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b/parse/train/5GihaaZKL4/5GihaaZKL4.md new file mode 100644 index 0000000000000000000000000000000000000000..8fc17b4cbcf461afbc0be59f9b346f9b19e4508f --- /dev/null +++ b/parse/train/5GihaaZKL4/5GihaaZKL4.md @@ -0,0 +1,394 @@ +# Pairwise Adjusted Mutual Information + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 A well-known metric for quantifying the similarity between two clusterings is +2 the adjusted mutual information. Compared to mutual information, a corrective +3 term based on random permutations of the labels is introduced, preventing two +4 clusterings being similar by chance. Unfortunately, this adjustment makes the +5 metric computationally expensive. In this paper, we propose a novel adjustment +6 based on pairwise label permutations instead of full label permutations. Specifically, +7 we consider permutations where only two samples, selected uniformly at random, +8 exchange their labels. We show that the corresponding adjusted metric, which +9 can be expressed explicitly, behaves similarly to the standard adjusted mutual +10 information for assessing the quality of a clustering, while having a much lower +11 time complexity. Both metrics are compared in terms of quality and performance +12 on experiments based on synthetic and real data. + +# 13 1 Introduction + +14 A well-known metric for quantifying the similarity between two clusterings of the same data is +15 the adjusted mutual information [Nguyen et al., 2009; Vinh et al., 2010]. Compared to mutual +16 information, this metric is adjusted against chance, meaning that the similarity cannot be due to +17 randomness but only to the structure of the dataset, appearing in both clusterings. This is the reason +18 why this metric is widely used in unsupervised learning, see [Zhang et al., 2013; Thirion et al., 2014; +19 Taha and Hanbury, 2015; Yang et al., 2016; Wang et al., 2017] for various applications. +20 The standard way of adjusting mutual information against chance is through random label permuta +21 tions of one of the clusterings [Vinh et al., 2010]. Unfortunately, this adjustment makes the metric +22 computationally expensive. Specifically, the time complexity of the metric is in $O ( \operatorname* { m a x } ( k , l ) n )$ , +23 where $k , l$ are the numbers of clusters in each clustering and $n$ is the number of samples [Romano et +24 al., 2014]. As a comparison, the time complexity of mutual information is equal to $O ( k l )$ given the +25 contingency matrix of the clusterings, i.e., the matrix counting the number of samples in each pair of +26 clusters, one per clustering. The additional computational effort required by adjustment is significant +27 as the number of samples $n$ is typically much larger than the numbers of clusters $k , l$ . +28 In this paper, we propose a novel adjustment based on pairwise permutations. That is, we consider +29 permutations where only two samples, selected uniformly at random, exchange their labels. We +30 show that the corresponding adjusted metric, we refer to as pairwise adjusted mutual information, +31 is as efficient as adjusted mutual information for assessing the quality of a clustering, with a much +32 lower time complexity. In particular, the time complexity is the same as that of mutual information. +33 The gain in complexity is significant, as the computation time is now independent of the number of +34 samples $n$ , given the contingency matrix. +35 The rest of the paper is organized as follows. We first provide the definition and key properties of +36 adjusted mutual information in the general setting of information theory. We then introduce mutual +37 information with pairwise adjustement and explain why the exact same properties are satisfied by +38 this new notion of adjusted mutual information. The application of both notions of adjustment to +39 clustering, including the explicit expressions of the corresponding metrics, is presented in section 4. +40 Experiments on both synthetic and real data are presented in section 5. Section 6 concludes the paper. + +# 41 2 Adjusted mutual information + +42 Let $P$ be the uniform probability measure on $\Omega = \{ 1 , \dots , n \}$ , for some positive integer $n$ . Let $X , Y$ +43 be random variables on the probability space $( \Omega , P )$ . Without any loss of generality, we assume that +44 $X$ and $Y$ are mapping from $\Omega$ to sets consisting of consecutive integers, starting from 1. Denoting by +45 $H$ the entropy, the mutual information between $X$ and $Y$ is defined by [Cover and Thomas, 1991]: + +$$ +I ( X , Y ) = H ( X ) + H ( Y ) - H ( X , Y ) . +$$ + +This is the information shared by $X$ and $Y$ , which is equal to 0 if $X$ and $Y$ are independent. A distance between $X$ and $Y$ can then be defined by: + +$$ +d ( X , Y ) = H ( X , Y ) - I ( X , Y ) = H ( X | Y ) + H ( Y | X ) . +$$ + +46 This distance, known as the variation of information, is a metric in the quotient space of random +47 variables under the equivalence relation $X \sim Y$ if and only if there is some bijection $\varphi$ such that +48 $X = \varphi ( Y )$ [Meila, 2003]. ˘ +49 Adjusted mutual information. The adjusted mutual information between $X$ and $Y$ , corresponding +50 to the mutual information between $X$ and $Y$ adjusted against chance, is defined by: + +$$ +\Delta I ( X , Y ) = I ( X , Y ) - \operatorname { E } ( I ( X , Y _ { \sigma } ) ) , +$$ + +51 where $Y _ { \sigma }$ is the random variable $Y \circ \sigma$ , for any permutation $\sigma$ of $\{ 1 , \ldots , n \}$ , and the expectation is +52 taken over all permutations $\sigma$ , chosen uniformly at random. +53 Remark 1 (Normalization). It is frequent to also normalize adjusted mutual information, so as to +54 get a score between $\boldsymbol { \theta }$ and 1 [Vinh et al., 2010; Romano et al., 2014]. In this paper, we only focus on +55 the adjustment step. Note that normalization can be equally applied to both considered notions of +56 adjustment and thus be studied separately. + +57 We have the equivalent definition: + +$$ +\begin{array} { l } { \displaystyle \Delta I ( X , Y ) = \mathrm { E } ( H ( X , Y _ { \sigma } ) ) - H ( X , Y ) , } \\ { \displaystyle = \frac { 1 } { 2 } ( \mathrm { E } ( d ( X , Y _ { \sigma } ) ) - d ( X , Y ) ) . } \end{array} +$$ + +58 This equivalence follows from Proposition 1 and the fact that the definition is symmetric in $X$ and $Y$ +59 All proofs are available in the supplementary material. + +60 Proposition 1. We have for any random variables $X$ and $Y$ : + +$$ +\begin{array} { r l } & { H ( X ) = \operatorname { E } ( H ( X _ { \sigma } ) ) , } \\ & { \operatorname { E } ( H ( X , Y _ { \sigma } ) ) = \operatorname { E } ( H ( X _ { \sigma } , Y ) ) , } \\ & { \operatorname { E } ( I ( X , Y _ { \sigma } ) ) = \operatorname { E } ( I ( X _ { \sigma } , Y ) ) . } \end{array} +$$ + +61 In view of (3), we expect $\Delta I ( X , Y )$ to be positive if $X$ and $Y$ share information, as $X$ is expected to +62 be closer to $Y$ (for the distance $d$ ) than to $Y _ { \sigma }$ , a randomized version of $Y$ . There are specific cases +63 where $\Delta I ( X , Y ) = 0$ , as stated in Proposition 2; these cases will be interpreted in terms of clustering +64 in section 4. +65 Proposition 2. We have $\Delta I ( X , Y ) = 0$ whenever $Y$ (or $X$ , by symmetry) is constant or equal to +66 some permutation of $\{ 1 , \ldots , n \}$ . + +Adjusted entropy. Observing that $H ( X ) = I ( X , X )$ , we define similarly the adjusted entropy of $X$ by: + +$$ +\Delta H ( X ) = \Delta I ( X , X ) = H ( X ) - \operatorname { E } ( I ( X , X _ { \sigma } ) ) . +$$ + +67 By (1), we get: + +$$ +\Delta H ( X ) = \operatorname { E } ( H ( X , X _ { \sigma } ) ) - H ( X ) = { \frac { 1 } { 2 } } \operatorname { E } ( d ( X , X _ { \sigma } ) ) . +$$ + +68 Since $d$ is a metric, this shows that the adjusted entropy of $X$ is non-negative. + +69 Proposition 3. We have $\Delta H ( X ) = 0$ if and only if X is constant or equal to some permutation of +70 $\{ 1 , \ldots , n \}$ . +71 Proposition 3 characterizes random variables with zero adjusted entropy. Again, this result will be +2 interpreted in terms of clustering in section 4. + +# 73 3 Pairwise adjustment + +74 In this section, we introduce pairwise adjusted mutual information. The definition is the same as +75 adjusted mutual information, except that the permutation $\sigma$ is now restricted to the set of pairwise +76 permutations. Specifically, we consider permutations $\sigma$ for which there exists $i , j \in \{ 1 , \ldots , n \}$ +77 such that $\sigma ( i ) \stackrel { \textstyle - } { = } j$ and $\overset { \cdot } { \boldsymbol { \sigma } ( j ) } = i$ , whereas $\sigma ( t ) = t$ for all $t \neq i , j$ . We consider the set of such +78 permutations $\sigma$ where the samples $i , j$ are drawn uniformly at random in the set $\{ 1 , \ldots , n \}$ . We +79 denote by $\sigma _ { \mathrm { p } }$ such a random permutation. Observe that $\sigma _ { \mathrm { p } }$ is the identity with probability $1 / n$ (the +80 probability that $i = j$ ). + +Pairwise adjusted mutual information. We define the pairwise adjusted mutual information as: + +$$ +\Delta _ { \mathrm { p } } I ( X , Y ) = I ( X , Y ) - \mathrm { E } ( I ( X , Y _ { \sigma _ { \mathrm { p } } } ) ) . +$$ + +81 This is exactly the same definition as the adjusted mutual information, except for the considered +82 permutations $\sigma _ { \mathrm { p } }$ . It can be readily verified that the same properties apply, with the exact same proofs, +83 a key property being that the random permutations $\sigma _ { \mathrm { p } }$ and $\sigma _ { \mathrm { p } } ^ { \mathrm { ~ - 1 ~ } }$ have the same distributions. In +84 particular, we have the analogue of (3): + +$$ +\begin{array} { l } { { \Delta _ { \mathrm { p } } I ( X , Y ) = \mathrm { E } ( H ( X , Y _ { \sigma _ { \mathrm { p } } } ) ) - H ( X , Y ) , } } \\ { { \ ~ = { \frac { 1 } { 2 } } ( \mathrm { E } ( d ( X , Y _ { \sigma _ { \mathrm { p } } } ) ) - d ( X , Y ) ) . } } \end{array} +$$ + +85 Moreover, $\Delta _ { \mathrm { p } } I ( X , Y ) = 0$ whenever $X$ or $Y$ is constant or equal to some permutation of $\{ 1 , \ldots , n \}$ + +Pairwise adjusted entropy. We also define the pairwise adjusted entropy as: + +$$ +\Delta _ { \mathrm { p } } H ( X ) = \Delta _ { \mathrm { p } } I ( X , X ) = H ( X ) - \mathrm { E } ( I ( X , X _ { \sigma _ { \mathrm { p } } } ) ) . +$$ + +86 We have $\Delta _ { \mathrm { p } } H ( X ) \geq 0$ , with equality if and only if $X$ is constant or equal to some permutation of +87 $\{ 1 , \ldots , n \}$ . + +# 88 4 Application to clustering + +89 Let $A = \{ A _ { 1 } , \ldots , A _ { k } \}$ and $B = \{ B _ { 1 } , \ldots , B _ { l } \}$ be two partitions of some finite set $\{ 1 , \ldots , n \}$ into $k$ +90 and $l$ clusters, respectively. Let $\Omega = \{ 1 , \dots , n \}$ and $\mathrm { P }$ be the uniform probability measure over $\Omega$ . +91 Consider the random variables $X$ and $Y$ defined on $( \Omega , \mathrm { P } )$ by $X ^ { - 1 } ( i ) \doteq A _ { i }$ for all $i = 1 , \ldots , k$ and +92 $Y ^ { - 1 } ( j ) = B _ { j }$ for all $j = 1 , \dots , l .$ . Note that $X ( \omega )$ and $Y ( \omega )$ can be interpreted as the labels $i$ and $j$ +93 of sample $\omega$ in clusterings $A$ and $B$ , for each $\omega \in \{ 1 , \ldots , n \}$ . +94 We denote by $a _ { i } ~ = ~ \left| A _ { i } \right|$ the size of cluster $A _ { i }$ , by $b _ { j } ~ = ~ | B _ { j } |$ the size of cluster $B _ { j }$ , and by +95 $n _ { i j } = | A _ { i } \cap B _ { j } |$ the number of samples both in cluster $A _ { i }$ and cluster $B _ { j }$ , for all $i = 1 , \ldots , k$ and +96 $j = 1 , \dots , l$ . The matrix $( n _ { i j } ) _ { 1 \leq i \leq k , 1 \leq j \leq l }$ is known as the contingency matrix. Note that $a _ { i }$ and $b _ { j }$ +97 are the sums of row $i$ and column $j$ of the contingency matrix, respectively. +98 Adjusted mutual information. A well-known metric for assessing the similarity $s ( A , B )$ between +99 clusterings $A$ and $B$ is the adjusted mutual information 1 $\Delta I ( X , { \bar { Y } } )$ between the corresponding +100 random variables $X$ and $Y$ . In words, this is the common information shared by clusterings $A$ and $B$ +101 not due to randomness. +102 By Proposition 2, we have $s ( A , B ) = 0$ whenever clustering $A$ (or $B$ , by symmetry) is trivial, that is, +103 it consists of a single cluster or of $n$ clusters (one per sample). This is a key property, showing the +104 interest of the adjustment. + +105 It is known that [Vinh et al., 2010]: + +$$ +\begin{array} { l } { { \displaystyle s ( A , B ) = - \sum _ { i = 1 } ^ { k } \sum _ { j = 1 } ^ { l } \frac { n _ { i j } } { n } \log \frac { n _ { i j } } { n } } } \\ { { \displaystyle + \sum _ { i = 1 } ^ { k } \sum _ { j = 1 } ^ { l } \sum _ { c = ( a _ { i } + b _ { j } - n ) ^ { + } } ^ { \operatorname* { m i n } ( a _ { i } , b _ { j } ) } \frac { a _ { i } ! b _ { j } ! ( n - a _ { i } ) ! ( n - b _ { j } ) ! } { n ! c ! ( a _ { i } - c ) ! ( b _ { j } - c ) ! ( n - a _ { i } - b _ { j } + c ) ! } \frac { c } { n } \log \frac { c } { n } } , } \end{array} +$$ + +with the notation 106 $( \cdot ) ^ { + } = \operatorname* { m a x } ( \cdot , 0 )$ . The time complexity of this formula, which is dominated by the 107 second term, is in $O ( \operatorname* { m a x } ( k , l ) n )$ [Romano et al., 2014]. In particular, it is linear in the number of 108 samples $n$ . + +109 Interestingly, we can similarly assess the quantity of information $q ( A )$ contained in clustering +110 $A$ through the adjusted entropy $\Delta H ( X )$ of the corresponding random variable $X$ . This is the +111 information contained in $A$ not due to randomness. We have $q ( A ) \geq 0$ and, by Proposition 3, +112 $q ( A ) = 0$ if and only if clustering $A$ is trivial, that is, it consists of a single cluster or of $n$ clusters +113 (one per sample). + +114 Since $q ( A ) = s ( A , A )$ , it follows from (6) that: + +$$ +\mathfrak { r } ( A ) = - \sum _ { i = 1 } ^ { k } \frac { a _ { i } } { n } \log \frac { a _ { i } } { n } + \sum _ { i , j = 1 } ^ { K } \sum _ { c = ( a _ { i } + a _ { j } - n ) ^ { + } } ^ { \operatorname* { m i n } ( a _ { i } , a _ { j } ) } + \frac { a _ { i } ! a _ { j } ! ( n - a _ { i } ) ! ( n - a _ { j } ) ! } { n ! c ! ( a _ { i } - c ) ! ( a _ { j } - c ) ! ( n - a _ { i } - a _ { j } + k ) ! } \frac { c } { n } \log \frac { c } { n } . +$$ + +115 The time complexity of this formula, also dominated by the second term, is in $O ( k n )$ . Again, this +116 complexity is linear in the number of samples $n$ . +117 Pairwise adjusted mutual information. The main contribution of the paper is the following new +118 measure of similarity $s _ { \mathrm { p } } ( A , B )$ between clusterings $A$ and $B$ , based on the pairwise adjusted mutual +119 information $\Delta _ { \mathrm { p } } I ( X , Y )$ between the corresponding random variables $X$ and $Y$ . We have an explicit +120 expression for this similarity: + +121 Theorem 1. We have for any clusterings $A , B$ + +$$ +\begin{array} { c } { { s _ { \mathrm { p } } ( A , B ) = 2 \displaystyle \sum _ { i = 1 } ^ { k } \displaystyle \sum _ { j = 1 } ^ { l } \displaystyle \frac { n _ { i j } ( n - a _ { i } - b _ { j } + n _ { i j } ) } { n ^ { 2 } } \left( \displaystyle \frac { n _ { i j } } { n } \log \displaystyle \frac { n _ { i j } } { n } - \displaystyle \frac { n _ { i j } - 1 } { n } \log \displaystyle \frac { n _ { i j } - 1 } { n } \right) } } \\ { { + 2 \displaystyle \sum _ { i = 1 } ^ { k } \displaystyle \sum _ { j = 1 } ^ { l } \displaystyle \frac { ( a _ { i } - n _ { i j } ) ( b _ { j } - n _ { i j } ) } { n ^ { 2 } } \left( \displaystyle \frac { n _ { i j } } { n } \log \displaystyle \frac { n _ { i j } } { n } - \displaystyle \frac { n _ { i j } + 1 } { n } \log \displaystyle \frac { n _ { i j } + 1 } { n } \right) . } } \end{array} +$$ + +122 The time complexity of this formula is in $O ( k l )$ , like mutual information. It is independent of the +123 number of samples $n$ , given the contingency matrix. Corollary 1 shows that the time complexity +124 reduces to $O ( m )$ , where $m$ is the number of non-zero entries of the contingency matrix, provided the +125 latter is stored in sparse format. + +126 Corollary 1. We have for any clusterings $A , B$ : + +$$ +{ \begin{array} { l } { { \displaystyle { \boldsymbol { s } } _ { \mathrm { p } } ( A , B ) = 2 \sum _ { i , j : n _ { i j } > 0 } { \frac { n _ { i j } ( n - a _ { i } - b _ { j } + n _ { i j } ) } { n ^ { 2 } } } \left( { \frac { n _ { i j } } { n } } \log { \frac { n _ { i j } } { n } } - { \frac { n _ { i j } - 1 } { n } } \log { \frac { n _ { i j } - 1 } { n } } \right) } } \\ { \displaystyle \qquad + 2 \sum _ { i , j : n _ { i j } > 0 } { \frac { ( a _ { i } - n _ { i j } ) ( b _ { j } - n _ { i j } ) } { n ^ { 2 } } } \left( { \frac { n _ { i j } } { n } } \log { \frac { n _ { i j } } { n } } - { \frac { n _ { i j } + 1 } { n } } \log { \frac { n _ { i j } + 1 } { n } } + { \frac { 1 } { n } } \log { \frac { 1 } { n } } \right) } \\ { \displaystyle \qquad - 2 \left( n ^ { 2 } - \sum _ { i = 1 } ^ { k } a _ { i } ^ { 2 } - \sum _ { j = 1 } ^ { l } b _ { i } ^ { 2 } + \sum _ { i , j : n _ { i j } > 0 } n _ { i j } ^ { 2 } \right) { \frac { 1 } { n } } \log { \frac { 1 } { n } } . } \end{array} } +$$ + +127 Similarly, we can define the quantity of information $q _ { \mathrm { p } } ( A )$ in clustering $A$ through the pairwise +128 adjusted entropy $\Delta _ { \mathrm { p } } H ( X )$ of the corresponding random variable $X$ . Again, $q _ { \mathrm { p } } ( A ) \geq 0$ , with +129 $q _ { \mathrm { p } } \overset { \cdot } { (} A ) = 0$ if and only if clustering $A$ is trivial. + +$$ +q _ { \mathrm { p } } ( A ) = 2 \sum _ { i = 1 } ^ { k } { \frac { a _ { i } ( n - a _ { i } ) } { n ^ { 2 } } } \left( { \frac { a _ { i } } { n } } \log { \frac { a _ { i } } { n } } - { \frac { a _ { i } - 1 } { n } } \log { \frac { a _ { i } - 1 } { n } } - { \frac { 1 } { n } } \log { \frac { 1 } { n } } \right) . +$$ + +131 Note that the time complexity of this formula in $O ( k )$ . It only depends on the number of clusters $k$ , +132 and not on the number of samples $n$ . + +# 133 5 Experiments + +134 In this section, we compare both notions of adjusted mutual information through experiments +135 involving synthetic and real data. The experiments are run on a computer equipped with an AMD +136 Ryzen Threadripper 1950X 16-Core Processor and 32 GB of RAM, with a a Debian $1 0 0 \mathrm { S }$ . All codes +137 and datasets used in the experiments are available in the supplementary material. +138 Synthetic data. We start with the simple case of $n = 1 0 0$ samples with clusters of even sizes, +139 consisting of consecutive samples. Specifically, we consider the set of clusterings $A ^ { ( s ) }$ , consisting of +140 clusters of size $s$ (except possibly the last one), for $s = 1 , 2 , \ldots , 1 0 0$ . In particular, both $A ^ { ( 1 ) }$ and +141 $A ^ { ( 1 0 0 ) }$ are trivial clusterings while $A ^ { ( 5 ) }$ consists of 20 clusters of size 5. +142 Figure 1 gives the similarity between clusterings $A ^ { ( 1 0 ) }$ and $A ^ { ( s ) }$ with respect to $s$ in terms of adjusted +143 mutual information, for both notions of adjustment, i.e., $s ( A ^ { ( 1 0 ) } , A ^ { ( s ) } )$ and $s _ { \mathrm { p } } ( A ^ { ( 1 0 ) } , A ^ { ( s ) } )$ . We +144 observe very close behaviors, suggesting that both notions of adjustment tend to capture the same +145 patterns in the clusterings. Note that the maximum similarity is attained for $s = 1 0$ in both cases, as +146 expected. The similarity is equal to 0 for $s \in \{ 1 , 1 0 0 \}$ for both cases, in agreement with Proposition +147 2. We also observe local peaks at $s = 2 0 , 3 0 , \ldots , 9 0$ , which can be interpreted by the fact that +148 clustering $A ^ { ( 1 0 ) }$ is a refinement of clustering $A ^ { ( s ) }$ for these values of $s$ ; similarly, the local peak at +149 $s = 5$ may be interpreted by the fact that clustering $A ^ { ( 5 ) }$ is a refinement of clustering $A ^ { ( 1 \bar { 0 } ) }$ . The +Spearman correlation between both metrics over all values of $s$ is equal to 0.99. +151 We now consider random clusterings. Specifically, we assign $n$ samples to $k$ clusters independently +152 at random, according to some probability distribution $p = ( p _ { 1 } , \dotsc , p _ { k } )$ , which is itself drawn at +153 random2. Consider three such random clusterings $A , B$ , $C$ (with the same parameters $n$ and $k$ , but +154 different probability distributions $p$ ). We would like to know whether $A$ is “closer" to $B$ or to $C$ . In +155 particular, we are interested in testing whether both notions of adjusted mutual information give the +156 same ordering in the sense that: + +![](images/6c42fe6957f17783f8c1554e8baaab5de933abc19aa87ac9cef5f942232debb2.jpg) +Figure 1: Comparison of metrics on synthetic data $\mathit { n } = 1 0 0 $ ). + +$$ +( s ( A , B ) - s ( A , C ) ) ( s _ { \mathrm { p } } ( A , B ) - s _ { \mathrm { p } } ( A , C ) ) \geq 0 . +$$ + +157 We compute the average precision score (fraction of triplets $A , B , C$ for which (7) is true) over 1 000 +158 independent samples of $A , B , C$ , for different values of $n$ and $k$ . We repeat the experiment 100 times +159 to get the mean and standard deviation. The results are given in Table 1. We observe a very high +160 precision score, always higher than $9 3 \%$ , showing that both notions of adjusted mutual information +161 tend to give the same ordering of these random clusterings. +162 For the performance gain, we compare the computation times of both versions of adjusted mutual +163 information for the similarity between clusterings $A$ and $B$ , where $A$ consists of $k = 1 0$ clusters +164 of same size and $B$ is a random clustering, drawn as in the previous experiment. Both versions of +165 adjusted mutual information are coded in Python, with the standard version imported from scikit-learn. +166 Figure 2 shows the computation time when the number of samples $n$ grows from $1 0 ^ { 2 }$ to $1 0 ^ { 7 }$ . The +167 performance gain brought by pairwise adjustement is significant. In particular, the computation time +168 becomes independent of the number of samples. + +
nkPrecision score
100 1002 50.972 ± 0.004 0.952 ± 0.007
100100.943 ± 0.006
100200.955 ± 0.008
500200.936 ± 0.007
1000 100020 500.933 ± 0.006
+ +![](images/b58a6502beb2f00f992bece4b886133981210c328e34993ded52113c1071e8e7.jpg) +Table 1: Precision score (mean $\pm$ standard deviation) +Figure 2: Computation time with respect to $n$ (mean $\pm$ standard deviation). + +Real data. We first consider the 79 datasets of the benchmark suite [Gagolewski,169 $2 0 2 0 ] ^ { 3 }$ . We apply 170 to each dataset each of the following clustering algorithms: + +• $k$ -means +• Affinity propagation +• Mean shift +• Spectral clustering +• Ward +• Agglomerative clustering +• DBSCAN +• OPTICS +• Birch +• Gaussian Mixture + +181 We use the scikit-learn4 implementation of these algorithms, with the corresponding default param +182 eters5. We get 10 clusterings per dataset. The quality of each clustering is assessed through the +183 similarity with the available ground-truth labels, using adjusted mutual information with either full +184 adjustment or pairwise adjustment. We then compute the Spearman correlation of the corresponding +185 similarities, a value of 1 meaning the exact same ordering of the 10 clusterings with full adjustment +186 and pairwise adjustment. The results are shown in Figure 3, together with the speed-up in computation +187 time due to pairwise adjustment. In both cases, the 79 datasets are ordered by the number of samples, +188 ranging from 105 to 105 600 [Gagolewski, 2020]. +189 We first observe that the correlation is very high, suggesting again that both notions of adjusted mutual +190 information tend to provide the same results. For 65 datasets among 79, the Spearman correlation is +191 higher than $9 5 \%$ . As for the computation time, we observe a significant performance gain, by one +192 order of magnitude for the largest datasets. +193 We have conducted the same experiments with OpenML [Vanschoren et al., 2013]6. We selected all +194 datasets with at least 1,000 but no more than 50,000 samples, at most 100 features (all numerical), no +195 missing data and ground-truth labels forming clusters of at least 5 samples on average. The results +196 are shown in Figure for the resulting 34 datasets. Again, the datasets are ordered by the number of +197 samples, here ranging from 1,188 to 45,918. The conclusions are similar. In particular, the Spearman +198 correlation is higher than $9 5 \%$ for 30 datasets among 34, and the performance gain exceeds 25 for the +199 largest datasets. + +![](images/4a3e4f946dd9c298bc0837f6334696c17a0313fb56ff03d594dce17266eda230.jpg) +Figure 3: Comparison of metrics on the Gagolewski benchmark. + +![](images/3056fac0045be17a9b229a2afba177861eb5726b163d2534a53a3d5166a34841.jpg) +Figure 4: Comparison of metrics on OpenML datasets. + +We have proposed another way of adjusting mutual information against chance, through pairwise label permutations. The novel metric, whose explicit expression is given in Theorem 1, has a much lower complexity than the usual adjusted mutual information. Interestingly, both metrics can also be used to assess the quantity of information contained in a clustering, which the common property of being equal to 0 if and only if the clustering is trivial, as stated in Proposition 3; again, the pairwise adjusted entropy, given in Corollary 2, has a much lower complexity. Experiments on synthetic and real data show that pairwise adjusted mutual information tends to provide the same results as the usual adjusted mutual information for comparing clusterings, while involving much less computations. + +For future work, we plan to extend this idea to other similarity metrics. While the practical interest is less obvious for the Adjusted Rand Index [Hubert and Arabie, 1985], due to the fact that the time complexity of this metric is already independent of the number of samples, it would be worth considering other versions of information theoretic measures, as those studied in [Romano et al., 2016]. + +# References + +Thomas M Cover and Joy A Thomas. Elements of Information Theory. Wiley, 1991. + +Marek Gagolewski. Benchmark suite for clustering algorithms – version 1, 2020. + +Lawrence Hubert and Phipps Arabie. Comparing partitions. Journal of classification, 2(1):193–218, 1985. + +Marina Meila. Comparing clusterings by the variation of information. In ˘ Learning theory and kernel machines, pages 173–187. Springer, 2003. + +Xuan Vinh Nguyen, Julien Epps, and James Bailey. Information theoretic measures for clusterings comparison: is a correction for chance necessary? In ICML, 2009. + +Simone Romano, James Bailey, Vinh Nguyen, and Karin Verspoor. Standardized mutual information for clustering comparisons: one step further in adjustment for chance. In International Conference on Machine Learning, pages 1143–1151, 2014. + +Simone Romano, Nguyen Xuan Vinh, James Bailey, and Karin Verspoor. Adjusting for chance clustering comparison measures. The Journal of Machine Learning Research, 17(1):4635–4666, 2016. + +Abdel Aziz Taha and Allan Hanbury. Metrics for evaluating 3d medical image segmentation: analysis, selection, and tool. BMC medical imaging, 15(1):29, 2015. + +Bertrand Thirion, Gaël Varoquaux, Elvis Dohmatob, and Jean-Baptiste Poline. Which fmri clustering gives good brain parcellations? Frontiers in neuroscience, 8:167, 2014. + +Joaquin Vanschoren, Jan N. van Rijn, Bernd Bischl, and Luis Torgo. Openml: Networked science in machine learning. SIGKDD Explorations, 15(2):49–60, 2013. + +Nguyen Xuan Vinh, Julien Epps, and James Bailey. Information theoretic measures for clusterings comparison: Variants, properties, normalization and correction for chance. The Journal of Machine Learning Research, 11:2837–2854, 2010. + +Bo Wang, Junjie Zhu, Emma Pierson, Daniele Ramazzotti, and Serafim Batzoglou. Visualization and analysis of single-cell rna-seq data by kernel-based similarity learning. Nature methods, 14(4):414–416, 2017. + +Zhao Yang, René Algesheimer, and Claudio J Tessone. A comparative analysis of community detection algorithms on artificial networks. Scientific reports, 6:30750, 2016. + +Jiajie Zhang, Paschalia Kapli, Pavlos Pavlidis, and Alexandros Stamatakis. A general species delimitation method with applications to phylogenetic placements. Bioinformatics, 29(22):2869– 2876, 2013. + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] A variant of adjusted mutual information. +(b) Did you describe the limitations of your work? [Yes] Pairwise adjustement only applied to mutual information in the present work, see Section 6. +(c) Did you discuss any potential negative societal impacts of your work? [N/A] +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] No specific assumption is required. Theorem 1, Corollary 2 and 3 give explicit expressions using notations defined at the beginning of Section 4. +(b) Did you include complete proofs of all theoretical results? [Yes] See the supplementary material. + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See the Jupyter notebooks in the supplementary material. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A] This is a metric for unsupervised learning. No data split required, no hyperparameter. +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Table 1 and Figure 2 (not applicable to other experiments). +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See the running times provided in the Figures; the resources used are detailed at the beginning of section 5. + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] See the references for the datasets. +(b) Did you mention the license of the assets? [N/A] +(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/train/5GihaaZKL4/5GihaaZKL4_content_list.json b/parse/train/5GihaaZKL4/5GihaaZKL4_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..808d03a948d5e057463a57a7401954a974be0855 --- /dev/null +++ b/parse/train/5GihaaZKL4/5GihaaZKL4_content_list.json @@ -0,0 +1,1272 @@ +[ + { + "type": "text", + "text": "Pairwise Adjusted Mutual Information ", + "text_level": 1, + "bbox": [ + 261, + 122, + 736, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 200, + 580, + 256 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 292, + 535, + 309 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 A well-known metric for quantifying the similarity between two clusterings is \n2 the adjusted mutual information. Compared to mutual information, a corrective \n3 term based on random permutations of the labels is introduced, preventing two \n4 clusterings being similar by chance. Unfortunately, this adjustment makes the \n5 metric computationally expensive. In this paper, we propose a novel adjustment \n6 based on pairwise label permutations instead of full label permutations. Specifically, \n7 we consider permutations where only two samples, selected uniformly at random, \n8 exchange their labels. We show that the corresponding adjusted metric, which \n9 can be expressed explicitly, behaves similarly to the standard adjusted mutual \n10 information for assessing the quality of a clustering, while having a much lower \n11 time complexity. Both metrics are compared in terms of quality and performance \n12 on experiments based on synthetic and real data. ", + "bbox": [ + 148, + 324, + 766, + 491 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "13 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 516, + 312, + 534 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "14 A well-known metric for quantifying the similarity between two clusterings of the same data is \n15 the adjusted mutual information [Nguyen et al., 2009; Vinh et al., 2010]. Compared to mutual \n16 information, this metric is adjusted against chance, meaning that the similarity cannot be due to \n17 randomness but only to the structure of the dataset, appearing in both clusterings. This is the reason \n18 why this metric is widely used in unsupervised learning, see [Zhang et al., 2013; Thirion et al., 2014; \n19 Taha and Hanbury, 2015; Yang et al., 2016; Wang et al., 2017] for various applications. \n20 The standard way of adjusting mutual information against chance is through random label permuta \n21 tions of one of the clusterings [Vinh et al., 2010]. Unfortunately, this adjustment makes the metric \n22 computationally expensive. Specifically, the time complexity of the metric is in $O ( \\operatorname* { m a x } ( k , l ) n )$ , \n23 where $k , l$ are the numbers of clusters in each clustering and $n$ is the number of samples [Romano et \n24 al., 2014]. As a comparison, the time complexity of mutual information is equal to $O ( k l )$ given the \n25 contingency matrix of the clusterings, i.e., the matrix counting the number of samples in each pair of \n26 clusters, one per clustering. The additional computational effort required by adjustment is significant \n27 as the number of samples $n$ is typically much larger than the numbers of clusters $k , l$ . \n28 In this paper, we propose a novel adjustment based on pairwise permutations. That is, we consider \n29 permutations where only two samples, selected uniformly at random, exchange their labels. We \n30 show that the corresponding adjusted metric, we refer to as pairwise adjusted mutual information, \n31 is as efficient as adjusted mutual information for assessing the quality of a clustering, with a much \n32 lower time complexity. In particular, the time complexity is the same as that of mutual information. \n33 The gain in complexity is significant, as the computation time is now independent of the number of \n34 samples $n$ , given the contingency matrix. \n35 The rest of the paper is organized as follows. We first provide the definition and key properties of \n36 adjusted mutual information in the general setting of information theory. We then introduce mutual \n37 information with pairwise adjustement and explain why the exact same properties are satisfied by \n38 this new notion of adjusted mutual information. The application of both notions of adjustment to \n39 clustering, including the explicit expressions of the corresponding metrics, is presented in section 4. \n40 Experiments on both synthetic and real data are presented in section 5. Section 6 concludes the paper. ", + "bbox": [ + 147, + 547, + 825, + 632 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 637, + 825, + 750 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 756, + 825, + 852 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 858, + 823, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 90, + 826, + 133 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "41 2 Adjusted mutual information ", + "text_level": 1, + "bbox": [ + 151, + 151, + 450, + 170 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "42 Let $P$ be the uniform probability measure on $\\Omega = \\{ 1 , \\dots , n \\}$ , for some positive integer $n$ . Let $X , Y$ \n43 be random variables on the probability space $( \\Omega , P )$ . Without any loss of generality, we assume that \n44 $X$ and $Y$ are mapping from $\\Omega$ to sets consisting of consecutive integers, starting from 1. Denoting by \n45 $H$ the entropy, the mutual information between $X$ and $Y$ is defined by [Cover and Thomas, 1991]: ", + "bbox": [ + 145, + 181, + 825, + 239 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/cf82846bb2627a2ca551c2cc2cb82b2fcfe5a7bb24ba0e3779aed226f151d84f.jpg", + "text": "$$\nI ( X , Y ) = H ( X ) + H ( Y ) - H ( X , Y ) .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 243, + 633, + 260 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This is the information shared by $X$ and $Y$ , which is equal to 0 if $X$ and $Y$ are independent. A distance between $X$ and $Y$ can then be defined by: ", + "bbox": [ + 169, + 263, + 825, + 292 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/a2ab30a9acd3c9e38212642424f70815e82a9eff40e5b1daf1f5e8a71f31364c.jpg", + "text": "$$\nd ( X , Y ) = H ( X , Y ) - I ( X , Y ) = H ( X | Y ) + H ( Y | X ) .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 296, + 691, + 314 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "46 This distance, known as the variation of information, is a metric in the quotient space of random \n47 variables under the equivalence relation $X \\sim Y$ if and only if there is some bijection $\\varphi$ such that \n48 $X = \\varphi ( Y )$ [Meila, 2003]. ˘ \n49 Adjusted mutual information. The adjusted mutual information between $X$ and $Y$ , corresponding \n50 to the mutual information between $X$ and $Y$ adjusted against chance, is defined by: ", + "bbox": [ + 147, + 316, + 825, + 359 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 373, + 826, + 402 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/d032a6e7cf0b34c7b4af347aa102c30d281a6d3dbdb878a15d55204d65c48c60.jpg", + "text": "$$\n\\Delta I ( X , Y ) = I ( X , Y ) - \\operatorname { E } ( I ( X , Y _ { \\sigma } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 406, + 627, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "51 where $Y _ { \\sigma }$ is the random variable $Y \\circ \\sigma$ , for any permutation $\\sigma$ of $\\{ 1 , \\ldots , n \\}$ , and the expectation is \n52 taken over all permutations $\\sigma$ , chosen uniformly at random. \n53 Remark 1 (Normalization). It is frequent to also normalize adjusted mutual information, so as to \n54 get a score between $\\boldsymbol { \\theta }$ and 1 [Vinh et al., 2010; Romano et al., 2014]. In this paper, we only focus on \n55 the adjustment step. Note that normalization can be equally applied to both considered notions of \n56 adjustment and thus be studied separately. ", + "bbox": [ + 155, + 428, + 823, + 455 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 458, + 826, + 515 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "57 We have the equivalent definition: ", + "bbox": [ + 148, + 525, + 397, + 539 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/7f15f2b449c961105904d498f389f8227de44b9e7ab7898e06c1f912aea8bb0d.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\Delta I ( X , Y ) = \\mathrm { E } ( H ( X , Y _ { \\sigma } ) ) - H ( X , Y ) , } \\\\ { \\displaystyle = \\frac { 1 } { 2 } ( \\mathrm { E } ( d ( X , Y _ { \\sigma } ) ) - d ( X , Y ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 541, + 640, + 593 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "58 This equivalence follows from Proposition 1 and the fact that the definition is symmetric in $X$ and $Y$ \n59 All proofs are available in the supplementary material. ", + "bbox": [ + 155, + 594, + 825, + 623 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "60 Proposition 1. We have for any random variables $X$ and $Y$ : ", + "bbox": [ + 145, + 626, + 576, + 641 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/077618321449a50c668ce3a0eb9ba65e513de9583f180784bcccbae6654290ad.jpg", + "text": "$$\n\\begin{array} { r l } & { H ( X ) = \\operatorname { E } ( H ( X _ { \\sigma } ) ) , } \\\\ & { \\operatorname { E } ( H ( X , Y _ { \\sigma } ) ) = \\operatorname { E } ( H ( X _ { \\sigma } , Y ) ) , } \\\\ & { \\operatorname { E } ( I ( X , Y _ { \\sigma } ) ) = \\operatorname { E } ( I ( X _ { \\sigma } , Y ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 392, + 643, + 604, + 699 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "61 In view of (3), we expect $\\Delta I ( X , Y )$ to be positive if $X$ and $Y$ share information, as $X$ is expected to \n62 be closer to $Y$ (for the distance $d$ ) than to $Y _ { \\sigma }$ , a randomized version of $Y$ . There are specific cases \n63 where $\\Delta I ( X , Y ) = 0$ , as stated in Proposition 2; these cases will be interpreted in terms of clustering \n64 in section 4. \n65 Proposition 2. We have $\\Delta I ( X , Y ) = 0$ whenever $Y$ (or $X$ , by symmetry) is constant or equal to \n66 some permutation of $\\{ 1 , \\ldots , n \\}$ . ", + "bbox": [ + 145, + 707, + 825, + 763 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 766, + 825, + 796 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Adjusted entropy. Observing that $H ( X ) = I ( X , X )$ , we define similarly the adjusted entropy of $X$ by: ", + "bbox": [ + 174, + 808, + 823, + 837 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/a8d03fbbb2b6e7c6d21917b87ecf52fcfc1496cefe5e1b514afda3c892d68e3d.jpg", + "text": "$$\n\\Delta H ( X ) = \\Delta I ( X , X ) = H ( X ) - \\operatorname { E } ( I ( X , X _ { \\sigma } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 835, + 663, + 853 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "67 By (1), we get: ", + "bbox": [ + 148, + 854, + 274, + 869 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/26992134bb67a03e7dca737a7ccbe7c8561dd01614e3842213ba431bb860555b.jpg", + "text": "$$\n\\Delta H ( X ) = \\operatorname { E } ( H ( X , X _ { \\sigma } ) ) - H ( X ) = { \\frac { 1 } { 2 } } \\operatorname { E } ( d ( X , X _ { \\sigma } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 866, + 681, + 896 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "68 Since $d$ is a metric, this shows that the adjusted entropy of $X$ is non-negative. ", + "bbox": [ + 143, + 897, + 681, + 912 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "69 Proposition 3. We have $\\Delta H ( X ) = 0$ if and only if X is constant or equal to some permutation of \n70 $\\{ 1 , \\ldots , n \\}$ . \n71 Proposition 3 characterizes random variables with zero adjusted entropy. Again, this result will be \n2 interpreted in terms of clustering in section 4. ", + "bbox": [ + 151, + 92, + 826, + 121 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 155, + 130, + 825, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "73 3 Pairwise adjustment ", + "text_level": 1, + "bbox": [ + 148, + 178, + 377, + 195 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "74 In this section, we introduce pairwise adjusted mutual information. The definition is the same as \n75 adjusted mutual information, except that the permutation $\\sigma$ is now restricted to the set of pairwise \n76 permutations. Specifically, we consider permutations $\\sigma$ for which there exists $i , j \\in \\{ 1 , \\ldots , n \\}$ \n77 such that $\\sigma ( i ) \\stackrel { \\textstyle - } { = } j$ and $\\overset { \\cdot } { \\boldsymbol { \\sigma } ( j ) } = i$ , whereas $\\sigma ( t ) = t$ for all $t \\neq i , j$ . We consider the set of such \n78 permutations $\\sigma$ where the samples $i , j$ are drawn uniformly at random in the set $\\{ 1 , \\ldots , n \\}$ . We \n79 denote by $\\sigma _ { \\mathrm { p } }$ such a random permutation. Observe that $\\sigma _ { \\mathrm { p } }$ is the identity with probability $1 / n$ (the \n80 probability that $i = j$ ). ", + "bbox": [ + 147, + 208, + 825, + 308 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Pairwise adjusted mutual information. We define the pairwise adjusted mutual information as: ", + "bbox": [ + 166, + 320, + 820, + 335 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/03ac58888fe785e0a9325b8d6691a8956ec44f48e5a35b5767dd36175b44ffcc.jpg", + "text": "$$\n\\Delta _ { \\mathrm { p } } I ( X , Y ) = I ( X , Y ) - \\mathrm { E } ( I ( X , Y _ { \\sigma _ { \\mathrm { p } } } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 342, + 633, + 361 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "81 This is exactly the same definition as the adjusted mutual information, except for the considered \n82 permutations $\\sigma _ { \\mathrm { p } }$ . It can be readily verified that the same properties apply, with the exact same proofs, \n83 a key property being that the random permutations $\\sigma _ { \\mathrm { p } }$ and $\\sigma _ { \\mathrm { p } } ^ { \\mathrm { ~ - 1 ~ } }$ have the same distributions. In \n84 particular, we have the analogue of (3): ", + "bbox": [ + 145, + 364, + 826, + 422 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1afde09627e6808d5d6b520dea7031e74a2b8e8680cd5dcb06d7125d106d5583.jpg", + "text": "$$\n\\begin{array} { l } { { \\Delta _ { \\mathrm { p } } I ( X , Y ) = \\mathrm { E } ( H ( X , Y _ { \\sigma _ { \\mathrm { p } } } ) ) - H ( X , Y ) , } } \\\\ { { \\ ~ = { \\frac { 1 } { 2 } } ( \\mathrm { E } ( d ( X , Y _ { \\sigma _ { \\mathrm { p } } } ) ) - d ( X , Y ) ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 428, + 648, + 478 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "85 Moreover, $\\Delta _ { \\mathrm { p } } I ( X , Y ) = 0$ whenever $X$ or $Y$ is constant or equal to some permutation of $\\{ 1 , \\ldots , n \\}$ ", + "bbox": [ + 150, + 483, + 825, + 500 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Pairwise adjusted entropy. We also define the pairwise adjusted entropy as: ", + "bbox": [ + 171, + 512, + 696, + 527 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/641176e6b078434f8fdb820b9a26d1cb01318c61780e4b971c80c95d85bc2105.jpg", + "text": "$$\n\\Delta _ { \\mathrm { p } } H ( X ) = \\Delta _ { \\mathrm { p } } I ( X , X ) = H ( X ) - \\mathrm { E } ( I ( X , X _ { \\sigma _ { \\mathrm { p } } } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 534, + 673, + 551 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "86 We have $\\Delta _ { \\mathrm { p } } H ( X ) \\geq 0$ , with equality if and only if $X$ is constant or equal to some permutation of \n87 $\\{ 1 , \\ldots , n \\}$ . ", + "bbox": [ + 147, + 558, + 826, + 588 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "88 4 Application to clustering ", + "text_level": 1, + "bbox": [ + 148, + 606, + 413, + 625 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "89 Let $A = \\{ A _ { 1 } , \\ldots , A _ { k } \\}$ and $B = \\{ B _ { 1 } , \\ldots , B _ { l } \\}$ be two partitions of some finite set $\\{ 1 , \\ldots , n \\}$ into $k$ \n90 and $l$ clusters, respectively. Let $\\Omega = \\{ 1 , \\dots , n \\}$ and $\\mathrm { P }$ be the uniform probability measure over $\\Omega$ . \n91 Consider the random variables $X$ and $Y$ defined on $( \\Omega , \\mathrm { P } )$ by $X ^ { - 1 } ( i ) \\doteq A _ { i }$ for all $i = 1 , \\ldots , k$ and \n92 $Y ^ { - 1 } ( j ) = B _ { j }$ for all $j = 1 , \\dots , l .$ . Note that $X ( \\omega )$ and $Y ( \\omega )$ can be interpreted as the labels $i$ and $j$ \n93 of sample $\\omega$ in clusterings $A$ and $B$ , for each $\\omega \\in \\{ 1 , \\ldots , n \\}$ . \n94 We denote by $a _ { i } ~ = ~ \\left| A _ { i } \\right|$ the size of cluster $A _ { i }$ , by $b _ { j } ~ = ~ | B _ { j } |$ the size of cluster $B _ { j }$ , and by \n95 $n _ { i j } = | A _ { i } \\cap B _ { j } |$ the number of samples both in cluster $A _ { i }$ and cluster $B _ { j }$ , for all $i = 1 , \\ldots , k$ and \n96 $j = 1 , \\dots , l$ . The matrix $( n _ { i j } ) _ { 1 \\leq i \\leq k , 1 \\leq j \\leq l }$ is known as the contingency matrix. Note that $a _ { i }$ and $b _ { j }$ \n97 are the sums of row $i$ and column $j$ of the contingency matrix, respectively. \n98 Adjusted mutual information. A well-known metric for assessing the similarity $s ( A , B )$ between \n99 clusterings $A$ and $B$ is the adjusted mutual information 1 $\\Delta I ( X , { \\bar { Y } } )$ between the corresponding \n100 random variables $X$ and $Y$ . In words, this is the common information shared by clusterings $A$ and $B$ \n101 not due to randomness. \n102 By Proposition 2, we have $s ( A , B ) = 0$ whenever clustering $A$ (or $B$ , by symmetry) is trivial, that is, \n103 it consists of a single cluster or of $n$ clusters (one per sample). This is a key property, showing the \n104 interest of the adjustment. ", + "bbox": [ + 147, + 636, + 825, + 708 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 713, + 825, + 770 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 784, + 825, + 839 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 844, + 825, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "105 It is known that [Vinh et al., 2010]: ", + "bbox": [ + 142, + 90, + 406, + 106 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d4273ff67d32c341f361e3ec3836ca9c9b2c2d629f4bda51b48ddd3d64ef10a8.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle s ( A , B ) = - \\sum _ { i = 1 } ^ { k } \\sum _ { j = 1 } ^ { l } \\frac { n _ { i j } } { n } \\log \\frac { n _ { i j } } { n } } } \\\\ { { \\displaystyle + \\sum _ { i = 1 } ^ { k } \\sum _ { j = 1 } ^ { l } \\sum _ { c = ( a _ { i } + b _ { j } - n ) ^ { + } } ^ { \\operatorname* { m i n } ( a _ { i } , b _ { j } ) } \\frac { a _ { i } ! b _ { j } ! ( n - a _ { i } ) ! ( n - b _ { j } ) ! } { n ! c ! ( a _ { i } - c ) ! ( b _ { j } - c ) ! ( n - a _ { i } - b _ { j } + c ) ! } \\frac { c } { n } \\log \\frac { c } { n } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 111, + 745, + 207 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "with the notation 106 $( \\cdot ) ^ { + } = \\operatorname* { m a x } ( \\cdot , 0 )$ . The time complexity of this formula, which is dominated by the 107 second term, is in $O ( \\operatorname* { m a x } ( k , l ) n )$ [Romano et al., 2014]. In particular, it is linear in the number of 108 samples $n$ . ", + "bbox": [ + 143, + 213, + 821, + 256 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "109 Interestingly, we can similarly assess the quantity of information $q ( A )$ contained in clustering \n110 $A$ through the adjusted entropy $\\Delta H ( X )$ of the corresponding random variable $X$ . This is the \n111 information contained in $A$ not due to randomness. We have $q ( A ) \\geq 0$ and, by Proposition 3, \n112 $q ( A ) = 0$ if and only if clustering $A$ is trivial, that is, it consists of a single cluster or of $n$ clusters \n113 (one per sample). ", + "bbox": [ + 140, + 261, + 825, + 332 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "114 Since $q ( A ) = s ( A , A )$ , it follows from (6) that: ", + "bbox": [ + 140, + 337, + 486, + 353 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/19bc010853f0e73a534237c66be212451a3aa3f4ee06ae491cb3ae3502c01b82.jpg", + "text": "$$\n\\mathfrak { r } ( A ) = - \\sum _ { i = 1 } ^ { k } \\frac { a _ { i } } { n } \\log \\frac { a _ { i } } { n } + \\sum _ { i , j = 1 } ^ { K } \\sum _ { c = ( a _ { i } + a _ { j } - n ) ^ { + } } ^ { \\operatorname* { m i n } ( a _ { i } , a _ { j } ) } + \\frac { a _ { i } ! a _ { j } ! ( n - a _ { i } ) ! ( n - a _ { j } ) ! } { n ! c ! ( a _ { i } - c ) ! ( a _ { j } - c ) ! ( n - a _ { i } - a _ { j } + k ) ! } \\frac { c } { n } \\log \\frac { c } { n } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 358, + 825, + 406 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "115 The time complexity of this formula, also dominated by the second term, is in $O ( k n )$ . Again, this \n116 complexity is linear in the number of samples $n$ . \n117 Pairwise adjusted mutual information. The main contribution of the paper is the following new \n118 measure of similarity $s _ { \\mathrm { p } } ( A , B )$ between clusterings $A$ and $B$ , based on the pairwise adjusted mutual \n119 information $\\Delta _ { \\mathrm { p } } I ( X , Y )$ between the corresponding random variables $X$ and $Y$ . We have an explicit \n120 expression for this similarity: ", + "bbox": [ + 140, + 412, + 826, + 443 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 455, + 826, + 512 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "121 Theorem 1. We have for any clusterings $A , B$ ", + "bbox": [ + 147, + 516, + 485, + 531 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e2e7b40893d8b24371f79bab22d698fd4de0ba985e1e4a34626d039849e8e53d.jpg", + "text": "$$\n\\begin{array} { c } { { s _ { \\mathrm { p } } ( A , B ) = 2 \\displaystyle \\sum _ { i = 1 } ^ { k } \\displaystyle \\sum _ { j = 1 } ^ { l } \\displaystyle \\frac { n _ { i j } ( n - a _ { i } - b _ { j } + n _ { i j } ) } { n ^ { 2 } } \\left( \\displaystyle \\frac { n _ { i j } } { n } \\log \\displaystyle \\frac { n _ { i j } } { n } - \\displaystyle \\frac { n _ { i j } - 1 } { n } \\log \\displaystyle \\frac { n _ { i j } - 1 } { n } \\right) } } \\\\ { { + 2 \\displaystyle \\sum _ { i = 1 } ^ { k } \\displaystyle \\sum _ { j = 1 } ^ { l } \\displaystyle \\frac { ( a _ { i } - n _ { i j } ) ( b _ { j } - n _ { i j } ) } { n ^ { 2 } } \\left( \\displaystyle \\frac { n _ { i j } } { n } \\log \\displaystyle \\frac { n _ { i j } } { n } - \\displaystyle \\frac { n _ { i j } + 1 } { n } \\log \\displaystyle \\frac { n _ { i j } + 1 } { n } \\right) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 217, + 537, + 781, + 630 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "122 The time complexity of this formula is in $O ( k l )$ , like mutual information. It is independent of the \n123 number of samples $n$ , given the contingency matrix. Corollary 1 shows that the time complexity \n124 reduces to $O ( m )$ , where $m$ is the number of non-zero entries of the contingency matrix, provided the \n125 latter is stored in sparse format. ", + "bbox": [ + 142, + 641, + 825, + 698 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "126 Corollary 1. We have for any clusterings $A , B$ : ", + "bbox": [ + 142, + 702, + 491, + 717 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b93e75c6e98c9c7222cea08072c0052698901b78fa5313073e7b0a1c076893ae.jpg", + "text": "$$\n{ \\begin{array} { l } { { \\displaystyle { \\boldsymbol { s } } _ { \\mathrm { p } } ( A , B ) = 2 \\sum _ { i , j : n _ { i j } > 0 } { \\frac { n _ { i j } ( n - a _ { i } - b _ { j } + n _ { i j } ) } { n ^ { 2 } } } \\left( { \\frac { n _ { i j } } { n } } \\log { \\frac { n _ { i j } } { n } } - { \\frac { n _ { i j } - 1 } { n } } \\log { \\frac { n _ { i j } - 1 } { n } } \\right) } } \\\\ { \\displaystyle \\qquad + 2 \\sum _ { i , j : n _ { i j } > 0 } { \\frac { ( a _ { i } - n _ { i j } ) ( b _ { j } - n _ { i j } ) } { n ^ { 2 } } } \\left( { \\frac { n _ { i j } } { n } } \\log { \\frac { n _ { i j } } { n } } - { \\frac { n _ { i j } + 1 } { n } } \\log { \\frac { n _ { i j } + 1 } { n } } + { \\frac { 1 } { n } } \\log { \\frac { 1 } { n } } \\right) } \\\\ { \\displaystyle \\qquad - 2 \\left( n ^ { 2 } - \\sum _ { i = 1 } ^ { k } a _ { i } ^ { 2 } - \\sum _ { j = 1 } ^ { l } b _ { i } ^ { 2 } + \\sum _ { i , j : n _ { i j } > 0 } n _ { i j } ^ { 2 } \\right) { \\frac { 1 } { n } } \\log { \\frac { 1 } { n } } . } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 186, + 722, + 812, + 858 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "127 Similarly, we can define the quantity of information $q _ { \\mathrm { p } } ( A )$ in clustering $A$ through the pairwise \n128 adjusted entropy $\\Delta _ { \\mathrm { p } } H ( X )$ of the corresponding random variable $X$ . Again, $q _ { \\mathrm { p } } ( A ) \\geq 0$ , with \n129 $q _ { \\mathrm { p } } \\overset { \\cdot } { (} A ) = 0$ if and only if clustering $A$ is trivial. ", + "bbox": [ + 142, + 868, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d6efe3eff4af1fbebe3291fcf15d84ca566233d02650744a19c42eac5fc67408.jpg", + "text": "$$\nq _ { \\mathrm { p } } ( A ) = 2 \\sum _ { i = 1 } ^ { k } { \\frac { a _ { i } ( n - a _ { i } ) } { n ^ { 2 } } } \\left( { \\frac { a _ { i } } { n } } \\log { \\frac { a _ { i } } { n } } - { \\frac { a _ { i } - 1 } { n } } \\log { \\frac { a _ { i } - 1 } { n } } - { \\frac { 1 } { n } } \\log { \\frac { 1 } { n } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 256, + 114, + 741, + 157 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "131 Note that the time complexity of this formula in $O ( k )$ . It only depends on the number of clusters $k$ , \n132 and not on the number of samples $n$ . ", + "bbox": [ + 142, + 171, + 828, + 200 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "133 5 Experiments ", + "text_level": 1, + "bbox": [ + 150, + 220, + 312, + 238 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "134 In this section, we compare both notions of adjusted mutual information through experiments \n135 involving synthetic and real data. The experiments are run on a computer equipped with an AMD \n136 Ryzen Threadripper 1950X 16-Core Processor and 32 GB of RAM, with a a Debian $1 0 0 \\mathrm { S }$ . All codes \n137 and datasets used in the experiments are available in the supplementary material. \n138 Synthetic data. We start with the simple case of $n = 1 0 0$ samples with clusters of even sizes, \n139 consisting of consecutive samples. Specifically, we consider the set of clusterings $A ^ { ( s ) }$ , consisting of \n140 clusters of size $s$ (except possibly the last one), for $s = 1 , 2 , \\ldots , 1 0 0$ . In particular, both $A ^ { ( 1 ) }$ and \n141 $A ^ { ( 1 0 0 ) }$ are trivial clusterings while $A ^ { ( 5 ) }$ consists of 20 clusters of size 5. \n142 Figure 1 gives the similarity between clusterings $A ^ { ( 1 0 ) }$ and $A ^ { ( s ) }$ with respect to $s$ in terms of adjusted \n143 mutual information, for both notions of adjustment, i.e., $s ( A ^ { ( 1 0 ) } , A ^ { ( s ) } )$ and $s _ { \\mathrm { p } } ( A ^ { ( 1 0 ) } , A ^ { ( s ) } )$ . We \n144 observe very close behaviors, suggesting that both notions of adjustment tend to capture the same \n145 patterns in the clusterings. Note that the maximum similarity is attained for $s = 1 0$ in both cases, as \n146 expected. The similarity is equal to 0 for $s \\in \\{ 1 , 1 0 0 \\}$ for both cases, in agreement with Proposition \n147 2. We also observe local peaks at $s = 2 0 , 3 0 , \\ldots , 9 0$ , which can be interpreted by the fact that \n148 clustering $A ^ { ( 1 0 ) }$ is a refinement of clustering $A ^ { ( s ) }$ for these values of $s$ ; similarly, the local peak at \n149 $s = 5$ may be interpreted by the fact that clustering $A ^ { ( 5 ) }$ is a refinement of clustering $A ^ { ( 1 \\bar { 0 } ) }$ . The \nSpearman correlation between both metrics over all values of $s$ is equal to 0.99. \n151 We now consider random clusterings. Specifically, we assign $n$ samples to $k$ clusters independently \n152 at random, according to some probability distribution $p = ( p _ { 1 } , \\dotsc , p _ { k } )$ , which is itself drawn at \n153 random2. Consider three such random clusterings $A , B$ , $C$ (with the same parameters $n$ and $k$ , but \n154 different probability distributions $p$ ). We would like to know whether $A$ is “closer\" to $B$ or to $C$ . In \n155 particular, we are interested in testing whether both notions of adjusted mutual information give the \n156 same ordering in the sense that: ", + "bbox": [ + 142, + 252, + 825, + 309 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 324, + 825, + 386 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 391, + 825, + 522 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/6c42fe6957f17783f8c1554e8baaab5de933abc19aa87ac9cef5f942232debb2.jpg", + "image_caption": [ + "Figure 1: Comparison of metrics on synthetic data $\\mathit { n } = 1 0 0 $ ). " + ], + "image_footnote": [], + "bbox": [ + 207, + 537, + 802, + 718 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 763, + 825, + 847 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/acc2081fdbfdc60e43d84da9a35a4d9d58b1916ec77a3d5a2674896dfedb08b3.jpg", + "text": "$$\n( s ( A , B ) - s ( A , C ) ) ( s _ { \\mathrm { p } } ( A , B ) - s _ { \\mathrm { p } } ( A , C ) ) \\geq 0 .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 856, + 665, + 873 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "157 We compute the average precision score (fraction of triplets $A , B , C$ for which (7) is true) over 1 000 \n158 independent samples of $A , B , C$ , for different values of $n$ and $k$ . We repeat the experiment 100 times \n159 to get the mean and standard deviation. The results are given in Table 1. We observe a very high \n160 precision score, always higher than $9 3 \\%$ , showing that both notions of adjusted mutual information \n161 tend to give the same ordering of these random clusterings. \n162 For the performance gain, we compare the computation times of both versions of adjusted mutual \n163 information for the similarity between clusterings $A$ and $B$ , where $A$ consists of $k = 1 0$ clusters \n164 of same size and $B$ is a random clustering, drawn as in the previous experiment. Both versions of \n165 adjusted mutual information are coded in Python, with the standard version imported from scikit-learn. \n166 Figure 2 shows the computation time when the number of samples $n$ grows from $1 0 ^ { 2 }$ to $1 0 ^ { 7 }$ . The \n167 performance gain brought by pairwise adjustement is significant. In particular, the computation time \n168 becomes independent of the number of samples. ", + "bbox": [ + 140, + 90, + 825, + 161 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/3f647e891dd722c2a713a1a2083b7a45c56c381599c24b0796c5b8f164efa7fd.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
nkPrecision score
100 1002 50.972 ± 0.004 0.952 ± 0.007
100100.943 ± 0.006
100200.955 ± 0.008
500200.936 ± 0.007
1000 100020 500.933 ± 0.006
", + "bbox": [ + 393, + 174, + 604, + 297 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 337, + 826, + 434 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/b58a6502beb2f00f992bece4b886133981210c328e34993ded52113c1071e8e7.jpg", + "image_caption": [ + "Table 1: Precision score (mean $\\pm$ standard deviation) ", + "Figure 2: Computation time with respect to $n$ (mean $\\pm$ standard deviation). " + ], + "image_footnote": [], + "bbox": [ + 339, + 450, + 656, + 613 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Real data. We first consider the 79 datasets of the benchmark suite [Gagolewski,169 $2 0 2 0 ] ^ { 3 }$ . We apply 170 to each dataset each of the following clustering algorithms: ", + "bbox": [ + 145, + 661, + 826, + 690 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• $k$ -means \n• Affinity propagation \n• Mean shift \n• Spectral clustering \n• Ward \n• Agglomerative clustering \n• DBSCAN \n• OPTICS \n• Birch \n• Gaussian Mixture ", + "bbox": [ + 215, + 702, + 405, + 893 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "181 We use the scikit-learn4 implementation of these algorithms, with the corresponding default param \n182 eters5. We get 10 clusterings per dataset. The quality of each clustering is assessed through the \n183 similarity with the available ground-truth labels, using adjusted mutual information with either full \n184 adjustment or pairwise adjustment. We then compute the Spearman correlation of the corresponding \n185 similarities, a value of 1 meaning the exact same ordering of the 10 clusterings with full adjustment \n186 and pairwise adjustment. The results are shown in Figure 3, together with the speed-up in computation \n187 time due to pairwise adjustment. In both cases, the 79 datasets are ordered by the number of samples, \n188 ranging from 105 to 105 600 [Gagolewski, 2020]. \n189 We first observe that the correlation is very high, suggesting again that both notions of adjusted mutual \n190 information tend to provide the same results. For 65 datasets among 79, the Spearman correlation is \n191 higher than $9 5 \\%$ . As for the computation time, we observe a significant performance gain, by one \n192 order of magnitude for the largest datasets. \n193 We have conducted the same experiments with OpenML [Vanschoren et al., 2013]6. We selected all \n194 datasets with at least 1,000 but no more than 50,000 samples, at most 100 features (all numerical), no \n195 missing data and ground-truth labels forming clusters of at least 5 samples on average. The results \n196 are shown in Figure for the resulting 34 datasets. Again, the datasets are ordered by the number of \n197 samples, here ranging from 1,188 to 45,918. The conclusions are similar. In particular, the Spearman \n198 correlation is higher than $9 5 \\%$ for 30 datasets among 34, and the performance gain exceeds 25 for the \n199 largest datasets. ", + "bbox": [ + 140, + 90, + 826, + 203 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/4a3e4f946dd9c298bc0837f6334696c17a0313fb56ff03d594dce17266eda230.jpg", + "image_caption": [ + "Figure 3: Comparison of metrics on the Gagolewski benchmark. " + ], + "image_footnote": [], + "bbox": [ + 207, + 222, + 802, + 402 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 444, + 825, + 500 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 506, + 825, + 603 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/3056fac0045be17a9b229a2afba177861eb5726b163d2534a53a3d5166a34841.jpg", + "image_caption": [ + "Figure 4: Comparison of metrics on OpenML datasets. " + ], + "image_footnote": [], + "bbox": [ + 207, + 623, + 802, + 803 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We have proposed another way of adjusting mutual information against chance, through pairwise label permutations. The novel metric, whose explicit expression is given in Theorem 1, has a much lower complexity than the usual adjusted mutual information. Interestingly, both metrics can also be used to assess the quantity of information contained in a clustering, which the common property of being equal to 0 if and only if the clustering is trivial, as stated in Proposition 3; again, the pairwise adjusted entropy, given in Corollary 2, has a much lower complexity. Experiments on synthetic and real data show that pairwise adjusted mutual information tends to provide the same results as the usual adjusted mutual information for comparing clusterings, while involving much less computations. ", + "bbox": [ + 171, + 121, + 825, + 231 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For future work, we plan to extend this idea to other similarity metrics. While the practical interest is less obvious for the Adjusted Rand Index [Hubert and Arabie, 1985], due to the fact that the time complexity of this metric is already independent of the number of samples, it would be worth considering other versions of information theoretic measures, as those studied in [Romano et al., 2016]. ", + "bbox": [ + 174, + 237, + 825, + 306 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 327, + 266, + 343 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Thomas M Cover and Joy A Thomas. Elements of Information Theory. Wiley, 1991. ", + "bbox": [ + 176, + 351, + 725, + 364 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Marek Gagolewski. Benchmark suite for clustering algorithms – version 1, 2020. ", + "bbox": [ + 179, + 373, + 707, + 388 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Lawrence Hubert and Phipps Arabie. Comparing partitions. Journal of classification, 2(1):193–218, 1985. ", + "bbox": [ + 178, + 397, + 825, + 426 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Marina Meila. Comparing clusterings by the variation of information. In ˘ Learning theory and kernel machines, pages 173–187. Springer, 2003. ", + "bbox": [ + 178, + 434, + 821, + 463 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Xuan Vinh Nguyen, Julien Epps, and James Bailey. Information theoretic measures for clusterings comparison: is a correction for chance necessary? In ICML, 2009. ", + "bbox": [ + 176, + 472, + 821, + 501 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Simone Romano, James Bailey, Vinh Nguyen, and Karin Verspoor. Standardized mutual information for clustering comparisons: one step further in adjustment for chance. In International Conference on Machine Learning, pages 1143–1151, 2014. ", + "bbox": [ + 176, + 508, + 823, + 551 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Simone Romano, Nguyen Xuan Vinh, James Bailey, and Karin Verspoor. Adjusting for chance clustering comparison measures. The Journal of Machine Learning Research, 17(1):4635–4666, 2016. ", + "bbox": [ + 178, + 560, + 825, + 602 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Abdel Aziz Taha and Allan Hanbury. Metrics for evaluating 3d medical image segmentation: analysis, selection, and tool. BMC medical imaging, 15(1):29, 2015. ", + "bbox": [ + 173, + 611, + 823, + 640 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Bertrand Thirion, Gaël Varoquaux, Elvis Dohmatob, and Jean-Baptiste Poline. Which fmri clustering gives good brain parcellations? Frontiers in neuroscience, 8:167, 2014. ", + "bbox": [ + 173, + 648, + 823, + 678 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Joaquin Vanschoren, Jan N. van Rijn, Bernd Bischl, and Luis Torgo. Openml: Networked science in machine learning. SIGKDD Explorations, 15(2):49–60, 2013. ", + "bbox": [ + 173, + 685, + 823, + 715 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Nguyen Xuan Vinh, Julien Epps, and James Bailey. Information theoretic measures for clusterings comparison: Variants, properties, normalization and correction for chance. The Journal of Machine Learning Research, 11:2837–2854, 2010. ", + "bbox": [ + 173, + 723, + 823, + 766 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Bo Wang, Junjie Zhu, Emma Pierson, Daniele Ramazzotti, and Serafim Batzoglou. Visualization and analysis of single-cell rna-seq data by kernel-based similarity learning. Nature methods, 14(4):414–416, 2017. ", + "bbox": [ + 168, + 775, + 823, + 816 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Zhao Yang, René Algesheimer, and Claudio J Tessone. A comparative analysis of community detection algorithms on artificial networks. Scientific reports, 6:30750, 2016. ", + "bbox": [ + 161, + 825, + 823, + 854 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Jiajie Zhang, Paschalia Kapli, Pavlos Pavlidis, and Alexandros Stamatakis. A general species delimitation method with applications to phylogenetic placements. Bioinformatics, 29(22):2869– 2876, 2013. ", + "bbox": [ + 163, + 863, + 825, + 905 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 116, + 339, + 130 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] A variant of adjusted mutual information. \n(b) Did you describe the limitations of your work? [Yes] Pairwise adjustement only applied to mutual information in the present work, see Section 6. \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 135, + 825, + 239 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 243, + 493, + 258 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] No specific assumption is required. Theorem 1, Corollary 2 and 3 give explicit expressions using notations defined at the beginning of Section 4. \n(b) Did you include complete proofs of all theoretical results? [Yes] See the supplementary material. ", + "bbox": [ + 238, + 262, + 825, + 334 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 339, + 393, + 354 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See the Jupyter notebooks in the supplementary material. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A] This is a metric for unsupervised learning. No data split required, no hyperparameter. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Table 1 and Figure 2 (not applicable to other experiments). \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See the running times provided in the Figures; the resources used are detailed at the beginning of section 5. ", + "bbox": [ + 238, + 358, + 825, + 531 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 209, + 536, + 823, + 551 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] See the references for the datasets. \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 554, + 825, + 689 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 694, + 705, + 708 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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Compared to mutual", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 89, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 89, + 459, + 100, + 468 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "information, this metric is adjusted against chance, meaning that the similarity cannot be due to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 90, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 90, + 470, + 100, + 479 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "randomness but only to the structure of the dataset, appearing in both clusterings. This is the reason", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 89, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 89, + 480, + 100, + 490 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "why this metric is widely used in unsupervised learning, see [Zhang et al., 2013; Thirion et al., 2014;", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 90, + 490, + 457, + 501 + ], + "spans": [ + { + "bbox": [ + 90, + 492, + 100, + 501 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 490, + 457, + 501 + ], + "score": 1.0, + "content": "Taha and Hanbury, 2015; Yang et al., 2016; Wang et al., 2017] for various applications.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 90, + 505, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 89, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 89, + 508, + 100, + 518 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "The standard way of adjusting mutual information against chance is through random label permuta-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 88, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 88, + 518, + 99, + 529 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 106, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "tions of one of the clusterings [Vinh et al., 2010]. Unfortunately, this adjustment makes the metric", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 89, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 89, + 529, + 100, + 539 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 440, + 540 + ], + "score": 1.0, + "content": "computationally expensive. Specifically, the time complexity of the metric is in", + "type": "text" + }, + { + "bbox": [ + 441, + 527, + 502, + 540 + ], + "score": 0.94, + "content": "O ( \\operatorname* { m a x } ( k , l ) n )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 527, + 506, + 540 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 88, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 88, + 540, + 100, + 551 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 104, + 537, + 132, + 551 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 538, + 147, + 550 + ], + "score": 0.87, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 537, + 346, + 551 + ], + "score": 1.0, + "content": "are the numbers of clusters in each clustering and", + "type": "text" + }, + { + "bbox": [ + 346, + 540, + 354, + 548 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "is the number of samples [Romano et", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 88, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 88, + 551, + 100, + 561 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 549, + 439, + 563 + ], + "score": 1.0, + "content": "al., 2014]. As a comparison, the time complexity of mutual information is equal to", + "type": "text" + }, + { + "bbox": [ + 439, + 550, + 465, + 561 + ], + "score": 0.92, + "content": "O ( k l )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "given the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 89, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 89, + 563, + 99, + 572 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "contingency matrix of the clusterings, i.e., the matrix counting the number of samples in each pair of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 88, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 88, + 573, + 100, + 583 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "clusters, one per clustering. The additional computational effort required by adjustment is significant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 89, + 582, + 449, + 596 + ], + "spans": [ + { + "bbox": [ + 89, + 585, + 100, + 594 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 104, + 582, + 210, + 596 + ], + "score": 1.0, + "content": "as the number of samples", + "type": "text" + }, + { + "bbox": [ + 210, + 584, + 218, + 592 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 582, + 430, + 596 + ], + "score": 1.0, + "content": "is typically much larger than the numbers of clusters", + "type": "text" + }, + { + "bbox": [ + 430, + 582, + 444, + 594 + ], + "score": 0.89, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 582, + 449, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 90, + 599, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 89, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 89, + 600, + 99, + 610 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "In this paper, we propose a novel adjustment based on pairwise permutations. 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Compared to mutual information, a corrective", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 278, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 93, + 282, + 99, + 290 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 141, + 278, + 470, + 292 + ], + "score": 1.0, + "content": "term based on random permutations of the labels is introduced, preventing two", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 290, + 470, + 302 + ], + "spans": [ + { + "bbox": [ + 93, + 293, + 99, + 301 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 142, + 290, + 470, + 302 + ], + "score": 1.0, + "content": "clusterings being similar by chance. Unfortunately, this adjustment makes the", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 301, + 470, + 315 + ], + "spans": [ + { + "bbox": [ + 93, + 304, + 99, + 312 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 141, + 301, + 470, + 315 + ], + "score": 1.0, + "content": "metric computationally expensive. In this paper, we propose a novel adjustment", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 311, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 92, + 315, + 100, + 323 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 141, + 311, + 470, + 325 + ], + "score": 1.0, + "content": "based on pairwise label permutations instead of full label permutations. Specifically,", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 323, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 92, + 325, + 99, + 334 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 141, + 323, + 470, + 335 + ], + "score": 1.0, + "content": "we consider permutations where only two samples, selected uniformly at random,", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 334, + 470, + 346 + ], + "spans": [ + { + "bbox": [ + 92, + 336, + 99, + 345 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 142, + 334, + 470, + 346 + ], + "score": 1.0, + "content": "exchange their labels. We show that the corresponding adjusted metric, which", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 345, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 92, + 347, + 100, + 356 + ], + "score": 1.0, + "content": "9", + "type": "text" + }, + { + "bbox": [ + 142, + 345, + 470, + 357 + ], + "score": 1.0, + "content": "can be expressed explicitly, behaves similarly to the standard adjusted mutual", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 355, + 470, + 369 + ], + "spans": [ + { + "bbox": [ + 90, + 357, + 100, + 368 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 141, + 355, + 470, + 369 + ], + "score": 1.0, + "content": "information for assessing the quality of a clustering, while having a much lower", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 367, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 90, + 369, + 99, + 378 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 142, + 367, + 470, + 379 + ], + "score": 1.0, + "content": "time complexity. Both metrics are compared in terms of quality and performance", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 378, + 338, + 390 + ], + "spans": [ + { + "bbox": [ + 90, + 380, + 99, + 389 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 142, + 378, + 338, + 390 + ], + "score": 1.0, + "content": "on experiments based on synthetic and real data.", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 11.5, + "bbox_fs": [ + 90, + 258, + 470, + 390 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 409, + 191, + 423 + ], + "lines": [ + { + "bbox": [ + 87, + 408, + 192, + 425 + ], + "spans": [ + { + "bbox": [ + 87, + 408, + 192, + 425 + ], + "score": 1.0, + "content": "13 1 Introduction", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "index", + "bbox": [ + 90, + 434, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 90, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 90, + 437, + 100, + 445 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "A well-known metric for quantifying the similarity between two clusterings of the same data is", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 89, + 448, + 100, + 457 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "the adjusted mutual information [Nguyen et al., 2009; Vinh et al., 2010]. Compared to mutual", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 89, + 459, + 100, + 468 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "information, this metric is adjusted against chance, meaning that the similarity cannot be due to", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 90, + 470, + 100, + 479 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "randomness but only to the structure of the dataset, appearing in both clusterings. This is the reason", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 89, + 480, + 100, + 490 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "why this metric is widely used in unsupervised learning, see [Zhang et al., 2013; Thirion et al., 2014;", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 490, + 457, + 501 + ], + "spans": [ + { + "bbox": [ + 90, + 492, + 100, + 501 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 490, + 457, + 501 + ], + "score": 1.0, + "content": "Taha and Hanbury, 2015; Yang et al., 2016; Wang et al., 2017] for various applications.", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 89, + 508, + 100, + 518 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "The standard way of adjusting mutual information against chance is through random label permuta-", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 88, + 518, + 99, + 529 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 106, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "tions of one of the clusterings [Vinh et al., 2010]. Unfortunately, this adjustment makes the metric", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 89, + 529, + 100, + 539 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 440, + 540 + ], + "score": 1.0, + "content": "computationally expensive. Specifically, the time complexity of the metric is in", + "type": "text" + }, + { + "bbox": [ + 441, + 527, + 502, + 540 + ], + "score": 0.94, + "content": "O ( \\operatorname* { m a x } ( k , l ) n )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 527, + 506, + 540 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 88, + 540, + 100, + 551 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 104, + 537, + 132, + 551 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 538, + 147, + 550 + ], + "score": 0.87, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 537, + 346, + 551 + ], + "score": 1.0, + "content": "are the numbers of clusters in each clustering and", + "type": "text" + }, + { + "bbox": [ + 346, + 540, + 354, + 548 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "is the number of samples [Romano et", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 88, + 551, + 100, + 561 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 549, + 439, + 563 + ], + "score": 1.0, + "content": "al., 2014]. As a comparison, the time complexity of mutual information is equal to", + "type": "text" + }, + { + "bbox": [ + 439, + 550, + 465, + 561 + ], + "score": 0.92, + "content": "O ( k l )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "given the", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 89, + 563, + 99, + 572 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "contingency matrix of the clusterings, i.e., the matrix counting the number of samples in each pair of", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 88, + 573, + 100, + 583 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "clusters, one per clustering. The additional computational effort required by adjustment is significant", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 582, + 449, + 596 + ], + "spans": [ + { + "bbox": [ + 89, + 585, + 100, + 594 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 104, + 582, + 210, + 596 + ], + "score": 1.0, + "content": "as the number of samples", + "type": "text" + }, + { + "bbox": [ + 210, + 584, + 218, + 592 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 582, + 430, + 596 + ], + "score": 1.0, + "content": "is typically much larger than the numbers of clusters", + "type": "text" + }, + { + "bbox": [ + 430, + 582, + 444, + 594 + ], + "score": 0.89, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 582, + 449, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 89, + 600, + 99, + 610 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "In this paper, we propose a novel adjustment based on pairwise permutations. That is, we consider", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 89, + 611, + 99, + 621 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 104, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "permutations where only two samples, selected uniformly at random, exchange their labels. We", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 619, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 89, + 622, + 100, + 632 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 619, + 506, + 634 + ], + "score": 1.0, + "content": "show that the corresponding adjusted metric, we refer to as pairwise adjusted mutual information,", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 88, + 633, + 99, + 643 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "is as efficient as adjusted mutual information for assessing the quality of a clustering, with a much", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 642, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 88, + 644, + 100, + 654 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 105, + 642, + 506, + 654 + ], + "score": 1.0, + "content": "lower time complexity. In particular, the time complexity is the same as that of mutual information.", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 89, + 655, + 100, + 665 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "The gain in complexity is significant, as the computation time is now independent of the number of", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 663, + 273, + 678 + ], + "spans": [ + { + "bbox": [ + 89, + 666, + 100, + 676 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 663, + 141, + 678 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 141, + 666, + 148, + 674 + ], + "score": 0.66, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 663, + 273, + 678 + ], + "score": 1.0, + "content": ", given the contingency matrix.", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 680, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 90, + 682, + 100, + 691 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 680, + 506, + 692 + ], + "score": 1.0, + "content": "The rest of the paper is organized as follows. 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Again, this result will be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 95, + 114, + 291, + 127 + ], + "spans": [ + { + "bbox": [ + 95, + 114, + 291, + 127 + ], + "score": 1.0, + "content": "2 interpreted in terms of clustering in section 4.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 91, + 141, + 231, + 155 + ], + "lines": [ + { + "bbox": [ + 87, + 139, + 232, + 158 + ], + "spans": [ + { + "bbox": [ + 87, + 139, + 232, + 158 + ], + "score": 1.0, + "content": "73 3 Pairwise adjustment", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 90, + 165, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 90, + 166, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 90, + 168, + 100, + 177 + ], + "score": 1.0, + "content": "74", + "type": "text" + }, + { + "bbox": [ + 104, + 166, + 506, + 178 + ], + "score": 1.0, + "content": "In this section, we introduce pairwise adjusted mutual information. 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In", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 90, + 324, + 264, + 336 + ], + "spans": [ + { + "bbox": [ + 90, + 326, + 100, + 334 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 106, + 324, + 264, + 336 + ], + "score": 1.0, + "content": "particular, we have the analogue of (3):", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 339, + 397, + 379 + ], + "lines": [ + { + "bbox": [ + 214, + 339, + 397, + 379 + ], + "spans": [ + { + "bbox": [ + 214, + 339, + 397, + 379 + ], + "score": 0.92, + "content": "\\begin{array} { l } { { \\Delta _ { \\mathrm { p } } I ( X , Y ) = \\mathrm { E } ( H ( X , Y _ { \\sigma _ { \\mathrm { p } } } ) ) - H ( X , Y ) , } } \\\\ { { \\ ~ = { \\frac { 1 } { 2 } } ( \\mathrm { E } ( d ( X , Y _ { \\sigma _ { \\mathrm { p } } } ) ) - d ( X , Y ) ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "1afde09627e6808d5d6b520dea7031e74a2b8e8680cd5dcb06d7125d106d5583.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 339, + 397, + 359.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 214, + 359.0, + 397, + 379.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 92, + 383, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 87, + 381, + 503, + 398 + ], + "spans": [ + { + "bbox": [ + 87, + 381, + 149, + 398 + ], + "score": 1.0, + "content": "85 Moreover,", + "type": "text" + }, + { + "bbox": [ + 149, + 383, + 216, + 396 + ], + "score": 0.92, + "content": "\\Delta _ { \\mathrm { p } } I ( X , Y ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 381, + 256, + 398 + ], + "score": 1.0, + "content": "whenever", + "type": "text" + }, + { + "bbox": [ + 257, + 384, + 267, + 393 + ], + "score": 0.86, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 381, + 278, + 398 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 279, + 384, + 288, + 393 + ], + "score": 0.86, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 381, + 459, + 398 + ], + "score": 1.0, + "content": "is constant or equal to some permutation of", + "type": "text" + }, + { + "bbox": [ + 460, + 383, + 503, + 396 + ], + "score": 0.93, + "content": "\\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 105, + 406, + 426, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 425, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 425, + 421 + ], + "score": 1.0, + "content": "Pairwise adjusted entropy. 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It can be readily verified that the same properties apply, with the exact same proofs,", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 89, + 314, + 100, + 325 + ], + "score": 1.0, + "content": "83", + "type": "text" + }, + { + "bbox": [ + 105, + 312, + 318, + 326 + ], + "score": 1.0, + "content": "a key property being that the random permutations", + "type": "text" + }, + { + "bbox": [ + 319, + 314, + 331, + 325 + ], + "score": 0.86, + "content": "\\sigma _ { \\mathrm { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 312, + 350, + 326 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 312, + 372, + 325 + ], + "score": 0.92, + "content": "\\sigma _ { \\mathrm { p } } ^ { \\mathrm { ~ - 1 ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "have the same distributions. In", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 324, + 264, + 336 + ], + "spans": [ + { + "bbox": [ + 90, + 326, + 100, + 334 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 106, + 324, + 264, + 336 + ], + "score": 1.0, + "content": "particular, we have the analogue of (3):", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 15.5, + "bbox_fs": [ + 88, + 290, + 506, + 336 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 339, + 397, + 379 + ], + "lines": [ + { + "bbox": [ + 214, + 339, + 397, + 379 + ], + "spans": [ + { + "bbox": [ + 214, + 339, + 397, + 379 + ], + "score": 0.92, + "content": "\\begin{array} { l } { { \\Delta _ { \\mathrm { p } } I ( X , Y ) = \\mathrm { E } ( H ( X , Y _ { \\sigma _ { \\mathrm { p } } } ) ) - H ( X , Y ) , } } \\\\ { { \\ ~ = { \\frac { 1 } { 2 } } ( \\mathrm { E } ( d ( X , Y _ { \\sigma _ { \\mathrm { p } } } ) ) - d ( X , Y ) ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "1afde09627e6808d5d6b520dea7031e74a2b8e8680cd5dcb06d7125d106d5583.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 339, + 397, + 359.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 214, + 359.0, + 397, + 379.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 92, + 383, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 87, + 381, + 503, + 398 + ], + "spans": [ + { + "bbox": [ + 87, + 381, + 149, + 398 + ], + "score": 1.0, + "content": "85 Moreover,", + "type": "text" + }, + { + "bbox": [ + 149, + 383, + 216, + 396 + ], + "score": 0.92, + "content": "\\Delta _ { \\mathrm { p } } I ( X , Y ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 381, + 256, + 398 + ], + "score": 1.0, + "content": "whenever", + "type": "text" + }, + { + "bbox": [ + 257, + 384, + 267, + 393 + ], + "score": 0.86, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 381, + 278, + 398 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 279, + 384, + 288, + 393 + ], + "score": 0.86, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 381, + 459, + 398 + ], + "score": 1.0, + "content": "is constant or equal to some permutation of", + "type": "text" + }, + { + "bbox": [ + 460, + 383, + 503, + 396 + ], + "score": 0.93, + "content": "\\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 87, + 381, + 503, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 406, + 426, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 425, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 425, + 421 + ], + "score": 1.0, + "content": "Pairwise adjusted entropy. 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In particular, both", + "type": "text" + }, + { + "bbox": [ + 467, + 281, + 486, + 292 + ], + "score": 0.9, + "content": "A ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 291, + 397, + 306 + ], + "spans": [ + { + "bbox": [ + 89, + 295, + 99, + 305 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 106, + 293, + 133, + 304 + ], + "score": 0.84, + "content": "A ^ { ( 1 0 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 291, + 246, + 306 + ], + "score": 1.0, + "content": "are trivial clusterings while", + "type": "text" + }, + { + "bbox": [ + 246, + 293, + 265, + 304 + ], + "score": 0.89, + "content": "A ^ { ( 5 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 291, + 397, + 306 + ], + "score": 1.0, + "content": "consists of 20 clusters of size 5.", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 86, + 313, + 100, + 324 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 104, + 309, + 298, + 324 + ], + "score": 1.0, + "content": "Figure 1 gives the similarity between clusterings", + "type": "text" + }, + { + "bbox": [ + 298, + 310, + 321, + 322 + ], + "score": 0.88, + "content": "A ^ { ( 1 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 309, + 338, + 324 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 310, + 357, + 321 + ], + "score": 0.9, + "content": "A ^ { ( s ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 309, + 418, + 324 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 419, + 314, + 425, + 321 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "in terms of adjusted", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 320, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 86, + 325, + 99, + 335 + ], + "score": 1.0, + "content": "143", + "type": "text" + }, + { + "bbox": [ + 104, + 320, + 342, + 337 + ], + "score": 1.0, + "content": "mutual information, for both notions of adjustment, i.e.,", + "type": "text" + }, + { + "bbox": [ + 342, + 322, + 399, + 335 + ], + "score": 0.92, + "content": "s ( A ^ { ( 1 0 ) } , A ^ { ( s ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 320, + 420, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 420, + 322, + 482, + 335 + ], + "score": 0.92, + "content": "s _ { \\mathrm { p } } ( A ^ { ( 1 0 ) } , A ^ { ( s ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 320, + 506, + 337 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 86, + 336, + 99, + 345 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "observe very close behaviors, suggesting that both notions of adjustment tend to capture the same", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 344, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 86, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 104, + 344, + 407, + 359 + ], + "score": 1.0, + "content": "patterns in the clusterings. Note that the maximum similarity is attained for", + "type": "text" + }, + { + "bbox": [ + 408, + 346, + 437, + 356 + ], + "score": 0.89, + "content": "s = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 344, + 506, + 359 + ], + "score": 1.0, + "content": "in both cases, as", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 367 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 356, + 270, + 370 + ], + "score": 1.0, + "content": "expected. The similarity is equal to 0 for", + "type": "text" + }, + { + "bbox": [ + 270, + 356, + 322, + 368 + ], + "score": 0.92, + "content": "s \\in \\{ 1 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "for both cases, in agreement with Proposition", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 369, + 100, + 379 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 366, + 250, + 380 + ], + "score": 1.0, + "content": "2. We also observe local peaks at", + "type": "text" + }, + { + "bbox": [ + 251, + 368, + 330, + 378 + ], + "score": 0.84, + "content": "s = 2 0 , 3 0 , \\ldots , 9 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 366, + 506, + 380 + ], + "score": 1.0, + "content": ", which can be interpreted by the fact that", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 377, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 86, + 381, + 99, + 390 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 149, + 392 + ], + "score": 1.0, + "content": "clustering", + "type": "text" + }, + { + "bbox": [ + 149, + 378, + 172, + 389 + ], + "score": 0.87, + "content": "A ^ { ( 1 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 377, + 289, + 392 + ], + "score": 1.0, + "content": "is a refinement of clustering", + "type": "text" + }, + { + "bbox": [ + 289, + 378, + 308, + 389 + ], + "score": 0.88, + "content": "A ^ { ( s ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 377, + 386, + 392 + ], + "score": 1.0, + "content": "for these values of", + "type": "text" + }, + { + "bbox": [ + 386, + 381, + 392, + 389 + ], + "score": 0.7, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 377, + 506, + 392 + ], + "score": 1.0, + "content": "; similarly, the local peak at", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 388, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 86, + 393, + 99, + 402 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 106, + 391, + 132, + 401 + ], + "score": 0.88, + "content": "s = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 388, + 320, + 405 + ], + "score": 1.0, + "content": "may be interpreted by the fact that clustering", + "type": "text" + }, + { + "bbox": [ + 320, + 390, + 339, + 401 + ], + "score": 0.89, + "content": "A ^ { ( 5 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 388, + 458, + 405 + ], + "score": 1.0, + "content": "is a refinement of clustering", + "type": "text" + }, + { + "bbox": [ + 458, + 390, + 481, + 401 + ], + "score": 0.87, + "content": "A ^ { ( 1 \\bar { 0 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 388, + 507, + 405 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 402, + 426, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 353, + 414 + ], + "score": 1.0, + "content": "Spearman correlation between both metrics over all values of", + "type": "text" + }, + { + "bbox": [ + 353, + 404, + 359, + 412 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 402, + 426, + 414 + ], + "score": 1.0, + "content": "is equal to 0.99.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 85, + 607, + 99, + 617 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 105, + 605, + 352, + 618 + ], + "score": 1.0, + "content": "We now consider random clusterings. Specifically, we assign", + "type": "text" + }, + { + "bbox": [ + 352, + 608, + 360, + 615 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 605, + 405, + 618 + ], + "score": 1.0, + "content": "samples to", + "type": "text" + }, + { + "bbox": [ + 406, + 606, + 412, + 615 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "clusters independently", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 616, + 507, + 630 + ], + "spans": [ + { + "bbox": [ + 85, + 618, + 100, + 628 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 105, + 616, + 330, + 630 + ], + "score": 1.0, + "content": "at random, according to some probability distribution", + "type": "text" + }, + { + "bbox": [ + 330, + 617, + 401, + 628 + ], + "score": 0.93, + "content": "p = ( p _ { 1 } , \\dotsc , p _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 616, + 507, + 630 + ], + "score": 1.0, + "content": ", which is itself drawn at", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 628, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 86, + 629, + 100, + 639 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 106, + 628, + 309, + 639 + ], + "score": 1.0, + "content": "random2. Consider three such random clusterings", + "type": "text" + }, + { + "bbox": [ + 309, + 628, + 331, + 638 + ], + "score": 0.3, + "content": "A , B", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 628, + 335, + 639 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 335, + 628, + 344, + 638 + ], + "score": 0.65, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 628, + 453, + 639 + ], + "score": 1.0, + "content": "(with the same parameters", + "type": "text" + }, + { + "bbox": [ + 454, + 630, + 461, + 637 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 628, + 479, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 479, + 628, + 486, + 637 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 628, + 505, + 639 + ], + "score": 1.0, + "content": ", but", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 85, + 640, + 100, + 650 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 105, + 638, + 241, + 650 + ], + "score": 1.0, + "content": "different probability distributions", + "type": "text" + }, + { + "bbox": [ + 241, + 640, + 248, + 650 + ], + "score": 0.72, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 638, + 385, + 650 + ], + "score": 1.0, + "content": "). We would like to know whether", + "type": "text" + }, + { + "bbox": [ + 385, + 639, + 394, + 648 + ], + "score": 0.83, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 638, + 449, + 650 + ], + "score": 1.0, + "content": "is “closer\" to", + "type": "text" + }, + { + "bbox": [ + 449, + 639, + 459, + 648 + ], + "score": 0.84, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 638, + 481, + 650 + ], + "score": 1.0, + "content": "or to", + "type": "text" + }, + { + "bbox": [ + 481, + 640, + 490, + 648 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 638, + 506, + 650 + ], + "score": 1.0, + "content": ". 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100 1002 50.972 ± 0.004 0.952 ± 0.007
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100200.955 ± 0.008
500200.936 ± 0.007
1000 100020 500.933 ± 0.006
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nkPrecision score
100 1002 50.972 ± 0.004 0.952 ± 0.007
100100.943 ± 0.006
100200.955 ± 0.008
500200.936 ± 0.007
1000 100020 500.933 ± 0.006
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The results are shown in Figure 3, together with the speed-up in computation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 136, + 507, + 152 + ], + "spans": [ + { + "bbox": [ + 86, + 140, + 100, + 150 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 105, + 136, + 507, + 152 + ], + "score": 1.0, + "content": "time due to pairwise adjustment. 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100100.943 ± 0.006
100200.955 ± 0.008
500200.936 ± 0.007
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sha256:899b1ddf3b906c351cb2017b2d492cbab8bfeecc78a3cd2add2c09a95a2caddc +size 59224 diff --git a/parse/train/9c-IsSptbmA/9c-IsSptbmA.md b/parse/train/9c-IsSptbmA/9c-IsSptbmA.md new file mode 100644 index 0000000000000000000000000000000000000000..464e42dc3e6dc71c039fa20aa32c18f3328f4b13 --- /dev/null +++ b/parse/train/9c-IsSptbmA/9c-IsSptbmA.md @@ -0,0 +1,257 @@ +# Be Confident! Towards Trustworthy Graph Neural Networks via Confidence Calibration + +Xiao Wang, Hongrui Liu, Chuan Shi∗, Cheng Yang + +School of Computer Science (National Pilot Software Engineering School) Beijing University of Posts and Telecommunications Beijing, China +{xiaowang, liuhongrui, shichuan, yangcheng}@bupt.edu.cn + +# Abstract + +Despite Graph Neural Networks (GNNs) have achieved remarkable accuracy, whether the results are trustworthy is still unexplored. Previous studies suggest that many modern neural networks are over-confident on the predictions, however, surprisingly, we discover that GNNs are primarily in the opposite direction, i.e., GNNs are under-confident. Therefore, the confidence calibration for GNNs is highly desired. In this paper, we propose a novel trustworthy GNN model by designing a topology-aware post-hoc calibration function. Specifically, we first verify that the confidence distribution in a graph has homophily property, and this finding inspires us to design a calibration GNN model (CaGCN) to learn the calibration function. CaGCN is able to obtain a unique transformation from logits of GNNs to the calibrated confidence for each node, meanwhile, such transformation is able to preserve the order between classes, satisfying the accuracypreserving property. Moreover, we apply the calibration GNN to self-training framework, showing that more trustworthy pseudo labels can be obtained with the calibrated confidence and further improve the performance. Extensive experiments demonstrate the effectiveness of our proposed model in terms of both calibration and accuracy. + +# 1 Introduction + +Graphs are ubiquitous in the real world, including social networks, e-commerce networks, traffic networks, and so on. Recently, Graph Neural Networks (GNNs), which are able to effectively learn the node representations based on the message-passing manner, have attracted considerable attention in dealing with graph data [16, 33, 39, 44, 15, 2, 34]. To date, GNNs have been applied to various applications and achieved remarkable accuracy, e.g., node classification [16, 33], link prediction [41] and graph classification [9]. + +However, it is well established that a model with good accuracy is not the only goal, but a trustworthy model is highly desired in many applications, especially in safety-critical fields [1]. Usually, a trustworthy model implies that it should know when it is likely to be incorrect, in other words, the probability, i.e., the confidence, associated with the predicted class label should reflect its ground truth correctness likelihood [12]. For example, in the scene of autonomous driving, the system will adopt the prediction given by the model only when the model has high confidence for its prediction. Otherwise, the decision-making power will be returned to the driver or the system adopts other safer strategies. Recently, the confidence calibration has attracted considerable attention in deep learning [12, 40, 19], which reveals that many modern neural network models are over-confident on the predictions, i.e., the prediction accuracy is lower than its confidence. However, it has not been studied in GNNs on the semi-supervised scenario, which gives rise to one fundamental question: will the current GNNs follow the same over-confident property as other neural networks? A well-informed answer can help us better understand GNNs and enable GNNs to be applied to various areas in a more reliable manner. + +As the first contribution of this study, we present experiments assessing the relationship between the confidence and the accuracy of Graph Convolutional Networks (GCNs) [16] and Graph Attention Networks (GAT) [33] in the node classification task (more details can be seen in Section 2), respectively. Surprisingly, we discover that existing GNNs are far distant from being well-calibrated, and more importantly, GNNs tend to be under-confident in their predictions, which is very different from other modern deep learning models that are often over-confident [12, 19]. GNNs being under-confident means that many predictions are distributed in the low-confidence range, and therefore, fewer predictions are available for safety-critical applications. Once the weakness is identified, another natural question is: how can we calibrate the confidence on predictions given by GNNs so as to make them more trustworthy? + +Essentially, the confidence calibration is to calibrate the outputs (also known logits) of original models (e.g., GNNs), therefore, a straightforward manner is to employ temperature scaling (TS) [12], OP-families [25] to learn calibration function using a held-out dataset in a post-hoc way. However, when being applied to graphs, they all ignore the effect of topology, which will inevitably make mistakes during calibration. For example, considering that the logits of two nodes $a$ and $^ b$ are the same, but node $a$ is similar to its neighbors while node $b$ is not. Apparently, the predictions of GCNs for $a$ should be more confident than $^ b$ , while the traditional calibration methods, e.g., TS, will learn the same confidence for $a$ and $^ b$ , because it does not consider the effect of topology. + +Moreover, most of them explore calibration functions only in the linear space [12, 18] while it is well known that non-linear space contains more complex function transformation which is able to calibrate networks with complicated landscapes well. Even if some works have explored the non-linear space such as Matrix Scaling [12], they generally degrade the classification accuracy of the original classifier, while a good accuracy is still a basic requirement by many applications. + +In this paper, we introduce a topology-aware post-hoc calibration method for GNNs. Specifically, for the logits given by the original classification GNNs, we employ another calibration GCN (CaGCN) to propagate confidence, naturally enabling that the confidence of topologically adjacent nodes becomes similar. CaGCN learns a unique temperature $t$ for each node for temperature scaling, thus preserving the accuracy of the original classification GCN. In addition, based on our finding that large numbers of high-accuracy predictions are distributed in the low-confidence range, we design a calibrated self-training model CaGCN-st in which the confidence is firstly calibrated then used to generate pseudo labels with high confidence. The contributions of this paper are three-fold: + +• We study the trustworthy problem of GNNs, and discover one unique characteristic of GNNs, i.e., the predictions made by GNNs are usually under-confident. +• We propose a novel trustworthy GNN model based on the confidence calibration. Our proposed calibration function has three features: topology-aware, non-linear, and accuracy-preserving. We further design a calibrated self-training GNN model, which can effectively utilize the predictions with high confidence. +• Extensive experiments demonstrate the effectiveness of our proposed models in terms of both calibration and accuracy. + +# 2 Notation and Preliminary Study + +In this paper, we focus on the calibration of semi-supervised node classification in an undirected attributed graph $G = \left( V , E \right)$ with the adjacent matrix $\mathbf { A } \in \mathbb { R } ^ { N \times N }$ and the node feature matrix $\mathbf { X } =$ $[ \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { N } ] ^ { \mathsf { T } }$ . $V$ is a set of nodes and $E \subseteq V \times V$ is a set of edges between nodes. $N = | V |$ is the number of nodes. Here we give the definition of perfect calibration of GNNs as follows: + +Definition 1. Given random variables A, $\mathbf { X }$ , $\mathbf { Y } \subseteq \{ 1 , \ldots , K \}$ and a GNN model $f _ { \theta }$ where $\theta$ is the learnable parameters, for node i with label $y _ { i } \in \mathbf { Y }$ , $\mathbf { z } _ { i } = f _ { \theta } ( \mathbf { x } _ { i } , \mathbf { A } ) = [ z _ { i , 1 } , \ldots , z _ { i , K } ] ^ { \mathsf { T } }$ is the output of GNNs (i.e., the prediction probability), and $\hat { y } _ { i } = \mathrm { a r g } \operatorname* { m a x } _ { k } z _ { i , k }$ and $\hat { p } _ { i } = \operatorname* { m a x } _ { k } z _ { i , k }$ are the prediction and the confidence respectively. Then we define $f _ { \theta }$ to be perfectly calibrated as: + +$$ +\mathbb { P } ( \hat { y } _ { i } = y _ { i } | \hat { p } _ { i } = p ) = p , \forall p \in [ 0 , 1 ] . +$$ + +![](images/b09bafe873b2b4009d9aab352efaf8bdb093b30da45eedaa1e5de35a56b8adff.jpg) +Figure 1: Reliability diagrams for GCN (top) and GAT (bottom) without confidence calibration. The diagram is expected to plot an identity function of accuracy with respect to confidence. Any deviation from a perfectly diagonal (i.e., the difference between blue and red histogram) represents the miscalibration. + +![](images/c742029dfe4c579e527d277d926e41da06ed21b6ba6d31bd18e006ec2ce69a5d.jpg) +Figure 2: Confidence distribution before calibration. + +According to Definition 1, GNN is perfectly calibrated only when the confidence $\hat { p } _ { i }$ is exactly equal to the true probability of getting a correct prediction for every node. + +Next, we take two representative GNNs (GCN [16] and GAT [33]) as examples to analyze whether they are perfectly calibrated. Specifically, we apply GCN and GAT to four widely used datasets Cora [29], Citeseer [29], Pubmed [29], CoraFull [3], and examine whether their results satisfy Definition 1. To provide more results, we select three label rates for training set (i.e., 20, 40, 60 labeled nodes per class). All the experimental settings follow [16, 33]. Since the true probability $p$ cannot be exactly known, we take an approximate way to evaluate the calibration as in [12]. In particular, we first partition the [0,1] range of confidence into 20 equal bins and then we group the nodes into corresponding bins according to their confidence. After that we calculate the average accuracy of each bin. We expect the average accuracy is equal to the average confidence of each bin, which means the model is approximately perfectly calibrated. For example, if the average confidence of nodes in the bin [0.95, 1.0] is 0.96, and then the classification accuracy in this bin should be $96 \%$ . + +We illustrate the results of label rate being 20 in Fig. 1 using Reliability Diagrams [23] here, where the $\mathbf { X }$ -axis is the confidence in 20 bins of equal size and y-axis is the average accuracy in each bin. The blue represents the classification accuracy of GCN and GAT while the red is our expectation. More results of label rate being 40, 60 and other GNN models can be seen in Fig. 8, Fig. 9, Fig. 12, Fig. 13 and Fig. 14 in the appendix. We can see that in all the datasets, the average accuracy of most bins is higher than the average confidence. In other words, these GNNs actually achieve remarkable performance, but they all output low confidence, i.e., the GNNs are usually under-confident. Please note that this phenomenon of GNNs is very different from other modern neural networks, which are generally known to be over-confident [12, 19]. Moreover, as shown in Fig. 2, we also visualize the confidence distribution of test nodes, where the $\mathbf { X }$ -axis is the confidence and y-axis is the density [28]. The histogram height multiplied by the width is equal to the frequency. The blue represents the confidence distribution of correct predictions while the yellow represents that of incorrect predictions. More results of label rate being 40 and 60 can be seen in Fig. 10 and Fig. 11 in the appendix. We can see that a large quantity of correct predictions are distributed in the low confidence range. The results above indicate that the current GNNs are far from perfect calibration, leading to unreliable confidence. + +# 3 Confidence Calibration on GCNs + +In this section, we propose our method to calibrate current GNNs. Given $\mathbf { A }$ and $\mathbf { X }$ , for a $l$ -layer GCN [16], the output of the GCN before the softmax layer can be obtained by: + +$$ +\begin{array} { r } { \mathbf { V } = \mathbf { A } \sigma ( \cdots \mathbf { A } \sigma ( \mathbf { A X W } ^ { ( 1 ) } ) \mathbf { W } ^ { ( 2 ) } \cdots ) \mathbf { W } ^ { ( l ) } = [ \mathbf { v } _ { 1 } , \cdots , \mathbf { v } _ { N } ] ^ { \mathsf { T } } , } \end{array} +$$ + +where $\mathbf { W } ^ { ( l ) }$ is the weight matrix of $l$ -th layer in GCN and $\sigma ( \cdot )$ is the activation function. For each node $i \in \{ 1 , \cdots , N \}$ , our goal is to learn a calibration function which is fed with $\mathbf { v } _ { i }$ (often known as the logit of node $i$ ) and outputs a calibrated confidence using a held-out dataset in a post-hoc way. The calibration function should satisfy three points below: (1) taking the network topology into account (2) non-linear (3) preserving the classification accuracy of the GCNs. + +# 3.1 CaGCN: GCNs as Calibration Function + +![](images/8a3f54c76241f96ccd4f9a15a0db471c74196c8af7f467a60b7c380c4cd01838.jpg) +Figure 3: The illustration of the confidence propagation. Different colors indicate different classes. + +Table 1: Summary of total variation of confidence before and after calibration (Bold: best). Uncal. is short for uncalibrated and TS is short for temperature scaling. + +
DatasetGCN
Uncal.TSOurs
Cora240.267172.346164.651
Citeseer128.145112.212108.684
Pubmed1299.331266.681113.41
CoraFull6014.324698.204500.30
+ +We assume that the ground-truth confidence distribution in a graph has homophily property, i.e., the confidence of neighboured nodes given by well-calibrated models should be similar, and thus we conduct an experiment to verify this. We employ the classic temperature scaling method [12] as our calibration function and use the total variation [27] of confidence as our evaluation, which sums the difference of confidence between all the neighboured nodes. We compare the total variation of confidence before and after confidence calibration, where the results are shown in Table 1. We can find that the total variation of confidence does decrease after temperature scaling, which verifies our assumption. This inspires us that if a GCN model is well-calibrated, then the confidence between neighbors should be more similar than before. + +To this end, we find that GCN itself can play the role of calibration function that meets above requirement since GCN is able to propagate node features along the network topology and smooth similar information between neighboured nodes. Therefore, we can employ another $l .$ -layer GCN (CaGCN) as our calibration function to propagate the confidence along the network topology. Specifically, given the output $\mathbf { V }$ of the classification GCN, the logit $\mathbf { v } _ { i } ^ { \prime }$ and confidence $\hat { p } _ { i }$ for node $i$ after calibration can be obtained by: + +$$ +\begin{array} { r l } & { { \mathbf { V } } ^ { \prime } = { \mathbf { A } } \sigma ( \cdots { \mathbf { A } } \sigma ( { \mathbf { A } } { \mathbf { V } } { \mathbf { W } } ^ { ( 1 ) } ) { \mathbf { W } } ^ { ( 2 ) } \cdots ) { \mathbf { W } } ^ { ( l ) } = [ { \mathbf { v } } ^ { \prime } _ { 1 } , \cdots , { \mathbf { v } } ^ { \prime } _ { N } ] ^ { \mathsf { T } } , } \\ & { { \mathbf { z } } _ { i } = [ \sigma _ { S M } ( \nu ^ { \prime } _ { i , 1 } ) , \cdots , \sigma _ { S M } ( \nu ^ { \prime } _ { i , K } ) ] ^ { \mathsf { T } } , \hat { p } _ { i } = \operatorname* { m a x } z _ { i , k } , } \end{array} +$$ + +where $\begin{array} { r } { \sigma _ { S M } ( \nu ^ { \prime } _ { i , \cdot } ) = \frac { \exp ( \nu ^ { \prime } _ { i , \cdot } ) } { \sum _ { j = 1 } ^ { K } \exp ( \nu ^ { \prime } _ { i , j } ) } } \end{array}$ is the softmax operation. Then the total variation of confidence will surely become lower and the original classification GCN will be calibrated. Please note that although temperature scaling can be directly applied here, compared with GCN, it does not take the network topology into account, which may cause mistakes mentioned in Section 1. Moreover, temperature scaling only employs a linear transformation, and GCN is able to learn a non-linear calibration function. + +For a comprehensive understanding of confidence propagation, we make a detailed and visible illustration here. As shown in Fig. 3, the logits of two nodes $a$ and $^ b$ are the same, but node $a$ is similar to its neighbors while node $b$ is not. Apparently, the predictions of GCNs for $a$ should be more confident than $b$ . Suppose that $a$ , $b$ and their neighbours are under-confident based on the observation above. If we continue to propagate their logits along the topology using another GCN, the logits of $a$ and its neighbors will tend to be the same. Therefore, if one or more of these nodes are calibrated during the calibration process, all of them will be calibrated as well. The confidence is propagated in this way. On the other hand, looking at another node $b$ , it is as difficult even for manual classification as it is for GCNs. Consequently, the confidence of $b$ should stay still even be lower. However, it will become higher because of the influence from $a$ if we use the traditional calibration method without considering the network topology. Instead, when the network topology is taken into account, the logit of $^ b$ will be averaged by its neighboured and each dimension tends to $1 / K$ . It will be correctly calibrated when other nodes in the same situation are well-calibrated. + +# 3.2 The Accuracy-Preserving Property + +Until now, we have proposed a non-linear calibration model CaGCN which can take the network topology into account, but the accuracy-preserving property cannot be satisfied. To address this problem, we firstly study the general accuracy-preserving calibration function. + +Proposition 1. Let $h : \mathbb { R } ^ { K } \to \mathbb { R } ^ { K }$ be a calibration function, $s : \mathbb { R } \to \mathbb { R }$ be a 1-D function and $\mathbf { v } _ { i } = [ \nu _ { i , 1 } , \cdots , \nu _ { i , K } ] ^ { \mathsf { T } }$ be the logit of node i. The calibration function $h$ preserves the classification accuracy of the original model if s is a strictly isotonic function and $h$ satisfies: + +$$ +\begin{array} { r } { h ( \mathbf { v } _ { i } ) = [ s ( \nu _ { i , 1 } ) , \dots , s ( \nu _ { i , K } ) ] ^ { \mathsf { T } } , \forall i \in \{ 1 , \cdots , N \} . } \end{array} +$$ + +Proof We set $\nu _ { i , 1 } < \nu _ { i , 2 } < \cdots < \nu _ { i , K }$ without loss of generality. Since $[ s ( \nu _ { i , 1 } ) , \ldots , s ( \nu _ { i , K } ) ] ^ { \mathsf { T } }$ shares the same order with $\mathbf { v } _ { i }$ as a result of the strictly isotonicity of $s$ , the order between classes of the logit $\mathbf { v } _ { i }$ is unchanged, hence the accuracy of the prediction is preserved.  + +Temperature scaling [12] is the simplest accuracy-preserving calibration method using a scalar parameter $t$ called temperature for all classes. Given the logit $\mathbf { v } _ { i }$ of node $i .$ , the confidence of the prediction is $\hat { p } _ { i } = \operatorname* { m a x } _ { k } \sigma _ { S M } ( \nu _ { i , k } / t ) ( t > 0 )$ . In temperature scaling, $h ( { \bf v } _ { i } ) = [ \nu _ { i , 1 } / t , \cdot \cdot \cdot , \nu _ { i , K } / t ] ^ { \top }$ is the calibration function and $s ( x ) = x / t$ is the strictly isotonic function. + +However, we can find that temperature scaling (TS) [40] only performs the same linear transformation for all the nodes using the same $t$ . As mentioned in Eq. 3, we propose to use CaGCN as our calibration function, while CaGCN is generally not isotonic, i.e., the order between classes of $\mathbf { v } _ { i }$ and $\mathbf { v } _ { i } ^ { \prime }$ is not the same, implying that after calibration by CaGCN, the accuracy of original GCN cannot be preserved. Instead, here we propose an improved CaGCN. Given the output $\mathbf { V }$ of the classification GCN, we firstly use a $l$ -layer GCN to learn a unique temperature $t _ { i }$ for each node $i .$ , then get a calibrated logit $\mathbf { v } _ { i } ^ { \prime }$ by transforming its original logit $\mathbf { v } _ { i }$ using $t _ { i }$ in a temperature-scaling way, and finally obtain calibrated confidence $\hat { p } _ { i }$ as follows: + +$$ +\begin{array} { r } { \mathbf { t } = \sigma ^ { + } ( \mathbf { A } \sigma ( \cdot \cdot \cdot \mathbf { A } \sigma ( \mathbf { A } \mathbf { V } \mathbf { W } ^ { ( 1 ) } ) \mathbf { W } ^ { ( 2 ) } \cdot \cdot \cdot ) \mathbf { W } ^ { ( l ) } ) = [ t _ { 1 } , \cdots , t _ { N } ] ^ { \top } ( t _ { i } > 0 , \forall i \in \{ 1 , \cdots , N \} ) , } \\ { \mathbf { v } _ { i } ^ { r } = h ( \mathbf { v } _ { i } , t _ { i } ) = [ \nu _ { i , 1 } / t _ { i } , \cdots , \nu _ { i , K } / t _ { i } ] ^ { \top } , \mathbf { z } _ { i } = [ \sigma _ { S M } ( \nu _ { i , 1 } ^ { \prime } ) , \cdots , \sigma _ { S M } ( \nu _ { i , K } ^ { \prime } ) ] ^ { \top } , \hat { p } _ { i } = \operatorname* { m a x } _ { { \boldsymbol { k } } } z _ { i , { \boldsymbol { k } } } , } \end{array} +$$ + +where $t _ { i } \in \mathbb { R }$ is a scalar greater than zero and $\sigma ^ { + } ( \mathbf { x } ) = l o g ( 1 + e x p ( \mathbf { x } ) )$ is an element-wise softplus activation [8]. The model proposed in Eq. 5 does not change the order between classes of $\mathbf { v } _ { i }$ and $\mathbf { v } _ { i } ^ { \prime }$ , implying that the accuracy of original GCN is preserved. Compared Eq. 5 with Eq. 3, we can find that Eq. 5 makes the same transformation on all the dimensions of $\mathbf { v } _ { i }$ , which will limit the learnable calibration function space. However, we will prove that actually Eq. 5 is the same with the model proposed in Eq. 3 on confidence calibration using the Proposition 2. Considering that for any logit $\mathbf { v } _ { i }$ , our expectation is in fact that the calibration model can output any confidence $\hat { p } _ { i } \in \big ( \frac { 1 } { K } , 1 \big )$ . Please note that $\hat { p } _ { i } \geq \frac { 1 } { K }$ , or the prediction will be changed. Since Eq. 3 has no limitation on the learnt calibration model, its output $\hat { p } _ { i }$ can take any value from $\frac { 1 } { K }$ to 1. Therefore, if we can prove the output $\hat { p } _ { i }$ in Eq. 5 can also traverse the interval $\textstyle { \big ( } { \frac { 1 } { K } } , 1 { \big ) }$ for any $\mathbf { v } _ { i }$ , the equality between Eq. 3 and Eq.5 can be proved. + +Proposition 2. Given the original logit $\mathbf { v } _ { i } = [ \nu _ { i , 1 } , \cdots , \nu _ { i , K } ] ^ { \mathsf { T } }$ of node $i ,$ assume $\nu _ { i , j }$ not approaching infinity for each $j \in \{ 1 , \cdots , K \}$ . The calibrated confidence $\hat { p } _ { i }$ in Eq. $5$ can traverse the interval $\textstyle { \big ( } { \frac { 1 } { K } } , 1 { \big ) }$ for node i. + +Proof We set $\nu _ { i , 1 } > \nu _ { i , 2 } > \cdots > \nu _ { i , K }$ without loss of generality. For any $\mathbf { v } _ { i } \in \mathbb { R } ^ { K }$ , with the assumption of $\mathbf { v } _ { i }$ not approaching infinity, we have that + +$$ +\operatorname* { l i m } _ { t \to 0 } \hat { p } _ { i } = \operatorname* { l i m } _ { t \to 0 } \frac { e x p ( \nu _ { i , 1 } / t _ { i } ) } { \sum _ { j = 1 } ^ { K } e x p ( \nu _ { i , j } / t _ { i } ) } = \operatorname* { l i m } _ { t \to 0 } \frac { e x p ( ( \nu _ { i , 1 } - \nu _ { i , 2 } ) / t _ { i } ) } { e x p ( ( \nu _ { i , 1 } - \nu _ { i , 2 } ) / t _ { i } ) + \sum _ { j = 2 } ^ { K } e x p ( ( \nu _ { i , j } - \nu _ { i , 2 } ) / t _ { i } ) } = 1 +$$ + +and + +$$ +\operatorname* { l i m } _ { t \to + \infty } \hat { p } _ { i } = \operatorname* { l i m } _ { t \to + \infty } \frac { e x p ( \nu _ { i , 1 } / t _ { i } ) } { \sum _ { j = 1 } ^ { K } e x p ( \nu _ { i , j } / t _ { i } ) } = \frac { 1 } { K } . +$$ + +Obviously, both $\sigma _ { S M } ( \nu _ { i , k } )$ and $\mathbf { v } _ { i } / t _ { i }$ are continuous, thus $\sigma _ { S M } ( \nu _ { i , k } / t _ { i } )$ is continuous. Therefore, $\begin{array} { r } { \hat { p } _ { i } = \operatorname* { m a x } _ { k } z _ { i , k } = \operatorname* { m a x } _ { k } \sigma _ { S M } ( \nu _ { i , k } / t _ { i } ) } \end{array}$ can traverse the interval $( 1 / K , 1 )$ . + +The assumption about $\mathbf { v } _ { i }$ is easy to be satisfied since the L2-norm in GCN drives the weight matrix $\mathbf { W }$ approaching zero matrix and each element in node feature matrix $\mathbf { X }$ is not infinity. Therefore, based on Eq. 2, each element $\nu _ { i , j }$ in $\mathbf { V }$ cannot approach infinity. From Proposition 2 we know that for any $\mathbf { v } _ { i }$ , there exactly exists such a unique temperature $t _ { i }$ that $\hat { p } _ { i }$ can take any value from $1 / K$ to 1. In other words, the model can be perfectly calibrated. + +# 3.3 Optimization Objective + +Since NLL loss [10] can be decomposed into calibration loss and refinement loss [21], minimizing NLL loss benefits for confidence calibration. Therefore, we employ the NLL loss as our objective function with an additional regularization term. We use the prediction probability $\mathbf { z } _ { i } \in \mathbb { R } ^ { K }$ in Eq. 5 to calculate the NLL loss. Denote the $K$ -class one-hot label for node $i$ as $\mathbf y _ { i } = [ y _ { i , 1 } , \cdots , y _ { i , K } ] ^ { \intercal }$ and suppose the size of the validation set is $| D _ { \nu a l } |$ . Then the NLL loss over all validation nodes is represented as ${ \mathcal { L } } _ { n l l }$ where: + +$$ +\mathcal { L } _ { n l l } = - \sum _ { i = 1 } ^ { | D _ { \nu a l } | } \sum _ { k = 1 } ^ { K } y _ { i , k } l o g ( z _ { i , k } ) . +$$ + +Due to the under-confidence of GCNs, our goal is to increase the confidence of correct predictions while decreasing that of incorrect predictions. Considering that for incorrect predictions, the NLL loss cannot directly reduce their confidence, therefore, we design a regularization term for NLL loss as follows: + +$$ +\mathcal { L } _ { c a l } = \frac { 1 } { n } ( \sum _ { i = 1 } ^ { | c o r | } 1 - z _ { i , m } ^ { ( c o r ) } + z _ { i , s } ^ { ( c o r ) } + \sum _ { i = 1 } ^ { | i n c | } z _ { i , m } ^ { ( i n c ) } - z _ { i , s } ^ { ( i n c ) } ) , +$$ + +where $\left| c o r \right|$ and $| i n c |$ are the number of nodes correctly and incorrectly predicted and $z _ { i , m }$ and $z _ { i , s }$ are the max and submax prediction probability. Intuitively, the confidence of incorrect predictions is decreased by reducing the gap between the max and the submax value of $\mathbf { z } _ { i }$ and vice versa. Combining ${ \mathcal { L } } _ { n l l }$ and $\mathcal { L } _ { c a l }$ , we have the following overall objective function: + +$$ +\begin{array} { r } { \mathcal { L } = \mathcal { L } _ { n l l } + \lambda \mathcal { L } _ { c a l } , } \end{array} +$$ + +where $\lambda$ is the parameter of the regularization term. With the guide of labeled data, we can optimize CaGCN via back propagation and learn the calibrated confidence. The overall framework of CaGCN is shown in Fig. 4. + +# 4 Self-training with Confidence Calibration + +Here we propose a practical application of confidence calibration to improve the performance of self-training in GCNs. Self-training is to predict the labels for unlabeled data, and then add them to the training set, so as to achieve better performance. When applying self-training to GCN, we firstly obtain the predictions $\hat { y } _ { i }$ and the confidence $\hat { p } _ { i }$ given by GCN and then add the most confident nodes to the training set with pseudo labels $\hat { y } _ { i }$ based on $\hat { p } _ { i }$ . We continue to train until convergence. However, existing self-training methods perform not as expected with higher label rates [30]. Considering the under-confidence of existing GCNs, motivated by [26], we argue that the under-performance of existing self-training methods originals from large numbers of high-accuracy predictions distributing in low-confidence intervals as shown in Fig. 2, causing that they cannot be added to the training set. + +![](images/14380d5b900b4d0eee8d6b970db34d8d09c36f136a97bfe1e293eb3e62c583da.jpg) +Figure 4: The overall framework of CaGCN. Solid lines represent that we can backpropagate gradient here while dashed lines represent we cannot. We firstly train a classification GCN using the training set to obtain the logit $\mathbf { V }$ of all the nodes. Then we feed $\mathbf { V }$ to $\mathrm { C a G C N }$ to get the temperature t and transform $\mathbf { V }$ using t into $\mathbf { V } ^ { \prime }$ . Finally, the loss can be obtained using $\mathbf { V } ^ { \prime }$ after softmax according to Eq. 10 and CaGCN can be optimized with the guide of the validation set. + +Table 2: ECE $\mathbf { \Gamma } ( \mathbf { M } { = } 2 0 )$ ) on different models and citation networks of various label rate (L/C) with and without calibration. Uncal. represents the uncalibrated model, (-) denotes this method cannot converge to a meaningful result and bold denotes the best result, the subscript of each result refers to the standard deviation $( \times 1 0 ^ { - 3 } )$ while the superscript refers to the results of paired t-test ( \* for 0.05 level and $^ { * * }$ for 0.01 level). + +
DatasetL/CGCNGAT
Uncal.TSMSCaGCNUncal.TSMSCaGCN
Cora200.13476.30.04885.50.04145.70.04016.70.15588.90.07179.80.05449.40.0450
400.11344.70.04177.20.03724.60.04075.40.13405.40.04857.70.04916.00.0365
600.09374.90.0355.40.03646.10.03764.40.12013.30.03936.10.04115.30.031332
Citeseer200.12487.10.06418.70.06443.70.05957.20.15345.00.09168.70.06339.80.05726.8
400.09577.70.06014.20.05385.70.05455.50.12528.70.07973.10.05905.40.05325.4
600.08066.40.05595.00.05216.40.05463.40.10905.90.06487.10.05199.10.05257.6
Pubmed200.05867.70.05413.80.04764.20.0405.00.08353.10.06564.60.05013.70.0356
400.04445.50.04466.30.04366.30.0402.00.08694.60.06586.50.05396.00.0308
600.04459.70.03676.00.03186.40.03114.80.09934.10.06696.30.04835.70.03082
CoraFull200.19866.10.10136.1=0.07760.21193.60.11015.1=0.07880
400.23215.40.11176.50.07010.24384.20.11338.30.07388
600.23374.00.09813.80.07680.24971.80.11335.20.0849
+ +Consequently, we design a self-training model CaGCN-st where confidence is firstly calibrated then employed to generate pseudo labels for unlabeled nodes. Specifically, given an unlabeled dataset $D _ { U }$ and a labeled dataset $D _ { L }$ which has been divided into three parts $D _ { t r a i n }$ , $D _ { \nu a l }$ and $D _ { t e s t }$ , we firstly train a classification GCN using $D _ { t r a i n }$ to get the logit of each node. Then all the logits will be fed into a CaGCN to train and we get a calibrated confidence for each node. It should be noted that instead of $D _ { \nu a l }$ , we still employ $D _ { t r a i n }$ to train our CaGCN. After that, the most confident predictions of $D _ { U }$ will be adopted as the pseudo labels according to a threshold $\tau$ and added to the label set. The $D _ { t r a i n }$ is enlarged in this way. The process above will be repeated $s$ stages until convergence. Please note that our classification GCN and CaGCN are re-initialized in each stage. + +# 5 Experiments + +In this section, we evaluate the performance of CaGCN on confidence calibration and CaGCN-st on self-training respectively. We choose the commonly used citation networks Cora [29], Citeseer [29], Pubmed [29] and CoraFull [3] for evaluation, and more detailed descriptions are in Appendix B. + +Table 3: Node classification accuracy and the standard deviation on GCN and its self-training variants. + +
DatasetL/CMethods
Orig.St.Ct.UnionInter.TS-stCaGCN-st
Cora2081.630.2482.270.3381.510.3081.850.6881.410.2882.680.2083.110.52
4083.990.2683.590.3483.660.2583.330.4183.380.3384.440.3584.370.38
6084.440.2984.980.3284.630.3185.030.3084.880.1885.600.2485.790.27
Citeseer2071.640.3273.240.4474.220.2974.600.3872.250.4574.200.2474.90040
4072.250.3274.700.3372.120.3974.790.3673.660.3275.620.1975.480.50
6073.200.3575.080.2973.210.3675.530.3075.230.2375.870.2476.430.20
Pubmed2079.570.3380.320.1879.670.3281.120.2979.590.2980.950.1881.16.36
4080.650.3982.200.3281.620.4081.840.2380.460.5582.280.3983.0821
6083.380.3483.350.2883.400.3683.320.3583.310.1783.260.3984.47023
CoraFull2060.450.4360.870.2860.120.4560.520.3561.010.5361.730.4162.190.49
4065.770.3765.830.4564.220.3564.330.4265.840.3766.110.6066.300.31
6066.520.2566.620.3066.640.2966.780.2966.820.3266.950.4567.600.40
+ +# 5.1 Confidence Calibration Evaluation + +Baselines. Since our CaGCN is a general calibration model for GNNs, here we choose GCN [16] and GAT [33] as our classification models. For comparison, we choose the classic post-hoc calibration methods temperature scaling (TS) [12] and matrix scaling with off-diagonal regularization (MS) [18] as our baselines. + +Experimental settings. For the base model GCN and GAT, i.e., the uncalibrated model, we follow parameters suggested by [16] and [33] and further carefully tune them to get optimal performance. For the post-hoc calibration technique, we follow the official implementation [12, 18]. For our CaGCN, we train a two-layer GCN with the hidden layer dimension to be 16. We set $\lambda = 0 . 5$ for all datasets, weight decay to be 5e-3 for Cora, Citeseer, Pubmed and 0.03 for CoraFull. Other parameters of CaGCN follows [16]. We evaluate the performance of confidence calibration by ECE [22], NLL [10] and Brier Score (BS) [4], which we expect are smaller, and we set the bin number $M = 2 0$ for ECE (more details can be seen in Appendix A). For all methods, we randomly run 10 times and report the average results. More detailed experimental settings can be seen in Appendix B. + +Results. Table 2 reports calibration results evaluated by ECE (more results on NLL and Brier Score are in Appendix C.1). We have the following observations: (1) Compared with uncalibrated models and other baselines, CaGCN is statistically significantly better at the $^ { * } 0 . 0 5$ level and $\ast \ast _ { } 0 . 0 1$ level. (2) The ECE values on uncalibrated models are generally the highest, implying that GCN and GAT are poorly calibrated. (3) MS behaves badly on datasets with many classes, e.g., CoraFull. This is because the number of parameters for matrix scaling scales quadratically with the number of classes while the size of the validation set keeps unchanged. Therefore, it will over-fit to the small validation set when dataset has a great number of classes. However, CaGCN does not have this problem. + +Additional analysis. In Section 2 we visualize the under-confidence problem of existing GNNs using reliability diagrams. Here we utilize the same visualization method to make a comparison before and after confidence calibration. As shown in Fig. 8, Fig. 9, Fig. 10 and Fig. 11 in the appendix, we can find that the confidence is well-calibrated after calibration. + +# 5.2 Classification Evaluation of Self-Training + +Baselines. Since self-training can be applied to any models, here we choose GCN and GAT as our base models, i.e., the original models (Orig.) without self-training, and we choose self-training (St.), co-training (Ct.), Union, Intersection (Inter.) methods proposed in [20] for comparison, which are commonly used as the baselines in self-training. Furthermore, we employ TS as the confidence calibration function in CaGCN-st as another baseline and we denote it by TS-st. + +Experimental settings. We set the learning rate $l r = 0 . 0 0 1$ for CaGCN-st and train our CaGCN-st 200 epochs for Cora, 150 epochs for Citeseer, 100 epochs for Pubmed and 500 epochs for CoraFull. We set the threshold $\tau \in \{ 0 . 8 , 0 . 8 5 , 0 . 9 , 0 . 9 5 , 0 . 9 9 \}$ and the maximum number of stage $s = 1 0$ . As for baselines, all the parameters follow [20] and we further carefully tune them to get optimal performance. For all methods, we randomly run 10 times and report the average results. + +Results. Table 3 summarizes the node classification accuracy on GCN and its self-training variants. More results on GAT can be seen in Appendix C.2. We have the following observation: (1) CaGCN-st consistently outperforms all the baselines on all the datasets and label rates at the $^ { * } 0 . 0 5$ level. (2) Compared with the base model, self-training methods generally achieve better results, which proves their effectiveness. (3) Self-training methods with confidence calibration (i.e., TS-st and CaGCN-st) have better performance, which implies that confidence calibration scales more correct predictions to the high confidence range while keeps incorrect predictions basically unchanged, which we believe is beneficial for self-training. + +Ablation study. CaGCN-st generates pseudo labels based on the calibrated confidence. Here we study the effectiveness of the confidence calibration function CaGCN in CaGCNst. We propose a variant GCN-st of CaGCNst, where CaGCN is removed from CaGCN-st while other parts are kept unchanged. All the experimental settings of GCN-st are the same as CaGCN-st. We report the results in Table 4, and we can observe that CaGCN-st consistently outperforms GCN-st on all the datasets, implying that self-training with calibrated confidence can generate more correct pseudo labels. + +Additional analysis. We also investigate the changing trends of accuracy with respect to the threshold $\tau$ in CaGCN-st in Appendix C.2 and study why GCNs are poorly calibrated in Appendix D. + +Table 4: Abaltion study on self-training + +
Dataset L/CGCNGAT
GCN-st CaGCN-stGCN-st CaGCN-st
Cora2082.2883.1184.08 84.08
4084.1084.37 85.5085.63
6085.16 85.7985.5786.26
Citeseer2074.1374.9073.73 74.34
4075.2875.4875.07 75.62
6075.85 76.4375.1376.08
Pubmed2081.0181.1680.34 81.17
4082.9083.0882.75 83.47
6083.44 84.4783.4683.95
CoraFull2061.3262.19 62.0965.46
4065.9666.3065.92 66.86
6066.43 67.6066.5467.45
+ +# 6 Related Work + +Graph Neural Networks. Modern GCNs mimics CNNs to learn the local and global structural patterns of graphs through designed convolution and readout functions. [5] generalizes CNNs to graph signal based on the spectrum of graph Laplacian. ChebNet [7] uses Chebyshev polynomials to approximate the $K$ -order localized graph filters and GCN [16] further employs the 1-order simplification of the Chebyshev filter. GAT [33] utilizes attention mechanisms to adaptively learn aggregation weights. GraphSAGE [13] uses various ways of pooling for aggregation. [20] introduces self-training to GCNs and [30] proposes a multi-stage self-supervised (M3S) self-training algorithm of GCNs. Both [20] and [30] focus on the few-shot learning and neither has ever explored self-training with higher label rates in GCNs. More works on GNNs can be found in surveys [36, 43], however, to the best of our knowledge, current GNNs have not considered the confidence calibration. + +Confidence Calibration. Confidence calibration has been studied for a long time in CV and NLP [12, 23, 25, 14, 19, 37, 40, 42]. [12] discovers modern neural networks are poorly calibrated and study factors influencing calibration. Platt scaling [24] is a simple post-hoc calibration method for binary models, which transforms the logit using scalar parameters. Temperature scaling is the simplest multi-class extension of Platt scaling and matrix and vector scaling are another two extensions of platt scaling. [40] proposes Mix-n-Match calibration strategies which mix parameter methods with non-parameter methods. [25] explores the non-linear space for post-hoc calibration function using a neural network. Moreover, [32] points out GNNs can be miscalibrated in the supervised scenario and mainly focus on miscalibration originated from the imbalanced class distribution. However, none of them have considered the confidence calibration in GNNs on the common semi-supervised scenario. + +# 7 Conclusion + +Current efforts on advancing GNNs mostly focus on classification accuracy. However, when deploying GNNs to real-world applications, especially safety-critical fields, whether the results of GNNs are trustworthy is another important factor that cannot be neglected. In this paper, we study the confidence calibration problem in GNNs and discover existing GNNs are under-confident on their predictions. To solve this problem, we propose a novel trustworthy GNN model CaGCN which respects the homophily property of confidence in GNNs and preserves the classification accuracy. Moreover, we propose a novel self-training method CaGCN-st where confidence is first calibrated by CaGCN and then used to generate pseudo labels. Extensive experiments demonstrate the effectiveness of our proposed model in terms of both calibration and accuracy. + +An interesting direction for future work is to extend CaGCN to other graph tasks, but more studies need to be conducted. We take the link prediction as an example, where we can regard the link prediction as a binary classification problem and the output as the confidence. Considering that the ground-truth confidence distribution for nodes should have the homophily property as is shown in Section 3.1, edges are likely to have the same property as well. As a result, we can employ CaGCN to propagate the confidence between edges by regarding the edges as the nodes. However, more exploration still needs to be conducted for the homophily property of edges. + +Broader impact. Current efforts on advancing GNNs mostly focus on classification accuracy. However, when deploying GNNs to real-world applications, especially safety-critical fields, whether the results of GNNs are trustworthy is another important factor. The demands for a trustworthy model are universal and extensive such as in the field of disease prediction [31], traffic states prediction [6] and object detection [11] for autonomous driving, where estimating the true probability of getting a correct prediction is necessary. We take the disease prediction [31] as an example, where GNNs are utilized to encode the information of different symptoms, users and diseases. In this scenario, accurately and comprehensively predicting diseases at an early stage will help patients receive prevention treatments in a timely manner. Otherwise, the misdiagnosis and missed diagnosis will endanger the health of patients. Therefore, a trustworthy model is urgently needed. Our CaGCN can make a trustworthy prediction based on its confidence, and as a result, decrease the risk of misdiagnosis and missed diagnosis. We hope our work can provide insights for future improvements in tackling the trustworthiness problem in other saftey-critical fields. + +Limitations. One potential issue of this work is that it provides a limited explanation to the underconfidence problem. We advocate peer researchers to look into this, making GNNs more reliable in different domains. Other than that, since this work is mostly on the discovery of the confidence calibration problem in GNNs and the theoretical aspect of improving calibration, we do not foresee any direct negative impacts on the society. + +# Acknowledgments and Disclosure of Funding + +This work is supported in part by the National Natural Science Foundation of China (No. 62172052, No. U20B2045, U1936104, 61772082, 61702296, 62002029). + +# References + +[1] Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané. Concrete problems in ai safety. arXiv preprint arXiv:1606.06565, 2016. +[2] Deyu Bo, Xiao Wang, Chuan Shi, and Huawei Shen. Beyond low-frequency information in graph convolutional networks. arXiv preprint arXiv:2101.00797, 2021. +[3] Aleksandar Bojchevski and Stephan Günnemann. Deep gaussian embedding of graphs: Unsupervised inductive learning via ranking. arXiv preprint arXiv:1707.03815, 2017. +[4] Glenn W Brier. 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How powerful are graph neural networks? In 7th International Conference on Learning Representations, ICLR 2019. +[39] Jiaxuan You, Zhitao Ying, and Jure Leskovec. Design space for graph neural networks. Advances in Neural Information Processing Systems, 33, 2020. +[40] Jize Zhang, Bhavya Kailkhura, and T Yong-Jin Han. Mix-n-match: Ensemble and compositional methods for uncertainty calibration in deep learning. In International Conference on Machine Learning, pages 11117–11128. PMLR, 2020. +[41] Muhan Zhang and Yixin Chen. Link prediction based on graph neural networks. arXiv preprint arXiv:1802.09691, 2018. +[42] Xujiang Zhao, Feng Chen, Shu Hu, and Jin-Hee Cho. Uncertainty aware semi-supervised learning on graph data. arXiv preprint arXiv:2010.12783, 2020. +[43] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. Graph neural networks: A review of methods and applications. AI Open, 1:57–81, 2020. +[44] Jiong Zhu, Yujun Yan, Lingxiao Zhao, Mark Heimann, Leman Akoglu, and Danai Koutra. Beyond homophily in graph neural networks: Current limitations and effective designs. Advances in Neural Information Processing Systems, 33, 2020. \ No newline at end of file diff --git a/parse/train/9c-IsSptbmA/9c-IsSptbmA_content_list.json b/parse/train/9c-IsSptbmA/9c-IsSptbmA_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..58cd45d808dbc4c1adf7817e786c439e7e1f955e --- /dev/null +++ b/parse/train/9c-IsSptbmA/9c-IsSptbmA_content_list.json @@ -0,0 +1,1163 @@ +[ + { + "type": "text", + "text": "Be Confident! Towards Trustworthy Graph Neural Networks via Confidence Calibration ", + "text_level": 1, + "bbox": [ + 191, + 122, + 810, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xiao Wang, Hongrui Liu, Chuan Shi∗, Cheng Yang ", + "bbox": [ + 325, + 220, + 674, + 236 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "School of Computer Science (National Pilot Software Engineering School) Beijing University of Posts and Telecommunications Beijing, China \n{xiaowang, liuhongrui, shichuan, yangcheng}@bupt.edu.cn ", + "bbox": [ + 256, + 237, + 741, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 327, + 535, + 343 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite Graph Neural Networks (GNNs) have achieved remarkable accuracy, whether the results are trustworthy is still unexplored. Previous studies suggest that many modern neural networks are over-confident on the predictions, however, surprisingly, we discover that GNNs are primarily in the opposite direction, i.e., GNNs are under-confident. Therefore, the confidence calibration for GNNs is highly desired. In this paper, we propose a novel trustworthy GNN model by designing a topology-aware post-hoc calibration function. Specifically, we first verify that the confidence distribution in a graph has homophily property, and this finding inspires us to design a calibration GNN model (CaGCN) to learn the calibration function. CaGCN is able to obtain a unique transformation from logits of GNNs to the calibrated confidence for each node, meanwhile, such transformation is able to preserve the order between classes, satisfying the accuracypreserving property. Moreover, we apply the calibration GNN to self-training framework, showing that more trustworthy pseudo labels can be obtained with the calibrated confidence and further improve the performance. Extensive experiments demonstrate the effectiveness of our proposed model in terms of both calibration and accuracy. ", + "bbox": [ + 233, + 358, + 766, + 593 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 176, + 618, + 312, + 636 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graphs are ubiquitous in the real world, including social networks, e-commerce networks, traffic networks, and so on. Recently, Graph Neural Networks (GNNs), which are able to effectively learn the node representations based on the message-passing manner, have attracted considerable attention in dealing with graph data [16, 33, 39, 44, 15, 2, 34]. To date, GNNs have been applied to various applications and achieved remarkable accuracy, e.g., node classification [16, 33], link prediction [41] and graph classification [9]. ", + "bbox": [ + 174, + 650, + 825, + 733 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, it is well established that a model with good accuracy is not the only goal, but a trustworthy model is highly desired in many applications, especially in safety-critical fields [1]. Usually, a trustworthy model implies that it should know when it is likely to be incorrect, in other words, the probability, i.e., the confidence, associated with the predicted class label should reflect its ground truth correctness likelihood [12]. For example, in the scene of autonomous driving, the system will adopt the prediction given by the model only when the model has high confidence for its prediction. Otherwise, the decision-making power will be returned to the driver or the system adopts other safer strategies. Recently, the confidence calibration has attracted considerable attention in deep learning [12, 40, 19], which reveals that many modern neural network models are over-confident on the predictions, i.e., the prediction accuracy is lower than its confidence. However, it has not been studied in GNNs on the semi-supervised scenario, which gives rise to one fundamental question: will the current GNNs follow the same over-confident property as other neural networks? A well-informed answer can help us better understand GNNs and enable GNNs to be applied to various areas in a more reliable manner. ", + "bbox": [ + 174, + 739, + 825, + 877 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As the first contribution of this study, we present experiments assessing the relationship between the confidence and the accuracy of Graph Convolutional Networks (GCNs) [16] and Graph Attention Networks (GAT) [33] in the node classification task (more details can be seen in Section 2), respectively. Surprisingly, we discover that existing GNNs are far distant from being well-calibrated, and more importantly, GNNs tend to be under-confident in their predictions, which is very different from other modern deep learning models that are often over-confident [12, 19]. GNNs being under-confident means that many predictions are distributed in the low-confidence range, and therefore, fewer predictions are available for safety-critical applications. Once the weakness is identified, another natural question is: how can we calibrate the confidence on predictions given by GNNs so as to make them more trustworthy? ", + "bbox": [ + 174, + 154, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Essentially, the confidence calibration is to calibrate the outputs (also known logits) of original models (e.g., GNNs), therefore, a straightforward manner is to employ temperature scaling (TS) [12], OP-families [25] to learn calibration function using a held-out dataset in a post-hoc way. However, when being applied to graphs, they all ignore the effect of topology, which will inevitably make mistakes during calibration. For example, considering that the logits of two nodes $a$ and $^ b$ are the same, but node $a$ is similar to its neighbors while node $b$ is not. Apparently, the predictions of GCNs for $a$ should be more confident than $^ b$ , while the traditional calibration methods, e.g., TS, will learn the same confidence for $a$ and $^ b$ , because it does not consider the effect of topology. ", + "bbox": [ + 174, + 297, + 825, + 409 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Moreover, most of them explore calibration functions only in the linear space [12, 18] while it is well known that non-linear space contains more complex function transformation which is able to calibrate networks with complicated landscapes well. Even if some works have explored the non-linear space such as Matrix Scaling [12], they generally degrade the classification accuracy of the original classifier, while a good accuracy is still a basic requirement by many applications. ", + "bbox": [ + 174, + 415, + 825, + 484 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we introduce a topology-aware post-hoc calibration method for GNNs. Specifically, for the logits given by the original classification GNNs, we employ another calibration GCN (CaGCN) to propagate confidence, naturally enabling that the confidence of topologically adjacent nodes becomes similar. CaGCN learns a unique temperature $t$ for each node for temperature scaling, thus preserving the accuracy of the original classification GCN. In addition, based on our finding that large numbers of high-accuracy predictions are distributed in the low-confidence range, we design a calibrated self-training model CaGCN-st in which the confidence is firstly calibrated then used to generate pseudo labels with high confidence. The contributions of this paper are three-fold: ", + "bbox": [ + 173, + 491, + 825, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We study the trustworthy problem of GNNs, and discover one unique characteristic of GNNs, i.e., the predictions made by GNNs are usually under-confident. \n• We propose a novel trustworthy GNN model based on the confidence calibration. Our proposed calibration function has three features: topology-aware, non-linear, and accuracy-preserving. We further design a calibrated self-training GNN model, which can effectively utilize the predictions with high confidence. \n• Extensive experiments demonstrate the effectiveness of our proposed models in terms of both calibration and accuracy. ", + "bbox": [ + 171, + 611, + 826, + 731 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Notation and Preliminary Study ", + "text_level": 1, + "bbox": [ + 173, + 747, + 473, + 765 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we focus on the calibration of semi-supervised node classification in an undirected attributed graph $G = \\left( V , E \\right)$ with the adjacent matrix $\\mathbf { A } \\in \\mathbb { R } ^ { N \\times N }$ and the node feature matrix $\\mathbf { X } =$ $[ \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { N } ] ^ { \\mathsf { T } }$ . $V$ is a set of nodes and $E \\subseteq V \\times V$ is a set of edges between nodes. $N = | V |$ is the number of nodes. Here we give the definition of perfect calibration of GNNs as follows: ", + "bbox": [ + 174, + 777, + 825, + 835 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Definition 1. Given random variables A, $\\mathbf { X }$ , $\\mathbf { Y } \\subseteq \\{ 1 , \\ldots , K \\}$ and a GNN model $f _ { \\theta }$ where $\\theta$ is the learnable parameters, for node i with label $y _ { i } \\in \\mathbf { Y }$ , $\\mathbf { z } _ { i } = f _ { \\theta } ( \\mathbf { x } _ { i } , \\mathbf { A } ) = [ z _ { i , 1 } , \\ldots , z _ { i , K } ] ^ { \\mathsf { T } }$ is the output of GNNs (i.e., the prediction probability), and $\\hat { y } _ { i } = \\mathrm { a r g } \\operatorname* { m a x } _ { k } z _ { i , k }$ and $\\hat { p } _ { i } = \\operatorname* { m a x } _ { k } z _ { i , k }$ are the prediction and the confidence respectively. Then we define $f _ { \\theta }$ to be perfectly calibrated as: ", + "bbox": [ + 174, + 837, + 825, + 895 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/a1d6d2847912d0278c04a6a6a38d4db990a5b4af3214b389f6597704627a4ce6.jpg", + "text": "$$\n\\mathbb { P } ( \\hat { y } _ { i } = y _ { i } | \\hat { p } _ { i } = p ) = p , \\forall p \\in [ 0 , 1 ] .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 896, + 611, + 912 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/b09bafe873b2b4009d9aab352efaf8bdb093b30da45eedaa1e5de35a56b8adff.jpg", + "image_caption": [ + "Figure 1: Reliability diagrams for GCN (top) and GAT (bottom) without confidence calibration. The diagram is expected to plot an identity function of accuracy with respect to confidence. Any deviation from a perfectly diagonal (i.e., the difference between blue and red histogram) represents the miscalibration. " + ], + "image_footnote": [], + "bbox": [ + 178, + 89, + 820, + 280 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/c742029dfe4c579e527d277d926e41da06ed21b6ba6d31bd18e006ec2ce69a5d.jpg", + "image_caption": [ + "Figure 2: Confidence distribution before calibration. " + ], + "image_footnote": [], + "bbox": [ + 178, + 362, + 818, + 558 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "According to Definition 1, GNN is perfectly calibrated only when the confidence $\\hat { p } _ { i }$ is exactly equal to the true probability of getting a correct prediction for every node. ", + "bbox": [ + 174, + 635, + 823, + 662 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Next, we take two representative GNNs (GCN [16] and GAT [33]) as examples to analyze whether they are perfectly calibrated. Specifically, we apply GCN and GAT to four widely used datasets Cora [29], Citeseer [29], Pubmed [29], CoraFull [3], and examine whether their results satisfy Definition 1. To provide more results, we select three label rates for training set (i.e., 20, 40, 60 labeled nodes per class). All the experimental settings follow [16, 33]. Since the true probability $p$ cannot be exactly known, we take an approximate way to evaluate the calibration as in [12]. In particular, we first partition the [0,1] range of confidence into 20 equal bins and then we group the nodes into corresponding bins according to their confidence. After that we calculate the average accuracy of each bin. We expect the average accuracy is equal to the average confidence of each bin, which means the model is approximately perfectly calibrated. For example, if the average confidence of nodes in the bin [0.95, 1.0] is 0.96, and then the classification accuracy in this bin should be $96 \\%$ . ", + "bbox": [ + 173, + 669, + 825, + 821 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We illustrate the results of label rate being 20 in Fig. 1 using Reliability Diagrams [23] here, where the $\\mathbf { X }$ -axis is the confidence in 20 bins of equal size and y-axis is the average accuracy in each bin. The blue represents the classification accuracy of GCN and GAT while the red is our expectation. More results of label rate being 40, 60 and other GNN models can be seen in Fig. 8, Fig. 9, Fig. 12, Fig. 13 and Fig. 14 in the appendix. We can see that in all the datasets, the average accuracy of most bins is higher than the average confidence. In other words, these GNNs actually achieve remarkable performance, but they all output low confidence, i.e., the GNNs are usually under-confident. Please note that this phenomenon of GNNs is very different from other modern neural networks, which are generally known to be over-confident [12, 19]. Moreover, as shown in Fig. 2, we also visualize the confidence distribution of test nodes, where the $\\mathbf { X }$ -axis is the confidence and y-axis is the density [28]. The histogram height multiplied by the width is equal to the frequency. The blue represents the confidence distribution of correct predictions while the yellow represents that of incorrect predictions. More results of label rate being 40 and 60 can be seen in Fig. 10 and Fig. 11 in the appendix. We can see that a large quantity of correct predictions are distributed in the low confidence range. The results above indicate that the current GNNs are far from perfect calibration, leading to unreliable confidence. ", + "bbox": [ + 174, + 828, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 826, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Confidence Calibration on GCNs ", + "text_level": 1, + "bbox": [ + 174, + 248, + 482, + 265 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we propose our method to calibrate current GNNs. Given $\\mathbf { A }$ and $\\mathbf { X }$ , for a $l$ -layer GCN [16], the output of the GCN before the softmax layer can be obtained by: ", + "bbox": [ + 173, + 279, + 823, + 308 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a7d5b8c7e66ef87756461de207f0e2c8249f54b9ba46cfb5f20e68d331a1c812.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { V } = \\mathbf { A } \\sigma ( \\cdots \\mathbf { A } \\sigma ( \\mathbf { A X W } ^ { ( 1 ) } ) \\mathbf { W } ^ { ( 2 ) } \\cdots ) \\mathbf { W } ^ { ( l ) } = [ \\mathbf { v } _ { 1 } , \\cdots , \\mathbf { v } _ { N } ] ^ { \\mathsf { T } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 310, + 686, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbf { W } ^ { ( l ) }$ is the weight matrix of $l$ -th layer in GCN and $\\sigma ( \\cdot )$ is the activation function. For each node $i \\in \\{ 1 , \\cdots , N \\}$ , our goal is to learn a calibration function which is fed with $\\mathbf { v } _ { i }$ (often known as the logit of node $i$ ) and outputs a calibrated confidence using a held-out dataset in a post-hoc way. The calibration function should satisfy three points below: (1) taking the network topology into account (2) non-linear (3) preserving the classification accuracy of the GCNs. ", + "bbox": [ + 173, + 334, + 825, + 405 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 CaGCN: GCNs as Calibration Function ", + "text_level": 1, + "bbox": [ + 173, + 419, + 491, + 434 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/8a3f54c76241f96ccd4f9a15a0db471c74196c8af7f467a60b7c380c4cd01838.jpg", + "image_caption": [ + "Figure 3: The illustration of the confidence propagation. Different colors indicate different classes. " + ], + "image_footnote": [], + "bbox": [ + 192, + 450, + 446, + 541 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/14b81ef663ebfbe06767368e359d87caf5ab62d3f4345793e5971617e1a09755.jpg", + "table_caption": [ + "Table 1: Summary of total variation of confidence before and after calibration (Bold: best). Uncal. is short for uncalibrated and TS is short for temperature scaling. " + ], + "table_footnote": [], + "table_body": "
DatasetGCN
Uncal.TSOurs
Cora240.267172.346164.651
Citeseer128.145112.212108.684
Pubmed1299.331266.681113.41
CoraFull6014.324698.204500.30
", + "bbox": [ + 503, + 506, + 821, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We assume that the ground-truth confidence distribution in a graph has homophily property, i.e., the confidence of neighboured nodes given by well-calibrated models should be similar, and thus we conduct an experiment to verify this. We employ the classic temperature scaling method [12] as our calibration function and use the total variation [27] of confidence as our evaluation, which sums the difference of confidence between all the neighboured nodes. We compare the total variation of confidence before and after confidence calibration, where the results are shown in Table 1. We can find that the total variation of confidence does decrease after temperature scaling, which verifies our assumption. This inspires us that if a GCN model is well-calibrated, then the confidence between neighbors should be more similar than before. ", + "bbox": [ + 173, + 604, + 825, + 729 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To this end, we find that GCN itself can play the role of calibration function that meets above requirement since GCN is able to propagate node features along the network topology and smooth similar information between neighboured nodes. Therefore, we can employ another $l .$ -layer GCN (CaGCN) as our calibration function to propagate the confidence along the network topology. Specifically, given the output $\\mathbf { V }$ of the classification GCN, the logit $\\mathbf { v } _ { i } ^ { \\prime }$ and confidence $\\hat { p } _ { i }$ for node $i$ after calibration can be obtained by: ", + "bbox": [ + 174, + 734, + 826, + 818 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6d5789c9c27fad970c919bcecb5593efa4d9d2909e80d36b0bd4fa29a9939f78.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\mathbf { V } } ^ { \\prime } = { \\mathbf { A } } \\sigma ( \\cdots { \\mathbf { A } } \\sigma ( { \\mathbf { A } } { \\mathbf { V } } { \\mathbf { W } } ^ { ( 1 ) } ) { \\mathbf { W } } ^ { ( 2 ) } \\cdots ) { \\mathbf { W } } ^ { ( l ) } = [ { \\mathbf { v } } ^ { \\prime } _ { 1 } , \\cdots , { \\mathbf { v } } ^ { \\prime } _ { N } ] ^ { \\mathsf { T } } , } \\\\ & { { \\mathbf { z } } _ { i } = [ \\sigma _ { S M } ( \\nu ^ { \\prime } _ { i , 1 } ) , \\cdots , \\sigma _ { S M } ( \\nu ^ { \\prime } _ { i , K } ) ] ^ { \\mathsf { T } } , \\hat { p } _ { i } = \\operatorname* { m a x } z _ { i , k } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 820, + 691, + 868 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\sigma _ { S M } ( \\nu ^ { \\prime } _ { i , \\cdot } ) = \\frac { \\exp ( \\nu ^ { \\prime } _ { i , \\cdot } ) } { \\sum _ { j = 1 } ^ { K } \\exp ( \\nu ^ { \\prime } _ { i , j } ) } } \\end{array}$ is the softmax operation. Then the total variation of confidence will surely become lower and the original classification GCN will be calibrated. Please note that although temperature scaling can be directly applied here, compared with GCN, it does not take the network topology into account, which may cause mistakes mentioned in Section 1. Moreover, temperature scaling only employs a linear transformation, and GCN is able to learn a non-linear calibration function. ", + "bbox": [ + 176, + 872, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For a comprehensive understanding of confidence propagation, we make a detailed and visible illustration here. As shown in Fig. 3, the logits of two nodes $a$ and $^ b$ are the same, but node $a$ is similar to its neighbors while node $b$ is not. Apparently, the predictions of GCNs for $a$ should be more confident than $b$ . Suppose that $a$ , $b$ and their neighbours are under-confident based on the observation above. If we continue to propagate their logits along the topology using another GCN, the logits of $a$ and its neighbors will tend to be the same. Therefore, if one or more of these nodes are calibrated during the calibration process, all of them will be calibrated as well. The confidence is propagated in this way. On the other hand, looking at another node $b$ , it is as difficult even for manual classification as it is for GCNs. Consequently, the confidence of $b$ should stay still even be lower. However, it will become higher because of the influence from $a$ if we use the traditional calibration method without considering the network topology. Instead, when the network topology is taken into account, the logit of $^ b$ will be averaged by its neighboured and each dimension tends to $1 / K$ . It will be correctly calibrated when other nodes in the same situation are well-calibrated. ", + "bbox": [ + 173, + 154, + 825, + 333 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 The Accuracy-Preserving Property ", + "text_level": 1, + "bbox": [ + 174, + 348, + 457, + 363 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Until now, we have proposed a non-linear calibration model CaGCN which can take the network topology into account, but the accuracy-preserving property cannot be satisfied. To address this problem, we firstly study the general accuracy-preserving calibration function. ", + "bbox": [ + 174, + 373, + 825, + 416 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 1. Let $h : \\mathbb { R } ^ { K } \\to \\mathbb { R } ^ { K }$ be a calibration function, $s : \\mathbb { R } \\to \\mathbb { R }$ be a 1-D function and $\\mathbf { v } _ { i } = [ \\nu _ { i , 1 } , \\cdots , \\nu _ { i , K } ] ^ { \\mathsf { T } }$ be the logit of node i. The calibration function $h$ preserves the classification accuracy of the original model if s is a strictly isotonic function and $h$ satisfies: ", + "bbox": [ + 173, + 417, + 823, + 460 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6368d8bf5e7d2b53e267659c9e6aa42c14a2d0325551be99767bff3fca4f24cc.jpg", + "text": "$$\n\\begin{array} { r } { h ( \\mathbf { v } _ { i } ) = [ s ( \\nu _ { i , 1 } ) , \\dots , s ( \\nu _ { i , K } ) ] ^ { \\mathsf { T } } , \\forall i \\in \\{ 1 , \\cdots , N \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 462, + 650, + 481 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof We set $\\nu _ { i , 1 } < \\nu _ { i , 2 } < \\cdots < \\nu _ { i , K }$ without loss of generality. Since $[ s ( \\nu _ { i , 1 } ) , \\ldots , s ( \\nu _ { i , K } ) ] ^ { \\mathsf { T } }$ shares the same order with $\\mathbf { v } _ { i }$ as a result of the strictly isotonicity of $s$ , the order between classes of the logit $\\mathbf { v } _ { i }$ is unchanged, hence the accuracy of the prediction is preserved. \u0004 ", + "bbox": [ + 174, + 489, + 825, + 534 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Temperature scaling [12] is the simplest accuracy-preserving calibration method using a scalar parameter $t$ called temperature for all classes. Given the logit $\\mathbf { v } _ { i }$ of node $i .$ , the confidence of the prediction is $\\hat { p } _ { i } = \\operatorname* { m a x } _ { k } \\sigma _ { S M } ( \\nu _ { i , k } / t ) ( t > 0 )$ . In temperature scaling, $h ( { \\bf v } _ { i } ) = [ \\nu _ { i , 1 } / t , \\cdot \\cdot \\cdot , \\nu _ { i , K } / t ] ^ { \\top }$ is the calibration function and $s ( x ) = x / t$ is the strictly isotonic function. ", + "bbox": [ + 174, + 539, + 825, + 594 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "However, we can find that temperature scaling (TS) [40] only performs the same linear transformation for all the nodes using the same $t$ . As mentioned in Eq. 3, we propose to use CaGCN as our calibration function, while CaGCN is generally not isotonic, i.e., the order between classes of $\\mathbf { v } _ { i }$ and $\\mathbf { v } _ { i } ^ { \\prime }$ is not the same, implying that after calibration by CaGCN, the accuracy of original GCN cannot be preserved. Instead, here we propose an improved CaGCN. Given the output $\\mathbf { V }$ of the classification GCN, we firstly use a $l$ -layer GCN to learn a unique temperature $t _ { i }$ for each node $i .$ , then get a calibrated logit $\\mathbf { v } _ { i } ^ { \\prime }$ by transforming its original logit $\\mathbf { v } _ { i }$ using $t _ { i }$ in a temperature-scaling way, and finally obtain calibrated confidence $\\hat { p } _ { i }$ as follows: ", + "bbox": [ + 173, + 599, + 826, + 712 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/71c235f59256bf1b1e6eb70bdeb69e6f20e43d4145d48cefd95364caeed78f65.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { t } = \\sigma ^ { + } ( \\mathbf { A } \\sigma ( \\cdot \\cdot \\cdot \\mathbf { A } \\sigma ( \\mathbf { A } \\mathbf { V } \\mathbf { W } ^ { ( 1 ) } ) \\mathbf { W } ^ { ( 2 ) } \\cdot \\cdot \\cdot ) \\mathbf { W } ^ { ( l ) } ) = [ t _ { 1 } , \\cdots , t _ { N } ] ^ { \\top } ( t _ { i } > 0 , \\forall i \\in \\{ 1 , \\cdots , N \\} ) , } \\\\ { \\mathbf { v } _ { i } ^ { r } = h ( \\mathbf { v } _ { i } , t _ { i } ) = [ \\nu _ { i , 1 } / t _ { i } , \\cdots , \\nu _ { i , K } / t _ { i } ] ^ { \\top } , \\mathbf { z } _ { i } = [ \\sigma _ { S M } ( \\nu _ { i , 1 } ^ { \\prime } ) , \\cdots , \\sigma _ { S M } ( \\nu _ { i , K } ^ { \\prime } ) ] ^ { \\top } , \\hat { p } _ { i } = \\operatorname* { m a x } _ { { \\boldsymbol { k } } } z _ { i , { \\boldsymbol { k } } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 217, + 712, + 779, + 758 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $t _ { i } \\in \\mathbb { R }$ is a scalar greater than zero and $\\sigma ^ { + } ( \\mathbf { x } ) = l o g ( 1 + e x p ( \\mathbf { x } ) )$ is an element-wise softplus activation [8]. The model proposed in Eq. 5 does not change the order between classes of $\\mathbf { v } _ { i }$ and $\\mathbf { v } _ { i } ^ { \\prime }$ , implying that the accuracy of original GCN is preserved. Compared Eq. 5 with Eq. 3, we can find that Eq. 5 makes the same transformation on all the dimensions of $\\mathbf { v } _ { i }$ , which will limit the learnable calibration function space. However, we will prove that actually Eq. 5 is the same with the model proposed in Eq. 3 on confidence calibration using the Proposition 2. Considering that for any logit $\\mathbf { v } _ { i }$ , our expectation is in fact that the calibration model can output any confidence $\\hat { p } _ { i } \\in \\big ( \\frac { 1 } { K } , 1 \\big )$ . Please note that $\\hat { p } _ { i } \\geq \\frac { 1 } { K }$ , or the prediction will be changed. Since Eq. 3 has no limitation on the learnt calibration model, its output $\\hat { p } _ { i }$ can take any value from $\\frac { 1 } { K }$ to 1. Therefore, if we can prove the output $\\hat { p } _ { i }$ in Eq. 5 can also traverse the interval $\\textstyle { \\big ( } { \\frac { 1 } { K } } , 1 { \\big ) }$ for any $\\mathbf { v } _ { i }$ , the equality between Eq. 3 and Eq.5 can be proved. ", + "bbox": [ + 173, + 761, + 826, + 912 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 2. Given the original logit $\\mathbf { v } _ { i } = [ \\nu _ { i , 1 } , \\cdots , \\nu _ { i , K } ] ^ { \\mathsf { T } }$ of node $i ,$ assume $\\nu _ { i , j }$ not approaching infinity for each $j \\in \\{ 1 , \\cdots , K \\}$ . The calibrated confidence $\\hat { p } _ { i }$ in Eq. $5$ can traverse the interval $\\textstyle { \\big ( } { \\frac { 1 } { K } } , 1 { \\big ) }$ for node i. ", + "bbox": [ + 173, + 89, + 825, + 136 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof We set $\\nu _ { i , 1 } > \\nu _ { i , 2 } > \\cdots > \\nu _ { i , K }$ without loss of generality. For any $\\mathbf { v } _ { i } \\in \\mathbb { R } ^ { K }$ , with the assumption of $\\mathbf { v } _ { i }$ not approaching infinity, we have that ", + "bbox": [ + 173, + 146, + 825, + 175 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d5c2beaad167f287a47ccc07ef36581b6a02eb27ebf24234a9ff41f9048e68ee.jpg", + "text": "$$\n\\operatorname* { l i m } _ { t \\to 0 } \\hat { p } _ { i } = \\operatorname* { l i m } _ { t \\to 0 } \\frac { e x p ( \\nu _ { i , 1 } / t _ { i } ) } { \\sum _ { j = 1 } ^ { K } e x p ( \\nu _ { i , j } / t _ { i } ) } = \\operatorname* { l i m } _ { t \\to 0 } \\frac { e x p ( ( \\nu _ { i , 1 } - \\nu _ { i , 2 } ) / t _ { i } ) } { e x p ( ( \\nu _ { i , 1 } - \\nu _ { i , 2 } ) / t _ { i } ) + \\sum _ { j = 2 } ^ { K } e x p ( ( \\nu _ { i , j } - \\nu _ { i , 2 } ) / t _ { i } ) } = 1\n$$", + "text_format": "latex", + "bbox": [ + 210, + 180, + 784, + 218 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 223, + 200, + 237 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/e8d5e62f66615d63c092dc9fb6aeec2668a046d8bc1c690bb926ed37c0535aac.jpg", + "text": "$$\n\\operatorname* { l i m } _ { t \\to + \\infty } \\hat { p } _ { i } = \\operatorname* { l i m } _ { t \\to + \\infty } \\frac { e x p ( \\nu _ { i , 1 } / t _ { i } ) } { \\sum _ { j = 1 } ^ { K } e x p ( \\nu _ { i , j } / t _ { i } ) } = \\frac { 1 } { K } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 232, + 629, + 270 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Obviously, both $\\sigma _ { S M } ( \\nu _ { i , k } )$ and $\\mathbf { v } _ { i } / t _ { i }$ are continuous, thus $\\sigma _ { S M } ( \\nu _ { i , k } / t _ { i } )$ is continuous. Therefore, $\\begin{array} { r } { \\hat { p } _ { i } = \\operatorname* { m a x } _ { k } z _ { i , k } = \\operatorname* { m a x } _ { k } \\sigma _ { S M } ( \\nu _ { i , k } / t _ { i } ) } \\end{array}$ can traverse the interval $( 1 / K , 1 )$ . ", + "bbox": [ + 173, + 272, + 826, + 301 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The assumption about $\\mathbf { v } _ { i }$ is easy to be satisfied since the L2-norm in GCN drives the weight matrix $\\mathbf { W }$ approaching zero matrix and each element in node feature matrix $\\mathbf { X }$ is not infinity. Therefore, based on Eq. 2, each element $\\nu _ { i , j }$ in $\\mathbf { V }$ cannot approach infinity. From Proposition 2 we know that for any $\\mathbf { v } _ { i }$ , there exactly exists such a unique temperature $t _ { i }$ that $\\hat { p } _ { i }$ can take any value from $1 / K$ to 1. In other words, the model can be perfectly calibrated. ", + "bbox": [ + 173, + 306, + 825, + 377 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 Optimization Objective ", + "text_level": 1, + "bbox": [ + 174, + 392, + 375, + 407 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Since NLL loss [10] can be decomposed into calibration loss and refinement loss [21], minimizing NLL loss benefits for confidence calibration. Therefore, we employ the NLL loss as our objective function with an additional regularization term. We use the prediction probability $\\mathbf { z } _ { i } \\in \\mathbb { R } ^ { K }$ in Eq. 5 to calculate the NLL loss. Denote the $K$ -class one-hot label for node $i$ as $\\mathbf y _ { i } = [ y _ { i , 1 } , \\cdots , y _ { i , K } ] ^ { \\intercal }$ and suppose the size of the validation set is $| D _ { \\nu a l } |$ . Then the NLL loss over all validation nodes is represented as ${ \\mathcal { L } } _ { n l l }$ where: ", + "bbox": [ + 173, + 416, + 826, + 501 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9965ba04f1a996b7a0d62019dff9ada34b67fafd3b5d951effcc6ad8c8c302cb.jpg", + "text": "$$\n\\mathcal { L } _ { n l l } = - \\sum _ { i = 1 } ^ { | D _ { \\nu a l } | } \\sum _ { k = 1 } ^ { K } y _ { i , k } l o g ( z _ { i , k } ) .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 500, + 598, + 541 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Due to the under-confidence of GCNs, our goal is to increase the confidence of correct predictions while decreasing that of incorrect predictions. Considering that for incorrect predictions, the NLL loss cannot directly reduce their confidence, therefore, we design a regularization term for NLL loss as follows: ", + "bbox": [ + 173, + 541, + 825, + 597 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6fb1a920474a4cb8b2389d8275f3a6f10ae164ec47befbc51ad413443af7f9de.jpg", + "text": "$$\n\\mathcal { L } _ { c a l } = \\frac { 1 } { n } ( \\sum _ { i = 1 } ^ { | c o r | } 1 - z _ { i , m } ^ { ( c o r ) } + z _ { i , s } ^ { ( c o r ) } + \\sum _ { i = 1 } ^ { | i n c | } z _ { i , m } ^ { ( i n c ) } - z _ { i , s } ^ { ( i n c ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 331, + 594, + 665, + 635 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\left| c o r \\right|$ and $| i n c |$ are the number of nodes correctly and incorrectly predicted and $z _ { i , m }$ and $z _ { i , s }$ are the max and submax prediction probability. Intuitively, the confidence of incorrect predictions is decreased by reducing the gap between the max and the submax value of $\\mathbf { z } _ { i }$ and vice versa. Combining ${ \\mathcal { L } } _ { n l l }$ and $\\mathcal { L } _ { c a l }$ , we have the following overall objective function: ", + "bbox": [ + 173, + 637, + 825, + 693 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5ef0b1a0b7867edb3bf1e51775a054a44299896e7de7a50143462ad4c3c41e14.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } = \\mathcal { L } _ { n l l } + \\lambda \\mathcal { L } _ { c a l } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 437, + 699, + 558, + 715 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\lambda$ is the parameter of the regularization term. With the guide of labeled data, we can optimize CaGCN via back propagation and learn the calibrated confidence. The overall framework of CaGCN is shown in Fig. 4. ", + "bbox": [ + 173, + 722, + 825, + 765 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Self-training with Confidence Calibration ", + "text_level": 1, + "bbox": [ + 174, + 782, + 552, + 800 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here we propose a practical application of confidence calibration to improve the performance of self-training in GCNs. Self-training is to predict the labels for unlabeled data, and then add them to the training set, so as to achieve better performance. When applying self-training to GCN, we firstly obtain the predictions $\\hat { y } _ { i }$ and the confidence $\\hat { p } _ { i }$ given by GCN and then add the most confident nodes to the training set with pseudo labels $\\hat { y } _ { i }$ based on $\\hat { p } _ { i }$ . We continue to train until convergence. However, existing self-training methods perform not as expected with higher label rates [30]. Considering the under-confidence of existing GCNs, motivated by [26], we argue that the under-performance of existing self-training methods originals from large numbers of high-accuracy predictions distributing in low-confidence intervals as shown in Fig. 2, causing that they cannot be added to the training set. ", + "bbox": [ + 173, + 813, + 826, + 912 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/14380d5b900b4d0eee8d6b970db34d8d09c36f136a97bfe1e293eb3e62c583da.jpg", + "image_caption": [ + "Figure 4: The overall framework of CaGCN. Solid lines represent that we can backpropagate gradient here while dashed lines represent we cannot. We firstly train a classification GCN using the training set to obtain the logit $\\mathbf { V }$ of all the nodes. Then we feed $\\mathbf { V }$ to $\\mathrm { C a G C N }$ to get the temperature t and transform $\\mathbf { V }$ using t into $\\mathbf { V } ^ { \\prime }$ . Finally, the loss can be obtained using $\\mathbf { V } ^ { \\prime }$ after softmax according to Eq. 10 and CaGCN can be optimized with the guide of the validation set. " + ], + "image_footnote": [], + "bbox": [ + 236, + 87, + 758, + 222 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f26ac82ceb817ff790784a35e9667852ceebac2a6b7820a560a1980b16e8bf77.jpg", + "table_caption": [ + "Table 2: ECE $\\mathbf { \\Gamma } ( \\mathbf { M } { = } 2 0 )$ ) on different models and citation networks of various label rate (L/C) with and without calibration. Uncal. represents the uncalibrated model, (-) denotes this method cannot converge to a meaningful result and bold denotes the best result, the subscript of each result refers to the standard deviation $( \\times 1 0 ^ { - 3 } )$ while the superscript refers to the results of paired t-test ( \\* for 0.05 level and $^ { * * }$ for 0.01 level). " + ], + "table_footnote": [], + "table_body": "
DatasetL/CGCNGAT
Uncal.TSMSCaGCNUncal.TSMSCaGCN
Cora200.13476.30.04885.50.04145.70.04016.70.15588.90.07179.80.05449.40.0450
400.11344.70.04177.20.03724.60.04075.40.13405.40.04857.70.04916.00.0365
600.09374.90.0355.40.03646.10.03764.40.12013.30.03936.10.04115.30.031332
Citeseer200.12487.10.06418.70.06443.70.05957.20.15345.00.09168.70.06339.80.05726.8
400.09577.70.06014.20.05385.70.05455.50.12528.70.07973.10.05905.40.05325.4
600.08066.40.05595.00.05216.40.05463.40.10905.90.06487.10.05199.10.05257.6
Pubmed200.05867.70.05413.80.04764.20.0405.00.08353.10.06564.60.05013.70.0356
400.04445.50.04466.30.04366.30.0402.00.08694.60.06586.50.05396.00.0308
600.04459.70.03676.00.03186.40.03114.80.09934.10.06696.30.04835.70.03082
CoraFull200.19866.10.10136.1=0.07760.21193.60.11015.1=0.07880
400.23215.40.11176.50.07010.24384.20.11338.30.07388
600.23374.00.09813.80.07680.24971.80.11335.20.0849
", + "bbox": [ + 174, + 392, + 825, + 628 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 656, + 823, + 684 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Consequently, we design a self-training model CaGCN-st where confidence is firstly calibrated then employed to generate pseudo labels for unlabeled nodes. Specifically, given an unlabeled dataset $D _ { U }$ and a labeled dataset $D _ { L }$ which has been divided into three parts $D _ { t r a i n }$ , $D _ { \\nu a l }$ and $D _ { t e s t }$ , we firstly train a classification GCN using $D _ { t r a i n }$ to get the logit of each node. Then all the logits will be fed into a CaGCN to train and we get a calibrated confidence for each node. It should be noted that instead of $D _ { \\nu a l }$ , we still employ $D _ { t r a i n }$ to train our CaGCN. After that, the most confident predictions of $D _ { U }$ will be adopted as the pseudo labels according to a threshold $\\tau$ and added to the label set. The $D _ { t r a i n }$ is enlarged in this way. The process above will be repeated $s$ stages until convergence. Please note that our classification GCN and CaGCN are re-initialized in each stage. ", + "bbox": [ + 173, + 690, + 825, + 815 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 837, + 312, + 853 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we evaluate the performance of CaGCN on confidence calibration and CaGCN-st on self-training respectively. We choose the commonly used citation networks Cora [29], Citeseer [29], Pubmed [29] and CoraFull [3] for evaluation, and more detailed descriptions are in Appendix B. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d8c33cf4a408cc06fa6f1aaf5cf9926679c8bec6d321b1f93eb2d0e57aba2a8d.jpg", + "table_caption": [ + "Table 3: Node classification accuracy and the standard deviation on GCN and its self-training variants. " + ], + "table_footnote": [], + "table_body": "
DatasetL/CMethods
Orig.St.Ct.UnionInter.TS-stCaGCN-st
Cora2081.630.2482.270.3381.510.3081.850.6881.410.2882.680.2083.110.52
4083.990.2683.590.3483.660.2583.330.4183.380.3384.440.3584.370.38
6084.440.2984.980.3284.630.3185.030.3084.880.1885.600.2485.790.27
Citeseer2071.640.3273.240.4474.220.2974.600.3872.250.4574.200.2474.90040
4072.250.3274.700.3372.120.3974.790.3673.660.3275.620.1975.480.50
6073.200.3575.080.2973.210.3675.530.3075.230.2375.870.2476.430.20
Pubmed2079.570.3380.320.1879.670.3281.120.2979.590.2980.950.1881.16.36
4080.650.3982.200.3281.620.4081.840.2380.460.5582.280.3983.0821
6083.380.3483.350.2883.400.3683.320.3583.310.1783.260.3984.47023
CoraFull2060.450.4360.870.2860.120.4560.520.3561.010.5361.730.4162.190.49
4065.770.3765.830.4564.220.3564.330.4265.840.3766.110.6066.300.31
6066.520.2566.620.3066.640.2966.780.2966.820.3266.950.4567.600.40
", + "bbox": [ + 173, + 109, + 826, + 332 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.1 Confidence Calibration Evaluation ", + "text_level": 1, + "bbox": [ + 176, + 358, + 455, + 373 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Baselines. Since our CaGCN is a general calibration model for GNNs, here we choose GCN [16] and GAT [33] as our classification models. For comparison, we choose the classic post-hoc calibration methods temperature scaling (TS) [12] and matrix scaling with off-diagonal regularization (MS) [18] as our baselines. ", + "bbox": [ + 174, + 383, + 825, + 440 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experimental settings. For the base model GCN and GAT, i.e., the uncalibrated model, we follow parameters suggested by [16] and [33] and further carefully tune them to get optimal performance. For the post-hoc calibration technique, we follow the official implementation [12, 18]. For our CaGCN, we train a two-layer GCN with the hidden layer dimension to be 16. We set $\\lambda = 0 . 5$ for all datasets, weight decay to be 5e-3 for Cora, Citeseer, Pubmed and 0.03 for CoraFull. Other parameters of CaGCN follows [16]. We evaluate the performance of confidence calibration by ECE [22], NLL [10] and Brier Score (BS) [4], which we expect are smaller, and we set the bin number $M = 2 0$ for ECE (more details can be seen in Appendix A). For all methods, we randomly run 10 times and report the average results. More detailed experimental settings can be seen in Appendix B. ", + "bbox": [ + 174, + 445, + 825, + 571 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results. Table 2 reports calibration results evaluated by ECE (more results on NLL and Brier Score are in Appendix C.1). We have the following observations: (1) Compared with uncalibrated models and other baselines, CaGCN is statistically significantly better at the $^ { * } 0 . 0 5$ level and $\\ast \\ast _ { } 0 . 0 1$ level. (2) The ECE values on uncalibrated models are generally the highest, implying that GCN and GAT are poorly calibrated. (3) MS behaves badly on datasets with many classes, e.g., CoraFull. This is because the number of parameters for matrix scaling scales quadratically with the number of classes while the size of the validation set keeps unchanged. Therefore, it will over-fit to the small validation set when dataset has a great number of classes. However, CaGCN does not have this problem. ", + "bbox": [ + 173, + 577, + 825, + 689 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Additional analysis. In Section 2 we visualize the under-confidence problem of existing GNNs using reliability diagrams. Here we utilize the same visualization method to make a comparison before and after confidence calibration. As shown in Fig. 8, Fig. 9, Fig. 10 and Fig. 11 in the appendix, we can find that the confidence is well-calibrated after calibration. ", + "bbox": [ + 174, + 694, + 825, + 751 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 Classification Evaluation of Self-Training ", + "text_level": 1, + "bbox": [ + 176, + 767, + 500, + 782 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Baselines. Since self-training can be applied to any models, here we choose GCN and GAT as our base models, i.e., the original models (Orig.) without self-training, and we choose self-training (St.), co-training (Ct.), Union, Intersection (Inter.) methods proposed in [20] for comparison, which are commonly used as the baselines in self-training. Furthermore, we employ TS as the confidence calibration function in CaGCN-st as another baseline and we denote it by TS-st. ", + "bbox": [ + 174, + 794, + 825, + 863 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experimental settings. We set the learning rate $l r = 0 . 0 0 1$ for CaGCN-st and train our CaGCN-st 200 epochs for Cora, 150 epochs for Citeseer, 100 epochs for Pubmed and 500 epochs for CoraFull. We set the threshold $\\tau \\in \\{ 0 . 8 , 0 . 8 5 , 0 . 9 , 0 . 9 5 , 0 . 9 9 \\}$ and the maximum number of stage $s = 1 0$ . As for baselines, all the parameters follow [20] and we further carefully tune them to get optimal performance. For all methods, we randomly run 10 times and report the average results. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 823, + 119 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Results. Table 3 summarizes the node classification accuracy on GCN and its self-training variants. More results on GAT can be seen in Appendix C.2. We have the following observation: (1) CaGCN-st consistently outperforms all the baselines on all the datasets and label rates at the $^ { * } 0 . 0 5$ level. (2) Compared with the base model, self-training methods generally achieve better results, which proves their effectiveness. (3) Self-training methods with confidence calibration (i.e., TS-st and CaGCN-st) have better performance, which implies that confidence calibration scales more correct predictions to the high confidence range while keeps incorrect predictions basically unchanged, which we believe is beneficial for self-training. ", + "bbox": [ + 174, + 126, + 825, + 237 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Ablation study. CaGCN-st generates pseudo labels based on the calibrated confidence. Here we study the effectiveness of the confidence calibration function CaGCN in CaGCNst. We propose a variant GCN-st of CaGCNst, where CaGCN is removed from CaGCN-st while other parts are kept unchanged. All the experimental settings of GCN-st are the same as CaGCN-st. We report the results in Table 4, and we can observe that CaGCN-st consistently outperforms GCN-st on all the datasets, implying that self-training with calibrated confidence can generate more correct pseudo labels. ", + "bbox": [ + 174, + 243, + 470, + 435 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Additional analysis. We also investigate the changing trends of accuracy with respect to the threshold $\\tau$ in CaGCN-st in Appendix C.2 and study why GCNs are poorly calibrated in Appendix D. ", + "bbox": [ + 174, + 443, + 472, + 484 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/20eff6104768a78e84f3da5ccc2ec04bc167b22aec0869a8b005ddcba1290c32.jpg", + "table_caption": [ + "Table 4: Abaltion study on self-training " + ], + "table_footnote": [], + "table_body": "
Dataset L/CGCNGAT
GCN-st CaGCN-stGCN-st CaGCN-st
Cora2082.2883.1184.08 84.08
4084.1084.37 85.5085.63
6085.16 85.7985.5786.26
Citeseer2074.1374.9073.73 74.34
4075.2875.4875.07 75.62
6075.85 76.4375.1376.08
Pubmed2081.0181.1680.34 81.17
4082.9083.0882.75 83.47
6083.44 84.4783.4683.95
CoraFull2061.3262.19 62.0965.46
4065.9666.3065.92 66.86
6066.43 67.6066.5467.45
", + "bbox": [ + 488, + 281, + 821, + 460 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 484, + 560, + 498 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 517, + 321, + 535 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Graph Neural Networks. Modern GCNs mimics CNNs to learn the local and global structural patterns of graphs through designed convolution and readout functions. [5] generalizes CNNs to graph signal based on the spectrum of graph Laplacian. ChebNet [7] uses Chebyshev polynomials to approximate the $K$ -order localized graph filters and GCN [16] further employs the 1-order simplification of the Chebyshev filter. GAT [33] utilizes attention mechanisms to adaptively learn aggregation weights. GraphSAGE [13] uses various ways of pooling for aggregation. [20] introduces self-training to GCNs and [30] proposes a multi-stage self-supervised (M3S) self-training algorithm of GCNs. Both [20] and [30] focus on the few-shot learning and neither has ever explored self-training with higher label rates in GCNs. More works on GNNs can be found in surveys [36, 43], however, to the best of our knowledge, current GNNs have not considered the confidence calibration. ", + "bbox": [ + 174, + 549, + 825, + 688 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Confidence Calibration. Confidence calibration has been studied for a long time in CV and NLP [12, 23, 25, 14, 19, 37, 40, 42]. [12] discovers modern neural networks are poorly calibrated and study factors influencing calibration. Platt scaling [24] is a simple post-hoc calibration method for binary models, which transforms the logit using scalar parameters. Temperature scaling is the simplest multi-class extension of Platt scaling and matrix and vector scaling are another two extensions of platt scaling. [40] proposes Mix-n-Match calibration strategies which mix parameter methods with non-parameter methods. [25] explores the non-linear space for post-hoc calibration function using a neural network. Moreover, [32] points out GNNs can be miscalibrated in the supervised scenario and mainly focus on miscalibration originated from the imbalanced class distribution. However, none of them have considered the confidence calibration in GNNs on the common semi-supervised scenario. ", + "bbox": [ + 174, + 693, + 825, + 832 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 852, + 297, + 868 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Current efforts on advancing GNNs mostly focus on classification accuracy. However, when deploying GNNs to real-world applications, especially safety-critical fields, whether the results of GNNs are trustworthy is another important factor that cannot be neglected. In this paper, we study the confidence calibration problem in GNNs and discover existing GNNs are under-confident on their predictions. To solve this problem, we propose a novel trustworthy GNN model CaGCN which respects the homophily property of confidence in GNNs and preserves the classification accuracy. Moreover, we propose a novel self-training method CaGCN-st where confidence is first calibrated by CaGCN and then used to generate pseudo labels. Extensive experiments demonstrate the effectiveness of our proposed model in terms of both calibration and accuracy. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 188 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "An interesting direction for future work is to extend CaGCN to other graph tasks, but more studies need to be conducted. We take the link prediction as an example, where we can regard the link prediction as a binary classification problem and the output as the confidence. Considering that the ground-truth confidence distribution for nodes should have the homophily property as is shown in Section 3.1, edges are likely to have the same property as well. As a result, we can employ CaGCN to propagate the confidence between edges by regarding the edges as the nodes. However, more exploration still needs to be conducted for the homophily property of edges. ", + "bbox": [ + 174, + 194, + 825, + 291 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Broader impact. Current efforts on advancing GNNs mostly focus on classification accuracy. However, when deploying GNNs to real-world applications, especially safety-critical fields, whether the results of GNNs are trustworthy is another important factor. The demands for a trustworthy model are universal and extensive such as in the field of disease prediction [31], traffic states prediction [6] and object detection [11] for autonomous driving, where estimating the true probability of getting a correct prediction is necessary. We take the disease prediction [31] as an example, where GNNs are utilized to encode the information of different symptoms, users and diseases. In this scenario, accurately and comprehensively predicting diseases at an early stage will help patients receive prevention treatments in a timely manner. Otherwise, the misdiagnosis and missed diagnosis will endanger the health of patients. Therefore, a trustworthy model is urgently needed. Our CaGCN can make a trustworthy prediction based on its confidence, and as a result, decrease the risk of misdiagnosis and missed diagnosis. We hope our work can provide insights for future improvements in tackling the trustworthiness problem in other saftey-critical fields. ", + "bbox": [ + 174, + 297, + 825, + 477 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Limitations. One potential issue of this work is that it provides a limited explanation to the underconfidence problem. We advocate peer researchers to look into this, making GNNs more reliable in different domains. Other than that, since this work is mostly on the discovery of the confidence calibration problem in GNNs and the theoretical aspect of improving calibration, we do not foresee any direct negative impacts on the society. ", + "bbox": [ + 174, + 484, + 825, + 554 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 573, + 553, + 590 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work is supported in part by the National Natural Science Foundation of China (No. 62172052, No. U20B2045, U1936104, 61772082, 61702296, 62002029). ", + "bbox": [ + 174, + 604, + 825, + 633 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 654, + 266, + 669 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané. Concrete problems in ai safety. arXiv preprint arXiv:1606.06565, 2016. \n[2] Deyu Bo, Xiao Wang, Chuan Shi, and Huawei Shen. 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Extensive experiments", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 448, + 470, + 461 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 470, + 461 + ], + "score": 1.0, + "content": "demonstrate the effectiveness of our proposed model in terms of both calibration", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 459, + 199, + 472 + ], + "spans": [ + { + "bbox": [ + 141, + 459, + 199, + 472 + ], + "score": 1.0, + "content": "and accuracy.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 284, + 471, + 472 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 490, + 191, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 192, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 192, + 506 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "Graphs are ubiquitous in the real world, including social networks, e-commerce networks, traffic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "networks, and so on. Recently, Graph Neural Networks (GNNs), which are able to effectively learn", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "the node representations based on the message-passing manner, have attracted considerable attention", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "in dealing with graph data [16, 33, 39, 44, 15, 2, 34]. To date, GNNs have been applied to various", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "applications and achieved remarkable accuracy, e.g., node classification [16, 33], link prediction [41]", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 220, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 220, + 583 + ], + "score": 1.0, + "content": "and graph classification [9].", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 515, + 506, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 599 + ], + "score": 1.0, + "content": "However, it is well established that a model with good accuracy is not the only goal, but a trustworthy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "model is highly desired in many applications, especially in safety-critical fields [1]. Usually, a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "score": 1.0, + "content": "trustworthy model implies that it should know when it is likely to be incorrect, in other words, the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 619, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 631 + ], + "score": 1.0, + "content": "probability, i.e., the confidence, associated with the predicted class label should reflect its ground", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "truth correctness likelihood [12]. For example, in the scene of autonomous driving, the system will", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "adopt the prediction given by the model only when the model has high confidence for its prediction.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "Otherwise, the decision-making power will be returned to the driver or the system adopts other safer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 661, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 677 + ], + "score": 1.0, + "content": "strategies. Recently, the confidence calibration has attracted considerable attention in deep learning", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 674, + 504, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 504, + 685 + ], + "score": 1.0, + "content": "[12, 40, 19], which reveals that many modern neural network models are over-confident on the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 685, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 696 + ], + "score": 1.0, + "content": "predictions, i.e., the prediction accuracy is lower than its confidence. However, it has not been studied", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "in GNNs on the semi-supervised scenario, which gives rise to one fundamental question: will the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "current GNNs follow the same over-confident property as other neural networks? A well-informed", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "score": 1.0, + "content": "answer can help us better understand GNNs and enable GNNs to be applied to various areas in a", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 197, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 197, + 117 + ], + "score": 1.0, + "content": "more reliable manner.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 585, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "in GNNs on the semi-supervised scenario, which gives rise to one fundamental question: will the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "current GNNs follow the same over-confident property as other neural networks? A well-informed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 507, + 107 + ], + "score": 1.0, + "content": "answer can help us better understand GNNs and enable GNNs to be applied to various areas in a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 197, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 197, + 117 + ], + "score": 1.0, + "content": "more reliable manner.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "score": 1.0, + "content": "As the first contribution of this study, we present experiments assessing the relationship between the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 144 + ], + "score": 1.0, + "content": "confidence and the accuracy of Graph Convolutional Networks (GCNs) [16] and Graph Attention Net-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "works (GAT) [33] in the node classification task (more details can be seen in Section 2), respectively.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "Surprisingly, we discover that existing GNNs are far distant from being well-calibrated, and more", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "importantly, GNNs tend to be under-confident in their predictions, which is very different from other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "modern deep learning models that are often over-confident [12, 19]. GNNs being under-confident", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "means that many predictions are distributed in the low-confidence range, and therefore, fewer predic-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "tions are available for safety-critical applications. Once the weakness is identified, another natural", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "question is: how can we calibrate the confidence on predictions given by GNNs so as to make them", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 182, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 182, + 232 + ], + "score": 1.0, + "content": "more trustworthy?", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "Essentially, the confidence calibration is to calibrate the outputs (also known logits) of original", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "models (e.g., GNNs), therefore, a straightforward manner is to employ temperature scaling (TS) [12],", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "OP-families [25] to learn calibration function using a held-out dataset in a post-hoc way. However,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "when being applied to graphs, they all ignore the effect of topology, which will inevitably make", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 442, + 293 + ], + "score": 1.0, + "content": "mistakes during calibration. 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Apparently, the predictions of GCNs", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 120, + 314 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 304, + 127, + 312 + ], + "score": 0.55, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 301, + 251, + 314 + ], + "score": 1.0, + "content": "should be more confident than", + "type": "text" + }, + { + "bbox": [ + 252, + 302, + 258, + 312 + ], + "score": 0.73, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 301, + 505, + 314 + ], + "score": 1.0, + "content": ", while the traditional calibration methods, e.g., TS, will learn", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 443, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 204, + 326 + ], + "score": 1.0, + "content": "the same confidence for", + "type": "text" + }, + { + "bbox": [ + 204, + 314, + 210, + 323 + ], + "score": 0.5, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 311, + 228, + 326 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 228, + 313, + 234, + 322 + ], + "score": 0.73, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 311, + 443, + 326 + ], + "score": 1.0, + "content": ", because it does not consider the effect of topology.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 341 + ], + "score": 1.0, + "content": "Moreover, most of them explore calibration functions only in the linear space [12, 18] while it is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "well known that non-linear space contains more complex function transformation which is able", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "to calibrate networks with complicated landscapes well. Even if some works have explored the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "score": 1.0, + "content": "non-linear space such as Matrix Scaling [12], they generally degrade the classification accuracy of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 483, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 483, + 386 + ], + "score": 1.0, + "content": "the original classifier, while a good accuracy is still a basic requirement by many applications.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "In this paper, we introduce a topology-aware post-hoc calibration method for GNNs. Specifically, for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "the logits given by the original classification GNNs, we employ another calibration GCN (CaGCN) to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "propagate confidence, naturally enabling that the confidence of topologically adjacent nodes becomes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 284, + 436 + ], + "score": 1.0, + "content": "similar. CaGCN learns a unique temperature", + "type": "text" + }, + { + "bbox": [ + 284, + 424, + 289, + 432 + ], + "score": 0.53, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 421, + 505, + 436 + ], + "score": 1.0, + "content": "for each node for temperature scaling, thus preserving", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "the accuracy of the original classification GCN. In addition, based on our finding that large numbers", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "of high-accuracy predictions are distributed in the low-confidence range, we design a calibrated", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "self-training model CaGCN-st in which the confidence is firstly calibrated then used to generate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 465, + 437, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 437, + 478 + ], + "score": 1.0, + "content": "pseudo labels with high confidence. The contributions of this paper are three-fold:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 105, + 484, + 506, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 507, + 497 + ], + "score": 1.0, + "content": "• We study the trustworthy problem of GNNs, and discover one unique characteristic of GNNs, i.e.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 496, + 354, + 507 + ], + "spans": [ + { + "bbox": [ + 114, + 496, + 354, + 507 + ], + "score": 1.0, + "content": "the predictions made by GNNs are usually under-confident.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "• We propose a novel trustworthy GNN model based on the confidence calibration. Our proposed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 113, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "calibration function has three features: topology-aware, non-linear, and accuracy-preserving. We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 113, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 113, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "further design a calibrated self-training GNN model, which can effectively utilize the predictions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 113, + 541, + 203, + 555 + ], + "spans": [ + { + "bbox": [ + 113, + 541, + 203, + 555 + ], + "score": 1.0, + "content": "with high confidence.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "• Extensive experiments demonstrate the effectiveness of our proposed models in terms of both", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 114, + 567, + 217, + 580 + ], + "spans": [ + { + "bbox": [ + 114, + 567, + 217, + 580 + ], + "score": 1.0, + "content": "calibration and accuracy.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 106, + 592, + 290, + 606 + ], + "lines": [ + { + "bbox": [ + 104, + 590, + 291, + 610 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 291, + 610 + ], + "score": 1.0, + "content": "2 Notation and Preliminary Study", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "In this paper, we focus on the calibration of semi-supervised node classification in an undirected", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 625, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 174, + 640 + ], + "score": 1.0, + "content": "attributed graph", + "type": "text" + }, + { + "bbox": [ + 174, + 628, + 219, + 640 + ], + "score": 0.93, + "content": "G = \\left( V , E \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 625, + 323, + 640 + ], + "score": 1.0, + "content": "with the adjacent matrix", + "type": "text" + }, + { + "bbox": [ + 323, + 627, + 367, + 638 + ], + "score": 0.93, + "content": "\\mathbf { A } \\in \\mathbb { R } ^ { N \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 625, + 485, + 640 + ], + "score": 1.0, + "content": "and the node feature matrix", + "type": "text" + }, + { + "bbox": [ + 485, + 628, + 505, + 639 + ], + "score": 0.84, + "content": "\\mathbf { X } =", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 159, + 651 + ], + "score": 0.9, + "content": "[ \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { N } ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 639, + 164, + 651 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 164, + 640, + 173, + 650 + ], + "score": 0.52, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 639, + 261, + 651 + ], + "score": 1.0, + "content": "is a set of nodes and", + "type": "text" + }, + { + "bbox": [ + 262, + 640, + 307, + 651 + ], + "score": 0.91, + "content": "E \\subseteq V \\times V", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 639, + 444, + 651 + ], + "score": 1.0, + "content": "is a set of edges between nodes.", + "type": "text" + }, + { + "bbox": [ + 445, + 640, + 479, + 652 + ], + "score": 0.92, + "content": "N = | V |", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 651, + 460, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 460, + 662 + ], + "score": 1.0, + "content": "number of nodes. Here we give the definition of perfect calibration of GNNs as follows:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5 + }, + { + "type": "text", + "bbox": [ + 107, + 663, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 277, + 676 + ], + "score": 1.0, + "content": "Definition 1. Given random variables A,", + "type": "text" + }, + { + "bbox": [ + 278, + 664, + 286, + 673 + ], + "score": 0.37, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 663, + 290, + 676 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 290, + 663, + 354, + 676 + ], + "score": 0.88, + "content": "\\mathbf { Y } \\subseteq \\{ 1 , \\ldots , K \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 663, + 432, + 676 + ], + "score": 1.0, + "content": "and a GNN model", + "type": "text" + }, + { + "bbox": [ + 432, + 664, + 443, + 675 + ], + "score": 0.77, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 663, + 471, + 676 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 472, + 664, + 479, + 674 + ], + "score": 0.48, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 663, + 505, + 676 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 675, + 507, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 284, + 689 + ], + "score": 1.0, + "content": "learnable parameters, for node i with label", + "type": "text" + }, + { + "bbox": [ + 284, + 676, + 310, + 687 + ], + "score": 0.87, + "content": "y _ { i } \\in \\mathbf { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 675, + 315, + 689 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 315, + 676, + 440, + 688 + ], + "score": 0.88, + "content": "\\mathbf { z } _ { i } = f _ { \\theta } ( \\mathbf { x } _ { i } , \\mathbf { A } ) = [ z _ { i , 1 } , \\ldots , z _ { i , K } ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 675, + 507, + 689 + ], + "score": 1.0, + "content": "is the output of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 285, + 699 + ], + "score": 1.0, + "content": "GNNs (i.e., the prediction probability), and", + "type": "text" + }, + { + "bbox": [ + 285, + 688, + 353, + 699 + ], + "score": 0.87, + "content": "\\hat { y } _ { i } = \\mathrm { a r g } \\operatorname* { m a x } _ { k } z _ { i , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 686, + 372, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 373, + 688, + 428, + 699 + ], + "score": 0.89, + "content": "\\hat { p } _ { i } = \\operatorname* { m a x } _ { k } z _ { i , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "are the prediction", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 698, + 428, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 298, + 710 + ], + "score": 1.0, + "content": "and the confidence respectively. Then we define", + "type": "text" + }, + { + "bbox": [ + 298, + 699, + 309, + 709 + ], + "score": 0.84, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 698, + 428, + 710 + ], + "score": 1.0, + "content": "to be perfectly calibrated as:", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49.5 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 710, + 374, + 723 + ], + "lines": [ + { + "bbox": [ + 237, + 710, + 374, + 723 + ], + "spans": [ + { + "bbox": [ + 237, + 710, + 374, + 723 + ], + "score": 0.89, + "content": "\\mathbb { P } ( \\hat { y } _ { i } = y _ { i } | \\hat { p } _ { i } = p ) = p , \\forall p \\in [ 0 , 1 ] .", + "type": "interline_equation", + "image_path": "a1d6d2847912d0278c04a6a6a38d4db990a5b4af3214b389f6597704627a4ce6.jpg" + } + ] + } + ], + "index": 52, + "virtual_lines": [ + { + "bbox": [ + 237, + 710, + 374, + 723 + ], + "spans": [], + "index": 52 + } + ] + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 73, + 507, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "score": 1.0, + "content": "As the first contribution of this study, we present experiments assessing the relationship between the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 144 + ], + "score": 1.0, + "content": "confidence and the accuracy of Graph Convolutional Networks (GCNs) [16] and Graph Attention Net-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "works (GAT) [33] in the node classification task (more details can be seen in Section 2), respectively.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "Surprisingly, we discover that existing GNNs are far distant from being well-calibrated, and more", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "importantly, GNNs tend to be under-confident in their predictions, which is very different from other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "modern deep learning models that are often over-confident [12, 19]. GNNs being under-confident", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "means that many predictions are distributed in the low-confidence range, and therefore, fewer predic-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "tions are available for safety-critical applications. Once the weakness is identified, another natural", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "question is: how can we calibrate the confidence on predictions given by GNNs so as to make them", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 182, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 182, + 232 + ], + "score": 1.0, + "content": "more trustworthy?", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 122, + 506, + 232 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "Essentially, the confidence calibration is to calibrate the outputs (also known logits) of original", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "models (e.g., GNNs), therefore, a straightforward manner is to employ temperature scaling (TS) [12],", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "OP-families [25] to learn calibration function using a held-out dataset in a post-hoc way. 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Even if some works have explored the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "score": 1.0, + "content": "non-linear space such as Matrix Scaling [12], they generally degrade the classification accuracy of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 483, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 483, + 386 + ], + "score": 1.0, + "content": "the original classifier, while a good accuracy is still a basic requirement by many applications.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 329, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "In this paper, we introduce a topology-aware post-hoc calibration method for GNNs. Specifically, for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "the logits given by the original classification GNNs, we employ another calibration GCN (CaGCN) to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "propagate confidence, naturally enabling that the confidence of topologically adjacent nodes becomes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 284, + 436 + ], + "score": 1.0, + "content": "similar. 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In addition, based on our finding that large numbers", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "of high-accuracy predictions are distributed in the low-confidence range, we design a calibrated", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "self-training model CaGCN-st in which the confidence is firstly calibrated then used to generate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 465, + 437, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 437, + 478 + ], + "score": 1.0, + "content": "pseudo labels with high confidence. The contributions of this paper are three-fold:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 389, + 506, + 478 + ] + }, + { + "type": "list", + "bbox": [ + 105, + 484, + 506, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 507, + 497 + ], + "score": 1.0, + "content": "• We study the trustworthy problem of GNNs, and discover one unique characteristic of GNNs, i.e.,", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 496, + 354, + 507 + ], + "spans": [ + { + "bbox": [ + 114, + 496, + 354, + 507 + ], + "score": 1.0, + "content": "the predictions made by GNNs are usually under-confident.", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "• We propose a novel trustworthy GNN model based on the confidence calibration. Our proposed", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 113, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 113, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "calibration function has three features: topology-aware, non-linear, and accuracy-preserving. We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 113, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 113, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "further design a calibrated self-training GNN model, which can effectively utilize the predictions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 113, + 541, + 203, + 555 + ], + "spans": [ + { + "bbox": [ + 113, + 541, + 203, + 555 + ], + "score": 1.0, + "content": "with high confidence.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 107, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "• Extensive experiments demonstrate the effectiveness of our proposed models in terms of both", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 567, + 217, + 580 + ], + "spans": [ + { + "bbox": [ + 114, + 567, + 217, + 580 + ], + "score": 1.0, + "content": "calibration and accuracy.", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 483, + 507, + 580 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 592, + 290, + 606 + ], + "lines": [ + { + "bbox": [ + 104, + 590, + 291, + 610 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 291, + 610 + ], + "score": 1.0, + "content": "2 Notation and Preliminary Study", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "In this paper, we focus on the calibration of semi-supervised node classification in an undirected", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 625, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 174, + 640 + ], + "score": 1.0, + "content": "attributed graph", + "type": "text" + }, + { + "bbox": [ + 174, + 628, + 219, + 640 + ], + "score": 0.93, + "content": "G = \\left( V , E \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 625, + 323, + 640 + ], + "score": 1.0, + "content": "with the adjacent matrix", + "type": "text" + }, + { + "bbox": [ + 323, + 627, + 367, + 638 + ], + "score": 0.93, + "content": "\\mathbf { A } \\in \\mathbb { R } ^ { N \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 625, + 485, + 640 + ], + "score": 1.0, + "content": "and the node feature matrix", + "type": "text" + }, + { + "bbox": [ + 485, + 628, + 505, + 639 + ], + "score": 0.84, + "content": "\\mathbf { X } =", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 159, + 651 + ], + "score": 0.9, + "content": "[ \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { N } ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 639, + 164, + 651 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 164, + 640, + 173, + 650 + ], + "score": 0.52, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 639, + 261, + 651 + ], + "score": 1.0, + "content": "is a set of nodes and", + "type": "text" + }, + { + "bbox": [ + 262, + 640, + 307, + 651 + ], + "score": 0.91, + "content": "E \\subseteq V \\times V", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 639, + 444, + 651 + ], + "score": 1.0, + "content": "is a set of edges between nodes.", + "type": "text" + }, + { + "bbox": [ + 445, + 640, + 479, + 652 + ], + "score": 0.92, + "content": "N = | V |", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 651, + 460, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 460, + 662 + ], + "score": 1.0, + "content": "number of nodes. Here we give the definition of perfect calibration of GNNs as follows:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 617, + 505, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 663, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 277, + 676 + ], + "score": 1.0, + "content": "Definition 1. 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Any", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 252, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 268 + ], + "score": 1.0, + "content": "deviation from a perfectly diagonal (i.e., the difference between blue and red histogram) represents", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 263, + 183, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 183, + 277 + ], + "score": 1.0, + "content": "the miscalibration.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 109, + 287, + 501, + 442 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 287, + 501, + 442 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 287, + 501, + 442 + ], + "spans": [ + { + "bbox": [ + 109, + 287, + 501, + 442 + ], + "score": 0.955, + "type": "image", + "image_path": "c742029dfe4c579e527d277d926e41da06ed21b6ba6d31bd18e006ec2ce69a5d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 109, + 287, + 501, + 338.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 109, + 338.6666666666667, + 501, + 390.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 109, + 390.33333333333337, + 501, + 442.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 200, + 452, + 410, + 463 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 199, + 451, + 411, + 465 + ], + "spans": [ + { + "bbox": [ + 199, + 451, + 411, + 465 + ], + "score": 1.0, + "content": "Figure 2: Confidence distribution before calibration.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 504, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 430, + 516 + ], + "score": 1.0, + "content": "According to Definition 1, GNN is perfectly calibrated only when the confidence", + "type": "text" + }, + { + "bbox": [ + 430, + 504, + 440, + 515 + ], + "score": 0.87, + "content": "\\hat { p } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "is exactly equal", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 514, + 378, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 378, + 526 + ], + "score": 1.0, + "content": "to the true probability of getting a correct prediction for every node.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "Next, we take two representative GNNs (GCN [16] and GAT [33]) as examples to analyze whether", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "they are perfectly calibrated. Specifically, we apply GCN and GAT to four widely used datasets Cora", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 553, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 504, + 565 + ], + "score": 1.0, + "content": "[29], Citeseer [29], Pubmed [29], CoraFull [3], and examine whether their results satisfy Definition", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "1. To provide more results, we select three label rates for training set (i.e., 20, 40, 60 labeled nodes", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 453, + 587 + ], + "score": 1.0, + "content": "per class). All the experimental settings follow [16, 33]. Since the true probability", + "type": "text" + }, + { + "bbox": [ + 453, + 576, + 461, + 586 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "cannot be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "exactly known, we take an approximate way to evaluate the calibration as in [12]. In particular, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "first partition the [0,1] range of confidence into 20 equal bins and then we group the nodes into", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "corresponding bins according to their confidence. After that we calculate the average accuracy of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "each bin. We expect the average accuracy is equal to the average confidence of each bin, which means", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "the model is approximately perfectly calibrated. For example, if the average confidence of nodes in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 640, + 462, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 438, + 652 + ], + "score": 1.0, + "content": "the bin [0.95, 1.0] is 0.96, and then the classification accuracy in this bin should be", + "type": "text" + }, + { + "bbox": [ + 438, + 640, + 458, + 650 + ], + "score": 0.86, + "content": "96 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 640, + 462, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "We illustrate the results of label rate being 20 in Fig. 1 using Reliability Diagrams [23] here, where", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 121, + 680 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 669, + 128, + 677 + ], + "score": 0.3, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "-axis is the confidence in 20 bins of equal size and y-axis is the average accuracy in each bin.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "The blue represents the classification accuracy of GCN and GAT while the red is our expectation.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "More results of label rate being 40, 60 and other GNN models can be seen in Fig. 8, Fig. 9, Fig. 12,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "Fig. 13 and Fig. 14 in the appendix. We can see that in all the datasets, the average accuracy of most", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "bins is higher than the average confidence. In other words, these GNNs actually achieve remarkable", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 303, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 71, + 502, + 222 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 71, + 502, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 71, + 502, + 222 + ], + "spans": [ + { + "bbox": [ + 109, + 71, + 502, + 222 + ], + "score": 0.882, + "type": "image", + "image_path": "b09bafe873b2b4009d9aab352efaf8bdb093b30da45eedaa1e5de35a56b8adff.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 71, + 502, + 121.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 121.33333333333334, + 502, + 171.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 171.66666666666669, + 502, + 222.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 232, + 506, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "Figure 1: Reliability diagrams for GCN (top) and GAT (bottom) without confidence calibration.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 242, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 257 + ], + "score": 1.0, + "content": "The diagram is expected to plot an identity function of accuracy with respect to confidence. Any", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 252, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 268 + ], + "score": 1.0, + "content": "deviation from a perfectly diagonal (i.e., the difference between blue and red histogram) represents", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 263, + 183, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 183, + 277 + ], + "score": 1.0, + "content": "the miscalibration.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 109, + 287, + 501, + 442 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 287, + 501, + 442 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 287, + 501, + 442 + ], + "spans": [ + { + "bbox": [ + 109, + 287, + 501, + 442 + ], + "score": 0.955, + "type": "image", + "image_path": "c742029dfe4c579e527d277d926e41da06ed21b6ba6d31bd18e006ec2ce69a5d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 109, + 287, + 501, + 338.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 109, + 338.6666666666667, + 501, + 390.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 109, + 390.33333333333337, + 501, + 442.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 200, + 452, + 410, + 463 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 199, + 451, + 411, + 465 + ], + "spans": [ + { + "bbox": [ + 199, + 451, + 411, + 465 + ], + "score": 1.0, + "content": "Figure 2: Confidence distribution before calibration.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 504, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 430, + 516 + ], + "score": 1.0, + "content": "According to Definition 1, GNN is perfectly calibrated only when the confidence", + "type": "text" + }, + { + "bbox": [ + 430, + 504, + 440, + 515 + ], + "score": 0.87, + "content": "\\hat { p } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "is exactly equal", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 514, + 378, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 378, + 526 + ], + "score": 1.0, + "content": "to the true probability of getting a correct prediction for every node.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 502, + 505, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "Next, we take two representative GNNs (GCN [16] and GAT [33]) as examples to analyze whether", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "they are perfectly calibrated. Specifically, we apply GCN and GAT to four widely used datasets Cora", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 553, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 504, + 565 + ], + "score": 1.0, + "content": "[29], Citeseer [29], Pubmed [29], CoraFull [3], and examine whether their results satisfy Definition", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "1. To provide more results, we select three label rates for training set (i.e., 20, 40, 60 labeled nodes", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 453, + 587 + ], + "score": 1.0, + "content": "per class). All the experimental settings follow [16, 33]. Since the true probability", + "type": "text" + }, + { + "bbox": [ + 453, + 576, + 461, + 586 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "cannot be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "exactly known, we take an approximate way to evaluate the calibration as in [12]. In particular, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "first partition the [0,1] range of confidence into 20 equal bins and then we group the nodes into", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "corresponding bins according to their confidence. After that we calculate the average accuracy of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "each bin. We expect the average accuracy is equal to the average confidence of each bin, which means", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "the model is approximately perfectly calibrated. For example, if the average confidence of nodes in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 640, + 462, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 438, + 652 + ], + "score": 1.0, + "content": "the bin [0.95, 1.0] is 0.96, and then the classification accuracy in this bin should be", + "type": "text" + }, + { + "bbox": [ + 438, + 640, + 458, + 650 + ], + "score": 0.86, + "content": "96 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 640, + 462, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 530, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "We illustrate the results of label rate being 20 in Fig. 1 using Reliability Diagrams [23] here, where", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 121, + 680 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 669, + 128, + 677 + ], + "score": 0.3, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "-axis is the confidence in 20 bins of equal size and y-axis is the average accuracy in each bin.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "The blue represents the classification accuracy of GCN and GAT while the red is our expectation.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "More results of label rate being 40, 60 and other GNN models can be seen in Fig. 8, Fig. 9, Fig. 12,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "Fig. 13 and Fig. 14 in the appendix. We can see that in all the datasets, the average accuracy of most", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "bins is higher than the average confidence. In other words, these GNNs actually achieve remarkable", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "performance, but they all output low confidence, i.e., the GNNs are usually under-confident. Please", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "note that this phenomenon of GNNs is very different from other modern neural networks, which are", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "generally known to be over-confident [12, 19]. Moreover, as shown in Fig. 2, we also visualize the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 303, + 118 + ], + "score": 1.0, + "content": "confidence distribution of test nodes, where the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 303, + 107, + 310, + 116 + ], + "score": 0.37, + "content": "\\mathbf { X }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 310, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "-axis is the confidence and y-axis is the density", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "[28]. The histogram height multiplied by the width is equal to the frequency. The blue represents the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "score": 1.0, + "content": "confidence distribution of correct predictions while the yellow represents that of incorrect predictions.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "More results of label rate being 40 and 60 can be seen in Fig. 10 and Fig. 11 in the appendix. We", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "score": 1.0, + "content": "can see that a large quantity of correct predictions are distributed in the low confidence range. The", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "results above indicate that the current GNNs are far from perfect calibration, leading to unreliable", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 155, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 155, + 182 + ], + "score": 1.0, + "content": "confidence.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 655, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "performance, but they all output low confidence, i.e., the GNNs are usually under-confident. Please", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "note that this phenomenon of GNNs is very different from other modern neural networks, which are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "generally known to be over-confident [12, 19]. Moreover, as shown in Fig. 2, we also visualize the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 303, + 118 + ], + "score": 1.0, + "content": "confidence distribution of test nodes, where the", + "type": "text" + }, + { + "bbox": [ + 303, + 107, + 310, + 116 + ], + "score": 0.37, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "-axis is the confidence and y-axis is the density", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "[28]. The histogram height multiplied by the width is equal to the frequency. The blue represents the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "score": 1.0, + "content": "confidence distribution of correct predictions while the yellow represents that of incorrect predictions.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "More results of label rate being 40 and 60 can be seen in Fig. 10 and Fig. 11 in the appendix. We", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "score": 1.0, + "content": "can see that a large quantity of correct predictions are distributed in the low confidence range. The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "results above indicate that the current GNNs are far from perfect calibration, leading to unreliable", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 155, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 155, + 182 + ], + "score": 1.0, + "content": "confidence.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 197, + 295, + 210 + ], + "lines": [ + { + "bbox": [ + 104, + 196, + 297, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 196, + 297, + 212 + ], + "score": 1.0, + "content": "3 Confidence Calibration on GCNs", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 504, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 393, + 234 + ], + "score": 1.0, + "content": "In this section, we propose our method to calibrate current GNNs. 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DatasetGCN
