
+ 562.0,
+ 1724.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 880.0,
+ 1710.0,
+ 905.0,
+ 1710.0,
+ 905.0,
+ 1725.0,
+ 880.0,
+ 1725.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 924.0,
+ 1704.0,
+ 1014.0,
+ 1704.0,
+ 1014.0,
+ 1727.0,
+ 924.0,
+ 1727.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 564.0,
+ 1719.0,
+ 625.0,
+ 1719.0,
+ 625.0,
+ 1739.0,
+ 564.0,
+ 1739.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 924.0,
+ 1718.0,
+ 987.0,
+ 1718.0,
+ 987.0,
+ 1741.0,
+ 924.0,
+ 1741.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1138.0,
+ 1728.0,
+ 1143.0,
+ 1728.0,
+ 1143.0,
+ 1733.0,
+ 1138.0,
+ 1733.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 518.0,
+ 1734.0,
+ 545.0,
+ 1734.0,
+ 545.0,
+ 1751.0,
+ 518.0,
+ 1751.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 880.0,
+ 1734.0,
+ 906.0,
+ 1734.0,
+ 906.0,
+ 1751.0,
+ 880.0,
+ 1751.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 752.0,
+ 1752.0,
+ 759.0,
+ 1752.0,
+ 759.0,
+ 1761.0,
+ 752.0,
+ 1761.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 519.0,
+ 1760.0,
+ 544.0,
+ 1760.0,
+ 544.0,
+ 1775.0,
+ 519.0,
+ 1775.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 880.0,
+ 1759.0,
+ 906.0,
+ 1759.0,
+ 906.0,
+ 1777.0,
+ 880.0,
+ 1777.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 505.0,
+ 1781.0,
+ 545.0,
+ 1781.0,
+ 545.0,
+ 1825.0,
+ 505.0,
+ 1825.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 716.0,
+ 1779.0,
+ 733.0,
+ 1779.0,
+ 733.0,
+ 1789.0,
+ 716.0,
+ 1789.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 859.0,
+ 1772.0,
+ 916.0,
+ 1772.0,
+ 916.0,
+ 1836.0,
+ 859.0,
+ 1836.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 521.0,
+ 1809.0,
+ 545.0,
+ 1809.0,
+ 545.0,
+ 1827.0,
+ 521.0,
+ 1827.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 884.0,
+ 1810.0,
+ 904.0,
+ 1810.0,
+ 904.0,
+ 1826.0,
+ 884.0,
+ 1826.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 521.0,
+ 1834.0,
+ 545.0,
+ 1834.0,
+ 545.0,
+ 1852.0,
+ 521.0,
+ 1852.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 884.0,
+ 1835.0,
+ 905.0,
+ 1835.0,
+ 905.0,
+ 1851.0,
+ 884.0,
+ 1851.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 522.0,
+ 1861.0,
+ 544.0,
+ 1861.0,
+ 544.0,
+ 1875.0,
+ 522.0,
+ 1875.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 883.0,
+ 1861.0,
+ 905.0,
+ 1861.0,
+ 905.0,
+ 1875.0,
+ 883.0,
+ 1875.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 531.0,
+ 1887.0,
+ 543.0,
+ 1887.0,
+ 543.0,
+ 1899.0,
+ 531.0,
+ 1899.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 894.0,
+ 1888.0,
+ 903.0,
+ 1888.0,
+ 903.0,
+ 1898.0,
+ 894.0,
+ 1898.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 531.0,
+ 1901.0,
+ 712.0,
+ 1901.0,
+ 712.0,
+ 1914.0,
+ 531.0,
+ 1914.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 715.0,
+ 1902.0,
+ 737.0,
+ 1902.0,
+ 737.0,
+ 1913.0,
+ 715.0,
+ 1913.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 742.0,
+ 1902.0,
+ 765.0,
+ 1902.0,
+ 765.0,
+ 1913.0,
+ 742.0,