Uncal.TSOurs
Cora240.267172.346164.651
Citeseer128.145112.212108.684
Pubmed1299.331266.681113.41
CoraFull6014.324698.204500.30
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Uncal. is short for uncalibrated and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 306, + 388, + 451, + 403 + ], + "spans": [ + { + "bbox": [ + 306, + 388, + 451, + 403 + ], + "score": 1.0, + "content": "TS is short for temperature scaling.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "table_body", + "bbox": [ + 308, + 401, + 503, + 475 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 308, + 401, + 503, + 475 + ], + "spans": [ + { + "bbox": [ + 308, + 401, + 503, + 475 + ], + "score": 0.98, + "html": "
DatasetGCN
Uncal.TSOurs
Cora240.267172.346164.651
Citeseer128.145112.212108.684
Pubmed1299.331266.681113.41
CoraFull6014.324698.204500.30
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Then the total variation of confidence will", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "surely become lower and the original classification GCN will be calibrated. Please note that although", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 72, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 84 + ], + "score": 1.0, + "content": "temperature scaling can be directly applied here, compared with GCN, it does not take the network", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "topology into account, which may cause mistakes mentioned in Section 1. Moreover, temperature", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "scaling only employs a linear transformation, and GCN is able to learn a non-linear calibration", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 145, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 145, + 117 + ], + "score": 1.0, + "content": "function.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 54.5, + "bbox_fs": [ + 105, + 689, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 84 + ], + "score": 1.0, + "content": "temperature scaling can be directly applied here, compared with GCN, it does not take the network", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "topology into account, which may cause mistakes mentioned in Section 1. Moreover, temperature", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "scaling only employs a linear transformation, and GCN is able to learn a non-linear calibration", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 145, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 145, + 117 + ], + "score": 1.0, + "content": "function.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "For a comprehensive understanding of confidence propagation, we make a detailed and visible", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 358, + 145 + ], + "score": 1.0, + "content": "illustration here. As shown in Fig. 3, the logits of two nodes", + "type": "text" + }, + { + "bbox": [ + 359, + 135, + 366, + 143 + ], + "score": 0.44, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 132, + 384, + 145 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 133, + 391, + 142 + ], + "score": 0.71, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 132, + 487, + 145 + ], + "score": 1.0, + "content": "are the same, but node", + "type": "text" + }, + { + "bbox": [ + 488, + 135, + 495, + 142 + ], + "score": 0.62, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 243, + 156 + ], + "score": 1.0, + "content": "similar to its neighbors while node", + "type": "text" + }, + { + "bbox": [ + 243, + 144, + 249, + 154 + ], + "score": 0.81, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 144, + 435, + 156 + ], + "score": 1.0, + "content": "is not. Apparently, the predictions of GCNs for", + "type": "text" + }, + { + "bbox": [ + 435, + 146, + 442, + 154 + ], + "score": 0.5, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "should be more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 164, + 167 + ], + "score": 1.0, + "content": "confident than", + "type": "text" + }, + { + "bbox": [ + 164, + 155, + 171, + 164 + ], + "score": 0.65, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 154, + 227, + 167 + ], + "score": 1.0, + "content": ". Suppose that", + "type": "text" + }, + { + "bbox": [ + 227, + 156, + 234, + 164 + ], + "score": 0.38, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 154, + 237, + 167 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 237, + 155, + 244, + 164 + ], + "score": 0.57, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "and their neighbours are under-confident based on the observation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 497, + 178 + ], + "score": 1.0, + "content": "above. If we continue to propagate their logits along the topology using another GCN, the logits of", + "type": "text" + }, + { + "bbox": [ + 498, + 167, + 504, + 175 + ], + "score": 0.59, + "content": "a", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "and its neighbors will tend to be the same. Therefore, if one or more of these nodes are calibrated", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "during the calibration process, all of them will be calibrated as well. The confidence is propagated in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 315, + 210 + ], + "score": 1.0, + "content": "this way. On the other hand, looking at another node", + "type": "text" + }, + { + "bbox": [ + 315, + 199, + 321, + 208 + ], + "score": 0.76, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 198, + 505, + 210 + ], + "score": 1.0, + "content": ", it is as difficult even for manual classification", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 307, + 221 + ], + "score": 1.0, + "content": "as it is for GCNs. Consequently, the confidence of", + "type": "text" + }, + { + "bbox": [ + 308, + 210, + 314, + 219 + ], + "score": 0.79, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "should stay still even be lower. However, it will", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 289, + 232 + ], + "score": 1.0, + "content": "become higher because of the influence from", + "type": "text" + }, + { + "bbox": [ + 289, + 222, + 296, + 230 + ], + "score": 0.6, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "if we use the traditional calibration method without", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "considering the network topology. Instead, when the network topology is taken into account, the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 138, + 254 + ], + "score": 1.0, + "content": "logit of", + "type": "text" + }, + { + "bbox": [ + 138, + 242, + 144, + 252 + ], + "score": 0.74, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 241, + 406, + 254 + ], + "score": 1.0, + "content": "will be averaged by its neighboured and each dimension tends to", + "type": "text" + }, + { + "bbox": [ + 406, + 241, + 425, + 253 + ], + "score": 0.85, + "content": "1 / K", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 241, + 505, + 254 + ], + "score": 1.0, + "content": ". It will be correctly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 385, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 385, + 264 + ], + "score": 1.0, + "content": "calibrated when other nodes in the same situation are well-calibrated.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 276, + 280, + 288 + ], + "lines": [ + { + "bbox": [ + 104, + 274, + 280, + 292 + ], + "spans": [ + { + "bbox": [ + 104, + 274, + 280, + 292 + ], + "score": 1.0, + "content": "3.2 The Accuracy-Preserving Property", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 296, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "Until now, we have proposed a non-linear calibration model CaGCN which can take the network", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "topology into account, but the accuracy-preserving property cannot be satisfied. To address this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 318, + 421, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 421, + 331 + ], + "score": 1.0, + "content": "problem, we firstly study the general accuracy-preserving calibration function.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 504, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 189, + 344 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 190, + 330, + 247, + 342 + ], + "score": 0.91, + "content": "h : \\mathbb { R } ^ { K } \\to \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 329, + 360, + 344 + ], + "score": 1.0, + "content": "be a calibration function,", + "type": "text" + }, + { + "bbox": [ + 361, + 331, + 406, + 342 + ], + "score": 0.89, + "content": "s : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 329, + 506, + 344 + ], + "score": 1.0, + "content": "be a 1-D function and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 186, + 354 + ], + "score": 0.9, + "content": "\\mathbf { v } _ { i } = [ \\nu _ { i , 1 } , \\cdots , \\nu _ { i , K } ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 341, + 383, + 355 + ], + "score": 1.0, + "content": "be the logit of node i. The calibration function", + "type": "text" + }, + { + "bbox": [ + 384, + 344, + 390, + 353 + ], + "score": 0.32, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "preserves the classification", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 424, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 379, + 366 + ], + "score": 1.0, + "content": "accuracy of the original model if s is a strictly isotonic function and", + "type": "text" + }, + { + "bbox": [ + 379, + 354, + 386, + 363 + ], + "score": 0.51, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 353, + 424, + 366 + ], + "score": 1.0, + "content": "satisfies:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 366, + 398, + 381 + ], + "lines": [ + { + "bbox": [ + 212, + 366, + 398, + 381 + ], + "spans": [ + { + "bbox": [ + 212, + 366, + 398, + 381 + ], + "score": 0.92, + "content": "\\begin{array} { r } { h ( \\mathbf { v } _ { i } ) = [ s ( \\nu _ { i , 1 } ) , \\dots , s ( \\nu _ { i , K } ) ] ^ { \\mathsf { T } } , \\forall i \\in \\{ 1 , \\cdots , N \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "6368d8bf5e7d2b53e267659c9e6aa42c14a2d0325551be99767bff3fca4f24cc.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 212, + 366, + 398, + 381 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 162, + 402 + ], + "score": 1.0, + "content": "Proof We set", + "type": "text" + }, + { + "bbox": [ + 163, + 390, + 250, + 401 + ], + "score": 0.89, + "content": "\\nu _ { i , 1 } < \\nu _ { i , 2 } < \\cdots < \\nu _ { i , K }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 388, + 380, + 402 + ], + "score": 1.0, + "content": "without loss of generality. Since", + "type": "text" + }, + { + "bbox": [ + 380, + 388, + 462, + 401 + ], + "score": 0.93, + "content": "[ s ( \\nu _ { i , 1 } ) , \\ldots , s ( \\nu _ { i , K } ) ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "shares the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 400, + 504, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 173, + 412 + ], + "score": 1.0, + "content": "same order with", + "type": "text" + }, + { + "bbox": [ + 173, + 402, + 182, + 411 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 400, + 336, + 412 + ], + "score": 1.0, + "content": "as a result of the strictly isotonicity of", + "type": "text" + }, + { + "bbox": [ + 336, + 402, + 342, + 410 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 400, + 495, + 412 + ], + "score": 1.0, + "content": ", the order between classes of the logit", + "type": "text" + }, + { + "bbox": [ + 495, + 402, + 504, + 411 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 411, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 363, + 423 + ], + "score": 1.0, + "content": "is unchanged, hence the accuracy of the prediction is preserved.", + "type": "text" + }, + { + "bbox": [ + 495, + 412, + 504, + 420 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "Temperature scaling [12] is the simplest accuracy-preserving calibration method using a scalar", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 149, + 450 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 150, + 440, + 155, + 448 + ], + "score": 0.6, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 438, + 363, + 450 + ], + "score": 1.0, + "content": "called temperature for all classes. Given the logit", + "type": "text" + }, + { + "bbox": [ + 363, + 439, + 372, + 449 + ], + "score": 0.81, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 438, + 407, + 450 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 408, + 439, + 412, + 448 + ], + "score": 0.6, + "content": "i .", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 438, + 505, + 450 + ], + "score": 1.0, + "content": ", the confidence of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 158, + 462 + ], + "score": 1.0, + "content": "prediction is", + "type": "text" + }, + { + "bbox": [ + 158, + 449, + 275, + 461 + ], + "score": 0.89, + "content": "\\hat { p } _ { i } = \\operatorname* { m a x } _ { k } \\sigma _ { S M } ( \\nu _ { i , k } / t ) ( t > 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 448, + 372, + 462 + ], + "score": 1.0, + "content": ". In temperature scaling,", + "type": "text" + }, + { + "bbox": [ + 372, + 448, + 480, + 461 + ], + "score": 0.91, + "content": "h ( { \\bf v } _ { i } ) = [ \\nu _ { i , 1 } / t , \\cdot \\cdot \\cdot , \\nu _ { i , K } / t ] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 374, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 204, + 472 + ], + "score": 1.0, + "content": "calibration function and", + "type": "text" + }, + { + "bbox": [ + 204, + 461, + 246, + 472 + ], + "score": 0.91, + "content": "s ( x ) = x / t", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 460, + 374, + 472 + ], + "score": 1.0, + "content": "is the strictly isotonic function.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "However, we can find that temperature scaling (TS) [40] only performs the same linear transformation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 230, + 500 + ], + "score": 1.0, + "content": "for all the nodes using the same", + "type": "text" + }, + { + "bbox": [ + 230, + 489, + 235, + 497 + ], + "score": 0.45, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 487, + 506, + 500 + ], + "score": 1.0, + "content": ". As mentioned in Eq. 3, we propose to use CaGCN as our calibration", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 428, + 511 + ], + "score": 1.0, + "content": "function, while CaGCN is generally not isotonic, i.e., the order between classes of", + "type": "text" + }, + { + "bbox": [ + 428, + 499, + 437, + 509 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 498, + 454, + 511 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 498, + 466, + 509 + ], + "score": 0.89, + "content": "\\mathbf { v } _ { i } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "is not the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "same, implying that after calibration by CaGCN, the accuracy of original GCN cannot be preserved.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 372, + 533 + ], + "score": 1.0, + "content": "Instead, here we propose an improved CaGCN. Given the output", + "type": "text" + }, + { + "bbox": [ + 372, + 520, + 381, + 530 + ], + "score": 0.62, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "of the classification GCN, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 530, + 504, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 530, + 153, + 544 + ], + "score": 1.0, + "content": "firstly use a", + "type": "text" + }, + { + "bbox": [ + 154, + 532, + 158, + 541 + ], + "score": 0.72, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 530, + 320, + 544 + ], + "score": 1.0, + "content": "-layer GCN to learn a unique temperature", + "type": "text" + }, + { + "bbox": [ + 321, + 533, + 327, + 542 + ], + "score": 0.84, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 530, + 383, + 544 + ], + "score": 1.0, + "content": "for each node", + "type": "text" + }, 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\\sigma ( \\mathbf { A } \\mathbf { V } \\mathbf { W } ^ { ( 1 ) } ) \\mathbf { W } ^ { ( 2 ) } \\cdot \\cdot \\cdot ) \\mathbf { W } ^ { ( l ) } ) = [ t _ { 1 } , \\cdots , t _ { N } ] ^ { \\top } ( t _ { i } > 0 , \\forall i \\in \\{ 1 , \\cdots , N \\} ) , } \\\\ { \\mathbf { v } _ { i } ^ { r } = h ( \\mathbf { v } _ { i } , t _ { i } ) = [ \\nu _ { i , 1 } / t _ { i } , \\cdots , \\nu _ { i , K } / t _ { i } ] ^ { \\top } , \\mathbf { z } _ { i } = [ \\sigma _ { S M } ( \\nu _ { i , 1 } ^ { \\prime } ) , \\cdots , \\sigma _ { S M } ( \\nu _ { i , K } ^ { \\prime } ) ] ^ { \\top } , \\hat { p } _ { i } = \\operatorname* { m a x } _ { { \\boldsymbol { k } } } z _ { i , { \\boldsymbol { k } } } , } \\end{array}", + "type": "interline_equation", + "image_path": "71c235f59256bf1b1e6eb70bdeb69e6f20e43d4145d48cefd95364caeed78f65.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 133, + 564, + 477, + 576.3333333333334 + ], + "spans": [], + "index": 40 + }, + { + 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The model proposed in Eq. 5 does not change the order between classes of", + "type": "text" + }, + { + "bbox": [ + 465, + 617, + 473, + 626 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 615, + 491, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 491, + 615, + 503, + 626 + ], + "score": 0.89, + "content": "\\mathbf { v } _ { i } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 615, + 506, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "implying that the accuracy of original GCN is preserved. Compared Eq. 5 with Eq. 3, we can find", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 372, + 649 + ], + "score": 1.0, + "content": "that Eq. 5 makes the same transformation on all the dimensions of", + "type": "text" + }, + { + "bbox": [ + 373, + 638, + 382, + 648 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 637, + 506, + 649 + ], + "score": 1.0, + "content": ", which will limit the learnable", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "calibration function space. However, we will prove that actually Eq. 5 is the same with the model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 658, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 494, + 672 + ], + "score": 1.0, + "content": "proposed in Eq. 3 on confidence calibration using the Proposition 2. Considering that for any logit", + "type": "text" + }, + { + "bbox": [ + 494, + 660, + 503, + 670 + ], + "score": 0.8, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 658, + 507, + 672 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 670, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 410, + 685 + ], + "score": 1.0, + "content": "our expectation is in fact that the calibration model can output any confidence", + "type": "text" + }, + { + "bbox": [ + 410, + 670, + 454, + 684 + ], + "score": 0.93, + "content": "\\hat { p } _ { i } \\in \\big ( \\frac { 1 } { K } , 1 \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 670, + 506, + 685 + ], + "score": 1.0, + "content": ". Please note", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 682, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 123, + 698 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 682, + 153, + 697 + ], + "score": 0.93, + "content": "\\hat { p } _ { i } \\geq \\frac { 1 } { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 684, + 506, + 698 + ], + "score": 1.0, + "content": ", or the prediction will be changed. 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Therefore, if we can prove the output", + "type": "text" + }, + { + "bbox": [ + 461, + 698, + 470, + 709 + ], + "score": 0.87, + "content": "\\hat { p } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 697, + 506, + 711 + ], + "score": 1.0, + "content": "in Eq. 5", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 500, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 223, + 724 + ], + "score": 1.0, + "content": "can also traverse the interval", + "type": "text" + }, + { + "bbox": [ + 223, + 709, + 248, + 724 + ], + "score": 0.92, + "content": "\\textstyle { \\big ( } { \\frac { 1 } { K } } , 1 { \\big ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 710, + 280, + 724 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 280, + 714, + 289, + 722 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 710, + 500, + 724 + ], + "score": 1.0, + "content": ", the equality between Eq. 3 and Eq.5 can be proved.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 47.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 506, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "For a comprehensive understanding of confidence propagation, we make a detailed and visible", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 358, + 145 + ], + "score": 1.0, + "content": "illustration here. As shown in Fig. 3, the logits of two nodes", + "type": "text" + }, + { + "bbox": [ + 359, + 135, + 366, + 143 + ], + "score": 0.44, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 132, + 384, + 145 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 133, + 391, + 142 + ], + "score": 0.71, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 132, + 487, + 145 + ], + "score": 1.0, + "content": "are the same, but node", + "type": "text" + }, + { + "bbox": [ + 488, + 135, + 495, + 142 + ], + "score": 0.62, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 243, + 156 + ], + "score": 1.0, + "content": "similar to its neighbors while node", + "type": "text" + }, + { + "bbox": [ + 243, + 144, + 249, + 154 + ], + "score": 0.81, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 144, + 435, + 156 + ], + "score": 1.0, + "content": "is not. Apparently, the predictions of GCNs for", + "type": "text" + }, + { + "bbox": [ + 435, + 146, + 442, + 154 + ], + "score": 0.5, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "should be more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 164, + 167 + ], + "score": 1.0, + "content": "confident than", + "type": "text" + }, + { + "bbox": [ + 164, + 155, + 171, + 164 + ], + "score": 0.65, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 154, + 227, + 167 + ], + "score": 1.0, + "content": ". Suppose that", + "type": "text" + }, + { + "bbox": [ + 227, + 156, + 234, + 164 + ], + "score": 0.38, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 154, + 237, + 167 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 237, + 155, + 244, + 164 + ], + "score": 0.57, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "and their neighbours are under-confident based on the observation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 497, + 178 + ], + "score": 1.0, + "content": "above. If we continue to propagate their logits along the topology using another GCN, the logits of", + "type": "text" + }, + { + "bbox": [ + 498, + 167, + 504, + 175 + ], + "score": 0.59, + "content": "a", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "and its neighbors will tend to be the same. Therefore, if one or more of these nodes are calibrated", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "during the calibration process, all of them will be calibrated as well. The confidence is propagated in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 315, + 210 + ], + "score": 1.0, + "content": "this way. On the other hand, looking at another node", + "type": "text" + }, + { + "bbox": [ + 315, + 199, + 321, + 208 + ], + "score": 0.76, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 198, + 505, + 210 + ], + "score": 1.0, + "content": ", it is as difficult even for manual classification", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 307, + 221 + ], + "score": 1.0, + "content": "as it is for GCNs. Consequently, the confidence of", + "type": "text" + }, + { + "bbox": [ + 308, + 210, + 314, + 219 + ], + "score": 0.79, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "should stay still even be lower. However, it will", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 289, + 232 + ], + "score": 1.0, + "content": "become higher because of the influence from", + "type": "text" + }, + { + "bbox": [ + 289, + 222, + 296, + 230 + ], + "score": 0.6, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "if we use the traditional calibration method without", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "considering the network topology. Instead, when the network topology is taken into account, the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 138, + 254 + ], + "score": 1.0, + "content": "logit of", + "type": "text" + }, + { + "bbox": [ + 138, + 242, + 144, + 252 + ], + "score": 0.74, + "content": "^ b", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 241, + 406, + 254 + ], + "score": 1.0, + "content": "will be averaged by its neighboured and each dimension tends to", + "type": "text" + }, + { + "bbox": [ + 406, + 241, + 425, + 253 + ], + "score": 0.85, + "content": "1 / K", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 241, + 505, + 254 + ], + "score": 1.0, + "content": ". It will be correctly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 385, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 385, + 264 + ], + "score": 1.0, + "content": "calibrated when other nodes in the same situation are well-calibrated.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 121, + 506, + 264 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 276, + 280, + 288 + ], + "lines": [ + { + "bbox": [ + 104, + 274, + 280, + 292 + ], + "spans": [ + { + "bbox": [ + 104, + 274, + 280, + 292 + ], + "score": 1.0, + "content": "3.2 The Accuracy-Preserving Property", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 296, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "Until now, we have proposed a non-linear calibration model CaGCN which can take the network", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "topology into account, but the accuracy-preserving property cannot be satisfied. To address this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 318, + 421, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 421, + 331 + ], + "score": 1.0, + "content": "problem, we firstly study the general accuracy-preserving calibration function.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 296, + 506, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 504, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 189, + 344 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 190, + 330, + 247, + 342 + ], + "score": 0.91, + "content": "h : \\mathbb { R } ^ { K } \\to \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 329, + 360, + 344 + ], + "score": 1.0, + "content": "be a calibration function,", + "type": "text" + }, + { + "bbox": [ + 361, + 331, + 406, + 342 + ], + "score": 0.89, + "content": "s : \\mathbb { R } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 329, + 506, + 344 + ], + "score": 1.0, + "content": "be a 1-D function and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 186, + 354 + ], + "score": 0.9, + "content": "\\mathbf { v } _ { i } = [ \\nu _ { i , 1 } , \\cdots , \\nu _ { i , K } ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 341, + 383, + 355 + ], + "score": 1.0, + "content": "be the logit of node i. 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Since", + "type": "text" + }, + { + "bbox": [ + 380, + 388, + 462, + 401 + ], + "score": 0.93, + "content": "[ s ( \\nu _ { i , 1 } ) , \\ldots , s ( \\nu _ { i , K } ) ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "shares the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 400, + 504, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 173, + 412 + ], + "score": 1.0, + "content": "same order with", + "type": "text" + }, + { + "bbox": [ + 173, + 402, + 182, + 411 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 400, + 336, + 412 + ], + "score": 1.0, + "content": "as a result of the strictly isotonicity of", + "type": "text" + }, + { + "bbox": [ + 336, + 402, + 342, + 410 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 400, + 495, + 412 + ], + "score": 1.0, + "content": ", the order between classes of the logit", + "type": "text" + }, + { + "bbox": [ + 495, + 402, + 504, + 411 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 411, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 363, + 423 + ], + "score": 1.0, + "content": "is unchanged, hence the accuracy of the prediction is preserved.", + "type": "text" + }, + { + "bbox": [ + 495, + 412, + 504, + 420 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 388, + 506, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "Temperature scaling [12] is the simplest accuracy-preserving calibration method using a scalar", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 149, + 450 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 150, + 440, + 155, + 448 + ], + "score": 0.6, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 438, + 363, + 450 + ], + "score": 1.0, + "content": "called temperature for all classes. 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In temperature scaling,", + "type": "text" + }, + { + "bbox": [ + 372, + 448, + 480, + 461 + ], + "score": 0.91, + "content": "h ( { \\bf v } _ { i } ) = [ \\nu _ { i , 1 } / t , \\cdot \\cdot \\cdot , \\nu _ { i , K } / t ] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 374, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 204, + 472 + ], + "score": 1.0, + "content": "calibration function and", + "type": "text" + }, + { + "bbox": [ + 204, + 461, + 246, + 472 + ], + "score": 0.91, + "content": "s ( x ) = x / t", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 460, + 374, + 472 + ], + "score": 1.0, + "content": "is the strictly isotonic function.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 428, + 505, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "However, we can find that temperature scaling (TS) [40] only performs the same linear transformation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 230, + 500 + ], + "score": 1.0, + "content": "for all the nodes using the same", + "type": "text" + }, + { + "bbox": [ + 230, + 489, + 235, + 497 + ], + "score": 0.45, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 487, + 506, + 500 + ], + "score": 1.0, + "content": ". 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\\sigma ( \\cdot \\cdot \\cdot \\mathbf { A } \\sigma ( \\mathbf { A } \\mathbf { V } \\mathbf { W } ^ { ( 1 ) } ) \\mathbf { W } ^ { ( 2 ) } \\cdot \\cdot \\cdot ) \\mathbf { W } ^ { ( l ) } ) = [ t _ { 1 } , \\cdots , t _ { N } ] ^ { \\top } ( t _ { i } > 0 , \\forall i \\in \\{ 1 , \\cdots , N \\} ) , } \\\\ { \\mathbf { v } _ { i } ^ { r } = h ( \\mathbf { v } _ { i } , t _ { i } ) = [ \\nu _ { i , 1 } / t _ { i } , \\cdots , \\nu _ { i , K } / t _ { i } ] ^ { \\top } , \\mathbf { z } _ { i } = [ \\sigma _ { S M } ( \\nu _ { i , 1 } ^ { \\prime } ) , \\cdots , \\sigma _ { S M } ( \\nu _ { i , K } ^ { \\prime } ) ] ^ { \\top } , \\hat { p } _ { i } = \\operatorname* { m a x } _ { { \\boldsymbol { k } } } z _ { i , { \\boldsymbol { k } } } , } \\end{array}", + "type": "interline_equation", + "image_path": "71c235f59256bf1b1e6eb70bdeb69e6f20e43d4145d48cefd95364caeed78f65.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 133, + 564, + 477, + 576.3333333333334 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The model proposed in Eq. 5 does not change the order between classes of", + "type": "text" + }, + { + "bbox": [ + 465, + 617, + 473, + 626 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 615, + 491, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 491, + 615, + 503, + 626 + ], + "score": 0.89, + "content": "\\mathbf { v } _ { i } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 615, + 506, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "implying that the accuracy of original GCN is preserved. Compared Eq. 5 with Eq. 3, we can find", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 372, + 649 + ], + "score": 1.0, + "content": "that Eq. 5 makes the same transformation on all the dimensions of", + "type": "text" + }, + { + "bbox": [ + 373, + 638, + 382, + 648 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 637, + 506, + 649 + ], + "score": 1.0, + "content": ", which will limit the learnable", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "calibration function space. However, we will prove that actually Eq. 5 is the same with the model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 658, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 494, + 672 + ], + "score": 1.0, + "content": "proposed in Eq. 3 on confidence calibration using the Proposition 2. Considering that for any logit", + "type": "text" + }, + { + "bbox": [ + 494, + 660, + 503, + 670 + ], + "score": 0.8, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 658, + 507, + 672 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 670, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 410, + 685 + ], + "score": 1.0, + "content": "our expectation is in fact that the calibration model can output any confidence", + "type": "text" + }, + { + "bbox": [ + 410, + 670, + 454, + 684 + ], + "score": 0.93, + "content": "\\hat { p } _ { i } \\in \\big ( \\frac { 1 } { K } , 1 \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 670, + 506, + 685 + ], + "score": 1.0, + "content": ". Please note", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 682, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 123, + 698 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 682, + 153, + 697 + ], + "score": 0.93, + "content": "\\hat { p } _ { i } \\geq \\frac { 1 } { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 684, + 506, + 698 + ], + "score": 1.0, + "content": ", or the prediction will be changed. Since Eq. 3 has no limitation on the learnt calibration", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 696, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 174, + 711 + ], + "score": 1.0, + "content": "model, its output", + "type": "text" + }, + { + "bbox": [ + 175, + 698, + 185, + 709 + ], + "score": 0.87, + "content": "\\hat { p } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 697, + 282, + 711 + ], + "score": 1.0, + "content": "can take any value from", + "type": "text" + }, + { + "bbox": [ + 282, + 696, + 290, + 711 + ], + "score": 0.88, + "content": "\\frac { 1 } { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 697, + 460, + 711 + ], + "score": 1.0, + "content": "to 1. Therefore, if we can prove the output", + "type": "text" + }, + { + "bbox": [ + 461, + 698, + 470, + 709 + ], + "score": 0.87, + "content": "\\hat { p } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 697, + 506, + 711 + ], + "score": 1.0, + "content": "in Eq. 5", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 500, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 223, + 724 + ], + "score": 1.0, + "content": "can also traverse the interval", + "type": "text" + }, + { + "bbox": [ + 223, + 709, + 248, + 724 + ], + "score": 0.92, + "content": "\\textstyle { \\big ( } { \\frac { 1 } { K } } , 1 { \\big ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 710, + 280, + 724 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 280, + 714, + 289, + 722 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 710, + 500, + 724 + ], + "score": 1.0, + "content": ", the equality between Eq. 3 and Eq.5 can be proved.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 604, + 507, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 71, + 505, + 108 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 268, + 86 + ], + "score": 1.0, + "content": "Proposition 2. Given the original logit", + "type": "text" + }, + { + "bbox": [ + 268, + 72, + 347, + 85 + ], + "score": 0.93, + "content": "\\mathbf { v } _ { i } = [ \\nu _ { i , 1 } , \\cdots , \\nu _ { i , K } ] ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 71, + 381, + 86 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 381, + 74, + 387, + 83 + ], + "score": 0.66, + "content": "i ,", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 71, + 421, + 86 + ], + "score": 1.0, + "content": "assume", + "type": "text" + }, + { + "bbox": [ + 421, + 75, + 435, + 85 + ], + "score": 0.86, + "content": "\\nu _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 71, + 505, + 86 + ], + "score": 1.0, + "content": "not approaching", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 504, + 100 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 171, + 100 + ], + "score": 1.0, + "content": "infinity for each", + "type": "text" + }, + { + "bbox": [ + 171, + 85, + 229, + 97 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , \\cdots , K \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 84, + 337, + 100 + ], + "score": 1.0, + "content": ". 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Therefore, based", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 200, + 279 + ], + "score": 1.0, + "content": "on Eq. 2, each element", + "type": "text" + }, + { + "bbox": [ + 201, + 267, + 214, + 277 + ], + "score": 0.88, + "content": "\\nu _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 264, + 225, + 279 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 226, + 266, + 235, + 276 + ], + "score": 0.64, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 264, + 506, + 279 + ], + "score": 1.0, + "content": "cannot approach infinity. From Proposition 2 we know that for any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 115, + 287 + ], + "score": 0.78, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 276, + 300, + 289 + ], + "score": 1.0, + "content": ", there exactly exists such a unique temperature", + "type": "text" + }, + { + "bbox": [ + 300, + 278, + 307, + 287 + ], + "score": 0.83, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 276, + 325, + 289 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 326, + 277, + 335, + 288 + ], + "score": 0.88, + "content": "\\hat { p } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 276, + 432, + 289 + ], + "score": 1.0, + "content": "can take any value from", + "type": "text" + }, + { + "bbox": [ + 432, + 276, + 451, + 288 + ], + "score": 0.89, + "content": "1 / K", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "to 1. In other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 287, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 287, + 299 + ], + "score": 1.0, + "content": "words, the model can be perfectly calibrated.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 311, + 230, + 323 + ], + "lines": [ + { + "bbox": [ + 104, + 309, + 231, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 231, + 326 + ], + "score": 1.0, + "content": "3.3 Optimization Objective", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 345 + ], + "score": 1.0, + "content": "Since NLL loss [10] can be decomposed into calibration loss and refinement loss [21], minimizing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "NLL loss benefits for confidence calibration. 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Denote the", + "type": "text" + }, + { + "bbox": [ + 276, + 365, + 285, + 374 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 362, + 405, + 378 + ], + "score": 1.0, + "content": "-class one-hot label for node", + "type": "text" + }, + { + "bbox": [ + 406, + 365, + 410, + 374 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 362, + 423, + 378 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 424, + 365, + 501, + 376 + ], + "score": 0.87, + "content": "\\mathbf y _ { i } = [ y _ { i , 1 } , \\cdots , y _ { i , K } ] ^ { \\intercal }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 285, + 388 + ], + "score": 1.0, + "content": "and suppose the size of the validation set is", + "type": "text" + }, + { + "bbox": [ + 285, + 375, + 309, + 387 + ], + "score": 0.91, + "content": "| D _ { \\nu a l } |", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 375, + 506, + 388 + ], + "score": 1.0, + "content": ". Then the NLL loss over all validation nodes is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 386, + 213, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 165, + 398 + ], + "score": 1.0, + "content": "represented as", + "type": "text" + }, + { + "bbox": [ + 166, + 386, + 183, + 397 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } _ { n l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 387, + 213, + 398 + ], + "score": 1.0, + "content": "where:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 396, + 366, + 429 + ], + "lines": [ + { + "bbox": [ + 245, + 396, + 366, + 429 + ], + "spans": [ + { + "bbox": [ + 245, + 396, + 366, + 429 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { n l l } = - \\sum _ { i = 1 } ^ { | D _ { \\nu a l } | } \\sum _ { k = 1 } ^ { K } y _ { i , k } l o g ( z _ { i , k } ) .", + "type": "interline_equation", + "image_path": "9965ba04f1a996b7a0d62019dff9ada34b67fafd3b5d951effcc6ad8c8c302cb.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 396, + 366, + 412.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 245, + 412.5, + 366, + 429.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Due to the under-confidence of GCNs, our goal is to increase the confidence of correct predictions", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "while decreasing that of incorrect predictions. 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Intuitively, the confidence of incorrect predictions is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 389, + 540 + ], + "score": 1.0, + "content": "decreased by reducing the gap between the max and the submax value of", + "type": "text" + }, + { + "bbox": [ + 389, + 528, + 397, + 538 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "and vice versa. Combining", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 538, + 361, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 538, + 124, + 549 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { n l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 538, + 142, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 538, + 159, + 549 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 538, + 361, + 550 + ], + "score": 1.0, + "content": ", we have the following overall objective function:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 554, + 342, + 567 + ], + "lines": [ + { + "bbox": [ + 268, + 554, + 342, + 567 + ], + "spans": [ + { + "bbox": [ + 268, + 554, + 342, + 567 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } = \\mathcal { L } _ { n l l } + \\lambda \\mathcal { L } _ { c a l } , } \\end{array}", + "type": "interline_equation", + "image_path": "5ef0b1a0b7867edb3bf1e51775a054a44299896e7de7a50143462ad4c3c41e14.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 268, + 554, + 342, + 567 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 132, + 584 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 572, + 140, + 582 + ], + "score": 0.68, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "is the parameter of the regularization term. With the guide of labeled data, we can optimize", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "CaGCN via back propagation and learn the calibrated confidence. The overall framework of CaGCN", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 595, + 183, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 183, + 606 + ], + "score": 1.0, + "content": "is shown in Fig. 4.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 620, + 338, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 339, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 339, + 636 + ], + "score": 1.0, + "content": "4 Self-training with Confidence Calibration", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 644, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "Here we propose a practical application of confidence calibration to improve the performance of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "self-training in GCNs. 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Considering", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "the under-confidence of existing GCNs, motivated by [26], we argue that the under-performance of", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 228, + 504, + 237 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 71, + 505, + 108 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 268, + 86 + ], + "score": 1.0, + "content": "Proposition 2. 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Therefore,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 381, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 244, + 240 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\hat { p } _ { i } = \\operatorname* { m a x } _ { k } z _ { i , k } = \\operatorname* { m a x } _ { k } \\sigma _ { S M } ( \\nu _ { i , k } / t _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 226, + 342, + 239 + ], + "score": 1.0, + "content": "can traverse the interval", + "type": "text" + }, + { + "bbox": [ + 343, + 227, + 376, + 239 + ], + "score": 0.91, + "content": "( 1 / K , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 226, + 381, + 239 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 214, + 506, + 240 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 194, + 255 + ], + "score": 1.0, + "content": "The assumption about", + "type": "text" + }, + { + "bbox": [ + 195, + 245, + 204, + 254 + ], + "score": 0.83, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 243, + 493, + 255 + ], + "score": 1.0, + "content": "is easy to be satisfied since the L2-norm in GCN drives the weight matrix", + "type": "text" + }, + { + "bbox": [ + 493, + 243, + 505, + 254 + ], + "score": 0.31, + "content": "\\mathbf { W }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 368, + 267 + ], + "score": 1.0, + "content": "approaching zero matrix and each element in node feature matrix", + "type": "text" + }, + { + "bbox": [ + 368, + 255, + 377, + 264 + ], + "score": 0.53, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "is not infinity. Therefore, based", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 200, + 279 + ], + "score": 1.0, + "content": "on Eq. 2, each element", + "type": "text" + }, + { + "bbox": [ + 201, + 267, + 214, + 277 + ], + "score": 0.88, + "content": "\\nu _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 264, + 225, + 279 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 226, + 266, + 235, + 276 + ], + "score": 0.64, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 264, + 506, + 279 + ], + "score": 1.0, + "content": "cannot approach infinity. From Proposition 2 we know that for any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 115, + 287 + ], + "score": 0.78, + "content": "\\mathbf { v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 276, + 300, + 289 + ], + "score": 1.0, + "content": ", there exactly exists such a unique temperature", + "type": "text" + }, + { + "bbox": [ + 300, + 278, + 307, + 287 + ], + "score": 0.83, + "content": "t _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 276, + 325, + 289 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 326, + 277, + 335, + 288 + ], + "score": 0.88, + "content": "\\hat { p } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 276, + 432, + 289 + ], + "score": 1.0, + "content": "can take any value from", + "type": "text" + }, + { + "bbox": [ + 432, + 276, + 451, + 288 + ], + "score": 0.89, + "content": "1 / K", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "to 1. In other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 287, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 287, + 299 + ], + "score": 1.0, + "content": "words, the model can be perfectly calibrated.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 243, + 506, + 299 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 311, + 230, + 323 + ], + "lines": [ + { + "bbox": [ + 104, + 309, + 231, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 231, + 326 + ], + "score": 1.0, + "content": "3.3 Optimization Objective", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 345 + ], + "score": 1.0, + "content": "Since NLL loss [10] can be decomposed into calibration loss and refinement loss [21], minimizing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "NLL loss benefits for confidence calibration. Therefore, we employ the NLL loss as our objective", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 351, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 444, + 367 + ], + "score": 1.0, + "content": "function with an additional regularization term. We use the prediction probability", + "type": "text" + }, + { + "bbox": [ + 445, + 353, + 478, + 364 + ], + "score": 0.89, + "content": "\\mathbf { z } _ { i } \\in \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 351, + 507, + 367 + ], + "score": 1.0, + "content": "in Eq.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 362, + 501, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 276, + 378 + ], + "score": 1.0, + "content": "5 to calculate the NLL loss. Denote the", + "type": "text" + }, + { + "bbox": [ + 276, + 365, + 285, + 374 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 362, + 405, + 378 + ], + "score": 1.0, + "content": "-class one-hot label for node", + "type": "text" + }, + { + "bbox": [ + 406, + 365, + 410, + 374 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 362, + 423, + 378 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 424, + 365, + 501, + 376 + ], + "score": 0.87, + "content": "\\mathbf y _ { i } = [ y _ { i , 1 } , \\cdots , y _ { i , K } ] ^ { \\intercal }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 285, + 388 + ], + "score": 1.0, + "content": "and suppose the size of the validation set is", + "type": "text" + }, + { + "bbox": [ + 285, + 375, + 309, + 387 + ], + "score": 0.91, + "content": "| D _ { \\nu a l } |", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 375, + 506, + 388 + ], + "score": 1.0, + "content": ". Then the NLL loss over all validation nodes is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 386, + 213, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 165, + 398 + ], + "score": 1.0, + "content": "represented as", + "type": "text" + }, + { + "bbox": [ + 166, + 386, + 183, + 397 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } _ { n l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 387, + 213, + 398 + ], + "score": 1.0, + "content": "where:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 329, + 507, + 398 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 396, + 366, + 429 + ], + "lines": [ + { + "bbox": [ + 245, + 396, + 366, + 429 + ], + "spans": [ + { + "bbox": [ + 245, + 396, + 366, + 429 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { n l l } = - \\sum _ { i = 1 } ^ { | D _ { \\nu a l } | } \\sum _ { k = 1 } ^ { K } y _ { i , k } l o g ( z _ { i , k } ) .", + "type": "interline_equation", + "image_path": "9965ba04f1a996b7a0d62019dff9ada34b67fafd3b5d951effcc6ad8c8c302cb.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 396, + 366, + 412.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 245, + 412.5, + 366, + 429.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Due to the under-confidence of GCNs, our goal is to increase the confidence of correct predictions", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "while decreasing that of incorrect predictions. Considering that for incorrect predictions, the NLL", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "loss cannot directly reduce their confidence, therefore, we design a regularization term for NLL loss", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 153, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 153, + 475 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 429, + 506, + 475 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 471, + 407, + 503 + ], + "lines": [ + { + "bbox": [ + 203, + 471, + 407, + 503 + ], + "spans": [ + { + "bbox": [ + 203, + 471, + 407, + 503 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { c a l } = \\frac { 1 } { n } ( \\sum _ { i = 1 } ^ { | c o r | } 1 - z _ { i , m } ^ { ( c o r ) } + z _ { i , s } ^ { ( c o r ) } + \\sum _ { i = 1 } ^ { | i n c | } z _ { i , m } ^ { ( i n c ) } - z _ { i , s } ^ { ( i n c ) } ) ,", + "type": "interline_equation", + "image_path": "6fb1a920474a4cb8b2389d8275f3a6f10ae164ec47befbc51ad413443af7f9de.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 471, + 407, + 487.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 203, + 487.0, + 407, + 503.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 133, + 518 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 505, + 153, + 517 + ], + "score": 0.91, + "content": "\\left| c o r \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 504, + 171, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 171, + 505, + 190, + 517 + ], + "score": 0.92, + "content": "| i n c |", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 504, + 443, + 518 + ], + "score": 1.0, + "content": "are the number of nodes correctly and incorrectly predicted and", + "type": "text" + }, + { + "bbox": [ + 443, + 507, + 459, + 517 + ], + "score": 0.87, + "content": "z _ { i , m }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 504, + 476, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 507, + 489, + 517 + ], + "score": 0.86, + "content": "z _ { i , s }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "the max and submax prediction probability. Intuitively, the confidence of incorrect predictions is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 389, + 540 + ], + "score": 1.0, + "content": "decreased by reducing the gap between the max and the submax value of", + "type": "text" + }, + { + "bbox": [ + 389, + 528, + 397, + 538 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "and vice versa. Combining", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 538, + 361, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 538, + 124, + 549 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { n l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 538, + 142, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 538, + 159, + 549 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 538, + 361, + 550 + ], + "score": 1.0, + "content": ", we have the following overall objective function:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 504, + 506, + 550 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 554, + 342, + 567 + ], + "lines": [ + { + "bbox": [ + 268, + 554, + 342, + 567 + ], + "spans": [ + { + "bbox": [ + 268, + 554, + 342, + 567 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } = \\mathcal { L } _ { n l l } + \\lambda \\mathcal { L } _ { c a l } , } \\end{array}", + "type": "interline_equation", + "image_path": "5ef0b1a0b7867edb3bf1e51775a054a44299896e7de7a50143462ad4c3c41e14.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 268, + 554, + 342, + 567 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 132, + 584 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 572, + 140, + 582 + ], + "score": 0.68, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "is the parameter of the regularization term. With the guide of labeled data, we can optimize", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "CaGCN via back propagation and learn the calibrated confidence. The overall framework of CaGCN", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 595, + 183, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 183, + 606 + ], + "score": 1.0, + "content": "is shown in Fig. 4.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 106, + 572, + 505, + 606 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 620, + 338, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 339, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 339, + 636 + ], + "score": 1.0, + "content": "4 Self-training with Confidence Calibration", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 644, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "Here we propose a practical application of confidence calibration to improve the performance of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "self-training in GCNs. 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We continue to train until convergence. However,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "existing self-training methods perform not as expected with higher label rates [30]. Considering", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "the under-confidence of existing GCNs, motivated by [26], we argue that the under-performance of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "existing self-training methods originals from large numbers of high-accuracy predictions distributing", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "in low-confidence intervals as shown in Fig. 2, causing that they cannot be added to the training set.", + "type": "text", + "cross_page": true + } + ], + "index": 17 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 645, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 145, + 69, + 464, + 176 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 69, + 464, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 69, + 464, + 176 + ], + "spans": [ + { + "bbox": [ + 145, + 69, + 464, + 176 + ], + "score": 0.969, + "type": "image", + "image_path": "14380d5b900b4d0eee8d6b970db34d8d09c36f136a97bfe1e293eb3e62c583da.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 145, + 69, + 464, + 104.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 145, + 104.66666666666666, + 464, + 140.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 145, + 140.33333333333331, + 464, + 175.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 184, + 505, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "score": 1.0, + "content": "Figure 4: The overall framework of CaGCN. 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DatasetL/CGCNGAT
Uncal.TSMSCaGCNUncal.TSMSCaGCN
Cora200.13476.30.04885.50.04145.70.04016.70.15588.90.07179.80.05449.40.0450
400.11344.70.04177.20.03724.60.04075.40.13405.40.04857.70.04916.00.0365
600.09374.90.0355.40.03646.10.03764.40.12013.30.03936.10.04115.30.031332
Citeseer200.12487.10.06418.70.06443.70.05957.20.15345.00.09168.70.06339.80.05726.8
400.09577.70.06014.20.05385.70.05455.50.12528.70.07973.10.05905.40.05325.4
600.08066.40.05595.00.05216.40.05463.40.10905.90.06487.10.05199.10.05257.6
Pubmed200.05867.70.05413.80.04764.20.0405.00.08353.10.06564.60.05013.70.0356
400.04445.50.04466.30.04366.30.0402.00.08694.60.06586.50.05396.00.0308
600.04459.70.03676.00.03186.40.03114.80.09934.10.06696.30.04835.70.03082
CoraFull200.19866.10.10136.1=0.07760.21193.60.11015.1=0.07880
400.23215.40.11176.50.07010.24384.20.11338.30.07388
600.23374.00.09813.80.07680.24971.80.11335.20.0849
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DatasetL/CGCNGAT
Uncal.TSMSCaGCNUncal.TSMSCaGCN
Cora200.13476.30.04885.50.04145.70.04016.70.15588.90.07179.80.05449.40.0450
400.11344.70.04177.20.03724.60.04075.40.13405.40.04857.70.04916.00.0365
600.09374.90.0355.40.03646.10.03764.40.12013.30.03936.10.04115.30.031332
Citeseer200.12487.10.06418.70.06443.70.05957.20.15345.00.09168.70.06339.80.05726.8
400.09577.70.06014.20.05385.70.05455.50.12528.70.07973.10.05905.40.05325.4
600.08066.40.05595.00.05216.40.05463.40.10905.90.06487.10.05199.10.05257.6
Pubmed200.05867.70.05413.80.04764.20.0405.00.08353.10.06564.60.05013.70.0356
400.04445.50.04466.30.04366.30.0402.00.08694.60.06586.50.05396.00.0308
600.04459.70.03676.00.03186.40.03114.80.09934.10.06696.30.04835.70.03082
CoraFull200.19866.10.10136.1=0.07760.21193.60.11015.1=0.07880
400.23215.40.11176.50.07010.24384.20.11338.30.07388
600.23374.00.09813.80.07680.24971.80.11335.20.0849
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The", + "type": "text" + }, + { + "bbox": [ + 470, + 614, + 495, + 624 + ], + "score": 0.9, + "content": "D _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 612, + 506, + 627 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 332, + 636 + ], + "score": 1.0, + "content": "enlarged in this way. The process above will be repeated", + "type": "text" + }, + { + "bbox": [ + 332, + 626, + 338, + 633 + ], + "score": 0.6, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "stages until convergence. Please note that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 633, + 376, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 376, + 647 + ], + "score": 1.0, + "content": "our classification GCN and CaGCN are re-initialized in each stage.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 547, + 506, + 647 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 663, + 191, + 676 + ], + "lines": [ + { + "bbox": [ + 103, + 660, + 193, + 680 + ], + "spans": [ + { + "bbox": [ + 103, + 660, + 193, + 680 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "In this section, we evaluate the performance of CaGCN on confidence calibration and CaGCN-st on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "self-training respectively. We choose the commonly used citation networks Cora [29], Citeseer [29],", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 711, + 492, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 492, + 723 + ], + "score": 1.0, + "content": "Pubmed [29] and CoraFull [3] for evaluation, and more detailed descriptions are in Appendix B.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 689, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 87, + 506, + 263 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 69, + 504, + 81 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 67, + 506, + 82 + ], + "spans": [ + { + "bbox": [ + 105, + 67, + 506, + 82 + ], + "score": 1.0, + "content": "Table 3: Node classification accuracy and the standard deviation on GCN and its self-training variants.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 87, + 506, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 87, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 87, + 506, + 263 + ], + "score": 0.986, + "html": "
DatasetL/CMethods
Orig.St.Ct.UnionInter.TS-stCaGCN-st
Cora2081.630.2482.270.3381.510.3081.850.6881.410.2882.680.2083.110.52
4083.990.2683.590.3483.660.2583.330.4183.380.3384.440.3584.370.38
6084.440.2984.980.3284.630.3185.030.3084.880.1885.600.2485.790.27
Citeseer2071.640.3273.240.4474.220.2974.600.3872.250.4574.200.2474.90040
4072.250.3274.700.3372.120.3974.790.3673.660.3275.620.1975.480.50
6073.200.3575.080.2973.210.3675.530.3075.230.2375.870.2476.430.20
Pubmed2079.570.3380.320.1879.670.3281.120.2979.590.2980.950.1881.16.36
4080.650.3982.200.3281.620.4081.840.2380.460.5582.280.3983.0821
6083.380.3483.350.2883.400.3683.320.3583.310.1783.260.3984.47023
CoraFull2060.450.4360.870.2860.120.4560.520.3561.010.5361.730.4162.190.49
4065.770.3765.830.4564.220.3564.330.4265.840.3766.110.6066.300.31
6066.520.2566.620.3066.640.2966.780.2966.820.3266.950.4567.600.40
", + "type": "table", + "image_path": "d8c33cf4a408cc06fa6f1aaf5cf9926679c8bec6d321b1f93eb2d0e57aba2a8d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 106, + 87, + 506, + 145.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 145.66666666666666, + 506, + 204.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 106, + 204.33333333333331, + 506, + 263.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "title", + "bbox": [ + 108, + 284, + 279, + 296 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 279, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 279, + 297 + ], + "score": 1.0, + "content": "5.1 Confidence Calibration Evaluation", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "Baselines. Since our CaGCN is a general calibration model for GNNs, here we choose GCN [16] and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "GAT [33] as our classification models. For comparison, we choose the classic post-hoc calibration", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "score": 1.0, + "content": "methods temperature scaling (TS) [12] and matrix scaling with off-diagonal regularization (MS) [18]", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 338, + 175, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 175, + 349 + ], + "score": 1.0, + "content": "as our baselines.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "Experimental settings. For the base model GCN and GAT, i.e., the uncalibrated model, we follow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 364, + 507, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 507, + 379 + ], + "score": 1.0, + "content": "parameters suggested by [16] and [33] and further carefully tune them to get optimal performance.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "For the post-hoc calibration technique, we follow the official implementation [12, 18]. For our", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 444, + 399 + ], + "score": 1.0, + "content": "CaGCN, we train a two-layer GCN with the hidden layer dimension to be 16. We set", + "type": "text" + }, + { + "bbox": [ + 444, + 387, + 477, + 398 + ], + "score": 0.9, + "content": "\\lambda = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "datasets, weight decay to be 5e-3 for Cora, Citeseer, Pubmed and 0.03 for CoraFull. Other parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "of CaGCN follows [16]. We evaluate the performance of confidence calibration by ECE [22], NLL", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 457, + 433 + ], + "score": 1.0, + "content": "[10] and Brier Score (BS) [4], which we expect are smaller, and we set the bin number", + "type": "text" + }, + { + "bbox": [ + 457, + 420, + 490, + 430 + ], + "score": 0.9, + "content": "M = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "ECE (more details can be seen in Appendix A). For all methods, we randomly run 10 times and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 442, + 470, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 470, + 455 + ], + "score": 1.0, + "content": "report the average results. More detailed experimental settings can be seen in Appendix B.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "Results. Table 2 reports calibration results evaluated by ECE (more results on NLL and Brier Score", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "are in Appendix C.1). We have the following observations: (1) Compared with uncalibrated models", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 479, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 383, + 492 + ], + "score": 1.0, + "content": "and other baselines, CaGCN is statistically significantly better at the", + "type": "text" + }, + { + "bbox": [ + 383, + 480, + 410, + 490 + ], + "score": 0.81, + "content": "^ { * } 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 479, + 450, + 492 + ], + "score": 1.0, + "content": "level and", + "type": "text" + }, + { + "bbox": [ + 450, + 480, + 481, + 490 + ], + "score": 0.84, + "content": "\\ast \\ast _ { } 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 479, + 507, + 492 + ], + "score": 1.0, + "content": "level.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 503 + ], + "score": 1.0, + "content": "(2) The ECE values on uncalibrated models are generally the highest, implying that GCN and GAT", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "are poorly calibrated. (3) MS behaves badly on datasets with many classes, e.g., CoraFull. This is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "because the number of parameters for matrix scaling scales quadratically with the number of classes", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "while the size of the validation set keeps unchanged. Therefore, it will over-fit to the small validation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 534, + 483, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 483, + 546 + ], + "score": 1.0, + "content": "set when dataset has a great number of classes. However, CaGCN does not have this problem.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "Additional analysis. In Section 2 we visualize the under-confidence problem of existing GNNs using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "reliability diagrams. Here we utilize the same visualization method to make a comparison before and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "after confidence calibration. As shown in Fig. 8, Fig. 9, Fig. 10 and Fig. 11 in the appendix, we can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 342, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 342, + 595 + ], + "score": 1.0, + "content": "find that the confidence is well-calibrated after calibration.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 306, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 308, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 308, + 623 + ], + "score": 1.0, + "content": "5.2 Classification Evaluation of Self-Training", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "Baselines. Since self-training can be applied to any models, here we choose GCN and GAT as our", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "base models, i.e., the original models (Orig.) without self-training, and we choose self-training (St.),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "co-training (Ct.), Union, Intersection (Inter.) methods proposed in [20] for comparison, which are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "commonly used as the baselines in self-training. Furthermore, we employ TS as the confidence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 673, + 427, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 427, + 685 + ], + "score": 1.0, + "content": "calibration function in CaGCN-st as another baseline and we denote it by TS-st.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 302, + 702 + ], + "score": 1.0, + "content": "Experimental settings. We set the learning rate", + "type": "text" + }, + { + "bbox": [ + 303, + 689, + 344, + 699 + ], + "score": 0.88, + "content": "l r = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "for CaGCN-st and train our CaGCN-st", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "200 epochs for Cora, 150 epochs for Citeseer, 100 epochs for Pubmed and 500 epochs for CoraFull.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 191, + 723 + ], + "score": 1.0, + "content": "We set the threshold", + "type": "text" + }, + { + "bbox": [ + 192, + 711, + 313, + 723 + ], + "score": 0.9, + "content": "\\tau \\in \\{ 0 . 8 , 0 . 8 5 , 0 . 9 , 0 . 9 5 , 0 . 9 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 710, + 459, + 723 + ], + "score": 1.0, + "content": "and the maximum number of stage", + "type": "text" + }, + { + "bbox": [ + 459, + 711, + 486, + 721 + ], + "score": 0.89, + "content": "s = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 710, + 505, + 723 + ], + "score": 1.0, + "content": ". As", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 87, + 506, + 263 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 69, + 504, + 81 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 67, + 506, + 82 + ], + "spans": [ + { + "bbox": [ + 105, + 67, + 506, + 82 + ], + "score": 1.0, + "content": "Table 3: Node classification accuracy and the standard deviation on GCN and its self-training variants.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 87, + 506, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 87, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 87, + 506, + 263 + ], + "score": 0.986, + "html": "
DatasetL/CMethods
Orig.St.Ct.UnionInter.TS-stCaGCN-st
Cora2081.630.2482.270.3381.510.3081.850.6881.410.2882.680.2083.110.52
4083.990.2683.590.3483.660.2583.330.4183.380.3384.440.3584.370.38
6084.440.2984.980.3284.630.3185.030.3084.880.1885.600.2485.790.27
Citeseer2071.640.3273.240.4474.220.2974.600.3872.250.4574.200.2474.90040
4072.250.3274.700.3372.120.3974.790.3673.660.3275.620.1975.480.50
6073.200.3575.080.2973.210.3675.530.3075.230.2375.870.2476.430.20
Pubmed2079.570.3380.320.1879.670.3281.120.2979.590.2980.950.1881.16.36
4080.650.3982.200.3281.620.4081.840.2380.460.5582.280.3983.0821
6083.380.3483.350.2883.400.3683.320.3583.310.1783.260.3984.47023
CoraFull2060.450.4360.870.2860.120.4560.520.3561.010.5361.730.4162.190.49
4065.770.3765.830.4564.220.3564.330.4265.840.3766.110.6066.300.31
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", + "type": "table", + "image_path": "d8c33cf4a408cc06fa6f1aaf5cf9926679c8bec6d321b1f93eb2d0e57aba2a8d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 106, + 87, + 506, + 145.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 145.66666666666666, + 506, + 204.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 106, + 204.33333333333331, + 506, + 263.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "title", + "bbox": [ + 108, + 284, + 279, + 296 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 279, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 279, + 297 + ], + "score": 1.0, + "content": "5.1 Confidence Calibration Evaluation", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "Baselines. Since our CaGCN is a general calibration model for GNNs, here we choose GCN [16] and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "GAT [33] as our classification models. For comparison, we choose the classic post-hoc calibration", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "score": 1.0, + "content": "methods temperature scaling (TS) [12] and matrix scaling with off-diagonal regularization (MS) [18]", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 338, + 175, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 175, + 349 + ], + "score": 1.0, + "content": "as our baselines.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 304, + 506, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "Experimental settings. For the base model GCN and GAT, i.e., the uncalibrated model, we follow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 364, + 507, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 507, + 379 + ], + "score": 1.0, + "content": "parameters suggested by [16] and [33] and further carefully tune them to get optimal performance.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "For the post-hoc calibration technique, we follow the official implementation [12, 18]. For our", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 444, + 399 + ], + "score": 1.0, + "content": "CaGCN, we train a two-layer GCN with the hidden layer dimension to be 16. We set", + "type": "text" + }, + { + "bbox": [ + 444, + 387, + 477, + 398 + ], + "score": 0.9, + "content": "\\lambda = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "datasets, weight decay to be 5e-3 for Cora, Citeseer, Pubmed and 0.03 for CoraFull. Other parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "of CaGCN follows [16]. We evaluate the performance of confidence calibration by ECE [22], NLL", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 457, + 433 + ], + "score": 1.0, + "content": "[10] and Brier Score (BS) [4], which we expect are smaller, and we set the bin number", + "type": "text" + }, + { + "bbox": [ + 457, + 420, + 490, + 430 + ], + "score": 0.9, + "content": "M = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "ECE (more details can be seen in Appendix A). For all methods, we randomly run 10 times and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 442, + 470, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 470, + 455 + ], + "score": 1.0, + "content": "report the average results. More detailed experimental settings can be seen in Appendix B.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 354, + 507, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "Results. Table 2 reports calibration results evaluated by ECE (more results on NLL and Brier Score", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "are in Appendix C.1). We have the following observations: (1) Compared with uncalibrated models", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 479, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 383, + 492 + ], + "score": 1.0, + "content": "and other baselines, CaGCN is statistically significantly better at the", + "type": "text" + }, + { + "bbox": [ + 383, + 480, + 410, + 490 + ], + "score": 0.81, + "content": "^ { * } 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 479, + 450, + 492 + ], + "score": 1.0, + "content": "level and", + "type": "text" + }, + { + "bbox": [ + 450, + 480, + 481, + 490 + ], + "score": 0.84, + "content": "\\ast \\ast _ { } 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 479, + 507, + 492 + ], + "score": 1.0, + "content": "level.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 503 + ], + "score": 1.0, + "content": "(2) The ECE values on uncalibrated models are generally the highest, implying that GCN and GAT", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "are poorly calibrated. (3) MS behaves badly on datasets with many classes, e.g., CoraFull. This is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "because the number of parameters for matrix scaling scales quadratically with the number of classes", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "while the size of the validation set keeps unchanged. Therefore, it will over-fit to the small validation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 534, + 483, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 483, + 546 + ], + "score": 1.0, + "content": "set when dataset has a great number of classes. However, CaGCN does not have this problem.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 458, + 507, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "Additional analysis. In Section 2 we visualize the under-confidence problem of existing GNNs using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "reliability diagrams. Here we utilize the same visualization method to make a comparison before and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "after confidence calibration. As shown in Fig. 8, Fig. 9, Fig. 10 and Fig. 11 in the appendix, we can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 342, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 342, + 595 + ], + "score": 1.0, + "content": "find that the confidence is well-calibrated after calibration.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 549, + 505, + 595 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 306, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 308, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 308, + 623 + ], + "score": 1.0, + "content": "5.2 Classification Evaluation of Self-Training", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "Baselines. Since self-training can be applied to any models, here we choose GCN and GAT as our", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "base models, i.e., the original models (Orig.) without self-training, and we choose self-training (St.),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "co-training (Ct.), Union, Intersection (Inter.) methods proposed in [20] for comparison, which are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "commonly used as the baselines in self-training. Furthermore, we employ TS as the confidence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 673, + 427, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 427, + 685 + ], + "score": 1.0, + "content": "calibration function in CaGCN-st as another baseline and we denote it by TS-st.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 628, + 506, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 302, + 702 + ], + "score": 1.0, + "content": "Experimental settings. We set the learning rate", + "type": "text" + }, + { + "bbox": [ + 303, + 689, + 344, + 699 + ], + "score": 0.88, + "content": "l r = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "for CaGCN-st and train our CaGCN-st", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "200 epochs for Cora, 150 epochs for Citeseer, 100 epochs for Pubmed and 500 epochs for CoraFull.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 191, + 723 + ], + "score": 1.0, + "content": "We set the threshold", + "type": "text" + }, + { + "bbox": [ + 192, + 711, + 313, + 723 + ], + "score": 0.9, + "content": "\\tau \\in \\{ 0 . 8 , 0 . 8 5 , 0 . 9 , 0 . 9 5 , 0 . 9 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 710, + 459, + 723 + ], + "score": 1.0, + "content": "and the maximum number of stage", + "type": "text" + }, + { + "bbox": [ + 459, + 711, + 486, + 721 + ], + "score": 0.89, + "content": "s = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 710, + 505, + 723 + ], + "score": 1.0, + "content": ". As", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "for baselines, all the parameters follow [20] and we further carefully tune them to get optimal", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 459, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 459, + 96 + ], + "score": 1.0, + "content": "performance. For all methods, we randomly run 10 times and report the average results.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 37, + "bbox_fs": [ + 106, + 688, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "for baselines, all the parameters follow [20] and we further carefully tune them to get optimal", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 459, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 459, + 96 + ], + "score": 1.0, + "content": "performance. For all methods, we randomly run 10 times and report the average results.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 507, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 507, + 113 + ], + "score": 1.0, + "content": "Results. Table 3 summarizes the node classification accuracy on GCN and its self-training variants.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "More results on GAT can be seen in Appendix C.2. We have the following observation: (1) CaGCN-st", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 438, + 135 + ], + "score": 1.0, + "content": "consistently outperforms all the baselines on all the datasets and label rates at the", + "type": "text" + }, + { + "bbox": [ + 438, + 122, + 465, + 132 + ], + "score": 0.77, + "content": "^ { * } 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "level. (2)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "Compared with the base model, self-training methods generally achieve better results, which proves", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "their effectiveness. (3) Self-training methods with confidence calibration (i.e., TS-st and CaGCN-st)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "have better performance, which implies that confidence calibration scales more correct predictions to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "the high confidence range while keeps incorrect predictions basically unchanged, which we believe is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 215, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 215, + 190 + ], + "score": 1.0, + "content": "beneficial for self-training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 288, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 288, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 288, + 205 + ], + "score": 1.0, + "content": "Ablation study. CaGCN-st generates pseudo", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 290, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 290, + 215 + ], + "score": 1.0, + "content": "labels based on the calibrated confidence.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 214, + 289, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 289, + 226 + ], + "score": 1.0, + "content": "Here we study the effectiveness of the confi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 289, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 289, + 236 + ], + "score": 1.0, + "content": "dence calibration function CaGCN in CaGCN-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 290, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 290, + 247 + ], + "score": 1.0, + "content": "st. We propose a variant GCN-st of CaGCN-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 248, + 289, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 289, + 258 + ], + "score": 1.0, + "content": "st, where CaGCN is removed from CaGCN-st", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 288, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 288, + 270 + ], + "score": 1.0, + "content": "while other parts are kept unchanged. All", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 288, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 288, + 280 + ], + "score": 1.0, + "content": "the experimental settings of GCN-st are the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 280, + 288, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 288, + 292 + ], + "score": 1.0, + "content": "same as CaGCN-st. We report the results in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 291, + 289, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 289, + 302 + ], + "score": 1.0, + "content": "Table 4, and we can observe that CaGCN-st", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 288, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 288, + 312 + ], + "score": 1.0, + "content": "consistently outperforms GCN-st on all the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 290, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 290, + 325 + ], + "score": 1.0, + "content": "datasets, implying that self-training with cal-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 289, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 289, + 336 + ], + "score": 1.0, + "content": "ibrated confidence can generate more correct", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 335, + 165, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 165, + 345 + ], + "score": 1.0, + "content": "pseudo labels.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 289, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 288, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 288, + 363 + ], + "score": 1.0, + "content": "Additional analysis. We also investigate the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 362, + 288, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 288, + 374 + ], + "score": 1.0, + "content": "changing trends of accuracy with respect to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 372, + 288, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 159, + 385 + ], + "score": 1.0, + "content": "the threshold", + "type": "text" + }, + { + "bbox": [ + 160, + 374, + 167, + 383 + ], + "score": 0.67, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 372, + 288, + 385 + ], + "score": 1.0, + "content": "in CaGCN-st in Appendix C.2", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "table", + "bbox": [ + 299, + 223, + 503, + 365 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 320, + 213, + 479, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 319, + 209, + 481, + 226 + ], + "spans": [ + { + "bbox": [ + 319, + 209, + 481, + 226 + ], + "score": 1.0, + "content": "Table 4: Abaltion study on self-training", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "table_body", + "bbox": [ + 299, + 223, + 503, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 299, + 223, + 503, + 365 + ], + "spans": [ + { + "bbox": [ + 299, + 223, + 503, + 365 + ], + "score": 0.983, + "html": "
Dataset L/CGCNGAT
GCN-st CaGCN-stGCN-st CaGCN-st
Cora2082.2883.1184.08 84.08
4084.1084.37 85.5085.63
6085.16 85.7985.5786.26
Citeseer2074.1374.9073.73 74.34
4075.2875.4875.07 75.62
6075.85 76.4375.1376.08
Pubmed2081.0181.1680.34 81.17
4082.9083.0882.75 83.47
6083.44 84.4783.4683.95
CoraFull2061.3262.19 62.0965.46
4065.9666.3065.92 66.86
6066.43 67.6066.5467.45
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Modern GCNs mimics CNNs to learn the local and global structural", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "patterns of graphs through designed convolution and readout functions. [5] generalizes CNNs to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "graph signal based on the spectrum of graph Laplacian. 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[20] introduces self-training", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "score": 1.0, + "content": "to GCNs and [30] proposes a multi-stage self-supervised (M3S) self-training algorithm of GCNs.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Both [20] and [30] focus on the few-shot learning and neither has ever explored self-training with", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "higher label rates in GCNs. 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[12] discovers modern neural networks are poorly calibrated and study", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 104, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "factors influencing calibration. Platt scaling [24] is a simple post-hoc calibration method for binary", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "models, which transforms the logit using scalar parameters. 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[25] explores the non-linear space for post-hoc calibration function using a", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 627, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 638 + ], + "score": 1.0, + "content": "neural network. Moreover, [32] points out GNNs can be miscalibrated in the supervised scenario and", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 638, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 649 + ], + "score": 1.0, + "content": "mainly focus on miscalibration originated from the imbalanced class distribution. However, none of", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 647, + 507, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 507, + 661 + ], + "score": 1.0, + "content": "them have considered the confidence calibration in GNNs on the common semi-supervised scenario.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 54.5 + }, + { + "type": "title", + "bbox": [ + 107, + 675, + 182, + 688 + ], + "lines": [ + { + "bbox": [ + 104, + 672, + 185, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 672, + 185, + 691 + ], + "score": 1.0, + "content": "7 Conclusion", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 60 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "score": 1.0, + "content": "Current efforts on advancing GNNs mostly focus on classification accuracy. However, when deploying", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "GNNs to real-world applications, especially safety-critical fields, whether the results of GNNs are", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 61.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 505, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 507, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 507, + 113 + ], + "score": 1.0, + "content": "Results. Table 3 summarizes the node classification accuracy on GCN and its self-training variants.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "More results on GAT can be seen in Appendix C.2. We have the following observation: (1) CaGCN-st", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 438, + 135 + ], + "score": 1.0, + "content": "consistently outperforms all the baselines on all the datasets and label rates at the", + "type": "text" + }, + { + "bbox": [ + 438, + 122, + 465, + 132 + ], + "score": 0.77, + "content": "^ { * } 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "level. (2)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "Compared with the base model, self-training methods generally achieve better results, which proves", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "their effectiveness. (3) Self-training methods with confidence calibration (i.e., TS-st and CaGCN-st)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "have better performance, which implies that confidence calibration scales more correct predictions to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "the high confidence range while keeps incorrect predictions basically unchanged, which we believe is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 215, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 215, + 190 + ], + "score": 1.0, + "content": "beneficial for self-training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 100, + 507, + 190 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 288, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 288, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 288, + 205 + ], + "score": 1.0, + "content": "Ablation study. CaGCN-st generates pseudo", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 290, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 290, + 215 + ], + "score": 1.0, + "content": "labels based on the calibrated confidence.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 214, + 289, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 289, + 226 + ], + "score": 1.0, + "content": "Here we study the effectiveness of the confi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 289, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 289, + 236 + ], + "score": 1.0, + "content": "dence calibration function CaGCN in CaGCN-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 290, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 290, + 247 + ], + "score": 1.0, + "content": "st. We propose a variant GCN-st of CaGCN-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 248, + 289, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 289, + 258 + ], + "score": 1.0, + "content": "st, where CaGCN is removed from CaGCN-st", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 288, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 288, + 270 + ], + "score": 1.0, + "content": "while other parts are kept unchanged. All", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 288, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 288, + 280 + ], + "score": 1.0, + "content": "the experimental settings of GCN-st are the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 280, + 288, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 288, + 292 + ], + "score": 1.0, + "content": "same as CaGCN-st. We report the results in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 291, + 289, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 289, + 302 + ], + "score": 1.0, + "content": "Table 4, and we can observe that CaGCN-st", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 288, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 288, + 312 + ], + "score": 1.0, + "content": "consistently outperforms GCN-st on all the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 290, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 290, + 325 + ], + "score": 1.0, + "content": "datasets, implying that self-training with cal-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 289, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 289, + 336 + ], + "score": 1.0, + "content": "ibrated confidence can generate more correct", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 335, + 165, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 165, + 345 + ], + "score": 1.0, + "content": "pseudo labels.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 192, + 290, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 289, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 288, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 288, + 363 + ], + "score": 1.0, + "content": "Additional analysis. We also investigate the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 362, + 288, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 288, + 374 + ], + "score": 1.0, + "content": "changing trends of accuracy with respect to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 372, + 288, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 159, + 385 + ], + "score": 1.0, + "content": "the threshold", + "type": "text" + }, + { + "bbox": [ + 160, + 374, + 167, + 383 + ], + "score": 0.67, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 372, + 288, + 385 + ], + "score": 1.0, + "content": "in CaGCN-st in Appendix C.2", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 383, + 343, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 343, + 397 + ], + "score": 1.0, + "content": "and study why GCNs are poorly calibrated in Appendix D.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 350, + 288, + 385 + ] + }, + { + "type": "table", + "bbox": [ + 299, + 223, + 503, + 365 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 320, + 213, + 479, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 319, + 209, + 481, + 226 + ], + "spans": [ + { + "bbox": [ + 319, + 209, + 481, + 226 + ], + "score": 1.0, + "content": "Table 4: Abaltion study on self-training", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "table_body", + "bbox": [ + 299, + 223, + 503, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 299, + 223, + 503, + 365 + ], + "spans": [ + { + "bbox": [ + 299, + 223, + 503, + 365 + ], + "score": 0.983, + "html": "
Dataset L/CGCNGAT
GCN-st CaGCN-stGCN-st CaGCN-st
Cora2082.2883.1184.08 84.08
4084.1084.37 85.5085.63
6085.16 85.7985.5786.26
Citeseer2074.1374.9073.73 74.34
4075.2875.4875.07 75.62
6075.85 76.4375.1376.08
Pubmed2081.0181.1680.34 81.17
4082.9083.0882.75 83.47
6083.44 84.4783.4683.95
CoraFull2061.3262.19 62.0965.46
4065.9666.3065.92 66.86
6066.43 67.6066.5467.45
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Modern GCNs mimics CNNs to learn the local and global structural", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "patterns of graphs through designed convolution and readout functions. [5] generalizes CNNs to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "graph signal based on the spectrum of graph Laplacian. 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GAT [33] utilizes attention mechanisms to adaptively learn aggregation", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "weights. GraphSAGE [13] uses various ways of pooling for aggregation. [20] introduces self-training", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "score": 1.0, + "content": "to GCNs and [30] proposes a multi-stage self-supervised (M3S) self-training algorithm of GCNs.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Both [20] and [30] focus on the few-shot learning and neither has ever explored self-training with", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "higher label rates in GCNs. More works on GNNs can be found in surveys [36, 43], however, to the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 534, + 448, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 448, + 545 + ], + "score": 1.0, + "content": "best of our knowledge, current GNNs have not considered the confidence calibration.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 435, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 561 + ], + "score": 1.0, + "content": "Confidence Calibration. Confidence calibration has been studied for a long time in CV and NLP", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "[12, 23, 25, 14, 19, 37, 40, 42]. [12] discovers modern neural networks are poorly calibrated and study", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 104, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "factors influencing calibration. Platt scaling [24] is a simple post-hoc calibration method for binary", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "models, which transforms the logit using scalar parameters. 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[25] explores the non-linear space for post-hoc calibration function using a", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 627, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 638 + ], + "score": 1.0, + "content": "neural network. Moreover, [32] points out GNNs can be miscalibrated in the supervised scenario and", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 638, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 649 + ], + "score": 1.0, + "content": "mainly focus on miscalibration originated from the imbalanced class distribution. However, none of", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 647, + 507, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 507, + 661 + ], + "score": 1.0, + "content": "them have considered the confidence calibration in GNNs on the common semi-supervised scenario.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 54.5, + "bbox_fs": [ + 104, + 550, + 507, + 661 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 675, + 182, + 688 + ], + "lines": [ + { + "bbox": [ + 104, + 672, + 185, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 672, + 185, + 691 + ], + "score": 1.0, + "content": "7 Conclusion", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 60 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "score": 1.0, + "content": "Current efforts on advancing GNNs mostly focus on classification accuracy. However, when deploying", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "GNNs to real-world applications, especially safety-critical fields, whether the results of GNNs are", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "trustworthy is another important factor that cannot be neglected. In this paper, we study the confidence", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 507, + 97 + ], + "score": 1.0, + "content": "calibration problem in GNNs and discover existing GNNs are under-confident on their predictions.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "To solve this problem, we propose a novel trustworthy GNN model CaGCN which respects the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "homophily property of confidence in GNNs and preserves the classification accuracy. Moreover, we", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "propose a novel self-training method CaGCN-st where confidence is first calibrated by CaGCN and", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "then used to generate pseudo labels. Extensive experiments demonstrate the effectiveness of our", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 341, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 341, + 151 + ], + "score": 1.0, + "content": "proposed model in terms of both calibration and accuracy.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 61.5, + "bbox_fs": [ + 105, + 698, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "trustworthy is another important factor that cannot be neglected. In this paper, we study the confidence", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 507, + 97 + ], + "score": 1.0, + "content": "calibration problem in GNNs and discover existing GNNs are under-confident on their predictions.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "To solve this problem, we propose a novel trustworthy GNN model CaGCN which respects the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "homophily property of confidence in GNNs and preserves the classification accuracy. Moreover, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "propose a novel self-training method CaGCN-st where confidence is first calibrated by CaGCN and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "then used to generate pseudo labels. Extensive experiments demonstrate the effectiveness of our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 341, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 341, + 151 + ], + "score": 1.0, + "content": "proposed model in terms of both calibration and accuracy.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "An interesting direction for future work is to extend CaGCN to other graph tasks, but more studies", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "need to be conducted. We take the link prediction as an example, where we can regard the link", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "prediction as a binary classification problem and the output as the confidence. Considering that the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "ground-truth confidence distribution for nodes should have the homophily property as is shown in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Section 3.1, edges are likely to have the same property as well. As a result, we can employ CaGCN", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "to propagate the confidence between edges by regarding the edges as the nodes. However, more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 411, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 411, + 233 + ], + "score": 1.0, + "content": "exploration still needs to be conducted for the homophily property of edges.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 507, + 250 + ], + "score": 1.0, + "content": "Broader impact. Current efforts on advancing GNNs mostly focus on classification accuracy.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "However, when deploying GNNs to real-world applications, especially safety-critical fields, whether", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 504, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 504, + 270 + ], + "score": 1.0, + "content": "the results of GNNs are trustworthy is another important factor. The demands for a trustworthy model", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "are universal and extensive such as in the field of disease prediction [31], traffic states prediction [6]", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 279, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 294 + ], + "score": 1.0, + "content": "and object detection [11] for autonomous driving, where estimating the true probability of getting", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 303 + ], + "score": 1.0, + "content": "a correct prediction is necessary. We take the disease prediction [31] as an example, where GNNs", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "are utilized to encode the information of different symptoms, users and diseases. In this scenario,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "accurately and comprehensively predicting diseases at an early stage will help patients receive", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "prevention treatments in a timely manner. Otherwise, the misdiagnosis and missed diagnosis will", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "endanger the health of patients. Therefore, a trustworthy model is urgently needed. Our CaGCN", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "can make a trustworthy prediction based on its confidence, and as a result, decrease the risk of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "misdiagnosis and missed diagnosis. We hope our work can provide insights for future improvements", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 368, + 381, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 381, + 380 + ], + "score": 1.0, + "content": "in tackling the trustworthiness problem in other saftey-critical fields.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "Limitations. One potential issue of this work is that it provides a limited explanation to the under-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "confidence problem. We advocate peer researchers to look into this, making GNNs more reliable", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "score": 1.0, + "content": "in different domains. 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Deep gaussian embedding of graphs: Unsu-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 127, + 609, + 452, + 622 + ], + "spans": [ + { + "bbox": [ + 127, + 609, + 452, + 622 + ], + "score": 1.0, + "content": "pervised inductive learning via ranking. arXiv preprint arXiv:1707.03815, 2017.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 110, + 627, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 110, + 627, + 505, + 642 + ], + "score": 1.0, + "content": "[4] Glenn W Brier. Verification of forecasts expressed in terms of probability. Monthly weather", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 127, + 639, + 228, + 652 + ], + "spans": [ + { + "bbox": [ + 127, + 639, + 228, + 652 + ], + "score": 1.0, + "content": "review, 78(1):1–3, 1950.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 110, + 658, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 110, + 658, + 505, + 673 + ], + "score": 1.0, + "content": "[5] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 127, + 670, + 412, + 682 + ], + "spans": [ + { + "bbox": [ + 127, + 670, + 412, + 682 + ], + "score": 1.0, + "content": "connected networks on graphs. arXiv preprint arXiv:1312.6203, 2013.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 110, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 110, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "[6] Zhiyong Cui, Kristian Henrickson, Ruimin Ke, and Yinhai Wang. Traffic graph convolutional", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 127, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 127, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "recurrent neural network: A deep learning framework for network-scale traffic learning and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 127, + 711, + 507, + 725 + ], + "spans": [ + { + "bbox": [ + 127, + 711, + 507, + 725 + ], + "score": 1.0, + "content": "forecasting. IEEE Transactions on Intelligent Transportation Systems, 21(11):4883–4894, 2019.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 73, + 507, + 151 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "An interesting direction for future work is to extend CaGCN to other graph tasks, but more studies", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "need to be conducted. 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Current efforts on advancing GNNs mostly focus on classification accuracy.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "However, when deploying GNNs to real-world applications, especially safety-critical fields, whether", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 504, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 504, + 270 + ], + "score": 1.0, + "content": "the results of GNNs are trustworthy is another important factor. 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Otherwise, the misdiagnosis and missed diagnosis will", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "endanger the health of patients. Therefore, a trustworthy model is urgently needed. Our CaGCN", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "can make a trustworthy prediction based on its confidence, and as a result, decrease the risk of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "misdiagnosis and missed diagnosis. 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Beyond low-frequency information in", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 127, + 579, + 415, + 591 + ], + "spans": [ + { + "bbox": [ + 127, + 579, + 415, + 591 + ], + "score": 1.0, + "content": "graph convolutional networks. arXiv preprint arXiv:2101.00797, 2021.", + "type": "text" + } + ], + "index": 39, + "is_list_end_line": true + }, + { + "bbox": [ + 111, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 111, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "[3] Aleksandar Bojchevski and Stephan Günnemann. 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DatasetGCN
Uncal.TSOurs
Cora240.267172.346164.651
Citeseer128.145112.212108.684
Pubmed1299.331266.681113.41
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DatasetL/CGCNGAT
Uncal.TSMSCaGCNUncal.TSMSCaGCN
Cora200.13476.30.04885.50.04145.70.04016.70.15588.90.07179.80.05449.40.0450
400.11344.70.04177.20.03724.60.04075.40.13405.40.04857.70.04916.00.0365
600.09374.90.0355.40.03646.10.03764.40.12013.30.03936.10.04115.30.031332
Citeseer200.12487.10.06418.70.06443.70.05957.20.15345.00.09168.70.06339.80.05726.8
400.09577.70.06014.20.05385.70.05455.50.12528.70.07973.10.05905.40.05325.4
600.08066.40.05595.00.05216.40.05463.40.10905.90.06487.10.05199.10.05257.6
Pubmed200.05867.70.05413.80.04764.20.0405.00.08353.10.06564.60.05013.70.0356
400.04445.50.04466.30.04366.30.0402.00.08694.60.06586.50.05396.00.0308
600.04459.70.03676.00.03186.40.03114.80.09934.10.06696.30.04835.70.03082
CoraFull200.19866.10.10136.1=0.07760.21193.60.11015.1=0.07880
400.23215.40.11176.50.07010.24384.20.11338.30.07388
600.23374.00.09813.80.07680.24971.80.11335.20.0849
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DatasetL/CMethods
Orig.St.Ct.UnionInter.TS-stCaGCN-st
Cora2081.630.2482.270.3381.510.3081.850.6881.410.2882.680.2083.110.52
4083.990.2683.590.3483.660.2583.330.4183.380.3384.440.3584.370.38
6084.440.2984.980.3284.630.3185.030.3084.880.1885.600.2485.790.27
Citeseer2071.640.3273.240.4474.220.2974.600.3872.250.4574.200.2474.90040
4072.250.3274.700.3372.120.3974.790.3673.660.3275.620.1975.480.50
6073.200.3575.080.2973.210.3675.530.3075.230.2375.870.2476.430.20
Pubmed2079.570.3380.320.1879.670.3281.120.2979.590.2980.950.1881.16.36
4080.650.3982.200.3281.620.4081.840.2380.460.5582.280.3983.0821
6083.380.3483.350.2883.400.3683.320.3583.310.1783.260.3984.47023
CoraFull2060.450.4360.870.2860.120.4560.520.3561.010.5361.730.4162.190.49
4065.770.3765.830.4564.220.3564.330.4265.840.3766.110.6066.300.31
6066.520.2566.620.3066.640.2966.780.2966.820.3266.950.4567.600.40
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Dataset L/CGCNGAT
GCN-st CaGCN-stGCN-st CaGCN-st
Cora2082.2883.1184.08 84.08
4084.1084.37 85.5085.63
6085.16 85.7985.5786.26
Citeseer2074.1374.9073.73 74.34
4075.2875.4875.07 75.62
6075.85 76.4375.1376.08
Pubmed2081.0181.1680.34 81.17
4082.9083.0882.75 83.47
6083.44 84.4783.4683.95
CoraFull2061.3262.19 62.0965.46
4065.9666.3065.92 66.86
6066.43 67.6066.5467.45
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sha256:364018732ad05554dd9942d7db2ee0a8a1f7e9a6b6edbdaee1def3a234b2dbd0 +size 5167 diff --git a/parse/train/B1lda1HtvB/B1lda1HtvB.md b/parse/train/B1lda1HtvB/B1lda1HtvB.md new file mode 100644 index 0000000000000000000000000000000000000000..5ccfd652c10aa21b1cd41de4b09e264581553ceb --- /dev/null +++ b/parse/train/B1lda1HtvB/B1lda1HtvB.md @@ -0,0 +1,600 @@ +# FEATURE SELECTION USING STOCHASTIC GATES + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Feature selection problems have been extensively studied in the setting of linear estimation, for instance LASSO, but less emphasis has been placed on feature selection for non-linear functions. In this study, we propose a method for feature selection in non-linear function estimation problems. The new procedure is based on directly penalizing the $\ell _ { 0 }$ norm of features, or the count of the number of selected features. Our $\ell _ { 0 }$ based regularization relies on a continuous relaxation of the Bernoulli distribution, which allows our model to learn the parameters of the approximate Bernoulli distributions via gradient descent. The proposed framework simultaneously learns a non-linear regression or classification function while selecting a small subset of features. We provide an information-theoretic justification for incorporating Bernoulli distribution into our approach. Furthermore, we evaluate our method using synthetic and real-life data and demonstrate that our approach outperforms other embedded methods in terms of predictive performance and feature selection. + +# 1 INTRODUCTION + +Feature selection is a fundamental task in machine learning and statistics. Feature selection leads to a number of potential benefits: reducing experimental costs Min et al. (2014), enhancing interpretability (Ribeiro et al., 2016), computational speed up and even improving model generalization on unseen data (Chandrashekar & Sahin, 2014). In biomedicine, scientists collect multitude datasets comprising of many biomarkers (e.g., genes or proteins) that require development of effective diagnostics or prognostics models. For instance, in Genome wide association studies (GWAS), feature selection can help identify such models and lead to improved risk assessment and reduced cost. + +Feature selection methods may be classified into three major categories: filter methods, wrapper methods, and embedded methods. Filter methods attempt to remove irrelevant features prior to learning a model. These methods filter features using a per-feature relevance score that is created based on some statistical measure (Battiti, 1994; Peng et al., 2005; Estévez et al., 2009; Song et al., 2007; 2012; Chen et al., 2017). Wrapper methods (Kohavi & John, 1997b; Stein et al., 2005; Zhu et al., 2007; Reunanen, 2003; Allen, 2013) use the outcome of a classifier to determine the relevance of each feature, which requires recomputing the classifier for each subset of features. This becomes computationally expensive for neural network based wrapper methods (Verikas & Bacauskiene, 2002; Kabir et al., 2010; Roy et al., 2015). + +Embedded methods aim to remove this burden by learning the model while simultaneously selecting the subset of relevant features. The Least Absolute Shrinkage and Selection Operator (LASSO) (Tibshirani, 1996) is a well-known embedded method, whose objective is to minimize the loss while enforcing an $\ell _ { 1 }$ constraint on the weights of the features. Although LASSO is scalable and widely used (Hans, 2009; Li et al., 2011; 2006), it is restricted to the domain of linear functions. To allow the model to capture nonlinear interaction, it is appealing to consider the non-convex extension of the LASSO formulation using neural networks. + +We develop a fully embedded feature selection method for nonlinear models. To the best of our knowledge it is the first $\ell _ { 0 }$ embedded feature selection method. Our method improves upon the LASSO formulation in two aspects: a) it captures nonlinear interactions between the features via neural network modeling and b) it employs an $\ell _ { 0 }$ -like regularization using gates whose weights are parametrized by a smooth variant of a Bernoulli distribution. Altogether these twofold improvements are formulated as a fully differentiable neural network. Specifically, our contributions are as follows: + +1. By utilizing a recent development of continuous and differentiable approximation to discrete distributions (Maddison et al., 2016), (Jang et al., 2017), (Louizos et al., 2017), we introduce a solution to the long standing problem of feature selection with an $\ell _ { 0 }$ regularization. +2. We present a novel simple relaxation of Bernoulli distribution to sparsify the input layer (the feature space) which we call stochastic gate (STG) and show its advantage over the distribution presented by (Louizos et al., 2017) both in performance and convergence time. +3. By applying these two relaxations to an input layer of a neural network, we perform embedded feature selection in classification, regression or survival analysis tasks and demonstrate its capabilities on artificial and real data sets. +4. We justify our probabilistic approach by analyzing the constrained Mutual Information maximization objective of feature selection. We demonstrate the applicability of our method using numerous examples (see Section 6 and Appendix). + +Notation: We refer to vectors as bold lowercase $_ { \textbf { \em x } }$ and random vectors as bold uppercase letters $\boldsymbol { X }$ . Scalars are non-bold case $y$ , while random variables are capital case $Y$ . A set is represented by script fonts $\mathcal { X } , \mathcal { Y } , \mathcal { S }$ . For example the $n ^ { t h }$ vector-valued observation is denoted as ${ \mathbf { \mathcal { x } } } _ { n }$ whereas $X _ { d }$ represents the $d ^ { t h }$ feature of the vector-valued random variable $\boldsymbol { X }$ . Let $[ n ] = 1 , 2 , \ldots , n$ . For a set $\bar { \mathcal { S } } \bar { \subset } [ D ]$ let the vector $\pmb { s } \in \{ 0 , 1 \} ^ { D }$ be the characteristic function for the set. That is $s _ { i } = 1$ if $i \in S$ and 0 otherwise. For two vectors $_ { \textbf { \em x } }$ and $_ { z }$ we denote $\pmb { x } \odot z$ to be the element-wise product between $_ { \textbf { \em x } }$ and $_ z$ . Thus, if we let $\pmb { s } \in \{ 0 , 1 \} ^ { D }$ be the characteristic vector of $s$ , then we may define $\mathbf { \Delta } \mathbf { x } _ { S } = \mathbf { \Delta } \mathbf { x } \odot \mathbf { \Delta } \mathbf { s }$ The $\ell _ { 1 }$ norm of a vector is denoted as $\begin{array} { r } { \| \pmb { x } \| _ { 1 } = \sum _ { i = 1 } ^ { D } | x _ { i } | } \end{array}$ . Finally, the $\ell _ { 0 }$ norm of a vector is denoted as $\| \pmb { x } \| _ { 0 }$ and counts the total number of non-zero entries in the vector $_ { \textbf { \em x } }$ . + +# 2 PROBLEM SETUP AND BACKGROUND + +Let $\boldsymbol { \mathcal { X } } ~ \in ~ \mathbb { R } ^ { D }$ be the input domain with corresponding response domain $\mathcal { V }$ . Given realizations from some unknown data distribution $P _ { X , Y }$ , the goal of embedded feature selection methods is to simultaneously find a subset of indices $\mathcal { S } \subset \{ 1 , . . . D \}$ and construct a model that predicts $Y$ based on the selected features $X _ { \mathcal { S } }$ . + +# 2.1 RISK MINIMIZATION OBJECTIVE + +We assume that we are given a family of functions $\mathcal { F }$ such that any function $f _ { \pmb { \theta } } \in \mathcal { F }$ is indexed by a set of parameters $\pmb \theta$ . Given some loss $L$ , a selection of features $s \subset [ D ]$ , and a choice of parameters $\pmb \theta$ , we denote the risk of our model as + +$$ +R ( \pmb \theta , \pmb s ) = \mathbb { E } _ { X , Y } L ( f _ { \boldsymbol \theta } ( \pmb X \odot \pmb s ) , Y ) , +$$ + +where we recall that $\pmb { s } = \{ 0 , 1 \} ^ { D }$ is a vector of indicator variables for the set $s$ and $\odot$ denotes the point-wise product. Thus, the goal of the feature selection problem is to find the parameters $\pmb \theta$ and $\pmb { s }$ that minimize $R ( \pmb \theta , s )$ such that $\| s \| _ { 0 }$ is small compared to $D$ . + +# 2.2 FEATURE SELECTION FOR LINEAR MODELS + +Before proceeding with our proposed method, we review the feature selection problem in the linear regression setting with the least squares loss. Thus, we restrict $\mathcal { F }$ to be the space of linear functions and the loss function to be the quadratic loss. + +Given observations $\{ x _ { n } , y _ { n } \} _ { n = 1 } ^ { N }$ we may consider the constrained empirical risk minimization problem + +$$ +\operatorname* { m i n } _ { \pmb { \theta } } \frac { 1 } { N } \sum _ { n = 1 } ^ { N } L ( \pmb { \theta } ^ { T } \pmb { x } _ { n } , y _ { n } ) \quad \mathrm { s . t . } \ \| \pmb { \theta } \| _ { 0 } \leq k . +$$ + +Since the above problem is intractable, a number of authors replace the $\ell _ { 0 }$ constraint with a surrogate function $\Omega ( \pmb \theta ) : \dot { \mathbb { R } } ^ { D } \mathbb { R } _ { + }$ designed to penalize the number of selected features in $\pmb \theta$ . A natural choice for $\Omega$ is the $\ell _ { 1 }$ norm, which yields a convex problem and more precisely the LASSO optimization problem. The $\ell _ { 1 }$ is known to be the closest convex relaxation to the $\ell _ { 0 }$ . In fact, in certain settings the + +LASSO optimization and the $\ell _ { 0 }$ based objective have the same solution. While the original LASSO problem focuses on the constrained optimization problem, the regularized least squares problem, which is often used in practice, yields the following minimization objective: + +$$ +\operatorname* { m i n } _ { \pmb { \theta } } \frac { 1 } { N } \sum _ { n = 1 } ^ { N } ( \pmb { \theta } ^ { T } \pmb { x } _ { n } - \pmb { y } _ { n } ) ^ { 2 } + \lambda \| \pmb { \theta } \| _ { 1 } . +$$ + +The hyperparameter $\lambda$ trades off the amount of regularization versus the fit of the objective1. The $\ell _ { 1 }$ regularized method is very effective for feature selection and prediction; however, it achieves this through shrinkage of the coefficients. As a result, (Fan & Li, 2001) have considered non-convex choices for $\Omega$ that perform well both theoretically and empirically for prediction and feature selection. + +Our goal is to apply such regularization techniques to perform feature selection while learning a non-linear function. Kernel methods have been considered (Yamada et al., 2014), but scale quadratically in the number of observations. To alleviate this burden, (Gregorová et al., 2018) use random Fourier features to approximate the kernel. (Li et al., 2016) and (Scardapane et al., 2017) take an alternative approach by modeling $f _ { \theta }$ using a neural network with $\ell _ { 1 }$ regularization on the input weights. However, in practice, introducing an $\ell _ { 1 }$ penalty into gradient descent does not provide sufficient sparsification. Below, we discuss our method that works to directly use an $\ell _ { 0 }$ penalty. + +# 3 PROPOSED METHOD + +We take a probabilistic approach to approximate the $\ell _ { 0 }$ norm, which can extend to non-linear models while remaining computationally efficient. To motivate such probabilistic approach, we provide theoretical support (see Section 4) based on a Mutual Information perspective of the feature selection problem. + +To view the $\ell _ { 0 }$ regularized version of the risk (Eq. 1) from a probabilistic perspective, one can introduce a Bernoulli random vector $\tilde { S }$ whose entries are independent and the $d ^ { t h }$ entry satisfies $\pi _ { d } = \mathbb { P } ( \tilde { S } _ { d } = 1 )$ for $d \in [ D ]$ . If we denote the empirical expectation over our observations as $\hat { \mathbb { E } } _ { X , Y }$ , then, the empirical regularized risk (Eq. 1) becomes + +$$ +\operatorname* { m i n } _ { \theta , \pi } \hat { R } ( \theta , \pi ) = \operatorname* { m i n } _ { \theta , \pi } \hat { \mathbb { E } } _ { X , Y } \mathbb { E } _ { \tilde { S } } \left[ L ( f _ { \theta } ( X \odot \tilde { S } ) , Y ) + \lambda | | \tilde { S } | | _ { 0 } \right] , +$$ + +where we have $\mathbb { E } _ { \tilde { S } } | | \tilde { S } | | _ { 0 } = \sum _ { d = 1 } ^ { D } \pi _ { d }$ and we constrain $\pi _ { d } \in \{ 0 , 1 \}$ . Clearly, this formulation is equivalent to Eq. 1, with a regularized penalty on cardinality rather than an explicit constraint. We may then relax the discrete constraint on $\pi _ { d }$ to be $\pi _ { d } \in [ 0 , 1 ]$ . + +Now, our goal is to find the model parameters $\pmb { \theta } ^ { * }$ and Bernoulli parameters $\pi ^ { * }$ that minimize the empirical risk $\hat { R } ( \pmb \theta , \pmb \pi )$ via gradient descent. However, an optimization of a loss function which includes discrete random variables suffers from high variance (see F in the Appendix for more details). Therefore, inspired by a recently developed continuous approximation for discrete random variables, suggested by (Jang et al., 2017; Maddison et al., 2016), we develop and use a novel and simple continuous distribution that is fully differentiable and suited for the task of feature selection. + +# 3.1 CONTINUOUS RELAXATION + +Our continuous relaxation for the Bernoulli variables ${ \cal \tilde { S } } _ { d }$ for $d \in \ [ D ]$ is termed stochastic gate (STG). The STG relies on the reparametrization trick, which is widely used for reducing the variance of gradient estimators (Miller et al., 2017; Figurnov et al., 2018). To construct a continuous approximation to Bernoulli random variable via the reparametrization trick, we define $z _ { d } = g ( \mu _ { d } + \epsilon _ { d } ) = \operatorname* { m a x } ( 0 , \operatorname* { m i n } ( 1 , \epsilon _ { d } + \mu _ { d } + 0 . 5 ) )$ where $\epsilon _ { d }$ is drawn from a Gaussian distribution $\mathcal { N } ( 0 , \sigma _ { d } )$ , where $\sigma _ { d }$ is fixed throughout training. This approximation can be viewed as a clipped, mean-shifted, Gaussian random vector. Furthermore, the gradient of the objective with respect to $\mu _ { d }$ can be computed via the chain rule. + +We can now rewrite the objective in Eq. 4 as + +$$ +\operatorname* { m i n } _ { \theta , \mu } \hat { R } ( \theta , \mu ) = \operatorname* { m i n } _ { \theta , \mu } \mathbb { E } _ { X , Y } \mathbb { E } _ { Z } \left[ L ( f _ { \theta } ( X \odot Z ) , Y ) + \lambda | | Z | | _ { 0 } \right] , +$$ + +where $z$ is a random vector with $D$ independent variables $z _ { d }$ for $[ D ]$ . To optimize the empirical surrogate of the objective (Eq. 5), we first differentiate it with respect to $\pmb { \mu }$ . Then, Monte Carlo sampling leads us to the following gradient estimator + +$$ +\frac { \partial } { \partial \mu _ { d } } \hat { R } ( \pmb \theta , \mu ) = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \bigg [ L ^ { \prime } ( z ^ { k } ) \frac { \partial z _ { d } ^ { k } } { \partial \mu _ { d } } \bigg ] + \frac { \lambda } { K } \frac { \partial } { \partial \mu _ { d } } \sum _ { k = 1 } ^ { K } \mathrm { P r } \{ z _ { d } ^ { k } > 0 \} , +$$ + +where $K$ is the number of Monte Carlo samples. Thus, we can update the parameters $\mu _ { d }$ for $[ D ]$ via gradient descent. We note that if we replace $\frac { \partial { z } _ { d } ^ { k } } { \partial { \mu } _ { d } }$ with 1, the above gradient estimator for $L ^ { \prime }$ is reduced to the Straight-Through estimator (Bengio et al., 2013). + +Under the continuous relaxation, the expected regularization term in the objective $\hat { R } ( \pmb \theta , \pmb \mu )$ is simply the sum of the probability that the gates $\{ z _ { d } \} _ { d = 1 } ^ { D }$ are active, which is equal to $\begin{array} { r } { \sum _ { d = 1 } ^ { D } \Phi \left( \frac { \mu _ { d } + \frac { 1 } { 2 } } { \sigma _ { d } } \right) } \end{array}$ where $\Phi$ is the standard Gaussian CDF. To conclude, we can now optimize the objective in Eq. 5 using gradient descent over the model parameters $\pmb \theta$ and the parameters $\pmb { \mu }$ representing the Gaussian’s mean (instead of the Bernoulli parameters $\pi$ ). + +After training, to remove the stochasticity from the learned gates, we set $\hat { z } _ { d } = \operatorname* { m a x } _ { \mathbf { \alpha } } ( 0 , \operatorname* { m i n } ( 1 , \mu _ { d } +$ 0.5)), which informs what features are selected. Note that when $\left| \mu _ { d } \right|$ is less than $\begin{array} { l } { { \frac { 1 } { 2 } } } \end{array}$ , $\hat { z } _ { d }$ returns the value between $( 0 , 1 )$ . In such a case, we can treat the value of $\hat { z } _ { d }$ as feature importance or employ an additional thresholding (i.e. 1 if $\hat { z } _ { d } > 0 . 5$ and 0 otherwise) depending on application-specific needs. In the Appendix, we provide the pseudo-code of our algorithm as well as the discussion of the choice of $\sigma _ { d }$ . + +# 4 CONNECTION TO MUTUAL INFORMATION + +In this section we show an equivalence between the Bernoulli formulation of the feature selection problem and the $\ell _ { 0 }$ regularized approach. + +# 4.1 MUTUAL INFORMATION BASED OBJECTIVE + +From an information theoretic perspective, the goal of feature selection is to find a subset of features $s$ that has the highest Mutual Information (MI) with the target variable $Y$ . Recall that the MI between two random variables can be defined as $I ( X ; Y ) = H ( Y ) - H ( Y | X )$ where $H ( Y ) , H ( Y | X )$ are the entropy of $p _ { Y } ( Y )$ and the conditional entropy of $p _ { Y | \mathbf { X } } ( Y | X )$ , respectively (Cover & Thomas, 2006). Then we can formulate the task as selecting $s$ such that the mutual information between $X _ { \mathcal { S } }$ and $Y$ are maximized: + +$$ +\operatorname* { m a x } _ { s } I ( X _ { s } , Y ) \quad { \mathrm { s . t . ~ } } | S | = k , +$$ + +where $k$ is the hypothesised number of relevant features. + +# 4.2 INTRODUCING RANDOMNESS + +We first demonstrate that under mild assumptions we can replace the deterministic search over the set $s$ (or corresponding indicator vector $\pmb { s }$ ), by a search over the parameters of the distributions that model $\pmb { s }$ . Our proposition is based on the following two assumptions: + +Assumption 1: There exists a subset of indices ${ \boldsymbol { S } } ^ { * }$ with cardinality equal to $k$ such that for any $i \in S ^ { * }$ we have $I ( X _ { i } ; Y | { \cal { X } } _ { \backslash \{ i \} } ) > 0$ . +Assumption 2: $I ( X _ { \cal S ^ { * c } } ; Y | X _ { \cal S ^ { * } } ) = 0$ . + +Discussion of assumptions: Assumption 1 that including an element from $S ^ { * }$ improves prediction accuracy. This assumption is equivalent to stating that feature $i$ is strongly relevant (Kohavi & John, 1997a; Brown et al., 2012). Assumption 2 simply states that $S ^ { * }$ is a superset of the Markov Blanket of the variable $Y$ (Brown et al., 2012). The assumptions are quite benign. For instance they are satisfied if $\boldsymbol { X }$ is drawn from a Gaussian with a non-degenerate covariance matrix and $Y = f ( X _ { \mathcal { S } ^ { * } } ) + w$ where $w$ is noise independent of $\boldsymbol { X }$ and $f$ is not degenerate. With these assumptions in hand, we may present our result. + +Proposition 1. Suppose that the above assumptions hold for the model. Then, solving the optimization in Eq. 6 is equivalent to solving the optimization + +$$ +\operatorname* { m a x } _ { \mathbf { 0 } \leq \pi \leq 1 } I ( X \odot \tilde { S } ; Y ) \quad s . t . \quad \sum _ { i } \mathbb { E } [ \tilde { S } _ { i } ] \leq k , +$$ + +where the coordinates $\tilde { S } _ { i }$ are drawn independently at random according to a Bernoulli distribution with parameter $\pi _ { i }$ . + +Due to length constraints, we leave the proof of this proposition and how it bridges the MI maximization (Eq. 6) and risk minimization (Eq. 2) to the Appendix (see Section C and D). + +# 5 RELATED WORK + +The two most related works to this study are (Louizos et al., 2017) and (Chen et al., 2018). In (Louizos et al., 2017), they introduce the Hard-Concrete (HC) distribution as a continuous surrogate for Bernoulli distributions in the context of model compression. The HC distribution is induced by applying a hard sigmoid $z = \operatorname* { m i n } ( 1 , \operatorname* { m a x } ( 0 , \bar { s } ) )$ to $\bar { s }$ , where we construct $\bar { s }$ by applying Sigmoid function to the logistic distribution as follows: + +$$ +u \sim U ( 0 , 1 ) , L = \log ( u ) - \log ( 1 - u ) , s = \frac { 1 } { 1 + \exp ( \frac { - ( \log \alpha + L ) } { \beta } ) } , \bar { s } = s ( \zeta - \tau ) + \tau +$$ + +The interval $( \tau , \zeta )$ , with $\tau < 0$ and $\zeta > 1$ , allows the distribution to have more probability mass on the edge of the support. The authors demonstrate how incorporating the HC in deep neural networks leads to fast convergence and improved generalization due to the sparsification effect. In this study, we introduce this type of sparsification for the long standing problem of feature selection with $\ell _ { 0 }$ regularization.We demonstrate that a simple relaxation of Bernoulli distributions is sufficient and works better than the HC distribution for feature selection. We find empirically that the HC yields high variance gradient estimates compared to the STG. This high-variance does not affect model sparsification as the sparsity pattern does not matter. Contrarily, for feature selection it is crucial as practitioners require a stable procedure. Furthermore, higher variance gradients will result in a slower convergence of the model (this has been demonstrated empirically in the supp. material). In (Chen et al., 2018), the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained models. Their method is focused on finding a subset of features given a particular instance, and therefore is not appropriate for general feature selection. + +Some studies tackle embedded feature selection problems by extending LASSO and group LASSO to neural network models. The authors (Li et al., 2016; Scardapane et al., 2017) and (Feng & Simon, 2017) have a similar goal as ours in performing feature selection, but instead rely on the $\ell _ { 1 }$ relaxation to the $\ell _ { 0 }$ . Our approach, which utilizes stochastic gates coupled with the $\ell _ { 0 }$ norm achieves substantially better empirical performance compared against other $\ell _ { 1 }$ based baselines. This point is demonstrated in the next section. + +# 6 EXPERIMENTS + +Here we provide empirical evaluation of our method in a wide range of settings. We begin with simple linear regression evaluating the capabilities of our approach to select a sparse set of relevant features (section 6.1). We then evaluate our method in nonlinear regression tasks (section 6.2). Next, we demonstrate the applicability of the method in a highly nonlinear classification task (section 6.3. Finally, we present two potential applications (section 7). + +We implement our method using both the STG and HC distributions and compare their applicability for feature selection. The hyperparamters of all the methods are optimized using validation sets via Optuna (Takuya Akiba & Koyama, 2019). In the Appendix, we provide all details of the Optuna based parameter tuning procedure. Additional experiments also appear in the Appendix, including high dimensional datasets and an extensive comparisons between the STG and HC (Louizos et al., 2017) distributions. + +# 6.1 SUPPORT RECOVERY IN LINEAR REGRESSION + +In the setting of noisy linear regression, (Wainwright, 2009) have analyzed the probability of LASSO to correctly identify a sparse subset of active variables. The problem is know as support recovery and is formulated as follows; let $\beta ^ { * } \in \mathbb { R } ^ { D }$ be a fixed sparse vector, such that $\beta _ { i } ^ { * } \in \{ - \overline { { 0 . 5 } } , 0 . 5 \}$ (with equal probability) if $i \in S$ , and $\beta _ { i } ^ { * } = 0$ otherwise. Suppose the cardinality of the support $| S | = k$ is known. Given a matrix of measurements $\pmb { X } \in \mathbb { R } ^ { N \times D }$ with values drawn independently from $N ( 0 , 1 )$ , the response $\textbf { { y } }$ is defined as + +$$ +\pmb { y } = \pmb { X } \beta ^ { * } + \pmb { w } , +$$ + +where the values of the noise $\pmb { w } _ { i } , i = 1 , . . . , N$ are drawn independently from $N ( 0 , 0 . 5 )$ + +Here, we reproduce this setting to evaluate the probability of the proposed approach to perfectly recover the support of $\beta ^ { * }$ . As suggested by (Wainwright, 2009), we use a sparsity that scales with $D$ such that $k = \lceil 0 . 4 D ^ { 0 . 7 5 } \rceil$ . For each number of samples $N$ in the range [10, 500], we run 200 simulations and count the portion of correctly recovered supports. We repeat this process for 2 different values of $D$ and compare our performance to LASSO. For LASSO, the regularization parameter was set to its optimal value $\begin{array} { r } { \alpha _ { N } = \sqrt { \frac { 2 \sigma ^ { 2 } \log ( D - k ) \log ( k ) } { N } } } \end{array}$ (Wainwright, 2009). For STG and HC we set $\lambda _ { N } = C \alpha _ { N }$ , such that $C$ is a constant, which is selected using a grid search in the range [0.1,10]. As evident from Fig. 1, even when restricting to linear functions our method has a clear advantage over LASSO. This implies that the $\ell _ { 0 }$ based penalty, although is non convex in nature, allows perfect recovery of the support using less samples. Furthermore, the proposed STG requires even less samples than the HC distribution and suffers from a smaller variance. + +![](images/b1fc1c69a9a18bb965657ede00fd6a171e90b89b1739aae91ad9b2b8a9d49603.jpg) +Figure 1: Probability of successfully recovering the support of a sparse signal $\boldsymbol { \beta } ^ { * } \in \mathbb { R } ^ { D }$ vs. the number of observations $N$ . Comparison between the proposed method, HC and LASSO under a linear additive noise model $\pmb { y } = \pmb { X } \beta ^ { * } + \pmb { w }$ using (a) $D = 6 4$ and (b) $D = 1 2 8$ . Central lines are the means while shaded are represent the standard deviation. + +# 6.2 NONLINEAR REGRESSION USING SYNTHETIC AND REAL DATASETS + +In this section, we evaluate our method for regression tasks against two other embedded feature selection methods: LASSO and Sparse Random Fourier Features (Gregorová et al., 2018). Following the same format as (Gregorová et al., 2018), the following functions are used to generate synthetic data: (SE1: 100/5) $y = \sin { ( x _ { 1 } + x _ { 3 } ) ^ { 2 } } \sin { ( x _ { 7 } x _ { 8 } x _ { 9 } ) } + \mathcal { N } ( 0 , 0 . 1 )$ . (SE2: 18/5) $\begin{array} { r } { y = \log ( ( \sum _ { s = 1 1 } ^ { \mathbf { \bar { 1 } } 5 } x _ { s } ) ^ { 2 } ) + } \end{array}$ $\mathcal { N } ( 0 , 0 . 1 )$ . (SE3: 1000/10) $y = 1 0 ( z _ { 1 } ^ { 2 } + z _ { 3 } ^ { 2 } ) e ^ { - 2 ( z _ { 1 } ^ { 2 } + z _ { 3 } ^ { 2 } ) } + \mathcal { N } ( 0 , 0 . 0 1 )$ , where each $x _ { i }$ is drawn from the standard Gaussian $\mathcal { N } ( 0 , 1 )$ . For (SE3), the five consecutive coordinates are generated by ${ x _ { 5 } } _ { ( j - 1 ) + i } = { z _ { j } } + \mathrm { { N } } ( 0 , 0 . 1 )$ , where $z _ { j } \sim \mathcal { N } ( 0 , 1 )$ for $i = 1 , . . . , 5$ and $j = 1 , . . . , 2 0 0$ . The numbers next to the experiment code indicate total dimensions/relevant dimensions in the feature space. We also evaluate our method using three real datasets: (RCP: 21/-), of computer systems activity, (REL 17/-) of F16 elevators and (RAI 39/-) of F16 ailernos all taken from the LIACC repository 2. For each dataset, we generate 30 different replications and randomly split the data into train, validation, and test set (see Appendix for more details). The root mean squared error on the test set averaged over 30 random replicated datasets are reported in Table 1. Our method outperforms all alternative methods for most cases. Note that (SE1) is generated using a sine function, which is in favor of the random Fourier feature based method (SRFF). + +Table 1: Regression performance comparison in terms of root mean squared error. The mean and standard deviation are computed across 30 resamples. $N$ is 1000 for SE1,2,3 and 6000 for RCP. The values for LASSO and SRFF are borrowed from (Gregorová et al., 2018). DNN represents a deep neural network without feature selection. + +
EXPLASSORFSG-L1-NNSRFFDNNHCSTG
SE10.29 (0.01)0.30 (0.01)0.29 (0.01)0.27 (0.01)0.29 (0.01)0.29 (0.01)0.29 (0.01)
SE22.22 (0.10)2.34 (0.17)2.35 (0.18)1.60 (0.10)2.05 (0.11)1.19 (0.31)0.87 (0.15)
SE30.68 (0.002)0.50(0.01)0.68 (0.01)0.48 (0.03)0.73 (0.02)0.33 (0.05)0.14 (0.10)
RCP9.69 (0.71)3.52 (0.12)9.64 (0.65)2.52 (0.18)2.89 (0.25)2.74 (0.75)2.44 (0.08)
REL0.47 (0.01)0.58 (0.01)0.44 (0.01)0.31 (0.03)0.61 (0.01)0.33 (0.04)0.27 (0.002)
RAI0.43 (0.02)0.48 (0.003)0.47(0.002)0.41(0.02)0.44 (0.01)0.41 (0.01)0.39 (0.01)
+ +6.3 NOISY BINARY XOR CLASSIFICATION + +In the following evaluation, we consider the problem of learning a binary XOR function for classification task. The first two coordinates $x _ { 1 } , x _ { 2 }$ are drawn from a binary "fair" Bernoulli distribution. The response variable is set as an XOR of the first coordinates, such that $y = x _ { 1 } \oplus x _ { 2 }$ . The coordinates $x _ { i } , i = 3 , . . . , D$ are nuisance features, also drawn from a binary "fair" Bernoulli distribution. The number of points we generate is $N = 1 , 5 0 0$ , of which $7 0 \%$ are reserved for test and $1 0 \%$ of the remaining training set was reserved for validation. We compare the proposed method to four embedded feature selection methods (LASSO (Tibshirani, 1996), C-support vectors (SVC) (Chang & Lin, 2011), deep feature selection (DFS) (Li et al., 2016), sparse group regularized NN (SG-L1-NN) (Scardapane et al., 2017)). To provide more benchmarks, we also compare our embedded method against three wrapper methods (Extremely Randomized Trees (Tree) (Rastogi & Shim, 2000), Random Forests (RF) (Strobl et al., 2008)) and Extreme Gradient Boosting (XGBOOST) (Chen & Guestrin, 2016). + +To evaluate the feature selection performance, we calculate the Informative Features Weight Ratio (IFWR). IFWR is defined as the sum of weights $W _ { d }$ over the informative features divided by the sum over all weights. In the case of binary weights the IFWR is in fact a recall measure for the relevant features (See the Appendix for more details.) + +The experiment is repeated 20 times for different values of $D$ , and the average test classification accuracy and standard deviation are presented in Fig. 2(a), followed by the IFWR in Fig. 2(b). The number of selected features affects the accuracy. Therefore, to treat all the methods in a fair manner, we tune the hyperparameter that controls the sparsity level using Optuna (Takuya Akiba & Koyama, 2019) which optimizes the overall accuracy across different $D \mathrm { s }$ . For instance, the wrapper methods (Tree, RF and XGBOOST) has a threshold value to retain features. We retrain them using only such features whose weight is higher than the threshold. In terms of feature ranking (see Fig. 2(c), only the tree based methods and the proposed (STG and HC based) provide the optimal median rank (which is 1.5) for the two relevant features. Nonetheless, the ranking provided by STG is the most stable comparing to all the alternative methods. + +![](images/06d3d26781acd541fbd45798a0a833a36dd7481d585a6b1bfa91e291fd85278b.jpg) +Figure 2: (a) Classification accuracy (mean and standard deviation) vs. the number of irrelevant noisy dimension $( D )$ for the XOR problem. (b) The Informative Features Weight Ratio (IFWR), central lines are the means while shaded are represent the standard deviation. IFWR is the sum of weights $W _ { d }$ over the informative features divided by the sum over all weights. (c) Box plots for the median rank of the two informative features. Black line and dashed red represent the median and mean of each method. Optimal median rank in this experiment is 1.5. + +# 7 APPLICATION + +# 7.1 PURIFIED POPULATIONS OF PERIPHERAL BLOOD MONOCYTES (PBMCS) + +In (Zheng et al., 2017), the authors have subjected more than 90, 000 purified cell populations of peripheral blood monocytes (PBMCs) to Single-cell RNA sequencing (scRNA-seq) analysis. Here we use this data and focus on classifying two subpopulations of T-cells, namely Naive and regulatory T-cells. We use the proposed method to select a subset of genes for which the network discriminates between Naive and regulatory T-cells. We first filter out the genes that are lowly expressed in the cells, which leaves us with $D = 2 5 3 8$ genes (features). The total number of cells in these two classes is $N = 2 0 7 4 2$ , of which we only use $1 \bar { 0 } \%$ of the data for training. We apply the proposed method for different values of $\lambda$ and report the number of selected features and classification accuracy on the test set. Here we compare our performance (STG and HC) to RF and LASSO. A least squares polynomial fit plot of the accuracy vs. number of selected features is presented in Fig. 3. The accuracy obtained by a NN without feature selection is $9 1 . 0 9 \%$ , which is comparable to what we achieve with a small fraction of the features. We have also evaluated the performance of the Hard-Concrete applied to all layers (HC-Full), following the procedure in (Louizos et al., 2017). Empirically we observed that using this type of regularization across all layers provides inferior capabilities in terms of feature selection. Moreover, when the HC-Full converges to a larger subset of features $( d > 5 0$ ), it does not generalize at all and the test accuracy is around 0.5. + +Table 2: Performance comparison of survival analysis on METABRIC. We run the same experiment 5 times with different train/test split and report the mean and the standard deviation on the test set. In (Katzman et al., 2018), it is reported that DeepSurv outperforms other existing survival analysis methods such as Random Survival Forest (RSF) (Ishwaran et al., 2008) and the original Cox Propotional Hazard Model. + +
DEEPSURVRSFCoX-LASSOCox-HCCoX-STG
C-INDEX0.612 (0.009)0.626 (0.006)0.580 (0.003)0.636 (0.007)0.633 (0.005)
# FEATURES221 (ALL)221 (ALL)44(0)8 (0.89)2（0)
+ +![](images/405bfb64f5f8487932868437d35db6bb0f8b6fbab6429900d03e75fc60c86ce7.jpg) +Figure 3: Classification of T-cells sub-populations. Accuracy vs. number of features selected by each method. Comparison of the proposed method (STG and HC) to Random Forests (RF) and LASSO. + +7.2 COX PROPORTIONAL HAZARD MODELS FOR SURVIVAL ANALYSIS + +We incorporate our STG into DeepSurv (Katzman et al., 2018) to examine how this procedure improves survival analysis. The input to this analysis are gene expression profiles (Curtis et al., 2012) along with additional commonly used clinical variables, the goal is to predict survival time. In this setting, identifying a small subset of predictive covariates is important as it may lead to cheaper and more stable medical assays. See the Appendix for additional description of the data and the experimental setup. + +We compared our method (Cox-STG and COX-HC) against three other methods: Cox model with $\ell _ { 1 }$ regularization (Cox-LASSO), Random Survival Forest (RSF) (Ishwaran et al., 2008) and the original DeepSurv. We evaluate the predictive ability of the learned models based on the concordance index (CI), a standard performance metric for model assessment in survival analysis, which measures the agreement between the rankings of the predicted and observed survival times. The performance in terms of the CI is reported in Table 2. We see that Cox-HC and Cox-STG outperform the alternative methods, indicating that our method shrinks the feature size while learning a predictive model. + +# 8 CONCLUSION + +In this paper, we propose a novel embedded feature selection method for neural networks and linear models based on stochastic gates. 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The loss $L$ is typically negative-log likelihood for classification and squared loss for regression. $N$ is the sample size, $D$ is the number of features, $K$ is the number of Monte Carlo samples. + +Input: $\pmb { X } \in \mathbb { R } ^ { N \times D }$ , target variables $\boldsymbol { y } \in \mathbb { R } ^ { N }$ , regularization parameter $\lambda$ , number of epochs $M$ , learning rate $\gamma$ , + +Output: Trained model $f _ { \pmb { \theta } }$ and parameter $\boldsymbol { \mu } \in \mathbb { R } ^ { D }$ + +. +1: Initialize the model parameter $\pmb { \theta }$ . Set $\pmb { \mu } = 0$ . +2: for $i = 1 , . . . , M$ do +3: for $n = 1 , . . . , N$ do +4: for $d = 1 , . . . , D$ do +5: for $k = 1 , . . . , K$ do +6: Sample (k)d ∼ $\epsilon _ { d } ^ { ( k ) } \sim N ( 0 , \sigma _ { d } )$ +7: Compute the gate z(kd $z _ { d } ^ { ( k ) } = \operatorname* { m a x } ( 0 , \operatorname* { m i n } ( 1 , \mu _ { d } + \epsilon _ { d } ^ { ( k ) } + 0 . 5 ) )$ +8: end for +9: $\begin{array} { r } { z _ { d } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } z _ { d } ^ { ( k ) } } \end{array}$ +10: end for +11: Set $\boldsymbol { z } = [ z _ { 1 } , . . . , z _ { D } ] ^ { T }$ +12: 13: end forCompute the loss $\begin{array} { r } { \hat { L } = \frac { 1 } { N } \sum _ { n = 1 } ^ { N } L ( f _ { \theta } ( \pmb { x } _ { n } \odot \pmb { z } ) , y _ { n } ) } \end{array}$ +14: N Compute the regularization term $\begin{array} { r } { R = \lambda \sum _ { d = 1 } ^ { D } \Phi ( \frac { \mu _ { d } + 0 . 5 } { \sigma _ { d } } ) } \end{array}$ +15: Update $\pmb { \theta } : = \pmb { \theta } - \gamma \nabla _ { \pmb { \theta } } \hat { L }$ and $\pmb { \mu } : = \pmb { \mu } - \gamma \nabla \pmb { \mu } ( \hat { L } + R )$ +16: end for + +17: + +Algorithm 1: STG: Feature selection using stochastic gates + +We empirically observe that setting the number of Monte Carlo samples $K = 1$ and the standard deviation of the Gaussian distribution $\sigma _ { d } = 0 . 5$ for $d = 1 , . . . , D$ suffices for feature selection in our experiments. See Section E in the Appendix for more details about the specific choice of $\sigma _ { d }$ . After training, the set of indices for selected features is: $\{ d : \operatorname* { m i n } ( 1 , \operatorname* { m a x } ( 0 , \mu _ { d } + 0 . 5 ) ) > 0 \}$ . + +# C PROOF OF PROPOSITION 1 + +We now provide a proof for Proposition 1, showing the equivalence between the stochastic optimization (Eq.7) and the deterministic one (Eq. 6). Let $\tilde { \cal S }$ be a subset such that $S ^ { * } \setminus \tilde { S } \ne \emptyset$ . That is there exists some element in $S ^ { * }$ that is not in $\tilde { \cal S }$ . For any such set $\tilde { \cal S }$ we have that $I ( X _ { \tilde { S } } ; Y ) < I ( X ; Y )$ . Indeed, if we let $i \in { \cal S } ^ { * } \cap \tilde { \cal S } ^ { c }$ then we have + +$$ +\begin{array} { r l } { I ( \boldsymbol { X } _ { \tilde { \mathcal { S } } } ; Y ) } & { \leq I ( \boldsymbol { X } _ { \backslash \{ i \} } ; Y ) } \\ & { = I ( \boldsymbol { X } ; Y ) - I ( \boldsymbol { X } _ { i } ; Y | \boldsymbol { X } _ { \backslash \{ i \} } ) } \\ & { < I ( \boldsymbol { X } ; Y ) , } \end{array} +$$ + +where the final inequality follows by Assumption 1. Assumption 2 also yields that for any set $\tilde { \cal S }$ such that $S ^ { * } \subset \tilde { S }$ , we have $I ( { \pmb X } _ { \tilde { S } } ; Y ) = I ( { \pmb X } ; Y )$ . Now, when we consider the Bernoulli optimization problem we have + +$$ +\operatorname* { m a x } _ { \pi } I ( X \odot S ; Y ) \quad \mathrm { s . t . } \quad \sum _ { l } \pi _ { l } \leq k \mathrm { a n d } 0 \leq \pi _ { l } \leq 1 . +$$ + +The mutual information can be expanded as + +$$ +I ( X \odot S ; Y ) = \sum _ { s } I ( X \odot s ; Y ) p _ { \pi } ( S = s ) , +$$ + +where we have used the fact that $_ { s }$ is independent of everything else. Recall that in optimization (Eq. 7) the coordinates of $\pmb { S }$ are sampled at random. Therefore, the distribution that is being optimized over $p _ { \pi }$ is a product distribution. Our goal is to understand the form of this distribution. To that end, we will consider a problem dropping the independence constraint. If we can show that the distribution found by solving this new optimization problem with less constraints is still a product distribution, then we obtain a solution to the original optimization (Eq. 6). + +Now, from above we know that the optimal value of the optimization is $I ( X \odot { \tilde { S } } ; Y )$ for any set $S ^ { * } \subset \tilde { S }$ . Hence, any unconstrained distribution should place all of its mass on such subsets in order to maximize the mutual information. As a result $\begin{array} { r } { \sum _ { l \in S ^ { * } } p ( S _ { l } = 1 ) = k } \end{array}$ . However, there is an optimization constraint that $\mathbb { E } [ \sum _ { l } S _ { l } ] \le k$ . Therefore, $\mathbb { E } [ S _ { l } ] = 0$ for any $l \notin S ^ { * }$ . Hence, the optimal solution is to select the distribution so that all of the mass is placed on the subset $S ^ { * }$ and no mass elsewhere. As this is also a product distribution, this complete the proof of the claim. + +# D BRIDGING THE TWO PERSPECTIVES + +To motivate the introduction of randomness into the risk, we have looked at the feature selection problem from a MI perspective. Based on the MI objective, we have observed that introducing randomness into the constrained maximization procedure, does not change the objective (Proposition 1). Here we provide a relation between the MI objective (Eq. 6) to the empirical risk (Eq. 1), which supports our proposed procedure. + +We first note that the MI maximization over the set $s$ can be reformulated as the minimization of the conditional entropy $H ( Y | \boldsymbol { X } _ { \mathcal { S } } )$ since $H ( Y )$ does not depend on $s$ : + +$$ +\displaystyle { \operatorname* { m a x } _ { \mathcal { S } } I ( \mathbf { \boldsymbol { X } } _ { \mathcal { S } } ; Y ) = \operatorname* { m a x } _ { \mathcal { S } } H ( Y ) - H ( Y | \mathbf { \boldsymbol { X } } _ { \mathcal { S } } ) \iff \operatorname* { m i n } _ { \mathcal { S } } H ( Y | \mathbf { \boldsymbol { X } } _ { \mathcal { S } } ) } . +$$ + +Recall that $X _ { \mathcal { S } } = X \odot \tilde { S }$ . By Proposition 1, we can rewrite the deterministic search over the set $s$ by a search over the Bernoulli parameters $\pi$ : + +$$ +\begin{array} { r l } { \displaystyle \operatorname* { m i n } _ { \pi } H ( \boldsymbol { Y } | \boldsymbol { X } \odot \tilde { \boldsymbol { S } } ) = \operatorname* { m i n } _ { \pi } \mathbb { E } _ { \boldsymbol { X } , \boldsymbol { Y } , \tilde { \boldsymbol { S } } } - \log P _ { \pmb { \theta } ^ { * } } ( \boldsymbol { Y } | \boldsymbol { X } \odot \tilde { \boldsymbol { S } } ) } & { } \\ { = \displaystyle \operatorname* { m i n } _ { \pmb { \theta } } \operatorname* { m i n } _ { \pi } \mathbb { E } _ { \boldsymbol { X } , \boldsymbol { Y } , \tilde { \boldsymbol { S } } } - \log P _ { \pmb { \theta } } ( \boldsymbol { Y } | \boldsymbol { X } \odot \tilde { \boldsymbol { S } } ) , } \end{array} +$$ + +where the expectation is over $X , Y \sim P _ { \pmb { \theta } _ { * } }$ , which is the true data distribution, and $\tilde { S } \sim B e r n ( \tilde { S } | \pi )$ . Put our model distribution as $P _ { \theta }$ , then we can rewrite the right hand side as: + +$$ +\begin{array} { r l } & { \mathbb { E } _ { X , Y , \tilde { S } } \log P _ { \theta ^ { * } } \big ( Y | X \odot \tilde { S } \big ) = \mathbb { E } _ { X , Y , \tilde { S } } \Bigg [ \log P _ { \theta ^ { * } } \big ( Y | X \odot \tilde { S } \big ) \frac { P _ { \theta } \big ( Y | X \odot \tilde { S } \big ) } { P _ { \theta } \big ( Y | X \odot \tilde { S } \big ) } \Bigg ] } \\ & { \qquad = \mathbb { E } _ { X , Y , \tilde { S } } \log \frac { P _ { \theta ^ { * } } \big ( Y | X \odot \tilde { S } \big ) } { P _ { \theta } \big ( Y | X \odot \tilde { S } \big ) } + \mathbb { E } _ { X , Y , \tilde { S } } \log P _ { \theta } \big ( Y | X \odot \tilde { S } \big ) . } \end{array} +$$ + +Since $\mathrm { K L } ( P _ { \theta ^ { * } } ( Y | X \odot \tilde { S } ) | | P _ { \theta } ( Y | X \odot \tilde { S } ) )$ is non-negative, $\mathbb { E } _ { \tilde { S } } \mathbf { K } \mathbf { L } ( P _ { \theta * } ( Y | \boldsymbol { X } \odot \tilde { \pmb { S } } ) | | P _ { \theta } ( Y | \boldsymbol { X } \odot \tilde { \pmb { S } } ) )$ is also non-negative because it is a weighted sum of non-negative terms. Noting that + +$$ +\mathbb { E } _ { \tilde { S } } \mathrm { K L } \big ( P _ { \theta * } ( Y | X \odot \tilde { S } ) | | P _ { \theta } ( Y | X \odot \tilde { S } ) \big ) = \mathbb { E } _ { X , Y , \tilde { S } } \log \frac { P _ { \theta * } ( Y | X \odot \tilde { S } ) } { P _ { \theta } ( Y | X \odot \tilde { S } ) } +$$ + +we can conclude that + +$$ +\mathbb { E } _ { X , Y , \tilde { S } } - \log P _ { \theta ^ { * } } ( Y | X \odot \tilde { S } ) \leq \mathbb { E } _ { X , Y , \tilde { S } } - \log P _ { \theta } ( Y | X \odot \tilde { S } ) . +$$ + +If we consider the negative log likelihood of the target given the observations (i.e. $- \log P _ { \theta } ( Y | X \odot \tilde { \pmb { S } } ) )$ as a loss function $L$ (which encodes the classification or regression function $f _ { \pmb { \theta } } ,$ ), then we see that minimizing the risk approximately maximizes the MI objective in Eq. 6. + +# E DETAILS OF REGULARIZATION TERM + +Here we provide the detail description of the regularization term. For the vector of stochastic gates $z \in \mathbb { R } ^ { D }$ , the regularization term is expressed as follows: + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } _ { Z } \| Z \| _ { 0 } = \displaystyle \sum _ { d = 1 } ^ { D } \mathbb { P } [ z _ { d } > 0 ] = \displaystyle \sum _ { d = 1 } ^ { D } \mathbb { P } [ \mu _ { d } + \sigma _ { d } \epsilon _ { d } > - \frac { 1 } { 2 } ] } \\ { = \displaystyle \sum _ { d = 1 } ^ { D } \{ 1 - \mathbb { P } [ \mu _ { d } + \sigma _ { d } \epsilon _ { d } \leq - \frac { 1 } { 2 } ] \} } \\ { = \displaystyle \sum _ { d = 1 } ^ { D } \{ 1 - \Phi ( - \frac { 1 } { \sigma _ { d } } ) \} } \\ { = \displaystyle \sum _ { d = 1 } ^ { D } \Phi ( \frac { \mu _ { d } + \frac { 1 } { 2 } } { \sigma _ { d } } ) . } \end{array} +$$ + +The derivative of the regularization term with respect to the distribution parameter $\mu _ { d }$ is simply the Gaussian PDF: + +$$ +\frac { \partial } { \partial \mu _ { d } } \mathbb { E } _ { Z } \left\| Z \right\| _ { 0 } = \frac { \partial } { \partial \mu _ { d } } \Phi ( \frac { \mu _ { d } + \frac { 1 } { 2 } } { \sigma _ { d } } ) = \frac { 1 } { \sqrt { 2 \pi \sigma _ { d } ^ { 2 } } } e ^ { - \frac { ( \mu _ { d } + \frac { 1 } { 2 } ) ^ { 2 } } { 2 \sigma _ { d } ^ { 2 } } } . +$$ + +The effect of $\sigma$ can be understood by looking at the value of $\frac { \partial } { \partial \mu _ { d } } \mathbb { E } _ { Z } | | Z | | _ { 0 }$ . In the first iteration during the training, $\mu _ { d }$ is 0. Therefore, during the initial phase of training, it is close to $\frac { \lambda } { \sqrt { 2 \pi \sigma _ { d } ^ { 2 } } } e ^ { - \frac { 1 } { 8 \sigma _ { d } ^ { 2 } } }$ . In order to remove irrelevant features, this term has to be greater than the derivative of the loss with respect to $\mu _ { d }$ because otherwise $\mu _ { d }$ is updated in the incorrect direction. To encourage such behavior, we set $\sigma = 0 . 5$ , which is around the maximum of the gradient during the initial phase as shown in Fig. 4. Although the point that attains the maximum moves as $\mu$ changes, we empirically observe that setting $\sigma = 0 . 5$ performs well in our experiments when the regularization parameter $\lambda$ is appropriately tuned (see Section I). + +![](images/35b45d282fb48659f375af3711fb64ad2e5ebd436d2ce1904752054ea3096810.jpg) +Figure 4: The plot of ∂∂µ EZ||Z||0|µ=0 = √ 12πσ2 e for $\sigma = [ 0 . 0 0 1 , 2 ]$ + +# F ISSUES IN GRADIENT ESTIMATION OF DISCRETE RANDOM VARIABLES + +In Section 3.1, we have introduced Bernoulli random $\widetilde { s } _ { d } , d = 1 , . . . , D$ variables with corresponding parameters $\pi _ { d }$ into the risk objective (Eq. 4). Taking the expectation over the $\ell _ { 0 }$ norm of $\tilde { s }$ boils down to the sum of the Bernoulli parameters $\pi _ { d }$ . However, the optimization of the resulting objective suffers from high variance due to the discrete nature of $\tilde { S }$ . Here, we attempt to convey this problem by analyzing the risk term in the objective in Eq. 4. Using the Bernoulli paramterization the empirical risk $\hat { R } ( \pmb \theta , \pmb \pi )$ is expressed as + +$$ +\sum _ { \mathscr { Z } : \{ 0 , 1 \} ^ { D } } \left[ \sum _ { n = 1 } ^ { N } [ L ( f _ { \theta } ( \mathscr { z } \odot \mathscr { x } _ { n } ) , \pmb { y } _ { n } ] \prod _ { d = 1 } ^ { D } \pi _ { d } ^ { \mathscr { z } _ { d } } ( 1 - \pi _ { d } ) ^ { 1 - \mathscr { z } _ { d } } \right] . +$$ + +In practice, as the outer sum involves enumerating $2 ^ { D }$ possibilities of the indicator variables, one can replace the outer sum with Monte Carlo samples from the product of Bernoulli distributions $B ( z | \pi )$ . However, a Monte Carlo estimate of $\frac { \partial } { \partial \pi _ { d } } \hat { R } ( \pmb \theta , \pmb \pi )$ suffers from high variance. To see this, consider the following exact gradient of the empirical risk with respect to $\pi _ { d }$ , which is + +$$ +\sum _ { z : \{ 0 , 1 \} ^ { D } , z _ { d } = 1 } \left[ L ( z ) p _ { z _ { i } \neq d } \right] - \sum _ { z : \{ 0 , 1 \} ^ { D } , z _ { d } = 0 } \left[ L ( z ) p _ { z _ { i } \neq d } \right] , +$$ + +where $p ( z _ { i \neq d } ) = \prod _ { i \neq d } ^ { D } \pi _ { i } ^ { z _ { i } } ( 1 - \pi _ { i } ) ^ { 1 - z _ { i } }$ , by absorbing the model $f _ { \theta } ( \cdot )$ and the data into $L ( \cdot )$ . Due to the discrete nature of $_ z$ , we see that even the sign of the gradient estimate becomes inaccurate if we can only access a small number of Monte Carlo samples. While a score-function estimator such as REINFORCE (Williams, 1992) can be used, it is known that the reparametrization trick reduces the variance more in practice. + +# G COMPARISON TO HARD-CONCRETE DISTRIBUTION + +To evaluate the strength of the proposed continuous relaxation of the Bernoulli distribution, described in Subsection 3.1, we compare it with the Hard-Concrete distribution (Louizos et al., 2017), another continuous surrogate for Bernoulli distributions, which was originally developed for neural network model compression. The details of the Hard-Concrete distribution is described below. + +# G.1 HARD-CONCRETE DISTRIBUTION + +The authors in (Louizos et al., 2017) introduce a modification of Binary Concrete, whose sampling procedure is as follows: + +$$ +\begin{array} { r } { u \sim U ( 0 , 1 ) , L = \log ( U ) - \log ( 1 - U ) } \\ { s = \frac { 1 } { 1 + \exp ( \frac { - ( \log { \alpha } + L ) } { \beta } ) } } \\ { \bar { s } = s ( \zeta - \tau ) + \tau } \\ { z = \operatorname* { m i n } ( 1 , \operatorname* { m a x } ( 0 , \bar { s } ) ) } \end{array} +$$ + +where $( \tau , \zeta )$ is an interval, with $\tau < 0$ and $\zeta > 1$ . This induces a new distribution, whose support is $[ 0 , 1 ]$ instead of $( 0 , 1 )$ . With $0 < \beta < 1$ , the probability density concentrates its mass near the end points, since values larger than $\frac { 1 - \tau } { \zeta - \tau }$ are rounded to one, whereas values smaller than $\frac { - \tau } { \zeta - \tau }$ are rounded to zero. + +The CDF of $s$ is + +$$ +Q _ { s } ( s | \beta , \log \alpha ) = \mathrm { { S i g m o i d } } ( ( \log s - \log ( 1 - s ) ) \beta - \log \alpha ) +$$ + +and so the CDF of $\bar { s }$ is + +$$ +Q _ { \bar { s } } ( \bar { s } | \phi ) = \mathrm { S i g m o i d } ( ( \log ( \frac { \bar { s } - \tau } { \zeta - \tau } ) - \log ( 1 - \frac { \bar { s } - \tau } { \zeta - \tau } ) ) \beta - \log \alpha +$$ + +where $\phi = ( \beta , \log \alpha , \zeta , \tau )$ + +Now, the probability of being the gate $z _ { i }$ being active is $1 - Q _ { \bar { s } } ( 0 | \phi )$ and can be written as + +$$ +1 - Q _ { \bar { s } } ( 0 | \phi ) = \mathrm { S i g m o i d } ( \log \alpha - \beta \log \frac { - \tau } { \zeta } ) +$$ + +# G.2 THE EFFECT OF THE HEAVY TAIL DISTRIBUTION + +The main difference between our proposed distribution (STG) and the Hard-Concrete (Louizos et al., 2017) distribution is that the latter is based on the logistic distribution, which has a heavier tail than the Gaussian distribution we have employed. As shown in Fig 5, the heavy-tailness results in instability during training. Furthermore, our method converges much faster and more reliably than the feature selection method using the Hard-Concrete distribution on a the two-moons, XOR and MADELON datasets (see Subsection J.1 and J.2). A similar phenomenon is also demonstrated in the MNIST experiment (see Subsection J.3). + +![](images/d25fe19f87e70d251501c821c2f367a6207558df2e41bbf126700c19922f444a.jpg) +Figure 5: Comparison between STG and Hard-Concrete on the Two-Moon dataset (a), XOR (b) and MADELON (c). The shaded area represents the standard deviation, calculated by running the same experiment 10 times with different random initializations. + +# H NON-CONVEX DETERMINISTIC REGULARIZATION + +Our proposed objective for feature selection (summarized in Eq. 5) consists of three major differences compared to the LASSO formulation. + +• Non-linearity, which is obtained by learning parameters of a multi-layer network with non linear activation’s. • Feature gates, these are a specific form on non convex regularization. • Stochasticity, which is achieved by injecting noise via the reparametrization trick. + +In section 6.1, we have demonstrated that even in the linear regime the proposed formulation outperforms the LASSO in the task of identifying active features (see Fig. 1(a)). + +The stochastic property of the gates proposed in this paper is justified using a MI prospective (see Section 4). Nonetheless, in this section we evaluate whether the practical performance gain of our method stems from the non-convex regularization or from it’s combination with the injected noise. + +We define a deterministic gate using $\tilde { z } _ { d } = g ( \mu _ { d } ) = \operatorname* { m a x } ( 0 , \operatorname* { m i n } ( 1 , \mu _ { d } + 0 . 5 ) )$ , where $\mu _ { d }$ is a parameter learned for each feature $d = 1 , . . . , D$ . In the linear regression setting we can define the non-convex deterministic (NCD) as + +$$ +\operatorname* { m i n } _ { \theta , \mu } \frac { 1 } { N } \sum _ { n = 1 } ^ { N } ( \theta ^ { T } x _ { n } \odot \tilde { z } - y _ { n } ) ^ { 2 } + \lambda \sum _ { d = 1 } ^ { D } \Phi \left( \frac { \mu _ { d } + 0 . 5 } { 0 . 5 } \right) , +$$ + +where $\Phi$ is the standard Gaussian CDF. Combined with $\tilde { z } _ { d }$ this non-convex regularized objective is deterministic and differentiable, and its solution can be approximated via gradient decent. We evaluate the applicability of such alternative objective for the task of feature selection in linear regression. The experimental setting follows the description in Section 6.1 using $D = 6 4$ . + +In Fig. 6(b), we compare the values of the deteministic gates (DNC) and stochastic gates (STG) using $N = 6 0$ . As depicted from this figure, the STG correctly identifies the active features (first 10) and sparsifies the non-active features. The deterministic version DNC does induce partial sparsity, but fails to remove some of the irrelevant features. Furthemore, it does not push the gates values of active features up to 1. Clearly, in the deterministic setting, shrinking the value of $\mu _ { d }$ while compensating with a large coefficient in $\pmb \theta$ is preferable for the objective in Eq. 12. + +To evaluate the probability of support recovery using DNC, we apply additional thresholding to the gates. Specifically, we use a threshold of 0.5 to define if a feature is active or not. A comparison with STG, HC and LASSO in this setting appears in Fig. 6(a). The DNC is superior to LASSO, but inferior to the HC and STG. Which means that there is a true advantage of stochacity in our formulation. + +One aspect that is crucially difference between the STG and DNC is what we call “second chance”. For the deterministic formulation, if a feature is zeroed out in an early training phase, the gradient of the feature vanishes for further training and it will never be pulled back. For the STG, even if the feature has been pulled to zero, the gate may still be pushed back up to one at a later training phase. This is due to the injected noise which allows to reevaluate the gradient of each gate. In Fig. 6(c), we demonstrate this “second chance” effect using $N = 6 0$ samples and presenting the gate’s values (throughout training) for an active feature. + +To conclude, this form of deterministic gates (Eq. 12) is not suitable for an embedded feature selection method. Using thresholding seems somewhat effective; however such thresholding was not required for the STG in any of our experiments. While the shrinkage phenomenon might be avoided using further regularization on $\pmb \theta$ this is out of the scope of the current study. + +![](images/2233abd145920d07fd52782d2454c31290b15af50d5ee60750b6ba242b8c7605.jpg) +Figure 6: Support recovery in the linear regression (see Section 6.1), the dimension $D = 6 4$ . (a) Phase transition comparison between the STG, HC, LASSO and the deterministic non convex (DNC). (b) An example of the output gates, STG $( z _ { d } )$ and DNC $( \tilde { z } _ { d } )$ for $N = 6 0$ . Each marker represents the gate value of the corresponding feature, of which the first 10 features are active. The deterministic gates are not pushed towards 1. (c) Demonstration of the “second chance” inherit to the STG. In the deterministic setting if a feature is eliminated its gradient vanishes for the rest of the training. The STG reevaluates this features and “recovers” after 4000 epochs. + +# I TUNING THE REGULARIZATION PARAMETER $\lambda$ + +There have been several studies (Lederer & Müller, 2015; Coen et al., 2006) on the problem of tuning regularization parameter in regression and classification. Perhaps the most related regularized objective is the LASSO formulation. Studies such as (Chand, 2012; Fan & Tang, 2013) develop methods for tuning the regularization parameter in LASSO. In Eq. 5 as in the LASSO formulation, increasing the value of $\lambda$ effectively forces a sparser solution with less features. For simplicity, we focus on one popular way for tuning $\lambda$ in the LASSO. The idea is to minimize the generalization error using a cross validation procedure over a range of $\lambda$ values. + +In this section, we demonstrate that this procedure is also applicable for the proposed approach. We apply the STG to the RAI dataset and evaluate the MSE using a 5-folds cross validation procedure. We compute the MSE on the validation data using 100 values of $\lambda$ logarithmically scaled in $[ 1 0 ^ { - 3 } , 1 0 ^ { 1 } ]$ As demonstrated in Fig. 7, using a cross validation procedure for tuning $\lambda$ provides a clear optimal value for this example. + +![](images/232c0999f711f8ccec17e4f6fbd02a77cb7878b79372594bf8c4c1568fd246d9.jpg) +Figure 7: Demonstrating a cross validation procedure on the RAI dataset (see section I for details). We perform 5-folds cross validation and evaluate the MSE on the test test. The optimal $\lambda = 0 . 1 3 5$ which seems stable in this example based on the low standard deviation. + +# J ADDITIONAL EXPERIMENTS + +# J.1 MADELON DATASET + +The MADELON dataset, first suggested for the NIPS 2003 feature selection problem, is a multivariate highly nonlinear binary classification problem. The MADELON dataset is generated using 32 groups of data points placed on a 5 dimensional hyper-cube and randomly labeling them by one of the two class labels. The first 5 informative features are then used to create 15 additional coordinates that are formed based on a random linear transformation of the first 5. A Gaussian noise $N ( 0 , 1 )$ is added to each feature. Next, additional 480 nuisance coordinates are added in the same manner. These features have no effect on the class label. Finally, $1 \%$ of the labels are flipped. To be consistent with the XOR and two moons experiments, we use 1, 500 points from this dataset, and evaluate our proposed method in terms of its predictive power and ability to detect the informative features. We vary the regularizaton parameter $\lambda$ in the range [0.01, 10] and evaluate the classification accuracy using 5 folds cross validation. In this example, we restrict our comparison to Random Forest and LASSO. We focus on Random Forest as it was the strongest competitor to our method in all of our experiments, while LASSO is evaluated because it is a widely used embedded feature selection method. + +As evident from Fig. 8(a), our method achieves the highest accuracy while using less features. Moreover, as depicted from this figure, peak performence occurs when selecting 5 features, thus, our method provides a clear indication to the true number of informative features. Both LASSO and RF on the other hand, do not provide a clear indication of the true number of relevant features. In Fig. 8(b), we evaluate the effect of $\lambda$ on the number and quality of selected features. As shown in this plot, there is a wide range of $\lambda$ ’s such that our method only selects relevant features. Finally, as evident from the plato on the right hand side of the red plot, there is a range of $\lambda$ ’s such that exactly 5 features are selected. + +![](images/f81c197fa97377e7b0126d38b6153864d82047273e824cbb916508ca6b386556.jpg) +Figure 8: (a) Classification accuracy on the MADELON data sets. We evaluate performance using 5-fold cross validation for different number of selected features. In this dataset, only the first 20 coordinates are informative. In that regime the proposed method outperforms RF and LASSO. (b) An empirical evaluation of the effect the regularization parameter $\lambda$ . The IFWR and the number of selected features are presented on both sides of the $y$ -axis of this plot. For both plots, the mean is presented as a solid/dashed line, while the standard deviation is marked as a shaded color around the mean. (c) Box plots for the median rank of the 5 original informative features. Black line and dashed red represent the median and mean of each method. Optimal median rank in this experiment is 3. + +# J.2 TWO MOONS CLASSIFICATION WITH NUISANCE FEATURES + +In this experiment, we construct a dataset based on "two moons" shape classes, concatenated with noisy features. The first two coordinates $x _ { 1 } , x _ { 2 }$ are generated by adding a Gaussian noise with zero mean and the variance of $\sigma _ { r } ^ { 2 } = 0 . 1$ onto two nested half circles, as presented in Fig. 9(a). Nuisance features $x _ { i } , i = 3 , . . . , D$ , are drawn from a Gaussian distribution with zero mean and variance of $\sigma _ { n } ^ { 2 } = 1$ . We reserve the $7 0 \%$ as a test set, and use $1 0 \%$ of the remaining training set as a validation set. We follow the same hyperparameter tuning procedure as in the XOR experiment. The classification accuracy is in Fig. 9(b). Based on the classification accuracies, it is evident that for a small number of nuisance dimensions all methods correctly identify the most relevant features. The proposed method (STG) and Random Forest (RF) are the only methods that achieve near perfect classification accuracy for a wide range of nuisance dimensions. The other NN based method (DFS) seem to converge to sub-optimal solutions. We note that the median rank for all the methods is 1.5. + +![](images/ba3a08516b182a43c04de57e2dc57232418b15b4618ba3a7fb4d6140a06cd2d5.jpg) +Figure 9: (a) Realizations from the "Two moons" shaped binary classification class. $X _ { 1 }$ and $X _ { 2 }$ are the relevant features, $X _ { i } , i = 3 , . . . , D$ are noisy features drawn from a Gaussian with zero mean and variance of 1. (b) Classification accuracy (mean and standard deviation based on 20 runs) vs. the number of irrelevant noisy dimension. + +# J.3 SPARSE HANDWRITTEN DIGITS CLASSIFICATION + +In the following toy example, we attempt to distinguish between images of handwritten digits of 3’s and 8’s using samples from MNIST (LeCun et al., 1998). The orientation and location of the digits is more or less the same throughout this dataset, therefore for these two classes (3’s and 8’s), we expect that some of the left side features (pixels) would be sufficient for the separation. The experiment is performed as followed. We reserve $9 0 \%$ of the data as the test set, and train on the remaining $1 0 \%$ . We then apply STG and evaluate the classification accuracy and the number of selected features. We use the architecture [200, 50, 10] with tanh activations. The experiment was repeated 10 times, the extracted features and accuracies were consistent over 20 trials. We noticed a relatively small number of selected features, which are positioned southwest and close to the center of the images, achieve very high classification accuracy. An example of 9 randomly selected samples overlaid with the weights of the selected features is presented in Fig. 10(a). Furthermore, we also evaluate the effect of $\lambda$ on the the number of selected features and accuracy of the method. We apply our method (STG) and its variant using the Hard-Concrete distribution to a randomly sampled training set of size $N = 1 5 0 0$ and vary $\lambda$ in the range of [0.001, 0.01]. In Fig. 10(b) we present the accuracy and sparsity level vs. the $\lambda$ parameter. This experiment demonstrates the improved performance of the proposed distribution compared to the Hard-Concrete (HC (Louizos et al., 2017)), which was designed for neural net model compression. Not only that the overall accuracy is superior, but also it seems that the transition as a function of $\lambda$ is smoother, which suggests that the method is less sensitive to the choice of $\lambda$ . + +# J.4 GISETTE DATASET + +The Gisette is a handwritten dataset also appears in the NIPS 2003 feature selection challenge. The data consists of 13, 500 handwritten digits of $\ ' 4 '$ and ’9’. The original digits have $2 8 \times 2 8$ pixels, which were modified for the feature selection challenge. Specifically, a random subset of the features are embedded in a 2500 dimensional space and additional 2500 irrelevant probes are added. The train, validation and test dimensions are $6 0 0 0 / 1 0 0 0 / 7 5 0 0$ respectively. We apply the proposed STG based feature selection method and compare its performance to Random Forests (RF) and Extremely Randomized Trees (Tree). As evident from Fig. 11, we obtain a high accuracy even for a dramatic reduction in the feature size. LASSO is not presented in this experiment as its performance is dramatically inferior to the alternatives (accuracy $< 0 . 6$ ). + +![](images/63dcdf118a319fb90f36df91f9fae0e284f578ca8325363eeec29265cc548456.jpg) +Figure 10: (a) Nine samples from MNIST (white) overlaid with the subset of 13 features (black) selected by STG. Based on only these features, the binary classification accuracy reaches $9 2 . 2 \%$ . For these nine randomly selected samples, all the 8’s have values within the support of the selected features, whereas for the 3’s there is no intersection. (b) The comparison of accuracy and sparsity level performance for $\lambda$ in the range of $[ 1 0 ^ { - 3 } , 1 0 ^ { - 2 } ]$ between using our proposed method (STG) and its variant using the Hard-Concrete (HC) distribution. + +![](images/5dd199d6e1cc192e49bf859e0f2459a44a82769f43d17f81306e036866dee8a9.jpg) +Figure 11: Classification of the binary noisy digits from the Gisette dataset. The total number of feature is 5000 of which 2500 are irrelevant probes. Here we compare the performance of the proposed approach to RF and Tree based classifier. The lines represent a least squares polynomial fit plot of the accuracy vs. number of selected features. + +# J.5 REUTERS CORPUS VOLUME I + +The Reuters Corpus Volume I (RCV1) consists of 800, 000 newswire stories manually labeled by 103 categories. This is a multilable regime, i.e. each story is assigned to multiple categories. Here we focus on a binary subset of this corpus, with 23, 203 stories. The total number of feature is 47, 236 and the train, validation and test portions are $1 0 \%$ , $8 . 5 \%$ , and $8 1 . 5 \%$ respectively. We evaluate the performance of our method using a 5-fold cross validation. A comparison the LASSO and RF appears in Fig. 12. This example demonstrates that our method is also effective in an extremely high dimensional regime of non linear function estimation. + +# J.6 DETAILS OF COX PROPORTIONAL HAZARD MODEL + +Survival times are assumed to follow a distribution, which is characterized by the survival function $S ( t ) = P ( T > t )$ . A hazard function, which measures the instantaneous rate of death, is defined by $\begin{array} { r } { h ( t ) = \operatorname* { l i m } _ { \Delta t 0 } \frac { P ( t < T \leq t + \Delta t | T > t ) } { \Delta t } = \frac { p ( t ) } { S ( t ) } } \end{array}$ . We can relate the two functions in the following way: S(t) = e− R t0 h(t)dt. + +Proportional hazard models assume a multiplicative effect of the covariates $x$ on the hazard function such that + +$$ +h ( t | \boldsymbol { x } ) = h _ { 0 } ( t ) e ^ { \theta ^ { T } \boldsymbol { x } } , +$$ + +![](images/bbdb8424abb8ba85469b699d8f13201af2a5fba3870fd125b2d054267de5fbd9.jpg) +Figure 12: Classification of the version of the RCV1 textual dataset. The total number of feature is 47, 236. Here we compare the performance of the proposed approach to RF and LASSO. The lines represent a least squares polynomial fit plot of the accuracy vs. number of selected features. + +where $h _ { 0 } ( t )$ is a baseline hazard function, which is often the exponential or Weibull distribution, and $\pmb \theta$ is the parameter of interests. + +One of the difficulties in estimating $\pmb { \theta }$ in survival analysis is that a large portion of the available data is censored. However, in order to obtain estimates, Cox observed that it is sufficient to maximize the partial-likelihood, which is defined as follows: $\begin{array} { r } { L ( \pmb \theta ) = \prod _ { T _ { i } \mathrm { u n c e n s o r e d } } \frac { e ^ { \pmb \theta ^ { T } \pmb x _ { i } } } { \sum _ { T _ { j } \geq T _ { i } } e ^ { \pmb \theta ^ { T } \pmb x _ { j } } } . } \end{array}$ + +In (Katzman et al., 2018), the authors propose DeepSurv, which uses a deep neural network model to replace the linear relations between the covariate $_ { \textbf { \em x } }$ and $\pmb { \theta }$ , demonstrating improvements of survival time prediction over existing models such as CPH and the random survival forest (Ishwaran $\&$ Kogalur, 2007), (Ishwaran et al., 2008). + +The Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) dataset consists of gene expression data and clinical features for 1, 980 patients, and $5 7 . 7 2 \%$ have an observed death due to breast cancer with a median survival time of 116 months. + +The METABRIC dataset involves 24,368 features (genes). Most genes are irrelevant for outcome prediction. To demonstrate the advantage of STG in the context of survival analysis, we selected 16 well-known genes relevant for survival (out of the 24,368 genes) that correspond to the Oncotype DX test, a gene panel used for treatment decision making. We also include five additional clinical features (hormone treatment indicator, radiotherapy indicator, chemotherapy indicator, ER-positive indicator, and age at diagnosis). We then added 200 additional irrelevant gene variables that we selected randomly from the remaining list of genes. + +After we omit the null values, the number of samples is 1969. We reserve the $2 0 \%$ for test, and use the $2 0 \%$ of the remaining training set as validation. (That is, Train: 1260, Valid: 315, Test: 394 samples.) + +Experimental Detail For DeepSurv, we manually select the architecture using the validation set so that we obtain a similar performance reported in (Katzman et al., 2018). The learning rate decay is set to 1. The learning rate and the regularization parameter are optimized via Optuna using the validation set, where the search range is set $L R : [ 1 e - 3 , 1 ]$ and $\lambda : [ 1 e - 3 , 1 ]$ . The hyperparameters used in the experiment are the following: architecture :[60, 20, 3], activation: selu (as suggested by (Katzman et al., 2018)), learning rate : 0.152, learning rate decay: 1.0, $\sigma : 0 . 5$ , $\lambda : 0 . 0 2 3$ , training epoch: 2000. Note that to see the effect of feature selection, we used the hyperparameters optimized for DeepSurv to test our method (STG-DeepSurv). + +# J.7 ADDITIONAL EXPERIMENTAL DETAILS + +Here we provide a full description of the procedures we have performed in the experimental parts of the paper. + +For synthetic datasets are first split into train, validation and test. Validation is always $1 0 \%$ of the train, while the exact ratios between train and test is detailed for each experiment separately. All the neural network weights are initialized by drawing from $\mathcal { N } ( 0 , 0 . 1 )$ and bias terms are set to 0. All the batch sizes are equal to the number of training samples. In table 3, we detail the search range of hyperparamters as well as the exact values used in our experiments. We set n-trials $= 1 0 0 0$ for Optuna, which is a define-by-run based hyperparameter optimization software equipped with efficient search and pruning strategies. We use SGD for all the experiments, except for the Cox model where we use Adam. All the experiments are conducted using Intel(R) Xeon(R) CPU E5-2620 v3 $( \underline { { { \omega } } } 2 . 4 \mathrm { G h z }$ $_ { \mathbf { X } 2 }$ (12 cores total). + +Table 3: List of the search range for the hyperparameters used in our expirements for XOR and Two-Moon + +
ParamSearch range
# dense layers #hidden units activation LR n-epoch (DFS, SG-L1-NN) α (SG-L1-NN) 入 (SG-L1-NN) 入 (STG,DFS) 入 (LASSO) n-est (RF, XGBoost, Tree) n-boost-round (XGBoost) Thresh (RF,XGBoost, Tree) max-depth (XGBoost) c (SVC)[1,3] [10,500] [tanh, relu, sigmoid] [le-4, 1e-1] [50,20000] [1e-3,1] [1e-7, 1] [1e-3, 1] [0.01, 1] [5,100] [1,100] [0.01,0.5] [0.01,0.5] [1e-7, 1]
+ +For the Phase Transition experiment, we use 0.1 as a learning rate. For all the experiments when we use Tree and RF, we use the default value for max-depth, so that nodes are expanded based on the purity. For the XOR problem, the exact architectures used for the NN based methods are: (STG/HC): [476, 490, 14] with Tanh, (DFS): [100, 10] with Tanh, (SG-L1-NN): [100, 10, 5] with Tanh. For the two moons we use (STG): [490, 406, 18] with Tanh, (DFS): [158, 27, 224] with Tanh, (SG-L1-NN): [88, 28, 27] with Tanh. For the XOR problem, we attempted to use Optuna to optimize parameters of DFS and SG-L1-NN, but we ended up using the architecture suggested by the authors (Li et al., 2016; Scardapane et al., 2017) as they outperform the values suggested by Optuna. The number of epochs used for the XOR problem is $2 0 K$ , 14K, 800 for STG/HC, DFS and SG-L1-NN respectively. Regularization parameters are $0 . 1 7 , 3 . 3 e \mathrm { ~ - ~ } 5$ and $3 e - 5$ respectively. The number of epochs used for the two-moons problem is $2 0 K$ , 1570, 708 for STG/HC, DFS and SG-L1-NN respectively. Regularization parameters are $0 . 4 8 , 9 e \mathrm { ~ - ~ } 3$ and $1 e - 3$ respectively. We note that the regularization parameters and learning procedure is different in nature, as we use an $\ell _ { 0 }$ type penalty. For the PBMC experiment, the architecture was hand-tuned using the validation set and set as [200, 100, 50, 10] with Tanh. Learning rate was 0.2 and the number of epochs 4000. The hyperparameter $\lambda$ varies in the range [0.001, 0.11] to achieve different levels of sparsity. For MADELON, we use the architecture optimized for the binary XOR classification. The number of epochs used is $2 0 K$ , the learning rate is 0.06 and $\lambda$ varies in the range [0.01, 10]. For regression, the learning rate and the $\lambda$ are optimized via Optuna using the search range $L R : [ 1 e - 4 , 1 ]$ and $\lambda : [ 1 e - 4 , 1 0 ]$ based on validation. The parameters used are the following: (SE1) architecture [600, 200, 100, 50, 1] with ReLu activations, num epochs: 5, $\lambda : 5$ , $L R : 0 . 0 0 0 1$ (SE2) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs $= 2 0 0 0$ , $\lambda : 5$ , $L R : 0 . 0 0 1$ (SE3) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs : 1000, $\lambda : 1$ , $L R : 0 . 0 0 5$ . (RCP) architecture [1000, 300, 150, 60, 20] with ReLu activation, num-epochs: 2000, $\lambda : 5 . 0$ , $L R : 0 . 0 0 1$ . (REL) architecture [26,91,63] with ReLu activation, num-epochs: 1600, $\lambda : 0 . 0 3 1$ , $L R : 0 . 0 0 7$ . (RAI) architecture [10,177] with ReLu activation, num-epochs: 1800, $\lambda : 0 . 0 1 9$ , $L R : 0 . 0 0 2$ . Architectures for SE1-SE3 and RCP where tuned manually. + +The ratio of train/test/valid split is 1:1:1 for synthetic data. For the real data (RCP and REL), the train size is 6000, the test and valid size is 1000 samples. For RAI the train size is 5000, the test and valid size is 1000 samples. + +In order to define the IFWR, for STG, the $d ^ { t h }$ feature weight is set to $\operatorname* { m a x } ( 0 , \operatorname* { m i n } ( 1 , \mu _ { d } + 0 . 5 ) )$ . For other neural net based methods, it is given by $\textstyle \sum _ { j } W _ { d j }$ , where $W$ is the weight matrix of the first layer. For other methods we just used the feature relevance returned by the trained model. Finally, the LASSO’s IFWR in the XOR experiment was omitted from the manuscript as it suffered from high variance. + +Regarding the comparison performed in the two-moons and XOR problem, we believe that adding IFWR along with classification accuracy versus number of feature selected provides a complementary perspective in demonstrating the efficacy of feature selection techniques. We emphasize that our goal is not to just rank features but select features by assigning the weight of 0 to irrelevant features while simultaneously obtaining good predictive accuracy. \ No newline at end of file diff --git a/parse/train/B1lda1HtvB/B1lda1HtvB_content_list.json b/parse/train/B1lda1HtvB/B1lda1HtvB_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e566396d5238fc936728779e54d03a1d0648f3ec --- /dev/null +++ b/parse/train/B1lda1HtvB/B1lda1HtvB_content_list.json @@ -0,0 +1,2986 @@ +[ + { + "type": "text", + "text": "FEATURE SELECTION USING STOCHASTIC GATES ", + "text_level": 1, + "bbox": [ + 176, + 98, + 764, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 145, + 400, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 210, + 544, + 224 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Feature selection problems have been extensively studied in the setting of linear estimation, for instance LASSO, but less emphasis has been placed on feature selection for non-linear functions. In this study, we propose a method for feature selection in non-linear function estimation problems. The new procedure is based on directly penalizing the $\\ell _ { 0 }$ norm of features, or the count of the number of selected features. Our $\\ell _ { 0 }$ based regularization relies on a continuous relaxation of the Bernoulli distribution, which allows our model to learn the parameters of the approximate Bernoulli distributions via gradient descent. The proposed framework simultaneously learns a non-linear regression or classification function while selecting a small subset of features. We provide an information-theoretic justification for incorporating Bernoulli distribution into our approach. Furthermore, we evaluate our method using synthetic and real-life data and demonstrate that our approach outperforms other embedded methods in terms of predictive performance and feature selection. ", + "bbox": [ + 233, + 242, + 766, + 435 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 467, + 336, + 482 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Feature selection is a fundamental task in machine learning and statistics. Feature selection leads to a number of potential benefits: reducing experimental costs Min et al. (2014), enhancing interpretability (Ribeiro et al., 2016), computational speed up and even improving model generalization on unseen data (Chandrashekar & Sahin, 2014). In biomedicine, scientists collect multitude datasets comprising of many biomarkers (e.g., genes or proteins) that require development of effective diagnostics or prognostics models. For instance, in Genome wide association studies (GWAS), feature selection can help identify such models and lead to improved risk assessment and reduced cost. ", + "bbox": [ + 174, + 498, + 825, + 597 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Feature selection methods may be classified into three major categories: filter methods, wrapper methods, and embedded methods. Filter methods attempt to remove irrelevant features prior to learning a model. These methods filter features using a per-feature relevance score that is created based on some statistical measure (Battiti, 1994; Peng et al., 2005; Estévez et al., 2009; Song et al., 2007; 2012; Chen et al., 2017). Wrapper methods (Kohavi & John, 1997b; Stein et al., 2005; Zhu et al., 2007; Reunanen, 2003; Allen, 2013) use the outcome of a classifier to determine the relevance of each feature, which requires recomputing the classifier for each subset of features. This becomes computationally expensive for neural network based wrapper methods (Verikas & Bacauskiene, 2002; Kabir et al., 2010; Roy et al., 2015). ", + "bbox": [ + 174, + 603, + 825, + 728 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Embedded methods aim to remove this burden by learning the model while simultaneously selecting the subset of relevant features. The Least Absolute Shrinkage and Selection Operator (LASSO) (Tibshirani, 1996) is a well-known embedded method, whose objective is to minimize the loss while enforcing an $\\ell _ { 1 }$ constraint on the weights of the features. Although LASSO is scalable and widely used (Hans, 2009; Li et al., 2011; 2006), it is restricted to the domain of linear functions. To allow the model to capture nonlinear interaction, it is appealing to consider the non-convex extension of the LASSO formulation using neural networks. ", + "bbox": [ + 174, + 736, + 825, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We develop a fully embedded feature selection method for nonlinear models. To the best of our knowledge it is the first $\\ell _ { 0 }$ embedded feature selection method. Our method improves upon the LASSO formulation in two aspects: a) it captures nonlinear interactions between the features via neural network modeling and b) it employs an $\\ell _ { 0 }$ -like regularization using gates whose weights are parametrized by a smooth variant of a Bernoulli distribution. Altogether these twofold improvements are formulated as a fully differentiable neural network. Specifically, our contributions are as follows: ", + "bbox": [ + 174, + 840, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1. By utilizing a recent development of continuous and differentiable approximation to discrete distributions (Maddison et al., 2016), (Jang et al., 2017), (Louizos et al., 2017), we introduce a solution to the long standing problem of feature selection with an $\\ell _ { 0 }$ regularization. \n2. We present a novel simple relaxation of Bernoulli distribution to sparsify the input layer (the feature space) which we call stochastic gate (STG) and show its advantage over the distribution presented by (Louizos et al., 2017) both in performance and convergence time. \n3. By applying these two relaxations to an input layer of a neural network, we perform embedded feature selection in classification, regression or survival analysis tasks and demonstrate its capabilities on artificial and real data sets. \n4. We justify our probabilistic approach by analyzing the constrained Mutual Information maximization objective of feature selection. We demonstrate the applicability of our method using numerous examples (see Section 6 and Appendix). ", + "bbox": [ + 210, + 103, + 826, + 287 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Notation: We refer to vectors as bold lowercase $_ { \\textbf { \\em x } }$ and random vectors as bold uppercase letters $\\boldsymbol { X }$ . Scalars are non-bold case $y$ , while random variables are capital case $Y$ . A set is represented by script fonts $\\mathcal { X } , \\mathcal { Y } , \\mathcal { S }$ . For example the $n ^ { t h }$ vector-valued observation is denoted as ${ \\mathbf { \\mathcal { x } } } _ { n }$ whereas $X _ { d }$ represents the $d ^ { t h }$ feature of the vector-valued random variable $\\boldsymbol { X }$ . Let $[ n ] = 1 , 2 , \\ldots , n$ . For a set $\\bar { \\mathcal { S } } \\bar { \\subset } [ D ]$ let the vector $\\pmb { s } \\in \\{ 0 , 1 \\} ^ { D }$ be the characteristic function for the set. That is $s _ { i } = 1$ if $i \\in S$ and 0 otherwise. For two vectors $_ { \\textbf { \\em x } }$ and $_ { z }$ we denote $\\pmb { x } \\odot z$ to be the element-wise product between $_ { \\textbf { \\em x } }$ and $_ z$ . Thus, if we let $\\pmb { s } \\in \\{ 0 , 1 \\} ^ { D }$ be the characteristic vector of $s$ , then we may define $\\mathbf { \\Delta } \\mathbf { x } _ { S } = \\mathbf { \\Delta } \\mathbf { x } \\odot \\mathbf { \\Delta } \\mathbf { s }$ The $\\ell _ { 1 }$ norm of a vector is denoted as $\\begin{array} { r } { \\| \\pmb { x } \\| _ { 1 } = \\sum _ { i = 1 } ^ { D } | x _ { i } | } \\end{array}$ . Finally, the $\\ell _ { 0 }$ norm of a vector is denoted as $\\| \\pmb { x } \\| _ { 0 }$ and counts the total number of non-zero entries in the vector $_ { \\textbf { \\em x } }$ . ", + "bbox": [ + 173, + 301, + 825, + 431 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PROBLEM SETUP AND BACKGROUND ", + "text_level": 1, + "bbox": [ + 174, + 452, + 516, + 468 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Let $\\boldsymbol { \\mathcal { X } } ~ \\in ~ \\mathbb { R } ^ { D }$ be the input domain with corresponding response domain $\\mathcal { V }$ . Given realizations from some unknown data distribution $P _ { X , Y }$ , the goal of embedded feature selection methods is to simultaneously find a subset of indices $\\mathcal { S } \\subset \\{ 1 , . . . D \\}$ and construct a model that predicts $Y$ based on the selected features $X _ { \\mathcal { S } }$ . ", + "bbox": [ + 174, + 482, + 825, + 540 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 RISK MINIMIZATION OBJECTIVE ", + "text_level": 1, + "bbox": [ + 176, + 556, + 436, + 571 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We assume that we are given a family of functions $\\mathcal { F }$ such that any function $f _ { \\pmb { \\theta } } \\in \\mathcal { F }$ is indexed by a set of parameters $\\pmb \\theta$ . Given some loss $L$ , a selection of features $s \\subset [ D ]$ , and a choice of parameters $\\pmb \\theta$ , we denote the risk of our model as ", + "bbox": [ + 174, + 582, + 825, + 626 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/dc229d0738ce2d5b008c6bb1f731f695117b4d441bcaefd9f2939b732d7fd21f.jpg", + "text": "$$\nR ( \\pmb \\theta , \\pmb s ) = \\mathbb { E } _ { X , Y } L ( f _ { \\boldsymbol \\theta } ( \\pmb X \\odot \\pmb s ) , Y ) ,\n$$", + "text_format": "latex", + "bbox": [ + 379, + 633, + 617, + 651 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where we recall that $\\pmb { s } = \\{ 0 , 1 \\} ^ { D }$ is a vector of indicator variables for the set $s$ and $\\odot$ denotes the point-wise product. Thus, the goal of the feature selection problem is to find the parameters $\\pmb \\theta$ and $\\pmb { s }$ that minimize $R ( \\pmb \\theta , s )$ such that $\\| s \\| _ { 0 }$ is small compared to $D$ . ", + "bbox": [ + 174, + 659, + 825, + 703 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 FEATURE SELECTION FOR LINEAR MODELS ", + "text_level": 1, + "bbox": [ + 174, + 719, + 521, + 734 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Before proceeding with our proposed method, we review the feature selection problem in the linear regression setting with the least squares loss. Thus, we restrict $\\mathcal { F }$ to be the space of linear functions and the loss function to be the quadratic loss. ", + "bbox": [ + 174, + 746, + 825, + 787 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given observations $\\{ x _ { n } , y _ { n } \\} _ { n = 1 } ^ { N }$ we may consider the constrained empirical risk minimization problem ", + "bbox": [ + 176, + 794, + 821, + 823 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/23437ec691bd541b2e649c6431a1721775a8ef2ed5065cb807c57dfc8ff7a1cf.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\pmb { \\theta } } \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } L ( \\pmb { \\theta } ^ { T } \\pmb { x } _ { n } , y _ { n } ) \\quad \\mathrm { s . t . } \\ \\| \\pmb { \\theta } \\| _ { 0 } \\leq k .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 821, + 640, + 864 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Since the above problem is intractable, a number of authors replace the $\\ell _ { 0 }$ constraint with a surrogate function $\\Omega ( \\pmb \\theta ) : \\dot { \\mathbb { R } } ^ { D } \\mathbb { R } _ { + }$ designed to penalize the number of selected features in $\\pmb \\theta$ . A natural choice for $\\Omega$ is the $\\ell _ { 1 }$ norm, which yields a convex problem and more precisely the LASSO optimization problem. The $\\ell _ { 1 }$ is known to be the closest convex relaxation to the $\\ell _ { 0 }$ . In fact, in certain settings the ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "LASSO optimization and the $\\ell _ { 0 }$ based objective have the same solution. While the original LASSO problem focuses on the constrained optimization problem, the regularized least squares problem, which is often used in practice, yields the following minimization objective: ", + "bbox": [ + 171, + 103, + 823, + 146 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/de6e21ecf18f27c84ee46c5fa6a38906cdc5e1dde8d56187b050cc77a30b6493.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\pmb { \\theta } } \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } ( \\pmb { \\theta } ^ { T } \\pmb { x } _ { n } - \\pmb { y } _ { n } ) ^ { 2 } + \\lambda \\| \\pmb { \\theta } \\| _ { 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 147, + 622, + 190 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The hyperparameter $\\lambda$ trades off the amount of regularization versus the fit of the objective1. The $\\ell _ { 1 }$ regularized method is very effective for feature selection and prediction; however, it achieves this through shrinkage of the coefficients. As a result, (Fan & Li, 2001) have considered non-convex choices for $\\Omega$ that perform well both theoretically and empirically for prediction and feature selection. ", + "bbox": [ + 173, + 191, + 825, + 248 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our goal is to apply such regularization techniques to perform feature selection while learning a non-linear function. Kernel methods have been considered (Yamada et al., 2014), but scale quadratically in the number of observations. To alleviate this burden, (Gregorová et al., 2018) use random Fourier features to approximate the kernel. (Li et al., 2016) and (Scardapane et al., 2017) take an alternative approach by modeling $f _ { \\theta }$ using a neural network with $\\ell _ { 1 }$ regularization on the input weights. However, in practice, introducing an $\\ell _ { 1 }$ penalty into gradient descent does not provide sufficient sparsification. Below, we discuss our method that works to directly use an $\\ell _ { 0 }$ penalty. ", + "bbox": [ + 173, + 253, + 825, + 353 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PROPOSED METHOD ", + "text_level": 1, + "bbox": [ + 176, + 371, + 377, + 387 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We take a probabilistic approach to approximate the $\\ell _ { 0 }$ norm, which can extend to non-linear models while remaining computationally efficient. To motivate such probabilistic approach, we provide theoretical support (see Section 4) based on a Mutual Information perspective of the feature selection problem. ", + "bbox": [ + 173, + 401, + 825, + 458 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To view the $\\ell _ { 0 }$ regularized version of the risk (Eq. 1) from a probabilistic perspective, one can introduce a Bernoulli random vector $\\tilde { S }$ whose entries are independent and the $d ^ { t h }$ entry satisfies $\\pi _ { d } = \\mathbb { P } ( \\tilde { S } _ { d } = 1 )$ for $d \\in [ D ]$ . If we denote the empirical expectation over our observations as $\\hat { \\mathbb { E } } _ { X , Y }$ , then, the empirical regularized risk (Eq. 1) becomes ", + "bbox": [ + 173, + 463, + 825, + 525 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f9502fc5e539e72ecf68bd2a5f9c6710031a89a69aecb4a50cb0c6c70e177a05.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta , \\pi } \\hat { R } ( \\theta , \\pi ) = \\operatorname* { m i n } _ { \\theta , \\pi } \\hat { \\mathbb { E } } _ { X , Y } \\mathbb { E } _ { \\tilde { S } } \\left[ L ( f _ { \\theta } ( X \\odot \\tilde { S } ) , Y ) + \\lambda | | \\tilde { S } | | _ { 0 } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 290, + 525, + 707, + 558 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where we have $\\mathbb { E } _ { \\tilde { S } } | | \\tilde { S } | | _ { 0 } = \\sum _ { d = 1 } ^ { D } \\pi _ { d }$ and we constrain $\\pi _ { d } \\in \\{ 0 , 1 \\}$ . Clearly, this formulation is equivalent to Eq. 1, with a regularized penalty on cardinality rather than an explicit constraint. We may then relax the discrete constraint on $\\pi _ { d }$ to be $\\pi _ { d } \\in [ 0 , 1 ]$ . ", + "bbox": [ + 174, + 560, + 825, + 606 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Now, our goal is to find the model parameters $\\pmb { \\theta } ^ { * }$ and Bernoulli parameters $\\pi ^ { * }$ that minimize the empirical risk $\\hat { R } ( \\pmb \\theta , \\pmb \\pi )$ via gradient descent. However, an optimization of a loss function which includes discrete random variables suffers from high variance (see F in the Appendix for more details). Therefore, inspired by a recently developed continuous approximation for discrete random variables, suggested by (Jang et al., 2017; Maddison et al., 2016), we develop and use a novel and simple continuous distribution that is fully differentiable and suited for the task of feature selection. ", + "bbox": [ + 173, + 611, + 826, + 698 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 CONTINUOUS RELAXATION ", + "text_level": 1, + "bbox": [ + 176, + 713, + 405, + 727 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our continuous relaxation for the Bernoulli variables ${ \\cal \\tilde { S } } _ { d }$ for $d \\in \\ [ D ]$ is termed stochastic gate (STG). The STG relies on the reparametrization trick, which is widely used for reducing the variance of gradient estimators (Miller et al., 2017; Figurnov et al., 2018). To construct a continuous approximation to Bernoulli random variable via the reparametrization trick, we define $z _ { d } = g ( \\mu _ { d } + \\epsilon _ { d } ) = \\operatorname* { m a x } ( 0 , \\operatorname* { m i n } ( 1 , \\epsilon _ { d } + \\mu _ { d } + 0 . 5 ) )$ where $\\epsilon _ { d }$ is drawn from a Gaussian distribution $\\mathcal { N } ( 0 , \\sigma _ { d } )$ , where $\\sigma _ { d }$ is fixed throughout training. This approximation can be viewed as a clipped, mean-shifted, Gaussian random vector. Furthermore, the gradient of the objective with respect to $\\mu _ { d }$ can be computed via the chain rule. ", + "bbox": [ + 173, + 737, + 826, + 851 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can now rewrite the objective in Eq. 4 as ", + "bbox": [ + 173, + 857, + 465, + 872 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9e88de04d931cda344da264b480603e7f443f2688879c02095a8029fd6541ef7.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta , \\mu } \\hat { R } ( \\theta , \\mu ) = \\operatorname* { m i n } _ { \\theta , \\mu } \\mathbb { E } _ { X , Y } \\mathbb { E } _ { Z } \\left[ L ( f _ { \\theta } ( X \\odot Z ) , Y ) + \\lambda | | Z | | _ { 0 } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 292, + 872, + 705, + 902 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $z$ is a random vector with $D$ independent variables $z _ { d }$ for $[ D ]$ . To optimize the empirical surrogate of the objective (Eq. 5), we first differentiate it with respect to $\\pmb { \\mu }$ . Then, Monte Carlo sampling leads us to the following gradient estimator ", + "bbox": [ + 174, + 102, + 825, + 146 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/972db09e3a82ea49b24731d587096c9c5c775a8ca434b9da074652c3463b4db7.jpg", + "text": "$$\n\\frac { \\partial } { \\partial \\mu _ { d } } \\hat { R } ( \\pmb \\theta , \\mu ) = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\bigg [ L ^ { \\prime } ( z ^ { k } ) \\frac { \\partial z _ { d } ^ { k } } { \\partial \\mu _ { d } } \\bigg ] + \\frac { \\lambda } { K } \\frac { \\partial } { \\partial \\mu _ { d } } \\sum _ { k = 1 } ^ { K } \\mathrm { P r } \\{ z _ { d } ^ { k } > 0 \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 148, + 714, + 193 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $K$ is the number of Monte Carlo samples. Thus, we can update the parameters $\\mu _ { d }$ for $[ D ]$ via gradient descent. We note that if we replace $\\frac { \\partial { z } _ { d } ^ { k } } { \\partial { \\mu } _ { d } }$ with 1, the above gradient estimator for $L ^ { \\prime }$ is reduced to the Straight-Through estimator (Bengio et al., 2013). ", + "bbox": [ + 174, + 194, + 826, + 244 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Under the continuous relaxation, the expected regularization term in the objective $\\hat { R } ( \\pmb \\theta , \\pmb \\mu )$ is simply the sum of the probability that the gates $\\{ z _ { d } \\} _ { d = 1 } ^ { D }$ are active, which is equal to $\\begin{array} { r } { \\sum _ { d = 1 } ^ { D } \\Phi \\left( \\frac { \\mu _ { d } + \\frac { 1 } { 2 } } { \\sigma _ { d } } \\right) } \\end{array}$ where $\\Phi$ is the standard Gaussian CDF. To conclude, we can now optimize the objective in Eq. 5 using gradient descent over the model parameters $\\pmb \\theta$ and the parameters $\\pmb { \\mu }$ representing the Gaussian’s mean (instead of the Bernoulli parameters $\\pi$ ). ", + "bbox": [ + 173, + 252, + 825, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "After training, to remove the stochasticity from the learned gates, we set $\\hat { z } _ { d } = \\operatorname* { m a x } _ { \\mathbf { \\alpha } } ( 0 , \\operatorname* { m i n } ( 1 , \\mu _ { d } +$ 0.5)), which informs what features are selected. Note that when $\\left| \\mu _ { d } \\right|$ is less than $\\begin{array} { l } { { \\frac { 1 } { 2 } } } \\end{array}$ , $\\hat { z } _ { d }$ returns the value between $( 0 , 1 )$ . In such a case, we can treat the value of $\\hat { z } _ { d }$ as feature importance or employ an additional thresholding (i.e. 1 if $\\hat { z } _ { d } > 0 . 5$ and 0 otherwise) depending on application-specific needs. In the Appendix, we provide the pseudo-code of our algorithm as well as the discussion of the choice of $\\sigma _ { d }$ . ", + "bbox": [ + 173, + 338, + 826, + 422 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 CONNECTION TO MUTUAL INFORMATION ", + "text_level": 1, + "bbox": [ + 174, + 441, + 552, + 458 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section we show an equivalence between the Bernoulli formulation of the feature selection problem and the $\\ell _ { 0 }$ regularized approach. ", + "bbox": [ + 173, + 473, + 823, + 502 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 MUTUAL INFORMATION BASED OBJECTIVE ", + "text_level": 1, + "bbox": [ + 176, + 517, + 511, + 531 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "From an information theoretic perspective, the goal of feature selection is to find a subset of features $s$ that has the highest Mutual Information (MI) with the target variable $Y$ . Recall that the MI between two random variables can be defined as $I ( X ; Y ) = H ( Y ) - H ( Y | X )$ where $H ( Y ) , H ( Y | X )$ are the entropy of $p _ { Y } ( Y )$ and the conditional entropy of $p _ { Y | \\mathbf { X } } ( Y | X )$ , respectively (Cover & Thomas, 2006). Then we can formulate the task as selecting $s$ such that the mutual information between $X _ { \\mathcal { S } }$ and $Y$ are maximized: ", + "bbox": [ + 173, + 542, + 825, + 626 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fe5d47089f80cca21aae1988abfc2bbb6cdeae059d319559dd9dcb143234f0e1.jpg", + "text": "$$\n\\operatorname* { m a x } _ { s } I ( X _ { s } , Y ) \\quad { \\mathrm { s . t . ~ } } | S | = k ,\n$$", + "text_format": "latex", + "bbox": [ + 398, + 625, + 599, + 647 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $k$ is the hypothesised number of relevant features. ", + "bbox": [ + 173, + 648, + 544, + 664 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 INTRODUCING RANDOMNESS ", + "text_level": 1, + "bbox": [ + 176, + 679, + 416, + 693 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We first demonstrate that under mild assumptions we can replace the deterministic search over the set $s$ (or corresponding indicator vector $\\pmb { s }$ ), by a search over the parameters of the distributions that model $\\pmb { s }$ . Our proposition is based on the following two assumptions: ", + "bbox": [ + 174, + 704, + 825, + 747 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assumption 1: There exists a subset of indices ${ \\boldsymbol { S } } ^ { * }$ with cardinality equal to $k$ such that for any $i \\in S ^ { * }$ we have $I ( X _ { i } ; Y | { \\cal { X } } _ { \\backslash \\{ i \\} } ) > 0$ . \nAssumption 2: $I ( X _ { \\cal S ^ { * c } } ; Y | X _ { \\cal S ^ { * } } ) = 0$ . ", + "bbox": [ + 173, + 761, + 825, + 806 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Discussion of assumptions: Assumption 1 that including an element from $S ^ { * }$ improves prediction accuracy. This assumption is equivalent to stating that feature $i$ is strongly relevant (Kohavi & John, 1997a; Brown et al., 2012). Assumption 2 simply states that $S ^ { * }$ is a superset of the Markov Blanket of the variable $Y$ (Brown et al., 2012). The assumptions are quite benign. For instance they are satisfied if $\\boldsymbol { X }$ is drawn from a Gaussian with a non-degenerate covariance matrix and $Y = f ( X _ { \\mathcal { S } ^ { * } } ) + w$ where $w$ is noise independent of $\\boldsymbol { X }$ and $f$ is not degenerate. With these assumptions in hand, we may present our result. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proposition 1. Suppose that the above assumptions hold for the model. Then, solving the optimization in Eq. 6 is equivalent to solving the optimization ", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1fb8300a27b3111f62f82c08d95ea22d446e3fe6e0b9b961e61617bd825c208e.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\mathbf { 0 } \\leq \\pi \\leq 1 } I ( X \\odot \\tilde { S } ; Y ) \\quad s . t . \\quad \\sum _ { i } \\mathbb { E } [ \\tilde { S } _ { i } ] \\leq k ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 140, + 650, + 174 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the coordinates $\\tilde { S } _ { i }$ are drawn independently at random according to a Bernoulli distribution with parameter $\\pi _ { i }$ . ", + "bbox": [ + 173, + 184, + 825, + 214 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Due to length constraints, we leave the proof of this proposition and how it bridges the MI maximization (Eq. 6) and risk minimization (Eq. 2) to the Appendix (see Section C and D). ", + "bbox": [ + 169, + 227, + 825, + 256 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 277, + 344, + 294 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The two most related works to this study are (Louizos et al., 2017) and (Chen et al., 2018). In (Louizos et al., 2017), they introduce the Hard-Concrete (HC) distribution as a continuous surrogate for Bernoulli distributions in the context of model compression. The HC distribution is induced by applying a hard sigmoid $z = \\operatorname* { m i n } ( 1 , \\operatorname* { m a x } ( 0 , \\bar { s } ) )$ to $\\bar { s }$ , where we construct $\\bar { s }$ by applying Sigmoid function to the logistic distribution as follows: ", + "bbox": [ + 174, + 310, + 825, + 381 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0e44c3d6a8ef774603f3aab46a09323b7288cb5db49591ed77636ef85a334a11.jpg", + "text": "$$\nu \\sim U ( 0 , 1 ) , L = \\log ( u ) - \\log ( 1 - u ) , s = \\frac { 1 } { 1 + \\exp ( \\frac { - ( \\log \\alpha + L ) } { \\beta } ) } , \\bar { s } = s ( \\zeta - \\tau ) + \\tau\n$$", + "text_format": "latex", + "bbox": [ + 218, + 388, + 779, + 428 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The interval $( \\tau , \\zeta )$ , with $\\tau < 0$ and $\\zeta > 1$ , allows the distribution to have more probability mass on the edge of the support. The authors demonstrate how incorporating the HC in deep neural networks leads to fast convergence and improved generalization due to the sparsification effect. In this study, we introduce this type of sparsification for the long standing problem of feature selection with $\\ell _ { 0 }$ regularization.We demonstrate that a simple relaxation of Bernoulli distributions is sufficient and works better than the HC distribution for feature selection. We find empirically that the HC yields high variance gradient estimates compared to the STG. This high-variance does not affect model sparsification as the sparsity pattern does not matter. Contrarily, for feature selection it is crucial as practitioners require a stable procedure. Furthermore, higher variance gradients will result in a slower convergence of the model (this has been demonstrated empirically in the supp. material). In (Chen et al., 2018), the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained models. Their method is focused on finding a subset of features given a particular instance, and therefore is not appropriate for general feature selection. ", + "bbox": [ + 173, + 435, + 825, + 617 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Some studies tackle embedded feature selection problems by extending LASSO and group LASSO to neural network models. The authors (Li et al., 2016; Scardapane et al., 2017) and (Feng & Simon, 2017) have a similar goal as ours in performing feature selection, but instead rely on the $\\ell _ { 1 }$ relaxation to the $\\ell _ { 0 }$ . Our approach, which utilizes stochastic gates coupled with the $\\ell _ { 0 }$ norm achieves substantially better empirical performance compared against other $\\ell _ { 1 }$ based baselines. This point is demonstrated in the next section. ", + "bbox": [ + 174, + 623, + 825, + 708 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 729, + 326, + 746 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here we provide empirical evaluation of our method in a wide range of settings. We begin with simple linear regression evaluating the capabilities of our approach to select a sparse set of relevant features (section 6.1). We then evaluate our method in nonlinear regression tasks (section 6.2). Next, we demonstrate the applicability of the method in a highly nonlinear classification task (section 6.3. Finally, we present two potential applications (section 7). ", + "bbox": [ + 174, + 762, + 825, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We implement our method using both the STG and HC distributions and compare their applicability for feature selection. The hyperparamters of all the methods are optimized using validation sets via Optuna (Takuya Akiba & Koyama, 2019). In the Appendix, we provide all details of the Optuna based parameter tuning procedure. Additional experiments also appear in the Appendix, including high dimensional datasets and an extensive comparisons between the STG and HC (Louizos et al., 2017) distributions. ", + "bbox": [ + 173, + 839, + 825, + 922 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "6.1 SUPPORT RECOVERY IN LINEAR REGRESSION ", + "text_level": 1, + "bbox": [ + 173, + 103, + 534, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the setting of noisy linear regression, (Wainwright, 2009) have analyzed the probability of LASSO to correctly identify a sparse subset of active variables. The problem is know as support recovery and is formulated as follows; let $\\beta ^ { * } \\in \\mathbb { R } ^ { D }$ be a fixed sparse vector, such that $\\beta _ { i } ^ { * } \\in \\{ - \\overline { { 0 . 5 } } , 0 . 5 \\}$ (with equal probability) if $i \\in S$ , and $\\beta _ { i } ^ { * } = 0$ otherwise. Suppose the cardinality of the support $| S | = k$ is known. Given a matrix of measurements $\\pmb { X } \\in \\mathbb { R } ^ { N \\times D }$ with values drawn independently from $N ( 0 , 1 )$ , the response $\\textbf { { y } }$ is defined as ", + "bbox": [ + 173, + 128, + 825, + 215 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/71701a6a4474f5f60cd370ee6839ee5cdb03e4dd32bf7b8a85336ef4e3c24cff.jpg", + "text": "$$\n\\pmb { y } = \\pmb { X } \\beta ^ { * } + \\pmb { w } ,\n$$", + "text_format": "latex", + "bbox": [ + 444, + 219, + 553, + 237 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where the values of the noise $\\pmb { w } _ { i } , i = 1 , . . . , N$ are drawn independently from $N ( 0 , 0 . 5 )$ ", + "bbox": [ + 171, + 242, + 745, + 258 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here, we reproduce this setting to evaluate the probability of the proposed approach to perfectly recover the support of $\\beta ^ { * }$ . As suggested by (Wainwright, 2009), we use a sparsity that scales with $D$ such that $k = \\lceil 0 . 4 D ^ { 0 . 7 5 } \\rceil$ . For each number of samples $N$ in the range [10, 500], we run 200 simulations and count the portion of correctly recovered supports. We repeat this process for 2 different values of $D$ and compare our performance to LASSO. For LASSO, the regularization parameter was set to its optimal value $\\begin{array} { r } { \\alpha _ { N } = \\sqrt { \\frac { 2 \\sigma ^ { 2 } \\log ( D - k ) \\log ( k ) } { N } } } \\end{array}$ (Wainwright, 2009). For STG and HC we set $\\lambda _ { N } = C \\alpha _ { N }$ , such that $C$ is a constant, which is selected using a grid search in the range [0.1,10]. As evident from Fig. 1, even when restricting to linear functions our method has a clear advantage over LASSO. This implies that the $\\ell _ { 0 }$ based penalty, although is non convex in nature, allows perfect recovery of the support using less samples. Furthermore, the proposed STG requires even less samples than the HC distribution and suffers from a smaller variance. ", + "bbox": [ + 173, + 263, + 825, + 426 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/b1fc1c69a9a18bb965657ede00fd6a171e90b89b1739aae91ad9b2b8a9d49603.jpg", + "image_caption": [ + "Figure 1: Probability of successfully recovering the support of a sparse signal $\\boldsymbol { \\beta } ^ { * } \\in \\mathbb { R } ^ { D }$ vs. the number of observations $N$ . Comparison between the proposed method, HC and LASSO under a linear additive noise model $\\pmb { y } = \\pmb { X } \\beta ^ { * } + \\pmb { w }$ using (a) $D = 6 4$ and (b) $D = 1 2 8$ . Central lines are the means while shaded are represent the standard deviation. " + ], + "image_footnote": [], + "bbox": [ + 189, + 436, + 794, + 590 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.2 NONLINEAR REGRESSION USING SYNTHETIC AND REAL DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 661, + 676, + 675 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we evaluate our method for regression tasks against two other embedded feature selection methods: LASSO and Sparse Random Fourier Features (Gregorová et al., 2018). Following the same format as (Gregorová et al., 2018), the following functions are used to generate synthetic data: (SE1: 100/5) $y = \\sin { ( x _ { 1 } + x _ { 3 } ) ^ { 2 } } \\sin { ( x _ { 7 } x _ { 8 } x _ { 9 } ) } + \\mathcal { N } ( 0 , 0 . 1 )$ . (SE2: 18/5) $\\begin{array} { r } { y = \\log ( ( \\sum _ { s = 1 1 } ^ { \\mathbf { \\bar { 1 } } 5 } x _ { s } ) ^ { 2 } ) + } \\end{array}$ $\\mathcal { N } ( 0 , 0 . 1 )$ . (SE3: 1000/10) $y = 1 0 ( z _ { 1 } ^ { 2 } + z _ { 3 } ^ { 2 } ) e ^ { - 2 ( z _ { 1 } ^ { 2 } + z _ { 3 } ^ { 2 } ) } + \\mathcal { N } ( 0 , 0 . 0 1 )$ , where each $x _ { i }$ is drawn from the standard Gaussian $\\mathcal { N } ( 0 , 1 )$ . For (SE3), the five consecutive coordinates are generated by ${ x _ { 5 } } _ { ( j - 1 ) + i } = { z _ { j } } + \\mathrm { { N } } ( 0 , 0 . 1 )$ , where $z _ { j } \\sim \\mathcal { N } ( 0 , 1 )$ for $i = 1 , . . . , 5$ and $j = 1 , . . . , 2 0 0$ . The numbers next to the experiment code indicate total dimensions/relevant dimensions in the feature space. We also evaluate our method using three real datasets: (RCP: 21/-), of computer systems activity, (REL 17/-) of F16 elevators and (RAI 39/-) of F16 ailernos all taken from the LIACC repository 2. For each dataset, we generate 30 different replications and randomly split the data into train, validation, and test set (see Appendix for more details). The root mean squared error on the test set averaged over 30 random replicated datasets are reported in Table 1. Our method outperforms all alternative methods for most cases. Note that (SE1) is generated using a sine function, which is in favor of the random Fourier feature based method (SRFF). ", + "bbox": [ + 173, + 686, + 826, + 900 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/df1bf63c38e0994800ae9a2375509a3b12386637fdb8d68617a85e22cf692989.jpg", + "table_caption": [ + "Table 1: Regression performance comparison in terms of root mean squared error. The mean and standard deviation are computed across 30 resamples. $N$ is 1000 for SE1,2,3 and 6000 for RCP. The values for LASSO and SRFF are borrowed from (Gregorová et al., 2018). DNN represents a deep neural network without feature selection. " + ], + "table_footnote": [ + "6.3 NOISY BINARY XOR CLASSIFICATION " + ], + "table_body": "
EXPLASSORFSG-L1-NNSRFFDNNHCSTG
SE10.29 (0.01)0.30 (0.01)0.29 (0.01)0.27 (0.01)0.29 (0.01)0.29 (0.01)0.29 (0.01)
SE22.22 (0.10)2.34 (0.17)2.35 (0.18)1.60 (0.10)2.05 (0.11)1.19 (0.31)0.87 (0.15)
SE30.68 (0.002)0.50(0.01)0.68 (0.01)0.48 (0.03)0.73 (0.02)0.33 (0.05)0.14 (0.10)
RCP9.69 (0.71)3.52 (0.12)9.64 (0.65)2.52 (0.18)2.89 (0.25)2.74 (0.75)2.44 (0.08)
REL0.47 (0.01)0.58 (0.01)0.44 (0.01)0.31 (0.03)0.61 (0.01)0.33 (0.04)0.27 (0.002)
RAI0.43 (0.02)0.48 (0.003)0.47(0.002)0.41(0.02)0.44 (0.01)0.41 (0.01)0.39 (0.01)
", + "bbox": [ + 173, + 121, + 893, + 226 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the following evaluation, we consider the problem of learning a binary XOR function for classification task. The first two coordinates $x _ { 1 } , x _ { 2 }$ are drawn from a binary \"fair\" Bernoulli distribution. The response variable is set as an XOR of the first coordinates, such that $y = x _ { 1 } \\oplus x _ { 2 }$ . The coordinates $x _ { i } , i = 3 , . . . , D$ are nuisance features, also drawn from a binary \"fair\" Bernoulli distribution. The number of points we generate is $N = 1 , 5 0 0$ , of which $7 0 \\%$ are reserved for test and $1 0 \\%$ of the remaining training set was reserved for validation. We compare the proposed method to four embedded feature selection methods (LASSO (Tibshirani, 1996), C-support vectors (SVC) (Chang & Lin, 2011), deep feature selection (DFS) (Li et al., 2016), sparse group regularized NN (SG-L1-NN) (Scardapane et al., 2017)). To provide more benchmarks, we also compare our embedded method against three wrapper methods (Extremely Randomized Trees (Tree) (Rastogi & Shim, 2000), Random Forests (RF) (Strobl et al., 2008)) and Extreme Gradient Boosting (XGBOOST) (Chen & Guestrin, 2016). ", + "bbox": [ + 173, + 252, + 826, + 406 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To evaluate the feature selection performance, we calculate the Informative Features Weight Ratio (IFWR). IFWR is defined as the sum of weights $W _ { d }$ over the informative features divided by the sum over all weights. In the case of binary weights the IFWR is in fact a recall measure for the relevant features (See the Appendix for more details.) ", + "bbox": [ + 174, + 412, + 825, + 468 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The experiment is repeated 20 times for different values of $D$ , and the average test classification accuracy and standard deviation are presented in Fig. 2(a), followed by the IFWR in Fig. 2(b). The number of selected features affects the accuracy. Therefore, to treat all the methods in a fair manner, we tune the hyperparameter that controls the sparsity level using Optuna (Takuya Akiba & Koyama, 2019) which optimizes the overall accuracy across different $D \\mathrm { s }$ . For instance, the wrapper methods (Tree, RF and XGBOOST) has a threshold value to retain features. We retrain them using only such features whose weight is higher than the threshold. In terms of feature ranking (see Fig. 2(c), only the tree based methods and the proposed (STG and HC based) provide the optimal median rank (which is 1.5) for the two relevant features. Nonetheless, the ranking provided by STG is the most stable comparing to all the alternative methods. ", + "bbox": [ + 173, + 474, + 825, + 613 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/06d3d26781acd541fbd45798a0a833a36dd7481d585a6b1bfa91e291fd85278b.jpg", + "image_caption": [ + "Figure 2: (a) Classification accuracy (mean and standard deviation) vs. the number of irrelevant noisy dimension $( D )$ for the XOR problem. (b) The Informative Features Weight Ratio (IFWR), central lines are the means while shaded are represent the standard deviation. IFWR is the sum of weights $W _ { d }$ over the informative features divided by the sum over all weights. (c) Box plots for the median rank of the two informative features. Black line and dashed red represent the median and mean of each method. Optimal median rank in this experiment is 1.5. " + ], + "image_footnote": [], + "bbox": [ + 184, + 617, + 820, + 728 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "7 APPLICATION ", + "text_level": 1, + "bbox": [ + 176, + 838, + 320, + 853 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "7.1 PURIFIED POPULATIONS OF PERIPHERAL BLOOD MONOCYTES (PBMCS) ", + "text_level": 1, + "bbox": [ + 174, + 871, + 712, + 883 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In (Zheng et al., 2017), the authors have subjected more than 90, 000 purified cell populations of peripheral blood monocytes (PBMCs) to Single-cell RNA sequencing (scRNA-seq) analysis. Here we use this data and focus on classifying two subpopulations of T-cells, namely Naive and regulatory T-cells. We use the proposed method to select a subset of genes for which the network discriminates between Naive and regulatory T-cells. We first filter out the genes that are lowly expressed in the cells, which leaves us with $D = 2 5 3 8$ genes (features). The total number of cells in these two classes is $N = 2 0 7 4 2$ , of which we only use $1 \\bar { 0 } \\%$ of the data for training. We apply the proposed method for different values of $\\lambda$ and report the number of selected features and classification accuracy on the test set. Here we compare our performance (STG and HC) to RF and LASSO. A least squares polynomial fit plot of the accuracy vs. number of selected features is presented in Fig. 3. The accuracy obtained by a NN without feature selection is $9 1 . 0 9 \\%$ , which is comparable to what we achieve with a small fraction of the features. We have also evaluated the performance of the Hard-Concrete applied to all layers (HC-Full), following the procedure in (Louizos et al., 2017). Empirically we observed that using this type of regularization across all layers provides inferior capabilities in terms of feature selection. Moreover, when the HC-Full converges to a larger subset of features $( d > 5 0$ ), it does not generalize at all and the test accuracy is around 0.5. ", + "bbox": [ + 173, + 895, + 820, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/66eca849473ebedb7251be9b39cd2bf28dee7b6f1b469ea80ea5be687bb65179.jpg", + "table_caption": [ + "Table 2: Performance comparison of survival analysis on METABRIC. We run the same experiment 5 times with different train/test split and report the mean and the standard deviation on the test set. In (Katzman et al., 2018), it is reported that DeepSurv outperforms other existing survival analysis methods such as Random Survival Forest (RSF) (Ishwaran et al., 2008) and the original Cox Propotional Hazard Model. " + ], + "table_footnote": [], + "table_body": "
DEEPSURVRSFCoX-LASSOCox-HCCoX-STG
C-INDEX0.612 (0.009)0.626 (0.006)0.580 (0.003)0.636 (0.007)0.633 (0.005)
# FEATURES221 (ALL)221 (ALL)44(0)8 (0.89)2（0)
", + "bbox": [ + 174, + 127, + 820, + 179 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 181, + 825, + 375 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/405bfb64f5f8487932868437d35db6bb0f8b6fbab6429900d03e75fc60c86ce7.jpg", + "image_caption": [ + "Figure 3: Classification of T-cells sub-populations. Accuracy vs. number of features selected by each method. Comparison of the proposed method (STG and HC) to Random Forests (RF) and LASSO. " + ], + "image_footnote": [], + "bbox": [ + 305, + 377, + 687, + 535 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7.2 COX PROPORTIONAL HAZARD MODELS FOR SURVIVAL ANALYSIS ", + "bbox": [ + 174, + 575, + 674, + 589 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We incorporate our STG into DeepSurv (Katzman et al., 2018) to examine how this procedure improves survival analysis. The input to this analysis are gene expression profiles (Curtis et al., 2012) along with additional commonly used clinical variables, the goal is to predict survival time. In this setting, identifying a small subset of predictive covariates is important as it may lead to cheaper and more stable medical assays. See the Appendix for additional description of the data and the experimental setup. ", + "bbox": [ + 174, + 601, + 825, + 685 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compared our method (Cox-STG and COX-HC) against three other methods: Cox model with $\\ell _ { 1 }$ regularization (Cox-LASSO), Random Survival Forest (RSF) (Ishwaran et al., 2008) and the original DeepSurv. We evaluate the predictive ability of the learned models based on the concordance index (CI), a standard performance metric for model assessment in survival analysis, which measures the agreement between the rankings of the predicted and observed survival times. The performance in terms of the CI is reported in Table 2. We see that Cox-HC and Cox-STG outperform the alternative methods, indicating that our method shrinks the feature size while learning a predictive model. ", + "bbox": [ + 174, + 691, + 825, + 790 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "8 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 809, + 318, + 825 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper, we propose a novel embedded feature selection method for neural networks and linear models based on stochastic gates. It has an advantage over previous $\\ell _ { 1 }$ based regularization methods in terms of achieving a high level of sparsity while learning effective non linear models. We justify our probabilistic feature selection framework from the information theoretic perspective. In experiments, we demonstrate that our method consistently outperforms existing embedded feature selection methods in both synthetic and real datasets. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 118 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Genevera I Allen. Automatic feature selection via weighted kernels and regularization. Journal of Computational and Graphical Statistics, 22(2):284–299, 2013. ", + "bbox": [ + 173, + 126, + 826, + 155 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Roberto Battiti. Using mutual information for selecting features in supervised neural net learning. IEEE Transactions on neural networks, 5(4):537–550, 1994. ", + "bbox": [ + 171, + 164, + 825, + 193 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Yoshua Bengio, Nicholas Léonard, and Aaron C. Courville. 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", + "bbox": [ + 171, + 726, + 825, + 755 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Makoto Yamada, Wittawat Jitkrittum, Leonid Sigal, Eric P Xing, and Masashi Sugiyama. Highdimensional feature selection by feature-wise kernelized lasso. Neural computation, 26(1):185–207, 2014. ", + "bbox": [ + 173, + 765, + 825, + 808 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Grace XY Zheng, Jessica M Terry, Phillip Belgrader, Paul Ryvkin, Zachary W Bent, Ryan Wilson, Solongo B Ziraldo, Tobias D Wheeler, Geoff P McDermott, Junjie Zhu, et al. Massively parallel digital transcriptional profiling of single cells. Nature communications, 8:14049, 2017. ", + "bbox": [ + 174, + 819, + 826, + 863 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Zexuan Zhu, Yew-Soon Ong, and Manoranjan Dash. Wrapper–filter feature selection algorithm using a memetic framework. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 37(1):70–76, 2007. ", + "bbox": [ + 173, + 875, + 826, + 916 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A APPENDIX B ALGORITHM ", + "bbox": [ + 173, + 101, + 316, + 152 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Here we present the pseudo code of our method, presented in Algorithm 1. The loss $L$ is typically negative-log likelihood for classification and squared loss for regression. $N$ is the sample size, $D$ is the number of features, $K$ is the number of Monte Carlo samples. ", + "bbox": [ + 174, + 166, + 825, + 209 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Input: $\\pmb { X } \\in \\mathbb { R } ^ { N \\times D }$ , target variables $\\boldsymbol { y } \\in \\mathbb { R } ^ { N }$ , regularization parameter $\\lambda$ , number of epochs $M$ , learning rate $\\gamma$ , ", + "bbox": [ + 173, + 224, + 808, + 253 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Output: Trained model $f _ { \\pmb { \\theta } }$ and parameter $\\boldsymbol { \\mu } \\in \\mathbb { R } ^ { D }$ ", + "bbox": [ + 178, + 253, + 516, + 268 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": ". \n1: Initialize the model parameter $\\pmb { \\theta }$ . Set $\\pmb { \\mu } = 0$ . \n2: for $i = 1 , . . . , M$ do \n3: for $n = 1 , . . . , N$ do \n4: for $d = 1 , . . . , D$ do \n5: for $k = 1 , . . . , K$ do \n6: Sample \u000f(k)d ∼ $\\epsilon _ { d } ^ { ( k ) } \\sim N ( 0 , \\sigma _ { d } )$ \n7: Compute the gate z(kd $z _ { d } ^ { ( k ) } = \\operatorname* { m a x } ( 0 , \\operatorname* { m i n } ( 1 , \\mu _ { d } + \\epsilon _ { d } ^ { ( k ) } + 0 . 5 ) )$ \n8: end for \n9: $\\begin{array} { r } { z _ { d } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } z _ { d } ^ { ( k ) } } \\end{array}$ \n10: end for \n11: Set $\\boldsymbol { z } = [ z _ { 1 } , . . . , z _ { D } ] ^ { T }$ \n12: 13: end forCompute the loss $\\begin{array} { r } { \\hat { L } = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } L ( f _ { \\theta } ( \\pmb { x } _ { n } \\odot \\pmb { z } ) , y _ { n } ) } \\end{array}$ \n14: N Compute the regularization term $\\begin{array} { r } { R = \\lambda \\sum _ { d = 1 } ^ { D } \\Phi ( \\frac { \\mu _ { d } + 0 . 5 } { \\sigma _ { d } } ) } \\end{array}$ \n15: Update $\\pmb { \\theta } : = \\pmb { \\theta } - \\gamma \\nabla _ { \\pmb { \\theta } } \\hat { L }$ and $\\pmb { \\mu } : = \\pmb { \\mu } - \\gamma \\nabla \\pmb { \\mu } ( \\hat { L } + R )$ \n16: end for ", + "bbox": [ + 178, + 257, + 653, + 517 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "17: ", + "bbox": [ + 176, + 508, + 196, + 520 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1: STG: Feature selection using stochastic gates ", + "bbox": [ + 302, + 520, + 696, + 535 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We empirically observe that setting the number of Monte Carlo samples $K = 1$ and the standard deviation of the Gaussian distribution $\\sigma _ { d } = 0 . 5$ for $d = 1 , . . . , D$ suffices for feature selection in our experiments. See Section E in the Appendix for more details about the specific choice of $\\sigma _ { d }$ . After training, the set of indices for selected features is: $\\{ d : \\operatorname* { m i n } ( 1 , \\operatorname* { m a x } ( 0 , \\mu _ { d } + 0 . 5 ) ) > 0 \\}$ . ", + "bbox": [ + 173, + 551, + 826, + 608 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 176, + 627, + 426, + 643 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We now provide a proof for Proposition 1, showing the equivalence between the stochastic optimization (Eq.7) and the deterministic one (Eq. 6). Let $\\tilde { \\cal S }$ be a subset such that $S ^ { * } \\setminus \\tilde { S } \\ne \\emptyset$ . That is there exists some element in $S ^ { * }$ that is not in $\\tilde { \\cal S }$ . For any such set $\\tilde { \\cal S }$ we have that $I ( X _ { \\tilde { S } } ; Y ) < I ( X ; Y )$ . Indeed, if we let $i \\in { \\cal S } ^ { * } \\cap \\tilde { \\cal S } ^ { c }$ then we have ", + "bbox": [ + 173, + 659, + 828, + 722 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/56d113d3cd6e3bf0dd9dae95bd9cb358f9fa5a3ec42ca0caab43c76862606bb1.jpg", + "text": "$$\n\\begin{array} { r l } { I ( \\boldsymbol { X } _ { \\tilde { \\mathcal { S } } } ; Y ) } & { \\leq I ( \\boldsymbol { X } _ { \\backslash \\{ i \\} } ; Y ) } \\\\ & { = I ( \\boldsymbol { X } ; Y ) - I ( \\boldsymbol { X } _ { i } ; Y | \\boldsymbol { X } _ { \\backslash \\{ i \\} } ) } \\\\ & { < I ( \\boldsymbol { X } ; Y ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 724, + 640, + 784 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where the final inequality follows by Assumption 1. Assumption 2 also yields that for any set $\\tilde { \\cal S }$ such that $S ^ { * } \\subset \\tilde { S }$ , we have $I ( { \\pmb X } _ { \\tilde { S } } ; Y ) = I ( { \\pmb X } ; Y )$ . Now, when we consider the Bernoulli optimization problem we have ", + "bbox": [ + 173, + 789, + 825, + 833 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/e5992f69b7584b344934b1d725018064d0708c8a19d4aac2e7f0f0626dd12b96.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\pi } I ( X \\odot S ; Y ) \\quad \\mathrm { s . t . } \\quad \\sum _ { l } \\pi _ { l } \\leq k \\mathrm { a n d } 0 \\leq \\pi _ { l } \\leq 1 .\n$$", + "text_format": "latex", + "bbox": [ + 312, + 838, + 683, + 872 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The mutual information can be expanded as ", + "bbox": [ + 174, + 876, + 464, + 891 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/ea1e2a0756538252da9f48d4e29f07413ec276c474fa106f23a0b0843e10ecff.jpg", + "text": "$$\nI ( X \\odot S ; Y ) = \\sum _ { s } I ( X \\odot s ; Y ) p _ { \\pi } ( S = s ) ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 895, + 651, + 928 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where we have used the fact that $_ { s }$ is independent of everything else. Recall that in optimization (Eq. 7) the coordinates of $\\pmb { S }$ are sampled at random. Therefore, the distribution that is being optimized over $p _ { \\pi }$ is a product distribution. Our goal is to understand the form of this distribution. To that end, we will consider a problem dropping the independence constraint. If we can show that the distribution found by solving this new optimization problem with less constraints is still a product distribution, then we obtain a solution to the original optimization (Eq. 6). ", + "bbox": [ + 173, + 103, + 826, + 188 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Now, from above we know that the optimal value of the optimization is $I ( X \\odot { \\tilde { S } } ; Y )$ for any set $S ^ { * } \\subset \\tilde { S }$ . Hence, any unconstrained distribution should place all of its mass on such subsets in order to maximize the mutual information. As a result $\\begin{array} { r } { \\sum _ { l \\in S ^ { * } } p ( S _ { l } = 1 ) = k } \\end{array}$ . However, there is an optimization constraint that $\\mathbb { E } [ \\sum _ { l } S _ { l } ] \\le k$ . Therefore, $\\mathbb { E } [ S _ { l } ] = 0$ for any $l \\notin S ^ { * }$ . Hence, the optimal solution is to select the distribution so that all of the mass is placed on the subset $S ^ { * }$ and no mass elsewhere. As this is also a product distribution, this complete the proof of the claim. ", + "bbox": [ + 173, + 195, + 825, + 284 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D BRIDGING THE TWO PERSPECTIVES ", + "text_level": 1, + "bbox": [ + 173, + 306, + 511, + 324 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To motivate the introduction of randomness into the risk, we have looked at the feature selection problem from a MI perspective. Based on the MI objective, we have observed that introducing randomness into the constrained maximization procedure, does not change the objective (Proposition 1). Here we provide a relation between the MI objective (Eq. 6) to the empirical risk (Eq. 1), which supports our proposed procedure. ", + "bbox": [ + 173, + 339, + 825, + 411 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We first note that the MI maximization over the set $s$ can be reformulated as the minimization of the conditional entropy $H ( Y | \\boldsymbol { X } _ { \\mathcal { S } } )$ since $H ( Y )$ does not depend on $s$ : ", + "bbox": [ + 171, + 416, + 823, + 446 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/4340acfbddae0dcd7ca93c3955ddda6b3e3d779bb5199cfe7113058d3e09b268.jpg", + "text": "$$\n\\displaystyle { \\operatorname* { m a x } _ { \\mathcal { S } } I ( \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ; Y ) = \\operatorname* { m a x } _ { \\mathcal { S } } H ( Y ) - H ( Y | \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ) \\iff \\operatorname* { m i n } _ { \\mathcal { S } } H ( Y | \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 455, + 722, + 478 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Recall that $X _ { \\mathcal { S } } = X \\odot \\tilde { S }$ . By Proposition 1, we can rewrite the deterministic search over the set $s$ by a search over the Bernoulli parameters $\\pi$ : ", + "bbox": [ + 173, + 497, + 823, + 527 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1281ade667d73948ce91bd121329f8e38779576e0e3752f862fc33710d9c20a8.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\operatorname* { m i n } _ { \\pi } H ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) = \\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { \\boldsymbol { X } , \\boldsymbol { Y } , \\tilde { \\boldsymbol { S } } } - \\log P _ { \\pmb { \\theta } ^ { * } } ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) } & { } \\\\ { = \\displaystyle \\operatorname* { m i n } _ { \\pmb { \\theta } } \\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { \\boldsymbol { X } , \\boldsymbol { Y } , \\tilde { \\boldsymbol { S } } } - \\log P _ { \\pmb { \\theta } } ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 535, + 700, + 590 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where the expectation is over $X , Y \\sim P _ { \\pmb { \\theta } _ { * } }$ , which is the true data distribution, and $\\tilde { S } \\sim B e r n ( \\tilde { S } | \\pi )$ . Put our model distribution as $P _ { \\theta }$ , then we can rewrite the right hand side as: ", + "bbox": [ + 173, + 599, + 825, + 631 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/44015d158845a22ad0a394062bebe138c8c4df0f42a85cde9cd840e24e907cb2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { X , Y , \\tilde { S } } \\log P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) = \\mathbb { E } _ { X , Y , \\tilde { S } } \\Bigg [ \\log P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) \\frac { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } \\Bigg ] } \\\\ & { \\qquad = \\mathbb { E } _ { X , Y , \\tilde { S } } \\log \\frac { P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) } { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } + \\mathbb { E } _ { X , Y , \\tilde { S } } \\log P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 640, + 789, + 723 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Since $\\mathrm { K L } ( P _ { \\theta ^ { * } } ( Y | X \\odot \\tilde { S } ) | | P _ { \\theta } ( Y | X \\odot \\tilde { S } ) )$ is non-negative, $\\mathbb { E } _ { \\tilde { S } } \\mathbf { K } \\mathbf { L } ( P _ { \\theta * } ( Y | \\boldsymbol { X } \\odot \\tilde { \\pmb { S } } ) | | P _ { \\theta } ( Y | \\boldsymbol { X } \\odot \\tilde { \\pmb { S } } ) )$ is also non-negative because it is a weighted sum of non-negative terms. Noting that ", + "bbox": [ + 173, + 738, + 826, + 770 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/d1fa31e8ca8ae1d6e011d190b67ac3220ff12e7468ce475f8644e0d4d8e05a72.jpg", + "text": "$$\n\\mathbb { E } _ { \\tilde { S } } \\mathrm { K L } \\big ( P _ { \\theta * } ( Y | X \\odot \\tilde { S } ) | | P _ { \\theta } ( Y | X \\odot \\tilde { S } ) \\big ) = \\mathbb { E } _ { X , Y , \\tilde { S } } \\log \\frac { P _ { \\theta * } ( Y | X \\odot \\tilde { S } ) } { P _ { \\theta } ( Y | X \\odot \\tilde { S } ) }\n$$", + "text_format": "latex", + "bbox": [ + 261, + 779, + 735, + 816 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "we can conclude that ", + "bbox": [ + 173, + 825, + 313, + 839 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/8919538185d34b2d5fb2cb4a0ee179a471e77ed1e9af6000ac6f176b4c5f18b5.jpg", + "text": "$$\n\\mathbb { E } _ { X , Y , \\tilde { S } } - \\log P _ { \\theta ^ { * } } ( Y | X \\odot \\tilde { S } ) \\leq \\mathbb { E } _ { X , Y , \\tilde { S } } - \\log P _ { \\theta } ( Y | X \\odot \\tilde { S } ) .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 848, + 704, + 869 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "If we consider the negative log likelihood of the target given the observations (i.e. $- \\log P _ { \\theta } ( Y | X \\odot \\tilde { \\pmb { S } } ) )$ as a loss function $L$ (which encodes the classification or regression function $f _ { \\pmb { \\theta } } ,$ ), then we see that minimizing the risk approximately maximizes the MI objective in Eq. 6. ", + "bbox": [ + 174, + 881, + 826, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "E DETAILS OF REGULARIZATION TERM ", + "text_level": 1, + "bbox": [ + 173, + 102, + 519, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we provide the detail description of the regularization term. For the vector of stochastic gates $z \\in \\mathbb { R } ^ { D }$ , the regularization term is expressed as follows: ", + "bbox": [ + 173, + 132, + 823, + 161 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e09a089146d547c94fd00e82f90c19adc399b73b47abc6c40a667e4041ef9598.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { Z } \\| Z \\| _ { 0 } = \\displaystyle \\sum _ { d = 1 } ^ { D } \\mathbb { P } [ z _ { d } > 0 ] = \\displaystyle \\sum _ { d = 1 } ^ { D } \\mathbb { P } [ \\mu _ { d } + \\sigma _ { d } \\epsilon _ { d } > - \\frac { 1 } { 2 } ] } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\{ 1 - \\mathbb { P } [ \\mu _ { d } + \\sigma _ { d } \\epsilon _ { d } \\leq - \\frac { 1 } { 2 } ] \\} } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\{ 1 - \\Phi ( - \\frac { 1 } { \\sigma _ { d } } ) \\} } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\Phi ( \\frac { \\mu _ { d } + \\frac { 1 } { 2 } } { \\sigma _ { d } } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 167, + 678, + 344 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The derivative of the regularization term with respect to the distribution parameter $\\mu _ { d }$ is simply the Gaussian PDF: ", + "bbox": [ + 173, + 348, + 826, + 377 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/f76a16a8b6b2e07af7151e1897e53751a8f07710893487f6ac9bde8d34920224.jpg", + "text": "$$\n\\frac { \\partial } { \\partial \\mu _ { d } } \\mathbb { E } _ { Z } \\left\\| Z \\right\\| _ { 0 } = \\frac { \\partial } { \\partial \\mu _ { d } } \\Phi ( \\frac { \\mu _ { d } + \\frac { 1 } { 2 } } { \\sigma _ { d } } ) = \\frac { 1 } { \\sqrt { 2 \\pi \\sigma _ { d } ^ { 2 } } } e ^ { - \\frac { ( \\mu _ { d } + \\frac { 1 } { 2 } ) ^ { 2 } } { 2 \\sigma _ { d } ^ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 380, + 686, + 421 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The effect of $\\sigma$ can be understood by looking at the value of $\\frac { \\partial } { \\partial \\mu _ { d } } \\mathbb { E } _ { Z } | | Z | | _ { 0 }$ . In the first iteration during the training, $\\mu _ { d }$ is 0. Therefore, during the initial phase of training, it is close to $\\frac { \\lambda } { \\sqrt { 2 \\pi \\sigma _ { d } ^ { 2 } } } e ^ { - \\frac { 1 } { 8 \\sigma _ { d } ^ { 2 } } }$ . In order to remove irrelevant features, this term has to be greater than the derivative of the loss with respect to $\\mu _ { d }$ because otherwise $\\mu _ { d }$ is updated in the incorrect direction. To encourage such behavior, we set $\\sigma = 0 . 5$ , which is around the maximum of the gradient during the initial phase as shown in Fig. 4. Although the point that attains the maximum moves as $\\mu$ changes, we empirically observe that setting $\\sigma = 0 . 5$ performs well in our experiments when the regularization parameter $\\lambda$ is appropriately tuned (see Section I). ", + "bbox": [ + 173, + 433, + 826, + 565 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/35b45d282fb48659f375af3711fb64ad2e5ebd436d2ce1904752054ea3096810.jpg", + "image_caption": [ + "Figure 4: The plot of ∂∂µ EZ||Z||0|µ=0 = √ 12πσ2 e for $\\sigma = [ 0 . 0 0 1 , 2 ]$ " + ], + "image_footnote": [], + "bbox": [ + 344, + 587, + 635, + 728 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F ISSUES IN GRADIENT ESTIMATION OF DISCRETE RANDOM VARIABLES ", + "text_level": 1, + "bbox": [ + 169, + 801, + 797, + 820 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In Section 3.1, we have introduced Bernoulli random $\\widetilde { s } _ { d } , d = 1 , . . . , D$ variables with corresponding parameters $\\pi _ { d }$ into the risk objective (Eq. 4). Taking the expectation over the $\\ell _ { 0 }$ norm of $\\tilde { s }$ boils down to the sum of the Bernoulli parameters $\\pi _ { d }$ . However, the optimization of the resulting objective suffers from high variance due to the discrete nature of $\\tilde { S }$ . Here, we attempt to convey this problem by analyzing the risk term in the objective in Eq. 4. Using the Bernoulli paramterization the empirical risk $\\hat { R } ( \\pmb \\theta , \\pmb \\pi )$ is expressed as ", + "bbox": [ + 173, + 833, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ea12c3bc505c5942512038ddca65e038b027aed3c11dccba910374c2c32a1bf4.jpg", + "text": "$$\n\\sum _ { \\mathscr { Z } : \\{ 0 , 1 \\} ^ { D } } \\left[ \\sum _ { n = 1 } ^ { N } [ L ( f _ { \\theta } ( \\mathscr { z } \\odot \\mathscr { x } _ { n } ) , \\pmb { y } _ { n } ] \\prod _ { d = 1 } ^ { D } \\pi _ { d } ^ { \\mathscr { z } _ { d } } ( 1 - \\pi _ { d } ) ^ { 1 - \\mathscr { z } _ { d } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 123, + 691, + 170 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In practice, as the outer sum involves enumerating $2 ^ { D }$ possibilities of the indicator variables, one can replace the outer sum with Monte Carlo samples from the product of Bernoulli distributions $B ( z | \\pi )$ . However, a Monte Carlo estimate of $\\frac { \\partial } { \\partial \\pi _ { d } } \\hat { R } ( \\pmb \\theta , \\pmb \\pi )$ suffers from high variance. To see this, consider the following exact gradient of the empirical risk with respect to $\\pi _ { d }$ , which is ", + "bbox": [ + 173, + 186, + 828, + 248 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d55d74c8420ee7bc7660cc28d659e4e1ff6de4c8af2a859336ca1e7055135e0e.jpg", + "text": "$$\n\\sum _ { z : \\{ 0 , 1 \\} ^ { D } , z _ { d } = 1 } \\left[ L ( z ) p _ { z _ { i } \\neq d } \\right] - \\sum _ { z : \\{ 0 , 1 \\} ^ { D } , z _ { d } = 0 } \\left[ L ( z ) p _ { z _ { i } \\neq d } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 308, + 255, + 686, + 292 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $p ( z _ { i \\neq d } ) = \\prod _ { i \\neq d } ^ { D } \\pi _ { i } ^ { z _ { i } } ( 1 - \\pi _ { i } ) ^ { 1 - z _ { i } }$ , by absorbing the model $f _ { \\theta } ( \\cdot )$ and the data into $L ( \\cdot )$ . Due to the discrete nature of $_ z$ , we see that even the sign of the gradient estimate becomes inaccurate if we can only access a small number of Monte Carlo samples. While a score-function estimator such as REINFORCE (Williams, 1992) can be used, it is known that the reparametrization trick reduces the variance more in practice. ", + "bbox": [ + 173, + 303, + 826, + 377 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "G COMPARISON TO HARD-CONCRETE DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 174, + 397, + 633, + 415 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To evaluate the strength of the proposed continuous relaxation of the Bernoulli distribution, described in Subsection 3.1, we compare it with the Hard-Concrete distribution (Louizos et al., 2017), another continuous surrogate for Bernoulli distributions, which was originally developed for neural network model compression. The details of the Hard-Concrete distribution is described below. ", + "bbox": [ + 173, + 430, + 826, + 487 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "G.1 HARD-CONCRETE DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 176, + 506, + 449, + 520 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The authors in (Louizos et al., 2017) introduce a modification of Binary Concrete, whose sampling procedure is as follows: ", + "bbox": [ + 173, + 531, + 823, + 560 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/27084ec3b004a76a1c3f90512c04c0c14b22235b2baff292f09be0fd00ba0425.jpg", + "text": "$$\n\\begin{array} { r } { u \\sim U ( 0 , 1 ) , L = \\log ( U ) - \\log ( 1 - U ) } \\\\ { s = \\frac { 1 } { 1 + \\exp ( \\frac { - ( \\log { \\alpha } + L ) } { \\beta } ) } } \\\\ { \\bar { s } = s ( \\zeta - \\tau ) + \\tau } \\\\ { z = \\operatorname* { m i n } ( 1 , \\operatorname* { m a x } ( 0 , \\bar { s } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 364, + 568, + 633, + 664 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $( \\tau , \\zeta )$ is an interval, with $\\tau < 0$ and $\\zeta > 1$ . This induces a new distribution, whose support is $[ 0 , 1 ]$ instead of $( 0 , 1 )$ . With $0 < \\beta < 1$ , the probability density concentrates its mass near the end points, since values larger than $\\frac { 1 - \\tau } { \\zeta - \\tau }$ are rounded to one, whereas values smaller than $\\frac { - \\tau } { \\zeta - \\tau }$ are rounded to zero. ", + "bbox": [ + 173, + 670, + 825, + 729 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The CDF of $s$ is ", + "bbox": [ + 174, + 734, + 282, + 751 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/dfc6a731c903d529ed2f486eadfa1ed8ad9a0887662fbf5aa5fa5e5ff378e9cf.jpg", + "text": "$$\nQ _ { s } ( s | \\beta , \\log \\alpha ) = \\mathrm { { S i g m o i d } } ( ( \\log s - \\log ( 1 - s ) ) \\beta - \\log \\alpha )\n$$", + "text_format": "latex", + "bbox": [ + 303, + 758, + 692, + 776 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "and so the CDF of $\\bar { s }$ is ", + "bbox": [ + 173, + 791, + 323, + 806 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/b790665ff906efecbc03b8f0a687bfa48b01ad9915f03d7d7e060840884e16e6.jpg", + "text": "$$\nQ _ { \\bar { s } } ( \\bar { s } | \\phi ) = \\mathrm { S i g m o i d } ( ( \\log ( \\frac { \\bar { s } - \\tau } { \\zeta - \\tau } ) - \\log ( 1 - \\frac { \\bar { s } - \\tau } { \\zeta - \\tau } ) ) \\beta - \\log \\alpha\n$$", + "text_format": "latex", + "bbox": [ + 290, + 811, + 707, + 844 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\phi = ( \\beta , \\log \\alpha , \\zeta , \\tau )$ ", + "bbox": [ + 173, + 852, + 348, + 867 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now, the probability of being the gate $z _ { i }$ being active is $1 - Q _ { \\bar { s } } ( 0 | \\phi )$ and can be written as ", + "bbox": [ + 169, + 872, + 766, + 890 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/709790339e131d7106c3874193b9a8845506d61ccb23d0e2b8ce48186ea672af.jpg", + "text": "$$\n1 - Q _ { \\bar { s } } ( 0 | \\phi ) = \\mathrm { S i g m o i d } ( \\log \\alpha - \\beta \\log \\frac { - \\tau } { \\zeta } )\n$$", + "text_format": "latex", + "bbox": [ + 352, + 897, + 645, + 929 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "G.2 THE EFFECT OF THE HEAVY TAIL DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 178, + 104, + 547, + 117 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The main difference between our proposed distribution (STG) and the Hard-Concrete (Louizos et al., 2017) distribution is that the latter is based on the logistic distribution, which has a heavier tail than the Gaussian distribution we have employed. As shown in Fig 5, the heavy-tailness results in instability during training. Furthermore, our method converges much faster and more reliably than the feature selection method using the Hard-Concrete distribution on a the two-moons, XOR and MADELON datasets (see Subsection J.1 and J.2). A similar phenomenon is also demonstrated in the MNIST experiment (see Subsection J.3). ", + "bbox": [ + 173, + 128, + 826, + 227 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/d25fe19f87e70d251501c821c2f367a6207558df2e41bbf126700c19922f444a.jpg", + "image_caption": [ + "Figure 5: Comparison between STG and Hard-Concrete on the Two-Moon dataset (a), XOR (b) and MADELON (c). The shaded area represents the standard deviation, calculated by running the same experiment 10 times with different random initializations. " + ], + "image_footnote": [], + "bbox": [ + 210, + 252, + 779, + 356 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "H NON-CONVEX DETERMINISTIC REGULARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 446, + 627, + 463 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Our proposed objective for feature selection (summarized in Eq. 5) consists of three major differences compared to the LASSO formulation. ", + "bbox": [ + 173, + 478, + 825, + 507 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "• Non-linearity, which is obtained by learning parameters of a multi-layer network with non linear activation’s. • Feature gates, these are a specific form on non convex regularization. • Stochasticity, which is achieved by injecting noise via the reparametrization trick. ", + "bbox": [ + 215, + 518, + 825, + 585 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In section 6.1, we have demonstrated that even in the linear regime the proposed formulation outperforms the LASSO in the task of identifying active features (see Fig. 1(a)). ", + "bbox": [ + 174, + 595, + 820, + 625 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The stochastic property of the gates proposed in this paper is justified using a MI prospective (see Section 4). Nonetheless, in this section we evaluate whether the practical performance gain of our method stems from the non-convex regularization or from it’s combination with the injected noise. ", + "bbox": [ + 174, + 631, + 825, + 674 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We define a deterministic gate using $\\tilde { z } _ { d } = g ( \\mu _ { d } ) = \\operatorname* { m a x } ( 0 , \\operatorname* { m i n } ( 1 , \\mu _ { d } + 0 . 5 ) )$ , where $\\mu _ { d }$ is a parameter learned for each feature $d = 1 , . . . , D$ . In the linear regression setting we can define the non-convex deterministic (NCD) as ", + "bbox": [ + 176, + 679, + 825, + 722 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/8ba2552f138179e6ad3578a4a71fb2d60e6c34f82b17702920967c6a64218668.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta , \\mu } \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } ( \\theta ^ { T } x _ { n } \\odot \\tilde { z } - y _ { n } ) ^ { 2 } + \\lambda \\sum _ { d = 1 } ^ { D } \\Phi \\left( \\frac { \\mu _ { d } + 0 . 5 } { 0 . 5 } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 313, + 728, + 686, + 772 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\Phi$ is the standard Gaussian CDF. Combined with $\\tilde { z } _ { d }$ this non-convex regularized objective is deterministic and differentiable, and its solution can be approximated via gradient decent. We evaluate the applicability of such alternative objective for the task of feature selection in linear regression. The experimental setting follows the description in Section 6.1 using $D = 6 4$ . ", + "bbox": [ + 173, + 776, + 825, + 833 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In Fig. 6(b), we compare the values of the deteministic gates (DNC) and stochastic gates (STG) using $N = 6 0$ . As depicted from this figure, the STG correctly identifies the active features (first 10) and sparsifies the non-active features. The deterministic version DNC does induce partial sparsity, but fails to remove some of the irrelevant features. Furthemore, it does not push the gates values of active features up to 1. Clearly, in the deterministic setting, shrinking the value of $\\mu _ { d }$ while compensating with a large coefficient in $\\pmb \\theta$ is preferable for the objective in Eq. 12. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "To evaluate the probability of support recovery using DNC, we apply additional thresholding to the gates. Specifically, we use a threshold of 0.5 to define if a feature is active or not. A comparison with STG, HC and LASSO in this setting appears in Fig. 6(a). The DNC is superior to LASSO, but inferior to the HC and STG. Which means that there is a true advantage of stochacity in our formulation. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "One aspect that is crucially difference between the STG and DNC is what we call “second chance”. For the deterministic formulation, if a feature is zeroed out in an early training phase, the gradient of the feature vanishes for further training and it will never be pulled back. For the STG, even if the feature has been pulled to zero, the gate may still be pushed back up to one at a later training phase. This is due to the injected noise which allows to reevaluate the gradient of each gate. In Fig. 6(c), we demonstrate this “second chance” effect using $N = 6 0$ samples and presenting the gate’s values (throughout training) for an active feature. ", + "bbox": [ + 173, + 180, + 826, + 277 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "To conclude, this form of deterministic gates (Eq. 12) is not suitable for an embedded feature selection method. Using thresholding seems somewhat effective; however such thresholding was not required for the STG in any of our experiments. While the shrinkage phenomenon might be avoided using further regularization on $\\pmb \\theta$ this is out of the scope of the current study. ", + "bbox": [ + 174, + 285, + 825, + 340 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/2233abd145920d07fd52782d2454c31290b15af50d5ee60750b6ba242b8c7605.jpg", + "image_caption": [ + "Figure 6: Support recovery in the linear regression (see Section 6.1), the dimension $D = 6 4$ . (a) Phase transition comparison between the STG, HC, LASSO and the deterministic non convex (DNC). (b) An example of the output gates, STG $( z _ { d } )$ and DNC $( \\tilde { z } _ { d } )$ for $N = 6 0$ . Each marker represents the gate value of the corresponding feature, of which the first 10 features are active. The deterministic gates are not pushed towards 1. (c) Demonstration of the “second chance” inherit to the STG. In the deterministic setting if a feature is eliminated its gradient vanishes for the rest of the training. The STG reevaluates this features and “recovers” after 4000 epochs. " + ], + "image_footnote": [], + "bbox": [ + 178, + 386, + 790, + 767 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "I TUNING THE REGULARIZATION PARAMETER $\\lambda$ ", + "text_level": 1, + "bbox": [ + 173, + 103, + 586, + 117 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "There have been several studies (Lederer & Müller, 2015; Coen et al., 2006) on the problem of tuning regularization parameter in regression and classification. Perhaps the most related regularized objective is the LASSO formulation. Studies such as (Chand, 2012; Fan & Tang, 2013) develop methods for tuning the regularization parameter in LASSO. In Eq. 5 as in the LASSO formulation, increasing the value of $\\lambda$ effectively forces a sparser solution with less features. For simplicity, we focus on one popular way for tuning $\\lambda$ in the LASSO. The idea is to minimize the generalization error using a cross validation procedure over a range of $\\lambda$ values. ", + "bbox": [ + 174, + 133, + 825, + 231 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In this section, we demonstrate that this procedure is also applicable for the proposed approach. We apply the STG to the RAI dataset and evaluate the MSE using a 5-folds cross validation procedure. We compute the MSE on the validation data using 100 values of $\\lambda$ logarithmically scaled in $[ 1 0 ^ { - 3 } , 1 0 ^ { 1 } ]$ As demonstrated in Fig. 7, using a cross validation procedure for tuning $\\lambda$ provides a clear optimal value for this example. ", + "bbox": [ + 174, + 238, + 825, + 308 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/232c0999f711f8ccec17e4f6fbd02a77cb7878b79372594bf8c4c1568fd246d9.jpg", + "image_caption": [ + "Figure 7: Demonstrating a cross validation procedure on the RAI dataset (see section I for details). We perform 5-folds cross validation and evaluate the MSE on the test test. The optimal $\\lambda = 0 . 1 3 5$ which seems stable in this example based on the low standard deviation. " + ], + "image_footnote": [], + "bbox": [ + 210, + 323, + 784, + 587 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "J ADDITIONAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 685, + 437, + 700 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "J.1 MADELON DATASET ", + "text_level": 1, + "bbox": [ + 174, + 717, + 366, + 732 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The MADELON dataset, first suggested for the NIPS 2003 feature selection problem, is a multivariate highly nonlinear binary classification problem. The MADELON dataset is generated using 32 groups of data points placed on a 5 dimensional hyper-cube and randomly labeling them by one of the two class labels. The first 5 informative features are then used to create 15 additional coordinates that are formed based on a random linear transformation of the first 5. A Gaussian noise $N ( 0 , 1 )$ is added to each feature. Next, additional 480 nuisance coordinates are added in the same manner. These features have no effect on the class label. Finally, $1 \\%$ of the labels are flipped. To be consistent with the XOR and two moons experiments, we use 1, 500 points from this dataset, and evaluate our proposed method in terms of its predictive power and ability to detect the informative features. We vary the regularizaton parameter $\\lambda$ in the range [0.01, 10] and evaluate the classification accuracy using 5 folds cross validation. In this example, we restrict our comparison to Random Forest and LASSO. We focus on Random Forest as it was the strongest competitor to our method in all of our experiments, while LASSO is evaluated because it is a widely used embedded feature selection method. ", + "bbox": [ + 173, + 743, + 825, + 922 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As evident from Fig. 8(a), our method achieves the highest accuracy while using less features. Moreover, as depicted from this figure, peak performence occurs when selecting 5 features, thus, our method provides a clear indication to the true number of informative features. Both LASSO and RF on the other hand, do not provide a clear indication of the true number of relevant features. In Fig. 8(b), we evaluate the effect of $\\lambda$ on the number and quality of selected features. As shown in this plot, there is a wide range of $\\lambda$ ’s such that our method only selects relevant features. Finally, as evident from the plato on the right hand side of the red plot, there is a range of $\\lambda$ ’s such that exactly 5 features are selected. ", + "bbox": [ + 173, + 103, + 826, + 214 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/f81c197fa97377e7b0126d38b6153864d82047273e824cbb916508ca6b386556.jpg", + "image_caption": [ + "Figure 8: (a) Classification accuracy on the MADELON data sets. We evaluate performance using 5-fold cross validation for different number of selected features. In this dataset, only the first 20 coordinates are informative. In that regime the proposed method outperforms RF and LASSO. (b) An empirical evaluation of the effect the regularization parameter $\\lambda$ . The IFWR and the number of selected features are presented on both sides of the $y$ -axis of this plot. For both plots, the mean is presented as a solid/dashed line, while the standard deviation is marked as a shaded color around the mean. (c) Box plots for the median rank of the 5 original informative features. Black line and dashed red represent the median and mean of each method. Optimal median rank in this experiment is 3. " + ], + "image_footnote": [], + "bbox": [ + 186, + 242, + 816, + 602 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "J.2 TWO MOONS CLASSIFICATION WITH NUISANCE FEATURES ", + "text_level": 1, + "bbox": [ + 174, + 772, + 616, + 785 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this experiment, we construct a dataset based on \"two moons\" shape classes, concatenated with noisy features. The first two coordinates $x _ { 1 } , x _ { 2 }$ are generated by adding a Gaussian noise with zero mean and the variance of $\\sigma _ { r } ^ { 2 } = 0 . 1$ onto two nested half circles, as presented in Fig. 9(a). Nuisance features $x _ { i } , i = 3 , . . . , D$ , are drawn from a Gaussian distribution with zero mean and variance of $\\sigma _ { n } ^ { 2 } = 1$ . We reserve the $7 0 \\%$ as a test set, and use $1 0 \\%$ of the remaining training set as a validation set. We follow the same hyperparameter tuning procedure as in the XOR experiment. The classification accuracy is in Fig. 9(b). Based on the classification accuracies, it is evident that for a small number of nuisance dimensions all methods correctly identify the most relevant features. The proposed method (STG) and Random Forest (RF) are the only methods that achieve near perfect classification accuracy for a wide range of nuisance dimensions. The other NN based method (DFS) seem to converge to sub-optimal solutions. We note that the median rank for all the methods is 1.5. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/ba3a08516b182a43c04de57e2dc57232418b15b4618ba3a7fb4d6140a06cd2d5.jpg", + "image_caption": [ + "Figure 9: (a) Realizations from the \"Two moons\" shaped binary classification class. $X _ { 1 }$ and $X _ { 2 }$ are the relevant features, $X _ { i } , i = 3 , . . . , D$ are noisy features drawn from a Gaussian with zero mean and variance of 1. (b) Classification accuracy (mean and standard deviation based on 20 runs) vs. the number of irrelevant noisy dimension. " + ], + "image_footnote": [], + "bbox": [ + 184, + 159, + 816, + 327 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "J.3 SPARSE HANDWRITTEN DIGITS CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 453, + 547, + 467 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In the following toy example, we attempt to distinguish between images of handwritten digits of 3’s and 8’s using samples from MNIST (LeCun et al., 1998). The orientation and location of the digits is more or less the same throughout this dataset, therefore for these two classes (3’s and 8’s), we expect that some of the left side features (pixels) would be sufficient for the separation. The experiment is performed as followed. We reserve $9 0 \\%$ of the data as the test set, and train on the remaining $1 0 \\%$ . We then apply STG and evaluate the classification accuracy and the number of selected features. We use the architecture [200, 50, 10] with tanh activations. The experiment was repeated 10 times, the extracted features and accuracies were consistent over 20 trials. We noticed a relatively small number of selected features, which are positioned southwest and close to the center of the images, achieve very high classification accuracy. An example of 9 randomly selected samples overlaid with the weights of the selected features is presented in Fig. 10(a). Furthermore, we also evaluate the effect of $\\lambda$ on the the number of selected features and accuracy of the method. We apply our method (STG) and its variant using the Hard-Concrete distribution to a randomly sampled training set of size $N = 1 5 0 0$ and vary $\\lambda$ in the range of [0.001, 0.01]. In Fig. 10(b) we present the accuracy and sparsity level vs. the $\\lambda$ parameter. This experiment demonstrates the improved performance of the proposed distribution compared to the Hard-Concrete (HC (Louizos et al., 2017)), which was designed for neural net model compression. Not only that the overall accuracy is superior, but also it seems that the transition as a function of $\\lambda$ is smoother, which suggests that the method is less sensitive to the choice of $\\lambda$ . ", + "bbox": [ + 173, + 481, + 826, + 744 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "J.4 GISETTE DATASET ", + "text_level": 1, + "bbox": [ + 174, + 770, + 339, + 784 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The Gisette is a handwritten dataset also appears in the NIPS 2003 feature selection challenge. The data consists of 13, 500 handwritten digits of $\\ ' 4 '$ and ’9’. The original digits have $2 8 \\times 2 8$ pixels, which were modified for the feature selection challenge. Specifically, a random subset of the features are embedded in a 2500 dimensional space and additional 2500 irrelevant probes are added. The train, validation and test dimensions are $6 0 0 0 / 1 0 0 0 / 7 5 0 0$ respectively. We apply the proposed STG based feature selection method and compare its performance to Random Forests (RF) and Extremely Randomized Trees (Tree). As evident from Fig. 11, we obtain a high accuracy even for a dramatic reduction in the feature size. LASSO is not presented in this experiment as its performance is dramatically inferior to the alternatives (accuracy $< 0 . 6$ ). ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/63dcdf118a319fb90f36df91f9fae0e284f578ca8325363eeec29265cc548456.jpg", + "image_caption": [ + "Figure 10: (a) Nine samples from MNIST (white) overlaid with the subset of 13 features (black) selected by STG. Based on only these features, the binary classification accuracy reaches $9 2 . 2 \\%$ . For these nine randomly selected samples, all the 8’s have values within the support of the selected features, whereas for the 3’s there is no intersection. (b) The comparison of accuracy and sparsity level performance for $\\lambda$ in the range of $[ 1 0 ^ { - 3 } , 1 0 ^ { - 2 } ]$ between using our proposed method (STG) and its variant using the Hard-Concrete (HC) distribution. " + ], + "image_footnote": [], + "bbox": [ + 186, + 99, + 807, + 296 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/5dd199d6e1cc192e49bf859e0f2459a44a82769f43d17f81306e036866dee8a9.jpg", + "image_caption": [ + "Figure 11: Classification of the binary noisy digits from the Gisette dataset. The total number of feature is 5000 of which 2500 are irrelevant probes. Here we compare the performance of the proposed approach to RF and Tree based classifier. The lines represent a least squares polynomial fit plot of the accuracy vs. number of selected features. " + ], + "image_footnote": [], + "bbox": [ + 320, + 417, + 656, + 556 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "J.5 REUTERS CORPUS VOLUME I ", + "text_level": 1, + "bbox": [ + 176, + 630, + 418, + 643 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The Reuters Corpus Volume I (RCV1) consists of 800, 000 newswire stories manually labeled by 103 categories. This is a multilable regime, i.e. each story is assigned to multiple categories. Here we focus on a binary subset of this corpus, with 23, 203 stories. The total number of feature is 47, 236 and the train, validation and test portions are $1 0 \\%$ , $8 . 5 \\%$ , and $8 1 . 5 \\%$ respectively. We evaluate the performance of our method using a 5-fold cross validation. A comparison the LASSO and RF appears in Fig. 12. This example demonstrates that our method is also effective in an extremely high dimensional regime of non linear function estimation. ", + "bbox": [ + 173, + 656, + 825, + 755 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "J.6 DETAILS OF COX PROPORTIONAL HAZARD MODEL ", + "text_level": 1, + "bbox": [ + 173, + 775, + 568, + 790 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Survival times are assumed to follow a distribution, which is characterized by the survival function $S ( t ) = P ( T > t )$ . A hazard function, which measures the instantaneous rate of death, is defined by $\\begin{array} { r } { h ( t ) = \\operatorname* { l i m } _ { \\Delta t 0 } \\frac { P ( t < T \\leq t + \\Delta t | T > t ) } { \\Delta t } = \\frac { p ( t ) } { S ( t ) } } \\end{array}$ . We can relate the two functions in the following way: S(t) = e− R t0 h(t)dt. ", + "bbox": [ + 173, + 801, + 826, + 867 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proportional hazard models assume a multiplicative effect of the covariates $x$ on the hazard function such that ", + "bbox": [ + 173, + 875, + 823, + 902 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/45e4bbe35d63b8e9b56a469a2b1f7ef34228d7d4a8acacfadc3fb0fb529df372.jpg", + "text": "$$\nh ( t | \\boldsymbol { x } ) = h _ { 0 } ( t ) e ^ { \\theta ^ { T } \\boldsymbol { x } } ,\n$$", + "text_format": "latex", + "bbox": [ + 426, + 902, + 570, + 925 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/bbdb8424abb8ba85469b699d8f13201af2a5fba3870fd125b2d054267de5fbd9.jpg", + "image_caption": [ + "Figure 12: Classification of the version of the RCV1 textual dataset. The total number of feature is 47, 236. Here we compare the performance of the proposed approach to RF and LASSO. The lines represent a least squares polynomial fit plot of the accuracy vs. number of selected features. " + ], + "image_footnote": [], + "bbox": [ + 320, + 101, + 656, + 239 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $h _ { 0 } ( t )$ is a baseline hazard function, which is often the exponential or Weibull distribution, and $\\pmb \\theta$ is the parameter of interests. ", + "bbox": [ + 173, + 296, + 823, + 325 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "One of the difficulties in estimating $\\pmb { \\theta }$ in survival analysis is that a large portion of the available data is censored. However, in order to obtain estimates, Cox observed that it is sufficient to maximize the partial-likelihood, which is defined as follows: $\\begin{array} { r } { L ( \\pmb \\theta ) = \\prod _ { T _ { i } \\mathrm { u n c e n s o r e d } } \\frac { e ^ { \\pmb \\theta ^ { T } \\pmb x _ { i } } } { \\sum _ { T _ { j } \\geq T _ { i } } e ^ { \\pmb \\theta ^ { T } \\pmb x _ { j } } } . } \\end{array}$ ", + "bbox": [ + 174, + 332, + 823, + 398 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In (Katzman et al., 2018), the authors propose DeepSurv, which uses a deep neural network model to replace the linear relations between the covariate $_ { \\textbf { \\em x } }$ and $\\pmb { \\theta }$ , demonstrating improvements of survival time prediction over existing models such as CPH and the random survival forest (Ishwaran $\\&$ Kogalur, 2007), (Ishwaran et al., 2008). ", + "bbox": [ + 174, + 402, + 825, + 459 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) dataset consists of gene expression data and clinical features for 1, 980 patients, and $5 7 . 7 2 \\%$ have an observed death due to breast cancer with a median survival time of 116 months. ", + "bbox": [ + 176, + 465, + 823, + 508 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The METABRIC dataset involves 24,368 features (genes). Most genes are irrelevant for outcome prediction. To demonstrate the advantage of STG in the context of survival analysis, we selected 16 well-known genes relevant for survival (out of the 24,368 genes) that correspond to the Oncotype DX test, a gene panel used for treatment decision making. We also include five additional clinical features (hormone treatment indicator, radiotherapy indicator, chemotherapy indicator, ER-positive indicator, and age at diagnosis). We then added 200 additional irrelevant gene variables that we selected randomly from the remaining list of genes. ", + "bbox": [ + 174, + 515, + 825, + 613 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "After we omit the null values, the number of samples is 1969. We reserve the $2 0 \\%$ for test, and use the $2 0 \\%$ of the remaining training set as validation. (That is, Train: 1260, Valid: 315, Test: 394 samples.) ", + "bbox": [ + 174, + 619, + 825, + 661 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Experimental Detail For DeepSurv, we manually select the architecture using the validation set so that we obtain a similar performance reported in (Katzman et al., 2018). The learning rate decay is set to 1. The learning rate and the regularization parameter are optimized via Optuna using the validation set, where the search range is set $L R : [ 1 e - 3 , 1 ]$ and $\\lambda : [ 1 e - 3 , 1 ]$ . The hyperparameters used in the experiment are the following: architecture :[60, 20, 3], activation: selu (as suggested by (Katzman et al., 2018)), learning rate : 0.152, learning rate decay: 1.0, $\\sigma : 0 . 5$ , $\\lambda : 0 . 0 2 3$ , training epoch: 2000. Note that to see the effect of feature selection, we used the hyperparameters optimized for DeepSurv to test our method (STG-DeepSurv). ", + "bbox": [ + 173, + 676, + 825, + 789 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "J.7 ADDITIONAL EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 806, + 483, + 820 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Here we provide a full description of the procedures we have performed in the experimental parts of the paper. ", + "bbox": [ + 174, + 832, + 823, + 861 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For synthetic datasets are first split into train, validation and test. Validation is always $1 0 \\%$ of the train, while the exact ratios between train and test is detailed for each experiment separately. All the neural network weights are initialized by drawing from $\\mathcal { N } ( 0 , 0 . 1 )$ and bias terms are set to 0. All the batch sizes are equal to the number of training samples. In table 3, we detail the search range of hyperparamters as well as the exact values used in our experiments. We set n-trials $= 1 0 0 0$ for Optuna, which is a define-by-run based hyperparameter optimization software equipped with efficient search and pruning strategies. We use SGD for all the experiments, except for the Cox model where we use Adam. All the experiments are conducted using Intel(R) Xeon(R) CPU E5-2620 v3 $( \\underline { { { \\omega } } } 2 . 4 \\mathrm { G h z }$ $_ { \\mathbf { X } 2 }$ (12 cores total). ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/c3f4901a0996737bac73de3f8916665e4fc2ca3fe4c8b45ebb01a219f2a1fd71.jpg", + "table_caption": [ + "Table 3: List of the search range for the hyperparameters used in our expirements for XOR and Two-Moon " + ], + "table_footnote": [], + "table_body": "
ParamSearch range
# dense layers #hidden units activation LR n-epoch (DFS, SG-L1-NN) α (SG-L1-NN) 入 (SG-L1-NN) 入 (STG,DFS) 入 (LASSO) n-est (RF, XGBoost, Tree) n-boost-round (XGBoost) Thresh (RF,XGBoost, Tree) max-depth (XGBoost) c (SVC)[1,3] [10,500] [tanh, relu, sigmoid] [le-4, 1e-1] [50,20000] [1e-3,1] [1e-7, 1] [1e-3, 1] [0.01, 1] [5,100] [1,100] [0.01,0.5] [0.01,0.5] [1e-7, 1]
", + "bbox": [ + 316, + 231, + 681, + 441 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "For the Phase Transition experiment, we use 0.1 as a learning rate. For all the experiments when we use Tree and RF, we use the default value for max-depth, so that nodes are expanded based on the purity. For the XOR problem, the exact architectures used for the NN based methods are: (STG/HC): [476, 490, 14] with Tanh, (DFS): [100, 10] with Tanh, (SG-L1-NN): [100, 10, 5] with Tanh. For the two moons we use (STG): [490, 406, 18] with Tanh, (DFS): [158, 27, 224] with Tanh, (SG-L1-NN): [88, 28, 27] with Tanh. For the XOR problem, we attempted to use Optuna to optimize parameters of DFS and SG-L1-NN, but we ended up using the architecture suggested by the authors (Li et al., 2016; Scardapane et al., 2017) as they outperform the values suggested by Optuna. The number of epochs used for the XOR problem is $2 0 K$ , 14K, 800 for STG/HC, DFS and SG-L1-NN respectively. Regularization parameters are $0 . 1 7 , 3 . 3 e \\mathrm { ~ - ~ } 5$ and $3 e - 5$ respectively. The number of epochs used for the two-moons problem is $2 0 K$ , 1570, 708 for STG/HC, DFS and SG-L1-NN respectively. Regularization parameters are $0 . 4 8 , 9 e \\mathrm { ~ - ~ } 3$ and $1 e - 3$ respectively. We note that the regularization parameters and learning procedure is different in nature, as we use an $\\ell _ { 0 }$ type penalty. For the PBMC experiment, the architecture was hand-tuned using the validation set and set as [200, 100, 50, 10] with Tanh. Learning rate was 0.2 and the number of epochs 4000. The hyperparameter $\\lambda$ varies in the range [0.001, 0.11] to achieve different levels of sparsity. For MADELON, we use the architecture optimized for the binary XOR classification. The number of epochs used is $2 0 K$ , the learning rate is 0.06 and $\\lambda$ varies in the range [0.01, 10]. For regression, the learning rate and the $\\lambda$ are optimized via Optuna using the search range $L R : [ 1 e - 4 , 1 ]$ and $\\lambda : [ 1 e - 4 , 1 0 ]$ based on validation. The parameters used are the following: (SE1) architecture [600, 200, 100, 50, 1] with ReLu activations, num epochs: 5, $\\lambda : 5$ , $L R : 0 . 0 0 0 1$ (SE2) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs $= 2 0 0 0$ , $\\lambda : 5$ , $L R : 0 . 0 0 1$ (SE3) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs : 1000, $\\lambda : 1$ , $L R : 0 . 0 0 5$ . (RCP) architecture [1000, 300, 150, 60, 20] with ReLu activation, num-epochs: 2000, $\\lambda : 5 . 0$ , $L R : 0 . 0 0 1$ . (REL) architecture [26,91,63] with ReLu activation, num-epochs: 1600, $\\lambda : 0 . 0 3 1$ , $L R : 0 . 0 0 7$ . (RAI) architecture [10,177] with ReLu activation, num-epochs: 1800, $\\lambda : 0 . 0 1 9$ , $L R : 0 . 0 0 2$ . Architectures for SE1-SE3 and RCP where tuned manually. ", + "bbox": [ + 173, + 465, + 826, + 839 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "The ratio of train/test/valid split is 1:1:1 for synthetic data. For the real data (RCP and REL), the train size is 6000, the test and valid size is 1000 samples. For RAI the train size is 5000, the test and valid size is 1000 samples. ", + "bbox": [ + 176, + 847, + 825, + 888 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "In order to define the IFWR, for STG, the $d ^ { t h }$ feature weight is set to $\\operatorname* { m a x } ( 0 , \\operatorname* { m i n } ( 1 , \\mu _ { d } + 0 . 5 ) )$ . For other neural net based methods, it is given by $\\textstyle \\sum _ { j } W _ { d j }$ , where $W$ is the weight matrix of the first layer. For other methods we just used the feature relevance returned by the trained model. Finally, the LASSO’s IFWR in the XOR experiment was omitted from the manuscript as it suffered from high variance. ", + "bbox": [ + 174, + 895, + 823, + 926 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Regarding the comparison performed in the two-moons and XOR problem, we believe that adding IFWR along with classification accuracy versus number of feature selected provides a complementary perspective in demonstrating the efficacy of feature selection techniques. We emphasize that our goal is not to just rank features but select features by assigning the weight of 0 to irrelevant features while simultaneously obtaining good predictive accuracy. 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Feature selection leads to a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "number of potential benefits: reducing experimental costs Min et al. (2014), enhancing interpretability", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "(Ribeiro et al., 2016), computational speed up and even improving model generalization on unseen", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "data (Chandrashekar & Sahin, 2014). In biomedicine, scientists collect multitude datasets comprising", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "of many biomarkers (e.g., genes or proteins) that require development of effective diagnostics or", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "prognostics models. 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This becomes", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "computationally expensive for neural network based wrapper methods (Verikas & Bacauskiene, 2002;", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 566, + 252, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 252, + 578 + ], + "score": 1.0, + "content": "Kabir et al., 2010; Roy et al., 2015).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "score": 1.0, + "content": "Embedded methods aim to remove this burden by learning the model while simultaneously selecting", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "the subset of relevant features. 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Our", + "type": "text" + }, + { + "bbox": [ + 226, + 248, + 236, + 258 + ], + "score": 0.87, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 247, + 470, + 260 + ], + "score": 1.0, + "content": "based regularization relies on a continuous relaxation of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 259, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 259, + 470, + 270 + ], + "score": 1.0, + "content": "the Bernoulli distribution, which allows our model to learn the parameters of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 270, + 470, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 270, + 470, + 281 + ], + "score": 1.0, + "content": "approximate Bernoulli distributions via gradient descent. The proposed framework", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 280, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 470, + 292 + ], + "score": 1.0, + "content": "simultaneously learns a non-linear regression or classification function while select-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "score": 1.0, + "content": "ing a small subset of features. We provide an information-theoretic justification for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 303, + 469, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 303, + 469, + 314 + ], + "score": 1.0, + "content": "incorporating Bernoulli distribution into our approach. Furthermore, we evaluate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "score": 1.0, + "content": "our method using synthetic and real-life data and demonstrate that our approach", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "score": 1.0, + "content": "outperforms other embedded methods in terms of predictive performance and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 335, + 213, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 213, + 347 + ], + "score": 1.0, + "content": "feature selection.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 10.5, + "bbox_fs": [ + 141, + 192, + 470, + 347 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 206, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 208, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 208, + 385 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "score": 1.0, + "content": "Feature selection is a fundamental task in machine learning and statistics. Feature selection leads to a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "number of potential benefits: reducing experimental costs Min et al. (2014), enhancing interpretability", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "(Ribeiro et al., 2016), computational speed up and even improving model generalization on unseen", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "data (Chandrashekar & Sahin, 2014). In biomedicine, scientists collect multitude datasets comprising", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "of many biomarkers (e.g., genes or proteins) that require development of effective diagnostics or", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "prognostics models. For instance, in Genome wide association studies (GWAS), feature selection can", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 434, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 434, + 474 + ], + "score": 1.0, + "content": "help identify such models and lead to improved risk assessment and reduced cost.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 396, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "Feature selection methods may be classified into three major categories: filter methods, wrapper", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "methods, and embedded methods. 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This becomes", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "computationally expensive for neural network based wrapper methods (Verikas & Bacauskiene, 2002;", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 566, + 252, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 252, + 578 + ], + "score": 1.0, + "content": "Kabir et al., 2010; Roy et al., 2015).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 478, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "score": 1.0, + "content": "Embedded methods aim to remove this burden by learning the model while simultaneously selecting", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "the subset of relevant features. 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Altogether these twofold improvements", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "are formulated as a fully differentiable neural network. Specifically, our contributions are as follows:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 129, + 82, + 506, + 228 + ], + "lines": [ + { + "bbox": [ + 130, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 130, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "1. By utilizing a recent development of continuous and differentiable approximation to discrete", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 142, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "distributions (Maddison et al., 2016), (Jang et al., 2017), (Louizos et al., 2017), we introduce", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 105, + 482, + 117 + ], + "spans": [ + { + "bbox": [ + 141, + 105, + 411, + 117 + ], + "score": 1.0, + "content": "a solution to the long standing problem of feature selection with an", + "type": "text" + }, + { + "bbox": [ + 412, + 105, + 422, + 116 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 105, + 482, + 117 + ], + "score": 1.0, + "content": "regularization.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 128, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 128, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "2. 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By utilizing a recent development of continuous and differentiable approximation to discrete", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 142, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "distributions (Maddison et al., 2016), (Jang et al., 2017), (Louizos et al., 2017), we introduce", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 105, + 482, + 117 + ], + "spans": [ + { + "bbox": [ + 141, + 105, + 411, + 117 + ], + "score": 1.0, + "content": "a solution to the long standing problem of feature selection with an", + "type": "text" + }, + { + "bbox": [ + 412, + 105, + 422, + 116 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 105, + 482, + 117 + ], + "score": 1.0, + "content": "regularization.", + "type": "text" + } + ], + "index": 2, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 128, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "2. 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The", + "type": "text" + }, + { + "bbox": [ + 495, + 153, + 505, + 163 + ], + "score": 0.85, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 175 + ], + "score": 1.0, + "content": "regularized method is very effective for feature selection and prediction; however, it achieves this", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "through shrinkage of the coefficients. 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Kernel methods have been considered (Yamada et al., 2014), but scale", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "quadratically in the number of observations. To alleviate this burden, (Gregorová et al., 2018) use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "random Fourier features to approximate the kernel. (Li et al., 2016) and (Scardapane et al., 2017)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 277, + 259 + ], + "score": 1.0, + "content": "take an alternative approach by modeling", + "type": "text" + }, + { + "bbox": [ + 277, + 246, + 289, + 259 + ], + "score": 0.8, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 246, + 407, + 259 + ], + "score": 1.0, + "content": "using a neural network with", + "type": "text" + }, + { + "bbox": [ + 407, + 246, + 417, + 257 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "regularization on the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 310, + 270 + ], + "score": 1.0, + "content": "input weights. 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Below, we discuss our method that works to directly use an", + "type": "text" + }, + { + "bbox": [ + 443, + 268, + 453, + 279 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 267, + 488, + 281 + ], + "score": 1.0, + "content": "penalty.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 294, + 231, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 232, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 232, + 309 + ], + "score": 1.0, + "content": "3 PROPOSED METHOD", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 313, + 331 + ], + "score": 1.0, + "content": "We take a probabilistic approach to approximate the", + "type": "text" + }, + { + "bbox": [ + 313, + 319, + 323, + 330 + ], + "score": 0.88, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "norm, which can extend to non-linear models", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "while remaining computationally efficient. To motivate such probabilistic approach, we provide", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "theoretical support (see Section 4) based on a Mutual Information perspective of the feature selection", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 351, + 145, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 145, + 364 + ], + "score": 1.0, + "content": "problem.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 158, + 381 + ], + "score": 1.0, + "content": "To view the", + "type": "text" + }, + { + "bbox": [ + 159, + 369, + 168, + 380 + ], + "score": 0.87, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "regularized version of the risk (Eq. 1) from a probabilistic perspective, one can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 259, + 393 + ], + "score": 1.0, + "content": "introduce a Bernoulli random vector", + "type": "text" + }, + { + "bbox": [ + 259, + 379, + 268, + 391 + ], + "score": 0.86, + "content": "\\tilde { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 380, + 430, + 393 + ], + "score": 1.0, + "content": "whose entries are independent and the", + "type": "text" + }, + { + "bbox": [ + 430, + 380, + 444, + 391 + ], + "score": 0.88, + "content": "d ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "entry satisfies", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 392, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 174, + 406 + ], + "score": 0.92, + "content": "\\pi _ { d } = \\mathbb { P } ( \\tilde { S } _ { d } = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 392, + 189, + 407 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 190, + 393, + 222, + 405 + ], + "score": 0.91, + "content": "d \\in [ D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 392, + 479, + 407 + ], + "score": 1.0, + "content": ". If we denote the empirical expectation over our observations as", + "type": "text" + }, + { + "bbox": [ + 479, + 392, + 502, + 406 + ], + "score": 0.92, + "content": "\\hat { \\mathbb { E } } _ { X , Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 392, + 506, + 407 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 404, + 316, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 316, + 416 + ], + "score": 1.0, + "content": "then, the empirical regularized risk (Eq. 1) becomes", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 416, + 433, + 442 + ], + "lines": [ + { + "bbox": [ + 178, + 416, + 433, + 442 + ], + "spans": [ + { + "bbox": [ + 178, + 416, + 433, + 442 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { \\theta , \\pi } \\hat { R } ( \\theta , \\pi ) = \\operatorname* { m i n } _ { \\theta , \\pi } \\hat { \\mathbb { E } } _ { X , Y } \\mathbb { E } _ { \\tilde { S } } \\left[ L ( f _ { \\theta } ( X \\odot \\tilde { S } ) , Y ) + \\lambda | | \\tilde { S } | | _ { 0 } \\right] ,", + "type": "interline_equation", + "image_path": "f9502fc5e539e72ecf68bd2a5f9c6710031a89a69aecb4a50cb0c6c70e177a05.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 178, + 416, + 433, + 442 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 104, + 443, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 443, + 171, + 462 + ], + "score": 1.0, + "content": "where we have", + "type": "text" + }, + { + "bbox": [ + 171, + 443, + 261, + 458 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\tilde { S } } | | \\tilde { S } | | _ { 0 } = \\sum _ { d = 1 } ^ { D } \\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 443, + 335, + 462 + ], + "score": 1.0, + "content": "and we constrain", + "type": "text" + }, + { + "bbox": [ + 335, + 446, + 385, + 458 + ], + "score": 0.93, + "content": "\\pi _ { d } \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 443, + 507, + 462 + ], + "score": 1.0, + "content": ". Clearly, this formulation is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "equivalent to Eq. 1, with a regularized penalty on cardinality rather than an explicit constraint. We", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 468, + 353, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 270, + 480 + ], + "score": 1.0, + "content": "may then relax the discrete constraint on", + "type": "text" + }, + { + "bbox": [ + 271, + 469, + 282, + 479 + ], + "score": 0.86, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 468, + 305, + 480 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 306, + 468, + 349, + 480 + ], + "score": 0.92, + "content": "\\pi _ { d } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 468, + 353, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 506, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 297, + 497 + ], + "score": 1.0, + "content": "Now, our goal is to find the model parameters", + "type": "text" + }, + { + "bbox": [ + 297, + 484, + 309, + 495 + ], + "score": 0.88, + "content": "\\pmb { \\theta } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 484, + 416, + 497 + ], + "score": 1.0, + "content": "and Bernoulli parameters", + "type": "text" + }, + { + "bbox": [ + 416, + 485, + 429, + 495 + ], + "score": 0.88, + "content": "\\pi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "that minimize the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 166, + 510 + ], + "score": 1.0, + "content": "empirical risk", + "type": "text" + }, + { + "bbox": [ + 167, + 496, + 200, + 509 + ], + "score": 0.93, + "content": "\\hat { R } ( \\pmb \\theta , \\pmb \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "via gradient descent. However, an optimization of a loss function which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "includes discrete random variables suffers from high variance (see F in the Appendix for more details).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "Therefore, inspired by a recently developed continuous approximation for discrete random variables,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "score": 1.0, + "content": "suggested by (Jang et al., 2017; Maddison et al., 2016), we develop and use a novel and simple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 476, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 476, + 553 + ], + "score": 1.0, + "content": "continuous distribution that is fully differentiable and suited for the task of feature selection.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 565, + 248, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 250, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 250, + 578 + ], + "score": 1.0, + "content": "3.1 CONTINUOUS RELAXATION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 506, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 331, + 599 + ], + "score": 1.0, + "content": "Our continuous relaxation for the Bernoulli variables", + "type": "text" + }, + { + "bbox": [ + 331, + 584, + 344, + 597 + ], + "score": 0.9, + "content": "{ \\cal \\tilde { S } } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 585, + 361, + 599 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 361, + 586, + 398, + 598 + ], + "score": 0.92, + "content": "d \\in \\ [ D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "is termed stochastic gate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "(STG). The STG relies on the reparametrization trick, which is widely used for reducing the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 619 + ], + "score": 1.0, + "content": "variance of gradient estimators (Miller et al., 2017; Figurnov et al., 2018). To construct a con-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "tinuous approximation to Bernoulli random variable via the reparametrization trick, we define", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 630, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 313, + 642 + ], + "score": 0.87, + "content": "z _ { d } = g ( \\mu _ { d } + \\epsilon _ { d } ) = \\operatorname* { m a x } ( 0 , \\operatorname* { m i n } ( 1 , \\epsilon _ { d } + \\mu _ { d } + 0 . 5 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 630, + 341, + 642 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 342, + 631, + 352, + 641 + ], + "score": 0.85, + "content": "\\epsilon _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 630, + 506, + 642 + ], + "score": 1.0, + "content": "is drawn from a Gaussian distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 145, + 652 + ], + "score": 0.9, + "content": "\\mathcal { N } ( 0 , \\sigma _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 639, + 176, + 653 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 177, + 643, + 189, + 651 + ], + "score": 0.85, + "content": "\\sigma _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "is fixed throughout training. This approximation can be viewed as a clipped,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 650, + 504, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 492, + 665 + ], + "score": 1.0, + "content": "mean-shifted, Gaussian random vector. 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While the original LASSO", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "problem focuses on the constrained optimization problem, the regularized least squares problem,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 411, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 411, + 117 + ], + "score": 1.0, + "content": "which is often used in practice, yields the following minimization objective:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 117, + 381, + 151 + ], + "lines": [ + { + "bbox": [ + 231, + 117, + 381, + 151 + ], + "spans": [ + { + "bbox": [ + 231, + 117, + 381, + 151 + ], + "score": 0.95, + "content": "\\operatorname* { m i n } _ { \\pmb { \\theta } } \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } ( \\pmb { \\theta } ^ { T } \\pmb { x } _ { n } - \\pmb { y } _ { n } ) ^ { 2 } + \\lambda \\| \\pmb { \\theta } \\| _ { 1 } .", + "type": "interline_equation", + "image_path": "de6e21ecf18f27c84ee46c5fa6a38906cdc5e1dde8d56187b050cc77a30b6493.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 117, + 381, + 134.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 231, + 134.0, + 381, + 151.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 152, + 505, + 197 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 189, + 165 + ], + "score": 1.0, + "content": "The hyperparameter", + "type": "text" + }, + { + "bbox": [ + 189, + 153, + 196, + 162 + ], + "score": 0.83, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 151, + 494, + 165 + ], + "score": 1.0, + "content": "trades off the amount of regularization versus the fit of the objective1. The", + "type": "text" + }, + { + "bbox": [ + 495, + 153, + 505, + 163 + ], + "score": 0.85, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 175 + ], + "score": 1.0, + "content": "regularized method is very effective for feature selection and prediction; however, it achieves this", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "through shrinkage of the coefficients. As a result, (Fan & Li, 2001) have considered non-convex", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 507, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 152, + 198 + ], + "score": 1.0, + "content": "choices for", + "type": "text" + }, + { + "bbox": [ + 152, + 186, + 160, + 195 + ], + "score": 0.82, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 185, + 507, + 198 + ], + "score": 1.0, + "content": "that perform well both theoretically and empirically for prediction and feature selection.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 151, + 507, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 215 + ], + "score": 1.0, + "content": "Our goal is to apply such regularization techniques to perform feature selection while learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "a non-linear function. Kernel methods have been considered (Yamada et al., 2014), but scale", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "quadratically in the number of observations. To alleviate this burden, (Gregorová et al., 2018) use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "random Fourier features to approximate the kernel. (Li et al., 2016) and (Scardapane et al., 2017)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 277, + 259 + ], + "score": 1.0, + "content": "take an alternative approach by modeling", + "type": "text" + }, + { + "bbox": [ + 277, + 246, + 289, + 259 + ], + "score": 0.8, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 246, + 407, + 259 + ], + "score": 1.0, + "content": "using a neural network with", + "type": "text" + }, + { + "bbox": [ + 407, + 246, + 417, + 257 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "regularization on the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 310, + 270 + ], + "score": 1.0, + "content": "input weights. However, in practice, introducing an", + "type": "text" + }, + { + "bbox": [ + 310, + 257, + 320, + 268 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "penalty into gradient descent does not provide", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 488, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 443, + 281 + ], + "score": 1.0, + "content": "sufficient sparsification. Below, we discuss our method that works to directly use an", + "type": "text" + }, + { + "bbox": [ + 443, + 268, + 453, + 279 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 267, + 488, + 281 + ], + "score": 1.0, + "content": "penalty.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 200, + 506, + 281 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 294, + 231, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 232, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 232, + 309 + ], + "score": 1.0, + "content": "3 PROPOSED METHOD", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 313, + 331 + ], + "score": 1.0, + "content": "We take a probabilistic approach to approximate the", + "type": "text" + }, + { + "bbox": [ + 313, + 319, + 323, + 330 + ], + "score": 0.88, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "norm, which can extend to non-linear models", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "while remaining computationally efficient. To motivate such probabilistic approach, we provide", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "theoretical support (see Section 4) based on a Mutual Information perspective of the feature selection", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 351, + 145, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 145, + 364 + ], + "score": 1.0, + "content": "problem.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 318, + 505, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 158, + 381 + ], + "score": 1.0, + "content": "To view the", + "type": "text" + }, + { + "bbox": [ + 159, + 369, + 168, + 380 + ], + "score": 0.87, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "regularized version of the risk (Eq. 1) from a probabilistic perspective, one can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 259, + 393 + ], + "score": 1.0, + "content": "introduce a Bernoulli random vector", + "type": "text" + }, + { + "bbox": [ + 259, + 379, + 268, + 391 + ], + "score": 0.86, + "content": "\\tilde { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 380, + 430, + 393 + ], + "score": 1.0, + "content": "whose entries are independent and the", + "type": "text" + }, + { + "bbox": [ + 430, + 380, + 444, + 391 + ], + "score": 0.88, + "content": "d ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "entry satisfies", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 392, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 174, + 406 + ], + "score": 0.92, + "content": "\\pi _ { d } = \\mathbb { P } ( \\tilde { S } _ { d } = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 392, + 189, + 407 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 190, + 393, + 222, + 405 + ], + "score": 0.91, + "content": "d \\in [ D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 392, + 479, + 407 + ], + "score": 1.0, + "content": ". If we denote the empirical expectation over our observations as", + "type": "text" + }, + { + "bbox": [ + 479, + 392, + 502, + 406 + ], + "score": 0.92, + "content": "\\hat { \\mathbb { E } } _ { X , Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 392, + 506, + 407 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 404, + 316, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 316, + 416 + ], + "score": 1.0, + "content": "then, the empirical regularized risk (Eq. 1) becomes", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 367, + 506, + 416 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 416, + 433, + 442 + ], + "lines": [ + { + "bbox": [ + 178, + 416, + 433, + 442 + ], + "spans": [ + { + "bbox": [ + 178, + 416, + 433, + 442 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { \\theta , \\pi } \\hat { R } ( \\theta , \\pi ) = \\operatorname* { m i n } _ { \\theta , \\pi } \\hat { \\mathbb { E } } _ { X , Y } \\mathbb { E } _ { \\tilde { S } } \\left[ L ( f _ { \\theta } ( X \\odot \\tilde { S } ) , Y ) + \\lambda | | \\tilde { S } | | _ { 0 } \\right] ,", + "type": "interline_equation", + "image_path": "f9502fc5e539e72ecf68bd2a5f9c6710031a89a69aecb4a50cb0c6c70e177a05.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 178, + 416, + 433, + 442 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 104, + 443, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 443, + 171, + 462 + ], + "score": 1.0, + "content": "where we have", + "type": "text" + }, + { + "bbox": [ + 171, + 443, + 261, + 458 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\tilde { S } } | | \\tilde { S } | | _ { 0 } = \\sum _ { d = 1 } ^ { D } \\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 443, + 335, + 462 + ], + "score": 1.0, + "content": "and we constrain", + "type": "text" + }, + { + "bbox": [ + 335, + 446, + 385, + 458 + ], + "score": 0.93, + "content": "\\pi _ { d } \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 443, + 507, + 462 + ], + "score": 1.0, + "content": ". Clearly, this formulation is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "equivalent to Eq. 1, with a regularized penalty on cardinality rather than an explicit constraint. We", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 468, + 353, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 270, + 480 + ], + "score": 1.0, + "content": "may then relax the discrete constraint on", + "type": "text" + }, + { + "bbox": [ + 271, + 469, + 282, + 479 + ], + "score": 0.86, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 468, + 305, + 480 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 306, + 468, + 349, + 480 + ], + "score": 0.92, + "content": "\\pi _ { d } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 468, + 353, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 443, + 507, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 506, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 297, + 497 + ], + "score": 1.0, + "content": "Now, our goal is to find the model parameters", + "type": "text" + }, + { + "bbox": [ + 297, + 484, + 309, + 495 + ], + "score": 0.88, + "content": "\\pmb { \\theta } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 484, + 416, + 497 + ], + "score": 1.0, + "content": "and Bernoulli parameters", + "type": "text" + }, + { + "bbox": [ + 416, + 485, + 429, + 495 + ], + "score": 0.88, + "content": "\\pi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "that minimize the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 166, + 510 + ], + "score": 1.0, + "content": "empirical risk", + "type": "text" + }, + { + "bbox": [ + 167, + 496, + 200, + 509 + ], + "score": 0.93, + "content": "\\hat { R } ( \\pmb \\theta , \\pmb \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "via gradient descent. However, an optimization of a loss function which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 521 + ], + "score": 1.0, + "content": "includes discrete random variables suffers from high variance (see F in the Appendix for more details).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "Therefore, inspired by a recently developed continuous approximation for discrete random variables,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "score": 1.0, + "content": "suggested by (Jang et al., 2017; Maddison et al., 2016), we develop and use a novel and simple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 476, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 476, + 553 + ], + "score": 1.0, + "content": "continuous distribution that is fully differentiable and suited for the task of feature selection.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 484, + 506, + 553 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 565, + 248, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 250, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 250, + 578 + ], + "score": 1.0, + "content": "3.1 CONTINUOUS RELAXATION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 506, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 331, + 599 + ], + "score": 1.0, + "content": "Our continuous relaxation for the Bernoulli variables", + "type": "text" + }, + { + "bbox": [ + 331, + 584, + 344, + 597 + ], + "score": 0.9, + "content": "{ \\cal \\tilde { S } } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 585, + 361, + 599 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 361, + 586, + 398, + 598 + ], + "score": 0.92, + "content": "d \\in \\ [ D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "is termed stochastic gate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "(STG). The STG relies on the reparametrization trick, which is widely used for reducing the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 619 + ], + "score": 1.0, + "content": "variance of gradient estimators (Miller et al., 2017; Figurnov et al., 2018). To construct a con-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "tinuous approximation to Bernoulli random variable via the reparametrization trick, we define", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 630, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 313, + 642 + ], + "score": 0.87, + "content": "z _ { d } = g ( \\mu _ { d } + \\epsilon _ { d } ) = \\operatorname* { m a x } ( 0 , \\operatorname* { m i n } ( 1 , \\epsilon _ { d } + \\mu _ { d } + 0 . 5 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 630, + 341, + 642 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 342, + 631, + 352, + 641 + ], + "score": 0.85, + "content": "\\epsilon _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 630, + 506, + 642 + ], + "score": 1.0, + "content": "is drawn from a Gaussian distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 145, + 652 + ], + "score": 0.9, + "content": "\\mathcal { N } ( 0 , \\sigma _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 639, + 176, + 653 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 177, + 643, + 189, + 651 + ], + "score": 0.85, + "content": "\\sigma _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "is fixed throughout training. This approximation can be viewed as a clipped,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 650, + 504, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 492, + 665 + ], + "score": 1.0, + "content": "mean-shifted, Gaussian random vector. Furthermore, the gradient of the objective with respect to", + "type": "text" + }, + { + "bbox": [ + 492, + 653, + 504, + 663 + ], + "score": 0.83, + "content": "\\mu _ { d }", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 662, + 251, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 251, + 675 + ], + "score": 1.0, + "content": "can be computed via the chain rule.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 584, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 679, + 285, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 286, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 286, + 693 + ], + "score": 1.0, + "content": "We can now rewrite the objective in Eq. 4 as", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 677, + 286, + 693 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 691, + 432, + 715 + ], + "lines": [ + { + "bbox": [ + 179, + 691, + 432, + 715 + ], + "spans": [ + { + "bbox": [ + 179, + 691, + 432, + 715 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { \\theta , \\mu } \\hat { R } ( \\theta , \\mu ) = \\operatorname* { m i n } _ { \\theta , \\mu } \\mathbb { E } _ { X , Y } \\mathbb { E } _ { Z } \\left[ L ( f _ { \\theta } ( X \\odot Z ) , Y ) + \\lambda | | Z | | _ { 0 } \\right] ,", + "type": "interline_equation", + "image_path": "9e88de04d931cda344da264b480603e7f443f2688879c02095a8029fd6541ef7.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 179, + 691, + 432, + 715 + ], + "spans": [], + "index": 45 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 134, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 144, + 93 + ], + "score": 0.79, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 82, + 245, + 95 + ], + "score": 1.0, + "content": "is a random vector with", + "type": "text" + }, + { + "bbox": [ + 246, + 83, + 255, + 92 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 82, + 348, + 95 + ], + "score": 1.0, + "content": "independent variables", + "type": "text" + }, + { + "bbox": [ + 349, + 84, + 360, + 93 + ], + "score": 0.85, + "content": "z _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 82, + 376, + 95 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 376, + 82, + 391, + 95 + ], + "score": 0.9, + "content": "[ D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". To optimize the empirical", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 409, + 106 + ], + "score": 1.0, + "content": "surrogate of the objective (Eq. 5), we first differentiate it with respect to", + "type": "text" + }, + { + "bbox": [ + 409, + 95, + 417, + 105 + ], + "score": 0.8, + "content": "\\pmb { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 94, + 505, + 106 + ], + "score": 1.0, + "content": ". 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In", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "(Louizos et al., 2017), they introduce the Hard-Concrete (HC) distribution as a continuous surrogate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "for Bernoulli distributions in the context of model compression. 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The authors demonstrate how incorporating the HC in deep neural networks", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "leads to fast convergence and improved generalization due to the sparsification effect. In this study,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 380, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 494, + 391 + ], + "score": 1.0, + "content": "we introduce this type of sparsification for the long standing problem of feature selection with", + "type": "text" + }, + { + "bbox": [ + 494, + 380, + 504, + 390 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "regularization.We demonstrate that a simple relaxation of Bernoulli distributions is sufficient and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "works better than the HC distribution for feature selection. We find empirically that the HC yields", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "high variance gradient estimates compared to the STG. This high-variance does not affect model", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "sparsification as the sparsity pattern does not matter. Contrarily, for feature selection it is crucial as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "practitioners require a stable procedure. Furthermore, higher variance gradients will result in a slower", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "convergence of the model (this has been demonstrated empirically in the supp. material). In (Chen", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "et al., 2018), the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "models. Their method is focused on finding a subset of features given a particular instance, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 477, + 334, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 334, + 491 + ], + "score": 1.0, + "content": "therefore is not appropriate for general feature selection.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "Some studies tackle embedded feature selection problems by extending LASSO and group LASSO", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "to neural network models. The authors (Li et al., 2016; Scardapane et al., 2017) and (Feng &", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 494, + 529 + ], + "score": 1.0, + "content": "Simon, 2017) have a similar goal as ours in performing feature selection, but instead rely on the", + "type": "text" + }, + { + "bbox": [ + 495, + 517, + 505, + 528 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 172, + 540 + ], + "score": 1.0, + "content": "relaxation to the", + "type": "text" + }, + { + "bbox": [ + 172, + 528, + 182, + 539 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 528, + 434, + 540 + ], + "score": 1.0, + "content": ". Our approach, which utilizes stochastic gates coupled with the", + "type": "text" + }, + { + "bbox": [ + 434, + 528, + 444, + 539 + ], + "score": 0.88, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "norm achieves", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 374, + 552 + ], + "score": 1.0, + "content": "substantially better empirical performance compared against other", + "type": "text" + }, + { + "bbox": [ + 374, + 539, + 384, + 550 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "based baselines. This point is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 550, + 240, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 240, + 561 + ], + "score": 1.0, + "content": "demonstrated in the next section.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 578, + 200, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 201, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 201, + 593 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "Here we provide empirical evaluation of our method in a wide range of settings. We begin with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "simple linear regression evaluating the capabilities of our approach to select a sparse set of relevant", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "features (section 6.1). We then evaluate our method in nonlinear regression tasks (section 6.2). Next,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "we demonstrate the applicability of the method in a highly nonlinear classification task (section 6.3.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 337, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 337, + 662 + ], + "score": 1.0, + "content": "Finally, we present two potential applications (section 7).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "We implement our method using both the STG and HC distributions and compare their applicability", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "for feature selection. The hyperparamters of all the methods are optimized using validation sets via", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Optuna (Takuya Akiba & Koyama, 2019). In the Appendix, we provide all details of the Optuna", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "based parameter tuning procedure. Additional experiments also appear in the Appendix, including", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 507, + 722 + ], + "score": 1.0, + "content": "high dimensional datasets and an extensive comparisons between the STG and HC (Louizos et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 187, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 187, + 733 + ], + "score": 1.0, + "content": "2017) distributions.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Proposition 1. Suppose that the above assumptions hold for the model. Then, solving the optimization", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 303, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 303, + 106 + ], + "score": 1.0, + "content": "in Eq. 6 is equivalent to solving the optimization", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 111, + 398, + 138 + ], + "lines": [ + { + "bbox": [ + 213, + 111, + 398, + 138 + ], + "spans": [ + { + "bbox": [ + 213, + 111, + 398, + 138 + ], + "score": 0.92, + "content": "\\operatorname* { m a x } _ { \\mathbf { 0 } \\leq \\pi \\leq 1 } I ( X \\odot \\tilde { S } ; Y ) \\quad s . t . \\quad \\sum _ { i } \\mathbb { E } [ \\tilde { S } _ { i } ] \\leq k ,", + "type": "interline_equation", + "image_path": "1fb8300a27b3111f62f82c08d95ea22d446e3fe6e0b9b961e61617bd825c208e.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 213, + 111, + 398, + 124.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 213, + 124.5, + 398, + 138.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 197, + 160 + ], + "score": 1.0, + "content": "where the coordinates", + "type": "text" + }, + { + "bbox": [ + 198, + 146, + 208, + 158 + ], + "score": 0.89, + "content": "\\tilde { S } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 146, + 506, + 160 + ], + "score": 1.0, + "content": "are drawn independently at random according to a Bernoulli distribution", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 184, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 170, + 172 + ], + "score": 1.0, + "content": "with parameter", + "type": "text" + }, + { + "bbox": [ + 170, + 160, + 180, + 169 + ], + "score": 0.85, + "content": "\\pi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 156, + 184, + 172 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 146, + 506, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 180, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "Due to length constraints, we leave the proof of this proposition and how it bridges the MI maximiza-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 191, + 435, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 435, + 204 + ], + "score": 1.0, + "content": "tion (Eq. 6) and risk minimization (Eq. 2) to the Appendix (see Section C and D).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 180, + 506, + 204 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 220, + 211, + 233 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 213, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 213, + 235 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "The two most related works to this study are (Louizos et al., 2017) and (Chen et al., 2018). In", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "(Louizos et al., 2017), they introduce the Hard-Concrete (HC) distribution as a continuous surrogate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "for Bernoulli distributions in the context of model compression. The HC distribution is induced by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 209, + 293 + ], + "score": 1.0, + "content": "applying a hard sigmoid", + "type": "text" + }, + { + "bbox": [ + 209, + 280, + 303, + 291 + ], + "score": 0.91, + "content": "z = \\operatorname* { m i n } ( 1 , \\operatorname* { m a x } ( 0 , \\bar { s } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 279, + 316, + 293 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 316, + 281, + 322, + 289 + ], + "score": 0.75, + "content": "\\bar { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 279, + 409, + 293 + ], + "score": 1.0, + "content": ", where we construct", + "type": "text" + }, + { + "bbox": [ + 409, + 281, + 415, + 289 + ], + "score": 0.61, + "content": "\\bar { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "by applying Sigmoid", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 291, + 293, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 293, + 303 + ], + "score": 1.0, + "content": "function to the logistic distribution as follows:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 246, + 506, + 303 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 308, + 477, + 339 + ], + "lines": [ + { + "bbox": [ + 134, + 308, + 477, + 339 + ], + "spans": [ + { + "bbox": [ + 134, + 308, + 477, + 339 + ], + "score": 0.92, + "content": "u \\sim U ( 0 , 1 ) , L = \\log ( u ) - \\log ( 1 - u ) , s = \\frac { 1 } { 1 + \\exp ( \\frac { - ( \\log \\alpha + L ) } { \\beta } ) } , \\bar { s } = s ( \\zeta - \\tau ) + \\tau", + "type": "interline_equation", + "image_path": "0e44c3d6a8ef774603f3aab46a09323b7288cb5db49591ed77636ef85a334a11.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 134, + 308, + 477, + 318.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 134, + 318.3333333333333, + 477, + 328.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 134, + 328.66666666666663, + 477, + 338.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 158, + 359 + ], + "score": 1.0, + "content": "The interval", + "type": "text" + }, + { + "bbox": [ + 158, + 346, + 180, + 358 + ], + "score": 0.92, + "content": "( \\tau , \\zeta )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 345, + 204, + 359 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 205, + 347, + 230, + 357 + ], + "score": 0.9, + "content": "\\tau < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 345, + 248, + 359 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 346, + 272, + 357 + ], + "score": 0.92, + "content": "\\zeta > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 345, + 506, + 359 + ], + "score": 1.0, + "content": ", allows the distribution to have more probability mass on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "the edge of the support. The authors demonstrate how incorporating the HC in deep neural networks", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "leads to fast convergence and improved generalization due to the sparsification effect. In this study,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 380, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 494, + 391 + ], + "score": 1.0, + "content": "we introduce this type of sparsification for the long standing problem of feature selection with", + "type": "text" + }, + { + "bbox": [ + 494, + 380, + 504, + 390 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "regularization.We demonstrate that a simple relaxation of Bernoulli distributions is sufficient and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "works better than the HC distribution for feature selection. We find empirically that the HC yields", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "high variance gradient estimates compared to the STG. This high-variance does not affect model", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "sparsification as the sparsity pattern does not matter. Contrarily, for feature selection it is crucial as", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "practitioners require a stable procedure. Furthermore, higher variance gradients will result in a slower", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "convergence of the model (this has been demonstrated empirically in the supp. material). In (Chen", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "et al., 2018), the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "models. Their method is focused on finding a subset of features given a particular instance, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 477, + 334, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 334, + 491 + ], + "score": 1.0, + "content": "therefore is not appropriate for general feature selection.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 345, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "Some studies tackle embedded feature selection problems by extending LASSO and group LASSO", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "to neural network models. The authors (Li et al., 2016; Scardapane et al., 2017) and (Feng &", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 494, + 529 + ], + "score": 1.0, + "content": "Simon, 2017) have a similar goal as ours in performing feature selection, but instead rely on the", + "type": "text" + }, + { + "bbox": [ + 495, + 517, + 505, + 528 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 172, + 540 + ], + "score": 1.0, + "content": "relaxation to the", + "type": "text" + }, + { + "bbox": [ + 172, + 528, + 182, + 539 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 528, + 434, + 540 + ], + "score": 1.0, + "content": ". Our approach, which utilizes stochastic gates coupled with the", + "type": "text" + }, + { + "bbox": [ + 434, + 528, + 444, + 539 + ], + "score": 0.88, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "norm achieves", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 374, + 552 + ], + "score": 1.0, + "content": "substantially better empirical performance compared against other", + "type": "text" + }, + { + "bbox": [ + 374, + 539, + 384, + 550 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "based baselines. This point is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 550, + 240, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 240, + 561 + ], + "score": 1.0, + "content": "demonstrated in the next section.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 495, + 506, + 561 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 578, + 200, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 201, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 201, + 593 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "Here we provide empirical evaluation of our method in a wide range of settings. We begin with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "simple linear regression evaluating the capabilities of our approach to select a sparse set of relevant", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "features (section 6.1). We then evaluate our method in nonlinear regression tasks (section 6.2). Next,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "we demonstrate the applicability of the method in a highly nonlinear classification task (section 6.3.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 337, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 337, + 662 + ], + "score": 1.0, + "content": "Finally, we present two potential applications (section 7).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 605, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "We implement our method using both the STG and HC distributions and compare their applicability", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "for feature selection. The hyperparamters of all the methods are optimized using validation sets via", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Optuna (Takuya Akiba & Koyama, 2019). In the Appendix, we provide all details of the Optuna", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "based parameter tuning procedure. Additional experiments also appear in the Appendix, including", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 507, + 722 + ], + "score": 1.0, + "content": "high dimensional datasets and an extensive comparisons between the STG and HC (Louizos et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 187, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 187, + 733 + ], + "score": 1.0, + "content": "2017) distributions.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 664, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 327, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 327, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 327, + 95 + ], + "score": 1.0, + "content": "6.1 SUPPORT RECOVERY IN LINEAR REGRESSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "In the setting of noisy linear regression, (Wainwright, 2009) have analyzed the probability of LASSO", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 113, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 505, + 128 + ], + "score": 1.0, + "content": "to correctly identify a sparse subset of active variables. 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Suppose the cardinality of the support", + "type": "text" + }, + { + "bbox": [ + 471, + 136, + 504, + 148 + ], + "score": 0.92, + "content": "| S | = k", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 144, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 289, + 163 + ], + "score": 1.0, + "content": "is known. Given a matrix of measurements", + "type": "text" + }, + { + "bbox": [ + 289, + 147, + 342, + 158 + ], + "score": 0.92, + "content": "\\pmb { X } \\in \\mathbb { R } ^ { N \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 144, + 506, + 163 + ], + "score": 1.0, + "content": "with values drawn independently from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 157, + 255, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 159, + 138, + 171 + ], + "score": 0.94, + "content": "N ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 157, + 195, + 173 + ], + "score": 1.0, + "content": ", the response", + "type": "text" + }, + { + "bbox": [ + 195, + 160, + 202, + 170 + ], + "score": 0.81, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 157, + 255, + 173 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 272, + 174, + 339, + 188 + ], + "lines": [ + { + "bbox": [ + 272, + 174, + 339, + 188 + ], + "spans": [ + { + "bbox": [ + 272, + 174, + 339, + 188 + ], + "score": 0.92, + "content": "\\pmb { y } = \\pmb { X } \\beta ^ { * } + \\pmb { w } ,", + "type": "interline_equation", + "image_path": "71701a6a4474f5f60cd370ee6839ee5cdb03e4dd32bf7b8a85336ef4e3c24cff.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 272, + 174, + 339, + 188 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 456, + 205 + ], + "lines": [ + { + "bbox": [ + 104, + 190, + 454, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 225, + 207 + ], + "score": 1.0, + "content": "where the values of the noise", + "type": "text" + }, + { + "bbox": [ + 225, + 193, + 290, + 205 + ], + "score": 0.92, + "content": "\\pmb { w } _ { i } , i = 1 , . . . , N", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 190, + 414, + 207 + ], + "score": 1.0, + "content": "are drawn independently from", + "type": "text" + }, + { + "bbox": [ + 414, + 193, + 454, + 205 + ], + "score": 0.92, + "content": "N ( 0 , 0 . 5 )", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 209, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "Here, we reproduce this setting to evaluate the probability of the proposed approach to perfectly", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 219, + 198, + 234 + ], + "score": 1.0, + "content": "recover the support of", + "type": "text" + }, + { + "bbox": [ + 198, + 221, + 210, + 232 + ], + "score": 0.88, + "content": "\\beta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 219, + 506, + 234 + ], + "score": 1.0, + "content": ". As suggested by (Wainwright, 2009), we use a sparsity that scales with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 229, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 107, + 232, + 116, + 242 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 229, + 158, + 246 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 158, + 232, + 225, + 244 + ], + "score": 0.92, + "content": "k = \\lceil 0 . 4 D ^ { 0 . 7 5 } \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 229, + 351, + 246 + ], + "score": 1.0, + "content": ". For each number of samples", + "type": "text" + }, + { + "bbox": [ + 352, + 232, + 362, + 242 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 229, + 506, + 246 + ], + "score": 1.0, + "content": "in the range [10, 500], we run 200", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "simulations and count the portion of correctly recovered supports. We repeat this process for 2", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 185, + 267 + ], + "score": 1.0, + "content": "different values of", + "type": "text" + }, + { + "bbox": [ + 186, + 254, + 195, + 264 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "and compare our performance to LASSO. For LASSO, the regularization", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 257, + 283 + ], + "score": 1.0, + "content": "parameter was set to its optimal value", + "type": "text" + }, + { + "bbox": [ + 257, + 265, + 367, + 285 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\alpha _ { N } = \\sqrt { \\frac { 2 \\sigma ^ { 2 } \\log ( D - k ) \\log ( k ) } { N } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 265, + 506, + 284 + ], + "score": 1.0, + "content": "(Wainwright, 2009). For STG and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 150, + 296 + ], + "score": 1.0, + "content": "HC we set", + "type": "text" + }, + { + "bbox": [ + 151, + 283, + 200, + 294 + ], + "score": 0.92, + "content": "\\lambda _ { N } = C \\alpha _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 282, + 242, + 296 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 243, + 284, + 252, + 293 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "is a constant, which is selected using a grid search in the range", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "[0.1,10]. As evident from Fig. 1, even when restricting to linear functions our method has a clear", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 305, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 295, + 317 + ], + "score": 1.0, + "content": "advantage over LASSO. This implies that the", + "type": "text" + }, + { + "bbox": [ + 295, + 305, + 305, + 316 + ], + "score": 0.87, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 306, + 506, + 317 + ], + "score": 1.0, + "content": "based penalty, although is non convex in nature,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "allows perfect recovery of the support using less samples. 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Comparison between the proposed method, HC and LASSO under a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 215, + 501 + ], + "score": 1.0, + "content": "linear additive noise model", + "type": "text" + }, + { + "bbox": [ + 215, + 489, + 279, + 501 + ], + "score": 0.93, + "content": "\\pmb { y } = \\pmb { X } \\beta ^ { * } + \\pmb { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 490, + 316, + 501 + ], + "score": 1.0, + "content": "using (a)", + "type": "text" + }, + { + "bbox": [ + 317, + 490, + 350, + 500 + ], + "score": 0.91, + "content": "D = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 490, + 381, + 501 + ], + "score": 1.0, + "content": "and (b)", + "type": "text" + }, + { + "bbox": [ + 381, + 490, + 419, + 500 + ], + "score": 0.89, + "content": "D = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 490, + 505, + 501 + ], + "score": 1.0, + "content": ". Central lines are the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 501, + 335, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 335, + 512 + ], + "score": 1.0, + "content": "means while shaded are represent the standard deviation.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + } + ], + "index": 22.75 + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 414, + 535 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 416, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 416, + 537 + ], + "score": 1.0, + "content": "6.2 NONLINEAR REGRESSION USING SYNTHETIC AND REAL DATASETS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 506, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "In this section, we evaluate our method for regression tasks against two other embedded feature", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "score": 1.0, + "content": "selection methods: LASSO and Sparse Random Fourier Features (Gregorová et al., 2018). Following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "the same format as (Gregorová et al., 2018), the following functions are used to generate synthetic data:", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 101, + 572, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 101, + 572, + 161, + 597 + ], + "score": 1.0, + "content": "(SE1: 100/5)", + "type": "text" + }, + { + "bbox": [ + 162, + 578, + 346, + 591 + ], + "score": 0.91, + "content": "y = \\sin { ( x _ { 1 } + x _ { 3 } ) ^ { 2 } } \\sin { ( x _ { 7 } x _ { 8 } x _ { 9 } ) } + \\mathcal { N } ( 0 , 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 572, + 402, + 597 + ], + "score": 1.0, + "content": ". (SE2: 18/5)", + "type": "text" + }, + { + "bbox": [ + 402, + 577, + 507, + 592 + ], + "score": 0.9, + "content": "\\begin{array} { r } { y = \\log ( ( \\sum _ { s = 1 1 } ^ { \\mathbf { \\bar { 1 } } 5 } x _ { s } ) ^ { 2 } ) + } \\end{array}", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 590, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 147, + 605 + ], + "score": 0.91, + "content": "\\mathcal { N } ( 0 , 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 590, + 223, + 606 + ], + "score": 1.0, + "content": ". 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For (SE3), the five consecutive coordinates are generated by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 107, + 614, + 221, + 628 + ], + "score": 0.92, + "content": "{ x _ { 5 } } _ { ( j - 1 ) + i } = { z _ { j } } + \\mathrm { { N } } ( 0 , 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 614, + 253, + 628 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 253, + 615, + 308, + 627 + ], + "score": 0.92, + "content": "z _ { j } \\sim \\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 614, + 324, + 628 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 324, + 615, + 369, + 626 + ], + "score": 0.91, + "content": "i = 1 , . . . , 5", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 614, + 388, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 388, + 615, + 444, + 626 + ], + "score": 0.9, + "content": "j = 1 , . . . , 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 614, + 506, + 628 + ], + "score": 1.0, + "content": ". The numbers", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 626, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 638 + ], + "score": 1.0, + "content": "next to the experiment code indicate total dimensions/relevant dimensions in the feature space. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "score": 1.0, + "content": "also evaluate our method using three real datasets: (RCP: 21/-), of computer systems activity, (REL", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "17/-) of F16 elevators and (RAI 39/-) of F16 ailernos all taken from the LIACC repository 2. 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Our method outperforms all alternative methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "for most cases. 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For LASSO, the regularization", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 257, + 283 + ], + "score": 1.0, + "content": "parameter was set to its optimal value", + "type": "text" + }, + { + "bbox": [ + 257, + 265, + 367, + 285 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\alpha _ { N } = \\sqrt { \\frac { 2 \\sigma ^ { 2 } \\log ( D - k ) \\log ( k ) } { N } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 265, + 506, + 284 + ], + "score": 1.0, + "content": "(Wainwright, 2009). 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As evident from Fig. 1, even when restricting to linear functions our method has a clear", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 305, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 295, + 317 + ], + "score": 1.0, + "content": "advantage over LASSO. This implies that the", + "type": "text" + }, + { + "bbox": [ + 295, + 305, + 305, + 316 + ], + "score": 0.87, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 306, + 506, + 317 + ], + "score": 1.0, + "content": "based penalty, although is non convex in nature,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "allows perfect recovery of the support using less samples. 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Comparison between the proposed method, HC and LASSO under a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 215, + 501 + ], + "score": 1.0, + "content": "linear additive noise model", + "type": "text" + }, + { + "bbox": [ + 215, + 489, + 279, + 501 + ], + "score": 0.93, + "content": "\\pmb { y } = \\pmb { X } \\beta ^ { * } + \\pmb { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 490, + 316, + 501 + ], + "score": 1.0, + "content": "using (a)", + "type": "text" + }, + { + "bbox": [ + 317, + 490, + 350, + 500 + ], + "score": 0.91, + "content": "D = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 490, + 381, + 501 + ], + "score": 1.0, + "content": "and (b)", + "type": "text" + }, + { + "bbox": [ + 381, + 490, + 419, + 500 + ], + "score": 0.89, + "content": "D = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 490, + 505, + 501 + ], + "score": 1.0, + "content": ". Central lines are the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 501, + 335, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 335, + 512 + ], + "score": 1.0, + "content": "means while shaded are represent the standard deviation.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + } + ], + "index": 22.75 + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 414, + 535 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 416, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 416, + 537 + ], + "score": 1.0, + "content": "6.2 NONLINEAR REGRESSION USING SYNTHETIC AND REAL DATASETS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 506, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "In this section, we evaluate our method for regression tasks against two other embedded feature", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "score": 1.0, + "content": "selection methods: LASSO and Sparse Random Fourier Features (Gregorová et al., 2018). Following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "the same format as (Gregorová et al., 2018), the following functions are used to generate synthetic data:", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 101, + 572, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 101, + 572, + 161, + 597 + ], + "score": 1.0, + "content": "(SE1: 100/5)", + "type": "text" + }, + { + "bbox": [ + 162, + 578, + 346, + 591 + ], + "score": 0.91, + "content": "y = \\sin { ( x _ { 1 } + x _ { 3 } ) ^ { 2 } } \\sin { ( x _ { 7 } x _ { 8 } x _ { 9 } ) } + \\mathcal { N } ( 0 , 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 572, + 402, + 597 + ], + "score": 1.0, + "content": ". 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DNN represents a deep", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 83, + 272, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 272, + 94 + ], + "score": 1.0, + "content": "neural network without feature selection.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 96, + 547, + 179 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 96, + 547, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 547, + 179 + ], + "score": 0.981, + "html": "
EXPLASSORFSG-L1-NNSRFFDNNHCSTG
SE10.29 (0.01)0.30 (0.01)0.29 (0.01)0.27 (0.01)0.29 (0.01)0.29 (0.01)0.29 (0.01)
SE22.22 (0.10)2.34 (0.17)2.35 (0.18)1.60 (0.10)2.05 (0.11)1.19 (0.31)0.87 (0.15)
SE30.68 (0.002)0.50(0.01)0.68 (0.01)0.48 (0.03)0.73 (0.02)0.33 (0.05)0.14 (0.10)
RCP9.69 (0.71)3.52 (0.12)9.64 (0.65)2.52 (0.18)2.89 (0.25)2.74 (0.75)2.44 (0.08)
REL0.47 (0.01)0.58 (0.01)0.44 (0.01)0.31 (0.03)0.61 (0.01)0.33 (0.04)0.27 (0.002)
RAI0.43 (0.02)0.48 (0.003)0.47(0.002)0.41(0.02)0.44 (0.01)0.41 (0.01)0.39 (0.01)
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The coordinates", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 169, + 245 + ], + "score": 0.92, + "content": "x _ { i } , i = 3 , . . . , D", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "are nuisance features, also drawn from a binary \"fair\" Bernoulli distribution. 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To provide more benchmarks, we also compare our embedded method against three", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "wrapper methods (Extremely Randomized Trees (Tree) (Rastogi & Shim, 2000), Random Forests", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 501, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 501, + 323 + ], + "score": 1.0, + "content": "(RF) (Strobl et al., 2008)) and Extreme Gradient Boosting (XGBOOST) (Chen & Guestrin, 2016).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "To evaluate the feature selection performance, we calculate the Informative Features Weight Ratio", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 296, + 349 + ], + "score": 1.0, + "content": "(IFWR). 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In the case of binary weights the IFWR is in fact a recall measure for the relevant", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 289, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 289, + 372 + ], + "score": 1.0, + "content": "features (See the Appendix for more details.)", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 351, + 390 + ], + "score": 1.0, + "content": "The experiment is repeated 20 times for different values of", + "type": "text" + }, + { + "bbox": [ + 351, + 378, + 361, + 387 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 377, + 505, + 390 + ], + "score": 1.0, + "content": ", and the average test classification", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "accuracy and standard deviation are presented in Fig. 2(a), followed by the IFWR in Fig. 2(b). The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "number of selected features affects the accuracy. Therefore, to treat all the methods in a fair manner,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "we tune the hyperparameter that controls the sparsity level using Optuna (Takuya Akiba & Koyama,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 347, + 434 + ], + "score": 1.0, + "content": "2019) which optimizes the overall accuracy across different", + "type": "text" + }, + { + "bbox": [ + 347, + 421, + 361, + 431 + ], + "score": 0.58, + "content": "D \\mathrm { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 420, + 505, + 434 + ], + "score": 1.0, + "content": ". For instance, the wrapper methods", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "(Tree, RF and XGBOOST) has a threshold value to retain features. We retrain them using only such", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "features whose weight is higher than the threshold. In terms of feature ranking (see Fig. 2(c), only the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "tree based methods and the proposed (STG and HC based) provide the optimal median rank (which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "is 1.5) for the two relevant features. Nonetheless, the ranking provided by STG is the most stable", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 271, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 271, + 487 + ], + "score": 1.0, + "content": "comparing to all the alternative methods.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "image", + "bbox": [ + 113, + 489, + 502, + 577 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 489, + 502, + 577 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 489, + 502, + 577 + ], + "spans": [ + { + "bbox": [ + 113, + 489, + 502, + 577 + ], + "score": 0.968, + "type": "image", + "image_path": "06d3d26781acd541fbd45798a0a833a36dd7481d585a6b1bfa91e291fd85278b.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 113, + 489, + 502, + 518.3333333333334 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 113, + 518.3333333333334, + 502, + 547.6666666666667 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 113, + 547.6666666666667, + 502, + 577.0000000000001 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 584, + 505, + 650 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "score": 1.0, + "content": "Figure 2: (a) Classification accuracy (mean and standard deviation) vs. the number of irrelevant noisy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 151, + 606 + ], + "score": 1.0, + "content": "dimension", + "type": "text" + }, + { + "bbox": [ + 152, + 596, + 168, + 606 + ], + "score": 0.71, + "content": "( D )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "for the XOR problem. 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EXPLASSORFSG-L1-NNSRFFDNNHCSTG
SE10.29 (0.01)0.30 (0.01)0.29 (0.01)0.27 (0.01)0.29 (0.01)0.29 (0.01)0.29 (0.01)
SE22.22 (0.10)2.34 (0.17)2.35 (0.18)1.60 (0.10)2.05 (0.11)1.19 (0.31)0.87 (0.15)
SE30.68 (0.002)0.50(0.01)0.68 (0.01)0.48 (0.03)0.73 (0.02)0.33 (0.05)0.14 (0.10)
RCP9.69 (0.71)3.52 (0.12)9.64 (0.65)2.52 (0.18)2.89 (0.25)2.74 (0.75)2.44 (0.08)
REL0.47 (0.01)0.58 (0.01)0.44 (0.01)0.31 (0.03)0.61 (0.01)0.33 (0.04)0.27 (0.002)
RAI0.43 (0.02)0.48 (0.003)0.47(0.002)0.41(0.02)0.44 (0.01)0.41 (0.01)0.39 (0.01)
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The coordinates", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 169, + 245 + ], + "score": 0.92, + "content": "x _ { i } , i = 3 , . . . , D", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "are nuisance features, also drawn from a binary \"fair\" Bernoulli distribution. 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To provide more benchmarks, we also compare our embedded method against three", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "wrapper methods (Extremely Randomized Trees (Tree) (Rastogi & Shim, 2000), Random Forests", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 501, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 501, + 323 + ], + "score": 1.0, + "content": "(RF) (Strobl et al., 2008)) and Extreme Gradient Boosting (XGBOOST) (Chen & Guestrin, 2016).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 200, + 507, + 323 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "To evaluate the feature selection performance, we calculate the Informative Features Weight Ratio", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 296, + 349 + ], + "score": 1.0, + "content": "(IFWR). 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In terms of feature ranking (see Fig. 2(c), only the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "tree based methods and the proposed (STG and HC based) provide the optimal median rank (which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "is 1.5) for the two relevant features. 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After", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 469, + 460, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 307, + 483 + ], + "score": 1.0, + "content": "training, the set of indices for selected features is:", + "type": "text" + }, + { + "bbox": [ + 308, + 470, + 455, + 482 + ], + "score": 0.91, + "content": "\\{ d : \\operatorname* { m i n } ( 1 , \\operatorname* { m a x } ( 0 , \\mu _ { d } + 0 . 5 ) ) > 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 469, + 460, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 437, + 505, + 483 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 497, + 261, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 262, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 262, + 512 + ], + "score": 1.0, + "content": "C PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 507, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 507, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 507, + 536 + ], + "score": 1.0, + "content": "We now provide a proof for Proposition 1, showing the equivalence between the stochastic optimiza-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 306, + 547 + ], + "score": 1.0, + "content": "tion (Eq.7) and the deterministic one (Eq. 6). Let", + "type": "text" + }, + { + "bbox": [ + 307, + 533, + 315, + 545 + ], + "score": 0.85, + "content": "\\tilde { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 534, + 401, + 547 + ], + "score": 1.0, + "content": "be a subset such that", + "type": "text" + }, + { + "bbox": [ + 401, + 533, + 448, + 547 + ], + "score": 0.93, + "content": "S ^ { * } \\setminus \\tilde { S } \\ne \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 534, + 506, + 547 + ], + "score": 1.0, + "content": ". That is there", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 201, + 560 + ], + "score": 1.0, + "content": "exists some element in", + "type": "text" + }, + { + "bbox": [ + 201, + 547, + 213, + 558 + ], + "score": 0.87, + "content": "S ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 546, + 268, + 560 + ], + "score": 1.0, + "content": "that is not in", + "type": "text" + }, + { + "bbox": [ + 268, + 546, + 276, + 557 + ], + "score": 0.82, + "content": "\\tilde { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 546, + 348, + 560 + ], + "score": 1.0, + "content": ". For any such set", + "type": "text" + }, + { + "bbox": [ + 349, + 546, + 357, + 557 + ], + "score": 0.84, + "content": "\\tilde { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 546, + 411, + 560 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + }, + { + "bbox": [ + 412, + 547, + 503, + 560 + ], + "score": 0.92, + "content": "I ( X _ { \\tilde { S } } ; Y ) < I ( X ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 546, + 506, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 559, + 281, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 174, + 573 + ], + "score": 1.0, + "content": "Indeed, if we let", + "type": "text" + }, + { + "bbox": [ + 174, + 559, + 224, + 571 + ], + "score": 0.92, + "content": "i \\in { \\cal S } ^ { * } \\cap \\tilde { \\cal S } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 559, + 281, + 573 + ], + "score": 1.0, + "content": "then we have", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 522, + 507, + 573 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 574, + 392, + 621 + ], + "lines": [ + { + "bbox": [ + 219, + 574, + 392, + 621 + ], + "spans": [ + { + "bbox": [ + 219, + 574, + 392, + 621 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { I ( \\boldsymbol { X } _ { \\tilde { \\mathcal { S } } } ; Y ) } & { \\leq I ( \\boldsymbol { X } _ { \\backslash \\{ i \\} } ; Y ) } \\\\ & { = I ( \\boldsymbol { X } ; Y ) - I ( \\boldsymbol { X } _ { i } ; Y | \\boldsymbol { X } _ { \\backslash \\{ i \\} } ) } \\\\ & { < I ( \\boldsymbol { X } ; Y ) , } \\end{array}", + "type": "interline_equation", + "image_path": "56d113d3cd6e3bf0dd9dae95bd9cb358f9fa5a3ec42ca0caab43c76862606bb1.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 219, + 574, + 392, + 589.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 219, + 589.6666666666666, + 392, + 605.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 219, + 605.3333333333333, + 392, + 620.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 625, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 475, + 639 + ], + "score": 1.0, + "content": "where the final inequality follows by Assumption 1. Assumption 2 also yields that for any set", + "type": "text" + }, + { + "bbox": [ + 475, + 624, + 484, + 636 + ], + "score": 0.84, + "content": "\\tilde { \\cal S }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 624, + 505, + 639 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 124, + 652 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 637, + 158, + 649 + ], + "score": 0.92, + "content": "S ^ { * } \\subset \\tilde { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 637, + 198, + 652 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 198, + 638, + 289, + 651 + ], + "score": 0.93, + "content": "I ( { \\pmb X } _ { \\tilde { S } } ; Y ) = I ( { \\pmb X } ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 637, + 505, + 652 + ], + "score": 1.0, + "content": ". Now, when we consider the Bernoulli optimization", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 178, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 178, + 662 + ], + "score": 1.0, + "content": "problem we have", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 624, + 505, + 662 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 664, + 418, + 691 + ], + "lines": [ + { + "bbox": [ + 191, + 664, + 418, + 691 + ], + "spans": [ + { + "bbox": [ + 191, + 664, + 418, + 691 + ], + "score": 0.92, + "content": "\\operatorname* { m a x } _ { \\pi } I ( X \\odot S ; Y ) \\quad \\mathrm { s . t . } \\quad \\sum _ { l } \\pi _ { l } \\leq k \\mathrm { a n d } 0 \\leq \\pi _ { l } \\leq 1 .", + "type": "interline_equation", + "image_path": "e5992f69b7584b344934b1d725018064d0708c8a19d4aac2e7f0f0626dd12b96.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 191, + 664, + 418, + 691 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 694, + 284, + 706 + ], + "lines": [ + { + "bbox": [ + 105, + 692, + 284, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 284, + 707 + ], + "score": 1.0, + "content": "The mutual information can be expanded as", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 692, + 284, + 707 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 709, + 399, + 735 + ], + "lines": [ + { + "bbox": [ + 210, + 709, + 399, + 735 + ], + "spans": [ + { + "bbox": [ + 210, + 709, + 399, + 735 + ], + "score": 0.94, + "content": "I ( X \\odot S ; Y ) = \\sum _ { s } I ( X \\odot s ; Y ) p _ { \\pi } ( S = s ) ,", + "type": "interline_equation", + "image_path": "ea1e2a0756538252da9f48d4e29f07413ec276c474fa106f23a0b0843e10ecff.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 210, + 709, + 399, + 735 + ], + "spans": [], + "index": 43 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 236, + 96 + ], + "score": 1.0, + "content": "where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 236, + 83, + 245, + 93 + ], + "score": 0.79, + "content": "_ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 81, + 507, + 96 + ], + "score": 1.0, + "content": "is independent of everything else. Recall that in optimization (Eq.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 192, + 106 + ], + "score": 1.0, + "content": "7) the coordinates of", + "type": "text" + }, + { + "bbox": [ + 193, + 94, + 201, + 104 + ], + "score": 0.79, + "content": "\\pmb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "are sampled at random. Therefore, the distribution that is being optimized", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 126, + 117 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 126, + 106, + 140, + 116 + ], + "score": 0.86, + "content": "p _ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "is a product distribution. Our goal is to understand the form of this distribution. To that end,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "we will consider a problem dropping the independence constraint. If we can show that the distribution", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "found by solving this new optimization problem with less constraints is still a product distribution,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 354, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 354, + 150 + ], + "score": 1.0, + "content": "then we obtain a solution to the original optimization (Eq. 6).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 401, + 168 + ], + "score": 1.0, + "content": "Now, from above we know that the optimal value of the optimization is", + "type": "text" + }, + { + "bbox": [ + 401, + 154, + 457, + 168 + ], + "score": 0.94, + "content": "I ( X \\odot { \\tilde { S } } ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 155, + 506, + 168 + ], + "score": 1.0, + "content": "for any set", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 167, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 139, + 179 + ], + "score": 0.9, + "content": "S ^ { * } \\subset \\tilde { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 167, + 506, + 182 + ], + "score": 1.0, + "content": ". Hence, any unconstrained distribution should place all of its mass on such subsets in order", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 177, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 177, + 312, + 195 + ], + "score": 1.0, + "content": "to maximize the mutual information. As a result", + "type": "text" + }, + { + "bbox": [ + 313, + 179, + 410, + 191 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { l \\in S ^ { * } } p ( S _ { l } = 1 ) = k } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 177, + 506, + 195 + ], + "score": 1.0, + "content": ". However, there is an", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 217, + 204 + ], + "score": 1.0, + "content": "optimization constraint that", + "type": "text" + }, + { + "bbox": [ + 217, + 191, + 275, + 204 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\sum _ { l } S _ { l } ] \\le k", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 190, + 322, + 204 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 322, + 192, + 365, + 203 + ], + "score": 0.92, + "content": "\\mathbb { E } [ S _ { l } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 190, + 396, + 204 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 396, + 191, + 424, + 203 + ], + "score": 0.9, + "content": "l \\notin S ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 190, + 506, + 204 + ], + "score": 1.0, + "content": ". Hence, the optimal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 203, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 438, + 214 + ], + "score": 1.0, + "content": "solution is to select the distribution so that all of the mass is placed on the subset", + "type": "text" + }, + { + "bbox": [ + 438, + 203, + 450, + 213 + ], + "score": 0.88, + "content": "S ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 203, + 505, + 214 + ], + "score": 1.0, + "content": "and no mass", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 448, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 448, + 226 + ], + "score": 1.0, + "content": "elsewhere. As this is also a product distribution, this complete the proof of the claim.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 106, + 243, + 313, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 313, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 313, + 258 + ], + "score": 1.0, + "content": "D BRIDGING THE TWO PERSPECTIVES", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 269, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "To motivate the introduction of randomness into the risk, we have looked at the feature selection", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "problem from a MI perspective. Based on the MI objective, we have observed that introducing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "randomness into the constrained maximization procedure, does not change the objective (Proposition", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "1). Here we provide a relation between the MI objective (Eq. 6) to the empirical risk (Eq. 1), which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 315, + 242, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 242, + 326 + ], + "score": 1.0, + "content": "supports our proposed procedure.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 105, + 330, + 504, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 309, + 343 + ], + "score": 1.0, + "content": "We first note that the MI maximization over the set", + "type": "text" + }, + { + "bbox": [ + 309, + 331, + 317, + 341 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "can be reformulated as the minimization of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 342, + 375, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 187, + 354 + ], + "score": 1.0, + "content": "conditional entropy", + "type": "text" + }, + { + "bbox": [ + 187, + 342, + 231, + 354 + ], + "score": 0.93, + "content": "H ( Y | \\boldsymbol { X } _ { \\mathcal { S } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 342, + 256, + 354 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 256, + 342, + 282, + 354 + ], + "score": 0.92, + "content": "H ( Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 342, + 362, + 354 + ], + "score": 1.0, + "content": "does not depend on", + "type": "text" + }, + { + "bbox": [ + 363, + 342, + 371, + 352 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 342, + 375, + 354 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 361, + 442, + 379 + ], + "lines": [ + { + "bbox": [ + 168, + 361, + 442, + 379 + ], + "spans": [ + { + "bbox": [ + 168, + 361, + 442, + 379 + ], + "score": 0.89, + "content": "\\displaystyle { \\operatorname* { m a x } _ { \\mathcal { S } } I ( \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ; Y ) = \\operatorname* { m a x } _ { \\mathcal { S } } H ( Y ) - H ( Y | \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ) \\iff \\operatorname* { m i n } _ { \\mathcal { S } } H ( Y | \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ) } .", + "type": "interline_equation", + "image_path": "4340acfbddae0dcd7ca93c3955ddda6b3e3d779bb5199cfe7113058d3e09b268.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 168, + 361, + 442, + 379 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 504, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 152, + 408 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + }, + { + "bbox": [ + 153, + 393, + 213, + 406 + ], + "score": 0.93, + "content": "X _ { \\mathcal { S } } = X \\odot \\tilde { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 393, + 495, + 408 + ], + "score": 1.0, + "content": ". By Proposition 1, we can rewrite the deterministic search over the set", + "type": "text" + }, + { + "bbox": [ + 496, + 396, + 504, + 405 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 406, + 288, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 275, + 419 + ], + "score": 1.0, + "content": "by a search over the Bernoulli parameters", + "type": "text" + }, + { + "bbox": [ + 275, + 408, + 283, + 416 + ], + "score": 0.78, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 406, + 288, + 419 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 424, + 429, + 468 + ], + "lines": [ + { + "bbox": [ + 181, + 424, + 429, + 468 + ], + "spans": [ + { + "bbox": [ + 181, + 424, + 429, + 468 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\displaystyle \\operatorname* { m i n } _ { \\pi } H ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) = \\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { \\boldsymbol { X } , \\boldsymbol { Y } , \\tilde { \\boldsymbol { S } } } - \\log P _ { \\pmb { \\theta } ^ { * } } ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) } & { } \\\\ { = \\displaystyle \\operatorname* { m i n } _ { \\pmb { \\theta } } \\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { \\boldsymbol { X } , \\boldsymbol { Y } , \\tilde { \\boldsymbol { S } } } - \\log P _ { \\pmb { \\theta } } ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "1281ade667d73948ce91bd121329f8e38779576e0e3752f862fc33710d9c20a8.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 181, + 424, + 429, + 438.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 181, + 438.6666666666667, + 429, + 453.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 181, + 453.33333333333337, + 429, + 468.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 223, + 489 + ], + "score": 1.0, + "content": "where the expectation is over", + "type": "text" + }, + { + "bbox": [ + 224, + 477, + 277, + 490 + ], + "score": 0.92, + "content": "X , Y \\sim P _ { \\pmb { \\theta } _ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 475, + 434, + 489 + ], + "score": 1.0, + "content": ", which is the true data distribution, and", + "type": "text" + }, + { + "bbox": [ + 434, + 475, + 503, + 489 + ], + "score": 0.93, + "content": "\\tilde { S } \\sim B e r n ( \\tilde { S } | \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 475, + 506, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 411, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 224, + 501 + ], + "score": 1.0, + "content": "Put our model distribution as", + "type": "text" + }, + { + "bbox": [ + 225, + 489, + 236, + 499 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 487, + 411, + 501 + ], + "score": 1.0, + "content": ", then we can rewrite the right hand side as:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 507, + 483, + 573 + ], + "lines": [ + { + "bbox": [ + 127, + 507, + 483, + 573 + ], + "spans": [ + { + "bbox": [ + 127, + 507, + 483, + 573 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { X , Y , \\tilde { S } } \\log P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) = \\mathbb { E } _ { X , Y , \\tilde { S } } \\Bigg [ \\log P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) \\frac { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } \\Bigg ] } \\\\ & { \\qquad = \\mathbb { E } _ { X , Y , \\tilde { S } } \\log \\frac { P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) } { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } + \\mathbb { E } _ { X , Y , \\tilde { S } } \\log P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "44015d158845a22ad0a394062bebe138c8c4df0f42a85cde9cd840e24e907cb2.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 127, + 507, + 483, + 529.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 127, + 529.0, + 483, + 551.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 127, + 551.0, + 483, + 573.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 585, + 506, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 131, + 600 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 585, + 278, + 599 + ], + "score": 0.92, + "content": "\\mathrm { K L } ( P _ { \\theta ^ { * } } ( Y | X \\odot \\tilde { S } ) | | P _ { \\theta } ( Y | X \\odot \\tilde { S } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 584, + 344, + 600 + ], + "score": 1.0, + "content": "is non-negative,", + "type": "text" + }, + { + "bbox": [ + 345, + 585, + 504, + 599 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\tilde { S } } \\mathbf { K } \\mathbf { L } ( P _ { \\theta * } ( Y | \\boldsymbol { X } \\odot \\tilde { \\pmb { S } } ) | | P _ { \\theta } ( Y | \\boldsymbol { X } \\odot \\tilde { \\pmb { S } } ) )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 598, + 445, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 445, + 610 + ], + "score": 1.0, + "content": "is also non-negative because it is a weighted sum of non-negative terms. Noting that", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 617, + 450, + 647 + ], + "lines": [ + { + "bbox": [ + 160, + 617, + 450, + 647 + ], + "spans": [ + { + "bbox": [ + 160, + 617, + 450, + 647 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\tilde { S } } \\mathrm { K L } \\big ( P _ { \\theta * } ( Y | X \\odot \\tilde { S } ) | | P _ { \\theta } ( Y | X \\odot \\tilde { S } ) \\big ) = \\mathbb { E } _ { X , Y , \\tilde { S } } \\log \\frac { P _ { \\theta * } ( Y | X \\odot \\tilde { S } ) } { P _ { \\theta } ( Y | X \\odot \\tilde { S } ) }", + "type": "interline_equation", + "image_path": "d1fa31e8ca8ae1d6e011d190b67ac3220ff12e7468ce475f8644e0d4d8e05a72.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 160, + 617, + 450, + 627.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 160, + 627.0, + 450, + 637.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 160, + 637.0, + 450, + 647.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 192, + 665 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 193, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 193, + 666 + ], + "score": 1.0, + "content": "we can conclude that", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 672, + 431, + 689 + ], + "lines": [ + { + "bbox": [ + 179, + 672, + 431, + 689 + ], + "spans": [ + { + "bbox": [ + 179, + 672, + 431, + 689 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { X , Y , \\tilde { S } } - \\log P _ { \\theta ^ { * } } ( Y | X \\odot \\tilde { S } ) \\leq \\mathbb { E } _ { X , Y , \\tilde { S } } - \\log P _ { \\theta } ( Y | X \\odot \\tilde { S } ) .", + "type": "interline_equation", + "image_path": "8919538185d34b2d5fb2cb4a0ee179a471e77ed1e9af6000ac6f176b4c5f18b5.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 179, + 672, + 431, + 689 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 421, + 712 + ], + "score": 1.0, + "content": "If we consider the negative log likelihood of the target given the observations (i.e.", + "type": "text" + }, + { + "bbox": [ + 421, + 697, + 504, + 711 + ], + "score": 0.89, + "content": "- \\log P _ { \\theta } ( Y | X \\odot \\tilde { \\pmb { S } } ) )", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 179, + 723 + ], + "score": 1.0, + "content": "as a loss function", + "type": "text" + }, + { + "bbox": [ + 180, + 710, + 187, + 720 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 708, + 417, + 723 + ], + "score": 1.0, + "content": "(which encodes the classification or regression function", + "type": "text" + }, + { + "bbox": [ + 417, + 710, + 431, + 723 + ], + "score": 0.84, + "content": "f _ { \\pmb { \\theta } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "), then we see that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 397, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 397, + 734 + ], + "score": 1.0, + "content": "minimizing the risk approximately maximizes the MI objective in Eq. 6.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 236, + 96 + ], + "score": 1.0, + "content": "where we have used the fact that", + "type": "text" + }, + { + "bbox": [ + 236, + 83, + 245, + 93 + ], + "score": 0.79, + "content": "_ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 81, + 507, + 96 + ], + "score": 1.0, + "content": "is independent of everything else. Recall that in optimization (Eq.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 192, + 106 + ], + "score": 1.0, + "content": "7) the coordinates of", + "type": "text" + }, + { + "bbox": [ + 193, + 94, + 201, + 104 + ], + "score": 0.79, + "content": "\\pmb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "are sampled at random. Therefore, the distribution that is being optimized", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 126, + 117 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 126, + 106, + 140, + 116 + ], + "score": 0.86, + "content": "p _ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "is a product distribution. Our goal is to understand the form of this distribution. To that end,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "we will consider a problem dropping the independence constraint. If we can show that the distribution", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "found by solving this new optimization problem with less constraints is still a product distribution,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 354, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 354, + 150 + ], + "score": 1.0, + "content": "then we obtain a solution to the original optimization (Eq. 6).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 81, + 507, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 401, + 168 + ], + "score": 1.0, + "content": "Now, from above we know that the optimal value of the optimization is", + "type": "text" + }, + { + "bbox": [ + 401, + 154, + 457, + 168 + ], + "score": 0.94, + "content": "I ( X \\odot { \\tilde { S } } ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 155, + 506, + 168 + ], + "score": 1.0, + "content": "for any set", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 167, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 139, + 179 + ], + "score": 0.9, + "content": "S ^ { * } \\subset \\tilde { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 167, + 506, + 182 + ], + "score": 1.0, + "content": ". Hence, any unconstrained distribution should place all of its mass on such subsets in order", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 177, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 177, + 312, + 195 + ], + "score": 1.0, + "content": "to maximize the mutual information. As a result", + "type": "text" + }, + { + "bbox": [ + 313, + 179, + 410, + 191 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { l \\in S ^ { * } } p ( S _ { l } = 1 ) = k } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 177, + 506, + 195 + ], + "score": 1.0, + "content": ". However, there is an", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 217, + 204 + ], + "score": 1.0, + "content": "optimization constraint that", + "type": "text" + }, + { + "bbox": [ + 217, + 191, + 275, + 204 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\sum _ { l } S _ { l } ] \\le k", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 190, + 322, + 204 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 322, + 192, + 365, + 203 + ], + "score": 0.92, + "content": "\\mathbb { E } [ S _ { l } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 190, + 396, + 204 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 396, + 191, + 424, + 203 + ], + "score": 0.9, + "content": "l \\notin S ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 190, + 506, + 204 + ], + "score": 1.0, + "content": ". Hence, the optimal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 203, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 438, + 214 + ], + "score": 1.0, + "content": "solution is to select the distribution so that all of the mass is placed on the subset", + "type": "text" + }, + { + "bbox": [ + 438, + 203, + 450, + 213 + ], + "score": 0.88, + "content": "S ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 203, + 505, + 214 + ], + "score": 1.0, + "content": "and no mass", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 448, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 448, + 226 + ], + "score": 1.0, + "content": "elsewhere. As this is also a product distribution, this complete the proof of the claim.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 154, + 506, + 226 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 243, + 313, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 313, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 313, + 258 + ], + "score": 1.0, + "content": "D BRIDGING THE TWO PERSPECTIVES", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 269, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "To motivate the introduction of randomness into the risk, we have looked at the feature selection", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "problem from a MI perspective. Based on the MI objective, we have observed that introducing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "randomness into the constrained maximization procedure, does not change the objective (Proposition", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "1). Here we provide a relation between the MI objective (Eq. 6) to the empirical risk (Eq. 1), which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 315, + 242, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 242, + 326 + ], + "score": 1.0, + "content": "supports our proposed procedure.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 270, + 506, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 330, + 504, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 309, + 343 + ], + "score": 1.0, + "content": "We first note that the MI maximization over the set", + "type": "text" + }, + { + "bbox": [ + 309, + 331, + 317, + 341 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "can be reformulated as the minimization of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 342, + 375, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 187, + 354 + ], + "score": 1.0, + "content": "conditional entropy", + "type": "text" + }, + { + "bbox": [ + 187, + 342, + 231, + 354 + ], + "score": 0.93, + "content": "H ( Y | \\boldsymbol { X } _ { \\mathcal { S } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 342, + 256, + 354 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 256, + 342, + 282, + 354 + ], + "score": 0.92, + "content": "H ( Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 342, + 362, + 354 + ], + "score": 1.0, + "content": "does not depend on", + "type": "text" + }, + { + "bbox": [ + 363, + 342, + 371, + 352 + ], + "score": 0.81, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 342, + 375, + 354 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 331, + 505, + 354 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 361, + 442, + 379 + ], + "lines": [ + { + "bbox": [ + 168, + 361, + 442, + 379 + ], + "spans": [ + { + "bbox": [ + 168, + 361, + 442, + 379 + ], + "score": 0.89, + "content": "\\displaystyle { \\operatorname* { m a x } _ { \\mathcal { S } } I ( \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ; Y ) = \\operatorname* { m a x } _ { \\mathcal { S } } H ( Y ) - H ( Y | \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ) \\iff \\operatorname* { m i n } _ { \\mathcal { S } } H ( Y | \\mathbf { \\boldsymbol { X } } _ { \\mathcal { S } } ) } .", + "type": "interline_equation", + "image_path": "4340acfbddae0dcd7ca93c3955ddda6b3e3d779bb5199cfe7113058d3e09b268.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 168, + 361, + 442, + 379 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 504, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 152, + 408 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + }, + { + "bbox": [ + 153, + 393, + 213, + 406 + ], + "score": 0.93, + "content": "X _ { \\mathcal { S } } = X \\odot \\tilde { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 393, + 495, + 408 + ], + "score": 1.0, + "content": ". By Proposition 1, we can rewrite the deterministic search over the set", + "type": "text" + }, + { + "bbox": [ + 496, + 396, + 504, + 405 + ], + "score": 0.8, + "content": "s", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 406, + 288, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 275, + 419 + ], + "score": 1.0, + "content": "by a search over the Bernoulli parameters", + "type": "text" + }, + { + "bbox": [ + 275, + 408, + 283, + 416 + ], + "score": 0.78, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 406, + 288, + 419 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 393, + 504, + 419 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 424, + 429, + 468 + ], + "lines": [ + { + "bbox": [ + 181, + 424, + 429, + 468 + ], + "spans": [ + { + "bbox": [ + 181, + 424, + 429, + 468 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\displaystyle \\operatorname* { m i n } _ { \\pi } H ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) = \\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { \\boldsymbol { X } , \\boldsymbol { Y } , \\tilde { \\boldsymbol { S } } } - \\log P _ { \\pmb { \\theta } ^ { * } } ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) } & { } \\\\ { = \\displaystyle \\operatorname* { m i n } _ { \\pmb { \\theta } } \\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { \\boldsymbol { X } , \\boldsymbol { Y } , \\tilde { \\boldsymbol { S } } } - \\log P _ { \\pmb { \\theta } } ( \\boldsymbol { Y } | \\boldsymbol { X } \\odot \\tilde { \\boldsymbol { S } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "1281ade667d73948ce91bd121329f8e38779576e0e3752f862fc33710d9c20a8.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 181, + 424, + 429, + 438.6666666666667 + ], 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"inline_equation" + }, + { + "bbox": [ + 503, + 475, + 506, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 411, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 224, + 501 + ], + "score": 1.0, + "content": "Put our model distribution as", + "type": "text" + }, + { + "bbox": [ + 225, + 489, + 236, + 499 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 487, + 411, + 501 + ], + "score": 1.0, + "content": ", then we can rewrite the right hand side as:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 475, + 506, + 501 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 507, + 483, + 573 + ], + "lines": [ + { + "bbox": [ + 127, + 507, + 483, + 573 + ], + "spans": [ + { + "bbox": [ + 127, + 507, + 483, + 573 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { X , Y , \\tilde { S } } \\log P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) = \\mathbb { E } _ { X , Y , \\tilde { S } } \\Bigg [ \\log P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) \\frac { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } \\Bigg ] } \\\\ & { \\qquad = \\mathbb { E } _ { X , Y , \\tilde { S } } \\log \\frac { P _ { \\theta ^ { * } } \\big ( Y | X \\odot \\tilde { S } \\big ) } { P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) } + \\mathbb { E } _ { X , Y , \\tilde { S } } \\log P _ { \\theta } \\big ( Y | X \\odot \\tilde { S } \\big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "44015d158845a22ad0a394062bebe138c8c4df0f42a85cde9cd840e24e907cb2.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 127, + 507, + 483, + 529.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 127, + 529.0, + 483, + 551.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 127, + 551.0, + 483, + 573.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 585, + 506, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 131, + 600 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 585, + 278, + 599 + ], + "score": 0.92, + "content": "\\mathrm { K L } ( P _ { \\theta ^ { * } } ( Y | X \\odot \\tilde { S } ) | | P _ { \\theta } ( Y | X \\odot \\tilde { S } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 584, + 344, + 600 + ], + "score": 1.0, + "content": "is non-negative,", + "type": "text" + }, + { + "bbox": [ + 345, + 585, + 504, + 599 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\tilde { S } } \\mathbf { K } \\mathbf { L } ( P _ { \\theta * } ( Y | \\boldsymbol { X } \\odot \\tilde { \\pmb { S } } ) | | P _ { \\theta } ( Y | \\boldsymbol { X } \\odot \\tilde { \\pmb { S } } ) )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 598, + 445, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 445, + 610 + ], + "score": 1.0, + "content": "is also non-negative because it is a weighted sum of non-negative terms. Noting that", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 584, + 504, + 610 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 617, + 450, + 647 + ], + "lines": [ + { + "bbox": [ + 160, + 617, + 450, + 647 + ], + "spans": [ + { + "bbox": [ + 160, + 617, + 450, + 647 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\tilde { S } } \\mathrm { K L } \\big ( P _ { \\theta * } ( Y | X \\odot \\tilde { S } ) | | P _ { \\theta } ( Y | X \\odot \\tilde { S } ) \\big ) = \\mathbb { E } _ { X , Y , \\tilde { S } } \\log \\frac { P _ { \\theta * } ( Y | X \\odot \\tilde { S } ) } { P _ { \\theta } ( Y | X \\odot \\tilde { S } ) }", + "type": "interline_equation", + "image_path": "d1fa31e8ca8ae1d6e011d190b67ac3220ff12e7468ce475f8644e0d4d8e05a72.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 160, + 617, + 450, + 627.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 160, + 627.0, + 450, + 637.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 160, + 637.0, + 450, + 647.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 192, + 665 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 193, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 193, + 666 + ], + "score": 1.0, + "content": "we can conclude that", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 654, + 193, + 666 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 672, + 431, + 689 + ], + "lines": [ + { + "bbox": [ + 179, + 672, + 431, + 689 + ], + "spans": [ + { + "bbox": [ + 179, + 672, + 431, + 689 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { X , Y , \\tilde { S } } - \\log P _ { \\theta ^ { * } } ( Y | X \\odot \\tilde { S } ) \\leq \\mathbb { E } _ { X , Y , \\tilde { S } } - \\log P _ { \\theta } ( Y | X \\odot \\tilde { S } ) .", + "type": "interline_equation", + "image_path": "8919538185d34b2d5fb2cb4a0ee179a471e77ed1e9af6000ac6f176b4c5f18b5.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 179, + 672, + 431, + 689 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 421, + 712 + ], + "score": 1.0, + "content": "If we consider the negative log likelihood of the target given the observations (i.e.", + "type": "text" + }, + { + "bbox": [ + 421, + 697, + 504, + 711 + ], + "score": 0.89, + "content": "- \\log P _ { \\theta } ( Y | X \\odot \\tilde { \\pmb { S } } ) )", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 179, + 723 + ], + "score": 1.0, + "content": "as a loss function", + "type": "text" + }, + { + "bbox": [ + 180, + 710, + 187, + 720 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 708, + 417, + 723 + ], + "score": 1.0, + "content": "(which encodes the classification or regression function", + "type": "text" + }, + { + "bbox": [ + 417, + 710, + 431, + 723 + ], + "score": 0.84, + "content": "f _ { \\pmb { \\theta } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "), then we see that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 397, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 397, + 734 + ], + "score": 1.0, + "content": "minimizing the risk approximately maximizes the MI objective in Eq. 6.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 697, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 318, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 319, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 319, + 95 + ], + "score": 1.0, + "content": "E DETAILS OF REGULARIZATION TERM", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 119 + ], + "score": 1.0, + "content": "Here we provide the detail description of the regularization term. For the vector of stochastic gates", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 117, + 333, + 130 + ], + "spans": [ + { + "bbox": [ + 107, + 117, + 140, + 127 + ], + "score": 0.91, + "content": "z \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 117, + 333, + 130 + ], + "score": 1.0, + "content": ", the regularization term is expressed as follows:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 133, + 415, + 273 + ], + "lines": [ + { + "bbox": [ + 195, + 133, + 415, + 273 + ], + "spans": [ + { + "bbox": [ + 195, + 133, + 415, + 273 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { Z } \\| Z \\| _ { 0 } = \\displaystyle \\sum _ { d = 1 } ^ { D } \\mathbb { P } [ z _ { d } > 0 ] = \\displaystyle \\sum _ { d = 1 } ^ { D } \\mathbb { P } [ \\mu _ { d } + \\sigma _ { d } \\epsilon _ { d } > - \\frac { 1 } { 2 } ] } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\{ 1 - \\mathbb { P } [ \\mu _ { d } + \\sigma _ { d } \\epsilon _ { d } \\leq - \\frac { 1 } { 2 } ] \\} } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\{ 1 - \\Phi ( - \\frac { 1 } { \\sigma _ { d } } ) \\} } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\Phi ( \\frac { \\mu _ { d } + \\frac { 1 } { 2 } } { \\sigma _ { d } } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e09a089146d547c94fd00e82f90c19adc399b73b47abc6c40a667e4041ef9598.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 133, + 415, + 147.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 195, + 147.0, + 415, + 161.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 195, + 161.0, + 415, + 175.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 195, + 175.0, + 415, + 189.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 195, + 189.0, + 415, + 203.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 195, + 203.0, + 415, + 217.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 195, + 217.0, + 415, + 231.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 195, + 231.0, + 415, + 245.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 195, + 245.0, + 415, + 259.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 195, + 259.0, + 415, + 273.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 438, + 289 + ], + "score": 1.0, + "content": "The derivative of the regularization term with respect to the distribution parameter", + "type": "text" + }, + { + "bbox": [ + 438, + 279, + 450, + 288 + ], + "score": 0.85, + "content": "\\mu _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "is simply the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 288, + 169, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 169, + 299 + ], + "score": 1.0, + "content": "Gaussian PDF:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 301, + 420, + 334 + ], + "lines": [ + { + "bbox": [ + 190, + 301, + 420, + 334 + ], + "spans": [ + { + "bbox": [ + 190, + 301, + 420, + 334 + ], + "score": 0.93, + "content": "\\frac { \\partial } { \\partial \\mu _ { d } } \\mathbb { E } _ { Z } \\left\\| Z \\right\\| _ { 0 } = \\frac { \\partial } { \\partial \\mu _ { d } } \\Phi ( \\frac { \\mu _ { d } + \\frac { 1 } { 2 } } { \\sigma _ { d } } ) = \\frac { 1 } { \\sqrt { 2 \\pi \\sigma _ { d } ^ { 2 } } } e ^ { - 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In the first iteration", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 358, + 508, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 184, + 378 + ], + "score": 1.0, + "content": "during the training,", + "type": "text" + }, + { + "bbox": [ + 184, + 366, + 196, + 376 + ], + "score": 0.85, + "content": "\\mu _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 364, + 448, + 378 + ], + "score": 1.0, + "content": "is 0. 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Using the Bernoulli paramterization the empirical", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 719, + 222, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 124, + 733 + ], + "score": 1.0, + "content": "risk", + "type": "text" + }, + { + "bbox": [ + 124, + 719, + 158, + 732 + ], + "score": 0.93, + "content": "\\hat { R } ( \\pmb \\theta , \\pmb \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 720, + 222, + 733 + ], + "score": 1.0, + "content": "is expressed as", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 318, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 319, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 319, + 95 + ], + "score": 1.0, + "content": "E DETAILS OF REGULARIZATION TERM", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 119 + ], + "score": 1.0, + "content": "Here we provide the detail description of the regularization term. For the vector of stochastic gates", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 117, + 333, + 130 + ], + "spans": [ + { + "bbox": [ + 107, + 117, + 140, + 127 + ], + "score": 0.91, + "content": "z \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 117, + 333, + 130 + ], + "score": 1.0, + "content": ", the regularization term is expressed as follows:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 104, + 505, + 130 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 133, + 415, + 273 + ], + "lines": [ + { + "bbox": [ + 195, + 133, + 415, + 273 + ], + "spans": [ + { + "bbox": [ + 195, + 133, + 415, + 273 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { Z } \\| Z \\| _ { 0 } = \\displaystyle \\sum _ { d = 1 } ^ { D } \\mathbb { P } [ z _ { d } > 0 ] = \\displaystyle \\sum _ { d = 1 } ^ { D } \\mathbb { P } [ \\mu _ { d } + \\sigma _ { d } \\epsilon _ { d } > - \\frac { 1 } { 2 } ] } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\{ 1 - \\mathbb { P } [ \\mu _ { d } + \\sigma _ { d } \\epsilon _ { d } \\leq - \\frac { 1 } { 2 } ] \\} } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\{ 1 - \\Phi ( - \\frac { 1 } { \\sigma _ { d } } ) \\} } \\\\ { = \\displaystyle \\sum _ { d = 1 } ^ { D } \\Phi ( \\frac { \\mu _ { d } + \\frac { 1 } { 2 } } { \\sigma _ { d } } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e09a089146d547c94fd00e82f90c19adc399b73b47abc6c40a667e4041ef9598.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 133, + 415, + 147.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 195, + 147.0, + 415, + 161.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 195, + 161.0, + 415, + 175.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 195, + 175.0, + 415, + 189.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 195, + 189.0, + 415, + 203.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 195, + 203.0, + 415, + 217.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 195, + 217.0, + 415, + 231.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 195, + 231.0, + 415, + 245.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 195, + 245.0, + 415, + 259.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 195, + 259.0, + 415, + 273.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 438, + 289 + ], + "score": 1.0, + "content": "The derivative of the regularization term with respect to the distribution parameter", + "type": "text" + }, + { + "bbox": [ + 438, + 279, + 450, + 288 + ], + "score": 0.85, + "content": "\\mu _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "is simply the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 288, + 169, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 169, + 299 + ], + "score": 1.0, + "content": "Gaussian PDF:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 275, + 505, + 299 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 301, + 420, + 334 + ], + "lines": [ + { + "bbox": [ + 190, + 301, + 420, + 334 + ], + "spans": [ + { + "bbox": [ + 190, + 301, + 420, + 334 + ], + "score": 0.93, + "content": "\\frac { \\partial } { \\partial \\mu _ { d } } \\mathbb { E } _ { Z } \\left\\| Z \\right\\| _ { 0 } = \\frac { \\partial } { \\partial \\mu _ { d } } \\Phi ( \\frac { \\mu _ { d } + \\frac { 1 } { 2 } } { \\sigma _ { d } } ) = \\frac { 1 } { \\sqrt { 2 \\pi \\sigma _ { d } ^ { 2 } } } e ^ { - 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In the first iteration", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 358, + 508, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 184, + 378 + ], + "score": 1.0, + "content": "during the training,", + "type": "text" + }, + { + "bbox": [ + 184, + 366, + 196, + 376 + ], + "score": 0.85, + "content": "\\mu _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 364, + 448, + 378 + ], + "score": 1.0, + "content": "is 0. 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As depicted from this figure, the STG correctly identifies the active features (first 10) and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "sparsifies the non-active features. The deterministic version DNC does induce partial sparsity, but", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "fails to remove some of the irrelevant features. Furthemore, it does not push the gates values of active", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 409, + 723 + ], + "score": 1.0, + "content": "features up to 1. 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As shown in Fig 5, the heavy-tailness results in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 148 + ], + "score": 1.0, + "content": "instability during training. Furthermore, our method converges much faster and more reliably than", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 158 + ], + "score": 1.0, + "content": "the feature selection method using the Hard-Concrete distribution on a the two-moons, XOR and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 170 + ], + "score": 1.0, + "content": "MADELON datasets (see Subsection J.1 and J.2). 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The shaded area represents the standard deviation, calculated by running the same", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 321, + 339, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 339, + 333 + ], + "score": 1.0, + "content": "experiment 10 times with different random initializations.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 354, + 384, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 386, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 386, + 369 + ], + "score": 1.0, + "content": "H NON-CONVEX DETERMINISTIC REGULARIZATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 393 + ], + "score": 1.0, + "content": "Our proposed objective for feature selection (summarized in Eq. 5) consists of three major differences", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 390, + 259, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 259, + 403 + ], + "score": 1.0, + "content": "compared to the LASSO formulation.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 378, + 505, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 411, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 131, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 131, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "• Non-linearity, which is obtained by learning parameters of a multi-layer network with non", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 423, + 218, + 433 + ], + "spans": [ + { + "bbox": [ + 142, + 423, + 218, + 433 + ], + "score": 1.0, + "content": "linear activation’s.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 131, + 436, + 420, + 449 + ], + "spans": [ + { + "bbox": [ + 131, + 436, + 420, + 449 + ], + "score": 1.0, + "content": "• Feature gates, these are a specific form on non convex regularization.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 132, + 452, + 470, + 464 + ], + "spans": [ + { + "bbox": [ + 132, + 452, + 470, + 464 + ], + "score": 1.0, + "content": "• Stochasticity, which is achieved by injecting noise via the reparametrization trick.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 131, + 411, + 505, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 502, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 504, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 504, + 484 + ], + "score": 1.0, + "content": "In section 6.1, we have demonstrated that even in the linear regime the proposed formulation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 483, + 429, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 429, + 497 + ], + "score": 1.0, + "content": "outperforms the LASSO in the task of identifying active features (see Fig. 1(a)).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 471, + 504, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "The stochastic property of the gates proposed in this paper is justified using a MI prospective (see", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Section 4). Nonetheless, in this section we evaluate whether the practical performance gain of our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 521, + 502, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 502, + 536 + ], + "score": 1.0, + "content": "method stems from the non-convex regularization or from it’s combination with the injected noise.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 500, + 505, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 538, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 263, + 552 + ], + "score": 1.0, + "content": "We define a deterministic gate using", + "type": "text" + }, + { + "bbox": [ + 263, + 539, + 438, + 551 + ], + "score": 0.92, + "content": "\\tilde { z } _ { d } = g ( \\mu _ { d } ) = \\operatorname* { m a x } ( 0 , \\operatorname* { m i n } ( 1 , \\mu _ { d } + 0 . 5 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 539, + 473, + 552 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 473, + 541, + 485, + 551 + ], + "score": 0.86, + "content": "\\mu _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 249, + 563 + ], + "score": 1.0, + "content": "parameter learned for each feature", + "type": "text" + }, + { + "bbox": [ + 249, + 551, + 300, + 561 + ], + "score": 0.92, + "content": "d = 1 , . . . , D", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 550, + 505, + 563 + ], + "score": 1.0, + "content": ". 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Combined with", + "type": "text" + }, + { + "bbox": [ + 331, + 616, + 343, + 627 + ], + "score": 0.88, + "content": "\\tilde { z } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "this non-convex regularized objective is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "deterministic and differentiable, and its solution can be approximated via gradient decent. We evaluate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "the applicability of such alternative objective for the task of feature selection in linear regression. The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 649, + 402, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 365, + 661 + ], + "score": 1.0, + "content": "experimental setting follows the description in Section 6.1 using", + "type": "text" + }, + { + "bbox": [ + 365, + 649, + 398, + 659 + ], + "score": 0.88, + "content": "D = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 649, + 402, + 661 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 615, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "In Fig. 6(b), we compare the values of the deteministic gates (DNC) and stochastic gates (STG) using", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 140, + 687 + ], + "score": 0.89, + "content": "N = 6 0", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 676, + 505, + 689 + ], + "score": 1.0, + "content": ". As depicted from this figure, the STG correctly identifies the active features (first 10) and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "sparsifies the non-active features. The deterministic version DNC does induce partial sparsity, but", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "fails to remove some of the irrelevant features. Furthemore, it does not push the gates values of active", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 409, + 723 + ], + "score": 1.0, + "content": "features up to 1. 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Specifically, we use a threshold of 0.5 to define if a feature is active or not. A comparison", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "with STG, HC and LASSO in this setting appears in Fig. 6(a). The DNC is superior to LASSO,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "but inferior to the HC and STG. Which means that there is a true advantage of stochacity in our", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 159, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 159, + 139 + ], + "score": 1.0, + "content": "formulation.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 506, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 155 + ], + "score": 1.0, + "content": "One aspect that is crucially difference between the STG and DNC is what we call “second chance”.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "For the deterministic formulation, if a feature is zeroed out in an early training phase, the gradient of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "the feature vanishes for further training and it will never be pulled back. 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A Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 426, + 633, + 458, + 645 + ], + "score": 0.93, + "content": "N ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "is added to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "each feature. Next, additional 480 nuisance coordinates are added in the same manner. These features", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 279, + 667 + ], + "score": 1.0, + "content": "have no effect on the class label. Finally,", + "type": "text" + }, + { + "bbox": [ + 279, + 655, + 294, + 666 + ], + "score": 0.9, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "of the labels are flipped. To be consistent with the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "XOR and two moons experiments, we use 1, 500 points from this dataset, and evaluate our proposed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "method in terms of its predictive power and ability to detect the informative features. We vary the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 201, + 700 + ], + "score": 1.0, + "content": "regularizaton parameter", + "type": "text" + }, + { + "bbox": [ + 201, + 688, + 209, + 698 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "in the range [0.01, 10] and evaluate the classification accuracy using 5 folds", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "cross validation. In this example, we restrict our comparison to Random Forest and LASSO. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "focus on Random Forest as it was the strongest competitor to our method in all of our experiments,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 469, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 469, + 733 + ], + "score": 1.0, + "content": "while LASSO is evaluated because it is a widely used embedded feature selection method.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 588, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "score": 1.0, + "content": "As evident from Fig. 8(a), our method achieves the highest accuracy while using less features.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "Moreover, as depicted from this figure, peak performence occurs when selecting 5 features, thus, our", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 116 + ], + "score": 1.0, + "content": "method provides a clear indication to the true number of informative features. Both LASSO and RF", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 507, + 129 + ], + "score": 1.0, + "content": "on the other hand, do not provide a clear indication of the true number of relevant features. In Fig.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 225, + 138 + ], + "score": 1.0, + "content": "8(b), we evaluate the effect of", + "type": "text" + }, + { + "bbox": [ + 225, + 127, + 232, + 136 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "on the number and quality of selected features. As shown in this plot,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 203, + 149 + ], + "score": 1.0, + "content": "there is a wide range of", + "type": "text" + }, + { + "bbox": [ + 204, + 138, + 211, + 147 + ], + "score": 0.72, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "’s such that our method only selects relevant features. 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We evaluate performance using", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "5-fold cross validation for different number of selected features. In this dataset, only the first 20", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 518, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 528 + ], + "score": 1.0, + "content": "coordinates are informative. In that regime the proposed method outperforms RF and LASSO. (b)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 371, + 539 + ], + "score": 1.0, + "content": "An empirical evaluation of the effect the regularization parameter", + "type": "text" + }, + { + "bbox": [ + 372, + 528, + 379, + 537 + ], + "score": 0.67, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 527, + 506, + 539 + ], + "score": 1.0, + "content": ". The IFWR and the number of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 316, + 551 + ], + "score": 1.0, + "content": "selected features are presented on both sides of the", + "type": "text" + }, + { + "bbox": [ + 316, + 540, + 322, + 550 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "-axis of this plot. For both plots, the mean is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "presented as a solid/dashed line, while the standard deviation is marked as a shaded color around the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "score": 1.0, + "content": "mean. (c) Box plots for the median rank of the 5 original informative features. Black line and dashed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 572, + 496, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 496, + 584 + ], + "score": 1.0, + "content": "red represent the median and mean of each method. 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The first two coordinates", + "type": "text" + }, + { + "bbox": [ + 270, + 645, + 296, + 655 + ], + "score": 0.89, + "content": "x _ { 1 } , x _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "are generated by adding a Gaussian noise with zero", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 209, + 668 + ], + "score": 1.0, + "content": "mean and the variance of", + "type": "text" + }, + { + "bbox": [ + 209, + 654, + 247, + 667 + ], + "score": 0.93, + "content": "\\sigma _ { r } ^ { 2 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "onto two nested half circles, as presented in Fig. 9(a). Nuisance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 141, + 678 + ], + "score": 1.0, + "content": "features", + "type": "text" + }, + { + "bbox": [ + 142, + 666, + 205, + 677 + ], + "score": 0.91, + "content": "x _ { i } , i = 3 , . . . , D", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", are drawn from a Gaussian distribution with zero mean and variance of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 675, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 676, + 137, + 689 + ], + "score": 0.92, + "content": "\\sigma _ { n } ^ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 675, + 200, + 691 + ], + "score": 1.0, + "content": ". We reserve the", + "type": "text" + }, + { + "bbox": [ + 201, + 677, + 220, + 687 + ], + "score": 0.88, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 675, + 300, + 691 + ], + "score": 1.0, + "content": "as a test set, and use", + "type": "text" + }, + { + "bbox": [ + 301, + 677, + 320, + 687 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 675, + 507, + 691 + ], + "score": 1.0, + "content": "of the remaining training set as a validation set.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "We follow the same hyperparameter tuning procedure as in the XOR experiment. The classification", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "accuracy is in Fig. 9(b). Based on the classification accuracies, it is evident that for a small number of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "nuisance dimensions all methods correctly identify the most relevant features. 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Both LASSO and RF", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 507, + 129 + ], + "score": 1.0, + "content": "on the other hand, do not provide a clear indication of the true number of relevant features. In Fig.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 225, + 138 + ], + "score": 1.0, + "content": "8(b), we evaluate the effect of", + "type": "text" + }, + { + "bbox": [ + 225, + 127, + 232, + 136 + ], + "score": 0.75, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "on the number and quality of selected features. As shown in this plot,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 203, + 149 + ], + "score": 1.0, + "content": "there is a wide range of", + "type": "text" + }, + { + "bbox": [ + 204, + 138, + 211, + 147 + ], + "score": 0.72, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "’s such that our method only selects relevant features. Finally, as evident", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 382, + 162 + ], + "score": 1.0, + "content": "from the plato on the right hand side of the red plot, there is a range of", + "type": "text" + }, + { + "bbox": [ + 382, + 149, + 389, + 158 + ], + "score": 0.7, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "’s such that exactly 5 features", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 158, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 158, + 170 + ], + "score": 1.0, + "content": "are selected.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 507, + 170 + ] + }, + { + "type": "image", + "bbox": [ + 114, + 192, + 500, + 477 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 192, + 500, + 477 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 192, + 500, + 477 + ], + "spans": [ + { + "bbox": [ + 114, + 192, + 500, + 477 + ], + "score": 0.975, + "type": "image", + "image_path": "f81c197fa97377e7b0126d38b6153864d82047273e824cbb916508ca6b386556.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 114, + 192, + 500, + 287.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 114, + 287.0, + 500, + 382.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 114, + 382.0, + 500, + 477.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 494, + 506, + 583 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 493, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 508 + ], + "score": 1.0, + "content": "Figure 8: (a) Classification accuracy on the MADELON data sets. We evaluate performance using", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "5-fold cross validation for different number of selected features. In this dataset, only the first 20", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 518, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 528 + ], + "score": 1.0, + "content": "coordinates are informative. In that regime the proposed method outperforms RF and LASSO. (b)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 527, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 371, + 539 + ], + "score": 1.0, + "content": "An empirical evaluation of the effect the regularization parameter", + "type": "text" + }, + { + "bbox": [ + 372, + 528, + 379, + 537 + ], + "score": 0.67, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 527, + 506, + 539 + ], + "score": 1.0, + "content": ". The IFWR and the number of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 316, + 551 + ], + "score": 1.0, + "content": "selected features are presented on both sides of the", + "type": "text" + }, + { + "bbox": [ + 316, + 540, + 322, + 550 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "-axis of this plot. For both plots, the mean is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "presented as a solid/dashed line, while the standard deviation is marked as a shaded color around the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 572 + ], + "score": 1.0, + "content": "mean. (c) Box plots for the median rank of the 5 original informative features. Black line and dashed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 572, + 496, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 496, + 584 + ], + "score": 1.0, + "content": "red represent the median and mean of each method. Optimal median rank in this experiment is 3.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + } + ], + "index": 11.75 + }, + { + "type": "title", + "bbox": [ + 107, + 612, + 377, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 379, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 379, + 624 + ], + "score": 1.0, + "content": "J.2 TWO MOONS CLASSIFICATION WITH NUISANCE FEATURES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In this experiment, we construct a dataset based on \"two moons\" shape classes, concatenated with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 270, + 657 + ], + "score": 1.0, + "content": "noisy features. The first two coordinates", + "type": "text" + }, + { + "bbox": [ + 270, + 645, + 296, + 655 + ], + "score": 0.89, + "content": "x _ { 1 } , x _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "are generated by adding a Gaussian noise with zero", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 209, + 668 + ], + "score": 1.0, + "content": "mean and the variance of", + "type": "text" + }, + { + "bbox": [ + 209, + 654, + 247, + 667 + ], + "score": 0.93, + "content": "\\sigma _ { r } ^ { 2 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "onto two nested half circles, as presented in Fig. 9(a). Nuisance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 141, + 678 + ], + "score": 1.0, + "content": "features", + "type": "text" + }, + { + "bbox": [ + 142, + 666, + 205, + 677 + ], + "score": 0.91, + "content": "x _ { i } , i = 3 , . . . , D", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", are drawn from a Gaussian distribution with zero mean and variance of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 675, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 676, + 137, + 689 + ], + "score": 0.92, + "content": "\\sigma _ { n } ^ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 675, + 200, + 691 + ], + "score": 1.0, + "content": ". We reserve the", + "type": "text" + }, + { + "bbox": [ + 201, + 677, + 220, + 687 + ], + "score": 0.88, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 675, + 300, + 691 + ], + "score": 1.0, + "content": "as a test set, and use", + "type": "text" + }, + { + "bbox": [ + 301, + 677, + 320, + 687 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 675, + 507, + 691 + ], + "score": 1.0, + "content": "of the remaining training set as a validation set.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "We follow the same hyperparameter tuning procedure as in the XOR experiment. The classification", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "accuracy is in Fig. 9(b). Based on the classification accuracies, it is evident that for a small number of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "nuisance dimensions all methods correctly identify the most relevant features. The proposed method", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "(STG) and Random Forest (RF) are the only methods that achieve near perfect classification accuracy", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "for a wide range of nuisance dimensions. The other NN based method (DFS) seem to converge to", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 422, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 422, + 105 + ], + "score": 1.0, + "content": "sub-optimal solutions. We note that the median rank for all the methods is 1.5.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 633, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "for a wide range of nuisance dimensions. The other NN based method (DFS) seem to converge to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 422, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 422, + 105 + ], + "score": 1.0, + "content": "sub-optimal solutions. 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(b) Classification accuracy (mean and standard deviation based on 20 runs) vs. the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 310, + 261, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 261, + 322 + ], + "score": 1.0, + "content": "number of irrelevant noisy dimension.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + } + ], + "index": 4.75 + }, + { + "type": "title", + "bbox": [ + 107, + 359, + 335, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 337, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 337, + 371 + ], + "score": 1.0, + "content": "J.3 SPARSE HANDWRITTEN DIGITS CLASSIFICATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 506, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "score": 1.0, + "content": "In the following toy example, we attempt to distinguish between images of handwritten digits of 3’s", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 393, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 405 + ], + "score": 1.0, + "content": "and 8’s using samples from MNIST (LeCun et al., 1998). The orientation and location of the digits is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "more or less the same throughout this dataset, therefore for these two classes (3’s and 8’s), we expect", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "that some of the left side features (pixels) would be sufficient for the separation. The experiment is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 250, + 438 + ], + "score": 1.0, + "content": "performed as followed. We reserve", + "type": "text" + }, + { + "bbox": [ + 250, + 425, + 270, + 436 + ], + "score": 0.91, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 425, + 483, + 438 + ], + "score": 1.0, + "content": "of the data as the test set, and train on the remaining", + "type": "text" + }, + { + "bbox": [ + 484, + 425, + 503, + 436 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 425, + 506, + 438 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 436, + 507, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 507, + 449 + ], + "score": 1.0, + "content": "We then apply STG and evaluate the classification accuracy and the number of selected features.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "We use the architecture [200, 50, 10] with tanh activations. The experiment was repeated 10 times,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "the extracted features and accuracies were consistent over 20 trials. We noticed a relatively small", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 468, + 507, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 507, + 483 + ], + "score": 1.0, + "content": "number of selected features, which are positioned southwest and close to the center of the images,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "achieve very high classification accuracy. An example of 9 randomly selected samples overlaid with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "the weights of the selected features is presented in Fig. 10(a). Furthermore, we also evaluate the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 142, + 515 + ], + "score": 1.0, + "content": "effect of", + "type": "text" + }, + { + "bbox": [ + 142, + 502, + 149, + 512 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "on the the number of selected features and accuracy of the method. We apply our method", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "(STG) and its variant using the Hard-Concrete distribution to a randomly sampled training set of size", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 107, + 524, + 150, + 534 + ], + "score": 0.9, + "content": "N = 1 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 523, + 186, + 537 + ], + "score": 1.0, + "content": "and vary", + "type": "text" + }, + { + "bbox": [ + 187, + 524, + 194, + 534 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "in the range of [0.001, 0.01]. In Fig. 10(b) we present the accuracy and sparsity", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 157, + 547 + ], + "score": 1.0, + "content": "level vs. the", + "type": "text" + }, + { + "bbox": [ + 157, + 536, + 164, + 545 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "parameter. 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The experiment was repeated 10 times,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "the extracted features and accuracies were consistent over 20 trials. We noticed a relatively small", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 468, + 507, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 507, + 483 + ], + "score": 1.0, + "content": "number of selected features, which are positioned southwest and close to the center of the images,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "achieve very high classification accuracy. An example of 9 randomly selected samples overlaid with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "the weights of the selected features is presented in Fig. 10(a). Furthermore, we also evaluate the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 142, + 515 + ], + "score": 1.0, + "content": "effect of", + "type": "text" + }, + { + "bbox": [ + 142, + 502, + 149, + 512 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "on the the number of selected features and accuracy of the method. 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This example demonstrates that our method is also effective in an extremely high", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 587, + 323, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 323, + 599 + ], + "score": 1.0, + "content": "dimensional regime of non linear function estimation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 106, + 614, + 348, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 349, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 349, + 626 + ], + "score": 1.0, + "content": "J.6 DETAILS OF COX PROPORTIONAL HAZARD MODEL", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 506, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "score": 1.0, + "content": "Survival times are assumed to follow a distribution, which is characterized by the survival function", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 107, + 646, + 178, + 659 + ], + "score": 0.91, + "content": "S ( t ) = P ( T > t )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 646, + 505, + 659 + ], + "score": 1.0, + "content": ". A hazard function, which measures the instantaneous rate of death, is defined by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 656, + 507, + 676 + ], + "spans": [ + { + "bbox": [ + 107, + 657, + 279, + 675 + ], + "score": 0.91, + "content": "\\begin{array} { r } { h ( t ) = \\operatorname* { l i m } _ { \\Delta t 0 } \\frac { P ( t < T \\leq t + \\Delta t | T > t ) } { \\Delta t } = \\frac { p ( t ) } { S ( t ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 656, + 507, + 676 + ], + "score": 1.0, + "content": ". 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This is a multilable regime, i.e. each story is assigned to multiple categories. Here we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "focus on a binary subset of this corpus, with 23, 203 stories. 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We evaluate the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 565, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 576 + ], + "score": 1.0, + "content": "performance of our method using a 5-fold cross validation. A comparison the LASSO and RF", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 576, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 589 + ], + "score": 1.0, + "content": "appears in Fig. 12. This example demonstrates that our method is also effective in an extremely high", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 587, + 323, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 323, + 599 + ], + "score": 1.0, + "content": "dimensional regime of non linear function estimation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 519, + 506, + 599 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 614, + 348, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 349, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 349, + 626 + ], + "score": 1.0, + "content": "J.6 DETAILS OF COX PROPORTIONAL HAZARD MODEL", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 506, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "score": 1.0, + "content": "Survival times are assumed to follow a distribution, which is characterized by the survival function", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 107, + 646, + 178, + 659 + ], + "score": 0.91, + "content": "S ( t ) = P ( T > t )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 646, + 505, + 659 + ], + "score": 1.0, + "content": ". 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The total number of feature is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 206, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 506, + 220 + ], + "score": 1.0, + "content": "47, 236. Here we compare the performance of the proposed approach to RF and LASSO. The lines", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 218, + 474, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 474, + 230 + ], + "score": 1.0, + "content": "represent a least squares polynomial fit plot of the accuracy vs. number of selected features.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + } + ], + "index": 6.25 + }, + { + "type": "text", + "bbox": [ + 106, + 235, + 504, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 132, + 247 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 235, + 155, + 248 + ], + "score": 0.94, + "content": "h _ { 0 } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "is a baseline hazard function, which is often the exponential or Weibull distribution, and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 245, + 231, + 259 + ], + "spans": [ + { + "bbox": [ + 107, + 247, + 114, + 257 + ], + "score": 0.77, + "content": "\\pmb \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 245, + 231, + 259 + ], + "score": 1.0, + "content": "is the parameter of interests.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 249, + 275 + ], + "score": 1.0, + "content": "One of the difficulties in estimating", + "type": "text" + }, + { + "bbox": [ + 249, + 264, + 257, + 273 + ], + "score": 0.71, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "in survival analysis is that a large portion of the available data", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "is censored. 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Most genes are irrelevant for outcome", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 431 + ], + "score": 1.0, + "content": "prediction. To demonstrate the advantage of STG in the context of survival analysis, we selected 16", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "well-known genes relevant for survival (out of the 24,368 genes) that correspond to the Oncotype", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "DX test, a gene panel used for treatment decision making. We also include five additional clinical", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "features (hormone treatment indicator, radiotherapy indicator, chemotherapy indicator, ER-positive", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "indicator, and age at diagnosis). We then added 200 additional irrelevant gene variables that we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 473, + 314, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 314, + 487 + ], + "score": 1.0, + "content": "selected randomly from the remaining list of genes.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 418, + 502 + ], + "score": 1.0, + "content": "After we omit the null values, the number of samples is 1969. 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Validation is always", + "type": "text" + }, + { + "bbox": [ + 458, + 687, + 478, + 698 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "train, while the exact ratios between train and test is detailed for each experiment separately. 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The total number of feature is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 206, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 506, + 220 + ], + "score": 1.0, + "content": "47, 236. Here we compare the performance of the proposed approach to RF and LASSO. 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However, in order to obtain estimates, Cox observed that it is sufficient to maximize the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 284, + 442, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 294, + 308 + ], + "score": 1.0, + "content": "partial-likelihood, which is defined as follows:", + "type": "text" + }, + { + "bbox": [ + 294, + 284, + 442, + 316 + ], + "score": 0.78, + "content": "\\begin{array} { r } { L ( \\pmb \\theta ) = \\prod _ { T _ { i } \\mathrm { u n c e n s o r e d } } \\frac { e ^ { \\pmb \\theta ^ { T } \\pmb x _ { i } } } { \\sum _ { T _ { j } \\geq T _ { i } } e ^ { \\pmb \\theta ^ { T } \\pmb x _ { j } } } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 263, + 505, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 319, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "In (Katzman et al., 2018), the authors propose DeepSurv, which uses a deep neural network model to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 331, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 304, + 342 + ], + "score": 1.0, + "content": "replace the linear relations between the covariate", + "type": "text" + }, + { + "bbox": [ + 304, + 332, + 312, + 340 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 331, + 330, + 342 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 331, + 337, + 340 + ], + "score": 0.7, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 331, + 505, + 342 + ], + "score": 1.0, + "content": ", demonstrating improvements of survival", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 495, + 353 + ], + "score": 1.0, + "content": "time prediction over existing models such as CPH and the random survival forest (Ishwaran", + "type": "text" + }, + { + "bbox": [ + 496, + 342, + 505, + 351 + ], + "score": 0.33, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 353, + 267, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 267, + 365 + ], + "score": 1.0, + "content": "Kogalur, 2007), (Ishwaran et al., 2008).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 318, + 505, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 369, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "The Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) dataset consists", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 377, + 392 + ], + "score": 1.0, + "content": "of gene expression data and clinical features for 1, 980 patients, and", + "type": "text" + }, + { + "bbox": [ + 377, + 380, + 410, + 391 + ], + "score": 0.89, + "content": "5 7 . 7 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "have an observed death", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 391, + 364, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 364, + 403 + ], + "score": 1.0, + "content": "due to breast cancer with a median survival time of 116 months.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 369, + 505, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "The METABRIC dataset involves 24,368 features (genes). 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We then added 200 additional irrelevant gene variables that we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 473, + 314, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 314, + 487 + ], + "score": 1.0, + "content": "selected randomly from the remaining list of genes.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 407, + 505, + 487 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 418, + 502 + ], + "score": 1.0, + "content": "After we omit the null values, the number of samples is 1969. We reserve the", + "type": "text" + }, + { + "bbox": [ + 418, + 491, + 438, + 501 + ], + "score": 0.88, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "for test, and use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 122, + 515 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 502, + 142, + 513 + ], + "score": 0.88, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "of the remaining training set as validation. 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The learning rate and the regularization parameter are optimized via Optuna using the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 277, + 582 + ], + "score": 1.0, + "content": "validation set, where the search range is set", + "type": "text" + }, + { + "bbox": [ + 277, + 570, + 342, + 582 + ], + "score": 0.92, + "content": "L R : [ 1 e - 3 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 570, + 359, + 582 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 360, + 570, + 415, + 582 + ], + "score": 0.91, + "content": "\\lambda : [ 1 e - 3 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 570, + 505, + 582 + ], + "score": 1.0, + "content": ". The hyperparameters", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "used in the experiment are the following: architecture :[60, 20, 3], activation: selu (as suggested by", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 590, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 395, + 605 + ], + "score": 1.0, + "content": "(Katzman et al., 2018)), learning rate : 0.152, learning rate decay: 1.0,", + "type": "text" + }, + { + "bbox": [ + 396, + 592, + 424, + 603 + ], + "score": 0.73, + "content": "\\sigma : 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 590, + 428, + 605 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 429, + 592, + 467, + 603 + ], + "score": 0.61, + "content": "\\lambda : 0 . 0 2 3", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 590, + 505, + 605 + ], + "score": 1.0, + "content": ", training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "epoch: 2000. Note that to see the effect of feature selection, we used the hyperparameters optimized", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 613, + 311, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 311, + 626 + ], + "score": 1.0, + "content": "for DeepSurv to test our method (STG-DeepSurv).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 537, + 506, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 639, + 296, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 297, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 297, + 651 + ], + "score": 1.0, + "content": "J.7 ADDITIONAL EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "Here we provide a full description of the procedures we have performed in the experimental parts of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 149, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 149, + 685 + ], + "score": 1.0, + "content": "the paper.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 659, + 506, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 458, + 700 + ], + "score": 1.0, + "content": "For synthetic datasets are first split into train, validation and test. Validation is always", + "type": "text" + }, + { + "bbox": [ + 458, + 687, + 478, + 698 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "train, while the exact ratios between train and test is detailed for each experiment separately. 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We set n-trials", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 458, + 83, + 490, + 93 + ], + "score": 0.83, + "content": "= 1 0 0 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 490, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "for", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "Optuna, which is a define-by-run based hyperparameter optimization software equipped with efficient", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "search and pruning strategies. We use SGD for all the experiments, except for the Cox model where", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 465, + 128 + ], + "score": 1.0, + "content": "we use Adam. 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We set n-trials", + "type": "text" + }, + { + "bbox": [ + 458, + 83, + 490, + 93 + ], + "score": 0.83, + "content": "= 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "Optuna, which is a define-by-run based hyperparameter optimization software equipped with efficient", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "search and pruning strategies. 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ParamSearch range