+ 1913.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 767.0,
+ 1901.0,
+ 818.0,
+ 1901.0,
+ 818.0,
+ 1914.0,
+ 767.0,
+ 1914.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 892.0,
+ 1900.0,
+ 995.0,
+ 1900.0,
+ 995.0,
+ 1914.0,
+ 892.0,
+ 1914.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 996.0,
+ 1901.0,
+ 1180.0,
+ 1901.0,
+ 1180.0,
+ 1914.0,
+ 996.0,
+ 1914.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 632.0,
+ 1910.0,
+ 715.0,
+ 1910.0,
+ 715.0,
+ 1928.0,
+ 632.0,
+ 1928.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 994.0,
+ 1910.0,
+ 1078.0,
+ 1910.0,
+ 1078.0,
+ 1928.0,
+ 994.0,
+ 1928.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1074.75,
+ 1775.5,
+ 1096.75,
+ 1775.5,
+ 1096.75,
+ 1789.5,
+ 1074.75,
+ 1789.5
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1948.0,
+ 1183.0,
+ 1948.0,
+ 1183.0,
+ 1996.0,
+ 291.0,
+ 1996.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1342.0,
+ 1948.0,
+ 1408.0,
+ 1948.0,
+ 1408.0,
+ 1996.0,
+ 1342.0,
+ 1996.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1984.0,
+ 1321.0,
+ 1984.0,
+ 1321.0,
+ 2020.0,
+ 293.0,
+ 2020.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 298.0,
+ 73.0,
+ 858.0,
+ 73.0,
+ 858.0,
+ 108.0,
+ 298.0,
+ 108.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 718.0,
+ 1405.0,
+ 718.0,
+ 1405.0,
+ 755.0,
+ 294.0,
+ 755.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 297.0,
+ 750.0,
+ 874.0,
+ 750.0,
+ 874.0,
+ 786.0,
+ 297.0,
+ 786.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 832.0,
+ 2085.0,
+ 868.0,
+ 2085.0,
+ 868.0,
+ 2124.0,
+ 832.0,
+ 2124.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 228.0,
+ 837.0,
+ 228.0,
+ 837.0,
+ 265.0,
+ 295.0,
+ 265.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 305.0,
+ 837.0,
+ 1013.0,
+ 837.0,
+ 1013.0,
+ 869.0,
+ 305.0,
+ 869.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1353.0,
+ 1403.0,
+ 1353.0,
+ 1403.0,
+ 1388.0,
+ 296.0,
+ 1388.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1384.0,
+ 709.0,
+ 1384.0,
+ 709.0,
+ 1419.0,
+ 295.0,
+ 1419.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 803.0,
+ 1384.0,
+ 1403.0,
+ 1384.0,
+ 1403.0,
+ 1419.0,
+ 803.0,
+ 1419.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1415.0,
+ 1403.0,
+ 1415.0,
+ 1403.0,
+ 1449.0,
+ 295.0,
+ 1449.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1446.0,
+ 1407.0,
+ 1446.0,
+ 1407.0,
+ 1480.0,
+ 293.0,
+ 1480.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1475.0,
+ 1405.0,
+ 1475.0,
+ 1405.0,
+ 1508.0,
+ 293.0,
+ 1508.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1504.0,
+ 916.0,
+ 1504.0,
+ 916.0,
+ 1540.0,
+ 293.0,
+ 1540.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1046.0,
+ 1504.0,
+ 1408.0,
+ 1504.0,
+ 1408.0,
+ 1540.0,
+ 1046.0,
+ 1540.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 1533.0,
+ 1236.0,
+ 1533.0,
+ 1236.0,
+ 1572.0,
+ 293.0,
+ 1572.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1121.0,
+ 1409.0,
+ 1121.0,
+ 1409.0,
+ 1159.0,
+ 294.0,