# dense layers #hidden units activation LR n-epoch (DFS, SG-L1-NN) α (SG-L1-NN) 入 (SG-L1-NN) 入 (STG,DFS) 入 (LASSO) n-est (RF, XGBoost, Tree) n-boost-round (XGBoost) Thresh (RF,XGBoost, Tree) max-depth (XGBoost) c (SVC)[1,3] [10,500] [tanh, relu, sigmoid] [le-4, 1e-1] [50,20000] [1e-3,1] [1e-7, 1] [1e-3, 1] [0.01, 1] [5,100] [1,100] [0.01,0.5] [0.01,0.5] [1e-7, 1]
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For all the experiments when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "we use Tree and RF, we use the default value for max-depth, so that nodes are expanded based", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "on the purity. For the XOR problem, the exact architectures used for the NN based methods", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "are: (STG/HC): [476, 490, 14] with Tanh, (DFS): [100, 10] with Tanh, (SG-L1-NN): [100, 10, 5]", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "with Tanh. For the two moons we use (STG): [490, 406, 18] with Tanh, (DFS): [158, 27, 224] with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "Tanh, (SG-L1-NN): [88, 28, 27] with Tanh. For the XOR problem, we attempted to use Optuna to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "optimize parameters of DFS and SG-L1-NN, but we ended up using the architecture suggested by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "the authors (Li et al., 2016; Scardapane et al., 2017) as they outperform the values suggested by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 363, + 469 + ], + "score": 1.0, + "content": "Optuna. The number of epochs used for the XOR problem is", + "type": "text" + }, + { + "bbox": [ + 363, + 457, + 383, + 467 + ], + "score": 0.4, + "content": "2 0 K", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 456, + 506, + 469 + ], + "score": 1.0, + "content": ", 14K, 800 for STG/HC, DFS", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 348, + 480 + ], + "score": 1.0, + "content": "and SG-L1-NN respectively. Regularization parameters are", + "type": "text" + }, + { + "bbox": [ + 348, + 468, + 406, + 478 + ], + "score": 0.83, + "content": "0 . 1 7 , 3 . 3 e \\mathrm { ~ - ~ } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 468, + 424, + 480 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 425, + 468, + 453, + 478 + ], + "score": 0.82, + "content": "3 e - 5", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 345, + 491 + ], + "score": 1.0, + "content": "The number of epochs used for the two-moons problem is", + "type": "text" + }, + { + "bbox": [ + 345, + 479, + 366, + 489 + ], + "score": 0.57, + "content": "2 0 K", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 477, + 506, + 491 + ], + "score": 1.0, + "content": ", 1570, 708 for STG/HC, DFS and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 335, + 502 + ], + "score": 1.0, + "content": "SG-L1-NN respectively. Regularization parameters are", + "type": "text" + }, + { + "bbox": [ + 335, + 490, + 386, + 501 + ], + "score": 0.83, + "content": "0 . 4 8 , 9 e \\mathrm { ~ - ~ } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 489, + 404, + 502 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 405, + 490, + 433, + 500 + ], + "score": 0.75, + "content": "1 e - 3", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "respectively. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "note that the regularization parameters and learning procedure is different in nature, as we use an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 107, + 511, + 117, + 523 + ], + "score": 0.86, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "type penalty. For the PBMC experiment, the architecture was hand-tuned using the validation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "set and set as [200, 100, 50, 10] with Tanh. 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The number of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 168, + 568 + ], + "score": 1.0, + "content": "epochs used is", + "type": "text" + }, + { + "bbox": [ + 169, + 555, + 189, + 565 + ], + "score": 0.85, + "content": "2 0 K", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 555, + 311, + 568 + ], + "score": 1.0, + "content": ", the learning rate is 0.06 and", + "type": "text" + }, + { + "bbox": [ + 311, + 556, + 318, + 565 + ], + "score": 0.74, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "varies in the range [0.01, 10]. 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ParamSearch range
# dense layers #hidden units activation LR n-epoch (DFS, SG-L1-NN) α (SG-L1-NN) 入 (SG-L1-NN) 入 (STG,DFS) 入 (LASSO) n-est (RF, XGBoost, Tree) n-boost-round (XGBoost) Thresh (RF,XGBoost, Tree) max-depth (XGBoost) c (SVC)[1,3] [10,500] [tanh, relu, sigmoid] [le-4, 1e-1] [50,20000] [1e-3,1] [1e-7, 1] [1e-3, 1] [0.01, 1] [5,100] [1,100] [0.01,0.5] [0.01,0.5] [1e-7, 1]
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For all the experiments when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "we use Tree and RF, we use the default value for max-depth, so that nodes are expanded based", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "on the purity. For the XOR problem, the exact architectures used for the NN based methods", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "are: (STG/HC): [476, 490, 14] with Tanh, (DFS): [100, 10] with Tanh, (SG-L1-NN): [100, 10, 5]", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "with Tanh. For the two moons we use (STG): [490, 406, 18] with Tanh, (DFS): [158, 27, 224] with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "Tanh, (SG-L1-NN): [88, 28, 27] with Tanh. For the XOR problem, we attempted to use Optuna to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "optimize parameters of DFS and SG-L1-NN, but we ended up using the architecture suggested by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "the authors (Li et al., 2016; Scardapane et al., 2017) as they outperform the values suggested by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 363, + 469 + ], + "score": 1.0, + "content": "Optuna. The number of epochs used for the XOR problem is", + "type": "text" + }, + { + "bbox": [ + 363, + 457, + 383, + 467 + ], + "score": 0.4, + "content": "2 0 K", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 456, + 506, + 469 + ], + "score": 1.0, + "content": ", 14K, 800 for STG/HC, DFS", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 348, + 480 + ], + "score": 1.0, + "content": "and SG-L1-NN respectively. Regularization parameters are", + "type": "text" + }, + { + "bbox": [ + 348, + 468, + 406, + 478 + ], + "score": 0.83, + "content": "0 . 1 7 , 3 . 3 e \\mathrm { ~ - ~ } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 468, + 424, + 480 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 425, + 468, + 453, + 478 + ], + "score": 0.82, + "content": "3 e - 5", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 345, + 491 + ], + "score": 1.0, + "content": "The number of epochs used for the two-moons problem is", + "type": "text" + }, + { + "bbox": [ + 345, + 479, + 366, + 489 + ], + "score": 0.57, + "content": "2 0 K", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 477, + 506, + 491 + ], + "score": 1.0, + "content": ", 1570, 708 for STG/HC, DFS and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 335, + 502 + ], + "score": 1.0, + "content": "SG-L1-NN respectively. 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RAI0.43 (0.02)0.48 (0.003)0.47(0.002)0.41(0.02)0.44 (0.01)0.41 (0.01)0.39 (0.01)
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ParamSearch range
# dense layers #hidden units activation LR n-epoch (DFS, SG-L1-NN) α (SG-L1-NN) 入 (SG-L1-NN) 入 (STG,DFS) 入 (LASSO) n-est (RF, XGBoost, Tree) n-boost-round (XGBoost) Thresh (RF,XGBoost, Tree) max-depth (XGBoost) c (SVC)[1,3] [10,500] [tanh, relu, sigmoid] [le-4, 1e-1] [50,20000] [1e-3,1] [1e-7, 1] [1e-3, 1] [0.01, 1] [5,100] [1,100] [0.01,0.5] [0.01,0.5] [1e-7, 1]
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sha256:0a3c2bcfc783de066386c95cd40e6607376387fcd1f7690ef0ebdfe874460eaa +size 3614 diff --git a/parse/train/BysZhEqee/BysZhEqee.md b/parse/train/BysZhEqee/BysZhEqee.md new file mode 100644 index 0000000000000000000000000000000000000000..fc8a1bae67669bbf39959204b22057a72c7ffd8a --- /dev/null +++ b/parse/train/BysZhEqee/BysZhEqee.md @@ -0,0 +1,277 @@ +# MARGINAL DEEP ARCHITECTURES: DEEP LEARNING FOR SMALL AND MIDDLE SCALE APPLICATIONS + +Yuchen Zheng, Guoqiang Zhong & Junyu Dong + +Department of Computer Science and Technology Ocean University of China ouczyc@outlook.com,{gqzhong, dongjunyu}@ouc.edu.cn + +# ABSTRACT + +In recent years, many deep architectures have been proposed in different fields. However, to obtain good results, most of the previous deep models need a large number of training data. In this paper, for small and middle scale applications, we propose a novel deep learning framework based on stacked feature learning models. Particularly, we stack marginal Fisher analysis (MFA) layer by layer for the initialization of the deep architecture and call it “Marginal Deep Architectures” (MDA). In the implementation of MDA, the weight matrices of MFA are first learned layer by layer, and then we exploit some deep learning techniques, such as back propagation, dropout and denoising to fine tune the network. To evaluate the effectiveness of MDA, we have compared it with some feature learning methods and deep learning models on 7 small and middle scale real-world applications, including handwritten digits recognition, speech recognition, historical document understanding, image classification, action recognition and so on. Extensive experiments demonstrate that MDA performs not only better than shallow feature learning models, but also state-of-the-art deep learning models in these applications. + +# 1 INTRODUCTION + +Deep learning methods have achieved desirable performance in many domains, such as image classification and detection, document analysis and recognition, natural language processing, video analysis (Krizhevsky et al., 2012; Chan et al., 2014; Ciresan et al., 2010; Collobert & Weston, 2008; Le et al., 2011). Deep learning methods learn the data representation by using multiple processing layers, which discover the intricate structure of high dimensional data with multiple levels of abstraction (LeCun et al., 2015). For example, for face recognition, the learned features of first layer may be the edges, directions and some local information. The second layer typically detects some object parts which are combination of edges and directions. The higher layers may further abstract the face image by combining the features of previous layers (outline of the eyes, nose, lips). This procedure is very similar with human visual and perceptual system. + +In recently years, many deep learning methods have been proposed (l. Boureau & others, 2008; Lee et al., 2009b;a; Hinton & Salakhutdinov, 2006). However, most models meet some difficult problems to solve, such as some parameters need to be randomly initialized, like the weight matrix of two successive layers in deep belief networks (DBNs) and the convolution kernel in convolutional neural networks (CNNs). In addition, traditional deep learning methods need a large scale training data to train the complex networks. It causes many problems in the training process. If we don’t initialize the parameters properly, the optimization procedure might need a long training time and fall into local minima. Alternatively, many feature learning models have been proposed to learn the intrinsic structures of high-dimensional data and avoid the curse of dimensionality. In particular, most of them can be trained with small and middle scale of data and their learning algorithms are generally based on closed-form solution or convex optimization. For instance, marginal Fisher analysis (MFA) (Yan et al., 2007; Zhong et al., 2013) is one of the feature learning models that is a supervised method based on the graph embedding framework. It utilizes an intrinsic graph to characterize the intraclass compactness, and another penalty graph to characterize the interclass separability. Its optimal solution can be learned by generalized eigenvalue decomposition. However, on the one hand, shallow feature learning models cannot work well on the data with highly nonlinear structure; on the other hand, few efforts are made to combine shallow feature learning models for the design of deep architectures. + +In order to simultaneously solve the existing problems in deep learning methods and combine the advantages of feature learning models, we proposed a novel deep learning method based on stacked feature learning models. Particularly, we stack marginal Fisher analysis (MFA) layer by layer for the initialization of the deep architecture and call it “Marginal Deep Architectures” (MDA). Firstly, the input data are mapped to higher dimensional space by using random weight matrix. Then we use MFA to learn the lower dimensional representation layer by layer. In the implementation of this architecture, we add some tricks in the training process, such as back propagation, dropout and denoising to fine tune the network. Finally, the softmax layer is connected to the last feature layer. We have compared our MDA with some feature learning methods and deep learning models on different domains of datasets (including handwritten digits recognition, speech recognition, historical document understanding, image classification, action recognition and so on). Extensive experiments demonstrate that MDA performs not only better than shallow feature learning models, but also state-of-the-art deep learning models in small and middle scale applications. + +The contributions of this work are highlighted as follows. + +1. We propose a novel structure to build a deep architecture. The first hidden layer has twice or quadruple neurons as the input layer. Then we can use some feature learning models layer by layer to learn the compact representations of data. Finally, we set the last layer as a softmax classifier. + +2. Traditional deep learning models in general need a large scale training data. Compared with traditional deep learning models, MDA can work better than traditional deep learning models in small and middle scale applications because the initialization of the weight matrices using MFA is much better than that using random initialization. + +3. Our MDA can work well in different domains of datasets, such as handwritten digits, spoken letters and natural images. Extensive experiments demonstrate that MDA is a general model to handel small and middle scale data. On the other hand, for large scale datasets, like CIFAR-10, MDA works comparatively with other deep learning methods. + +The rest of this paper is organized as follows: In Section 2, we give a brief overview of related work. In Section 3, we present the marginal Fisher analysis (MFA) and the proposed marginal deep architectures (MDA) in detail. The experimental settings and results are reported in Section 4, while Section 5 concludes this paper with remarks and future work. + +# 2 RELATED WORK + +With the development of deep learning methods, many deep networks have been proposed in recent years (Donahue et al., 2013; Krizhevsky et al., 2012; Long et al., 2015; Zhou et al., 2014). These deep learning models show their powerful performance in various fields, such as image classification and analysis, document analysis and recognition, natural language processing et al. In the area of image analysis, Hinton et al. proposed a large, deep convolutional neural network (Alex net) to classify the 1.2 million high-resolution images in the ImageNet. It uses efficient GPU to speed their method. The results show that a large, deep convolutional neural network is capable of achieving recordbreaking results on a highly challenging dataset using purely supervised learning (Krizhevsky et al., 2012). In order to popularize the deep convolutional neural network, Donahue ea al. proposed DeCAF (Deep Convolutional Activation Feature) which is trained in a fully supervised fashion on a large, fixed set of object recognition tasks (Donahue et al., 2013). DeCAF provides a uniform framework for researchers who can improve and change this framework on some specific tasks. However, its performance at scene recognition has not attained the same level of success. In order to handle this problem, Zhou et al. introduce a new scene-centric database called Places with over 7 million labeled pictures of scenes. Then, they learn the deep features for scene recognition tasks by using the same architecture as ImageNet, and establish new state-of-the-art results on several scenecentric datasets (Zhou et al., 2014). However, these methods based on convolutional operation need very large scale training samples and a long training time. They can not work well on small and middle scale applications. + +In other domains, deep learning methods also achieve good performance. Hinton et al. represent the shared views of four research groups that have had recent successes in using DNNs for automatic speech recognition (ASR). The DNNs that contain many layers of nonlinear hidden units and a very large output layer can outperform Gaussian mixture models (GMMs) at acoustic modeling for speech recognition on a variety of data sets (Hinton et al., 2012a). In the area of genetics, Xiong et al. use “deep learning” computer algorithms to derive a computational model that takes as input DNA sequences and applies general rules to predict splicing in human tissues (Xiong et al., 2015). It reveals the genetic origins of disease and how strongly genetic variants affect RNA splicing. In the area of natural language understanding, deep learning models have delivered strong results on topic classification, sentiment analysis et al. Sutskever et al. proposed a general approach, the Long Short-Term Memory (LSTM) architecture which can solve the general sequence to sequence problems better than before (Sutskever et al., 2014). In addition, Hinton et al. proposed autoencoder (AE) networks that is an effective way to learn the low-dimensional codes of high-dimensional data. Based on autoencoder, there are also have many excellent works to handle various tasks. Vincent et al. proposed a denoising autoencoder (DAE) which maked the learned representations robust to partial corruption of the input data (Vincent et al., 2008). The denoising autoencoder which initialize the deep architectures layer by layer is very similar with human visual system. Hinton et al. introduced random ‘dropout’ to prevent the overfitting which improve many benchmark tasks and obtain new records for speech and object recognition (Hinton et al., 2012b). Then, Vincent et al. proposed stacked denoising autoencoders (SDAE) which based on stacking layers of stacked denoising autoencoders (Vincent et al., 2010). It is very useful to learn the higher level representations and work well on natural images and handwritten digits. However, for the same reason, they also need a large scale training set and a long training time. They have no advantages to handle the small and middle scale applications. + +Moreover, in the field of feature learning models, dimensionality reduction plays a crucial role to handle the problems for compressing, visualizing high-dimensional data and avoiding the “curse of dimensionality” (van der Maaten et al., 2009; van der Maaten, 2007). Traditional dimensionality reduction mainly can be classified into three types: linear or nonlinear, like principal components analysis (PCA) (Jolliffe, 2002) and linearity preserving projection (LPP) (Niyogi, 2004) are linear methods, stochastic neighbor embedding (SNE) (Hinton & Roweis, 2002) is a nonlinear method; supervised or unsupervised, such as marginal Fisher analysis (MFA) (Yan et al., 2007; Zhong et al., 2013) and linear discriminant analysis (LDA) (Fisher, 1936) are supervised methods, PCA is an unsupervised method; local or global, like MFA and SNE are local methods, PCA is a global method. Many feature learning models based on geometry theory provide different solutions to the problem of dimensionality reduction. Yan et al. proposed a general formulation about graph embedding framework can exploit new dimensionality reduction algorithms (Yan et al., 2007). If only directly use some feature learning models to extract the good representation from original data, it often eventually couldn’t get a good outcome. Considering this situation, we try to choose some excellent feature learning models and combine them with some deep learning algorithms. MFA is one special formulation of the graph embedding models based on this framework. It utilizes an intrinsic graph to characterize the intraclass compactness, and another penalty graph to characterize the interclass separability. Our motivation is to combine the advantage of MFA and deep architectures and propose a new initialization method for deep learning algorithms. + +There are also have some excellent works about feature learning models combined the deep architectures (Yuan et al.; George et al., 2014; Ngiam et al., 2011). Yuan et al. proposed an improved multilayer learning model to solve the scene recognition task (Yuan et al.). This model overcome the limitation of shallow, one-layer representations for scene recognition. Trigeorgis et al proposed deep Semi-NMF, that is able to learn such hidden representations from different, unknown attributes of a given dataset (George et al., 2014). Ngiam proposed a deep architectures to learn features over multiple modalities (Ngiam et al., 2011). They showed that multi-modality feature learning is better than one modality and achieved good performance on video and audio datasets. However, in general, we can only obtain data from one modality. In this work, we combine the advantages of MFA and deep architectures, which based on stacked feature learning models (Zheng et al., 2014; 2015), then we use some deep learning tricks, like back propagation, denoising and dropout to fine tuning the network. The advantage of this deep architecture is that we can learn the desirable weight matrix even if the training data is not large enough. And compared with traditional deep learning models and shallow feature learning models, our MDA achieved state-of-the-art results in most cases. + +![](images/c274b80803e2cd5bc8944e1881ad315fd04a306b1afaf87b2cfcee74878a65ee.jpg) +Figure 1: The brief representation for MDA. $\mathbf { W } _ { r _ { 1 } }$ represents the first layer random weight matrix, ${ \bf W } _ { M F A _ { 2 } }$ and ${ \bf W } _ { M F A _ { 3 } }$ represent the weight matrixes learned by MFA. The dotted red lines represent the dropout operation, the dotted red circle is the ‘dropout’ node, and the cross nodes are corrupted. The denoising and dropout operation are completely random. For simplicity, we have omitted bias terms. + +# 3 MARGINAL DEEP ARCHITECTURES (MDA) + +In this section, we firstly introduce a novel framework of deep architectures, then we introduce marginal Fisher analysis (MFA) and the proposed marginal deep architectures (MDA) in detail. In addition, we also present some deep learning tricks that we used in the MDA model, including back propagation, denoising and dropout. + +# 3.1 A NOVEL FRAMEWORK OF DEEP ARCHITECTURES + +The feature learning problem is generally formulated as follow. Given $n$ data, $\{ \mathbf { x } _ { 1 } ^ { T } , \hdots , \mathbf { x } _ { n } ^ { T } \} \in \Re ^ { D }$ , where $D$ is the dimensionality of the data space, we seeks the compact representations of these data, i.e., $\{ \mathbf { y } _ { 1 } ^ { T } , \ldots , \mathbf { y } _ { n } ^ { T } \} \in \Re ^ { d }$ , where $d$ is the dimensionality of the low dimensional embeddings. + +In order to improve the accuracy of shallow feature learning models, we use stacked feature learning models to construct the deep architectures (Zheng et al., 2014; 2015), which is a general framework for different applications. In this case, the mapping of data from the original $D$ -dimensional space to the resulted $d$ -dimensional space can be described as + +$$ +D \Longrightarrow D _ { 1 } \Longrightarrow \cdots \Longrightarrow D _ { i } \Longrightarrow \cdots \Longrightarrow D _ { p - 1 } \Longrightarrow d , +$$ + +where $D _ { 1 }$ is the first higher dimensional space, the number of the node is twice or quadruple as the input layer. $D _ { i }$ represents the dimensionality of the $i$ -th intermediate representation space, and $p$ is the total steps of mappings. Here, we can use different feature learning models for the learning of each layer. As the feature learning models are optimized layer by layer, we can obtain the mapping functions between successive layers. The first hidden layer is random by $\mathbf { W } _ { r 1 }$ , and the representation is, + +$$ +\mathbf { a } ^ { 1 } = g ( \mathbf { W } _ { r _ { 1 } } ^ { T } \mathbf { x } + \mathbf { b } ) +$$ + +where, $g ( . )$ is a non-linear activation or transfer function. Then, we can use some feature learning models to initialize the next layers. The representations of next hidden layers are, + +$$ +\mathbf { a } ^ { k } = g ( \mathbf { W } _ { F _ { k - 1 } } ^ { T } \mathbf { a } ^ { k - 1 } + \mathbf { b } ) +$$ + +where, $\mathbf { W } _ { F _ { k - 1 } }$ is the weight matrix of the $k - 1$ th layer learned from feature learning models. + +# 3.2 MARGINAL FISHER ANALYSIS (MFA) + +Based on our novel framework of deep architecture, we introduce Marginal Fisher Analysis (MFA) to build MDA. Here, many traditional feature learning models, such as linear discriminant analysis (LDA), can be used as building blocks of MDA. Take LDA as an example. It assumes that the data of each class follow a Gaussian distribution. However, this assumption is not often satisfied in the real world. Without this assumption, LDA can not work well to separate the data with nonlinear structure. Alternatively, MFA can solve this problem effectively. Hence, considering the learning capability, we choose MFA as the build blocks of MDA in our work. MFA used the graph embedding framework to set up an intrinsic graph that characterizes the intraclass compactness and another penalty graph which characterizes the interclass separability. The marginal Fisher criterion is defined as + +$$ +\mathbf { W } ^ { * } = \operatorname * { a r g m i n } _ { \mathbf { W } } \frac { \operatorname { t r } ( \mathbf { W } ^ { T } \mathbf { X } ( \mathbf { D } - \mathbf { A } ) \mathbf { X } ^ { T } \mathbf { W } ) } { \operatorname { t r } ( \mathbf { W } ^ { T } \mathbf { X } ( \mathbf { D } ^ { p } - \mathbf { A } ^ { p } ) \mathbf { X } ^ { T } \mathbf { W } ) } +$$ + +where $\mathbf { D }$ and $\mathbf { D } ^ { p }$ are diagonal matrices with elements $\begin{array} { r } { \mathbf { D } _ { i i } = \sum _ { j } \mathbf { A } _ { i j } } \end{array}$ , and $\begin{array} { r } { \mathbf { D } _ { i j } ^ { p } = \sum _ { j } \mathbf { A } _ { i j } ^ { p } } \end{array}$ , respectively. Then we can learn the projection matrix to multiply PCA’s projection and marginal Fisher projection, + +$$ +\mathbf { W } _ { M F A } = \mathbf { W } _ { P C A } \mathbf { W } ^ { * } +$$ + +# 3.3 MARGINAL DEEP ARCHITECTURES (MDA) + +In order to combine the advantages of MFA and proposed deep architectures, we propose the marginal deep architectures (or MDA). The MDA inherited from the proposed novel framework of deep architectures is shown in Fig. 1. As an input vector $\mathbf { x } \in [ 0 , 1 \bar { ] } ^ { d }$ , we first map it to higher dimensional space by a random weight matrix $\mathbf { W } _ { r 1 }$ . The representation of first hidden layer is computed as + +$$ +\mathbf { a } ^ { 1 } = s ( \mathbf { W } _ { r _ { 1 } } ^ { T } \mathbf { x } + \mathbf { b } ) +$$ + +where, $s ( . )$ is the sigmoid function $\begin{array} { r } { s ( x ) = \frac { 1 } { 1 + e ^ { - x } } } \end{array}$ , $\mathbf { b }$ is the bias terms, $\mathbf { a } ^ { 1 }$ is the output of first layer. From second layer to $( n - 1 )$ -th layer, we use the weight matrices learned from MFA to map layer by layer. + +$$ +\mathbf { a } ^ { k } = s ( \mathbf { W } _ { M F A _ { k - 1 } } ^ { T } \mathbf { a } ^ { k - 1 } + \mathbf { b } ) +$$ + +The last layer is a softmax regression layer and the number of neuron is the number of category. The cost function is defined as, + +$$ +J ( \mathbf { w } ) = - \frac { 1 } { N } ( \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { K } \mathbf { I } ( y _ { i } = j ) \log \frac { \exp ( \mathbf { w } _ { j } ^ { T } \mathbf { a } _ { i } ^ { n - 1 } ) } { \sum _ { l = 1 } ^ { K } \exp ( \mathbf { w } _ { l } ^ { T } \mathbf { a } _ { i } ^ { n - 1 } ) } ) +$$ + +where, $\mathbf { I } ( x )$ is the indicator function, $\mathbf { I } ( x ) = 1$ if $x$ is true, else $\mathbf { I } ( x ) = 0$ . $y _ { i }$ is the label corresponding to $\mathbf { x } _ { i }$ . Then the probability that $\mathbf { x } _ { i }$ is classified to $j$ is, + +$$ +p ( y _ { i } = j | \mathbf { x } _ { i } , \mathbf { w } ) = \frac { \exp ( \mathbf { w } _ { j } ^ { T } \mathbf { a } _ { i } ^ { n - 1 } ) } { \sum _ { l = 1 } ^ { K } \exp ( \mathbf { w } _ { l } ^ { T } \mathbf { a } _ { i } ^ { n - 1 } ) } +$$ + +Taking derivatives, one can show that the gradient is, + +$$ +\nabla J ( \mathbf { w } ) = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } [ \mathbf { x } _ { i } ( \mathbf { I } ( y _ { i } = j ) - p ( y _ { i } = j | \mathbf { x } _ { i } , \mathbf { w } ) ] +$$ + +If the $n { - } 1$ layer’s neurons are more than the last layer, we can continue using MFA to map it. On the contrary, If the $n - 1$ layer’s neurons are less than last layer, we can randomly initialize the weight matrix between this two layers. Next, in order to improve the MDA, we introduce back propagation, denoising and dropout operation. + +# 3.4 BACK PROPAGATION + +In order to adjust the network, we use back propagation (Rumelhart et al., 1986) to compute partial derivative and stochastic gradient descent to update the weight matrixes and the bias terms. For each node $i$ in output layer $n$ -th layer), we compute an error term as + +$$ +\delta _ { i } ^ { n } = \nabla J ( \mathbf { w } ) +$$ + +where, $J ( \mathbf { w } )$ is the cost function computed from Equ.8 and $\nabla J ( \mathbf { w } )$ computed from Equ.10. For each node $i$ in $( n - 1 )$ -th to second layer, the error term is computed as, + +$$ +\delta _ { i } ^ { k } = ( \sum _ { j = 1 } ^ { k + 1 } w _ { j i } ^ { k } \delta _ { j } ^ { k + 1 } ) s ^ { \prime } ( z _ { i } ^ { k } ) +$$ + +The back propagation procedure relies on computing the gradient of an objective function with respect to the weights of a multilayer stacked modules. It starting from the output at the top and end to the input at the bottom. + +# 3.5 DENOISING OPERATION + +Vincent et al. proposed the denoising autoencoder to improve the robustness of autoencoder (Vincent et al., 2008). It’s very similar with the regularization methods and avoids the “overfitting” problem. The basic idea is to corrupt partial input data by the desired proportion of $\nu$ “destruction”. for each input $\mathbf { x }$ , a fixed number $\nu d$ of components are chosen at random, and their value is forced to 0, while the others are left untouched. The initial input $\mathbf { x }$ to get a partially destroyed version $\tilde { \mathbf { x } }$ by means of a stochastic mapping, + +$$ +\tilde { \mathbf { x } } \sim q _ { D } ( \tilde { \mathbf { x } } | \mathbf { x } ) +$$ + +where, $q _ { D } ( \tilde { \mathbf { x } } | \mathbf { x } )$ is the unknown distribution. Then, for a hidden representation $\mathbf { h }$ , + +$$ +\mathbf { h } = s ( \mathbf { W } ^ { T } \widetilde { \mathbf { x } } + \mathbf { b } ) +$$ + +In our MDA, we use this idea to improve the network, please refer to Fig. 1 to find clear sight. For the input layer, the output of first hidden layer is represented as, + +$$ +\mathbf { a } ^ { 2 } = s ( \mathbf { W } _ { r _ { 1 } } ^ { T } \tilde { \mathbf { x } } + \mathbf { b } _ { 1 } ) +$$ + +where, $\mathbf { W } _ { r _ { 1 } }$ is the first layer random weight matrix, $\mathbf { b } _ { 1 }$ is the bias term of first layer. The “denoising” operation is established to a hypothetical additional specific criterion: robustness to partial destruction of the input, which means a good intermediate representation is learned from unknown distribution of its observed input. This operation helps for learning more stable structure and avoids the overfitting problem in most cases. + +# 3.6 DROPOUT + +As the same reason with denoising operation, dropout is a trick to prevent overfitting (Hinton et al., 2012b). When a large feedforward neural network is trained on a small training set, dropout performed well on test set. In order to prevent the complex co-adaptations on the training data, the basic idea of dropout is that each hidden node is randomly omitted from the network with a probability of $\beta$ , so a hidden node can’t rely on other hidden node. In another view, dropout is as a very efficient way of performing model averaging with neural networks. On test set, we train many separate networks and then to apply each of these networks to the test data. Dropout operation can save the train time and then we average the predictions produced by a very large number of different networks. Fig. 1 shows the dropout operation in our MDA. + +# 4 EXPERIMENTS + +# 4.1 DATESET DESCRIPTIONS + +We evaluate the performance of MDA on five benchmark data sets. The detail of the data is showed in Tab 1. The USPS 1 data set is a handwritten digits image data set includes 7291 training samples and 2007 test samples from 10 classes with 256 dimensional features. This task is to recognize the digits 0 to 9. The Isolet 2 data set is a collection of audio feature vectors of spoken letters from the English alphabet. It includes 6238 training samples and 1559 test samples from 26 classes with 614 dimensional features. The task is to identify which letter is spoken based on the recorded (and pre-processed) audio signal. Sensor 3 is a sensorless drive diagnosis data set includes 46816 training samples and 11693 test samples from 11 classes with 48 dimensional features. The features are extracted from electric current drive signals. The task is to classify 11 different classes with different conditions of the drive which has intact and defective components. Covertype 4 contains geological and map-based data from four wilderness areas located in the Roosevelt National Forest of northern Colorado. It includes 15120 training samples and 565892 test samples from 7 classes with 54 dimensional features. The task is to identify forest cover type from cartographic variables. For the IbnSina 5 ancient Arabic document data set, we use 50 pages of the manuscript for training (17543 training samples) and 10 pages for testing (3125 test samples). The data samples belong to 174 classes of subwords and are of dimensionality 200. + +Table 1: Characteristics of datasets used in evaluation. + +
DATASETSTATISTICS
datasetntraintest[Vd(target)
USPS92987291200710256(32)
Isolet77976238155926614(308)
Sensor5850946816116931148(24)
Covertype58101215120565892754(27)
Ibnsina20668175433125174200(100)
CIFAR-10600005000010000103072(64)
CMU49445393(24)
+ +In addition, we also use a large scale dataset CIFAR-10 6 to test our MDA on large scale applications. The CIFAR-10 dataset consists of $6 0 0 0 0 3 2 \times 3 2$ colour images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. We also test our MDA on a specific task which use the CMU motion capture (CMU mocap) data set 7. The CMU mocap data set includes three categories, namely, jumping, running and walking. We choose 49 video sequences from four subjects. For each sequence, the features are generated using Lawrences method 8, with dimensionality 93 (Zhong et al., 2010). By reason of the few samples of CMU, we adopt 10-fold cross-validation in our experiments and use the average error rate and standard deviation to evaluate the performance. + +# 4.2 CLASSIFICATION ON FIVE BENCHMARK DATA SETS + +# 4.2.1 BASELINE METHODS + +In order to evaluate the performance of MDA, we compared our MDA with 5 deep learning models include autoencoder (AE) (Hinton & Salakhutdinov, 2006), stacked autoencoders, denoising autoencoders (Vincent et al., 2008), stacked denoising autoencoders (Vincent et al., 2010) and stacked denoising autoencoders with dropout, 2 feature learning models, MFA (Zhong et al., 2013; Yan et al., 2007) and PCA (Jolliffe, 2002), PCA deep architecture base on our uniform framework and the classification accuracy on original space. + +# 4.2.2 EXPERIMENTAL SETTINGS + +All of the deep learning methods have the same settings. The size of minibatch was set to 100, the learning rate and momentum were the default value 1 and 0.5, the number of epoch was set to 400, the dropout rate and denoising rate $\nu$ were set to 0.1. For the AE and SAE, weight penalty of the $L 2$ norm was set to $1 0 ^ { - 4 }$ . For MFA, the number of nearest neighbors for constructing the intrinsic graph was set to 5, while that for constructing the penalty graph was set to 20. The target spaces of MFA and PCA on different data sets were showed in Tab 1. For the USPS data set, The architecture was set to $2 5 6 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 - 3 2$ . For the Isolet data set ,the architecture was set to $6 1 7 - 1 3 2 4 - 6 1 7 - 3 0 8$ . For the Sensor data set, the architecture was set to $4 8 - 9 6 - 4 8 - 2 4$ . + +Table 2: The classification accuracy on different datasets. “ORIG” represents the results obtained in the original data space. ‘PDA’ represents the PCA deep architecture. ‘MDA’ represents the MFA deep architecture. The best reslut is highlighted with boldface. + +
MethodORIGPCAMFAAESAEDAE(dropout)DAESDAEPDAMDA
USPS0.83660.94020.93920.94020.94020.95810.95320.94520.95860.9601
Isolet0.94670.92370.92690.95190.95060.95960.95190.95430.95840.9622
Sensor0.81510.80420.82340.79950.83250.81780.77640.78700.85820.8558
Covertype0.55760.55960.60570.74050.55760.70930.73970.74400.74580.7589
Ibnsina0.89570.91900.92060.93630.91840.94020.93700.92610.94210.9491
+ +Table 3: The structures on 5 data sets. “None” represents without second layer in MDA. “Twice” means the second layer’s nodes are as twice as the input layer. “Quadruple” represents the second layer’s nodes are as quadruple as the input layer. “Octuple” represents the second layer’s nodes are as octuple as the input layer. + +
DatasetNoneTwiceQuadrupleOctuple
USPS256-128-64-32256-512-256-128-64-32256-1024-512-256-128-64-32256-2048-1024-512-256-128-64-32
Isolet617-308617-1324-617-308617-2648-1324-617-308617-5296-2648-1324-617-308
Sensor48-2448-96-2448-192-96-2448-384-192-96-24
Covertype54-2754-108-2754-216-108-54-2754-432-216-108-54-27
Ibnsina200-100200-400-200-100200-800-400-200-100200-1600-800-400-200-100
+ +For the Covertype data set, we set the architecture to $5 4 - 2 1 6 - 1 0 8 - 5 4 - 2 7$ . Finally, for Ibnsina data set, the architecture was set to $2 0 0 - 4 0 0 - 2 0 0 - 1 0 0$ . + +# 4.2.3 CLASSIFICATION RESULTS + +The experimental results are shown in Tab. 2. We can see that our MDA achieves the best results on four dataset except the Sensor dataset, but MDA achieves the second best result on Sensor data set and only below the PDA. The PDA achieves the best result on Sensor data set and the second best results on other data sets. These results demonstrate that our uniform deep architectures achieve the good performance in most case. In addition, MDA not only outperform the traditional deep learning models, but also the shallow feature learning models. It shows that our deep architectures based on stacked some feature learning models can learn the better feature than shallow feature learning models. + +# 4.3 EVALUATION + +# 4.3.1 DIFFERENT STRUCTURES FOR MDA + +In order to evaluate the desired structures of MDA, we changed the node’s number of the second layer. For USPS data set, we get rid of the second layer and the architecture was $2 5 6 - 1 2 8 - 6 4 - 3 2$ . Then, we set the number of node of the second layer was as twice as the input layer, the architecture was $2 5 6 - 1 2 8 - 6 4 - 3 2$ . Next, the number of node was as quadruple as the input layer, the architecture was $2 5 6 - 1 0 2 4 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 - 3 2$ . Finally, the node’s number is as octuple as the input layer, the architecture was $2 5 6 \mathrm { ~ - ~ } 2 0 4 8 \mathrm { ~ - ~ } 1 0 2 4 \mathrm { ~ - ~ } 5 1 2 \mathrm { ~ - ~ } 2 5 6 \mathrm { ~ - ~ } 1 2 8 \mathrm { ~ - ~ } 6 4 \mathrm { ~ - ~ } 3 2$ . The structures of other data sets are shown in Tab. 3. + +The experimental results are shown in Tab. 4. When the the number of nodes of the second layer is as twice as the input layer, MDA achieved the minimum classification error on all data sets except the Covertype data set. When the number of nodes of the second layer is as quadruple as the input layer, MDA get the worst result on Covertype data set. We can conclude that MDA can work well when the number of nodes of the second layer is as twice or quadruple as the input layer. + +Table 4: The classification error with different structures on 5 data sets. The best results (minimum error) are highlighted with boldface. + +
DatasetNoneTwiceQuadrupleOctuple
USPS0.04630.03990.04330.0453
Isolet0.03980.03780.04170.0430
Sensor0.21720.14420.15590.7856
Covertype0.38760.38780.24110.3806
Ibnsina0.06430.05090.06140.0858
+ +Table 5: The classification error on 5 datasets with different number of hidden layers. + +
The number of hidden layers1234567
USPS0.057800.050320.050820.051820.039900.057300.05132
Isolet0.039770.041690.037850.038490.054520.042340.04683
Covertype0.271710.256010.241100.267310.27491
Sensor0.204570.155220.144200.170270.15567
Ibnsina0.056960.051840.050880.060160.06720
+ +Table 6: The classification error on gray-CIFAR10 and CMU mocap data sets. + +
(a）gray-CIFAR10(b) CMU mocap
MethodErrorMethodError
AE0.5117AE0.3970±0.1343
SAE0.5252SAE0.4106 ± 0.1648
DAE(dropout)0.5090DAE(dropout)0.3970±0.1343
DAE0.5176DAE0.4061 ± 0.1540
SDAE0.5113SDAE0.3970 ± 0.1343
PDA0.5085PDA0.3591 ± 0.0815
MDA0.4947MDA0.3636 ± 0.0958
+ +# 4.3.2 DIFFERENT NUMBER OF HIDDEN LAYERS FOR MDA + +In order to evaluate how many hidden layers adapt to different datasets, we designed some experiments which have different number of hidden layers. We used $1 \sim 7$ hidden layers on USPS and Isolet datasets and $1 \sim 5$ hidden layers on Covertype, Sensor and Ibnsina datasets. The experimental settings were same as previous experiments. + +Tab. 5 shows the classification error on 5 datasets with different hidden layers. All the datasets achieved the best results when hidden layer’s number is 3 except USPS dataset. The USPS dataset achieved the best result when hidden layer’s number is 5. As $1 \sim 3$ hidden layers, with the increase of the number of layers, the classification error is decreasing on all datasets. As small and middle scale applications, we don’t need very deep architectures to handle it. As large scale applications, we can design deeper architectures to achieve better performance. + +# 4.4 CLASSIFICATION ON LARGE SCALE DATASET CIFAR-10 + +The previous section introduced the advantages of MDA on small and middle scale applications. In order to evaluate the universality of MDA, we chose a relatively large scale dataset CIFAR-10 to test the performance of MDA. + +In our experiments, we first transformed the color images to gray images in order to reduce the dimensionality of input. Then we took one sample as a 1024 dimensional vector which is the input of our MDA. So, we can call this data set gray-CIFAR10. The architecture was set to $1 0 2 4 - 2 0 4 8 -$ $1 0 2 4 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 .$ the minibatch’s size was set to 100, the dropout ratio and denoising ratio were set to 0.1, the number of epoch was set to 400, the learning rate was set to 1, the momentum was set to 0.5. We compared our MDA with previous 6 methods. + +Table. 6(a) shows the classification error on gray-CIFAR10, we can see that PDA and MDA achieved the best results in these 7 methods. However, all of the methods on this framework didn’t perform well because we use the gray operation. + +# 4.5 CLASSIFICATION ON CMU MOCAP DATA SET + +CMU mocap data set is a very small dataset that only has 49 samples. Traditional deep learning methods didn’t work well in these kind of applications. We test our MDA and PDA and compared them with other 5 deep learning models. The architectures for all deep models (except the PDA) were set to $9 3 - 1 8 6 - 9 3 - 4 7 - 2 4$ . Specially, since the CMU mocap data set only has 49 samples, the PCA method only reduce the dimensionality to 49 at most, so the architecture of PDA was set to + +$9 3 - 1 8 6 - 2 4 $ . The denoising ratio and dropout ratio were set to 0.1 on DAE, DAE with dropout, SDAE, SAE, PDA and MDA. The weight penalty on AE was set to $1 0 ^ { - 4 }$ . The learning rate was set to 0.01, the momentum was set to 0.5 and the number of epoch is set to 600. The experiment was test on 10-fold cross validation. The experimental results are shown in Tab. 6(b). + +In Tab. 6(b), our PDA and MDA achieved the best results in this dataset and have lower standard deviation than other deep learning models. It demonstrates that our PDA and MDA are more stable than other deep learning models. The traditional autoencoder, SDAE, DAE with dropout achieved the same result in this dataset and better than SAE and DAE. + +# 5 CONCLUSION + +In this paper, we proposed a novel deep learning framework that based on stacked some feature learning models to handle small or middle data sets. Then we introduce MFA in this framework, called MDA. The deep learning tricks like backpropagation, denoising and dropout operation are applied on MDA to improve its performance. 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In NIPS, pp. 487–495, 2014. \ No newline at end of file diff --git a/parse/train/BysZhEqee/BysZhEqee_content_list.json b/parse/train/BysZhEqee/BysZhEqee_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b4497c3e06e77b1172c82f9e0af93e6bc5b13a09 --- /dev/null +++ b/parse/train/BysZhEqee/BysZhEqee_content_list.json @@ -0,0 +1,1256 @@ +[ + { + "type": "text", + "text": "MARGINAL DEEP ARCHITECTURES: DEEP LEARNING FOR SMALL AND MIDDLE SCALE APPLICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yuchen Zheng, Guoqiang Zhong & Junyu Dong ", + "bbox": [ + 184, + 170, + 519, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Computer Science and Technology Ocean University of China ouczyc@outlook.com,{gqzhong, dongjunyu}@ouc.edu.cn ", + "bbox": [ + 184, + 186, + 669, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 262, + 544, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years, many deep architectures have been proposed in different fields. However, to obtain good results, most of the previous deep models need a large number of training data. In this paper, for small and middle scale applications, we propose a novel deep learning framework based on stacked feature learning models. Particularly, we stack marginal Fisher analysis (MFA) layer by layer for the initialization of the deep architecture and call it “Marginal Deep Architectures” (MDA). In the implementation of MDA, the weight matrices of MFA are first learned layer by layer, and then we exploit some deep learning techniques, such as back propagation, dropout and denoising to fine tune the network. To evaluate the effectiveness of MDA, we have compared it with some feature learning methods and deep learning models on 7 small and middle scale real-world applications, including handwritten digits recognition, speech recognition, historical document understanding, image classification, action recognition and so on. Extensive experiments demonstrate that MDA performs not only better than shallow feature learning models, but also state-of-the-art deep learning models in these applications. ", + "bbox": [ + 233, + 292, + 764, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 539, + 336, + 554 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep learning methods have achieved desirable performance in many domains, such as image classification and detection, document analysis and recognition, natural language processing, video analysis (Krizhevsky et al., 2012; Chan et al., 2014; Ciresan et al., 2010; Collobert & Weston, 2008; Le et al., 2011). Deep learning methods learn the data representation by using multiple processing layers, which discover the intricate structure of high dimensional data with multiple levels of abstraction (LeCun et al., 2015). For example, for face recognition, the learned features of first layer may be the edges, directions and some local information. The second layer typically detects some object parts which are combination of edges and directions. The higher layers may further abstract the face image by combining the features of previous layers (outline of the eyes, nose, lips). This procedure is very similar with human visual and perceptual system. ", + "bbox": [ + 174, + 569, + 825, + 708 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recently years, many deep learning methods have been proposed (l. Boureau & others, 2008; Lee et al., 2009b;a; Hinton & Salakhutdinov, 2006). However, most models meet some difficult problems to solve, such as some parameters need to be randomly initialized, like the weight matrix of two successive layers in deep belief networks (DBNs) and the convolution kernel in convolutional neural networks (CNNs). In addition, traditional deep learning methods need a large scale training data to train the complex networks. It causes many problems in the training process. If we don’t initialize the parameters properly, the optimization procedure might need a long training time and fall into local minima. Alternatively, many feature learning models have been proposed to learn the intrinsic structures of high-dimensional data and avoid the curse of dimensionality. In particular, most of them can be trained with small and middle scale of data and their learning algorithms are generally based on closed-form solution or convex optimization. For instance, marginal Fisher analysis (MFA) (Yan et al., 2007; Zhong et al., 2013) is one of the feature learning models that is a supervised method based on the graph embedding framework. It utilizes an intrinsic graph to characterize the intraclass compactness, and another penalty graph to characterize the interclass separability. Its optimal solution can be learned by generalized eigenvalue decomposition. However, on the one hand, shallow feature learning models cannot work well on the data with highly nonlinear structure; on the other hand, few efforts are made to combine shallow feature learning models for the design of deep architectures. ", + "bbox": [ + 174, + 715, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 145 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In order to simultaneously solve the existing problems in deep learning methods and combine the advantages of feature learning models, we proposed a novel deep learning method based on stacked feature learning models. Particularly, we stack marginal Fisher analysis (MFA) layer by layer for the initialization of the deep architecture and call it “Marginal Deep Architectures” (MDA). Firstly, the input data are mapped to higher dimensional space by using random weight matrix. Then we use MFA to learn the lower dimensional representation layer by layer. In the implementation of this architecture, we add some tricks in the training process, such as back propagation, dropout and denoising to fine tune the network. Finally, the softmax layer is connected to the last feature layer. We have compared our MDA with some feature learning methods and deep learning models on different domains of datasets (including handwritten digits recognition, speech recognition, historical document understanding, image classification, action recognition and so on). Extensive experiments demonstrate that MDA performs not only better than shallow feature learning models, but also state-of-the-art deep learning models in small and middle scale applications. ", + "bbox": [ + 174, + 154, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The contributions of this work are highlighted as follows. ", + "bbox": [ + 174, + 340, + 549, + 354 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We propose a novel structure to build a deep architecture. The first hidden layer has twice or quadruple neurons as the input layer. Then we can use some feature learning models layer by layer to learn the compact representations of data. Finally, we set the last layer as a softmax classifier. ", + "bbox": [ + 176, + 361, + 825, + 404 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2. Traditional deep learning models in general need a large scale training data. Compared with traditional deep learning models, MDA can work better than traditional deep learning models in small and middle scale applications because the initialization of the weight matrices using MFA is much better than that using random initialization. ", + "bbox": [ + 174, + 410, + 823, + 467 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3. Our MDA can work well in different domains of datasets, such as handwritten digits, spoken letters and natural images. Extensive experiments demonstrate that MDA is a general model to handel small and middle scale data. On the other hand, for large scale datasets, like CIFAR-10, MDA works comparatively with other deep learning methods. ", + "bbox": [ + 174, + 473, + 823, + 529 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The rest of this paper is organized as follows: In Section 2, we give a brief overview of related work. In Section 3, we present the marginal Fisher analysis (MFA) and the proposed marginal deep architectures (MDA) in detail. The experimental settings and results are reported in Section 4, while Section 5 concludes this paper with remarks and future work. ", + "bbox": [ + 176, + 536, + 823, + 592 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 622, + 343, + 638 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "With the development of deep learning methods, many deep networks have been proposed in recent years (Donahue et al., 2013; Krizhevsky et al., 2012; Long et al., 2015; Zhou et al., 2014). These deep learning models show their powerful performance in various fields, such as image classification and analysis, document analysis and recognition, natural language processing et al. In the area of image analysis, Hinton et al. proposed a large, deep convolutional neural network (Alex net) to classify the 1.2 million high-resolution images in the ImageNet. It uses efficient GPU to speed their method. The results show that a large, deep convolutional neural network is capable of achieving recordbreaking results on a highly challenging dataset using purely supervised learning (Krizhevsky et al., 2012). In order to popularize the deep convolutional neural network, Donahue ea al. proposed DeCAF (Deep Convolutional Activation Feature) which is trained in a fully supervised fashion on a large, fixed set of object recognition tasks (Donahue et al., 2013). DeCAF provides a uniform framework for researchers who can improve and change this framework on some specific tasks. However, its performance at scene recognition has not attained the same level of success. In order to handle this problem, Zhou et al. introduce a new scene-centric database called Places with over 7 million labeled pictures of scenes. Then, they learn the deep features for scene recognition tasks by using the same architecture as ImageNet, and establish new state-of-the-art results on several scenecentric datasets (Zhou et al., 2014). However, these methods based on convolutional operation need very large scale training samples and a long training time. They can not work well on small and middle scale applications. ", + "bbox": [ + 173, + 659, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In other domains, deep learning methods also achieve good performance. Hinton et al. represent the shared views of four research groups that have had recent successes in using DNNs for automatic speech recognition (ASR). The DNNs that contain many layers of nonlinear hidden units and a very large output layer can outperform Gaussian mixture models (GMMs) at acoustic modeling for speech recognition on a variety of data sets (Hinton et al., 2012a). In the area of genetics, Xiong et al. use “deep learning” computer algorithms to derive a computational model that takes as input DNA sequences and applies general rules to predict splicing in human tissues (Xiong et al., 2015). It reveals the genetic origins of disease and how strongly genetic variants affect RNA splicing. In the area of natural language understanding, deep learning models have delivered strong results on topic classification, sentiment analysis et al. Sutskever et al. proposed a general approach, the Long Short-Term Memory (LSTM) architecture which can solve the general sequence to sequence problems better than before (Sutskever et al., 2014). In addition, Hinton et al. proposed autoencoder (AE) networks that is an effective way to learn the low-dimensional codes of high-dimensional data. Based on autoencoder, there are also have many excellent works to handle various tasks. Vincent et al. proposed a denoising autoencoder (DAE) which maked the learned representations robust to partial corruption of the input data (Vincent et al., 2008). The denoising autoencoder which initialize the deep architectures layer by layer is very similar with human visual system. Hinton et al. introduced random ‘dropout’ to prevent the overfitting which improve many benchmark tasks and obtain new records for speech and object recognition (Hinton et al., 2012b). Then, Vincent et al. proposed stacked denoising autoencoders (SDAE) which based on stacking layers of stacked denoising autoencoders (Vincent et al., 2010). It is very useful to learn the higher level representations and work well on natural images and handwritten digits. However, for the same reason, they also need a large scale training set and a long training time. They have no advantages to handle the small and middle scale applications. ", + "bbox": [ + 173, + 103, + 825, + 435 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Moreover, in the field of feature learning models, dimensionality reduction plays a crucial role to handle the problems for compressing, visualizing high-dimensional data and avoiding the “curse of dimensionality” (van der Maaten et al., 2009; van der Maaten, 2007). Traditional dimensionality reduction mainly can be classified into three types: linear or nonlinear, like principal components analysis (PCA) (Jolliffe, 2002) and linearity preserving projection (LPP) (Niyogi, 2004) are linear methods, stochastic neighbor embedding (SNE) (Hinton & Roweis, 2002) is a nonlinear method; supervised or unsupervised, such as marginal Fisher analysis (MFA) (Yan et al., 2007; Zhong et al., 2013) and linear discriminant analysis (LDA) (Fisher, 1936) are supervised methods, PCA is an unsupervised method; local or global, like MFA and SNE are local methods, PCA is a global method. Many feature learning models based on geometry theory provide different solutions to the problem of dimensionality reduction. Yan et al. proposed a general formulation about graph embedding framework can exploit new dimensionality reduction algorithms (Yan et al., 2007). If only directly use some feature learning models to extract the good representation from original data, it often eventually couldn’t get a good outcome. Considering this situation, we try to choose some excellent feature learning models and combine them with some deep learning algorithms. MFA is one special formulation of the graph embedding models based on this framework. It utilizes an intrinsic graph to characterize the intraclass compactness, and another penalty graph to characterize the interclass separability. Our motivation is to combine the advantage of MFA and deep architectures and propose a new initialization method for deep learning algorithms. ", + "bbox": [ + 174, + 443, + 825, + 707 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "There are also have some excellent works about feature learning models combined the deep architectures (Yuan et al.; George et al., 2014; Ngiam et al., 2011). Yuan et al. proposed an improved multilayer learning model to solve the scene recognition task (Yuan et al.). This model overcome the limitation of shallow, one-layer representations for scene recognition. Trigeorgis et al proposed deep Semi-NMF, that is able to learn such hidden representations from different, unknown attributes of a given dataset (George et al., 2014). Ngiam proposed a deep architectures to learn features over multiple modalities (Ngiam et al., 2011). They showed that multi-modality feature learning is better than one modality and achieved good performance on video and audio datasets. However, in general, we can only obtain data from one modality. In this work, we combine the advantages of MFA and deep architectures, which based on stacked feature learning models (Zheng et al., 2014; 2015), then we use some deep learning tricks, like back propagation, denoising and dropout to fine tuning the network. The advantage of this deep architecture is that we can learn the desirable weight matrix even if the training data is not large enough. And compared with traditional deep learning models and shallow feature learning models, our MDA achieved state-of-the-art results in most cases. ", + "bbox": [ + 174, + 714, + 825, + 907 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/c274b80803e2cd5bc8944e1881ad315fd04a306b1afaf87b2cfcee74878a65ee.jpg", + "image_caption": [ + "Figure 1: The brief representation for MDA. $\\mathbf { W } _ { r _ { 1 } }$ represents the first layer random weight matrix, ${ \\bf W } _ { M F A _ { 2 } }$ and ${ \\bf W } _ { M F A _ { 3 } }$ represent the weight matrixes learned by MFA. The dotted red lines represent the dropout operation, the dotted red circle is the ‘dropout’ node, and the cross nodes are corrupted. The denoising and dropout operation are completely random. For simplicity, we have omitted bias terms. " + ], + "image_footnote": [], + "bbox": [ + 192, + 113, + 818, + 327 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 MARGINAL DEEP ARCHITECTURES (MDA) ", + "text_level": 1, + "bbox": [ + 174, + 428, + 562, + 444 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we firstly introduce a novel framework of deep architectures, then we introduce marginal Fisher analysis (MFA) and the proposed marginal deep architectures (MDA) in detail. In addition, we also present some deep learning tricks that we used in the MDA model, including back propagation, denoising and dropout. ", + "bbox": [ + 174, + 458, + 825, + 515 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 A NOVEL FRAMEWORK OF DEEP ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 173, + 530, + 568, + 545 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The feature learning problem is generally formulated as follow. Given $n$ data, $\\{ \\mathbf { x } _ { 1 } ^ { T } , \\hdots , \\mathbf { x } _ { n } ^ { T } \\} \\in \\Re ^ { D }$ , where $D$ is the dimensionality of the data space, we seeks the compact representations of these data, i.e., $\\{ \\mathbf { y } _ { 1 } ^ { T } , \\ldots , \\mathbf { y } _ { n } ^ { T } \\} \\in \\Re ^ { d }$ , where $d$ is the dimensionality of the low dimensional embeddings. ", + "bbox": [ + 173, + 556, + 825, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to improve the accuracy of shallow feature learning models, we use stacked feature learning models to construct the deep architectures (Zheng et al., 2014; 2015), which is a general framework for different applications. In this case, the mapping of data from the original $D$ -dimensional space to the resulted $d$ -dimensional space can be described as ", + "bbox": [ + 173, + 606, + 825, + 661 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e48bd81154f91a5ea5dda9d89cf734f6617fbeb8c5f71d072f0e2781ba053ad9.jpg", + "text": "$$\nD \\Longrightarrow D _ { 1 } \\Longrightarrow \\cdots \\Longrightarrow D _ { i } \\Longrightarrow \\cdots \\Longrightarrow D _ { p - 1 } \\Longrightarrow d ,\n$$", + "text_format": "latex", + "bbox": [ + 318, + 666, + 676, + 683 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $D _ { 1 }$ is the first higher dimensional space, the number of the node is twice or quadruple as the input layer. $D _ { i }$ represents the dimensionality of the $i$ -th intermediate representation space, and $p$ is the total steps of mappings. Here, we can use different feature learning models for the learning of each layer. As the feature learning models are optimized layer by layer, we can obtain the mapping functions between successive layers. The first hidden layer is random by $\\mathbf { W } _ { r 1 }$ , and the representation is, ", + "bbox": [ + 173, + 685, + 825, + 768 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/cde999a4e35091bf4f23246488c718422f995fa7bcccdb30139c8571a632047b.jpg", + "text": "$$\n\\mathbf { a } ^ { 1 } = g ( \\mathbf { W } _ { r _ { 1 } } ^ { T } \\mathbf { x } + \\mathbf { b } )\n$$", + "text_format": "latex", + "bbox": [ + 429, + 767, + 566, + 787 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where, $g ( . )$ is a non-linear activation or transfer function. Then, we can use some feature learning models to initialize the next layers. The representations of next hidden layers are, ", + "bbox": [ + 169, + 787, + 825, + 815 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ef93d321cc2c811face57d2e0c41b146db24ba676248fda4b98931bbe4502d77.jpg", + "text": "$$\n\\mathbf { a } ^ { k } = g ( \\mathbf { W } _ { F _ { k - 1 } } ^ { T } \\mathbf { a } ^ { k - 1 } + \\mathbf { b } )\n$$", + "text_format": "latex", + "bbox": [ + 408, + 818, + 588, + 839 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where, $\\mathbf { W } _ { F _ { k - 1 } }$ is the weight matrix of the $k - 1$ th layer learned from feature learning models. ", + "bbox": [ + 173, + 842, + 790, + 858 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 MARGINAL FISHER ANALYSIS (MFA) ", + "text_level": 1, + "bbox": [ + 174, + 869, + 478, + 883 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Based on our novel framework of deep architecture, we introduce Marginal Fisher Analysis (MFA) to build MDA. Here, many traditional feature learning models, such as linear discriminant analysis (LDA), can be used as building blocks of MDA. Take LDA as an example. It assumes that the data of each class follow a Gaussian distribution. However, this assumption is not often satisfied in the real world. Without this assumption, LDA can not work well to separate the data with nonlinear structure. Alternatively, MFA can solve this problem effectively. Hence, considering the learning capability, we choose MFA as the build blocks of MDA in our work. MFA used the graph embedding framework to set up an intrinsic graph that characterizes the intraclass compactness and another penalty graph which characterizes the interclass separability. The marginal Fisher criterion is defined as ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 213 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/877247b3c98ed967435eb6df93fad7f2ab1ca9e45bf0cad44746747987e6211c.jpg", + "text": "$$\n\\mathbf { W } ^ { * } = \\operatorname * { a r g m i n } _ { \\mathbf { W } } \\frac { \\operatorname { t r } ( \\mathbf { W } ^ { T } \\mathbf { X } ( \\mathbf { D } - \\mathbf { A } ) \\mathbf { X } ^ { T } \\mathbf { W } ) } { \\operatorname { t r } ( \\mathbf { W } ^ { T } \\mathbf { X } ( \\mathbf { D } ^ { p } - \\mathbf { A } ^ { p } ) \\mathbf { X } ^ { T } \\mathbf { W } ) }\n$$", + "text_format": "latex", + "bbox": [ + 348, + 209, + 648, + 246 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathbf { D }$ and $\\mathbf { D } ^ { p }$ are diagonal matrices with elements $\\begin{array} { r } { \\mathbf { D } _ { i i } = \\sum _ { j } \\mathbf { A } _ { i j } } \\end{array}$ , and $\\begin{array} { r } { \\mathbf { D } _ { i j } ^ { p } = \\sum _ { j } \\mathbf { A } _ { i j } ^ { p } } \\end{array}$ , respectively. Then we can learn the projection matrix to multiply PCA’s projection and marginal Fisher projection, ", + "bbox": [ + 173, + 248, + 825, + 292 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3beaafa107bacc9eeea8541278f196e0ffbb797dcd6e9b96c541d52922f7d76f.jpg", + "text": "$$\n\\mathbf { W } _ { M F A } = \\mathbf { W } _ { P C A } \\mathbf { W } ^ { * }\n$$", + "text_format": "latex", + "bbox": [ + 419, + 311, + 576, + 329 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 MARGINAL DEEP ARCHITECTURES (MDA) ", + "text_level": 1, + "bbox": [ + 174, + 344, + 516, + 359 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to combine the advantages of MFA and proposed deep architectures, we propose the marginal deep architectures (or MDA). The MDA inherited from the proposed novel framework of deep architectures is shown in Fig. 1. As an input vector $\\mathbf { x } \\in [ 0 , 1 \\bar { ] } ^ { d }$ , we first map it to higher dimensional space by a random weight matrix $\\mathbf { W } _ { r 1 }$ . The representation of first hidden layer is computed as ", + "bbox": [ + 173, + 369, + 825, + 439 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e6cd3eab47c7ae7e7455ddd4365ebc67bc752a93d63e0a628b31c4d7038f3171.jpg", + "text": "$$\n\\mathbf { a } ^ { 1 } = s ( \\mathbf { W } _ { r _ { 1 } } ^ { T } \\mathbf { x } + \\mathbf { b } )\n$$", + "text_format": "latex", + "bbox": [ + 429, + 438, + 566, + 458 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where, $s ( . )$ is the sigmoid function $\\begin{array} { r } { s ( x ) = \\frac { 1 } { 1 + e ^ { - x } } } \\end{array}$ , $\\mathbf { b }$ is the bias terms, $\\mathbf { a } ^ { 1 }$ is the output of first layer. From second layer to $( n - 1 )$ -th layer, we use the weight matrices learned from MFA to map layer by layer. ", + "bbox": [ + 174, + 460, + 825, + 506 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1189ed79fa4401abcd45a76620911de11fa1bc279e216f1fa03b9138cb39d9e7.jpg", + "text": "$$\n\\mathbf { a } ^ { k } = s ( \\mathbf { W } _ { M F A _ { k - 1 } } ^ { T } \\mathbf { a } ^ { k - 1 } + \\mathbf { b } )\n$$", + "text_format": "latex", + "bbox": [ + 397, + 503, + 601, + 525 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The last layer is a softmax regression layer and the number of neuron is the number of category. The cost function is defined as, ", + "bbox": [ + 173, + 526, + 823, + 555 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/aca09428b093e3b718e83768ae88a16df7cac33700175114ff8693a00ea72c67.jpg", + "text": "$$\nJ ( \\mathbf { w } ) = - \\frac { 1 } { N } ( \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { K } \\mathbf { I } ( y _ { i } = j ) \\log \\frac { \\exp ( \\mathbf { w } _ { j } ^ { T } \\mathbf { a } _ { i } ^ { n - 1 } ) } { \\sum _ { l = 1 } ^ { K } \\exp ( \\mathbf { w } _ { l } ^ { T } \\mathbf { a } _ { i } ^ { n - 1 } ) } )\n$$", + "text_format": "latex", + "bbox": [ + 302, + 560, + 694, + 606 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where, $\\mathbf { I } ( x )$ is the indicator function, $\\mathbf { I } ( x ) = 1$ if $x$ is true, else $\\mathbf { I } ( x ) = 0$ . $y _ { i }$ is the label corresponding to $\\mathbf { x } _ { i }$ . Then the probability that $\\mathbf { x } _ { i }$ is classified to $j$ is, ", + "bbox": [ + 173, + 611, + 821, + 640 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7c8e7b3cd3d343f1d93a97a03620a9e10e2cbb326348bf9af64be250161f783a.jpg", + "text": "$$\np ( y _ { i } = j | \\mathbf { x } _ { i } , \\mathbf { w } ) = \\frac { \\exp ( \\mathbf { w } _ { j } ^ { T } \\mathbf { a } _ { i } ^ { n - 1 } ) } { \\sum _ { l = 1 } ^ { K } \\exp ( \\mathbf { w } _ { l } ^ { T } \\mathbf { a } _ { i } ^ { n - 1 } ) }\n$$", + "text_format": "latex", + "bbox": [ + 362, + 645, + 635, + 688 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Taking derivatives, one can show that the gradient is, ", + "bbox": [ + 178, + 691, + 519, + 707 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/651eeda402594bab59a62af664c8bcb72c85deaea2f041e5e6df5e3431ec8c43.jpg", + "text": "$$\n\\nabla J ( \\mathbf { w } ) = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } [ \\mathbf { x } _ { i } ( \\mathbf { I } ( y _ { i } = j ) - p ( y _ { i } = j | \\mathbf { x } _ { i } , \\mathbf { w } ) ]\n$$", + "text_format": "latex", + "bbox": [ + 321, + 712, + 674, + 756 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "If the $n { - } 1$ layer’s neurons are more than the last layer, we can continue using MFA to map it. On the contrary, If the $n - 1$ layer’s neurons are less than last layer, we can randomly initialize the weight matrix between this two layers. Next, in order to improve the MDA, we introduce back propagation, denoising and dropout operation. ", + "bbox": [ + 174, + 761, + 825, + 818 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 BACK PROPAGATION ", + "text_level": 1, + "bbox": [ + 176, + 833, + 359, + 848 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to adjust the network, we use back propagation (Rumelhart et al., 1986) to compute partial derivative and stochastic gradient descent to update the weight matrixes and the bias terms. For each node $i$ in output layer $n$ -th layer), we compute an error term as ", + "bbox": [ + 174, + 859, + 825, + 902 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e2f5605fef6b0c66ef3ccb53a515120dcfd4ccbc82923f12f0d6e6728d687359.jpg", + "text": "$$\n\\delta _ { i } ^ { n } = \\nabla J ( \\mathbf { w } )\n$$", + "text_format": "latex", + "bbox": [ + 452, + 909, + 545, + 925 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where, $J ( \\mathbf { w } )$ is the cost function computed from Equ.8 and $\\nabla J ( \\mathbf { w } )$ computed from Equ.10. For each node $i$ in $( n - 1 )$ -th to second layer, the error term is computed as, ", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/837f96fbb1c3ece17ecba3070adae005c99a8dfcf08dc71cdfbe9524b7261a97.jpg", + "text": "$$\n\\delta _ { i } ^ { k } = ( \\sum _ { j = 1 } ^ { k + 1 } w _ { j i } ^ { k } \\delta _ { j } ^ { k + 1 } ) s ^ { \\prime } ( z _ { i } ^ { k } )\n$$", + "text_format": "latex", + "bbox": [ + 410, + 138, + 588, + 183 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The back propagation procedure relies on computing the gradient of an objective function with respect to the weights of a multilayer stacked modules. It starting from the output at the top and end to the input at the bottom. ", + "bbox": [ + 174, + 196, + 825, + 238 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.5 DENOISING OPERATION", + "text_level": 1, + "bbox": [ + 176, + 256, + 377, + 270 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Vincent et al. proposed the denoising autoencoder to improve the robustness of autoencoder (Vincent et al., 2008). It’s very similar with the regularization methods and avoids the “overfitting” problem. The basic idea is to corrupt partial input data by the desired proportion of $\\nu$ “destruction”. for each input $\\mathbf { x }$ , a fixed number $\\nu d$ of components are chosen at random, and their value is forced to 0, while the others are left untouched. The initial input $\\mathbf { x }$ to get a partially destroyed version $\\tilde { \\mathbf { x } }$ by means of a stochastic mapping, ", + "bbox": [ + 173, + 281, + 825, + 364 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/660408f503aacbc3a2faea47dddffc9516366d804de1fdf8b86d07fa813229e5.jpg", + "text": "$$\n\\tilde { \\mathbf { x } } \\sim q _ { D } ( \\tilde { \\mathbf { x } } | \\mathbf { x } )\n$$", + "text_format": "latex", + "bbox": [ + 452, + 363, + 544, + 381 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where, $q _ { D } ( \\tilde { \\mathbf { x } } | \\mathbf { x } )$ is the unknown distribution. Then, for a hidden representation $\\mathbf { h }$ , ", + "bbox": [ + 173, + 382, + 707, + 397 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/14e1cf54d4eeeaafbd34e9ed1efcd8466844e3b19b8692b713167e797267b53a.jpg", + "text": "$$\n\\mathbf { h } = s ( \\mathbf { W } ^ { T } \\widetilde { \\mathbf { x } } + \\mathbf { b } )\n$$", + "text_format": "latex", + "bbox": [ + 434, + 404, + 562, + 422 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In our MDA, we use this idea to improve the network, please refer to Fig. 1 to find clear sight. For the input layer, the output of first hidden layer is represented as, ", + "bbox": [ + 169, + 428, + 825, + 457 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5581d79a33c0453d7c2c2faec48e37f6e1a76f7f4561ca2366afeef1a4f8a152.jpg", + "text": "$$\n\\mathbf { a } ^ { 2 } = s ( \\mathbf { W } _ { r _ { 1 } } ^ { T } \\tilde { \\mathbf { x } } + \\mathbf { b } _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 426, + 462, + 570, + 482 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where, $\\mathbf { W } _ { r _ { 1 } }$ is the first layer random weight matrix, $\\mathbf { b } _ { 1 }$ is the bias term of first layer. The “denoising” operation is established to a hypothetical additional specific criterion: robustness to partial destruction of the input, which means a good intermediate representation is learned from unknown distribution of its observed input. This operation helps for learning more stable structure and avoids the overfitting problem in most cases. ", + "bbox": [ + 174, + 487, + 823, + 558 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.6 DROPOUT ", + "text_level": 1, + "bbox": [ + 176, + 574, + 284, + 588 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As the same reason with denoising operation, dropout is a trick to prevent overfitting (Hinton et al., 2012b). When a large feedforward neural network is trained on a small training set, dropout performed well on test set. In order to prevent the complex co-adaptations on the training data, the basic idea of dropout is that each hidden node is randomly omitted from the network with a probability of $\\beta$ , so a hidden node can’t rely on other hidden node. In another view, dropout is as a very efficient way of performing model averaging with neural networks. On test set, we train many separate networks and then to apply each of these networks to the test data. Dropout operation can save the train time and then we average the predictions produced by a very large number of different networks. Fig. 1 shows the dropout operation in our MDA. ", + "bbox": [ + 174, + 599, + 825, + 724 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 746, + 326, + 761 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 DATESET DESCRIPTIONS ", + "text_level": 1, + "bbox": [ + 174, + 776, + 387, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate the performance of MDA on five benchmark data sets. The detail of the data is showed in Tab 1. The USPS 1 data set is a handwritten digits image data set includes 7291 training samples and 2007 test samples from 10 classes with 256 dimensional features. This task is to recognize the digits 0 to 9. The Isolet 2 data set is a collection of audio feature vectors of spoken letters from the English alphabet. It includes 6238 training samples and 1559 test samples from 26 classes with 614 dimensional features. The task is to identify which letter is spoken based on the recorded (and pre-processed) audio signal. Sensor 3 is a sensorless drive diagnosis data set includes 46816 training samples and 11693 test samples from 11 classes with 48 dimensional features. The features are extracted from electric current drive signals. The task is to classify 11 different classes with different conditions of the drive which has intact and defective components. Covertype 4 contains geological and map-based data from four wilderness areas located in the Roosevelt National Forest of northern Colorado. It includes 15120 training samples and 565892 test samples from 7 classes with 54 dimensional features. The task is to identify forest cover type from cartographic variables. For the IbnSina 5 ancient Arabic document data set, we use 50 pages of the manuscript for training (17543 training samples) and 10 pages for testing (3125 test samples). The data samples belong to 174 classes of subwords and are of dimensionality 200. ", + "bbox": [ + 174, + 803, + 825, + 886 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/6b056ca116756f30d622430404ff53ec91ca830095ce7a1e7fcc6d86b902104f.jpg", + "table_caption": [ + "Table 1: Characteristics of datasets used in evaluation. " + ], + "table_footnote": [], + "table_body": "
DATASETSTATISTICS
datasetntraintest[Vd(target)
USPS92987291200710256(32)
Isolet77976238155926614(308)
Sensor5850946816116931148(24)
Covertype58101215120565892754(27)
Ibnsina20668175433125174200(100)
CIFAR-10600005000010000103072(64)
CMU49445393(24)
", + "bbox": [ + 336, + 127, + 661, + 223 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 248, + 825, + 388 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In addition, we also use a large scale dataset CIFAR-10 6 to test our MDA on large scale applications. The CIFAR-10 dataset consists of $6 0 0 0 0 3 2 \\times 3 2$ colour images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. We also test our MDA on a specific task which use the CMU motion capture (CMU mocap) data set 7. The CMU mocap data set includes three categories, namely, jumping, running and walking. We choose 49 video sequences from four subjects. For each sequence, the features are generated using Lawrences method 8, with dimensionality 93 (Zhong et al., 2010). By reason of the few samples of CMU, we adopt 10-fold cross-validation in our experiments and use the average error rate and standard deviation to evaluate the performance. ", + "bbox": [ + 173, + 395, + 825, + 520 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 CLASSIFICATION ON FIVE BENCHMARK DATA SETS ", + "text_level": 1, + "bbox": [ + 174, + 539, + 575, + 554 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2.1 BASELINE METHODS", + "text_level": 1, + "bbox": [ + 176, + 565, + 375, + 580 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In order to evaluate the performance of MDA, we compared our MDA with 5 deep learning models include autoencoder (AE) (Hinton & Salakhutdinov, 2006), stacked autoencoders, denoising autoencoders (Vincent et al., 2008), stacked denoising autoencoders (Vincent et al., 2010) and stacked denoising autoencoders with dropout, 2 feature learning models, MFA (Zhong et al., 2013; Yan et al., 2007) and PCA (Jolliffe, 2002), PCA deep architecture base on our uniform framework and the classification accuracy on original space. ", + "bbox": [ + 173, + 592, + 825, + 675 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2.2 EXPERIMENTAL SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 691, + 413, + 707 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "All of the deep learning methods have the same settings. The size of minibatch was set to 100, the learning rate and momentum were the default value 1 and 0.5, the number of epoch was set to 400, the dropout rate and denoising rate $\\nu$ were set to 0.1. For the AE and SAE, weight penalty of the $L 2$ norm was set to $1 0 ^ { - 4 }$ . For MFA, the number of nearest neighbors for constructing the intrinsic graph was set to 5, while that for constructing the penalty graph was set to 20. The target spaces of MFA and PCA on different data sets were showed in Tab 1. For the USPS data set, The architecture was set to $2 5 6 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 - 3 2$ . For the Isolet data set ,the architecture was set to $6 1 7 - 1 3 2 4 - 6 1 7 - 3 0 8$ . For the Sensor data set, the architecture was set to $4 8 - 9 6 - 4 8 - 2 4$ . ", + "bbox": [ + 173, + 717, + 826, + 829 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/9adbca5cce73440fd70beb482bcde824429198a700f2ec2662ed1e6874cb15f8.jpg", + "table_caption": [ + "Table 2: The classification accuracy on different datasets. “ORIG” represents the results obtained in the original data space. ‘PDA’ represents the PCA deep architecture. ‘MDA’ represents the MFA deep architecture. The best reslut is highlighted with boldface. " + ], + "table_footnote": [], + "table_body": "
MethodORIGPCAMFAAESAEDAE(dropout)DAESDAEPDAMDA
USPS0.83660.94020.93920.94020.94020.95810.95320.94520.95860.9601
Isolet0.94670.92370.92690.95190.95060.95960.95190.95430.95840.9622
Sensor0.81510.80420.82340.79950.83250.81780.77640.78700.85820.8558
Covertype0.55760.55960.60570.74050.55760.70930.73970.74400.74580.7589
Ibnsina0.89570.91900.92060.93630.91840.94020.93700.92610.94210.9491
", + "bbox": [ + 187, + 154, + 810, + 218 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/66f1156099018b785d5a9adce50bbad4d34ca3098c87590ea3123c8ccaf7e990.jpg", + "table_caption": [ + "Table 3: The structures on 5 data sets. “None” represents without second layer in MDA. “Twice” means the second layer’s nodes are as twice as the input layer. “Quadruple” represents the second layer’s nodes are as quadruple as the input layer. “Octuple” represents the second layer’s nodes are as octuple as the input layer. " + ], + "table_footnote": [], + "table_body": "
DatasetNoneTwiceQuadrupleOctuple
USPS256-128-64-32256-512-256-128-64-32256-1024-512-256-128-64-32256-2048-1024-512-256-128-64-32
Isolet617-308617-1324-617-308617-2648-1324-617-308617-5296-2648-1324-617-308
Sensor48-2448-96-2448-192-96-2448-384-192-96-24
Covertype54-2754-108-2754-216-108-54-2754-432-216-108-54-27
Ibnsina200-100200-400-200-100200-800-400-200-100200-1600-800-400-200-100
", + "bbox": [ + 181, + 299, + 815, + 363 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For the Covertype data set, we set the architecture to $5 4 - 2 1 6 - 1 0 8 - 5 4 - 2 7$ . Finally, for Ibnsina data set, the architecture was set to $2 0 0 - 4 0 0 - 2 0 0 - 1 0 0$ . ", + "bbox": [ + 174, + 393, + 823, + 422 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2.3 CLASSIFICATION RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 439, + 413, + 454 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The experimental results are shown in Tab. 2. We can see that our MDA achieves the best results on four dataset except the Sensor dataset, but MDA achieves the second best result on Sensor data set and only below the PDA. The PDA achieves the best result on Sensor data set and the second best results on other data sets. These results demonstrate that our uniform deep architectures achieve the good performance in most case. In addition, MDA not only outperform the traditional deep learning models, but also the shallow feature learning models. It shows that our deep architectures based on stacked some feature learning models can learn the better feature than shallow feature learning models. ", + "bbox": [ + 174, + 463, + 825, + 575 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 594, + 305, + 609 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3.1 DIFFERENT STRUCTURES FOR MDA ", + "text_level": 1, + "bbox": [ + 176, + 621, + 478, + 636 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In order to evaluate the desired structures of MDA, we changed the node’s number of the second layer. For USPS data set, we get rid of the second layer and the architecture was $2 5 6 - 1 2 8 - 6 4 - 3 2$ . Then, we set the number of node of the second layer was as twice as the input layer, the architecture was $2 5 6 - 1 2 8 - 6 4 - 3 2$ . Next, the number of node was as quadruple as the input layer, the architecture was $2 5 6 - 1 0 2 4 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 - 3 2$ . Finally, the node’s number is as octuple as the input layer, the architecture was $2 5 6 \\mathrm { ~ - ~ } 2 0 4 8 \\mathrm { ~ - ~ } 1 0 2 4 \\mathrm { ~ - ~ } 5 1 2 \\mathrm { ~ - ~ } 2 5 6 \\mathrm { ~ - ~ } 1 2 8 \\mathrm { ~ - ~ } 6 4 \\mathrm { ~ - ~ } 3 2$ . The structures of other data sets are shown in Tab. 3. ", + "bbox": [ + 174, + 646, + 825, + 744 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The experimental results are shown in Tab. 4. When the the number of nodes of the second layer is as twice as the input layer, MDA achieved the minimum classification error on all data sets except the Covertype data set. When the number of nodes of the second layer is as quadruple as the input layer, MDA get the worst result on Covertype data set. We can conclude that MDA can work well when the number of nodes of the second layer is as twice or quadruple as the input layer. ", + "bbox": [ + 174, + 751, + 823, + 794 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/60aec4c3f4ad42f7dc90f4da29387160bb6eef06ae6a7de9fb93248997827318.jpg", + "table_caption": [ + "Table 4: The classification error with different structures on 5 data sets. The best results (minimum error) are highlighted with boldface. " + ], + "table_footnote": [], + "table_body": "
DatasetNoneTwiceQuadrupleOctuple
USPS0.04630.03990.04330.0453
Isolet0.03980.03780.04170.0430
Sensor0.21720.14420.15590.7856
Covertype0.38760.38780.24110.3806
Ibnsina0.06430.05090.06140.0858
", + "bbox": [ + 351, + 857, + 647, + 921 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/4ef4c442ae01a5b0cdcde9a9c30947726f91fc535e57d3200e67c4fa22087d55.jpg", + "table_caption": [ + "Table 5: The classification error on 5 datasets with different number of hidden layers. " + ], + "table_footnote": [], + "table_body": "
The number of hidden layers1234567
USPS0.057800.050320.050820.051820.039900.057300.05132
Isolet0.039770.041690.037850.038490.054520.042340.04683
Covertype0.271710.256010.241100.267310.27491
Sensor0.204570.155220.144200.170270.15567
Ibnsina0.056960.051840.050880.060160.06720
", + "bbox": [ + 223, + 126, + 776, + 190 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/76ff48557289169df8189f267fd8fae9c931b480f34793e4a94220a36556df0c.jpg", + "table_caption": [ + "Table 6: The classification error on gray-CIFAR10 and CMU mocap data sets. " + ], + "table_footnote": [], + "table_body": "
(a）gray-CIFAR10(b) CMU mocap
MethodErrorMethodError
AE0.5117AE0.3970±0.1343
SAE0.5252SAE0.4106 ± 0.1648
DAE(dropout)0.5090DAE(dropout)0.3970±0.1343
DAE0.5176DAE0.4061 ± 0.1540
SDAE0.5113SDAE0.3970 ± 0.1343
PDA0.5085PDA0.3591 ± 0.0815
MDA0.4947MDA0.3636 ± 0.0958
", + "bbox": [ + 308, + 233, + 686, + 335 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 367, + 823, + 396 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3.2 DIFFERENT NUMBER OF HIDDEN LAYERS FOR MDA ", + "text_level": 1, + "bbox": [ + 174, + 411, + 593, + 426 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In order to evaluate how many hidden layers adapt to different datasets, we designed some experiments which have different number of hidden layers. We used $1 \\sim 7$ hidden layers on USPS and Isolet datasets and $1 \\sim 5$ hidden layers on Covertype, Sensor and Ibnsina datasets. The experimental settings were same as previous experiments. ", + "bbox": [ + 174, + 436, + 825, + 492 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Tab. 5 shows the classification error on 5 datasets with different hidden layers. All the datasets achieved the best results when hidden layer’s number is 3 except USPS dataset. The USPS dataset achieved the best result when hidden layer’s number is 5. As $1 \\sim 3$ hidden layers, with the increase of the number of layers, the classification error is decreasing on all datasets. As small and middle scale applications, we don’t need very deep architectures to handle it. As large scale applications, we can design deeper architectures to achieve better performance. ", + "bbox": [ + 174, + 500, + 825, + 583 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 CLASSIFICATION ON LARGE SCALE DATASET CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 601, + 604, + 616 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The previous section introduced the advantages of MDA on small and middle scale applications. In order to evaluate the universality of MDA, we chose a relatively large scale dataset CIFAR-10 to test the performance of MDA. ", + "bbox": [ + 176, + 627, + 821, + 670 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In our experiments, we first transformed the color images to gray images in order to reduce the dimensionality of input. Then we took one sample as a 1024 dimensional vector which is the input of our MDA. So, we can call this data set gray-CIFAR10. The architecture was set to $1 0 2 4 - 2 0 4 8 -$ $1 0 2 4 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 .$ the minibatch’s size was set to 100, the dropout ratio and denoising ratio were set to 0.1, the number of epoch was set to 400, the learning rate was set to 1, the momentum was set to 0.5. We compared our MDA with previous 6 methods. ", + "bbox": [ + 174, + 676, + 825, + 761 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table. 6(a) shows the classification error on gray-CIFAR10, we can see that PDA and MDA achieved the best results in these 7 methods. However, all of the methods on this framework didn’t perform well because we use the gray operation. ", + "bbox": [ + 176, + 767, + 823, + 809 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 CLASSIFICATION ON CMU MOCAP DATA SET ", + "text_level": 1, + "bbox": [ + 174, + 827, + 529, + 842 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "CMU mocap data set is a very small dataset that only has 49 samples. Traditional deep learning methods didn’t work well in these kind of applications. We test our MDA and PDA and compared them with other 5 deep learning models. The architectures for all deep models (except the PDA) were set to $9 3 - 1 8 6 - 9 3 - 4 7 - 2 4$ . Specially, since the CMU mocap data set only has 49 samples, the PCA method only reduce the dimensionality to 49 at most, so the architecture of PDA was set to ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "$9 3 - 1 8 6 - 2 4 $ . The denoising ratio and dropout ratio were set to 0.1 on DAE, DAE with dropout, SDAE, SAE, PDA and MDA. The weight penalty on AE was set to $1 0 ^ { - 4 }$ . The learning rate was set to 0.01, the momentum was set to 0.5 and the number of epoch is set to 600. The experiment was test on 10-fold cross validation. The experimental results are shown in Tab. 6(b). ", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In Tab. 6(b), our PDA and MDA achieved the best results in this dataset and have lower standard deviation than other deep learning models. It demonstrates that our PDA and MDA are more stable than other deep learning models. The traditional autoencoder, SDAE, DAE with dropout achieved the same result in this dataset and better than SAE and DAE. ", + "bbox": [ + 174, + 166, + 825, + 222 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 242, + 318, + 258 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we proposed a novel deep learning framework that based on stacked some feature learning models to handle small or middle data sets. Then we introduce MFA in this framework, called MDA. The deep learning tricks like backpropagation, denoising and dropout operation are applied on MDA to improve its performance. Extensive experiments on 7 different type data sets demonstrate that MDA performs not only better than shallow feature learning models, but also stateof-the-art deep learning models on small and middle scale applications. The evaluation of MDA show that how to adjust the parameters make the MDA work well. For future work, we plan to try other feature learning models and explore the different structures for this novel deep learning model. In addition, we plan to explore new deep architectures based on this framework to handle the large scale datasets. ", + "bbox": [ + 174, + 273, + 825, + 412 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 434, + 285, + 449 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "T.-H. Chan, K. Jia, S. Gao, J. Lu, Z. Zeng, and Y. Ma. PCANet: A simple deep learning baseline for image classification? arXiv preprint arXiv:1404.3606, 2014. \nD. C. Ciresan, U. Meier, L. M. Gambardella, and J. Schmidhuber. Deep, big, simple neural nets for handwritten digit recognition. Neural computation, 22(12):3207–3220, 2010. \nR. Collobert and J. Weston. A unified architecture for natural language processing: Deep neural networks with multitask learning. In ICML, pp. 160–167, 2008. \nJ. Donahue, Y. 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Ex-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 374, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 374, + 469, + 387 + ], + "score": 1.0, + "content": "tensive experiments demonstrate that MDA performs not only better than shallow", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 386, + 469, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 469, + 398 + ], + "score": 1.0, + "content": "feature learning models, but also state-of-the-art deep learning models in these", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 195, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 195, + 408 + ], + "score": 1.0, + "content": "applications.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 206, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 208, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 208, + 442 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Deep learning methods have achieved desirable performance in many domains, such as image classi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "fication and detection, document analysis and recognition, natural language processing, video anal-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "ysis (Krizhevsky et al., 2012; Chan et al., 2014; Ciresan et al., 2010; Collobert & Weston, 2008;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 506, + 498 + ], + "score": 1.0, + "content": "Le et al., 2011). Deep learning methods learn the data representation by using multiple processing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "layers, which discover the intricate structure of high dimensional data with multiple levels of ab-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "straction (LeCun et al., 2015). For example, for face recognition, the learned features of first layer", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "may be the edges, directions and some local information. 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This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 378, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 378, + 563 + ], + "score": 1.0, + "content": "procedure is very similar with human visual and perceptual system.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "In recently years, many deep learning methods have been proposed (l. Boureau & others, 2008;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "Lee et al., 2009b;a; Hinton & Salakhutdinov, 2006). 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In addition, traditional deep learning methods need a large scale training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "data to train the complex networks. It causes many problems in the training process. If we don’t", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "initialize the parameters properly, the optimization procedure might need a long training time and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "fall into local minima. Alternatively, many feature learning models have been proposed to learn the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "intrinsic structures of high-dimensional data and avoid the curse of dimensionality. In particular,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "most of them can be trained with small and middle scale of data and their learning algorithms", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "are generally based on closed-form solution or convex optimization. For instance, marginal Fisher", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "analysis (MFA) (Yan et al., 2007; Zhong et al., 2013) is one of the feature learning models that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "is a supervised method based on the graph embedding framework. It utilizes an intrinsic graph", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "to characterize the intraclass compactness, and another penalty graph to characterize the interclass", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "score": 1.0, + "content": "separability. Its optimal solution can be learned by generalized eigenvalue decomposition. 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In this paper, for small and middle scale applications, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 470, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 470, + 278 + ], + "score": 1.0, + "content": "propose a novel deep learning framework based on stacked feature learning mod-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "els. Particularly, we stack marginal Fisher analysis (MFA) layer by layer for the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 286, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 286, + 470, + 299 + ], + "score": 1.0, + "content": "initialization of the deep architecture and call it “Marginal Deep Architectures”", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 298, + 470, + 309 + ], + "spans": [ + { + "bbox": [ + 142, + 298, + 470, + 309 + ], + "score": 1.0, + "content": "(MDA). In the implementation of MDA, the weight matrices of MFA are first", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 309, + 470, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 470, + 321 + ], + "score": 1.0, + "content": "learned layer by layer, and then we exploit some deep learning techniques, such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 470, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 470, + 332 + ], + "score": 1.0, + "content": "as back propagation, dropout and denoising to fine tune the network. To evalu-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 470, + 344 + ], + "score": 1.0, + "content": "ate the effectiveness of MDA, we have compared it with some feature learning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 341, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 341, + 470, + 354 + ], + "score": 1.0, + "content": "methods and deep learning models on 7 small and middle scale real-world ap-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "plications, including handwritten digits recognition, speech recognition, historical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "score": 1.0, + "content": "document understanding, image classification, action recognition and so on. Ex-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 374, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 374, + 469, + 387 + ], + "score": 1.0, + "content": "tensive experiments demonstrate that MDA performs not only better than shallow", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 386, + 469, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 469, + 398 + ], + "score": 1.0, + "content": "feature learning models, but also state-of-the-art deep learning models in these", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 397, + 195, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 195, + 408 + ], + "score": 1.0, + "content": "applications.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14.5, + "bbox_fs": [ + 141, + 232, + 470, + 408 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 206, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 208, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 208, + 442 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "Deep learning methods have achieved desirable performance in many domains, such as image classi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "fication and detection, document analysis and recognition, natural language processing, video anal-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "ysis (Krizhevsky et al., 2012; Chan et al., 2014; Ciresan et al., 2010; Collobert & Weston, 2008;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 506, + 498 + ], + "score": 1.0, + "content": "Le et al., 2011). Deep learning methods learn the data representation by using multiple processing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "layers, which discover the intricate structure of high dimensional data with multiple levels of ab-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "straction (LeCun et al., 2015). For example, for face recognition, the learned features of first layer", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "may be the edges, directions and some local information. The second layer typically detects some", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "object parts which are combination of edges and directions. The higher layers may further abstract", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "the face image by combining the features of previous layers (outline of the eyes, nose, lips). This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 378, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 378, + 563 + ], + "score": 1.0, + "content": "procedure is very similar with human visual and perceptual system.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 451, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "In recently years, many deep learning methods have been proposed (l. Boureau & others, 2008;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "Lee et al., 2009b;a; Hinton & Salakhutdinov, 2006). However, most models meet some difficult", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "problems to solve, such as some parameters need to be randomly initialized, like the weight matrix", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "of two successive layers in deep belief networks (DBNs) and the convolution kernel in convolutional", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "neural networks (CNNs). In addition, traditional deep learning methods need a large scale training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "data to train the complex networks. It causes many problems in the training process. If we don’t", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "initialize the parameters properly, the optimization procedure might need a long training time and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "fall into local minima. 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For instance, marginal Fisher", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "analysis (MFA) (Yan et al., 2007; Zhong et al., 2013) is one of the feature learning models that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "is a supervised method based on the graph embedding framework. It utilizes an intrinsic graph", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "to characterize the intraclass compactness, and another penalty graph to characterize the interclass", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 735 + ], + "score": 1.0, + "content": "separability. Its optimal solution can be learned by generalized eigenvalue decomposition. 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We have compared our MDA with some feature learning methods and deep learning mod-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "score": 1.0, + "content": "els on different domains of datasets (including handwritten digits recognition, speech recognition,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "historical document understanding, image classification, action recognition and so on). Extensive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "experiments demonstrate that MDA performs not only better than shallow feature learning models,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 445, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 445, + 264 + ], + "score": 1.0, + "content": "but also state-of-the-art deep learning models in small and middle scale applications.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 336, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 338, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 338, + 283 + ], + "score": 1.0, + "content": "The contributions of this work are highlighted as follows.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 286, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "1. We propose a novel structure to build a deep architecture. The first hidden layer has twice or", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "quadruple neurons as the input layer. Then we can use some feature learning models layer by layer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 491, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 491, + 321 + ], + "score": 1.0, + "content": "to learn the compact representations of data. Finally, we set the last layer as a softmax classifier.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 504, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "2. Traditional deep learning models in general need a large scale training data. Compared with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "traditional deep learning models, MDA can work better than traditional deep learning models in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "small and middle scale applications because the initialization of the weight matrices using MFA is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 359, + 305, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 305, + 371 + ], + "score": 1.0, + "content": "much better than that using random initialization.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 504, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "3. Our MDA can work well in different domains of datasets, such as handwritten digits, spoken", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 398 + ], + "score": 1.0, + "content": "letters and natural images. Extensive experiments demonstrate that MDA is a general model to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "handel small and middle scale data. On the other hand, for large scale datasets, like CIFAR-10,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 408, + 356, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 356, + 420 + ], + "score": 1.0, + "content": "MDA works comparatively with other deep learning methods.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 425, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 437 + ], + "score": 1.0, + "content": "The rest of this paper is organized as follows: In Section 2, we give a brief overview of related", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "work. In Section 3, we present the marginal Fisher analysis (MFA) and the proposed marginal deep", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "score": 1.0, + "content": "architectures (MDA) in detail. The experimental settings and results are reported in Section 4, while", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 458, + 353, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 353, + 470 + ], + "score": 1.0, + "content": "Section 5 concludes this paper with remarks and future work.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 210, + 506 + ], + "lines": [ + { + "bbox": [ + 104, + 492, + 213, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 213, + 509 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "With the development of deep learning methods, many deep networks have been proposed in recent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "years (Donahue et al., 2013; Krizhevsky et al., 2012; Long et al., 2015; Zhou et al., 2014). These", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "deep learning models show their powerful performance in various fields, such as image classification", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "and analysis, document analysis and recognition, natural language processing et al. 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The results show that a large, deep convolutional neural network is capable of achieving", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "recordbreaking results on a highly challenging dataset using purely supervised learning (Krizhevsky", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "et al., 2012). In order to popularize the deep convolutional neural network, Donahue ea al. proposed", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "DeCAF (Deep Convolutional Activation Feature) which is trained in a fully supervised fashion on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "a large, fixed set of object recognition tasks (Donahue et al., 2013). DeCAF provides a uniform", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "score": 1.0, + "content": "framework for researchers who can improve and change this framework on some specific tasks.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "However, its performance at scene recognition has not attained the same level of success. In order", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "to handle this problem, Zhou et al. introduce a new scene-centric database called Places with over 7", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "million labeled pictures of scenes. Then, they learn the deep features for scene recognition tasks by", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "using the same architecture as ImageNet, and establish new state-of-the-art results on several scene-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "centric datasets (Zhou et al., 2014). However, these methods based on convolutional operation need", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "very large scale training samples and a long training time. 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We have compared our MDA with some feature learning methods and deep learning mod-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "score": 1.0, + "content": "els on different domains of datasets (including handwritten digits recognition, speech recognition,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "historical document understanding, image classification, action recognition and so on). Extensive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "experiments demonstrate that MDA performs not only better than shallow feature learning models,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 445, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 445, + 264 + ], + "score": 1.0, + "content": "but also state-of-the-art deep learning models in small and middle scale applications.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 121, + 506, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 336, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 338, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 338, + 283 + ], + "score": 1.0, + "content": "The contributions of this work are highlighted as follows.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 268, + 338, + 283 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 286, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "1. We propose a novel structure to build a deep architecture. The first hidden layer has twice or", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "quadruple neurons as the input layer. Then we can use some feature learning models layer by layer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 491, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 491, + 321 + ], + "score": 1.0, + "content": "to learn the compact representations of data. Finally, we set the last layer as a softmax classifier.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 286, + 505, + 321 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 504, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "2. Traditional deep learning models in general need a large scale training data. Compared with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "traditional deep learning models, MDA can work better than traditional deep learning models in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "small and middle scale applications because the initialization of the weight matrices using MFA is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 359, + 305, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 305, + 371 + ], + "score": 1.0, + "content": "much better than that using random initialization.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 325, + 506, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 504, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "3. Our MDA can work well in different domains of datasets, such as handwritten digits, spoken", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 398 + ], + "score": 1.0, + "content": "letters and natural images. Extensive experiments demonstrate that MDA is a general model to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "handel small and middle scale data. On the other hand, for large scale datasets, like CIFAR-10,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 408, + 356, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 356, + 420 + ], + "score": 1.0, + "content": "MDA works comparatively with other deep learning methods.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 375, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 425, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 437 + ], + "score": 1.0, + "content": "The rest of this paper is organized as follows: In Section 2, we give a brief overview of related", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "work. In Section 3, we present the marginal Fisher analysis (MFA) and the proposed marginal deep", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "score": 1.0, + "content": "architectures (MDA) in detail. The experimental settings and results are reported in Section 4, while", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 458, + 353, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 353, + 470 + ], + "score": 1.0, + "content": "Section 5 concludes this paper with remarks and future work.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 425, + 506, + 470 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 210, + 506 + ], + "lines": [ + { + "bbox": [ + 104, + 492, + 213, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 213, + 509 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "With the development of deep learning methods, many deep networks have been proposed in recent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "years (Donahue et al., 2013; Krizhevsky et al., 2012; Long et al., 2015; Zhou et al., 2014). These", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "deep learning models show their powerful performance in various fields, such as image classification", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "and analysis, document analysis and recognition, natural language processing et al. In the area of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "image analysis, Hinton et al. proposed a large, deep convolutional neural network (Alex net) to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "classify the 1.2 million high-resolution images in the ImageNet. It uses efficient GPU to speed their", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 603 + ], + "score": 1.0, + "content": "method. The results show that a large, deep convolutional neural network is capable of achieving", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "recordbreaking results on a highly challenging dataset using purely supervised learning (Krizhevsky", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "et al., 2012). In order to popularize the deep convolutional neural network, Donahue ea al. proposed", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "DeCAF (Deep Convolutional Activation Feature) which is trained in a fully supervised fashion on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "a large, fixed set of object recognition tasks (Donahue et al., 2013). DeCAF provides a uniform", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 504, + 656 + ], + "score": 1.0, + "content": "framework for researchers who can improve and change this framework on some specific tasks.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "However, its performance at scene recognition has not attained the same level of success. In order", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "to handle this problem, Zhou et al. introduce a new scene-centric database called Places with over 7", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "million labeled pictures of scenes. Then, they learn the deep features for scene recognition tasks by", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "using the same architecture as ImageNet, and establish new state-of-the-art results on several scene-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "centric datasets (Zhou et al., 2014). However, these methods based on convolutional operation need", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "very large scale training samples and a long training time. They can not work well on small and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 212, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 212, + 734 + ], + "score": 1.0, + "content": "middle scale applications.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 522, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "In other domains, deep learning methods also achieve good performance. Hinton et al. represent", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 504, + 106 + ], + "score": 1.0, + "content": "the shared views of four research groups that have had recent successes in using DNNs for automat-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "ic speech recognition (ASR). The DNNs that contain many layers of nonlinear hidden units and a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "very large output layer can outperform Gaussian mixture models (GMMs) at acoustic modeling for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "speech recognition on a variety of data sets (Hinton et al., 2012a). In the area of genetics, Xiong", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "et al. use “deep learning” computer algorithms to derive a computational model that takes as input", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "DNA sequences and applies general rules to predict splicing in human tissues (Xiong et al., 2015).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "It reveals the genetic origins of disease and how strongly genetic variants affect RNA splicing. In", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "score": 1.0, + "content": "the area of natural language understanding, deep learning models have delivered strong results on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "topic classification, sentiment analysis et al. Sutskever et al. proposed a general approach, the Long", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Short-Term Memory (LSTM) architecture which can solve the general sequence to sequence prob-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "lems better than before (Sutskever et al., 2014). In addition, Hinton et al. proposed autoencoder", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "(AE) networks that is an effective way to learn the low-dimensional codes of high-dimensional data.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "Based on autoencoder, there are also have many excellent works to handle various tasks. Vincent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "et al. proposed a denoising autoencoder (DAE) which maked the learned representations robust to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 246, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 260 + ], + "score": 1.0, + "content": "partial corruption of the input data (Vincent et al., 2008). The denoising autoencoder which initialize", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "the deep architectures layer by layer is very similar with human visual system. Hinton et al. intro-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "duced random ‘dropout’ to prevent the overfitting which improve many benchmark tasks and obtain", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "new records for speech and object recognition (Hinton et al., 2012b). Then, Vincent et al. proposed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "stacked denoising autoencoders (SDAE) which based on stacking layers of stacked denoising au-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "toencoders (Vincent et al., 2010). It is very useful to learn the higher level representations and work", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "well on natural images and handwritten digits. However, for the same reason, they also need a large", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "scale training set and a long training time. They have no advantages to handle the small and middle", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 335, + 181, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 181, + 347 + ], + "score": 1.0, + "content": "scale applications.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "Moreover, in the field of feature learning models, dimensionality reduction plays a crucial role to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "handle the problems for compressing, visualizing high-dimensional data and avoiding the “curse of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "score": 1.0, + "content": "dimensionality” (van der Maaten et al., 2009; van der Maaten, 2007). Traditional dimensionality", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "reduction mainly can be classified into three types: linear or nonlinear, like principal components", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "analysis (PCA) (Jolliffe, 2002) and linearity preserving projection (LPP) (Niyogi, 2004) are linear", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "methods, stochastic neighbor embedding (SNE) (Hinton & Roweis, 2002) is a nonlinear method;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "score": 1.0, + "content": "supervised or unsupervised, such as marginal Fisher analysis (MFA) (Yan et al., 2007; Zhong et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "2013) and linear discriminant analysis (LDA) (Fisher, 1936) are supervised methods, PCA is an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "unsupervised method; local or global, like MFA and SNE are local methods, PCA is a global method.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "Many feature learning models based on geometry theory provide different solutions to the problem", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "of dimensionality reduction. Yan et al. proposed a general formulation about graph embedding", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "score": 1.0, + "content": "framework can exploit new dimensionality reduction algorithms (Yan et al., 2007). If only directly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "use some feature learning models to extract the good representation from original data, it often", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "eventually couldn’t get a good outcome. Considering this situation, we try to choose some excellent", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "feature learning models and combine them with some deep learning algorithms. MFA is one special", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "formulation of the graph embedding models based on this framework. It utilizes an intrinsic graph", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 528, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 539 + ], + "score": 1.0, + "content": "to characterize the intraclass compactness, and another penalty graph to characterize the interclass", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "separability. Our motivation is to combine the advantage of MFA and deep architectures and propose", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 550, + 335, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 335, + 561 + ], + "score": 1.0, + "content": "a new initialization method for deep learning algorithms.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 719 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 504, + 577 + ], + "score": 1.0, + "content": "There are also have some excellent works about feature learning models combined the deep archi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "tectures (Yuan et al.; George et al., 2014; Ngiam et al., 2011). Yuan et al. proposed an improved", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "multilayer learning model to solve the scene recognition task (Yuan et al.). This model overcome", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "the limitation of shallow, one-layer representations for scene recognition. Trigeorgis et al proposed", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "deep Semi-NMF, that is able to learn such hidden representations from different, unknown attributes", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "of a given dataset (George et al., 2014). Ngiam proposed a deep architectures to learn features over", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "multiple modalities (Ngiam et al., 2011). They showed that multi-modality feature learning is better", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "score": 1.0, + "content": "than one modality and achieved good performance on video and audio datasets. However, in gen-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "eral, we can only obtain data from one modality. In this work, we combine the advantages of MFA", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "and deep architectures, which based on stacked feature learning models (Zheng et al., 2014; 2015),", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 673, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 690 + ], + "score": 1.0, + "content": "then we use some deep learning tricks, like back propagation, denoising and dropout to fine tuning", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "the network. The advantage of this deep architecture is that we can learn the desirable weight matrix", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "even if the training data is not large enough. And compared with traditional deep learning models", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 708, + 483, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 483, + 721 + ], + "score": 1.0, + "content": "and shallow feature learning models, our MDA achieved state-of-the-art results in most cases.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 49.5 + } + ], + "page_idx": 2, + "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 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "In other domains, deep learning methods also achieve good performance. Hinton et al. represent", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 504, + 106 + ], + "score": 1.0, + "content": "the shared views of four research groups that have had recent successes in using DNNs for automat-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "ic speech recognition (ASR). The DNNs that contain many layers of nonlinear hidden units and a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "very large output layer can outperform Gaussian mixture models (GMMs) at acoustic modeling for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "speech recognition on a variety of data sets (Hinton et al., 2012a). In the area of genetics, Xiong", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "et al. use “deep learning” computer algorithms to derive a computational model that takes as input", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "DNA sequences and applies general rules to predict splicing in human tissues (Xiong et al., 2015).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "It reveals the genetic origins of disease and how strongly genetic variants affect RNA splicing. In", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 184 + ], + "score": 1.0, + "content": "the area of natural language understanding, deep learning models have delivered strong results on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "topic classification, sentiment analysis et al. Sutskever et al. proposed a general approach, the Long", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Short-Term Memory (LSTM) architecture which can solve the general sequence to sequence prob-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "lems better than before (Sutskever et al., 2014). In addition, Hinton et al. proposed autoencoder", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "(AE) networks that is an effective way to learn the low-dimensional codes of high-dimensional data.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "Based on autoencoder, there are also have many excellent works to handle various tasks. Vincent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "et al. proposed a denoising autoencoder (DAE) which maked the learned representations robust to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 246, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 260 + ], + "score": 1.0, + "content": "partial corruption of the input data (Vincent et al., 2008). The denoising autoencoder which initialize", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "the deep architectures layer by layer is very similar with human visual system. Hinton et al. intro-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "duced random ‘dropout’ to prevent the overfitting which improve many benchmark tasks and obtain", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "new records for speech and object recognition (Hinton et al., 2012b). Then, Vincent et al. proposed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "stacked denoising autoencoders (SDAE) which based on stacking layers of stacked denoising au-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "toencoders (Vincent et al., 2010). It is very useful to learn the higher level representations and work", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "well on natural images and handwritten digits. However, for the same reason, they also need a large", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "scale training set and a long training time. They have no advantages to handle the small and middle", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 335, + 181, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 181, + 347 + ], + "score": 1.0, + "content": "scale applications.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 81, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "Moreover, in the field of feature learning models, dimensionality reduction plays a crucial role to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "handle the problems for compressing, visualizing high-dimensional data and avoiding the “curse of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 386 + ], + "score": 1.0, + "content": "dimensionality” (van der Maaten et al., 2009; van der Maaten, 2007). Traditional dimensionality", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "reduction mainly can be classified into three types: linear or nonlinear, like principal components", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "analysis (PCA) (Jolliffe, 2002) and linearity preserving projection (LPP) (Niyogi, 2004) are linear", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "methods, stochastic neighbor embedding (SNE) (Hinton & Roweis, 2002) is a nonlinear method;", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "score": 1.0, + "content": "supervised or unsupervised, such as marginal Fisher analysis (MFA) (Yan et al., 2007; Zhong et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "2013) and linear discriminant analysis (LDA) (Fisher, 1936) are supervised methods, PCA is an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "unsupervised method; local or global, like MFA and SNE are local methods, PCA is a global method.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "Many feature learning models based on geometry theory provide different solutions to the problem", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "of dimensionality reduction. Yan et al. proposed a general formulation about graph embedding", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "score": 1.0, + "content": "framework can exploit new dimensionality reduction algorithms (Yan et al., 2007). If only directly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "use some feature learning models to extract the good representation from original data, it often", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "eventually couldn’t get a good outcome. Considering this situation, we try to choose some excellent", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "feature learning models and combine them with some deep learning algorithms. MFA is one special", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "formulation of the graph embedding models based on this framework. 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Our motivation is to combine the advantage of MFA and deep architectures and propose", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 550, + 335, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 335, + 561 + ], + "score": 1.0, + "content": "a new initialization method for deep learning algorithms.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 351, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 719 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 504, + 577 + ], + "score": 1.0, + "content": "There are also have some excellent works about feature learning models combined the deep archi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "tectures (Yuan et al.; George et al., 2014; Ngiam et al., 2011). Yuan et al. proposed an improved", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "multilayer learning model to solve the scene recognition task (Yuan et al.). This model overcome", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "the limitation of shallow, one-layer representations for scene recognition. Trigeorgis et al proposed", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "deep Semi-NMF, that is able to learn such hidden representations from different, unknown attributes", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "of a given dataset (George et al., 2014). Ngiam proposed a deep architectures to learn features over", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "multiple modalities (Ngiam et al., 2011). They showed that multi-modality feature learning is better", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "score": 1.0, + "content": "than one modality and achieved good performance on video and audio datasets. However, in gen-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "eral, we can only obtain data from one modality. In this work, we combine the advantages of MFA", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "and deep architectures, which based on stacked feature learning models (Zheng et al., 2014; 2015),", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 673, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 690 + ], + "score": 1.0, + "content": "then we use some deep learning tricks, like back propagation, denoising and dropout to fine tuning", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "the network. The advantage of this deep architecture is that we can learn the desirable weight matrix", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "even if the training data is not large enough. 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This operation helps for learning more stable structure and avoids", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 430, + 258, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 258, + 443 + ], + "score": 1.0, + "content": "the overfitting problem in most cases.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 385, + 506, + 443 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 455, + 174, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 176, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 176, + 468 + ], + "score": 1.0, + "content": "3.6 DROPOUT", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "As the same reason with denoising operation, dropout is a trick to prevent overfitting (Hinton et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 500 + ], + "score": 1.0, + "content": "2012b). 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Covertype 4 contains", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "geological and map-based data from four wilderness areas located in the Roosevelt National Forest", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "of northern Colorado. It includes 15120 training samples and 565892 test samples from 7 classes", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "with 54 dimensional features. 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DATASETSTATISTICS
datasetntraintest[Vd(target)
USPS92987291200710256(32)
Isolet77976238155926614(308)
Sensor5850946816116931148(24)
Covertype58101215120565892754(27)
Ibnsina20668175433125174200(100)
CIFAR-10600005000010000103072(64)
CMU49445393(24)
", + "type": "table", + "image_path": "6b056ca116756f30d622430404ff53ec91ca830095ce7a1e7fcc6d86b902104f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 206, + 101, + 405, + 116.2 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 206, + 116.2, + 405, + 131.4 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 206, + 131.4, + 405, + 146.6 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 206, + 146.6, + 405, + 161.79999999999998 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 206, + 161.79999999999998, + 405, + 176.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 197, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 196, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 506, + 210 + ], + "score": 1.0, + "content": "(and pre-processed) audio signal. 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Covertype 4 contains", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "geological and map-based data from four wilderness areas located in the Roosevelt National Forest", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "of northern Colorado. It includes 15120 training samples and 565892 test samples from 7 classes", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "with 54 dimensional features. The task is to identify forest cover type from cartographic variables.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "score": 1.0, + "content": "For the IbnSina 5 ancient Arabic document data set, we use 50 pages of the manuscript for training", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "(17543 training samples) and 10 pages for testing (3125 test samples). 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The CMU mocap data", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "set includes three categories, namely, jumping, running and walking. We choose 49 video sequences", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "from four subjects. For each sequence, the features are generated using Lawrences method 8, with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "dimensionality 93 (Zhong et al., 2010). By reason of the few samples of CMU, we adopt 10-fold", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "cross-validation in our experiments and use the average error rate and standard deviation to evaluate", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 402, + 176, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 176, + 414 + ], + "score": 1.0, + "content": "the performance.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 427, + 352, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 353, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 353, + 440 + ], + "score": 1.0, + "content": "4.2 CLASSIFICATION ON FIVE BENCHMARK DATA SETS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 230, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 232, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 232, + 462 + ], + "score": 1.0, + "content": "4.2.1 BASELINE METHODS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 469, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "In order to evaluate the performance of MDA, we compared our MDA with 5 deep learning models", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "include autoencoder (AE) (Hinton & Salakhutdinov, 2006), stacked autoencoders, denoising au-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "toencoders (Vincent et al., 2008), stacked denoising autoencoders (Vincent et al., 2010) and stacked", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 504, + 514 + ], + "score": 1.0, + "content": "denoising autoencoders with dropout, 2 feature learning models, MFA (Zhong et al., 2013; Yan", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "et al., 2007) and PCA (Jolliffe, 2002), PCA deep architecture base on our uniform framework and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 286, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 286, + 536 + ], + "score": 1.0, + "content": "the classification accuracy on original space.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 548, + 253, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 254, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 254, + 561 + ], + "score": 1.0, + "content": "4.2.2 EXPERIMENTAL SETTINGS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "All of the deep learning methods have the same settings. 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DATASETSTATISTICS
datasetntraintest[Vd(target)
USPS92987291200710256(32)
Isolet77976238155926614(308)
Sensor5850946816116931148(24)
Covertype58101215120565892754(27)
Ibnsina20668175433125174200(100)
CIFAR-10600005000010000103072(64)
CMU49445393(24)
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The CIFAR-10 dataset consists of", + "type": "text" + }, + { + "bbox": [ + 282, + 325, + 344, + 335 + ], + "score": 0.87, + "content": "6 0 0 0 0 3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "colour images in 10 classes, with 6000", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "images per class. There are 50000 training images and 10000 test images. We also test our MDA on", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "a specific task which use the CMU motion capture (CMU mocap) data set 7. The CMU mocap data", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "set includes three categories, namely, jumping, running and walking. We choose 49 video sequences", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "from four subjects. For each sequence, the features are generated using Lawrences method 8, with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "dimensionality 93 (Zhong et al., 2010). By reason of the few samples of CMU, we adopt 10-fold", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "cross-validation in our experiments and use the average error rate and standard deviation to evaluate", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 402, + 176, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 176, + 414 + ], + "score": 1.0, + "content": "the performance.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 313, + 506, + 414 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 427, + 352, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 353, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 353, + 440 + ], + "score": 1.0, + "content": "4.2 CLASSIFICATION ON FIVE BENCHMARK DATA SETS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 230, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 232, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 232, + 462 + ], + "score": 1.0, + "content": "4.2.1 BASELINE METHODS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 469, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "In order to evaluate the performance of MDA, we compared our MDA with 5 deep learning models", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "include autoencoder (AE) (Hinton & Salakhutdinov, 2006), stacked autoencoders, denoising au-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "toencoders (Vincent et al., 2008), stacked denoising autoencoders (Vincent et al., 2010) and stacked", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 504, + 514 + ], + "score": 1.0, + "content": "denoising autoencoders with dropout, 2 feature learning models, MFA (Zhong et al., 2013; Yan", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "et al., 2007) and PCA (Jolliffe, 2002), PCA deep architecture base on our uniform framework and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 286, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 286, + 536 + ], + "score": 1.0, + "content": "the classification accuracy on original space.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 468, + 506, + 536 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 548, + 253, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 254, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 254, + 561 + ], + "score": 1.0, + "content": "4.2.2 EXPERIMENTAL SETTINGS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "All of the deep learning methods have the same settings. The size of minibatch was set to 100, the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "learning rate and momentum were the default value 1 and 0.5, the number of epoch was set to 400,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 250, + 603 + ], + "score": 1.0, + "content": "the dropout rate and denoising rate", + "type": "text" + }, + { + "bbox": [ + 251, + 592, + 258, + 600 + ], + "score": 0.72, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "were set to 0.1. For the AE and SAE, weight penalty of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 120, + 612 + ], + "score": 0.58, + "content": "L 2", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 600, + 187, + 614 + ], + "score": 1.0, + "content": "norm was set to", + "type": "text" + }, + { + "bbox": [ + 187, + 601, + 209, + 612 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 600, + 506, + 614 + ], + "score": 1.0, + "content": ". For MFA, the number of nearest neighbors for constructing the intrinsic", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "graph was set to 5, while that for constructing the penalty graph was set to 20. The target spaces of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "MFA and PCA on different data sets were showed in Tab 1. For the USPS data set, The architecture", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 150, + 646 + ], + "score": 1.0, + "content": "was set to", + "type": "text" + }, + { + "bbox": [ + 150, + 634, + 297, + 645 + ], + "score": 0.9, + "content": "2 5 6 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 - 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 634, + 506, + 646 + ], + "score": 1.0, + "content": ". For the Isolet data set ,the architecture was set to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 211, + 656 + ], + "score": 0.82, + "content": "6 1 7 - 1 3 2 4 - 6 1 7 - 3 0 8", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 644, + 422, + 657 + ], + "score": 1.0, + "content": ". For the Sensor data set, the architecture was set to", + "type": "text" + }, + { + "bbox": [ + 422, + 645, + 501, + 656 + ], + "score": 0.88, + "content": "4 8 - 9 6 - 4 8 - 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 569, + 506, + 657 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 115, + 122, + 496, + 173 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 89, + 504, + 122 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "score": 1.0, + "content": "Table 2: The classification accuracy on different datasets. “ORIG” represents the results obtained", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 112 + ], + "score": 1.0, + "content": "in the original data space. ‘PDA’ represents the PCA deep architecture. ‘MDA’ represents the MFA", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 111, + 358, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 358, + 123 + ], + "score": 1.0, + "content": "deep architecture. The best reslut is highlighted with boldface.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 115, + 122, + 496, + 173 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 122, + 496, + 173 + ], + "spans": [ + { + "bbox": [ + 115, + 122, + 496, + 173 + ], + "score": 0.976, + "html": "
MethodORIGPCAMFAAESAEDAE(dropout)DAESDAEPDAMDA
USPS0.83660.94020.93920.94020.94020.95810.95320.94520.95860.9601
Isolet0.94670.92370.92690.95190.95060.95960.95190.95430.95840.9622
Sensor0.81510.80420.82340.79950.83250.81780.77640.78700.85820.8558
Covertype0.55760.55960.60570.74050.55760.70930.73970.74400.74580.7589
Ibnsina0.89570.91900.92060.93630.91840.94020.93700.92610.94210.9491
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DatasetNoneTwiceQuadrupleOctuple
USPS256-128-64-32256-512-256-128-64-32256-1024-512-256-128-64-32256-2048-1024-512-256-128-64-32
Isolet617-308617-1324-617-308617-2648-1324-617-308617-5296-2648-1324-617-308
Sensor48-2448-96-2448-192-96-2448-384-192-96-24
Covertype54-2754-108-2754-216-108-54-2754-432-216-108-54-27
Ibnsina200-100200-400-200-100200-800-400-200-100200-1600-800-400-200-100
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Finally, for Ibnsina", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 324, + 349, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 247, + 335 + ], + "score": 1.0, + "content": "data set, the architecture was set to", + "type": "text" + }, + { + "bbox": [ + 248, + 324, + 345, + 335 + ], + "score": 0.84, + "content": "2 0 0 - 4 0 0 - 2 0 0 - 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 324, + 349, + 335 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 348, + 253, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 254, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 254, + 361 + ], + "score": 1.0, + "content": "4.2.3 CLASSIFICATION RESULTS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "The experimental results are shown in Tab. 2. We can see that our MDA achieves the best results on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 380, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 391 + ], + "score": 1.0, + "content": "four dataset except the Sensor dataset, but MDA achieves the second best result on Sensor data set", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "score": 1.0, + "content": "and only below the PDA. The PDA achieves the best result on Sensor data set and the second best", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "results on other data sets. These results demonstrate that our uniform deep architectures achieve the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 426 + ], + "score": 1.0, + "content": "good performance in most case. In addition, MDA not only outperform the traditional deep learning", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 435 + ], + "score": 1.0, + "content": "models, but also the shallow feature learning models. It shows that our deep architectures based", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "on stacked some feature learning models can learn the better feature than shallow feature learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 445, + 141, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 141, + 457 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 471, + 187, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 189, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 189, + 485 + ], + "score": 1.0, + "content": "4.3 EVALUATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 293, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 295, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 295, + 505 + ], + "score": 1.0, + "content": "4.3.1 DIFFERENT STRUCTURES FOR MDA", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "In order to evaluate the desired structures of MDA, we changed the node’s number of the second", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 523, + 504, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 422, + 535 + ], + "score": 1.0, + "content": "layer. 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DatasetNoneTwiceQuadrupleOctuple
USPS0.04630.03990.04330.0453
Isolet0.03980.03780.04170.0430
Sensor0.21720.14420.15590.7856
Covertype0.38760.38780.24110.3806
Ibnsina0.06430.05090.06140.0858
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MethodORIGPCAMFAAESAEDAE(dropout)DAESDAEPDAMDA
USPS0.83660.94020.93920.94020.94020.95810.95320.94520.95860.9601
Isolet0.94670.92370.92690.95190.95060.95960.95190.95430.95840.9622
Sensor0.81510.80420.82340.79950.83250.81780.77640.78700.85820.8558
Covertype0.55760.55960.60570.74050.55760.70930.73970.74400.74580.7589
Ibnsina0.89570.91900.92060.93630.91840.94020.93700.92610.94210.9491
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DatasetNoneTwiceQuadrupleOctuple
USPS256-128-64-32256-512-256-128-64-32256-1024-512-256-128-64-32256-2048-1024-512-256-128-64-32
Isolet617-308617-1324-617-308617-2648-1324-617-308617-5296-2648-1324-617-308
Sensor48-2448-96-2448-192-96-2448-384-192-96-24
Covertype54-2754-108-2754-216-108-54-2754-432-216-108-54-27
Ibnsina200-100200-400-200-100200-800-400-200-100200-1600-800-400-200-100
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The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 579, + 300, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 300, + 589 + ], + "score": 1.0, + "content": "structures of other data sets are shown in Tab. 3.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 512, + 505, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 504, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "The experimental results are shown in Tab. 4. When the the number of nodes of the second layer is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 504, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 504, + 618 + ], + "score": 1.0, + "content": "as twice as the input layer, MDA achieved the minimum classification error on all data sets except", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "the Covertype data set. When the number of nodes of the second layer is as quadruple as the input", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "layer, MDA get the worst result on Covertype data set. We can conclude that MDA can work well", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 302, + 462, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 462, + 315 + ], + "score": 1.0, + "content": "when the number of nodes of the second layer is as twice or quadruple as the input layer.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 595, + 506, + 630 + ] + }, + { + "type": "table", + "bbox": [ + 215, + 679, + 396, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 657, + 503, + 678 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 671 + ], + "score": 1.0, + "content": "Table 4: The classification error with different structures on 5 data sets. The best results (minimum", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 668, + 253, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 253, + 679 + ], + "score": 1.0, + "content": "error) are highlighted with boldface.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "table_body", + "bbox": [ + 215, + 679, + 396, + 730 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 215, + 679, + 396, + 730 + ], + "spans": [ + { + "bbox": [ + 215, + 679, + 396, + 730 + ], + "score": 0.97, + "html": "
DatasetNoneTwiceQuadrupleOctuple
USPS0.04630.03990.04330.0453
Isolet0.03980.03780.04170.0430
Sensor0.21720.14420.15590.7856
Covertype0.38760.38780.24110.3806
Ibnsina0.06430.05090.06140.0858
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The number of hidden layers1234567
USPS0.057800.050320.050820.051820.039900.057300.05132
Isolet0.039770.041690.037850.038490.054520.042340.04683
Covertype0.271710.256010.241100.267310.27491
Sensor0.204570.155220.144200.170270.15567
Ibnsina0.056960.051840.050880.060160.06720
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(a）gray-CIFAR10(b) CMU mocap
MethodErrorMethodError
AE0.5117AE0.3970±0.1343
SAE0.5252SAE0.4106 ± 0.1648
DAE(dropout)0.5090DAE(dropout)0.3970±0.1343
DAE0.5176DAE0.4061 ± 0.1540
SDAE0.5113SDAE0.3970 ± 0.1343
PDA0.5085PDA0.3591 ± 0.0815
MDA0.4947MDA0.3636 ± 0.0958
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We can conclude that MDA can work well", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 302, + 462, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 462, + 315 + ], + "score": 1.0, + "content": "when the number of nodes of the second layer is as twice or quadruple as the input layer.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 326, + 363, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 365, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 365, + 339 + ], + "score": 1.0, + "content": "4.3.2 DIFFERENT NUMBER OF HIDDEN LAYERS FOR MDA", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "In order to evaluate how many hidden layers adapt to different datasets, we designed some experi-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 362, + 369 + ], + "score": 1.0, + "content": "ments which have different number of hidden layers. We used", + "type": "text" + }, + { + "bbox": [ + 363, + 358, + 390, + 368 + ], + "score": 0.87, + "content": "1 \\sim 7", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "hidden layers on USPS and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 181, + 381 + ], + "score": 1.0, + "content": "Isolet datasets and", + "type": "text" + }, + { + "bbox": [ + 181, + 369, + 205, + 379 + ], + "score": 0.87, + "content": "1 \\sim 5", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "hidden layers on Covertype, Sensor and Ibnsina datasets. The experimental", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 380, + 285, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 285, + 392 + ], + "score": 1.0, + "content": "settings were same as previous experiments.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "Tab. 5 shows the classification error on 5 datasets with different hidden layers. All the datasets", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "achieved the best results when hidden layer’s number is 3 except USPS dataset. The USPS dataset", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 351, + 430 + ], + "score": 1.0, + "content": "achieved the best result when hidden layer’s number is 5. As", + "type": "text" + }, + { + "bbox": [ + 351, + 419, + 376, + 428 + ], + "score": 0.88, + "content": "1 \\sim 3", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "hidden layers, with the increase", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "of the number of layers, the classification error is decreasing on all datasets. As small and middle", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "scale applications, we don’t need very deep architectures to handle it. As large scale applications,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 451, + 371, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 371, + 464 + ], + "score": 1.0, + "content": "we can design deeper architectures to achieve better performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 476, + 370, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 371, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 371, + 489 + ], + "score": 1.0, + "content": "4.4 CLASSIFICATION ON LARGE SCALE DATASET CIFAR-10", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 503, + 531 + ], + "lines": [ + { + "bbox": [ + 107, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "The previous section introduced the advantages of MDA on small and middle scale applications. In", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "order to evaluate the universality of MDA, we chose a relatively large scale dataset CIFAR-10 to test", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 519, + 213, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 213, + 532 + ], + "score": 1.0, + "content": "the performance of MDA.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "In our experiments, we first transformed the color images to gray images in order to reduce the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "dimensionality of input. Then we took one sample as a 1024 dimensional vector which is the input", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 559, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 443, + 570 + ], + "score": 1.0, + "content": "of our MDA. So, we can call this data set gray-CIFAR10. The architecture was set to", + "type": "text" + }, + { + "bbox": [ + 444, + 559, + 504, + 569 + ], + "score": 0.87, + "content": "1 0 2 4 - 2 0 4 8 -", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 217, + 580 + ], + "score": 0.59, + "content": "1 0 2 4 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 .", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "the minibatch’s size was set to 100, the dropout ratio and denoising ratio", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "were set to 0.1, the number of epoch was set to 400, the learning rate was set to 1, the momentum", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 369, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 369, + 604 + ], + "score": 1.0, + "content": "was set to 0.5. We compared our MDA with previous 6 methods.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 108, + 608, + 504, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "Table. 6(a) shows the classification error on gray-CIFAR10, we can see that PDA and MDA achieved", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 620, + 504, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 620, + 504, + 631 + ], + "score": 1.0, + "content": "the best results in these 7 methods. However, all of the methods on this framework didn’t perform", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 630, + 267, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 267, + 642 + ], + "score": 1.0, + "content": "well because we use the gray operation.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 107, + 655, + 324, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 325, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 325, + 668 + ], + "score": 1.0, + "content": "4.5 CLASSIFICATION ON CMU MOCAP DATA SET", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "CMU mocap data set is a very small dataset that only has 49 samples. Traditional deep learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "methods didn’t work well in these kind of applications. We test our MDA and PDA and compared", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "them with other 5 deep learning models. The architectures for all deep models (except the PDA)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 151, + 723 + ], + "score": 1.0, + "content": "were set to", + "type": "text" + }, + { + "bbox": [ + 152, + 710, + 251, + 720 + ], + "score": 0.89, + "content": "9 3 - 1 8 6 - 9 3 - 4 7 - 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 709, + 505, + 723 + ], + "score": 1.0, + "content": ". Specially, since the CMU mocap data set only has 49 samples,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "the PCA method only reduce the dimensionality to 49 at most, so the architecture of PDA was set to", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "page_idx": 8, + "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 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 137, + 100, + 475, + 151 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 134, + 89, + 475, + 100 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 87, + 478, + 102 + ], + "spans": [ + { + "bbox": [ + 133, + 87, + 478, + 102 + ], + "score": 1.0, + "content": "Table 5: The classification error on 5 datasets with different number of hidden layers.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 137, + 100, + 475, + 151 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 100, + 475, + 151 + ], + "spans": [ + { + "bbox": [ + 137, + 100, + 475, + 151 + ], + "score": 0.976, + "html": "
The number of hidden layers1234567
USPS0.057800.050320.050820.051820.039900.057300.05132
Isolet0.039770.041690.037850.038490.054520.042340.04683
Covertype0.271710.256010.241100.267310.27491
Sensor0.204570.155220.144200.170270.15567
Ibnsina0.056960.051840.050880.060160.06720
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(a）gray-CIFAR10(b) CMU mocap
MethodErrorMethodError
AE0.5117AE0.3970±0.1343
SAE0.5252SAE0.4106 ± 0.1648
DAE(dropout)0.5090DAE(dropout)0.3970±0.1343
DAE0.5176DAE0.4061 ± 0.1540
SDAE0.5113SDAE0.3970 ± 0.1343
PDA0.5085PDA0.3591 ± 0.0815
MDA0.4947MDA0.3636 ± 0.0958
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We used", + "type": "text" + }, + { + "bbox": [ + 363, + 358, + 390, + 368 + ], + "score": 0.87, + "content": "1 \\sim 7", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "hidden layers on USPS and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 181, + 381 + ], + "score": 1.0, + "content": "Isolet datasets and", + "type": "text" + }, + { + "bbox": [ + 181, + 369, + 205, + 379 + ], + "score": 0.87, + "content": "1 \\sim 5", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "hidden layers on Covertype, Sensor and Ibnsina datasets. The experimental", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 380, + 285, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 285, + 392 + ], + "score": 1.0, + "content": "settings were same as previous experiments.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 347, + 505, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "Tab. 5 shows the classification error on 5 datasets with different hidden layers. All the datasets", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "achieved the best results when hidden layer’s number is 3 except USPS dataset. The USPS dataset", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 351, + 430 + ], + "score": 1.0, + "content": "achieved the best result when hidden layer’s number is 5. As", + "type": "text" + }, + { + "bbox": [ + 351, + 419, + 376, + 428 + ], + "score": 0.88, + "content": "1 \\sim 3", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "hidden layers, with the increase", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "of the number of layers, the classification error is decreasing on all datasets. As small and middle", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "scale applications, we don’t need very deep architectures to handle it. As large scale applications,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 451, + 371, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 371, + 464 + ], + "score": 1.0, + "content": "we can design deeper architectures to achieve better performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 395, + 506, + 464 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 476, + 370, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 371, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 371, + 489 + ], + "score": 1.0, + "content": "4.4 CLASSIFICATION ON LARGE SCALE DATASET CIFAR-10", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 503, + 531 + ], + "lines": [ + { + "bbox": [ + 107, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 107, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "The previous section introduced the advantages of MDA on small and middle scale applications. In", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "order to evaluate the universality of MDA, we chose a relatively large scale dataset CIFAR-10 to test", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 519, + 213, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 213, + 532 + ], + "score": 1.0, + "content": "the performance of MDA.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 497, + 505, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "In our experiments, we first transformed the color images to gray images in order to reduce the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "dimensionality of input. Then we took one sample as a 1024 dimensional vector which is the input", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 559, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 443, + 570 + ], + "score": 1.0, + "content": "of our MDA. So, we can call this data set gray-CIFAR10. The architecture was set to", + "type": "text" + }, + { + "bbox": [ + 444, + 559, + 504, + 569 + ], + "score": 0.87, + "content": "1 0 2 4 - 2 0 4 8 -", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 217, + 580 + ], + "score": 0.59, + "content": "1 0 2 4 - 5 1 2 - 2 5 6 - 1 2 8 - 6 4 .", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "the minibatch’s size was set to 100, the dropout ratio and denoising ratio", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "were set to 0.1, the number of epoch was set to 400, the learning rate was set to 1, the momentum", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 369, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 369, + 604 + ], + "score": 1.0, + "content": "was set to 0.5. We compared our MDA with previous 6 methods.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 537, + 505, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 608, + 504, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 506, + 621 + ], + "score": 1.0, + "content": "Table. 6(a) shows the classification error on gray-CIFAR10, we can see that PDA and MDA achieved", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 620, + 504, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 620, + 504, + 631 + ], + "score": 1.0, + "content": "the best results in these 7 methods. However, all of the methods on this framework didn’t perform", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 630, + 267, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 267, + 642 + ], + "score": 1.0, + "content": "well because we use the gray operation.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 607, + 506, + 642 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 655, + 324, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 325, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 325, + 668 + ], + "score": 1.0, + "content": "4.5 CLASSIFICATION ON CMU MOCAP DATA SET", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "CMU mocap data set is a very small dataset that only has 49 samples. Traditional deep learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "methods didn’t work well in these kind of applications. We test our MDA and PDA and compared", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "them with other 5 deep learning models. The architectures for all deep models (except the PDA)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 151, + 723 + ], + "score": 1.0, + "content": "were set to", + "type": "text" + }, + { + "bbox": [ + 152, + 710, + 251, + 720 + ], + "score": 0.89, + "content": "9 3 - 1 8 6 - 9 3 - 4 7 - 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 709, + 505, + 723 + ], + "score": 1.0, + "content": ". Specially, since the CMU mocap data set only has 49 samples,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "the PCA method only reduce the dimensionality to 49 at most, so the architecture of PDA was set to", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 676, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 168, + 93 + ], + "score": 0.51, + "content": "9 3 - 1 8 6 - 2 4 ", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 82, + 506, + 95 + ], + "score": 1.0, + "content": ". The denoising ratio and dropout ratio were set to 0.1 on DAE, DAE with dropout,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 377, + 106 + ], + "score": 1.0, + "content": "SDAE, SAE, PDA and MDA. The weight penalty on AE was set to", + "type": "text" + }, + { + "bbox": [ + 377, + 93, + 398, + 104 + ], + "score": 0.91, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 92, + 506, + 106 + ], + "score": 1.0, + "content": ". The learning rate was set", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "to 0.01, the momentum was set to 0.5 and the number of epoch is set to 600. The experiment was", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 430, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 430, + 127 + ], + "score": 1.0, + "content": "test on 10-fold cross validation. The experimental results are shown in Tab. 6(b).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "In Tab. 6(b), our PDA and MDA achieved the best results in this dataset and have lower standard", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "deviation than other deep learning models. It demonstrates that our PDA and MDA are more stable", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "than other deep learning models. The traditional autoencoder, SDAE, DAE with dropout achieved", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 351, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 351, + 176 + ], + "score": 1.0, + "content": "the same result in this dataset and better than SAE and DAE.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 192, + 195, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 197, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 197, + 208 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "In this paper, we proposed a novel deep learning framework that based on stacked some feature", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "learning models to handle small or middle data sets. Then we introduce MFA in this framework,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 253 + ], + "score": 1.0, + "content": "called MDA. The deep learning tricks like backpropagation, denoising and dropout operation are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "applied on MDA to improve its performance. Extensive experiments on 7 different type data sets", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "score": 1.0, + "content": "demonstrate that MDA performs not only better than shallow feature learning models, but also state-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 284 + ], + "score": 1.0, + "content": "of-the-art deep learning models on small and middle scale applications. The evaluation of MDA", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 297 + ], + "score": 1.0, + "content": "show that how to adjust the parameters make the MDA work well. 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DATASETSTATISTICS
datasetntraintest[Vd(target)
USPS92987291200710256(32)
Isolet77976238155926614(308)
Sensor5850946816116931148(24)
Covertype58101215120565892754(27)
Ibnsina20668175433125174200(100)
CIFAR-10600005000010000103072(64)
CMU49445393(24)
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MethodORIGPCAMFAAESAEDAE(dropout)DAESDAEPDAMDA
USPS0.83660.94020.93920.94020.94020.95810.95320.94520.95860.9601
Isolet0.94670.92370.92690.95190.95060.95960.95190.95430.95840.9622
Sensor0.81510.80420.82340.79950.83250.81780.77640.78700.85820.8558
Covertype0.55760.55960.60570.74050.55760.70930.73970.74400.74580.7589
Ibnsina0.89570.91900.92060.93630.91840.94020.93700.92610.94210.9491
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DatasetNoneTwiceQuadrupleOctuple
USPS0.04630.03990.04330.0453
Isolet0.03980.03780.04170.0430
Sensor0.21720.14420.15590.7856
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Ibnsina0.06430.05090.06140.0858
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DatasetNoneTwiceQuadrupleOctuple
USPS256-128-64-32256-512-256-128-64-32256-1024-512-256-128-64-32256-2048-1024-512-256-128-64-32
Isolet617-308617-1324-617-308617-2648-1324-617-308617-5296-2648-1324-617-308
Sensor48-2448-96-2448-192-96-2448-384-192-96-24
Covertype54-2754-108-2754-216-108-54-2754-432-216-108-54-27
Ibnsina200-100200-400-200-100200-800-400-200-100200-1600-800-400-200-100
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The number of hidden layers1234567
USPS0.057800.050320.050820.051820.039900.057300.05132
Isolet0.039770.041690.037850.038490.054520.042340.04683
Covertype0.271710.256010.241100.267310.27491
Sensor0.204570.155220.144200.170270.15567
Ibnsina0.056960.051840.050880.060160.06720
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(a）gray-CIFAR10(b) CMU mocap
MethodErrorMethodError
AE0.5117AE0.3970±0.1343
SAE0.5252SAE0.4106 ± 0.1648
DAE(dropout)0.5090DAE(dropout)0.3970±0.1343
DAE0.5176DAE0.4061 ± 0.1540
SDAE0.5113SDAE0.3970 ± 0.1343
PDA0.5085PDA0.3591 ± 0.0815
MDA0.4947MDA0.3636 ± 0.0958
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0000000000000000000000000000000000000000..5654479c81c950e201cc0634d83d80f0545b759b --- /dev/null +++ b/parse/train/HkuGJ3kCb/HkuGJ3kCb.md @@ -0,0 +1,688 @@ +# ALL-BUT-THE-TOP: SIMPLE AND EFFECTIVE POSTPROCESSING FOR WORD REPRESENTATIONS + +Jiaqi Mu, Pramod Viswanath University of Illinois at Urbana Champaign {jiaqimu2, pramodv}@illinois.edu + +# ABSTRACT + +Real-valued word representations have transformed NLP applications; popular examples are word2vec and GloVe, recognized for their ability to capture linguistic regularities. In this paper, we demonstrate a very simple, and yet counter-intuitive, postprocessing technique – eliminate the common mean vector and a few top dominating directions from the word vectors – that renders off-the-shelf representations even stronger. The postprocessing is empirically validated on a variety of lexical-level intrinsic tasks (word similarity, concept categorization, word analogy) and sentence-level tasks (semantic textural similarity and text classification) on multiple datasets and with a variety of representation methods and hyperparameter choices in multiple languages; in each case, the processed representations are consistently better than the original ones. + +# 1 INTRODUCTION + +Words and their interactions (as sentences) are the basic units of natural language. Although words are readily modeled as discrete atomic units, this is unable to capture the relation between the words. Recent distributional real-valued representations of words (examples: word2vec, GloVe) have transformed the landscape of NLP applications – for instance, text classification (Socher et al., 2013b; Maas et al., 2011; Kim, 2014), machine translation (Sutskever et al., 2014; Bahdanau et al., 2014) and knowledge base completion (Bordes et al., 2013; Socher et al., 2013a). The success comes from the geometry of the representations that efficiently captures linguistic regularities: the semantic similarity of words is well captured by the similarity of the corresponding vector representations. + +A variety of approaches have been proposed in recent years to learn the word representations: Collobert et al. (2011); Turian et al. (2010) learn the representations via semi-supervised learning by jointly training the language model and downstream applications; Bengio et al. (2003); Mikolov et al. (2010); Huang et al. (2012) do so by fitting the data into a neural network language model; Mikolov et al. (2013); Mnih & Hinton (2007) by log-linear models; and Dhillon et al. (2012); Pennington et al. (2014); Levy & Goldberg (2014); Stratos et al. (2015); Arora et al. (2016) by producing a lowdimensional representation of the cooccurrence statistics. Despite the wide disparity of algorithms to induce word representations, the performance of several of the recent methods is roughly similar on a variety of intrinsic and extrinsic evaluation testbeds. + +In this paper, we find that a simple processing renders the off-the-shelf existing representations even stronger. The proposed algorithm is motivated by the following observation. + +Observation Every representation we tested, in many languages, has the following properties: + +• The word representations have non-zero mean – indeed, word vectors share a large common vector (with norm up to a half of the average norm of word vector). • After removing the common mean vector, the representations are far from isotropic – indeed, much of the energy of most word vectors is contained in a very low dimensional subspace (say, 8 dimensions out of 300). + +Implication Since all words share the same common vector and have the same dominating directions, and such vector and directions strongly influence the word representations in the same way, we propose to eliminate them by: (a) removing the nonzero mean vector from all word vectors, effectively reducing the energy; (b) projecting the representations away from the dominating $D$ directions, effectively reducing the dimension. Experiments suggest that $D$ depends on the representations (for example, the dimension of the representation, the training methods and their specific hyperparameters, the training corpus) and also depends on the downstream applications. Nevertheless, a rule of thumb of choosing $D$ around $d / 1 0 0$ , where $d$ is the dimension of the word representations, works uniformly well across multiple languages and multiple representations and multiple test scenarios. + +We emphasize that the proposed postprocessing is counter intuitive – typically denoising by dimensionality reduction is done by eliminating the weakest directions (in a singular value decomposition of the stacked word vectors), and not the dominating ones. Yet, such postprocessing yields a “purified” and more “isotropic” word representation as seen in our elaborate experiments. + +Experiments By postprocessing the word representation by eliminating the common parts, we find the processed word representations to capture stronger linguistic regularities. We demonstrate this quantitatively, by comparing the performance of both the original word representations and the processed ones on three canonical lexical-level tasks: + +• word similarity task tests the extent to which the representations capture the similarity between two words – the processed representations are consistently better on seven different datasets, on average by $1 . 7 \%$ ; concept categorization task tests the extent to which the clusters of word representations capture the word semantics – the processed representations are consistently better on three different datasets, by $2 . 8 \%$ , $4 . 5 \%$ and $4 . 3 \%$ ; +word analogy task tests the extent to which the difference of two representations captures a latent linguistic relation – again, the performance is consistently improved (by $0 . 5 \%$ on semantic analogies, $0 . 2 \%$ on syntactic analogies and $0 . 4 \%$ in total). Since part of the dominant components are inherently canceled due to the subtraction operation while solving the analogy, we posit that the performance improvement is not as pronounced as earlier. + +Extrinsic evaluations provide a way to test the goodness of representations in specific downstream tasks. We evaluate the effect of postprocessing on a standardized and important extrinsic evaluation task on sentence modeling: semantic textual similarity task – where we represent a sentence by its averaged word vectors and score the similarity between a pair of sentences by the cosine similarity between the corresponding sentence representation. Postprocessing improves the performance consistently and significantly over 21 different datasets (average improvement of $4 \%$ ). + +Word representations have been particularly successful in NLP applications involving supervisedlearning, especially in conjunction with neural network architecture. Indeed, we see the power of postprocessing in an experiment on a standard text classification task using a well established convoluntional neural network (CNN) classifier (Kim, 2014) and three RNN classifiers (with vanilla RNN, GRU (Chung et al., 2015) and LSTM Greff et al. (2016) as recurrent units). Across two different pre-trained word vectors, five datasets and four different architectures, the performance with processing improves on a majority of instances (34 out of 40) by a good margin $2 . 8 5 \%$ on average), and the two performances with and without processing are comparable in the remaining ones. + +Related Work. Our work is directly related to word representation algorithms, most of which have been elaborately cited. + +Aspects similar to our postprocessing algorithm have appeared in specific NLP contexts very recently in (Sahlgren et al., 2016) (centering the mean) and (Arora et al., 2017) (nulling away only the first principal component). Although there is a superficial similarity between our work and (Arora et al. 2017), the nulling directions we take and the one they take are fundamentally different. Specifically, in Arora et al. (2017), the first dominating vector is \*dataset-specific\*, i.e., they first compute the sentence representation for the entire semantic textual similarity dataset, then extract the top direction from those sentence representations and finally project the sentence representation away from it. By doing so, the top direction will inherently encode the common information across the entire dataset, the top direction for the "headlines" dataset may encode common information about news articles while the top direction for "Twitter’15" may encode the common information about tweets. In contrast, our dominating vectors are over the entire vocabulary of the language. + +More generally, the idea of removing the top principal components has been studied in the context of positive-valued, high-dimensional data matrix analysis (Bullinaria & Levy, 2012; Price et al., 2006). Bullinaria & Levy (2012) posits that the highest variance components of the cooccurrence matrix are corrupted by information other than lexical semantics, thus heuristically justifying the removal of the top principal components. A similar idea appears in the context of population matrix analysis (Price et al., 2006), where the entries are also all positive. Our postprocessing operation is on dense low-dimensional representations (with both positive and negative entries). + +We posit that the postprocessing operation makes the representations more “isotropic” with stronger self-normalization properties – discussed in detail in Section 2 and Appendix A. Our main point is that this isotropy condition can be explicitly enforced to come up with new embedding algorithms (of which our proposed post-processing is a simple and practical version). + +# 2 POSTPROCESSING + +We test our observations on various word representations: four publicly available word representations (WORD2VEC1 (Mikolov et al., 2013) trained using Google News, $\mathrm { G L O V E } ^ { 2 }$ (Pennington et al., 2014) trained using Common Crawl, RAND-WALK (Arora et al., 2016) trained using Wikipedia and TSCCA3 trained using English Gigaword) and two self-trained word representations using CBOW and Skip-gram (Mikolov et al., 2013) on the 2010 Wikipedia corpus from (Al-Rfou et al., 2013). The detailed statistics for all representations are listed in Table 1. For completeness, we also consider the representations on other languages: a detailed study is provided in Appendix C.2. + +Table 1: A detailed description for the embeddings in this paper. + +
LanguageCorpusdimvocab sizeavg.u(w)2μl2
WORD2VECEnglishGoogle News3003,000,0002.040.69
GLOVEEnglishCommon Crawl3002,196,0178.303.15
RAND-WALKEnglishWikipedia30068,4302.270.70
CBOWEnglishWikipedia3001,028,9611.140.29
Skip-GramEnglishWikipedia3001,028,9612.321.25
+ +Let $v ( w ) \in \mathbb { R } ^ { d }$ be a word representation for a given word $w$ in the vocabulary $\nu$ . We observe the following two phenomena in each of the word representations listed above: + +![](images/ae50d675d2b50730ceab0ff862f138686a254dcfb15311df36d02c58a5bef8e0.jpg) +Figure 1: The decay of the normalized singular values of word representation. + +• $\{ v ( w ) : w \in \mathcal { V } \}$ are not of zero-mean: i.e., all $v ( w )$ share a non-zero common vector, $v ( w ) = \tilde { v } ( w ) + \mu$ , where $\mu$ is the average of all $v ( w )$ ’s, i.e., $\begin{array} { r } { \mu = 1 / | \mathcal { V } | \sum _ { w \in \mathcal { V } } v ( w ) } \end{array}$ . The norm of $\mu$ is approximately $1 / 6$ to 1/2 of the average norm of all $v ( w )$ (cf. Table 1). + +• $\{ \tilde { v } ( w ) : w \in \mathcal { V } \}$ are not isotropic: Let $u _ { 1 } , . . . , u _ { d }$ be the first to the last components recovered by the principal component analysis (PCA) of $\{ \tilde { v } ( w ) : w \in \mathcal { V } \}$ , and $\sigma _ { 1 } , . . . , \sigma _ { d }$ be the corresponding combinations of we observe that normalized variance ratio. Each $u$ $\begin{array} { r } { \tilde { v } ( w ) = \sum _ { i = 1 } ^ { d } \alpha _ { i } ( w ) u _ { i } } \end{array}$ $\tilde { v } ( w )$ can be written as a linear . As shown in Figure 1,lly for small values of $\sigma _ { i }$ $i$ and remains roughly constant over the later ones. This suggests there exists $D$ such that $\alpha _ { i } \gg \alpha _ { j }$ for all $i \leq D$ and $j \gg D$ ; from Figure 1 one observes that $D$ is roughly 10 with dimension $d = 3 0 0$ . + +Angular Asymmetry of Representations A modern understanding of word representations involves either PMI-based (including word2vec (Mikolov et al., 2010; Levy & Goldberg, 2014) and GloVe (Pennington et al., 2014)) or CCA-based spectral factorization approaches. While CCA-based spectral factorization methods have long been understood from a probabilistic (i.e., generative model) view point (Browne, 1979; Hotelling, 1936) and recently in the NLP context (Stratos et al., 2015), a corresponding effort for the PMI-based methods has only recently been conducted in an inspired work (Arora et al., 2016). + +Arora et al. (2016) propose a generative model (named RAND-WALK) of sentences, where every word is parameterized by a $d$ -dimensional vector. With a key postulate that the word vectors are angularly uniform (“isotropic"), the family of PMI-based word representations can be explained under the RAND-WALK model in terms of the maximum likelihood rule. Our observation that word vectors learnt through PMI-based approaches are not of zero-mean and are not isotropic (c.f. Section 2) contradicts with this postulate. The isotropy conditions are relaxed in Section 2.2 of (Arora et al., 2016), but the match with the spectral properties observed in Figure 1 is not immediate. + +This contradiction is explicitly resloved by relaxing the constraints on the word vectors to directly fit the observed spectral properties. The relaxed conditions are: the word vectors should be isotropic around a point (whose distance to the origin is a small fraction of the average norm of word vectors) lying on a low dimensional subspace. Our main result is to show that even with this enlarged parameter-space, the maximum likelihood rule continues to be close to the PMI-based spectral factorization methods. A brief summary of RAND-WALK, and the mathematical connection between our work and theirs, are explored in detail in Appendix A. + +# 2.1 ALGORITHM + +Since all word representations share the same common vector $\mu$ and have the same dominating directions and such vector and directions strongly influence the word representations in the same way, we propose to eliminate them, as formally achieved as Algorithm 1. + +
Algorithm 1: Postprocessing algorithm on word representations.
Input:Word representations {v(w),w ∈V},a threshold parameter D, 1 Compute the mean of {ν(w),w²∈V},μ← 高Σwev v(w),u(w) ← v(w)- μ
2 Compute the PCA components: u1,., ud ← PCA({u(w),w ∈ V}).
3 Preprocessthe representations: v’'(w) ← u(w)-∑=1 (uT v(w)) ui
Output :Processed representations v'(w).
+ +Significance of Nulled Vectors Consider the representation of the words as viewed in terms of the top $D$ PCA coefficients $\alpha _ { \ell } ( w )$ , for $1 \le \ell \le D$ . We find that these few coefficients encode the frequency of the word to a significant degree; Figure 2 illustrates the relation between the $( \alpha _ { 1 } ( w ) , \alpha _ { 2 } ( w ) )$ and the unigram probabilty $p ( w )$ , where the correlation is geometrically visible. + +![](images/4b9dd24523f3afb1815c7678268e8ebcf8a834a9d1a81a98d3663f0b43d21250.jpg) +Figure 2: The top two PCA directions (i.e, $\alpha _ { 1 } ( w )$ and $\alpha _ { 2 } ( w ) )$ ) encode frequency. + +Discussion In our proposed processing algorithm, the number of components to be nulled, $D$ , is the only hyperparameter that needs to be tuned. We find that a good rule of thumb is to choose $D$ approximately to be $d / 1 0 0$ , where $d$ is the dimension of a word representation. This is empirically justified in the experiments of the following section where $d = 3 0 0$ is standard for published word representations. We trained word representations for higher values of $d$ using the WORD2VEC and + +GLOVE algorithms and repeated these experiments; we see corresponding consistent improvements due to postprocessing in Appendix C. + +# 2.2 POSTPROCESSING AS A “ROUNDING” TOWARDS ISOTROPY + +The idea of isotropy comes from the partition function defined in (Arora et al., 2016), + +$$ +Z ( c ) = \sum _ { w \in \mathcal { V } } \exp \left( c ^ { \top } v ( w ) \right) , +$$ + +where $Z ( c )$ should approximately be a constant with any unit vector $c$ (c.f. Lemma 2.1 in (Arora et al., 2016)). Hence, we mathematically define a measure of isotropy as follows, + +$$ +I ( \{ v ( w ) \} ) = \frac { \operatorname* { m i n } _ { \| c \| = 1 } Z ( c ) } { \operatorname* { m a x } _ { \| c \| = 1 } Z ( c ) } , +$$ + +where $I ( \{ v ( w ) \} )$ ranges from 0 to 1, and $I ( \{ v ( w ) \} )$ closer to 1 indicates that $\{ v ( w ) \}$ is more isotropic. The intuition behind our postprocessing algorithm can also be motivated by letting $I ( \{ v ( \bar { w } ) \} ) 1$ . + +Let $V$ be the matrix stacked by all word vectors, where the rows correspond to word vectors, and $1 _ { | \nu | }$ be the $| \nu |$ -dimensional vectors with all entries equal to one, $Z ( c )$ can be equivalently defined as follows, + +$$ +Z ( c ) = | \mathcal { V } | + 1 _ { | | } \mathcal { V } | ^ { \top } V c + \frac { 1 } { 2 } c ^ { \top } V ^ { \top } V c + \sum _ { k = 3 } ^ { \infty } \frac { 1 } { k ! } \sum _ { w \in \mathcal { V } } ( c ^ { \top } v ( w ) ) ^ { k } . +$$ + +$I ( \{ v ( w ) \} )$ is, therefore, can be very coarsely approximated by, + +# • A first order approximation: + +$$ +I ( \{ v ( w ) \} ) \approx \frac { | \mathcal { V } | + \operatorname* { m i n } _ { \| c \| = 1 } 1 _ { | \mathcal { V } | } ^ { \top } V c } { | \mathcal { V } | + \operatorname* { m a x } _ { \| c \| = 1 } 1 _ { | \mathcal { V } | } ^ { \top } V c } = \frac { | \mathcal { V } | - \| 1 _ { | \mathcal { V } | } ^ { \top } V \| } { | \mathcal { V } | + \| 1 _ { | \mathcal { V } | } ^ { \top } V \| } . +$$ + +Letting $I ( \{ v ( w ) \} ) = 1$ yields $\| \boldsymbol { 1 } _ { | \nu | } ^ { \top } V \| = 0$ , which is equivalent to $\begin{array} { r } { \sum _ { w \in \mathcal { V } } v ( w ) = 0 } \end{array}$ . The intuition behind the first order approximation matches with the first step of the proposed algorithm, where we enforce $v ( w )$ to have a zero mean. + +# • A second order approximation: + +$$ +I ( \{ v ( w ) \} ) \approx \frac { | \mathcal V | + \operatorname* { m i n } _ { \| c \| = 1 } 1 _ { | \mathcal V | } ^ { \top } V c + \operatorname* { m i n } _ { \| c \| = 1 } \frac { 1 } { 2 } c ^ { \top } V ^ { \top } V c } { | \mathcal V | + \operatorname* { m a x } _ { \| c \| = 1 } 1 _ { | \mathcal V | } ^ { \top } V c + \operatorname* { m a x } _ { \| c \| = 1 } \frac { 1 } { 2 } c ^ { \top } V ^ { \top } V c } = \frac { | \mathcal V | - \| 1 _ { | \mathcal V | } ^ { \top } V \| + \frac { 1 } { 2 } \sigma _ { \operatorname* { m i n } } ^ { 2 } } { | \mathcal V | + \| 1 _ { | \mathcal V | } ^ { \top } V \| + \frac { 1 } { 2 } \sigma _ { \operatorname* { m a x } } ^ { 2 } } , +$$ + +where $\sigma _ { \mathrm { m i n } }$ and $\sigma _ { \mathrm { m a x } }$ are the smallest and largest singular value of $V$ , respectively. Letting $I ( \{ v ( w ) \} ) = 1$ yields $\| \boldsymbol { 1 } _ { | \nu | } ^ { \top } V \| = 0$ and $\sigma _ { \mathrm { m i n } } = \sigma _ { \mathrm { m a x } }$ . The fact that $\sigma _ { \mathrm { m i n } } = \sigma _ { \mathrm { m a x } }$ suggests the spectrum of $v ( w )$ ’s should be flat. The second step of the proposed algorithm removes the highest singular values, and therefore explicitly flatten the spectrum of $V$ . + +Empirical Verification Indeed, we empirically validate the effect of postprocessing of on $I ( \{ v ( w ) \} )$ . Since there is no closed-form solution for arg $\operatorname* { m a x } _ { \| c \| = 1 } Z ( c )$ or $\mathrm { a r g m i n } _ { \| c \| = 1 } Z ( c )$ , and it is impossible to enumerate all $c$ ’s, we estimate the measure by, + +$$ +I ( \{ v ( w ) \} ) \approx \frac { \operatorname* { m i n } _ { c \in C } Z ( c ) } { \operatorname* { m a x } _ { c \in C } Z ( c ) } , +$$ + +where $C$ is the set of eigenvectors of $V ^ { \top } V$ . The value of $I ( \{ v ( w ) \} )$ for the original vectors and processed ones are reported in Table 2, where we can observe that the degree of isotropy vastly increases in terms of this measure. + +
beforeafter
WORD2VEC0.70.95
GLOVE0.0650.6
+ +A formal way to verify the isotropy property is to directly check if the “self-normalization" property (i.e., $\dot { Z } ( c )$ is a constant, independent of $c$ (Andreas & Klein, 2015)) holds more strongly. Such a validation is seen diagrammatically in Figure 3 where we randomly sampled $1 { , } 0 0 0 c$ ’s as (Arora et al., + +![](images/624b7de02a6a5c554a4d7a959dc4fff52056e9a11d06419f52c96f1f9ad3f5ff.jpg) +Table 2: Before-After on the measure of isotropy. +Figure 3: The histogram of $Z ( c )$ for 1,000 randomly sampled vectors $c$ of unit norm, where $x$ -axis is normalized by the mean of all values and $D = 2$ for GLOVE and $D = 3$ for WORD2VEC. + +# 3 EXPERIMENTS + +Given the popularity and widespread use of WORD2VEC (Mikolov et al., 2013) and GLOVE (Pennington et al., 2014), we use their publicly available pre-trained reprepsentations in the following experiments. We choose $D = 3$ for WORD2VEC and $D = 2$ for GLOVE. The key underlying principle behind word representations is that similar words should have similar representations. Following the tradition of evaluating word representations (Schnabel et al., 2015; Baroni et al., 2014), we perform three canonical lexical-level tasks: (a) word similarity; (b) concept categorization; (c) word analogy; and one sentence-level task: (d) semantic textual similarity. The processed representations consistently improve performance on all three of them, and especially strongly on the first two. + +Word Similarity The word similarity task is as follows: given a pair of words, the algorithm assigns a “similarity" score – if the pair of words are highly related then the score should also be high and vice versa. The algorithm is evaluated in terms of Spearman’s rank correlation compared to (a gold set of) human judgements. + +Table 3: Before-After results $\left( \mathbf { x } 1 0 0 \right)$ on word similarity task on seven datasets. + +
WORD2VECGLOVE
orig.proc.orig.proc.
RG6576.0878.3476.9674.36
WS68.2969.0573.7976.79
RW53.7454.3346.4152.04
MEN78.2079.0880.4981.78
MTurk68.2369.3569.2970.85
SimLex44.2045.1040 8344.97
SimVerb36.3536.5028.3332.23
+ +For this experiment, we use seven standard datasets: the first published RG65 dataset (Rubenstein & Goodenough, 1965); the widely used WordSim-353 (WS) dataset (Finkelstein et al., 2001) which contains 353 pairs of commonly used verbs and nouns; the rare-words (RW) dataset (Lu + +ong et al., 2013) composed of rarely used words; the MEN dataset (Bruni et al., 2014) where the 3000 pairs of words are rated by crowdsourced participants; the MTurk dataset (Radinsky et al., 2011) where the 287 pairs of words are rated in terms of relatedness; the SimLex-999 (SimLex) dataset (Hill et al., 2016) where the score measures “genuine" similarity; and lastly the SimVerb-3500 (SimVerb) dataset (Gerz et al., 2016), a newly released large dataset focusing on similarity of verbs. + +In our experiment, the algorithm scores the similarity between two words by the cosine similarity between the two corresponding word vectors $( \mathrm { C o s S i m } ( v _ { 1 } , v _ { 2 } ) = v _ { 1 } ^ { \top } v _ { 2 } / \Vert \dot { v _ { 1 } } \Vert \Vert v _ { 2 } \Vert )$ . The detailed performance on the seven datasets is reported in Table 3, where we see a consistent and significant performance improvement due to postprocessing, across all seven datasets. These statistics (average improvement of $2 . 3 \%$ ) suggest that by removing the common parts, the remaining word representations are able to capture stronger semantic relatedness/similarity between words. + +Table 4: Before-After results $\left( \mathbf { x } 1 0 0 \right)$ on the categorization task. + +
WORD2VECGLOVE
orig.proc.orig.proc.
ap54.4357.7264 .1865.42
esslli75.0084.0981.8281.82
battig71.9781.7186.5986.59
+ +Concept Categorization This task is an indirect evaluation of the similarity principle: given a set of concepts, the algorithm needs to group them into different categories (for example, “bear” and “cat” are both animals and “city” and “country” are both related to districts). The clustering performance is then evaluated in terms of purity (Manning et al., 2008) – the fraction of the total number of the objects that were classified correctly. + +We conduct this task on three different datasets: the Almuhareb-Poesio (ap) dataset (Almuhareb, 2006) contains 402 concepts which fall into 21 categories; the ESSLLI 2008 Distributional Semantic Workshop shared-task dataset (Baroni et al., 2008) that contains 44 concepts in 6 categories; and the Battig test set (Baroni & Lenci, 2010) that contains 83 words in 10 categories. + +Here we follow the setting and the proposed algorithm in (Baroni et al., 2014; Schnabel et al., 2015) – we cluster words (via their representations) using the classical $k$ -Means algorithm (with fixed $k$ ). Again, the processed vectors perform consistently better on all three datasets (with average improvement of $2 . 5 \%$ ); the full details are in Table 4. + +Table 5: Before-After results $( \mathrm { x } 1 0 0 )$ on the word analogy task. + +
WORD2VECGLOVE
orig.proc.orig.proc.
syntax73.4673.5074.9575.40
semantics72.2873.3679.2279.25
all72.9373.4476.8977.15
+ +Word Analogy The analogy task tests to what extent the word representations can encode latent linguistic relations between a pair of words. Given three words $w _ { 1 }$ , $w _ { 2 }$ , and $w _ { 3 }$ , the analogy task requires the algorithm to find the word $w _ { 4 }$ such that $w _ { 4 }$ is to $w _ { 3 }$ as $w _ { 2 }$ is to $w _ { 1 }$ . + +We use the analogy dataset introduced in (Mikolov et al., 2013). The dataset can be divided into two parts: (a) the semantic part containing around $9 \mathrm { k }$ + +questions, focusing on the latent semantic relation between pairs of words (for example, what is to Chicago as Texas is to Houston); and (b) the syntatic one containing roughly $1 0 . 5 \mathrm { k }$ questions, focusing on the latent syntatic relation between pairs of words (for example, what is to “amazing” as “apprently” is to “apparent”). + +In our setting, we use the original algorithm introduced in (Mikolov et al., 2013) to solve this problem, i.e., $w _ { 4 }$ is the word that maximize the cosine similarity between $v ( w _ { 4 } )$ and $v ( w _ { 2 } ) - v ( w _ { 1 } ) \bar { + } v ( w _ { 3 } )$ . The average performance on the analogy task is provided in Table 5 (with a detailed performance provided in Table 19 in Appendix D). It can be noticed that while postprocessing continues to improve the performance, the improvement is not as pronounced as earlier. We hypothesize that this is because the mean and some dominant components get canceled during the subtraction of $v ( w _ { 2 } )$ from $v ( w _ { 1 } )$ , and therefore the effect of postprocessing is less relevant. + +Table 6: Before-After results $\left( \mathbf { x } 1 0 0 \right)$ on the semantic textual similarity tasks. + +
WORD2VECGLOVE
orig.proc.orig.proc.
201257.2257.6748.2754.06
201356.8157.9844.8357.71
201462.8963.3051.1159.23
201562.7463.3547.2357.29
SICK70.1070 2065.1467.85
all60.8861.4549.1956.76
+ +Semantic Textual Similarity Extrinsic evaluations measure the contribution of a word representation to specific downstream tasks; below, we study the effect of postprocessing on a standard sentence modeling task – semantic textual similarity (STS) which aims at testing the degree to which the algorithm can capture the semantic equivalence between two sentences. For each pair of sentences, the algorithm needs to measure how similar the two sentences are. The degree to which the measure matches with human judgment (in terms of Pearson correlation) is an index of the algorithm’s performance. We test the word representations on 20 textual similarity datasets from the 2012-2015 SemEval STS tasks (Agirre et al., 2012; 2013; 2014; 2015), and the 2012 SemEval Semantic Related task (SICK) (Marelli et al., 2014). + +Representing sentences by the average of their constituent word representations is surprisingly effective in encoding the semantic information of sentences (Wieting et al., 2015; Adi et al., 2016) and close to the state-of-the-art in these datasets. We follow this rubric and represent a sentence $s$ based on its averaged word representation, i.e., $\begin{array} { r } { \boldsymbol { v } ( \boldsymbol { s } ) = \frac { 1 } { | \boldsymbol { s } | } \sum _ { w \in \boldsymbol { s } } \boldsymbol { v } ( w ) } \end{array}$ , and then compute the similarity between two sentences via the cosine similarity between the two representations. The average performance of the original and processed representations is itemized in Table 6 (with a detailed performance in Table 20 in Appendix E) – we see a consistent and significant improvement in performance because of postprocessing (on average $4 \%$ improvement). + +# 4 POSTPROCESSING AND SUPERVISED CLASSIFICATION + +Supervised downstream NLP applications have greatly improved their performances in recent years by combining the discriminative learning powers of neural networks in conjunction with the word representations. We evaluate the performance of a variety of neural network architectures on a standard and important NLP application: text classification, with sentiment analysis being a particularly important and popular example. The task is defined as follows: given a sentence, the algorithm needs to decide which category it falls into. The categories can be either binary (e.g., positive/negative) or can be more fine-grained (e.g. very positive, positive, neutral, negative, and very negative). + +We evaluate the word representations (with and without postprocessing) using four different neural network architectures (CNN, vanilla-RNN, GRU-RNN and LSTM-RNN) on five benchmarks: (a) the movie review (MR) dataset (Pang & Lee, 2005); (b) the subjectivity (SUBJ) dataset (Pang & Lee, 2004); (c) the TREC question dataset (Li & Roth, 2002); (d) the IMDb dataset (Maas et al., 2011); (e) the stanford sentiment treebank (SST) dataset (Socher et al., 2013a). A detailed description of these standard datasets, their training/test parameters and the cross validation methods adopted is in Appendix F. Specifically, we allow the parameter $D$ (i.e., the number of nulled components) to vary between 0 and 4, and the best performance of the four neural network architectures with the now-standard CNN-based text classification algorithm (Kim, 2014) (implemented using tensorflow4) is itemized in Table 7. The key observation is that the performance of postprocessing is better in a majority (34 out of 40) of the instances by $2 . 3 2 \%$ on average, and in the rest the instances the two performances (with and without postprocessing) are comparable. + +
vanill-RNNGRURNNLSTMRNN
WORI2VECGLOVEWORI2VECGLVEWORI2VECGLDVEWORI2VECGLVE
orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.
MR70.8071.2771.0171.1174.9574.0171.1472.5677.8678.2674.9875.1375.6977.3472.0271.84
SUBJ87.1487.3386.9887.2582.8587.6081.4587.3790.9691.1091.1691.8590.2390.5490.7490.82
TREC87.8089.0087.6089.0080.6089.2085.2089.0091.6092.4091.6093.0088.0091.2085.8091.20
SST38.4638.3338.8237.8342.0839.9141.4541.9041.8645.0236.5237.6943.0842.0837.5138.05
IMDb86.6887.1287.2787.1050.1553.1452.7676.0782.9683.4781.5082.4481.2982.6079.1081.33
+ +Table 7: Before-After results $( \mathrm { x } 1 0 0 )$ on the text classification task using CNN (Kim, 2014) and vanilla RNN, GRU-RNN and LSTM-RNN. + +A further validation of the postprocessing operation in a variety of downstream applications (eg: named entity recognition, syntactic parsers, machine translation) and classification methods (eg: random forests, neural network architectures) is of active research interest. Of particular interest is the impact of the postprocessing on the rate of convergence and generalization capabilities of the classifiers. Such a systematic study would entail a concerted and large-scale effort by the research community and is left to future research. + +Discussion All neural network architectures, ranging from feedforward to recurrent (either vanilla or GRU or LSTM), implement at least linear processing of hidden/input state vectors at each of their nodes; thus the postprocessing operation suggested in this paper can in principle be automatically “learnt” by the neural network, if such internal learning is in-line with the end-to-end training examples. Yet, in practice this is complicated due to limitations of optimization procedures (SGD) and sample noise. We conduct a preliminary experiment in Appendix B and show that subtracting the mean (i.e., the first step of postprocessing) is “effectively learnt" by neural networks within their nodes. + +# 5 CONCLUSION + +We present a simple postprocessing operation that renders word representations even stronger, by eliminating the top principal components of all words. Such an simple operation could be used for word embeddings in downstream tasks or as intializations for training task-specific embeddings. Due to their popularity, we have used the published representations of WORD2VEC and GLOVE in English in the main text of this paper; postprocessing continues to be successful for other representations and in multilingual settings – the detailed empirical results are tabulated in Appendix C. + +# REFERENCES + +Yossi Adi, Einat Kermany, Yonatan Belinkov, Ofer Lavi, and Yoav Goldberg. 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While CCA-based spectral factorization methods have long been understood from a probabilistic (i.e., generative model) view point (Browne, 1979; Hotelling, 1936) and recently in the NLP context (Stratos et al., 2015), a corresponding effort for the PMI-based methods has only recently been conducted in an inspired work (Arora et al., 2016). + +(Arora et al., 2016) propose a generative model (named RAND-WALK) of sentences, where every word is parameterized by a $d$ -dimensional vector. With a key postulate that the word vectors are angularly uniform (“isotropic"), the family of PMI-based word representations can be explained under the RAND-WALK model in terms of the maximum likelihood rule. Our observation that word vectors learnt through PMI-based approaches are not of zero-mean and are not isotropic (c.f. Section 2) contradicts with this postulate. The isotropy conditions are relaxed in Section 2.2 of (Arora et al., 2016), but the match with the spectral properties observed in Figure 1 is not immediate. + +In this section, we resolve this by explicitly relaxing the constraints on the word vectors to directly fit the observed spectral properties. The relaxed conditions are: the word vectors should be isotropic around a point (whose distance to the origin is a small fraction of the average norm of word vectors) lying on a low dimensional subspace. Our main result is to show that even with this enlarged parameter-space, the maximum likelihood rule continues to be close to the PMI-based spectral factorization methods. Formally, the model, the original constraints of (Arora et al., 2016) and the enlarged constraints on the word vectors are listed below: + +• A generative model of sentences: the word at time $t$ , denoted by $w _ { t }$ , is generated via a log-linear model with a latent discourse variable $c _ { t }$ (Arora et al., 2016), i.e., + +$$ +p ( w _ { t } | c _ { t } ) = \frac { 1 } { Z ( c _ { t } ) } \exp \left( c _ { t } ^ { \top } v ( w _ { t } ) \right) , +$$ + +where $v ( w ) \in \mathbb { R } ^ { d }$ is the vector representation for a word $w$ in the vocabulary $V$ , $c _ { t }$ is the latent variable which forms a “slowly moving" random walk, and the partition function: $\begin{array} { r } { Z ( c ) = \sum _ { w \in \mathcal { V } } \exp \left( c ^ { \top } v ( w ) \right) } \end{array}$ . + +• Constraints on the word vectors: (Arora et al., 2016) suppose that there is a Bayesian priori on the word vectors: + +The ensemble of word vectors consists of i.i.d. draws generated by $\boldsymbol { v } = \boldsymbol { s } \cdot \boldsymbol { \hat { v } }$ where $\hat { v }$ is from the spherical Gaussian distribution, and $s$ is a scalar random variable. + +A deterministic version of this prior is discussed in Section 2.2 of (Arora et al., 2016), but part of these (relaxed) conditions on the word vectors are specifically meant for Theorem 4.1 and not the main theorem (Theorem 2.2). The geometry of the word representations is only evaluated via the ratio of the quadratic mean of the singular values to the smallest one being small enough. This meets the relaxed conditions, but not sufficient to validate the proof approach of the main result (Theorem 2.2); what would be needed is that the ratio of the largest singular value to the smallest one be small. + +• Revised conditions: We revise the Bayesian prior postulate (and in a deterministic fashion) formally as follows: there is a mean vector $\mu$ , $D$ orthonormal vectors $u _ { 1 } , \ldots , u _ { D }$ (that are orthogonal and of unit norm), such that every word vector $v ( w )$ can be represented by, + +$$ +v ( w ) = \mu + \sum _ { i = 1 } ^ { D } \alpha _ { i } ( w ) u _ { i } + \tilde { v } ( w ) , +$$ + +where $\mu$ is bounded, $\alpha _ { i }$ is bounded by $A$ , $D$ is bounded by $D A ^ { 2 } \ = \ o ( d ) , \ \tilde { v } ( u$ ) are statistically isotropic. By statistical isotropy, we mean: for high-dimensional rectangles $R$ $\begin{array} { r } { \frac { 1 } { | \mathcal { V } | } \sum _ { w \in \mathcal { V } } \mathbf { 1 } ( \tilde { v } ( w ) \in R ) \int _ { R } f ( \tilde { v } ) d \tilde { v } } \end{array}$ , as $| \nu | \to \infty$ , where $f$ is an angle-independent pdf, i.e., $f ( \tilde { v } )$ is a function of $\| \tilde { v } \|$ . + +The revised postulate differs from the original one in two ways: (a) it imposes a formal deterministic constraint on the word vectors; (b) the revised postulate allows the word vectors to be angularly asymmetric: as long as the energy in the direction of $u _ { 1 } , \ldots , u _ { D }$ is bounded, there is no constraint on the coefficients. Indeed, note that there is no constraint on $\tilde { v } ( w )$ to be orthogonal to $u _ { 1 } , \ldots , u _ { D }$ . + +Empirical Validation We can verify that the enlarged conditions are met by the existing word representations. Specifically, the natural choice for $\mu$ is the mean of the word representations and $u _ { 1 } \ldots u _ { D }$ are the singular vectors associated with the top $D$ singular values of the matrix of word vectors. We pick $D = 2 0$ for WORD2VEC and $D = 1 0$ for GLOVE, and the corresponding value of $D A ^ { 2 }$ for WORD2VEC and GLOVE vectors are both roughly 40, respectively; both values are small compared to $d = 3 0 0$ . + +![](images/4b8cef86f46030d6cc4853b73297993875e11f99c5c37b3bb9b91793e95189e5.jpg) +Figure 4: Spectrum of the published WORD2VEC and GLOVE and random Gaussian matrices, ignoring the top $D$ components; $D = 1 0$ for GLOVE and $D = 2 0$ for WORD2VEC. + +This leaves us to check the statistical isotropy of the “remaining" vectors $\tilde { v } ( w )$ for words $w$ in the vocabulary. We do this by plotting the remaining spectrum (i.e. the $( D + 1 )$ -th, ..., 300th singular values) for the published WORD2VEC and GLOVE vectors in Figure 4. As a comparison, the empirical spectrum of a random Gaussian matrix is also plotted in Figure 4. We see that both spectra are flat (since the vocabulary size is much larger than the dimension $d = 3 0 0$ ). Thus the postprocessing operation can also be viewed as a way of making the vectors “more isotropic”. + +Mathematical Contribution Under the revised postulate, we show that the main theorem in (Arora et al., 2016) (c.f. Theorem 2.2) still holds. Formally: + +Theorem A.1 Suppose the word vectors satisfy the constraints. Then + +$$ +\mathrm { P M I } ( w _ { 1 } , w _ { 2 } ) \stackrel { \mathrm { d e f } } { = } \log \frac { p ( w _ { 1 } , w _ { 2 } ) } { p ( w _ { 1 } ) p ( w _ { 2 } ) } \frac { v ( w _ { 1 } ) ^ { \top } v ( w _ { 2 } ) } { d } , \quad a s | \mathcal { V } | \infty , +$$ + +where $p ( w )$ is the unigram distribution induced from the model (2), and $p ( w _ { 1 } , w _ { 2 } )$ is the probability that two words $w _ { 1 }$ and $w _ { 2 }$ occur with each other within distance $q$ . + +The proof is in Appendix G. Theorem A.1 suggests that the RAND-WALK generative model and its properties proposed by (Arora et al., 2016) can be generalized to a broader setting (with a relaxed restriction on the geometry of word representations) – relevantly, this relaxation on the geometry of word representations is empirically satisfied by the vectors learnt as part of the maximum likelihood rule. + +# B NEURAL NETWORKS LEARN TO POSTPROCESS + +Every neural network family posseses the ability to conduct linear processing inside their nodes; this includes feedforward and recurrent and convolutional neural network models. Thus, in principle, the postprocessing operation can be “learnt and implemented" within the parameters of the neural network. On the other hand, due to the large number of parameters within the neural network, it is unclear how to verify such a process, even if it were learnt (only one of the layers might be implementing the postprocessing operation or via a combination of multiple effects). + +To address this issue, we have adopted a comparative approach in the rest of this section. The comparative approach involves adding an extra layer interposed in between the inputs (which are word vectors) and the rest of the neural network. This extra layer involves only linear processing. Next we compare the results of the final parameters of the extra layer (trained jointly with the rest of tne neural network parameters, using the end-to-end training examples) with and without preprocessing of the word vectors. Such a comparative approach allows us to separate the effect of the postprocessing operation on the word vectors from the complicated “semantics” of the neural network parameters. + +
vanillaGRULSTM
W.G.W.G.W.G.
MR82.0749.2381.3547.6377.9544.92
SUBJ84.0249.9483.1550.6083.3948.95
TREC81.6852.9982.6850.4280.8046.77
SST79.6446.5978.0643.2177.7242.82
IMDb93.4866.3794.4955.2487.2746.74
+ +![](images/0000e8dbd070394e21308164cbb1d856e0bac9486f36e22091e032d44078c442.jpg) +Figure 6: The cosine similarity (x100) between + +Figure 5: Time-expanded RNN architecture withan appended layer involving linear bias. $b _ { \mathrm { p r o c . } } + \mu$ and EC $b _ { \mathrm { o r i g . } }$ , where W. and G. stand forLOVE respectively. WORD2V and G + +Experiment We construct a modified neural network by explicitly adding a “postprocessing unit" as the first layer of the RNN architecture (as in Figure 5, where the appended layer is used to test the first step (i.e., remove the mean vector)) of the postprocessing algorithm. + +In the modified neural network, the input word vectors are now $\boldsymbol { v } ( \boldsymbol { w } ) - \boldsymbol { b }$ instead of $v ( w )$ . Here $b$ is a bias vector trained jointly with the rest of the neural network parameters. Note that this is only a relabeling of the parameters from the perspective of the RNN architecture: the nonlinear activation function of the node is now operated on $\overset { \triangledown } { \boldsymbol { A } } ( \boldsymbol { v } ( \boldsymbol { w } ) - \boldsymbol { b } ) + \boldsymbol { b } ^ { \prime } = \boldsymbol { A } \boldsymbol { v } ( \boldsymbol { w } ) + ( \boldsymbol { b } ^ { \prime } - \boldsymbol { A } \boldsymbol { b } )$ instead of the previous $A v ( w ) + b ^ { \prime }$ . Let $b _ { \mathrm { p r o c } }$ . and $b _ { \mathrm { o r i g } }$ . be the inferred biases when using the processed and original word representations, respectively. + +We itemize the cosine similarity between $b _ { \mathrm { p r o c . } } + \mu$ and $b _ { \mathrm { o r i g . } }$ . in Table 6 for the 5 different datasets and 3 different neural network architectures. In each case, the cosine similarity is remarkably large (on average 0.66, in 300 dimensions) – in other words, trained neural networks implicitly postprocess the word vectors nearly exactly as we proposed. This agenda is successfully implemented in the context of verifying the removal of the mean vector. + +The second step of our postprocessing algorithm (i.e., nulling away from the top principal components) is equivalent to applying a projection matrix $\begin{array} { r } { P = I - \sum _ { i = 1 } ^ { D } u _ { i } u _ { i } ^ { \top } } \end{array}$ on the word vectors, where $u _ { i }$ is the $i$ -th principal component and $D$ is the number of the removed directions. A comparative analysis effort for the second step (nulling the dominant PCA directions) is the following. Instead of applying a bias term $b$ , we multiply by a matrix $Q$ to simulate the projection operation. The input word vectors are now $Q _ { \mathrm { o r i g . } } v ( w )$ instead of $v ( w )$ for the original word vectors, and $Q _ { \mathrm { p r o c . } } P v ( \bar { w } )$ instead of $P v ( w )$ for the processed vectors. Testing the similarity between $Q _ { \mathrm { o r i g . } } P$ and $Q _ { \mathrm { p r o c . } }$ , allows us to verify if the neural network learns to conduct the projection operation as proposed. + +In our experiment, we found that such a result cannot be inferred. One possibility is that there are too many parameters in both $Q _ { \mathrm { p r o c } }$ . and $Q _ { \mathrm { o r i g . } }$ , which adds randomness to the experiment. Alternatively, the neural network weights may not be able to learn the second step of the postprocessing operation (indeed, in our experiments postprocessing significantly boosted end-performance of neural network architectures). A more careful experimental setup to test whether the second step of the postprocessing operation is learnt is left as a future research direction. + +# C EXPERIMENTS ON VARIOUS REPRESENTATIONS + +In the main text, we have reported empirical results for two published word representations: WORD2VEC and GLOVE, each in 300 dimensions. In this section, we report the results of the same experiments in a variety of other settings to show the generalization capability of the postprocessing operation: representations trained via WORD2VEC and GLOVE algorithms in dimensions other than 300, other representations algorithms (specifically TSCCA and RAND-WALK) and in multiple languages. + +# C.1 STATISTICS OF MULTILINGUAL WORD REPRESENTATIONS + +We use the publicly available TSCCA representations (Dhillon et al., 2012) on German, French, Spanish, Italian, Dutch and Chinese. The detailed statistics can be found in Table 8 and the decay of their singular values are plotted in Figure 7. + +Table 8: A detailed description for the TSCCA embeddings in this paper. + +
LanguageCorpusdimvocab sizeavg.u(w)l2μl2
TSCCA-EnEnglishGigawords200300,0004.380.78
TSCCA-DeGermanNewswire200300,0004.520.79
TSCCA-FrFrenchGigaword200300,0004.340.81
TSCCA-EsSpanishGigaword200300.0004.170.79
TSCCA-ItItalianNewswire+Wiki200300,0004.340.79
TSCCA-N1DutchNewswire+Wiki200300,0004.460.72
TSCCA-ZhChineseGigaword200300.0004.510.89
+ +![](images/c31c5918d39e7c9069410fa5d0e535e28a69290bac0643aab8717f578613d0db.jpg) +Figure 7: The decay of the normalized singular values of multilingual word representation. + +# C.2 MULTILINGUAL GENERALIZATION + +In this section, we perform the word similarity task with the original and the processed TSCCA word representations in German and Spanish on three German similarity datasets (GUR65 – a German version of the RG65 dataset, GUR350, and ZG222 in terms of relatedness) (Zesch & Gurevych, 2006) and the Spanish version of RG65 dataset (Camacho-Collados et al., 2015). The choice of $D$ is 2 for both German and Spanish. + +The experiment setup and the similarity scoring algorithm are the same as those in Section 3. The detailed experiment results are provided in Table 9, from which we observe that the processed representations are consistently better than the original ones. This provides evidence to the generalization capabilities of the postprocessing operation to multiple languages (similarity datasets in Spanish and German were the only ones we could locate). + +# C.3 GENERALIZATION TO DIFFERENT REPRESENTATION ALGORITHMS + +Given the popularity and widespread use of WORD2VEC (Mikolov et al., 2013) and GLOVE (Pennington et al., 2014), the main text has solely focused on their published publicly avalable + +Table 9: Before-After results (x100) on the word similarity task in multiple languages. + +
languageTSCCA
orig.proc.
RG65Spanish60.3360.37
GUR65German61.7564.39
GUR350German44.9146.59
ZG222German30.3732.92
+ +300-dimension representations. In this section, we show that the proposed postprocessing algorithm generalizes to other representation methods. Specifically, we demonstrate this on RAND-WALK (obtained via personal communication) and TSCCA (publicly available) on all the experiments of Section 3. The choice of $D$ is 2 for both RAND-WALK and TSCCA. + +In summary, the performance improvements on the similarity task, the concept categorization task, the analogy task, and the semantic textual similarity dataset are on average $2 . 2 3 \%$ , $2 . 3 9 \%$ , $0 . 1 1 \%$ and $0 . 6 1 \%$ , respectively. The detailed statistics are provided in Table 10, Table 11, Table 12 and Table 13, respectively. These results are a testament to the generalization capabilities of the postprocessing algorithm to other representation algorithms. + +Table 10: Before-After results $( \mathrm { x } 1 0 0 )$ on the word similarity task on seven datasets. + +
RAND-WALKTSCCA
orig.proc.orig.proc.
RG6580.6682.9647.5347.67
WS65.8974.3754.2154.35
RW45.1151.2343.9643.72
MEN73.5677.2265.4865.62
MTurk64.3566.1159.6560.03
SimLex34.0536.5534.8634.91
SimVerb16.0521.8423.7923.83
+ +Table 11: Before-After results $( \mathrm { x } 1 0 0 )$ on the categorization task. + +
RAND-WALKTSCCA
orig.proc.orig.proc.
ap59.8362.3660.0063.42
esslli72.7372.7368.1870.45
battig75.7381.8270.7370.73
+ +Table 12: Before-After results (x100) on the word analogy task. + +
RAND-WALKTSCCA
orig.proc.orig.proc.
syn.60.3960.4837.7237.80
sem.83.5583.8214.5414.55
all70.5070.6727.3027.35
+ +# C.4 ROLE OF DIMENSIONS + +The main text has focused on the dimension choice of $d = 3 0 0$ , due to its popularity. In this section we explore the role of the dimension in terms of both choice of $D$ and the performance of the postprocessing operation – we do this by using skip-gram model on the 2010 snapshot of Wikipedia corpus (Al-Rfou et al., 2013) to train word representations. We first observe that the two phenomena of Section 2 continue to hold: + +• From Table 14 we observe that the ratio between the norm of $\mu$ and the norm average of all $v ( w )$ spans from 1/3 to $1 / 4$ ; + +Table 13: Before-After results (x100) on the semantic textual similarity tasks. + +
RAND-WALKTSCCA
orig.proc.orig.proc.
201238.0337.6644.5144.63
201337.4736.8543.2142.74
201446.0648.3252.8552.87
201547.8251.7656.2256.14
SICK51.5851.7656.1556.11
all43.4844.6750.0150.23
+ +
dim3004005006007008009001000
avg.u(w)l24.515.175.916.226.496.736.957.15
μ221.741.761.771.781.791.801.811.83
+ +Table 14: Statistics on word representation of dimensions 300, 400, ..., and 1000 using the skip-gram model. + +• From Figure 8 we observe that the decay of the variance ratios $\sigma _ { i }$ is near exponential for small values of $i$ and remains roughly constant over the later ones. + +![](images/144178c36d042206487ea6fb8b7dd93a93da2d7a6f83423558362445e9f29eda.jpg) +Figure 8: The decay of the normalized singular values of word representations. + +A rule of thumb choice of $D$ is around $d / 1 0 0$ . We validate this claim empirically by performing the tasks in Section 3 on word representations of higher dimensions, ranging from 300 to 1000, where we set the parameter $D = d / \bar { 1 } 0 0$ . In summary, the performance improvement on the four itemized tasks of Section 3 are $2 . 2 7 \%$ , $3 . 3 7 \%$ , 0.01 and $1 . 9 2 \%$ respectively; the detailed results can be found in Table 15, Table 16, Table 17, and Table 18. Again, note that the improvement for analogy tasks is marginal. These experimental results justify the rule-of-thumb setting of $D = d / 1 0 0$ , although we emphasize that the improvements can be further accentuated by tuning the choice of $D$ based on the specific setting. + +# D EXPERIMENTS ON WORD ANALOGY TASK + +The detailed performance on the analogy task is provided in Table 19. + +# E EXPERIMENTS ON SEMANTIC TEXTUAL SIMILARITY TASK + +The detailed performance on the semantic textual similarity is provided in Table 20. + +Table 15: Before-After results $( \mathrm { x } 1 0 0 )$ on word similarity task on seven datasets. + +
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
RG6573.5774.7275.6479.8777.7281.9777.5980.7
WS70.2571.9570.872.8870.3972.7371.6474.04
RW46.2549.1145.9747.6346.648.5945.747.81
MEN75.6677.5976.0777.8975.978.1575.8878.15
Mturk75.6677.5967.6868.1166.8968.2567.667.87
SimLex34.0236.1935.1737.135.7337.6535.7638.04
SimVerb22.2224.9822.9125.3223.0325.8223.3525.97
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
RG6577.381.0777.5281.0779.7582.3478.1879.07
WS70.3173.0271.5274.6571.1973.0671.574.78
RW45.8648.444.964944.4449.2244.549.03
MEN75.8478.2175.8477.9676.1678.3576.7278.1
Mturk67.4767.7967.676867.9868.8768.3469.44
SimLex35.337.5936.5437.8536.6238.4436.6738.58
SimVerb22.8125.623.4825.5723.6825.7623.2426.58
+ +Table 16: Before-After results $( \mathrm { x } 1 0 0 )$ on the categorization task. + +
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
ap46.148.6142.5745.3446.8550.8840.345.84
esslli68.1872.7364.282.7264.265.4365.9172.73
battig71.677.7868.187568.1870.4546.9166.67
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
ap38.0441.3134.7639.834.7627.4627.9628.21
esslli54.5554.5568.1856.8272.7372.7352.2752.27
battig62.9666.6767.969.1449.3859.2651.8546.91
+ +
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
syn.60.4860.5261.6161.4560.9360.8461.6661.57
sem.74.5174.5477.1177.3676.3976.8977.2877.61
all.66.8666.8768.6668.6967.8868.1168.7768.81
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
syn.60.9461.0268.3868.3460.4760.3067.5667.30
sem.77.2477.2677.2477.3576.7676.9076.7176.51
all.68.3668.4168.3868.5067.9167.6767.5667.30
+ +Table 17: Before-After results (x100) on the word analogy task. + +# F STATISTICS OF TEXT CLASSIFICATION DATASETS + +We evaluate the word representations (with and without postprocessing) using four different neural network architectures (CNN, vanilla-RNN, GRU-RNN and LSTM-RNN) on five benchmarks: + +• the movie review (MR) dataset (Pang & Lee, 2005) where each review is composed by only one sentence; +• the subjectivity (SUBJ) dataset (Pang & Lee, 2004) where the algorithm needs to decide whether a sentence is subjective or objective; +• the TREC question dataset (Li & Roth, 2002) where all the questions in this dataset has to be partitioned into six categories; +• the IMDb dataset (Maas et al., 2011) – each review consists of several sentences; +• the Stanford sentiment treebank (SST) dataset (Socher et al., 2013a), where we only use the full sentences as the training data. + +Table 18: Before-After results (x100) on the semantic textual similarity tasks. + +
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
201254.5154.9554.3154.5755.1356.2355.3556.03
201356.5857.8956.3557.3557.5559.3857.4359.00
201459.661.9259.5761.6261.1964.3861.1063.86
201559.6561.4859.6961.1961.6364.7761.4264.04
SICK68.8970.7960.670.2768.6371.0068.5870.57
all58.3259.9158.2559.5559.6162.0259.5761.55
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
201255.5256.4954.4754.8554.6955.1854.3454.78
201357.6159.3156.7557.6256.9858.2656.7857.73
201461.5764.7760.5162.8360.8963.3460.7863.03
201562.0565.4560.7462.8461.0963.4860.9263.03
SICK68.3870.6367.9469.5967.8669.567.5869.16
all59.9662.3458.8760.3959.1660.8858.9460.48
+ +Table 19: Before-After results (x100) on the word analogy task. + +
WORD2VECGLOVE
orig.proc.orig.proc.
capital-common-countries82.0183.6095.0695.96
capital-world78.3880.0891.8992.31
city-in-state69.5669.8869.5670.45
currency32.4332.9221.5921.36
family84.9884.5995.8495.65
gram1-adjective-to-adverb28.0227.7240.4239.21
gram2-opposite40.1440.5131.6530.91
gram3-comparative89.1989.2686.9387.09
gram4-superlative82.7183.3390.4690.59
gram5-present-participle79.3679.6482.9582.76
gram6-nationality-adjective90.2490.3690.2490.24
gram7-past-tense66.0366.5363.9164.87
gram8-plural91.0790.6195.2795.36
gram9-plural-verbs68.7467.5867.2468.05
+ +In TREC, SST and IMDb, the datasets have already been split into train/test sets. Otherwise we use 10-fold cross validation in the remaining datasets (i.e., MR and SUBJ). Detailed statistics of various features of each of the datasets are provided in Table 21. + +# G PROOF OF THEOREM A.1 + +Given the similarity between the setup in Theorem 2.2 in (Arora et al., 2016) and Theorem A.1, many parts of the original proof can be reused except one key aspect – the concentration of $Z ( c )$ . We summarize this part in the following lemma: + +Table 20: Before-After results (x100) on the semantic textual similarity tasks. + +
WORD2VECGLOVE
orig.proc.orig.proc.
2012.MSRpar42.1243.8544.5444.09
2012.MSRvid72.0772.1664.4768.05
2012.OnWN69.3869.4853.0765.67
2012.SMTeuroparl53.1554.3241.7445.28
2012.SMTnews49.3748.5337.5447.22
2013.FNWN40.7041.9637.5439.34
2013.OnWN67.8768.1747.2258.60
2013.headlines61.8863.8149.7357.20
2014.OnWN74.6174.7857.4167.56
2014.deft-forum32.1933.2621.5529.39
2014.deft-news66.8365.9665.1471.45
2014.headlines58.0159.5847.0552.60
2014.images73.7574.1757.2268.28
2014.tweet-news71.9272.0758.3266.13
2015.answers-forum46.3546.8030.0239.86
2015.answers-students68.0767.9949.2062.38
2015.belief59.7260.4244.0557.68
2015.headlines61.4763.4546.2253.31
2015.images78.0978.0866.6373.20
SICK70.1070.2065.1467.85
+ +Table 21: Statistics for the five datasets after tokenization: c represents the number of classes; $l$ represents the average sentence length; Train represents the size of the training set; and Test represent the size of the test set. + +
C1TrainTest
MR22010,66210-fold cross validation
SUBJ22310,00010-fold cross validation
TREC6105,952500
SST51811,8552,210
IMDb210025,00025,000
+ +Lemma G.1 Let c be a random variable uniformly distributed over the unit sphere, we prove that with high probability, $Z ( c ) / | \nu |$ converges to a constant $Z$ : + +$$ +\begin{array} { r } { p ( ( 1 - \epsilon _ { z } ) Z \le Z ( c ) \le ( 1 + \epsilon _ { z } ) Z ) \ge 1 - \delta , } \end{array} +$$ + +where $\epsilon _ { z } = \Omega ( ( D + 1 ) / | \mathcal { V } | )$ and $\delta = \Omega ( ( D A ^ { 2 } + \| \mu \| ^ { 2 } ) / d )$ . + +Our proof differs from the one in (Arora et al., 2016) in two ways: (a) we treat $v ( w )$ as deterministic parameters instead of random variables and prove the Lemma by showing a certain concentration of measure; (b) the asymmetric parts $\mu$ and $u _ { 1 } , . . . , u _ { D }$ , (which did not exist in the original proof), need to be carefully addressed to complete the proof. + +# G.1 PROOF OF LEMMA G.1 + +Given the constraints on the word vectors (3), the partition function $Z ( c )$ can be rewritten as, + +$$ +\begin{array} { r l } & { Z ( c ) = \displaystyle \sum _ { v \in \mathcal { V } } \exp ( c ^ { \top } v ( w ) ) } \\ & { \quad \quad = \displaystyle \sum _ { v \in \mathcal { V } } \exp \left( c ^ { \top } \left( \mu + \sum _ { i = 1 } ^ { D } \alpha _ { i } ( w ) u _ { i } + \tilde { v } ( w ) \right) \right) } \\ & { \quad \quad = \displaystyle \sum _ { v \in \mathcal { V } } \exp ( c ^ { \top } \mu ) \left[ \displaystyle \prod _ { i = 1 } ^ { D } \exp ( \alpha _ { i } ( w ) c ^ { \top } u _ { i } ) \right] \exp \left( c ^ { \top } \tilde { v } ( w ) \right) . } \end{array} +$$ + +The equation above suggests that we can divide the proof into five parts. + +Step 1: for every unit vector $c$ , one has, + +$$ +\frac { 1 } { | \mathcal { V } | } \sum _ { w \in \mathcal { V } } \exp ( c ^ { \top } \tilde { v } ( w ) ) \mathbb { E } _ { f } ( \exp ( c ^ { \top } \tilde { v } ) ) , \ \mathrm { a s } \ | \mathcal { V } | \infty . +$$ + +Proof Let $M , N$ be a positive integer, and let $A _ { M } \subset \mathbb { R } ^ { d }$ such that, + +$$ +A _ { M , N } = \left\{ \widetilde v \in \mathbb { R } ^ { d } : \frac { M - 1 } { N } < \exp ( c ^ { \top } \widetilde v ) \leq \frac { M } { N } \right\} . +$$ + +Since $A _ { M , N }$ can be represented by a union of countable disjoint rectangles, we know that for every $M , N \in \mathbb { N } _ { + }$ , + +$$ +\frac { 1 } { | \mathcal { V } | } \sum _ { w \in \mathcal { V } } \mathbf { 1 } ( \tilde { v } ( w ) \in A _ { M , N } ) = \int _ { A _ { M , N } } f ( \tilde { v } ) d \tilde { v } . +$$ + +Further, since $A _ { M , N }$ are disjoint for different $M$ ’s and $\mathbb { R } ^ { d } = \cup _ { M = 1 } ^ { \infty } A _ { M , N }$ , one has, + +$$ +\begin{array} { r l } { \displaystyle \frac { 1 } { | \mathcal { V } | } \sum _ { w \in \mathcal { V } } \exp \big ( c ^ { \top } \tilde { v } ( w ) \big ) = \displaystyle \sum _ { M = 1 } ^ { \infty } \frac { 1 } { | \mathcal { V } | } \sum _ { w \in \mathcal { V } } \mathbf { 1 } \big ( \tilde { v } ( w ) \in A _ { M , N } \big ) \exp \big ( c ^ { \top } \tilde { v } ( w ) \big ) } & { } \\ { \displaystyle \leq \sum _ { M = 1 } ^ { \infty } \frac { 1 } { | \mathcal { V } | } \sum _ { w \in \mathcal { V } } \mathbf { 1 } \big ( \tilde { v } ( w ) \in A _ { M , N } \big ) \frac { M } { N } } & { } \\ { \displaystyle \to \sum _ { M = 1 } ^ { \infty } \frac { M } { N } \int _ { A _ { M , N } } f ( \tilde { v } ) d \tilde { v } . } \end{array} +$$ + +The above statement holds for every $N$ . Let $N \to \infty$ , by definition of integration, one has, + +$$ +\operatorname* { l i m } _ { N \infty } \sum _ { M = 1 } ^ { \infty } \frac { M } { N } \int _ { A _ { M , N } } f ( \tilde { v } ) d \tilde { v } = \mathbb { E } _ { f } ( \exp ( c ^ { \top } \tilde { v } ) ) , +$$ + +which yields, + +$$ +\frac { 1 } { | \mathcal { V } | } \sum _ { w \in \mathcal { V } } \exp ( c ^ { \top } \tilde { v } ( w ) ) \leq \mathbb { E } _ { f } ( \exp ( c ^ { \top } \tilde { v } ) ) , \ \mathrm { a s \ } | \mathcal { V } | \infty . +$$ + +Similarly, one has, + +$$ +\begin{array} { r } { \frac { 1 } { | \mathcal { V } | } \displaystyle \sum _ { w \in \mathcal { V } } \exp \left( c ^ { \top } \tilde { v } ( w ) \right) \geq \displaystyle \operatorname* { l i m } _ { N \to \infty } \sum _ { M = 1 } ^ { \infty } \frac { M - 1 } { N } \int _ { A _ { M , N } } f ( \tilde { v } ) d \tilde { v } } \\ { = \mathbb { E } _ { f } \left( \exp \left( c ^ { \top } \tilde { v } \right) \right) , \ \mathrm { a s } \ | \mathcal { V } | \to \infty . } \end{array} +$$ + +Putting (6) and (7) proves (5). + +Step 2: the expected value, $\mathbb { E } _ { f } \left( \exp \left( c ^ { \top } \tilde { v } \right) \right)$ is a constant independent of $c$ : + +$$ +\mathbb { E } _ { f } \left( \exp \left( \boldsymbol { c } ^ { \top } \tilde { \boldsymbol { v } } \right) \right) = Z _ { 0 } . +$$ + +Proof Let $Q \in \mathbb { R } ^ { d \times d }$ be a orthonormal matrix such that $Q ^ { \top } c _ { 0 } = c$ where $c _ { 0 } = ( 1 , 0 , . . . , 0 ) ^ { \top }$ and $\operatorname* { d e t } ( Q ) = 1$ , then we have $f ( \tilde { v } ) = f ( Q \tilde { v } )$ , and, + +$$ +\begin{array} { r l } & { \mathbb { E } _ { f } \left( \exp \left( c _ { 0 } ^ { \top } \tilde { v } \right) \right) = \displaystyle \int _ { \tilde { v } } f ( \tilde { v } ) \exp \left( c _ { 0 } ^ { \top } \tilde { v } \right) d \tilde { v } } \\ & { \quad \quad \quad = \displaystyle \int _ { \tilde { v } } f ( Q \tilde { v } ) \exp \left( c ^ { \top } Q \tilde { v } \right) \operatorname* { d e t } ( Q ) d \tilde { v } } \\ & { \quad \quad = \displaystyle \int _ { \tilde { v } ^ { \prime } } f ( \tilde { v } ^ { \prime } ) \exp \left( c ^ { \top } \tilde { v } ^ { \prime } \right) d \tilde { v } ^ { \prime } = \mathbb { E } _ { f } \left( \exp \left( c ^ { \top } \tilde { v } \right) \right) , } \end{array} +$$ + +which proves (8). + +Step 3: for any vector $\mu$ , one has the following concentration property, + +$$ +p \left( \big | \exp \left( c ^ { \top } \mu \right) - 1 \big | > k \right) \leq 2 \left[ \exp \left( - \frac { 1 } { 4 } \right) + \frac { \| \mu \| ^ { 2 } } { d - 1 } \frac { 1 } { \log ^ { 2 } ( 1 - k ) } \right] +$$ + +Proof Let $c _ { 1 } , . . . , c _ { d }$ be i.i.d. $\mathcal { N } ( 0 , 1 )$ , and let $\textstyle C = \sum _ { i = 1 } ^ { d } c _ { i } ^ { 2 }$ , then $c = ( c _ { 1 } , . . . , c _ { d } ) / \sqrt { C }$ is uniform over unit sphere. Since $c$ is uniform, then without loss of generality we can consider $\boldsymbol { \mu } = ( \lVert \boldsymbol { \mu } \rVert , 0 , . . . , 0 )$ . Thus it suffices to bound $\exp \left( \| \boldsymbol { \mu } \| c _ { 1 } / \sqrt { C } \right)$ . We divide the proof into the following steps: + +• $C$ follows chi-square distribution with the degree of freedom of $d$ , thus $C$ can be bounded by (Laurent & Massart, 2000), + +$$ +\begin{array} { r } { p ( C \geq d + 2 \sqrt { d x } + 2 x ) \leq \exp ( - x ) , \forall x > 0 . } \\ { p ( C \leq d - 2 \sqrt { d x } ) \leq \exp ( - x ) , \forall x > 0 . } \end{array} +$$ + +• Therefore for any $x > 0$ , one has, + +$$ +p \left( | C - d | \geq 2 { \sqrt { d x } } \right) \leq \exp ( - x ) +$$ + +Let $x = 1 / 4 d$ , one has, + +$$ +\begin{array} { l } { { p ( C > d + 1 ) \le \exp \left( - \displaystyle \frac { 1 } { 4 d } \right) , } } \\ { { \displaystyle p ( C < d - 1 ) \le \exp \left( - \displaystyle \frac { 1 } { 4 d } \right) . } } \end{array} +$$ + +• Since $c _ { 1 }$ is a Gaussian random variable with variance 1, by Chebyshev’s inequality, one has, + +$$ +p \left( \boldsymbol { y } c _ { i } \ge k \right) \le \frac { \boldsymbol { y } ^ { 2 } } { k ^ { 2 } } , \qquad p \left( \boldsymbol { y } c _ { i } \le - k \right) \le \frac { \boldsymbol { y } ^ { 2 } } { k ^ { 2 } } , \forall k > 0 +$$ + +and therefore thus, + +$$ +\begin{array} { r l } & { p ( \exp ( y c _ { i } ) - 1 > k ) \leq \cfrac { y ^ { 2 } } { \log ^ { 2 } ( 1 + k ) } , } \\ & { p ( \exp ( y c _ { i } ) - 1 < - k ) \leq \cfrac { y ^ { 2 } } { \log ( 1 - k ) ^ { 2 } } , \forall k > 0 . } \end{array} +$$ + +• Therefore we can bound $\exp \left( \| \boldsymbol { \mu } \| c _ { 1 } / \sqrt { C } \right)$ by, + +$$ +\begin{array} { r l } & { p \left( \exp \left( \frac { \| \mu \| c _ { 1 } } { \sqrt { C } } \right) - 1 > k \right) \leq p \left( C > d + 1 \right) } \\ & { \qquad + p \left( \exp \left( \frac { \| \mu \| c _ { 1 } } { \sqrt { C } } \right) - 1 > k \bigg | C < d + 1 \right) p \left( C < d + 1 \right) } \\ & { \qquad \leq \exp \left( - \frac { 1 } { 4 d } \right) + p \left( \exp \left( \frac { \| \mu \| c _ { 1 } } { \sqrt { d + 1 } } \right) - 1 > k \right) } \\ & { \qquad = \exp \left( - \frac { 1 } { 4 d } \right) + \frac { \| \mu \| ^ { 2 } } { d + 1 } \frac { 1 } { \log \left( 1 - k \right) ^ { 2 } } . } \\ & { p \left( \exp \left( \frac { \| \mu \| c _ { 1 } } { \sqrt { C } } \right) - 1 < - k \right) \leq \exp \left( - \frac { 1 } { 4 d } \right) + \frac { \| \mu \| ^ { 2 } } { d - 1 } \frac { 1 } { \log ^ { 2 } \left( 1 + k \right) } . } \end{array} +$$ + +Combining the two inequalities above, one has (9) proved. + +Step 4: We are now ready to prove convergence of $Z ( c )$ . With (9), let $\mathcal { C } \subset \mathbb { R } ^ { d }$ such that, + +$$ +\mathcal { C } = \left\{ c : \left| \exp ( c ^ { \top } \mu ) - 1 \right| < k , \left| \exp ( A c ^ { \top } u _ { i } ) - 1 \right| < k , \left| \exp ( - A c ^ { \top } u _ { i } ) - 1 \right| < k \forall i = 1 , . . . , D \right\} +$$ + +Then we can bound the probability on $\mathcal { C }$ by, + +$$ +\begin{array} { l } { p ( \mathcal { C } ) \geq p \left( \left| \exp ( c ^ { \top } \mu ) - 1 \right| < k \right) + \displaystyle \sum _ { i = 1 } ^ { D } p ( \left| \exp ( A c ^ { \top } u _ { i } ) - 1 \right| < k ) - 2 D } \\ { \geq 1 - ( 2 D + 1 ) \exp \left( - \displaystyle \frac { 1 } { 4 d } \right) - \displaystyle \frac { 2 D A ^ { 2 } } { d - 1 } \displaystyle \frac { 1 } { \log ^ { 2 } ( 1 - k ) } - \displaystyle \frac { \| \mu \| ^ { 2 } } { d - 1 } \displaystyle \frac { 1 } { \log ^ { 2 } ( 1 - k ) } . } \end{array} +$$ + +Next, we need to show that for every $w$ , the corresponding $\mathcal { C } ( w )$ , i.e., + +$$ +\mathcal { C } ( \boldsymbol { w } ) = \left\{ c : \left| \exp ( c ^ { \top } \boldsymbol { \mu } ) - 1 \right| < k , \left| \exp ( \alpha _ { i } ( \boldsymbol { w } ) c ^ { \top } \boldsymbol { u } _ { i } ) - 1 \right| < k , \forall i = 1 , . . . , D \right\} +$$ + +We observe that $\alpha _ { i } ( w )$ is bounded by $A$ , therefore for any $c$ that, + +$\begin{array} { r } { \operatorname* { m i n } ( \exp ( - A c ^ { \top } u _ { i } ) , \exp ( A c ^ { \top } u _ { i } ) ) \leq \exp ( \alpha _ { i } c ^ { \top } u _ { i } ) \leq \operatorname* { m a x } ( \exp ( - A c ^ { \top } u _ { i } ) , \exp ( A c ^ { \top } u _ { i } ) ) , } \end{array}$ and thus, 1 $\mathrm { n i n } ( \exp ( - A c ^ { \top } u _ { i } ) , \exp ( A c ^ { \top } u _ { i } ) ) - 1 \leq \exp ( \alpha _ { i } c ^ { \top } u _ { i } ) - 1 \leq \operatorname* { m a x } ( \exp ( - A c ^ { \top } u _ { i } ) , \exp ( A c ^ { \top } u _ { i } ) ) - 1$ , which yields, + +$$ +| \exp ( \alpha _ { i } c ^ { \top } u _ { i } ) - 1 | \leq \operatorname* { m a x } ( | \exp ( - A c ^ { \top } u _ { i } ) - 1 | , | \exp ( A c ^ { \top } u _ { i } ) - 1 | ) < k . +$$ + +Therefore we prove $c ( w ) \supset { \mathcal { C } }$ . Assembling everything together, one has, + +$$ +\begin{array} { r l } & { p \left( \left| \exp ( c ^ { \top } \mu ) \prod _ { i = 1 } ^ { D } \exp ( \alpha _ { i } ( w ) c ^ { \top } u _ { i } ) - 1 \right| > ( D + 1 ) k , \forall i = 1 , . . . , D , \forall w \in \mathscr { V } \right) } \\ & { \leq p ( \bar { \mathscr { C } } ) } \\ & { \leq ( 2 D + 1 ) \exp ( - \frac { 1 } { 4 d } ) + \frac { 2 D A ^ { 2 } } { d - 1 } \frac { 1 } { \log ^ { 2 } ( 1 - k ) } + \frac { \| \mu \| ^ { 2 } } { d - 1 } \frac { 1 } { \log ^ { 2 } ( 1 - k ) } } \end{array} +$$ + +For every $c \in { \mathcal { C } }$ , one has, + +$$ +\frac { 1 } { | \mathcal { V } | } | Z ( c ) - Z _ { 0 } | \leq \frac { ( D + 1 ) k } { | \mathcal { V } | } Z _ { 0 } . +$$ + +Let $Z = | \nu | Z _ { 0 }$ , one can conclude that, + +$$ +\begin{array} { r } { p ( ( 1 - \epsilon _ { z } ) Z \le Z ( c ) \le ( 1 + \epsilon _ { z } ) Z ) \ge 1 - \delta , } \end{array} +$$ + +where $\epsilon _ { z } = \Omega ( ( D + 1 ) / | \mathcal { V } | )$ and $\delta = \Omega ( D A ^ { 2 } / d )$ . + +# G.2 PROOF OF THEOREM A.1 + +Having Lemma G.1 ready, we can follow the same proof as in (Arora et al., 2016) that both $p ( w )$ and $p ( w , w ^ { \prime } )$ are correlated with $\lVert \boldsymbol v ( \boldsymbol w ) \rVert$ , formally + +$$ +\begin{array} { c } { \displaystyle \log p ( w ) \to \frac { \| v ( w ) \| ^ { 2 } } { 2 d } - \log Z , ~ \mathrm { a s } ~ | \mathcal { V } | \to \infty , } \\ { \displaystyle \log p ( w , w ^ { \prime } ) \to \frac { \| v ( w ) + v ( w ^ { \prime } ) \| ^ { 2 } } { 2 d } - \log Z , ~ \mathrm { a s } ~ | \mathcal { V } | \to \infty . } \end{array} +$$ + +Therefore, the inference presented in (Arora et al., 2016) (i.e., (4)) is obvious by assembling (12) and (13) together: + +$$ +\operatorname { P M I } ( w , w ^ { \prime } ) \to \frac { v ( w ) ^ { \top } v ( w ^ { \prime } ) } { d } , ~ \mathrm { a s } ~ \vert \mathcal { V } \vert \to \infty . +$$ \ No newline at end of file diff --git a/parse/train/HkuGJ3kCb/HkuGJ3kCb_content_list.json b/parse/train/HkuGJ3kCb/HkuGJ3kCb_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..34cc49637a0aa4e7eec7ff5853dd87f9b61355ea --- /dev/null +++ b/parse/train/HkuGJ3kCb/HkuGJ3kCb_content_list.json @@ -0,0 +1,3456 @@ +[ + { + "type": "text", + "text": "ALL-BUT-THE-TOP: SIMPLE AND EFFECTIVE POSTPROCESSING FOR WORD REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 826, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jiaqi Mu, Pramod Viswanath University of Illinois at Urbana Champaign {jiaqimu2, pramodv}@illinois.edu ", + "bbox": [ + 183, + 170, + 498, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 250, + 544, + 263 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Real-valued word representations have transformed NLP applications; popular examples are word2vec and GloVe, recognized for their ability to capture linguistic regularities. In this paper, we demonstrate a very simple, and yet counter-intuitive, postprocessing technique – eliminate the common mean vector and a few top dominating directions from the word vectors – that renders off-the-shelf representations even stronger. The postprocessing is empirically validated on a variety of lexical-level intrinsic tasks (word similarity, concept categorization, word analogy) and sentence-level tasks (semantic textural similarity and text classification) on multiple datasets and with a variety of representation methods and hyperparameter choices in multiple languages; in each case, the processed representations are consistently better than the original ones. ", + "bbox": [ + 233, + 279, + 766, + 431 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 457, + 334, + 472 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Words and their interactions (as sentences) are the basic units of natural language. Although words are readily modeled as discrete atomic units, this is unable to capture the relation between the words. Recent distributional real-valued representations of words (examples: word2vec, GloVe) have transformed the landscape of NLP applications – for instance, text classification (Socher et al., 2013b; Maas et al., 2011; Kim, 2014), machine translation (Sutskever et al., 2014; Bahdanau et al., 2014) and knowledge base completion (Bordes et al., 2013; Socher et al., 2013a). The success comes from the geometry of the representations that efficiently captures linguistic regularities: the semantic similarity of words is well captured by the similarity of the corresponding vector representations. ", + "bbox": [ + 174, + 487, + 825, + 599 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A variety of approaches have been proposed in recent years to learn the word representations: Collobert et al. (2011); Turian et al. (2010) learn the representations via semi-supervised learning by jointly training the language model and downstream applications; Bengio et al. (2003); Mikolov et al. (2010); Huang et al. (2012) do so by fitting the data into a neural network language model; Mikolov et al. (2013); Mnih & Hinton (2007) by log-linear models; and Dhillon et al. (2012); Pennington et al. (2014); Levy & Goldberg (2014); Stratos et al. (2015); Arora et al. (2016) by producing a lowdimensional representation of the cooccurrence statistics. Despite the wide disparity of algorithms to induce word representations, the performance of several of the recent methods is roughly similar on a variety of intrinsic and extrinsic evaluation testbeds. ", + "bbox": [ + 174, + 606, + 825, + 731 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we find that a simple processing renders the off-the-shelf existing representations even stronger. The proposed algorithm is motivated by the following observation. ", + "bbox": [ + 176, + 738, + 821, + 766 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Observation Every representation we tested, in many languages, has the following properties: ", + "bbox": [ + 178, + 781, + 802, + 795 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• The word representations have non-zero mean – indeed, word vectors share a large common vector (with norm up to a half of the average norm of word vector). • After removing the common mean vector, the representations are far from isotropic – indeed, much of the energy of most word vectors is contained in a very low dimensional subspace (say, 8 dimensions out of 300). ", + "bbox": [ + 215, + 806, + 825, + 880 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Implication Since all words share the same common vector and have the same dominating directions, and such vector and directions strongly influence the word representations in the same way, we propose to eliminate them by: (a) removing the nonzero mean vector from all word vectors, effectively reducing the energy; (b) projecting the representations away from the dominating $D$ directions, effectively reducing the dimension. Experiments suggest that $D$ depends on the representations (for example, the dimension of the representation, the training methods and their specific hyperparameters, the training corpus) and also depends on the downstream applications. Nevertheless, a rule of thumb of choosing $D$ around $d / 1 0 0$ , where $d$ is the dimension of the word representations, works uniformly well across multiple languages and multiple representations and multiple test scenarios. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We emphasize that the proposed postprocessing is counter intuitive – typically denoising by dimensionality reduction is done by eliminating the weakest directions (in a singular value decomposition of the stacked word vectors), and not the dominating ones. Yet, such postprocessing yields a “purified” and more “isotropic” word representation as seen in our elaborate experiments. ", + "bbox": [ + 174, + 208, + 825, + 263 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Experiments By postprocessing the word representation by eliminating the common parts, we find the processed word representations to capture stronger linguistic regularities. We demonstrate this quantitatively, by comparing the performance of both the original word representations and the processed ones on three canonical lexical-level tasks: ", + "bbox": [ + 176, + 279, + 825, + 334 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• word similarity task tests the extent to which the representations capture the similarity between two words – the processed representations are consistently better on seven different datasets, on average by $1 . 7 \\%$ ; concept categorization task tests the extent to which the clusters of word representations capture the word semantics – the processed representations are consistently better on three different datasets, by $2 . 8 \\%$ , $4 . 5 \\%$ and $4 . 3 \\%$ ; \nword analogy task tests the extent to which the difference of two representations captures a latent linguistic relation – again, the performance is consistently improved (by $0 . 5 \\%$ on semantic analogies, $0 . 2 \\%$ on syntactic analogies and $0 . 4 \\%$ in total). Since part of the dominant components are inherently canceled due to the subtraction operation while solving the analogy, we posit that the performance improvement is not as pronounced as earlier. ", + "bbox": [ + 215, + 345, + 825, + 507 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Extrinsic evaluations provide a way to test the goodness of representations in specific downstream tasks. We evaluate the effect of postprocessing on a standardized and important extrinsic evaluation task on sentence modeling: semantic textual similarity task – where we represent a sentence by its averaged word vectors and score the similarity between a pair of sentences by the cosine similarity between the corresponding sentence representation. Postprocessing improves the performance consistently and significantly over 21 different datasets (average improvement of $4 \\%$ ). ", + "bbox": [ + 174, + 518, + 825, + 603 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Word representations have been particularly successful in NLP applications involving supervisedlearning, especially in conjunction with neural network architecture. Indeed, we see the power of postprocessing in an experiment on a standard text classification task using a well established convoluntional neural network (CNN) classifier (Kim, 2014) and three RNN classifiers (with vanilla RNN, GRU (Chung et al., 2015) and LSTM Greff et al. (2016) as recurrent units). Across two different pre-trained word vectors, five datasets and four different architectures, the performance with processing improves on a majority of instances (34 out of 40) by a good margin $2 . 8 5 \\%$ on average), and the two performances with and without processing are comparable in the remaining ones. ", + "bbox": [ + 174, + 609, + 825, + 722 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Related Work. Our work is directly related to word representation algorithms, most of which have been elaborately cited. ", + "bbox": [ + 176, + 736, + 821, + 763 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Aspects similar to our postprocessing algorithm have appeared in specific NLP contexts very recently in (Sahlgren et al., 2016) (centering the mean) and (Arora et al., 2017) (nulling away only the first principal component). Although there is a superficial similarity between our work and (Arora et al. 2017), the nulling directions we take and the one they take are fundamentally different. Specifically, in Arora et al. (2017), the first dominating vector is \\*dataset-specific\\*, i.e., they first compute the sentence representation for the entire semantic textual similarity dataset, then extract the top direction from those sentence representations and finally project the sentence representation away from it. By doing so, the top direction will inherently encode the common information across the entire dataset, the top direction for the \"headlines\" dataset may encode common information about news articles while the top direction for \"Twitter’15\" may encode the common information about tweets. In contrast, our dominating vectors are over the entire vocabulary of the language. ", + "bbox": [ + 174, + 771, + 826, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "More generally, the idea of removing the top principal components has been studied in the context of positive-valued, high-dimensional data matrix analysis (Bullinaria & Levy, 2012; Price et al., 2006). Bullinaria & Levy (2012) posits that the highest variance components of the cooccurrence matrix are corrupted by information other than lexical semantics, thus heuristically justifying the removal of the top principal components. A similar idea appears in the context of population matrix analysis (Price et al., 2006), where the entries are also all positive. Our postprocessing operation is on dense low-dimensional representations (with both positive and negative entries). ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We posit that the postprocessing operation makes the representations more “isotropic” with stronger self-normalization properties – discussed in detail in Section 2 and Appendix A. Our main point is that this isotropy condition can be explicitly enforced to come up with new embedding algorithms (of which our proposed post-processing is a simple and practical version). ", + "bbox": [ + 174, + 208, + 825, + 263 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 POSTPROCESSING ", + "text_level": 1, + "bbox": [ + 176, + 285, + 354, + 300 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We test our observations on various word representations: four publicly available word representations (WORD2VEC1 (Mikolov et al., 2013) trained using Google News, $\\mathrm { G L O V E } ^ { 2 }$ (Pennington et al., 2014) trained using Common Crawl, RAND-WALK (Arora et al., 2016) trained using Wikipedia and TSCCA3 trained using English Gigaword) and two self-trained word representations using CBOW and Skip-gram (Mikolov et al., 2013) on the 2010 Wikipedia corpus from (Al-Rfou et al., 2013). The detailed statistics for all representations are listed in Table 1. For completeness, we also consider the representations on other languages: a detailed study is provided in Appendix C.2. ", + "bbox": [ + 173, + 316, + 825, + 415 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/8fb51e0bb35704de5680f511479b84a9e047eaf0c290a6f0829284432a2d777b.jpg", + "table_caption": [ + "Table 1: A detailed description for the embeddings in this paper. " + ], + "table_footnote": [], + "table_body": "
LanguageCorpusdimvocab sizeavg.u(w)2μl2
WORD2VECEnglishGoogle News3003,000,0002.040.69
GLOVEEnglishCommon Crawl3002,196,0178.303.15
RAND-WALKEnglishWikipedia30068,4302.270.70
CBOWEnglishWikipedia3001,028,9611.140.29
Skip-GramEnglishWikipedia3001,028,9612.321.25
", + "bbox": [ + 183, + 428, + 815, + 515 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $v ( w ) \\in \\mathbb { R } ^ { d }$ be a word representation for a given word $w$ in the vocabulary $\\nu$ . We observe the following two phenomena in each of the word representations listed above: ", + "bbox": [ + 383, + 560, + 823, + 603 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/ae50d675d2b50730ceab0ff862f138686a254dcfb15311df36d02c58a5bef8e0.jpg", + "image_caption": [ + "Figure 1: The decay of the normalized singular values of word representation. " + ], + "image_footnote": [], + "bbox": [ + 173, + 577, + 369, + 690 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• $\\{ v ( w ) : w \\in \\mathcal { V } \\}$ are not of zero-mean: i.e., all $v ( w )$ share a non-zero common vector, $v ( w ) = \\tilde { v } ( w ) + \\mu$ , where $\\mu$ is the average of all $v ( w )$ ’s, i.e., $\\begin{array} { r } { \\mu = 1 / | \\mathcal { V } | \\sum _ { w \\in \\mathcal { V } } v ( w ) } \\end{array}$ . The norm of $\\mu$ is approximately $1 / 6$ to 1/2 of the average norm of all $v ( w )$ (cf. Table 1). ", + "bbox": [ + 385, + 609, + 825, + 680 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• $\\{ \\tilde { v } ( w ) : w \\in \\mathcal { V } \\}$ are not isotropic: Let $u _ { 1 } , . . . , u _ { d }$ be the first to the last components recovered by the principal component analysis (PCA) of $\\{ \\tilde { v } ( w ) : w \\in \\mathcal { V } \\}$ , and $\\sigma _ { 1 } , . . . , \\sigma _ { d }$ be the corresponding combinations of we observe that normalized variance ratio. Each $u$ $\\begin{array} { r } { \\tilde { v } ( w ) = \\sum _ { i = 1 } ^ { d } \\alpha _ { i } ( w ) u _ { i } } \\end{array}$ $\\tilde { v } ( w )$ can be written as a linear . As shown in Figure 1,lly for small values of $\\sigma _ { i }$ $i$ and remains roughly constant over the later ones. This suggests there exists $D$ such that $\\alpha _ { i } \\gg \\alpha _ { j }$ for all $i \\leq D$ and $j \\gg D$ ; from Figure 1 one observes that $D$ is roughly 10 with dimension $d = 3 0 0$ . ", + "bbox": [ + 385, + 685, + 825, + 785 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 186, + 786, + 821, + 813 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Angular Asymmetry of Representations A modern understanding of word representations involves either PMI-based (including word2vec (Mikolov et al., 2010; Levy & Goldberg, 2014) and GloVe (Pennington et al., 2014)) or CCA-based spectral factorization approaches. While CCA-based spectral factorization methods have long been understood from a probabilistic (i.e., generative model) view point (Browne, 1979; Hotelling, 1936) and recently in the NLP context (Stratos et al., 2015), a corresponding effort for the PMI-based methods has only recently been conducted in an inspired work (Arora et al., 2016). ", + "bbox": [ + 176, + 829, + 825, + 872 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Arora et al. (2016) propose a generative model (named RAND-WALK) of sentences, where every word is parameterized by a $d$ -dimensional vector. With a key postulate that the word vectors are angularly uniform (“isotropic\"), the family of PMI-based word representations can be explained under the RAND-WALK model in terms of the maximum likelihood rule. Our observation that word vectors learnt through PMI-based approaches are not of zero-mean and are not isotropic (c.f. Section 2) contradicts with this postulate. The isotropy conditions are relaxed in Section 2.2 of (Arora et al., 2016), but the match with the spectral properties observed in Figure 1 is not immediate. ", + "bbox": [ + 174, + 166, + 825, + 265 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This contradiction is explicitly resloved by relaxing the constraints on the word vectors to directly fit the observed spectral properties. The relaxed conditions are: the word vectors should be isotropic around a point (whose distance to the origin is a small fraction of the average norm of word vectors) lying on a low dimensional subspace. Our main result is to show that even with this enlarged parameter-space, the maximum likelihood rule continues to be close to the PMI-based spectral factorization methods. A brief summary of RAND-WALK, and the mathematical connection between our work and theirs, are explored in detail in Appendix A. ", + "bbox": [ + 174, + 270, + 825, + 369 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.1 ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 386, + 302, + 400 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Since all word representations share the same common vector $\\mu$ and have the same dominating directions and such vector and directions strongly influence the word representations in the same way, we propose to eliminate them, as formally achieved as Algorithm 1. ", + "bbox": [ + 174, + 411, + 825, + 454 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/b0e539acdeefe1ed5282257f1b418e9378a289e40ce71a811baec47ce5e5512f.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm 1: Postprocessing algorithm on word representations.
Input:Word representations {v(w),w ∈V},a threshold parameter D, 1 Compute the mean of {ν(w),w²∈V},μ← 高Σwev v(w),u(w) ← v(w)- μ
2 Compute the PCA components: u1,., ud ← PCA({u(w),w ∈ V}).
3 Preprocessthe representations: v’'(w) ← u(w)-∑=1 (uT v(w)) ui
Output :Processed representations v'(w).
", + "bbox": [ + 158, + 470, + 823, + 571 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Significance of Nulled Vectors Consider the representation of the words as viewed in terms of the top $D$ PCA coefficients $\\alpha _ { \\ell } ( w )$ , for $1 \\le \\ell \\le D$ . We find that these few coefficients encode the frequency of the word to a significant degree; Figure 2 illustrates the relation between the $( \\alpha _ { 1 } ( w ) , \\alpha _ { 2 } ( w ) )$ and the unigram probabilty $p ( w )$ , where the correlation is geometrically visible. ", + "bbox": [ + 173, + 598, + 825, + 655 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4b9dd24523f3afb1815c7678268e8ebcf8a834a9d1a81a98d3663f0b43d21250.jpg", + "image_caption": [ + "Figure 2: The top two PCA directions (i.e, $\\alpha _ { 1 } ( w )$ and $\\alpha _ { 2 } ( w ) )$ ) encode frequency. " + ], + "image_footnote": [], + "bbox": [ + 184, + 662, + 810, + 815 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Discussion In our proposed processing algorithm, the number of components to be nulled, $D$ , is the only hyperparameter that needs to be tuned. We find that a good rule of thumb is to choose $D$ approximately to be $d / 1 0 0$ , where $d$ is the dimension of a word representation. This is empirically justified in the experiments of the following section where $d = 3 0 0$ is standard for published word representations. We trained word representations for higher values of $d$ using the WORD2VEC and ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "GLOVE algorithms and repeated these experiments; we see corresponding consistent improvements due to postprocessing in Appendix C. ", + "bbox": [ + 171, + 103, + 825, + 133 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.2 POSTPROCESSING AS A “ROUNDING” TOWARDS ISOTROPY ", + "text_level": 1, + "bbox": [ + 173, + 151, + 620, + 166 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The idea of isotropy comes from the partition function defined in (Arora et al., 2016), ", + "bbox": [ + 174, + 178, + 733, + 194 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2d545eb51c4755a624d28f2729e5e239d313c39114d6e92e3e30c750c05dc68c.jpg", + "text": "$$\nZ ( c ) = \\sum _ { w \\in \\mathcal { V } } \\exp \\left( c ^ { \\top } v ( w ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 403, + 202, + 593, + 237 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $Z ( c )$ should approximately be a constant with any unit vector $c$ (c.f. Lemma 2.1 in (Arora et al., 2016)). Hence, we mathematically define a measure of isotropy as follows, ", + "bbox": [ + 173, + 246, + 825, + 275 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3e0bb740fd179a3cf7fd9fc9eb21a36367e9acdd2b4ddfa716d8ce5c68d5aa49.jpg", + "text": "$$\nI ( \\{ v ( w ) \\} ) = \\frac { \\operatorname* { m i n } _ { \\| c \\| = 1 } Z ( c ) } { \\operatorname* { m a x } _ { \\| c \\| = 1 } Z ( c ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 395, + 284, + 602, + 320 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $I ( \\{ v ( w ) \\} )$ ranges from 0 to 1, and $I ( \\{ v ( w ) \\} )$ closer to 1 indicates that $\\{ v ( w ) \\}$ is more isotropic. The intuition behind our postprocessing algorithm can also be motivated by letting $I ( \\{ v ( \\bar { w } ) \\} ) 1$ . ", + "bbox": [ + 174, + 328, + 825, + 372 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let $V$ be the matrix stacked by all word vectors, where the rows correspond to word vectors, and $1 _ { | \\nu | }$ be the $| \\nu |$ -dimensional vectors with all entries equal to one, $Z ( c )$ can be equivalently defined as follows, ", + "bbox": [ + 173, + 377, + 825, + 421 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d658c75ea4b0c14eec340546cc71808ddb512dab40b68dc5ecb59f341acea0b6.jpg", + "text": "$$\nZ ( c ) = | \\mathcal { V } | + 1 _ { | | } \\mathcal { V } | ^ { \\top } V c + \\frac { 1 } { 2 } c ^ { \\top } V ^ { \\top } V c + \\sum _ { k = 3 } ^ { \\infty } \\frac { 1 } { k ! } \\sum _ { w \\in \\mathcal { V } } ( c ^ { \\top } v ( w ) ) ^ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 428, + 710, + 470 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$I ( \\{ v ( w ) \\} )$ is, therefore, can be very coarsely approximated by, ", + "bbox": [ + 173, + 479, + 588, + 496 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• A first order approximation: ", + "text_level": 1, + "bbox": [ + 217, + 507, + 433, + 522 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6221e30826da28f13f4b2f82d46caaaf5c965693c461fd70c11761ce68e39d24.jpg", + "text": "$$\nI ( \\{ v ( w ) \\} ) \\approx \\frac { | \\mathcal { V } | + \\operatorname* { m i n } _ { \\| c \\| = 1 } 1 _ { | \\mathcal { V } | } ^ { \\top } V c } { | \\mathcal { V } | + \\operatorname* { m a x } _ { \\| c \\| = 1 } 1 _ { | \\mathcal { V } | } ^ { \\top } V c } = \\frac { | \\mathcal { V } | - \\| 1 _ { | \\mathcal { V } | } ^ { \\top } V \\| } { | \\mathcal { V } | + \\| 1 _ { | \\mathcal { V } | } ^ { \\top } V \\| } .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 530, + 720, + 573 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Letting $I ( \\{ v ( w ) \\} ) = 1$ yields $\\| \\boldsymbol { 1 } _ { | \\nu | } ^ { \\top } V \\| = 0$ , which is equivalent to $\\begin{array} { r } { \\sum _ { w \\in \\mathcal { V } } v ( w ) = 0 } \\end{array}$ . The intuition behind the first order approximation matches with the first step of the proposed algorithm, where we enforce $v ( w )$ to have a zero mean. ", + "bbox": [ + 230, + 582, + 825, + 628 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• A second order approximation: ", + "text_level": 1, + "bbox": [ + 217, + 635, + 450, + 650 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e96b859500a41e2d502b106f74d12e34a46e706a40b9f1e488bb23d1d5227684.jpg", + "text": "$$\nI ( \\{ v ( w ) \\} ) \\approx \\frac { | \\mathcal V | + \\operatorname* { m i n } _ { \\| c \\| = 1 } 1 _ { | \\mathcal V | } ^ { \\top } V c + \\operatorname* { m i n } _ { \\| c \\| = 1 } \\frac { 1 } { 2 } c ^ { \\top } V ^ { \\top } V c } { | \\mathcal V | + \\operatorname* { m a x } _ { \\| c \\| = 1 } 1 _ { | \\mathcal V | } ^ { \\top } V c + \\operatorname* { m a x } _ { \\| c \\| = 1 } \\frac { 1 } { 2 } c ^ { \\top } V ^ { \\top } V c } = \\frac { | \\mathcal V | - \\| 1 _ { | \\mathcal V | } ^ { \\top } V \\| + \\frac { 1 } { 2 } \\sigma _ { \\operatorname* { m i n } } ^ { 2 } } { | \\mathcal V | + \\| 1 _ { | \\mathcal V | } ^ { \\top } V \\| + \\frac { 1 } { 2 } \\sigma _ { \\operatorname* { m a x } } ^ { 2 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 230, + 657, + 839, + 700 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\sigma _ { \\mathrm { m i n } }$ and $\\sigma _ { \\mathrm { m a x } }$ are the smallest and largest singular value of $V$ , respectively. Letting $I ( \\{ v ( w ) \\} ) = 1$ yields $\\| \\boldsymbol { 1 } _ { | \\nu | } ^ { \\top } V \\| = 0$ and $\\sigma _ { \\mathrm { m i n } } = \\sigma _ { \\mathrm { m a x } }$ . The fact that $\\sigma _ { \\mathrm { m i n } } = \\sigma _ { \\mathrm { m a x } }$ suggests the spectrum of $v ( w )$ ’s should be flat. The second step of the proposed algorithm removes the highest singular values, and therefore explicitly flatten the spectrum of $V$ . ", + "bbox": [ + 232, + 707, + 828, + 768 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Empirical Verification Indeed, we empirically validate the effect of postprocessing of on $I ( \\{ v ( w ) \\} )$ . Since there is no closed-form solution for arg $\\operatorname* { m a x } _ { \\| c \\| = 1 } Z ( c )$ or $\\mathrm { a r g m i n } _ { \\| c \\| = 1 } Z ( c )$ , and it is impossible to enumerate all $c$ ’s, we estimate the measure by, ", + "bbox": [ + 176, + 785, + 826, + 828 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8dc7b299cad75604f18a659e7a26d70bc38d87b3ed9a87a47b5327fde39901e7.jpg", + "text": "$$\nI ( \\{ v ( w ) \\} ) \\approx \\frac { \\operatorname* { m i n } _ { c \\in C } Z ( c ) } { \\operatorname* { m a x } _ { c \\in C } Z ( c ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 837, + 596, + 871 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $C$ is the set of eigenvectors of $V ^ { \\top } V$ . The value of $I ( \\{ v ( w ) \\} )$ for the original vectors and processed ones are reported in Table 2, where we can observe that the degree of isotropy vastly increases in terms of this measure. ", + "bbox": [ + 174, + 880, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/739242b54b19ad0cf931163e81b7b97cd7450366a9e99494378249584070cca0.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
beforeafter
WORD2VEC0.70.95
GLOVE0.0650.6
", + "bbox": [ + 218, + 106, + 442, + 151 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A formal way to verify the isotropy property is to directly check if the “self-normalization\" property (i.e., $\\dot { Z } ( c )$ is a constant, independent of $c$ (Andreas & Klein, 2015)) holds more strongly. Such a validation is seen diagrammatically in Figure 3 where we randomly sampled $1 { , } 0 0 0 c$ ’s as (Arora et al., ", + "bbox": [ + 501, + 103, + 825, + 199 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/624b7de02a6a5c554a4d7a959dc4fff52056e9a11d06419f52c96f1f9ad3f5ff.jpg", + "image_caption": [ + "Table 2: Before-After on the measure of isotropy. ", + "Figure 3: The histogram of $Z ( c )$ for 1,000 randomly sampled vectors $c$ of unit norm, where $x$ -axis is normalized by the mean of all values and $D = 2$ for GLOVE and $D = 3$ for WORD2VEC. " + ], + "image_footnote": [], + "bbox": [ + 232, + 189, + 764, + 359 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 431, + 326, + 448 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Given the popularity and widespread use of WORD2VEC (Mikolov et al., 2013) and GLOVE (Pennington et al., 2014), we use their publicly available pre-trained reprepsentations in the following experiments. We choose $D = 3$ for WORD2VEC and $D = 2$ for GLOVE. The key underlying principle behind word representations is that similar words should have similar representations. Following the tradition of evaluating word representations (Schnabel et al., 2015; Baroni et al., 2014), we perform three canonical lexical-level tasks: (a) word similarity; (b) concept categorization; (c) word analogy; and one sentence-level task: (d) semantic textual similarity. The processed representations consistently improve performance on all three of them, and especially strongly on the first two. ", + "bbox": [ + 173, + 462, + 826, + 588 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Word Similarity The word similarity task is as follows: given a pair of words, the algorithm assigns a “similarity\" score – if the pair of words are highly related then the score should also be high and vice versa. The algorithm is evaluated in terms of Spearman’s rank correlation compared to (a gold set of) human judgements. ", + "bbox": [ + 501, + 603, + 825, + 702 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/a1d581f7d6a350c99974758213d780ad882efe839df464c3829396a11eff80af.jpg", + "table_caption": [ + "Table 3: Before-After results $\\left( \\mathbf { x } 1 0 0 \\right)$ on word similarity task on seven datasets. " + ], + "table_footnote": [], + "table_body": "
WORD2VECGLOVE
orig.proc.orig.proc.
RG6576.0878.3476.9674.36
WS68.2969.0573.7976.79
RW53.7454.3346.4152.04
MEN78.2079.0880.4981.78
MTurk68.2369.3569.2970.85
SimLex44.2045.1040 8344.97
SimVerb36.3536.5028.3332.23
", + "bbox": [ + 174, + 608, + 483, + 739 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For this experiment, we use seven standard datasets: the first published RG65 dataset (Rubenstein & Goodenough, 1965); the widely used WordSim-353 (WS) dataset (Finkelstein et al., 2001) which contains 353 pairs of commonly used verbs and nouns; the rare-words (RW) dataset (Lu", + "bbox": [ + 501, + 708, + 825, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "ong et al., 2013) composed of rarely used words; the MEN dataset (Bruni et al., 2014) where the 3000 pairs of words are rated by crowdsourced participants; the MTurk dataset (Radinsky et al., 2011) where the 287 pairs of words are rated in terms of relatedness; the SimLex-999 (SimLex) dataset (Hill et al., 2016) where the score measures “genuine\" similarity; and lastly the SimVerb-3500 (SimVerb) dataset (Gerz et al., 2016), a newly released large dataset focusing on similarity of verbs. ", + "bbox": [ + 173, + 791, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In our experiment, the algorithm scores the similarity between two words by the cosine similarity between the two corresponding word vectors $( \\mathrm { C o s S i m } ( v _ { 1 } , v _ { 2 } ) = v _ { 1 } ^ { \\top } v _ { 2 } / \\Vert \\dot { v _ { 1 } } \\Vert \\Vert v _ { 2 } \\Vert )$ . The detailed performance on the seven datasets is reported in Table 3, where we see a consistent and significant performance improvement due to postprocessing, across all seven datasets. These statistics (average improvement of $2 . 3 \\%$ ) suggest that by removing the common parts, the remaining word representations are able to capture stronger semantic relatedness/similarity between words. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/00b30bf69bfa4b26715cd094aab0f81832f8970f1737258373ca7e6e9e03a1c3.jpg", + "table_caption": [ + "Table 4: Before-After results $\\left( \\mathbf { x } 1 0 0 \\right)$ on the categorization task. " + ], + "table_footnote": [], + "table_body": "
WORD2VECGLOVE
orig.proc.orig.proc.
ap54.4357.7264 .1865.42
esslli75.0084.0981.8281.82
battig71.9781.7186.5986.59
", + "bbox": [ + 173, + 159, + 468, + 233 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Concept Categorization This task is an indirect evaluation of the similarity principle: given a set of concepts, the algorithm needs to group them into different categories (for example, “bear” and “cat” are both animals and “city” and “country” are both related to districts). The clustering performance is then evaluated in terms of purity (Manning et al., 2008) – the fraction of the total number of the objects that were classified correctly. ", + "bbox": [ + 482, + 155, + 825, + 280 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We conduct this task on three different datasets: the Almuhareb-Poesio (ap) dataset (Almuhareb, 2006) contains 402 concepts which fall into 21 categories; the ESSLLI 2008 Distributional Semantic Workshop shared-task dataset (Baroni et al., 2008) that contains 44 concepts in 6 categories; and the Battig test set (Baroni & Lenci, 2010) that contains 83 words in 10 categories. ", + "bbox": [ + 173, + 286, + 826, + 343 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Here we follow the setting and the proposed algorithm in (Baroni et al., 2014; Schnabel et al., 2015) – we cluster words (via their representations) using the classical $k$ -Means algorithm (with fixed $k$ ). Again, the processed vectors perform consistently better on all three datasets (with average improvement of $2 . 5 \\%$ ); the full details are in Table 4. ", + "bbox": [ + 171, + 349, + 825, + 405 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/64065ed2fa0d3b79058122005af3acd6ba51c4b5c36f57962d479ce69015020b.jpg", + "table_caption": [ + "Table 5: Before-After results $( \\mathrm { x } 1 0 0 )$ on the word analogy task. " + ], + "table_footnote": [], + "table_body": "
WORD2VECGLOVE
orig.proc.orig.proc.
syntax73.4673.5074.9575.40
semantics72.2873.3679.2279.25
all72.9373.4476.8977.15
", + "bbox": [ + 174, + 433, + 491, + 507 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Word Analogy The analogy task tests to what extent the word representations can encode latent linguistic relations between a pair of words. Given three words $w _ { 1 }$ , $w _ { 2 }$ , and $w _ { 3 }$ , the analogy task requires the algorithm to find the word $w _ { 4 }$ such that $w _ { 4 }$ is to $w _ { 3 }$ as $w _ { 2 }$ is to $w _ { 1 }$ . ", + "bbox": [ + 501, + 429, + 825, + 512 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use the analogy dataset introduced in (Mikolov et al., 2013). The dataset can be divided into two parts: (a) the semantic part containing around $9 \\mathrm { k }$ ", + "bbox": [ + 500, + 518, + 823, + 560 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "questions, focusing on the latent semantic relation between pairs of words (for example, what is to Chicago as Texas is to Houston); and (b) the syntatic one containing roughly $1 0 . 5 \\mathrm { k }$ questions, focusing on the latent syntatic relation between pairs of words (for example, what is to “amazing” as “apprently” is to “apparent”). ", + "bbox": [ + 174, + 561, + 825, + 616 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In our setting, we use the original algorithm introduced in (Mikolov et al., 2013) to solve this problem, i.e., $w _ { 4 }$ is the word that maximize the cosine similarity between $v ( w _ { 4 } )$ and $v ( w _ { 2 } ) - v ( w _ { 1 } ) \\bar { + } v ( w _ { 3 } )$ . The average performance on the analogy task is provided in Table 5 (with a detailed performance provided in Table 19 in Appendix D). It can be noticed that while postprocessing continues to improve the performance, the improvement is not as pronounced as earlier. We hypothesize that this is because the mean and some dominant components get canceled during the subtraction of $v ( w _ { 2 } )$ from $v ( w _ { 1 } )$ , and therefore the effect of postprocessing is less relevant. ", + "bbox": [ + 173, + 622, + 826, + 722 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/6a57e72c25f4f18fae93b1387c280c2f1aac9605592c82fa3c54ca9d9adcc93b.jpg", + "table_caption": [ + "Table 6: Before-After results $\\left( \\mathbf { x } 1 0 0 \\right)$ on the semantic textual similarity tasks. " + ], + "table_footnote": [], + "table_body": "
WORD2VECGLOVE
orig.proc.orig.proc.
201257.2257.6748.2754.06
201356.8157.9844.8357.71
201462.8963.3051.1159.23
201562.7463.3547.2357.29
SICK70.1070 2065.1467.85
all60.8861.4549.1956.76
", + "bbox": [ + 173, + 747, + 464, + 864 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Semantic Textual Similarity Extrinsic evaluations measure the contribution of a word representation to specific downstream tasks; below, we study the effect of postprocessing on a standard sentence modeling task – semantic textual similarity (STS) which aims at testing the degree to which the algorithm can capture the semantic equivalence between two sentences. For each pair of sentences, the algorithm needs to measure how similar the two sentences are. The degree to which the measure matches with human judgment (in terms of Pearson correlation) is an index of the algorithm’s performance. We test the word representations on 20 textual similarity datasets from the 2012-2015 SemEval STS tasks (Agirre et al., 2012; 2013; 2014; 2015), and the 2012 SemEval Semantic Related task (SICK) (Marelli et al., 2014). ", + "bbox": [ + 482, + 743, + 825, + 922 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Representing sentences by the average of their constituent word representations is surprisingly effective in encoding the semantic information of sentences (Wieting et al., 2015; Adi et al., 2016) and close to the state-of-the-art in these datasets. We follow this rubric and represent a sentence $s$ based on its averaged word representation, i.e., $\\begin{array} { r } { \\boldsymbol { v } ( \\boldsymbol { s } ) = \\frac { 1 } { | \\boldsymbol { s } | } \\sum _ { w \\in \\boldsymbol { s } } \\boldsymbol { v } ( w ) } \\end{array}$ , and then compute the similarity between two sentences via the cosine similarity between the two representations. The average performance of the original and processed representations is itemized in Table 6 (with a detailed performance in Table 20 in Appendix E) – we see a consistent and significant improvement in performance because of postprocessing (on average $4 \\%$ improvement). ", + "bbox": [ + 174, + 138, + 825, + 252 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 POSTPROCESSING AND SUPERVISED CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 275, + 651, + 291 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Supervised downstream NLP applications have greatly improved their performances in recent years by combining the discriminative learning powers of neural networks in conjunction with the word representations. We evaluate the performance of a variety of neural network architectures on a standard and important NLP application: text classification, with sentiment analysis being a particularly important and popular example. The task is defined as follows: given a sentence, the algorithm needs to decide which category it falls into. The categories can be either binary (e.g., positive/negative) or can be more fine-grained (e.g. very positive, positive, neutral, negative, and very negative). ", + "bbox": [ + 174, + 306, + 825, + 405 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We evaluate the word representations (with and without postprocessing) using four different neural network architectures (CNN, vanilla-RNN, GRU-RNN and LSTM-RNN) on five benchmarks: (a) the movie review (MR) dataset (Pang & Lee, 2005); (b) the subjectivity (SUBJ) dataset (Pang & Lee, 2004); (c) the TREC question dataset (Li & Roth, 2002); (d) the IMDb dataset (Maas et al., 2011); (e) the stanford sentiment treebank (SST) dataset (Socher et al., 2013a). A detailed description of these standard datasets, their training/test parameters and the cross validation methods adopted is in Appendix F. Specifically, we allow the parameter $D$ (i.e., the number of nulled components) to vary between 0 and 4, and the best performance of the four neural network architectures with the now-standard CNN-based text classification algorithm (Kim, 2014) (implemented using tensorflow4) is itemized in Table 7. The key observation is that the performance of postprocessing is better in a majority (34 out of 40) of the instances by $2 . 3 2 \\%$ on average, and in the rest the instances the two performances (with and without postprocessing) are comparable. ", + "bbox": [ + 174, + 411, + 825, + 578 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/413b0e6e42d6b9abdf04fe77f0f476ea9e3bfe3a9e3a5ff370fbae9e23dd75f5.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
vanill-RNNGRURNNLSTMRNN
WORI2VECGLOVEWORI2VECGLVEWORI2VECGLDVEWORI2VECGLVE
orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.orig.proc.
MR70.8071.2771.0171.1174.9574.0171.1472.5677.8678.2674.9875.1375.6977.3472.0271.84
SUBJ87.1487.3386.9887.2582.8587.6081.4587.3790.9691.1091.1691.8590.2390.5490.7490.82
TREC87.8089.0087.6089.0080.6089.2085.2089.0091.6092.4091.6093.0088.0091.2085.8091.20
SST38.4638.3338.8237.8342.0839.9141.4541.9041.8645.0236.5237.6943.0842.0837.5138.05
IMDb86.6887.1287.2787.1050.1553.1452.7676.0782.9683.4781.5082.4481.2982.6079.1081.33
", + "bbox": [ + 176, + 593, + 825, + 672 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 7: Before-After results $( \\mathrm { x } 1 0 0 )$ on the text classification task using CNN (Kim, 2014) and vanilla RNN, GRU-RNN and LSTM-RNN. ", + "bbox": [ + 173, + 683, + 823, + 710 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A further validation of the postprocessing operation in a variety of downstream applications (eg: named entity recognition, syntactic parsers, machine translation) and classification methods (eg: random forests, neural network architectures) is of active research interest. Of particular interest is the impact of the postprocessing on the rate of convergence and generalization capabilities of the classifiers. Such a systematic study would entail a concerted and large-scale effort by the research community and is left to future research. ", + "bbox": [ + 173, + 726, + 825, + 809 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Discussion All neural network architectures, ranging from feedforward to recurrent (either vanilla or GRU or LSTM), implement at least linear processing of hidden/input state vectors at each of their nodes; thus the postprocessing operation suggested in this paper can in principle be automatically “learnt” by the neural network, if such internal learning is in-line with the end-to-end training examples. Yet, in practice this is complicated due to limitations of optimization procedures (SGD) and sample noise. We conduct a preliminary experiment in Appendix B and show that subtracting the mean (i.e., the first step of postprocessing) is “effectively learnt\" by neural networks within their nodes. ", + "bbox": [ + 173, + 827, + 825, + 897 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 152, + 318, + 169 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We present a simple postprocessing operation that renders word representations even stronger, by eliminating the top principal components of all words. Such an simple operation could be used for word embeddings in downstream tasks or as intializations for training task-specific embeddings. Due to their popularity, we have used the published representations of WORD2VEC and GLOVE in English in the main text of this paper; postprocessing continues to be successful for other representations and in multilingual settings – the detailed empirical results are tabulated in Appendix C. 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", + "bbox": [ + 173, + 295, + 823, + 324 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Torsten Zesch and Iryna Gurevych. Automatically creating datasets for measures of semantic relatedness. In Proceedings of the Workshop on Linguistic Distances, pp. 16–24. Association for Computational Linguistics, 2006. ", + "bbox": [ + 176, + 333, + 825, + 376 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Appendix: All-but-the-Top: Simple and Effective postprocessing for Word Representations ", + "text_level": 1, + "bbox": [ + 178, + 98, + 821, + 138 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A ANGULAR ASYMMETRY OF REPRESENTATIONS ", + "bbox": [ + 174, + 157, + 602, + 174 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A modern understanding of word representations involves either PMI-based (including word2vec (Mikolov et al., 2010; Levy & Goldberg, 2014) and GloVe (Pennington et al., 2014)) or CCA-based spectral factorization approaches. While CCA-based spectral factorization methods have long been understood from a probabilistic (i.e., generative model) view point (Browne, 1979; Hotelling, 1936) and recently in the NLP context (Stratos et al., 2015), a corresponding effort for the PMI-based methods has only recently been conducted in an inspired work (Arora et al., 2016). ", + "bbox": [ + 174, + 189, + 825, + 273 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(Arora et al., 2016) propose a generative model (named RAND-WALK) of sentences, where every word is parameterized by a $d$ -dimensional vector. With a key postulate that the word vectors are angularly uniform (“isotropic\"), the family of PMI-based word representations can be explained under the RAND-WALK model in terms of the maximum likelihood rule. Our observation that word vectors learnt through PMI-based approaches are not of zero-mean and are not isotropic (c.f. Section 2) contradicts with this postulate. The isotropy conditions are relaxed in Section 2.2 of (Arora et al., 2016), but the match with the spectral properties observed in Figure 1 is not immediate. ", + "bbox": [ + 174, + 280, + 825, + 378 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section, we resolve this by explicitly relaxing the constraints on the word vectors to directly fit the observed spectral properties. The relaxed conditions are: the word vectors should be isotropic around a point (whose distance to the origin is a small fraction of the average norm of word vectors) lying on a low dimensional subspace. Our main result is to show that even with this enlarged parameter-space, the maximum likelihood rule continues to be close to the PMI-based spectral factorization methods. Formally, the model, the original constraints of (Arora et al., 2016) and the enlarged constraints on the word vectors are listed below: ", + "bbox": [ + 174, + 385, + 825, + 483 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• A generative model of sentences: the word at time $t$ , denoted by $w _ { t }$ , is generated via a log-linear model with a latent discourse variable $c _ { t }$ (Arora et al., 2016), i.e., ", + "bbox": [ + 215, + 494, + 823, + 522 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/f05c56f4e83000d5ae7638aa1a9ad42ff2998db7b5aa0dce450c30a95a313ed0.jpg", + "text": "$$\np ( w _ { t } | c _ { t } ) = \\frac { 1 } { Z ( c _ { t } ) } \\exp \\left( c _ { t } ^ { \\top } v ( w _ { t } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 411, + 529, + 643, + 561 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $v ( w ) \\in \\mathbb { R } ^ { d }$ is the vector representation for a word $w$ in the vocabulary $V$ , $c _ { t }$ is the latent variable which forms a “slowly moving\" random walk, and the partition function: $\\begin{array} { r } { Z ( c ) = \\sum _ { w \\in \\mathcal { V } } \\exp \\left( c ^ { \\top } v ( w ) \\right) } \\end{array}$ . ", + "bbox": [ + 228, + 569, + 825, + 613 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• Constraints on the word vectors: (Arora et al., 2016) suppose that there is a Bayesian priori on the word vectors: ", + "bbox": [ + 215, + 616, + 823, + 643 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The ensemble of word vectors consists of i.i.d. draws generated by $\\boldsymbol { v } = \\boldsymbol { s } \\cdot \\boldsymbol { \\hat { v } }$ where $\\hat { v }$ is from the spherical Gaussian distribution, and $s$ is a scalar random variable. ", + "bbox": [ + 264, + 647, + 792, + 689 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A deterministic version of this prior is discussed in Section 2.2 of (Arora et al., 2016), but part of these (relaxed) conditions on the word vectors are specifically meant for Theorem 4.1 and not the main theorem (Theorem 2.2). The geometry of the word representations is only evaluated via the ratio of the quadratic mean of the singular values to the smallest one being small enough. This meets the relaxed conditions, but not sufficient to validate the proof approach of the main result (Theorem 2.2); what would be needed is that the ratio of the largest singular value to the smallest one be small. ", + "bbox": [ + 228, + 694, + 825, + 792 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• Revised conditions: We revise the Bayesian prior postulate (and in a deterministic fashion) formally as follows: there is a mean vector $\\mu$ , $D$ orthonormal vectors $u _ { 1 } , \\ldots , u _ { D }$ (that are orthogonal and of unit norm), such that every word vector $v ( w )$ can be represented by, ", + "bbox": [ + 218, + 796, + 825, + 838 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7bcfe9fb855fcd09affc8d60a25b70aff6aa18f0b2a05ffe7dec84a25bfc9126.jpg", + "text": "$$\nv ( w ) = \\mu + \\sum _ { i = 1 } ^ { D } \\alpha _ { i } ( w ) u _ { i } + \\tilde { v } ( w ) ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 844, + 640, + 888 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $\\mu$ is bounded, $\\alpha _ { i }$ is bounded by $A$ , $D$ is bounded by $D A ^ { 2 } \\ = \\ o ( d ) , \\ \\tilde { v } ( u$ ) are statistically isotropic. By statistical isotropy, we mean: for high-dimensional rectangles $R$ $\\begin{array} { r } { \\frac { 1 } { | \\mathcal { V } | } \\sum _ { w \\in \\mathcal { V } } \\mathbf { 1 } ( \\tilde { v } ( w ) \\in R ) \\int _ { R } f ( \\tilde { v } ) d \\tilde { v } } \\end{array}$ , as $| \\nu | \\to \\infty$ , where $f$ is an angle-independent pdf, i.e., $f ( \\tilde { v } )$ is a function of $\\| \\tilde { v } \\|$ . ", + "bbox": [ + 233, + 895, + 823, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 232, + 102, + 825, + 136 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The revised postulate differs from the original one in two ways: (a) it imposes a formal deterministic constraint on the word vectors; (b) the revised postulate allows the word vectors to be angularly asymmetric: as long as the energy in the direction of $u _ { 1 } , \\ldots , u _ { D }$ is bounded, there is no constraint on the coefficients. Indeed, note that there is no constraint on $\\tilde { v } ( w )$ to be orthogonal to $u _ { 1 } , \\ldots , u _ { D }$ . ", + "bbox": [ + 173, + 147, + 825, + 204 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Empirical Validation We can verify that the enlarged conditions are met by the existing word representations. Specifically, the natural choice for $\\mu$ is the mean of the word representations and $u _ { 1 } \\ldots u _ { D }$ are the singular vectors associated with the top $D$ singular values of the matrix of word vectors. We pick $D = 2 0$ for WORD2VEC and $D = 1 0$ for GLOVE, and the corresponding value of $D A ^ { 2 }$ for WORD2VEC and GLOVE vectors are both roughly 40, respectively; both values are small compared to $d = 3 0 0$ . ", + "bbox": [ + 173, + 218, + 825, + 303 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/4b8cef86f46030d6cc4853b73297993875e11f99c5c37b3bb9b91793e95189e5.jpg", + "image_caption": [ + "Figure 4: Spectrum of the published WORD2VEC and GLOVE and random Gaussian matrices, ignoring the top $D$ components; $D = 1 0$ for GLOVE and $D = 2 0$ for WORD2VEC. " + ], + "image_footnote": [], + "bbox": [ + 383, + 314, + 614, + 445 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This leaves us to check the statistical isotropy of the “remaining\" vectors $\\tilde { v } ( w )$ for words $w$ in the vocabulary. We do this by plotting the remaining spectrum (i.e. the $( D + 1 )$ -th, ..., 300th singular values) for the published WORD2VEC and GLOVE vectors in Figure 4. As a comparison, the empirical spectrum of a random Gaussian matrix is also plotted in Figure 4. We see that both spectra are flat (since the vocabulary size is much larger than the dimension $d = 3 0 0$ ). Thus the postprocessing operation can also be viewed as a way of making the vectors “more isotropic”. ", + "bbox": [ + 174, + 502, + 825, + 587 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Mathematical Contribution Under the revised postulate, we show that the main theorem in (Arora et al., 2016) (c.f. Theorem 2.2) still holds. Formally: ", + "bbox": [ + 174, + 601, + 823, + 631 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Theorem A.1 Suppose the word vectors satisfy the constraints. Then ", + "bbox": [ + 174, + 641, + 630, + 656 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/2abe32081a93d834cb5ce8227391e11e0e994692fef6afb07b3ffc7b130649a6.jpg", + "text": "$$\n\\mathrm { P M I } ( w _ { 1 } , w _ { 2 } ) \\stackrel { \\mathrm { d e f } } { = } \\log \\frac { p ( w _ { 1 } , w _ { 2 } ) } { p ( w _ { 1 } ) p ( w _ { 2 } ) } \\frac { v ( w _ { 1 } ) ^ { \\top } v ( w _ { 2 } ) } { d } , \\quad a s | \\mathcal { V } | \\infty ,\n$$", + "text_format": "latex", + "bbox": [ + 267, + 662, + 728, + 699 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $p ( w )$ is the unigram distribution induced from the model (2), and $p ( w _ { 1 } , w _ { 2 } )$ is the probability that two words $w _ { 1 }$ and $w _ { 2 }$ occur with each other within distance $q$ . ", + "bbox": [ + 174, + 705, + 823, + 734 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The proof is in Appendix G. Theorem A.1 suggests that the RAND-WALK generative model and its properties proposed by (Arora et al., 2016) can be generalized to a broader setting (with a relaxed restriction on the geometry of word representations) – relevantly, this relaxation on the geometry of word representations is empirically satisfied by the vectors learnt as part of the maximum likelihood rule. ", + "bbox": [ + 173, + 744, + 825, + 815 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B NEURAL NETWORKS LEARN TO POSTPROCESS ", + "text_level": 1, + "bbox": [ + 174, + 835, + 598, + 852 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Every neural network family posseses the ability to conduct linear processing inside their nodes; this includes feedforward and recurrent and convolutional neural network models. Thus, in principle, the postprocessing operation can be “learnt and implemented\" within the parameters of the neural network. On the other hand, due to the large number of parameters within the neural network, it is unclear how to verify such a process, even if it were learnt (only one of the layers might be implementing the postprocessing operation or via a combination of multiple effects). ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To address this issue, we have adopted a comparative approach in the rest of this section. The comparative approach involves adding an extra layer interposed in between the inputs (which are word vectors) and the rest of the neural network. This extra layer involves only linear processing. Next we compare the results of the final parameters of the extra layer (trained jointly with the rest of tne neural network parameters, using the end-to-end training examples) with and without preprocessing of the word vectors. Such a comparative approach allows us to separate the effect of the postprocessing operation on the word vectors from the complicated “semantics” of the neural network parameters. ", + "bbox": [ + 173, + 138, + 825, + 251 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/e668727ff6a7c1bcf2a788998dc91788d94dc84b09f6188760cca060207c6458.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
vanillaGRULSTM
W.G.W.G.W.G.
MR82.0749.2381.3547.6377.9544.92
SUBJ84.0249.9483.1550.6083.3948.95
TREC81.6852.9982.6850.4280.8046.77
SST79.6446.5978.0643.2177.7242.82
IMDb93.4866.3794.4955.2487.2746.74
", + "bbox": [ + 490, + 292, + 802, + 372 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/0000e8dbd070394e21308164cbb1d856e0bac9486f36e22091e032d44078c442.jpg", + "image_caption": [ + "Figure 6: The cosine similarity (x100) between " + ], + "image_footnote": [], + "bbox": [ + 218, + 285, + 442, + 391 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Figure 5: Time-expanded RNN architecture withan appended layer involving linear bias. $b _ { \\mathrm { p r o c . } } + \\mu$ and EC $b _ { \\mathrm { o r i g . } }$ , where W. and G. stand forLOVE respectively. WORD2V and G ", + "bbox": [ + 171, + 398, + 805, + 433 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Experiment We construct a modified neural network by explicitly adding a “postprocessing unit\" as the first layer of the RNN architecture (as in Figure 5, where the appended layer is used to test the first step (i.e., remove the mean vector)) of the postprocessing algorithm. ", + "bbox": [ + 174, + 502, + 823, + 545 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In the modified neural network, the input word vectors are now $\\boldsymbol { v } ( \\boldsymbol { w } ) - \\boldsymbol { b }$ instead of $v ( w )$ . Here $b$ is a bias vector trained jointly with the rest of the neural network parameters. Note that this is only a relabeling of the parameters from the perspective of the RNN architecture: the nonlinear activation function of the node is now operated on $\\overset { \\triangledown } { \\boldsymbol { A } } ( \\boldsymbol { v } ( \\boldsymbol { w } ) - \\boldsymbol { b } ) + \\boldsymbol { b } ^ { \\prime } = \\boldsymbol { A } \\boldsymbol { v } ( \\boldsymbol { w } ) + ( \\boldsymbol { b } ^ { \\prime } - \\boldsymbol { A } \\boldsymbol { b } )$ instead of the previous $A v ( w ) + b ^ { \\prime }$ . Let $b _ { \\mathrm { p r o c } }$ . and $b _ { \\mathrm { o r i g } }$ . be the inferred biases when using the processed and original word representations, respectively. ", + "bbox": [ + 174, + 551, + 825, + 636 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We itemize the cosine similarity between $b _ { \\mathrm { p r o c . } } + \\mu$ and $b _ { \\mathrm { o r i g . } }$ . in Table 6 for the 5 different datasets and 3 different neural network architectures. In each case, the cosine similarity is remarkably large (on average 0.66, in 300 dimensions) – in other words, trained neural networks implicitly postprocess the word vectors nearly exactly as we proposed. This agenda is successfully implemented in the context of verifying the removal of the mean vector. ", + "bbox": [ + 174, + 642, + 825, + 713 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The second step of our postprocessing algorithm (i.e., nulling away from the top principal components) is equivalent to applying a projection matrix $\\begin{array} { r } { P = I - \\sum _ { i = 1 } ^ { D } u _ { i } u _ { i } ^ { \\top } } \\end{array}$ on the word vectors, where $u _ { i }$ is the $i$ -th principal component and $D$ is the number of the removed directions. A comparative analysis effort for the second step (nulling the dominant PCA directions) is the following. Instead of applying a bias term $b$ , we multiply by a matrix $Q$ to simulate the projection operation. The input word vectors are now $Q _ { \\mathrm { o r i g . } } v ( w )$ instead of $v ( w )$ for the original word vectors, and $Q _ { \\mathrm { p r o c . } } P v ( \\bar { w } )$ instead of $P v ( w )$ for the processed vectors. Testing the similarity between $Q _ { \\mathrm { o r i g . } } P$ and $Q _ { \\mathrm { p r o c . } }$ , allows us to verify if the neural network learns to conduct the projection operation as proposed. ", + "bbox": [ + 173, + 719, + 825, + 833 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In our experiment, we found that such a result cannot be inferred. One possibility is that there are too many parameters in both $Q _ { \\mathrm { p r o c } }$ . and $Q _ { \\mathrm { o r i g . } }$ , which adds randomness to the experiment. Alternatively, the neural network weights may not be able to learn the second step of the postprocessing operation (indeed, in our experiments postprocessing significantly boosted end-performance of neural network architectures). A more careful experimental setup to test whether the second step of the postprocessing operation is learnt is left as a future research direction. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C EXPERIMENTS ON VARIOUS REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 606, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In the main text, we have reported empirical results for two published word representations: WORD2VEC and GLOVE, each in 300 dimensions. In this section, we report the results of the same experiments in a variety of other settings to show the generalization capability of the postprocessing operation: representations trained via WORD2VEC and GLOVE algorithms in dimensions other than 300, other representations algorithms (specifically TSCCA and RAND-WALK) and in multiple languages. ", + "bbox": [ + 174, + 132, + 825, + 217 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.1 STATISTICS OF MULTILINGUAL WORD REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 233, + 620, + 248 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use the publicly available TSCCA representations (Dhillon et al., 2012) on German, French, Spanish, Italian, Dutch and Chinese. The detailed statistics can be found in Table 8 and the decay of their singular values are plotted in Figure 7. ", + "bbox": [ + 176, + 258, + 825, + 303 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/e19ce574e1018b25a148a22c17caa89edd1352f76fa389b43c9378a3c7e8d53b.jpg", + "table_caption": [ + "Table 8: A detailed description for the TSCCA embeddings in this paper. " + ], + "table_footnote": [], + "table_body": "
LanguageCorpusdimvocab sizeavg.u(w)l2μl2
TSCCA-EnEnglishGigawords200300,0004.380.78
TSCCA-DeGermanNewswire200300,0004.520.79
TSCCA-FrFrenchGigaword200300,0004.340.81
TSCCA-EsSpanishGigaword200300.0004.170.79
TSCCA-ItItalianNewswire+Wiki200300,0004.340.79
TSCCA-N1DutchNewswire+Wiki200300,0004.460.72
TSCCA-ZhChineseGigaword200300.0004.510.89
", + "bbox": [ + 192, + 314, + 805, + 429 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/c31c5918d39e7c9069410fa5d0e535e28a69290bac0643aab8717f578613d0db.jpg", + "image_caption": [ + "Figure 7: The decay of the normalized singular values of multilingual word representation. " + ], + "image_footnote": [], + "bbox": [ + 366, + 478, + 629, + 628 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.2 MULTILINGUAL GENERALIZATION ", + "text_level": 1, + "bbox": [ + 176, + 680, + 457, + 694 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this section, we perform the word similarity task with the original and the processed TSCCA word representations in German and Spanish on three German similarity datasets (GUR65 – a German version of the RG65 dataset, GUR350, and ZG222 in terms of relatedness) (Zesch & Gurevych, 2006) and the Spanish version of RG65 dataset (Camacho-Collados et al., 2015). The choice of $D$ is 2 for both German and Spanish. ", + "bbox": [ + 174, + 705, + 825, + 776 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The experiment setup and the similarity scoring algorithm are the same as those in Section 3. The detailed experiment results are provided in Table 9, from which we observe that the processed representations are consistently better than the original ones. This provides evidence to the generalization capabilities of the postprocessing operation to multiple languages (similarity datasets in Spanish and German were the only ones we could locate). ", + "bbox": [ + 174, + 782, + 825, + 853 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.3 GENERALIZATION TO DIFFERENT REPRESENTATION ALGORITHMS ", + "text_level": 1, + "bbox": [ + 178, + 869, + 678, + 883 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Given the popularity and widespread use of WORD2VEC (Mikolov et al., 2013) and GLOVE (Pennington et al., 2014), the main text has solely focused on their published publicly avalable ", + "bbox": [ + 174, + 895, + 820, + 924 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/37d35c74119fe22bf29389d772fed16faa748e7f508f609f1ec2ef626ef530e1.jpg", + "table_caption": [ + "Table 9: Before-After results (x100) on the word similarity task in multiple languages. " + ], + "table_footnote": [], + "table_body": "
languageTSCCA
orig.proc.
RG65Spanish60.3360.37
GUR65German61.7564.39
GUR350German44.9146.59
ZG222German30.3732.92
", + "bbox": [ + 362, + 101, + 635, + 189 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "300-dimension representations. In this section, we show that the proposed postprocessing algorithm generalizes to other representation methods. Specifically, we demonstrate this on RAND-WALK (obtained via personal communication) and TSCCA (publicly available) on all the experiments of Section 3. The choice of $D$ is 2 for both RAND-WALK and TSCCA. ", + "bbox": [ + 174, + 236, + 825, + 291 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In summary, the performance improvements on the similarity task, the concept categorization task, the analogy task, and the semantic textual similarity dataset are on average $2 . 2 3 \\%$ , $2 . 3 9 \\%$ , $0 . 1 1 \\%$ and $0 . 6 1 \\%$ , respectively. The detailed statistics are provided in Table 10, Table 11, Table 12 and Table 13, respectively. These results are a testament to the generalization capabilities of the postprocessing algorithm to other representation algorithms. ", + "bbox": [ + 173, + 299, + 825, + 369 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/7562f79b92f4f10789626c15831b2497a2015a086aa67dd4f8fe53c075e62cb9.jpg", + "table_caption": [ + "Table 10: Before-After results $( \\mathrm { x } 1 0 0 )$ on the word similarity task on seven datasets. " + ], + "table_footnote": [], + "table_body": "
RAND-WALKTSCCA
orig.proc.orig.proc.
RG6580.6682.9647.5347.67
WS65.8974.3754.2154.35
RW45.1151.2343.9643.72
MEN73.5677.2265.4865.62
MTurk64.3566.1159.6560.03
SimLex34.0536.5534.8634.91
SimVerb16.0521.8423.7923.83
", + "bbox": [ + 341, + 380, + 656, + 512 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/76452710f53ec39de71afc9710e727d75643e2e64811088cfa485ac1894a6321.jpg", + "table_caption": [ + "Table 11: Before-After results $( \\mathrm { x } 1 0 0 )$ on the categorization task. " + ], + "table_footnote": [], + "table_body": "
RAND-WALKTSCCA
orig.proc.orig.proc.
ap59.8362.3660.0063.42
esslli72.7372.7368.1870.45
battig75.7381.8270.7370.73
", + "bbox": [ + 351, + 553, + 648, + 627 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/4b1eb9292350a55c058b5f41ea5b5e7e11f04c3fbdb29d7a0453d704da6d8f0b.jpg", + "table_caption": [ + "Table 12: Before-After results (x100) on the word analogy task. " + ], + "table_footnote": [], + "table_body": "
RAND-WALKTSCCA
orig.proc.orig.proc.
syn.60.3960.4837.7237.80
sem.83.5583.8214.5414.55
all70.5070.6727.3027.35
", + "bbox": [ + 354, + 669, + 643, + 743 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.4 ROLE OF DIMENSIONS ", + "text_level": 1, + "bbox": [ + 174, + 789, + 374, + 804 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The main text has focused on the dimension choice of $d = 3 0 0$ , due to its popularity. In this section we explore the role of the dimension in terms of both choice of $D$ and the performance of the postprocessing operation – we do this by using skip-gram model on the 2010 snapshot of Wikipedia corpus (Al-Rfou et al., 2013) to train word representations. We first observe that the two phenomena of Section 2 continue to hold: ", + "bbox": [ + 174, + 815, + 825, + 886 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "• From Table 14 we observe that the ratio between the norm of $\\mu$ and the norm average of all $v ( w )$ spans from 1/3 to $1 / 4$ ; ", + "bbox": [ + 215, + 895, + 823, + 924 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/64dabd6e186b8e9a8a23c25e2ea8ee8efdb139fac143cb2964701137398a61bf.jpg", + "table_caption": [ + "Table 13: Before-After results (x100) on the semantic textual similarity tasks. " + ], + "table_footnote": [], + "table_body": "
RAND-WALKTSCCA
orig.proc.orig.proc.
201238.0337.6644.5144.63
201337.4736.8543.2142.74
201446.0648.3252.8552.87
201547.8251.7656.2256.14
SICK51.5851.7656.1556.11
all43.4844.6750.0150.23
", + "bbox": [ + 351, + 101, + 647, + 218 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/80053da6839a3fe2cfe3f66132d6f179997ea99c8f30743a75be339d3689d56e.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
dim3004005006007008009001000
avg.u(w)l24.515.175.916.226.496.736.957.15
μ221.741.761.771.781.791.801.811.83
", + "bbox": [ + 246, + 262, + 751, + 308 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Table 14: Statistics on word representation of dimensions 300, 400, ..., and 1000 using the skip-gram model. ", + "bbox": [ + 171, + 318, + 825, + 347 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "• From Figure 8 we observe that the decay of the variance ratios $\\sigma _ { i }$ is near exponential for small values of $i$ and remains roughly constant over the later ones. ", + "bbox": [ + 214, + 376, + 823, + 405 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/144178c36d042206487ea6fb8b7dd93a93da2d7a6f83423558362445e9f29eda.jpg", + "image_caption": [ + "Figure 8: The decay of the normalized singular values of word representations. " + ], + "image_footnote": [], + "bbox": [ + 351, + 422, + 647, + 592 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A rule of thumb choice of $D$ is around $d / 1 0 0$ . We validate this claim empirically by performing the tasks in Section 3 on word representations of higher dimensions, ranging from 300 to 1000, where we set the parameter $D = d / \\bar { 1 } 0 0$ . In summary, the performance improvement on the four itemized tasks of Section 3 are $2 . 2 7 \\%$ , $3 . 3 7 \\%$ , 0.01 and $1 . 9 2 \\%$ respectively; the detailed results can be found in Table 15, Table 16, Table 17, and Table 18. Again, note that the improvement for analogy tasks is marginal. These experimental results justify the rule-of-thumb setting of $D = d / 1 0 0$ , although we emphasize that the improvements can be further accentuated by tuning the choice of $D$ based on the specific setting. ", + "bbox": [ + 173, + 655, + 825, + 768 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D EXPERIMENTS ON WORD ANALOGY TASK ", + "text_level": 1, + "bbox": [ + 174, + 795, + 562, + 811 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The detailed performance on the analogy task is provided in Table 19. ", + "bbox": [ + 174, + 832, + 630, + 847 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E EXPERIMENTS ON SEMANTIC TEXTUAL SIMILARITY TASK ", + "text_level": 1, + "bbox": [ + 173, + 872, + 696, + 890 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The detailed performance on the semantic textual similarity is provided in Table 20. ", + "bbox": [ + 173, + 909, + 722, + 924 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/483017946bac6d3bb7ed267e2ae28d165dc9e4ad64c30918c69b006ecdd65ac8.jpg", + "table_caption": [ + "Table 15: Before-After results $( \\mathrm { x } 1 0 0 )$ on word similarity task on seven datasets. " + ], + "table_footnote": [], + "table_body": "
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
RG6573.5774.7275.6479.8777.7281.9777.5980.7
WS70.2571.9570.872.8870.3972.7371.6474.04
RW46.2549.1145.9747.6346.648.5945.747.81
MEN75.6677.5976.0777.8975.978.1575.8878.15
Mturk75.6677.5967.6868.1166.8968.2567.667.87
SimLex34.0236.1935.1737.135.7337.6535.7638.04
SimVerb22.2224.9822.9125.3223.0325.8223.3525.97
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
RG6577.381.0777.5281.0779.7582.3478.1879.07
WS70.3173.0271.5274.6571.1973.0671.574.78
RW45.8648.444.964944.4449.2244.549.03
MEN75.8478.2175.8477.9676.1678.3576.7278.1
Mturk67.4767.7967.676867.9868.8768.3469.44
SimLex35.337.5936.5437.8536.6238.4436.6738.58
SimVerb22.8125.623.4825.5723.6825.7623.2426.58
", + "bbox": [ + 233, + 102, + 764, + 361 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/d8a3cdabb9216480778875f77f23e2348c21530f91d832911d6c4dfc819ff01b.jpg", + "table_caption": [ + "Table 16: Before-After results $( \\mathrm { x } 1 0 0 )$ on the categorization task. " + ], + "table_footnote": [], + "table_body": "
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
ap46.148.6142.5745.3446.8550.8840.345.84
esslli68.1872.7364.282.7264.265.4365.9172.73
battig71.677.7868.187568.1870.4546.9166.67
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
ap38.0441.3134.7639.834.7627.4627.9628.21
esslli54.5554.5568.1856.8272.7372.7352.2752.27
battig62.9666.6767.969.1449.3859.2651.8546.91
", + "bbox": [ + 245, + 401, + 753, + 546 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/913926dce57c75c958222d014cbf1c722d4592856a00e89bc3b1221f3f77ec1f.jpg", + "table_caption": [], + "table_footnote": [ + "Table 17: Before-After results (x100) on the word analogy task. " + ], + "table_body": "
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
syn.60.4860.5261.6161.4560.9360.8461.6661.57
sem.74.5174.5477.1177.3676.3976.8977.2877.61
all.66.8666.8768.6668.6967.8868.1168.7768.81
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
syn.60.9461.0268.3868.3460.4760.3067.5667.30
sem.77.2477.2677.2477.3576.7676.9076.7176.51
all.68.3668.4168.3868.5067.9167.6767.5667.30
", + "bbox": [ + 246, + 588, + 750, + 733 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "F STATISTICS OF TEXT CLASSIFICATION DATASETS ", + "text_level": 1, + "bbox": [ + 173, + 784, + 617, + 801 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We evaluate the word representations (with and without postprocessing) using four different neural network architectures (CNN, vanilla-RNN, GRU-RNN and LSTM-RNN) on five benchmarks: ", + "bbox": [ + 173, + 818, + 825, + 847 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• the movie review (MR) dataset (Pang & Lee, 2005) where each review is composed by only one sentence; \n• the subjectivity (SUBJ) dataset (Pang & Lee, 2004) where the algorithm needs to decide whether a sentence is subjective or objective; \n• the TREC question dataset (Li & Roth, 2002) where all the questions in this dataset has to be partitioned into six categories; \n• the IMDb dataset (Maas et al., 2011) – each review consists of several sentences; \n• the Stanford sentiment treebank (SST) dataset (Socher et al., 2013a), where we only use the full sentences as the training data. ", + "bbox": [ + 217, + 859, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/e13f316ab026c86fa81be06f63c3700230a6d89fe9f8bebc72dfb1b4d6e0256e.jpg", + "table_caption": [ + "Table 18: Before-After results (x100) on the semantic textual similarity tasks. " + ], + "table_footnote": [], + "table_body": "
Dim300400500600
orig.proc.orig.proc.orig.proc.orig.proc.
201254.5154.9554.3154.5755.1356.2355.3556.03
201356.5857.8956.3557.3557.5559.3857.4359.00
201459.661.9259.5761.6261.1964.3861.1063.86
201559.6561.4859.6961.1961.6364.7761.4264.04
SICK68.8970.7960.670.2768.6371.0068.5870.57
all58.3259.9158.2559.5559.6162.0259.5761.55
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
201255.5256.4954.4754.8554.6955.1854.3454.78
201357.6159.3156.7557.6256.9858.2656.7857.73
201461.5764.7760.5162.8360.8963.3460.7863.03
201562.0565.4560.7462.8461.0963.4860.9263.03
SICK68.3870.6367.9469.5967.8669.567.5869.16
all59.9662.3458.8760.3959.1660.8858.9460.48
", + "bbox": [ + 245, + 101, + 753, + 332 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/ec2c7fe7148223f6a70c4aaafb14a0325e7e17e14de30f9ba8c504a590b94c21.jpg", + "table_caption": [ + "Table 19: Before-After results (x100) on the word analogy task. " + ], + "table_footnote": [], + "table_body": "
WORD2VECGLOVE
orig.proc.orig.proc.
capital-common-countries82.0183.6095.0695.96
capital-world78.3880.0891.8992.31
city-in-state69.5669.8869.5670.45
currency32.4332.9221.5921.36
family84.9884.5995.8495.65
gram1-adjective-to-adverb28.0227.7240.4239.21
gram2-opposite40.1440.5131.6530.91
gram3-comparative89.1989.2686.9387.09
gram4-superlative82.7183.3390.4690.59
gram5-present-participle79.3679.6482.9582.76
gram6-nationality-adjective90.2490.3690.2490.24
gram7-past-tense66.0366.5363.9164.87
gram8-plural91.0790.6195.2795.36
gram9-plural-verbs68.7467.5867.2468.05
", + "bbox": [ + 282, + 376, + 715, + 607 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 215, + 661, + 826, + 753 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In TREC, SST and IMDb, the datasets have already been split into train/test sets. Otherwise we use 10-fold cross validation in the remaining datasets (i.e., MR and SUBJ). Detailed statistics of various features of each of the datasets are provided in Table 21. ", + "bbox": [ + 178, + 767, + 825, + 810 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "G PROOF OF THEOREM A.1 ", + "text_level": 1, + "bbox": [ + 176, + 835, + 418, + 852 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Given the similarity between the setup in Theorem 2.2 in (Arora et al., 2016) and Theorem A.1, many parts of the original proof can be reused except one key aspect – the concentration of $Z ( c )$ . We summarize this part in the following lemma: ", + "bbox": [ + 174, + 871, + 825, + 914 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/83aca756737460fd798b69d3c009436b680978c439d1c4a8b9c1aa11c615b099.jpg", + "table_caption": [ + "Table 20: Before-After results (x100) on the semantic textual similarity tasks. " + ], + "table_footnote": [], + "table_body": "
WORD2VECGLOVE
orig.proc.orig.proc.
2012.MSRpar42.1243.8544.5444.09
2012.MSRvid72.0772.1664.4768.05
2012.OnWN69.3869.4853.0765.67
2012.SMTeuroparl53.1554.3241.7445.28
2012.SMTnews49.3748.5337.5447.22
2013.FNWN40.7041.9637.5439.34
2013.OnWN67.8768.1747.2258.60
2013.headlines61.8863.8149.7357.20
2014.OnWN74.6174.7857.4167.56
2014.deft-forum32.1933.2621.5529.39
2014.deft-news66.8365.9665.1471.45
2014.headlines58.0159.5847.0552.60
2014.images73.7574.1757.2268.28
2014.tweet-news71.9272.0758.3266.13
2015.answers-forum46.3546.8030.0239.86
2015.answers-students68.0767.9949.2062.38
2015.belief59.7260.4244.0557.68
2015.headlines61.4763.4546.2253.31
2015.images78.0978.0866.6373.20
SICK70.1070.2065.1467.85
", + "bbox": [ + 297, + 101, + 699, + 417 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/dce90ada14a2feac02934ff5a8ade3bd82d453791fb4273c1fb0abec01119c91.jpg", + "table_caption": [ + "Table 21: Statistics for the five datasets after tokenization: c represents the number of classes; $l$ represents the average sentence length; Train represents the size of the training set; and Test represent the size of the test set. " + ], + "table_footnote": [], + "table_body": "
C1TrainTest
MR22010,66210-fold cross validation
SUBJ22310,00010-fold cross validation
TREC6105,952500
SST51811,8552,210
IMDb210025,00025,000
", + "bbox": [ + 312, + 492, + 684, + 580 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Lemma G.1 Let c be a random variable uniformly distributed over the unit sphere, we prove that with high probability, $Z ( c ) / | \\nu |$ converges to a constant $Z$ : ", + "bbox": [ + 169, + 691, + 825, + 722 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/7939e7aa0b75e2c32cf89b214e10656644e8caa36d078243fbdf4e184cbd0f8d.jpg", + "text": "$$\n\\begin{array} { r } { p ( ( 1 - \\epsilon _ { z } ) Z \\le Z ( c ) \\le ( 1 + \\epsilon _ { z } ) Z ) \\ge 1 - \\delta , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 351, + 762, + 645, + 780 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $\\epsilon _ { z } = \\Omega ( ( D + 1 ) / | \\mathcal { V } | )$ and $\\delta = \\Omega ( ( D A ^ { 2 } + \\| \\mu \\| ^ { 2 } ) / d )$ . ", + "bbox": [ + 173, + 821, + 570, + 840 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Our proof differs from the one in (Arora et al., 2016) in two ways: (a) we treat $v ( w )$ as deterministic parameters instead of random variables and prove the Lemma by showing a certain concentration of measure; (b) the asymmetric parts $\\mu$ and $u _ { 1 } , . . . , u _ { D }$ , (which did not exist in the original proof), need to be carefully addressed to complete the proof. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "G.1 PROOF OF LEMMA G.1 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 377, + 118 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Given the constraints on the word vectors (3), the partition function $Z ( c )$ can be rewritten as, ", + "bbox": [ + 171, + 128, + 782, + 145 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/84e6f7cd1b249da6bf0177959ce06bb94589aaca6f8e7bce0c3666f586e59fb5.jpg", + "text": "$$\n\\begin{array} { r l } & { Z ( c ) = \\displaystyle \\sum _ { v \\in \\mathcal { V } } \\exp ( c ^ { \\top } v ( w ) ) } \\\\ & { \\quad \\quad = \\displaystyle \\sum _ { v \\in \\mathcal { V } } \\exp \\left( c ^ { \\top } \\left( \\mu + \\sum _ { i = 1 } ^ { D } \\alpha _ { i } ( w ) u _ { i } + \\tilde { v } ( w ) \\right) \\right) } \\\\ & { \\quad \\quad = \\displaystyle \\sum _ { v \\in \\mathcal { V } } \\exp ( c ^ { \\top } \\mu ) \\left[ \\displaystyle \\prod _ { i = 1 } ^ { D } \\exp ( \\alpha _ { i } ( w ) c ^ { \\top } u _ { i } ) \\right] \\exp \\left( c ^ { \\top } \\tilde { v } ( w ) \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 151, + 705, + 277 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The equation above suggests that we can divide the proof into five parts. ", + "bbox": [ + 173, + 282, + 647, + 297 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Step 1: for every unit vector $c$ , one has, ", + "bbox": [ + 173, + 313, + 446, + 328 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/96418a7eb81e6b6f6fec61f7fcf9b1261c05453b28da48cc1ceae0f67f3f4711.jpg", + "text": "$$\n\\frac { 1 } { | \\mathcal { V } | } \\sum _ { w \\in \\mathcal { V } } \\exp ( c ^ { \\top } \\tilde { v } ( w ) ) \\mathbb { E } _ { f } ( \\exp ( c ^ { \\top } \\tilde { v } ) ) , \\ \\mathrm { a s } \\ | \\mathcal { V } | \\infty .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 333, + 692, + 372 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof Let $M , N$ be a positive integer, and let $A _ { M } \\subset \\mathbb { R } ^ { d }$ such that, ", + "bbox": [ + 173, + 383, + 612, + 400 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/74545bfdaad9b5d4f1acbff1707e39de3b5d410099e57d9ad31eca1a2ce9db4f.jpg", + "text": "$$\nA _ { M , N } = \\left\\{ \\widetilde v \\in \\mathbb { R } ^ { d } : \\frac { M - 1 } { N } < \\exp ( c ^ { \\top } \\widetilde v ) \\leq \\frac { M } { N } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 405, + 668, + 440 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Since $A _ { M , N }$ can be represented by a union of countable disjoint rectangles, we know that for every $M , N \\in \\mathbb { N } _ { + }$ , ", + "bbox": [ + 173, + 445, + 823, + 476 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/33331d80926862e44d139723369d8a45c0460e554f469ddd962e58dc1913fbda.jpg", + "text": "$$\n\\frac { 1 } { | \\mathcal { V } | } \\sum _ { w \\in \\mathcal { V } } \\mathbf { 1 } ( \\tilde { v } ( w ) \\in A _ { M , N } ) = \\int _ { A _ { M , N } } f ( \\tilde { v } ) d \\tilde { v } .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 481, + 648, + 520 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Further, since $A _ { M , N }$ are disjoint for different $M$ ’s and $\\mathbb { R } ^ { d } = \\cup _ { M = 1 } ^ { \\infty } A _ { M , N }$ , one has, ", + "bbox": [ + 173, + 527, + 722, + 545 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/f976fb1102c5a416bc150de2eb26f33cdb78ec2b0c0a2c81eee8354a4891fa42.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\frac { 1 } { | \\mathcal { V } | } \\sum _ { w \\in \\mathcal { V } } \\exp \\big ( c ^ { \\top } \\tilde { v } ( w ) \\big ) = \\displaystyle \\sum _ { M = 1 } ^ { \\infty } \\frac { 1 } { | \\mathcal { V } | } \\sum _ { w \\in \\mathcal { V } } \\mathbf { 1 } \\big ( \\tilde { v } ( w ) \\in A _ { M , N } \\big ) \\exp \\big ( c ^ { \\top } \\tilde { v } ( w ) \\big ) } & { } \\\\ { \\displaystyle \\leq \\sum _ { M = 1 } ^ { \\infty } \\frac { 1 } { | \\mathcal { V } | } \\sum _ { w \\in \\mathcal { V } } \\mathbf { 1 } \\big ( \\tilde { v } ( w ) \\in A _ { M , N } \\big ) \\frac { M } { N } } & { } \\\\ { \\displaystyle \\to \\sum _ { M = 1 } ^ { \\infty } \\frac { M } { N } \\int _ { A _ { M , N } } f ( \\tilde { v } ) d \\tilde { v } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 551, + 736, + 679 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The above statement holds for every $N$ . Let $N \\to \\infty$ , by definition of integration, one has, ", + "bbox": [ + 169, + 683, + 766, + 699 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/624dede8e965d512f85cff2e312a93c77a35b7f64c72c5352ea00d4336750915.jpg", + "text": "$$\n\\operatorname* { l i m } _ { N \\infty } \\sum _ { M = 1 } ^ { \\infty } \\frac { M } { N } \\int _ { A _ { M , N } } f ( \\tilde { v } ) d \\tilde { v } = \\mathbb { E } _ { f } ( \\exp ( c ^ { \\top } \\tilde { v } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 705, + 663, + 747 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "which yields, ", + "bbox": [ + 173, + 753, + 263, + 768 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/ee647ea356bba5d273aa923fca5b4a93467ffa7f964278c0316cf7a612001c32.jpg", + "text": "$$\n\\frac { 1 } { | \\mathcal { V } | } \\sum _ { w \\in \\mathcal { V } } \\exp ( c ^ { \\top } \\tilde { v } ( w ) ) \\leq \\mathbb { E } _ { f } ( \\exp ( c ^ { \\top } \\tilde { v } ) ) , \\ \\mathrm { a s \\ } | \\mathcal { V } | \\infty .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 775, + 691, + 813 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Similarly, one has, ", + "bbox": [ + 174, + 819, + 297, + 834 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/a23ff63c4fcbb63686ed361af17a7bf72584361ac960d48efbd58161236de53b.jpg", + "text": "$$\n\\begin{array} { r } { \\frac { 1 } { | \\mathcal { V } | } \\displaystyle \\sum _ { w \\in \\mathcal { V } } \\exp \\left( c ^ { \\top } \\tilde { v } ( w ) \\right) \\geq \\displaystyle \\operatorname* { l i m } _ { N \\to \\infty } \\sum _ { M = 1 } ^ { \\infty } \\frac { M - 1 } { N } \\int _ { A _ { M , N } } f ( \\tilde { v } ) d \\tilde { v } } \\\\ { = \\mathbb { E } _ { f } \\left( \\exp \\left( c ^ { \\top } \\tilde { v } \\right) \\right) , \\ \\mathrm { a s } \\ | \\mathcal { V } | \\to \\infty . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 840, + 699, + 905 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Putting (6) and (7) proves (5). ", + "bbox": [ + 173, + 909, + 370, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Step 2: the expected value, $\\mathbb { E } _ { f } \\left( \\exp \\left( c ^ { \\top } \\tilde { v } \\right) \\right)$ is a constant independent of $c$ : ", + "bbox": [ + 171, + 101, + 678, + 121 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/a147ab957622716e02e851c1ad6fae05d04c02598593dbecac4a7543da7a0f81.jpg", + "text": "$$\n\\mathbb { E } _ { f } \\left( \\exp \\left( \\boldsymbol { c } ^ { \\top } \\tilde { \\boldsymbol { v } } \\right) \\right) = Z _ { 0 } .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 130, + 576, + 150 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof Let $Q \\in \\mathbb { R } ^ { d \\times d }$ be a orthonormal matrix such that $Q ^ { \\top } c _ { 0 } = c$ where $c _ { 0 } = ( 1 , 0 , . . . , 0 ) ^ { \\top }$ and $\\operatorname* { d e t } ( Q ) = 1$ , then we have $f ( \\tilde { v } ) = f ( Q \\tilde { v } )$ , and, ", + "bbox": [ + 171, + 161, + 826, + 194 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/3d8c1e2d74e9790d3728fa368a75b5d808c8c9940e44fa961dbbd9704d9b32ba.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { f } \\left( \\exp \\left( c _ { 0 } ^ { \\top } \\tilde { v } \\right) \\right) = \\displaystyle \\int _ { \\tilde { v } } f ( \\tilde { v } ) \\exp \\left( c _ { 0 } ^ { \\top } \\tilde { v } \\right) d \\tilde { v } } \\\\ & { \\quad \\quad \\quad = \\displaystyle \\int _ { \\tilde { v } } f ( Q \\tilde { v } ) \\exp \\left( c ^ { \\top } Q \\tilde { v } \\right) \\operatorname* { d e t } ( Q ) d \\tilde { v } } \\\\ & { \\quad \\quad = \\displaystyle \\int _ { \\tilde { v } ^ { \\prime } } f ( \\tilde { v } ^ { \\prime } ) \\exp \\left( c ^ { \\top } \\tilde { v } ^ { \\prime } \\right) d \\tilde { v } ^ { \\prime } = \\mathbb { E } _ { f } \\left( \\exp \\left( c ^ { \\top } \\tilde { v } \\right) \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 202, + 712, + 305 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "which proves (8). ", + "bbox": [ + 173, + 313, + 289, + 328 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Step 3: for any vector $\\mu$ , one has the following concentration property, ", + "bbox": [ + 173, + 345, + 650, + 362 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/58e5db9c661477cc0fd6562807d0e25d0df6cba575c29f681d105bc09cecc04b.jpg", + "text": "$$\np \\left( \\big | \\exp \\left( c ^ { \\top } \\mu \\right) - 1 \\big | > k \\right) \\leq 2 \\left[ \\exp \\left( - \\frac { 1 } { 4 } \\right) + \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 279, + 371, + 718, + 407 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof Let $c _ { 1 } , . . . , c _ { d }$ be i.i.d. $\\mathcal { N } ( 0 , 1 )$ , and let $\\textstyle C = \\sum _ { i = 1 } ^ { d } c _ { i } ^ { 2 }$ , then $c = ( c _ { 1 } , . . . , c _ { d } ) / \\sqrt { C }$ is uniform over unit sphere. Since $c$ is uniform, then without loss of generality we can consider $\\boldsymbol { \\mu } = ( \\lVert \\boldsymbol { \\mu } \\rVert , 0 , . . . , 0 )$ . Thus it suffices to bound $\\exp \\left( \\| \\boldsymbol { \\mu } \\| c _ { 1 } / \\sqrt { C } \\right)$ . We divide the proof into the following steps: ", + "bbox": [ + 173, + 420, + 826, + 476 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• $C$ follows chi-square distribution with the degree of freedom of $d$ , thus $C$ can be bounded by (Laurent & Massart, 2000), ", + "bbox": [ + 217, + 487, + 825, + 517 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/dc60314495b8ba9a9ba0041ac620a14abb7b6480fd1e9c9a268d064585ab61a3.jpg", + "text": "$$\n\\begin{array} { r } { p ( C \\geq d + 2 \\sqrt { d x } + 2 x ) \\leq \\exp ( - x ) , \\forall x > 0 . } \\\\ { p ( C \\leq d - 2 \\sqrt { d x } ) \\leq \\exp ( - x ) , \\forall x > 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 374, + 525, + 681, + 570 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Therefore for any $x > 0$ , one has, ", + "bbox": [ + 217, + 585, + 455, + 601 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/3c51eb2659f6e1121bbfbb555b60c2cca7fb7da1d4c6d773623463b09ce0de2a.jpg", + "text": "$$\np \\left( | C - d | \\geq 2 { \\sqrt { d x } } \\right) \\leq \\exp ( - x )\n$$", + "text_format": "latex", + "bbox": [ + 413, + 611, + 640, + 638 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Let $x = 1 / 4 d$ , one has, ", + "bbox": [ + 232, + 648, + 385, + 662 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/f925bb6c44e7c1e535f189b76963c5deb9bf294bc6d549799a82d29fafdb3ed7.jpg", + "text": "$$\n\\begin{array} { l } { { p ( C > d + 1 ) \\le \\exp \\left( - \\displaystyle \\frac { 1 } { 4 d } \\right) , } } \\\\ { { \\displaystyle p ( C < d - 1 ) \\le \\exp \\left( - \\displaystyle \\frac { 1 } { 4 d } \\right) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 421, + 672, + 633, + 746 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Since $c _ { 1 }$ is a Gaussian random variable with variance 1, by Chebyshev’s inequality, one has, ", + "bbox": [ + 214, + 760, + 826, + 776 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/e2309a397d75cbd1bcb7fb7dcd3111c131f4d2f7994ef511fadf9c11eaec4f0b.jpg", + "text": "$$\np \\left( \\boldsymbol { y } c _ { i } \\ge k \\right) \\le \\frac { \\boldsymbol { y } ^ { 2 } } { k ^ { 2 } } , \\qquad p \\left( \\boldsymbol { y } c _ { i } \\le - k \\right) \\le \\frac { \\boldsymbol { y } ^ { 2 } } { k ^ { 2 } } , \\forall k > 0\n$$", + "text_format": "latex", + "bbox": [ + 354, + 785, + 700, + 818 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "and therefore thus, ", + "bbox": [ + 232, + 827, + 356, + 840 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/b4da1afa08a60e16c7e438f9414a618326fbc535f7c7a8e67b82bc5da8515d05.jpg", + "text": "$$\n\\begin{array} { r l } & { p ( \\exp ( y c _ { i } ) - 1 > k ) \\leq \\cfrac { y ^ { 2 } } { \\log ^ { 2 } ( 1 + k ) } , } \\\\ & { p ( \\exp ( y c _ { i } ) - 1 < - k ) \\leq \\cfrac { y ^ { 2 } } { \\log ( 1 - k ) ^ { 2 } } , \\forall k > 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 849, + 689, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Therefore we can bound $\\exp \\left( \\| \\boldsymbol { \\mu } \\| c _ { 1 } / \\sqrt { C } \\right)$ by, ", + "bbox": [ + 215, + 101, + 545, + 125 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/3b0a9645bd49c6d733939d3b9c8b15f792b068a5fe13b8ce5518999cac5d235f.jpg", + "text": "$$\n\\begin{array} { r l } & { p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { C } } \\right) - 1 > k \\right) \\leq p \\left( C > d + 1 \\right) } \\\\ & { \\qquad + p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { C } } \\right) - 1 > k \\bigg | C < d + 1 \\right) p \\left( C < d + 1 \\right) } \\\\ & { \\qquad \\leq \\exp \\left( - \\frac { 1 } { 4 d } \\right) + p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { d + 1 } } \\right) - 1 > k \\right) } \\\\ & { \\qquad = \\exp \\left( - \\frac { 1 } { 4 d } \\right) + \\frac { \\| \\mu \\| ^ { 2 } } { d + 1 } \\frac { 1 } { \\log \\left( 1 - k \\right) ^ { 2 } } . } \\\\ & { p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { C } } \\right) - 1 < - k \\right) \\leq \\exp \\left( - \\frac { 1 } { 4 d } \\right) + \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\frac { 1 } { \\log ^ { 2 } \\left( 1 + k \\right) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 133, + 846, + 311 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Combining the two inequalities above, one has (9) proved. ", + "bbox": [ + 230, + 314, + 616, + 329 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Step 4: We are now ready to prove convergence of $Z ( c )$ . With (9), let $\\mathcal { C } \\subset \\mathbb { R } ^ { d }$ such that, ", + "bbox": [ + 173, + 342, + 767, + 359 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/bc6fd5dae6ac59e4f43b3e41a1dacd8efdff4d74c0b36f3c80f06005596ddc0d.jpg", + "text": "$$\n\\mathcal { C } = \\left\\{ c : \\left| \\exp ( c ^ { \\top } \\mu ) - 1 \\right| < k , \\left| \\exp ( A c ^ { \\top } u _ { i } ) - 1 \\right| < k , \\left| \\exp ( - A c ^ { \\top } u _ { i } ) - 1 \\right| < k \\forall i = 1 , . . . , D \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 183, + 364, + 816, + 385 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Then we can bound the probability on $\\mathcal { C }$ by, ", + "bbox": [ + 174, + 390, + 462, + 405 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/7653c2509a7bd7b4b32b168c0d11d8afd1f50e6939efb2402524a181bff56f90.jpg", + "text": "$$\n\\begin{array} { l } { p ( \\mathcal { C } ) \\geq p \\left( \\left| \\exp ( c ^ { \\top } \\mu ) - 1 \\right| < k \\right) + \\displaystyle \\sum _ { i = 1 } ^ { D } p ( \\left| \\exp ( A c ^ { \\top } u _ { i } ) - 1 \\right| < k ) - 2 D } \\\\ { \\geq 1 - ( 2 D + 1 ) \\exp \\left( - \\displaystyle \\frac { 1 } { 4 d } \\right) - \\displaystyle \\frac { 2 D A ^ { 2 } } { d - 1 } \\displaystyle \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } - \\displaystyle \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\displaystyle \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 410, + 764, + 491 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Next, we need to show that for every $w$ , the corresponding $\\mathcal { C } ( w )$ , i.e., ", + "bbox": [ + 174, + 496, + 627, + 511 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/7093bc60a266f621c33b344c3d39aaae796feab9f14b8cfccc2f5ce11900af0f.jpg", + "text": "$$\n\\mathcal { C } ( \\boldsymbol { w } ) = \\left\\{ c : \\left| \\exp ( c ^ { \\top } \\boldsymbol { \\mu } ) - 1 \\right| < k , \\left| \\exp ( \\alpha _ { i } ( \\boldsymbol { w } ) c ^ { \\top } \\boldsymbol { u } _ { i } ) - 1 \\right| < k , \\forall i = 1 , . . . , D \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 516, + 759, + 537 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We observe that $\\alpha _ { i } ( w )$ is bounded by $A$ , therefore for any $c$ that, ", + "bbox": [ + 173, + 542, + 599, + 558 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\operatorname* { m i n } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) \\leq \\exp ( \\alpha _ { i } c ^ { \\top } u _ { i } ) \\leq \\operatorname* { m a x } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) , } \\end{array}$ and thus, 1 $\\mathrm { n i n } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) - 1 \\leq \\exp ( \\alpha _ { i } c ^ { \\top } u _ { i } ) - 1 \\leq \\operatorname* { m a x } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) - 1$ , which yields, ", + "bbox": [ + 171, + 563, + 851, + 647 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/7b9fd1defdcae810f366488d5c12ea318fbb3d451fd3aea042142dc2e69f71e4.jpg", + "text": "$$\n| \\exp ( \\alpha _ { i } c ^ { \\top } u _ { i } ) - 1 | \\leq \\operatorname* { m a x } ( | \\exp ( - A c ^ { \\top } u _ { i } ) - 1 | , | \\exp ( A c ^ { \\top } u _ { i } ) - 1 | ) < k .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 652, + 746, + 671 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Therefore we prove $c ( w ) \\supset { \\mathcal { C } }$ . Assembling everything together, one has, ", + "bbox": [ + 171, + 676, + 651, + 693 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/73ccadb4e5dc5dfa52fe3a82c2887e9756c6f5cfc8bb0a668a6452bb7d6e7155.jpg", + "text": "$$\n\\begin{array} { r l } & { p \\left( \\left| \\exp ( c ^ { \\top } \\mu ) \\prod _ { i = 1 } ^ { D } \\exp ( \\alpha _ { i } ( w ) c ^ { \\top } u _ { i } ) - 1 \\right| > ( D + 1 ) k , \\forall i = 1 , . . . , D , \\forall w \\in \\mathscr { V } \\right) } \\\\ & { \\leq p ( \\bar { \\mathscr { C } } ) } \\\\ & { \\leq ( 2 D + 1 ) \\exp ( - \\frac { 1 } { 4 d } ) + \\frac { 2 D A ^ { 2 } } { d - 1 } \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } + \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 699, + 764, + 799 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For every $c \\in { \\mathcal { C } }$ , one has, ", + "bbox": [ + 173, + 803, + 341, + 818 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/a149ffe7948aa3bf02a702508b145daeb5f78ac2e47eccad12028656f54e0450.jpg", + "text": "$$\n\\frac { 1 } { | \\mathcal { V } | } | Z ( c ) - Z _ { 0 } | \\leq \\frac { ( D + 1 ) k } { | \\mathcal { V } | } Z _ { 0 } .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 824, + 609, + 858 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Let $Z = | \\nu | Z _ { 0 }$ , one can conclude that, ", + "bbox": [ + 173, + 863, + 429, + 878 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/6218036f66d6c575e42d6ba85951398c9b62c25239b9e56e12a548025fec97aa.jpg", + "text": "$$\n\\begin{array} { r } { p ( ( 1 - \\epsilon _ { z } ) Z \\le Z ( c ) \\le ( 1 + \\epsilon _ { z } ) Z ) \\ge 1 - \\delta , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 883, + 645, + 901 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where $\\epsilon _ { z } = \\Omega ( ( D + 1 ) / | \\mathcal { V } | )$ and $\\delta = \\Omega ( D A ^ { 2 } / d )$ . ", + "bbox": [ + 173, + 907, + 503, + 925 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G.2 PROOF OF THEOREM A.1 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 393, + 118 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Having Lemma G.1 ready, we can follow the same proof as in (Arora et al., 2016) that both $p ( w )$ and $p ( w , w ^ { \\prime } )$ are correlated with $\\lVert \\boldsymbol v ( \\boldsymbol w ) \\rVert$ , formally ", + "bbox": [ + 171, + 128, + 823, + 159 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/378b9fd2634143229d3ab9022b54e2d5bb37bc2ac441f4aabcee064fc6d470a7.jpg", + "text": "$$\n\\begin{array} { c } { \\displaystyle \\log p ( w ) \\to \\frac { \\| v ( w ) \\| ^ { 2 } } { 2 d } - \\log Z , ~ \\mathrm { a s } ~ | \\mathcal { V } | \\to \\infty , } \\\\ { \\displaystyle \\log p ( w , w ^ { \\prime } ) \\to \\frac { \\| v ( w ) + v ( w ^ { \\prime } ) \\| ^ { 2 } } { 2 d } - \\log Z , ~ \\mathrm { a s } ~ | \\mathcal { V } | \\to \\infty . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 165, + 689, + 232 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Therefore, the inference presented in (Arora et al., 2016) (i.e., (4)) is obvious by assembling (12) and (13) together: ", + "bbox": [ + 173, + 234, + 826, + 263 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/5834139df253fec6ce7f333557055450a17d126a860011c348a6478069964ce2.jpg", + "text": "$$\n\\operatorname { P M I } ( w , w ^ { \\prime } ) \\to \\frac { v ( w ) ^ { \\top } v ( w ^ { \\prime } ) } { d } , ~ \\mathrm { a s } ~ \\vert \\mathcal { V } \\vert \\to \\infty .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 268, + 643, + 303 + ], + "page_idx": 24 + } +] \ No newline at end of file diff --git a/parse/train/HkuGJ3kCb/HkuGJ3kCb_middle.json b/parse/train/HkuGJ3kCb/HkuGJ3kCb_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..1292ed490e8de568fbb4ab9098441be455cdd118 --- /dev/null +++ b/parse/train/HkuGJ3kCb/HkuGJ3kCb_middle.json @@ -0,0 +1,64772 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 507, + 97 + ], + "score": 1.0, + "content": "ALL-BUT-THE-TOP: SIMPLE AND EFFECTIVE POST-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 432, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 432, + 117 + ], + "score": 1.0, + "content": "PROCESSING FOR WORD REPRESENTATIONS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 305, + 168 + ], + "lines": [ + { + "bbox": [ + 112, + 135, + 240, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 135, + 240, + 147 + ], + "score": 1.0, + "content": "Jiaqi Mu, Pramod Viswanath", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 288, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 288, + 159 + ], + "score": 1.0, + "content": "University of Illinois at Urbana Champaign", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 113, + 158, + 306, + 169 + ], + "spans": [ + { + "bbox": [ + 113, + 158, + 306, + 169 + ], + "score": 1.0, + "content": "{jiaqimu2, pramodv}@illinois.edu", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 209 + ], + "lines": [ + { + "bbox": [ + 276, + 196, + 336, + 212 + ], + "spans": [ + { + "bbox": [ + 276, + 196, + 336, + 212 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 221, + 469, + 342 + ], + "lines": [ + { + "bbox": [ + 141, + 221, + 470, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 470, + 234 + ], + "score": 1.0, + "content": "Real-valued word representations have transformed NLP applications; popular", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 469, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 469, + 245 + ], + "score": 1.0, + "content": "examples are word2vec and GloVe, recognized for their ability to capture linguistic", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 470, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 470, + 256 + ], + "score": 1.0, + "content": "regularities. 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In this paper, we demonstrate a very simple, and yet counter-intuitive,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 470, + 267 + ], + "score": 1.0, + "content": "postprocessing technique – eliminate the common mean vector and a few top", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 470, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 470, + 277 + ], + "score": 1.0, + "content": "dominating directions from the word vectors – that renders off-the-shelf represen-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 471, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 471, + 289 + ], + "score": 1.0, + "content": "tations even stronger. The postprocessing is empirically validated on a variety of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 286, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 286, + 470, + 300 + ], + "score": 1.0, + "content": "lexical-level intrinsic tasks (word similarity, concept categorization, word analogy)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 469, + 311 + ], + "score": 1.0, + "content": "and sentence-level tasks (semantic textural similarity and text classification) on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "score": 1.0, + "content": "multiple datasets and with a variety of representation methods and hyperparameter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 469, + 333 + ], + "score": 1.0, + "content": "choices in multiple languages; in each case, the processed representations are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 309, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 309, + 344 + ], + "score": 1.0, + "content": "consistently better than the original ones.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 221, + 471, + 344 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 205, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 208, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 208, + 378 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "Words and their interactions (as sentences) are the basic units of natural language. Although words", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "are readily modeled as discrete atomic units, this is unable to capture the relation between the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "words. Recent distributional real-valued representations of words (examples: word2vec, GloVe) have", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "transformed the landscape of NLP applications – for instance, text classification (Socher et al., 2013b;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "Maas et al., 2011; Kim, 2014), machine translation (Sutskever et al., 2014; Bahdanau et al., 2014) and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "knowledge base completion (Bordes et al., 2013; Socher et al., 2013a). The success comes from the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "geometry of the representations that efficiently captures linguistic regularities: the semantic similarity", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 464, + 455, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 455, + 476 + ], + "score": 1.0, + "content": "of words is well captured by the similarity of the corresponding vector representations.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 385, + 506, + 476 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 507, + 493 + ], + "score": 1.0, + "content": "A variety of approaches have been proposed in recent years to learn the word representations:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 505 + ], + "score": 1.0, + "content": "Collobert et al. 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Experiments suggest that", + "type": "text" + }, + { + "bbox": [ + 352, + 105, + 362, + 114 + ], + "score": 0.74, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "depends on the representations (for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "example, the dimension of the representation, the training methods and their specific hyperparameters,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "the training corpus) and also depends on the downstream applications. Nevertheless, a rule of thumb", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 155, + 150 + ], + "score": 1.0, + "content": "of choosing", + "type": "text" + }, + { + "bbox": [ + 155, + 138, + 165, + 147 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 137, + 195, + 150 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 196, + 137, + 222, + 149 + ], + "score": 0.89, + "content": "d / 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 137, + 252, + 150 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 252, + 138, + 258, + 147 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "is the dimension of the word representations, works uniformly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 458, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 458, + 161 + ], + "score": 1.0, + "content": "well across multiple languages and multiple representations and multiple test scenarios.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 507, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 507, + 179 + ], + "score": 1.0, + "content": "We emphasize that the proposed postprocessing is counter intuitive – typically denoising by dimen-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "sionality reduction is done by eliminating the weakest directions (in a singular value decomposition", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 201 + ], + "score": 1.0, + "content": "of the stacked word vectors), and not the dominating ones. Yet, such postprocessing yields a “purified”", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 424, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 424, + 211 + ], + "score": 1.0, + "content": "and more “isotropic” word representation as seen in our elaborate experiments.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 221, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "Experiments By postprocessing the word representation by eliminating the common parts, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "find the processed word representations to capture stronger linguistic regularities. We demonstrate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "this quantitatively, by comparing the performance of both the original word representations and the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 255, + 321, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 321, + 266 + ], + "score": 1.0, + "content": "processed ones on three canonical lexical-level tasks:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 132, + 274, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 132, + 274, + 504, + 287 + ], + "spans": [ + { + "bbox": [ + 132, + 274, + 504, + 287 + ], + "score": 1.0, + "content": "• word similarity task tests the extent to which the representations capture the similarity", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "between two words – the processed representations are consistently better on seven different", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 297, + 263, + 308 + ], + "spans": [ + { + "bbox": [ + 142, + 297, + 237, + 308 + ], + "score": 1.0, + "content": "datasets, on average by", + "type": "text" + }, + { + "bbox": [ + 237, + 297, + 259, + 307 + ], + "score": 0.85, + "content": "1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 297, + 263, + 308 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "concept categorization task tests the extent to which the clusters of word representations", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "capture the word semantics – the processed representations are consistently better on three", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 332, + 320, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 227, + 345 + ], + "score": 1.0, + "content": "different datasets, by", + "type": "text" + }, + { + "bbox": [ + 228, + 333, + 250, + 344 + ], + "score": 0.82, + "content": "2 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 332, + 253, + 345 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 253, + 333, + 276, + 343 + ], + "score": 0.82, + "content": "4 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 332, + 293, + 345 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 294, + 333, + 316, + 343 + ], + "score": 0.85, + "content": "4 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 332, + 320, + 345 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 137, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 137, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "word analogy task tests the extent to which the difference of two representations captures", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 140, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 140, + 358, + 482, + 371 + ], + "score": 1.0, + "content": "a latent linguistic relation – again, the performance is consistently improved (by", + "type": "text" + }, + { + "bbox": [ + 482, + 358, + 505, + 369 + ], + "score": 0.85, + "content": "0 . 5 \\%", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 370, + 237, + 382 + ], + "score": 1.0, + "content": "on semantic analogies,", + "type": "text" + }, + { + "bbox": [ + 238, + 370, + 261, + 380 + ], + "score": 0.87, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 370, + 372, + 382 + ], + "score": 1.0, + "content": "on syntactic analogies and", + "type": "text" + }, + { + "bbox": [ + 372, + 370, + 395, + 380 + ], + "score": 0.87, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "in total). Since part of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 141, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "dominant components are inherently canceled due to the subtraction operation while solving", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 392, + 494, + 404 + ], + "spans": [ + { + "bbox": [ + 142, + 392, + 494, + 404 + ], + "score": 1.0, + "content": "the analogy, we posit that the performance improvement is not as pronounced as earlier.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "Extrinsic evaluations provide a way to test the goodness of representations in specific downstream", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "tasks. We evaluate the effect of postprocessing on a standardized and important extrinsic evaluation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "task on sentence modeling: semantic textual similarity task – where we represent a sentence by its", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "averaged word vectors and score the similarity between a pair of sentences by the cosine similarity", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "between the corresponding sentence representation. Postprocessing improves the performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 452, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 430, + 479 + ], + "score": 1.0, + "content": "consistently and significantly over 21 different datasets (average improvement of", + "type": "text" + }, + { + "bbox": [ + 431, + 466, + 445, + 477 + ], + "score": 0.84, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 452, + 479 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 507, + 496 + ], + "score": 1.0, + "content": "Word representations have been particularly successful in NLP applications involving supervised-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "learning, especially in conjunction with neural network architecture. Indeed, we see the power", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "of postprocessing in an experiment on a standard text classification task using a well established", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "convoluntional neural network (CNN) classifier (Kim, 2014) and three RNN classifiers (with vanilla", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "RNN, GRU (Chung et al., 2015) and LSTM Greff et al. (2016) as recurrent units). Across two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "different pre-trained word vectors, five datasets and four different architectures, the performance with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 549, + 507, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 426, + 562 + ], + "score": 1.0, + "content": "processing improves on a majority of instances (34 out of 40) by a good margin", + "type": "text" + }, + { + "bbox": [ + 427, + 549, + 454, + 560 + ], + "score": 0.85, + "content": "2 . 8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 549, + 507, + 562 + ], + "score": 1.0, + "content": "on average),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 482, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 482, + 573 + ], + "score": 1.0, + "content": "and the two performances with and without processing are comparable in the remaining ones.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 108, + 583, + 503, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "Related Work. Our work is directly related to word representation algorithms, most of which have", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 199, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 199, + 606 + ], + "score": 1.0, + "content": "been elaborately cited.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Aspects similar to our postprocessing algorithm have appeared in specific NLP contexts very recently", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "in (Sahlgren et al., 2016) (centering the mean) and (Arora et al., 2017) (nulling away only the first", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "principal component). Although there is a superficial similarity between our work and (Arora et al.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "score": 1.0, + "content": "2017), the nulling directions we take and the one they take are fundamentally different. Specifically,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "in Arora et al. (2017), the first dominating vector is *dataset-specific*, i.e., they first compute the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "sentence representation for the entire semantic textual similarity dataset, then extract the top direction", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "from those sentence representations and finally project the sentence representation away from it.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "By doing so, the top direction will inherently encode the common information across the entire", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "dataset, the top direction for the \"headlines\" dataset may encode common information about news", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "articles while the top direction for \"Twitter’15\" may encode the common information about tweets.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 436, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 436, + 734 + ], + "score": 1.0, + "content": "In contrast, our dominating vectors are over the entire vocabulary of the language.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 47 + } + ], + "page_idx": 1, + "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, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 82, + 507, + 161 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 507, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 507, + 179 + ], + "score": 1.0, + "content": "We emphasize that the proposed postprocessing is counter intuitive – typically denoising by dimen-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "sionality reduction is done by eliminating the weakest directions (in a singular value decomposition", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 201 + ], + "score": 1.0, + "content": "of the stacked word vectors), and not the dominating ones. Yet, such postprocessing yields a “purified”", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 424, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 424, + 211 + ], + "score": 1.0, + "content": "and more “isotropic” word representation as seen in our elaborate experiments.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 165, + 507, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 221, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "Experiments By postprocessing the word representation by eliminating the common parts, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "find the processed word representations to capture stronger linguistic regularities. We demonstrate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "this quantitatively, by comparing the performance of both the original word representations and the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 255, + 321, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 321, + 266 + ], + "score": 1.0, + "content": "processed ones on three canonical lexical-level tasks:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 221, + 505, + 266 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 274, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 132, + 274, + 504, + 287 + ], + "spans": [ + { + "bbox": [ + 132, + 274, + 504, + 287 + ], + "score": 1.0, + "content": "• word similarity task tests the extent to which the representations capture the similarity", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "between two words – the processed representations are consistently better on seven different", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 297, + 263, + 308 + ], + "spans": [ + { + "bbox": [ + 142, + 297, + 237, + 308 + ], + "score": 1.0, + "content": "datasets, on average by", + "type": "text" + }, + { + "bbox": [ + 237, + 297, + 259, + 307 + ], + "score": 0.85, + "content": "1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 297, + 263, + 308 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 17, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "concept categorization task tests the extent to which the clusters of word representations", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "capture the word semantics – the processed representations are consistently better on three", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 332, + 320, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 227, + 345 + ], + "score": 1.0, + "content": "different datasets, by", + "type": "text" + }, + { + "bbox": [ + 228, + 333, + 250, + 344 + ], + "score": 0.82, + "content": "2 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 332, + 253, + 345 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 253, + 333, + 276, + 343 + ], + "score": 0.82, + "content": "4 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 332, + 293, + 345 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 294, + 333, + 316, + 343 + ], + "score": 0.85, + "content": "4 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 332, + 320, + 345 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + }, + { + "bbox": [ + 137, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 137, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "word analogy task tests the extent to which the difference of two representations captures", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 140, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 140, + 358, + 482, + 371 + ], + "score": 1.0, + "content": "a latent linguistic relation – again, the performance is consistently improved (by", + "type": "text" + }, + { + "bbox": [ + 482, + 358, + 505, + 369 + ], + "score": 0.85, + "content": "0 . 5 \\%", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 370, + 237, + 382 + ], + "score": 1.0, + "content": "on semantic analogies,", + "type": "text" + }, + { + "bbox": [ + 238, + 370, + 261, + 380 + ], + "score": 0.87, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 370, + 372, + 382 + ], + "score": 1.0, + "content": "on syntactic analogies and", + "type": "text" + }, + { + "bbox": [ + 372, + 370, + 395, + 380 + ], + "score": 0.87, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "in total). Since part of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 141, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "dominant components are inherently canceled due to the subtraction operation while solving", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 392, + 494, + 404 + ], + "spans": [ + { + "bbox": [ + 142, + 392, + 494, + 404 + ], + "score": 1.0, + "content": "the analogy, we posit that the performance improvement is not as pronounced as earlier.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20, + "bbox_fs": [ + 132, + 274, + 506, + 404 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "Extrinsic evaluations provide a way to test the goodness of representations in specific downstream", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "tasks. We evaluate the effect of postprocessing on a standardized and important extrinsic evaluation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "task on sentence modeling: semantic textual similarity task – where we represent a sentence by its", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "averaged word vectors and score the similarity between a pair of sentences by the cosine similarity", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "between the corresponding sentence representation. Postprocessing improves the performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 452, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 430, + 479 + ], + "score": 1.0, + "content": "consistently and significantly over 21 different datasets (average improvement of", + "type": "text" + }, + { + "bbox": [ + 431, + 466, + 445, + 477 + ], + "score": 0.84, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 452, + 479 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 410, + 505, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 507, + 496 + ], + "score": 1.0, + "content": "Word representations have been particularly successful in NLP applications involving supervised-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "learning, especially in conjunction with neural network architecture. Indeed, we see the power", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "of postprocessing in an experiment on a standard text classification task using a well established", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "convoluntional neural network (CNN) classifier (Kim, 2014) and three RNN classifiers (with vanilla", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "RNN, GRU (Chung et al., 2015) and LSTM Greff et al. (2016) as recurrent units). Across two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "different pre-trained word vectors, five datasets and four different architectures, the performance with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 549, + 507, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 426, + 562 + ], + "score": 1.0, + "content": "processing improves on a majority of instances (34 out of 40) by a good margin", + "type": "text" + }, + { + "bbox": [ + 427, + 549, + 454, + 560 + ], + "score": 0.85, + "content": "2 . 8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 549, + 507, + 562 + ], + "score": 1.0, + "content": "on average),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 482, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 482, + 573 + ], + "score": 1.0, + "content": "and the two performances with and without processing are comparable in the remaining ones.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 483, + 507, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 583, + 503, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "Related Work. Our work is directly related to word representation algorithms, most of which have", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 199, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 199, + 606 + ], + "score": 1.0, + "content": "been elaborately cited.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 582, + 505, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Aspects similar to our postprocessing algorithm have appeared in specific NLP contexts very recently", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "in (Sahlgren et al., 2016) (centering the mean) and (Arora et al., 2017) (nulling away only the first", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "principal component). Although there is a superficial similarity between our work and (Arora et al.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "score": 1.0, + "content": "2017), the nulling directions we take and the one they take are fundamentally different. Specifically,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "in Arora et al. (2017), the first dominating vector is *dataset-specific*, i.e., they first compute the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "sentence representation for the entire semantic textual similarity dataset, then extract the top direction", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "from those sentence representations and finally project the sentence representation away from it.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "By doing so, the top direction will inherently encode the common information across the entire", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "dataset, the top direction for the \"headlines\" dataset may encode common information about news", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "articles while the top direction for \"Twitter’15\" may encode the common information about tweets.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 436, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 436, + 734 + ], + "score": 1.0, + "content": "In contrast, our dominating vectors are over the entire vocabulary of the language.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 611, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "More generally, the idea of removing the top principal components has been studied in the context of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "positive-valued, high-dimensional data matrix analysis (Bullinaria & Levy, 2012; Price et al., 2006).", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "Bullinaria & Levy (2012) posits that the highest variance components of the cooccurrence matrix are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "corrupted by information other than lexical semantics, thus heuristically justifying the removal of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "the top principal components. A similar idea appears in the context of population matrix analysis", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "(Price et al., 2006), where the entries are also all positive. 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The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "detailed statistics for all representations are listed in Table 1. For completeness, we also consider the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 317, + 434, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 434, + 330 + ], + "score": 1.0, + "content": "representations on other languages: a detailed study is provided in Appendix C.2.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "table", + "bbox": [ + 112, + 339, + 499, + 408 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 112, + 339, + 499, + 408 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 339, + 499, + 408 + ], + "spans": [ + { + "bbox": [ + 112, + 339, + 499, + 408 + ], + "score": 0.978, + "html": "
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RAND-WALKEnglishWikipedia30068,4302.270.70
CBOWEnglishWikipedia3001,028,9611.140.29
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Algorithm 1: Postprocessing algorithm on word representations.
Input:Word representations {v(w),w ∈V},a threshold parameter D, 1 Compute the mean of {ν(w),w²∈V},μ← 高Σwev v(w),u(w) ← v(w)- μ
2 Compute the PCA components: u1,., ud ← PCA({u(w),w ∈ V}).
3 Preprocessthe representations: v’'(w) ← u(w)-∑=1 (uT v(w)) ui
Output :Processed representations v'(w).
", + "type": "table", + "image_path": "b0e539acdeefe1ed5282257f1b418e9378a289e40ce71a811baec47ce5e5512f.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 97, + 373, + 504, + 399.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 97, + 399.6666666666667, + 504, + 426.33333333333337 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 97, + 426.33333333333337, + 504, + 453.00000000000006 + ], + "spans": [], + "index": 24 + } + ] + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "Significance of Nulled Vectors Consider the representation of the words as viewed in terms of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 138, + 498 + ], + "score": 1.0, + "content": "the top", + "type": "text" + }, + { + "bbox": [ + 138, + 486, + 148, + 496 + ], + "score": 0.7, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 485, + 223, + 498 + ], + "score": 1.0, + "content": "PCA coefficients", + "type": "text" + }, + { + "bbox": [ + 223, + 486, + 249, + 497 + ], + "score": 0.92, + "content": "\\alpha _ { \\ell } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 485, + 269, + 498 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 270, + 486, + 321, + 497 + ], + "score": 0.92, + "content": "1 \\le \\ell \\le D", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 485, + 506, + 498 + ], + "score": 1.0, + "content": ". 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Algorithm 1: Postprocessing algorithm on word representations.
Input:Word representations {v(w),w ∈V},a threshold parameter D, 1 Compute the mean of {ν(w),w²∈V},μ← 高Σwev v(w),u(w) ← v(w)- μ
2 Compute the PCA components: u1,., ud ← PCA({u(w),w ∈ V}).
3 Preprocessthe representations: v’'(w) ← u(w)-∑=1 (uT v(w)) ui
Output :Processed representations v'(w).
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WORD2VECGLOVE
orig.proc.orig.proc.
RG6576.0878.3476.9674.36
WS68.2969.0573.7976.79
RW53.7454.3346.4152.04
MEN78.2079.0880.4981.78
MTurk68.2369.3569.2970.85
SimLex44.2045.1040 8344.97
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WORD2VECGLOVE
orig.proc.orig.proc.
RG6576.0878.3476.9674.36
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RW53.7454.3346.4152.04
MEN78.2079.0880.4981.78
MTurk68.2369.3569.2970.85
SimLex44.2045.1040 8344.97
SimVerb36.3536.5028.3332.23
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The detailed", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "performance on the seven datasets is reported in Table 3, where we see a consistent and signifi-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "cant performance improvement due to postprocessing, across all seven datasets. These statistics", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 211, + 95 + ], + "score": 1.0, + "content": "(average improvement of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 211, + 83, + 234, + 93 + ], + "score": 0.85, + "content": "2 . 3 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 234, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ") suggest that by removing the common parts, the remaining word", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 471, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 471, + 106 + ], + "score": 1.0, + "content": "representations are able to capture stronger semantic relatedness/similarity between words.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 53.5, + "bbox_fs": [ + 105, + 688, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 211, + 95 + ], + "score": 1.0, + "content": "(average improvement of", + "type": "text" + }, + { + "bbox": [ + 211, + 83, + 234, + 93 + ], + "score": 0.85, + "content": "2 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ") suggest that by removing the common parts, the remaining word", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 471, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 471, + 106 + ], + "score": 1.0, + "content": "representations are able to capture stronger semantic relatedness/similarity between words.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table", + "bbox": [ + 106, + 126, + 287, + 185 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 126, + 287, + 185 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 126, + 287, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 287, + 185 + ], + "score": 0.979, + "html": "
WORD2VECGLOVE
orig.proc.orig.proc.
ap54.4357.7264 .1865.42
esslli75.0084.0981.8281.82
battig71.9781.7186.5986.59
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The clustering performance is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 294, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 294, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "then evaluated in terms of purity (Manning et al.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 294, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 294, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "2008) – the fraction of the total number of the objects", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 294, + 209, + 414, + 223 + ], + "spans": [ + { + "bbox": [ + 294, + 209, + 414, + 223 + ], + "score": 1.0, + "content": "that were classified correctly.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 506, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "We conduct this task on three different datasets: the Almuhareb-Poesio (ap) dataset (Almuhareb,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "2006) contains 402 concepts which fall into 21 categories; the ESSLLI 2008 Distributional Semantic", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "Workshop shared-task dataset (Baroni et al., 2008) that contains 44 concepts in 6 categories; and the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 261, + 420, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 420, + 272 + ], + "score": 1.0, + "content": "Battig test set (Baroni & Lenci, 2010) that contains 83 words in 10 categories.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 105, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 506, + 289 + ], + "score": 1.0, + "content": "Here we follow the setting and the proposed algorithm in (Baroni et al., 2014; Schnabel et al., 2015)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 375, + 299 + ], + "score": 1.0, + "content": "– we cluster words (via their representations) using the classical", + "type": "text" + }, + { + "bbox": [ + 375, + 289, + 382, + 298 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "-Means algorithm (with fixed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 114, + 309 + ], + "score": 0.58, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "). 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WORD2VECGLOVE
orig.proc.orig.proc.
syntax73.4673.5074.9575.40
semantics72.2873.3679.2279.25
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The clustering performance is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 294, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 294, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "then evaluated in terms of purity (Manning et al.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 294, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 294, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "2008) – the fraction of the total number of the objects", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 294, + 209, + 414, + 223 + ], + "spans": [ + { + "bbox": [ + 294, + 209, + 414, + 223 + ], + "score": 1.0, + "content": "that were classified correctly.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 294, + 123, + 507, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 506, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "We conduct this task on three different datasets: the Almuhareb-Poesio (ap) dataset (Almuhareb,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "2006) contains 402 concepts which fall into 21 categories; the ESSLLI 2008 Distributional Semantic", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "Workshop shared-task dataset (Baroni et al., 2008) that contains 44 concepts in 6 categories; and the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 261, + 420, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 420, + 272 + ], + "score": 1.0, + "content": "Battig test set (Baroni & Lenci, 2010) that contains 83 words in 10 categories.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 227, + 506, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 277, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 506, + 289 + ], + "score": 1.0, + "content": "Here we follow the setting and the proposed algorithm in (Baroni et al., 2014; Schnabel et al., 2015)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 375, + 299 + ], + "score": 1.0, + "content": "– we cluster words (via their representations) using the classical", + "type": "text" + }, + { + "bbox": [ + 375, + 289, + 382, + 298 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "-Means algorithm (with fixed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 114, + 309 + ], + "score": 0.58, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "). 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WORD2VECGLOVE
orig.proc.orig.proc.
syntax73.4673.5074.9575.40
semantics72.2873.3679.2279.25
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It can be noticed that while postprocessing continues to improve", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 538, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 506, + 550 + ], + "score": 1.0, + "content": "the performance, the improvement is not as pronounced as earlier. 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WORD2VECGLOVE
orig.proc.orig.proc.
201257.2257.6748.2754.06
201356.8157.9844.8357.71
201462.8963.3051.1159.23
201562.7463.3547.2357.29
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We test the", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 294, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 294, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "word representations on 20 textual similarity datasets", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "from the 2012-2015 SemEval STS tasks (Agirre et al., 2012; 2013; 2014; 2015), and the 2012", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 354, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 354, + 105 + ], + "score": 1.0, + "content": "SemEval Semantic Related task (SICK) (Marelli et al., 2014).", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 66, + "bbox_fs": [ + 294, + 588, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "from the 2012-2015 SemEval STS tasks (Agirre et al., 2012; 2013; 2014; 2015), and the 2012", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 354, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 354, + 105 + ], + "score": 1.0, + "content": "SemEval Semantic Related task (SICK) (Marelli et al., 2014).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "Representing sentences by the average of their constituent word representations is surprisingly", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "score": 1.0, + "content": "effective in encoding the semantic information of sentences (Wieting et al., 2015; Adi et al., 2016)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "and close to the state-of-the-art in these datasets. We follow this rubric and represent a sentence", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 142, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 107, + 146, + 113, + 153 + ], + "score": 0.45, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 142, + 313, + 159 + ], + "score": 1.0, + "content": "based on its averaged word representation, i.e.,", + "type": "text" + }, + { + "bbox": [ + 313, + 142, + 408, + 158 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\boldsymbol { v } ( \\boldsymbol { s } ) = \\frac { 1 } { | \\boldsymbol { s } | } \\sum _ { w \\in \\boldsymbol { s } } \\boldsymbol { v } ( w ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 142, + 506, + 159 + ], + "score": 1.0, + "content": ", and then compute the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "similarity between two sentences via the cosine similarity between the two representations. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "average performance of the original and processed representations is itemized in Table 6 (with a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "detailed performance in Table 20 in Appendix E) – we see a consistent and significant improvement", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 402, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 325, + 202 + ], + "score": 1.0, + "content": "in performance because of postprocessing (on average", + "type": "text" + }, + { + "bbox": [ + 325, + 189, + 340, + 199 + ], + "score": 0.85, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 188, + 402, + 202 + ], + "score": 1.0, + "content": "improvement).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 218, + 399, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 401, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 401, + 232 + ], + "score": 1.0, + "content": "4 POSTPROCESSING AND SUPERVISED CLASSIFICATION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "Supervised downstream NLP applications have greatly improved their performances in recent years", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "by combining the discriminative learning powers of neural networks in conjunction with the word", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "representations. We evaluate the performance of a variety of neural network architectures on a standard", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "and important NLP application: text classification, with sentiment analysis being a particularly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "important and popular example. The task is defined as follows: given a sentence, the algorithm needs", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "to decide which category it falls into. The categories can be either binary (e.g., positive/negative) or", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 470, + 322 + ], + "score": 1.0, + "content": "can be more fine-grained (e.g. very positive, positive, neutral, negative, and very negative).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "We evaluate the word representations (with and without postprocessing) using four different neural", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "network architectures (CNN, vanilla-RNN, GRU-RNN and LSTM-RNN) on five benchmarks: (a) the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "movie review (MR) dataset (Pang & Lee, 2005); (b) the subjectivity (SUBJ) dataset (Pang & Lee,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "2004); (c) the TREC question dataset (Li & Roth, 2002); (d) the IMDb dataset (Maas et al., 2011);", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "(e) the stanford sentiment treebank (SST) dataset (Socher et al., 2013a). A detailed description of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "these standard datasets, their training/test parameters and the cross validation methods adopted is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 321, + 406 + ], + "score": 1.0, + "content": "in Appendix F. 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SST38.4638.3338.8237.8342.0839.9141.4541.9041.8645.0236.5237.6943.0842.0837.5138.05
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Of particular interest is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 608, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 504, + 621 + ], + "score": 1.0, + "content": "the impact of the postprocessing on the rate of convergence and generalization capabilities of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 620, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 631 + ], + "score": 1.0, + "content": "classifiers. 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We follow this rubric and represent a sentence", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 142, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 107, + 146, + 113, + 153 + ], + "score": 0.45, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 142, + 313, + 159 + ], + "score": 1.0, + "content": "based on its averaged word representation, i.e.,", + "type": "text" + }, + { + "bbox": [ + 313, + 142, + 408, + 158 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\boldsymbol { v } ( \\boldsymbol { s } ) = \\frac { 1 } { | \\boldsymbol { s } | } \\sum _ { w \\in \\boldsymbol { s } } \\boldsymbol { v } ( w ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 142, + 506, + 159 + ], + "score": 1.0, + "content": ", and then compute the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "similarity between two sentences via the cosine similarity between the two representations. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "average performance of the original and processed representations is itemized in Table 6 (with a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "detailed performance in Table 20 in Appendix E) – we see a consistent and significant improvement", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 402, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 325, + 202 + ], + "score": 1.0, + "content": "in performance because of postprocessing (on average", + "type": "text" + }, + { + "bbox": [ + 325, + 189, + 340, + 199 + ], + "score": 0.85, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 188, + 402, + 202 + ], + "score": 1.0, + "content": "improvement).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 110, + 506, + 202 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 218, + 399, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 401, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 401, + 232 + ], + "score": 1.0, + "content": "4 POSTPROCESSING AND SUPERVISED CLASSIFICATION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "Supervised downstream NLP applications have greatly improved their performances in recent years", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "by combining the discriminative learning powers of neural networks in conjunction with the word", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "representations. We evaluate the performance of a variety of neural network architectures on a standard", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "and important NLP application: text classification, with sentiment analysis being a particularly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "important and popular example. The task is defined as follows: given a sentence, the algorithm needs", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "to decide which category it falls into. The categories can be either binary (e.g., positive/negative) or", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 470, + 322 + ], + "score": 1.0, + "content": "can be more fine-grained (e.g. very positive, positive, neutral, negative, and very negative).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 244, + 506, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "We evaluate the word representations (with and without postprocessing) using four different neural", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "network architectures (CNN, vanilla-RNN, GRU-RNN and LSTM-RNN) on five benchmarks: (a) the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "movie review (MR) dataset (Pang & Lee, 2005); (b) the subjectivity (SUBJ) dataset (Pang & Lee,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "2004); (c) the TREC question dataset (Li & Roth, 2002); (d) the IMDb dataset (Maas et al., 2011);", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "(e) the stanford sentiment treebank (SST) dataset (Socher et al., 2013a). 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We pick", + "type": "text" + }, + { + "bbox": [ + 174, + 207, + 208, + 217 + ], + "score": 0.9, + "content": "D = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 207, + 295, + 219 + ], + "score": 1.0, + "content": "for WORD2VEC and", + "type": "text" + }, + { + "bbox": [ + 296, + 207, + 329, + 217 + ], + "score": 0.91, + "content": "D = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "for GLOVE, and the corresponding value of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 217, + 128, + 228 + ], + "score": 0.89, + "content": "D A ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "for WORD2VEC and GLOVE vectors are both roughly 40, respectively; both values are small", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 197, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 158, + 241 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + }, + { + "bbox": [ + 159, + 229, + 193, + 239 + ], + "score": 0.89, + "content": "d = 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 228, + 197, + 241 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "image", + "bbox": [ + 235, + 249, + 376, + 353 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 235, + 249, + 376, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 235, + 249, + 376, + 353 + ], + "spans": [ + { + "bbox": [ + 235, + 249, + 376, + 353 + ], + "score": 0.971, + "type": "image", + "image_path": "4b8cef86f46030d6cc4853b73297993875e11f99c5c37b3bb9b91793e95189e5.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 249, + 376, + 301.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 235, + 301.0, + 376, + 353.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 363, + 504, + 387 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "score": 1.0, + "content": "Figure 4: Spectrum of the published WORD2VEC and GLOVE and random Gaussian matrices,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 375, + 447, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 172, + 388 + ], + "score": 1.0, + "content": "ignoring the top", + "type": "text" + }, + { + "bbox": [ + 173, + 376, + 183, + 385 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 375, + 237, + 388 + ], + "score": 1.0, + "content": "components;", + "type": "text" + }, + { + "bbox": [ + 237, + 375, + 270, + 385 + ], + "score": 0.9, + "content": "D = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 375, + 338, + 388 + ], + "score": 1.0, + "content": "for GLOVE and", + "type": "text" + }, + { + "bbox": [ + 338, + 375, + 371, + 385 + ], + "score": 0.9, + "content": "D = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 375, + 447, + 388 + ], + "score": 1.0, + "content": "for WORD2VEC.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 405, + 411 + ], + "score": 1.0, + "content": "This leaves us to check the statistical isotropy of the “remaining\" vectors", + "type": "text" + }, + { + "bbox": [ + 405, + 398, + 426, + 410 + ], + "score": 0.91, + "content": "\\tilde { v } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 397, + 469, + 411 + ], + "score": 1.0, + "content": "for words", + "type": "text" + }, + { + "bbox": [ + 470, + 400, + 478, + 408 + ], + "score": 0.71, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 397, + 505, + 411 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 382, + 422 + ], + "score": 1.0, + "content": "vocabulary. We do this by plotting the remaining spectrum (i.e. the", + "type": "text" + }, + { + "bbox": [ + 382, + 409, + 416, + 421 + ], + "score": 0.9, + "content": "( D + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 408, + 506, + 422 + ], + "score": 1.0, + "content": "-th, ..., 300th singular", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "values) for the published WORD2VEC and GLOVE vectors in Figure 4. As a comparison, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "empirical spectrum of a random Gaussian matrix is also plotted in Figure 4. We see that both", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 421, + 455 + ], + "score": 1.0, + "content": "spectra are flat (since the vocabulary size is much larger than the dimension", + "type": "text" + }, + { + "bbox": [ + 421, + 442, + 458, + 453 + ], + "score": 0.89, + "content": "d = 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "). Thus the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 453, + 483, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 483, + 466 + ], + "score": 1.0, + "content": "postprocessing operation can also be viewed as a way of making the vectors “more isotropic”.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 504, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "Mathematical Contribution Under the revised postulate, we show that the main theorem in (Arora", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 487, + 318, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 318, + 500 + ], + "score": 1.0, + "content": "et al., 2016) (c.f. Theorem 2.2) still holds. Formally:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 508, + 386, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 387, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 387, + 523 + ], + "score": 1.0, + "content": "Theorem A.1 Suppose the word vectors satisfy the constraints. Then", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 525, + 446, + 554 + ], + "lines": [ + { + "bbox": [ + 164, + 525, + 446, + 554 + ], + "spans": [ + { + "bbox": [ + 164, + 525, + 446, + 554 + ], + "score": 0.94, + "content": "\\mathrm { P M I } ( w _ { 1 } , w _ { 2 } ) \\stackrel { \\mathrm { d e f } } { = } \\log \\frac { p ( w _ { 1 } , w _ { 2 } ) } { p ( w _ { 1 } ) p ( w _ { 2 } ) } \\frac { v ( w _ { 1 } ) ^ { \\top } v ( w _ { 2 } ) } { d } , \\quad a s | \\mathcal { V } | \\infty ,", + "type": "interline_equation", + "image_path": "2abe32081a93d834cb5ce8227391e11e0e994692fef6afb07b3ffc7b130649a6.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 164, + 525, + 446, + 534.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 164, + 534.6666666666666, + 446, + 544.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 164, + 544.3333333333333, + 446, + 553.9999999999999 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 132, + 572 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 559, + 154, + 571 + ], + "score": 0.92, + "content": "p ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 558, + 393, + 572 + ], + "score": 1.0, + "content": "is the unigram distribution induced from the model (2), and", + "type": "text" + }, + { + "bbox": [ + 393, + 559, + 434, + 571 + ], + "score": 0.93, + "content": "p ( w _ { 1 } , w _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "is the probability", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 569, + 376, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 168, + 583 + ], + "score": 1.0, + "content": "that two words", + "type": "text" + }, + { + "bbox": [ + 168, + 572, + 180, + 581 + ], + "score": 0.85, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 569, + 199, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 572, + 212, + 581 + ], + "score": 0.83, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 569, + 366, + 583 + ], + "score": 1.0, + "content": "occur with each other within distance", + "type": "text" + }, + { + "bbox": [ + 366, + 573, + 372, + 581 + ], + "score": 0.75, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 569, + 376, + 583 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "The proof is in Appendix G. Theorem A.1 suggests that the RAND-WALK generative model and its", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "properties proposed by (Arora et al., 2016) can be generalized to a broader setting (with a relaxed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "restriction on the geometry of word representations) – relevantly, this relaxation on the geometry of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "word representations is empirically satisfied by the vectors learnt as part of the maximum likelihood", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 128, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 128, + 647 + ], + "score": 1.0, + "content": "rule.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 366, + 675 + ], + "lines": [ + { + "bbox": [ + 104, + 661, + 367, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 661, + 367, + 677 + ], + "score": 1.0, + "content": "B NEURAL NETWORKS LEARN TO POSTPROCESS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Every neural network family posseses the ability to conduct linear processing inside their nodes; this", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "includes feedforward and recurrent and convolutional neural network models. Thus, in principle,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "the postprocessing operation can be “learnt and implemented\" within the parameters of the neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "network. On the other hand, due to the large number of parameters within the neural network, it", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 142, + 81, + 505, + 108 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 141, + 80, + 507, + 110 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 117, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "The revised postulate differs from the original one in two ways: (a) it imposes a formal deterministic", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "constraint on the word vectors; (b) the revised postulate allows the word vectors to be angularly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 317, + 151 + ], + "score": 1.0, + "content": "asymmetric: as long as the energy in the direction of", + "type": "text" + }, + { + "bbox": [ + 317, + 140, + 358, + 150 + ], + "score": 0.89, + "content": "u _ { 1 } , \\ldots , u _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "is bounded, there is no constraint on", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 487, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 340, + 164 + ], + "score": 1.0, + "content": "the coefficients. Indeed, note that there is no constraint on", + "type": "text" + }, + { + "bbox": [ + 340, + 150, + 361, + 162 + ], + "score": 0.92, + "content": "\\tilde { v } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 149, + 441, + 164 + ], + "score": 1.0, + "content": "to be orthogonal to", + "type": "text" + }, + { + "bbox": [ + 441, + 152, + 482, + 162 + ], + "score": 0.88, + "content": "u _ { 1 } , \\ldots , u _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 149, + 487, + 164 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 117, + 506, + 164 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "Empirical Validation We can verify that the enlarged conditions are met by the existing word", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 185, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 316, + 197 + ], + "score": 1.0, + "content": "representations. Specifically, the natural choice for", + "type": "text" + }, + { + "bbox": [ + 316, + 186, + 324, + 196 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 185, + 506, + 197 + ], + "score": 1.0, + "content": "is the mean of the word representations and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 196, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 107, + 198, + 145, + 207 + ], + "score": 0.88, + "content": "u _ { 1 } \\ldots u _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 196, + 340, + 208 + ], + "score": 1.0, + "content": "are the singular vectors associated with the top", + "type": "text" + }, + { + "bbox": [ + 340, + 196, + 350, + 206 + ], + "score": 0.76, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 196, + 506, + 208 + ], + "score": 1.0, + "content": "singular values of the matrix of word", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 174, + 219 + ], + "score": 1.0, + "content": "vectors. We pick", + "type": "text" + }, + { + "bbox": [ + 174, + 207, + 208, + 217 + ], + "score": 0.9, + "content": "D = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 207, + 295, + 219 + ], + "score": 1.0, + "content": "for WORD2VEC and", + "type": "text" + }, + { + "bbox": [ + 296, + 207, + 329, + 217 + ], + "score": 0.91, + "content": "D = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "for GLOVE, and the corresponding value of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 217, + 128, + 228 + ], + "score": 0.89, + "content": "D A ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "for WORD2VEC and GLOVE vectors are both roughly 40, respectively; both values are small", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 197, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 158, + 241 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + }, + { + "bbox": [ + 159, + 229, + 193, + 239 + ], + "score": 0.89, + "content": "d = 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 228, + 197, + 241 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 174, + 506, + 241 + ] + }, + { + "type": "image", + "bbox": [ + 235, + 249, + 376, + 353 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 235, + 249, + 376, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 235, + 249, + 376, + 353 + ], + "spans": [ + { + "bbox": [ + 235, + 249, + 376, + 353 + ], + "score": 0.971, + "type": "image", + "image_path": "4b8cef86f46030d6cc4853b73297993875e11f99c5c37b3bb9b91793e95189e5.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 249, + 376, + 301.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 235, + 301.0, + 376, + 353.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 363, + 504, + 387 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "score": 1.0, + "content": "Figure 4: Spectrum of the published WORD2VEC and GLOVE and random Gaussian matrices,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 375, + 447, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 172, + 388 + ], + "score": 1.0, + "content": "ignoring the top", + "type": "text" + }, + { + "bbox": [ + 173, + 376, + 183, + 385 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 375, + 237, + 388 + ], + "score": 1.0, + "content": "components;", + "type": "text" + }, + { + "bbox": [ + 237, + 375, + 270, + 385 + ], + "score": 0.9, + "content": "D = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 375, + 338, + 388 + ], + "score": 1.0, + "content": "for GLOVE and", + "type": "text" + }, + { + "bbox": [ + 338, + 375, + 371, + 385 + ], + "score": 0.9, + "content": "D = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 375, + 447, + 388 + ], + "score": 1.0, + "content": "for WORD2VEC.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 405, + 411 + ], + "score": 1.0, + "content": "This leaves us to check the statistical isotropy of the “remaining\" vectors", + "type": "text" + }, + { + "bbox": [ + 405, + 398, + 426, + 410 + ], + "score": 0.91, + "content": "\\tilde { v } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 397, + 469, + 411 + ], + "score": 1.0, + "content": "for words", + "type": "text" + }, + { + "bbox": [ + 470, + 400, + 478, + 408 + ], + "score": 0.71, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 397, + 505, + 411 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 382, + 422 + ], + "score": 1.0, + "content": "vocabulary. We do this by plotting the remaining spectrum (i.e. the", + "type": "text" + }, + { + "bbox": [ + 382, + 409, + 416, + 421 + ], + "score": 0.9, + "content": "( D + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 408, + 506, + 422 + ], + "score": 1.0, + "content": "-th, ..., 300th singular", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "values) for the published WORD2VEC and GLOVE vectors in Figure 4. As a comparison, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "empirical spectrum of a random Gaussian matrix is also plotted in Figure 4. We see that both", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 421, + 455 + ], + "score": 1.0, + "content": "spectra are flat (since the vocabulary size is much larger than the dimension", + "type": "text" + }, + { + "bbox": [ + 421, + 442, + 458, + 453 + ], + "score": 0.89, + "content": "d = 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "). Thus the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 453, + 483, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 483, + 466 + ], + "score": 1.0, + "content": "postprocessing operation can also be viewed as a way of making the vectors “more isotropic”.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 397, + 506, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 504, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "Mathematical Contribution Under the revised postulate, we show that the main theorem in (Arora", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 487, + 318, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 318, + 500 + ], + "score": 1.0, + "content": "et al., 2016) (c.f. Theorem 2.2) still holds. Formally:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 475, + 505, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 508, + 386, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 387, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 387, + 523 + ], + "score": 1.0, + "content": "Theorem A.1 Suppose the word vectors satisfy the constraints. Then", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 507, + 387, + 523 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 525, + 446, + 554 + ], + "lines": [ + { + "bbox": [ + 164, + 525, + 446, + 554 + ], + "spans": [ + { + "bbox": [ + 164, + 525, + 446, + 554 + ], + "score": 0.94, + "content": "\\mathrm { P M I } ( w _ { 1 } , w _ { 2 } ) \\stackrel { \\mathrm { d e f } } { = } \\log \\frac { p ( w _ { 1 } , w _ { 2 } ) } { p ( w _ { 1 } ) p ( w _ { 2 } ) } \\frac { v ( w _ { 1 } ) ^ { \\top } v ( w _ { 2 } ) } { d } , \\quad a s | \\mathcal { V } | \\infty ,", + "type": "interline_equation", + "image_path": "2abe32081a93d834cb5ce8227391e11e0e994692fef6afb07b3ffc7b130649a6.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 164, + 525, + 446, + 534.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 164, + 534.6666666666666, + 446, + 544.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 164, + 544.3333333333333, + 446, + 553.9999999999999 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 132, + 572 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 559, + 154, + 571 + ], + "score": 0.92, + "content": "p ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 558, + 393, + 572 + ], + "score": 1.0, + "content": "is the unigram distribution induced from the model (2), and", + "type": "text" + }, + { + "bbox": [ + 393, + 559, + 434, + 571 + ], + "score": 0.93, + "content": "p ( w _ { 1 } , w _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "is the probability", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 569, + 376, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 168, + 583 + ], + "score": 1.0, + "content": "that two words", + "type": "text" + }, + { + "bbox": [ + 168, + 572, + 180, + 581 + ], + "score": 0.85, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 569, + 199, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 572, + 212, + 581 + ], + "score": 0.83, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 569, + 366, + 583 + ], + "score": 1.0, + "content": "occur with each other within distance", + "type": "text" + }, + { + "bbox": [ + 366, + 573, + 372, + 581 + ], + "score": 0.75, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 569, + 376, + 583 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 558, + 505, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "The proof is in Appendix G. Theorem A.1 suggests that the RAND-WALK generative model and its", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "properties proposed by (Arora et al., 2016) can be generalized to a broader setting (with a relaxed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "restriction on the geometry of word representations) – relevantly, this relaxation on the geometry of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "word representations is empirically satisfied by the vectors learnt as part of the maximum likelihood", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 128, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 128, + 647 + ], + "score": 1.0, + "content": "rule.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 591, + 506, + 647 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 366, + 675 + ], + "lines": [ + { + "bbox": [ + 104, + 661, + 367, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 661, + 367, + 677 + ], + "score": 1.0, + "content": "B NEURAL NETWORKS LEARN TO POSTPROCESS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Every neural network family posseses the ability to conduct linear processing inside their nodes; this", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "includes feedforward and recurrent and convolutional neural network models. Thus, in principle,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "the postprocessing operation can be “learnt and implemented\" within the parameters of the neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "network. On the other hand, due to the large number of parameters within the neural network, it", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "is unclear how to verify such a process, even if it were learnt (only one of the layers might be", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 447, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 447, + 106 + ], + "score": 1.0, + "content": "implementing the postprocessing operation or via a combination of multiple effects).", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "is unclear how to verify such a process, even if it were learnt (only one of the layers might be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 447, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 447, + 106 + ], + "score": 1.0, + "content": "implementing the postprocessing operation or via a combination of multiple effects).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 123 + ], + "score": 1.0, + "content": "To address this issue, we have adopted a comparative approach in the rest of this section. The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "comparative approach involves adding an extra layer interposed in between the inputs (which are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "word vectors) and the rest of the neural network. This extra layer involves only linear processing.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "Next we compare the results of the final parameters of the extra layer (trained jointly with the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "rest of tne neural network parameters, using the end-to-end training examples) with and without", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "preprocessing of the word vectors. Such a comparative approach allows us to separate the effect of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "the postprocessing operation on the word vectors from the complicated “semantics” of the neural", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 190, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 190, + 199 + ], + "score": 1.0, + "content": "network parameters.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "table", + "bbox": [ + 300, + 232, + 491, + 295 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 300, + 232, + 491, + 295 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 300, + 232, + 491, + 295 + ], + "spans": [ + { + "bbox": [ + 300, + 232, + 491, + 295 + ], + "score": 0.856, + "html": "
vanillaGRULSTM
W.G.W.G.W.G.
MR82.0749.2381.3547.6377.9544.92
SUBJ84.0249.9483.1550.6083.3948.95
TREC81.6852.9982.6850.4280.8046.77
SST79.6446.5978.0643.2177.7242.82
IMDb93.4866.3794.4955.2487.2746.74
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The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "comparative approach involves adding an extra layer interposed in between the inputs (which are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "word vectors) and the rest of the neural network. 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vanillaGRULSTM
W.G.W.G.W.G.
MR82.0749.2381.3547.6377.9544.92
SUBJ84.0249.9483.1550.6083.3948.95
TREC81.6852.9982.6850.4280.8046.77
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LanguageCorpusdimvocab sizeavg.u(w)l2μl2
TSCCA-EnEnglishGigawords200300,0004.380.78
TSCCA-DeGermanNewswire200300,0004.520.79
TSCCA-FrFrenchGigaword200300,0004.340.81
TSCCA-EsSpanishGigaword200300.0004.170.79
TSCCA-ItItalianNewswire+Wiki200300,0004.340.79
TSCCA-N1DutchNewswire+Wiki200300,0004.460.72
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LanguageCorpusdimvocab sizeavg.u(w)l2μl2
TSCCA-EnEnglishGigawords200300,0004.380.78
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TSCCA-FrFrenchGigaword200300,0004.340.81
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TSCCA-N1DutchNewswire+Wiki200300,0004.460.72
TSCCA-ZhChineseGigaword200300.0004.510.89
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languageTSCCA
orig.proc.
RG65Spanish60.3360.37
GUR65German61.7564.39
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languageTSCCA
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RAND-WALKTSCCA
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RG6580.6682.9647.5347.67
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WORD2VECGLOVE
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capital-world78.3880.0891.8992.31
city-in-state69.5669.8869.5670.45
currency32.4332.9221.5921.36
family84.9884.5995.8495.65
gram1-adjective-to-adverb28.0227.7240.4239.21
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gram3-comparative89.1989.2686.9387.09
gram4-superlative82.7183.3390.4690.59
gram5-present-participle79.3679.6482.9582.76
gram6-nationality-adjective90.2490.3690.2490.24
gram7-past-tense66.0366.5363.9164.87
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Dim300400500600
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201254.5154.9554.3154.5755.1356.2355.3556.03
201356.5857.8956.3557.3557.5559.3857.4359.00
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all58.3259.9158.2559.5559.6162.0259.5761.55
Dim7008009001000
orig.proc.orig.proc.orig.proc.orig.proc.
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201357.6159.3156.7557.6256.9858.2656.7857.73
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SICK68.3870.6367.9469.5967.8669.567.5869.16
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WORD2VECGLOVE
orig.proc.orig.proc.
capital-common-countries82.0183.6095.0695.96
capital-world78.3880.0891.8992.31
city-in-state69.5669.8869.5670.45
currency32.4332.9221.5921.36
family84.9884.5995.8495.65
gram1-adjective-to-adverb28.0227.7240.4239.21
gram2-opposite40.1440.5131.6530.91
gram3-comparative89.1989.2686.9387.09
gram4-superlative82.7183.3390.4690.59
gram5-present-participle79.3679.6482.9582.76
gram6-nationality-adjective90.2490.3690.2490.24
gram7-past-tense66.0366.5363.9164.87
gram8-plural91.0790.6195.2795.36
gram9-plural-verbs68.7467.5867.2468.05
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WORD2VECGLOVE
orig.proc.orig.proc.
2012.MSRpar42.1243.8544.5444.09
2012.MSRvid72.0772.1664.4768.05
2012.OnWN69.3869.4853.0765.67
2012.SMTeuroparl53.1554.3241.7445.28
2012.SMTnews49.3748.5337.5447.22
2013.FNWN40.7041.9637.5439.34
2013.OnWN67.8768.1747.2258.60
2013.headlines61.8863.8149.7357.20
2014.OnWN74.6174.7857.4167.56
2014.deft-forum32.1933.2621.5529.39
2014.deft-news66.8365.9665.1471.45
2014.headlines58.0159.5847.0552.60
2014.images73.7574.1757.2268.28
2014.tweet-news71.9272.0758.3266.13
2015.answers-forum46.3546.8030.0239.86
2015.answers-students68.0767.9949.2062.38
2015.belief59.7260.4244.0557.68
2015.headlines61.4763.4546.2253.31
2015.images78.0978.0866.6373.20
SICK70.1070.2065.1467.85
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C1TrainTest
MR22010,66210-fold cross validation
SUBJ22310,00010-fold cross validation
TREC6105,952500
SST51811,8552,210
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C1TrainTest
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SUBJ22310,00010-fold cross validation
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i.i.d.", + "type": "text" + }, + { + "bbox": [ + 214, + 335, + 247, + 347 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 330, + 279, + 352 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 280, + 333, + 336, + 348 + ], + "score": 0.93, + "content": "\\textstyle C = \\sum _ { i = 1 } ^ { d } c _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 330, + 359, + 352 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 360, + 334, + 441, + 347 + ], + "score": 0.93, + "content": "c = ( c _ { 1 } , . . . , c _ { d } ) / \\sqrt { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 330, + 509, + 352 + ], + "score": 1.0, + "content": "is uniform over", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 345, + 507, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 181, + 360 + ], + "score": 1.0, + 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Since", + "type": "text" + }, + { + "bbox": [ + 182, + 349, + 187, + 356 + ], + "score": 0.76, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 345, + 427, + 360 + ], + "score": 1.0, + "content": "is uniform, then without loss of generality we can consider", + "type": "text" + }, + { + "bbox": [ + 428, + 346, + 502, + 358 + ], + "score": 0.93, + "content": "\\boldsymbol { \\mu } = ( \\lVert \\boldsymbol { \\mu } \\rVert , 0 , . . . , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 345, + 507, + 360 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 207, + 376 + ], + "score": 1.0, + "content": "Thus it suffices to bound", + "type": "text" + }, + { + "bbox": [ + 207, + 357, + 282, + 378 + ], + "score": 0.92, + "content": "\\exp \\left( \\| \\boldsymbol { \\mu } \\| c _ { 1 } / \\sqrt { C } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 359, + 469, + 376 + ], + "score": 1.0, + "content": ". 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1 > k \\right) \\leq p \\left( C > d + 1 \\right) } \\\\ & { \\qquad + p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { C } } \\right) - 1 > k \\bigg | C < d + 1 \\right) p \\left( C < d + 1 \\right) } \\\\ & { \\qquad \\leq \\exp \\left( - \\frac { 1 } { 4 d } \\right) + p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { d + 1 } } \\right) - 1 > k \\right) } \\\\ & { \\qquad = \\exp \\left( - \\frac { 1 } { 4 d } \\right) + \\frac { \\| \\mu \\| ^ { 2 } } { d + 1 } \\frac { 1 } { \\log \\left( 1 - k \\right) ^ { 2 } } . } \\\\ & { p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { C } } \\right) - 1 < - k \\right) \\leq \\exp \\left( - \\frac { 1 } { 4 d } \\right) + \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\frac { 1 } { \\log ^ { 2 } \\left( 1 + k \\right) } . } \\end{array}", + "type": "interline_equation", + "image_path": "3b0a9645bd49c6d733939d3b9c8b15f792b068a5fe13b8ce5518999cac5d235f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 142, + 106, + 518, + 153.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 142, + 153.0, + 518, + 200.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 142, + 200.0, + 518, + 247.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 249, + 377, + 261 + ], + "lines": [ + { + "bbox": [ + 142, + 248, + 378, + 263 + ], + "spans": [ + { + "bbox": [ + 142, + 248, + 378, + 263 + ], + "score": 1.0, + "content": "Combining the two inequalities above, one has (9) proved.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 470, + 285 + ], + "lines": [ + { + "bbox": [ + 104, + 271, + 469, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 318, + 286 + ], + "score": 1.0, + "content": "Step 4: We are now ready to prove convergence of", + "type": "text" + }, + { + "bbox": [ + 318, + 272, + 339, + 285 + ], + "score": 0.91, + "content": "Z ( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 271, + 394, + 286 + ], + "score": 1.0, + "content": ". With (9), let", + "type": "text" + }, + { + "bbox": [ + 395, + 272, + 426, + 283 + ], + "score": 0.91, + "content": "\\mathcal { C } \\subset \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 271, + 469, + 286 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 289, + 500, + 305 + ], + "lines": [ + { + "bbox": [ + 112, + 289, + 500, + 305 + ], + "spans": [ + { + "bbox": [ + 112, + 289, + 500, + 305 + ], + "score": 0.91, + "content": "\\mathcal { C } = \\left\\{ c : \\left| \\exp ( c ^ { \\top } \\mu ) - 1 \\right| < k , \\left| \\exp ( A c ^ { \\top } u _ { i } ) - 1 \\right| < k , \\left| \\exp ( - A c ^ { \\top } u _ { i } ) - 1 \\right| < k \\forall i = 1 , . . . , D \\right\\}", + "type": "interline_equation", + "image_path": "bc6fd5dae6ac59e4f43b3e41a1dacd8efdff4d74c0b36f3c80f06005596ddc0d.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 112, + 289, + 500, + 305 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 283, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 283, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 260, + 324 + ], + "score": 1.0, + "content": "Then we can bound the probability on", + "type": "text" + }, + { + "bbox": [ + 261, + 309, + 267, + 319 + ], + "score": 0.81, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 306, + 283, + 324 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 325, + 468, + 389 + ], + "lines": [ + { + "bbox": [ + 142, + 325, + 468, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 325, + 468, + 389 + ], + "score": 0.93, + "content": "\\begin{array} { l } { p ( \\mathcal { C } ) \\geq p \\left( \\left| \\exp ( c ^ { \\top } \\mu ) - 1 \\right| < k \\right) + \\displaystyle \\sum _ { i = 1 } ^ { D } p ( \\left| \\exp ( A c ^ { \\top } u _ { i } ) - 1 \\right| < k ) - 2 D } \\\\ { \\geq 1 - ( 2 D + 1 ) \\exp \\left( - \\displaystyle \\frac { 1 } { 4 d } \\right) - \\displaystyle \\frac { 2 D A ^ { 2 } } { d - 1 } \\displaystyle \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } - \\displaystyle \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\displaystyle \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "7653c2509a7bd7b4b32b168c0d11d8afd1f50e6939efb2402524a181bff56f90.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 142, + 325, + 468, + 346.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 142, + 346.3333333333333, + 468, + 367.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 142, + 367.66666666666663, + 468, + 388.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 384, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 385, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 255, + 408 + ], + "score": 1.0, + "content": "Next, we need to show that for every", + "type": "text" + }, + { + "bbox": [ + 255, + 396, + 264, + 403 + ], + "score": 0.8, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 390, + 342, + 408 + ], + "score": 1.0, + "content": ", the corresponding", + "type": "text" + }, + { + "bbox": [ + 342, + 393, + 364, + 405 + ], + "score": 0.91, + "content": "\\mathcal { C } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 390, + 385, + 408 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 409, + 465, + 426 + ], + "lines": [ + { + "bbox": [ + 145, + 409, + 465, + 426 + ], + "spans": [ + { + "bbox": [ + 145, + 409, + 465, + 426 + ], + "score": 0.91, + "content": "\\mathcal { C } ( \\boldsymbol { w } ) = \\left\\{ c : \\left| \\exp ( c ^ { \\top } \\boldsymbol { \\mu } ) - 1 \\right| < k , \\left| \\exp ( \\alpha _ { i } ( \\boldsymbol { w } ) c ^ { \\top } \\boldsymbol { u } _ { i } ) - 1 \\right| < k , \\forall i = 1 , . . . , D \\right\\}", + "type": "interline_equation", + "image_path": "7093bc60a266f621c33b344c3d39aaae796feab9f14b8cfccc2f5ce11900af0f.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 145, + 409, + 465, + 426 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 367, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 367, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 172, + 444 + ], + "score": 1.0, + "content": "We observe that", + "type": "text" + }, + { + "bbox": [ + 172, + 430, + 198, + 442 + ], + "score": 0.93, + "content": "\\alpha _ { i } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 428, + 258, + 444 + ], + "score": 1.0, + "content": "is bounded by", + "type": "text" + }, + { + "bbox": [ + 258, + 431, + 267, + 440 + ], + "score": 0.84, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 428, + 340, + 444 + ], + "score": 1.0, + "content": ", therefore for any", + "type": "text" + }, + { + "bbox": [ + 340, + 433, + 345, + 440 + ], + "score": 0.77, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 428, + 367, + 444 + ], + "score": 1.0, + "content": "that,", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 105, + 446, + 521, + 513 + ], + "lines": [ + { + "bbox": [ + 124, + 446, + 486, + 462 + ], + "spans": [ + { + "bbox": [ + 124, + 446, + 486, + 462 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\operatorname* { m i n } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) \\leq \\exp ( \\alpha _ { i } c ^ { \\top } u _ { i } ) \\leq \\operatorname* { m a x } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) , } \\end{array}", + "type": "inline_equation", + "image_path": "0ae602f12a5dc8b9cc40b38523a2ce0de28d9a219f107c486edbebc1a144eaf2.jpg" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 465, + 147, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 147, + 480 + ], + "score": 1.0, + "content": "and thus,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 481, + 520, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 111, + 498 + ], + "score": 0.992, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 111, + 481, + 514, + 497 + ], + "score": 0.81, + "content": "\\mathrm { n i n } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) - 1 \\leq \\exp ( \\alpha _ { i } c ^ { \\top } u _ { i } ) - 1 \\leq \\operatorname* { m a x } ( \\exp ( - A c ^ { \\top } u _ { i } ) , \\exp ( A c ^ { \\top } u _ { i } ) ) - 1", + "type": "inline_equation", + "image_path": "c2d0c76c8ddc9811c73bdced41886d7d8b2de7f7e59c5470f044be878244670c.jpg" + }, + { + "bbox": [ + 514, + 481, + 520, + 498 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 500, + 164, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 164, + 515 + ], + "score": 1.0, + "content": "which yields,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 517, + 457, + 532 + ], + "lines": [ + { + "bbox": [ + 154, + 517, + 457, + 532 + ], + "spans": [ + { + "bbox": [ + 154, + 517, + 457, + 532 + ], + "score": 0.89, + "content": "| \\exp ( \\alpha _ { i } c ^ { \\top } u _ { i } ) - 1 | \\leq \\operatorname* { m a x } ( | \\exp ( - A c ^ { \\top } u _ { i } ) - 1 | , | \\exp ( A c ^ { \\top } u _ { i } ) - 1 | ) < k .", + "type": "interline_equation", + "image_path": "7b9fd1defdcae810f366488d5c12ea318fbb3d451fd3aea042142dc2e69f71e4.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 154, + 517, + 457, + 532 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 536, + 399, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 401, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 187, + 552 + ], + "score": 1.0, + "content": "Therefore we prove", + "type": "text" + }, + { + "bbox": [ + 187, + 537, + 228, + 549 + ], + "score": 0.93, + "content": "c ( w ) \\supset { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 534, + 401, + 552 + ], + "score": 1.0, + "content": ". 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Z _ { 0 } | \\leq \\frac { ( D + 1 ) k } { | \\mathcal { V } | } Z _ { 0 } .", + "type": "interline_equation", + "image_path": "a149ffe7948aa3bf02a702508b145daeb5f78ac2e47eccad12028656f54e0450.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 237, + 653, + 373, + 680 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 684, + 263, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 264, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 122, + 698 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 684, + 168, + 696 + ], + "score": 0.93, + "content": "Z = | \\nu | Z _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 682, + 264, + 698 + ], + "score": 1.0, + "content": ", one can conclude that,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 700, + 395, + 714 + ], + "lines": [ + { + "bbox": [ + 214, + 700, + 395, + 714 + ], + "spans": [ + { + "bbox": [ + 214, + 700, + 395, + 714 + ], + "score": 0.89, + "content": "\\begin{array} { r } { p ( ( 1 - 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1 > k \\right) \\leq p \\left( C > d + 1 \\right) } \\\\ & { \\qquad + p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { C } } \\right) - 1 > k \\bigg | C < d + 1 \\right) p \\left( C < d + 1 \\right) } \\\\ & { \\qquad \\leq \\exp \\left( - \\frac { 1 } { 4 d } \\right) + p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { d + 1 } } \\right) - 1 > k \\right) } \\\\ & { \\qquad = \\exp \\left( - \\frac { 1 } { 4 d } \\right) + \\frac { \\| \\mu \\| ^ { 2 } } { d + 1 } \\frac { 1 } { \\log \\left( 1 - k \\right) ^ { 2 } } . } \\\\ & { p \\left( \\exp \\left( \\frac { \\| \\mu \\| c _ { 1 } } { \\sqrt { C } } \\right) - 1 < - k \\right) \\leq \\exp \\left( - \\frac { 1 } { 4 d } \\right) + \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\frac { 1 } { \\log ^ { 2 } \\left( 1 + k \\right) } . } \\end{array}", + "type": "interline_equation", + "image_path": "3b0a9645bd49c6d733939d3b9c8b15f792b068a5fe13b8ce5518999cac5d235f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 142, + 106, + 518, + 153.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 142, + 153.0, + 518, + 200.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 142, + 200.0, + 518, + 247.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 249, + 377, + 261 + ], + "lines": [ + { + "bbox": [ + 142, + 248, + 378, + 263 + ], + "spans": [ + { + "bbox": [ + 142, + 248, + 378, + 263 + ], + "score": 1.0, + "content": "Combining the two inequalities above, one has (9) proved.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 142, + 248, + 378, + 263 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 470, + 285 + ], + "lines": [ + { + "bbox": [ + 104, + 271, + 469, + 286 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 318, + 286 + ], + "score": 1.0, + "content": "Step 4: We are now ready to prove convergence of", + "type": "text" + }, + { + "bbox": [ + 318, + 272, + 339, + 285 + ], + "score": 0.91, + "content": "Z ( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 271, + 394, + 286 + ], + "score": 1.0, + "content": ". With (9), let", + "type": "text" + }, + { + "bbox": [ + 395, + 272, + 426, + 283 + ], + "score": 0.91, + "content": "\\mathcal { C } \\subset \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 271, + 469, + 286 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 271, + 469, + 286 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 289, + 500, + 305 + ], + "lines": [ + { + "bbox": [ + 112, + 289, + 500, + 305 + ], + "spans": [ + { + "bbox": [ + 112, + 289, + 500, + 305 + ], + "score": 0.91, + "content": "\\mathcal { C } = \\left\\{ c : \\left| \\exp ( c ^ { \\top } \\mu ) - 1 \\right| < k , \\left| \\exp ( A c ^ { \\top } u _ { i } ) - 1 \\right| < k , \\left| \\exp ( - A c ^ { \\top } u _ { i } ) - 1 \\right| < k \\forall i = 1 , . . . , D \\right\\}", + "type": "interline_equation", + "image_path": "bc6fd5dae6ac59e4f43b3e41a1dacd8efdff4d74c0b36f3c80f06005596ddc0d.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 112, + 289, + 500, + 305 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 283, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 283, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 260, + 324 + ], + "score": 1.0, + "content": "Then we can bound the probability on", + "type": "text" + }, + { + "bbox": [ + 261, + 309, + 267, + 319 + ], + "score": 0.81, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 306, + 283, + 324 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 306, + 283, + 324 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 325, + 468, + 389 + ], + "lines": [ + { + "bbox": [ + 142, + 325, + 468, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 325, + 468, + 389 + ], + "score": 0.93, + "content": "\\begin{array} { l } { p ( \\mathcal { C } ) \\geq p \\left( \\left| \\exp ( c ^ { \\top } \\mu ) - 1 \\right| < k \\right) + \\displaystyle \\sum _ { i = 1 } ^ { D } p ( \\left| \\exp ( A c ^ { \\top } u _ { i } ) - 1 \\right| < k ) - 2 D } \\\\ { \\geq 1 - ( 2 D + 1 ) \\exp \\left( - \\displaystyle \\frac { 1 } { 4 d } \\right) - \\displaystyle \\frac { 2 D A ^ { 2 } } { d - 1 } \\displaystyle \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } - \\displaystyle \\frac { \\| \\mu \\| ^ { 2 } } { d - 1 } \\displaystyle \\frac { 1 } { \\log ^ { 2 } ( 1 - k ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "7653c2509a7bd7b4b32b168c0d11d8afd1f50e6939efb2402524a181bff56f90.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 142, + 325, + 468, + 346.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 142, + 346.3333333333333, + 468, + 367.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 142, + 367.66666666666663, + 468, + 388.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 384, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 385, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 255, + 408 + ], + "score": 1.0, + "content": "Next, we need to show that for every", + "type": "text" + }, + { + "bbox": [ + 255, + 396, + 264, + 403 + ], + "score": 0.8, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 390, + 342, + 408 + ], + "score": 1.0, + "content": ", the corresponding", + "type": "text" + }, + { + "bbox": [ + 342, + 393, + 364, + 405 + ], + "score": 0.91, + "content": "\\mathcal { C } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 390, + 385, + 408 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 390, + 385, + 408 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 409, + 465, + 426 + ], + "lines": [ + { + "bbox": [ + 145, + 409, + 465, + 426 + ], + "spans": [ + { + "bbox": [ + 145, + 409, + 465, + 426 + ], + "score": 0.91, + "content": "\\mathcal { C } ( \\boldsymbol { w } ) = \\left\\{ c : \\left| \\exp ( c ^ { \\top } \\boldsymbol { \\mu } ) - 1 \\right| < k , \\left| \\exp ( \\alpha _ { i } ( \\boldsymbol { w } ) c ^ { \\top } \\boldsymbol { u } _ { i } ) - 1 \\right| < k , \\forall i = 1 , . . . , D \\right\\}", + "type": "interline_equation", + "image_path": "7093bc60a266f621c33b344c3d39aaae796feab9f14b8cfccc2f5ce11900af0f.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 145, + 409, + 465, + 426 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 367, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 367, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 172, + 444 + ], + "score": 1.0, + "content": "We observe that", + "type": "text" + }, + { + "bbox": [ + 172, + 430, + 198, + 442 + ], + "score": 0.93, + "content": "\\alpha _ { i } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 428, + 258, + 444 + ], + "score": 1.0, + "content": "is bounded by", + "type": "text" + }, + { + "bbox": [ + 258, + 431, + 267, + 440 + ], + "score": 0.84, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 428, + 340, + 444 + ], + "score": 1.0, + "content": ", therefore for any", + "type": "text" + }, + { + "bbox": [ + 340, + 433, + 345, + 440 + ], + "score": 0.77, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 428, + 367, + 444 + ], + "score": 1.0, + "content": "that,", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 428, + 367, + 444 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 446, + 521, + 513 + ], + "lines": [ + { + "bbox": [ + 124, + 446, + 486, + 462 + ], + "spans": [ + { + "bbox": [ + 124, + 446, + 486, + 462 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\operatorname* { m i n } ( \\exp ( - 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1 | , | \\exp ( A c ^ { \\top } u _ { i } ) - 1 | ) < k .", + "type": "interline_equation", + "image_path": "7b9fd1defdcae810f366488d5c12ea318fbb3d451fd3aea042142dc2e69f71e4.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 154, + 517, + 457, + 532 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 536, + 399, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 401, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 187, + 552 + ], + "score": 1.0, + "content": "Therefore we prove", + "type": "text" + }, + { + "bbox": [ + 187, + 537, + 228, + 549 + ], + "score": 0.93, + "content": "c ( w ) \\supset { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 534, + 401, + 552 + ], + "score": 1.0, + "content": ". 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Algorithm 1: Postprocessing algorithm on word representations.