+ 1159.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1154.0,
+ 1404.0,
+ 1154.0,
+ 1404.0,
+ 1187.0,
+ 294.0,
+ 1187.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1185.0,
+ 1408.0,
+ 1185.0,
+ 1408.0,
+ 1220.0,
+ 295.0,
+ 1220.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 1211.0,
+ 1411.0,
+ 1211.0,
+ 1411.0,
+ 1251.0,
+ 292.0,
+ 1251.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1245.0,
+ 1018.0,
+ 1245.0,
+ 1018.0,
+ 1279.0,
+ 295.0,
+ 1279.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1145.0,
+ 1245.0,
+ 1408.0,
+ 1245.0,
+ 1408.0,
+ 1279.0,
+ 1145.0,
+ 1279.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 291.0,
+ 1273.0,
+ 891.0,
+ 1273.0,
+ 891.0,
+ 1311.0,
+ 291.0,
+ 1311.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 1018.0,
+ 1273.0,
+ 1407.0,
+ 1273.0,
+ 1407.0,
+ 1311.0,
+ 1018.0,
+ 1311.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 1306.0,
+ 1280.0,
+ 1306.0,
+ 1280.0,
+ 1341.0,
+ 294.0,
+ 1341.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 891.0,
+ 1405.0,
+ 891.0,
+ 1405.0,
+ 927.0,
+ 294.0,
+ 927.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 926.0,
+ 1403.0,
+ 926.0,
+ 1403.0,
+ 957.0,
+ 295.0,
+ 957.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 294.0,
+ 956.0,
+ 1403.0,
+ 956.0,
+ 1403.0,
+ 986.0,
+ 294.0,
+ 986.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 988.0,
+ 1405.0,
+ 988.0,
+ 1405.0,
+ 1019.0,
+ 296.0,
+ 1019.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1017.0,
+ 1403.0,
+ 1017.0,
+ 1403.0,
+ 1048.0,
+ 296.0,
+ 1048.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1047.0,
+ 1405.0,
+ 1047.0,
+ 1405.0,
+ 1081.0,
+ 295.0,
+ 1081.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 1079.0,
+ 1342.0,
+ 1079.0,
+ 1342.0,
+ 1110.0,
+ 295.0,
+ 1110.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 286.0,
+ 1404.0,
+ 286.0,
+ 1404.0,
+ 321.0,
+ 293.0,
+ 321.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 319.0,
+ 1409.0,
+ 319.0,
+ 1409.0,
+ 352.0,
+ 295.0,
+ 352.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 293.0,
+ 347.0,
+ 1410.0,
+ 347.0,
+ 1410.0,
+ 382.0,
+ 293.0,
+ 382.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 292.0,
+ 376.0,
+ 1137.0,
+ 376.0,
+ 1137.0,
+ 415.0,
+ 292.0,
+ 415.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1582.0,
+ 1405.0,
+ 1582.0,
+ 1405.0,
+ 1618.0,
+ 296.0,
+ 1618.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 296.0,
+ 1613.0,
+ 791.0,
+ 1613.0,
+ 791.0,
+ 1645.0,
+ 296.0,
+ 1645.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 305.0,
+ 837.0,
+ 1013.0,
+ 837.0,
+ 1013.0,
+ 869.0,
+ 305.0,
+ 869.0
+ ],
+ "score": 1.0,
+ "text": ""
+ },
+ {
+ "category_id": 15,
+ "poly": [
+ 295.0,
+ 228.0,
+ 837.0,
+ 228.0,
+ 837.0,
+ 265.0,
+ 295.0,
+ 265.0
+ ],
+ "score": 1.0,
+ "text": ""
+ }
+ ],
+ "page_info": {
+ "page_no": 11,
+ "width": 1700,
+ "height": 2200
+ }
+ },
+ {
+ "layout_dets": [
+ {
+ "category_id": 5,
+ "poly": [
+ 422,
+ 452,
+ 1267,
+ 452,
+ 1267,
+ 671,
+ 422,
+ 671
+ ],
+ "score": 0.977,
+ "html": "