Input:Word representations {v(w),w ∈V},a threshold parameter D, 1 Compute the mean of {ν(w),w²∈V},μ← 高Σwev v(w),u(w) ← v(w)- μ
2 Compute the PCA components: u1,., ud ← PCA({u(w),w ∈ V}).
3 Preprocessthe representations: v’'(w) ← u(w)-∑=1 (uT v(w)) ui
Output :Processed representations v'(w).
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LanguageCorpusdimvocab sizeavg.u(w)l2μl2
TSCCA-EnEnglishGigawords200300,0004.380.78
TSCCA-DeGermanNewswire200300,0004.520.79
TSCCA-FrFrenchGigaword200300,0004.340.81
TSCCA-EsSpanishGigaword200300.0004.170.79
TSCCA-ItItalianNewswire+Wiki200300,0004.340.79
TSCCA-N1DutchNewswire+Wiki200300,0004.460.72
TSCCA-ZhChineseGigaword200300.0004.510.89
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RAND-WALKTSCCA
orig.proc.orig.proc.
syn.60.3960.4837.7237.80
sem.83.5583.8214.5414.55
all70.5070.6727.3027.35
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RAND-WALKTSCCA
orig.proc.orig.proc.
RG6580.6682.9647.5347.67
WS65.8974.3754.2154.35
RW45.1151.2343.9643.72
MEN73.5677.2265.4865.62
MTurk64.3566.1159.6560.03
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orig.proc.orig.proc.
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2012.OnWN69.3869.4853.0765.67
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2014.OnWN74.6174.7857.4167.56
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2015.answers-forum46.3546.8030.0239.86
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2015.belief59.7260.4244.0557.68
2015.headlines61.4763.4546.2253.31
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SICK70.1070.2065.1467.85
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C1TrainTest
MR22010,66210-fold cross validation
SUBJ22310,00010-fold cross validation
TREC6105,952500
SST51811,8552,210
IMDb210025,00025,000
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0000000000000000000000000000000000000000..a6d7fdbecfb475d358e92128af8cfacbc4a9eeb3 --- /dev/null +++ b/parse/train/PKubaeJkw3/PKubaeJkw3.md @@ -0,0 +1,396 @@ +# RETHINKING ARCHITECTURE SELECTION IN DIFFERENTIABLE NAS + +Ruochen Wang1, Minhao Cheng1, Xiangning Chen1, Xiaocheng Tang2, Cho-Jui Hsieh1 +1Department of Computer Science, UCLA, 2DiDi AI Labs +{ruocwang, mhcheng}@ucla.edu {xiangning, chohsieh}@cs.ucla.edu +xiaochengtang@didiglobal.com + +# ABSTRACT + +Differentiable Neural Architecture Search is one of the most popular Neural Architecture Search (NAS) methods for its search efficiency and simplicity, accomplished by jointly optimizing the model weight and architecture parameters in a weight-sharing supernet via gradient-based algorithms. At the end of the search phase, the operations with the largest architecture parameters will be selected to form the final architecture, with the implicit assumption that the values of architecture parameters reflect the operation strength. While much has been discussed about the supernet’s optimization, the architecture selection process has received little attention. We provide empirical and theoretical analysis to show that the magnitude of architecture parameters does not necessarily indicate how much the operation contributes to the supernet’s performance. We propose an alternative perturbation-based architecture selection that directly measures each operation’s influence on the supernet. We re-evaluate several differentiable NAS methods with the proposed architecture selection and find that it is able to extract significantly improved architectures from the underlying supernets consistently. Furthermore, we find that several failure modes of DARTS can be greatly alleviated with the proposed selection method, indicating that much of the poor generalization observed in DARTS can be attributed to the failure of magnitude-based architecture selection rather than entirely the optimization of its supernet. + +# 1 INTRODUCTION + +Neural Architecture Search (NAS) has been drawing increasing attention in both academia and industry for its potential to automatize the process of discovering high-performance architectures, which have long been handcrafted. Early works on NAS deploy Evolutionary Algorithm (Stanley & Miikkulainen, 2002; Real et al., 2017; Liu et al., 2017) and Reinforcement Learning (Zoph & Le, 2017; Pham et al., 2018; Zhong et al., 2018) to guide the architecture discovery process. Recently, several one-shot methods have been proposed that significantly improve the search efficiency (Brock et al., 2018; Guo et al., 2019; Bender et al., 2018). + +As a particularly popular instance of one-shot methods, DARTS (Liu et al., 2019) enables the search process to be performed with a gradient-based optimizer in an end-to-end manner. It applies continuous relaxation that transforms the categorical choice of architectures into continuous architecture parameters $\alpha$ . The resulting supernet can be optimized via gradient-based methods, and the operations associated with the largest architecture parameters are selected to form the final architecture. Despite its simplicity, several works cast doubt on the effectiveness of DARTS. For example, a simple randomized search (Li & Talwalkar, 2019) outperforms the original DARTS; Zela et al. (2020) observes that DARTS degenerates to networks filled with parametric-free operations such as the skip connection or even random noise, leading to the poor performance of the selected architecture. + +While the majority of previous research attributes the failure of DARTS to its supernet optimization (Zela et al., 2020; Chen & Hsieh, 2020; Chen et al., 2021), little has been discussed about the validity of another important assumption: the value of α reflects the strength of the underlying operations. In this paper, we conduct an in-depth analysis of this problem. Surprisingly, we find that in many cases, $\alpha$ does not really indicate the operation importance in a supernet. Firstly, the operation associated with larger $\alpha$ does not necessarily result in higher validation accuracy after discretization. Secondly, as an important example, we show mathematically that the domination of skip connection observed in DARTS (i.e. $\alpha _ { s k i p }$ becomes larger than other operations.) is in fact a reasonable outcome of the supernet’s optimization but becomes problematic when we rely on $\alpha$ to select the best operation. + +If $\alpha$ is not a good indicator of operation strength, how should we select the final architecture from a pretrained supernet? Our analysis indicates that the strength of each operation should be evaluated based on its contribution to the supernet performance instead. To this end, we propose an alternative perturbation-based architecture selection method. Given a pretrained supernet, the best operation on an edge is selected and discretized based on how much it perturbs the supernet accuracy; The final architecture is derived edge by edge, with fine-tuning in between so that the supernet remains converged for every operation decision. We re-evaluate several differentiable NAS methods (DARTS (Liu et al., 2019), SDARTS (Chen & Hsieh, 2020), SGAS (Li et al., 2020)) and show that the proposed selection method is able to consistently extract significantly improved architectures from the supernets than magnitude-based counterparts. Furthermore, we find that the robustness issues of DARTS can be greatly alleviated by replacing the magnitude-based selection with the proposed perturbation-based selection method. + +# 2 BACKGROUND AND RELATED WORK + +Preliminaries of Differentiable Architecture Search (DARTS) We start by reviewing the formulation of DARTS. DARTS’ search space consists of repetitions of cell-based microstructures. Every cell can be viewed as a DAG with N nodes and $\mathrm { E }$ edges, where each node represents a latent feature map $x ^ { i }$ , and each edge is associated with an operation $o$ (e.g. skip connect, sep conv $3 x 3$ ) from the search space $\mathcal { O }$ . Continuous relaxation is then applied to this search space. Concretely, every operation on an edge is activated during the search phase, with their outputs mixed by the architecture parameter α to form the final mixed output of that edge m¯ (xi) = Po∈O P exp αoo0 exp αo0 . This particular formulation allows the architecture search to be performed in a differentiable manner: DARTS jointly optimizes $\alpha$ and model weight $w$ with the following bilevel objective via alternative gradient updates: + +$$ +\operatorname* { m i n } _ { \alpha } \mathcal { L } _ { v a l } ( w ^ { * } , \alpha ) \mathrm { s . t . } w ^ { * } = \arg \operatorname* { m i n } _ { w } \mathcal { L } _ { t r a i n } ( w , \alpha ) . +$$ + +We refer to the continuous relaxed network used in the search phase as the supernet of DARTS. At the end of the search phase, the operation associated with the largest $\alpha _ { o }$ on each edge will be selected from the supernet to form the final architecture. + +Failure mode analysis of DARTS Several works cast doubt on the robustness of DARTS. Zela et al. (2020) tests DARTS on four different search spaces and observes significantly degenerated performance. They empirically find that the selected architectures perform poorly when DARTS’ supernet falls into high curvature areas of validation loss (captured by large dominant eigenvalues of the Hessian $\nabla _ { \alpha , \alpha } ^ { 2 } \mathcal { L } _ { v a l } ( w , \alpha ) )$ . While Zela et al. (2020) relates this problem to the failure of supernet training in DARTS, we examine it from the architecture selection aspects of DARTS, and show that much of DARTS’ robustness issue can be alleviated by a better architecture selection method. + +Progressive search space shrinking There is a line of research on NAS that focuses on reducing the search cost and aligning the model sizes of the search and evaluation phases via progressive search space shrinking (Liu et al., 2018; Li et al., 2019; Chen et al., 2021; Li et al., 2020). The general scheme of these methods is to prune out weak operations and edges sequentially during the search phase, based on the magnitude of $\alpha$ following DARTS. Our method is orthogonal to them in this respect, since we select operations based on how much it contributes to the supernet’s performance rather than the $\alpha$ value. Although we also discretize edges greedily and fine-tune the network in between, the purpose is to let the supernet recover from the loss of accuracy after discretization to accurately evaluate operation strength on the next edge, rather than to reduce the search cost. + +# 3 THE PITFALL OF MAGNITUDE-BASED ARCHITECTURE SELECTION IN DARTS + +In this section, we put forward the opinion that the architecture parameter $\alpha$ does not necessarily represent the strength of the underlying operation in general, backed by both empirical and theoretical evidence. As an important example, we mathematically justify that the skip connection domination phenomenon observed in DARTS is reasonable by itself, and becomes problematic when combined with the magnitude-based architecture selection. + +# 3.1 $\alpha$ MAY NOT REPRESENT THE OPERATION STRENGTH + +![](images/edb10b3c13ea4154d37633f3cbc076def08cf6dd2ef065dd8b384fd9afd746f5.jpg) +Figure 1: $\alpha$ vs discretization accuracy at convergence of all operations on 3 randomly selected edges from a pretrained DARTS supernet (one subplot per edge). The magnitude of $\alpha$ for each operation does not necessarily agree with its relative discretization accuracy at convergence. + +![](images/77e533953356028536aaa448409ba1cfccb71639465d6c5e104d9488c7bdad81.jpg) +Figure 2: Operation strength on each edge of S2 (skip connect, sep conv ${ _ { - 3 x 3 } }$ ). (a). Operations associated with the largest $\alpha$ . (b). Operations that result in the highest discretization validation accuracy at convergence. Parameterized operations are marked red. + +Following DARTS, existing differentiable NAS methods use the value of architecture parameters $\alpha$ to select the final architecture from the supernet, with the implicit assumption that $\alpha$ represents the strength of the underlying operations. In this section, we study the validity of this assumption in detail. + +Consider one edge on a pretrained supernet; the strength of an operation on the edge can be naturally defined as the supernet accuracy after we discretize to this operation and fine-tune the remaining network until it converges again; we refer to this as ”discretization accuracy at convergence” for short. The operation that achieves the best discretization accuracy at convergence can be considered as the best operation for the given edge. Figure 1 shows the comparison of $\alpha$ (blue) and operation strength (orange) of randomly select edges on DARTS supernet. As we can see, the magnitude of $\alpha$ for each operation does not necessarily agree with their relative strength measured by discretization accuracy at convergence. Moreover, operations assigned with small $\alpha \mathbf { s }$ are sometimes strong ones that lead to high discretization accuracy at convergence. To further verify the mismatch, we investigate the operation strength on search space S2, where DARTS fails dramatically due to excessive skip connections (Zela et al., 2020). S2 is a variant of DARTS search space that only contains two operations per edge (skip connect, sep conv ${ _ { 3 x 3 } }$ ). Figure 2 shows the selected operations based on $\alpha$ (left) and operation strength (right) on all edges on S2. From Figure 2a, we can see that αskip connect $> \alpha _ { s e p . c o n v . 3 x 3 }$ on 12 of 14 edges. Consequently, the derived child architecture will lack representation ability and perform poorly due to too many skip connections. However, as shown in Figure 2b, the supernet benefits more from discretizing to sep conv 3x3 than skip connect on half of the edges. + +# 3.2 A CASE STUDY: SKIP CONNECTION + +Several works point out that DARTS tends to assign large $\alpha$ to skip connections, resulting in shallow architectures with poor generability (Zela et al., 2020; Liang et al., 2019; Bi et al., 2019). This ”skip connection domination” issue is generally attributed to the failure of DARTS’ supernet optimization. In contrast, we draw inspiration from research on ResNet (He et al., 2016) and show that this phenomenon by itself is a reasonable outcome while DARTS refines its estimation of the optimal feature map, rendering $\alpha _ { s k i p }$ ineffective in the architecture selection. + +In vanilla networks (e.g., VGG), each layer computes a new level of feature map from the output feature map of the predecessor layer; thus, reordering layers at test time would dramatically hurt the performance (Veit et al., 2016). Unlike vanilla networks, Greff et al. (2017) and Veit et al. (2016) discover that successive layers in ResNet with compatible channel sizes are in fact estimating the same optimal feature map so that the outputs of these layers + +Table 1: Test accuracy before and after layer (edge) shuffling on cifar10. For ResNet and VGG, we randomly swap two layers in each stage (defined as successive layers between two downsampling blocks. For DARTS supernet, we randomly swap two edges in every cell. + +
VGGResNetDARTS
Before92.6993.8688.44
After9.83 ± 0.3383.2015 ± 2.0381.09 ± 1.87
+ +stay relatively close to each other at convergence; As a result, ResNet’s test accuracy remains robust under layer reordering. Greff et al. (2017) refers to this unique way of feature map estimation in ResNet as the ”unrolled estimation.” + +DARTS’ supernet resembles ResNet, rather than vanilla networks like VGG, in both appearance and behavior. Appearance-wise, within a cell of DARTS’ supernet, edges with skip connection are in direct correspondence with the successive residual layers in ResNet. Behavior-wise, DARTS’ supernet also exhibits a high degree of robustness under edge shuffling. As shown in Table 1, randomly reordering edges on a pretrained DARTS’ supernet at test time also has little effect on its performance. This evidence indicates that DARTS performs unrolled estimation like ResNet as well, i.e., edges within a cell share the same optimal feature map that they try to estimate. In the following proposition, we apply this finding and provide the optimal solution of $\alpha$ in the sense of minimizing the variance of feature map estimation. + +Proposition 1. 1 Without loss of generality, consider one cell from a simplified search space consists of two operations: (skip, conv). Let $m ^ { * }$ denotes the optimal feature map, which is shared across all edges according to the unrolled estimation view (Greff et al., 2017). Let $o _ { e } ( x _ { e } )$ be the output of convolution operation, and let $x _ { e }$ be the skip connection (i.e., the input feature map of edge e). Assume $m ^ { * }$ , $o _ { e } ( x _ { e } )$ and $x _ { e }$ are normalized to the same scale. The current estimation of $m ^ { * }$ can then be written as: + +$$ +\overline { { { m } } } _ { e } ( x _ { e } ) = \frac { \exp ( \alpha _ { c o n v } ) } { \exp ( \alpha _ { c o n v } ) + \exp ( \alpha _ { s k i p } ) } o _ { e } ( x _ { e } ) + \frac { \exp ( \alpha _ { s k i p } ) } { \exp ( \alpha _ { c o n v } ) + \exp ( \alpha _ { s k i p } ) } x _ { e } , +$$ + +where $\alpha _ { c o n v }$ and $\alpha _ { s k i p }$ are the architecture parameters defined in DARTS. The optimal $\alpha _ { c o n v } ^ { * }$ and $\alpha _ { s k i p } ^ { * }$ minimizing v $a r \mathbf { \hat { ( } } \overline { { m } } _ { e } ( x _ { e } ) - m ^ { * } )$ , the variance of the difference between the optimal feature map $m ^ { * }$ and its current estimation $\overline { { m } } _ { e } ( x _ { e } )$ , are given by: + +$$ +\begin{array} { l } { \alpha _ { c o n v } ^ { * } \propto v a r ( x _ { e } - m ^ { * } ) } \\ { \alpha _ { s k i p } ^ { * } \propto v a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) . } \end{array} +$$ + +We refer the reader to Appendix A.4 for detailed proof. From eq. (3) and eq. (4), we can see that the relative magnitudes of $\alpha _ { s k i p }$ and $\alpha _ { c o n v }$ come down to which one of $x _ { e }$ or $o _ { e } ( x _ { e } )$ is closer to $m ^ { * }$ in variance: + +• $o _ { e } ( x _ { e } )$ is the output of a single convolution operation instead of the complete mixed output of edge $e$ , so it will deviate from $m ^ { * }$ even at convergence. + +Therefore, in a well-optimized supernet, $x _ { e }$ will naturally be closer to $m ^ { * }$ than $o _ { e } ( x _ { e } )$ , causing $\alpha _ { s k i p }$ to be greater than $\alpha _ { c o n v }$ . + +Our analysis above indicates that the better the supernet, the larger the $( \alpha _ { s k i p } - \alpha _ { c o n v } )$ gap (softmaxed) will become since $x _ { e }$ gets closer and closer to $m ^ { * }$ as the supernet is optimized. This result is evidenced in Figure 3, where $m e a n ( \alpha _ { s k i p } - \alpha _ { c o n v } )$ continues to grow as the supernet gets better. In this case, although $\alpha _ { s k i p } > \alpha _ { c o n v }$ is reasonable by itself, it becomes an inductive bias to NAS if we were to select the final architecture based on $\alpha$ . + +# 4 PERTURBATION-BASED ARCHITECTURE SELECTION + +![](images/79a873ee67039a51da8c30bc3160564d6e0226f824e4320c9a3517871f5adb3b.jpg) +Figure 3: $m e a n ( \alpha _ { s k i p } - \alpha _ { c o n v } )$ (softmaxed) v.s. supernet’s validation accuracy. The gap of $( \alpha _ { s k i p } - \alpha _ { c o n v } )$ increases as supernet gets better. + +Instead of relying on the $\alpha$ value to select the best operation, we propose to directly evaluate operation strength in + +terms of its contribution to the supernet’s performance. The operation selection criterion is laid out in section 4.1. In section 4.2, we describe the entire architecture selection process. + +# 4.1 EVALUATING THE STRENGTH OF EACH OPERATION + +In section 3.1, we define the strength of each operation on a given edge as how much it contributes to the performance of the supernet, measured by discretization accuracy. To avoid inaccurate evaluation due to large disturbance of the supernet during discretization, we fine-tune the remaining supernet until it converges again, and then compute its validation accuracy (discretization accuracy at convergence). The fine-tuning process needs to be carried out for evaluating each operation on an edge, leading to substantial computation costs. + +To alleviate the computational overhead, we consider a more practical measure of operation strength: for each operation on a given edge, we mask it out while keeping all other operations, and re-evaluate the supernet. The one that results in the largest drop in the supernet’s validation accuracy will be considered as the most important operation on that edge. This alternative criterion incurs much less perturbation to the supernet than discretization since it only deletes one operation from the supernet at a time. As a result, the supernet’s validation accuracy after deletion stays close to the unmodified supernet, and thus it alleviates the requirement of tuning the remaining supernet to convergence. Therefore, we implement this measurement for the operation selection in this work. + +Input: A pretrained supernet $S$ , Set of edges $\mathcal { E }$ from $S$ , Set of nodes $\mathcal { N }$ from $S$ +Result: Set of selected operations $\{ o _ { e } ^ { * } \} _ { e \in \mathcal { E } }$ +while $| \mathcal { E } | > 0$ do randomly select an edge $e \in { \mathcal { E } }$ (and remove it from $\mathcal { E }$ ); forall operation o on edge $e$ do evaluate the validation accuracy of $S$ when $o$ is removed $( A C C _ { \backslash o } )$ ; end select the best operation for $e$ : $\begin{array} { r } { o _ { e } ^ { * } \gets \operatorname * { a r g m i n } _ { o } A C C _ { \backslash o } } \end{array}$ ; discretize edge $e$ to $o _ { e } ^ { * }$ and tune the remaining supernet for a few epochs; +end + +# 4.2 THE COMPLETE ARCHITECTURE SELECTION PROCESS + +Our method operates directly on top of DARTS’ pretrained supernet. Given a supernet, we randomly iterate over all of its edges. We evaluate each operation on an edge, and select the best one to be discretized based on the measurement described in section 4.1. After that, we tune the supernet for a few epochs to recover the accuracy lost during discretization. The above steps are repeated until all edges are decided. Algorithm 1 summarizes the operation selection process. The cell topology is decided in a similar fashion. We refer the reader to Appendix A.3 for the full algorithm, including deciding the cell topology. This simple method is termed ”perturbation-based architecture selection (PT)” in the following sections. + +# 5 EXPERIMENTAL RESULTS + +In this section, we demonstrate that the perturbation-based architecture selection method is able to consistently find better architectures than those selected based on the values of $\alpha$ . The evaluation is based on the search space of DARTS and NAS-Bench-201 (Dong & Yang, 2020), and we show that the perturbation-based architecture selection method can be applied to several variants of DARTS. + +# 5.1 RESULTS ON DARTS’ CNN SEARCH SPACE + +We keep all the search and retrain settings identical to DARTS since our method only modifies the architecture selection part. After the search phase, we perform perturbation-based architecture selection following Algorithm 1 on the pretrained supernet. We tune the supernet for 5 epochs between two selections as it is enough for the supernet to recover from the drop of accuracy after discretization. We run the search and architecture selection phase with four random seeds and report both the best and average test errors of the obtained architectures. + +As shown in Table 2, the proposed method (DARTS $+ \mathrm { P T }$ ) improves DARTS’ test error from $3 . 0 0 \%$ to $2 . 6 1 \%$ , with manageable search cost (0.8 GPU days). Note that by only changing the architecture selection method, DARTS performs significantly better than many other differentiable NAS methods that enjoy carefully designed optimization process of the supernet, such as GDAS (Dong & Yang, 2019) and SNAS (Xie et al., 2019). This empirical result suggests that architecture selection is crucial to DARTS: with the proper selection algorithm, DARTS remains a very competitive method. + +Our method is also able to improve the performance of other variants of DARTS. To show this, we evaluate our method on SDARTS(rs) and SGAS (Chen & Hsieh, 2020; Li et al., 2020). SDARTS(rs) is a variant of DARTS that regularizes the search phase by applying Gaussian perturbation to $\alpha$ . Unlike DARTS and SDARTS, SGAS performs progressive search space shrinking. Concretely, SGAS progressively discretizes its edges with the order from most to least important, based on a novel edge importance score. For a fair comparison, we keep its unique search space shrinking process unmodified and only replace its magnitude-based operation selection with ours. As we can see from Table 2, our method consistently achieves better average test errors than its magnitudebased counterpart. Concretely, the proposed method improves SDARTS’ test error from $2 . 6 7 \%$ to $2 . 5 4 \%$ and SGAS’ test error from $2 . 6 6 \%$ to $2 . 5 6 \%$ . Moreover, the best architecture discovered in our experiments achieves a test error of $2 . 4 4 \%$ , ranked top among other NAS methods. + +![](images/7d4b6638faec392f9f4d135d85de588ec6564e674791a16ea64c8e91e9c9bcca.jpg) +5.2 PERFORMANCE ON NAS-BENCH-201 SEARCH SPACE +Figure 4: Trajectory of test accuracy on space NAS-Bench-201 and three datasets (Left: cifar10, Middle: cifar100, Right: Imagenet16-120). The test accuracy of our method is plotted by taking the snapshots of DARTS’ supernet at corresponding epochs and run our selection method on top of it. + +To further verify the effectiveness of the proposed perturbation-based architecture selection, we conduct experiments on NAS-Bench-201. NAS-Bench-201 provides a unified cell-based search space similar to DARTS. Every architecture in the search space is trained under the same protocol on three datasets (cifar10, cifar100, and imagenet16-120), and their performance can be obtained by querying the database. As in section 5.1, we take the pretrained supernet from DARTS and apply our method on top of it. All other settings are kept unmodified. Figure 4 shows the performance trajectory of DARTS+PT compared with DARTS. While the architectures found by magnitude-based selection degenerates over time, the perturbation-based method is able to extract better architectures from the same underlying supernets stably. The result implies that the DARTS’ degenerated performance comes from the failure of magnitude based architecture selection. + +Table 2: Comparison with state-of-the-art image classifiers on CIFAR-10. + +
ArchitectureTest Error (%)Params (M)Search Cost (GPU days)Search Method
DenseNet-BC (Huang etal.,2017)3.4625.6-manual
NASNet-A (Zoph et al.,2018)2.653.32000RL
AmoebaNet-A (Real etal.,2019)3.34± 0.063.23150evolution
AmoebaNet-B (Real et al.,2019)2.55 ± 0.052.83150evolution
PNAS (Liu et al.,2018)*3.41 ± 0.093.2225SMBO
ENAS (Pham et al.,2018)2.894.60.5RL
NAONet (Luo et al.,2018)3.533.10.4NAO
SNAS (moderate) (Xie et al.,2019)2.85±0.022.81.5gradient
GDAS(Dong& Yang,2019)2.933.40.3gradient
BayesNAS (Zhou et al.,2019)2.81± 0.043.40.2gradient
ProxylessNAS (Cai et al.,2019)†2.085.74.0gradient
NASP(Yao et al.,2020)2.83±0.093.30.1gradient
P-DARTS (Chen et al.,2019)2.503.40.3gradient
PC-DARTS (Xu et al.,2020)2.57 ± 0.073.60.1gradient
R-DARTS (L2) Zela et al. (2020)2.95 ± 0.21-1.6gradient
DARTS (Liu et al.,2019)3.00±0.143.30.4gradient
SDARTS-RS (Chen& Hsieh,2020)2.67± 0.033.40.4gradient
SGAS(Cri 1.avg) (Li et al.,2020)2.66 ± 0.243.70.25gradient
DARTS+PT (avg)*2.61±0.083.00.8tgradient
DARTS+PT (best)2.483.30.8tgradient
SDARTS-RS+PT (avg)*2.54 ± 0.103.30.8t
SDARTS-RS+PT(best)2.443.20.8tgradient
SGAS+PT (Crit.1 avg)*2.56 ± 0.103.90.29tgradient gradient
SGAS+PT (Crit.1 best)2.463.90.29tgradient
+ +† Obtained on a different space with PyramidNet (Han et al., 2017) as the backbone. ‡ Recorded on a single GTX 1080Ti GPU. ∗ Obtained by running the search and retrain phase under four different seeds and computing the average test error of the derived architectures. + +# 6 ANALYSIS + +# 6.1 ISSUE WITH THE ROBUSTNESS OF DARTS + +Zela et al. (2020) observes that DARTS tends to yield degenerate architectures with abysmal performance. We conjecture that this robustness issue of DARTS can be explained by the failure of magnitude-based architecture selection. + +To show this, we test DARTS’ performance with perturbation-based architecture selection on four spaces proposed by Zela et al. (2020) (S1-S4). The complete specifications of these spaces can be found in Appendix A.2. Given a supernet, the architecture selected based on $\alpha$ performs poorly across spaces and datasets (column 3 in Table 3). However, our method is able to consistently extract meaningful architectures with significantly improved performance (Column 4 in Table 3). + +Table 3: DARTS+PT on S1-S4 (test error $( \% ) _ { . }$ + +
DatasetSpaceDARTSDARTS+PT(Ours)DARTS+PT(fixα)*
C10S13.843.502.86
S24.852.792.59
S33.342.492.52
S47.202.642.58
C100S129.4624.4824.40
S226.0523.1623.30
S328.9022.03
S422.8520.8021.94 20.66
SVHNS14.582.622.39
S23.532.532.32
S33.412.422.32
S43.052.422.39
+ +? This column will be explained later in Section 6.3 + +Notably, DART $\mathrm { S + P T }$ is able to find meaningful architecture on S2 (skip connect, sep conv ${ } _ { - 3 x 3 }$ ) and S4 (noise, sep conv $3 x 3$ ), where DARTS failed dramatically. As shown in Figure 5, on S2, while magnitude-based selection degenerates to architectures filled with skip connections, DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ is able to find architecture with 4 convolutions; On S4, DART $\mathrm { \ s + P T }$ consistently favors sep conv 3x3 on edges where $\alpha$ selects noise. + +![](images/6f35a1fec263ceaf499587658509937b12ee3ee4f90c4afd7b5d47144b0dc9a6.jpg) +Figure 5: Comparison of normal cells found on S2 and S4. The perturbation-based architecture selection (DARTS $+ \mathrm { P T }$ ) is able to find reasonable architectures in cases where the magnitude-based method (DARTS) fails dramatically. The complete architecture can be found in Appendix A.9. Non-trivial operations are marked red. + +# 6.2 PROGRESSIVE TUNING + +In addition to operation selection, we also tune the supernet after an edge is discretized so that the supernet could regain the lost accuracy. To measure the effectiveness of our operation selection criterion alone, we conduct an ablation study on the progressive tuning part. Concretely, we test a baseline by combining progressive tuning with magnitude-based operation selection instead of our selection criterion, which we code-named DARTS $+$ PT-Mag. Figure 6 plots the change of validation accuracy of DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ and DARTS $+$ PT-Mag during the operation selection phase. As we can see, DART $\mathsf { s } + \mathsf { P T }$ is able to identify better operations that lead to higher validation accuracy than the magnitude-based alternative, revealing the effectiveness of our operation selection criteria. Moreover, DARTS+PT-Mag is only able to obtain a test error of $2 . 8 5 \%$ on DARTS space + +![](images/7a67b64fd4ef3e701516214be42db891468961f0e71afdc6dc6fc9c6d6766051.jpg) +Figure 6: The trajectory of validation accuracy in the operation selection phase on S2. DARTS $+ \mathrm { P T }$ is able to select better operations that lead to higher accuracy of the supernet than DARTS $\mathrm { \Phi + P T }$ - Mag. + +on cifar10, much worse than DART $\mathsf { S } { + } \mathsf { P T }$ $( 2 . 6 1 \% )$ , indicating that the operation selection part plays a crucial role in our method. + +# 6.3 FIXING $\alpha$ AS UNIFORM + +Since the proposed method does not rely on $\alpha$ for architecture selection, a natural question is whether it is necessary to optimize a stand-alone $\alpha$ . We find that by fixing $\alpha = 0$ (uniform weights for all the operations) while training supernet and applying perturbation-based architecture se + +Table 4: DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ v.s. DART $\mathrm { S } { + } \mathrm { P T }$ (fixed $\alpha$ ) on more spaces (test error $\%$ ) on cifar10. + +
SpaceDARTSDARTS+PTDARTS+PT (fix α)
DARTS Space3.002.612.87
NAS-Bench-20145.711.896.20
+ +lection, the resulting method performs on-par with DART $\mathrm { \Delta } \mathrm { S } { + } \mathrm { P T } ,$ , and in some cases even better. For example, DARTS $\mathrm { \Phi } _ { \mathrm { + P T } }$ (fix $\alpha$ ) achieves better performance than DARTS $\mathrm { \Phi + P T }$ on NAS-Bench-201. On DARTS’ search space and its variants S1-S4, DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ (fix $\alpha$ ) performs similarly to DART $\mathrm { \bf { S } } { + } \mathrm { \bf { P } } \mathrm { \bf { T } }$ . The results can be found in Table 3 and Table 4. This surprising finding suggests that even the most naive approach, simply training a supernet without $\alpha$ , will be a competitive method when combining with the proposed perturbation-based architecture selection. + +# 7 CONCLUSION AND DISCUSSION + +This paper attempts to understand Differentiable NAS methods from the architecture selection perspective. We re-examine the magnitude-based architecture selection process of DARTS and provide empirical and theoretical evidence on why it does not indicate the underlying operation strength. We introduce an alternative perturbation-based architecture selection method that directly measures the operation strength via its contribution to the supernet performance. The proposed selection method is able to consistently extract improved architecture from supernets trained identically to the respective base methods on several spaces and datasets. + +Our method brings more freedom in supernet training as it does not rely on $\alpha$ to derive the final architecture. 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URL http://dx.doi.org/10. 1109/CVPR.2018.00907. + +A APPENDIX + +A.1 DESCRIPTION ABOUT OUR BASELINE MODELS + +# A.1.1 DARTS + +DARTS (Liu et al., 2019) is a pioneering work that introduces the general differentiable NAS framework, which we reviewed in section 2. In DARTS, the topology and operation are searched together. Concretely, at the end of the search, it selects one operation for every edge in the normal (reduction) cell based on the architecture parameter $\alpha$ . Then it selects two input edges for each node in the cell by comparing the largest $\alpha$ of every input edge. The final architecture consists of one operation on each of the eight edges in the normal (reduction) cell. The operation on an edge will be selected from a pool of seven candidates: skip connection, avg pool ${ } _ { - 3 \mathrm { x } 3 }$ , max pool ${ } _ { - 3 \mathrm { x } 3 }$ , sep conv 3x3, sep conv 5x5, dil conv 3x3, and dil conv 5x5. In addition to these operations, DARTS also maintains a ”none” op, which is used exclusively for determining the topology rather than treated as an operation (Liu et al., 2019). Since the main focus of our paper is operation assignment, we omit none op when applying the proposed selection method on DARTS. + +# A.1.2 SDARTS + +SDARTS (Chen & Hsieh, 2020) is a variant of DARTS aiming at regularizing the bilevel optimization process in DARTS via random Gaussian perturbation, inspired by the recent finding that regularizing DARTS’ supernet leads to improved performance. While the optimization of architecture parameter $\alpha$ in SDARTS is identical to DARTS, it distorts the architecture parameters $\alpha$ with random Gaussian noise while training the model weights $w$ . This simple yet effective regularizer is able to consistently improve the robustness and performance of DARTS. + +# A.1.3 SGAS + +SGAS (Li et al., 2020) represents a line of research on improving the search efficiency of differentiable NAS by progressively shrinking the search space. It first trains the model weights $w$ alone for 10 epochs. After that, it selects one edge from the supernet and then selects the best operation on that edge based on $\alpha$ to be discretized. The edge selection is based on the ranking of the proposed edge importance score. The process stops after all eight edges of the final architecture are decided. + +# A.2 MICROARCHITECTURE OF SPACE S1 - S4 + +Zela et al. (2020) introduces four variants of the DARTS’ space (S1, S2, S3, and S4) to study the robustness of DARTS. These spaces differ from DARTS’ original space only in the number and types of operations on each edge. Apart from that, everything else is the same. + +• S1 is a pre-optimized search space consisting of top2 operations selected by DARTS. As a result, each edge contains a different set of operations to be searched from. • S2 consists of two operations: skip connect and sep conv 3x3. • S3 consists of three operations: none, skip connect and sep conv 3x3. • S4 consists of two operations: noise and sep conv 3x3. The noise operation outputs a random Gaussian noise $\mathcal { N } ( 0 , 1 )$ regardless of the input. This operation generally hurts the performance of discretized architecture, and should be avoided by NAS algorithm. + +# Algorithm 2: Perturbation-based Architecture Selection + +Input: A pretrained Supernet $S$ , Set of Edges $\mathcal { E }$ from $s$ , Set of Nodes $\mathcal { N }$ from $s$ +Result: Set of selected operations $\{ o _ { e } ^ { * } \} _ { e \in \mathcal { E } }$ , and top2 input edges for each node $\{ ( e _ { n } ^ { ( 1 ) * } , e _ { n } ^ { ( 2 ) * } \} _ { n \in \mathcal { N } }$ +while $| \mathcal { E } | > 0 \mathbf { d } \mathbf { 0 } \ / / \ \mathrm { o } $ peration selection phase randomly select an edge $e \in { \mathcal { E } }$ (and remove it from $\mathcal { E }$ ); forall operation o on edge $e$ do evaluate the validation accuracy of $S$ when $o$ is removed $( A C C _ { \backslash o } )$ ; end select the best operation on $e$ : $\begin{array} { r } { o _ { e } ^ { * } \gets \operatorname * { a r g m i n } _ { o } A C C _ { \backslash o } } \end{array}$ ; discretize edge $e$ to $o _ { e } ^ { * }$ and train the remaining supernet until it converges again; +end +while $| { \mathcal { N } } | > 0$ do $/ /$ topology selection phase randomly select a node $n \in \mathcal N$ (and remove it from $\mathcal { N }$ ); forall input edge e of node $n$ do evaluate the validation accuracy of $S$ when $e$ is removed $( A C C _ { \backslash e } )$ ; end set top2 edges on $n ( e _ { n } ^ { ( 1 ) * } , e _ { n } ^ { ( 2 ) * } )$ to be the ones with lowest and second lowest $A C C _ { \backslash e }$ ; prune out all other edges of $n$ and train the remaining supernet until it converges again; +end + +# A.4 PROOF OF PROPOSITION 1 + +Proof. Let $\theta _ { s k i p } = S o f t m a x ( \alpha _ { s k i p } )$ and $\theta _ { c o n v } = S o f t m a x ( \alpha _ { c o n v } )$ . Then the mixed operation can be written as $\overline { { m } } _ { e } ( x _ { e } ) = \theta _ { c o n v } o _ { e } ( x _ { e } ) + \theta _ { s k i p } x _ { e }$ . We formally formulate the objective to be: + +$$ +\begin{array} { r l } { \underset { \theta _ { s k i p } , \theta _ { c o n v } } { \operatorname* { m i n } } V a r ( \overline { { m } } _ { e } ( x _ { e } ) - m ^ { * } ) } & { { } } \\ { s . t . } & { { } \theta _ { s k i p } + \theta _ { c o n v } = 1 } \end{array} +$$ + +This constraint optimization problem can be solved with Lagrangian multipliers: + +$$ +\begin{array} { r l } { L ( \theta _ { s k i p } , \theta _ { c o n s } , \lambda ) = V a r ( \overline { { m } } _ { e } ( x _ { e } ) - m ^ { * } ) - \lambda ( \theta _ { s k i p } + \theta _ { c o n s } - 1 ) } \\ & { = V a r ( \theta _ { c o n s } \theta _ { e } ( x _ { e } ) + \theta _ { s k i p } x _ { e } - m ^ { * } ) - \lambda ( \theta _ { s k i p } + \theta _ { c o n v } - 1 ) } \\ & { = V a r ( \theta _ { c o n v } ( \theta _ { e } ( x _ { e } ) - m ^ { * } ) + \theta _ { s k i p } ( x _ { e } - m ^ { * } ) ) } \\ & { \quad - \lambda ( \theta _ { s k i p } + \theta _ { c o n v } - 1 ) } \\ & { = V a r ( \theta _ { c o n v } ( \theta _ { e } ( x _ { e } ) - m ^ { * } ) ) + V a r ( \theta _ { s k i p } ( x _ { e } - m ^ { * } ) ) } \\ & { \quad + 2 C o v ( \theta _ { c o n v } ( \theta _ { c o } ( x _ { e } ) - m ^ { * } ) , \theta _ { s k i p } ( x _ { e } - m ^ { * } ) ) } \\ & { \quad - \lambda ( \theta _ { s k i p } + \theta _ { c o n v } - 1 ) } \\ & { = \theta _ { c o n v } ^ { 2 } V a r ( \theta _ { c } ( x _ { e } ) - m ^ { * } ) + \theta _ { s k i p } ^ { 2 } V a r ( x _ { e } - m ^ { * } ) } \\ & { \quad + 2 \theta _ { c o n v } \theta _ { s k i p } C o v ( \theta _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } \\ & { \quad - \lambda ( \theta _ { s k i p } + \theta _ { c o n v } - 1 ) } \end{array} +$$ + +Setting: + +$$ +\begin{array} { c } { { \displaystyle \frac { \partial { \cal L } } { \partial \lambda } = \theta _ { c o n v } + \theta _ { s k i p } - 1 = 0 } } \\ { { \displaystyle \frac { \partial { \cal L } } { \partial \theta _ { c o n v } } = 2 \theta _ { c o n v } V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) + 2 \theta _ { s k i p } C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } } \\ { { - \lambda = 0 } } \\ { { \displaystyle \frac { \partial { \cal L } } { \partial \theta _ { s k i p } } = 2 \theta _ { c o n v } C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) + 2 \theta _ { s k i p } V a r ( x _ { e } - m ^ { * } ) } } \\ { { - \lambda = 0 } } \end{array} +$$ + +Solving the above equations will give us: + +$$ +\begin{array} { r } { \theta _ { c o n v } ^ { * } = \frac { V a r ( x _ { e } - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } { Z } } \\ { \theta _ { s k i p } ^ { * } = \frac { V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } { Z } } \end{array} +$$ + +Where $Z = V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) + V a r ( x _ { e } - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } )$ $m ^ { * } , x _ { e } - m ^ { * } )$ . Aligning basis with DARTS, we get: + +$$ +\begin{array} { r l } & { \alpha _ { c o n v } ^ { * } = \log \left[ V a r ( x _ { e } - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) \right] + C } \\ & { \alpha _ { s k i p } ^ { * } = \log \left[ V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) \right] + C } \end{array} +$$ + +The only term that differentiates $\alpha _ { s k i p }$ from $\alpha _ { c o n v }$ is the first term inside the logarithm, therefore: + +$$ +\begin{array} { l l } { \alpha _ { c o n v } ^ { * } \propto v a r ( x _ { e } - m ^ { * } ) } \\ { \alpha _ { s k i p } ^ { * } \propto v a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) } \end{array} +$$ + +# A.5 MORE FIGURES ON $\alpha$ AND DISCRETIZATION ACCURACY AT CONVERGENCE + +We provide extra figures similar to Figure 1 to take into account the randomness of supernet’s training. We first train 6 supernets with different seeds, and then randomly select 1 edge from each of them. We can see that the results are consistent with Figure 1. As shown in Figure 7, the magnitude of $\alpha$ for each operation does not necessarily agree with its relative discretization accuracy at convergence. + +![](images/4769e09f6f49a48005b1e0fe8c7f4c5167355953dbc8e51a55a7e2fb302fa270.jpg) +Figure 7: $\alpha$ v.s. discretization accuracy at convergence of all the operations on 6 randomly selected edges from DARTS’ supernet trained with different seeds (one subfigure for each edge). + +# A.6 PERFORMANCE TRAJECTORY OF DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ (FIX $\alpha$ ) ON NAS-BENCH-201 + +We plot the performance trajectory of DARTS $\therefore \mathrm { { P T } }$ (fix $\alpha$ ) on NAS-Bench-201 similar to Figure 4. As shown in Figure 8, it consistently achieves strong performance without training $\alpha$ at all, indicating that the extra freedom of supernet training without $\alpha$ can be explored to develop improved search algorithm in the future. + +![](images/3a8cc463ff98b77d226b10b964925ce1c6793d1d334bac2cc7482d37bf05baab.jpg) +Figure 8: Trajectory of test accuracy of architectures found by DART $\mathrm { S + P T }$ (fix $\alpha$ ) on space NAS Bench-201 and 3 datasets (Left: cifar10, Middle: cifar100, Right: Imagenet16-120). + +Table 5: Evaluation of the derived architecture on ImageNet in the mobile setting. + +
ArchitectureTest Error(%)Params (M)Search Method
top-1top-5
DARTS (Liu et al.,2019)26.78.74.7gradient
DARTS+PT25.58.04.6gradient
+ +# A.7 ABLATION STUDY ON THE NUMBER OF FINE-TUNING EPOCHS + +As described in section 4.2, we perform fine-tuning between two edge decisions to recover supernet from the accuracy drop after discretization. The number of fine-tuning epochs is set to 5 for all experiments because empirically we find that it is enough for the supernet to converge again. In this section, we conduct an ablation study on the effect of the number of fine-tuning epochs. As shown in Figure 9, the gain from tuning the supernet longer than 5 epochs is marginal. + +![](images/fe9e8f5e0a0bf1215f29118846a7a8ad9cbe04e487bea75cdec798872d0c79e5.jpg) +Figure 9: Performance of DARTS $+ \mathrm { P T }$ under different number of fine-tuning epochs on NAS-Bench201. Tuning the supernet longer results in marginal improvement on the proposed method. + +# A.8 ARCHITECTURE TRANSFERABILITY EVALUATION ON IMAGENET + +We further evaluate the performance of the derived architecture on ImageNet. We strictly follow the training protocals as well as the hyperparameter settings of DARTS (Liu et al., 2019) for this experiment. As shown in Table 5, the proposed method improves the top1 performance of DARTS by $1 . 2 \%$ . + +A.9 SEARCHED ARCHITECTURES + +![](images/ba6fb9451f32ea0dff3966a31009b1cbf94a107522565f61c27af9cf40613427.jpg) + +![](images/b18d222162af78be6d23ee560cdc672d2af15f5a33c95e49ebebababa6cec8a7.jpg) +Figure 11: Normal and Reduction cells discovered by SDARTS $\mathrm { \Phi + P T }$ on cifar10 + +![](images/b7a93ff110cd4f6914e6e6f08aca6ee9bd6d5ef8603780bf4a5a1327eed570de.jpg) +Figure 12: Normal and Reduction cells discovered by SGAS-PT on cifar10 + +![](images/30ec24fb20a517db7d1ab52fd5a2eb71b9e9bc84e6533896d7ae68b7ae0e1772.jpg) +Figure 13: Normal and Reduction cells discovered by DART $\mathrm { \Phi } _ { \mathrm { { + P T } } }$ on cifar10 on Space S1 + +![](images/c6ad613dc19587883600fb74625b38bdd4a59430fd55d9af92dcbd9a1dd69fe0.jpg) +Figure 14: Normal and Reduction cells discovered by DART $\mathrm { S + P T }$ on cifar10 on Space S2 + +![](images/00acd5ee54fc703993fadb077c732ffab60238b7157e8e2748054bc3bcdefdc5.jpg) +Figure 15: Normal and Reduction cells discovered by DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ on cifar10 on Space S3 + +![](images/235735e0f44865ab7324cae91246430c9e94d460774cc09630f20d6d4dfa43d5.jpg) +Figure 16: Normal and Reduction cells discovered by DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ on cifar10 on Space S4 + +![](images/09533ad4ab7c7b374db480b1fe2a0cbe7af449757c0c2d59123e77ffae1607b7.jpg) +Figure 17: Normal and Reduction cells discovered by DART $\mathrm { S + P T }$ on cifar100 on Space S1 + +![](images/004c922fac1299be31ace0dce05e7f12a23a9649584d78931b627713fc4aa282.jpg) +Figure 18: Normal and Reduction cells discovered by DART $\mathrm { \bf { S } } { + } \mathrm { \bf { P } } \mathrm { \bf { T } }$ on cifar100 on Space S2 + +![](images/622085a0604fde209f398b616ffeb7ea48221fa621782a74d7db85726aadc0bf.jpg) +Figure 19: Normal and Reduction cells discovered by DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ on cifar100 on Space S3 + +![](images/9023cc585111b9f761131ac24f8ad24e97a1c28f94d532927a03715974a98e0d.jpg) +Figure 20: Normal and Reduction cells discovered by DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ on cifar100 on Space S4 + +![](images/5fe05e2fccc2b40c05aaa469399108ac1ef910f1f13c9c992bbd5fcafca4df29.jpg) +Figure 21: Normal and Reduction cells discovered by DART $\mathrm { \bf { S } } { + } \mathrm { \bf { P } } \mathrm { \bf { T } }$ on svhn on Space S1 + +![](images/ebbe4d8de8ea38bed16167fd402d5681be1114fa488fddca8402bb9bb923efe7.jpg) +Figure 22: Normal and Reduction cells discovered by DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ on svhn on Space S2 + +![](images/ff2a5f93379ef0610a6416fba97ee7701b0b076b6239d08bacb840610bdc6a9f.jpg) +Figure 23: Normal and Reduction cells discovered by DART $\mathrm { \Phi } _ { \mathrm { { + P T } } }$ on svhn on Space S3 + +![](images/53bc7a1aaa5388c94d2d694b5fb2fa8d5e53784b7062958cbca1edabc7a5f76a.jpg) +Figure 24: Normal and Reduction cells discovered by DART $\mathrm { \bf S } { + } \mathrm { \bf P T }$ on svhn on Space S4 \ No newline at end of file diff --git a/parse/train/PKubaeJkw3/PKubaeJkw3_content_list.json b/parse/train/PKubaeJkw3/PKubaeJkw3_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d7cc097b40e42a639db27ef3124c971d3db02fff --- /dev/null +++ b/parse/train/PKubaeJkw3/PKubaeJkw3_content_list.json @@ -0,0 +1,2043 @@ +[ + { + "type": "text", + "text": "RETHINKING ARCHITECTURE SELECTION IN DIFFERENTIABLE NAS ", + "text_level": 1, + "bbox": [ + 174, + 99, + 820, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ruochen Wang1, Minhao Cheng1, Xiangning Chen1, Xiaocheng Tang2, Cho-Jui Hsieh1 \n1Department of Computer Science, UCLA, 2DiDi AI Labs \n{ruocwang, mhcheng}@ucla.edu {xiangning, chohsieh}@cs.ucla.edu \nxiaochengtang@didiglobal.com ", + "bbox": [ + 186, + 170, + 754, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 263, + 544, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Differentiable Neural Architecture Search is one of the most popular Neural Architecture Search (NAS) methods for its search efficiency and simplicity, accomplished by jointly optimizing the model weight and architecture parameters in a weight-sharing supernet via gradient-based algorithms. At the end of the search phase, the operations with the largest architecture parameters will be selected to form the final architecture, with the implicit assumption that the values of architecture parameters reflect the operation strength. While much has been discussed about the supernet’s optimization, the architecture selection process has received little attention. We provide empirical and theoretical analysis to show that the magnitude of architecture parameters does not necessarily indicate how much the operation contributes to the supernet’s performance. We propose an alternative perturbation-based architecture selection that directly measures each operation’s influence on the supernet. We re-evaluate several differentiable NAS methods with the proposed architecture selection and find that it is able to extract significantly improved architectures from the underlying supernets consistently. Furthermore, we find that several failure modes of DARTS can be greatly alleviated with the proposed selection method, indicating that much of the poor generalization observed in DARTS can be attributed to the failure of magnitude-based architecture selection rather than entirely the optimization of its supernet. ", + "bbox": [ + 233, + 295, + 764, + 559 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 585, + 336, + 602 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural Architecture Search (NAS) has been drawing increasing attention in both academia and industry for its potential to automatize the process of discovering high-performance architectures, which have long been handcrafted. Early works on NAS deploy Evolutionary Algorithm (Stanley & Miikkulainen, 2002; Real et al., 2017; Liu et al., 2017) and Reinforcement Learning (Zoph & Le, 2017; Pham et al., 2018; Zhong et al., 2018) to guide the architecture discovery process. Recently, several one-shot methods have been proposed that significantly improve the search efficiency (Brock et al., 2018; Guo et al., 2019; Bender et al., 2018). ", + "bbox": [ + 174, + 617, + 825, + 714 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As a particularly popular instance of one-shot methods, DARTS (Liu et al., 2019) enables the search process to be performed with a gradient-based optimizer in an end-to-end manner. It applies continuous relaxation that transforms the categorical choice of architectures into continuous architecture parameters $\\alpha$ . The resulting supernet can be optimized via gradient-based methods, and the operations associated with the largest architecture parameters are selected to form the final architecture. Despite its simplicity, several works cast doubt on the effectiveness of DARTS. For example, a simple randomized search (Li & Talwalkar, 2019) outperforms the original DARTS; Zela et al. (2020) observes that DARTS degenerates to networks filled with parametric-free operations such as the skip connection or even random noise, leading to the poor performance of the selected architecture. ", + "bbox": [ + 174, + 722, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While the majority of previous research attributes the failure of DARTS to its supernet optimization (Zela et al., 2020; Chen & Hsieh, 2020; Chen et al., 2021), little has been discussed about the validity of another important assumption: the value of α reflects the strength of the underlying operations. In this paper, we conduct an in-depth analysis of this problem. Surprisingly, we find that in many cases, $\\alpha$ does not really indicate the operation importance in a supernet. Firstly, the operation associated with larger $\\alpha$ does not necessarily result in higher validation accuracy after discretization. Secondly, as an important example, we show mathematically that the domination of skip connection observed in DARTS (i.e. $\\alpha _ { s k i p }$ becomes larger than other operations.) is in fact a reasonable outcome of the supernet’s optimization but becomes problematic when we rely on $\\alpha$ to select the best operation. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "If $\\alpha$ is not a good indicator of operation strength, how should we select the final architecture from a pretrained supernet? Our analysis indicates that the strength of each operation should be evaluated based on its contribution to the supernet performance instead. To this end, we propose an alternative perturbation-based architecture selection method. Given a pretrained supernet, the best operation on an edge is selected and discretized based on how much it perturbs the supernet accuracy; The final architecture is derived edge by edge, with fine-tuning in between so that the supernet remains converged for every operation decision. We re-evaluate several differentiable NAS methods (DARTS (Liu et al., 2019), SDARTS (Chen & Hsieh, 2020), SGAS (Li et al., 2020)) and show that the proposed selection method is able to consistently extract significantly improved architectures from the supernets than magnitude-based counterparts. Furthermore, we find that the robustness issues of DARTS can be greatly alleviated by replacing the magnitude-based selection with the proposed perturbation-based selection method. ", + "bbox": [ + 173, + 166, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND AND RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 363, + 504, + 378 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Preliminaries of Differentiable Architecture Search (DARTS) We start by reviewing the formulation of DARTS. DARTS’ search space consists of repetitions of cell-based microstructures. Every cell can be viewed as a DAG with N nodes and $\\mathrm { E }$ edges, where each node represents a latent feature map $x ^ { i }$ , and each edge is associated with an operation $o$ (e.g. skip connect, sep conv $3 x 3$ ) from the search space $\\mathcal { O }$ . Continuous relaxation is then applied to this search space. Concretely, every operation on an edge is activated during the search phase, with their outputs mixed by the architecture parameter α to form the final mixed output of that edge m¯ (xi) = Po∈O P exp αoo0 exp αo0 . This particular formulation allows the architecture search to be performed in a differentiable manner: DARTS jointly optimizes $\\alpha$ and model weight $w$ with the following bilevel objective via alternative gradient updates: ", + "bbox": [ + 173, + 398, + 825, + 542 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/46a32003b05afc1af4dae4e935b96e101afe441a49e7f8b2e02efba180101e64.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\alpha } \\mathcal { L } _ { v a l } ( w ^ { * } , \\alpha ) \\mathrm { s . t . } w ^ { * } = \\arg \\operatorname* { m i n } _ { w } \\mathcal { L } _ { t r a i n } ( w , \\alpha ) .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 556, + 687, + 582 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We refer to the continuous relaxed network used in the search phase as the supernet of DARTS. At the end of the search phase, the operation associated with the largest $\\alpha _ { o }$ on each edge will be selected from the supernet to form the final architecture. ", + "bbox": [ + 174, + 595, + 825, + 638 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Failure mode analysis of DARTS Several works cast doubt on the robustness of DARTS. Zela et al. (2020) tests DARTS on four different search spaces and observes significantly degenerated performance. They empirically find that the selected architectures perform poorly when DARTS’ supernet falls into high curvature areas of validation loss (captured by large dominant eigenvalues of the Hessian $\\nabla _ { \\alpha , \\alpha } ^ { 2 } \\mathcal { L } _ { v a l } ( w , \\alpha ) )$ . While Zela et al. (2020) relates this problem to the failure of supernet training in DARTS, we examine it from the architecture selection aspects of DARTS, and show that much of DARTS’ robustness issue can be alleviated by a better architecture selection method. ", + "bbox": [ + 173, + 661, + 825, + 761 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Progressive search space shrinking There is a line of research on NAS that focuses on reducing the search cost and aligning the model sizes of the search and evaluation phases via progressive search space shrinking (Liu et al., 2018; Li et al., 2019; Chen et al., 2021; Li et al., 2020). The general scheme of these methods is to prune out weak operations and edges sequentially during the search phase, based on the magnitude of $\\alpha$ following DARTS. Our method is orthogonal to them in this respect, since we select operations based on how much it contributes to the supernet’s performance rather than the $\\alpha$ value. Although we also discretize edges greedily and fine-tune the network in between, the purpose is to let the supernet recover from the loss of accuracy after discretization to accurately evaluate operation strength on the next edge, rather than to reduce the search cost. ", + "bbox": [ + 174, + 784, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 THE PITFALL OF MAGNITUDE-BASED ARCHITECTURE SELECTION IN DARTS ", + "text_level": 1, + "bbox": [ + 169, + 103, + 769, + 135 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we put forward the opinion that the architecture parameter $\\alpha$ does not necessarily represent the strength of the underlying operation in general, backed by both empirical and theoretical evidence. As an important example, we mathematically justify that the skip connection domination phenomenon observed in DARTS is reasonable by itself, and becomes problematic when combined with the magnitude-based architecture selection. ", + "bbox": [ + 173, + 151, + 825, + 220 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 $\\alpha$ MAY NOT REPRESENT THE OPERATION STRENGTH", + "text_level": 1, + "bbox": [ + 173, + 238, + 573, + 251 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/edb10b3c13ea4154d37633f3cbc076def08cf6dd2ef065dd8b384fd9afd746f5.jpg", + "image_caption": [ + "Figure 1: $\\alpha$ vs discretization accuracy at convergence of all operations on 3 randomly selected edges from a pretrained DARTS supernet (one subplot per edge). The magnitude of $\\alpha$ for each operation does not necessarily agree with its relative discretization accuracy at convergence. " + ], + "image_footnote": [], + "bbox": [ + 176, + 280, + 815, + 401 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/77e533953356028536aaa448409ba1cfccb71639465d6c5e104d9488c7bdad81.jpg", + "image_caption": [ + "Figure 2: Operation strength on each edge of S2 (skip connect, sep conv ${ _ { - 3 x 3 } }$ ). (a). Operations associated with the largest $\\alpha$ . (b). Operations that result in the highest discretization validation accuracy at convergence. Parameterized operations are marked red. " + ], + "image_footnote": [], + "bbox": [ + 176, + 474, + 821, + 590 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Following DARTS, existing differentiable NAS methods use the value of architecture parameters $\\alpha$ to select the final architecture from the supernet, with the implicit assumption that $\\alpha$ represents the strength of the underlying operations. In this section, we study the validity of this assumption in detail. ", + "bbox": [ + 174, + 652, + 825, + 708 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Consider one edge on a pretrained supernet; the strength of an operation on the edge can be naturally defined as the supernet accuracy after we discretize to this operation and fine-tune the remaining network until it converges again; we refer to this as ”discretization accuracy at convergence” for short. The operation that achieves the best discretization accuracy at convergence can be considered as the best operation for the given edge. Figure 1 shows the comparison of $\\alpha$ (blue) and operation strength (orange) of randomly select edges on DARTS supernet. As we can see, the magnitude of $\\alpha$ for each operation does not necessarily agree with their relative strength measured by discretization accuracy at convergence. Moreover, operations assigned with small $\\alpha \\mathbf { s }$ are sometimes strong ones that lead to high discretization accuracy at convergence. To further verify the mismatch, we investigate the operation strength on search space S2, where DARTS fails dramatically due to excessive skip connections (Zela et al., 2020). S2 is a variant of DARTS search space that only contains two operations per edge (skip connect, sep conv ${ _ { 3 x 3 } }$ ). Figure 2 shows the selected operations based on $\\alpha$ (left) and operation strength (right) on all edges on S2. From Figure 2a, we can see that αskip connect $> \\alpha _ { s e p . c o n v . 3 x 3 }$ on 12 of 14 edges. Consequently, the derived child architecture will lack representation ability and perform poorly due to too many skip connections. However, as shown in Figure 2b, the supernet benefits more from discretizing to sep conv 3x3 than skip connect on half of the edges. ", + "bbox": [ + 174, + 715, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 A CASE STUDY: SKIP CONNECTION ", + "text_level": 1, + "bbox": [ + 176, + 148, + 455, + 162 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Several works point out that DARTS tends to assign large $\\alpha$ to skip connections, resulting in shallow architectures with poor generability (Zela et al., 2020; Liang et al., 2019; Bi et al., 2019). This ”skip connection domination” issue is generally attributed to the failure of DARTS’ supernet optimization. In contrast, we draw inspiration from research on ResNet (He et al., 2016) and show that this phenomenon by itself is a reasonable outcome while DARTS refines its estimation of the optimal feature map, rendering $\\alpha _ { s k i p }$ ineffective in the architecture selection. ", + "bbox": [ + 173, + 174, + 825, + 258 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In vanilla networks (e.g., VGG), each layer computes a new level of feature map from the output feature map of the predecessor layer; thus, reordering layers at test time would dramatically hurt the performance (Veit et al., 2016). Unlike vanilla networks, Greff et al. (2017) and Veit et al. (2016) discover that successive layers in ResNet with compatible channel sizes are in fact estimating the same optimal feature map so that the outputs of these layers ", + "bbox": [ + 174, + 265, + 483, + 404 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/2e29115b52e9ffde6b2e20413ea5292ec2bfbf4730935c9f3371d0f2784c7d82.jpg", + "table_caption": [ + "Table 1: Test accuracy before and after layer (edge) shuffling on cifar10. For ResNet and VGG, we randomly swap two layers in each stage (defined as successive layers between two downsampling blocks. For DARTS supernet, we randomly swap two edges in every cell. " + ], + "table_footnote": [], + "table_body": "
VGGResNetDARTS
Before92.6993.8688.44
After9.83 ± 0.3383.2015 ± 2.0381.09 ± 1.87
", + "bbox": [ + 500, + 354, + 823, + 392 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "stay relatively close to each other at convergence; As a result, ResNet’s test accuracy remains robust under layer reordering. Greff et al. (2017) refers to this unique way of feature map estimation in ResNet as the ”unrolled estimation.” ", + "bbox": [ + 176, + 404, + 823, + 445 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "DARTS’ supernet resembles ResNet, rather than vanilla networks like VGG, in both appearance and behavior. Appearance-wise, within a cell of DARTS’ supernet, edges with skip connection are in direct correspondence with the successive residual layers in ResNet. Behavior-wise, DARTS’ supernet also exhibits a high degree of robustness under edge shuffling. As shown in Table 1, randomly reordering edges on a pretrained DARTS’ supernet at test time also has little effect on its performance. This evidence indicates that DARTS performs unrolled estimation like ResNet as well, i.e., edges within a cell share the same optimal feature map that they try to estimate. In the following proposition, we apply this finding and provide the optimal solution of $\\alpha$ in the sense of minimizing the variance of feature map estimation. ", + "bbox": [ + 173, + 452, + 825, + 578 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proposition 1. 1 Without loss of generality, consider one cell from a simplified search space consists of two operations: (skip, conv). Let $m ^ { * }$ denotes the optimal feature map, which is shared across all edges according to the unrolled estimation view (Greff et al., 2017). Let $o _ { e } ( x _ { e } )$ be the output of convolution operation, and let $x _ { e }$ be the skip connection (i.e., the input feature map of edge e). Assume $m ^ { * }$ , $o _ { e } ( x _ { e } )$ and $x _ { e }$ are normalized to the same scale. The current estimation of $m ^ { * }$ can then be written as: ", + "bbox": [ + 173, + 580, + 825, + 664 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/231bd00f2e5a4d6de9f7f318ca96bcfb35350202adc05b39603a2349c93e6e86.jpg", + "text": "$$\n\\overline { { { m } } } _ { e } ( x _ { e } ) = \\frac { \\exp ( \\alpha _ { c o n v } ) } { \\exp ( \\alpha _ { c o n v } ) + \\exp ( \\alpha _ { s k i p } ) } o _ { e } ( x _ { e } ) + \\frac { \\exp ( \\alpha _ { s k i p } ) } { \\exp ( \\alpha _ { c o n v } ) + \\exp ( \\alpha _ { s k i p } ) } x _ { e } ,\n$$", + "text_format": "latex", + "bbox": [ + 245, + 665, + 751, + 699 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\alpha _ { c o n v }$ and $\\alpha _ { s k i p }$ are the architecture parameters defined in DARTS. The optimal $\\alpha _ { c o n v } ^ { * }$ and $\\alpha _ { s k i p } ^ { * }$ minimizing v $a r \\mathbf { \\hat { ( } } \\overline { { m } } _ { e } ( x _ { e } ) - m ^ { * } )$ , the variance of the difference between the optimal feature map $m ^ { * }$ and its current estimation $\\overline { { m } } _ { e } ( x _ { e } )$ , are given by: ", + "bbox": [ + 173, + 702, + 825, + 747 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3e7e80bc404a78212c472e3135df580f736186dfa87731d59e12e6c1ef653a74.jpg", + "text": "$$\n\\begin{array} { l } { \\alpha _ { c o n v } ^ { * } \\propto v a r ( x _ { e } - m ^ { * } ) } \\\\ { \\alpha _ { s k i p } ^ { * } \\propto v a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 403, + 750, + 593, + 787 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We refer the reader to Appendix A.4 for detailed proof. From eq. (3) and eq. (4), we can see that the relative magnitudes of $\\alpha _ { s k i p }$ and $\\alpha _ { c o n v }$ come down to which one of $x _ { e }$ or $o _ { e } ( x _ { e } )$ is closer to $m ^ { * }$ in variance: ", + "bbox": [ + 174, + 795, + 825, + 837 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• $o _ { e } ( x _ { e } )$ is the output of a single convolution operation instead of the complete mixed output of edge $e$ , so it will deviate from $m ^ { * }$ even at convergence. ", + "bbox": [ + 210, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Therefore, in a well-optimized supernet, $x _ { e }$ will naturally be closer to $m ^ { * }$ than $o _ { e } ( x _ { e } )$ , causing $\\alpha _ { s k i p }$ to be greater than $\\alpha _ { c o n v }$ . ", + "bbox": [ + 174, + 142, + 821, + 171 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our analysis above indicates that the better the supernet, the larger the $( \\alpha _ { s k i p } - \\alpha _ { c o n v } )$ gap (softmaxed) will become since $x _ { e }$ gets closer and closer to $m ^ { * }$ as the supernet is optimized. This result is evidenced in Figure 3, where $m e a n ( \\alpha _ { s k i p } - \\alpha _ { c o n v } )$ continues to grow as the supernet gets better. In this case, although $\\alpha _ { s k i p } > \\alpha _ { c o n v }$ is reasonable by itself, it becomes an inductive bias to NAS if we were to select the final architecture based on $\\alpha$ . ", + "bbox": [ + 174, + 178, + 549, + 289 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 PERTURBATION-BASED ARCHITECTURE SELECTION ", + "text_level": 1, + "bbox": [ + 173, + 309, + 405, + 343 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/79a873ee67039a51da8c30bc3160564d6e0226f824e4320c9a3517871f5adb3b.jpg", + "image_caption": [ + "Figure 3: $m e a n ( \\alpha _ { s k i p } - \\alpha _ { c o n v } )$ (softmaxed) v.s. supernet’s validation accuracy. The gap of $( \\alpha _ { s k i p } - \\alpha _ { c o n v } )$ increases as supernet gets better. " + ], + "image_footnote": [], + "bbox": [ + 568, + 172, + 805, + 308 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Instead of relying on the $\\alpha$ value to select the best operation, we propose to directly evaluate operation strength in ", + "bbox": [ + 173, + 358, + 547, + 386 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "terms of its contribution to the supernet’s performance. The operation selection criterion is laid out in section 4.1. In section 4.2, we describe the entire architecture selection process. ", + "bbox": [ + 176, + 386, + 825, + 414 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 EVALUATING THE STRENGTH OF EACH OPERATION", + "text_level": 1, + "bbox": [ + 173, + 431, + 565, + 444 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In section 3.1, we define the strength of each operation on a given edge as how much it contributes to the performance of the supernet, measured by discretization accuracy. To avoid inaccurate evaluation due to large disturbance of the supernet during discretization, we fine-tune the remaining supernet until it converges again, and then compute its validation accuracy (discretization accuracy at convergence). The fine-tuning process needs to be carried out for evaluating each operation on an edge, leading to substantial computation costs. ", + "bbox": [ + 174, + 457, + 825, + 540 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To alleviate the computational overhead, we consider a more practical measure of operation strength: for each operation on a given edge, we mask it out while keeping all other operations, and re-evaluate the supernet. The one that results in the largest drop in the supernet’s validation accuracy will be considered as the most important operation on that edge. This alternative criterion incurs much less perturbation to the supernet than discretization since it only deletes one operation from the supernet at a time. As a result, the supernet’s validation accuracy after deletion stays close to the unmodified supernet, and thus it alleviates the requirement of tuning the remaining supernet to convergence. Therefore, we implement this measurement for the operation selection in this work. ", + "bbox": [ + 173, + 546, + 825, + 659 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Input: A pretrained supernet $S$ , Set of edges $\\mathcal { E }$ from $S$ , Set of nodes $\\mathcal { N }$ from $S$ \nResult: Set of selected operations $\\{ o _ { e } ^ { * } \\} _ { e \\in \\mathcal { E } }$ \nwhile $| \\mathcal { E } | > 0$ do randomly select an edge $e \\in { \\mathcal { E } }$ (and remove it from $\\mathcal { E }$ ); forall operation o on edge $e$ do evaluate the validation accuracy of $S$ when $o$ is removed $( A C C _ { \\backslash o } )$ ; end select the best operation for $e$ : $\\begin{array} { r } { o _ { e } ^ { * } \\gets \\operatorname * { a r g m i n } _ { o } A C C _ { \\backslash o } } \\end{array}$ ; discretize edge $e$ to $o _ { e } ^ { * }$ and tune the remaining supernet for a few epochs; \nend ", + "bbox": [ + 173, + 688, + 694, + 835 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 THE COMPLETE ARCHITECTURE SELECTION PROCESS", + "text_level": 1, + "bbox": [ + 174, + 856, + 584, + 869 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our method operates directly on top of DARTS’ pretrained supernet. Given a supernet, we randomly iterate over all of its edges. We evaluate each operation on an edge, and select the best one to be discretized based on the measurement described in section 4.1. After that, we tune the supernet for a few epochs to recover the accuracy lost during discretization. The above steps are repeated until all edges are decided. Algorithm 1 summarizes the operation selection process. The cell topology is decided in a similar fashion. We refer the reader to Appendix A.3 for the full algorithm, including deciding the cell topology. This simple method is termed ”perturbation-based architecture selection (PT)” in the following sections. ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 194, + 418, + 209 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we demonstrate that the perturbation-based architecture selection method is able to consistently find better architectures than those selected based on the values of $\\alpha$ . The evaluation is based on the search space of DARTS and NAS-Bench-201 (Dong & Yang, 2020), and we show that the perturbation-based architecture selection method can be applied to several variants of DARTS. ", + "bbox": [ + 174, + 226, + 825, + 281 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 RESULTS ON DARTS’ CNN SEARCH SPACE ", + "text_level": 1, + "bbox": [ + 174, + 299, + 517, + 313 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We keep all the search and retrain settings identical to DARTS since our method only modifies the architecture selection part. After the search phase, we perform perturbation-based architecture selection following Algorithm 1 on the pretrained supernet. We tune the supernet for 5 epochs between two selections as it is enough for the supernet to recover from the drop of accuracy after discretization. We run the search and architecture selection phase with four random seeds and report both the best and average test errors of the obtained architectures. ", + "bbox": [ + 174, + 324, + 825, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As shown in Table 2, the proposed method (DARTS $+ \\mathrm { P T }$ ) improves DARTS’ test error from $3 . 0 0 \\%$ to $2 . 6 1 \\%$ , with manageable search cost (0.8 GPU days). Note that by only changing the architecture selection method, DARTS performs significantly better than many other differentiable NAS methods that enjoy carefully designed optimization process of the supernet, such as GDAS (Dong & Yang, 2019) and SNAS (Xie et al., 2019). This empirical result suggests that architecture selection is crucial to DARTS: with the proper selection algorithm, DARTS remains a very competitive method. ", + "bbox": [ + 174, + 415, + 825, + 500 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our method is also able to improve the performance of other variants of DARTS. To show this, we evaluate our method on SDARTS(rs) and SGAS (Chen & Hsieh, 2020; Li et al., 2020). SDARTS(rs) is a variant of DARTS that regularizes the search phase by applying Gaussian perturbation to $\\alpha$ . Unlike DARTS and SDARTS, SGAS performs progressive search space shrinking. Concretely, SGAS progressively discretizes its edges with the order from most to least important, based on a novel edge importance score. For a fair comparison, we keep its unique search space shrinking process unmodified and only replace its magnitude-based operation selection with ours. As we can see from Table 2, our method consistently achieves better average test errors than its magnitudebased counterpart. Concretely, the proposed method improves SDARTS’ test error from $2 . 6 7 \\%$ to $2 . 5 4 \\%$ and SGAS’ test error from $2 . 6 6 \\%$ to $2 . 5 6 \\%$ . Moreover, the best architecture discovered in our experiments achieves a test error of $2 . 4 4 \\%$ , ranked top among other NAS methods. ", + "bbox": [ + 174, + 506, + 825, + 659 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/7d4b6638faec392f9f4d135d85de588ec6564e674791a16ea64c8e91e9c9bcca.jpg", + "image_caption": [ + "5.2 PERFORMANCE ON NAS-BENCH-201 SEARCH SPACE ", + "Figure 4: Trajectory of test accuracy on space NAS-Bench-201 and three datasets (Left: cifar10, Middle: cifar100, Right: Imagenet16-120). The test accuracy of our method is plotted by taking the snapshots of DARTS’ supernet at corresponding epochs and run our selection method on top of it. " + ], + "image_footnote": [], + "bbox": [ + 176, + 710, + 810, + 825 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To further verify the effectiveness of the proposed perturbation-based architecture selection, we conduct experiments on NAS-Bench-201. NAS-Bench-201 provides a unified cell-based search space similar to DARTS. Every architecture in the search space is trained under the same protocol on three datasets (cifar10, cifar100, and imagenet16-120), and their performance can be obtained by querying the database. As in section 5.1, we take the pretrained supernet from DARTS and apply our method on top of it. All other settings are kept unmodified. Figure 4 shows the performance trajectory of DARTS+PT compared with DARTS. While the architectures found by magnitude-based selection degenerates over time, the perturbation-based method is able to extract better architectures from the same underlying supernets stably. The result implies that the DARTS’ degenerated performance comes from the failure of magnitude based architecture selection. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/43c2f4febbef3dc135c3a0f6e3a9aa1924c8cf2e32fcf01dc1000bc5b83cffd0.jpg", + "table_caption": [ + "Table 2: Comparison with state-of-the-art image classifiers on CIFAR-10. " + ], + "table_footnote": [ + "† Obtained on a different space with PyramidNet (Han et al., 2017) as the backbone. ‡ Recorded on a single GTX 1080Ti GPU. ∗ Obtained by running the search and retrain phase under four different seeds and computing the average test error of the derived architectures. " + ], + "table_body": "
ArchitectureTest Error (%)Params (M)Search Cost (GPU days)Search Method
DenseNet-BC (Huang etal.,2017)3.4625.6-manual
NASNet-A (Zoph et al.,2018)2.653.32000RL
AmoebaNet-A (Real etal.,2019)3.34± 0.063.23150evolution
AmoebaNet-B (Real et al.,2019)2.55 ± 0.052.83150evolution
PNAS (Liu et al.,2018)*3.41 ± 0.093.2225SMBO
ENAS (Pham et al.,2018)2.894.60.5RL
NAONet (Luo et al.,2018)3.533.10.4NAO
SNAS (moderate) (Xie et al.,2019)2.85±0.022.81.5gradient
GDAS(Dong& Yang,2019)2.933.40.3gradient
BayesNAS (Zhou et al.,2019)2.81± 0.043.40.2gradient
ProxylessNAS (Cai et al.,2019)†2.085.74.0gradient
NASP(Yao et al.,2020)2.83±0.093.30.1gradient
P-DARTS (Chen et al.,2019)2.503.40.3gradient
PC-DARTS (Xu et al.,2020)2.57 ± 0.073.60.1gradient
R-DARTS (L2) Zela et al. (2020)2.95 ± 0.21-1.6gradient
DARTS (Liu et al.,2019)3.00±0.143.30.4gradient
SDARTS-RS (Chen& Hsieh,2020)2.67± 0.033.40.4gradient
SGAS(Cri 1.avg) (Li et al.,2020)2.66 ± 0.243.70.25gradient
DARTS+PT (avg)*2.61±0.083.00.8tgradient
DARTS+PT (best)2.483.30.8tgradient
SDARTS-RS+PT (avg)*2.54 ± 0.103.30.8t
SDARTS-RS+PT(best)2.443.20.8tgradient
SGAS+PT (Crit.1 avg)*2.56 ± 0.103.90.29tgradient gradient
SGAS+PT (Crit.1 best)2.463.90.29tgradient
", + "bbox": [ + 263, + 118, + 732, + 402 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 469, + 825, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 594, + 290, + 611 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 ISSUE WITH THE ROBUSTNESS OF DARTS ", + "text_level": 1, + "bbox": [ + 176, + 630, + 504, + 645 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Zela et al. (2020) observes that DARTS tends to yield degenerate architectures with abysmal performance. We conjecture that this robustness issue of DARTS can be explained by the failure of magnitude-based architecture selection. ", + "bbox": [ + 174, + 659, + 826, + 700 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To show this, we test DARTS’ performance with perturbation-based architecture selection on four spaces proposed by Zela et al. (2020) (S1-S4). The complete specifications of these spaces can be found in Appendix A.2. Given a supernet, the architecture selected based on $\\alpha$ performs poorly across spaces and datasets (column 3 in Table 3). However, our method is able to consistently extract meaningful architectures with significantly improved performance (Column 4 in Table 3). ", + "bbox": [ + 174, + 708, + 450, + 875 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/8527d8fe5556581230991e8c4fc0a25565223426e83a745d81503812896689cc.jpg", + "table_caption": [ + "Table 3: DARTS+PT on S1-S4 (test error $( \\% ) _ { . }$ " + ], + "table_footnote": [ + "? This column will be explained later in Section 6.3 " + ], + "table_body": "
DatasetSpaceDARTSDARTS+PT(Ours)DARTS+PT(fixα)*
C10S13.843.502.86
S24.852.792.59
S33.342.492.52
S47.202.642.58
C100S129.4624.4824.40
S226.0523.1623.30
S328.9022.03
S422.8520.8021.94 20.66
SVHNS14.582.622.39
S23.532.532.32
S33.412.422.32
S43.052.422.39
", + "bbox": [ + 467, + 718, + 823, + 845 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Notably, DART $\\mathrm { S + P T }$ is able to find meaningful architecture on S2 (skip connect, sep conv ${ } _ { - 3 x 3 }$ ) and S4 (noise, sep conv $3 x 3$ ), where DARTS failed dramatically. As shown in Figure 5, on S2, while magnitude-based selection degenerates to architectures filled with skip connections, DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ is able to find architecture with 4 convolutions; On S4, DART $\\mathrm { \\ s + P T }$ consistently favors sep conv 3x3 on edges where $\\alpha$ selects noise. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/6f35a1fec263ceaf499587658509937b12ee3ee4f90c4afd7b5d47144b0dc9a6.jpg", + "image_caption": [ + "Figure 5: Comparison of normal cells found on S2 and S4. The perturbation-based architecture selection (DARTS $+ \\mathrm { P T }$ ) is able to find reasonable architectures in cases where the magnitude-based method (DARTS) fails dramatically. The complete architecture can be found in Appendix A.9. Non-trivial operations are marked red. " + ], + "image_footnote": [], + "bbox": [ + 181, + 156, + 818, + 280 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 PROGRESSIVE TUNING ", + "text_level": 1, + "bbox": [ + 174, + 369, + 370, + 382 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In addition to operation selection, we also tune the supernet after an edge is discretized so that the supernet could regain the lost accuracy. To measure the effectiveness of our operation selection criterion alone, we conduct an ablation study on the progressive tuning part. Concretely, we test a baseline by combining progressive tuning with magnitude-based operation selection instead of our selection criterion, which we code-named DARTS $+$ PT-Mag. Figure 6 plots the change of validation accuracy of DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ and DARTS $+$ PT-Mag during the operation selection phase. As we can see, DART $\\mathsf { s } + \\mathsf { P T }$ is able to identify better operations that lead to higher validation accuracy than the magnitude-based alternative, revealing the effectiveness of our operation selection criteria. Moreover, DARTS+PT-Mag is only able to obtain a test error of $2 . 8 5 \\%$ on DARTS space ", + "bbox": [ + 174, + 398, + 483, + 660 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/7a67b64fd4ef3e701516214be42db891468961f0e71afdc6dc6fc9c6d6766051.jpg", + "image_caption": [ + "Figure 6: The trajectory of validation accuracy in the operation selection phase on S2. DARTS $+ \\mathrm { P T }$ is able to select better operations that lead to higher accuracy of the supernet than DARTS $\\mathrm { \\Phi + P T }$ - Mag. " + ], + "image_footnote": [], + "bbox": [ + 509, + 393, + 792, + 558 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "on cifar10, much worse than DART $\\mathsf { S } { + } \\mathsf { P T }$ $( 2 . 6 1 \\% )$ , indicating that the operation selection part plays a crucial role in our method. ", + "bbox": [ + 173, + 661, + 823, + 689 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.3 FIXING $\\alpha$ AS UNIFORM ", + "text_level": 1, + "bbox": [ + 176, + 714, + 375, + 728 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Since the proposed method does not rely on $\\alpha$ for architecture selection, a natural question is whether it is necessary to optimize a stand-alone $\\alpha$ . We find that by fixing $\\alpha = 0$ (uniform weights for all the operations) while training supernet and applying perturbation-based architecture se", + "bbox": [ + 174, + 743, + 450, + 840 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/986863bceecd8aab11108b0d57b9176c3ba55c1b1f812ac1d440d0666430f5a3.jpg", + "table_caption": [ + "Table 4: DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ v.s. DART $\\mathrm { S } { + } \\mathrm { P T }$ (fixed $\\alpha$ ) on more spaces (test error $\\%$ ) on cifar10. " + ], + "table_footnote": [], + "table_body": "
SpaceDARTSDARTS+PTDARTS+PT (fix α)
DARTS Space3.002.612.87
NAS-Bench-20145.711.896.20
", + "bbox": [ + 467, + 791, + 823, + 827 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "lection, the resulting method performs on-par with DART $\\mathrm { \\Delta } \\mathrm { S } { + } \\mathrm { P T } ,$ , and in some cases even better. For example, DARTS $\\mathrm { \\Phi } _ { \\mathrm { + P T } }$ (fix $\\alpha$ ) achieves better performance than DARTS $\\mathrm { \\Phi + P T }$ on NAS-Bench-201. On DARTS’ search space and its variants S1-S4, DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ (fix $\\alpha$ ) performs similarly to DART $\\mathrm { \\bf { S } } { + } \\mathrm { \\bf { P } } \\mathrm { \\bf { T } }$ . The results can be found in Table 3 and Table 4. This surprising finding suggests that even the most naive approach, simply training a supernet without $\\alpha$ , will be a competitive method when combining with the proposed perturbation-based architecture selection. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 CONCLUSION AND DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 465, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This paper attempts to understand Differentiable NAS methods from the architecture selection perspective. We re-examine the magnitude-based architecture selection process of DARTS and provide empirical and theoretical evidence on why it does not indicate the underlying operation strength. We introduce an alternative perturbation-based architecture selection method that directly measures the operation strength via its contribution to the supernet performance. The proposed selection method is able to consistently extract improved architecture from supernets trained identically to the respective base methods on several spaces and datasets. ", + "bbox": [ + 174, + 132, + 825, + 229 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our method brings more freedom in supernet training as it does not rely on $\\alpha$ to derive the final architecture. We hope the perturbation-based architecture selection can bring a new perspective to the NAS community to rethink the role of $\\alpha$ in Differential NAS. ", + "bbox": [ + 176, + 237, + 825, + 279 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENT ", + "text_level": 1, + "bbox": [ + 176, + 300, + 357, + 315 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work is supported in part by NSF under IIS-1901527, IIS-2008173, IIS-2048280 and by Army Research Laboratory under agreement number W911NF-20-2-0158. 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", + "bbox": [ + 174, + 516, + 825, + 571 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A APPENDIX ", + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 DESCRIPTION ABOUT OUR BASELINE MODELS ", + "bbox": [ + 174, + 143, + 537, + 157 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1.1 DARTS ", + "text_level": 1, + "bbox": [ + 176, + 174, + 287, + 189 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "DARTS (Liu et al., 2019) is a pioneering work that introduces the general differentiable NAS framework, which we reviewed in section 2. In DARTS, the topology and operation are searched together. Concretely, at the end of the search, it selects one operation for every edge in the normal (reduction) cell based on the architecture parameter $\\alpha$ . Then it selects two input edges for each node in the cell by comparing the largest $\\alpha$ of every input edge. The final architecture consists of one operation on each of the eight edges in the normal (reduction) cell. The operation on an edge will be selected from a pool of seven candidates: skip connection, avg pool ${ } _ { - 3 \\mathrm { x } 3 }$ , max pool ${ } _ { - 3 \\mathrm { x } 3 }$ , sep conv 3x3, sep conv 5x5, dil conv 3x3, and dil conv 5x5. In addition to these operations, DARTS also maintains a ”none” op, which is used exclusively for determining the topology rather than treated as an operation (Liu et al., 2019). Since the main focus of our paper is operation assignment, we omit none op when applying the proposed selection method on DARTS. ", + "bbox": [ + 174, + 207, + 825, + 358 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1.2 SDARTS ", + "text_level": 1, + "bbox": [ + 176, + 390, + 297, + 405 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "SDARTS (Chen & Hsieh, 2020) is a variant of DARTS aiming at regularizing the bilevel optimization process in DARTS via random Gaussian perturbation, inspired by the recent finding that regularizing DARTS’ supernet leads to improved performance. While the optimization of architecture parameter $\\alpha$ in SDARTS is identical to DARTS, it distorts the architecture parameters $\\alpha$ with random Gaussian noise while training the model weights $w$ . This simple yet effective regularizer is able to consistently improve the robustness and performance of DARTS. ", + "bbox": [ + 174, + 420, + 825, + 505 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1.3 SGAS ", + "text_level": 1, + "bbox": [ + 176, + 536, + 277, + 550 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "SGAS (Li et al., 2020) represents a line of research on improving the search efficiency of differentiable NAS by progressively shrinking the search space. It first trains the model weights $w$ alone for 10 epochs. After that, it selects one edge from the supernet and then selects the best operation on that edge based on $\\alpha$ to be discretized. The edge selection is based on the ranking of the proposed edge importance score. The process stops after all eight edges of the final architecture are decided. ", + "bbox": [ + 176, + 566, + 825, + 637 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 MICROARCHITECTURE OF SPACE S1 - S4 ", + "text_level": 1, + "bbox": [ + 174, + 670, + 501, + 684 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Zela et al. (2020) introduces four variants of the DARTS’ space (S1, S2, S3, and S4) to study the robustness of DARTS. These spaces differ from DARTS’ original space only in the number and types of operations on each edge. Apart from that, everything else is the same. ", + "bbox": [ + 176, + 702, + 825, + 744 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• S1 is a pre-optimized search space consisting of top2 operations selected by DARTS. As a result, each edge contains a different set of operations to be searched from. • S2 consists of two operations: skip connect and sep conv 3x3. • S3 consists of three operations: none, skip connect and sep conv 3x3. • S4 consists of two operations: noise and sep conv 3x3. The noise operation outputs a random Gaussian noise $\\mathcal { N } ( 0 , 1 )$ regardless of the input. This operation generally hurts the performance of discretized architecture, and should be avoided by NAS algorithm. ", + "bbox": [ + 210, + 767, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 2: Perturbation-based Architecture Selection ", + "text_level": 1, + "bbox": [ + 174, + 133, + 544, + 147 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Input: A pretrained Supernet $S$ , Set of Edges $\\mathcal { E }$ from $s$ , Set of Nodes $\\mathcal { N }$ from $s$ \nResult: Set of selected operations $\\{ o _ { e } ^ { * } \\} _ { e \\in \\mathcal { E } }$ , and top2 input edges for each node $\\{ ( e _ { n } ^ { ( 1 ) * } , e _ { n } ^ { ( 2 ) * } \\} _ { n \\in \\mathcal { N } }$ \nwhile $| \\mathcal { E } | > 0 \\mathbf { d } \\mathbf { 0 } \\ / / \\ \\mathrm { o } $ peration selection phase randomly select an edge $e \\in { \\mathcal { E } }$ (and remove it from $\\mathcal { E }$ ); forall operation o on edge $e$ do evaluate the validation accuracy of $S$ when $o$ is removed $( A C C _ { \\backslash o } )$ ; end select the best operation on $e$ : $\\begin{array} { r } { o _ { e } ^ { * } \\gets \\operatorname * { a r g m i n } _ { o } A C C _ { \\backslash o } } \\end{array}$ ; discretize edge $e$ to $o _ { e } ^ { * }$ and train the remaining supernet until it converges again; \nend \nwhile $| { \\mathcal { N } } | > 0$ do $/ /$ topology selection phase randomly select a node $n \\in \\mathcal N$ (and remove it from $\\mathcal { N }$ ); forall input edge e of node $n$ do evaluate the validation accuracy of $S$ when $e$ is removed $( A C C _ { \\backslash e } )$ ; end set top2 edges on $n ( e _ { n } ^ { ( 1 ) * } , e _ { n } ^ { ( 2 ) * } )$ to be the ones with lowest and second lowest $A C C _ { \\backslash e }$ ; prune out all other edges of $n$ and train the remaining supernet until it converges again; \nend ", + "bbox": [ + 171, + 141, + 805, + 433 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.4 PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 176, + 454, + 400, + 469 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. Let $\\theta _ { s k i p } = S o f t m a x ( \\alpha _ { s k i p } )$ and $\\theta _ { c o n v } = S o f t m a x ( \\alpha _ { c o n v } )$ . Then the mixed operation can be written as $\\overline { { m } } _ { e } ( x _ { e } ) = \\theta _ { c o n v } o _ { e } ( x _ { e } ) + \\theta _ { s k i p } x _ { e }$ . We formally formulate the objective to be: ", + "bbox": [ + 173, + 479, + 823, + 510 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7f11cd93810649a5aa5e41ed8d01b97cd31c1718108e5c10b5437d3fd7954410.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { \\theta _ { s k i p } , \\theta _ { c o n v } } { \\operatorname* { m i n } } V a r ( \\overline { { m } } _ { e } ( x _ { e } ) - m ^ { * } ) } & { { } } \\\\ { s . t . } & { { } \\theta _ { s k i p } + \\theta _ { c o n v } = 1 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 397, + 512, + 602, + 558 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This constraint optimization problem can be solved with Lagrangian multipliers: ", + "bbox": [ + 173, + 560, + 700, + 575 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/2e5f9d8153572b80275ad079b48472cf9050dadadd602a499d5aed6cf1fd6932.jpg", + "text": "$$\n\\begin{array} { r l } { L ( \\theta _ { s k i p } , \\theta _ { c o n s } , \\lambda ) = V a r ( \\overline { { m } } _ { e } ( x _ { e } ) - m ^ { * } ) - \\lambda ( \\theta _ { s k i p } + \\theta _ { c o n s } - 1 ) } \\\\ & { = V a r ( \\theta _ { c o n s } \\theta _ { e } ( x _ { e } ) + \\theta _ { s k i p } x _ { e } - m ^ { * } ) - \\lambda ( \\theta _ { s k i p } + \\theta _ { c o n v } - 1 ) } \\\\ & { = V a r ( \\theta _ { c o n v } ( \\theta _ { e } ( x _ { e } ) - m ^ { * } ) + \\theta _ { s k i p } ( x _ { e } - m ^ { * } ) ) } \\\\ & { \\quad - \\lambda ( \\theta _ { s k i p } + \\theta _ { c o n v } - 1 ) } \\\\ & { = V a r ( \\theta _ { c o n v } ( \\theta _ { e } ( x _ { e } ) - m ^ { * } ) ) + V a r ( \\theta _ { s k i p } ( x _ { e } - m ^ { * } ) ) } \\\\ & { \\quad + 2 C o v ( \\theta _ { c o n v } ( \\theta _ { c o } ( x _ { e } ) - m ^ { * } ) , \\theta _ { s k i p } ( x _ { e } - m ^ { * } ) ) } \\\\ & { \\quad - \\lambda ( \\theta _ { s k i p } + \\theta _ { c o n v } - 1 ) } \\\\ & { = \\theta _ { c o n v } ^ { 2 } V a r ( \\theta _ { c } ( x _ { e } ) - m ^ { * } ) + \\theta _ { s k i p } ^ { 2 } V a r ( x _ { e } - m ^ { * } ) } \\\\ & { \\quad + 2 \\theta _ { c o n v } \\theta _ { s k i p } C o v ( \\theta _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } \\\\ & { \\quad - \\lambda ( \\theta _ { s k i p } + \\theta _ { c o n v } - 1 ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 577, + 767, + 762 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Setting: ", + "bbox": [ + 173, + 768, + 227, + 784 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/8b66ff3e7032889cdbf0f53adea192d144b65b09ea02cd24625f16c7b782940e.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac { \\partial { \\cal L } } { \\partial \\lambda } = \\theta _ { c o n v } + \\theta _ { s k i p } - 1 = 0 } } \\\\ { { \\displaystyle \\frac { \\partial { \\cal L } } { \\partial \\theta _ { c o n v } } = 2 \\theta _ { c o n v } V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) + 2 \\theta _ { s k i p } C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } } \\\\ { { - \\lambda = 0 } } \\\\ { { \\displaystyle \\frac { \\partial { \\cal L } } { \\partial \\theta _ { s k i p } } = 2 \\theta _ { c o n v } C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) + 2 \\theta _ { s k i p } V a r ( x _ { e } - m ^ { * } ) } } \\\\ { { - \\lambda = 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 787, + 748, + 922 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Solving the above equations will give us: ", + "bbox": [ + 174, + 103, + 444, + 118 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a12f6755d926a8eb707d04d1f2b0b7d4710704c7ba35210040c8c90a2b84c720.jpg", + "text": "$$\n\\begin{array} { r } { \\theta _ { c o n v } ^ { * } = \\frac { V a r ( x _ { e } - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } { Z } } \\\\ { \\theta _ { s k i p } ^ { * } = \\frac { V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) } { Z } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 140, + 699, + 207 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Where $Z = V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) + V a r ( x _ { e } - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } )$ $m ^ { * } , x _ { e } - m ^ { * } )$ . Aligning basis with DARTS, we get: ", + "bbox": [ + 171, + 215, + 823, + 244 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/486844352645efed38db76fc2a4baedc2b70e6313fb2d4ab7fb698fe91ff7933.jpg", + "text": "$$\n\\begin{array} { r l } & { \\alpha _ { c o n v } ^ { * } = \\log \\left[ V a r ( x _ { e } - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) \\right] + C } \\\\ & { \\alpha _ { s k i p } ^ { * } = \\log \\left[ V a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) - C o v ( o _ { e } ( x _ { e } ) - m ^ { * } , x _ { e } - m ^ { * } ) \\right] + C } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 247, + 736, + 291 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The only term that differentiates $\\alpha _ { s k i p }$ from $\\alpha _ { c o n v }$ is the first term inside the logarithm, therefore: ", + "bbox": [ + 174, + 299, + 813, + 314 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c54cf0da455eb70f2ef0ac20a78d536b43daccf150490adbc30c888c56851de4.jpg", + "text": "$$\n\\begin{array} { l l } { \\alpha _ { c o n v } ^ { * } \\propto v a r ( x _ { e } - m ^ { * } ) } \\\\ { \\alpha _ { s k i p } ^ { * } \\propto v a r ( o _ { e } ( x _ { e } ) - m ^ { * } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 405, + 318, + 591, + 356 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.5 MORE FIGURES ON $\\alpha$ AND DISCRETIZATION ACCURACY AT CONVERGENCE ", + "text_level": 1, + "bbox": [ + 176, + 395, + 741, + 411 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We provide extra figures similar to Figure 1 to take into account the randomness of supernet’s training. We first train 6 supernets with different seeds, and then randomly select 1 edge from each of them. We can see that the results are consistent with Figure 1. As shown in Figure 7, the magnitude of $\\alpha$ for each operation does not necessarily agree with its relative discretization accuracy at convergence. ", + "bbox": [ + 173, + 421, + 825, + 492 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/4769e09f6f49a48005b1e0fe8c7f4c5167355953dbc8e51a55a7e2fb302fa270.jpg", + "image_caption": [ + "Figure 7: $\\alpha$ v.s. discretization accuracy at convergence of all the operations on 6 randomly selected edges from DARTS’ supernet trained with different seeds (one subfigure for each edge). " + ], + "image_footnote": [], + "bbox": [ + 174, + 517, + 820, + 785 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.6 PERFORMANCE TRAJECTORY OF DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ (FIX $\\alpha$ ) ON NAS-BENCH-201 ", + "text_level": 1, + "bbox": [ + 174, + 840, + 740, + 857 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We plot the performance trajectory of DARTS $\\therefore \\mathrm { { P T } }$ (fix $\\alpha$ ) on NAS-Bench-201 similar to Figure 4. As shown in Figure 8, it consistently achieves strong performance without training $\\alpha$ at all, indicating that the extra freedom of supernet training without $\\alpha$ can be explored to develop improved search algorithm in the future. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/3a8cc463ff98b77d226b10b964925ce1c6793d1d334bac2cc7482d37bf05baab.jpg", + "image_caption": [ + "Figure 8: Trajectory of test accuracy of architectures found by DART $\\mathrm { S + P T }$ (fix $\\alpha$ ) on space NAS Bench-201 and 3 datasets (Left: cifar10, Middle: cifar100, Right: Imagenet16-120). " + ], + "image_footnote": [], + "bbox": [ + 176, + 106, + 812, + 220 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/be87d9249d976f75e0c33d3f9eb203ce89c5dfbdb69772bd1b559c164b204dc5.jpg", + "table_caption": [ + "Table 5: Evaluation of the derived architecture on ImageNet in the mobile setting. " + ], + "table_footnote": [], + "table_body": "
ArchitectureTest Error(%)Params (M)Search Method
top-1top-5
DARTS (Liu et al.,2019)26.78.74.7gradient
DARTS+PT25.58.04.6gradient
", + "bbox": [ + 282, + 276, + 709, + 332 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.7 ABLATION STUDY ON THE NUMBER OF FINE-TUNING EPOCHS ", + "text_level": 1, + "bbox": [ + 174, + 347, + 650, + 362 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "As described in section 4.2, we perform fine-tuning between two edge decisions to recover supernet from the accuracy drop after discretization. The number of fine-tuning epochs is set to 5 for all experiments because empirically we find that it is enough for the supernet to converge again. In this section, we conduct an ablation study on the effect of the number of fine-tuning epochs. As shown in Figure 9, the gain from tuning the supernet longer than 5 epochs is marginal. ", + "bbox": [ + 173, + 373, + 826, + 444 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/fe9e8f5e0a0bf1215f29118846a7a8ad9cbe04e487bea75cdec798872d0c79e5.jpg", + "image_caption": [ + "Figure 9: Performance of DARTS $+ \\mathrm { P T }$ under different number of fine-tuning epochs on NAS-Bench201. Tuning the supernet longer results in marginal improvement on the proposed method. " + ], + "image_footnote": [], + "bbox": [ + 352, + 467, + 622, + 626 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.8 ARCHITECTURE TRANSFERABILITY EVALUATION ON IMAGENET ", + "text_level": 1, + "bbox": [ + 176, + 683, + 669, + 698 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We further evaluate the performance of the derived architecture on ImageNet. We strictly follow the training protocals as well as the hyperparameter settings of DARTS (Liu et al., 2019) for this experiment. As shown in Table 5, the proposed method improves the top1 performance of DARTS by $1 . 2 \\%$ . ", + "bbox": [ + 174, + 709, + 825, + 765 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.9 SEARCHED ARCHITECTURES ", + "bbox": [ + 176, + 782, + 419, + 796 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/ba6fb9451f32ea0dff3966a31009b1cbf94a107522565f61c27af9cf40613427.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 192, + 113, + 810, + 227 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/b18d222162af78be6d23ee560cdc672d2af15f5a33c95e49ebebababa6cec8a7.jpg", + "image_caption": [ + "Figure 11: Normal and Reduction cells discovered by SDARTS $\\mathrm { \\Phi + P T }$ on cifar10 " + ], + "image_footnote": [], + "bbox": [ + 187, + 261, + 808, + 348 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/b7a93ff110cd4f6914e6e6f08aca6ee9bd6d5ef8603780bf4a5a1327eed570de.jpg", + "image_caption": [ + "Figure 12: Normal and Reduction cells discovered by SGAS-PT on cifar10 " + ], + "image_footnote": [], + "bbox": [ + 192, + 388, + 808, + 491 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/30ec24fb20a517db7d1ab52fd5a2eb71b9e9bc84e6533896d7ae68b7ae0e1772.jpg", + "image_caption": [ + "Figure 13: Normal and Reduction cells discovered by DART $\\mathrm { \\Phi } _ { \\mathrm { { + P T } } }$ on cifar10 on Space S1 " + ], + "image_footnote": [], + "bbox": [ + 187, + 540, + 807, + 614 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/c6ad613dc19587883600fb74625b38bdd4a59430fd55d9af92dcbd9a1dd69fe0.jpg", + "image_caption": [ + "Figure 14: Normal and Reduction cells discovered by DART $\\mathrm { S + P T }$ on cifar10 on Space S2 " + ], + "image_footnote": [], + "bbox": [ + 187, + 662, + 807, + 738 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/00acd5ee54fc703993fadb077c732ffab60238b7157e8e2748054bc3bcdefdc5.jpg", + "image_caption": [ + "Figure 15: Normal and Reduction cells discovered by DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ on cifar10 on Space S3 " + ], + "image_footnote": [], + "bbox": [ + 194, + 780, + 810, + 900 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/235735e0f44865ab7324cae91246430c9e94d460774cc09630f20d6d4dfa43d5.jpg", + "image_caption": [ + "Figure 16: Normal and Reduction cells discovered by DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ on cifar10 on Space S4 " + ], + "image_footnote": [], + "bbox": [ + 192, + 136, + 805, + 258 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/09533ad4ab7c7b374db480b1fe2a0cbe7af449757c0c2d59123e77ffae1607b7.jpg", + "image_caption": [ + "Figure 17: Normal and Reduction cells discovered by DART $\\mathrm { S + P T }$ on cifar100 on Space S1 " + ], + "image_footnote": [], + "bbox": [ + 189, + 325, + 808, + 424 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/004c922fac1299be31ace0dce05e7f12a23a9649584d78931b627713fc4aa282.jpg", + "image_caption": [ + "Figure 18: Normal and Reduction cells discovered by DART $\\mathrm { \\bf { S } } { + } \\mathrm { \\bf { P } } \\mathrm { \\bf { T } }$ on cifar100 on Space S2 " + ], + "image_footnote": [], + "bbox": [ + 186, + 468, + 810, + 589 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/622085a0604fde209f398b616ffeb7ea48221fa621782a74d7db85726aadc0bf.jpg", + "image_caption": [ + "Figure 19: Normal and Reduction cells discovered by DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ on cifar100 on Space S3 " + ], + "image_footnote": [], + "bbox": [ + 202, + 647, + 810, + 878 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/9023cc585111b9f761131ac24f8ad24e97a1c28f94d532927a03715974a98e0d.jpg", + "image_caption": [ + "Figure 20: Normal and Reduction cells discovered by DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ on cifar100 on Space S4 " + ], + "image_footnote": [], + "bbox": [ + 200, + 138, + 795, + 349 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/5fe05e2fccc2b40c05aaa469399108ac1ef910f1f13c9c992bbd5fcafca4df29.jpg", + "image_caption": [ + "Figure 21: Normal and Reduction cells discovered by DART $\\mathrm { \\bf { S } } { + } \\mathrm { \\bf { P } } \\mathrm { \\bf { T } }$ on svhn on Space S1 " + ], + "image_footnote": [], + "bbox": [ + 187, + 409, + 810, + 473 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/ebbe4d8de8ea38bed16167fd402d5681be1114fa488fddca8402bb9bb923efe7.jpg", + "image_caption": [ + "Figure 22: Normal and Reduction cells discovered by DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ on svhn on Space S2 " + ], + "image_footnote": [], + "bbox": [ + 187, + 535, + 807, + 621 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/ff2a5f93379ef0610a6416fba97ee7701b0b076b6239d08bacb840610bdc6a9f.jpg", + "image_caption": [ + "Figure 23: Normal and Reduction cells discovered by DART $\\mathrm { \\Phi } _ { \\mathrm { { + P T } } }$ on svhn on Space S3 " + ], + "image_footnote": [], + "bbox": [ + 191, + 669, + 810, + 750 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/53bc7a1aaa5388c94d2d694b5fb2fa8d5e53784b7062958cbca1edabc7a5f76a.jpg", + "image_caption": [ + "Figure 24: Normal and Reduction cells discovered by DART $\\mathrm { \\bf S } { + } \\mathrm { \\bf P T }$ on svhn on Space S4 " + ], + "image_footnote": [], + "bbox": [ + 191, + 808, + 807, + 895 + ], + "page_idx": 17 + } +] \ No newline at end of file diff --git a/parse/train/PKubaeJkw3/PKubaeJkw3_middle.json b/parse/train/PKubaeJkw3/PKubaeJkw3_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..31c97027da34d84f2662c491db6bd4df5cfb1eae --- /dev/null +++ b/parse/train/PKubaeJkw3/PKubaeJkw3_middle.json @@ -0,0 +1,47186 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 502, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 97 + ], + "spans": [ + { + 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We propose an alternative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "score": 1.0, + "content": "perturbation-based architecture selection that directly measures each operation’s", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 366, + 470, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 470, + 378 + ], + "score": 1.0, + "content": "influence on the supernet. We re-evaluate several differentiable NAS methods with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 378, + 469, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 469, + 389 + ], + "score": 1.0, + "content": "the proposed architecture selection and find that it is able to extract significantly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 470, + 400 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 400 + ], + "score": 1.0, + "content": "improved architectures from the underlying supernets consistently. Furthermore,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 398, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 142, + 398, + 469, + 411 + ], + "score": 1.0, + "content": "we find that several failure modes of DARTS can be greatly alleviated with the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 469, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 469, + 422 + ], + "score": 1.0, + "content": "proposed selection method, indicating that much of the poor generalization ob-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 421, + 469, + 433 + ], + "spans": [ + { + "bbox": [ + 141, + 421, + 469, + 433 + ], + "score": 1.0, + "content": "served in DARTS can be attributed to the failure of magnitude-based architecture", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 432, + 387, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 432, + 387, + 444 + ], + "score": 1.0, + "content": "selection rather than entirely the optimization of its supernet.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 234, + 470, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 464, + 206, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 208, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 208, + 480 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "Neural Architecture Search (NAS) has been drawing increasing attention in both academia and in-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "dustry for its potential to automatize the process of discovering high-performance architectures,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "which have long been handcrafted. Early works on NAS deploy Evolutionary Algorithm (Stanley", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "& Miikkulainen, 2002; Real et al., 2017; Liu et al., 2017) and Reinforcement Learning (Zoph & Le,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "2017; Pham et al., 2018; Zhong et al., 2018) to guide the architecture discovery process. Recently,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "several one-shot methods have been proposed that significantly improve the search efficiency (Brock", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 556, + 309, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 309, + 567 + ], + "score": 1.0, + "content": "et al., 2018; Guo et al., 2019; Bender et al., 2018).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 488, + 506, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "As a particularly popular instance of one-shot methods, DARTS (Liu et al., 2019) enables the search", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "process to be performed with a gradient-based optimizer in an end-to-end manner. It applies contin-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "uous relaxation that transforms the categorical choice of architectures into continuous architecture", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 153, + 619 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 153, + 607, + 161, + 615 + ], + "score": 0.67, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 604, + 506, + 619 + ], + "score": 1.0, + "content": ". The resulting supernet can be optimized via gradient-based methods, and the opera-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "tions associated with the largest architecture parameters are selected to form the final architecture.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "Despite its simplicity, several works cast doubt on the effectiveness of DARTS. For example, a sim-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "ple randomized search (Li & Talwalkar, 2019) outperforms the original DARTS; Zela et al. (2020)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "observes that DARTS degenerates to networks filled with parametric-free operations such as the skip", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 486, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 486, + 673 + ], + "score": 1.0, + "content": "connection or even random noise, leading to the poor performance of the selected architecture.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 572, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "While the majority of previous research attributes the failure of DARTS to its supernet optimization", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "(Zela et al., 2020; Chen & Hsieh, 2020; Chen et al., 2021), little has been discussed about the validity", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "of another important assumption: the value of α reflects the strength of the underlying operations. In", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "this paper, we conduct an in-depth analysis of this problem. Surprisingly, we find that in many cases,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 722, + 114, + 730 + ], + "score": 0.7, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "does not really indicate the operation importance in a supernet. Firstly, the operation associated", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 151, + 95 + ], + "score": 1.0, + "content": "with larger", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 152, + 84, + 160, + 92 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 160, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "does not necessarily result in higher validation accuracy after discretization. Secondly,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "as an important example, we show mathematically that the domination of skip connection observed", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 170, + 118 + ], + "score": 1.0, + "content": "in DARTS (i.e.", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 171, + 106, + 194, + 117 + ], + "score": 0.88, + "content": "\\alpha _ { s k i p }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 194, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "becomes larger than other operations.) is in fact a reasonable outcome of the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 493, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 373, + 128 + ], + "score": 1.0, + "content": "supernet’s optimization but becomes problematic when we rely on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 374, + 117, + 381, + 125 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 381, + 116, + 493, + 128 + ], + "score": 1.0, + "content": "to select the best operation.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 676, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 151, + 95 + ], + "score": 1.0, + "content": "with larger", + "type": "text" + }, + { + "bbox": [ + 152, + 84, + 160, + 92 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "does not necessarily result in higher validation accuracy after discretization. Secondly,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "as an important example, we show mathematically that the domination of skip connection observed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 170, + 118 + ], + "score": 1.0, + "content": "in DARTS (i.e.", + "type": "text" + }, + { + "bbox": [ + 171, + 106, + 194, + 117 + ], + "score": 0.88, + "content": "\\alpha _ { s k i p }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "becomes larger than other operations.) is in fact a reasonable outcome of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 493, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 373, + 128 + ], + "score": 1.0, + "content": "supernet’s optimization but becomes problematic when we rely on", + "type": "text" + }, + { + "bbox": [ + 374, + 117, + 381, + 125 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 116, + 493, + 128 + ], + "score": 1.0, + "content": "to select the best operation.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 115, + 144 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 134, + 124, + 142 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "is not a good indicator of operation strength, how should we select the final architecture from a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "pretrained supernet? Our analysis indicates that the strength of each operation should be evaluated", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "based on its contribution to the supernet performance instead. To this end, we propose an alternative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 104, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "perturbation-based architecture selection method. Given a pretrained supernet, the best operation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "on an edge is selected and discretized based on how much it perturbs the supernet accuracy; The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "final architecture is derived edge by edge, with fine-tuning in between so that the supernet remains", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 209 + ], + "score": 1.0, + "content": "converged for every operation decision. We re-evaluate several differentiable NAS methods (DARTS", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "(Liu et al., 2019), SDARTS (Chen & Hsieh, 2020), SGAS (Li et al., 2020)) and show that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "proposed selection method is able to consistently extract significantly improved architectures from", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "the supernets than magnitude-based counterparts. Furthermore, we find that the robustness issues", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "of DARTS can be greatly alleviated by replacing the magnitude-based selection with the proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 257, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 257, + 264 + ], + "score": 1.0, + "content": "perturbation-based selection method.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 288, + 309, + 300 + ], + "lines": [ + { + "bbox": [ + 104, + 286, + 312, + 302 + ], + "spans": [ + { + "bbox": [ + 104, + 286, + 312, + 302 + ], + "score": 1.0, + "content": "2 BACKGROUND AND RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "Preliminaries of Differentiable Architecture Search (DARTS) We start by reviewing the for-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "mulation of DARTS. DARTS’ search space consists of repetitions of cell-based microstructures.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 324, + 351 + ], + "score": 1.0, + "content": "Every cell can be viewed as a DAG with N nodes and", + "type": "text" + }, + { + "bbox": [ + 324, + 339, + 332, + 349 + ], + "score": 0.32, + "content": "\\mathrm { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "edges, where each node represents a latent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 157, + 363 + ], + "score": 1.0, + "content": "feature map", + "type": "text" + }, + { + "bbox": [ + 157, + 349, + 167, + 360 + ], + "score": 0.86, + "content": "x ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 349, + 360, + 363 + ], + "score": 1.0, + "content": ", and each edge is associated with an operation", + "type": "text" + }, + { + "bbox": [ + 360, + 352, + 366, + 360 + ], + "score": 0.62, + "content": "o", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 349, + 483, + 363 + ], + "score": 1.0, + "content": "(e.g. skip connect, sep conv", + "type": "text" + }, + { + "bbox": [ + 483, + 350, + 501, + 361 + ], + "score": 0.32, + "content": "3 x 3", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 349, + 505, + 363 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 198, + 374 + ], + "score": 1.0, + "content": "from the search space", + "type": "text" + }, + { + "bbox": [ + 199, + 361, + 208, + 371 + ], + "score": 0.69, + "content": "\\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 361, + 505, + 374 + ], + "score": 1.0, + "content": ". Continuous relaxation is then applied to this search space. Concretely,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "every operation on an edge is activated during the search phase, with their outputs mixed by the ar-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 102, + 378, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 102, + 378, + 482, + 401 + ], + "score": 1.0, + "content": "chitecture parameter α to form the final mixed output of that edge m¯ (xi) = Po∈O P exp αoo0 exp αo0", + "type": "text" + }, + { + "bbox": [ + 501, + 382, + 505, + 396 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "This particular formulation allows the architecture search to be performed in a differentiable manner:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 210, + 421 + ], + "score": 1.0, + "content": "DARTS jointly optimizes", + "type": "text" + }, + { + "bbox": [ + 210, + 409, + 218, + 417 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 406, + 293, + 421 + ], + "score": 1.0, + "content": "and model weight", + "type": "text" + }, + { + "bbox": [ + 293, + 409, + 302, + 417 + ], + "score": 0.79, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 406, + 506, + 421 + ], + "score": 1.0, + "content": "with the following bilevel objective via alternative", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 177, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 177, + 431 + ], + "score": 1.0, + "content": "gradient updates:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 441, + 421, + 461 + ], + "lines": [ + { + "bbox": [ + 188, + 441, + 421, + 461 + ], + "spans": [ + { + "bbox": [ + 188, + 441, + 421, + 461 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { \\alpha } \\mathcal { L } _ { v a l } ( w ^ { * } , \\alpha ) \\mathrm { s . t . } w ^ { * } = \\arg \\operatorname* { m i n } _ { w } \\mathcal { L } _ { t r a i n } ( w , \\alpha ) .", + "type": "interline_equation", + "image_path": "46a32003b05afc1af4dae4e935b96e101afe441a49e7f8b2e02efba180101e64.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 188, + 441, + 421, + 461 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "score": 1.0, + "content": "We refer to the continuous relaxed network used in the search phase as the supernet of DARTS.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 403, + 496 + ], + "score": 1.0, + "content": "At the end of the search phase, the operation associated with the largest", + "type": "text" + }, + { + "bbox": [ + 403, + 485, + 416, + 495 + ], + "score": 0.85, + "content": "\\alpha _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "on each edge will be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 495, + 333, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 333, + 507 + ], + "score": 1.0, + "content": "selected from the supernet to form the final architecture.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "Failure mode analysis of DARTS Several works cast doubt on the robustness of DARTS. Zela", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "et al. (2020) tests DARTS on four different search spaces and observes significantly degenerated", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "performance. They empirically find that the selected architectures perform poorly when DARTS’", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "supernet falls into high curvature areas of validation loss (captured by large dominant eigenvalues of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 568, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 155, + 584 + ], + "score": 1.0, + "content": "the Hessian", + "type": "text" + }, + { + "bbox": [ + 155, + 569, + 225, + 582 + ], + "score": 0.92, + "content": "\\nabla _ { \\alpha , \\alpha } ^ { 2 } \\mathcal { L } _ { v a l } ( w , \\alpha ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 568, + 507, + 584 + ], + "score": 1.0, + "content": ". While Zela et al. (2020) relates this problem to the failure of supernet", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "training in DARTS, we examine it from the architecture selection aspects of DARTS, and show that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "score": 1.0, + "content": "much of DARTS’ robustness issue can be alleviated by a better architecture selection method.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "Progressive search space shrinking There is a line of research on NAS that focuses on reducing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "the search cost and aligning the model sizes of the search and evaluation phases via progressive", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "search space shrinking (Liu et al., 2018; Li et al., 2019; Chen et al., 2021; Li et al., 2020). The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "score": 1.0, + "content": "general scheme of these methods is to prune out weak operations and edges sequentially during", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 294, + 678 + ], + "score": 1.0, + "content": "the search phase, based on the magnitude of", + "type": "text" + }, + { + "bbox": [ + 294, + 667, + 302, + 676 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "following DARTS. Our method is orthogonal to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "them in this respect, since we select operations based on how much it contributes to the supernet’s", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 225, + 700 + ], + "score": 1.0, + "content": "performance rather than the", + "type": "text" + }, + { + "bbox": [ + 225, + 690, + 233, + 698 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "value. 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Our analysis indicates that the strength of each operation should be evaluated", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "based on its contribution to the supernet performance instead. To this end, we propose an alternative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 104, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "perturbation-based architecture selection method. Given a pretrained supernet, the best operation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "on an edge is selected and discretized based on how much it perturbs the supernet accuracy; The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "final architecture is derived edge by edge, with fine-tuning in between so that the supernet remains", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 209 + ], + "score": 1.0, + "content": "converged for every operation decision. We re-evaluate several differentiable NAS methods (DARTS", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "(Liu et al., 2019), SDARTS (Chen & Hsieh, 2020), SGAS (Li et al., 2020)) and show that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "proposed selection method is able to consistently extract significantly improved architectures from", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "the supernets than magnitude-based counterparts. Furthermore, we find that the robustness issues", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "of DARTS can be greatly alleviated by replacing the magnitude-based selection with the proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 257, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 257, + 264 + ], + "score": 1.0, + "content": "perturbation-based selection method.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 132, + 506, + 264 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 288, + 309, + 300 + ], + "lines": [ + { + "bbox": [ + 104, + 286, + 312, + 302 + ], + "spans": [ + { + "bbox": [ + 104, + 286, + 312, + 302 + ], + "score": 1.0, + "content": "2 BACKGROUND AND RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "Preliminaries of Differentiable Architecture Search (DARTS) We start by reviewing the for-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "mulation of DARTS. DARTS’ search space consists of repetitions of cell-based microstructures.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 324, + 351 + ], + "score": 1.0, + "content": "Every cell can be viewed as a DAG with N nodes and", + "type": "text" + }, + { + "bbox": [ + 324, + 339, + 332, + 349 + ], + "score": 0.32, + "content": "\\mathrm { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "edges, where each node represents a latent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 157, + 363 + ], + "score": 1.0, + "content": "feature map", + "type": "text" + }, + { + "bbox": [ + 157, + 349, + 167, + 360 + ], + "score": 0.86, + "content": "x ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 349, + 360, + 363 + ], + "score": 1.0, + "content": ", and each edge is associated with an operation", + "type": "text" + }, + { + "bbox": [ + 360, + 352, + 366, + 360 + ], + "score": 0.62, + "content": "o", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 349, + 483, + 363 + ], + "score": 1.0, + "content": "(e.g. skip connect, sep conv", + "type": "text" + }, + { + "bbox": [ + 483, + 350, + 501, + 361 + ], + "score": 0.32, + "content": "3 x 3", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 349, + 505, + 363 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 198, + 374 + ], + "score": 1.0, + "content": "from the search space", + "type": "text" + }, + { + "bbox": [ + 199, + 361, + 208, + 371 + ], + "score": 0.69, + "content": "\\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 361, + 505, + 374 + ], + "score": 1.0, + "content": ". Continuous relaxation is then applied to this search space. Concretely,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "every operation on an edge is activated during the search phase, with their outputs mixed by the ar-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 102, + 378, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 102, + 378, + 482, + 401 + ], + "score": 1.0, + "content": "chitecture parameter α to form the final mixed output of that edge m¯ (xi) = Po∈O P exp αoo0 exp αo0", + "type": "text" + }, + { + "bbox": [ + 501, + 382, + 505, + 396 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "This particular formulation allows the architecture search to be performed in a differentiable manner:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 210, + 421 + ], + "score": 1.0, + "content": "DARTS jointly optimizes", + "type": "text" + }, + { + "bbox": [ + 210, + 409, + 218, + 417 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 406, + 293, + 421 + ], + "score": 1.0, + "content": "and model weight", + "type": "text" + }, + { + "bbox": [ + 293, + 409, + 302, + 417 + ], + "score": 0.79, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 406, + 506, + 421 + ], + "score": 1.0, + "content": "with the following bilevel objective via alternative", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 177, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 177, + 431 + ], + "score": 1.0, + "content": "gradient updates:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5, + "bbox_fs": [ + 102, + 316, + 506, + 431 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 441, + 421, + 461 + ], + "lines": [ + { + "bbox": [ + 188, + 441, + 421, + 461 + ], + "spans": [ + { + "bbox": [ + 188, + 441, + 421, + 461 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { \\alpha } \\mathcal { L } _ { v a l } ( w ^ { * } , \\alpha ) \\mathrm { s . t . } w ^ { * } = \\arg \\operatorname* { m i n } _ { w } \\mathcal { L } _ { t r a i n } ( w , \\alpha ) .", + "type": "interline_equation", + "image_path": "46a32003b05afc1af4dae4e935b96e101afe441a49e7f8b2e02efba180101e64.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 188, + 441, + 421, + 461 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 504, + 484 + ], + "score": 1.0, + "content": "We refer to the continuous relaxed network used in the search phase as the supernet of DARTS.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 403, + 496 + ], + "score": 1.0, + "content": "At the end of the search phase, the operation associated with the largest", + "type": "text" + }, + { + "bbox": [ + 403, + 485, + 416, + 495 + ], + "score": 0.85, + "content": "\\alpha _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "on each edge will be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 495, + 333, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 333, + 507 + ], + "score": 1.0, + "content": "selected from the supernet to form the final architecture.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 472, + 505, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "Failure mode analysis of DARTS Several works cast doubt on the robustness of DARTS. Zela", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "et al. (2020) tests DARTS on four different search spaces and observes significantly degenerated", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "performance. They empirically find that the selected architectures perform poorly when DARTS’", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "supernet falls into high curvature areas of validation loss (captured by large dominant eigenvalues of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 568, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 155, + 584 + ], + "score": 1.0, + "content": "the Hessian", + "type": "text" + }, + { + "bbox": [ + 155, + 569, + 225, + 582 + ], + "score": 0.92, + "content": "\\nabla _ { \\alpha , \\alpha } ^ { 2 } \\mathcal { L } _ { v a l } ( w , \\alpha ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 568, + 507, + 584 + ], + "score": 1.0, + "content": ". While Zela et al. (2020) relates this problem to the failure of supernet", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "training in DARTS, we examine it from the architecture selection aspects of DARTS, and show that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "score": 1.0, + "content": "much of DARTS’ robustness issue can be alleviated by a better architecture selection method.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 525, + 507, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "Progressive search space shrinking There is a line of research on NAS that focuses on reducing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "the search cost and aligning the model sizes of the search and evaluation phases via progressive", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "search space shrinking (Liu et al., 2018; Li et al., 2019; Chen et al., 2021; Li et al., 2020). The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 669 + ], + "score": 1.0, + "content": "general scheme of these methods is to prune out weak operations and edges sequentially during", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 294, + 678 + ], + "score": 1.0, + "content": "the search phase, based on the magnitude of", + "type": "text" + }, + { + "bbox": [ + 294, + 667, + 302, + 676 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "following DARTS. Our method is orthogonal to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "them in this respect, since we select operations based on how much it contributes to the supernet’s", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 225, + 700 + ], + "score": 1.0, + "content": "performance rather than the", + "type": "text" + }, + { + "bbox": [ + 225, + 690, + 233, + 698 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "value. Although we also discretize edges greedily and fine-tune", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the network in between, the purpose is to let the supernet recover from the loss of accuracy after", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "discretization to accurately evaluate operation strength on the next edge, rather than to reduce the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 155, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 155, + 732 + ], + "score": 1.0, + "content": "search cost.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 621, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 104, + 82, + 471, + 107 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 473, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 473, + 96 + ], + "score": 1.0, + "content": "3 THE PITFALL OF MAGNITUDE-BASED ARCHITECTURE SELECTION IN", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 124, + 95, + 168, + 109 + ], + "spans": [ + { + "bbox": [ + 124, + 95, + 168, + 109 + ], + "score": 1.0, + "content": "DARTS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 120, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 504, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 397, + 132 + ], + "score": 1.0, + "content": "In this section, we put forward the opinion that the architecture parameter", + "type": "text" + }, + { + "bbox": [ + 397, + 122, + 405, + 130 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 120, + 504, + 132 + ], + "score": 1.0, + "content": "does not necessarily rep-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 505, + 144 + ], + "score": 1.0, + "content": "resent the strength of the underlying operation in general, backed by both empirical and theoretical", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "evidence. As an important example, we mathematically justify that the skip connection domination", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 152, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 506, + 165 + ], + "score": 1.0, + "content": "phenomenon observed in DARTS is reasonable by itself, and becomes problematic when combined", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 301, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 301, + 176 + ], + "score": 1.0, + "content": "with the magnitude-based architecture selection.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 106, + 189, + 351, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 188, + 352, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 130, + 201 + ], + "score": 1.0, + "content": "3.1", + "type": "text" + }, + { + "bbox": [ + 130, + 190, + 138, + 198 + ], + "score": 0.58, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 188, + 352, + 201 + ], + "score": 1.0, + "content": "MAY NOT REPRESENT THE OPERATION STRENGTH", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "image", + "bbox": [ + 108, + 222, + 499, + 318 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 222, + 499, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 222, + 499, + 318 + ], + "spans": [ + { + "bbox": [ + 108, + 222, + 499, + 318 + ], + "score": 0.967, + "type": "image", + "image_path": "edb10b3c13ea4154d37633f3cbc076def08cf6dd2ef065dd8b384fd9afd746f5.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 108, + 222, + 499, + 254.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 108, + 254.0, + 499, + 286.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 108, + 286.0, + 499, + 318.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 322, + 505, + 356 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 145, + 335 + ], + "score": 1.0, + "content": "Figure 1:", + "type": "text" + }, + { + "bbox": [ + 146, + 324, + 153, + 332 + ], + "score": 0.68, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "vs discretization accuracy at convergence of all operations on 3 randomly selected edges", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 420, + 345 + ], + "score": 1.0, + "content": "from a pretrained DARTS supernet (one subplot per edge). The magnitude of", + "type": "text" + }, + { + "bbox": [ + 420, + 335, + 428, + 343 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 333, + 504, + 345 + ], + "score": 1.0, + "content": "for each operation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 344, + 435, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 435, + 357 + ], + "score": 1.0, + "content": "does not necessarily agree with its relative discretization accuracy at convergence.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "image", + "bbox": [ + 108, + 376, + 503, + 468 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 376, + 503, + 468 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 376, + 503, + 468 + ], + "spans": [ + { + "bbox": [ + 108, + 376, + 503, + 468 + ], + "score": 0.965, + "type": "image", + "image_path": "77e533953356028536aaa448409ba1cfccb71639465d6c5e104d9488c7bdad81.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 108, + 376, + 503, + 406.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 108, + 406.6666666666667, + 503, + 437.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 108, + 437.33333333333337, + 503, + 468.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 470, + 505, + 503 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 409, + 483 + ], + "score": 1.0, + "content": "Figure 2: Operation strength on each edge of S2 (skip connect, sep conv", + "type": "text" + }, + { + "bbox": [ + 410, + 470, + 428, + 481 + ], + "score": 0.3, + "content": "{ _ { - 3 x 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "). (a). Operations", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 218, + 493 + ], + "score": 1.0, + "content": "associated with the largest", + "type": "text" + }, + { + "bbox": [ + 218, + 483, + 226, + 491 + ], + "score": 0.58, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ". (b). Operations that result in the highest discretization validation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 492, + 377, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 377, + 504 + ], + "score": 1.0, + "content": "accuracy at convergence. Parameterized operations are marked red.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 496, + 530 + ], + "score": 1.0, + "content": "Following DARTS, existing differentiable NAS methods use the value of architecture parameters", + "type": "text" + }, + { + "bbox": [ + 496, + 520, + 504, + 527 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 438, + 540 + ], + "score": 1.0, + "content": "to select the final architecture from the supernet, with the implicit assumption that", + "type": "text" + }, + { + "bbox": [ + 438, + 530, + 446, + 538 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "strength of the underlying operations. In this section, we study the validity of this assumption in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 550, + 135, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 135, + 562 + ], + "score": 1.0, + "content": "detail.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "Consider one edge on a pretrained supernet; the strength of an operation on the edge can be naturally", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "defined as the supernet accuracy after we discretize to this operation and fine-tune the remaining net-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "work until it converges again; we refer to this as ”discretization accuracy at convergence” for short.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "The operation that achieves the best discretization accuracy at convergence can be considered as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 410, + 624 + ], + "score": 1.0, + "content": "the best operation for the given edge. Figure 1 shows the comparison of", + "type": "text" + }, + { + "bbox": [ + 410, + 613, + 418, + 621 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "(blue) and operation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 622, + 504, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 496, + 635 + ], + "score": 1.0, + "content": "strength (orange) of randomly select edges on DARTS supernet. As we can see, the magnitude of", + "type": "text" + }, + { + "bbox": [ + 496, + 623, + 504, + 632 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "for each operation does not necessarily agree with their relative strength measured by discretization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 382, + 657 + ], + "score": 1.0, + "content": "accuracy at convergence. Moreover, operations assigned with small", + "type": "text" + }, + { + "bbox": [ + 383, + 645, + 394, + 654 + ], + "score": 0.29, + "content": "\\alpha \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "are sometimes strong ones", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "that lead to high discretization accuracy at convergence. To further verify the mismatch, we inves-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "tigate the operation strength on search space S2, where DARTS fails dramatically due to excessive", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "skip connections (Zela et al., 2020). S2 is a variant of DARTS search space that only contains two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 287, + 700 + ], + "score": 1.0, + "content": "operations per edge (skip connect, sep conv", + "type": "text" + }, + { + "bbox": [ + 288, + 688, + 306, + 699 + ], + "score": 0.43, + "content": "{ _ { 3 x 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "). Figure 2 shows the selected operations based", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 120, + 713 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 701, + 128, + 709 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "(left) and operation strength (right) on all edges on S2. From Figure 2a, we can see that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 161, + 721 + ], + "score": 1.0, + "content": "αskip connect", + "type": "text" + }, + { + "bbox": [ + 161, + 711, + 227, + 722 + ], + "score": 0.73, + "content": "> \\alpha _ { s e p . c o n v . 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "on 12 of 14 edges. Consequently, the derived child architecture will", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "lack representation ability and perform poorly due to too many skip connections. However, as shown", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 104, + 82, + 471, + 107 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 473, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 473, + 96 + ], + "score": 1.0, + "content": "3 THE PITFALL OF MAGNITUDE-BASED ARCHITECTURE SELECTION IN", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 124, + 95, + 168, + 109 + ], + "spans": [ + { + "bbox": [ + 124, + 95, + 168, + 109 + ], + "score": 1.0, + "content": "DARTS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 120, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 504, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 397, + 132 + ], + "score": 1.0, + "content": "In this section, we put forward the opinion that the architecture parameter", + "type": "text" + }, + { + "bbox": [ + 397, + 122, + 405, + 130 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 120, + 504, + 132 + ], + "score": 1.0, + "content": "does not necessarily rep-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 505, + 144 + ], + "score": 1.0, + "content": "resent the strength of the underlying operation in general, backed by both empirical and theoretical", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "evidence. As an important example, we mathematically justify that the skip connection domination", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 152, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 506, + 165 + ], + "score": 1.0, + "content": "phenomenon observed in DARTS is reasonable by itself, and becomes problematic when combined", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 301, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 301, + 176 + ], + "score": 1.0, + "content": "with the magnitude-based architecture selection.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 120, + 506, + 176 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 189, + 351, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 188, + 352, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 130, + 201 + ], + "score": 1.0, + "content": "3.1", + "type": "text" + }, + { + "bbox": [ + 130, + 190, + 138, + 198 + ], + "score": 0.58, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 188, + 352, + 201 + ], + "score": 1.0, + "content": "MAY NOT REPRESENT THE OPERATION STRENGTH", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "image", + "bbox": [ + 108, + 222, + 499, + 318 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 222, + 499, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 222, + 499, + 318 + ], + "spans": [ + { + "bbox": [ + 108, + 222, + 499, + 318 + ], + "score": 0.967, + "type": "image", + "image_path": "edb10b3c13ea4154d37633f3cbc076def08cf6dd2ef065dd8b384fd9afd746f5.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 108, + 222, + 499, + 254.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 108, + 254.0, + 499, + 286.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 108, + 286.0, + 499, + 318.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 322, + 505, + 356 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 145, + 335 + ], + "score": 1.0, + "content": "Figure 1:", + "type": "text" + }, + { + "bbox": [ + 146, + 324, + 153, + 332 + ], + "score": 0.68, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "vs discretization accuracy at convergence of all operations on 3 randomly selected edges", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 420, + 345 + ], + "score": 1.0, + "content": "from a pretrained DARTS supernet (one subplot per edge). The magnitude of", + "type": "text" + }, + { + "bbox": [ + 420, + 335, + 428, + 343 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 333, + 504, + 345 + ], + "score": 1.0, + "content": "for each operation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 344, + 435, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 435, + 357 + ], + "score": 1.0, + "content": "does not necessarily agree with its relative discretization accuracy at convergence.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "image", + "bbox": [ + 108, + 376, + 503, + 468 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 376, + 503, + 468 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 376, + 503, + 468 + ], + "spans": [ + { + "bbox": [ + 108, + 376, + 503, + 468 + ], + "score": 0.965, + "type": "image", + "image_path": "77e533953356028536aaa448409ba1cfccb71639465d6c5e104d9488c7bdad81.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 108, + 376, + 503, + 406.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 108, + 406.6666666666667, + 503, + 437.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 108, + 437.33333333333337, + 503, + 468.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 470, + 505, + 503 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 409, + 483 + ], + "score": 1.0, + "content": "Figure 2: Operation strength on each edge of S2 (skip connect, sep conv", + "type": "text" + }, + { + "bbox": [ + 410, + 470, + 428, + 481 + ], + "score": 0.3, + "content": "{ _ { - 3 x 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "). (a). Operations", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 218, + 493 + ], + "score": 1.0, + "content": "associated with the largest", + "type": "text" + }, + { + "bbox": [ + 218, + 483, + 226, + 491 + ], + "score": 0.58, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ". (b). Operations that result in the highest discretization validation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 492, + 377, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 377, + 504 + ], + "score": 1.0, + "content": "accuracy at convergence. Parameterized operations are marked red.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 496, + 530 + ], + "score": 1.0, + "content": "Following DARTS, existing differentiable NAS methods use the value of architecture parameters", + "type": "text" + }, + { + "bbox": [ + 496, + 520, + 504, + 527 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 438, + 540 + ], + "score": 1.0, + "content": "to select the final architecture from the supernet, with the implicit assumption that", + "type": "text" + }, + { + "bbox": [ + 438, + 530, + 446, + 538 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "strength of the underlying operations. In this section, we study the validity of this assumption in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 550, + 135, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 135, + 562 + ], + "score": 1.0, + "content": "detail.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 516, + 506, + 562 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "Consider one edge on a pretrained supernet; the strength of an operation on the edge can be naturally", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "defined as the supernet accuracy after we discretize to this operation and fine-tune the remaining net-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "work until it converges again; we refer to this as ”discretization accuracy at convergence” for short.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "The operation that achieves the best discretization accuracy at convergence can be considered as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 410, + 624 + ], + "score": 1.0, + "content": "the best operation for the given edge. Figure 1 shows the comparison of", + "type": "text" + }, + { + "bbox": [ + 410, + 613, + 418, + 621 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "(blue) and operation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 622, + 504, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 496, + 635 + ], + "score": 1.0, + "content": "strength (orange) of randomly select edges on DARTS supernet. As we can see, the magnitude of", + "type": "text" + }, + { + "bbox": [ + 496, + 623, + 504, + 632 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "for each operation does not necessarily agree with their relative strength measured by discretization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 382, + 657 + ], + "score": 1.0, + "content": "accuracy at convergence. Moreover, operations assigned with small", + "type": "text" + }, + { + "bbox": [ + 383, + 645, + 394, + 654 + ], + "score": 0.29, + "content": "\\alpha \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "are sometimes strong ones", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "that lead to high discretization accuracy at convergence. To further verify the mismatch, we inves-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "tigate the operation strength on search space S2, where DARTS fails dramatically due to excessive", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "skip connections (Zela et al., 2020). S2 is a variant of DARTS search space that only contains two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 287, + 700 + ], + "score": 1.0, + "content": "operations per edge (skip connect, sep conv", + "type": "text" + }, + { + "bbox": [ + 288, + 688, + 306, + 699 + ], + "score": 0.43, + "content": "{ _ { 3 x 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "). Figure 2 shows the selected operations based", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 120, + 713 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 701, + 128, + 709 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "(left) and operation strength (right) on all edges on S2. From Figure 2a, we can see that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 161, + 721 + ], + "score": 1.0, + "content": "αskip connect", + "type": "text" + }, + { + "bbox": [ + 161, + 711, + 227, + 722 + ], + "score": 0.73, + "content": "> \\alpha _ { s e p . c o n v . 3 x 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "on 12 of 14 edges. Consequently, the derived child architecture will", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "lack representation ability and perform poorly due to too many skip connections. However, as shown", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "in Figure 2b, the supernet benefits more from discretizing to sep conv 3x3 than skip connect on half", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 160, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 160, + 107 + ], + "score": 1.0, + "content": "of the edges.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 567, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "in Figure 2b, the supernet benefits more from discretizing to sep conv 3x3 than skip connect on half", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 160, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 160, + 107 + ], + "score": 1.0, + "content": "of the edges.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 108, + 118, + 279, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 118, + 281, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 281, + 131 + ], + "score": 1.0, + "content": "3.2 A CASE STUDY: SKIP CONNECTION", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 335, + 150 + ], + "score": 1.0, + "content": "Several works point out that DARTS tends to assign large", + "type": "text" + }, + { + "bbox": [ + 335, + 141, + 343, + 149 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 139, + 505, + 150 + ], + "score": 1.0, + "content": "to skip connections, resulting in shallow", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 148, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 505, + 163 + ], + "score": 1.0, + "content": "architectures with poor generability (Zela et al., 2020; Liang et al., 2019; Bi et al., 2019). This ”skip", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "connection domination” issue is generally attributed to the failure of DARTS’ supernet optimiza-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 504, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 504, + 183 + ], + "score": 1.0, + "content": "tion. In contrast, we draw inspiration from research on ResNet (He et al., 2016) and show that this", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "phenomenon by itself is a reasonable outcome while DARTS refines its estimation of the optimal", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 193, + 383, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 199, + 206 + ], + "score": 1.0, + "content": "feature map, rendering", + "type": "text" + }, + { + "bbox": [ + 199, + 195, + 223, + 206 + ], + "score": 0.89, + "content": "\\alpha _ { s k i p }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 193, + 383, + 206 + ], + "score": 1.0, + "content": "ineffective in the architecture selection.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 296, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 298, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 298, + 223 + ], + "score": 1.0, + "content": "In vanilla networks (e.g., VGG), each layer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 297, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 297, + 232 + ], + "score": 1.0, + "content": "computes a new level of feature map from the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 232, + 297, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 297, + 245 + ], + "score": 1.0, + "content": "output feature map of the predecessor layer;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 297, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 297, + 254 + ], + "score": 1.0, + "content": "thus, reordering layers at test time would dra-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 297, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 297, + 266 + ], + "score": 1.0, + "content": "matically hurt the performance (Veit et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 297, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 297, + 277 + ], + "score": 1.0, + "content": "2016). Unlike vanilla networks, Greff et al.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 276, + 297, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 297, + 288 + ], + "score": 1.0, + "content": "(2017) and Veit et al. (2016) discover that suc-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 297, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 297, + 299 + ], + "score": 1.0, + "content": "cessive layers in ResNet with compatible chan-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 298, + 297, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 297, + 310 + ], + "score": 1.0, + "content": "nel sizes are in fact estimating the same optimal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 297, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 297, + 321 + ], + "score": 1.0, + "content": "feature map so that the outputs of these layers", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "table", + "bbox": [ + 306, + 281, + 504, + 311 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 212, + 504, + 279 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 211, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 303, + 211, + 506, + 224 + ], + "score": 1.0, + "content": "Table 1: Test accuracy before and after layer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 304, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 304, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "(edge) shuffling on cifar10. For ResNet and VGG,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 303, + 234, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 303, + 234, + 505, + 245 + ], + "score": 1.0, + "content": "we randomly swap two layers in each stage (de-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 303, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 303, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "fined as successive layers between two downsam-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 303, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 303, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "pling blocks. For DARTS supernet, we randomly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 303, + 268, + 424, + 279 + ], + "spans": [ + { + "bbox": [ + 303, + 268, + 424, + 279 + ], + "score": 1.0, + "content": "swap two edges in every cell.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "table_body", + "bbox": [ + 306, + 281, + 504, + 311 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 306, + 281, + 504, + 311 + ], + "spans": [ + { + "bbox": [ + 306, + 281, + 504, + 311 + ], + "score": 0.958, + "html": "
VGGResNetDARTS
Before92.6993.8688.44
After9.83 ± 0.3383.2015 ± 2.0381.09 ± 1.87
", + "type": "table", + "image_path": "2e29115b52e9ffde6b2e20413ea5292ec2bfbf4730935c9f3371d0f2784c7d82.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 306, + 281, + 504, + 296.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 306, + 296.0, + 504, + 311.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 108, + 320, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "stay relatively close to each other at convergence; As a result, ResNet’s test accuracy remains robust", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "under layer reordering. Greff et al. (2017) refers to this unique way of feature map estimation in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 340, + 255, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 255, + 353 + ], + "score": 1.0, + "content": "ResNet as the ”unrolled estimation.”", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "score": 1.0, + "content": "DARTS’ supernet resembles ResNet, rather than vanilla networks like VGG, in both appearance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "and behavior. Appearance-wise, within a cell of DARTS’ supernet, edges with skip connection are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 381, + 504, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 504, + 392 + ], + "score": 1.0, + "content": "in direct correspondence with the successive residual layers in ResNet. Behavior-wise, DARTS’", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 390, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 506, + 405 + ], + "score": 1.0, + "content": "supernet also exhibits a high degree of robustness under edge shuffling. As shown in Table 1,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "randomly reordering edges on a pretrained DARTS’ supernet at test time also has little effect on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "its performance. This evidence indicates that DARTS performs unrolled estimation like ResNet as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "well, i.e., edges within a cell share the same optimal feature map that they try to estimate. In the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 434, + 507, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 434, + 449 + ], + "score": 1.0, + "content": "following proposition, we apply this finding and provide the optimal solution of", + "type": "text" + }, + { + "bbox": [ + 434, + 437, + 442, + 445 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 434, + 507, + 449 + ], + "score": 1.0, + "content": "in the sense of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 447, + 312, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 312, + 459 + ], + "score": 1.0, + "content": "minimizing the variance of feature map estimation.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "Proposition 1. 1 Without loss of generality, consider one cell from a simplified search space consists", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 250, + 483 + ], + "score": 1.0, + "content": "of two operations: (skip, conv). Let", + "type": "text" + }, + { + "bbox": [ + 250, + 472, + 264, + 482 + ], + "score": 0.85, + "content": "m ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "denotes the optimal feature map, which is shared across all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 407, + 495 + ], + "score": 1.0, + "content": "edges according to the unrolled estimation view (Greff et al., 2017). 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", + "type": "table", + "image_path": "2e29115b52e9ffde6b2e20413ea5292ec2bfbf4730935c9f3371d0f2784c7d82.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 306, + 281, + 504, + 296.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 306, + 296.0, + 504, + 311.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 108, + 320, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "stay relatively close to each other at convergence; As a result, ResNet’s test accuracy remains robust", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "under layer reordering. Greff et al. (2017) refers to this unique way of feature map estimation in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 340, + 255, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 255, + 353 + ], + "score": 1.0, + "content": "ResNet as the ”unrolled estimation.”", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 320, + 506, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "score": 1.0, + "content": "DARTS’ supernet resembles ResNet, rather than vanilla networks like VGG, in both appearance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "and behavior. Appearance-wise, within a cell of DARTS’ supernet, edges with skip connection are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 381, + 504, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 504, + 392 + ], + "score": 1.0, + "content": "in direct correspondence with the successive residual layers in ResNet. Behavior-wise, DARTS’", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 390, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 506, + 405 + ], + "score": 1.0, + "content": "supernet also exhibits a high degree of robustness under edge shuffling. As shown in Table 1,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "randomly reordering edges on a pretrained DARTS’ supernet at test time also has little effect on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "its performance. This evidence indicates that DARTS performs unrolled estimation like ResNet as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "well, i.e., edges within a cell share the same optimal feature map that they try to estimate. In the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 434, + 507, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 434, + 449 + ], + "score": 1.0, + "content": "following proposition, we apply this finding and provide the optimal solution of", + "type": "text" + }, + { + "bbox": [ + 434, + 437, + 442, + 445 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 434, + 507, + 449 + ], + "score": 1.0, + "content": "in the sense of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 447, + 312, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 312, + 459 + ], + "score": 1.0, + "content": "minimizing the variance of feature map estimation.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 357, + 507, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "Proposition 1. 1 Without loss of generality, consider one cell from a simplified search space consists", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 250, + 483 + ], + "score": 1.0, + "content": "of two operations: (skip, conv). Let", + "type": "text" + }, + { + "bbox": [ + 250, + 472, + 264, + 482 + ], + "score": 0.85, + "content": "m ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "denotes the optimal feature map, which is shared across all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 407, + 495 + ], + "score": 1.0, + "content": "edges according to the unrolled estimation view (Greff et al., 2017). Let", + "type": "text" + }, + { + "bbox": [ + 408, + 483, + 435, + 494 + ], + "score": 0.92, + "content": "o _ { e } ( x _ { e } )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "be the output of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 234, + 506 + ], + "score": 1.0, + "content": "convolution operation, and let", + "type": "text" + }, + { + "bbox": [ + 234, + 494, + 245, + 504 + ], + "score": 0.85, + "content": "x _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "be the skip connection (i.e., the input feature map of edge e).", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 139, + 516 + ], + "score": 1.0, + "content": "Assume", + "type": "text" + }, + { + "bbox": [ + 140, + 505, + 153, + 514 + ], + "score": 0.82, + "content": "m ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 504, + 157, + 516 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 158, + 505, + 185, + 516 + ], + "score": 0.91, + "content": "o _ { e } ( x _ { e } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 504, + 203, + 516 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 505, + 214, + 515 + ], + "score": 0.84, + "content": "x _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 504, + 453, + 516 + ], + "score": 1.0, + "content": "are normalized to the same scale. 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In this case, although", + "type": "text" + }, + { + "bbox": [ + 245, + 197, + 307, + 209 + ], + "score": 0.89, + "content": "\\alpha _ { s k i p } > \\alpha _ { c o n v }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 195, + 338, + 210 + ], + "score": 1.0, + "content": "is rea-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 338, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 338, + 218 + ], + "score": 1.0, + "content": "sonable by itself, it becomes an inductive bias to NAS if", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 218, + 312, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 300, + 230 + ], + "score": 1.0, + "content": "we were to select the final architecture based on", + "type": "text" + }, + { + "bbox": [ + 300, + 221, + 307, + 228 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 218, + 312, + 230 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 106, + 245, + 248, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 245, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 245, + 259 + ], + "score": 1.0, + "content": "4 PERTURBATION-BASED", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 261, + 249, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 249, + 273 + ], + "score": 1.0, + "content": "ARCHITECTURE SELECTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "image", + "bbox": [ + 348, + 137, + 493, + 244 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 348, + 137, + 493, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 348, + 137, + 493, + 244 + ], + "spans": [ + { + "bbox": [ + 348, + 137, + 493, + 244 + ], + "score": 0.946, + "type": "image", + "image_path": "79a873ee67039a51da8c30bc3160564d6e0226f824e4320c9a3517871f5adb3b.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 348, + 137, + 493, + 190.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 348, + 190.5, + 493, + 244.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 344, + 246, + 504, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 342, + 245, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 342, + 245, + 385, + 260 + ], + "score": 1.0, + "content": "Figure 3:", + "type": "text" + }, + { + "bbox": [ + 386, + 247, + 479, + 259 + ], + "score": 0.81, + "content": "m e a n ( \\alpha _ { s k i p } - \\alpha _ { c o n v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 245, + 506, + 260 + ], + "score": 1.0, + "content": "(soft-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 343, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 343, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "maxed) v.s. supernet’s validation accu-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 342, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 342, + 268, + 421, + 282 + ], + "score": 1.0, + "content": "racy. The gap of", + "type": "text" + }, + { + "bbox": [ + 421, + 269, + 488, + 281 + ], + "score": 0.88, + "content": "( \\alpha _ { s k i p } - \\alpha _ { c o n v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "in-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 343, + 280, + 468, + 292 + ], + "spans": [ + { + "bbox": [ + 343, + 280, + 468, + 292 + ], + "score": 1.0, + "content": "creases as supernet gets better.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 335, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 336, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 207, + 296 + ], + "score": 1.0, + "content": "Instead of relying on the", + "type": "text" + }, + { + "bbox": [ + 207, + 286, + 215, + 294 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 283, + 336, + 296 + ], + "score": 1.0, + "content": "value to select the best opera-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 336, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 336, + 307 + ], + "score": 1.0, + "content": "tion, we propose to directly evaluate operation strength in", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 108, + 306, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "terms of its contribution to the supernet’s performance. The operation selection criterion is laid out", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 315, + 437, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 437, + 329 + ], + "score": 1.0, + "content": "in section 4.1. In section 4.2, we describe the entire architecture selection process.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 106, + 342, + 346, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 347, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 347, + 353 + ], + "score": 1.0, + "content": "4.1 EVALUATING THE STRENGTH OF EACH OPERATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "In section 3.1, we define the strength of each operation on a given edge as how much it contributes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 373, + 504, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 504, + 385 + ], + "score": 1.0, + "content": "to the performance of the supernet, measured by discretization accuracy. To avoid inaccurate eval-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 382, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 397 + ], + "score": 1.0, + "content": "uation due to large disturbance of the supernet during discretization, we fine-tune the remaining", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 394, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 409 + ], + "score": 1.0, + "content": "supernet until it converges again, and then compute its validation accuracy (discretization accuracy", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "at convergence). The fine-tuning process needs to be carried out for evaluating each operation on an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 417, + 294, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 294, + 429 + ], + "score": 1.0, + "content": "edge, leading to substantial computation costs.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 504, + 446 + ], + "score": 1.0, + "content": "To alleviate the computational overhead, we consider a more practical measure of operation strength:", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "for each operation on a given edge, we mask it out while keeping all other operations, and re-evaluate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "the supernet. The one that results in the largest drop in the supernet’s validation accuracy will be", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "considered as the most important operation on that edge. This alternative criterion incurs much less", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "perturbation to the supernet than discretization since it only deletes one operation from the supernet", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "at a time. As a result, the supernet’s validation accuracy after deletion stays close to the unmodified", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "supernet, and thus it alleviates the requirement of tuning the remaining supernet to convergence.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 511, + 442, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 442, + 523 + ], + "score": 1.0, + "content": "Therefore, we implement this measurement for the operation selection in this work.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 545, + 425, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 424, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 225, + 558 + ], + "score": 1.0, + "content": "Input: A pretrained supernet", + "type": "text" + }, + { + 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After that, we tune the supernet for", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "text", + "bbox": [ + 129, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 132, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 132, + 81, + 142, + 96 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 82, + 170, + 95 + ], + "score": 0.92, + "content": "o _ { e } ( x _ { e } )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "is the output of a single convolution operation instead of the complete mixed output", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 92, + 376, + 107 + ], + "spans": [ + { + "bbox": [ + 141, + 92, + 174, + 107 + ], + "score": 1.0, + "content": "of edge", + "type": "text" + }, + { + "bbox": [ + 174, + 96, + 180, + 104 + ], + "score": 0.58, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 92, + 274, + 107 + ], + "score": 1.0, + "content": ", so it will deviate from", + "type": "text" + }, + { + "bbox": [ + 274, + 94, + 289, + 104 + ], + "score": 0.87, + "content": "m ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 92, + 376, + 107 + ], + "score": 1.0, + "content": "even at convergence.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 132, + 81, + 505, + 107 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 113, + 503, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 504, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 267, + 129 + ], + "score": 1.0, + "content": "Therefore, in a well-optimized supernet,", + "type": "text" + }, + { + "bbox": [ + 268, + 115, + 279, + 124 + ], + "score": 0.85, + "content": "x _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 109, + 381, + 129 + ], + "score": 1.0, + "content": "will naturally be closer to", + "type": "text" + }, + { + "bbox": [ + 382, + 114, + 396, + 124 + ], + "score": 0.88, + "content": "m ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 109, + 416, + 129 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 416, + 113, + 444, + 126 + ], + "score": 0.92, + "content": "o _ { e } ( x _ { e } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 109, + 480, + 129 + ], + "score": 1.0, + "content": ", causing", + "type": "text" + }, + { + "bbox": [ + 480, + 115, + 504, + 126 + ], + "score": 0.66, + "content": "\\alpha _ { s k i p }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 207, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 178, + 138 + ], + "score": 1.0, + "content": "to be greater than", + "type": "text" + }, + { + "bbox": [ + 178, + 126, + 203, + 136 + ], + "score": 0.87, + "content": "\\alpha _ { c o n v }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 123, + 207, + 138 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 109, + 504, + 138 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 336, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 337, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 337, + 154 + ], + "score": 1.0, + "content": "Our analysis above indicates that the better the supernet,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 337, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 164, + 166 + ], + "score": 1.0, + "content": "the larger the", + "type": "text" + }, + { + "bbox": [ + 164, + 153, + 231, + 164 + ], + "score": 0.88, + "content": "( \\alpha _ { s k i p } - 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The gap of", + "type": "text" + }, + { + "bbox": [ + 421, + 269, + 488, + 281 + ], + "score": 0.88, + "content": "( \\alpha _ { s k i p } - \\alpha _ { c o n v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "in-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 343, + 280, + 468, + 292 + ], + "spans": [ + { + "bbox": [ + 343, + 280, + 468, + 292 + ], + "score": 1.0, + "content": "creases as supernet gets better.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 335, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 336, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 207, + 296 + ], + "score": 1.0, + "content": "Instead of relying on the", + "type": "text" + }, + { + "bbox": [ + 207, + 286, + 215, + 294 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 283, + 336, + 296 + ], + "score": 1.0, + "content": "value to select the best opera-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 336, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 336, + 307 + ], + "score": 1.0, + "content": "tion, we propose to directly evaluate operation strength in", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 283, + 336, + 307 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 306, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "terms of its contribution to the supernet’s performance. The operation selection criterion is laid out", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 315, + 437, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 437, + 329 + ], + "score": 1.0, + "content": "in section 4.1. 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The fine-tuning process needs to be carried out for evaluating each operation on an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 417, + 294, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 294, + 429 + ], + "score": 1.0, + "content": "edge, leading to substantial computation costs.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 361, + 506, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 504, + 446 + ], + "score": 1.0, + "content": "To alleviate the computational overhead, we consider a more practical measure of operation strength:", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "for each operation on a given edge, we mask it out while keeping all other operations, and re-evaluate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "the supernet. 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"content": "and tune the remaining supernet for a few epochs;", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 647, + 126, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 126, + 660 + ], + "score": 1.0, + "content": "end", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 545, + 424, + 660 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 678, + 358, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 359, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 359, + 691 + ], + "score": 1.0, + "content": "4.2 THE COMPLETE ARCHITECTURE SELECTION PROCESS", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Our method operates directly on top of DARTS’ pretrained supernet. Given a supernet, we randomly", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "iterate over all of its edges. We evaluate each operation on an edge, and select the best one to be", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "discretized based on the measurement described in section 4.1. After that, we tune the supernet for", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "a few epochs to recover the accuracy lost during discretization. The above steps are repeated until", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "all edges are decided. Algorithm 1 summarizes the operation selection process. The cell topology is", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 118 + ], + "score": 1.0, + "content": "decided in a similar fashion. We refer the reader to Appendix A.3 for the full algorithm, including", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "deciding the cell topology. This simple method is termed ”perturbation-based architecture selection", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 235, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 235, + 139 + ], + "score": 1.0, + "content": "(PT)” in the following sections.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 699, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "a few epochs to recover the accuracy lost during discretization. The above steps are repeated until", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "all edges are decided. Algorithm 1 summarizes the operation selection process. The cell topology is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 118 + ], + "score": 1.0, + "content": "decided in a similar fashion. We refer the reader to Appendix A.3 for the full algorithm, including", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "deciding the cell topology. This simple method is termed ”perturbation-based architecture selection", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 235, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 235, + 139 + ], + "score": 1.0, + "content": "(PT)” in the following sections.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 154, + 256, + 166 + ], + "lines": [ + { + "bbox": [ + 104, + 152, + 258, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 152, + 258, + 168 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 179, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "In this section, we demonstrate that the perturbation-based architecture selection method is able to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 191, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 422, + 202 + ], + "score": 1.0, + "content": "consistently find better architectures than those selected based on the values of", + "type": "text" + }, + { + "bbox": [ + 423, + 192, + 430, + 200 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 191, + 505, + 202 + ], + "score": 1.0, + "content": ". The evaluation is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "based on the search space of DARTS and NAS-Bench-201 (Dong & Yang, 2020), and we show that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 500, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 500, + 224 + ], + "score": 1.0, + "content": "the perturbation-based architecture selection method can be applied to several variants of DARTS.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 237, + 317, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 318, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 318, + 250 + ], + "score": 1.0, + "content": "5.1 RESULTS ON DARTS’ CNN SEARCH SPACE", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "We keep all the search and retrain settings identical to DARTS since our method only modifies", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "the architecture selection part. After the search phase, we perform perturbation-based architecture", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "selection following Algorithm 1 on the pretrained supernet. We tune the supernet for 5 epochs", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "between two selections as it is enough for the supernet to recover from the drop of accuracy after", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "discretization. We run the search and architecture selection phase with four random seeds and report", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 313, + 370, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 370, + 324 + ], + "score": 1.0, + "content": "both the best and average test errors of the obtained architectures.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 316, + 342 + ], + "score": 1.0, + "content": "As shown in Table 2, the proposed method (DARTS", + "type": "text" + }, + { + "bbox": [ + 316, + 330, + 334, + 340 + ], + "score": 0.33, + "content": "+ \\mathrm { P T }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 329, + 477, + 342 + ], + "score": 1.0, + "content": ") improves DARTS’ test error from", + "type": "text" + }, + { + "bbox": [ + 477, + 329, + 504, + 340 + ], + "score": 0.83, + "content": "3 . 0 0 \\%", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 116, + 353 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 340, + 144, + 351 + ], + "score": 0.88, + "content": "2 . 6 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 340, + 505, + 353 + ], + "score": 1.0, + "content": ", with manageable search cost (0.8 GPU days). Note that by only changing the architecture", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "selection method, DARTS performs significantly better than many other differentiable NAS methods", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 376 + ], + "score": 1.0, + "content": "that enjoy carefully designed optimization process of the supernet, such as GDAS (Dong & Yang,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "2019) and SNAS (Xie et al., 2019). This empirical result suggests that architecture selection is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "crucial to DARTS: with the proper selection algorithm, DARTS remains a very competitive method.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "Our method is also able to improve the performance of other variants of DARTS. To show this, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "evaluate our method on SDARTS(rs) and SGAS (Chen & Hsieh, 2020; Li et al., 2020). SDARTS(rs)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 493, + 437 + ], + "score": 1.0, + "content": "is a variant of DARTS that regularizes the search phase by applying Gaussian perturbation to", + "type": "text" + }, + { + "bbox": [ + 493, + 425, + 501, + 433 + ], + "score": 0.68, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 422, + 506, + 437 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "score": 1.0, + "content": "Unlike DARTS and SDARTS, SGAS performs progressive search space shrinking. Concretely,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "SGAS progressively discretizes its edges with the order from most to least important, based on a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "novel edge importance score. For a fair comparison, we keep its unique search space shrinking", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "process unmodified and only replace its magnitude-based operation selection with ours. As we can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "see from Table 2, our method consistently achieves better average test errors than its magnitude-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 465, + 502 + ], + "score": 1.0, + "content": "based counterpart. Concretely, the proposed method improves SDARTS’ test error from", + "type": "text" + }, + { + "bbox": [ + 466, + 489, + 493, + 500 + ], + "score": 0.87, + "content": "2 . 6 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 133, + 510 + ], + "score": 0.91, + "content": "2 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 500, + 246, + 512 + ], + "score": 1.0, + "content": "and SGAS’ test error from", + "type": "text" + }, + { + "bbox": [ + 246, + 500, + 274, + 510 + ], + "score": 0.88, + "content": "2 . 6 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 500, + 286, + 512 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 286, + 500, + 313, + 510 + ], + "score": 0.88, + "content": "2 . 5 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 500, + 506, + 512 + ], + "score": 1.0, + "content": ". Moreover, the best architecture discovered in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 511, + 454, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 265, + 523 + ], + "score": 1.0, + "content": "our experiments achieves a test error of", + "type": "text" + }, + { + "bbox": [ + 266, + 511, + 292, + 522 + ], + "score": 0.86, + "content": "2 . 4 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 511, + 454, + 523 + ], + "score": 1.0, + "content": ", ranked top among other NAS methods.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "image", + "bbox": [ + 108, + 563, + 496, + 654 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 106, + 536, + 357, + 547 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 534, + 358, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 358, + 549 + ], + "score": 1.0, + "content": "5.2 PERFORMANCE ON NAS-BENCH-201 SEARCH SPACE", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "image_body", + "bbox": [ + 108, + 563, + 496, + 654 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 563, + 496, + 654 + ], + "spans": [ + { + "bbox": [ + 108, + 563, + 496, + 654 + ], + "score": 0.964, + "type": "image", + "image_path": "7d4b6638faec392f9f4d135d85de588ec6564e674791a16ea64c8e91e9c9bcca.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 108, + 563, + 496, + 593.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 108, + 593.3333333333334, + 496, + 623.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 108, + 623.6666666666667, + 496, + 654.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 656, + 503, + 690 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Figure 4: Trajectory of test accuracy on space NAS-Bench-201 and three datasets (Left: cifar10,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "Middle: cifar100, Right: Imagenet16-120). The test accuracy of our method is plotted by taking the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 500, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 500, + 690 + ], + "score": 1.0, + "content": "snapshots of DARTS’ supernet at corresponding epochs and run our selection method on top of it.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "To further verify the effectiveness of the proposed perturbation-based architecture selection, we con-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "duct experiments on NAS-Bench-201. NAS-Bench-201 provides a unified cell-based search space", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "similar to DARTS. Every architecture in the search space is trained under the same protocol on three", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 139 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 154, + 256, + 166 + ], + "lines": [ + { + "bbox": [ + 104, + 152, + 258, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 152, + 258, + 168 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 179, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "In this section, we demonstrate that the perturbation-based architecture selection method is able to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 191, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 422, + 202 + ], + "score": 1.0, + "content": "consistently find better architectures than those selected based on the values of", + "type": "text" + }, + { + "bbox": [ + 423, + 192, + 430, + 200 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 191, + 505, + 202 + ], + "score": 1.0, + "content": ". The evaluation is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "based on the search space of DARTS and NAS-Bench-201 (Dong & Yang, 2020), and we show that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 500, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 500, + 224 + ], + "score": 1.0, + "content": "the perturbation-based architecture selection method can be applied to several variants of DARTS.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 178, + 506, + 224 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 237, + 317, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 318, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 318, + 250 + ], + "score": 1.0, + "content": "5.1 RESULTS ON DARTS’ CNN SEARCH SPACE", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "We keep all the search and retrain settings identical to DARTS since our method only modifies", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "the architecture selection part. After the search phase, we perform perturbation-based architecture", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "selection following Algorithm 1 on the pretrained supernet. We tune the supernet for 5 epochs", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "between two selections as it is enough for the supernet to recover from the drop of accuracy after", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "discretization. We run the search and architecture selection phase with four random seeds and report", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 313, + 370, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 370, + 324 + ], + "score": 1.0, + "content": "both the best and average test errors of the obtained architectures.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 258, + 505, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 316, + 342 + ], + "score": 1.0, + "content": "As shown in Table 2, the proposed method (DARTS", + "type": "text" + }, + { + "bbox": [ + 316, + 330, + 334, + 340 + ], + "score": 0.33, + "content": "+ \\mathrm { P T }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 329, + 477, + 342 + ], + "score": 1.0, + "content": ") improves DARTS’ test error from", + "type": "text" + }, + { + "bbox": [ + 477, + 329, + 504, + 340 + ], + "score": 0.83, + "content": "3 . 0 0 \\%", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 116, + 353 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 340, + 144, + 351 + ], + "score": 0.88, + "content": "2 . 6 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 340, + 505, + 353 + ], + "score": 1.0, + "content": ", with manageable search cost (0.8 GPU days). Note that by only changing the architecture", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "selection method, DARTS performs significantly better than many other differentiable NAS methods", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 376 + ], + "score": 1.0, + "content": "that enjoy carefully designed optimization process of the supernet, such as GDAS (Dong & Yang,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "2019) and SNAS (Xie et al., 2019). This empirical result suggests that architecture selection is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "crucial to DARTS: with the proper selection algorithm, DARTS remains a very competitive method.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 329, + 506, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "Our method is also able to improve the performance of other variants of DARTS. To show this, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "evaluate our method on SDARTS(rs) and SGAS (Chen & Hsieh, 2020; Li et al., 2020). SDARTS(rs)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 493, + 437 + ], + "score": 1.0, + "content": "is a variant of DARTS that regularizes the search phase by applying Gaussian perturbation to", + "type": "text" + }, + { + "bbox": [ + 493, + 425, + 501, + 433 + ], + "score": 0.68, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 422, + 506, + 437 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "score": 1.0, + "content": "Unlike DARTS and SDARTS, SGAS performs progressive search space shrinking. Concretely,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "SGAS progressively discretizes its edges with the order from most to least important, based on a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "novel edge importance score. For a fair comparison, we keep its unique search space shrinking", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "process unmodified and only replace its magnitude-based operation selection with ours. As we can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "see from Table 2, our method consistently achieves better average test errors than its magnitude-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 465, + 502 + ], + "score": 1.0, + "content": "based counterpart. Concretely, the proposed method improves SDARTS’ test error from", + "type": "text" + }, + { + "bbox": [ + 466, + 489, + 493, + 500 + ], + "score": 0.87, + "content": "2 . 6 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 133, + 510 + ], + "score": 0.91, + "content": "2 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 500, + 246, + 512 + ], + "score": 1.0, + "content": "and SGAS’ test error from", + "type": "text" + }, + { + "bbox": [ + 246, + 500, + 274, + 510 + ], + "score": 0.88, + "content": "2 . 6 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 500, + 286, + 512 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 286, + 500, + 313, + 510 + ], + "score": 0.88, + "content": "2 . 5 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 500, + 506, + 512 + ], + "score": 1.0, + "content": ". Moreover, the best architecture discovered in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 511, + 454, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 265, + 523 + ], + "score": 1.0, + "content": "our experiments achieves a test error of", + "type": "text" + }, + { + "bbox": [ + 266, + 511, + 292, + 522 + ], + "score": 0.86, + "content": "2 . 4 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 511, + 454, + 523 + ], + "score": 1.0, + "content": ", ranked top among other NAS methods.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 401, + 506, + 523 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 563, + 496, + 654 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 106, + 536, + 357, + 547 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 534, + 358, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 358, + 549 + ], + "score": 1.0, + "content": "5.2 PERFORMANCE ON NAS-BENCH-201 SEARCH SPACE", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "image_body", + "bbox": [ + 108, + 563, + 496, + 654 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 563, + 496, + 654 + ], + "spans": [ + { + "bbox": [ + 108, + 563, + 496, + 654 + ], + "score": 0.964, + "type": "image", + "image_path": "7d4b6638faec392f9f4d135d85de588ec6564e674791a16ea64c8e91e9c9bcca.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 108, + 563, + 496, + 593.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 108, + 593.3333333333334, + 496, + 623.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 108, + 623.6666666666667, + 496, + 654.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 656, + 503, + 690 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Figure 4: Trajectory of test accuracy on space NAS-Bench-201 and three datasets (Left: cifar10,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "Middle: cifar100, Right: Imagenet16-120). The test accuracy of our method is plotted by taking the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 500, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 500, + 690 + ], + "score": 1.0, + "content": "snapshots of DARTS’ supernet at corresponding epochs and run our selection method on top of it.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "To further verify the effectiveness of the proposed perturbation-based architecture selection, we con-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "duct experiments on NAS-Bench-201. 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As in section 5.1, we take the pretrained supernet from DARTS and apply our method", + "type": "text", + "cross_page": true + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "on top of it. All other settings are kept unmodified. Figure 4 shows the performance trajectory of", + "type": "text", + "cross_page": true + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "DARTS+PT compared with DARTS. While the architectures found by magnitude-based selection", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "degenerates over time, the perturbation-based method is able to extract better architectures from the", + "type": "text", + "cross_page": true + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "same underlying supernets stably. The result implies that the DARTS’ degenerated performance", + "type": "text", + "cross_page": true + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 438, + 370, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 370, + 451 + ], + "score": 1.0, + "content": "comes from the failure of magnitude based architecture selection.", + "type": "text", + "cross_page": true + } + ], + "index": 29 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 699, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 161, + 94, + 448, + 319 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 157, + 81, + 453, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 156, + 79, + 453, + 94 + ], + "spans": [ + { + "bbox": [ + 156, + 79, + 453, + 94 + ], + "score": 1.0, + "content": "Table 2: Comparison with state-of-the-art image classifiers on CIFAR-10.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 161, + 94, + 448, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 161, + 94, + 448, + 319 + ], + "spans": [ + { + "bbox": [ + 161, + 94, + 448, + 319 + ], + "score": 0.983, + "html": "
ArchitectureTest Error (%)Params (M)Search Cost (GPU days)Search Method
DenseNet-BC (Huang etal.,2017)3.4625.6-manual
NASNet-A (Zoph et al.,2018)2.653.32000RL
AmoebaNet-A (Real etal.,2019)3.34± 0.063.23150evolution
AmoebaNet-B (Real et al.,2019)2.55 ± 0.052.83150evolution
PNAS (Liu et al.,2018)*3.41 ± 0.093.2225SMBO
ENAS (Pham et al.,2018)2.894.60.5RL
NAONet (Luo et al.,2018)3.533.10.4NAO
SNAS (moderate) (Xie et al.,2019)2.85±0.022.81.5gradient
GDAS(Dong& Yang,2019)2.933.40.3gradient
BayesNAS (Zhou et al.,2019)2.81± 0.043.40.2gradient
ProxylessNAS (Cai et al.,2019)†2.085.74.0gradient
NASP(Yao et al.,2020)2.83±0.093.30.1gradient
P-DARTS (Chen et al.,2019)2.503.40.3gradient
PC-DARTS (Xu et al.,2020)2.57 ± 0.073.60.1gradient
R-DARTS (L2) Zela et al. (2020)2.95 ± 0.21-1.6gradient
DARTS (Liu et al.,2019)3.00±0.143.30.4gradient
SDARTS-RS (Chen& Hsieh,2020)2.67± 0.033.40.4gradient
SGAS(Cri 1.avg) (Li et al.,2020)2.66 ± 0.243.70.25gradient
DARTS+PT (avg)*2.61±0.083.00.8tgradient
DARTS+PT (best)2.483.30.8tgradient
SDARTS-RS+PT (avg)*2.54 ± 0.103.30.8t
SDARTS-RS+PT(best)2.443.20.8tgradient
SGAS+PT (Crit.1 avg)*2.56 ± 0.103.90.29tgradient gradient
SGAS+PT (Crit.1 best)2.463.90.29tgradient
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As in section 5.1, we take the pretrained supernet from DARTS and apply our method", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "on top of it. All other settings are kept unmodified. Figure 4 shows the performance trajectory of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "DARTS+PT compared with DARTS. While the architectures found by magnitude-based selection", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "degenerates over time, the perturbation-based method is able to extract better architectures from the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "same underlying supernets stably. The result implies that the DARTS’ degenerated performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 438, + 370, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 370, + 451 + ], + "score": 1.0, + "content": "comes from the failure of magnitude based architecture selection.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 107, + 471, + 178, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 181, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 181, + 486 + ], + "score": 1.0, + "content": "6 ANALYSIS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 499, + 309, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 311, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 311, + 512 + ], + "score": 1.0, + "content": "6.1 ISSUE WITH THE ROBUSTNESS OF DARTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 506, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "Zela et al. (2020) observes that DARTS tends to yield degenerate architectures with abysmal per-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "formance. 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(2020) (S1-S4). The complete", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 605, + 277, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 277, + 617 + ], + "score": 1.0, + "content": "specifications of these spaces can be found", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 616, + 277, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 277, + 628 + ], + "score": 1.0, + "content": "in Appendix A.2. Given a supernet, the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 627, + 277, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 229, + 639 + ], + "score": 1.0, + "content": "architecture selected based on", + "type": "text" + }, + { + "bbox": [ + 230, + 629, + 237, + 637 + ], + "score": 0.7, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 627, + 277, + 639 + ], + "score": 1.0, + "content": "performs", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 638, + 277, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 277, + 650 + ], + "score": 1.0, + "content": "poorly across spaces and datasets (column", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 648, + 277, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 277, + 660 + ], + "score": 1.0, + "content": "3 in Table 3). 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DatasetSpaceDARTSDARTS+PT(Ours)DARTS+PT(fixα)*
C10S13.843.502.86
S24.852.792.59
S33.342.492.52
S47.202.642.58
C100S129.4624.4824.40
S226.0523.1623.30
S328.9022.03
S422.8520.8021.94 20.66
SVHNS14.582.622.39
S23.532.532.32
S33.412.422.32
S43.052.422.39
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ArchitectureTest Error (%)Params (M)Search Cost (GPU days)Search Method
DenseNet-BC (Huang etal.,2017)3.4625.6-manual
NASNet-A (Zoph et al.,2018)2.653.32000RL
AmoebaNet-A (Real etal.,2019)3.34± 0.063.23150evolution
AmoebaNet-B (Real et al.,2019)2.55 ± 0.052.83150evolution
PNAS (Liu et al.,2018)*3.41 ± 0.093.2225SMBO
ENAS (Pham et al.,2018)2.894.60.5RL
NAONet (Luo et al.,2018)3.533.10.4NAO
SNAS (moderate) (Xie et al.,2019)2.85±0.022.81.5gradient
GDAS(Dong& Yang,2019)2.933.40.3gradient
BayesNAS (Zhou et al.,2019)2.81± 0.043.40.2gradient
ProxylessNAS (Cai et al.,2019)†2.085.74.0gradient
NASP(Yao et al.,2020)2.83±0.093.30.1gradient
P-DARTS (Chen et al.,2019)2.503.40.3gradient
PC-DARTS (Xu et al.,2020)2.57 ± 0.073.60.1gradient
R-DARTS (L2) Zela et al. (2020)2.95 ± 0.21-1.6gradient
DARTS (Liu et al.,2019)3.00±0.143.30.4gradient
SDARTS-RS (Chen& Hsieh,2020)2.67± 0.033.40.4gradient
SGAS(Cri 1.avg) (Li et al.,2020)2.66 ± 0.243.70.25gradient
DARTS+PT (avg)*2.61±0.083.00.8tgradient
DARTS+PT (best)2.483.30.8tgradient
SDARTS-RS+PT (avg)*2.54 ± 0.103.30.8t
SDARTS-RS+PT(best)2.443.20.8tgradient
SGAS+PT (Crit.1 avg)*2.56 ± 0.103.90.29tgradient gradient
SGAS+PT (Crit.1 best)2.463.90.29tgradient
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DatasetSpaceDARTSDARTS+PT(Ours)DARTS+PT(fixα)*
C10S13.843.502.86
S24.852.792.59
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S47.202.642.58
C100S129.4624.4824.40
S226.0523.1623.30
S328.9022.03
S422.8520.8021.94 20.66
SVHNS14.582.622.39
S23.532.532.32
S33.412.422.32
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The complete architecture can be found in Appendix A.9.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 256, + 262, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 262, + 269 + ], + "score": 1.0, + "content": "Non-trivial operations are marked red.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + } + ], + "index": 4.75 + }, + { + "type": "title", + "bbox": [ + 107, + 293, + 227, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 229, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 229, + 305 + ], + "score": 1.0, + "content": "6.2 PROGRESSIVE TUNING", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 296, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 297, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 297, + 328 + ], + "score": 1.0, + "content": "In addition to operation selection, we also tune", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 327, + 297, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 297, + 339 + ], + "score": 1.0, + "content": "the supernet after an edge is discretized so", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 337, + 297, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 297, + 350 + ], + "score": 1.0, + "content": "that the supernet could regain the lost accu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 347, + 297, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 297, + 362 + ], + "score": 1.0, + "content": "racy. 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Learning transferable architectures", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 115, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "for scalable image recognition. 2018 IEEE/CVF Conference on Computer Vision and Pattern", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 432, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 115, + 432, + 504, + 443 + ], + "score": 1.0, + "content": "Recognition, Jun 2018. doi: 10.1109/cvpr.2018.00907. URL http://dx.doi.org/10.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 116, + 442, + 241, + 453 + ], + "spans": [ + { + "bbox": [ + 116, + 442, + 241, + 453 + ], + "score": 1.0, + "content": "1109/CVPR.2018.00907.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 409, + 505, + 453 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 81, + 182, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "score": 1.0, + "content": "A APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 114, + 329, + 125 + ], + "lines": [ + { + "bbox": [ + 106, + 114, + 330, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 330, + 126 + ], + "score": 1.0, + "content": "A.1 DESCRIPTION ABOUT OUR BASELINE MODELS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 138, + 176, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 178, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 178, + 152 + ], + "score": 1.0, + "content": "A.1.1 DARTS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 176 + ], + "score": 1.0, + "content": "DARTS (Liu et al., 2019) is a pioneering work that introduces the general differentiable NAS frame-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 174, + 504, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 504, + 187 + ], + "score": 1.0, + "content": "work, which we reviewed in section 2. In DARTS, the topology and operation are searched together.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "Concretely, at the end of the search, it selects one operation for every edge in the normal (reduction)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 197, + 504, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 268, + 208 + ], + "score": 1.0, + "content": "cell based on the architecture parameter", + "type": "text" + }, + { + "bbox": [ + 268, + 199, + 276, + 207 + ], + "score": 0.67, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 197, + 504, + 208 + ], + "score": 1.0, + "content": ". Then it selects two input edges for each node in the cell", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 208, + 220 + ], + "score": 1.0, + "content": "by comparing the largest", + "type": "text" + }, + { + "bbox": [ + 209, + 209, + 217, + 217 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "of every input edge. The final architecture consists of one operation on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 219, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 231 + ], + "score": 1.0, + "content": "each of the eight edges in the normal (reduction) cell. The operation on an edge will be selected", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 358, + 242 + ], + "score": 1.0, + "content": "from a pool of seven candidates: skip connection, avg pool", + "type": "text" + }, + { + "bbox": [ + 358, + 230, + 376, + 240 + ], + "score": 0.26, + "content": "{ } _ { - 3 \\mathrm { x } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 229, + 421, + 242 + ], + "score": 1.0, + "content": ", max pool", + "type": "text" + }, + { + "bbox": [ + 422, + 230, + 440, + 240 + ], + "score": 0.28, + "content": "{ } _ { - 3 \\mathrm { x } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 229, + 505, + 242 + ], + "score": 1.0, + "content": ", sep conv 3x3,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "sep conv 5x5, dil conv 3x3, and dil conv 5x5. In addition to these operations, DARTS also main-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "tains a ”none” op, which is used exclusively for determining the topology rather than treated as an", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "operation (Liu et al., 2019). Since the main focus of our paper is operation assignment, we omit", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 376, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 376, + 286 + ], + "score": 1.0, + "content": "none op when applying the proposed selection method on DARTS.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 309, + 182, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 183, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 183, + 323 + ], + "score": 1.0, + "content": "A.1.2 SDARTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "SDARTS (Chen & Hsieh, 2020) is a variant of DARTS aiming at regularizing the bilevel opti-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "mization process in DARTS via random Gaussian perturbation, inspired by the recent finding that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "regularizing DARTS’ supernet leads to improved performance. While the optimization of architec-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 167, + 379 + ], + "score": 1.0, + "content": "ture parameter", + "type": "text" + }, + { + "bbox": [ + 168, + 369, + 176, + 377 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 367, + 475, + 379 + ], + "score": 1.0, + "content": "in SDARTS is identical to DARTS, it distorts the architecture parameters", + "type": "text" + }, + { + "bbox": [ + 475, + 369, + 483, + 377 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 336, + 390 + ], + "score": 1.0, + "content": "random Gaussian noise while training the model weights", + "type": "text" + }, + { + "bbox": [ + 336, + 379, + 345, + 388 + ], + "score": 0.68, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 377, + 505, + 390 + ], + "score": 1.0, + "content": ". This simple yet effective regularizer is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 389, + 398, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 398, + 402 + ], + "score": 1.0, + "content": "able to consistently improve the robustness and performance of DARTS.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 170, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 171, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 171, + 438 + ], + "score": 1.0, + "content": "A.1.3 SGAS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 108, + 449, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "SGAS (Li et al., 2020) represents a line of research on improving the search efficiency of differen-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 457, + 473 + ], + "score": 1.0, + "content": "tiable NAS by progressively shrinking the search space. 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The edge selection is based on the ranking of the proposed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 502, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 502, + 505 + ], + "score": 1.0, + "content": "edge importance score. The process stops after all eight edges of the final architecture are decided.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 307, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 308, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 308, + 543 + ], + "score": 1.0, + "content": "A.2 MICROARCHITECTURE OF SPACE S1 - S4", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 556, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "Zela et al. (2020) introduces four variants of the DARTS’ space (S1, S2, S3, and S4) to study the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "robustness of DARTS. These spaces differ from DARTS’ original space only in the number and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 578, + 422, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 422, + 592 + ], + "score": 1.0, + "content": "types of operations on each edge. Apart from that, everything else is the same.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 129, + 608, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 132, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 132, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "• S1 is a pre-optimized search space consisting of top2 operations selected by DARTS. 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The number of fine-tuning epochs is set to 5 for all", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "experiments because empirically we find that it is enough for the supernet to converge again. In this", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "section, we conduct an ablation study on the effect of the number of fine-tuning epochs. 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ArchitectureTest Error (%)Params (M)Search Cost (GPU days)Search Method
DenseNet-BC (Huang etal.,2017)3.4625.6-manual
NASNet-A (Zoph et al.,2018)2.653.32000RL
AmoebaNet-A (Real etal.,2019)3.34± 0.063.23150evolution
AmoebaNet-B (Real et al.,2019)2.55 ± 0.052.83150evolution
PNAS (Liu et al.,2018)*3.41 ± 0.093.2225SMBO
ENAS (Pham et al.,2018)2.894.60.5RL
NAONet (Luo et al.,2018)3.533.10.4NAO
SNAS (moderate) (Xie et al.,2019)2.85±0.022.81.5gradient
GDAS(Dong& Yang,2019)2.933.40.3gradient
BayesNAS (Zhou et al.,2019)2.81± 0.043.40.2gradient
ProxylessNAS (Cai et al.,2019)†2.085.74.0gradient
NASP(Yao et al.,2020)2.83±0.093.30.1gradient
P-DARTS (Chen et al.,2019)2.503.40.3gradient
PC-DARTS (Xu et al.,2020)2.57 ± 0.073.60.1gradient
R-DARTS (L2) Zela et al. (2020)2.95 ± 0.21-1.6gradient
DARTS (Liu et al.,2019)3.00±0.143.30.4gradient
SDARTS-RS (Chen& Hsieh,2020)2.67± 0.033.40.4gradient
SGAS(Cri 1.avg) (Li et al.,2020)2.66 ± 0.243.70.25gradient
DARTS+PT (avg)*2.61±0.083.00.8tgradient
DARTS+PT (best)2.483.30.8tgradient
SDARTS-RS+PT (avg)*2.54 ± 0.103.30.8t
SDARTS-RS+PT(best)2.443.20.8tgradient
SGAS+PT (Crit.1 avg)*2.56 ± 0.103.90.29tgradient gradient
SGAS+PT (Crit.1 best)2.463.90.29tgradient
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DatasetSpaceDARTSDARTS+PT(Ours)DARTS+PT(fixα)*
C10S13.843.502.86
S24.852.792.59
S33.342.492.52
S47.202.642.58
C100S129.4624.4824.40
S226.0523.1623.30
S328.9022.03
S422.8520.8021.94 20.66
SVHNS14.582.622.39
S23.532.532.32
S33.412.422.32
S43.052.422.39
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SpaceDARTSDARTS+PTDARTS+PT (fix α)
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each new task that an agent needs to learn. In other machine learning fields, such as natural language processing or computer vision, pre-training on large, previously collected datasets to bootstrap learning for new tasks has emerged as a powerful paradigm to reduce data requirements when learning a new task. In this paper, we ask the following question: how can we enable similarly useful pre-training for RL agents? We propose a method for pre-training behavioral priors that can capture complex input-output relationships observed in successful trials from a wide range of previously seen tasks, and we show how this learned prior can be used for rapidly learning new tasks without impeding the RL agent’s ability to try out novel behaviors. We demonstrate the effectiveness of our approach in challenging robotic manipulation domains involving image observations and sparse reward functions, where our method outperforms prior works by a substantial margin. Additional materials can be found on our project website: https://sites.google.com/view/parrot-rl + +# 1 INTRODUCTION + +Reinforcement Learning (RL) is an attractive paradigm for robotic learning because of its flexibility in being able to learn a diverse range of skills and its capacity to continuously improve. However, RL algorithms typically require a large amount of data to solve each individual task, including simple ones. Since an RL agent is generally initialized without any prior knowledge, it must try many largely unproductive behaviors before it discovers a high-reward outcome. In contrast, humans rarely attempt to solve new tasks in this way: they draw on their prior experience of what is useful when they attempt a new task, which substantially shrinks the task search space. For example, faced with a new task involving objects on a table, a person might grasp an object, stack multiple objects, or explore other object rearrangements, rather than re-learning how to move their arms and fingers. + +Can we endow RL agents with a similar sort of behavioral prior from past experience? In other fields of machine learning, the use of large prior datasets to bootstrap acquisition of new capabilities has been studied extensively to good effect. For example, language models trained on large, diverse datasets offer representations that drastically improve the efficiency of learning downstream tasks (Devlin et al., 2019). What would be the analogue of this kind of pre-training in robotics and RL? One way we can approach this problem is to leverage successful trials from a wide range of previously seen tasks to improve learning for new tasks. The data could come from previously learned policies, from human demonstrations, or even unstructured teleoperation of robots (Lynch et al., 2019). In this paper, we show that behavioral priors can be obtained through representation learning, and the representation in question must not only be a representation of inputs, but actually a representation of input-output relationships – a space of possible and likely mappings from states to actions among which the learning process can interpolate when confronted with a new task. + +What makes for a good representation for RL? Given a new task, a good representation must (a) provide an effective exploration strategy, (b) simplify the policy learning problem for the RL algorithm, and (c) allow the RL agent to retain full control over the environment. In this paper, we address + +![](images/3cf924a289e0fcdc95a90d419c43c1d5e5e637b657ba0f9ca4072d113a28f90e.jpg) +Figure 1: Our problem setting. Our training dataset consists of near-optimal state-action trajectories (without reward labels) from a wide range of tasks. Each task might involve interacting with a different set of objects. Even for the same set of objects, the task can be different depending on our objective. For example, in the upper right corner, the objective could be picking up a cup, or it could be to place the bottle on the yellow cube. We learn a behavioral prior from this multi-task dataset capable of trying many different useful behaviors when placed in a new environment, and can aid an RL agent to quickly learn a specific task in this new environment. + +Transfer Behavioral all of these challenges through learning an invertible function that maps noise vectors to complex, Prior Learning New Taskshigh-dimensional environment actions. Building on prior work in normalizing flows (Dinh et al., 2017), we train this mapping to maximize the (conditional) log-likelihood of actions observed in Trial 2successful trials from past tasks. When dropped into a new MDP, the RL agent can now sample from a unit Gaussian, and use the learned mapping (which we refer to as the behavioral prior) to generate likely environment actions, conditional on the current observation. This learned mapping essentially transforms the original MDP into a simpler one for the RL agent, as long as the original Trial n-1 MDP shares (partial) structure with previously seen MDPs (see Section 3). Furthermore, since this mapping is invertible, the RL agent still retains full control over the original MDP: for every possible Trial n environment action, there exists a point within the support of the Gaussian distribution that maps to that action. This allows the RL agent to still try out new behaviors that are distinct from what was previously observed. + +Our main contribution is a framework for pre-training in RL from a diverse multi-task dataset, which produces a behavioral prior that accelerates acquisition of new skills. We present an instantiation of this framework in robotic manipulation, where we utilize manipulation data from a diverse range of prior tasks to train our behavioral prior, and then use it to bootstrap exploration for new tasks. By making it possible to pre-train action representations on large prior datasets for robotics and RL, we hope that our method provides a path toward leveraging large datasets in the RL and robotics settings, much like language models can leverage large text corpora in NLP and unsupervised pretraining can leverage large image datasets in computer vision. Our method, which we call Prior AcceleRated ReinfOrcemenT (PARROT), is able to quickly learn tasks that involve manipulating previously unseen objects, from image observations and sparse rewards, in settings where RL from scratch fails to learn a policy at all. We also compare against prior works that incorporate prior data for RL, and show that PARROT substantially outperforms these prior works. + +# 2 RELATED WORK + +Combining RL with demonstrations. Our work is related to methods for learning from demonstrations (Pomerleau, 1989; Schaal et al., 2003; Ratliff et al., 2007; Pastor et al., 2009; Ho & Ermon, 2016; Finn et al., 2017b; Giusti et al., 2016; Sun et al., 2017; Zhang et al., 2017; Lynch et al., 2019). While demonstrations can also be used to speed up RL (Schaal, 1996; Peters & Schaal, 2006; Kormushev et al., 2010; Hester et al., 2017; Vecer´ık et al., 2017; Nair et al., 2018; Rajeswaran et al., 2018; Silver et al., 2018; Peng et al., 2018; Johannink et al., 2019; Gupta et al., 2019), this usually requires collecting demonstrations for the specific task that is being learned. In contrast, we use data from a wide range of other prior tasks to speed up RL for a new task. As we show in our experiments, PARROT is better suited to this problem setting when compared to prior methods that combine imitation and RL for the same task. + +Generative modeling and RL. Several prior works model multi-modal action distributions using expert trajectories from different tasks. IntentionGAN (Hausman et al.) and InfoGAIL (Li et al., 2017) learn multi-modal policies via interaction with an environment using an adversarial imitation approach (Ho & Ermon, 2016), but we learn these distributions only from data. Other works learn these distributions from data (Xie et al., 2019; Rhinehart et al., 2020) and utilize them for planning at test time to optimize a user-provided cost function. In contrast, we use the behavioral prior to augment model-free RL of a new task. This allows us to learn policies for new tasks that may be substantially different from prior tasks, since we can collect data specific to the new task at hand, and we do not explicitly need to model the environment, which can be complicated for high-dimensional state and action spaces, such as when performing continuous control from images observations. Another line of work (Ghadirzadeh et al., 2017; Ham¨ al¨ ainen et al. ¨ , 2019; Ghadirzadeh et al., 2020) explores using generative models for RL, using a variational autoencoder (Kingma & Welling, 2014) to model entire trajectories in an observation-independent manner, and then learning an open-loop, single-step policy using RL to solve the downstream task. Our approach differs in several key aspects: (1) our model is observation-conditioned, allowing it to prioritize actions that are relevant to the current scene or environment, (2) our model allows for closed-loop feedback control, and (3) our model is invertible, allowing the high-level policy to retain full control over the action space. Our experiments demonstrate these aspects are crucial for solving harder tasks. + +Hierarchical learning. Our method can be interpreted as training a hierarchical model: the lowlevel policy is the behavioral prior trained on prior data, while the high-level policy is trained using RL and controls the low-level policy. This structure is similar to prior work in hierarchical RL (Dayan & Hinton, 1992; Parr & Russell, 1997; Dietterich, 1998; Sutton et al., 1999; Kulkarni et al., 2016). We divide prior work in hierarchical learning into two categories: methods that seek to learn both the low-level and high-level policies through active interaction with an environment (Kupcsik et al., 2013; Heess et al., 2016; Bacon et al., 2017; Florensa et al., 2017; Haarnoja et al., 2018a; Nachum et al., 2018; Chandak et al., 2019; Peng et al., 2019), and methods that learn temporally extended actions, also known as options, from demonstrations, and then recompose them to perform long-horizon tasks through RL or planning (Fox et al., 2017; Krishnan et al., 2017; Kipf et al., 2019; Shankar et al., 2020; Shankar & Gupta, 2020). Our work shares similarities with the data-driven approach of the latter methods, but work on options focuses on modeling the temporal structure in demonstrations for a small number of long-horizon tasks, while our behavioral prior is not concerned with temporally-extended abstractions, but rather with transforming the original MDP into one where potentially useful behaviors are more likely, and useless behaviors are less likely. + +Meta-learning. Our goal in this paper is to utilize data from previously seen tasks to speed up RL for new tasks. Meta-RL (Duan et al., 2016; Wang et al., 2016; Finn et al., 2017a; Mishra et al., 2017; Rakelly et al., 2019; Mendonca et al., 2019; Zintgraf et al., 2020; Fakoor et al., 2020) and meta-imitation methods (Duan et al., 2017; Finn et al., 2017c; Huang et al., 2018; James et al., 2018; Paine et al., 2018; Yu et al., 2018; Huang et al., 2019; Zhou et al., 2020) also seek to speed up learning for new tasks by leveraging experience from previously seen tasks. While meta-learning provides an appealing and principled framework to accelerate acquisition of future tasks, we focus on a more lightweight approach with relaxed assumptions that make our method more practically applicable, and we discuss these assumptions in detail in the next section. + +# 3 PROBLEM SETUP + +Our goal is to improve an agent’s ability to learn new tasks by incorporating a behavioral prior, which it can acquire from previously seen tasks. Each task can be considered a Markov decision process (MDP), which is defined by a tuple $( S , \mathcal { A } , \mathrm { T } , r , \gamma )$ , where $s$ and $\mathcal { A }$ represent state and action spaces, $\mathrm { T } ( s ^ { \prime } | s , a )$ and $r ( s , a )$ represent the dynamics and reward functions, and $\gamma \in ( 0 , 1 )$ represents the discount factor. Let $p ( M )$ denote a distribution over such MDPs, with the constraint that the state and action spaces are fixed. In our experiments, we treat high-dimensional images as $s$ , which means that this constraint is not very restrictive in practice. In order for the behavioral prior to be able to accelerate the acquisition of new skills, we assume the behavioral prior is trained on data that structurally resembles potential optimal policies for all or part of the new task. For example, if the new task requires placing a bottle in a tray, the prior data might include some behaviors that involve picking up objects. There are many ways to formalize this assumption. One way to state this formally is to assume that prior data consists of executions of near-optimal policies for MDPs drawn according to $M \sim p ( M )$ , and the new task $M ^ { \star }$ is likewise drawn from $p ( M )$ . In this case, the generative process for the prior data can be expressed as: + +![](images/4f170d5f78598944c4582be21171bfae4492ee08968d8d88f79989f97c7ab666.jpg) +Figure 2: PARROT. Using successful trials from a large variety of tasks, we learn an invertible mapping $f _ { \phi }$ s that maps noise $z$ to useful actions $a$ Z0 Z1 Z2 Z3=a . This mapping is conditioned on the current observation, which in our case is an RGB image. The image is passed through a stack of convolutional layers and flattened to obtain an image encoding $\psi ( s )$ , and this image encoding is then used to condition each individual transformation $f _ { i }$ pi(z|s)of our overall mapping function $f _ { \phi }$ . The parameters of the mapping (including the convolutional encoder) are a learned through maximizing the conditional log-likelihood of state-action pairs observed in the dataset. When learning a new task, this mapping can simplify the MDP for an RL agent by mapping actions sampled from a randomly initialized policy to actions that are likely to lead to useful behavior in the current scene. Since the mapping is invertible, the RL agent still retains full control over the action space of the original MDP, simply the likelihood of executing a useful action is increased through use of the pre-trained mapping. + +$$ +M \sim p ( M ) , \quad \pi _ { M } ( \tau ) = \arg \operatorname* { m a x } _ { \pi } \mathbb { E } _ { \pi , M } [ R _ { M } ] , \quad \tau _ { M } \sim \pi _ { M } ( \tau ) , +$$ + +where $\tau _ { M } = ( s _ { 1 } , a _ { 1 } , s _ { 2 } , a _ { 2 } , . . . , s _ { T } , a _ { T } )$ is a sequence of state and actions, $\pi _ { M } ( \tau )$ denotes a nearoptimal policy (Kearns & Singh, 2002) for MDP $M$ and $\textstyle R _ { M } = \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t }$ . When incorporating the behavioral prior for learning a new task $M ^ { \star }$ , our goal is the same as standard RL: to find a policy $\pi$ that maximizes the expected return ar $\operatorname { \mu } _ { 5 } \operatorname* { m a x } _ { \pi } \mathbb { E } _ { \pi , M ^ { \star } } [ R _ { M ^ { \star } } ]$ . Our assumption on tasks being drawn from a distribution $p ( M )$ shares similarities with the meta-RL problem (Wang et al., 2016; Duan et al., 2016), but our setup is different: it does not require accessing any task in $p ( M )$ except the new task we are learning, $M ^ { \star }$ . Meta-RL methods need to interact with the tasks in $p ( M )$ during meta-training, with access to rewards and additional samples, whereas we learn our behavioral prior simply from data, without even requiring this data to be labeled with rewards. This is of particular importance for real-world problem settings such as robotics: it is much easier to store data from prior tasks (e.g., different environments) than to have a robot physically revisit those prior settings and retry those tasks, and not requiring known rewards makes it possible to use data from a variety of sources, including human-provided demonstrations. In our setting, RL is performed in only one environment, while the prior data can come from many environments. + +Our setting is related to meta-imitation learning (Duan et al., 2017; Finn et al., 2017c), as we speed up learning new tasks using data collected from past tasks. However, meta-imitation learning methods require at least one demonstration for each new task, whereas our method can learn new tasks without any demonstrations. Further, our data requirements are less stringent: meta-imitation learning methods require all demonstrations to be optimal, require all trajectories in the dataset to have a task label, and requires “paired demonstrations”, i.e. at least two demonstrations for each task (since meta-imitation methods maximize the likelihood of actions from one demonstration after conditioning the policy on another demonstration from the same task). Relaxing these requirements increases the scalability of our method: we can incorporate data from a wider range of sources, and we do not need to explicitly organize it into specific tasks. + +# 4 BEHAVIORAL PRIORS FOR REINFORCEMENT LEARNING + +Our method learns a behavioral prior for downstream RL by utilizing a dataset $\mathcal { D }$ of (near-optimal) state-action pairs from previously seen tasks. We do so by learning a state-conditioned mapping $f _ { \phi } \colon { \mathcal { Z } } \times { \mathcal { S } } \to { \mathcal { A } }$ (where $\phi$ denotes learnable parameters) that transforms a noise vector $z$ into an action $a$ that is likely to be useful in the current state $s$ . This removes the need for exploring via “meaningless” random behavior, and instead enables an exploration process where the agent attempts behaviors that have been shown to be useful in previously seen domains. For example, if a robotic arm is placed in front of several objects, randomly sampling $z$ (from a simple distribution, such as the unit Gaussian) and applying the mapping $a = f _ { \phi } ( z ; s )$ should result in actions that, when executed, result in meaningful interactions with the objects. This learned mapping essentially transforms the MDP experienced by the RL agent into a simpler one, where every random action executed in this transformed MDP is much more likely to lead to a useful behavior. + +How can we learn such a mapping? In this paper, we propose to learn this mapping through stateconditioned generative modeling of the actions observed in the original dataset $\mathcal { D }$ , and we refer to this state-conditioned distribution over actions as the behavioral prior $p _ { \mathrm { p r i o r } } ( a | s )$ . A deep generative model takes noise as input, and outputs a plausible sample from the target distribution, i.e. it can represent $p _ { \mathrm { p r i o r } } ( a | s )$ as a distribution over noise $z$ using a deterministic mapping $f _ { \phi } : \mathcal { Z } \times \mathcal { S } \mapsto A$ When learning a new task, we can use this mapping to reparametrize the action space of the RL agent: if the action chosen by the randomly initialized neural network policy is $z$ , then we execute the action $a = f _ { \phi } ( z ; s )$ in the original MDP, and learn a policy $\pi ( \boldsymbol { z } | \boldsymbol { s } )$ that maximizes the task reward through learning to control the inputs to the mapping $f _ { \phi }$ . The training of the behavioral prior and the task-specific policy is decoupled, allowing us to mix and match RL algorithms and generative models to best suit the application of interest. An overview of our overall architecture is depicted in Figure 2. In the next subsection, we discuss what properties we would like the behavioral prior to satisfy, and present one particular choice for learning a prior that satisfies all of these properties. + +# 4.1 LEARNING A BEHAVIORAL PRIOR WITH NORMALIZING FLOWS + +For the behavioral prior to be effective, it needs to satisfy certain properties. Since we learn the prior from a multi-task dataset, containing several different behaviors even for the same initial state, the learned prior should be capable of representing complex, multi-modal distributions. Second, it should provide a mapping for generating “useful” actions from noise samples when learning a new task. Third, the prior should be state-conditioned, so that only actions that are relevant to the current state are sampled. And finally, the learned mapping should allow easier learning in the reparameterized action space without hindering the RL agent’s ability to attempt novel behaviors, including actions that might not have been observed in the dataset $\mathcal { D }$ . Generative models based on normalizing flows (Dinh et al., 2017) satisfy all of these properties well: they allow maximizing the model’s exact log-likelihood of observed examples, and learn a deterministic, invertible mapping that transforms samples from a simple distribution $p _ { z }$ to examples observed in the training dataset. In particular, the real-valued non-volume preserving (real NVP) architecture introduced by Dinh et al. (2017) allows using deep neural networks to parameterize this mapping (making it expressive) While the original real NVP work modelled unconditional distributions, follow-up work has found that it can be easily extended to incorporate conditioning information (Ardizzone et al., 2019). We refer the reader to prior work (Dinh et al., 2017) for a complete description of real NVPs, and summarize its key features here. Given an invertible mapping $a = f _ { \phi } ( z ; s )$ , the change of variable formula allows expressing the likelihood of the observed actions using samples from $\mathcal { D }$ in the following way: + +$$ +p _ { \mathrm { p r i o r } } ( a | s ) = p _ { z } \left( f _ { \phi } ^ { - 1 } ( a ; s ) \right) \left| \operatorname* { d e t } \left( \partial f _ { \phi } ^ { - 1 } ( a ; s ) / \partial a \right) \right| +$$ + +Dinh et al. (2017) propose a particular (unconditioned) form of the invertible mapping $f _ { \phi }$ , called an affine coupling layer, that maintains tractability of the likelihood term above, while still allowing the mapping $f _ { \phi }$ to be expressive. Several coupling layers can be composed together to transform simple noise vectors into samples from complex distributions, and each layer can be conditioned on other variables, as shown in Figure 2. + +# 4.2 ACCELERATED REINFORCEMENT LEARNING VIA BEHAVIORAL PRIORS + +After we obtain the mapping $f _ { \phi } ( z ; s )$ from the behavioral prior learned by maximizing the likelihood term in Equation 2, we would like to use it to accelerate RL when solving a new task. Instead of learning a policy $\pi _ { \boldsymbol { \theta } } ( a | s )$ that directly executes its actions in the original MDP, we learn a policy $\pi _ { \boldsymbol { \theta } } { \left( z | \boldsymbol { s } \right) }$ , and execute an action in the environment according to $a = f _ { \phi } ( z ; s )$ . As shown in Figure 2, this essentially transforms the MDP experienced for the RL agent into one where random actions $z \sim p _ { z }$ (where $p _ { z }$ is the base distribution used for training the mapping $f _ { \phi }$ ) are much more likely to result in useful behaviors. To enable effective exploration at the start of the learning period, we initialize the RL policy to the base distribution used for training the prior, so that at the beginning of training, $\pi _ { \theta } ( z | s ) \mathbf { \bar { \theta } } { : = } \bar { p _ { z } } ( z )$ . Since the mapping $f _ { \phi }$ is invertible, the RL agent still retains full control over the action space: for any given $a$ , it can always find a $z$ that generates $z = f _ { \phi } ^ { - 1 } ( a ; s )$ in the original MDP. The learned mapping increases the likelihood of useful actions without crippling the RL agent, making it ideal for fine-tuning from task-specific data. Our complete method is described in Algorithm 1 in Appendix A. Note that we need to learn the mapping $f _ { \phi }$ only once, and it can be used for accelerated learning of any new task. + +# 4.3 IMPLEMENTATION DETAILS + +We use real NVP to learn $f _ { \phi }$ , and as shown in Figure 2, each coupling layer in the real NVP takes as input the the output of the previous coupling layer, and the conditioning information. The conditioning information in our case corresponds to RGB image observations, which allows us to train a single behavioral prior across a wide variety of tasks, even when the tasks might have different underlying states (for example, different objects). We train a real NVP model with four coupling layers; the exact architecture, and other hyperparameters, are detailed in Appendix B. The behavioral prior can be combined with any RL algorithm that is suitable for continuous action spaces, and we chose to use the soft actor-critic (Haarnoja et al., 2018b) due to its stability and ease of use. + +# 5 EXPERIMENTS + +Our experiments seek to answer: (1) Can the behavioral prior accelerate learning of new tasks? (2) How does PARROT compare to prior works that accelerate RL with demonstrations? (3) How does PARROT compare to prior methods that combine hierarchical imitation with RL? + +Domains. We evaluate our method on a suite of challenging robotic manipulation tasks, a subset of which are depicted in Figure 3. Each task involves controlling a 6-DoF robotic arm and its gripper, with a 7D action space. The observation is a $4 8 \times 4 8$ RGB image, which allows us to use the same observation representation across tasks, even though each underlying task might have a different underlying state (e.g., different objects). No other observations (such as joint angles or end-effector positions) are provided. In each task, the robot needs to interact with one or two objects in the scene to achieve its objective, and there are three objects in each scene. Note that all of the objects in the test scenes are novel – the dataset $\mathcal { D }$ contains no interactions with these objects. The object positions at the start of each trial are randomized, and the policy must infer these positions from image observations in order to successfully solve the task. A reward of $+ 1$ is provided when the objective for the task is achieved, and the reward is zero otherwise. Detailed information on the objective for each task and example rollouts are provided in Appendix C.1, and on our anonymous project website1. + +Data collection. Our behavioral prior is trained on a diverse dataset of trajectories from a wide range of tasks, and then utilized to accelerate reinforcement learning of new tasks. As discussed in Section 3, for the prior to be effective, it needs to be trained on a dataset that structurally resembles the behaviors that might be optimal for the new tasks. In our case, all of the behaviors involve repositioning objects (i.e., picking up objects and moving them to new locations), which represents a very general class of tasks that might be performed by a robotic arm. While the dataset can be collected in many ways, such as from human demonstrations or prior tasks solved by the robot, we collect it using a set of randomized scripted policies, see Appendix C.2 for details. Since the policies are randomized, not every execution of such a policy results in a useful behavior, and we decide to keep or discard a collected trajectory based on a simple predefined rule: if the trajectory collected ends with a successful grasp or rearrangement of any one of the objects in the scene, we add this trajectory to our dataset. We collect a dataset of 50K trajectories, where each trajectory is of length 25 timesteps ${ \approx } 5 – 6$ seconds), for a total of $1 . 2 5 \mathrm { m }$ observation-action pairs. The observation is a $4 8 \times 4 8$ RGB image, while the actions are continuous 7D vectors. Data collection involves interactions with over 50 everyday objects (see Appendix C.3); the diversity of this dataset enables learning priors that can produce useful behavior when interacting with a new object. + +![](images/b36fce4e54479f5d06fd57353f9add4588c8c16905918bb6288c534601591ccb.jpg) +Figure 3: Tasks. A subset of our evaluation tasks, with one task shown in each row. In the first task (first row), the objective is to pick up a can and place it in the pan. In the second task, the robot must pick up the vase and put it in the basket. In the third task, the goal is to place the chair on top of the checkerboard. In the fourth task, the robot must pick up the mug and hold it above a certain height. Initial positions of all objects are randomized, and must be inferred from visual observations. Not all objects in the scene are relevant to the current task. + +# 5.1 RESULTS, COMPARISONS AND ANALYSIS + +To answer the questions posed at the start of this section, we compare PARROT against a number of prior works, as well as ablations of our method. Additional implementation details and hyperparameters can be found in Appendix B. + +Soft-Actor Critic (SAC). For a basic RL comparison, we compare against the vanilla soft-actor critic algorithm (Haarnoja et al., 2018b), which does not incorporate any previously collected data. + +SAC with demonstrations (BC-SAC). We compare against a method that incorporates demonstrations to speed up learning of new tasks. In particular, we initialize the SAC policy by performing behavioral cloning on the entire dataset $\mathcal { D }$ , and then fine-tune it using SAC. This approach is similar to what has been used in prior work (Rajeswaran et al., 2018), except we use SAC as our RL algorithm. While we did test other methods that are designed to use demonstration data with RL such as DDPGfD (Vecer´ık et al., 2017) and AWAC (Nair et al., 2020), we found that our simple $\mathrm { B C } + \mathrm { S A C }$ variant performed better. This somewhat contradicts the results reported in prior work (Nair et al., 2020), but we believe that this is because the prior data is not labeled with rewards (all transitions are assigned a reward of 0), and more powerful demonstration $+ \mathrm { R L }$ methods require access to these rewards, and subsequently struggle due to the reward misspecification. + +Transfer Learning via Feature Learning (VAE-features). We compare against prior methods for transfer learning (in RL) that involve learning a robust representation of the input observation. Similar to Higgins et al. (2017b), we train a $\beta$ -VAE using the observations in our training set, and train a policy on top of the features learned by this VAE when learning downstream tasks. + +Trajectory modeling and RL (TrajRL). Ghadirzadeh et al. (2020) model entire trajectories using a VAE, and learn a one-step policy on top of the VAE to solve tasks using RL. Our implementation of this method uses a VAE architecture identical to the original paper’s, and we then train a policy using SAC to solve new tasks with the action space induced by the VAE. We performed additional hyperparameter tuning for this comparison, the details of which can be found in Appendix B. + +Hierarchical imitation and RL (HIRL). Prior works in hierarchical imitation learning (Fox et al., 2017; Shankar & Gupta, 2020) train latent variable models over expert demonstrations to discover options, and later utilize these options to learn longhorizon tasks using RL. While PARROT can also be extended to model the temporal structure in trajectories through conditioning on past states and actions, by modeling $p _ { \mathrm { p r i o r } } ( a _ { t } , \vert s _ { t } , s _ { t - 1 } , . . . , a _ { t - 1 } , . . . , a _ { 0 } )$ instead of $p _ { \mathrm { p r i o r } } ( a _ { t } | s _ { t } )$ , we focus on a simpler version of the model in this paper that does not condition on the past. In order to provide a fair comparison, we modify the model proposed by Shankar & Gupta (2020) to remove the past conditioning, which then reduces to training a conditional VAE, and performing RL on the action space induced by the latent space of this VAE. This comparison is similar to our proposed approach, but with one crucial difference: the mapping we learn is invertible, and allows the + +![](images/133c093c644f1ff87690827386b4e12b15f59b8f9313eac9f9a617f7c0e23011.jpg) +Figure 4: We plot trajectories from executing a random policy, with and without the behavioral prior. We see that the behavioral prior substantially increases the likelihood of executing an action that is likely to lead to a meaningful interaction with an object, while still exploring a diverse set of actions. + +RL agent to retain full control over the final actions in the environment (since for every $a \in { \mathcal { A } }$ , there exist some $z = f _ { \phi } ^ { - 1 } ( a ; s )$ , while a latent space learned by a VAE provides no such guarantee). + +![](images/71e3fea3c3c288d5e0d54f0e6c8949e4991fbf2dbad22df8c1e81bceefc36855.jpg) +Figure 5: Results. The lines represent average performance across multiple random seeds, and the shaded areas represent the standard deviation. PARROT is able to learn much faster than prior methods on a majority of the tasks, and shows little variance across runs (all experiments were run with three random seeds, computational constraints of image-based RL make it difficult to run more seeds). Note that some methods that failed to make any progress on certain tasks (such as “Place Sculpture in Basket”) overlap each other with a success rate of zero. SAC and VAE-features fail to make progress on any of the tasks. + +Exploration via behavioral prior (Prior-explore). We also run experiments with an ablation of our method: instead of using a behavioral prior to transform the MDP being experienced by the RL agent, we use it to simply aid the exploration process. While collecting data, an action is executed from the prior with probability $\epsilon$ , else an action is executed from the learned policy. We experimented with $\epsilon = 0 . 1 , 0 . 3 , 0 . 7 , 0 . 9$ , and found 0.9 to perform best. + +Main results. Our results are summarised in Figure 5. We see that PARROT is able to solve all of the tasks substantially faster and achieve substantially higher final returns than other methods. The SAC baseline (which does not use any prior data) fails to make progress on any of the tasks, which we suspect is due to the challenge of exploring in sparse reward settings with a randomly initialized policy. Figure 4 illustrates a comparison between using a behavioral prior and a random policy for exploration. The VAE-features baseline similarly fails to make any progress, and due to the same reason: the difficulty of exploration in a sparse reward setting. Initializing the SAC policy with behavior cloning allows it to make progress on only two of the tasks, which is not surprising: a Gaussian policy learned through a behavior cloning loss is not expressive enough to represent the complex, multi-modal action distributions observed in dataset $\mathcal { D }$ . Both TrajRL and HIRL perform much better than any of the other baselines, but their performance plateaus a lot earlier than PARROT. While the initial exploration performance of our learned behavioral prior is not substantially better from these methods (denoted by the initial success rate in the learning curves), the flexibility of the representation it offers (through learning an invertible mapping) allows the RL agent to improve far beyond its initial performance. Prior-explore, an ablation of our method, is able to make progress on most tasks, but is unable to learn as fast as our method, and also demonstrates unstable learning on some of the tasks. We suspect this is due to the following reason: while off-policy RL methods like SAC aim to learn from data collected by any policy, they are in practice quite sensitive to the data distribution, and can run into issues if the data collection policy differs substantially from the policy being learned (Kumar et al., 2019). + +Impact of dataset size on performance. We conducted additional experiments on a subset of our tasks to evaluate how final performance is impacted as a result of dataset size, results from which are shown in Figure 6. As one might expect, the size of the dataset positively correlates with performance, but about 10K trajectories are sufficient for obtaining good performance, and collecting additional data yields diminishing returns. Note that initializing with even a smaller dataset size (like 5K trajectories) yields much better performance than learning from scratch. + +![](images/87515611fe55cabe6c38f511cdcd9334dd7d7d4eba6406b801d418d6ab1d9432.jpg) +Figure 6: Impact of dataset size on performance. We observe that training on 10K, 25K or 50K trajectories yields similar performance. + +Mismatch between train and test tasks. We ran experiments in which we deliberately bias the training dataset so that the training tasks and test tasks are functionally different (i.e. involve substantially different actions), the results from which are shown in Figure 7. We observe that if the prior is trained on pick and place tasks alone, it can still solve downstream grasping tasks well. However, if the prior is trained only on grasping, it is unable to perform well when solving pick and place tasks. We suspect this is due to the fact that pick and place tasks involve a completely new action (that of opening the gripper), which is never observed by the prior if it is trained only on grasping, making it difficult to learn this behavior from scratch for downstream tasks. + +![](images/1c702e4fba9af88ce0efc764e1ece6595eae7d6955825e3481ac3f1004538d6b.jpg) +Figure 7: Impact of train/test mismatch on performance. Each plot shows results for four tasks. Note that for the pick and place tasks, the performance is close to zero, and the curves mostly overlap each other on the $\mathbf { X }$ -axis. + +# 6 CONCLUSION + +We presented PARROT, a method for learning behavioral priors using successful trials from a wide range of tasks. Learning from priors accelerates RL on new tasks–including manipulating previously unseen objects from high-dimensional image observations–which RL from scratch often fails to learn. Our method also compares favorably to other prior works that use prior data to bootstrap learning for new tasks. While our method learns faster and performs better than prior work in learning novel tasks, it still requires thousands of trials to attain high success rates. Improving this efficiency even further, perhaps inspired by ideas in meta-learning, could be a promising direction for future work. Our work opens the possibility for several exciting future directions. PARROT provides a mapping for executing actions in new environments structurally similar to those of prior tasks. While we primarily utilized this mapping to accelerate learning of new tasks, future work could investigate how it can also enable safe exploration of new environments (Hunt et al., 2020; Rhinehart et al., 2020). While the invertibility of our learned mapping ensures that it is theoretically possible for the RL policy to execute any action in the original MDP, the probability of executing an action can become very low if this action was never seen in the training set. This can be an issue if there is a significant mismatch between the training dataset and the downstream task (as shown in our experiments), and tackling this issue would make for an interesting problem. Since our method speeds up learning using a problem setup that takes into account real world considerations (no rewards for prior data, no need to revisit prior tasks, etc.), we are also excited about its future application to domains like real world robotics. + +# REFERENCES + +Lynton Ardizzone, Jakob Kruse, Carsten Rother, and Ullrich Kothe. Analyzing inverse problems ¨ with invertible neural networks. In ICLR, 2019. + +Pierre-Luc Bacon, Jean Harb, and Doina Precup. The option-critic architecture. 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In ICLR, 2020. + +# Appendices + +A ALGORITHM + +# Algorithm 1 RL with Behavioral Priors + +1: Input: Dataset $\mathcal { D }$ of state-action pairs $( s , a )$ from previous tasks, new task $M ^ { \star }$ +2: Learn $f _ { \phi }$ by maximizing the likelihood term in Equation 2 +3: for step $k$ in $\{ 1 , . . . , \Nu \}$ do +4: $s $ current observation +5: Sample $z \sim \pi _ { \theta } ( z | s )$ +6: $a \gets f _ { \phi } ( z ; s )$ +7: $s ^ { \prime } , r \gets$ Execute $a$ in $M ^ { \star }$ +8: Update $\pi _ { \boldsymbol { \theta } } { \left( z | \boldsymbol { s } \right) }$ with $( s , z , s ^ { \prime } , r )$ +9: end for +10: Return: Policy $\pi _ { \boldsymbol { \theta } } { \left( z | \boldsymbol { s } \right) }$ for task $M ^ { \star }$ . + +# B IMPLEMENTATION DETAILS AND HYPERPARAMETER TUNING + +We now provide details of the neural network architectures and other hyperparameters used in our experiments. + +Behavioral prior. We use a conditional real NVP with four affine coupling layers as our behavioral prior. The architecture for a single coupling layer is shown in Figure 8. We use a learning rate of $1 e { - 4 }$ and the Adam (Kingma & Ba, 2015) optimizer to train the behavioral prior for 500K steps. + +![](images/d8259f7032d0a26269de84982caf5fb5ae835fd2fcb2703cc2856a97cfa68b01.jpg) +Figure 8: Coupling layer architecture. A computation graph for a single affine coupling layer is shown in (a). Given an input noise $z$ , the coupling layers transform it into $z ^ { \prime }$ through the following operations: $z _ { 1 : d } ^ { \prime } = z _ { 1 : d }$ and $z _ { d + 1 : D } ^ { \prime } = z _ { d + 1 : D } \odot \exp ( v ( z _ { 1 : d } ; \phi ( s ) ) ) + t ( z _ { 1 : d } ; \phi ( s ) )$ , where the $v$ , $t$ and $\psi$ are functions implemented using neural networks whose architectures are shown in (b) and (c). Since $v$ and $t$ have the same input, they are implemented using a single fully connected neural network (shown in (b)), and the output of this network is split into two. The image encoder, $\psi ( s )$ is implemented using a convolutional neural network with parameters shown in (c). + +TrajVAE. For this comparison, we use the same architecture as Ghadirzadeh et al. (2020). The decoder consists of three fully connected layers with 128, 256, and 512 units respectively. BatchNorm (Ioffe & Szegedy, 2015) and ReLU nonlinearity are applied after each layer. The encoder is symmetric: 512, 256, and 128 layers, respectively. The size of the latent space is 8 (same as the behavioral prior). We sweep the following values for the $\beta$ parameter (Higgins et al., 2017a): 0.1, 0.01, 0.005, 0.001, 0.0005, and find 0.001 to be optimal. We initialize $\beta$ to zero at the start of training, and anneal it to the target $\beta$ value using a logistic function, achieving half of the target value in + +25K steps. We use a learning rate of $1 e { - 4 }$ and the Adam (Kingma & Ba, 2015) optimizer to train this model for 500K steps. + +HIRL. This comparison is implemented using a conditional variational autoencoder, and uses an architecture that is similar to the one used by the TrajVAE, but with two differences: since this comparison uses image conditioning, we use the same convolutional network $\psi$ as the behavioral prior to encode the image (shown in Figure 8), and pass it as conditioning information to both the encoder and decoder networks. Second, instead of modeling the entire trajectory in a single forward pass, it instead models individual actions, allowing the high-level policy to perform closed-loop control, similar to the behavioral prior model. We sweep the following values for the $\beta$ parameter: 0.1, 0.01, 0.005, 0.001, 0.0005, and find 0.001 to be optimal. We found the annealing process to be essential for obtaining good RL performance using this method. We use a learning rate of $1 e { - 4 }$ and the Adam (Kingma & Ba, 2015) optimizer to train this model for 500K steps. + +Behavior cloning (BC). We implement behavior cloning via maximum likelihood with a Gaussian policy (and entropy regularization (Haarnoja et al., 2017)). For both behavior cloning and RL with SAC, we used the same policy network architecture as shown in Figure 9. We train this model for 2M steps, using Adam with a learning rate of $3 e ^ { - 4 }$ . + +VAE-features. For this comparison, we use the standard VAE architecture used for CIFAR-10 experiments (van den Oord et al., 2017). The encoder consists of two strided convolutional layers (stride 2, window size $4 \times 4$ ), which is followed by two residual $3 \times 3$ blocks, all of which have 256 hidden units. Each residual block is implemented as ReLU, 3x3 conv, ReLU, 1x1 conv. The decoder is symmetric to the encoder. We train this model for $1 . 5 \mathbf { M }$ steps, using Adam with a learning rate of $1 e ^ { - 3 }$ and a batch size of 128. + +Soft Actor Critic (SAC). We use the soft actor critic method (Haarnoja et al., 2018b) as our RL algorithm, with the hyperparameters shown in Table 1. We use the same hyperparameters for all of our RL experiments (our method, HIRL, TrajRL, $\mathrm { B C + S A C }$ , SAC). + +Table 1: Hyperparameters for soft-actor critic (SAC) + +
Hyperparametervalue used
Target network update period1000 steps
discount factor y0.99
policy learning rate3e-4
Q-function learning rate3e-4
reward scale1.0
automatic entropy tuning number of update steps per env stepenabled 1
+ +![](images/a212ba77eae356406c1ef304168eef8040fcaeb7683fbbcf7e6973fa18cb81fb.jpg) +Figure 9: Policy and Q-function network architectures. We use a convolutional neural network to represent the Q-function for SAC, shown in this figure. The policy network is identical, except it does not take in an action as an input and outputs a 7D action instead of a scalar Q-value. + +# C EXPERIMENTAL SETUP + +# C.1 TASKS + +We provided a visual depiction of 4 of our 8 evaluations tasks in Figure 3, and the remaining tasks are shown here in Figure 10. + +![](images/997596562173eeaa1e0aa5481037930eb5a85fac3177b215921df467e5ec1da1.jpg) +Figure 10: In the first row, the objective is to grasp a can and lift it above a certain height. Rows two and three are similar, except the objective is to grasp a vase and a baseball cap, respectively. The final row depicts a task where the goal is to pick the baseball cap and place it on the marble cube. + +# C.2 DATA COLLECTION + +We collected our dataset using scripted policies detailed in Algorithms 2 and 3. + +
Algorithm 2 Scripted GraspingAlgorithm3 Scripted Pick and Place
1:threshold ←0.021:threshold ← 0.02
2:1 numTimesteps ←252:numTimesteps ←25
3:targetPoint ← object position3:placeAttempted ←False
4: for tin(O,numTimesteps) do4: dropPos ←point above container
5:eePos ← end effector position5: for t in (O, numTimesteps) do
6:targetEEDist ← distance(targetPoint,eePos)6: eePos ← end effector position
7:if targetEEDist > threshold then7: objectDropDist ← distance(eePos,dropPos)
8:action ← targetPoint-eePosif placeAttempted then
9:else if gripperOpened then8: 9: action ←0
10:action ← close gripper10: else if object not grasped AND objectDropDist
11:else if object not raised high enough then> threshold then Execute grasp using Algorithm 2
12:action ← lift upward11:
13:else12: else if objectDropDist >threshold then
14:action←013: action ← dropPos－eePos
15:end if14: else
16:noise ~ N(0,0.1)15: action ← open gripper
17:action ←action+noise16: placeAttempted ←True 17:
18:s'←env.step(action)else
19: end foraction←0
20:19:end if
20:noise ~ N(0,0.1)
21:action ← action+noise
22:s'← env.step(action)
23: end for
+ +# C.3 SIMULATION OBJECTS + +To collect data in diverse environments, we used 3D object models from the ShapeNet dataset (Chang et al., 2015) and the PyBullet (Coumans & Bai, 2016) object libraries. + +![](images/70a13307ae9387287d751006ca299f8b6ba9fd3ca9f1601621f68eb785c66bdd.jpg) +Figure 11: Train objects. + +![](images/009106603d4e59ab9c688d634d9b6ddb184158e1951ddad365613029261b79d7.jpg) +Figure 12: Test objects \ No newline at end of file diff --git a/parse/train/Ysuv-WOFeKR/Ysuv-WOFeKR_middle.json b/parse/train/Ysuv-WOFeKR/Ysuv-WOFeKR_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..17138664f1f406cd0c9d83bc571513171d76148e --- /dev/null +++ b/parse/train/Ysuv-WOFeKR/Ysuv-WOFeKR_middle.json @@ -0,0 +1,45584 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 97 + ], + "score": 1.0, + "content": "PARROT: DATA-DRIVEN BEHAVIORAL PRIORS FOR", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 322, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 322, + 118 + ], + "score": 1.0, + "content": "REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 114, + 134, + 480, + 157 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 480, + 148 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 195, + 148 + ], + "score": 1.0, + "content": "Avi Singh∗, Huihan", + "type": "text" + }, + { + "bbox": [ + 196, + 135, + 217, + 146 + ], + "score": 0.45, + "content": "\\mathbf { L i u } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 134, + 480, + 148 + ], + "score": 1.0, + "content": ", Gaoyue Zhou, Albert Yu, Nicholas Rhinehart, Sergey Levine", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 145, + 250, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 145, + 250, + 159 + ], + "score": 1.0, + "content": "University of California, Berkeley", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 198 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 199 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 199 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 142, + 212, + 468, + 388 + ], + "lines": [ + { + "bbox": [ + 141, + 212, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 212, + 469, + 223 + ], + "score": 1.0, + "content": "Reinforcement learning provides a general framework for flexible decision mak-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 223, + 469, + 235 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 469, + 235 + ], + "score": 1.0, + "content": "ing and control, but requires extensive data collection for each new task that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "score": 1.0, + "content": "an agent needs to learn. 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We propose a method for pre-training behavioral priors that can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 312 + ], + "score": 1.0, + "content": "capture complex input-output relationships observed in successful trials from a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 311, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 142, + 311, + 469, + 322 + ], + "score": 1.0, + "content": "wide range of previously seen tasks, and we show how this learned prior can", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "score": 1.0, + "content": "be used for rapidly learning new tasks without impeding the RL agent’s abil-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 332, + 470, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 470, + 346 + ], + "score": 1.0, + "content": "ity to try out novel behaviors. We demonstrate the effectiveness of our ap-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "score": 1.0, + "content": "proach in challenging robotic manipulation domains involving image observa-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "score": 1.0, + "content": "tions and sparse reward functions, where our method outperforms prior works by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 365, + 469, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 469, + 377 + ], + "score": 1.0, + "content": "a substantial margin. Additional materials can be found on our project website:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 377, + 378, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 377, + 378, + 389 + ], + "score": 1.0, + "content": "https://sites.google.com/view/parrot-rl", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 140, + 212, + 470, + 389 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 411, + 205, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 208, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 208, + 426 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Reinforcement Learning (RL) is an attractive paradigm for robotic learning because of its flexibility", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "in being able to learn a diverse range of skills and its capacity to continuously improve. However, RL", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "algorithms typically require a large amount of data to solve each individual task, including simple", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 468, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 505, + 484 + ], + "score": 1.0, + "content": "ones. Since an RL agent is generally initialized without any prior knowledge, it must try many", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "largely unproductive behaviors before it discovers a high-reward outcome. In contrast, humans", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "rarely attempt to solve new tasks in this way: they draw on their prior experience of what is useful", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "when they attempt a new task, which substantially shrinks the task search space. For example, faced", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "with a new task involving objects on a table, a person might grasp an object, stack multiple objects,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 525, + 503, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 503, + 538 + ], + "score": 1.0, + "content": "or explore other object rearrangements, rather than re-learning how to move their arms and fingers.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 436, + 506, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "Can we endow RL agents with a similar sort of behavioral prior from past experience? In other", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "score": 1.0, + "content": "fields of machine learning, the use of large prior datasets to bootstrap acquisition of new capabil-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 577 + ], + "score": 1.0, + "content": "ities has been studied extensively to good effect. For example, language models trained on large,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "diverse datasets offer representations that drastically improve the efficiency of learning downstream", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "tasks (Devlin et al., 2019). What would be the analogue of this kind of pre-training in robotics", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "and RL? One way we can approach this problem is to leverage successful trials from a wide range", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "of previously seen tasks to improve learning for new tasks. The data could come from previously", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "learned policies, from human demonstrations, or even unstructured teleoperation of robots (Lynch", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "et al., 2019). In this paper, we show that behavioral priors can be obtained through representation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "learning, and the representation in question must not only be a representation of inputs, but actually", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "a representation of input-output relationships – a space of possible and likely mappings from states", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 662, + 484, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 484, + 673 + ], + "score": 1.0, + "content": "to actions among which the learning process can interpolate when confronted with a new task.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 542, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 692 + ], + "score": 1.0, + "content": "What makes for a good representation for RL? Given a new task, a good representation must (a) pro-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "vide an effective exploration strategy, (b) simplify the policy learning problem for the RL algorithm,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "and (c) allow the RL agent to retain full control over the environment. In this paper, we address", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 678, + 505, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 80, + 502, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 80, + 502, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 80, + 502, + 214 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 502, + 214 + ], + "score": 0.974, + "type": "image", + "image_path": "3cf924a289e0fcdc95a90d419c43c1d5e5e637b657ba0f9ca4072d113a28f90e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 80, + 502, + 124.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 124.66666666666666, + 502, + 169.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 169.33333333333331, + 502, + 213.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 222, + 505, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "Figure 1: Our problem setting. Our training dataset consists of near-optimal state-action trajectories (without", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "reward labels) from a wide range of tasks. Each task might involve interacting with a different set of objects.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "Even for the same set of objects, the task can be different depending on our objective. For example, in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "upper right corner, the objective could be picking up a cup, or it could be to place the bottle on the yellow cube.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 262, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 272 + ], + "score": 1.0, + "content": "We learn a behavioral prior from this multi-task dataset capable of trying many different useful behaviors when", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "placed in a new environment, and can aid an RL agent to quickly learn a specific task in this new environment.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "score": 1.0, + "content": "Transfer Behavioral all of these challenges through learning an invertible function that maps noise vectors to complex,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 311, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 504, + 324 + ], + "score": 1.0, + "content": "Prior Learning New Taskshigh-dimensional environment actions. Building on prior work in normalizing flows (Dinh et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "score": 1.0, + "content": "2017), we train this mapping to maximize the (conditional) log-likelihood of actions observed in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "score": 1.0, + "content": "Trial 2successful trials from past tasks. When dropped into a new MDP, the RL agent can now sample", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "from a unit Gaussian, and use the learned mapping (which we refer to as the behavioral prior) to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "score": 1.0, + "content": "generate likely environment actions, conditional on the current observation. This learned mapping", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "essentially transforms the original MDP into a simpler one for the RL agent, as long as the original", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "Trial n-1 MDP shares (partial) structure with previously seen MDPs (see Section 3). Furthermore, since this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "mapping is invertible, the RL agent still retains full control over the original MDP: for every possible", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "Trial n environment action, there exists a point within the support of the Gaussian distribution that maps to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "that action. This allows the RL agent to still try out new behaviors that are distinct from what was", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 421, + 192, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 192, + 433 + ], + "score": 1.0, + "content": "previously observed.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 438, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 451 + ], + "score": 1.0, + "content": "Our main contribution is a framework for pre-training in RL from a diverse multi-task dataset, which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "produces a behavioral prior that accelerates acquisition of new skills. We present an instantiation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "score": 1.0, + "content": "of this framework in robotic manipulation, where we utilize manipulation data from a diverse range", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "of prior tasks to train our behavioral prior, and then use it to bootstrap exploration for new tasks.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "By making it possible to pre-train action representations on large prior datasets for robotics and RL,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "we hope that our method provides a path toward leveraging large datasets in the RL and robotics", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 504, + 516 + ], + "score": 1.0, + "content": "settings, much like language models can leverage large text corpora in NLP and unsupervised pre-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "training can leverage large image datasets in computer vision. Our method, which we call Prior", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 540 + ], + "score": 1.0, + "content": "AcceleRated ReinfOrcemenT (PARROT), is able to quickly learn tasks that involve manipulating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "previously unseen objects, from image observations and sparse rewards, in settings where RL from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "scratch fails to learn a policy at all. We also compare against prior works that incorporate prior data", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 414, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 414, + 571 + ], + "score": 1.0, + "content": "for RL, and show that PARROT substantially outperforms these prior works.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 592, + 210, + 605 + ], + "lines": [ + { + "bbox": [ + 104, + 591, + 213, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 591, + 213, + 608 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Combining RL with demonstrations. Our work is related to methods for learning from demon-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "strations (Pomerleau, 1989; Schaal et al., 2003; Ratliff et al., 2007; Pastor et al., 2009; Ho & Ermon,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 644, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 504, + 655 + ], + "score": 1.0, + "content": "2016; Finn et al., 2017b; Giusti et al., 2016; Sun et al., 2017; Zhang et al., 2017; Lynch et al., 2019).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 504, + 667 + ], + "score": 1.0, + "content": "While demonstrations can also be used to speed up RL (Schaal, 1996; Peters & Schaal, 2006; Ko-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "rmushev et al., 2010; Hester et al., 2017; Vecer´ık et al., 2017; Nair et al., 2018; Rajeswaran et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "2018; Silver et al., 2018; Peng et al., 2018; Johannink et al., 2019; Gupta et al., 2019), this usually", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "requires collecting demonstrations for the specific task that is being learned. In contrast, we use", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "data from a wide range of other prior tasks to speed up RL for a new task. As we show in our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "experiments, PARROT is better suited to this problem setting when compared to prior methods that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 286, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 286, + 732 + ], + "score": 1.0, + "content": "combine imitation and RL for the same task.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 80, + 502, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 80, + 502, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 80, + 502, + 214 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 502, + 214 + ], + "score": 0.974, + "type": "image", + "image_path": "3cf924a289e0fcdc95a90d419c43c1d5e5e637b657ba0f9ca4072d113a28f90e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 80, + 502, + 124.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 124.66666666666666, + 502, + 169.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 169.33333333333331, + 502, + 213.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 222, + 505, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "Figure 1: Our problem setting. Our training dataset consists of near-optimal state-action trajectories (without", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "reward labels) from a wide range of tasks. Each task might involve interacting with a different set of objects.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "Even for the same set of objects, the task can be different depending on our objective. For example, in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "upper right corner, the objective could be picking up a cup, or it could be to place the bottle on the yellow cube.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 262, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 272 + ], + "score": 1.0, + "content": "We learn a behavioral prior from this multi-task dataset capable of trying many different useful behaviors when", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "placed in a new environment, and can aid an RL agent to quickly learn a specific task in this new environment.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "score": 1.0, + "content": "Transfer Behavioral all of these challenges through learning an invertible function that maps noise vectors to complex,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 311, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 504, + 324 + ], + "score": 1.0, + "content": "Prior Learning New Taskshigh-dimensional environment actions. Building on prior work in normalizing flows (Dinh et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 335 + ], + "score": 1.0, + "content": "2017), we train this mapping to maximize the (conditional) log-likelihood of actions observed in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "score": 1.0, + "content": "Trial 2successful trials from past tasks. When dropped into a new MDP, the RL agent can now sample", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "from a unit Gaussian, and use the learned mapping (which we refer to as the behavioral prior) to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 370 + ], + "score": 1.0, + "content": "generate likely environment actions, conditional on the current observation. This learned mapping", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "essentially transforms the original MDP into a simpler one for the RL agent, as long as the original", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "Trial n-1 MDP shares (partial) structure with previously seen MDPs (see Section 3). Furthermore, since this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "mapping is invertible, the RL agent still retains full control over the original MDP: for every possible", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "Trial n environment action, there exists a point within the support of the Gaussian distribution that maps to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "that action. This allows the RL agent to still try out new behaviors that are distinct from what was", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 421, + 192, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 192, + 433 + ], + "score": 1.0, + "content": "previously observed.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 301, + 506, + 433 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 438, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 451 + ], + "score": 1.0, + "content": "Our main contribution is a framework for pre-training in RL from a diverse multi-task dataset, which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "produces a behavioral prior that accelerates acquisition of new skills. We present an instantiation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "score": 1.0, + "content": "of this framework in robotic manipulation, where we utilize manipulation data from a diverse range", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "of prior tasks to train our behavioral prior, and then use it to bootstrap exploration for new tasks.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "By making it possible to pre-train action representations on large prior datasets for robotics and RL,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "we hope that our method provides a path toward leveraging large datasets in the RL and robotics", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 504, + 516 + ], + "score": 1.0, + "content": "settings, much like language models can leverage large text corpora in NLP and unsupervised pre-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "training can leverage large image datasets in computer vision. Our method, which we call Prior", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 540 + ], + "score": 1.0, + "content": "AcceleRated ReinfOrcemenT (PARROT), is able to quickly learn tasks that involve manipulating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "previously unseen objects, from image observations and sparse rewards, in settings where RL from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "scratch fails to learn a policy at all. We also compare against prior works that incorporate prior data", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 414, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 414, + 571 + ], + "score": 1.0, + "content": "for RL, and show that PARROT substantially outperforms these prior works.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 438, + 506, + 571 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 592, + 210, + 605 + ], + "lines": [ + { + "bbox": [ + 104, + 591, + 213, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 591, + 213, + 608 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Combining RL with demonstrations. Our work is related to methods for learning from demon-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "strations (Pomerleau, 1989; Schaal et al., 2003; Ratliff et al., 2007; Pastor et al., 2009; Ho & Ermon,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 644, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 504, + 655 + ], + "score": 1.0, + "content": "2016; Finn et al., 2017b; Giusti et al., 2016; Sun et al., 2017; Zhang et al., 2017; Lynch et al., 2019).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 504, + 667 + ], + "score": 1.0, + "content": "While demonstrations can also be used to speed up RL (Schaal, 1996; Peters & Schaal, 2006; Ko-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "rmushev et al., 2010; Hester et al., 2017; Vecer´ık et al., 2017; Nair et al., 2018; Rajeswaran et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "2018; Silver et al., 2018; Peng et al., 2018; Johannink et al., 2019; Gupta et al., 2019), this usually", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "requires collecting demonstrations for the specific task that is being learned. In contrast, we use", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "data from a wide range of other prior tasks to speed up RL for a new task. As we show in our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "experiments, PARROT is better suited to this problem setting when compared to prior methods that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 286, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 286, + 732 + ], + "score": 1.0, + "content": "combine imitation and RL for the same task.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 621, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Generative modeling and RL. Several prior works model multi-modal action distributions using", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "expert trajectories from different tasks. IntentionGAN (Hausman et al.) and InfoGAIL (Li et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "2017) learn multi-modal policies via interaction with an environment using an adversarial imitation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "approach (Ho & Ermon, 2016), but we learn these distributions only from data. Other works learn", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "score": 1.0, + "content": "these distributions from data (Xie et al., 2019; Rhinehart et al., 2020) and utilize them for planning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "at test time to optimize a user-provided cost function. In contrast, we use the behavioral prior to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "augment model-free RL of a new task. This allows us to learn policies for new tasks that may be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "substantially different from prior tasks, since we can collect data specific to the new task at hand, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "we do not explicitly need to model the environment, which can be complicated for high-dimensional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "state and action spaces, such as when performing continuous control from images observations.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "Another line of work (Ghadirzadeh et al., 2017; Ham¨ al¨ ainen et al. ¨ , 2019; Ghadirzadeh et al., 2020)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "explores using generative models for RL, using a variational autoencoder (Kingma & Welling, 2014)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "to model entire trajectories in an observation-independent manner, and then learning an open-loop,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "single-step policy using RL to solve the downstream task. Our approach differs in several key", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "aspects: (1) our model is observation-conditioned, allowing it to prioritize actions that are relevant", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "to the current scene or environment, (2) our model allows for closed-loop feedback control, and (3)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "our model is invertible, allowing the high-level policy to retain full control over the action space.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 270, + 424, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 424, + 281 + ], + "score": 1.0, + "content": "Our experiments demonstrate these aspects are crucial for solving harder tasks.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "Hierarchical learning. Our method can be interpreted as training a hierarchical model: the low-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 504, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 504, + 308 + ], + "score": 1.0, + "content": "level policy is the behavioral prior trained on prior data, while the high-level policy is trained us-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "ing RL and controls the low-level policy. This structure is similar to prior work in hierarchical", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "RL (Dayan & Hinton, 1992; Parr & Russell, 1997; Dietterich, 1998; Sutton et al., 1999; Kulka-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "rni et al., 2016). We divide prior work in hierarchical learning into two categories: methods that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 341, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 352 + ], + "score": 1.0, + "content": "seek to learn both the low-level and high-level policies through active interaction with an environ-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 365 + ], + "score": 1.0, + "content": "ment (Kupcsik et al., 2013; Heess et al., 2016; Bacon et al., 2017; Florensa et al., 2017; Haarnoja", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "et al., 2018a; Nachum et al., 2018; Chandak et al., 2019; Peng et al., 2019), and methods that learn", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "score": 1.0, + "content": "temporally extended actions, also known as options, from demonstrations, and then recompose them", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 385, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 396 + ], + "score": 1.0, + "content": "to perform long-horizon tasks through RL or planning (Fox et al., 2017; Krishnan et al., 2017; Kipf", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "et al., 2019; Shankar et al., 2020; Shankar & Gupta, 2020). Our work shares similarities with the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "data-driven approach of the latter methods, but work on options focuses on modeling the temporal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "structure in demonstrations for a small number of long-horizon tasks, while our behavioral prior is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "not concerned with temporally-extended abstractions, but rather with transforming the original MDP", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 439, + 493, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 493, + 452 + ], + "score": 1.0, + "content": "into one where potentially useful behaviors are more likely, and useless behaviors are less likely.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "Meta-learning. Our goal in this paper is to utilize data from previously seen tasks to speed up RL", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "score": 1.0, + "content": "for new tasks. Meta-RL (Duan et al., 2016; Wang et al., 2016; Finn et al., 2017a; Mishra et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "2017; Rakelly et al., 2019; Mendonca et al., 2019; Zintgraf et al., 2020; Fakoor et al., 2020) and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "meta-imitation methods (Duan et al., 2017; Finn et al., 2017c; Huang et al., 2018; James et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "2018; Paine et al., 2018; Yu et al., 2018; Huang et al., 2019; Zhou et al., 2020) also seek to speed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "up learning for new tasks by leveraging experience from previously seen tasks. While meta-learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "provides an appealing and principled framework to accelerate acquisition of future tasks, we focus", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "on a more lightweight approach with relaxed assumptions that make our method more practically", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 544, + 401, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 401, + 556 + ], + "score": 1.0, + "content": "applicable, and we discuss these assumptions in detail in the next section.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 573, + 213, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 215, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 215, + 588 + ], + "score": 1.0, + "content": "3 PROBLEM SETUP", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Our goal is to improve an agent’s ability to learn new tasks by incorporating a behavioral prior,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "which it can acquire from previously seen tasks. Each task can be considered a Markov decision", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 293, + 635 + ], + "score": 1.0, + "content": "process (MDP), which is defined by a tuple", + "type": "text" + }, + { + "bbox": [ + 293, + 622, + 351, + 634 + ], + "score": 0.92, + "content": "( S , \\mathcal { A } , \\mathrm { T } , r , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 621, + 384, + 635 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 385, + 622, + 393, + 632 + ], + "score": 0.77, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 621, + 414, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 622, + 423, + 632 + ], + "score": 0.68, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "represent state and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 165, + 645 + ], + "score": 1.0, + "content": "action spaces,", + "type": "text" + }, + { + "bbox": [ + 166, + 633, + 206, + 645 + ], + "score": 0.92, + "content": "\\mathrm { T } ( s ^ { \\prime } | s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 633, + 225, + 645 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 633, + 254, + 645 + ], + "score": 0.93, + "content": "r ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 633, + 460, + 645 + ], + "score": 1.0, + "content": "represent the dynamics and reward functions, and", + "type": "text" + }, + { + "bbox": [ + 460, + 633, + 504, + 645 + ], + "score": 0.93, + "content": "\\gamma \\in ( 0 , 1 )", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 245, + 656 + ], + "score": 1.0, + "content": "represents the discount factor. Let", + "type": "text" + }, + { + "bbox": [ + 245, + 644, + 270, + 656 + ], + "score": 0.92, + "content": "p ( M )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "denote a distribution over such MDPs, with the constraint", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 495, + 668 + ], + "score": 1.0, + "content": "that the state and action spaces are fixed. In our experiments, we treat high-dimensional images as", + "type": "text" + }, + { + "bbox": [ + 496, + 657, + 501, + 665 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 654, + 506, + 668 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "which means that this constraint is not very restrictive in practice. In order for the behavioral prior to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "be able to accelerate the acquisition of new skills, we assume the behavioral prior is trained on data", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "that structurally resembles potential optimal policies for all or part of the new task. For example,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "if the new task requires placing a bottle in a tray, the prior data might include some behaviors that", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "involve picking up objects. There are many ways to formalize this assumption. One way to state", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "this formally is to assume that prior data consists of executions of near-optimal policies for MDPs", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 48.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Generative modeling and RL. Several prior works model multi-modal action distributions using", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "expert trajectories from different tasks. IntentionGAN (Hausman et al.) and InfoGAIL (Li et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "2017) learn multi-modal policies via interaction with an environment using an adversarial imitation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "approach (Ho & Ermon, 2016), but we learn these distributions only from data. Other works learn", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "score": 1.0, + "content": "these distributions from data (Xie et al., 2019; Rhinehart et al., 2020) and utilize them for planning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "at test time to optimize a user-provided cost function. In contrast, we use the behavioral prior to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "augment model-free RL of a new task. This allows us to learn policies for new tasks that may be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "substantially different from prior tasks, since we can collect data specific to the new task at hand, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "we do not explicitly need to model the environment, which can be complicated for high-dimensional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "state and action spaces, such as when performing continuous control from images observations.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "Another line of work (Ghadirzadeh et al., 2017; Ham¨ al¨ ainen et al. ¨ , 2019; Ghadirzadeh et al., 2020)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "explores using generative models for RL, using a variational autoencoder (Kingma & Welling, 2014)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "to model entire trajectories in an observation-independent manner, and then learning an open-loop,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "single-step policy using RL to solve the downstream task. Our approach differs in several key", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "aspects: (1) our model is observation-conditioned, allowing it to prioritize actions that are relevant", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "to the current scene or environment, (2) our model allows for closed-loop feedback control, and (3)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "our model is invertible, allowing the high-level policy to retain full control over the action space.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 270, + 424, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 424, + 281 + ], + "score": 1.0, + "content": "Our experiments demonstrate these aspects are crucial for solving harder tasks.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 82, + 506, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "Hierarchical learning. Our method can be interpreted as training a hierarchical model: the low-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 504, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 504, + 308 + ], + "score": 1.0, + "content": "level policy is the behavioral prior trained on prior data, while the high-level policy is trained us-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "ing RL and controls the low-level policy. This structure is similar to prior work in hierarchical", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "RL (Dayan & Hinton, 1992; Parr & Russell, 1997; Dietterich, 1998; Sutton et al., 1999; Kulka-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "rni et al., 2016). We divide prior work in hierarchical learning into two categories: methods that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 341, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 352 + ], + "score": 1.0, + "content": "seek to learn both the low-level and high-level policies through active interaction with an environ-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 365 + ], + "score": 1.0, + "content": "ment (Kupcsik et al., 2013; Heess et al., 2016; Bacon et al., 2017; Florensa et al., 2017; Haarnoja", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "et al., 2018a; Nachum et al., 2018; Chandak et al., 2019; Peng et al., 2019), and methods that learn", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "score": 1.0, + "content": "temporally extended actions, also known as options, from demonstrations, and then recompose them", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 385, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 396 + ], + "score": 1.0, + "content": "to perform long-horizon tasks through RL or planning (Fox et al., 2017; Krishnan et al., 2017; Kipf", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "et al., 2019; Shankar et al., 2020; Shankar & Gupta, 2020). Our work shares similarities with the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "data-driven approach of the latter methods, but work on options focuses on modeling the temporal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "structure in demonstrations for a small number of long-horizon tasks, while our behavioral prior is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "not concerned with temporally-extended abstractions, but rather with transforming the original MDP", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 439, + 493, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 493, + 452 + ], + "score": 1.0, + "content": "into one where potentially useful behaviors are more likely, and useless behaviors are less likely.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 285, + 505, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "Meta-learning. Our goal in this paper is to utilize data from previously seen tasks to speed up RL", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 504, + 478 + ], + "score": 1.0, + "content": "for new tasks. Meta-RL (Duan et al., 2016; Wang et al., 2016; Finn et al., 2017a; Mishra et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "2017; Rakelly et al., 2019; Mendonca et al., 2019; Zintgraf et al., 2020; Fakoor et al., 2020) and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "meta-imitation methods (Duan et al., 2017; Finn et al., 2017c; Huang et al., 2018; James et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "2018; Paine et al., 2018; Yu et al., 2018; Huang et al., 2019; Zhou et al., 2020) also seek to speed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "up learning for new tasks by leveraging experience from previously seen tasks. While meta-learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "provides an appealing and principled framework to accelerate acquisition of future tasks, we focus", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "on a more lightweight approach with relaxed assumptions that make our method more practically", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 544, + 401, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 401, + 556 + ], + "score": 1.0, + "content": "applicable, and we discuss these assumptions in detail in the next section.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 455, + 506, + 556 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 573, + 213, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 215, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 215, + 588 + ], + "score": 1.0, + "content": "3 PROBLEM SETUP", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Our goal is to improve an agent’s ability to learn new tasks by incorporating a behavioral prior,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "which it can acquire from previously seen tasks. Each task can be considered a Markov decision", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 293, + 635 + ], + "score": 1.0, + "content": "process (MDP), which is defined by a tuple", + "type": "text" + }, + { + "bbox": [ + 293, + 622, + 351, + 634 + ], + "score": 0.92, + "content": "( S , \\mathcal { A } , \\mathrm { T } , r , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 621, + 384, + 635 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 385, + 622, + 393, + 632 + ], + "score": 0.77, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 621, + 414, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 622, + 423, + 632 + ], + "score": 0.68, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "represent state and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 165, + 645 + ], + "score": 1.0, + "content": "action spaces,", + "type": "text" + }, + { + "bbox": [ + 166, + 633, + 206, + 645 + ], + "score": 0.92, + "content": "\\mathrm { T } ( s ^ { \\prime } | s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 633, + 225, + 645 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 633, + 254, + 645 + ], + "score": 0.93, + "content": "r ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 633, + 460, + 645 + ], + "score": 1.0, + "content": "represent the dynamics and reward functions, and", + "type": "text" + }, + { + "bbox": [ + 460, + 633, + 504, + 645 + ], + "score": 0.93, + "content": "\\gamma \\in ( 0 , 1 )", + "type": "inline_equation" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 245, + 656 + ], + "score": 1.0, + "content": "represents the discount factor. Let", + "type": "text" + }, + { + "bbox": [ + 245, + 644, + 270, + 656 + ], + "score": 0.92, + "content": "p ( M )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "denote a distribution over such MDPs, with the constraint", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 495, + 668 + ], + "score": 1.0, + "content": "that the state and action spaces are fixed. In our experiments, we treat high-dimensional images as", + "type": "text" + }, + { + "bbox": [ + 496, + 657, + 501, + 665 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 654, + 506, + 668 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "which means that this constraint is not very restrictive in practice. In order for the behavioral prior to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "be able to accelerate the acquisition of new skills, we assume the behavioral prior is trained on data", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "that structurally resembles potential optimal policies for all or part of the new task. 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This learned mapping essentially", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "transforms the MDP experienced by the RL agent into a simpler one, where every random action", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 440, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 440, + 139 + ], + "score": 1.0, + "content": "executed in this transformed MDP is much more likely to lead to a useful behavior.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "How can we learn such a mapping? In this paper, we propose to learn this mapping through state-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 428, + 166 + ], + "score": 1.0, + "content": "conditioned generative modeling of the actions observed in the original dataset", + "type": "text" + }, + { + "bbox": [ + 428, + 155, + 437, + 164 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 155, + 505, + 166 + ], + "score": 1.0, + "content": ", and we refer to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 385, + 179 + ], + "score": 1.0, + "content": "this state-conditioned distribution over actions as the behavioral prior", + "type": "text" + }, + { + "bbox": [ + 385, + 165, + 426, + 177 + ], + "score": 0.92, + "content": "p _ { \\mathrm { p r i o r } } ( a | s )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 165, + 506, + 179 + ], + "score": 1.0, + "content": ". A deep generative", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "model takes noise as input, and outputs a plausible sample from the target distribution, i.e. it can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 187, + 504, + 201 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 145, + 201 + ], + "score": 1.0, + "content": "represent", + "type": "text" + }, + { + "bbox": [ + 146, + 187, + 186, + 199 + ], + "score": 0.93, + "content": "p _ { \\mathrm { p r i o r } } ( a | s )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 187, + 299, + 201 + ], + "score": 1.0, + "content": "as a distribution over noise", + "type": "text" + }, + { + "bbox": [ + 299, + 189, + 306, + 198 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 187, + 431, + 201 + ], + "score": 1.0, + "content": "using a deterministic mapping", + "type": "text" + }, + { + "bbox": [ + 432, + 187, + 504, + 200 + ], + "score": 0.91, + "content": "f _ { \\phi } : \\mathcal { Z } \\times \\mathcal { S } \\mapsto A", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "When learning a new task, we can use this mapping to reparametrize the action space of the RL", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 427, + 222 + ], + "score": 1.0, + "content": "agent: if the action chosen by the randomly initialized neural network policy is", + "type": "text" + }, + { + "bbox": [ + 428, + 212, + 434, + 219 + ], + "score": 0.72, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 209, + 506, + 222 + ], + "score": 1.0, + "content": ", then we execute", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 147, + 233 + ], + "score": 1.0, + "content": "the action", + "type": "text" + }, + { + "bbox": [ + 147, + 220, + 199, + 232 + ], + "score": 0.93, + "content": "a = f _ { \\phi } ( z ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 220, + 352, + 233 + ], + "score": 1.0, + "content": "in the original MDP, and learn a policy", + "type": "text" + }, + { + "bbox": [ + 353, + 220, + 380, + 232 + ], + "score": 0.93, + "content": "\\pi ( \\boldsymbol { z } | \\boldsymbol { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "that maximizes the task reward", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 324, + 245 + ], + "score": 1.0, + "content": "through learning to control the inputs to the mapping", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 336, + 243 + ], + "score": 0.88, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 231, + 506, + 245 + ], + "score": 1.0, + "content": ". The training of the behavioral prior and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "the task-specific policy is decoupled, allowing us to mix and match RL algorithms and generative", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "models to best suit the application of interest. An overview of our overall architecture is depicted in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "Figure 2. In the next subsection, we discuss what properties we would like the behavioral prior to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 492, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 492, + 288 + ], + "score": 1.0, + "content": "satisfy, and present one particular choice for learning a prior that satisfies all of these properties.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 299, + 401, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 402, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 402, + 312 + ], + "score": 1.0, + "content": "4.1 LEARNING A BEHAVIORAL PRIOR WITH NORMALIZING FLOWS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "For the behavioral prior to be effective, it needs to satisfy certain properties. Since we learn the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "prior from a multi-task dataset, containing several different behaviors even for the same initial state,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "the learned prior should be capable of representing complex, multi-modal distributions. Second,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "score": 1.0, + "content": "it should provide a mapping for generating “useful” actions from noise samples when learning a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 365, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 376 + ], + "score": 1.0, + "content": "new task. Third, the prior should be state-conditioned, so that only actions that are relevant to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 387 + ], + "score": 1.0, + "content": "current state are sampled. And finally, the learned mapping should allow easier learning in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "reparameterized action space without hindering the RL agent’s ability to attempt novel behaviors,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 383, + 408 + ], + "score": 1.0, + "content": "including actions that might not have been observed in the dataset", + "type": "text" + }, + { + "bbox": [ + 384, + 397, + 393, + 407 + ], + "score": 0.76, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 397, + 505, + 408 + ], + "score": 1.0, + "content": ". Generative models based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 406, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 506, + 422 + ], + "score": 1.0, + "content": "on normalizing flows (Dinh et al., 2017) satisfy all of these properties well: they allow maximizing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 417, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 433 + ], + "score": 1.0, + "content": "the model’s exact log-likelihood of observed examples, and learn a deterministic, invertible mapping", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 306, + 443 + ], + "score": 1.0, + "content": "that transforms samples from a simple distribution", + "type": "text" + }, + { + "bbox": [ + 306, + 432, + 317, + 441 + ], + "score": 0.86, + "content": "p _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "to examples observed in the training dataset. In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 440, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 504, + 453 + ], + "score": 1.0, + "content": "particular, the real-valued non-volume preserving (real NVP) architecture introduced by Dinh et al.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "(2017) allows using deep neural networks to parameterize this mapping (making it expressive) While", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "the original real NVP work modelled unconditional distributions, follow-up work has found that it", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "can be easily extended to incorporate conditioning information (Ardizzone et al., 2019). We refer the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "reader to prior work (Dinh et al., 2017) for a complete description of real NVPs, and summarize its", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 296, + 508 + ], + "score": 1.0, + "content": "key features here. Given an invertible mapping", + "type": "text" + }, + { + "bbox": [ + 297, + 496, + 348, + 508 + ], + "score": 0.94, + "content": "a = f _ { \\phi } ( z ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 496, + 506, + 508 + ], + "score": 1.0, + "content": ", the change of variable formula allows", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 506, + 483, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 384, + 520 + ], + "score": 1.0, + "content": "expressing the likelihood of the observed actions using samples from", + "type": "text" + }, + { + "bbox": [ + 384, + 509, + 393, + 516 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 506, + 483, + 520 + ], + "score": 1.0, + "content": "in the following way:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 523, + 402, + 539 + ], + "lines": [ + { + "bbox": [ + 209, + 523, + 402, + 539 + ], + "spans": [ + { + "bbox": [ + 209, + 523, + 402, + 539 + ], + "score": 0.93, + "content": "p _ { \\mathrm { p r i o r } } ( a | s ) = p _ { z } \\left( f _ { \\phi } ^ { - 1 } ( a ; s ) \\right) \\left| \\operatorname* { d e t } \\left( \\partial f _ { \\phi } ^ { - 1 } ( a ; s ) / \\partial a \\right) \\right|", + "type": "interline_equation", + "image_path": "3e994494eba57fd08f15f3299fe9ce0123822cc54c9bccd6ca4dd506e614f05a.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 209, + 523, + 402, + 539 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 452, + 556 + ], + "score": 1.0, + "content": "Dinh et al. (2017) propose a particular (unconditioned) form of the invertible mapping", + "type": "text" + }, + { + "bbox": [ + 452, + 544, + 463, + 556 + ], + "score": 0.86, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 543, + 506, + 556 + ], + "score": 1.0, + "content": ", called an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "affine coupling layer, that maintains tractability of the likelihood term above, while still allowing the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 144, + 578 + ], + "score": 1.0, + "content": "mapping", + "type": "text" + }, + { + "bbox": [ + 144, + 566, + 155, + 578 + ], + "score": 0.89, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "to be expressive. Several coupling layers can be composed together to transform simple", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "noise vectors into samples from complex distributions, and each layer can be conditioned on other", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 587, + 235, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 235, + 600 + ], + "score": 1.0, + "content": "variables, as shown in Figure 2.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 107, + 612, + 432, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 433, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 433, + 624 + ], + "score": 1.0, + "content": "4.2 ACCELERATED REINFORCEMENT LEARNING VIA BEHAVIORAL PRIORS", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 221, + 646 + ], + "score": 1.0, + "content": "After we obtain the mapping", + "type": "text" + }, + { + "bbox": [ + 221, + 633, + 254, + 645 + ], + "score": 0.93, + "content": "f _ { \\phi } ( z ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "from the behavioral prior learned by maximizing the likelihood", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "term in Equation 2, we would like to use it to accelerate RL when solving a new task. Instead of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 178, + 668 + ], + "score": 1.0, + "content": "learning a policy", + "type": "text" + }, + { + "bbox": [ + 178, + 655, + 210, + 667 + ], + "score": 0.93, + "content": "\\pi _ { \\boldsymbol { \\theta } } ( a | s )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "that directly executes its actions in the original MDP, we learn a policy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 107, + 666, + 138, + 678 + ], + "score": 0.93, + "content": "\\pi _ { \\boldsymbol { \\theta } } { \\left( z | \\boldsymbol { s } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 666, + 359, + 679 + ], + "score": 1.0, + "content": ", and execute an action in the environment according to", + "type": "text" + }, + { + "bbox": [ + 359, + 666, + 410, + 678 + ], + "score": 0.94, + "content": "a = f _ { \\phi } ( z ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 666, + 505, + 679 + ], + "score": 1.0, + "content": ". As shown in Figure 2,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "this essentially transforms the MDP experienced for the RL agent into one where random actions", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 689, + 137, + 699 + ], + "score": 0.9, + "content": "z \\sim p _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 687, + 169, + 700 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 170, + 689, + 181, + 699 + ], + "score": 0.86, + "content": "p _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 687, + 399, + 700 + ], + "score": 1.0, + "content": "is the base distribution used for training the mapping", + "type": "text" + }, + { + "bbox": [ + 400, + 688, + 411, + 700 + ], + "score": 0.83, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 687, + 504, + 700 + ], + "score": 1.0, + "content": ") are much more likely", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "to result in useful behaviors. To enable effective exploration at the start of the learning period, we", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "initialize the RL policy to the base distribution used for training the prior, so that at the beginning of", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 142, + 734 + ], + "score": 1.0, + "content": "training,", + "type": "text" + }, + { + "bbox": [ + 143, + 720, + 212, + 732 + ], + "score": 0.93, + "content": "\\pi _ { \\theta } ( z | s ) \\mathbf { \\bar { \\theta } } { : = } \\bar { p _ { z } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 720, + 294, + 734 + ], + "score": 1.0, + "content": ". Since the mapping", + "type": "text" + }, + { + "bbox": [ + 294, + 722, + 304, + 733 + ], + "score": 0.88, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "is invertible, the RL agent still retains full control", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 139 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "How can we learn such a mapping? In this paper, we propose to learn this mapping through state-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 428, + 166 + ], + "score": 1.0, + "content": "conditioned generative modeling of the actions observed in the original dataset", + "type": "text" + }, + { + "bbox": [ + 428, + 155, + 437, + 164 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 155, + 505, + 166 + ], + "score": 1.0, + "content": ", and we refer to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 385, + 179 + ], + "score": 1.0, + "content": "this state-conditioned distribution over actions as the behavioral prior", + "type": "text" + }, + { + "bbox": [ + 385, + 165, + 426, + 177 + ], + "score": 0.92, + "content": "p _ { \\mathrm { p r i o r } } ( a | s )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 165, + 506, + 179 + ], + "score": 1.0, + "content": ". A deep generative", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "model takes noise as input, and outputs a plausible sample from the target distribution, i.e. it can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 187, + 504, + 201 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 145, + 201 + ], + "score": 1.0, + "content": "represent", + "type": "text" + }, + { + "bbox": [ + 146, + 187, + 186, + 199 + ], + "score": 0.93, + "content": "p _ { \\mathrm { p r i o r } } ( a | s )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 187, + 299, + 201 + ], + "score": 1.0, + "content": "as a distribution over noise", + "type": "text" + }, + { + "bbox": [ + 299, + 189, + 306, + 198 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 187, + 431, + 201 + ], + "score": 1.0, + "content": "using a deterministic mapping", + "type": "text" + }, + { + "bbox": [ + 432, + 187, + 504, + 200 + ], + "score": 0.91, + "content": "f _ { \\phi } : \\mathcal { Z } \\times \\mathcal { S } \\mapsto A", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "When learning a new task, we can use this mapping to reparametrize the action space of the RL", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 427, + 222 + ], + "score": 1.0, + "content": "agent: if the action chosen by the randomly initialized neural network policy is", + "type": "text" + }, + { + "bbox": [ + 428, + 212, + 434, + 219 + ], + "score": 0.72, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 209, + 506, + 222 + ], + "score": 1.0, + "content": ", then we execute", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 147, + 233 + ], + "score": 1.0, + "content": "the action", + "type": "text" + }, + { + "bbox": [ + 147, + 220, + 199, + 232 + ], + "score": 0.93, + "content": "a = f _ { \\phi } ( z ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 220, + 352, + 233 + ], + "score": 1.0, + "content": "in the original MDP, and learn a policy", + "type": "text" + }, + { + "bbox": [ + 353, + 220, + 380, + 232 + ], + "score": 0.93, + "content": "\\pi ( \\boldsymbol { z } | \\boldsymbol { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "that maximizes the task reward", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 324, + 245 + ], + "score": 1.0, + "content": "through learning to control the inputs to the mapping", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 336, + 243 + ], + "score": 0.88, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 231, + 506, + 245 + ], + "score": 1.0, + "content": ". The training of the behavioral prior and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "the task-specific policy is decoupled, allowing us to mix and match RL algorithms and generative", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "models to best suit the application of interest. An overview of our overall architecture is depicted in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "Figure 2. In the next subsection, we discuss what properties we would like the behavioral prior to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 492, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 492, + 288 + ], + "score": 1.0, + "content": "satisfy, and present one particular choice for learning a prior that satisfies all of these properties.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 143, + 506, + 288 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 299, + 401, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 402, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 402, + 312 + ], + "score": 1.0, + "content": "4.1 LEARNING A BEHAVIORAL PRIOR WITH NORMALIZING FLOWS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "For the behavioral prior to be effective, it needs to satisfy certain properties. Since we learn the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "prior from a multi-task dataset, containing several different behaviors even for the same initial state,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "the learned prior should be capable of representing complex, multi-modal distributions. Second,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "score": 1.0, + "content": "it should provide a mapping for generating “useful” actions from noise samples when learning a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 365, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 376 + ], + "score": 1.0, + "content": "new task. Third, the prior should be state-conditioned, so that only actions that are relevant to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 387 + ], + "score": 1.0, + "content": "current state are sampled. And finally, the learned mapping should allow easier learning in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "reparameterized action space without hindering the RL agent’s ability to attempt novel behaviors,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 383, + 408 + ], + "score": 1.0, + "content": "including actions that might not have been observed in the dataset", + "type": "text" + }, + { + "bbox": [ + 384, + 397, + 393, + 407 + ], + "score": 0.76, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 397, + 505, + 408 + ], + "score": 1.0, + "content": ". 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In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 440, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 504, + 453 + ], + "score": 1.0, + "content": "particular, the real-valued non-volume preserving (real NVP) architecture introduced by Dinh et al.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "(2017) allows using deep neural networks to parameterize this mapping (making it expressive) While", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "the original real NVP work modelled unconditional distributions, follow-up work has found that it", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "can be easily extended to incorporate conditioning information (Ardizzone et al., 2019). We refer the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "reader to prior work (Dinh et al., 2017) for a complete description of real NVPs, and summarize its", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 296, + 508 + ], + "score": 1.0, + "content": "key features here. Given an invertible mapping", + "type": "text" + }, + { + "bbox": [ + 297, + 496, + 348, + 508 + ], + "score": 0.94, + "content": "a = f _ { \\phi } ( z ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 496, + 506, + 508 + ], + "score": 1.0, + "content": ", the change of variable formula allows", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 506, + 483, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 384, + 520 + ], + "score": 1.0, + "content": "expressing the likelihood of the observed actions using samples from", + "type": "text" + }, + { + "bbox": [ + 384, + 509, + 393, + 516 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 506, + 483, + 520 + ], + "score": 1.0, + "content": "in the following way:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 320, + 507, + 520 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 523, + 402, + 539 + ], + "lines": [ + { + "bbox": [ + 209, + 523, + 402, + 539 + ], + "spans": [ + { + "bbox": [ + 209, + 523, + 402, + 539 + ], + "score": 0.93, + "content": "p _ { \\mathrm { p r i o r } } ( a | s ) = p _ { z } \\left( f _ { \\phi } ^ { - 1 } ( a ; s ) \\right) \\left| \\operatorname* { d e t } \\left( \\partial f _ { \\phi } ^ { - 1 } ( a ; s ) / \\partial a \\right) \\right|", + "type": "interline_equation", + "image_path": "3e994494eba57fd08f15f3299fe9ce0123822cc54c9bccd6ca4dd506e614f05a.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 209, + 523, + 402, + 539 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 452, + 556 + ], + "score": 1.0, + "content": "Dinh et al. (2017) propose a particular (unconditioned) form of the invertible mapping", + "type": "text" + }, + { + "bbox": [ + 452, + 544, + 463, + 556 + ], + "score": 0.86, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 543, + 506, + 556 + ], + "score": 1.0, + "content": ", called an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "affine coupling layer, that maintains tractability of the likelihood term above, while still allowing the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 144, + 578 + ], + "score": 1.0, + "content": "mapping", + "type": "text" + }, + { + "bbox": [ + 144, + 566, + 155, + 578 + ], + "score": 0.89, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "to be expressive. 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Instead of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 178, + 668 + ], + "score": 1.0, + "content": "learning a policy", + "type": "text" + }, + { + "bbox": [ + 178, + 655, + 210, + 667 + ], + "score": 0.93, + "content": "\\pi _ { \\boldsymbol { \\theta } } ( a | s )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "that directly executes its actions in the original MDP, we learn a policy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 107, + 666, + 138, + 678 + ], + "score": 0.93, + "content": "\\pi _ { \\boldsymbol { \\theta } } { \\left( z | \\boldsymbol { s } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 666, + 359, + 679 + ], + "score": 1.0, + "content": ", and execute an action in the environment according to", + "type": "text" + }, + { + "bbox": [ + 359, + 666, + 410, + 678 + ], + "score": 0.94, + "content": "a = f _ { \\phi } ( z ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 666, + 505, + 679 + ], + "score": 1.0, + "content": ". As shown in Figure 2,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "this essentially transforms the MDP experienced for the RL agent into one where random actions", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 689, + 137, + 699 + ], + "score": 0.9, + "content": "z \\sim p _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 687, + 169, + 700 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 170, + 689, + 181, + 699 + ], + "score": 0.86, + "content": "p _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 687, + 399, + 700 + ], + "score": 1.0, + "content": "is the base distribution used for training the mapping", + "type": "text" + }, + { + "bbox": [ + 400, + 688, + 411, + 700 + ], + "score": 0.83, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 687, + 504, + 700 + ], + "score": 1.0, + "content": ") are much more likely", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "to result in useful behaviors. To enable effective exploration at the start of the learning period, we", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "initialize the RL policy to the base distribution used for training the prior, so that at the beginning of", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 142, + 734 + ], + "score": 1.0, + "content": "training,", + "type": "text" + }, + { + "bbox": [ + 143, + 720, + 212, + 732 + ], + "score": 0.93, + "content": "\\pi _ { \\theta } ( z | s ) \\mathbf { \\bar { \\theta } } { : = } \\bar { p _ { z } } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 720, + 294, + 734 + ], + "score": 1.0, + "content": ". 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The learned mapping increases the likelihood of useful actions without crippling the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "RL agent, making it ideal for fine-tuning from task-specific data. Our complete method is described", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 394, + 130 + ], + "score": 1.0, + "content": "in Algorithm 1 in Appendix A. Note that we need to learn the mapping", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 394, + 117, + 406, + 129 + ], + "score": 0.88, + "content": "f _ { \\phi }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 406, + 116, + 505, + 130 + ], + "score": 1.0, + "content": "only once, and it can be", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 128, + 291, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 291, + 140 + ], + "score": 1.0, + "content": "used for accelerated learning of any new task.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 632, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 256, + 97 + ], + "score": 1.0, + "content": "over the action space: for any given", + "type": "text" + }, + { + "bbox": [ + 256, + 85, + 263, + 92 + ], + "score": 0.57, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 81, + 349, + 97 + ], + "score": 1.0, + "content": ", it can always find a", + "type": "text" + }, + { + "bbox": [ + 349, + 84, + 356, + 92 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 81, + 416, + 97 + ], + "score": 1.0, + "content": "that generates", + "type": "text" + }, + { + "bbox": [ + 417, + 81, + 477, + 96 + ], + "score": 0.94, + "content": "z = f _ { \\phi } ^ { - 1 } ( a ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 81, + 506, + 97 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "original MDP. The learned mapping increases the likelihood of useful actions without crippling the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "RL agent, making it ideal for fine-tuning from task-specific data. Our complete method is described", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 394, + 130 + ], + "score": 1.0, + "content": "in Algorithm 1 in Appendix A. Note that we need to learn the mapping", + "type": "text" + }, + { + "bbox": [ + 394, + 117, + 406, + 129 + ], + "score": 0.88, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 116, + 505, + 130 + ], + "score": 1.0, + "content": "only once, and it can be", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 128, + 291, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 291, + 140 + ], + "score": 1.0, + "content": "used for accelerated learning of any new task.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 153, + 248, + 164 + ], + "lines": [ + { + "bbox": [ + 106, + 153, + 250, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 250, + 165 + ], + "score": 1.0, + "content": "4.3 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 174, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 212, + 186 + ], + "score": 1.0, + "content": "We use real NVP to learn", + "type": "text" + }, + { + "bbox": [ + 212, + 174, + 223, + 186 + ], + "score": 0.88, + "content": "f _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 174, + 505, + 186 + ], + "score": 1.0, + "content": ", and as shown in Figure 2, each coupling layer in the real NVP takes", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "as input the the output of the previous coupling layer, and the conditioning information. The con-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "ditioning information in our case corresponds to RGB image observations, which allows us to train", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 207, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 104, + 207, + 506, + 219 + ], + "score": 1.0, + "content": "a single behavioral prior across a wide variety of tasks, even when the tasks might have different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "underlying states (for example, different objects). We train a real NVP model with four coupling", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "layers; the exact architecture, and other hyperparameters, are detailed in Appendix B. The behav-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "ioral prior can be combined with any RL algorithm that is suitable for continuous action spaces, and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 486, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 486, + 263 + ], + "score": 1.0, + "content": "we chose to use the soft actor-critic (Haarnoja et al., 2018b) due to its stability and ease of use.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 279, + 200, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 201, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 201, + 294 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "Our experiments seek to answer: (1) Can the behavioral prior accelerate learning of new tasks? (2)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "How does PARROT compare to prior works that accelerate RL with demonstrations? (3) How does", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 434, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 434, + 338 + ], + "score": 1.0, + "content": "PARROT compare to prior methods that combine hierarchical imitation with RL?", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 312, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 313, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 313, + 355 + ], + "score": 1.0, + "content": "Domains. We evaluate our method on a suite of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 354, + 314, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 314, + 366 + ], + "score": 1.0, + "content": "challenging robotic manipulation tasks, a subset of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 365, + 313, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 313, + 377 + ], + "score": 1.0, + "content": "which are depicted in Figure 3. Each task involves", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 376, + 313, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 313, + 389 + ], + "score": 1.0, + "content": "controlling a 6-DoF robotic arm and its gripper,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 387, + 312, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 282, + 399 + ], + "score": 1.0, + "content": "with a 7D action space. The observation is a", + "type": "text" + }, + { + "bbox": [ + 282, + 388, + 312, + 398 + ], + "score": 0.89, + "content": "4 8 \\times 4 8", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 398, + 313, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 313, + 410 + ], + "score": 1.0, + "content": "RGB image, which allows us to use the same ob-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 410, + 313, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 313, + 421 + ], + "score": 1.0, + "content": "servation representation across tasks, even though", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 420, + 313, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 313, + 432 + ], + "score": 1.0, + "content": "each underlying task might have a different under-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 431, + 313, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 313, + 443 + ], + "score": 1.0, + "content": "lying state (e.g., different objects). 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(2)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "How does PARROT compare to prior works that accelerate RL with demonstrations? (3) How does", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 434, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 434, + 338 + ], + "score": 1.0, + "content": "PARROT compare to prior methods that combine hierarchical imitation with RL?", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 304, + 505, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 312, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 313, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 313, + 355 + ], + "score": 1.0, + "content": "Domains. 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Additional implementation details and hyperpa-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 231, + 260, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 260, + 243 + ], + "score": 1.0, + "content": "rameters can be found in Appendix B.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 247, + 504, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "Soft-Actor Critic (SAC). 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This approach is similar", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "to what has been used in prior work (Rajeswaran et al., 2018), except we use SAC as our RL algo-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "rithm. While we did test other methods that are designed to use demonstration data with RL such as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 330, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 459, + 342 + ], + "score": 1.0, + "content": "DDPGfD (Vecer´ık et al., 2017) and AWAC (Nair et al., 2020), we found that our simple", + "type": "text" + }, + { + "bbox": [ + 459, + 330, + 504, + 341 + ], + "score": 0.81, + "content": "\\mathrm { B C } + \\mathrm { S A C }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "variant performed better. This somewhat contradicts the results reported in prior work (Nair et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "2020), but we believe that this is because the prior data is not labeled with rewards (all transitions", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 354, + 376 + ], + "score": 1.0, + "content": "are assigned a reward of 0), and more powerful demonstration", + "type": "text" + }, + { + "bbox": [ + 354, + 363, + 377, + 374 + ], + "score": 0.78, + "content": "+ \\mathrm { R L }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "methods require access to these", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 389, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 389, + 386 + ], + "score": 1.0, + "content": "rewards, and subsequently struggle due to the reward misspecification.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "Transfer Learning via Feature Learning (VAE-features). We compare against prior methods", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "for transfer learning (in RL) that involve learning a robust representation of the input observation.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 286, + 425 + ], + "score": 1.0, + "content": "Similar to Higgins et al. (2017b), we train a", + "type": "text" + }, + { + "bbox": [ + 287, + 413, + 294, + 424 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "-VAE using the observations in our training set, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 467, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 467, + 437 + ], + "score": 1.0, + "content": "train a policy on top of the features learned by this VAE when learning downstream tasks.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 454 + ], + "score": 1.0, + "content": "Trajectory modeling and RL (TrajRL). Ghadirzadeh et al. (2020) model entire trajectories using", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "a VAE, and learn a one-step policy on top of the VAE to solve tasks using RL. Our implementation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "of this method uses a VAE architecture identical to the original paper’s, and we then train a policy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "using SAC to solve new tasks with the action space induced by the VAE. We performed additional", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 480, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 480, + 497 + ], + "score": 1.0, + "content": "hyperparameter tuning for this comparison, the details of which can be found in Appendix B.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 316, + 711 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 317, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 317, + 514 + ], + "score": 1.0, + "content": "Hierarchical imitation and RL (HIRL). Prior", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 512, + 316, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 316, + 524 + ], + "score": 1.0, + "content": "works in hierarchical imitation learning (Fox et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 522, + 317, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 317, + 535 + ], + "score": 1.0, + "content": "2017; Shankar & Gupta, 2020) train latent variable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 316, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 316, + 547 + ], + "score": 1.0, + "content": "models over expert demonstrations to discover op-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 544, + 316, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 316, + 558 + ], + "score": 1.0, + "content": "tions, and later utilize these options to learn long-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 317, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 317, + 568 + ], + "score": 1.0, + "content": "horizon tasks using RL. While PARROT can also be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 567, + 317, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 317, + 579 + ], + "score": 1.0, + "content": "extended to model the temporal structure in trajecto-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 577, + 317, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 317, + 591 + ], + "score": 1.0, + "content": "ries through conditioning on past states and actions,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 589, + 317, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 162, + 603 + ], + "score": 1.0, + "content": "by modeling", + "type": "text" + }, + { + "bbox": [ + 162, + 589, + 300, + 601 + ], + "score": 0.91, + "content": "p _ { \\mathrm { p r i o r } } ( a _ { t } , \\vert s _ { t } , s _ { t - 1 } , . . . , a _ { t - 1 } , . . . , a _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 589, + 317, + 603 + ], + "score": 1.0, + "content": "in-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 317, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 141, + 613 + ], + "score": 1.0, + "content": "stead of", + "type": "text" + }, + { + "bbox": [ + 141, + 601, + 189, + 612 + ], + "score": 0.93, + "content": "p _ { \\mathrm { p r i o r } } ( a _ { t } | s _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 600, + 317, + 613 + ], + "score": 1.0, + "content": ", we focus on a simpler version", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 611, + 317, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 317, + 623 + ], + "score": 1.0, + "content": "of the model in this paper that does not condition", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 622, + 317, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 317, + 634 + ], + "score": 1.0, + "content": "on the past. In order to provide a fair comparison,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 633, + 317, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 317, + 645 + ], + "score": 1.0, + "content": "we modify the model proposed by Shankar & Gupta", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 644, + 317, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 317, + 656 + ], + "score": 1.0, + "content": "(2020) to remove the past conditioning, which then", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 655, + 316, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 316, + 667 + ], + "score": 1.0, + "content": "reduces to training a conditional VAE, and perform-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 317, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 317, + 677 + ], + "score": 1.0, + "content": "ing RL on the action space induced by the latent", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 317, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 317, + 689 + ], + "score": 1.0, + "content": "space of this VAE. 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We see that the behavioral prior substan-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 323, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 323, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "tially increases the likelihood of executing an ac-", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 323, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 323, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "tion that is likely to lead to a meaningful interac-", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 324, + 675, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 324, + 675, + 505, + 686 + ], + "score": 1.0, + "content": "tion with an object, while still exploring a diverse", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 324, + 686, + 376, + 695 + ], + "spans": [ + { + "bbox": [ + 324, + 686, + 376, + 695 + ], + "score": 1.0, + "content": "set of actions.", + "type": "text" + } + ], + "index": 65 + } + ], + "index": 62 + } + ], + "index": 58.5 + }, + { + "type": "text", + "bbox": [ + 106, + 710, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 452, + 722 + ], + "score": 1.0, + "content": "RL agent to retain full control over the final actions in the environment (since for every", + "type": "text" + }, + { + "bbox": [ + 452, + 710, + 479, + 720 + ], + "score": 0.9, + "content": "a \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ", there", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 720, + 484, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 151, + 734 + ], + "score": 1.0, + "content": "exist some", + "type": "text" + }, + { + "bbox": [ + 151, + 720, + 209, + 735 + ], + "score": 0.94, + "content": "z = f _ { \\phi } ^ { - 1 } ( a ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 720, + 484, + 734 + ], + "score": 1.0, + "content": ", while a latent space learned by a VAE provides no such guarantee).", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 66.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [], + "index": 3.5, + "bbox_fs": [ + 105, + 83, + 506, + 172 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 187, + 306, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 307, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 307, + 199 + ], + "score": 1.0, + "content": "5.1 RESULTS, COMPARISONS AND ANALYSIS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "To answer the questions posed at the start of this section, we compare PARROT against a number", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "of prior works, as well as ablations of our method. Additional implementation details and hyperpa-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 231, + 260, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 260, + 243 + ], + "score": 1.0, + "content": "rameters can be found in Appendix B.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 208, + 505, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 247, + 504, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "Soft-Actor Critic (SAC). 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This approach is similar", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "to what has been used in prior work (Rajeswaran et al., 2018), except we use SAC as our RL algo-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "rithm. While we did test other methods that are designed to use demonstration data with RL such as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 330, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 459, + 342 + ], + "score": 1.0, + "content": "DDPGfD (Vecer´ık et al., 2017) and AWAC (Nair et al., 2020), we found that our simple", + "type": "text" + }, + { + "bbox": [ + 459, + 330, + 504, + 341 + ], + "score": 0.81, + "content": "\\mathrm { B C } + \\mathrm { S A C }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "variant performed better. This somewhat contradicts the results reported in prior work (Nair et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "2020), but we believe that this is because the prior data is not labeled with rewards (all transitions", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 354, + 376 + ], + "score": 1.0, + "content": "are assigned a reward of 0), and more powerful demonstration", + "type": "text" + }, + { + "bbox": [ + 354, + 363, + 377, + 374 + ], + "score": 0.78, + "content": "+ \\mathrm { R L }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "methods require access to these", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 389, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 389, + 386 + ], + "score": 1.0, + "content": "rewards, and subsequently struggle due to the reward misspecification.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 274, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "Transfer Learning via Feature Learning (VAE-features). We compare against prior methods", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "for transfer learning (in RL) that involve learning a robust representation of the input observation.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 286, + 425 + ], + "score": 1.0, + "content": "Similar to Higgins et al. (2017b), we train a", + "type": "text" + }, + { + "bbox": [ + 287, + 413, + 294, + 424 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "-VAE using the observations in our training set, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 467, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 467, + 437 + ], + "score": 1.0, + "content": "train a policy on top of the features learned by this VAE when learning downstream tasks.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 390, + 505, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 454 + ], + "score": 1.0, + "content": "Trajectory modeling and RL (TrajRL). Ghadirzadeh et al. (2020) model entire trajectories using", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "a VAE, and learn a one-step policy on top of the VAE to solve tasks using RL. Our implementation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "of this method uses a VAE architecture identical to the original paper’s, and we then train a policy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "using SAC to solve new tasks with the action space induced by the VAE. We performed additional", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 480, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 480, + 497 + ], + "score": 1.0, + "content": "hyperparameter tuning for this comparison, the details of which can be found in Appendix B.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 439, + 505, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 316, + 711 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 317, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 317, + 514 + ], + "score": 1.0, + "content": "Hierarchical imitation and RL (HIRL). Prior", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 512, + 316, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 316, + 524 + ], + "score": 1.0, + "content": "works in hierarchical imitation learning (Fox et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 522, + 317, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 317, + 535 + ], + "score": 1.0, + "content": "2017; Shankar & Gupta, 2020) train latent variable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 316, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 316, + 547 + ], + "score": 1.0, + "content": "models over expert demonstrations to discover op-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 544, + 316, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 316, + 558 + ], + "score": 1.0, + "content": "tions, and later utilize these options to learn long-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 317, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 317, + 568 + ], + "score": 1.0, + "content": "horizon tasks using RL. While PARROT can also be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 567, + 317, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 317, + 579 + ], + "score": 1.0, + "content": "extended to model the temporal structure in trajecto-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 577, + 317, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 317, + 591 + ], + "score": 1.0, + "content": "ries through conditioning on past states and actions,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 589, + 317, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 162, + 603 + ], + "score": 1.0, + "content": "by modeling", + "type": "text" + }, + { + "bbox": [ + 162, + 589, + 300, + 601 + ], + "score": 0.91, + "content": "p _ { \\mathrm { p r i o r } } ( a _ { t } , \\vert s _ { t } , s _ { t - 1 } , . . . , a _ { t - 1 } , . . . , a _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 589, + 317, + 603 + ], + "score": 1.0, + "content": "in-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 317, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 141, + 613 + ], + "score": 1.0, + "content": "stead of", + "type": "text" + }, + { + "bbox": [ + 141, + 601, + 189, + 612 + ], + "score": 0.93, + "content": "p _ { \\mathrm { p r i o r } } ( a _ { t } | s _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 600, + 317, + 613 + ], + "score": 1.0, + "content": ", we focus on a simpler version", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 611, + 317, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 317, + 623 + ], + "score": 1.0, + "content": "of the model in this paper that does not condition", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 622, + 317, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 317, + 634 + ], + "score": 1.0, + "content": "on the past. In order to provide a fair comparison,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 633, + 317, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 317, + 645 + ], + "score": 1.0, + "content": "we modify the model proposed by Shankar & Gupta", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 644, + 317, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 317, + 656 + ], + "score": 1.0, + "content": "(2020) to remove the past conditioning, which then", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 655, + 316, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 316, + 667 + ], + "score": 1.0, + "content": "reduces to training a conditional VAE, and perform-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 317, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 317, + 677 + ], + "score": 1.0, + "content": "ing RL on the action space induced by the latent", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 317, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 317, + 689 + ], + "score": 1.0, + "content": "space of this VAE. This comparison is similar to our", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 317, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 317, + 700 + ], + "score": 1.0, + "content": "proposed approach, but with one crucial difference:", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 317, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 317, + 711 + ], + "score": 1.0, + "content": "the mapping we learn is invertible, and allows the", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 501, + 317, + 711 + ] + }, + { + "type": "image", + "bbox": [ + 329, + 520, + 499, + 612 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 329, + 520, + 499, + 612 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 329, + 520, + 499, + 612 + ], + "spans": [ + { + "bbox": [ + 329, + 520, + 499, + 612 + ], + "score": 0.93, + "type": "image", + "image_path": "133c093c644f1ff87690827386b4e12b15f59b8f9313eac9f9a617f7c0e23011.jpg" + } + ] + } + ], + "index": 55, + "virtual_lines": [ + { + "bbox": [ + 329, + 520, + 499, + 533.1428571428571 + ], + "spans": [], + "index": 52 + }, + { + "bbox": [ + 329, + 533.1428571428571, + 499, + 546.2857142857142 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 329, + 546.2857142857142, + 499, + 559.4285714285713 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 329, + 559.4285714285713, + 499, + 572.5714285714284 + ], + "spans": [], + "index": 55 + }, + { + "bbox": [ + 329, + 572.5714285714284, + 499, + 585.7142857142856 + ], + "spans": [], + "index": 56 + }, + { + "bbox": [ + 329, + 585.7142857142856, + 499, + 598.8571428571427 + ], + "spans": [], + "index": 57 + }, + { + "bbox": [ + 329, + 598.8571428571427, + 499, + 611.9999999999998 + ], + "spans": [], + "index": 58 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 324, + 625, + 504, + 695 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 324, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "Figure 4: We plot trajectories from executing a", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 323, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 323, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "random policy, with and without the behavioral", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 323, + 645, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 323, + 645, + 505, + 656 + ], + "score": 1.0, + "content": "prior. We see that the behavioral prior substan-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 323, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 323, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "tially increases the likelihood of executing an ac-", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 323, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 323, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "tion that is likely to lead to a meaningful interac-", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 324, + 675, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 324, + 675, + 505, + 686 + ], + "score": 1.0, + "content": "tion with an object, while still exploring a diverse", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 324, + 686, + 376, + 695 + ], + "spans": [ + { + "bbox": [ + 324, + 686, + 376, + 695 + ], + "score": 1.0, + "content": "set of actions.", + "type": "text" + } + ], + "index": 65 + } + ], + "index": 62 + } + ], + "index": 58.5 + }, + { + "type": "text", + "bbox": [ + 106, + 710, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 452, + 722 + ], + "score": 1.0, + "content": "RL agent to retain full control over the final actions in the environment (since for every", + "type": "text" + }, + { + "bbox": [ + 452, + 710, + 479, + 720 + ], + "score": 0.9, + "content": "a \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ", there", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 720, + 484, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 151, + 734 + ], + "score": 1.0, + "content": "exist some", + "type": "text" + }, + { + "bbox": [ + 151, + 720, + 209, + 735 + ], + "score": 0.94, + "content": "z = f _ { \\phi } ^ { - 1 } ( a ; s )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 720, + 484, + 734 + ], + "score": 1.0, + "content": ", while a latent space learned by a VAE provides no such guarantee).", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 66.5, + "bbox_fs": [ + 105, + 709, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 120, + 80, + 503, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 80, + 503, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 80, + 503, + 283 + ], + "spans": [ + { + "bbox": [ + 120, + 80, + 503, + 283 + ], + "score": 0.976, + "type": "image", + "image_path": "71e3fea3c3c288d5e0d54f0e6c8949e4991fbf2dbad22df8c1e81bceefc36855.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 120, + 80, + 503, + 147.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 147.66666666666669, + 503, + 215.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 215.33333333333337, + 503, + 283.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 294, + 505, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "Figure 5: Results. The lines represent average performance across multiple random seeds, and the shaded", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "areas represent the standard deviation. PARROT is able to learn much faster than prior methods on a majority", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "of the tasks, and shows little variance across runs (all experiments were run with three random seeds, compu-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "tational constraints of image-based RL make it difficult to run more seeds). Note that some methods that failed", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 334, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 344 + ], + "score": 1.0, + "content": "to make any progress on certain tasks (such as “Place Sculpture in Basket”) overlap each other with a success", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 344, + 387, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 387, + 355 + ], + "score": 1.0, + "content": "rate of zero. SAC and VAE-features fail to make progress on any of the tasks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "Exploration via behavioral prior (Prior-explore). We also run experiments with an ablation of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "our method: instead of using a behavioral prior to transform the MDP being experienced by the RL", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "agent, we use it to simply aid the exploration process. While collecting data, an action is executed", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 228, + 447 + ], + "score": 1.0, + "content": "from the prior with probability", + "type": "text" + }, + { + "bbox": [ + 228, + 437, + 234, + 444 + ], + "score": 0.38, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 435, + 505, + 447 + ], + "score": 1.0, + "content": ", else an action is executed from the learned policy. We experimented", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 446, + 336, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 127, + 457 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 446, + 209, + 457 + ], + "score": 0.85, + "content": "\\epsilon = 0 . 1 , 0 . 3 , 0 . 7 , 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 446, + 336, + 457 + ], + "score": 1.0, + "content": ", and found 0.9 to perform best.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Main results. Our results are summarised in Figure 5. We see that PARROT is able to solve all", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "of the tasks substantially faster and achieve substantially higher final returns than other methods.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "The SAC baseline (which does not use any prior data) fails to make progress on any of the tasks,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "which we suspect is due to the challenge of exploring in sparse reward settings with a randomly", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "initialized policy. Figure 4 illustrates a comparison between using a behavioral prior and a random", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "policy for exploration. The VAE-features baseline similarly fails to make any progress, and due to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "the same reason: the difficulty of exploration in a sparse reward setting. Initializing the SAC policy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "score": 1.0, + "content": "with behavior cloning allows it to make progress on only two of the tasks, which is not surprising:", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 551, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 504, + 562 + ], + "score": 1.0, + "content": "a Gaussian policy learned through a behavior cloning loss is not expressive enough to represent the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 355, + 573 + ], + "score": 1.0, + "content": "complex, multi-modal action distributions observed in dataset", + "type": "text" + }, + { + "bbox": [ + 355, + 562, + 364, + 571 + ], + "score": 0.7, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 561, + 505, + 573 + ], + "score": 1.0, + "content": ". Both TrajRL and HIRL perform", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 573, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 504, + 584 + ], + "score": 1.0, + "content": "much better than any of the other baselines, but their performance plateaus a lot earlier than PAR-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "ROT. While the initial exploration performance of our learned behavioral prior is not substantially", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 594, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 504, + 607 + ], + "score": 1.0, + "content": "better from these methods (denoted by the initial success rate in the learning curves), the flexibility", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "of the representation it offers (through learning an invertible mapping) allows the RL agent to im-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "prove far beyond its initial performance. Prior-explore, an ablation of our method, is able to make", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "progress on most tasks, but is unable to learn as fast as our method, and also demonstrates unstable", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "learning on some of the tasks. We suspect this is due to the following reason: while off-policy RL", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "methods like SAC aim to learn from data collected by any policy, they are in practice quite sensitive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "to the data distribution, and can run into issues if the data collection policy differs substantially from", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 671, + 290, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 290, + 683 + ], + "score": 1.0, + "content": "the policy being learned (Kumar et al., 2019).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "Impact of dataset size on performance. We conducted additional experiments on a subset of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "our tasks to evaluate how final performance is impacted as a result of dataset size, results from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "which are shown in Figure 6. As one might expect, the size of the dataset positively correlates", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "with performance, but about 10K trajectories are sufficient for obtaining good performance, and", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": [ + 120, + 80, + 503, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 80, + 503, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 80, + 503, + 283 + ], + "spans": [ + { + "bbox": [ + 120, + 80, + 503, + 283 + ], + "score": 0.976, + "type": "image", + "image_path": "71e3fea3c3c288d5e0d54f0e6c8949e4991fbf2dbad22df8c1e81bceefc36855.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 120, + 80, + 503, + 147.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 147.66666666666669, + 503, + 215.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 215.33333333333337, + 503, + 283.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 294, + 505, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "Figure 5: Results. The lines represent average performance across multiple random seeds, and the shaded", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "areas represent the standard deviation. PARROT is able to learn much faster than prior methods on a majority", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "of the tasks, and shows little variance across runs (all experiments were run with three random seeds, compu-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "tational constraints of image-based RL make it difficult to run more seeds). Note that some methods that failed", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 334, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 344 + ], + "score": 1.0, + "content": "to make any progress on certain tasks (such as “Place Sculpture in Basket”) overlap each other with a success", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 344, + 387, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 387, + 355 + ], + "score": 1.0, + "content": "rate of zero. SAC and VAE-features fail to make progress on any of the tasks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "Exploration via behavioral prior (Prior-explore). We also run experiments with an ablation of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "our method: instead of using a behavioral prior to transform the MDP being experienced by the RL", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "agent, we use it to simply aid the exploration process. While collecting data, an action is executed", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 228, + 447 + ], + "score": 1.0, + "content": "from the prior with probability", + "type": "text" + }, + { + "bbox": [ + 228, + 437, + 234, + 444 + ], + "score": 0.38, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 435, + 505, + 447 + ], + "score": 1.0, + "content": ", else an action is executed from the learned policy. We experimented", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 446, + 336, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 127, + 457 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 446, + 209, + 457 + ], + "score": 0.85, + "content": "\\epsilon = 0 . 1 , 0 . 3 , 0 . 7 , 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 446, + 336, + 457 + ], + "score": 1.0, + "content": ", and found 0.9 to perform best.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 401, + 506, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Main results. Our results are summarised in Figure 5. We see that PARROT is able to solve all", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "of the tasks substantially faster and achieve substantially higher final returns than other methods.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "The SAC baseline (which does not use any prior data) fails to make progress on any of the tasks,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "which we suspect is due to the challenge of exploring in sparse reward settings with a randomly", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "initialized policy. Figure 4 illustrates a comparison between using a behavioral prior and a random", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "policy for exploration. The VAE-features baseline similarly fails to make any progress, and due to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "the same reason: the difficulty of exploration in a sparse reward setting. Initializing the SAC policy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "score": 1.0, + "content": "with behavior cloning allows it to make progress on only two of the tasks, which is not surprising:", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 551, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 504, + 562 + ], + "score": 1.0, + "content": "a Gaussian policy learned through a behavior cloning loss is not expressive enough to represent the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 355, + 573 + ], + "score": 1.0, + "content": "complex, multi-modal action distributions observed in dataset", + "type": "text" + }, + { + "bbox": [ + 355, + 562, + 364, + 571 + ], + "score": 0.7, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 561, + 505, + 573 + ], + "score": 1.0, + "content": ". Both TrajRL and HIRL perform", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 573, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 504, + 584 + ], + "score": 1.0, + "content": "much better than any of the other baselines, but their performance plateaus a lot earlier than PAR-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "ROT. While the initial exploration performance of our learned behavioral prior is not substantially", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 594, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 504, + 607 + ], + "score": 1.0, + "content": "better from these methods (denoted by the initial success rate in the learning curves), the flexibility", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "of the representation it offers (through learning an invertible mapping) allows the RL agent to im-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "prove far beyond its initial performance. Prior-explore, an ablation of our method, is able to make", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "progress on most tasks, but is unable to learn as fast as our method, and also demonstrates unstable", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "learning on some of the tasks. We suspect this is due to the following reason: while off-policy RL", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "methods like SAC aim to learn from data collected by any policy, they are in practice quite sensitive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "to the data distribution, and can run into issues if the data collection policy differs substantially from", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 671, + 290, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 290, + 683 + ], + "score": 1.0, + "content": "the policy being learned (Kumar et al., 2019).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 462, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "Impact of dataset size on performance. We conducted additional experiments on a subset of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "our tasks to evaluate how final performance is impacted as a result of dataset size, results from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "which are shown in Figure 6. As one might expect, the size of the dataset positively correlates", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "with performance, but about 10K trajectories are sufficient for obtaining good performance, and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "collecting additional data yields diminishing returns. Note that initializing with even a smaller", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 269, + 478, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 478, + 281 + ], + "score": 1.0, + "content": "dataset size (like 5K trajectories) yields much better performance than learning from scratch.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 688, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 79, + 502, + 199 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 79, + 502, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 502, + 199 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 502, + 199 + ], + "score": 0.97, + "type": "image", + "image_path": "87515611fe55cabe6c38f511cdcd9334dd7d7d4eba6406b801d418d6ab1d9432.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 79, + 502, + 119.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 119.0, + 502, + 159.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 159.0, + 502, + 199.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 210, + 504, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "Figure 6: Impact of dataset size on performance. We observe that training on 10K, 25K or 50K trajectories", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 220, + 208, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 208, + 232 + ], + "score": 1.0, + "content": "yields similar performance.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "collecting additional data yields diminishing returns. Note that initializing with even a smaller", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 269, + 478, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 478, + 281 + ], + "score": 1.0, + "content": "dataset size (like 5K trajectories) yields much better performance than learning from scratch.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 296, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 297, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 297, + 297 + ], + "score": 1.0, + "content": "Mismatch between train and test tasks. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 297, + 297, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 297, + 308 + ], + "score": 1.0, + "content": "ran experiments in which we deliberately bias", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 307, + 297, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 297, + 320 + ], + "score": 1.0, + "content": "the training dataset so that the training tasks and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 318, + 297, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 297, + 330 + ], + "score": 1.0, + "content": "test tasks are functionally different (i.e. involve", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 329, + 297, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 297, + 341 + ], + "score": 1.0, + "content": "substantially different actions), the results from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 340, + 297, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 297, + 352 + ], + "score": 1.0, + "content": "which are shown in Figure 7. We observe that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 351, + 297, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 297, + 363 + ], + "score": 1.0, + "content": "if the prior is trained on pick and place tasks", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 362, + 297, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 297, + 375 + ], + "score": 1.0, + "content": "alone, it can still solve downstream grasping", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 374, + 297, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 297, + 385 + ], + "score": 1.0, + "content": "tasks well. 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We suspect this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 406, + 297, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 297, + 417 + ], + "score": 1.0, + "content": "is due to the fact that pick and place tasks in-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 417, + 297, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 297, + 429 + ], + "score": 1.0, + "content": "volve a completely new action (that of open-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 428, + 297, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 297, + 439 + ], + "score": 1.0, + "content": "ing the gripper), which is never observed by the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 439, + 297, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 297, + 451 + ], + "score": 1.0, + "content": "prior if it is trained only on grasping, making it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 450, + 297, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 297, + 461 + ], + "score": 1.0, + "content": "difficult to learn this behavior from scratch for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 182, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 182, + 472 + ], + "score": 1.0, + "content": "downstream tasks.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 15 + }, + { + "type": "image", + "bbox": [ + 306, + 302, + 502, + 404 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 306, + 302, + 502, + 404 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 306, + 302, + 502, + 404 + ], + "spans": [ + { + "bbox": [ + 306, + 302, + 502, + 404 + ], + "score": 0.961, + "type": "image", + "image_path": "1c702e4fba9af88ce0efc764e1ece6595eae7d6955825e3481ac3f1004538d6b.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 306, + 302, + 502, + 314.75 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 306, + 314.75, + 502, + 327.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 306, + 327.5, + 502, + 340.25 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 306, + 340.25, + 502, + 353.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 306, + 353.0, + 502, + 365.75 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 306, + 365.75, + 502, + 378.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 306, + 378.5, + 502, + 391.25 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 306, + 391.25, + 502, + 404.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 418, + 505, + 469 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 304, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 304, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "Figure 7: Impact of train/test mismatch on perfor-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 304, + 429, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 304, + 429, + 505, + 439 + ], + "score": 1.0, + "content": "mance. 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PARROT", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "provides a mapping for executing actions in new environments structurally similar to those of prior", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "tasks. While we primarily utilized this mapping to accelerate learning of new tasks, future work", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "could investigate how it can also enable safe exploration of new environments (Hunt et al., 2020;", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "score": 1.0, + "content": "Rhinehart et al., 2020). 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This can be an issue", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "if there is a significant mismatch between the training dataset and the downstream task (as shown", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "in our experiments), and tackling this issue would make for an interesting problem. 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We observe that training on 10K, 25K or 50K trajectories", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 220, + 208, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 208, + 232 + ], + "score": 1.0, + "content": "yields similar performance.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 280 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 106, + 257, + 505, + 281 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 296, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 297, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 297, + 297 + ], + "score": 1.0, + "content": "Mismatch between train and test tasks. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 297, + 297, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 297, + 308 + ], + "score": 1.0, + "content": "ran experiments in which we deliberately bias", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 307, + 297, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 297, + 320 + ], + "score": 1.0, + "content": "the training dataset so that the training tasks and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 318, + 297, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 297, + 330 + ], + "score": 1.0, + "content": "test tasks are functionally different (i.e. involve", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 329, + 297, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 297, + 341 + ], + "score": 1.0, + "content": "substantially different actions), the results from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 340, + 297, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 297, + 352 + ], + "score": 1.0, + "content": "which are shown in Figure 7. We observe that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 351, + 297, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 297, + 363 + ], + "score": 1.0, + "content": "if the prior is trained on pick and place tasks", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 362, + 297, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 297, + 375 + ], + "score": 1.0, + "content": "alone, it can still solve downstream grasping", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 374, + 297, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 297, + 385 + ], + "score": 1.0, + "content": "tasks well. 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We suspect this", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 406, + 297, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 297, + 417 + ], + "score": 1.0, + "content": "is due to the fact that pick and place tasks in-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 417, + 297, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 297, + 429 + ], + "score": 1.0, + "content": "volve a completely new action (that of open-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 428, + 297, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 297, + 439 + ], + "score": 1.0, + "content": "ing the gripper), which is never observed by the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 439, + 297, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 297, + 451 + ], + "score": 1.0, + "content": "prior if it is trained only on grasping, making it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 450, + 297, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 297, + 461 + ], + "score": 1.0, + "content": "difficult to learn this behavior from scratch for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 182, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 182, + 472 + ], + "score": 1.0, + "content": "downstream tasks.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 286, + 297, + 472 + ] + }, + { + "type": "image", + "bbox": [ + 306, + 302, + 502, + 404 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 306, + 302, + 502, + 404 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 306, + 302, + 502, + 404 + ], + "spans": [ + { + "bbox": [ + 306, + 302, + 502, + 404 + ], + "score": 0.961, + "type": "image", + "image_path": "1c702e4fba9af88ce0efc764e1ece6595eae7d6955825e3481ac3f1004538d6b.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 306, + 302, + 502, + 314.75 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 306, + 314.75, + 502, + 327.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 306, + 327.5, + 502, + 340.25 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 306, + 340.25, + 502, + 353.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 306, + 353.0, + 502, + 365.75 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 306, + 365.75, + 502, + 378.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 306, + 378.5, + 502, + 391.25 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 306, + 391.25, + 502, + 404.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 418, + 505, + 469 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 304, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 304, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "Figure 7: Impact of train/test mismatch on perfor-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 304, + 429, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 304, + 429, + 505, + 439 + ], + "score": 1.0, + "content": "mance. Each plot shows results for four tasks. Note", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 304, + 438, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 304, + 438, + 505, + 449 + ], + "score": 1.0, + "content": "that for the pick and place tasks, the performance is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 304, + 448, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 304, + 448, + 505, + 459 + ], + "score": 1.0, + "content": "close to zero, and the curves mostly overlap each other", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 304, + 459, + 355, + 468 + ], + "spans": [ + { + "bbox": [ + 304, + 459, + 329, + 468 + ], + "score": 1.0, + "content": "on the", + "type": "text" + }, + { + "bbox": [ + 329, + 460, + 335, + 468 + ], + "score": 0.43, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 459, + 355, + 468 + ], + "score": 1.0, + "content": "-axis.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + } + ], + "index": 29.25 + }, + { + "type": "title", + "bbox": [ + 108, + 488, + 195, + 501 + ], + "lines": [ + { + "bbox": [ + 104, + 486, + 197, + 504 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 197, + 504 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "We presented PARROT, a method for learning behavioral priors using successful trials from a wide", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "range of tasks. Learning from priors accelerates RL on new tasks–including manipulating previously", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 536, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 506, + 548 + ], + "score": 1.0, + "content": "unseen objects from high-dimensional image observations–which RL from scratch often fails to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 545, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 545, + 505, + 560 + ], + "score": 1.0, + "content": "learn. Our method also compares favorably to other prior works that use prior data to bootstrap", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "learning for new tasks. While our method learns faster and performs better than prior work in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "learning novel tasks, it still requires thousands of trials to attain high success rates. Improving this", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "efficiency even further, perhaps inspired by ideas in meta-learning, could be a promising direction", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "for future work. Our work opens the possibility for several exciting future directions. PARROT", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "provides a mapping for executing actions in new environments structurally similar to those of prior", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "tasks. While we primarily utilized this mapping to accelerate learning of new tasks, future work", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "could investigate how it can also enable safe exploration of new environments (Hunt et al., 2020;", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "score": 1.0, + "content": "Rhinehart et al., 2020). While the invertibility of our learned mapping ensures that it is theoretically", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "possible for the RL policy to execute any action in the original MDP, the probability of executing", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "an action can become very low if this action was never seen in the training set. This can be an issue", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "if there is a significant mismatch between the training dataset and the downstream task (as shown", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "in our experiments), and tackling this issue would make for an interesting problem. Since our", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "method speeds up learning using a problem setup that takes into account real world considerations", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "(no rewards for prior data, no need to revisit prior tasks, etc.), we are also excited about its future", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 711, + 298, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 298, + 723 + ], + "score": 1.0, + "content": "application to domains like real world robotics.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 47, + "bbox_fs": [ + 104, + 513, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 176, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 506, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 114 + ], + "score": 1.0, + "content": "Lynton Ardizzone, Jakob Kruse, Carsten Rother, and Ullrich Kothe. 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The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "decoder consists of three fully connected layers with 128, 256, and 512 units respectively. Batch-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Norm (Ioffe & Szegedy, 2015) and ReLU nonlinearity are applied after each layer. The encoder is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "symmetric: 512, 256, and 128 layers, respectively. The size of the latent space is 8 (same as the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 338, + 711 + ], + "score": 1.0, + "content": "behavioral prior). We sweep the following values for the", + "type": "text" + }, + { + "bbox": [ + 339, + 699, + 346, + 710 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "parameter (Higgins et al., 2017a): 0.1,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 387, + 722 + ], + "score": 1.0, + "content": "0.01, 0.005, 0.001, 0.0005, and find 0.001 to be optimal. We initialize", + "type": "text" + }, + { + "bbox": [ + 387, + 711, + 395, + 721 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "to zero at the start of train-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 227, + 733 + ], + "score": 1.0, + "content": "ing, and anneal it to the target", + "type": "text" + }, + { + "bbox": [ + 228, + 721, + 235, + 732 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "value using a logistic function, achieving half of the target value in", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 655, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 257, + 95 + ], + "score": 1.0, + "content": "25K steps. 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This comparison is implemented using a conditional variational autoencoder, and uses an", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "architecture that is similar to the one used by the TrajVAE, but with two differences: since this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 423, + 144 + ], + "score": 1.0, + "content": "comparison uses image conditioning, we use the same convolutional network", + "type": "text" + }, + { + "bbox": [ + 424, + 133, + 432, + 144 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "as the behavioral", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "prior to encode the image (shown in Figure 8), and pass it as conditioning information to both the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "encoder and decoder networks. Second, instead of modeling the entire trajectory in a single forward", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "pass, it instead models individual actions, allowing the high-level policy to perform closed-loop", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 451, + 189 + ], + "score": 1.0, + "content": "control, similar to the behavioral prior model. 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Hyperparametervalue used
Target network update period1000 steps
discount factor y0.99
policy learning rate3e-4
Q-function learning rate3e-4
reward scale1.0
automatic entropy tuning number of update steps per env stepenabled 1
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We use a convolutional neural network to represent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 630, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 641 + ], + "score": 1.0, + "content": "the Q-function for SAC, shown in this figure. 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This comparison is implemented using a conditional variational autoencoder, and uses an", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "architecture that is similar to the one used by the TrajVAE, but with two differences: since this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 423, + 144 + ], + "score": 1.0, + "content": "comparison uses image conditioning, we use the same convolutional network", + "type": "text" + }, + { + "bbox": [ + 424, + 133, + 432, + 144 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "as the behavioral", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "prior to encode the image (shown in Figure 8), and pass it as conditioning information to both the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "encoder and decoder networks. Second, instead of modeling the entire trajectory in a single forward", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "pass, it instead models individual actions, allowing the high-level policy to perform closed-loop", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 451, + 189 + ], + "score": 1.0, + "content": "control, similar to the behavioral prior model. We sweep the following values for the", + "type": "text" + }, + { + "bbox": [ + 451, + 177, + 459, + 188 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "parameter:", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 504, + 199 + ], + "score": 1.0, + "content": "0.1, 0.01, 0.005, 0.001, 0.0005, and find 0.001 to be optimal. We found the annealing process to be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 464, + 210 + ], + "score": 1.0, + "content": "essential for obtaining good RL performance using this method. We use a learning rate of", + "type": "text" + }, + { + "bbox": [ + 464, + 199, + 487, + 209 + ], + "score": 0.78, + "content": "1 e { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 417, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 417, + 222 + ], + "score": 1.0, + "content": "the Adam (Kingma & Ba, 2015) optimizer to train this model for 500K steps.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 111, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Behavior cloning (BC). We implement behavior cloning via maximum likelihood with a Gaussian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "policy (and entropy regularization (Haarnoja et al., 2017)). For both behavior cloning and RL with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "SAC, we used the same policy network architecture as shown in Figure 9. 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We train this model for", + "type": "text" + }, + { + "bbox": [ + 316, + 320, + 339, + 330 + ], + "score": 0.39, + "content": "1 . 5 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "steps, using Adam with a learning rate of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 329, + 225, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 128, + 341 + ], + "score": 0.89, + "content": "1 e ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 329, + 225, + 342 + ], + "score": 1.0, + "content": "and a batch size of 128.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 276, + 506, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "Soft Actor Critic (SAC). 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Hyperparametervalue used
Target network update period1000 steps
discount factor y0.99
policy learning rate3e-4
Q-function learning rate3e-4
reward scale1.0
automatic entropy tuning number of update steps per env stepenabled 1
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Algorithm 2 Scripted GraspingAlgorithm3 Scripted Pick and Place
1:threshold ←0.021:threshold ← 0.02
2:1 numTimesteps ←252:numTimesteps ←25
3:targetPoint ← object position3:placeAttempted ←False
4: for tin(O,numTimesteps) do4: dropPos ←point above container
5:eePos ← end effector position5: for t in (O, numTimesteps) do
6:targetEEDist ← distance(targetPoint,eePos)6: eePos ← end effector position
7:if targetEEDist > threshold then7: objectDropDist ← distance(eePos,dropPos)
8:action ← targetPoint-eePosif placeAttempted then
9:else if gripperOpened then8: 9: action ←0
10:action ← close gripper10: else if object not grasped AND objectDropDist
11:else if object not raised high enough then> threshold then Execute grasp using Algorithm 2
12:action ← lift upward11:
13:else12: else if objectDropDist >threshold then
14:action←013: action ← dropPos－eePos
15:end if14: else
16:noise ~ N(0,0.1)15: action ← open gripper
17:action ←action+noise16: placeAttempted ←True 17:
18:s'←env.step(action)else
19: end foraction←0
20:19:end if
20:noise ~ N(0,0.1)
21:action ← action+noise
22:s'← env.step(action)
23: end for
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Algorithm 2 Scripted GraspingAlgorithm3 Scripted Pick and Place
1:threshold ←0.021:threshold ← 0.02
2:1 numTimesteps ←252:numTimesteps ←25
3:targetPoint ← object position3:placeAttempted ←False
4: for tin(O,numTimesteps) do4: dropPos ←point above container
5:eePos ← end effector position5: for t in (O, numTimesteps) do
6:targetEEDist ← distance(targetPoint,eePos)6: eePos ← end effector position
7:if targetEEDist > threshold then7: objectDropDist ← distance(eePos,dropPos)
8:action ← targetPoint-eePosif placeAttempted then
9:else if gripperOpened then8: 9: action ←0
10:action ← close gripper10: else if object not grasped AND objectDropDist
11:else if object not raised high enough then> threshold then Execute grasp using Algorithm 2
12:action ← lift upward11:
13:else12: else if objectDropDist >threshold then
14:action←013: action ← dropPos－eePos
15:end if14: else
16:noise ~ N(0,0.1)15: action ← open gripper
17:action ←action+noise16: placeAttempted ←True 17:
18:s'←env.step(action)else
19: end foraction←0
20:19:end if
20:noise ~ N(0,0.1)
21:action ← action+noise
22:s'← env.step(action)
23: end for
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Hyperparametervalue used
Target network update period1000 steps
discount factor y0.99
policy learning rate3e-4
Q-function learning rate3e-4
reward scale1.0
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Algorithm 2 Scripted GraspingAlgorithm3 Scripted Pick and Place
1:threshold ←0.021:threshold ← 0.02
2:1 numTimesteps ←252:numTimesteps ←25
3:targetPoint ← object position3:placeAttempted ←False
4: for tin(O,numTimesteps) do4: dropPos ←point above container
5:eePos ← end effector position5: for t in (O, numTimesteps) do
6:targetEEDist ← distance(targetPoint,eePos)6: eePos ← end effector position
7:if targetEEDist > threshold then7: objectDropDist ← distance(eePos,dropPos)
8:action ← targetPoint-eePosif placeAttempted then
9:else if gripperOpened then8: 9: action ←0
10:action ← close gripper10: else if object not grasped AND objectDropDist
11:else if object not raised high enough then> threshold then Execute grasp using Algorithm 2
12:action ← lift upward11:
13:else12: else if objectDropDist >threshold then
14:action←013: action ← dropPos－eePos
15:end if14: else
16:noise ~ N(0,0.1)15: action ← open gripper
17:action ←action+noise16: placeAttempted ←True 17:
18:s'←env.step(action)else
19: end foraction←0
20:19:end if
20:noise ~ N(0,0.1)
21:action ← action+noise
22:s'← env.step(action)
23: end for
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sha256:716d417a54a36c4378fd65937d790af716479be2734421c09626716dd545309e +size 13998 diff --git a/parse/train/lxHgXYN4bwl/lxHgXYN4bwl.md b/parse/train/lxHgXYN4bwl/lxHgXYN4bwl.md new file mode 100644 index 0000000000000000000000000000000000000000..571a760b3179c9576045fb6ad1115883d33eabd8 --- /dev/null +++ b/parse/train/lxHgXYN4bwl/lxHgXYN4bwl.md @@ -0,0 +1,1801 @@ +# EXPRESSIVE POWER OF INVARIANT AND EQUIVARIANT GRAPH NEURAL NETWORKS + +Waïss Azizian ENS, PSL University, Paris, France waiss.azizian@ens.fr + +Marc Lelarge +INRIA & ENS, PSL University, Paris, France +marc.lelarge@ens.fr + +# ABSTRACT + +Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power of these GNN architectures. The current universality theorems only apply to intractable classes of GNNs. Here, we prove the first approximation guarantees for practical GNNs, paving the way for a better understanding of their generalization. Our theoretical results are proved for invariant GNNs computing a graph embedding (permutation of the nodes of the input graph does not affect the output) and equivariant GNNs computing an embedding of the nodes (permutation of the input permutes the output). We show that Folklore Graph Neural Networks (FGNN), which are tensor based GNNs augmented with matrix multiplication are the most expressive architectures proposed so far for a given tensor order. We illustrate our results on the Quadratic Assignment Problem (a NP-Hard combinatorial problem) by showing that FGNNs are able to learn how to solve the problem, leading to much better average performances than existing algorithms (based on spectral, SDP or other GNNs architectures). On a practical side, we also implement masked tensors to handle batches of graphs of varying sizes. + +# 1 INTRODUCTION + +Graph Neural Networks (GNN) are designed to deal with graph structured data. Since a graph is not changed by permutation of its nodes, GNNs should be either invariant if they return a result that must not depend on the representation of the input (typically when building a graph embedding) or equivariant if the output must be permuted when the input is permuted (typically when building an embedding of the nodes). More fundamentally, incorporating symmetries in machine learning is a fundamental problem as it allows to reduce the number of degree of freedom to be learned. + +Deep learning on graphs. This paper focuses on learning deep representation of graphs with network architectures, namely GNN, designed to be invariant to permutation or equivariant by permutation. From a practical perspective, various message passing GNNs have been proposed, see Dwivedi et al. (2020) for a recent survey and benchmarking on learning tasks. In this paper, we study 3 architectures: Message passing GNN (MGNN) which is probably the most popular architecture used in practice, order- $k$ Linear GNN $k$ -LGNN) proposed in Maron et al. (2018) and order- $k$ Folklore GNN $k$ - FGNN) first introduced by Maron et al. (2019a). MGNN layers are local thus highly parallelizable on GPUs which make them scalable for large sparse graphs. $k$ -LGNN and $k$ -FGNN are dealing with representations of graphs as tensors of order $k$ which make them of little practical use for $k \geq 3$ . + +In order to compare these architectures, the separating power of these networks has been compared to a hierarchy of graph invariants developed for the graph isomorphism problem. Namely, for $k \geq 2$ $k { \mathrm { - W L } } ( G )$ are invariants based on the Weisfeiler-Lehman tests (described in Section 4.1). For each $k \geq 2$ , $( \dot { k } + 1 )$ -WL has strictly more separating power than $k$ -WL (in the sense that there is a pair of non-isomorphic graphs distinguishable by $( k + 1 )$ -WL and not by $k$ -WL). GIN (which are invariant MGNN) introduced in $\mathrm { X u }$ et al. (2018) are shown to be as powerful as 2-WL. In Maron et al. (2019a), Geerts (2020b) and Geerts (2020a), $k$ -LGNN are shown to be as powerful as $k$ -WL and 2-FGNN is shown to be as powerful as 3-WL. In this paper, we extend this last result about $k$ -FGNN to general values of $k$ . So in term of separating power, when restricted to tensors of order $k$ , $k$ -FGNN is the most powerful architecture among the ones considered in this work. This means that for a given pair of graphs $G$ and $G ^ { \prime }$ , if $( k + { \bar { 1 } } ) – \mathbf { W L } ( G ) \neq ( k + 1 ) – \mathbf { W L } ( G ^ { \prime } )$ , then there exists a $k$ -FGNN, say $\mathbf { G N N } _ { G , G ^ { \prime } }$ such that $\mathbf { G } \mathbf { N } \mathbf { N } _ { G , G ^ { \prime } } ( G ) \neq \mathbf { G } \mathbf { N } \mathbf { N } _ { G , G ^ { \prime } } ( G ^ { \prime } )$ . + +Approximation results for GNNs. Results on the separating power of GNNs only deal with pairwise comparison of graphs: we need a priori a different GNN for each pair of graphs in order to distinguish them. Such results are of little help in a practical learning scenario. Our main contribution in this paper overcomes this issue and we show that a single GNN can give a meaningful representation for all graphs. More precisely, we characterize the set of functions that can be approximated by MGNNs, $k$ -LGNNs and $k$ -FGNNs respectively. Standard Stone-Weierstrass theorem shows that if an algebra $\mathcal { A }$ of real continuous functions separates points, then $\mathcal { A }$ is dense in the set of continuous function on a compact set. Here we extend such a theorem to general functions with symmetries and apply it to invariant and equivaraint functions to get our main result for GNNs. As a consequence, we show that $k$ -FGNNs have the best approximation power among architectures dealing with tensors of order $k$ . + +Universality results for GNNs. Universal approximation theorems (similar to Cybenko (1989) for multi-layers perceptron) have been proved for linear GNNs in Maron et al. (2019b); Keriven & Peyré (2019); Chen et al. (2019). They show that some classes of GNNs can approximate any function defined on graphs. To be able to approximate any invariant function, they require the use of very complex networks, namely $k$ -LGNN where $k$ tends to infinity with $n$ the number of nodes. Since we prove that any invariant function less powerful than $( k + 1 )$ -WL can be approximated by a $k$ -FGNN, letting $k$ tends to infinity directly implies universality. Universality results for $k$ -FGNN is another contribution of our work. + +Equivariant GNNs. Our second set of results extends previous analysis from invariant functions to equivariant functions. There are much less results about equivariant GNNs: Keriven & Peyré (2019) proves the universality of linear equivariant GNNs, and Maehara & Hoang (2019) shows the universality of a new class of networks they introduced. Here, we consider a natural equivariant extension of $k$ -WL and prove that equivariant $( k + 1 )$ -LGNNs and $k$ -FGNN can approximate any equivariant function less powerful than this equivariant $( k + 1 )$ -WL for $k \geq 1$ . At this stage, we should note that all universality results for GNNs by Maron et al. (2019b); Keriven & Peyré (2019); Chen et al. (2019) are easily recovered from our main results. Also our analysis is valid for graphs of varying sizes. + +Empirical results for the Quadratic Assigment Problem (QAP). To validate our theoretical contributions, we empirically show that 2-FGNN outperforms classical MGNN. Indeed, Maron et al. (2019a) already demonstrate state of the art results for the invariant version of 2-FGNNs (for graph classification or graph regression). Here we consider the graph alignment problem and show that the equivariant 2-FGNN is able to learn a node embedding which beats by a large margin other algorithms (based on spectral method, SDP or GNNs). + +Outline and contribution. After reviewing more previous works and notations in the next section, we define the various classes of GNNs studied in this paper in Section $3 :$ message passing GNN, linear GNN and folklore GNN. Section 4 contains our main theoretical results for GNNs. First in Section 4.2 we describe the separating power of each GNN architecture with respect to the WeisfeilerLehman test. In Section 4.3, we give approximation guarantees for MGNNs, LGNNs and FGNNs at fixed order of tensor. They cover both the invariant and equivariant cases and are our main theoretical contributions. For these, we develop in Section D a fine-grained Stone-Weierstrass approximation theorem for vector-valued functions with symmetries. Our theorem handles both invariant and equivariant cases and is inspired by recent works in approximation theory. In Section 6, we illustrate our theoretical results on a practical application: the graph alignment problem, a well-known NP-hard problem. We highlight a previously overlooked implementation question: the handling of batches of graphs of varying sizes. A PyTorch implementation of the code necessary to reproduce the results is available at https://github.com/mlelarge/graph_neural_net + +# 2 RELATED WORK + +The pioneering works that applied neural networks to graphs are Gori et al. (2005) and Scarselli et al. (2009) that learn node representation with recurrent neural networks. More recent message passing architectures make use of non-linear functions of the adjacency matrix (Kipf & Welling, 2016), for example polynomials (Defferrard et al., 2016). For regular-grid graphs, they match classical convolutional networks which by design can only approximate translation-invariant functions and hence have limited expressive power. In this paper, we focus instead on more expressive architectures. + +Following the recent surge in interest in graph neural networks, some works have tried to extend the pioneering work of Cybenko (1989); Hornik et al. (1989) for various GNN architectures. Among the first ones is Scarselli et al. (2009), which studied invariant message-passing GNNs. They showed that such networks can approximate, in a weak sense, all functions whose discriminatory power is weaker than 1-WL. Yarotsky (2018) described universal architectures which are invariant or equivariant to some group action. These models rely on polynomial intermediate layers of arbitrary degrees, which would be prohibitive in practice. Maron et al. (2019b) leveraged classical results about the polynomials invariant to a group action to show that $k$ -LGNN are universal as $k$ tends to infinity with the number of nodes. Keriven & Peyré (2019) derived a similar result, in the more complicated equivariant case by introducing a new Stone-Weierstrass theorem. Similarly to Maron et al. (2019b), they require the order of tensors to go to infinity. Another route towards universality is the one of Chen et al. (2019). In the invariant setting, they show for a class of GNN that universality is equivalent to being able to discriminate between (non-isomorphic) graphs. However, the only way to achieve such discriminatory power is to use tensors of arbitrary high order, see also Ravanbakhsh (2020). Our work encompass and precise these results using high-order tensors as it yields approximation guarantees even at fixed order of tensor. + +CPNGNN in Sato et al. (2019) and DimeNet in Klicpera et al. (2020) are message passing GNN incorporating more information than those studied here. Partial results about their separating power follows from Garg et al. (2020) which provides impossibility results to decide graph properties including girth, circumference, diameter, radius, conjoint cycle, total number of cycles, and $k$ -cliques. Chen et al. (2020) studies the ability of GNNs to count graph substructures. Though our theorems are much more general, note that their results are improved by the present work. Note also, that if the nodes are given distinct features, MGNNs become much more expressive Loukas (2019) but looses their invariant or equivariant properties. Averaging i.e. relational pooling (RP) has been proposed to recover these properties Murphy et al. (2019a). However, the ideal RP, leading to a universal approximation, cannot be used for large graphs due to its complexity of $O ( | V | ! )$ . Regarding the other classes of RPGNN i.e. the $k$ -ary pooling (Murphy et al., 2019b), we will show how our general theorems in the invariant case can be applied to characterize their approximation power (see Section 5). + +Note that for neural networks on sets, the situation is a bit simpler. Efficient architectures such as DeepSets (Zaheer et al., 2017) or PointNet (Qi et al., 2017) have been shown to be invariant universal. Similar results exist in the equivariant case (Segol & Lipman, 2020; Maron et al., 2020), whose proofs rely on polynomial arguments. Though this is not our main motivation, our approximation theorems could also be applied in this context see Sections D.3 and D.4. + +# 2.1 NOTATIONS: GRAPHS AS TENSORS + +We denote by $\mathbb { F } , \mathbb { F } _ { 0 } , \mathbb { F } _ { 1 / 2 } , \mathbb { F } _ { 1 } , \dots .$ arbitrary finite-dimensional spaces of the form $\mathbb { R } ^ { p }$ (for various values of $p$ ) typically representing the space of features. Product of vectors in $\mathbb { R } ^ { p }$ always refer to component-wise product. There are two ways to see graphs with features. First, graphs can be seen as tensors of order $k$ : $G \in \mathbb { F } ^ { n ^ { k } }$ . The classical representation of a graph by its (weighted) adjacency matrix for $k = 2$ is a tensor of order 2 in $\mathbb { R } ^ { n ^ { 2 } }$ . This case allows for features on edges by replacing $\mathbb { R } ^ { n ^ { 2 } }$ with $\mathbb { F } ^ { n ^ { 2 } }$ where $\mathbb { F }$ is some $\mathbb { R } ^ { p }$ . Second, graphs can also be represented by their discrete structure with an additional feature vector. More exactly, denote by ${ \mathcal { G } } _ { n }$ the set of discrete graphs $G = ( V , E )$ with $n$ nodes $V = [ n ]$ and edges $E \subseteq V ^ { 2 }$ (with no weights on edges). Such a $G \in \mathcal G _ { n }$ with a vector $h ^ { 0 } \in \mathbb { F } ^ { n }$ represents a graphs with features on the vertices. + +# 2.2 DEFINITIONS: INVARIANT AND EQUIVARIANT OPERATORS + +Let $[ n ] = \{ 1 , \dots , n \}$ . The set of permutations on $[ n ]$ is denoted by $S _ { n }$ . For $G \in \mathbb { F } ^ { n ^ { k } }$ and $\sigma \in S _ { n }$ , we define: $( \sigma \star G ) _ { \sigma ( i _ { 1 } ) , \ldots , \sigma ( i _ { k } ) } = G _ { i _ { 1 } , \ldots , i _ { k } }$ . Note that the $\star$ operation is valid between a permutation in $S _ { n }$ and a graph $G$ as soon as the number of nodes of $G$ is $n$ , i.e. it is valid for any order $k$ tensor representation of the graph. Two graphs $G _ { 1 } , G _ { 2 }$ are said isomorphic if they have the same number of nodes and there exists a permutation $\sigma$ such that $G _ { 1 } = \sigma \star G _ { 2 }$ . + +Definition 1. A function $f : \mathbb { F } _ { 0 } ^ { n ^ { k } } \to \mathbb { F } _ { 1 }$ is said to be invariant if $f ( \sigma \star G ) = f ( G )$ for every permutation $\sigma \in S _ { n }$ and every $G \in \mathbb { F } _ { 0 } ^ { n ^ { k } }$ . A function $f : \mathbb { F } _ { 0 } ^ { n ^ { k } } \to \mathbb { F } _ { 1 } ^ { n ^ { \ell } }$ is said to be equivariant if $f ( \sigma \star G ) = \sigma \star f ( G )$ for every permutation $\sigma \in S _ { n }$ and every $G \in \mathbb { F } _ { 0 } ^ { n ^ { k } }$ . + +Note that composing an equivariant function with an invariant function gives an invariant function. For $k \geq 1$ , we define the invariant summation layer $S ^ { k } : \mathbb { F } ^ { n ^ { k } } \mathbb { F }$ by $\begin{array} { r } { S ^ { k } ( G ) = \sum _ { \mathbf i \in [ n ] ^ { k } } G _ { \mathbf i } } \end{array}$ for $G \in \mathbb { F } ^ { n ^ { k } }$ . We also define the equivariant reduction layer $S _ { 1 } ^ { k } : \mathbb { F } ^ { n ^ { k } } \mathbb { F } ^ { n }$ as follows: $S _ { 1 } ^ { k } ( G ) _ { i } =$ $\sum _ { 1 \leq i _ { 2 } . . . i _ { k } \leq n } G _ { i , i _ { 2 } , . . . i _ { k } }$ . For message passing GNN, we will use the equivariant layer $\mathrm { I d } + \lambda S ^ { 1 }$ : $\mathbb { F } ^ { n } \to \mathbb { F } ^ { n }$ defined by, $( \operatorname { I d } + \lambda S ^ { 1 } ) ( G ) _ { i } = G _ { i } + \lambda S ^ { 1 } ( G )$ , where $\lambda \in \mathbb { R }$ is a learnable parameter. + +In the sequel, we will need a mapping $I ^ { k }$ lifting the input graph to a higher order tensor. We denote by $I ^ { k } : \mathbb { F } _ { 0 } ^ { n ^ { 2 } } \to \mathbb { F } _ { 1 } ^ { n ^ { k } }$ the initialization function mapping for a given graph each $k$ -tuple to its isomorphism type. We refer to the appendix Section C.3 for a precise description of this linear equivariant function. Note at this stage that $I ^ { 2 }$ is given by, for $G \in \mathbb { F } ^ { n ^ { 2 } }$ , $I ( G ) _ { i , j } = ( G _ { i , j } , \delta _ { i , j } )$ where $\delta _ { i , j }$ is 0 if $i \neq j$ and 1 otherwise. Indeed for a pair of nodes $i , j$ in a graph (without features), there are only three isomorphism types: $i = j ; i \neq j$ and $( i , j )$ is an edge; $i \neq j$ but $( i , j )$ is not an edge. + +# 3 GNN DEFINITIONS + +In this section, we define the various GNN architectures studied in this paper. In all architectures, there is a main building block or layer mapping $\mathbb { F } _ { t } ^ { n ^ { k } }$ k to F n kt +1 where $\mathbb { F } _ { t } ^ { n ^ { k } }$ can be seen as the space for the representation of the graph at layer . We will define three different types of layers for message passing GNN, linear GNN and folklore GNN.The case $k = 2$ is probably the most interesting case from a practical point view and corresponds to a case where a layer takes as input a graph (with features on nodes and edges) and produces as output a graph (with new features on nodes and edges). For each type of GNNs, thwill share the last function: $m _ { I } : \mathbb { F } _ { T + 1 } \mathbb { F }$ variant and an equivaraint for the invariant case and $m _ { E } : \mathbb { F } _ { T + 1 } ^ { n } \mathbb { F } ^ { n }$ ecturesfor the equivariant case which are continuous functions. It is typically modeled by a Multi Layer Perceptron, which is applied on each component for the equivariant case. In words, each network takes as input a graph $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ , produces in the invariant case a graph embedding in $\mathbb { F } _ { T + 1 }$ and in the equivaraint case a node embedding in $\mathbb { F } _ { T + 1 } ^ { n }$ , then these embeddings are passed through the function $m _ { I }$ or $m _ { E }$ respectively to get a feature in $\mathbb { F }$ or $\mathbb { F } ^ { n }$ for the learning task. + +# 3.1 MESSAGE PASSING GNN + +Message passing GNN (MGNN) are defined for classical graphs $G$ with features on the nodes. More exactly they take as input a discrete graph $G = ( V , E ) \in \mathcal { G } _ { n }$ and features on the nodes $h ^ { 0 } \in \mathbb { F } ^ { n }$ . MGNN are then defined inductively as follows: let $h _ { i } ^ { \ell } \in \mathbb { F } _ { \ell }$ denote the feature at layer $\ell$ associated with node $i$ , the updated features $h _ { i } ^ { \ell + 1 }$ are obtained as: $h _ { i } ^ { \ell + 1 } = f \left( h _ { i } ^ { \ell } , \left\{ \left\{ h _ { j } ^ { \ell } \right\} \right\} _ { j \sim i } \right)$ , where $j \sim i$ means that nodes $j$ and $i$ are neighbors in the graph $G$ , i.e. $( i , j ) \in E$ , and the function $f$ is a learnable function taking as input the feature vector of the center vertex $h _ { i } ^ { \ell }$ and the multiset of features of the neighboring vertices $\left\{ \left\{ h _ { j } ^ { \ell } \right\} \right\} _ { j \sim i }$ . Indeed, it follows from Lem. 33 in Appendix, that any such function $f$ can be approximated by a layer of the form, + +$$ +h _ { i } ^ { \ell + 1 } = f _ { 0 } \left( h _ { i } ^ { \ell } , \sum _ { j \sim i } f _ { 1 } \left( h _ { i } ^ { \ell } , h _ { j } ^ { \ell } \right) \right) , +$$ + +where $f _ { 0 } : \mathbb { F } _ { \ell } \times \mathbb { F } _ { \ell + 1 / 2 } \to \mathbb { F } _ { \ell + 1 }$ and $f _ { 1 } : \mathbb { F } _ { \ell } \times \mathbb { F } _ { \ell } \to \mathbb { F } _ { \ell + 1 / 2 }$ , so that $\mathbb { F } _ { \ell }$ is the field for the features at the $\ell$ -th layer. We call such a function a message passing layer and denote it by $F : \mathbb { F } _ { \ell } ^ { n } \to \mathbb { F } _ { \ell + 1 } ^ { n }$ (note that $F$ depends implicitly from the graph). Then an equivariant message passing GNN is simply obtained by the composition of message passing layers: $F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 }$ , where each $F _ { i }$ is a message passing layer. Clearly since each $F _ { i }$ is equivariant, this message passing GNN is also equivariant and produces features on each node in the space $\mathbb { F } _ { T }$ . In order to obtain an invariant GNN, we apply an invariant function from $\mathbb { F } _ { T } ^ { n } \to \mathbb { F } _ { T + 1 }$ on the output of an equivariant message passing GNN. In practice, a symmetric function is applied on the vectors of features indexed by the nodes, typically the sum of the features $\begin{array} { r } { \sum _ { i } ( F _ { T } \circ . . . F _ { 2 } \circ F _ { 1 } ( G ) ) _ { i } } \end{array}$ is taken as an invariant feature for the graph $G$ . With our notation, $S ^ { 1 } \circ F _ { T } \circ . . . F _ { 2 } \circ F _ { 1 }$ (where $S ^ { 1 }$ was defined in Section2.2) defines an invariant message passing GNN. + +Hence, we define the sets of message passing GNNs as follows: + +$$ +\begin{array} { r c l } { \mathbf { M G N N } _ { I } } & { = } & { \{ m _ { I } \circ S ^ { 1 } \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } , \forall T \} } \\ { \mathbf { M G N N } _ { E } } & { = } & { \{ m _ { E } \circ ( \mathrm { I d } + \lambda S ^ { 1 } ) \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } , \forall T \} } \end{array} +$$ + +where $F _ { t } : \mathbb { F } _ { t } ^ { n } \to \mathbb { F } _ { t + 1 } ^ { n }$ are message passing layers. + +# 3.2 LINEAR GNN + +We define the linear graph layer of order $k$ as $F : \mathbb { F } _ { \ell } ^ { n ^ { k } } \to \mathbb { F } _ { \ell + 1 } ^ { n ^ { k } }$ , where for all $G \in \mathbb { F } _ { \ell } ^ { n ^ { k } }$ , $F ( G ) =$ $f \left( L [ G ] \right)$ where $L : \mathbb { F } _ { \ell } ^ { n ^ { k } } \to \mathbb { F } _ { \ell } ^ { n ^ { k } }$ is a linear equivariant function, and $f : \mathbb { F } _ { \ell } \to \mathbb { F } _ { \ell + 1 }$ is a learnable function applied on each of the $n ^ { k }$ features and $\mathbb { F } _ { \ell }$ is the field for the features at the $\ell \cdot$ -th layer. + +We then define the sets of linear GNNs as follows: + +$$ +\begin{array} { r l r } { k \mathrm { - } \mathrm { L G N N } _ { I } } & { = } & { \{ m _ { I } \circ S ^ { k } \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k } , \forall T \} } \\ { k \mathrm { - } \mathrm { L G N N } _ { E } } & { = } & { \{ m _ { E } \circ S _ { 1 } ^ { k } \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k } , \forall T \} } \end{array} +$$ + +where $I ^ { k } : \mathbb { F } _ { 0 } ^ { n ^ { 2 } } \to \mathbb { F } _ { 1 } ^ { n ^ { k } }$ is defined in $\ S 2 . 2$ and for $t \geq 1$ , $F _ { t } : \mathbb { F } _ { t } ^ { n ^ { k } } \to \mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ are linear equivariant layers. + +# 3.3 FOLKLORE GNN + +The main building block of Folklore GNN (FGNN) is what we call the folklore graph layer (FGL) of order k defined as follows: for k ≥ 1, F : Fnk\` $F : \mathbb { F } _ { \ell } ^ { n ^ { k } } \to \mathbb { F } _ { \ell + 1 } ^ { n ^ { k } }$ 1 where for all G ∈ Fnk\` a nd all $\mathbf { i } \in [ n ] ^ { k }$ , + +$$ +F ( G ) _ { \mathbf { i } } = f _ { 0 } \left( G _ { \mathbf { i } } , \sum _ { j = 1 } ^ { n } \prod _ { w = 1 } ^ { k } f _ { w } \left( G _ { i _ { 1 } , \dots , i _ { w - 1 } , j , i _ { w + 1 } , \dots , i _ { k } } \right) \right) , +$$ + +where $f _ { 0 } : \mathbb { F } _ { \ell } \times \mathbb { F } _ { \ell + 1 / 2 } \to \mathbb { F } _ { \ell + 1 }$ and $f _ { k } : \mathbb { F } _ { \ell } \to \mathbb { F } _ { \ell + 1 / 2 }$ are learnable functions. As shown in Lem. 33 in Appendix, FGL is an equivariant function which is indeed very expressive. + +F t : F n 2t $F _ { t } : \mathbb { F } _ { t } ^ { n ^ { 2 } } \mathbb { F } _ { t + 1 } ^ { n ^ { 2 } }$ phs , so $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ $F _ { T } \circ \dots F _ { 1 } \circ F _ { 0 }$ w define 2-FGNN by composing folklore gris an equivariant GNN producing a graph in $\mathbb { F } _ { T + 1 } ^ { n ^ { 2 } }$ yers. To obtain an invariant feature of the graph, we use the summation layer $S ^ { 2 }$ defined in Section 2.2 so that $S ^ { 2 } \circ F _ { T } \circ . . . F _ { 1 } \circ F _ { 0 }$ is now an invariant 2-FGNN. In order to define general $k$ -FGNN, we first need to lift the classical graph to a tensor in $\mathbb { F } ^ { n ^ { k } }$ , then we apply folklore graph layers of order $k$ and finally we need to project the tensor in $\mathbb { F } ^ { n ^ { k } }$ to a tensor in $\mathbb { F } ^ { n }$ for the equivariant version and to a tensor in $\mathbb { F }$ for the invariant version. The first step is done with the linear equivariant function $I ^ { k } : \mathbb { F } _ { 0 } ^ { n ^ { 2 } } \to \mathbb { F } _ { 1 } ^ { n ^ { k } }$ defined in Section 2.2. The last step is done with the reduction layer $S _ { 1 } ^ { k }$ for the equivariant case and the summation layer $S ^ { k }$ for the invariant case, both defined in Section 2.2. + +We define the sets of folklore GNNs as follows: + +$$ +\begin{array} { r l r } { k \mathrm { – F G N N } _ { I } } & { = } & { \{ m _ { I } \circ S ^ { k } \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k } , \forall T \} } \\ { k \mathrm { – F G N N } _ { E } } & { = } & { \{ m _ { E } \circ S _ { 1 } ^ { k } \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k } , \forall T \} } \end{array} +$$ + +where $F _ { t } : \mathbb { F } _ { t } ^ { n ^ { k } } \to \mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ are FGLs. + +# 4 THEORETICAL RESULTS FOR GNNS + +# 4.1 WEISFEILER-LEHMAN INVARIANT AND EQUIVARIANT VERSIONS + +We introduce a family of functions on graphs parametrized by integers $k \geq 2$ developed for the graph isomorphism problem and working with tuples of $k$ vertices. Each $k$ -tuple $\mathbf { i } \in V ^ { \dot { k } } = [ n ] ^ { k }$ is given a color $c ^ { 0 } ( \mathbf { i } )$ corresponding to its isomorphism type (see Section B.2). The $k$ -WL test relies on the following notion of neighborhood, defined by, for any $w \in [ k ]$ , and $\mathbf { i } = ( i _ { 1 } , \dots , i _ { k } ) \in V ^ { k }$ , $N _ { w } ( { \bf i } ) =$ $\{ ( i _ { 1 } , \dotsc , i _ { w - 1 } , j , i _ { w + 1 } , \dotsc , i _ { k } ) : j \in V \}$ . Then, the colors of the $k$ -tuples are refined as follows, $c ^ { t + 1 } ( \mathbf { i } ) = \operatorname { L e x } \left( c ^ { t } ( \mathbf { i } ) , ( C _ { 1 } ^ { t } ( \mathbf { i } ) , \dots , C _ { k } ^ { t } ( \mathbf { i } ) ) \right)$ where, for $w \in [ k ]$ $[ k ] , C _ { w } ^ { t } ( \mathbf { i } ) = \Big \{ \Big \{ c ^ { t } ( \tilde { \mathbf { i } } ) : \tilde { \mathbf { i } } \in N _ { w } ( \mathbf { i } ) \Big \} \Big \}$ and the function Lex means that all occuring colors are lexicographically ordered and replaced by an initial segment of the natural numbers. + +For a graph $G$ , let $k { \mathrm { - W L } } _ { I } ^ { T } ( G )$ denote the multiset of colors of the $k$ -WL algorithm at the $T ^ { t h }$ iteration. After a finite number of steps (which depends on the number of vertices in the graph), the algorithm stops because a stable coloring is reached (no color class of $k$ -tuples is further divided). We denote by $k { \mathrm { - W L } } _ { I } ( G )$ the multiset of colors in the stable coloring. This is a graph invariant that is usually used to test if graphs are isomorphic. The power of this invariant increases with $k$ Cai et al. (1989). + +We now define an equivariant version of $k$ -WL test to express the discriminatory power of equivariant architectures For this, we construct a coloring of the vertices from the coloring of the $k$ -tuples given by the standard $k$ -WL algorithm. Formally, define $k – \mathbf { W L } _ { E } ^ { T } : \mathbb { F } _ { 0 } ^ { n ^ { 2 } } \to \mathbb { F } ^ { n }$ by, for $i \in V$ : $k { \mathrm { - W L } } _ { E } ^ { T } ( G ) _ { i } =$ $\left\{ \left\{ c ^ { T } ( \mathbf { i } ) : \mathbf { i } \in V ^ { k } , i _ { 1 } = i \right\} \right\}$ . Similarly, define $k { \mathrm { - } } \mathbf { W } \mathbf { L } _ { E } ( G ) = \left\{ \left\{ c ( \mathbf { i } ) : \mathbf { i } \in V ^ { k } , i _ { 1 } = i \right\} \right\}$ where $c ( \mathbf { i } )$ is the stable coloring obtained by the algorithm. + +# 4.2 SEPARATING POWER OF GNNS + +We formulate our results using the equivalence relation introduced by Timofte (2005), which characterizes the separating power of a set of functions. + +Definition 2. Let $\mathcal { F }$ be a set of functions $f$ defined on a set $X$ , where each $f$ takes its values in some $Y _ { f }$ . The equivalence relation $\rho \left( \mathcal { F } \right)$ defined by $\mathcal { F }$ on $X$ is: for any $x , x ^ { \prime } \in X$ , + +$$ +( x , x ^ { \prime } ) \in \rho \left( { \mathcal { F } } \right) \iff \forall f \in { \mathcal { F } } , f ( x ) = f ( x ^ { \prime } ) . +$$ + +Given two sets of functions $\mathcal { F }$ and $\mathcal { E }$ , we say that $\mathcal { F }$ is more separating (resp. strictly more separating) than $\mathcal { E }$ if $\rho \left( { \mathcal { F } } \right) \subseteq \rho \left( { \mathcal { E } } \right)$ (resp. $\rho \left( { \mathcal { F } } \right) \subsetneq \bar { \rho ( \mathcal { E } ) } )$ . Note that all the functions in $\mathcal { F }$ and $\mathcal { E }$ need to be defined on the same set but can take values in different sets. For example, we can easily see that for the $k$ -WL algorithm defined above, the equivariant version is more separating than the invariant one. + +Some properties of the WL hierarchy of tests can be rephrased with the notion of separating power. In particular, Cai et al. (1989) showed that $( k + 1 ) \ – \mathbf { W } \mathbf { L } _ { I }$ distinguishes strictly more than $k { \mathrm { - } } \mathbf { W } \mathbf { L } _ { I }$ , which can be rewritten simply as (for a function $f$ , we write $\rho \left( f \right)$ for $\rho \left( \left\{ f \right\} \right) ,$ + +$$ +\rho \left( { ( k + 1 ) } { - } \mathbf { W } \mathbf { L } _ { I } \right) \subsetneq \rho \left( k { - } \mathbf { W } \mathbf { L } _ { I } \right) . +$$ + +This notion of separating power enables us to concisely summarize the current knowledge about the discriminatory power of classes of GNN. + +Proposition 3. We have, for $k \geq 2$ , + +$$ +\begin{array} { r l } { \rho \left( M G N N _ { I } \right) = \rho \left( 2 \ – W L _ { I } \right) \qquad } & { { } \rho \left( M G N N _ { E } \right) = \rho \left( 2 \ – W L _ { E } \right) } \\ { \rho \left( k \ – L G N N _ { I } \right) = \rho \left( k \ – W L _ { I } \right) \qquad } & { { } \rho \left( k \ – L G N N _ { E } \right) \subseteq \rho \left( k \ – W L _ { E } \right) } \\ { \rho \left( k \ – F G N N _ { I } \right) = \rho \left( ( k + 1 ) \ – W L _ { I } \right) \qquad } & { { } \rho \left( k \ – F G N N _ { E } \right) = \rho \left( ( k + 1 ) \ – W L _ { E } \right) } \end{array} +$$ + +Only results about the invariant cases were previously known: (4) comes from $\mathrm { X u }$ et al. (2018), (5) from Maron et al. (2018) Geerts (2020a) and one inclusion of (6) comes from Maron et al. (2019a). The equality in (6) for general $k \geq 2$ is proved in Section C. + +Note that for $k = 2$ , all GNNs are dealing with tensors of order 2 i.e. with the adjacency matrix of the graph. However, the complexities of the various layers are quite different: for the message passing GNN, all computations are local (scaling with the maximum degree in the graph) and can be done in parallel; for the linear layer, there are only 15 linear functions from $\mathbb { R } ^ { n ^ { 2 } } \to \mathbb { R } ^ { n ^ { 2 } }$ for all values of $n$ (Maron et al., 2018); the folklore layer involves a (dense) matrix multiplication of shape $n \times n$ . If 2-FGNN is the most complex architecture, we see that it has the best separating power among all architectures proposed so far dealing with tensors of order 2. + +For $X , Y$ finite-dimensional spaces, let us denote by $\mathcal { C } _ { I } ( X , Y ) , \mathcal { C } _ { E } ( X , Y )$ , , the set of invariant, respectively equivariant, continuous functions from $X$ to $Y$ . The closure of a class of function $\mathcal { F }$ for the uniform norm is denoted by $\overline { { \mathcal { F } } }$ . Our result extend easily to graphs of varying sizes but this is deferred to Section F.2 for clarity. + +The theorem below states in particular that the class $k$ -FGNN can approximate any continuous function that is less separating than $( k + 1 )$ -WL in the invariant and in the equivariant cases. + +Theorem 4. Let $K _ { d i s c r } \subseteq \mathcal G _ { n } \times \mathbb { F } _ { 0 } ^ { n }$ , $K \subseteq \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ be compact sets. For the invariant case, we have: + +$$ +\begin{array} { r l r } & { \overline { { M G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K _ { d i s c r } , \mathbb { F } ) : \rho \left( 2 \mathrm { - } W L _ { I } \right) \subseteq \rho \left( f \right) \} } & \\ & { \overline { { k \mathrm { - } L G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k \mathrm { - } W L _ { I } \right) \subseteq \rho \left( f \right) \} } & \\ & { \overline { { k \mathrm { - } F G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( ( k + 1 ) \mathrm { - } W L _ { I } \right) \subseteq \rho \left( f \right) \} } & \end{array} +$$ + +For the equivariant case, we have: + +$$ +\begin{array} { r c l } { \underline { { M G N N _ { E } } } } & { = } & { \{ f \in { \mathcal C } _ { E } ( K _ { d i s c r } , \mathbb F ^ { n } ) : \rho \left( 2 ^ { - } W L _ { E } \right) \subseteq \rho \left( f \right) \} } \\ { \underline { { \cdot } L G N N _ { E } } } & { = } & { \{ f \in { \mathcal C } _ { E } ( K , \mathbb F ^ { n } ) : \rho \left( k \cdot L G N N _ { E } \right) \subseteq \rho \left( f \right) \} \supset \{ f \in { \mathcal C } _ { E } ( K , \mathbb F ^ { n } ) : \rho \left( k \cdot W L _ { E } \right) \subseteq \rho \left( f \right) \} } \\ { \cdot \overline { { F G N N _ { E } } } } & { = } & { \{ f \in { \mathcal C } _ { E } ( K , \mathbb F ^ { n } ) : \rho \left( ( k + 1 ) \cdot W L _ { E } \right) \subseteq \rho \left( f \right) \} } \end{array} +$$ + +In the invariant case for $k = 2$ ,we have $\mathbf { M G N N } _ { I } = \overline { { 2 \mathrm { - } \mathbf { L G N N } _ { I } } } \subsetneq \overline { { 2 \mathrm { - } \mathbf { F G N N } _ { I } } }$ where the strictness of the last inclusion comes from (3). In other words, ${ \mathrm { 2 - F G N N } } _ { I }$ has a better power of approximation than the other architectures working with tensors of order 2. We already knew by Proposition 3 that ${ \mathrm { 2 - F G N N } } _ { I }$ is the best separating architecture among those studied in this paper, dealing with tensor of order 2 and our theorem implies that this is also the case for the approximation power. + +To clarify the meaning of these statements, we explain why the inclusions $\mathbf { \check { \Psi } } \subseteq \mathbf { \Psi }$ are actually straightforward. For concreteness, we focus on $\overline { { k \mathrm { - } \mathrm { F G N N } _ { I } } } \ \subseteq \ \left\{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( ( k + 1 ) \mathrm { - } \mathrm { W L } _ { I } \right) \subseteq \rho \left( f \right) \right\}$ . Take $h ~ \in ~ \overline { { k \mathrm { - } \mathrm { F G N N } _ { I } } }$ , this means that there is a sequence $\mathbf { G } \mathbf { N } \mathbf { N } _ { j } \in k \mathrm { \mathrm { - } } \mathbf { F } \mathbf { G } \mathbf { N } \mathbf { N } _ { I }$ such that, $\mathrm { s u p } _ { G \in K } \| h ( G ) - \mathbf { G } \mathbf { N } \mathbf { N } _ { j } ( G ) \|$ goes to zero when $j$ goes to infinity. + +Therefore, $h$ is continuous and constant on each $\rho \left( k { \mathrm { - F G N N } } _ { I } \right)$ -class. Indeed, for any $( G , G ^ { \prime } ) \in$ $\rho \left( k { \mathrm { - } } \mathrm { F G N N } _ { I } \right)$ , $\mathbf { G N N } _ { j } ( G ) = \mathbf { G N N } _ { j } ( G ^ { \prime } )$ so that $h ( G ) = \mathrm { l i m } _ { i } { \bf G } { \bf N } { \bf N } _ { j } ( G ) = \mathrm { l i m } _ { j } { \bf G } { \bf N } { \bf N } _ { j } ( G ^ { \prime } ) =$ $h ( G ^ { \prime } )$ . Hence we have $\rho \left( k { \mathrm { - F G N N } } _ { I } \right) \subseteq \rho \left( h \right)$ and by Prop. 3, $\rho \left( k { \mathrm { - } } \mathrm { F G N N } _ { I } \right) = \rho \left( ( k + 1 ) { \mathrm { - } } \mathrm { W L } _ { I } \right)$ , allowing us to get the inclusion above. + +On the contrary, the reverse inclusions $\cdot \bigcirc ^ { \prime \prime }$ are much more intricate but they are also the most valuable. For instance, consider the inclusion $\overline { { k \mathrm { - } \mathrm { F G N N } _ { I } } } \supset \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( ( k + 1 ) \mathrm { - } \mathrm { W L } _ { I } \right) \subseteq \rho \left( f \right) \}$ . If one wishes to learn a function $h \in { \mathcal { C } } _ { I } ( K , \mathbb { F } )$ with $k { \mathrm { - } } \mathrm { F G N N } _ { I }$ , this function must at least be approximable by the class of $k { \mathrm { - F G N N } } _ { I }$ . Our theorem precisely guarantees that if $h$ is less separating that $k { \mathrm { - } } \mathbf { W } \mathbf { L } _ { I }$ , it can be approximated by $k { \mathrm { - F G N N } } _ { I }$ : + +$$ +\forall \epsilon > 0 , \exists \mathbf { G } \mathbf { N } \mathbf { N } \in k \mathbf { \mathrm { \mathrm { - } } } \mathbf { F } \mathbf { G } \mathbf { N } \mathbf { N } _ { I } , \mathbf { \Pi } _ { G \in K } \| h ( G ) - \mathbf { G } \mathbf { N } \mathbf { N } ( G ) \| \leq \epsilon . +$$ + +For this, we show a much more general version of the famous Stone-Weierstrass theorem (see Section D) which relates the separating power with the approximation power. Following the elegant idea of Maehara & Hoang (2019), we augment the input space to transform vector-valued equivariant functions into scalar invariant maps. Then, we apply a fine-grained approximation theorem from Timofte (2005).We also provide specialized versions of our abstract theorem in Section 5, which can be easily used to determine the approximation capabilities of any deep learning architecture. + +Our theorem has also implications for universality results like Maron et al. (2019b); Keriven & Peyré (2019). A class of GNN is said to be universal if its closure on a compact set $K$ is the whole $\mathcal { C } _ { I } ( K , \mathbb { F } )$ (or $\mathcal { C } _ { E } ( K , \mathbb { F } ^ { n } ) )$ ). In particular, Thm. 4 implies that $n$ -LGNN and $n$ -FGNN are universal as $n$ -WL distinguishes non-isomorphic graphs of size $n$ . This recovers a result of Ravanbakhsh (2020) for LGNN. Moreover, we can leverage the extensive literature on the WL tests to give more subtle√ results. For instance, Cai et al. (1989, $\ S 8 . 2 )$ show that, for planar graphs, $O ( { \sqrt { n } } )$ -WL can distinguish non-isomoprhic instances. Therefore, $O ( \sqrt { n } )$ -LGNN or $\mathbf { \bar { O } } ( { \sqrt { n } } ) $ -FGNN achieve universality in the particular, yet common, case of planar graphs. On a more practical side, Fürer (2010, Thm. 4.5) shows that the spectrum of a graph is less separating than 3-WL so that functions of the spectrum can actually be well approximated by 2-FGNN. + +# 5 EXPRESSIVENESS OF GNNS + +We now state the general theorems which are our main tools in proving our approximation guarantees for GNNs. Theirs proofs are deferred to Section D.9 which contains our generalization of the StoneWeierstrass theorem with symmetries. We need to first introduce more general definitions: If $G$ is a finite group acting on some topological space $X$ , we say that $G$ acts continuously on $X$ if, for all $g \in G$ , $x \mapsto g \cdot x$ is continuous. If $G$ is a finite group acting on some compact set $X$ and some topological space $Y$ , we define the sets of equivariant and invariant continuous functions by, + +$$ +\begin{array} { r l } & { \mathcal { C } _ { E } ( X , Y ) = \{ f \in \mathcal { C } ( X , Y ) : \forall x \in X , \forall g \in G , f ( g \cdot x ) = g \cdot f ( x ) \} } \\ & { \mathcal { C } _ { I } ( X , Y ) = \{ f \in \mathcal { C } ( X , Y ) : \forall x \in X , \forall g \in G , f ( g \cdot x ) = f ( x ) \} } \end{array} +$$ + +Note that these definitions extend Definition 1 to a general group. + +Theorem 5. Let $X$ be a compact space, $\mathbb { F } = \mathbb { R } ^ { p }$ be some finite-dimensional vector space, $G$ be a finite group acting (continuously) on $X$ . + +Let $\textstyle { \mathcal { F } } _ { 0 } \subseteq \bigcup _ { h = 1 } ^ { \infty } { \mathcal { C } } _ { I } ( X , \mathbb { R } ^ { h } )$ be a non-empty set of invariant functions, stable by concatenation, and consider, + +$$ +\mathcal { F } = \{ m \circ f : f \in \mathcal { F } _ { 0 } \cap \mathcal { C } ( X , \mathbb { R } ^ { h } ) , m : \mathbb { R } ^ { h } \to \mathbb { F } M L P , h \geq 1 \} \subseteq \mathcal { C } ( X , \mathbb { F } ) . +$$ + +Then the closure of $\mathcal { F }$ is, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \{ f \in \mathcal { C } _ { I } ( X , \mathbb { F } ) : \rho \left( \mathcal { F } _ { 0 } \right) \subseteq \rho \left( f \right) \} . } \end{array} +$$ + +We can apply Theorem 5 to the class of $k$ -ary relational pooling GNN introduced in Murphy et al. (2019a). As a result, we get that this class of invariant $k$ -RP GNN can approximate any continuous function $f$ with $\rho ( k - \mathrm { R P G N N } ) \subseteq \rho ( f )$ but to the best of our knowledge, $\rho ( k - \mathrm { R P G N N } )$ is not known and only $\rho ( k - \mathrm { R P G N N } ) \subset \rho ( 2 - W L _ { I } )$ is proved in Murphy et al. (2019a). We now state our general theorem for the equivariant case: + +Theorem 6. Let $X$ be a compact space, $\mathbb { F } = \mathbb { R } ^ { p }$ and $G = S _ { n }$ the permutation group, acting (continuously) on $X$ and acting on $\mathbb { F } ^ { n }$ by, for $\sigma \in S _ { n }$ , $x \in \mathbb { F } ^ { n }$ , + +$$ +\forall i \in \{ 1 , \ldots , p \} , ( \sigma \cdot x ) _ { i } = x _ { \sigma ^ { - 1 } ( i ) } , +$$ + +Let ${ \mathcal { F } } _ { 0 } \subseteq \bigcup _ { h = 1 } ^ { \infty } { \mathcal { C } } _ { E } \left( X , ( \mathbb { R } ^ { h } ) ^ { n } \right)$ be a non-empty set of equivariant functions, stable by concatenation, and consider, + +$$ +{ \mathcal { F } } = \{ x \mapsto ( m ( f ( x ) _ { 1 } ) , \ldots , m ( f ( x ) _ { n } ) ) : f \in { \mathcal { F } } _ { 0 } \cap { \mathcal { C } } \left( X , ( \mathbb { R } ^ { h } ) ^ { n } \right) , m : \mathbb { R } ^ { h } \to \mathbb { F } M L P , h \geq 1 \} +$$ + +Assume, that, if $f \in \mathcal { F } _ { 0 }$ , then, + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \in { \mathcal { F } } _ { 0 } . +$$ + +Then the closure of $\mathcal { F }$ is, + +$$ +\overline { { \mathcal { F } } } = \{ f \in \mathcal { C } _ { E } ( X , \mathbb { F } ^ { n } ) : \rho \left( \mathcal { F } _ { 0 } \right) \subseteq \rho \left( f \right) \} \ . +$$ + +Applications of these theorems fo the case of Pointnet Qi et al. (2017) are provided in Section D.9 + +# 6 QUADRATIC ASSIGNMENT PROBLEM + +To empirically evaluate our results, we study the Quadratic Assignment Problem (QAP), a classical problem in combinatorial optimization. For $A , B n \times n$ symmetric matrices, it consists in solving + +$$ +{ \mathrm { m a x i m i z e ~ t r a c e } } ( A X B X ^ { \top } ) , { \mathrm { s u b j e c t ~ t o ~ } } X \in \Pi , +$$ + +where $\Pi$ is the set of $n \times n$ permutation matrices. Many optimization problems can be formulated as QAP. An example is the network alignment problem, which consists in finding the best matching between two graphs, represented by their adjacency matrices $A$ and $B$ . Though QAP is known to be NP-hard, recent works such as Nowak et al. (2018) have investigated whether it can be solved efficiently w.r.t. a fixed input distribution. More precisely, Nowak et al. (2018) studied whether one can learn to solve this problem using a MGNN trained on a dataset of already solved instances. However, as shown below, both the baselines and their approach fail on regular graphs, a class of graph considered as particularly hard for isomorphism testing. + +To remedy this weakness, we consider $2 { \mathrm { - } } \mathrm { F G N N } _ { E }$ . We then follow the siamese method of (Nowak et al., 2018): given two graphs, our system produces an embedding in $\mathbb { F } ^ { n }$ for each graph, where $n$ is the number of nodes, which are then multiplied together to obtain a $n \times n$ similarity matrix on nodes. A permutation is finally computed by solving a Linear Assignment Problem (LAP) with this resulting $n \times n$ as cost matrix. We tested our architecture on two distribution: the Erdos–Rényi model and ˝ random regular graphs. The accuracy in matching the graphs is much improved compare to previous works. The experimental setup is described more precisely in Section A.1. + +![](images/3de1a936c4bf2c8959509d03db9c9edaefbda37bf89260168626ddac22c6232f.jpg) +Figure 1: Fraction of matched nodes for pairs of correlated graphs (with edge density 0.2) as a function of the noise, see Section A.1 for details. + +# 7 CONCLUSION + +We derived the expressive power of various practical GNN architectures: message passing GNN, linear GNN and folklore GNN; both for their invariant and equivariant counterparts. Our results unify and extend the recent works in this direction. In particular, we are able to recover all the universality results proved for GNNs so far. Similarly to existing results in the literature, we do not deal here with the sizes of the embeddings constructed at different layers, i.e. the sizes of the spaces $\mathbb { F } _ { \ell }$ , and these sizes are supposed to grow to infinity with the number of nodes $n$ in the graph. Obtaining bounds on the scaling of the sizes of the features to ensure that the results presented here are still valid is an interesting open question. We show that folklore GNNs have the best power of approximation among all GNNs studied here dealing with tensors of order 2. From a practical perspective, we demonstrate their improved performance on the QAP with a significant gap in performances compared to other approaches. + +# ACKNOWLEDGMENTS + +This work was supported in part by the French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR19-P3IA-0001 (PRAIRIE 3IA Institute). M.L. thanks Google for Google Cloud Platform research credits and NVIDIA for a NVIDIA GPU Grant. + +# REFERENCES + +J.-Y. Cai, M. Furer, and N. Immerman. An optimal lower bound on the number of variables for graph identification. 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Vishwanathan, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pp. 3391–3401, 2017. + +# A Experimental results 14 + +A.1 Details on the experimental setup 14 +A.2 Experimental results on graphs of varying size . 14 +A.3 Generalization for regular graphs . 15 + +# B Weisfeiler-Lehman tests 16 + +B.1 Weisfeiler-Lehman test on vertices 16 +B.2 Isomorphism type . 16 +B.3 Weisfeiler-Lehman and Folklore Weisfeiler-Lehman tests of order $k \geq 2$ . 17 + +# C Separating power of GNN 18 + +C.1 Mutli-linear perceptrons 18 +C.2 Augmented separating power 18 +C.3 Initialization layer . 19 +C.4 Known results about the separating power of some GNN classes 20 +C.5 Bounding the separating power of $k$ -FGNN 20 +C.6 Conclusion 21 + +# D Stone-Weierstrass theorem with symmetries 21 + +D.1 General notations . 22 +D.2 Separating power 22 +D.3 Approximation theorems for real-valued functions . 23 +D.4 The equivariant approximation theorem 24 +D.5 A preliminary version of the equivariant approximation theorem 26 +D.6 Characterizing the subalgebras of $\mathbb { R } ^ { p }$ 27 +D.7 Proof of the main equivariant approximation theorem 29 +D.8 Practical reductions 30 +D.9 Reductions for GNNs 33 + +# E Proofs for expressiveness of GNNs 34 + +E.1 Expressivity of GNN layers . 34 +E.2 Approximation theorems for GNNs 35 + +# F Extension to graphs of varying sizes 37 + +F.1 Extension to disconnected input spaces . . 37 +F.2 Approximation theorem with varying graph size . . . 39 + +# A EXPERIMENTAL RESULTS + +# A.1 DETAILS ON THE EXPERIMENTAL SETUP + +We consider a $2 { \mathrm { - } } \mathrm { F G N N } _ { E }$ and train it to solve random planted problem instances of the QAP. Given a pair of graphs $G _ { 1 } , G _ { 2 }$ with $n$ nodes each, we consider the siamese $2 { \mathrm { - } } \mathrm { F G N N } _ { E }$ encoder producing embeddings $E _ { 1 } , E _ { 2 } \in \mathbb { R } ^ { n \times k }$ . Those embeddings are used to predict a matching as follows: we first compute the outer product $E _ { 1 } E _ { 2 } ^ { T }$ , then we take a softmax along each row and use standard cross-entropy loss to predict the corresponding permutation index. We used $2 { \mathrm { - } } \mathrm { F G N N } _ { E }$ with 2 layers, each MLP having depth 3 and hidden states of size 64. We trained for 25 epochs with batches of size 32, a learning rate of 1e-4 and Adam optimizer. The PyTorch code is available in the supplementary material. + +For each experiment, the dataset was made of 20000 graphs for the train set, 1000 for the validation set and 1000 for the test set. For the experiment with Erdos–Rényi random graphs, we consider ˝ $G _ { 1 }$ to be a random Erdos–Rényi graph with edge density˝ $p _ { e } = 0 . 2$ and $n = 5 0$ vertices. The graph $G _ { 2 }$ is a small perturbation of $G _ { 1 }$ according to the following error model considered in Feizi et al. (2016): + +$$ +G _ { 2 } = G _ { 1 } \odot ( 1 - Q ) + ( 1 - G _ { 1 } ) \odot Q ^ { \prime } , +$$ + +where $Q$ and $Q ^ { \prime }$ are Erdos–Rényi random graphs with edge density ˝ $p _ { 1 }$ and $p _ { 2 } = p _ { 1 } p _ { e } / ( 1 - p _ { e } )$ respectively, so that $G _ { 2 }$ has the same expected degree as $G _ { 1 }$ . The noise level is the parameter $p _ { 1 }$ . For regular graphs, we followed the same experimental setup but now $G _ { 1 }$ is a random regular graph with degree $d = 1 0$ . Regular graphs are interesting example as they tend to be considered harder to align due to their more symmetric structure. + +# A.2 EXPERIMENTAL RESULTS ON GRAPHS OF VARYING SIZE + +![](images/68e538d254f564e7cec0985a6c326e6ed430e58c68d1e39f3b9e6b48208c6669.jpg) +Figure 2: Fraction of matched nodes for pairs of correlated graphs (with edge density 0.2) as a function of the noise, see Section A.1 for details. + +We tested our models on dataset of graphs of varying size, as this setting is also encompassed by our theory. + +However, contrary to message-passing GNN, GNN based on tensors do not work well with batches of graphs of varying size. Previous implementations, such as the one of Maron et al. (2019a), group the graphs in the dataset by size, enabling the GNN to only deal with batches of graphs on the same size. + +![](images/030b6bc0ced87c1a9d34f138c1bc1289d003a176c8c11c4142fe35d0382d1684.jpg) +Figure 3: Each line corresponds to a model trained at a given noise level and shows its accuracy across all noise levels. + +Instead, we use masking, which is a standard practice in recurrent neural networks. A batch of $b$ tensors of sizes $n _ { 1 } \times n _ { 1 } , n _ { 2 } \times n _ { 2 } , . . . , n _ { b } \times n _ { b }$ is represented as a tensor $b \times n _ { m a x } \times n _ { m a x }$ where $n _ { m a x } = \mathrm { m a x } _ { i = 1 , \dots , b } n _ { i }$ . A mask is created at initialization and is used to ensure that the operations on the full tensor translates to valid operations on each of the individual tensor. + +We implemented this functionnality as a class MaskedTensors. Thanks to the newest improvements of PyTorch (Paszke et al., 2019), MaskedTensors act as a subclass of fundamental Tensor class. Thus they almost seamlessly integrate into standard PyTorch code. We refer the reader to the code for more details: https://github.com/mlelarge/graph_neural_net + +Results of our architecture and implementation with graphs of varying size are shown below on Figure 2. The only difference with the setting described above is that the number of nodes is now random. The number of vertices of a graph is indeed chosen randomly according to binomial distribution of parameters $n = 5 0$ and $p _ { n } = 0 . 9$ . + +# A.3 GENERALIZATION FOR REGULAR GRAPHS + +We made the following experiment with the same general setting as in Section A.1 with regular graphs. We trained different models for all noise levels between 0 and 0.22 but in Figure 3, we plot the accuracy of each model across all noise levels. We observe that a majority of the models actually generalize to settings with noise level on which they were not trained. Indeed, the model trained with noise level $\approx 0 . 1$ is performing best among all models across all noise levels! + +# B WEISFEILER-LEHMAN TESTS + +Here we describe more precisely this hierarchy of tests, which will be used extensively to characterize the discirminatory power of classes of GNN. See Douglas (2011); Grohe (2017); Fürer (2017) for graph-theoretic introductions to these algorithms. + +# B.1 WEISFEILER-LEHMAN TEST ON VERTICES + +We now present the initial vertex coloring algorithm. + +Input. This algorithm takes as input a discrete graph structure $G = ( V , E ) \in \mathcal { G } _ { n }$ with $V = [ n ]$ , $E \subseteq V ^ { 2 }$ and $h \in \mathbb { F } _ { 0 } ^ { n }$ features on the vertices. + +Initialization. Each vertex $s \in V$ is given a color $c _ { \mathrm { W L } } ^ { 0 } ( G , h ) _ { s } = h _ { s }$ corresponding to its features vector. + +Refining the coloring. The colors of the vertices are updated as follows, + +$$ +c _ { \mathrm { W L } } ^ { t + 1 } ( G , h ) _ { s } = \mathrm { L e x } \left( c _ { \mathrm { W L } } ^ { t } ( G , h ) _ { s } , \{ \{ c _ { \mathrm { W L } } ^ { t } ( G , h ) ( \tilde { s } ) : \tilde { s } \sim s \} \} \right) , +$$ + +and the function Lex means that all occuring colors are lexicographically ordered. + +For each graph $G \in \mathcal G _ { n }$ and each vector of features $h \in \mathbb { F } _ { 0 } ^ { n }$ , there exists a time $T ( G , h )$ from which the sequence of colorings $( c _ { \mathrm { W L } } ^ { t } ( G , h ) ) _ { t \geq 0 }$ is stationary. More exactly, the colorings are not refined anymore: for any $t \geq T ( G , h ) , s , s ^ { \prime } \in \bar { V }$ , + +$$ +c _ { \mathrm { W L } } ^ { t } ( G , h ) _ { s } = c _ { \mathrm { W L } } ^ { t } ( G , h ) _ { s ^ { \prime } } \longleftrightarrow c _ { \mathrm { W L } } ^ { T ( G , h ) } ( G , h ) _ { s } = c _ { \mathrm { W L } } ^ { T ( G , h ) } ( G , h ) _ { s ^ { \prime } } . +$$ + +Denote the resulting coloring cT (G,h)WL (G, h) by simply cWL(G, h). cWL is now a mapping from $\mathcal { G } _ { n } \times \mathbb { F } _ { 0 } ^ { n } \to Z ^ { n }$ for some space of colors $Z$ . + +Invariant tests The proper Weisfeiler-Lehman test is invariant and is defined by, for $t \geq 0$ and $( G , h ) \in \mathcal { G } _ { n } \times \mathbb { F } _ { 0 } ^ { n }$ a graph, + +$$ +\begin{array} { r l } & { \operatorname { W L } _ { I } ^ { t } ( G ) = \{ \{ c _ { \mathrm { W L } } ^ { t } ( G ) _ { s } : s \in V \} \} } \\ & { \operatorname { W L } _ { I } ( G ) = \{ \{ c _ { \mathrm { W L } } ( G ) _ { s } : s \in V \} \} } \end{array} +$$ + +Equivariant tests For the vertex coloring algorithm, $c _ { \mathrm { W L } }$ is already an equiavriant mapping so we define, for $t \geq 0$ and $( G , h ) \in \mathcal { G } _ { n } \times \mathbb { F } _ { 0 } ^ { n }$ a graph, + +$$ +\begin{array} { r } { \mathrm { W L } _ { E } ^ { t } = c _ { \mathrm { W L } } ^ { t } } \\ { \mathrm { W L } _ { E } = c _ { \mathrm { W L } } } \end{array} +$$ + +# B.2 ISOMORPHISM TYPE + +The initialization of the higher-order variants of the Weisfeiler-Lehman test is slightly more intricate. For this we need to define the isomorphism type of a $k$ -tuple w.r.t. a graph described by a tensor $\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ + +A $k$ -tuple $( i _ { 1 } , \dots , i _ { k } ) \in [ n ] ^ { k }$ in a graph $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ and a $k$ -tuple $( j _ { 1 } , \ldots , j _ { k } ) \in [ n ] ^ { k }$ in a graph $H \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ are said to have the same isomoprhism type if the mapping $i _ { w } \mapsto j _ { w }$ is a well-defined partial isormophism. Explicitly, this means that, + +$$ +\begin{array} { r l } & { \bullet \forall w , w ^ { \prime } \in [ k ] , i _ { w } = i _ { w ^ { \prime } } \iff j _ { w } = j _ { w ^ { \prime } } . } \\ & { \bullet \forall w , w ^ { \prime } \in [ k ] , G _ { i _ { w } , i _ { w ^ { \prime } } } = H _ { j _ { w } , j _ { w ^ { \prime } } } . } \end{array} +$$ + +Denote by $\operatorname { i s o } ( G ) _ { i _ { 1 } , \dots , i _ { k } }$ the isomorphism type of the $k$ -tuple $( i _ { 1 } , \dots , i _ { k } ) \in [ n ] ^ { k }$ in a graph $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ + +B.3 WEISFEILER-LEHMAN AND FOLKLORE WEISFEILER-LEHMAN TESTS OF ORDER $k \geq 2$ + +We now present the folklore version of the Weisfeiler-Lehman test of order $k$ ( $k$ -FWL), for $k \geq 2$ , along with $k$ -WL for clarity. For both, we follow the presentation of Maron et al. (2019a) (except for the equivariant tests). + +Input. These algorithms take as input a graph $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ which can be seen as a coloring on the pair of nodes. + +Initialization. Each $k$ -tuple $s \in V ^ { k }$ is given a color $c _ { k \mathrm { - } \mathrm { W L } } ^ { 0 } ( G ) _ { s } = c _ { k \mathrm { - } \mathrm { F W L } } ^ { 0 } ( G ) _ { s }$ corresponding to its isomorphism type. + +$k$ -WL. The $k$ -WL test relies on the following notion of neighborhood, defined by, for any $w \in [ k ]$ , and $s = ( i _ { 1 } , \ldots , i _ { k } ) \in V ^ { k }$ , + +$$ +N _ { w } ( s ) = \left\{ ( i _ { 1 } , \dots , i _ { w - 1 } , j , i _ { w + 1 } , \dots , i _ { k } ) : j \in V \right\} . +$$ + +Then, the colors of the $k$ -tuples $s \in V ^ { k }$ are refined as follows, + +$$ +c _ { \mathrm { W L } } ^ { t + 1 } ( G ) _ { s } = \mathrm { L e x } \left( c _ { k \mathrm { - } \mathrm { W L } } ^ { t } ( G ) _ { s } , ( C _ { 1 } ^ { t } ( s ) , \ldots , C _ { k } ^ { t } ( s ) ) \right) . +$$ + +where, for $w \in [ k ]$ , + +$$ +C _ { w } ^ { t } ( s ) = \left\{ \left\{ c _ { k - \mathrm { W L } } ^ { t } ( G ) _ { \tilde { s } } : \tilde { s } \in N _ { w } ( s ) \right\} \right\} . +$$ + +For each graph $G ~ \in ~ \mathbb { F } _ { 0 } ^ { n }$ , there exists a time $T ( G )$ from which the sequence of colorings $( c _ { k - \mathrm { W L } } ^ { t } ( G ) ) _ { t \geq 0 }$ is stationary. More exactly, the colorings are not refined anymore: for any $t \geq T ( G )$ , $s , s ^ { \prime } \in V ^ { k }$ , + +$$ +c _ { k \mathrm { - } \mathrm { W L } } ^ { t } ( G ) _ { s } = c _ { k \mathrm { - } \mathrm { W L } } ^ { t } ( G ) _ { s ^ { \prime } } \iff c _ { k \mathrm { - } \mathrm { W L } } ^ { T ( G ) } ( G ) _ { s } = c _ { k \mathrm { - } \mathrm { W L } } ^ { T ( G ) } ( G ) _ { s ^ { \prime } } . +$$ + +Denote the resulting coloring $c _ { k \mathrm { - W L } } ^ { T ( G ) } ( G )$ by simply $c _ { k - \mathrm { W L } } ( G )$ + +$k$ -FWL. For $k$ -FWL, the corresponding notion of neighborhood is defined by, for any $j \in V$ , and $s = ( i _ { 1 } , \ldots , i _ { k } ) \in V ^ { k }$ , + +$$ +N _ { j } ^ { F } ( s ) = \{ ( j , i _ { 2 } , \ldots , i _ { k } ) , ( i _ { 1 } , j , i _ { 3 } , \ldots , i _ { k } ) , \ldots , ( i _ { 1 } , i _ { 2 } , \ldots , i _ { k - 1 } , j ) \} +$$ + +Then, the colors of the $k$ -tuples $s \in V ^ { k }$ are refined as follows, + +$$ +c _ { k - \mathrm { W L } } ^ { t + 1 } ( G ) _ { s } = \mathrm { L e x } \left( c _ { k - \mathrm { F W L } } ^ { t } ( G ) _ { s } , \left\{ \left\{ C _ { j } ^ { t } ( s ) : j \in V \right\} \right\} \right) , +$$ + +where, for $j \in V$ , + +$$ +C _ { j } ^ { t } ( s ) = \big ( c _ { k \mathrm { \scriptsize - F W L } } ^ { t } ( G ) _ { \tilde { s } } : \tilde { s } \in N _ { j } ^ { F } ( s ) \big ) \ . +$$ + +Like $k$ -WL, for each graph $G \in \mathbb { F } _ { 0 } ^ { n }$ , there exists a time $T ( G )$ from which the sequence of colorings $( c _ { k - \mathrm { W L } } ^ { t } ( G ) ) _ { t \geq 0 }$ is stationary. Similarly, denote the resulting coloring $c _ { k \mathrm { - W L } } ^ { T ( G ) } ( G )$ by simply $c _ { k - \mathrm { W L } } ( G )$ . The colors $c _ { k - \mathrm { F W L } } ^ { t }$ and $c _ { k - \mathrm { F W L } } ^ { t }$ at iteration $t$ define a mapping from $\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ to space of colorings of $k$ -tuples, $Z ^ { n ^ { k } }$ for some space $Z$ . + +Invariant tests The standard versions of the Weisfeiler-Lehman tests are invariant and can be defined by, for $t \geq 0$ and $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ a graph, + +$$ +\begin{array} { r l } & { k { \mathrm { – } } \mathbf { W } \mathbf { L } _ { I } ^ { t } ( G ) = \left\{ \left\{ c _ { k { \mathrm { - } } \mathbf { W } \mathbf { L } } ^ { t } ( G ) _ { s } : s \in V ^ { k } \right\} \right\} } \\ & { k { \mathrm { – } } \mathbf { W } \mathbf { L } _ { I } ( G ) = \left\{ \left\{ c _ { k { \mathrm { - } } \mathbf { W } \mathbf { L } } ( G ) _ { s } : s \in V ^ { k } \right\} \right\} } \\ & { k { \mathrm { – } } \mathbf { F } \mathbf { W } \mathbf { L } _ { I } ^ { t } ( G ) = \left\{ \left\{ c _ { k { \mathrm { - } } \mathbf { F } \mathbf { W } \mathbf { L } } ^ { t } ( G ) _ { s } : s \in V ^ { k } \right\} \right\} } \\ & { k { \mathrm { – } } \mathbf { F } \mathbf { W } \mathbf { L } _ { I } ( G ) = \left\{ \left\{ c _ { k { \mathrm { - } } \mathbf { F } \mathbf { W } \mathbf { L } } ( G ) _ { s } : s \in V ^ { k } \right\} \right\} \ . } \end{array} +$$ + +Equivariant tests We now introduce the equivariant version of these tests. Many extensions are possible, we chose this one for its simplicity. For $t \geq 0$ , $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ a graph, $i \in V$ , + +$$ +\begin{array} { r l } & { k \mathrm { - } { \mathbb { W } } \mathrm { L } _ { E } ^ { t } ( G ) _ { i } = \{ \{ c _ { k - \mathrm { W L } } ^ { t } ( G ) _ { s } : s \in V ^ { k } , s _ { 1 } = i \} \} } \\ & { k \mathrm { - } { \mathbb { W } } \mathrm { L } _ { E } ( G ) _ { i } = \{ \{ c _ { k - \mathrm { W L } } ( G ) _ { s } : s \in V ^ { k } , s _ { 1 } = i \} \} } \\ & { k \mathrm { - } { \mathrm { F W L } } _ { E } ^ { t } ( G ) _ { i } = \{ \{ c _ { k - \mathrm { F W L } } ^ { t } ( G ) _ { s } : s \in V ^ { k } , s _ { 1 } = i \} \} } \\ & { k \mathrm { - } { \mathrm { F W L } } _ { E } ( G ) _ { i } = \{ \{ c _ { k - \mathrm { F W L } } ( G ) _ { s } : s \in V ^ { k } , s _ { 1 } = i \} \} \ . } \end{array} +$$ + +# C SEPARATING POWER OF GNN + +The goal of this section is to prove, + +Proposition 3. We have, for $k \geq 2$ , + +$$ +\begin{array} { r l } { \rho \left( M G N N _ { I } \right) = \rho \left( 2 \ – W L _ { I } \right) \qquad } & { { } \rho \left( M G N N _ { E } \right) = \rho \left( 2 \ – W L _ { E } \right) } \\ { \rho \left( k \ – L G N N _ { I } \right) = \rho \left( k \ – W L _ { I } \right) \qquad } & { { } \rho \left( k \ – L G N N _ { E } \right) \subseteq \rho \left( k \ – W L _ { E } \right) } \\ { \rho \left( k \ – F G N N _ { I } \right) = \rho \left( ( k + 1 ) \ – W L _ { I } \right) \qquad } & { { } \rho \left( k \ – F G N N _ { E } \right) = \rho \left( ( k + 1 ) \ – W L _ { E } \right) } \end{array} +$$ + +# C.1 MUTLI-LINEAR PERCEPTRONS + +In the following we will use extensively multi-linear perceptrons (MLP) and their universality properties. Yet, for the sake of simplicity, we do not define precisely what we mean by MLP. + +Given two finite-dimensional feature spaces $\mathbb { F } _ { 0 }$ and $\mathbb { F } _ { 1 }$ , we only assume we are given a class of (continuous) MLP from $\mathbb { F } _ { 0 }$ to $\mathbb { F } _ { 1 }$ which is large enough to be dense in $\mathcal { C } ( \mathbb { F } _ { 0 } , \mathbb { F } _ { 1 } )$ . See for instance Hornik et al. (1989); Cybenko (1989) for precise conditions for MLP to be universal. + +# C.2 AUGMENTED SEPARATING POWER + +To factor the proof, we introduce another notion of separating power. + +Definition 7. For $n , m \geq 1$ fixed, let $X$ be a set, $\mathbb { F }$ be a some space and $\mathcal { F }$ be a set of functions from $X$ to $Y = \mathbb { F } ^ { n \times m }$ . Then, the augmented separating power of $\mathcal { F }$ is, + +$$ +\rho _ { n , m } ^ { a u g m } ( \mathcal { F } ) = \rho ( \{ ( x , i , j ) \in X \times \{ 1 , \ldots , n \} \times \{ 1 , \ldots , m \} \mapsto f ( x ) _ { i , j } : f \in \mathcal { F } \} ) . +$$ + +Explicitly, for $x , y \in X , i , j \in \{ 1 , \dots , n \}$ + +$$ +( x , i , j , y , k , l ) \in \rho _ { n , m } ^ { a u g m } ( \mathcal { F } ) \iff \forall f \in \mathcal { F } , f ( x ) _ { i , j } = f ( y ) _ { k , l } . +$$ + +Note that when $n = m = 1$ , the augmented separating power is exactly the same as the original separating power, so we identify $\rho \left( . \right)$ with $\rho _ { 1 , 1 } ^ { a u g m } ( . )$ . We also identify ρaugmn,1 (.) with ρaug1,n $\rho _ { 1 , n } ^ { a u g m } \left( . \right)$ , that we denote by $\rho _ { n } ^ { a u g m } \left( . \right)$ . + +First, it is easy to see that this notion is more precise than the separating power. + +Lemma 8. If $\mathcal { F }$ and $\mathcal { G }$ are set of functions from $X$ to $\mathbb { F } ^ { n \times m }$ , + +and, in particular, + +$$ +\begin{array} { r l } & { \rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right) \subseteq \rho _ { n , m } ^ { a u g m } \left( \mathcal { G } \right) \implies \rho \left( \mathcal { F } \right) \subseteq \rho \left( \mathcal { G } \right) , } \\ & { } \\ & { \rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right) = \rho _ { n , m } ^ { a u g m } \left( \mathcal { G } \right) \implies \rho \left( \mathcal { F } \right) = \rho \left( \mathcal { G } \right) , } \end{array} +$$ + +The interest in this notion is justified by the following lemma, which shows that this notion behaves well under composition with “reduction layers”. + +Lemma 9. For $n , m \geq 1$ fixed, let $X$ be a compact topological space, $Y = \mathbb { F } ^ { n \times m }$ , $\mathbb { F }$ some finitedimensional space, ${ \mathcal { F } } \subseteq { \mathcal { C } } ( X , Y )$ , and $\tau : X Z ^ { n \times m } a$ function for some space $Z$ . Define ${ \mathcal { F } } \subseteq { \mathcal { C } } ( X , \mathbb { F } ^ { n } )$ by, + +$$ +\tilde { \boldsymbol { x } } = \{ x \in X \longmapsto ( \sum _ { j = 1 } ^ { m } h ( f ( x ) _ { 1 , j } ) , \sum _ { j = 1 } ^ { m } h ( f ( x ) _ { 2 , j } ) , \ldots , \sum _ { i = 1 } ^ { m } h ( f ( x ) _ { n , j } ) ) : h : \mathbb { F } \to \mathbb { F } M L P , \ f \in \mathcal { R } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } \to \mathbb { F } \mathrm { ~ a n d ~ } \mathbb { F } +$$ + +and $\widetilde { \tau } : X \to \widetilde { Z } ^ { n }$ by, for $x \in X$ , + +$$ +\forall i \in \{ 1 , \ldots , n \} , \ : \widetilde { \tau } ( x ) _ { i } = \{ \{ \tau ( x ) _ { i , j } : 1 \leq j \leq m \} \} . +$$ + +Then, + +$$ +\begin{array} { r l } & { \rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right) \subseteq \rho _ { n , m } ^ { a u g m } \left( \tau \right) \implies \rho _ { n , m } ^ { a u g m } \left( \widetilde { \mathcal { F } } \right) \subseteq \rho _ { n , m } ^ { a u g m } \left( \widetilde { \tau } \right) } \\ & { \rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right) \supset \rho _ { n , m } ^ { a u g m } \left( \tau \right) \implies \rho _ { n , m } ^ { a u g m } \left( \widetilde { \mathcal { F } } \right) \supset \rho _ { n , m } ^ { a u g m } \left( \widetilde { \tau } \right) . } \end{array} +$$ + +Note that, in the statement, we implicitly see $\widetilde { \mathcal F }$ as a set of functions from $X$ to $\mathbb { F } ^ { n \times 1 }$ to fit the definition of augmented separating power. + +Proof. We show the two inclusions independently. + +$( \subseteqright)$ We first show that, + +$$ +\rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right) \subseteq \rho _ { n , m } ^ { a u g m } \left( \tau \right) \implies \rho _ { n , m } ^ { a u g m } \left( \widetilde { \mathcal { F } } \right) \subseteq \rho _ { n , m } ^ { a u g m } \left( \widetilde { \tau } \right) +$$ + +Take $( x , i , y , k ) \in \rho _ { n , m } ^ { a u g m } \left( \widetilde { \mathcal { F } } \right)$ . This means that, for any $h : \mathbb { F } \to \mathbb { F } , f \in { \mathcal { F } } .$ + +$$ +\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { i , j } ) = \sum _ { j = 1 } ^ { m } h ( f ( y ) _ { k , j } ) . +$$ + +By Lem. 31 and the universality of MLP, there exists a permutation $\sigma \in S _ { m }$ such that $( { \dot { f } } ( x ) _ { i , \sigma ( 1 ) } , \dots , f ( x ) _ { i , \sigma ( m ) } ) = ( f ( y ) _ { k , 1 } , \dots , f ( y ) _ { k , m } )$ . By definition of the augmented separating power, this means that, for any $j \in \{ 1 , \dots , m \}$ , $( x , i , \sigma ( j ) , y , k , j ) \in \rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right)$ Hence, by assumption, for any $j ~ \in ~ \{ 1 , \dots , m \}$ , $( x , i , \sigma ( j ) , y , k , j ) \ \in \ \rho _ { n , m } ^ { a u g m } \left( \tau \right) .$ , i.e. $\tau ( x ) _ { i , \sigma ( j ) } = \tau ( y ) _ { k , j }$ . But this exactly means that $\widetilde { \tau } ( x ) _ { i } = \widetilde { \tau } ( y ) _ { k }$ so that $( x , i , y , k ) \in$ $\rho _ { n , m } ^ { a u g m } \left( \tilde { \tau } \right)$ as required. + +$( \supset )$ We now show the other inclusion, + +$$ +\rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right) \supset \rho _ { n , m } ^ { a u g m } \left( \tau \right) \implies \rho _ { n , m } ^ { a u g m } \left( \widetilde { \mathcal { F } } \right) \supset \rho _ { n , m } ^ { a u g m } \left( \widetilde { \tau } \right) . +$$ + +Take $( x , i , y , k ) \in \rho _ { n , m } ^ { a u g m } \left( \widetilde { \tau } \right)$ . By definition of $\widetilde { \tau }$ , this means that there exists $\sigma \in S _ { m }$ such that $\tau ( x ) _ { i , \sigma ( j ) } = \tau ( y ) _ { k , j }$ eso that $( x , i , \sigma ( j ) , y , k , j ) \in \rho _ { n , m } ^ { a u g m } \left( \tau \right) \subseteq \rho _ { n , m } ^ { a u g m } \left( \mathcal { F } \right)$ . Hence, for any $f \in { \mathcal { F } }$ , $( f ( x ) _ { i , \sigma ( 1 ) } , \ldots , f ( x ) _ { i , \sigma ( m ) } ) = ( f ( y ) _ { k , 1 } , \ldots , f ( y ) _ { k , m } )$ , and so, for any $h : \mathbb { F } \mathbb { F }$ , + +$$ +\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { i , j } ) = \sum _ { j = 1 } ^ { m } h ( f ( y ) _ { k , j } ) . +$$ + +Therefore, $( x , i , y , k ) \in \rho _ { n , m } ^ { a u g m } \left( \widetilde { \mathcal { F } } \right)$ , which concludes the proof. + +# C.3 INITIALIZATION LAYER + +We use the same initialization layer as Maron et al. (2019a); Chen et al. (2020) and recall it below. The initial graph is a tensor of the form $G \in \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ with $\mathbb { F } _ { 0 } = \mathbb { R } ^ { e + 2 }$ ; the last channel of $G _ { : , : , e + 1 }$ encodes the adjacency matrix of the graph and the first $e$ channels $G _ { : , : , 1 : e }$ are zero outside the diagonal and $G _ { i , i , 1 : e } \in \mathbb { R } ^ { e }$ is the color of vertex $v _ { i } \in V$ . We then define $I ^ { k } : \mathbb { F } _ { 0 } ^ { n ^ { 2 } } \to \mathbb { F } _ { 1 } ^ { n ^ { k } }$ with $\mathbb { F } _ { 1 } = \mathbb { R } ^ { k ^ { 2 } \times ( e + 2 ) }$ as follows: + +$$ +\begin{array} { r c l } { { \displaystyle { \cal I } ^ { k } ( G ) _ { { \bf i } , r , s , w } } } & { { = } } & { { G _ { i _ { r } , i _ { s } , w } , w \in [ e + 1 ] , } } \\ { { \displaystyle { \cal I } ^ { k } ( G ) _ { { \bf i } , r , s , e + 2 } } } & { { = } } & { { { \bf 1 } ( i _ { r } = i _ { s } ) , } } \end{array} +$$ + +for $\mathbf { i } \in [ n ] ^ { k }$ and $r , s \in [ k ]$ . This linear equivariant layer has the same separating power as the isormorphism type, which is defined in Section B.2. + +exists I k : Fn20 Lemma 10 ((Maron et al., 2019a, C.1)). For $I ^ { k } : \mathbb { F } _ { 0 } ^ { n ^ { 2 } } \to \mathbb { F } _ { 1 } ^ { n ^ { k } }$ such that, $k \geq 2$ , $p _ { 0 } \geq 1$ , $\mathbb { F } _ { 0 } = \mathbb { R } ^ { p _ { 0 } }$ , $\mathbb { F } _ { 1 } = \mathbb { R } ^ { k ^ { 2 } \times ( p _ { 0 } + 1 ) }$ , there + +$$ +\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( I ^ { k } \right) = \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( \mathrm { i s o } \right) . +$$ + +# C.4 KNOWN RESULTS ABOUT THE SEPARATING POWER OF SOME GNN CLASSES + +First, we need to define some classes of GNN from which both the invariant and equivaraint GNN we considered are built. See Section 3 for details about the different layers. + +$\begin{array} { r } { \mathbf { M G N N } _ { e m b } = \left\{ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } : F _ { t } : \mathbb { F } _ { t } ^ { n } \to \mathbb { F } _ { t + 1 } ^ { n } \right. } \end{array}$ message passing layer, $t = 1 , \ldots , T , T \geq 1 \}$ $\mathfrak { k } \mathrm { - } \mathrm { L G N N } _ { e m b } = \{ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k } : \mathbb { F } _ { t } ^ { n } \to \mathbb { F } _ { t + 1 } ^ { n }$ linear equivariant layer, $t = 1 , \ldots , T , T \geq 1 \}$ $\begin{array} { r } { k \mathrm { { \mathrm { - } } F G N N } _ { e m b } = \{ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k } : \mathbb { F } _ { t } ^ { n } \mathbb { F } _ { t + 1 } ^ { n } \mathrm { { F G L } } , t = 1 , \dots , T , T \geq 1 \} . } \end{array}$ + +Then, the precise results from the literature can be rephrased as, + +Lemma 11 $\mathrm { { X u } }$ et al. (2018),Maron et al. (2018),Maron et al. (2019a)). For $k \geq 2$ , + +$$ +\begin{array} { r l } & { \qquad \rho _ { n } ^ { a u g m } \left( M G N N _ { e m b } \right) = \rho _ { n } ^ { a u g m } \left( c _ { W L } \right) } \\ & { \qquad \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( k { - } L G N N _ { e m b } \right) \subseteq \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( c _ { k - W L } \right) } \\ & { \qquad \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( k { - } F G N N _ { e m b } \right) \subseteq \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( c _ { k - F W L } \right) } \end{array} +$$ + +(8) comes from $\mathrm { X u }$ et al. (2018, §A, $\ S \mathbf { B }$ ), (9) and (10) from Maron et al. (2019a, §C, $\ S _ { \mathbf { D } }$ ). + +# C.5 BOUNDING THE SEPARATING POWER OF $k$ -FGNN + +We complete the results of the literature with a bound on the separating power of $k$ -FGNN. Note that the particular case of $k = 2$ is already proven in Geerts (2020a). + +Lemma 12. For any $k \geq 2$ + +$$ +\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( k – F G N N _ { e m b } \right) \supset \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( c _ { k - F W L } \right) , +$$ + +so that + +$$ +\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( k – F G N N _ { e m b } \right) = \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( c _ { k - F W L } \right) . +$$ + +Proof. Define, + +$$ +k \mathrm { \mathrm { - } F G N N } _ { e m b } ^ { T } = \{ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k } : \mathbb { F } _ { t } ^ { n } \to \mathbb { F } _ { t + 1 } ^ { n } \mathrm { \ F G L } , t = 1 , \dots , T \} , +$$ + +the set of functions defined by exactly $T$ FGL layers. We show by induction that, for any $T \geq 0$ + +$$ +\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( k \mathrm { – F G N N } _ { e m b } ^ { T } \right) \supset \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( c _ { k \mathrm { – F W L } } ^ { T } \right) \ . +$$ + +For $T = 0$ , this is immediate by the definition of $I ^ { k }$ in Section C.3. + +Assume now that this inclusion holds at $T - 1 \geq 0$ . We show that it also holds at $T$ . Take $G , G ^ { \prime } \in \mathbb { F } _ { 0 } ^ { n ^ { k } }$ and $s , s ^ { \prime } \in [ n ] ^ { k }$ such that, + +$$ +c _ { k \mathrm { - F W L } } ^ { T } ( G ) _ { s } = c _ { k \mathrm { - F W L } } ^ { T } ( G ^ { \prime } ) _ { s ^ { \prime } } . +$$ + +$f \in k \mathrm { - F G N N } _ { e m b } ^ { T }$ + +$$ +f ( G ) _ { s } = f ( G ^ { \prime } ) _ { s ^ { \prime } } . +$$ + +But, by definition of the update rule $k$ -FWL, the equality of the colors of $s$ and $s ^ { \prime }$ above implies that, + +$$ +c _ { k \mathrm { - F W L } } ^ { T - 1 } ( G ) _ { s } = c _ { k \mathrm { - F W L } } ^ { T - 1 } ( G ^ { \prime } ) _ { s ^ { \prime } } , +$$ + +and that there exists $\sigma \in S _ { n }$ such that, for any $j \in [ n ]$ , + +$$ +\left( c _ { k \mathrm { - } \mathrm { F W L } } ^ { T - 1 } ( G ) _ { \tilde { s } } : \tilde { s } \in N _ { j } ^ { F } ( s ) \right) = \left( c _ { k \mathrm { - } \mathrm { F W L } } ^ { T - 1 } ( G ^ { \prime } ) _ { \tilde { s } } : \tilde { s } \in N _ { \sigma ( j ) } ^ { F } ( s ^ { \prime } ) \right) . +$$ + +Let $s = ( i _ { 1 } , \ldots , i _ { k } )$ and $s ^ { \prime } = ( j _ { 1 } , \ldots , j _ { k } )$ . Then this implies that, for any $w \in [ k ] , j \in [ n ]$ , + +$$ +c _ { k \mathrm { - } \mathrm { F W L } } ^ { T - 1 } ( G ) _ { i _ { 1 } , \dots , i _ { w - 1 } , j , i _ { w + 1 } , \dots , i _ { k } } = c _ { k \mathrm { - } \mathrm { F W L } } ^ { T - 1 } ( G ^ { \prime } ) _ { j _ { 1 } , \dots , j _ { w - 1 } , \sigma ( j ) , j _ { w + 1 } , \dots , j _ { k } } . +$$ + +We now use the induction hypothesis, i.e. that, + +$$ +\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( k \mathrm { – F G N N } _ { e m b } ^ { T - 1 } \right) \supset \rho _ { n , n ^ { k - 1 } } ^ { a u g m } \left( c _ { k \mathrm { – F W L } } ^ { T - 1 } \right) \ . +$$ + +Take any $f _ { T - 1 } \in k – \mathrm { F G N N } _ { e m b } ^ { T - 1 } .$ . By (11), + +$$ +f _ { T - 1 } ( G ) _ { s } = f _ { T - 1 } ( G ^ { \prime } ) _ { s ^ { \prime } } . +$$ + +By (12), for any $w \in [ k ] , j \in [ n ]$ , + +$$ +\begin{array} { r } { f _ { T - 1 } ( G ) _ { i _ { 1 } , \dots , i _ { w - 1 } , j , i _ { w + 1 } , \dots , i _ { k } } = f ^ { T - 1 } ( G ^ { \prime } ) _ { j _ { 1 } , \dots , j _ { w - 1 } , \sigma ( j ) , j _ { w + 1 } , \dots , j _ { k } } . } \end{array} +$$ + +By the definition of FGL Section 3.3, for any $F _ { T } : \mathbb { F } _ { T } ^ { n ^ { k } } \to \mathbb { F } _ { T + 1 } ^ { n ^ { k } }$ FGL, $F _ { T } \circ f _ { T - 1 } ( G ) _ { s } = F _ { T } \circ$ $f _ { T - 1 } ( G ^ { \prime } ) _ { s ^ { \prime } }$ . Therefore, for any $f \in k \mathrm { - F G N N } _ { e m b } ^ { T }$ , + +$$ +f ( G ) _ { s } = f ( G ^ { \prime } ) _ { s ^ { \prime } } , +$$ + +which concludes the proof. + +# C.6 CONCLUSION + +Proposition 13. We have, for $k \geq 2$ + +$$ +\begin{array} { r l } { \rho \left( M G N N _ { I } \right) = \rho \left( 2 \ – W L _ { I } \right) \qquad } & { { } \rho \left( M G N N _ { E } \right) = \rho \left( 2 \ – W L _ { E } \right) } \\ { \rho \left( 2 \ – L G N N _ { I } \right) = \rho \left( 2 \ – W L _ { I } \right) \qquad } & { { } } \\ { \rho \left( k \ – L G N N _ { I } \right) \subseteq \rho \left( k \ – W L _ { I } \right) \qquad } & { { } \rho \left( k \ – L G N N _ { E } \right) \subseteq \rho \left( k \ – W L _ { E } \right) } \\ { \rho \left( k \ – F G N N _ { I } \right) = \rho \left( k \ – F W L _ { I } \right) \qquad } & { { } \rho \left( k \ – F G N N _ { E } \right) = \rho \left( k \ – F W L _ { E } \right) } \end{array} +$$ + +Proof. Most of the statements come from the literature or are direct consequences of the lemmas above. + +• Proof of (13). The invariant case is proven in $\mathrm { X u }$ et al. (2018, Lem. 2, Thm. 3). The equivariant case comes from Lem. 11, Lem. 8, the fact that the layers $m _ { E } \circ ( \mathrm { I d } + \lambda S ^ { 1 } )$ does not change the separating power and recalling that simply $\mathrm { W L } _ { E } = c _ { W L }$ . +• (14) is the exact result Chen et al. (2020, Thm. 6). +• Proof of (15). The invariant case is exactly Maron et al. (2019a, Thm. 1). The equivariant case comes from Lem. 11, Lem. 9, Lem. 8 and the fact that the layer $m _ { E }$ does not change the separating power. +• Proof of (16). The direct inclusion of the invariant case corresponds to Maron et al. (2019a, Thm. 2). The other cases are a consequence of Lem. 11, Lem. 9, Lem. 8 and the fact that the layers $m _ { I }$ and $m _ { E }$ does not change the separating power. + +# D STONE-WEIERSTRASS THEOREM WITH SYMMETRIES + +This section presents our extension of the Stone-Weierstrass theorem dealing with functions with symmetries. The scope of this section is not restricted to graphs or even tensors and we will deal with general spaces and general symmetries. To illustrate it, we will present applications for the PointNet architecture Qi et al. (2017). Our approximation results for GNNs (Theorems Thm. 5 and Thm. 6) are then obtained from these theoretical results applied to tensors and the symmetric group in Section D.9 + +As explained above, we are dealing in this section with a much larger scope than graphs and permutations. We first need to extend the notations introduced above. The notations introduced below will make this section self-contained. + +If $X$ is some topological space, and $F \subseteq X$ , denote by $\overline { F }$ its closure. + +If $X$ is a topological space and $Y = \mathbb { R } ^ { p }$ some finite-dimensional space, denote by ${ \mathcal { C } } ( X , Y )$ the set of continuous functions from $X$ to $Y$ . + +Moreover, if $X$ is compact, we endow ${ \mathcal { C } } ( X , Y )$ with the topology of uniform convergence, which is defined by the norm, ${ \bar { f } } \mapsto \operatorname* { s u p } _ { x \in X } \| f ( x ) \|$ for some norm $\intercal \Vert . \Vert$ on $Y$ . + +If $G$ is a finite group acting on some topological space $X$ , we say that $G$ acts continuously on $X$ if, for all $g \in G$ , $x \mapsto g \cdot x$ is continuous. + +If $G$ is a finite group acting on some compact set $X$ and some topological space $Y$ , we define the sets of equivariant and invariant continuous functions by, + +$$ +\begin{array} { r l } & { \mathcal { C } _ { E } ( X , Y ) = \{ f \in \mathcal { C } ( X , Y ) : \forall x \in X , \forall g \in G , f ( g \cdot x ) = g \cdot f ( x ) \} } \\ & { \mathcal { C } _ { I } ( X , Y ) = \{ f \in \mathcal { C } ( X , Y ) : \forall x \in X , \forall g \in G , f ( g \cdot x ) = f ( x ) \} } \end{array} +$$ + +Note that these definitions extend Definition 1 to a general group. + +If $Y = \mathbb { R } ^ { p }$ , we denote the coordinate-wise multiplication, or Hadamard product of $y , y ^ { \prime } \in Y$ simply by $y y ^ { \prime } = ( y _ { 1 } y _ { 1 } ^ { \prime } , \dots , y _ { p } y _ { p } ^ { \prime } ) \in Y$ . We say that a subset $A \subseteq \mathbb { R } ^ { p }$ is a subalgebra of $\mathbb { R } ^ { p }$ if it is both a linear space and stable by multiplication. + +This product in turn defines a product on ${ \mathcal { C } } ( X , Y )$ with $Y = \mathbb { R } ^ { p }$ by, for $f , g \in { \mathcal { C } } ( X , Y ) , f g : x \mapsto$ $f ( x ) g ( x )$ . + +In addition, we also extend the scalar-vector product of $Y = \mathbb { R } ^ { p }$ to functions: if $g \in { \mathcal { C } } ( X , \mathbb { R } )$ and $f \in { \mathcal { C } } ( X , Y )$ , their product $g f$ is the function $g f : x \mapsto g ( x ) f ( x )$ . Given a set of scalar functions ${ \mathcal { S } } \subseteq { \mathcal { C } } ( X , \mathbb { R } )$ and a set of vector-valued functions ${ \mathcal { F } } \subseteq { \mathcal { C } } ( X , Y )$ , the set of products of functions of these two sets will be denoted by, + +$$ +S \cdot { \mathcal { F } } = \left\{ g f : g \in { \mathcal { S } } , f \in { \mathcal { F } } \right\} . +$$ + +Moreover, we denote by 1 the continuous function from some $X$ to $\mathbb { R } ^ { p }$ defined by $x \mapsto ( 1 , \ldots , 1 )$ . In particular, if $f$ is a function from $X$ to $\mathbb { R }$ , $f \mathbf { 1 }$ denotes the function $x \mapsto ( f ( x ) , \ldots , f ( x ) )$ which goes from $X$ to $Y = \mathbb { R } ^ { p }$ . Finally, we say that ${ \mathcal { F } } \subseteq { \mathcal { C } } ( X , Y )$ with $Y$ being some $\mathbb { R } ^ { p }$ is a subalgebra if it is a linear space which is also stable by multiplication. + +# D.2 SEPARATING POWER + +We recall the definition of separating power that we introduced above: + +Definition 14. Let $\mathcal { F }$ be a set of functions $f$ defined on a set $X$ , where each $f$ takes its values in some $Y _ { f }$ . The equivalence relation $\rho \left( \mathcal { F } \right)$ defined by $\mathcal { F }$ on $X$ is: for any $x , x ^ { \prime } \in X$ , + +$$ +( x , x ^ { \prime } ) \in \rho \left( { \mathcal { F } } \right) \iff \forall f \in { \mathcal { F } } , f ( x ) = f ( x ^ { \prime } ) . +$$ + +For a function $f$ , we write $\rho \left( f \right)$ for $\rho \left( \{ f \} \right)$ . + +Separating power is stable by closure: + +Lemma 15. Let $X$ be a compact topological space, $Y$ be some finite-dimensional and ${ \mathcal { F } } \subseteq { \mathcal { C } } ( X , Y )$ . Then, + +$$ +\rho \left( \mathcal { F } \right) = \rho \left( \overline { { \mathcal { F } } } \right) +$$ + +Proof. As ${ \mathcal { F } } \subseteq { \overline { { \mathcal { F } } } }$ , $\overline { { \mathcal { F } } }$ is more separating than $\mathcal { F }$ , i.e. $\rho \left( { \overline { { \mathcal { F } } } } \right) \subseteq \rho \left( { \mathcal { F } } \right)$ . + +Conversely, take $( x , y ) \notin \rho \left( { \overline { { \mathcal { F } } } } \right)$ . By definition, there exists $h \in { \overline { { \mathcal { F } } } }$ such that $h ( x ) \neq h ( y )$ so that if $\epsilon = \| h ( x ) - h ( y ) \|$ , $\epsilon > 0$ (for some norm $\left. . \right.$ on $Y$ ). As $h \in { \overline { { \mathcal { F } } } }$ , there is some $f \in { \mathcal { F } }$ such that + +$\begin{array} { r } { \operatorname* { s u p } _ { X } \| h - f \| \leq \frac { \epsilon } { 3 } } \end{array}$ . Therefore, by the triangular inequality, + +$$ += \| h ( x ) - h ( y ) \| \leq \| f ( x ) - h ( x ) \| + \| f ( x ) - f ( y ) \| + \| f ( y ) - h ( y ) \| \leq { \frac { 2 } { 3 } } \epsilon + \| f ( x ) - f ( y ) \| . +$$ + +It follows that $\| f ( x ) - f ( y ) \| \ge \frac { \epsilon } { 3 } > 0$ so that $f ( x ) \neq f ( y )$ and $( x , y ) \notin \rho \left( { \mathcal { F } } \right)$ . + +D.3 APPROXIMATION THEOREMS FOR REAL-VALUED FUNCTIONS + +We start by recalling Stone-Weierstrass theorem, see Rudin (1991, Thm. 5.7). + +Theorem 16 (Stone-Weierstrass). Let $X$ be a compact space, and $\mathcal { F }$ be a subalgebra of ${ \mathcal { C } } ( X , \mathbb { R } )$ the space of real-valued continuous functions of $X$ , which contains the constant function 1. If $\mathcal { F }$ separates points, i.e. $\rho \left( { \mathcal { F } } \right) = \{ \left( x , x \right) : x \in X \}$ , then $\mathcal { F }$ is dense in ${ \mathcal { C } } ( X , \mathbb { R } )$ . + +We now prove an extension of this classical result due to Timofte (2005) allowing us to deal with much smaller $\mathcal { F }$ by dropping the requirement that $\mathcal { F }$ separates points. + +Corollary 17. Let $X$ be a compact space, and $\mathcal { F }$ be a subalgebra of ${ \mathcal { C } } ( X , \mathbb { R } )$ the space of real-valued continuous functions of $X$ , which contains the constant function 1. Then, + +$$ +\overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } ( X , \mathbb { R } ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) \right\} . +$$ + +Note that if $\mathcal { F }$ separates points, we get back the classical result as every function satisfies $\{ ( x , x ) :$ $x \in X \} \subseteq \rho \left( f \right)$ . + +Example 18. The invariant version of PointNet is able to learn functions of the form $\textstyle \sum _ { i } f ( x _ { i } )$ for $f \in { \mathcal { C } } ( \mathbb { R } ^ { p } , \mathbb { R } )$ . We can apply Corollary 17 to this setting. Consider the case where $X$ is a compact subset of $\mathbb { R } ^ { p }$ and $\mathcal { F } = \textstyle \{ \hat { x } \mapsto g ( \sum _ { i = 1 } ^ { n } \bar { f } ( x _ { i } ) )$ , $f \in \mathcal { C } ( \mathbf { \breve { X } } , \mathbb { R } ^ { h } ) , g \in \mathcal { C } ( \mathbb { R } ^ { h } , \mathbb { R } ) \}$ . $\mathcal { F }$ is a subalgebra of ${ \mathcal { C } } ( X , \mathbb { R } )$ (indeed of ${ \mathcal { C } } _ { I } ( X , \mathbb { R } ) ,$ ) which contains the constant function 1. Then, it easy to see that $\rho \left( { \mathcal { F } } \right) = \{ ( x , \sigma \star x ) , \sigma \in S _ { n } \}$ , where $\sigma \star x$ is defined by $( \sigma \star x ) _ { \sigma ( i ) } = x _ { i }$ for all $i$ (see Lem. 31 for a formal statement). + +Now note that for a function $f \in { \mathcal { C } } ( X , \mathbb { R } )$ , the condition $\left\{ ( x , \sigma \star x ) , \sigma \in S _ { n } \right\} \subseteq \rho \left( f \right)$ is equivalent to $f \in { \mathcal { C } } _ { I } ( X , \mathbb { R } )$ . So that Corollary 17 implies that $\overline { { \mathcal { F } } } = \mathcal { C } _ { I } ( X , \mathbb { R } )$ which means that PointNet is universal for approximating invariant functions. This was already proved in Qi et al. (2017) . + +We now provide a proof of Corollary 17 for completeness. + +Proof. The first inclusion ${ \overline { { \mathcal { F } } } } \subseteq \{ f \in { \mathcal { C } } ( X , \mathbb { R } ) : \rho \left( { \mathcal { F } } \right) \subseteq \rho \left( f \right) \}$ follows from the same argument as the one given below Theorem 4 so we focus on the other one. + +For every $x \in X$ , let $x _ { \mathcal { F } }$ denote its $\rho \left( \mathcal { F } \right)$ -class. The quotient set and the canonical surjection are: $X _ { \mathcal { F } } = X / \rho \left( \mathcal { F } \right) = \left\{ x _ { \mathcal { F } } , x \in X \right\}$ and $\pi _ { \mathcal { F } } : X \to X _ { \mathcal { F } }$ , $\bar { \pi } _ { \mathcal { F } } ( x ) = x _ { \mathcal { F } }$ . + +A function $g : X \to \mathbb { R }$ factorizes as $g = \hat { g } \circ \pi _ { \mathcal { F } }$ for some $\hat { g } : X _ { \mathcal { F } } \mathbb { R }$ if and only if $\rho \left( { \mathcal { F } } \right) \subseteq \rho \left( g \right)$ . In this case $\hat { g }$ is unique, since $\pi _ { \mathcal { F } }$ is a surjection. In particular. every $f \in { \mathcal { F } }$ factorizes uniquely as $f = \hat { f } \circ \bar { \pi } _ { \mathcal { F } }$ , $\hat { f } : X _ { \mathcal { F } } \mathbb { R }$ , and $\hat { \mathcal { F } } = \{ \hat { f } , f \in \mathcal { F } \}$ clearly separates points on $X _ { \mathcal { F } }$ . We refer to Munkres (2000, $\ S 2 2 _ { , }$ ) for the properties of the quotient topology. In particular, by the properties of the quotient topology on $X _ { \mathcal { F } }$ , $\hat { \mathcal { F } }$ is a subalgebra of $\mathcal { C } ( X _ { \mathcal { F } } , \mathbb { R } )$ and $X _ { \mathcal { F } }$ is compact. Hence, we can apply Theorem 16 to $\hat { \mathcal { F } }$ and $\hat { \mathcal { F } }$ is dense in $\mathcal { C } ( X _ { \mathcal { F } } , \mathbb { R } )$ . + +Now take $f \in { \mathcal { C } } ( X , \mathbb { R } )$ with $\rho \left( { \mathcal { F } } \right) \subseteq \rho \left( f \right)$ and we show that $f \in { \overline { { \mathcal { F } } } }$ . Again, $\rho \left( { \mathcal { F } } \right) \subseteq \rho \left( f \right)$ implies that $f = \hat { f } \circ \pi _ { \mathcal { F } }$ . Let $\epsilon > 0$ . By density of $\hat { \mathcal F }$ , there is some $\hat { h } \in \hat { \mathcal F }$ such that $\begin{array} { r } { \operatorname* { s u p } _ { x _ { \mathcal { F } } } | \hat { h } ( x _ { \mathcal { F } } ) ) - } \end{array}$ $\hat { f } ( x _ { \mathcal { F } } ) | \leq \epsilon$ . But, by construction of $\hat { F }$ , there exits $h \in { \mathcal { F } }$ such that $\hat { h } \circ \pi _ { \mathcal F } = h$ . Thus, + +$$ +\operatorname* { s u p } _ { x \in X } \left| h ( x ) - f ( x ) \right| = \operatorname* { s u p } _ { x \in X } \left| \hat { h } ( \pi _ { \mathcal { F } } ( x ) ) - \hat { f } ( \pi _ { \mathcal { F } } ( x ) ) \right| = \operatorname* { s u p } _ { x _ { \mathcal { F } } \in X _ { \mathcal { F } } } \left| \hat { h } ( x _ { \mathcal { F } } ) ) - \hat { f } ( x _ { \mathcal { F } } ) \right| \leq \epsilon . +$$ + +As this holds for any $\epsilon > 0$ , we have proven that $f \in { \overline { { \mathcal { F } } } }$ . + +We first need to extend Corollary 17 to vector-valued functions. For this, we need to have a vectorvalued version of the Stone-Weierstrass theorem and as shown by the example below additional assumptions have to be made. + +Example 19. We consider now the equivariant version of PointNet corresponding to the particular case where $X$ is a compact subset of $( \mathbb { R } ^ { p } ) ^ { n }$ , $Y = \mathbb { R } ^ { n }$ and ${ \mathcal { F } } = \{ x \mapsto ( f ( x _ { 1 } ) , \ldots , f ( x _ { n } ) )$ , $f \in$ ${ \mathcal { C } } ( \mathbb { R } , \mathbb { R } ) \}$ . Then clearly $\mathcal { F }$ is a subalgebra of ${ \mathcal { C } } _ { E } ( X , Y )$ containing the constant function 1 and $\rho \left( { \mathcal { F } } \right) = \{ ( x , x ) , x \in X \}$ . Hence if Corollary 17 would be true with vector-valued functions instead of real-valued functions, we would have that $\mathcal { F }$ is dense in ${ \mathcal { C } } ( X , Y )$ . But this can clearly not be true as ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { E } ( X , Y )$ which is clearly not dense in ${ \mathcal { C } } ( X , Y )$ . + +We now present an extension of Corollary 17 also due to Timofte (2005): + +Proposition 20. Let $X$ be a compact space, $Y = \mathbb { R } ^ { p }$ for some $p \geq 1$ . Let ${ \mathcal { F } } \subseteq { \mathcal { C } } ( X , Y )$ . If there exists a nonempty subset $S \subseteq { \mathcal { C } } ( X , \mathbb { R } )$ such that: + +$$ +S \cdot { \mathcal { F } } \subseteq { \mathcal { F } } a n d , \rho \left( S \right) \subseteq \rho \left( { \mathcal { F } } \right) . +$$ + +Then we have + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } ( X , Y ) , \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) , f ( x ) \in \overline { { \mathcal { F } ( x ) } } \right\} , } \end{array} +$$ + +where ${ \mathcal { F } } ( x ) = \{ f ( x ) , f \in { \mathcal { F } } \}$ . Moreover in (18), we can replace $\rho \left( \mathcal { F } \right)$ by $\rho \left( S \right)$ . + +Note that in the particular case $Y = \mathbb { R }$ and if $\mathcal { F }$ is a subalgebra of ${ \mathcal { C } } ( X , \mathbb { R } )$ , then we can take $S = \mathcal { F }$ in (17) and if the constant function 1 is in $\mathcal { F }$ , then $\mathcal { F } ( x ) = \mathbb { R }$ , so that we recover Corollary 17. + +Now consider the case where $\mathcal { F }$ is a subalgebra of $\mathcal { C } ( X , \mathbb { R } ^ { p } )$ . We need to find a set $S \in { \mathcal { C } } ( X , \mathbb { R } )$ satisfying (17) i.e. with a better separating power than $\mathcal { F }$ but containing real-valued functions such that $s f \in \mathcal { F }$ for all $s \in S$ and $f \in { \mathcal { F } }$ . In the sequel, we will consider the set $\mathcal { F } _ { s c a l } =$ $\{ f \in { \mathcal { C } } ( X , \mathbb { R } ) : f { \bf 1 } \in { \mathcal { F } } \}$ . We clearly have $\mathcal { F } _ { s c a l } \cdot \mathcal { F } \subseteq \mathcal { F }$ since $\mathcal { F }$ is a subalgebra. Hence, in this setting, Prop. 20 can be rewritten as follows: + +Corollary 21. Let $X$ be a compact space, $Y = \mathbb { R } ^ { p }$ for some $p$ , $G$ be a finite group acting (continuously) on $X$ and ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { I } ( X , Y )$ a (non-empty) set of invariant functions associated to $G$ . + +Consider the following assumptions, + +1. $\mathcal { F }$ is a sub-algebra of ${ \mathcal { C } } ( X , Y )$ and the constant function 1 is in $\mathcal { F }$ . + +2. The set of functions $\mathcal { F } _ { s c a l } \subseteq \mathcal { C } ( X , \mathbb { R } )$ defined by, + +$$ +\mathcal { F } _ { s c a l } = \{ f \in \mathcal { C } ( X , \mathbb { R } ) : f { \bf 1 } \in \mathcal { F } \} +$$ + +satisfy, + +$$ +\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \mathcal { F } \right) . +$$ + +3. For any $x \in X$ , there exists $f \in { \mathcal { F } }$ such that $f ( x )$ has pairwise distinct coordinates, i.e., for any indices $i , j \in \{ 1 , \ldots , p \}$ with $i \neq j$ , $f ( x ) _ { i } \not = f ( x ) _ { j }$ . + +Then the closure of $\mathcal { F }$ (for the topology of uniform convergence) is, + +$$ +{ \overline { { \mathcal { F } } } } = \{ f \in { \mathcal { C } } _ { I } ( X , Y ) : \rho \left( { \mathcal { F } } \right) \subseteq \rho \left( f \right) \} ~ . +$$ + +Note that Assumptions 1 and 2 ensures that (17) is valid, while Assumption 3 ensures that $\mathcal { F } ( \boldsymbol { x } ) = \mathbb { R } ^ { p }$ . Unfortunately, in the equivariant case, the condition $\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \mathcal { F } \right)$ is too strong and we now explain how we will relax it. + +For the sake of simplicity, we consider here the particular setting adapted to graphs: let $n \geq 1$ be a fixed number (corresponding to the number of nodes), $X$ be a compact set of graphs in $\mathbb { R } ^ { n ^ { 2 } }$ and $Y = \mathbb { F } ^ { n }$ with $\mathbb { F } = \mathbb { R } ^ { p }$ for some $p \geq 1$ . We define the action of the symmetric group $S _ { n }$ on $X$ by $( \sigma \star x ) _ { \sigma ( i ) , \sigma ( j ) } = x _ { i . j }$ and on $Y$ by $( \sigma \star y ) _ { \sigma ( i ) } = y _ { i } \in \mathbb { R } ^ { p }$ . Hence the set of continuous equivariant functions ${ \mathcal { C } } _ { E } ( X , Y )$ agrees with Definition 1. + +Now consider the case where ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { E } ( X , Y )$ is a subalgebra of equivariant functions. Then, $f \in \mathcal { F } _ { s c a l }$ needs to be invariant in order for $f \mathbf { 1 }$ to be equivariant and hence in $\mathcal { F }$ . As a result, we see that $\mathcal { F } _ { s c a l }$ will not separate points of $X$ in the same orbit, i.e. $x$ and $\sigma \star x$ . But these points will typically be separated by $\mathcal { F }$ , since for any $f \in { \mathcal { F } }$ , we have $f ( \sigma \star x ) = \sigma \star f ( x )$ which is not equal to $f ( x )$ unless $f$ is invariant. + +We see that we need somehow to require a weaker separating power for $\mathcal { F }$ . More formally, two isomorphic graphs will have permuted outputs through an equivariant function, but should not be considered as separated. Let $O r b ( x ) = \{ \bar { \sigma } \star x , \sigma \in S _ { n } \}$ and $O r b ( y ) = \{ \sigma \star y , \sigma \in S _ { n } \}$ . For any equivariant function $f \in { \mathcal { C } } _ { E } ( X , Y )$ , for any $z \in O r b ( x )$ , we have $f ( z ) \in O r b ( f ( x ) )$ . Then let $\pi : Y \to Y / S _ { n }$ be the canonical projection $\pi ( y ) = O r b ( y )$ . We define + +$$ +\begin{array} { r } { ( x , x ^ { \prime } ) \in \rho \left( \pi \circ \mathcal { F } \right) \Leftrightarrow \forall f \in \mathcal { F } , O r b ( f ( x ) ) = O r b ( f ( x ^ { \prime } ) ) } \\ { \Leftrightarrow \forall f \in \mathcal { F } , \exists \sigma \in { \cal S } _ { n } , f ( \sigma \star x ) = f ( x ^ { \prime } ) . } \end{array} +$$ + +In particular, we see that if $x ^ { \prime } \in O r b ( x )$ then $( \boldsymbol { x } , \boldsymbol { x } ^ { \prime } ) \in \rho \left( \pi \circ \mathcal { F } \right)$ for any ${ \mathcal { F } } \in { \mathcal { C } } _ { E } ( X , Y )$ . Moreover, two graphs $x$ and $x ^ { \prime }$ are $\rho \left( \pi \circ { \mathcal { F } } \right)$ -distinct if there exists a function $f \in { \mathcal { F } }$ such that $\forall \sigma$ , $f ( \sigma \star x ) \neq$ $f ( x ^ { \prime } )$ , i.e. the function $f$ discriminates $O r b ( x )$ from $O r b ( x ^ { \prime } )$ in the sense that for any $z \in O r b ( x )$ and $z ^ { \prime } \in O r b ( x ^ { \prime } )$ , we have $f ( z ) \neq f ( z ^ { \prime } )$ . + +To obtain an equivalent of Proposition 20 with ${ \mathcal { C } } _ { E } ( X , Y )$ replacing ${ \mathcal { C } } ( X , Y )$ , we are able to relax assumption (17) to $\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \pi \circ \mathcal { F } \right)$ . Our main general result in this direction is the following theorem (proved in Section D.7) which might be of independent interest: + +Theorem 22. Let $X$ be a compact space, $Y = \mathbb { R } ^ { p }$ for some $p$ , $G$ be a finite group acting (continuously) on $X$ and $Y$ and ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { E } ( X , Y )$ a (non-empty) set of equivariant functions. + +Denote by $\pi : Y \longrightarrow Y / G$ the canonical projection on the quotient space $Y / G$ . Consider the following assumptions, + +1. $\mathcal { F }$ is a sub-algebra of ${ \mathcal { C } } ( X , Y )$ and the constant function $\mathbf { 1 }$ is in $\mathcal { F }$ . + +2. The set of functions $\mathcal { F } _ { s c a l } \subseteq \mathcal { C } ( X , \mathbb { R } )$ defined by, + +$$ +\mathcal { F } _ { s c a l } = \{ f \in \mathcal { C } ( X , \mathbb { R } ) : f \mathbf { 1 } \in \mathcal { F } \} +$$ + +satisfy, + +$$ +\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \pi \circ \mathcal { F } \right) . +$$ + +Then the closure of $\mathcal { F }$ (for the topology of uniform convergence) is, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } _ { E } ( X , Y ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) , \forall x \in X , f ( x ) \in \mathcal { F } ( x ) \right\} , } \end{array} +$$ + +where ${ \mathcal { F } } ( x ) = \{ f ( x ) , f \in { \mathcal { F } } \}$ . Moreover, i ${ } ^ { F } I ( x ) = \{ ( i , j ) \in [ p ] ^ { 2 } : \forall y \in { \mathcal { F } } ( x ) , y _ { i } = y _ { j } \} ,$ , then we have: + +$$ +{ \mathcal { F } } ( x ) = \left\{ y \in \mathbb { R } ^ { p } : \forall ( i , j ) \in I ( x ) , y _ { i } = y _ { j } \right\} . +$$ + +Example 23. We now demonstrate how Theorem 22 can be used to recover the universality results in Segol & Lipman (2020). In this paper, the authors study equivariant neural network architectures working with unordered sets, corresponding in our case to $X = Y = \mathbb { R } ^ { n }$ and the group being the symmetric group $S _ { n }$ . They show that the PointNet architecture cannot approximate any (continuous) equivariant function and that adding a single so-called transmission layer is enough to make this architecture universal. + +Indeed, PointNet can only learn maps of the form $x \in \mathbb { R } ^ { n } \mapsto ( f ( x _ { 1 } ) \dots f ( x _ { n } ) )$ , which are not universal in the class of equivariant functions, as shown by Segol & Lipman (2020, Lem. 3). Now, their transmission layer is a map of the form $x \in \mathbb { R } ^ { n } \mapsto ( \mathbf { 1 } ^ { T } \dot { x } ) \mathbf { 1 }$ . Therefore, in PointNetST, adding such a layer precisely adds a large class of functions to ${ \mathcal { F } } = \{ ( f ( x _ { 1 } , \sum _ { i } g ( x _ { i } ) ) , \dots , f ( x _ { n } , \sum _ { i } g ( x _ { i } { \bar { ) } } ) )$ , $f \in$ $\mathcal { C } ( \mathbb { R } \times \mathbb { R } ^ { h } , \mathbb { R } )$ , $g \in \mathcal { C } ( \mathbb { R } , \mathbb { R } ^ { h } ) , h \geq 1 \}$ . $\mathcal { F }$ is still an algebra and as shown in Example 19, we have $\rho \left( { \mathcal { F } } \right) = \{ ( x , x ) , x \in X \}$ . Moreover, we have $\mathcal { F } _ { s c a l } = \mathcal { C } _ { I } ( X , \mathbb { R } )$ by Lem. 33 in particular, we get $\rho \left( \mathcal { F } _ { s c a l } \right) = \{ ( x , \sigma \star x )$ , $x \in X \}$ , so that we obviously have $\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \pi \circ \mathcal { F } \right)$ . In summary, Theorem 22 implies the universality of PointNetST in $\mathcal { C } _ { E } ( X , Y )$ . + +# D.5 A PRELIMINARY VERSION OF THE EQUIVARIANT APPROXIMATION THEOREM + +We start by proving a version of Theorem 22 with a slightly weaker condition: + +Proposition 24. Let $X$ be a compact space, $Y ~ = ~ \mathbb { R } ^ { p }$ for some $p _ { ; }$ , $G$ be a finite group acting (continuously) on $X$ and $Y$ and ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { E } ( X , Y )$ a (non-empty) set of equivariant functions. + +Consider the following assumptions, + +1. $\mathcal { F }$ is a subalgebra ${ \mathcal { C } } _ { E } ( X , Y )$ . + +2. The set of real-valued functions $\mathcal { F } _ { s c a l } \subseteq \mathcal { C } ( X , \mathbb { R } )$ defined by, + +$$ +\mathcal { F } _ { s c a l } = \{ f \in \mathcal { C } ( X , \mathbb { R } ) : f \mathbf { 1 } \in \mathcal { F } \} +$$ + +satisfies, + +$$ +\begin{array} { r } { \rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \left\{ \left( x , x ^ { \prime } \right) \in X \times X : \exists g \in G , \left( g \cdot x , x ^ { \prime } \right) \in \rho \left( \mathcal { F } \right) \right\} . } \end{array} +$$ + +Then the closure of $\mathcal { F }$ (for the topology of uniform convergence) is, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } _ { E } ( X , Y ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) , \forall x \in X , f ( x ) \in \overline { { \mathcal { F } ( x ) } } \right\} , } \end{array} +$$ + +where ${ \mathcal { F } } ( x ) = \{ f ( x ) , f \in { \mathcal { F } } \}$ . + +The proof of this theorem relies on two main ingredients. First, following the elegant idea of Maehara & Hoang (2019), we augment the input space to transform the vector-valued equivariant functions into scalar maps. Second, we apply the fine-grained approximation result Cor. 17. + +Proof. As uniform convergence implies point-wise convergence, the first inclusion is immediate, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } \subseteq \left\{ f \in \mathcal { C } _ { E } ( X , Y ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) , \forall x \in X , f ( x ) \in \overline { { \mathcal { F } ( x ) } } \right\} . } \end{array} +$$ + +The rest of the proof is devoted to the other direction. + +For convenience, denote by $\Phi$ the family of linear forms associated to the canonical basis of $\mathbb { R } ^ { p }$ , i.e., + +$$ +\Phi = \{ y \mapsto y _ { i } : 1 \leq i \leq p \} \subseteq { \mathcal { C } } ( Y , \mathbb { R } ) . +$$ + +Define our augmented input space as ${ \tilde { X } } = X \times \Phi$ . As $\Phi$ is finite and $X$ is compact, $\tilde { X }$ is still a compact space. We now transform $\mathcal { F }$ , a class of equivariant functions from $X$ to $Y$ , into $\tilde { \mathcal { F } }$ a class of maps from $\tilde { X }$ to $\mathbb { R }$ . Define + +$$ +{ \tilde { \mathcal { F } } } = \{ ( x , \varphi ) \mapsto \varphi ( f ( x ) ) : f \in { \mathcal { F } } \} \ . +$$ + +We check that $\tilde { \mathcal { F } }$ is indeed a subset of $\mathcal { C } ( \tilde { X } , \mathbb { R } )$ . Indeed, as $\Phi$ is finite, it is equipped with the discrete topology. Hence, each singleton $\{ \varphi \}$ for $\varphi \in \Phi$ is open in $\Phi$ and it suffices to check the continuity in the first variable with $\varphi$ fixed. But, if $f \in { \mathcal { F } }$ , $x \stackrel { \textstyle - } { \mapsto } \varphi ( f ( x ) )$ is continuous as a composition of continuous maps. + +We can now apply Cor. 17 to $\tilde { \mathcal { F } } \subseteq \mathcal { C } ( \tilde { X } , \mathbb { R } )$ . Therefore, the closure of $\tilde { F }$ in $\mathcal { C } ( \tilde { X } , \mathbb { R } )$ is, + +$$ +\begin{array} { r } { \overline { { \tilde { \mathcal { F } } } } = \left\{ v \in \mathcal { C } ( \tilde { X } , \mathbb { R } ) : \rho \left( \tilde { \mathcal { F } } \right) \subseteq \rho \left( v \right) , \forall ( x , \varphi ) \in \tilde { X } , v ( x , \varphi ) \in \overline { { \tilde { \mathcal { F } } ( x , \varphi ) } } \right\} . } \end{array} +$$ + +We now show the equality of (19). Take $h$ in the right-hand side of (19), i.e. $h \in { \mathcal { C } } _ { E } ( X , Y )$ such that $\rho \left( { \mathcal { F } } \right) \subseteq \rho \left( h \right)$ and $h ( x ) \in { \overline { { \mathcal { F } ( x ) } } }$ for all $x \in X$ . We show that $\tilde { h }$ , defined by ${ \tilde { h } } : ( x , \varphi ) \mapsto \varphi ( h ( x ) )$ , belongs to $\overline { { \tilde { F } } }$ using the result above. + +• As $h$ is continuous, by the same argument as above, $( x , \varphi ) \mapsto \varphi ( h ( x ) )$ is continuous on $\tilde { X }$ + +• We check that $\rho \left( \tilde { \mathcal { F } } \right) \subseteq \rho \left( \tilde { h } \right)$ + +Take $( x , \varphi ) , ( y , \psi ) \in \tilde { X }$ such that, for all $f \in { \mathcal { F } }$ , + +$$ +\varphi ( f ( x ) ) = \psi ( f ( y ) ) , +$$ + +and we aim at showing that $\varphi ( h ( x ) ) = \psi ( h ( y ) )$ . + +To gain more information from (20), we apply it to functions of the form $f \mathbf { 1 }$ with $f \in \mathcal { F } _ { s c a l }$ By definition of $\Phi$ , this translates to $f ( x ) = f ( y )$ , for any $f \in \mathcal { F } _ { s c a l }$ . Therefore $( x , y ) \in$ $\rho \left( \mathcal { F } _ { s c a l } \right)$ and so there exists $g \in G$ such that $( g \cdot { \dot { x } } , y ) \in \rho \left( { \mathcal { F } } \right)$ . Plugging this into (20) and using the equivariance of $f$ , + +$$ +\forall f \in { \mathcal { F } } , \varphi ( f ( x ) ) = \psi ( g \cdot f ( x ) ) . +$$ + +As $G$ acts continuously on $Y$ , both $\varphi$ and $z \mapsto \psi ( g \cdot z )$ are continuous and, as a consequence of the equality above, coincide on $\overline { { \mathcal { F } ( \boldsymbol { x } ) } }$ . But we assumed that $h ( x ) \in { \overline { { \mathcal { F } ( x ) } } }$ and therefore the equality also holds for $h$ , i.e. $\varphi ( h ( x ) ) = \psi ( g \cdot h ( x ) )$ . + +Finally, recalls that, by assumption, $\rho \left( \mathcal { F } \right) \subseteq \rho \left( h \right)$ . Therefore $( g \cdot x , y ) \in \rho \left( \mathcal { F } \right)$ implies that $h ( g \cdot x ) = h ( y )$ and, combined with the result above, $\varphi ( h ( x ) ) = \psi ( h ( y ) )$ . + +• We verify that, for $x \in X$ , $\varphi \in \Phi$ , $\varphi ( h ( x ) )$ belongs to $\scriptstyle { \overline { { { \tilde { \mathcal { F } } } ( x ) } } }$ . Indeed, recall that $h ( x ) \in$ $\overline { { \mathcal { F } ( \boldsymbol { x } ) } }$ . Therefore, as $\varphi$ is continuous, $\varphi ( h ( x ) )$ is in $\overline { { \varphi ( \mathcal { F } ( x ) ) } }$ which is included in $\overline { { { \tilde { \mathcal { F } } } ( \boldsymbol { x } ) } }$ . + +This shows that ${ \tilde { h } } : ( x , \varphi ) \mapsto \varphi ( h ( x ) )$ is in $\overrightharpoon { \tilde { \mathcal { F } } }$ . Consequently, for any $\epsilon > 0$ , there exists $f \in { \mathcal { F } }$ such that, + +$$ +\forall x \in X , \forall \varphi \in \Phi , | \varphi ( h ( x ) ) - \varphi ( f ( x ) ) | \leq \epsilon . +$$ + +If $Y = \mathbb { R } ^ { p }$ is endowed with the infinity norms on coordinates, by definition of $\Phi$ , this means that, + +$$ +\forall x \in X , \ \| h ( x ) - f ( x ) \| \leq \epsilon . +$$ + +Remark 25. In the particular case of $G \subseteq S _ { n }$ being a group of permutations acting on $\mathbb { R } ^ { p }$ by, for $g \in G , x \in \mathbb { R } ^ { p }$ , + +$$ +\forall i \in \{ 1 , . . . , p \} , ( g \cdot x ) _ { i } = x _ { g ^ { - 1 } ( i ) } , +$$ + +the functions of $\tilde { \mathcal { F } }$ are indeed invariant, as shown by Maehara & Hoang (2019). For this, a left action on $\Phi$ is defined by, for $g \in G , \varphi \in \Phi$ , + +$$ +\forall x \in \mathbb { R } ^ { p } , ( g \cdot \varphi ) ( x ) = \varphi ( g ^ { - 1 } \cdot x ) . +$$ + +In other words, the action of $g$ on the linear form associated to the $i ^ { t h }$ coordinate yields the linear form associated to the $g ( i ) ^ { t h }$ coordinate. One can now check that the functions $\tilde { \mathcal { F } }$ are invariant. + +# D.6 CHARACTERIZING THE SUBALGEBRAS OF $\mathbb { R } ^ { p }$ + +Before moving to our general result, we need to study the structure of the subalgebras of $\mathbb { R } ^ { p }$ . For this, we will use the following simple lemma. + +In the following lemma, $\mathbb { R } [ X _ { 1 } , \ldots , X _ { p } ]$ denotes the set of multivariate polynomials with $p$ indeterminates (and real coefficients). + +Lemma 26. Let $C \subseteq \mathbb { R } ^ { p }$ be a finite subset of $\mathbb { R } ^ { p }$ . There exists $P \in \mathbb { R } [ X _ { 1 } , \ldots , X _ { p } ]$ such that $P _ { \mid C }$ the restriction of $P$ to $C$ , is an injective map. + +Proof. Let $x ^ { 1 } , \ldots , x ^ { m } \in \mathbb { R } ^ { p }$ be distinct vectors such that $\{ x ^ { 1 } , \ldots , x ^ { m } \} = C$ . Similarly to Lagrange polynomials, define, + +$$ +P ( X _ { 1 } , \dots , X _ { p } ) = \sum _ { i = 1 } ^ { m } i \prod _ { j \neq i } { \frac { \sum _ { l = 1 } ^ { p } ( X _ { l } - x _ { l } ^ { j } ) ^ { 2 } } { \| x ^ { i } - x ^ { j } \| _ { 2 } ^ { 2 } } } , +$$ + +which is a well-defined multivariate polynomial. Note that, seeing $X = ( X _ { 1 } , \ldots , X _ { p } )$ as a vector in $\mathbb { R } ^ { p }$ , it can also be written as, + +$$ +P ( X _ { 1 } , \dots , X _ { p } ) = \sum _ { i = 1 } ^ { m } i \prod _ { j \neq i } { \frac { \| X - x ^ { j } \| _ { 2 } ^ { 2 } } { \| x ^ { i } - x ^ { j } \| _ { 2 } ^ { 2 } } } . +$$ + +By construction, $P ( x ^ { i } ) = i$ and therefore $P$ is an injective map on $C$ + +Lemma 27. For a subalgebra $\mathcal { A }$ of $\mathbb { R } ^ { p }$ , we define: + +$$ +\begin{array} { l } { J = \left\{ j \in \left\{ 1 , \dots , p \right\} : \forall x \in A , x _ { j } = 0 \right\} } \\ { I = \left\{ ( i , j ) \in ( J ^ { C } ) ^ { 2 } : \forall x \in A , x _ { i } = x _ { j } \right\} . } \end{array} +$$ + +Then, we have : + +$$ +\begin{array} { r } { \mathcal { A } = \{ x \in \mathbb { R } ^ { p } : \forall ( i , j ) \in I , x _ { i } = x _ { j } , \forall j \in J , x _ { j } = 0 \} . } \end{array} +$$ + +Proof. Before proving the general case, we focus on the situation where $\mathcal { A }$ will turn out to be the whole $\mathbb { R } ^ { p }$ . + +Assume that the two following conditions holds, + +$$ +\begin{array} { l } { \forall i \neq j , \exists x \in A , x _ { i } \neq x _ { j } } \\ { \forall i , \exists x \in A , x _ { i } \neq 0 } \end{array} +$$ + +Our goal is to show that, under these additional assumptions, $\ b { A } = \mathbb { R } ^ { p }$ . We divide the proof in three parts, first we show that $\mathbf { 1 } \in { \mathcal { A } }$ using (24), giving us that $\mathcal { A }$ is closed under polynomials, then that there is $x \in { \mathcal { A } }$ with pairwise distinct coordinates thanks to (23) and finally that this implies that $\mathcal { A }$ is the whole space. + +Note that if $p = 1$ , (24) and the linear space property of $\mathcal { A }$ immediately give the result. + +• Here we prove that (24) implies that $\mathbf { 1 } \in { \mathcal { A } }$ . + +– First, we construct by induction $x \in { \mathcal { A } }$ such that $x _ { i } \neq 0$ for any index $i$ . More precisely, our induction hypothesis at step $j \in \{ 1 , \ldots , p \}$ is, + +$$ +\exists x \in A , \forall 1 \leq i \leq j , x _ { i } \neq 0 . +$$ + +By (24), this holds for $j = 1$ . + +Now, assume that it holds at $j - 1$ for some $p \ge j \ge 2$ and take $x \in { \mathcal { A } }$ such that $x _ { i } \neq 0$ for any $1 \leq i \leq j - 1$ and $y \in { \mathcal { A } }$ such that $j _ { j } \neq 0$ by (24). + +By definition of $x$ , the set, + +$$ +\left\{ \lambda \in \mathbb { R } : \exists 1 \leq i \leq j - 1 , \lambda x _ { i } + y _ { i } = 0 \right\} , +$$ + +is finite. Thus, there exists $\lambda \in \mathbb { R }$ such that $\lambda x _ { i } + y _ { i } \neq 0$ for any $1 \leq i \leq j - 1$ and for $i = j$ too as $x _ { j } = 0$ and $y _ { j } \neq 0$ . As $\mathcal { A }$ is a subalgebra, $\lambda x + y \in A$ and this concludes the induction step. + +– Let $x \in { \mathcal { A } }$ be the vector constructed, i.e. such that $x _ { i } \neq 0$ for every index $i$ . We prove that $\mathbf { 1 } \in { \mathcal { A } }$ by constructing 1 from $x$ . + +Indeed, using Lagrange interpolation, take $P \in \mathbb { R } [ X ]$ such that $\textstyle P ( x _ { i } ) = { \frac { 1 } { x _ { i } } }$ for every $i$ (Note that this noes not matter if some $x _ { i }$ are equal, as the $\frac { 1 } { x _ { i } }$ would also be the same.) + +Finally, as $\mathcal { A }$ is a subalgebra, $x P ( x )$ , which is to be understood coordinate-wise, is in $\mathcal { A }$ and so is $\mathbf { 1 } = x P ( x )$ . + +• We show that the previous point and (23) imply that there exists a vector in $\mathcal { A }$ with pairwise distinct coordinates, i.e. that there exists $x \in { \mathcal { A } }$ such that, for any $i \neq j$ , $x _ { i } \neq x _ { j }$ . Using (23), for any $i < j$ , there exists $x ^ { i j } \in { \mathcal { A } }$ such that $x _ { i } ^ { i j } \neq x _ { j } ^ { i j }$ . We wish to combine the family $( x ^ { i j } ) _ { i < j }$ into a single vector. + +For this we use Lem. 26. Seeing each collection $( x _ { k } ^ { i j } ) _ { i < j }$ as vector of $\mathbb { R } ^ { p ( p - 1 ) / 2 }$ , we define $C = \{ ( x _ { k } ^ { i j } ) _ { i < j } : 1 \leq k \leq p \}$ , which is a finite subset (of cardinal $p$ ) of $\mathbb { R } ^ { p ( p - 1 ) / 2 }$ . By Lem. 26, there exists $P \in \mathbb { R } [ \dot { X } _ { 1 } , \dots , X _ { p ( p - 1 ) / 2 } ]$ such that $P$ is an injective map on $C$ . + +As $\mathcal { A }$ is a subalgebra and $\mathbf { 1 } \in { \mathcal { A } }$ , the vector, + +$$ +P \left( ( x ^ { i j } ) _ { i < j } \right) = \left( \begin{array} { c } { { P \left( ( x _ { 1 } ^ { i j } ) _ { i < j } \right) } } \\ { { \vdots } } \\ { { P \left( ( x _ { p } ^ { i j } ) _ { i < j } \right) } } \end{array} \right) , +$$ + +is in $\mathcal { A }$ too. + +We now check that this vector has pairwise distinct coordinates. Let $l < k$ , then $x _ { l } ^ { l k } \neq x _ { k } ^ { l k }$ and therefore $( x _ { l } ^ { i j } ) _ { i < j } \neq ( x _ { k } ^ { i j } ) _ { i < j }$ . By construction of $P$ , $P \left( ( x _ { l } ^ { i j } ) _ { i < j } \right) \neq P \left( ( { \bar { x } } _ { k } ^ { i j } ) _ { i < j } \right)$ , i.e. $P \left( ( x ^ { i j } ) _ { i < j } \right) _ { l } \neq P \left( ( x ^ { i j } ) _ { i < j } \right) _ { k }$ . Thus, $P \left( ( x ^ { i j } ) _ { i < j } \right) \in { \mathcal { A } }$ has pairwise distinct coordinates as required. + +• Finally, we show that $\ b { \mathcal { A } } = \mathbb { R } ^ { p }$ . This is a direct consequence of the point above and of Lagrange interpolation. Indeed, take any $y \in \mathbb { R } ^ { p }$ and denote by $x \in { \mathcal { A } }$ the vector that we just constructed with pairwise distinct coordinates. Therefore, by Lagrange interpolation, there exists $P \in \mathbb { R } [ X ]$ such that $P ( x _ { i } ) = y _ { i }$ for every $i \in \{ 1 , \ldots , p \}$ . As $\mathcal { A }$ is a sublagebra and $\mathbf { 1 } \in { \mathcal { A } }$ , $P ( x ) \in { \mathcal { A } }$ and hence $y \in { \mathcal { A } }$ . + +Finally, we return to the general case. We introduce the set of indexes of $I$ and $J$ which appear in the result and use them to reduce the situation to the previous case. Define, + +$$ +\begin{array} { l } { J = \left\{ j \in \left\{ 1 , \dots , p \right\} : \forall x \in A , x _ { j } = 0 \right\} } \\ { I = \left\{ ( i , j ) \in ( J ^ { C } ) ^ { 2 } : \forall x \in A , x _ { i } = x _ { j } \right\} . } \end{array} +$$ + +and denote by $\mathcal { A } ^ { \prime } = \{ x \in \mathbb { R } ^ { p } : \forall ( i , j ) \in I$ , $x _ { i } = x _ { j }$ , $\forall j \in J$ , $x _ { j } = 0 \}$ , which is also a subalgebra. +By definition, it holds that ${ \mathcal { A } } \subseteq { \mathcal { A } } ^ { \prime }$ . + +By construction, $I$ is an equivalence relation on $J ^ { C }$ and denote by $J ^ { C } / I$ its equivalence classes. Let $p ^ { \prime } = | J ^ { C } / I |$ and choose $i _ { 1 } , \ldots , i _ { p ^ { \prime } }$ representatives of the equivalence classes. Consider the map, + +$$ +\varphi : \left\{ \begin{array} { l l } { \mathbb { R } ^ { p } \longrightarrow \mathbb { R } ^ { p ^ { \prime } } } \\ { x \longmapsto \left( x _ { i _ { 1 } } , \ldots , x _ { i _ { p ^ { \prime } } } \right) . } \end{array} \right. +$$ + +$\varphi$ is an algebra homomoprhism so that $\varphi ( A )$ is a subalgebra of $\mathbb { R } ^ { p ^ { \prime } }$ . But, by construction of $I$ and $J$ , $\varphi ( A )$ satisfies (23) and (24). Whence, by our result in this particular case, $\varphi ( \mathcal { A } ) = \mathbb { R } ^ { p ^ { \prime } }$ . + +However, ${ \mathcal { A } } \subseteq { \mathcal { A } } ^ { \prime }$ implies that $\varphi ( \mathcal { A } ) \subseteq \varphi ( \mathcal { A } ^ { \prime } ) \subseteq \mathbb { R } ^ { p ^ { \prime } }$ . Therefore, $\mathbb { R } ^ { p ^ { \prime } } = \varphi ( \mathcal { A } ) = \varphi ( \mathcal { A } ^ { \prime } )$ . But, by construction of $\varphi$ , $\varphi$ is actually an injective map on $\mathcal { A } ^ { \prime }$ . Therefore, we deduce from $\varphi ( \mathcal { A } ) = \varphi ( \mathcal { A } ^ { \prime } )$ that $A = A ^ { \prime }$ , concluding the proof. □ + +# D.7 PROOF OF THE MAIN EQUIVARIANT APPROXIMATION THEOREM + +We can now fully exploit the structure of subalgebra of $\mathcal { F }$ thanks to the results above, and in particular relax the second assumption of Prop. 24 to give our main theorem 22. We first prove the following lemma. + +Lemma 28. Under the assumptions of Thm. 22, for any $H \subseteq G$ subgroup, for any $x , y \in X$ + +$$ +\forall f \in { \mathcal { F } } , \ \exists g \in H , \ f ( g \cdot x ) = f ( y ) \iff \exists g \in H , \forall f \in { \mathcal { F } } , \ f ( g \cdot x ) = f ( y ) \ . +$$ + +In particular, + +$$ +( x , y ) \in \rho \left( \pi \circ { \mathcal { F } } \right) \iff \exists g \in G , ( g \cdot x , y ) \in \rho \left( { \mathcal { F } } \right) . +$$ + +Proof. The reverse implication is immediate so we focus on the direct one and prove its contraposition, i.e., + +$$ +\forall g \in H , \exists f \in { \mathcal { F } } , f ( g \cdot x ) \neq f ( y ) \implies \exists f \in { \mathcal { F } } , \forall g \in H , f ( g \cdot x ) \neq f ( y ) . +$$ + +rove this, we take advantage of $\mathcal { F }$ being a subalgebra and $H$ being finite. Let $H = \{ g _ { 1 } , \dots , g _ { h } \}$ define, + +$$ +{ \mathcal { A } } = \left\{ ( f ( g _ { 1 } \cdot x ) , \ldots , f ( g _ { h } \cdot x ) , f ( y ) ) : f \in { \mathcal { F } } \right\} . +$$ + +As $\mathcal { F }$ is a subalgebra of ${ \mathcal { C } } ( X , \mathbb { R } ^ { p } )$ , $\mathcal { A }$ is a subalgebra of $\mathbb { R } ^ { p ^ { \prime } }$ with $p ^ { \prime } = p ( h + 1 )$ . By Lem. 27, + +$$ +\begin{array} { r } { A = \{ z \in \mathbb { R } ^ { p ^ { \prime } } : \forall ( i , j ) \in I , z _ { i } = z _ { j } , \forall j \in J , z _ { j } = 0 \} . } \end{array} +$$ + +where $I$ and $J$ can be chosen to be,1 + +$$ +\begin{array} { r l } & { J = \left\{ j \in \{ 1 , \dots , p \} : \forall z \in A , z _ { j } = 0 \right\} } \\ & { I = \left\{ ( i , j ) \in \{ 1 , \dots , p \} ^ { 2 } : \forall z \in A , z _ { i } = z _ { j } \right\} . } \end{array} +$$ + +As $I$ is an equivalence relation, an element of $\mathcal { A }$ is uniquely defined by its coordinates on equivalence classes of $I$ . Therefore, one can choose $z \in { \mathcal { A } }$ such that $z _ { i } = z _ { j } \iff ( i , j ) \in I$ . By definition of $\mathcal { A }$ , there exists $f ^ { * } \in { \mathcal { F } }$ such that $( f ^ { * } ( g _ { 1 } \cdot x ) , \ldots , f ^ { * } ( g _ { h } \cdot x ) , ^ { ^ { \scriptstyle * } } f ^ { * } ( y ) ) = z$ . We now check that $f ^ { * }$ is indeed appropriate. Take $l \in \{ 1 , \ldots , h \}$ , we want to show that $f ^ { * } ( g _ { l } \cdot x ) \neq f ^ { * } ( y )$ . By assumption, there exists $f \in { \mathcal { F } }$ such that $f ( g _ { l } \cdot x ) \neq f ( y )$ , i.e. there exists $i \in \{ 1 , \ldots , p \}$ such that $f ( g _ { l } \cdot \bar { x } ) _ { i } \neq f ( y ) _ { i }$ . Therefore, $( ( l - 1 ) p + i , h p + i )$ cannot be in $I$ so that $z _ { ( l - 1 ) p + i } \neq z _ { h p + 1 }$ , i.e. $f ^ { * } ( g _ { l } \cdot x ) _ { i } \neq f ^ { * } ( y ) _ { i }$ . □ + +We now prove our main abstract theorem. + +Theorem 22. Let $X$ be a compact space, $Y = \mathbb { R } ^ { p }$ for some $p , G$ be a finite group acting (continuously) on $X$ and $Y$ and ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { E } ( X , Y )$ a (non-empty) set of equivariant functions. + +Denote by $\pi : Y \longrightarrow Y / G$ the canonical projection on the quotient space $Y / G$ . Consider the following assumptions, + +1. $\mathcal { F }$ is a sub-algebra of ${ \mathcal { C } } ( X , Y )$ and the constant function 1 is in $\mathcal { F }$ . + +2. The set of functions $\mathcal { F } _ { s c a l } \subseteq \mathcal { C } ( X , \mathbb { R } )$ defined by, + +$$ +\mathcal { F } _ { s c a l } = \{ f \in \mathcal { C } ( X , \mathbb { R } ) : f \mathbf { 1 } \in \mathcal { F } \} +$$ + +satisfy, + +$$ +\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \pi \circ \mathcal { F } \right) . +$$ + +Then the closure of $\mathcal { F }$ (for the topology of uniform convergence) is, + +$$ +\begin{array} { r } { \mathcal { F } = \{ f \in \mathcal { C } _ { E } ( X , Y ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) , \forall x \in X , f ( x ) \in \mathcal { F } ( x ) \} , } \end{array} +$$ + +where ${ \mathcal { F } } ( x ) = \{ f ( x ) , f \in { \mathcal { F } } \}$ . Moreover, $i f I ( x ) = \{ ( i , j ) \in [ p ] ^ { 2 } : \forall y \in \mathcal { F } ( x ) , y _ { i } = y _ { j } \} ,$ , then we have: + +$$ +{ \mathcal { F } } ( x ) = \left\{ y \in \mathbb { R } ^ { p } : \forall ( i , j ) \in I ( x ) , y _ { i } = y _ { j } \right\} . +$$ + +Proof of Thm. 22. By Lem. 28, the second assumption of Prop. 24 is also satisfied. To get the conclusion of Thm. 22, note that $\mathcal { F } ( x )$ is now a linear subspace of a finite-dimensional vector space and therefore it is closed. Thus, $\overline { { \mathcal { F } ( \boldsymbol { x } ) } } = \mathcal { F } ( \boldsymbol { x } )$ which is a subalgebra. Applying Lem. 27 to $\mathcal { F } ( x )$ and noting that, necessarily $J = \emptyset$ as $\mathbf { 1 } \in { \mathcal { F } } ( x )$ by assumption, gives the result of Thm. 22. □ + +# D.8 PRACTICAL REDUCTIONS + +Though the results we proved above were formulated using classic hypotheses, such as requiring $\mathcal { F }$ to be a subalgebra, we can give much more compact versions for our setting. We also reduce the assumption that $\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \pi \circ \mathcal { F } \right)$ to a more practical one. + +We start with the invariant case. + +Corollary 29. Let $X$ be a compact space, $Y = \mathbb { F } = \mathbb { R } ^ { p }$ be some finite-dimensional vector space, $G$ be a finite group acting (continuously) on $X$ and ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { I } ( X , Y )$ a (non-empty) set of invariant functions. + +Assume that, for any $h \in \mathcal { C } ( \mathbb { F } ^ { 2 } , \mathbb { F } )$ and $f , g \in { \mathcal { F } }$ , + +$$ +x \mapsto h ( f ( x ) , g ( x ) ) \in \mathcal { F } . +$$ + +Then the closure of $\mathcal { F }$ is, + +$$ +{ \overline { { \mathcal { F } } } } = \{ f \in { \mathcal { C } } _ { I } ( X , Y ) : \rho \left( { \mathcal { F } } \right) \subseteq \rho \left( f \right) \} ~ . +$$ + +Proof. We wish to apply Thm. 22 but for this we need $G$ to act on $Y$ . Define a (trivial) action of $G$ on $Y$ by, + +$$ +\forall g \in G , \forall y \in Y , g \cdot y = y . +$$ + +With this action on $Y$ , $\mathcal { C } _ { E } ( X , Y ) = \mathcal { C } _ { I } ( X , Y )$ . Moreover, $Y / G = Y$ , $\pi : Y \to Y / G$ is the identity so that $\rho \left( \pi \circ { \mathcal { F } } \right) = \rho \left( { \mathcal { F } } \right)$ . + +Our assumption clearly ensure that $\mathcal { F }$ is indeed a subalgebra and contains the constant function $\nVdash$ + +All that is left to show to apply Thm. 22 is that the set of functions $\mathcal { F } _ { s c a l } \subseteq \mathcal { C } ( X , \mathbb { R } )$ defined by, + +$$ +\mathcal { F } _ { s c a l } = \{ f \in \mathcal { C } ( X , \mathbb { R } ) : f { \bf 1 } \in \mathcal { F } \} +$$ + +satisfies, + +$$ +\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \mathcal { F } \right) . +$$ + +Take $( x , y ) \notin \rho \left( { \mathcal { F } } \right)$ and we show that $( x , y ) \notin \rho \left( \mathcal { F } _ { s c a l } \right)$ . Indeed, by definition there exists $f \in { \mathcal { F } }$ , $i \in \{ 1 , \ldots , p \}$ such that $f ( x ) _ { i } \neq f ( y ) _ { i }$ . Let $l \in \mathcal { C } ( \mathbb { F } , \mathbb { R } )$ defined by $l ( z ) = z _ { i }$ and $h \in { \mathcal { C } } ( \mathbb { F } , \mathbb { F } )$ defined by, for $z \in \mathbb { F }$ , $h ( z ) = ( l ( z ) , \ldots , l ( z ) ) = l ( z ) \mathcal { k }$ . Then, by assumption $h \circ f \in { \mathcal { F } }$ . But $h \circ f$ is $( l \circ f ) \mathbf { 1 }$ with $l \circ f \in { \mathcal { C } } ( X , \mathbb { R } )$ , so that, by definition, $l \circ f \in \mathcal { F } _ { s c a l }$ . Moreover, $l \circ f ( x ) \neq l \circ f ( y )$ . Therefore, $( x , y ) \notin \rho \left( \mathcal { F } _ { s c a l } \right)$ . Therefore, we can apply Thm. 22. We get that, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } _ { E } ( X , Y ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) , \forall x \in X , f ( x ) \in \mathcal { F } ( x ) \right\} , } \end{array} +$$ + +and + +$$ +\mathcal { F } ( x ) = \left\{ y \in \mathbb { F } ^ { p } : \forall ( i , j ) \in I ( x ) , y _ { i } = y _ { j } \right\} , +$$ + +with $I ( x )$ given by, + +$$ +I ( x ) = \{ ( i , j ) \in \{ 1 , \ldots , p \} ^ { 2 } : \forall y \in { \mathcal { F } } ( x ) , y _ { i } = y _ { j } \} . +$$ + +To conclude the proof, we now show that $I ( x ) = \{ ( i , i ) : i \in \{ 1 , . . . , p \} \}$ , which will imply that $\mathcal { F } ( x ) = \mathbb { R } ^ { p } = Y$ . + +Indeed, the constant function $z \mapsto ( 1 , 2 , . . . , p )$ is in $\mathcal { F }$ by assumption. Therefore, $I ( x )$ is reduced to $\{ ( i , i ) : i \in \{ 1 , . . . , p \} \}$ . + +In our previous version of our approximation result for node embedding, we did not allow features in the output as it would have made the statement and the proof a bit convoluted. With this new assumption, this is much easier. + +Corollary 30. Let $X$ be a compact space, $Y = \mathbb { F } ^ { n }$ , with $\mathbb { F } = \mathbb { R } ^ { p }$ and $G = S _ { n }$ the permutation group, acting (continuously) on $X$ and acting on $\mathbb { F } ^ { n }$ by, for $\sigma \in S _ { n }$ , $x \in \mathbb { F } ^ { n }$ , + +$$ +\forall i \in \{ 1 , \ldots , p \} , ( \sigma \cdot x ) _ { i } = x _ { \sigma ^ { - 1 } ( i ) } , +$$ + +Let ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { E } ( X , \mathbb { F } ^ { n } )$ be a (non-empty) set of equivariant functions. + +Consider the following assumptions, + +1. For any $h \in { \mathcal { C } } ( \mathbb { F } ^ { 2 } , \mathbb { F } ) , f , g \in { \mathcal { F } } ,$ , + +$$ +x \mapsto \left( h ( f ( x ) _ { 1 } , g ( x ) _ { 1 } ) , \ldots , h ( f ( x ) _ { n } , g ( x ) _ { n } ) \right) \in { \mathcal { F } } . +$$ + +2. If $f \in { \mathcal { F } }$ , + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \in { \mathcal { F } } . +$$ + +Then the closure of $\mathcal { F }$ (for the topology of uniform convergence) is, + +$$ +\overline { { \mathcal { F } } } = \{ f \in \mathcal { C } _ { E } ( X , \mathbb { F } ^ { n } ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) \} \ . +$$ + +For this we need a handy lemma, whose proof relies on a result about multi-symmetric polynomials from Maron et al. (2019a). + +Lemma 31. Let $\mathbb { F } = \mathbb { R } ^ { p }$ be some finite-dimensional space. Take $x _ { 1 } , \ldots , x _ { n } , y _ { 1 } , \ldots , y _ { n } \in \mathbb { F }$ such that, for any $\sigma \in S _ { n }$ , $( x _ { 1 } , \ldots , x _ { n } ) \neq ( y _ { \sigma ( 1 ) } , \ldots , y _ { \sigma ( n ) } )$ . Then, there exists $h \in { \mathcal { C } } ( \mathbb { F } , \mathbb { R } )$ such that, + +$$ +\sum _ { i = 1 } ^ { n } h ( x _ { i } ) \neq \sum _ { i = 1 } ^ { n } h ( y _ { i } ) . +$$ + +Moreover, $h$ can be written as $h ( \boldsymbol { x } ) = h _ { 1 } ( x _ { 1 } ) h _ { 2 } ( x _ { 2 } ) \ldots h _ { p } ( x _ { p } )$ for any $x \in \mathbb { F }$ with $h _ { 1 } , \ldots , h _ { p } \in$ ${ \mathcal { C } } ( \mathbb { R } , \mathbb { R } )$ . + +Proof. By Maron et al. (2019a, Prop. 1), there exists $\alpha _ { 1 } , \dotsc , \alpha _ { p }$ non-negative integers such that $\textstyle \sum _ { i = 1 } ^ { n } x _ { i 1 } ^ { \alpha _ { 1 } } x _ { i 2 } ^ { \alpha _ { 2 } } \dotsc x _ { i p } ^ { \alpha _ { p } } \neq \textstyle \sum _ { i = 1 } ^ { n } y _ { i 1 } ^ { \alpha _ { 1 } } y _ { i 2 } ^ { \alpha _ { 2 } } \dotsc x _ { i p } ^ { \alpha _ { p } }$ . Taking $h : x \mapsto x _ { i 1 } ^ { \alpha _ { 1 } } x _ { i 2 } ^ { \alpha _ { 2 } } \ldots x _ { i p } ^ { \alpha _ { p } }$ yields the result. □ + +Proof of Cor. 30. We want to apply Thm. 22. With our first assumption, the first assumption of Thm. 22 is easily verified. + +We now focus on the second one. As in the statement of Thm. 22, define $\mathcal { F } _ { s c a l } \subseteq \mathcal { C } ( X , \mathbb { R } )$ by, + +$$ +\mathcal { F } _ { s c a l } = \{ f \in \mathcal { C } ( X , \mathbb { R } ) : f \mathbf { 1 } \in \mathcal { F } \} \ . +$$ + +We have to show that $\rho \left( \mathcal { F } _ { s c a l } \right) \subseteq \rho \left( \pi \circ \mathcal { F } \right)$ . For this, take $x , y \notin \rho \left( \pi \circ { \mathcal { F } } \right)$ . There exists $f \in { \mathcal { F } }$ such that for any $\sigma \in S _ { n }$ , $\sigma \cdot f ( x ) \neq f ( y )$ . We have to find $l \in \mathcal { F } _ { s c a l }$ such that $l ( x ) \neq l ( y )$ . In other words, from a function in $\mathbb { F } ^ { n }$ which discriminates between $x$ and $y$ we have to build a function in $\mathbb { R }$ . + +First, we exhibit a function which discriminates between $x$ and $y$ . Apply Lem. 31 to the vectors $f ( x )$ and $f ( y )$ : there exists $h _ { 0 } \in \mathcal { C } ( \mathbb { F } , \mathbb { R } )$ such that $\textstyle \sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( x _ { i } ) ) \neq \sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( y _ { i } ) )$ . + +To fit the assumptions, we build $h \in { \mathcal { C } } ( \mathbb { F } , \mathbb { F } )$ from $h _ { 0 }$ by $h : x \in \mathbb { F } \mapsto ( h _ { 0 } ( x ) , \ldots , h _ { 0 } ( x ) ) \in \mathbb { F }$ . $g \in \mathcal { C } ( \mathbb { F } , \mathbb { F } )$ $g ( z ) = ( z _ { 1 } , \dots , z _ { 1 } )$ r any . All $z \in \mathbb { F }$ and e to d $l \in \mathcal { C } ( X , \mathbb { R } )$ dt , for, i.e., $w \in X$ $\begin{array} { r } { l ( w ) = \sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( w _ { i } ) ) } \end{array}$ $l ( x ) \neq l ( y )$ $l \in \mathcal { F } _ { s c a l }$ $l \mathbf { 1 } \in \mathcal { F }$ . This is where the two assumptions we made come into play. Indeed, the first one implies that $h \circ f \in { \mathcal { F } }$ and the second gives + +$$ +z \mapsto \left( \sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) , \ldots , \sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) \right) \in { \mathcal { F } } . +$$ + +Finally, the first assumption ensure that, + +$$ +z \mapsto \left( g \left( \sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) \right) , \ldots , g \left( \sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) \right) \right) \in \mathcal { F } . +$$ + +But this last function is none other than $l \mathbf { 1 }$ , which shows that $l \in \mathcal { F } _ { s c a l }$ as required. + +We have successfully verified the hypothesis of Thm. 22. Therefore, the closure of $\mathcal { F }$ is, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } _ { E } ( X , \mathbb { F } ^ { n } ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) , \forall x \in X , f ( x ) \in \mathcal { F } ( x ) \right\} , } \end{array} +$$ + +with + +$$ +\mathcal { F } ( x ) = \{ y \in \mathbb { F } ^ { n } : \forall ( i , i ^ { \prime } , j , j ^ { \prime } ) \in I ( x ) , y _ { i , i ^ { \prime } } = y _ { j , j ^ { \prime } } \} , +$$ + +and + +$$ +I ( x ) = \{ ( i , i ^ { \prime } , j , j ^ { \prime } ) \in ( \{ 1 , \dots , n \} \times \{ 1 , \dots , p \} ) ^ { 2 } : \forall y \in \mathcal { F } ( x ) , y _ { i , i ^ { \prime } } = y _ { j , j ^ { \prime } } \} . +$$ + +To get the desired result, we need to get rid of the condition ${ \bf \nabla } ^ { \bullet } f ( x ) \in \mathcal { F } ( x ) ^ { \mathfrak { V } }$ in the description of $\scriptstyle { \overline { { \mathcal { F } } } }$ Fix $x \in X$ . + +First, we show that + +$$ +{ \mathcal { F } } ( x ) = \left\{ y \in \mathbb { F } ^ { n } : \forall ( i , j ) \in J ( x ) , y _ { i } = y _ { j } \right\} , +$$ + +with + +$$ +J ( x ) = \left\{ ( i , j ) \in \left\{ 1 , \dots , n \right\} ^ { 2 } : y _ { i } = y _ { j } \right\} , +$$ + +(note that the equalities here are not in $\mathbb { R }$ anymore but in $\mathbb { F }$ ). The direct inclusion ${ \bf \ " } \subseteq { \bf \ " }$ is immediate by construction of $J ( x )$ so we focus on the reverse direction. For this, we show that the 4-tuples $( i , i ^ { \prime } , j , j ^ { \prime } )$ of $I ( x )$ necessarily satisfy $i ^ { \prime } = j ^ { \prime }$ and $( i , j ) \in J ( x )$ . + +First note that, by the first assumption, the vector $\boldsymbol { y } ^ { 0 } \in \mathbb { F } ^ { n }$ such that $y _ { i } ^ { 0 } = ( 1 , 2 , \ldots , p )$ for $i \in$ $\{ 1 , \ldots , n \}$ is in $\mathcal { F } ( x )$ . Indeed, take the constant function always equal to $( 1 , 2 , \ldots , p )$ as $h$ . Now, consider a 4-tuple $( i , i ^ { \prime } , j , j ^ { \prime } )$ of $I ( x )$ and we show that, actually, $( i , j ) \in { \dot { J } } ( x )$ and $i ^ { \prime } = j ^ { \prime }$ . As $y ^ { 0 }$ in $\mathcal { F } ( x )$ , and $y _ { i , i ^ { \prime } } ^ { 0 } = \dot { i } ^ { \prime }$ , $y _ { j , j ^ { \prime } } ^ { 0 } = j ^ { \prime }$ , $( i , i ^ { \prime } , j , j ^ { \prime } ) \in I ( x )$ implies that $j ^ { \prime } = i ^ { \prime }$ . Consider, $k \in \{ 1 , \ldots , p \}$ . We show that, for any $y \in { \mathcal { F } } ( x )$ , $y _ { i , k } = y _ { j , k }$ . But such a $y$ can be written as $y = f ( x )$ for some $f \in { \mathcal { F } }$ . Consider, the function $h \in { \mathcal { C } } ( \mathbb { F } , \mathbb { F } )$ associated to the permutation $( i ^ { \prime } k )$ , defined by, + +$$ +z \mapsto \left( z _ { ( i ^ { \prime } k ) ( 1 ) } , \dots , z _ { ( i ^ { \prime } k ) ( n ) } \right) = \left( z _ { 1 } , \dots , z _ { i ^ { \prime } - 1 } , z _ { k } , z _ { i ^ { \prime } + 1 } , \dots , z _ { k - 1 } , z _ { i ^ { \prime } } , z _ { k + 1 } , \dots , z _ { p } \right) . +$$ + +By our first assumption, $z \mapsto ( h ( f ( z ) _ { 1 } ) , \ldots , h ( f ( z ) _ { n } ) ) \in \mathcal { F }$ so that $( h ( y _ { 1 } ) , \dots , h ( y _ { n } ) ) \in { \mathcal { F } } ( x )$ . In particular, as $( i , i ^ { \prime } , j , i ^ { \prime } ) \in I ( x ) , h ( y _ { i } ) _ { i ^ { \prime } } = h ( y _ { j } ) _ { i ^ { \prime } }$ , i.e. $y _ { i , k } = y _ { j , k }$ . Therefore, $( i , j ) \in J ( x )$ . + +Finally, we can conclude that ${ \mathcal { F } } ( x ) ~ \supset ~ \{ y ~ \in ~ \mathbb { F } ^ { n } ~ : ~ \forall ( i , j ) ~ \in ~ J ( x )$ , $y _ { i } ~ = ~ y _ { j } \}$ . Indeed, take $y \in \mathbb { F } ^ { \cdot } : \forall ( i , j ) \in J ( x )$ , $y _ { i } = y _ { j }$ . We show that all the constraints of $I ( x )$ are satisfied. Indeed, take $( i , i ^ { \prime } , j , j ^ { \prime } ) \in I ( x )$ . We have shown that $( i , j ) \in J ( x )$ and $i ^ { \prime } = j ^ { \prime }$ so that $y _ { i } = y _ { j }$ and in particular $y _ { i , i ^ { \prime } } = y _ { j , j ^ { \prime } }$ . Therefore, this finishes the proof of $\mathcal { F } ( x ) \supset \{ y \in \mathbb { F } ^ { n } : \forall ( i , j ) \in J ( \bar { x } ) , y$ $y _ { i } = y _ { j } \}$ . + +Thus, with + +$$ +\begin{array} { r l } & { { \mathcal F } ( x ) = \left\{ y \in \mathbb { F } ^ { n } : \forall ( i , j ) \in J ( x ) , y _ { i } = y _ { j } \right\} , } \\ & { } \\ & { J ( x ) = \left\{ ( i , j ) \in \{ 1 , \dots , n \} ^ { 2 } : y _ { i } = y _ { j } \right\} . } \end{array} +$$ + +We have proven so far, that, + +$$ +\begin{array} { r } { \overline { { \boldsymbol { \varepsilon } } } = \{ f \in \mathcal { C } _ { E } ( X , \mathbb { F } ^ { n } ) : \rho ( \mathcal { F } ) \subseteq \rho ( f ) , \forall x \in X , f ( x ) \in \mathcal { F } ( x ) \} \subseteq \{ f \in \mathcal { C } _ { E } ( X , \mathbb { F } ^ { n } ) : \rho ( \mathcal { F } ) \subseteq \rho ( f ) \} } \end{array} +$$ + +Take $h \in { \mathcal { C } } _ { E } ( X , \mathbb { F } ^ { n } )$ such that $\rho \left( \mathcal { F } \right) \subseteq \rho \left( h \right)$ and fix $x \in X$ . Our goal is to show that, for any $( i , j ) \in J ( x ) , h ( x ) _ { i } = h ( x ) _ { j }$ so that $h ( x ) \in { \mathcal { F } } ( x )$ . But, if $( i , j ) \in J ( x )$ , then for any $f \in { \mathcal { F } }$ , $f ( x ) _ { i } = f ( x ) _ { j }$ so that $( i j ) \cdot { \dot { f } } ( x ) = f ( x )$ , where $( i j )$ denotes the permutation which exchanges $i$ and $j$ . Moreover, as $( i j ) \in S _ { n }$ , by equivariance, this means that $f ( ( i \dot { \boldsymbol { j } } ) \cdot \boldsymbol { x } ) = f ( \boldsymbol { x } )$ for every $f \in { \mathcal { F } }$ and therefore that $( ( i j ) \cdot x , x ) \in \rho \left( { \mathcal { F } } \right)$ . By assumption, we infer that $( ( i j ) \cdot x , x ) \in \rho \left( h \right)$ too, i.e. that $h ( ( i j ) \cdot x ) = \ddot { h } ( \dot { x } )$ and so that $h ( \dot { x } ) _ { i } = h ( x ) _ { j }$ by equivariance, which concludes our proof. + +# D.9 REDUCTIONS FOR GNNS + +We now present a lemma which explains how to instantiate the two corollaries above in the case of GNNs by replacing continuous functions with MLPs. + +Lemma 32. Fix $X$ some compact space, $n \geq 1$ and $\mathbb { F }$ a finite-dimensional feature space. Let $\textstyle { \mathcal { F } } _ { 0 } \subseteq \bigcup _ { h = 1 } ^ { \infty } { \mathcal { C } } ( X , \mathbb { R } ^ { h } )$ be stable by concatenation and consider, + +$$ +{ \mathbf { \Psi } } = \{ x \mapsto ( m ( f ( x ) _ { 1 } ) , \dots , m ( f ( x ) _ { n } ) ) : f \in \mathcal { F } _ { 0 } \cap \mathcal { C } ( X , \mathbb { R } ^ { h } ) , m : \mathbb { R } ^ { h } \to \mathbb { F } M L P , h \geq 1 \} \subseteq \mathcal { C } ( X , \mathbb { F } ) . +$$ + +Then, i ${ \mathcal { \cdot } } { \mathcal { E } } ( { \mathcal { F } } ) \subseteq { \mathcal { C } } ( X , \mathbb { F } )$ is the set of functions obtained by replacing the MLP m in the definition of $\mathcal { F }$ by an arbitrary continuous function, $\mathcal { E } ( \mathcal { F } )$ satisfies, + +2. $\rho \left( \mathcal { F } \right) = \rho \left( \mathcal { E } ( \mathcal { F } ) \right)$ + +3. For any $h \in { \mathcal { C } } ( \mathbb { F } ^ { 2 } , \mathbb { F } ) , f , g \in { \mathcal { E } } ( { \mathcal { F } } )$ + +$$ +x \mapsto \left( h ( f ( x ) _ { 1 } , g ( x ) _ { 1 } ) , \ldots , h ( f ( x ) _ { n } , g ( x ) _ { n } ) \right) \in \mathcal { E } ( \mathcal { F } ) . +$$ + +4. If, for any $f \in { \mathcal { F } }$ , + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \in { \mathcal { F } } , +$$ + +then, for any $f \in { \mathcal { E } } ( { \mathcal { F } } )$ , + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \in { \mathcal { E } } ( { \mathcal { F } } ) . +$$ + +5. If $\mathcal { F }$ is equivariant w.r.t. the action described in Cor. $3 0 _ { ; }$ , so is $\mathcal { E } ( \mathcal { F } )$ . + +Proof. Define, + +$$ +\begin{array} { r } { \mathcal { E } ( \mathcal { F } ) = \left\{ x \mapsto ( m ( f ( x ) _ { 1 } ) , \ldots , m ( f ( x ) _ { n } ) ) : f \in \mathcal { F } _ { 0 } \cap \mathcal { C } ( X , \mathbb { R } ^ { h } ) , m \in \mathcal { C } ( \mathbb { R } ^ { h } , \mathbb { F } ) , h \geq 1 \right\} . } \end{array} +$$ + +As MLP are continuous and by the universality of MLP on a compact set (see Section C.1), + +$$ +{ \mathcal { F } } \subseteq { \mathcal { E } } ( { \mathcal { F } } ) \subseteq { \overline { { \mathcal { F } } } } . +$$ + +This already implies 1. and that $\rho \left( \overline { { \mathcal { F } } } \right) \subseteq \rho \left( \mathcal { E } ( \mathcal { F } ) \right) \subseteq \rho \left( \mathcal { F } \right)$ . Using Lem. 15 yields 2. + +We now show 3. Take $h \in \mathcal { C } ( \mathbb { F } ^ { 2 } , \mathbb { F } ) , f , g \in \mathcal { F } _ { 0 } , f \in \mathcal { C } ( X , \mathbb { R } ^ { h _ { f } } ) , g \in \mathcal { C } ( X , \mathbb { R } ^ { h _ { g } } )$ and $m \in \mathcal { C } ( \mathbb { R } ^ { h _ { f } } , \mathbb { F } )$ $l \in \mathcal { C } ( \mathbb { R } ^ { h _ { g } } , \mathbb { F } )$ . All we have to show is that, + +$$ +x \mapsto \left( h ( m ( f ( x ) _ { 1 } ) , l ( g ( x ) _ { 1 } ) ) , \ldots , h ( m ( f ( x ) _ { n } ) , l ( g ( x ) _ { n } ) ) \right) \in \mathcal { E } ( \mathcal { F } ) . +$$ + +But as $\mathcal { F } _ { 0 }$ is stable by concatenation, $x \ \mapsto \ ( f ( x ) , g ( x ) ) \ \in \ \mathbb { R } ^ { h _ { f } + h _ { g } } .$ is still in $\mathcal { F } _ { 0 }$ . Moreover, $y \in \mathbb { R } ^ { h _ { f } + h _ { g } } \mapsto h ( m ( \dot { y } _ { 1 } , \dots , y _ { h _ { f } } ) , l ( y _ { h _ { f } + 1 } , \dots , y _ { h _ { f } + h _ { g } } )$ is also in $\mathcal { C } ( \mathbb { R } ^ { h _ { f } + h _ { g } } , \mathbb { F } )$ which shows that the map above is indeed in $\mathcal { E } ( \mathcal { F } )$ . The last two points are immediate consequences of the definition of $\mathcal { E } ( \mathcal { F } )$ . □ + +Thm. 5 and Thm. 6 are now obtained by combining Lem. 32 with Cor. 29 and Cor. 30. + +# E PROOFS FOR EXPRESSIVENESS OF GNNS + +Note that in the Theorem 6, the additional stability assumption is “almost” necessary to obtain the result. Indeed, if the result holds, i.e., + +$$ +\overline { { \mathcal { F } } } = \{ f \in \mathcal { C } _ { E } ( X , \mathbb { F } ^ { n } ) : \rho \left( \mathcal { F } _ { 0 } \right) \subseteq \rho \left( f \right) \} , +$$ + +then one can show, that, if $f \in { \mathcal { F } }$ , then + +$$ +{ \tilde { f } } : x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \in { \overline { { \mathcal { F } } } } . +$$ + +Indeed, $f \in { \mathcal { F } }$ so that it has a weaker discriminating power than $\mathcal { F }$ , so that $\rho \left( \mathcal { F } \right) = \rho \left( \mathcal { F } _ { 0 } \right) \subseteq \rho \left( f \right)$ . +But, by construction of $\tilde { f } , \rho \left( f \right) \subseteq \rho \left( \tilde { f } \right)$ so that $\rho \left( \mathcal { F } _ { 0 } \right) \subseteq \rho \left( \tilde { f } \right)$ . As $\tilde { f }$ is also in $\mathcal { C } _ { E } ( X , \mathbb { F } ^ { n } )$ , $\tilde { f } \in \overline { { \mathcal { F } } }$ . + +# E.1 EXPRESSIVITY OF GNN LAYERS + +Lemma 33. Fix $\mathbb { F } _ { 0 } , \mathbb { F } _ { 1 }$ (non-trivial) finite dimensionnal vector spaces. Consider the action of $G = S _ { n }$ on $\mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k }$ defined by, + +$$ +\forall \sigma \in { \mathcal { S } } _ { n } , \forall x ^ { 0 } \in \mathbb { F } _ { 0 } , \forall x \in \mathbb { F } ^ { n \times k } , \sigma \cdot ( x ^ { 0 } , x ) = ( x ^ { 0 } , x _ { \sigma ^ { - 1 } ( 1 ) } , \dots , x _ { \sigma ^ { - 1 } ( n ) } ) +$$ + +Let $K \subseteq \mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k }$ be a compact set. Then, the set of functions from $K \subseteq \mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k } t o$ $\mathbb { F } _ { 1 }$ of the form, + +$$ +( x ^ { 0 } , x ) \longmapsto f _ { 0 } \left( x ^ { 0 } , \sum _ { j = 1 } ^ { n } \prod _ { w = 1 } ^ { k } f _ { w } ( x _ { j , w } ) \right) +$$ + +where $f _ { 0 } : \mathbb { F } _ { 0 } \times \mathbb { R } ^ { h } \to \mathbb { F } _ { 1 }$ , $f _ { j } : \mathbb { F } _ { 0 } \mathbb { R } ^ { h }$ , $j = 1 , \dots , k$ are multi-linear perceptrons and $h \geq 1$ , is dense in $\mathcal { C } _ { I } ( K , \mathbb { F } _ { 1 } )$ . + +Proof. Denote by $\mathcal { F } \subseteq \mathcal { C } _ { I } ( K , \mathbb { F } _ { 1 } )$ the set of such functions. To prove that $\overline { { \mathcal { F } } } = \mathcal { C } _ { I } ( K , \mathbb { F } _ { 1 } )$ , we first apply Thm. 5. + +We get, that, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } _ { I } ( \mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k } , \mathbb { F } _ { 1 } ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) \right\} . } \end{array} +$$ + +We now characterize $\rho \left( \mathcal { F } \right)$ . Actually, it is equal to $\rho \left( i n v \right) = \{ ( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \in \left( \mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k } \right) ^ { 2 }$ : $\exists \sigma \in S _ { n }$ , $( x ^ { 0 } , x ) = \sigma \cdot ( y ^ { 0 } , y ) \}$ . As the functions of $\mathcal { F }$ are invariant, $\rho \left( i n v \right) \subseteq \rho \left( { \mathcal { F } } \right)$ . We now ke . I $( x _ { \hphantom { } , \hphantom { } } ^ { 0 } , x ) , ( y ^ { 0 } , y ) \in \mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k }$ at there does notMLP such that $\sigma \in S _ { n }$ uch thatso that $( x ^ { 0 } , x ) = \sigma \cdot ( y ^ { 0 } , y )$ $x ^ { 0 } \neq y ^ { 0 }$ $f _ { 0 } : \mathbb { F } _ { 0 } \to \mathbb { F } _ { 1 }$ $f ( x ^ { 0 } ) \neq f ( y ^ { 0 } )$ $( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \notin \rho \left( { \mathcal F } \right)$ . + +Otherwise, there does not exists $\sigma ~ \in ~ S _ { n }$ such that $( x _ { \sigma ^ { - 1 } ( 1 ) } , \ldots , x _ { \sigma ^ { - 1 } ( n ) } ) \ = \ ( y _ { 1 } , \ldots , y _ { n } )$ by definition of the action of $G \ = \ S _ { n }$ . Now, apply Lem. 31 with $\mathbb { F } \mathbb { F } _ { 0 } ^ { k }$ , the universality of MLP and use the decomposition given to get $f _ { j } : \mathbb { F } _ { 0 } \ \to \ \mathbb { R } ^ { h }$ , $j ~ = ~ 1 , \ldots , k$ such that $\begin{array} { r } { \sum _ { j = 1 } ^ { n } \prod _ { w = 1 } ^ { k } f _ { w } ( x _ { j , w } ) \neq \sum _ { j = 1 } ^ { n } \prod _ { w = 1 } ^ { k } f _ { w } ( y _ { j , w } ) } \end{array}$ . Choosing an appropriate MLP $f _ { 0 } : \mathbb { R } ^ { h } \mathbb { F } _ { 1 }$ yields that $( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \notin \rho \left( { \mathcal F } \right)$ . + +Hence, we have shown that, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } _ { I } ( \mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k } , \mathbb { F } _ { 1 } ) : \rho \left( i n v \right) \subseteq \rho \left( f \right) \right\} = \mathcal { C } _ { I } ( \mathbb { F } _ { 0 } \times \mathbb { F } _ { 0 } ^ { n \times k } , \mathbb { F } _ { 1 } ) . } \end{array} +$$ + +# E.2 APPROXIMATION THEOREMS FOR GNNS + +We now have all the tools to finally prove our main result. + +Theorem 34. Let $K _ { d i s c r } \subseteq \mathcal G _ { n } \times \mathbb { F } _ { 0 } ^ { n }$ , $K \subseteq \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ be compact sets. For the invariant case, we have: + +$$ +\begin{array} { r l } & { \overline { { M G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K _ { d i s c r } , \mathbb { F } ) : \rho \left( 2 \overline { { N L _ { I } } } \right) \subseteq \rho \left( f \right) \} } \\ & { \overline { { 2 - L G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( 2 \overline { { N L _ { I } } } \right) \subseteq \rho \left( f \right) \} } \\ & { \overline { { k - L G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k { - } L G N N _ { I } \right) \subseteq \rho \left( f \right) \} \supset \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k { - } W L _ { I } \right) \subseteq \rho \left( f \right) \} } \\ & { \overline { { k - F G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k { - } F W L _ { I } \right) \subseteq \rho \left( f \right) \} } \end{array} +$$ + +For the equivariant case, we have: + +$$ +\begin{array} { r c l } { { \underline { { M G N N _ { E } } } } } & { { = } } & { { \{ f \in { \mathcal C } _ { E } ( K _ { d i s c r } , \mathbb F ^ { n } ) : \rho \left( 2 ^ { - } { \cal W } L _ { E } \right) \subseteq \rho \left( f \right) \} } } \\ { { \boldsymbol { \cdot } L G N N _ { E } } } & { { = } } & { { \{ f \in { \mathcal C } _ { E } ( K , \mathbb F ^ { n } ) : \rho \left( k { \cal - } L G N N _ { E } \right) \subseteq \rho \left( f \right) \} \supset \{ f \in { \mathcal C } _ { E } ( K , \mathbb F ^ { n } ) : \rho \left( k { \cal - } W L _ { E } \right) \subseteq \rho \left( f \right) \} } } \\ { { \boldsymbol { \cdot } R G N N _ { E } } } & { { = } } & { { \{ f \in { \mathcal C } _ { E } ( K , \mathbb F ^ { n } ) : \rho \left( k { \cal - } F W L _ { E } \right) \subseteq \rho \left( f \right) \} } } \end{array} +$$ + +We decompose the proof with an additional lemma. + +Lemma 35. Let $K _ { d i s c r } \subseteq \mathcal G _ { n } \times \mathbb { F } _ { 0 } ^ { n }$ , $K \subseteq \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ be compact sets. For the invariant case, and any $k \geq 2$ , we have, + +$$ +\begin{array} { r l r } & { \overline { { M G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K _ { d i s c r } , \mathbb { F } ) : \rho \left( M G N N _ { I } \right) \subseteq \rho \left( f \right) \} } & \\ & { \overline { { k \ – L G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k \ – L G N N _ { I } \right) \subseteq \rho \left( f \right) \} } & \\ & { \overline { { k \ – F G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k \ – F G N N _ { I } \right) \subseteq \rho \left( f \right) \} } & \end{array} +$$ + +For the equivariant case, and any $k \geq 2$ , we have: + +$$ +\begin{array} { r l } { \overline { { M G N N _ { E } } } } & { = \{ f \in \mathcal { C } _ { E } ( K _ { d i s c r } , \mathbb { F } ) : \rho \left( M G N N _ { E } \right) \subseteq \rho \left( f \right) \} } \\ { \overline { { k \ – L G N N _ { E } } } } & { = \{ f \in \mathcal { C } _ { E } ( K , \mathbb { F } ) : \rho \left( k \ – L G N N _ { E } \right) \subseteq \rho \left( f \right) \} } \\ { \overline { { k \ – F G N N _ { E } } } } & { = \{ f \in \mathcal { C } _ { E } ( K , \mathbb { F } ) : \rho \left( k \ – F G N N _ { E } \right) \subseteq \rho \left( f \right) \} } \end{array} +$$ + +Proof of Thm. 34. The theorem is now a direct consequence of Prop. 13 and Lem. 35. + +We now move to the proof of Lem. 35. + +Proof of Lem. 35. First, focus on the invariant case. Let $\mathcal { F }$ denote ${ \bf M G N N } _ { I }$ , $k { \mathrm { - } } \mathrm { L G N N } _ { I }$ or $k { \mathrm { - } } \mathrm { F G N N } _ { I }$ and $X$ be either $K _ { d i s c r }$ or $K$ so that $X$ is compact and ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { I } ( X , \mathbb { F } )$ . Applying Thm. 5 directly gives, + +$$ +\overline { { \mathcal { F } } } = \{ f \in \mathcal { C } _ { I } ( X , \mathbb { F } ) : \rho \left( \mathcal { F } \right) \subseteq \rho \left( f \right) \} \mathrm { ~ , ~ } +$$ + +which is the desired result. + +We now move to the equivariant case. First, let us replace $\mathbf { M G N N } _ { E }$ by another class, which is slightly simpler to analyze. Define, + +$$ +\begin{array} { r l } & { \mathbf { M G N N } _ { E ^ { \prime } } = \left\{ m _ { E } \circ \left( \left( 1 - \lambda \right) \mathrm { I d } + \lambda S ^ { 1 } \right) \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } : F _ { t } : } \\ & { \qquad \mathbb { F } _ { t } ^ { n } \to \mathbb { F } _ { t + 1 } ^ { n } \mathrm { m e s s a g e ~ p a s s i n g ~ l a y e r } , t = 1 , \dots , T , T \geq 1 , \lambda \in \left\{ 0 , 1 \right\} \right\} . } \end{array} +$$ + +It holds that $\rho \left( \mathbf { M G N N } _ { E ^ { \prime } } \right) = \rho \left( \mathbf { M G N N } _ { E } \right)$ and, + +$$ +\begin{array} { r } { \mathbf { M G N N } _ { E ^ { \prime } } \subseteq \mathbf { \overline { { M G N N } } } _ { E } \subseteq \{ f \in \mathcal { C } _ { E } ( K _ { d i s c r } , \mathbb { F } ^ { n } ) : \rho \left( \mathbf { M G N N } _ { E } \right) \subseteq \rho \left( f \right) \} } \\ { = \{ f \in \mathcal { C } _ { E } ( K _ { d i s c r } , \mathbb { F } ^ { n } ) : \rho \left( \mathbf { M G N N } _ { E ^ { \prime } } \right) \subseteq \rho \left( f \right) \} } \end{array} +$$ + +Therefore, if we show that, + +$$ +\overline { { \mathbf { M G N N } _ { E ^ { \prime } } } } = \left\{ f \in \mathcal { C } _ { E } \big ( K _ { d i s c r } , \mathbb { F } ^ { n } \big ) : \rho \left( \mathbf { M G N N } _ { E ^ { \prime } } \right) \subseteq \rho \left( f \right) \right\} , +$$ + +we will have the desired result. + +Let $\mathcal { F }$ denote $\mathbf { M G N N } _ { E ^ { \prime } }$ , $k { \mathrm { - } } \mathrm { L G N N } _ { E }$ or $k { \mathrm { - } } \mathrm { F G N N } _ { E }$ and $X$ be either $K _ { d i s c r }$ or $K$ so that $X$ is compact and ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { E } ( X , \mathbb { F } ^ { n } )$ . We now wish to apply Thm. 6, but we need to verify the stability assumption first. Thus, we show that, for any $f \in { \mathcal { F } }$ , + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \in { \mathcal { F } } , +$$ + +• If $\mathcal { F } = \mathbf { M G N N } _ { E ^ { \prime } }$ .Take $f \in \mathbf { M G N N } _ { E ^ { \prime } }$ . $f$ is of the form, + +$$ +m _ { E } \circ ( ( 1 - \lambda ) \operatorname { I d } + \lambda S ^ { 1 } ) \circ F _ { T } \circ . . . F _ { 2 } \circ F _ { 1 } , +$$ + +where $F _ { t } : \mathbb { F } _ { t } ^ { n } \to \mathbb { F } _ { t + 1 } ^ { n }$ are message passing layers, $\mathbb { F } _ { T + 1 } = \mathbb { F }$ and $\lambda \in \{ 0 , 1 \}$ . We need to show that there is a $\mathrm { M G N N } _ { E ^ { \prime } }$ which implements, + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) ~ . +$$ + +If $\lambda = 1$ , $f$ is exactly the function of (26). Otherwise, if $\lambda = 0$ , we build another $\mathbf { M G N N } _ { E ^ { \prime } }$ implementing this function. Denote by $F _ { T + 1 } : \mathbb { F } ^ { n } \mathbb { F } ^ { n }$ the (simple) message passing layer defined by $F _ { T + 1 } ( h ) _ { i } = m _ { E } ( h _ { i } )$ for $i \in [ n ]$ and any $h \in \mathbb { F } ^ { n }$ . Then, the $\mathbf { M G N N } _ { E ^ { \prime } }$ $( 1 - \lambda ^ { \prime } ) \operatorname { I d } + \lambda ^ { \prime } S ^ { 1 } ) \circ F _ { T + 1 } \circ F _ { T } \circ . . . F _ { 2 } \circ F _ { 1 }$ with $\lambda ^ { \prime } = 1$ exactly implements (26). + +• If $\mathcal { F } = k \mathrm { - } \mathrm { L G N N } _ { E }$ . A function $f$ of this class is of the form, + +$$ +m _ { E } \circ S _ { 1 } ^ { k } \circ F _ { T } \circ . . . F _ { 2 } \circ F _ { 1 } \circ I ^ { k } , +$$ + +where $F _ { t } : \mathbb { F } _ { t } ^ { n ^ { k } } \to \mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ linear graph layers and $\mathbb { F } _ { T + 1 } = \mathbb { F }$ . Our goal is to show that there is a GNN of $k { \mathrm { - } } \mathrm { L G N N } _ { E }$ + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \ , +$$ + +By definition of linear graph layers, the map $F _ { T + 1 } : \mathbb { F } ^ { n ^ { k } } \mathbb { F } ^ { n ^ { k } }$ defined by, for $G \in \mathbb { F } ^ { n ^ { k } }$ , + +$$ +\forall ( i _ { 1 } , \ldots , i _ { k } ) \in [ n ] ^ { k } , F _ { T + 1 } ( G ) _ { i _ { 1 } , \ldots , i _ { k } } = m _ { E } ( S _ { 1 } ^ { k } ( G ) _ { i } ) , +$$ + +is a linear graph layer as defined in Section 3. Now, consider, the linear graph layer, $F _ { T + 2 } : \mathbb { F } ^ { n ^ { k } } \mathbb { F } ^ { n ^ { k } }$ defined by, for $G \in \mathbb { F } ^ { n ^ { k } }$ , + +$$ +\forall ( i _ { 1 } , \dots , i _ { k } ) \in [ n ] ^ { k } , F _ { T + 1 } ( G ) _ { i _ { 1 } , \dots , i _ { k } } = \sum _ { i = 1 } ^ { n } G _ { i , i _ { 2 } , \dots , i _ { k } } . +$$ + +Then, the $k$ -LGNN $F _ { T + 2 } \circ F _ { T + 1 } \circ F _ { T } \circ \dots F _ { 2 } \circ F _ { 1 } \circ I ^ { k }$ exactly implements (27). + +• If $\mathcal { F } = k \mathrm { - F G N N } _ { E }$ . A function $f$ of this class is of the form, + +$$ +m _ { E } \circ S _ { 1 } ^ { k } \circ F _ { T } \circ . . . F _ { 2 } \circ F _ { 1 } \circ I ^ { k } , +$$ + +where Ft : Fnkt $F _ { t } : \mathbb { F } _ { t } ^ { n ^ { k } } \to \mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ → Fnkt+1 a re FGL (see Section 3) and $\mathbb { F } _ { T + 1 } = \mathbb { F }$ . We build a GNN of which implements, + +$$ +x \mapsto \left( \sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \ldots , \sum _ { i = 1 } ^ { n } f ( x ) _ { i } \right) \ , +$$ + +$w \in [ k ]$ $H _ { w } : \mathbb { F } ^ { n ^ { k } } \mathbb { F } ^ { n ^ { k } }$ + +$$ +\forall G \in \mathbb { F } ^ { n ^ { k } } , \forall ( i _ { 1 } , \dots , i _ { k } ) \in [ n ] ^ { k } , H _ { w } ( G ) _ { i _ { 1 } , \dots , i _ { k } } = \sum _ { j = 1 } ^ { n } G _ { i _ { 1 } , \dots , i _ { w - 1 } , j , i _ { w + 1 } , \dots , i _ { k } } . +$$ + +Then $H _ { 2 } \circ \cdot \cdot \cdot \circ H _ { k } : \mathbb { F } ^ { n ^ { k } } \to \mathbb { F } ^ { n ^ { k } }$ computes the sum of the elements of the input tensor over the last $k - 1$ dimensions like to $S _ { 1 } ^ { k } : \mathbb { F } ^ { n ^ { k } } \to \mathbb { F } ^ { n }$ and $H _ { 1 } \circ H _ { 2 } \circ \cdots \circ H _ { k } : \mathbb { F } ^ { n ^ { k } } \to \mathbb { F } ^ { n ^ { k } }$ computes the full sum of the elements of the input tensor like to $S ^ { k } : \mathbb { F } ^ { n ^ { k } } \mathbb { F }$ . Finally, consider the FGL $F _ { T + 1 } : \mathbb { F } ^ { n ^ { k } } \mathbb { F } ^ { n ^ { k } }$ associated to $m _ { E }$ , i.e. such that, for any $G \in \mathbb { F } ^ { n ^ { k } }$ , + +$$ +\forall ( i _ { 1 } , \dots , i _ { k } ) \in [ n ] ^ { k } , F _ { T + 1 } ( G ) _ { i _ { 1 } , \dots , i _ { k } } = m _ { E } ( G _ { i _ { 1 } , \dots , i _ { k } } ) . +$$ + +Now, the $k$ -FGNN + +$$ +H _ { 1 } \circ H _ { 2 } \circ \cdot \cdot \cdot \circ H _ { k } \circ F _ { T + 1 } \circ H _ { 2 } \circ \cdot \cdot \cdot \circ H _ { k } \circ F _ { T + 1 } \circ F _ { T } \circ . . . F _ { 2 } \circ F _ { 1 } \circ I ^ { k } +$$ + +exactly implements (28). + +# F EXTENSION TO GRAPHS OF VARYING SIZES + +# F.1 EXTENSION TO DISCONNECTED INPUT SPACES + +Here we show that our results can be extended to graphs of varying sizes similarly to Keriven & Peyré (2019). There are two ways to do it. The first would be to directly adapt all the proofs but it would make them more cumbersome. Instead, we extend them with a simple argument presented below. As a side benefit, this general lemma makes it possible to extend almost all the approximation results for graph neural networks from the literature. + +The abstract setting is the following. Given a compact input space $X$ , assume that there is some finite set $A$ , and $X _ { \alpha }$ , $\alpha \in A$ a family of pairwise disjoints compact sets such that $\textstyle X = \bigcup _ { \alpha \in A } X _ { \alpha }$ Crucially, we will assume that the $X _ { \alpha }$ are in distinct connected components. Intuitively, they do not "touch" each other. Similarly, assume that the output space $Y$ can be written as $\textstyle Y = \operatorname { L } _ { \alpha \in A } Y _ { \alpha }$ , with $Y _ { \alpha }$ , $\alpha \in A$ a family of (pairwise disjoints) real vector spaces. + +Before moving to the results, we need a last definition. Informally, we need that the functions $f$ that we will consider do not change the number of nodes of their inputs. Formally, we say that $f : X \longrightarrow Y$ is adapted if $f ( X _ { \alpha } ) \subseteq Y _ { \alpha }$ for each $\alpha \in A$ , and denote by $\mathcal { C } _ { a d } ( X , Y )$ the of continuous and adapted functions from $X$ to $Y$ . + +We can now state our lemma to adapt our results to graphs with varying node sizes. + +Lemma 36. Let $X$ be a compact space, $Y$ be a topological space and $A$ a finite set. Assume that there exists $X _ { \alpha }$ , $\alpha \in A$ a family of pairwise disjoints compact sets such that $\textstyle X = \bigcup _ { \alpha \in A } X _ { \alpha }$ and the $X _ { \alpha }$ ’s are in distinct connected components. Also assume that there is $Y _ { \alpha }$ , $\alpha \in A$ a family of (pairwise disjoints) real vector spaces such that $\textstyle Y = \operatorname { L } _ { \alpha \in A } Y _ { \alpha }$ . Consider ${ \mathcal { F } } \subseteq { \mathcal { C } } _ { a d } ( X , Y )$ a (non-empty) set of adapted functions and, for each $\alpha \in A$ , define $\mathcal { F } _ { | X _ { \alpha } } = \{ f _ { | X _ { \alpha } } : f \in \mathcal { F } \} \subseteq \mathcal { C } ( X _ { \alpha } , Y _ { \alpha } )$ the restriction of the functions of $\mathcal { F }$ to $X _ { \alpha }$ . + +Assume that the following holds, + +1. Each $\mathcal { F } _ { | X _ { \alpha } }$ is a sub-algebra of $\mathcal { C } ( X _ { \alpha } , Y _ { \alpha } )$ which contains the constant function ${ \mathbf { 1 } } _ { Y _ { \alpha } }$ + +2. There is a set of functions $\mathcal { F } _ { s c a l } \subseteq \mathcal { C } ( X , \mathbb { R } )$ such that $\mathcal { F } _ { s c a l } . \mathcal { F } \subseteq \mathcal { F }$ and it discriminates between the $X _ { \alpha }$ ’s, i.e. $\textstyle \rho ( { \mathcal { F } } _ { s c a l } ) \subseteq \sqcup _ { \alpha \in A } X _ { \alpha } ^ { 2 }$ . + +Then the closure of $\mathcal { F }$ is, + +$$ +\begin{array} { r } { \overline { { \mathcal { F } } } = \left\{ f \in \mathcal { C } _ { a d } ( X , Y ) : \forall \alpha \in A , f _ { | X _ { \alpha } } \in \overline { { \mathcal { F } _ { | X _ { \alpha } } } } \right\} , } \end{array} +$$ + +for the distance on $\mathcal { C } _ { a d } ( X , Y )$ defined by, for $f , g \in \mathcal { C } _ { a d } ( X , Y )$ , + +$$ +d ( f , g ) = \operatorname* { m a x } _ { \alpha \in A } \operatorname* { s u p } _ { X _ { \alpha } } \| f - g \| _ { Y _ { \alpha } } . +$$ + +Proof. Define the set of functions $S \subseteq { \mathcal { C } } ( X , \mathbb { R } )$ by, + +$$ +S = \left\{ f \in \mathcal { C } ( X , \mathbb { R } ) : f . \mathcal { F } \subseteq \mathcal { F } \right\} . +$$ + +By definition, $\mathcal { F } _ { s c a l } \subseteq S$ . Moreover, as each $\mathcal { F } _ { | X _ { \alpha } }$ is a subalgebra, $S$ is also a subalgebra of ${ \mathcal { C } } ( X , \mathbb { R } )$ + +As $X$ is compact, we can apply Cor. 17 to $S$ to get that, in ${ \mathcal { C } } ( X , \mathbb { R } )$ , + +$$ +{ \overline { { S } } } = \left\{ f \in { \mathcal { C } } ( X , \mathbb { R } ) : \rho \left( S \right) \subseteq \rho \left( f \right) \right\} . +$$ + +In the first part of the proof, we check that, for each $\alpha$ , the function $f _ { \alpha } : X \longrightarrow [ 0 , 1 ]$ defined by + +$$ +\begin{array} { r } { f _ { \alpha | X _ { \alpha } } = 1 } \\ { \forall \beta \neq \alpha , f _ { \alpha | X _ { \beta } } = 0 . } \end{array} +$$ + +is continuous and satisfy $\rho \left( S \right) \subseteq \rho \left( f _ { \alpha } \right)$ . This will means that such functions belong to $\overline { { \mathcal { F } _ { s c a l } } }$ . + +As the $X _ { \beta }$ ’s are in different connected components, it is enough to check its continuity on each $X _ { \beta }$ . +Indeed, each $f _ { \alpha | X _ { \beta } }$ is constant so continuous. The second fact comes from the second assumption. +Indeed, take $( x , y ) \in X$ such that $( x , y ) \in \rho \left( S \right)$ . As $\mathcal { F } _ { s c a l } \subseteq S$ , in particular $( x , y ) \in \rho \left( \mathcal { F } _ { s c a l } \right)$ . +But, by assumption, $\textstyle \rho ( { \mathcal { F } } _ { s c a l } ) \subseteq \sqcup _ { \beta \in A } X _ { \beta } ^ { 2 }$ . Thus, necessarily $x$ and $y$ belong to the same $X _ { \beta }$ so that $f _ { \alpha } ( x ) = f _ { \alpha } ( y )$ . + +Therefore, we conclude that $f _ { \alpha } \in \overline { { S } }$ . + +We now prove the announced equality. By the definition of the distance, the first inclusion, $\overline { { \mathcal { F } } } \subseteq$ $\{ f \in \mathcal { C } _ { a d } ( X , Y ) : f _ { | X _ { \alpha } } \in \overline { { \mathcal { F } _ { \alpha } } } \}$ is immediate. We focus on the other way. Take $h \in \mathcal { C } _ { a d } ( X , Y )$ such that $h _ { | X _ { \alpha } } \in \overline { { \mathcal { F } _ { \alpha } } }$ for every $\alpha \in A$ and $\epsilon > 0$ . By definition of $h$ , there exists, for each $\alpha \in A$ , $g _ { \alpha } \in \mathcal { F }$ such that, + +$$ +\underset { X _ { \alpha } } { \operatorname* { s u p } } \Vert h - g _ { \alpha } \Vert _ { Y _ { \alpha } } \leq \epsilon , +$$ + +As the $X _ { \alpha }$ ’s are compact and the $g _ { \alpha }$ ’s are continuous, $\begin{array} { r } { \operatorname* { m a x } _ { \alpha \in A } \operatorname* { s u p } _ { X _ { \alpha } } \| g _ { \alpha } \| < + \infty } \end{array}$ and denote by $M > 0$ a bound on this quantity. We have shown above that each $f _ { \alpha }$ is in $\overline { S }$ so there exists, for each $\alpha \in A$ , $l _ { \alpha } \in S$ such that, + +$$ +\operatorname* { s u p } _ { X } | f _ { \alpha } - l _ { \alpha } | \leq \frac { \epsilon } { M } . +$$ + +By definition of $S$ and, as $\mathcal { F }$ is a subalgebra, $\textstyle \sum _ { \alpha \in A } l _ { \alpha } g _ { \alpha } \in { \mathcal { F } }$ and, for each $\beta \in A$ , + +$$ +\begin{array} { r l r } { { \operatorname* { s u p } _ { X _ { \beta } } \| h - \sum _ { \alpha \in A } l _ { \alpha } g _ { \alpha } \in \mathcal { F } \| \le \operatorname* { s u p } _ { X _ { \beta } } \| h - g _ { \beta } \| + \| g _ { \beta } - l _ { \beta } g _ { \beta } \| + \| \sum _ { \alpha \neq \beta } l _ { \alpha } g _ { \alpha } \| } } \\ & { } & { \le \epsilon + M \times \frac { \epsilon } { M } + ( | A | - 1 ) M \times \frac { \epsilon } { M } = ( | A | + 1 ) \epsilon , } \end{array} +$$ + +which concludes the proof. + +We now state our theorem in the case of varying graph size like Keriven & Peyré (2019). With Lem. 36, it is indeed straightforward to extend any approximation result initially proven for a class of graphs of fixed size. However, as the complete proof would require new notations again, we only give a sketch of proof. + +Fix $N \geq 1$ and consider the space of graphs (described by tensors) of size less than $N$ , $\mathbb { F } _ { 0 } ^ { \le N ^ { 2 } } =$ $\textstyle \bigcup _ { n = 1 } ^ { N } \mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ . Equip this space with the final topology or, equivalently, the graph edit distance. Then, the last thing to check to apply Lem. 36 is that the classes of GNN that we consider indeed discriminate between graphs of different sizes, which is immediate. + +Likewise, for message passing GNN, consider $\begin{array} { r } { \mathcal { G } _ { \leq N } \times \mathbb { F } _ { 0 } ^ { \leq N } = \bigcup _ { n = 1 } ^ { N } \mathcal { G } _ { n } \times \mathbb { F } _ { 0 } ^ { n } } \end{array}$ , with a similar topology or the graph edit distance. + +Corollary 37. Let $K _ { d i s c r } \subseteq \mathcal { G } _ { \leq N } \times \mathbb { F } _ { 0 } ^ { \leq N }$ , $K \subseteq \mathbb { F } _ { 0 } ^ { \leq N ^ { 2 } }$ be compact sets. For the invariant case, we + +$$ +\begin{array} { r l } & { \overline { { M G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K _ { d i s c r } , \mathbb { F } ) : \rho \left( 2 \overline { { N L _ { I } } } \right) \subseteq \rho \left( f \right) \} } \\ & { \overline { { 2 - L G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( 2 \overline { { N L _ { I } } } \right) \subseteq \rho \left( f \right) \} } \\ & { \overline { { k - L G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k { - } L G N N _ { I } \right) \subseteq \rho \left( f \right) \} \supset \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k { - } W L _ { I } \right) \subseteq \rho \left( f \right) \} } \\ & { \overline { { k - F G N N _ { I } } } = \{ f \in \mathcal { C } _ { I } ( K , \mathbb { F } ) : \rho \left( k { - } F W L _ { I } \right) \subseteq \rho \left( f \right) \} } \end{array} +$$ + +For the equivariant case, we have: + +$$ +\begin{array} { r c l } { \overline { { M G N N _ { E } } } } & { = } & { \left\{ f \in \mathcal { C } _ { E } ( K _ { d i s c r } , \mathbb { F } ^ { \leq N } ) : \rho \left( 2 \overline { { { \cdot } } } W L _ { E } \right) \subseteq \rho \left( f \right) \right\} } \\ { \widehat { \cdot } L G N N _ { E } } & { = } & { \left\{ f \in \mathcal { C } _ { E } ( K , \mathbb { F } ^ { \leq N } ) : \rho \left( k { - } L G N N _ { E } \right) \subseteq \rho \left( f \right) \right\} \supset \left\{ f \in \mathcal { C } _ { E } ( K , \mathbb { F } ^ { \leq N } ) : \rho \left( k { - } W L _ { E } \right) \subseteq \rho \left( f \right) \right\} } \\ { \cdot { F G N N _ { E } } } & { = } & { \left\{ f \in \mathcal { C } _ { E } ( K , \mathbb { F } ^ { \leq N } ) : \rho \left( k { - } F W L _ { E } \right) \subseteq \rho \left( f \right) \right\} } \end{array} +$$ + +Proof. This corollary is a direct consequence of Thm. 34 and Lem. 36, using Lem. 32 to satisfy the sub-algebra assumption of Lem. 36. □ \ No newline at end of file diff --git a/parse/train/lxHgXYN4bwl/lxHgXYN4bwl_content_list.json b/parse/train/lxHgXYN4bwl/lxHgXYN4bwl_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2881b766adb25cdcdbb9f01fb5cf821d3e18e168 --- /dev/null +++ b/parse/train/lxHgXYN4bwl/lxHgXYN4bwl_content_list.json @@ -0,0 +1,8276 @@ +[ + { + "type": "text", + "text": "EXPRESSIVE POWER OF INVARIANT AND EQUIVARIANT GRAPH NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 99, + 699, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Waïss Azizian ENS, PSL University, Paris, France waiss.azizian@ens.fr ", + "bbox": [ + 183, + 172, + 416, + 214 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Marc Lelarge \nINRIA & ENS, PSL University, Paris, France \nmarc.lelarge@ens.fr ", + "bbox": [ + 473, + 172, + 772, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 252, + 544, + 267 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power of these GNN architectures. The current universality theorems only apply to intractable classes of GNNs. Here, we prove the first approximation guarantees for practical GNNs, paving the way for a better understanding of their generalization. Our theoretical results are proved for invariant GNNs computing a graph embedding (permutation of the nodes of the input graph does not affect the output) and equivariant GNNs computing an embedding of the nodes (permutation of the input permutes the output). We show that Folklore Graph Neural Networks (FGNN), which are tensor based GNNs augmented with matrix multiplication are the most expressive architectures proposed so far for a given tensor order. We illustrate our results on the Quadratic Assignment Problem (a NP-Hard combinatorial problem) by showing that FGNNs are able to learn how to solve the problem, leading to much better average performances than existing algorithms (based on spectral, SDP or other GNNs architectures). On a practical side, we also implement masked tensors to handle batches of graphs of varying sizes. ", + "bbox": [ + 233, + 282, + 766, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 545, + 334, + 560 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph Neural Networks (GNN) are designed to deal with graph structured data. Since a graph is not changed by permutation of its nodes, GNNs should be either invariant if they return a result that must not depend on the representation of the input (typically when building a graph embedding) or equivariant if the output must be permuted when the input is permuted (typically when building an embedding of the nodes). More fundamentally, incorporating symmetries in machine learning is a fundamental problem as it allows to reduce the number of degree of freedom to be learned. ", + "bbox": [ + 174, + 575, + 825, + 659 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep learning on graphs. This paper focuses on learning deep representation of graphs with network architectures, namely GNN, designed to be invariant to permutation or equivariant by permutation. From a practical perspective, various message passing GNNs have been proposed, see Dwivedi et al. (2020) for a recent survey and benchmarking on learning tasks. In this paper, we study 3 architectures: Message passing GNN (MGNN) which is probably the most popular architecture used in practice, order- $k$ Linear GNN $k$ -LGNN) proposed in Maron et al. (2018) and order- $k$ Folklore GNN $k$ - FGNN) first introduced by Maron et al. (2019a). MGNN layers are local thus highly parallelizable on GPUs which make them scalable for large sparse graphs. $k$ -LGNN and $k$ -FGNN are dealing with representations of graphs as tensors of order $k$ which make them of little practical use for $k \\geq 3$ . ", + "bbox": [ + 174, + 666, + 825, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In order to compare these architectures, the separating power of these networks has been compared to a hierarchy of graph invariants developed for the graph isomorphism problem. Namely, for $k \\geq 2$ $k { \\mathrm { - W L } } ( G )$ are invariants based on the Weisfeiler-Lehman tests (described in Section 4.1). For each $k \\geq 2$ , $( \\dot { k } + 1 )$ -WL has strictly more separating power than $k$ -WL (in the sense that there is a pair of non-isomorphic graphs distinguishable by $( k + 1 )$ -WL and not by $k$ -WL). GIN (which are invariant MGNN) introduced in $\\mathrm { X u }$ et al. (2018) are shown to be as powerful as 2-WL. In Maron et al. (2019a), Geerts (2020b) and Geerts (2020a), $k$ -LGNN are shown to be as powerful as $k$ -WL and 2-FGNN is shown to be as powerful as 3-WL. In this paper, we extend this last result about $k$ -FGNN to general values of $k$ . So in term of separating power, when restricted to tensors of order $k$ , $k$ -FGNN is the most powerful architecture among the ones considered in this work. This means that for a given pair of graphs $G$ and $G ^ { \\prime }$ , if $( k + { \\bar { 1 } } ) – \\mathbf { W L } ( G ) \\neq ( k + 1 ) – \\mathbf { W L } ( G ^ { \\prime } )$ , then there exists a $k$ -FGNN, say $\\mathbf { G N N } _ { G , G ^ { \\prime } }$ such that $\\mathbf { G } \\mathbf { N } \\mathbf { N } _ { G , G ^ { \\prime } } ( G ) \\neq \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { G , G ^ { \\prime } } ( G ^ { \\prime } )$ . ", + "bbox": [ + 174, + 799, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 147 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Approximation results for GNNs. Results on the separating power of GNNs only deal with pairwise comparison of graphs: we need a priori a different GNN for each pair of graphs in order to distinguish them. Such results are of little help in a practical learning scenario. Our main contribution in this paper overcomes this issue and we show that a single GNN can give a meaningful representation for all graphs. More precisely, we characterize the set of functions that can be approximated by MGNNs, $k$ -LGNNs and $k$ -FGNNs respectively. Standard Stone-Weierstrass theorem shows that if an algebra $\\mathcal { A }$ of real continuous functions separates points, then $\\mathcal { A }$ is dense in the set of continuous function on a compact set. Here we extend such a theorem to general functions with symmetries and apply it to invariant and equivaraint functions to get our main result for GNNs. As a consequence, we show that $k$ -FGNNs have the best approximation power among architectures dealing with tensors of order $k$ . ", + "bbox": [ + 173, + 152, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Universality results for GNNs. Universal approximation theorems (similar to Cybenko (1989) for multi-layers perceptron) have been proved for linear GNNs in Maron et al. (2019b); Keriven & Peyré (2019); Chen et al. (2019). They show that some classes of GNNs can approximate any function defined on graphs. To be able to approximate any invariant function, they require the use of very complex networks, namely $k$ -LGNN where $k$ tends to infinity with $n$ the number of nodes. Since we prove that any invariant function less powerful than $( k + 1 )$ -WL can be approximated by a $k$ -FGNN, letting $k$ tends to infinity directly implies universality. Universality results for $k$ -FGNN is another contribution of our work. ", + "bbox": [ + 173, + 297, + 825, + 410 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Equivariant GNNs. Our second set of results extends previous analysis from invariant functions to equivariant functions. There are much less results about equivariant GNNs: Keriven & Peyré (2019) proves the universality of linear equivariant GNNs, and Maehara & Hoang (2019) shows the universality of a new class of networks they introduced. Here, we consider a natural equivariant extension of $k$ -WL and prove that equivariant $( k + 1 )$ -LGNNs and $k$ -FGNN can approximate any equivariant function less powerful than this equivariant $( k + 1 )$ -WL for $k \\geq 1$ . At this stage, we should note that all universality results for GNNs by Maron et al. (2019b); Keriven & Peyré (2019); Chen et al. (2019) are easily recovered from our main results. Also our analysis is valid for graphs of varying sizes. ", + "bbox": [ + 173, + 416, + 825, + 542 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Empirical results for the Quadratic Assigment Problem (QAP). To validate our theoretical contributions, we empirically show that 2-FGNN outperforms classical MGNN. Indeed, Maron et al. (2019a) already demonstrate state of the art results for the invariant version of 2-FGNNs (for graph classification or graph regression). Here we consider the graph alignment problem and show that the equivariant 2-FGNN is able to learn a node embedding which beats by a large margin other algorithms (based on spectral method, SDP or GNNs). ", + "bbox": [ + 173, + 549, + 825, + 632 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Outline and contribution. After reviewing more previous works and notations in the next section, we define the various classes of GNNs studied in this paper in Section $3 :$ message passing GNN, linear GNN and folklore GNN. Section 4 contains our main theoretical results for GNNs. First in Section 4.2 we describe the separating power of each GNN architecture with respect to the WeisfeilerLehman test. In Section 4.3, we give approximation guarantees for MGNNs, LGNNs and FGNNs at fixed order of tensor. They cover both the invariant and equivariant cases and are our main theoretical contributions. For these, we develop in Section D a fine-grained Stone-Weierstrass approximation theorem for vector-valued functions with symmetries. Our theorem handles both invariant and equivariant cases and is inspired by recent works in approximation theory. In Section 6, we illustrate our theoretical results on a practical application: the graph alignment problem, a well-known NP-hard problem. We highlight a previously overlooked implementation question: the handling of batches of graphs of varying sizes. A PyTorch implementation of the code necessary to reproduce the results is available at https://github.com/mlelarge/graph_neural_net ", + "bbox": [ + 173, + 640, + 825, + 820 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 847, + 341, + 862 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The pioneering works that applied neural networks to graphs are Gori et al. (2005) and Scarselli et al. (2009) that learn node representation with recurrent neural networks. More recent message passing architectures make use of non-linear functions of the adjacency matrix (Kipf & Welling, 2016), for example polynomials (Defferrard et al., 2016). For regular-grid graphs, they match classical convolutional networks which by design can only approximate translation-invariant functions and hence have limited expressive power. In this paper, we focus instead on more expressive architectures. ", + "bbox": [ + 174, + 882, + 825, + 922 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Following the recent surge in interest in graph neural networks, some works have tried to extend the pioneering work of Cybenko (1989); Hornik et al. (1989) for various GNN architectures. Among the first ones is Scarselli et al. (2009), which studied invariant message-passing GNNs. They showed that such networks can approximate, in a weak sense, all functions whose discriminatory power is weaker than 1-WL. Yarotsky (2018) described universal architectures which are invariant or equivariant to some group action. These models rely on polynomial intermediate layers of arbitrary degrees, which would be prohibitive in practice. Maron et al. (2019b) leveraged classical results about the polynomials invariant to a group action to show that $k$ -LGNN are universal as $k$ tends to infinity with the number of nodes. Keriven & Peyré (2019) derived a similar result, in the more complicated equivariant case by introducing a new Stone-Weierstrass theorem. Similarly to Maron et al. (2019b), they require the order of tensors to go to infinity. Another route towards universality is the one of Chen et al. (2019). In the invariant setting, they show for a class of GNN that universality is equivalent to being able to discriminate between (non-isomorphic) graphs. However, the only way to achieve such discriminatory power is to use tensors of arbitrary high order, see also Ravanbakhsh (2020). Our work encompass and precise these results using high-order tensors as it yields approximation guarantees even at fixed order of tensor. ", + "bbox": [ + 174, + 154, + 825, + 375 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "CPNGNN in Sato et al. (2019) and DimeNet in Klicpera et al. (2020) are message passing GNN incorporating more information than those studied here. Partial results about their separating power follows from Garg et al. (2020) which provides impossibility results to decide graph properties including girth, circumference, diameter, radius, conjoint cycle, total number of cycles, and $k$ -cliques. Chen et al. (2020) studies the ability of GNNs to count graph substructures. Though our theorems are much more general, note that their results are improved by the present work. Note also, that if the nodes are given distinct features, MGNNs become much more expressive Loukas (2019) but looses their invariant or equivariant properties. Averaging i.e. relational pooling (RP) has been proposed to recover these properties Murphy et al. (2019a). However, the ideal RP, leading to a universal approximation, cannot be used for large graphs due to its complexity of $O ( | V | ! )$ . Regarding the other classes of RPGNN i.e. the $k$ -ary pooling (Murphy et al., 2019b), we will show how our general theorems in the invariant case can be applied to characterize their approximation power (see Section 5). ", + "bbox": [ + 174, + 381, + 825, + 563 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Note that for neural networks on sets, the situation is a bit simpler. Efficient architectures such as DeepSets (Zaheer et al., 2017) or PointNet (Qi et al., 2017) have been shown to be invariant universal. Similar results exist in the equivariant case (Segol & Lipman, 2020; Maron et al., 2020), whose proofs rely on polynomial arguments. Though this is not our main motivation, our approximation theorems could also be applied in this context see Sections D.3 and D.4. ", + "bbox": [ + 174, + 569, + 825, + 638 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 NOTATIONS: GRAPHS AS TENSORS ", + "text_level": 1, + "bbox": [ + 176, + 664, + 450, + 678 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We denote by $\\mathbb { F } , \\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 / 2 } , \\mathbb { F } _ { 1 } , \\dots .$ arbitrary finite-dimensional spaces of the form $\\mathbb { R } ^ { p }$ (for various values of $p$ ) typically representing the space of features. Product of vectors in $\\mathbb { R } ^ { p }$ always refer to component-wise product. There are two ways to see graphs with features. First, graphs can be seen as tensors of order $k$ : $G \\in \\mathbb { F } ^ { n ^ { k } }$ . The classical representation of a graph by its (weighted) adjacency matrix for $k = 2$ is a tensor of order 2 in $\\mathbb { R } ^ { n ^ { 2 } }$ . This case allows for features on edges by replacing $\\mathbb { R } ^ { n ^ { 2 } }$ with $\\mathbb { F } ^ { n ^ { 2 } }$ where $\\mathbb { F }$ is some $\\mathbb { R } ^ { p }$ . Second, graphs can also be represented by their discrete structure with an additional feature vector. More exactly, denote by ${ \\mathcal { G } } _ { n }$ the set of discrete graphs $G = ( V , E )$ with $n$ nodes $V = [ n ]$ and edges $E \\subseteq V ^ { 2 }$ (with no weights on edges). Such a $G \\in \\mathcal G _ { n }$ with a vector $h ^ { 0 } \\in \\mathbb { F } ^ { n }$ represents a graphs with features on the vertices. ", + "bbox": [ + 173, + 693, + 825, + 825 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 DEFINITIONS: INVARIANT AND EQUIVARIANT OPERATORS", + "text_level": 1, + "bbox": [ + 173, + 851, + 617, + 864 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $[ n ] = \\{ 1 , \\dots , n \\}$ . The set of permutations on $[ n ]$ is denoted by $S _ { n }$ . For $G \\in \\mathbb { F } ^ { n ^ { k } }$ and $\\sigma \\in S _ { n }$ , we define: $( \\sigma \\star G ) _ { \\sigma ( i _ { 1 } ) , \\ldots , \\sigma ( i _ { k } ) } = G _ { i _ { 1 } , \\ldots , i _ { k } }$ . Note that the $\\star$ operation is valid between a permutation in $S _ { n }$ and a graph $G$ as soon as the number of nodes of $G$ is $n$ , i.e. it is valid for any order $k$ tensor representation of the graph. Two graphs $G _ { 1 } , G _ { 2 }$ are said isomorphic if they have the same number of nodes and there exists a permutation $\\sigma$ such that $G _ { 1 } = \\sigma \\star G _ { 2 }$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 1. A function $f : \\mathbb { F } _ { 0 } ^ { n ^ { k } } \\to \\mathbb { F } _ { 1 }$ is said to be invariant if $f ( \\sigma \\star G ) = f ( G )$ for every permutation $\\sigma \\in S _ { n }$ and every $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { k } }$ . A function $f : \\mathbb { F } _ { 0 } ^ { n ^ { k } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { \\ell } }$ is said to be equivariant if $f ( \\sigma \\star G ) = \\sigma \\star f ( G )$ for every permutation $\\sigma \\in S _ { n }$ and every $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { k } }$ . ", + "bbox": [ + 174, + 136, + 825, + 188 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that composing an equivariant function with an invariant function gives an invariant function. For $k \\geq 1$ , we define the invariant summation layer $S ^ { k } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F }$ by $\\begin{array} { r } { S ^ { k } ( G ) = \\sum _ { \\mathbf i \\in [ n ] ^ { k } } G _ { \\mathbf i } } \\end{array}$ for $G \\in \\mathbb { F } ^ { n ^ { k } }$ . We also define the equivariant reduction layer $S _ { 1 } ^ { k } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F } ^ { n }$ as follows: $S _ { 1 } ^ { k } ( G ) _ { i } =$ $\\sum _ { 1 \\leq i _ { 2 } . . . i _ { k } \\leq n } G _ { i , i _ { 2 } , . . . i _ { k } }$ . For message passing GNN, we will use the equivariant layer $\\mathrm { I d } + \\lambda S ^ { 1 }$ : $\\mathbb { F } ^ { n } \\to \\mathbb { F } ^ { n }$ defined by, $( \\operatorname { I d } + \\lambda S ^ { 1 } ) ( G ) _ { i } = G _ { i } + \\lambda S ^ { 1 } ( G )$ , where $\\lambda \\in \\mathbb { R }$ is a learnable parameter. ", + "bbox": [ + 173, + 196, + 826, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the sequel, we will need a mapping $I ^ { k }$ lifting the input graph to a higher order tensor. We denote by $I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }$ the initialization function mapping for a given graph each $k$ -tuple to its isomorphism type. We refer to the appendix Section C.3 for a precise description of this linear equivariant function. Note at this stage that $I ^ { 2 }$ is given by, for $G \\in \\mathbb { F } ^ { n ^ { 2 } }$ , $I ( G ) _ { i , j } = ( G _ { i , j } , \\delta _ { i , j } )$ where $\\delta _ { i , j }$ is 0 if $i \\neq j$ and 1 otherwise. Indeed for a pair of nodes $i , j$ in a graph (without features), there are only three isomorphism types: $i = j ; i \\neq j$ and $( i , j )$ is an edge; $i \\neq j$ but $( i , j )$ is not an edge. ", + "bbox": [ + 173, + 286, + 826, + 377 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 GNN DEFINITIONS ", + "text_level": 1, + "bbox": [ + 176, + 396, + 362, + 412 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we define the various GNN architectures studied in this paper. In all architectures, there is a main building block or layer mapping $\\mathbb { F } _ { t } ^ { n ^ { k } }$ k to F n kt +1 where $\\mathbb { F } _ { t } ^ { n ^ { k } }$ can be seen as the space for the representation of the graph at layer . We will define three different types of layers for message passing GNN, linear GNN and folklore GNN.The case $k = 2$ is probably the most interesting case from a practical point view and corresponds to a case where a layer takes as input a graph (with features on nodes and edges) and produces as output a graph (with new features on nodes and edges). For each type of GNNs, thwill share the last function: $m _ { I } : \\mathbb { F } _ { T + 1 } \\mathbb { F }$ variant and an equivaraint for the invariant case and $m _ { E } : \\mathbb { F } _ { T + 1 } ^ { n } \\mathbb { F } ^ { n }$ ecturesfor the equivariant case which are continuous functions. It is typically modeled by a Multi Layer Perceptron, which is applied on each component for the equivariant case. In words, each network takes as input a graph $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ , produces in the invariant case a graph embedding in $\\mathbb { F } _ { T + 1 }$ and in the equivaraint case a node embedding in $\\mathbb { F } _ { T + 1 } ^ { n }$ , then these embeddings are passed through the function $m _ { I }$ or $m _ { E }$ respectively to get a feature in $\\mathbb { F }$ or $\\mathbb { F } ^ { n }$ for the learning task. ", + "bbox": [ + 173, + 426, + 826, + 613 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 MESSAGE PASSING GNN ", + "text_level": 1, + "bbox": [ + 174, + 630, + 388, + 643 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Message passing GNN (MGNN) are defined for classical graphs $G$ with features on the nodes. More exactly they take as input a discrete graph $G = ( V , E ) \\in \\mathcal { G } _ { n }$ and features on the nodes $h ^ { 0 } \\in \\mathbb { F } ^ { n }$ . MGNN are then defined inductively as follows: let $h _ { i } ^ { \\ell } \\in \\mathbb { F } _ { \\ell }$ denote the feature at layer $\\ell$ associated with node $i$ , the updated features $h _ { i } ^ { \\ell + 1 }$ are obtained as: $h _ { i } ^ { \\ell + 1 } = f \\left( h _ { i } ^ { \\ell } , \\left\\{ \\left\\{ h _ { j } ^ { \\ell } \\right\\} \\right\\} _ { j \\sim i } \\right)$ , where $j \\sim i$ means that nodes $j$ and $i$ are neighbors in the graph $G$ , i.e. $( i , j ) \\in E$ , and the function $f$ is a learnable function taking as input the feature vector of the center vertex $h _ { i } ^ { \\ell }$ and the multiset of features of the neighboring vertices $\\left\\{ \\left\\{ h _ { j } ^ { \\ell } \\right\\} \\right\\} _ { j \\sim i }$ . Indeed, it follows from Lem. 33 in Appendix, that any such function $f$ can be approximated by a layer of the form, ", + "bbox": [ + 173, + 655, + 826, + 782 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1bb03554d0d65080bb33276b96b7d764bbf43b33c0f880c1c408ab867ad2ad78.jpg", + "text": "$$\nh _ { i } ^ { \\ell + 1 } = f _ { 0 } \\left( h _ { i } ^ { \\ell } , \\sum _ { j \\sim i } f _ { 1 } \\left( h _ { i } ^ { \\ell } , h _ { j } ^ { \\ell } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 786, + 598, + 837 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f _ { 0 } : \\mathbb { F } _ { \\ell } \\times \\mathbb { F } _ { \\ell + 1 / 2 } \\to \\mathbb { F } _ { \\ell + 1 }$ and $f _ { 1 } : \\mathbb { F } _ { \\ell } \\times \\mathbb { F } _ { \\ell } \\to \\mathbb { F } _ { \\ell + 1 / 2 }$ , so that $\\mathbb { F } _ { \\ell }$ is the field for the features at the $\\ell$ -th layer. We call such a function a message passing layer and denote it by $F : \\mathbb { F } _ { \\ell } ^ { n } \\to \\mathbb { F } _ { \\ell + 1 } ^ { n }$ (note that $F$ depends implicitly from the graph). Then an equivariant message passing GNN is simply obtained by the composition of message passing layers: $F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 }$ , where each $F _ { i }$ is a message passing layer. Clearly since each $F _ { i }$ is equivariant, this message passing GNN is also equivariant and produces features on each node in the space $\\mathbb { F } _ { T }$ . In order to obtain an invariant GNN, we apply an invariant function from $\\mathbb { F } _ { T } ^ { n } \\to \\mathbb { F } _ { T + 1 }$ on the output of an equivariant message passing GNN. In practice, a symmetric function is applied on the vectors of features indexed by the nodes, typically the sum of the features $\\begin{array} { r } { \\sum _ { i } ( F _ { T } \\circ . . . F _ { 2 } \\circ F _ { 1 } ( G ) ) _ { i } } \\end{array}$ is taken as an invariant feature for the graph $G$ . With our notation, $S ^ { 1 } \\circ F _ { T } \\circ . . . F _ { 2 } \\circ F _ { 1 }$ (where $S ^ { 1 }$ was defined in Section2.2) defines an invariant message passing GNN. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 176 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Hence, we define the sets of message passing GNNs as follows: ", + "bbox": [ + 176, + 181, + 591, + 196 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e7f169e6504f63723c1f47d39a41bf5bd9fa81cc6a41e797a3ecf7c27a72da97.jpg", + "text": "$$\n\\begin{array} { r c l } { \\mathbf { M G N N } _ { I } } & { = } & { \\{ m _ { I } \\circ S ^ { 1 } \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } , \\forall T \\} } \\\\ { \\mathbf { M G N N } _ { E } } & { = } & { \\{ m _ { E } \\circ ( \\mathrm { I d } + \\lambda S ^ { 1 } ) \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } , \\forall T \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 196, + 696, + 236 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $F _ { t } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n }$ are message passing layers. ", + "bbox": [ + 174, + 236, + 511, + 252 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 LINEAR GNN ", + "text_level": 1, + "bbox": [ + 174, + 266, + 312, + 281 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We define the linear graph layer of order $k$ as $F : \\mathbb { F } _ { \\ell } ^ { n ^ { k } } \\to \\mathbb { F } _ { \\ell + 1 } ^ { n ^ { k } }$ , where for all $G \\in \\mathbb { F } _ { \\ell } ^ { n ^ { k } }$ , $F ( G ) =$ $f \\left( L [ G ] \\right)$ where $L : \\mathbb { F } _ { \\ell } ^ { n ^ { k } } \\to \\mathbb { F } _ { \\ell } ^ { n ^ { k } }$ is a linear equivariant function, and $f : \\mathbb { F } _ { \\ell } \\to \\mathbb { F } _ { \\ell + 1 }$ is a learnable function applied on each of the $n ^ { k }$ features and $\\mathbb { F } _ { \\ell }$ is the field for the features at the $\\ell \\cdot$ -th layer. ", + "bbox": [ + 174, + 290, + 825, + 342 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We then define the sets of linear GNNs as follows: ", + "bbox": [ + 174, + 348, + 504, + 363 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d8381016aafebf76d992d5b76e58d33f7c2290a92a4e77a4225d806d358a9f5a.jpg", + "text": "$$\n\\begin{array} { r l r } { k \\mathrm { - } \\mathrm { L G N N } _ { I } } & { = } & { \\{ m _ { I } \\circ S ^ { k } \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } , \\forall T \\} } \\\\ { k \\mathrm { - } \\mathrm { L G N N } _ { E } } & { = } & { \\{ m _ { E } \\circ S _ { 1 } ^ { k } \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } , \\forall T \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 363, + 691, + 404 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }$ is defined in $\\ S 2 . 2$ and for $t \\geq 1$ , $F _ { t } : \\mathbb { F } _ { t } ^ { n ^ { k } } \\to \\mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ are linear equivariant layers. ", + "bbox": [ + 176, + 404, + 820, + 435 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 FOLKLORE GNN ", + "text_level": 1, + "bbox": [ + 174, + 450, + 333, + 465 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The main building block of Folklore GNN (FGNN) is what we call the folklore graph layer (FGL) of order k defined as follows: for k ≥ 1, F : Fnk\\` $F : \\mathbb { F } _ { \\ell } ^ { n ^ { k } } \\to \\mathbb { F } _ { \\ell + 1 } ^ { n ^ { k } }$ 1 where for all G ∈ Fnk\\` a nd all $\\mathbf { i } \\in [ n ] ^ { k }$ , ", + "bbox": [ + 171, + 476, + 825, + 510 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e7e658ed32fab7ca9c5488b2b5c884554f5bcc4343b0e5a5b7dd411fa9736613.jpg", + "text": "$$\nF ( G ) _ { \\mathbf { i } } = f _ { 0 } \\left( G _ { \\mathbf { i } } , \\sum _ { j = 1 } ^ { n } \\prod _ { w = 1 } ^ { k } f _ { w } \\left( G _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 290, + 512, + 673, + 561 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $f _ { 0 } : \\mathbb { F } _ { \\ell } \\times \\mathbb { F } _ { \\ell + 1 / 2 } \\to \\mathbb { F } _ { \\ell + 1 }$ and $f _ { k } : \\mathbb { F } _ { \\ell } \\to \\mathbb { F } _ { \\ell + 1 / 2 }$ are learnable functions. As shown in Lem. 33 in Appendix, FGL is an equivariant function which is indeed very expressive. ", + "bbox": [ + 169, + 563, + 821, + 592 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "F t : F n 2t $F _ { t } : \\mathbb { F } _ { t } ^ { n ^ { 2 } } \\mathbb { F } _ { t + 1 } ^ { n ^ { 2 } }$ phs , so $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ $F _ { T } \\circ \\dots F _ { 1 } \\circ F _ { 0 }$ w define 2-FGNN by composing folklore gris an equivariant GNN producing a graph in $\\mathbb { F } _ { T + 1 } ^ { n ^ { 2 } }$ yers. To obtain an invariant feature of the graph, we use the summation layer $S ^ { 2 }$ defined in Section 2.2 so that $S ^ { 2 } \\circ F _ { T } \\circ . . . F _ { 1 } \\circ F _ { 0 }$ is now an invariant 2-FGNN. In order to define general $k$ -FGNN, we first need to lift the classical graph to a tensor in $\\mathbb { F } ^ { n ^ { k } }$ , then we apply folklore graph layers of order $k$ and finally we need to project the tensor in $\\mathbb { F } ^ { n ^ { k } }$ to a tensor in $\\mathbb { F } ^ { n }$ for the equivariant version and to a tensor in $\\mathbb { F }$ for the invariant version. The first step is done with the linear equivariant function $I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }$ defined in Section 2.2. The last step is done with the reduction layer $S _ { 1 } ^ { k }$ for the equivariant case and the summation layer $S ^ { k }$ for the invariant case, both defined in Section 2.2. ", + "bbox": [ + 173, + 597, + 825, + 741 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We define the sets of folklore GNNs as follows: ", + "bbox": [ + 173, + 746, + 486, + 761 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a744d7e4a4adf2ec749aff5abe599b5fcc6e10b89f079e591e2d095b175ee188.jpg", + "text": "$$\n\\begin{array} { r l r } { k \\mathrm { – F G N N } _ { I } } & { = } & { \\{ m _ { I } \\circ S ^ { k } \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } , \\forall T \\} } \\\\ { k \\mathrm { – F G N N } _ { E } } & { = } & { \\{ m _ { E } \\circ S _ { 1 } ^ { k } \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } , \\forall T \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 761, + 689, + 801 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $F _ { t } : \\mathbb { F } _ { t } ^ { n ^ { k } } \\to \\mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ are FGLs. ", + "bbox": [ + 174, + 801, + 406, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 THEORETICAL RESULTS FOR GNNS ", + "text_level": 1, + "bbox": [ + 173, + 838, + 503, + 856 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 WEISFEILER-LEHMAN INVARIANT AND EQUIVARIANT VERSIONS ", + "text_level": 1, + "bbox": [ + 174, + 869, + 663, + 885 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We introduce a family of functions on graphs parametrized by integers $k \\geq 2$ developed for the graph isomorphism problem and working with tuples of $k$ vertices. Each $k$ -tuple $\\mathbf { i } \\in V ^ { \\dot { k } } = [ n ] ^ { k }$ is given a color $c ^ { 0 } ( \\mathbf { i } )$ corresponding to its isomorphism type (see Section B.2). The $k$ -WL test relies on the following notion of neighborhood, defined by, for any $w \\in [ k ]$ , and $\\mathbf { i } = ( i _ { 1 } , \\dots , i _ { k } ) \\in V ^ { k }$ , $N _ { w } ( { \\bf i } ) =$ $\\{ ( i _ { 1 } , \\dotsc , i _ { w - 1 } , j , i _ { w + 1 } , \\dotsc , i _ { k } ) : j \\in V \\}$ . Then, the colors of the $k$ -tuples are refined as follows, $c ^ { t + 1 } ( \\mathbf { i } ) = \\operatorname { L e x } \\left( c ^ { t } ( \\mathbf { i } ) , ( C _ { 1 } ^ { t } ( \\mathbf { i } ) , \\dots , C _ { k } ^ { t } ( \\mathbf { i } ) ) \\right)$ where, for $w \\in [ k ]$ $[ k ] , C _ { w } ^ { t } ( \\mathbf { i } ) = \\Big \\{ \\Big \\{ c ^ { t } ( \\tilde { \\mathbf { i } } ) : \\tilde { \\mathbf { i } } \\in N _ { w } ( \\mathbf { i } ) \\Big \\} \\Big \\}$ and the function Lex means that all occuring colors are lexicographically ordered and replaced by an initial segment of the natural numbers. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 196 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For a graph $G$ , let $k { \\mathrm { - W L } } _ { I } ^ { T } ( G )$ denote the multiset of colors of the $k$ -WL algorithm at the $T ^ { t h }$ iteration. After a finite number of steps (which depends on the number of vertices in the graph), the algorithm stops because a stable coloring is reached (no color class of $k$ -tuples is further divided). We denote by $k { \\mathrm { - W L } } _ { I } ( G )$ the multiset of colors in the stable coloring. This is a graph invariant that is usually used to test if graphs are isomorphic. The power of this invariant increases with $k$ Cai et al. (1989). ", + "bbox": [ + 173, + 202, + 825, + 273 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now define an equivariant version of $k$ -WL test to express the discriminatory power of equivariant architectures For this, we construct a coloring of the vertices from the coloring of the $k$ -tuples given by the standard $k$ -WL algorithm. Formally, define $k – \\mathbf { W L } _ { E } ^ { T } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } ^ { n }$ by, for $i \\in V$ : $k { \\mathrm { - W L } } _ { E } ^ { T } ( G ) _ { i } =$ $\\left\\{ \\left\\{ c ^ { T } ( \\mathbf { i } ) : \\mathbf { i } \\in V ^ { k } , i _ { 1 } = i \\right\\} \\right\\}$ . Similarly, define $k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } _ { E } ( G ) = \\left\\{ \\left\\{ c ( \\mathbf { i } ) : \\mathbf { i } \\in V ^ { k } , i _ { 1 } = i \\right\\} \\right\\}$ where $c ( \\mathbf { i } )$ is the stable coloring obtained by the algorithm. ", + "bbox": [ + 173, + 279, + 825, + 354 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 SEPARATING POWER OF GNNS ", + "text_level": 1, + "bbox": [ + 176, + 373, + 428, + 388 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We formulate our results using the equivalence relation introduced by Timofte (2005), which characterizes the separating power of a set of functions. ", + "bbox": [ + 173, + 400, + 823, + 429 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition 2. Let $\\mathcal { F }$ be a set of functions $f$ defined on a set $X$ , where each $f$ takes its values in some $Y _ { f }$ . The equivalence relation $\\rho \\left( \\mathcal { F } \\right)$ defined by $\\mathcal { F }$ on $X$ is: for any $x , x ^ { \\prime } \\in X$ , ", + "bbox": [ + 173, + 434, + 825, + 463 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/aa3dc273a72061739bf6fbbce56cc676e2f02887d47b6c736a399519af9f32a5.jpg", + "text": "$$\n( x , x ^ { \\prime } ) \\in \\rho \\left( { \\mathcal { F } } \\right) \\iff \\forall f \\in { \\mathcal { F } } , f ( x ) = f ( x ^ { \\prime } ) .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 472, + 655, + 491 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Given two sets of functions $\\mathcal { F }$ and $\\mathcal { E }$ , we say that $\\mathcal { F }$ is more separating (resp. strictly more separating) than $\\mathcal { E }$ if $\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( { \\mathcal { E } } \\right)$ (resp. $\\rho \\left( { \\mathcal { F } } \\right) \\subsetneq \\bar { \\rho ( \\mathcal { E } ) } )$ . Note that all the functions in $\\mathcal { F }$ and $\\mathcal { E }$ need to be defined on the same set but can take values in different sets. For example, we can easily see that for the $k$ -WL algorithm defined above, the equivariant version is more separating than the invariant one. ", + "bbox": [ + 174, + 506, + 825, + 563 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Some properties of the WL hierarchy of tests can be rephrased with the notion of separating power. In particular, Cai et al. (1989) showed that $( k + 1 ) \\ – \\mathbf { W } \\mathbf { L } _ { I }$ distinguishes strictly more than $k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } _ { I }$ , which can be rewritten simply as (for a function $f$ , we write $\\rho \\left( f \\right)$ for $\\rho \\left( \\left\\{ f \\right\\} \\right) ,$ ", + "bbox": [ + 174, + 569, + 826, + 612 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9baf958ba7bd01950cd74622f6d97fced58c1f937e57c1ae4de760f6a0fed1ee.jpg", + "text": "$$\n\\rho \\left( { ( k + 1 ) } { - } \\mathbf { W } \\mathbf { L } _ { I } \\right) \\subsetneq \\rho \\left( k { - } \\mathbf { W } \\mathbf { L } _ { I } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 621, + 609, + 638 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This notion of separating power enables us to concisely summarize the current knowledge about the discriminatory power of classes of GNN. ", + "bbox": [ + 176, + 647, + 825, + 675 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 3. We have, for $k \\geq 2$ , ", + "bbox": [ + 174, + 681, + 408, + 696 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0ad3c4570af4e4e7eccbc889a029a90cb3a7dc28ac37a8b37f26d70ae2528aa1.jpg", + "text": "$$\n\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) = \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { E } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 704, + 758, + 760 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Only results about the invariant cases were previously known: (4) comes from $\\mathrm { X u }$ et al. (2018), (5) from Maron et al. (2018) Geerts (2020a) and one inclusion of (6) comes from Maron et al. (2019a). The equality in (6) for general $k \\geq 2$ is proved in Section C. ", + "bbox": [ + 176, + 773, + 825, + 818 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that for $k = 2$ , all GNNs are dealing with tensors of order 2 i.e. with the adjacency matrix of the graph. However, the complexities of the various layers are quite different: for the message passing GNN, all computations are local (scaling with the maximum degree in the graph) and can be done in parallel; for the linear layer, there are only 15 linear functions from $\\mathbb { R } ^ { n ^ { 2 } } \\to \\mathbb { R } ^ { n ^ { 2 } }$ for all values of $n$ (Maron et al., 2018); the folklore layer involves a (dense) matrix multiplication of shape $n \\times n$ . If 2-FGNN is the most complex architecture, we see that it has the best separating power among all architectures proposed so far dealing with tensors of order 2. ", + "bbox": [ + 173, + 823, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For $X , Y$ finite-dimensional spaces, let us denote by $\\mathcal { C } _ { I } ( X , Y ) , \\mathcal { C } _ { E } ( X , Y )$ , , the set of invariant, respectively equivariant, continuous functions from $X$ to $Y$ . The closure of a class of function $\\mathcal { F }$ for the uniform norm is denoted by $\\overline { { \\mathcal { F } } }$ . Our result extend easily to graphs of varying sizes but this is deferred to Section F.2 for clarity. ", + "bbox": [ + 173, + 128, + 826, + 185 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The theorem below states in particular that the class $k$ -FGNN can approximate any continuous function that is less separating than $( k + 1 )$ -WL in the invariant and in the equivariant cases. ", + "bbox": [ + 173, + 191, + 823, + 222 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 4. Let $K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }$ , $K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ be compact sets. For the invariant case, we have: ", + "bbox": [ + 174, + 223, + 807, + 242 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/16f667e67876d4cc8a9da81a88fff721b73209be35db8a2b53edba1076f7a893.jpg", + "text": "$$\n\\begin{array} { r l r } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( 2 \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\mathrm { - } L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\mathrm { - } F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 243, + 691, + 304 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the equivariant case, we have: ", + "bbox": [ + 173, + 304, + 400, + 319 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/e1989546bf6a79c24c53ac33c959e55f6831fa898bb4bf80d0a859a679de07db.jpg", + "text": "$$\n\\begin{array} { r c l } { \\underline { { M G N N _ { E } } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K _ { d i s c r } , \\mathbb F ^ { n } ) : \\rho \\left( 2 ^ { - } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\underline { { \\cdot } L G N N _ { E } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k \\cdot L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} \\supset \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k \\cdot W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\cdot \\overline { { F G N N _ { E } } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( ( k + 1 ) \\cdot W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 319, + 839, + 380 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the invariant case for $k = 2$ ,we have $\\mathbf { M G N N } _ { I } = \\overline { { 2 \\mathrm { - } \\mathbf { L G N N } _ { I } } } \\subsetneq \\overline { { 2 \\mathrm { - } \\mathbf { F G N N } _ { I } } }$ where the strictness of the last inclusion comes from (3). In other words, ${ \\mathrm { 2 - F G N N } } _ { I }$ has a better power of approximation than the other architectures working with tensors of order 2. We already knew by Proposition 3 that ${ \\mathrm { 2 - F G N N } } _ { I }$ is the best separating architecture among those studied in this paper, dealing with tensor of order 2 and our theorem implies that this is also the case for the approximation power. ", + "bbox": [ + 173, + 386, + 826, + 458 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To clarify the meaning of these statements, we explain why the inclusions $\\mathbf { \\check { \\Psi } } \\subseteq \\mathbf { \\Psi }$ are actually straightforward. For concreteness, we focus on $\\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } } \\ \\subseteq \\ \\left\\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } \\mathrm { W L } _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\right\\}$ . Take $h ~ \\in ~ \\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } }$ , this means that there is a sequence $\\mathbf { G } \\mathbf { N } \\mathbf { N } _ { j } \\in k \\mathrm { \\mathrm { - } } \\mathbf { F } \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { I }$ such that, $\\mathrm { s u p } _ { G \\in K } \\| h ( G ) - \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { j } ( G ) \\|$ goes to zero when $j$ goes to infinity. ", + "bbox": [ + 173, + 464, + 826, + 525 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Therefore, $h$ is continuous and constant on each $\\rho \\left( k { \\mathrm { - F G N N } } _ { I } \\right)$ -class. Indeed, for any $( G , G ^ { \\prime } ) \\in$ $\\rho \\left( k { \\mathrm { - } } \\mathrm { F G N N } _ { I } \\right)$ , $\\mathbf { G N N } _ { j } ( G ) = \\mathbf { G N N } _ { j } ( G ^ { \\prime } )$ so that $h ( G ) = \\mathrm { l i m } _ { i } { \\bf G } { \\bf N } { \\bf N } _ { j } ( G ) = \\mathrm { l i m } _ { j } { \\bf G } { \\bf N } { \\bf N } _ { j } ( G ^ { \\prime } ) =$ $h ( G ^ { \\prime } )$ . Hence we have $\\rho \\left( k { \\mathrm { - F G N N } } _ { I } \\right) \\subseteq \\rho \\left( h \\right)$ and by Prop. 3, $\\rho \\left( k { \\mathrm { - } } \\mathrm { F G N N } _ { I } \\right) = \\rho \\left( ( k + 1 ) { \\mathrm { - } } \\mathrm { W L } _ { I } \\right)$ , allowing us to get the inclusion above. ", + "bbox": [ + 174, + 530, + 825, + 587 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "On the contrary, the reverse inclusions $\\cdot \\bigcirc ^ { \\prime \\prime }$ are much more intricate but they are also the most valuable. For instance, consider the inclusion $\\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } } \\supset \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } \\mathrm { W L } _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\}$ . If one wishes to learn a function $h \\in { \\mathcal { C } } _ { I } ( K , \\mathbb { F } )$ with $k { \\mathrm { - } } \\mathrm { F G N N } _ { I }$ , this function must at least be approximable by the class of $k { \\mathrm { - F G N N } } _ { I }$ . Our theorem precisely guarantees that if $h$ is less separating that $k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } _ { I }$ , it can be approximated by $k { \\mathrm { - F G N N } } _ { I }$ : ", + "bbox": [ + 173, + 593, + 826, + 665 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/ff04799ea56c8034a8553f9bbd7340bb36bf1df903d57bf77f028e4df5c1159f.jpg", + "text": "$$\n\\forall \\epsilon > 0 , \\exists \\mathbf { G } \\mathbf { N } \\mathbf { N } \\in k \\mathbf { \\mathrm { \\mathrm { - } } } \\mathbf { F } \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { I } , \\mathbf { \\Pi } _ { G \\in K } \\| h ( G ) - \\mathbf { G } \\mathbf { N } \\mathbf { N } ( G ) \\| \\leq \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 666, + 709, + 693 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For this, we show a much more general version of the famous Stone-Weierstrass theorem (see Section D) which relates the separating power with the approximation power. Following the elegant idea of Maehara & Hoang (2019), we augment the input space to transform vector-valued equivariant functions into scalar invariant maps. Then, we apply a fine-grained approximation theorem from Timofte (2005).We also provide specialized versions of our abstract theorem in Section 5, which can be easily used to determine the approximation capabilities of any deep learning architecture. ", + "bbox": [ + 173, + 694, + 825, + 779 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our theorem has also implications for universality results like Maron et al. (2019b); Keriven & Peyré (2019). A class of GNN is said to be universal if its closure on a compact set $K$ is the whole $\\mathcal { C } _ { I } ( K , \\mathbb { F } )$ (or $\\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { n } ) )$ ). In particular, Thm. 4 implies that $n$ -LGNN and $n$ -FGNN are universal as $n$ -WL distinguishes non-isomorphic graphs of size $n$ . This recovers a result of Ravanbakhsh (2020) for LGNN. Moreover, we can leverage the extensive literature on the WL tests to give more subtle√ results. For instance, Cai et al. (1989, $\\ S 8 . 2 )$ show that, for planar graphs, $O ( { \\sqrt { n } } )$ -WL can distinguish non-isomoprhic instances. Therefore, $O ( \\sqrt { n } )$ -LGNN or $\\mathbf { \\bar { O } } ( { \\sqrt { n } } ) $ -FGNN achieve universality in the particular, yet common, case of planar graphs. On a more practical side, Fürer (2010, Thm. 4.5) shows that the spectrum of a graph is less separating than 3-WL so that functions of the spectrum can actually be well approximated by 2-FGNN. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPRESSIVENESS OF GNNS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 441, + 118 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now state the general theorems which are our main tools in proving our approximation guarantees for GNNs. Theirs proofs are deferred to Section D.9 which contains our generalization of the StoneWeierstrass theorem with symmetries. We need to first introduce more general definitions: If $G$ is a finite group acting on some topological space $X$ , we say that $G$ acts continuously on $X$ if, for all $g \\in G$ , $x \\mapsto g \\cdot x$ is continuous. If $G$ is a finite group acting on some compact set $X$ and some topological space $Y$ , we define the sets of equivariant and invariant continuous functions by, ", + "bbox": [ + 173, + 132, + 826, + 218 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/2ac7d4cb09dc16302a9ddf1068259e35c1e0efc44f1ac821016dc286cbf91083.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { C } _ { E } ( X , Y ) = \\{ f \\in \\mathcal { C } ( X , Y ) : \\forall x \\in X , \\forall g \\in G , f ( g \\cdot x ) = g \\cdot f ( x ) \\} } \\\\ & { \\mathcal { C } _ { I } ( X , Y ) = \\{ f \\in \\mathcal { C } ( X , Y ) : \\forall x \\in X , \\forall g \\in G , f ( g \\cdot x ) = f ( x ) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 222, + 728, + 260 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Note that these definitions extend Definition 1 to a general group. ", + "bbox": [ + 173, + 263, + 602, + 279 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 5. Let $X$ be a compact space, $\\mathbb { F } = \\mathbb { R } ^ { p }$ be some finite-dimensional vector space, $G$ be a finite group acting (continuously) on $X$ . ", + "bbox": [ + 171, + 282, + 823, + 311 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Let $\\textstyle { \\mathcal { F } } _ { 0 } \\subseteq \\bigcup _ { h = 1 } ^ { \\infty } { \\mathcal { C } } _ { I } ( X , \\mathbb { R } ^ { h } )$ be a non-empty set of invariant functions, stable by concatenation, and consider, ", + "bbox": [ + 171, + 318, + 825, + 347 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/8d34581ed3a7abea357c426b47d31508df5c1f5fb95a6affd0874c735b33d5a8.jpg", + "text": "$$\n\\mathcal { F } = \\{ m \\circ f : f \\in \\mathcal { F } _ { 0 } \\cap \\mathcal { C } ( X , \\mathbb { R } ^ { h } ) , m : \\mathbb { R } ^ { h } \\to \\mathbb { F } M L P , h \\geq 1 \\} \\subseteq \\mathcal { C } ( X , \\mathbb { F } ) .\n$$", + "text_format": "latex", + "bbox": [ + 248, + 351, + 746, + 371 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ is, ", + "bbox": [ + 173, + 383, + 336, + 398 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/08c9a6f5fed1d948af94274f1e35938867f3f5bbdfdea98e54744a5fd142b60a.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\{ f \\in \\mathcal { C } _ { I } ( X , \\mathbb { F } ) : \\rho \\left( \\mathcal { F } _ { 0 } \\right) \\subseteq \\rho \\left( f \\right) \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 404, + 632, + 421 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We can apply Theorem 5 to the class of $k$ -ary relational pooling GNN introduced in Murphy et al. (2019a). As a result, we get that this class of invariant $k$ -RP GNN can approximate any continuous function $f$ with $\\rho ( k - \\mathrm { R P G N N } ) \\subseteq \\rho ( f )$ but to the best of our knowledge, $\\rho ( k - \\mathrm { R P G N N } )$ is not known and only $\\rho ( k - \\mathrm { R P G N N } ) \\subset \\rho ( 2 - W L _ { I } )$ is proved in Murphy et al. (2019a). We now state our general theorem for the equivariant case: ", + "bbox": [ + 173, + 434, + 825, + 506 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 6. Let $X$ be a compact space, $\\mathbb { F } = \\mathbb { R } ^ { p }$ and $G = S _ { n }$ the permutation group, acting (continuously) on $X$ and acting on $\\mathbb { F } ^ { n }$ by, for $\\sigma \\in S _ { n }$ , $x \\in \\mathbb { F } ^ { n }$ , ", + "bbox": [ + 171, + 508, + 823, + 537 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/a8e830db7e3ec8a0726ee2f7ffad843ffcbe9401fc5b0a150f44e1b9d5529bc2.jpg", + "text": "$$\n\\forall i \\in \\{ 1 , \\ldots , p \\} , ( \\sigma \\cdot x ) _ { i } = x _ { \\sigma ^ { - 1 } ( i ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 542, + 619, + 563 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Let ${ \\mathcal { F } } _ { 0 } \\subseteq \\bigcup _ { h = 1 } ^ { \\infty } { \\mathcal { C } } _ { E } \\left( X , ( \\mathbb { R } ^ { h } ) ^ { n } \\right)$ be a non-empty set of equivariant functions, stable by concatenation, and consider, ", + "bbox": [ + 173, + 575, + 825, + 606 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/42300d20a8fdf58f334964d84b54e06ffa96164da13265bcdcb1e45afbc8524e.jpg", + "text": "$$\n{ \\mathcal { F } } = \\{ x \\mapsto ( m ( f ( x ) _ { 1 } ) , \\ldots , m ( f ( x ) _ { n } ) ) : f \\in { \\mathcal { F } } _ { 0 } \\cap { \\mathcal { C } } \\left( X , ( \\mathbb { R } ^ { h } ) ^ { n } \\right) , m : \\mathbb { R } ^ { h } \\to \\mathbb { F } M L P , h \\geq 1 \\}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 609, + 810, + 630 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Assume, that, if $f \\in \\mathcal { F } _ { 0 }$ , then, ", + "bbox": [ + 173, + 642, + 367, + 657 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/a71f9a7ca3bc8d68beddac024ecfd07ec2f6199a9abeb4c1f4c2320994819fec.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\in { \\mathcal { F } } _ { 0 } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 662, + 671, + 705 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ is, ", + "bbox": [ + 173, + 718, + 336, + 733 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/284ecee200741021c750dd9f75de53de55aa44c2476eda6bef0a662350bba65e.jpg", + "text": "$$\n\\overline { { \\mathcal { F } } } = \\{ f \\in \\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } ) : \\rho \\left( \\mathcal { F } _ { 0 } \\right) \\subseteq \\rho \\left( f \\right) \\} \\ .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 738, + 637, + 757 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Applications of these theorems fo the case of Pointnet Qi et al. (2017) are provided in Section D.9 ", + "bbox": [ + 171, + 768, + 813, + 785 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 QUADRATIC ASSIGNMENT PROBLEM ", + "text_level": 1, + "bbox": [ + 174, + 804, + 514, + 821 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To empirically evaluate our results, we study the Quadratic Assignment Problem (QAP), a classical problem in combinatorial optimization. For $A , B n \\times n$ symmetric matrices, it consists in solving ", + "bbox": [ + 173, + 835, + 823, + 866 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/5334911e11d0f8b7a3f1099520a79f1d58ee1c90cbdc28a009f40d34130160e5.jpg", + "text": "$$\n{ \\mathrm { m a x i m i z e ~ t r a c e } } ( A X B X ^ { \\top } ) , { \\mathrm { s u b j e c t ~ t o ~ } } X \\in \\Pi ,\n$$", + "text_format": "latex", + "bbox": [ + 323, + 871, + 637, + 890 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\Pi$ is the set of $n \\times n$ permutation matrices. Many optimization problems can be formulated as QAP. An example is the network alignment problem, which consists in finding the best matching between two graphs, represented by their adjacency matrices $A$ and $B$ . Though QAP is known to be NP-hard, recent works such as Nowak et al. (2018) have investigated whether it can be solved efficiently w.r.t. a fixed input distribution. More precisely, Nowak et al. (2018) studied whether one can learn to solve this problem using a MGNN trained on a dataset of already solved instances. However, as shown below, both the baselines and their approach fail on regular graphs, a class of graph considered as particularly hard for isomorphism testing. ", + "bbox": [ + 171, + 895, + 826, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To remedy this weakness, we consider $2 { \\mathrm { - } } \\mathrm { F G N N } _ { E }$ . We then follow the siamese method of (Nowak et al., 2018): given two graphs, our system produces an embedding in $\\mathbb { F } ^ { n }$ for each graph, where $n$ is the number of nodes, which are then multiplied together to obtain a $n \\times n$ similarity matrix on nodes. A permutation is finally computed by solving a Linear Assignment Problem (LAP) with this resulting $n \\times n$ as cost matrix. We tested our architecture on two distribution: the Erdos–Rényi model and ˝ random regular graphs. The accuracy in matching the graphs is much improved compare to previous works. The experimental setup is described more precisely in Section A.1. ", + "bbox": [ + 173, + 194, + 825, + 292 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/3de1a936c4bf2c8959509d03db9c9edaefbda37bf89260168626ddac22c6232f.jpg", + "image_caption": [ + "Figure 1: Fraction of matched nodes for pairs of correlated graphs (with edge density 0.2) as a function of the noise, see Section A.1 for details. " + ], + "image_footnote": [], + "bbox": [ + 183, + 306, + 828, + 563 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 631, + 318, + 647 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We derived the expressive power of various practical GNN architectures: message passing GNN, linear GNN and folklore GNN; both for their invariant and equivariant counterparts. Our results unify and extend the recent works in this direction. In particular, we are able to recover all the universality results proved for GNNs so far. Similarly to existing results in the literature, we do not deal here with the sizes of the embeddings constructed at different layers, i.e. the sizes of the spaces $\\mathbb { F } _ { \\ell }$ , and these sizes are supposed to grow to infinity with the number of nodes $n$ in the graph. Obtaining bounds on the scaling of the sizes of the features to ensure that the results presented here are still valid is an interesting open question. We show that folklore GNNs have the best power of approximation among all GNNs studied here dealing with tensors of order 2. From a practical perspective, we demonstrate their improved performance on the QAP with a significant gap in performances compared to other approaches. ", + "bbox": [ + 173, + 662, + 825, + 815 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 837, + 356, + 852 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work was supported in part by the French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR19-P3IA-0001 (PRAIRIE 3IA Institute). M.L. thanks Google for Google Cloud Platform research credits and NVIDIA for a NVIDIA GPU Grant. 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", + "bbox": [ + 173, + 395, + 825, + 465 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A Experimental results 14 ", + "text_level": 1, + "bbox": [ + 176, + 103, + 825, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 Details on the experimental setup 14 \nA.2 Experimental results on graphs of varying size . 14 \nA.3 Generalization for regular graphs . 15 ", + "bbox": [ + 197, + 126, + 826, + 190 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B Weisfeiler-Lehman tests 16 ", + "text_level": 1, + "bbox": [ + 174, + 218, + 825, + 233 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.1 Weisfeiler-Lehman test on vertices 16 \nB.2 Isomorphism type . 16 \nB.3 Weisfeiler-Lehman and Folklore Weisfeiler-Lehman tests of order $k \\geq 2$ . 17 ", + "bbox": [ + 196, + 241, + 826, + 304 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C Separating power of GNN 18 ", + "text_level": 1, + "bbox": [ + 173, + 330, + 823, + 347 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C.1 Mutli-linear perceptrons 18 \nC.2 Augmented separating power 18 \nC.3 Initialization layer . 19 \nC.4 Known results about the separating power of some GNN classes 20 \nC.5 Bounding the separating power of $k$ -FGNN 20 \nC.6 Conclusion 21 ", + "bbox": [ + 197, + 356, + 826, + 488 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D Stone-Weierstrass theorem with symmetries 21 ", + "text_level": 1, + "bbox": [ + 173, + 516, + 823, + 531 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D.1 General notations . 22 \nD.2 Separating power 22 \nD.3 Approximation theorems for real-valued functions . 23 \nD.4 The equivariant approximation theorem 24 \nD.5 A preliminary version of the equivariant approximation theorem 26 \nD.6 Characterizing the subalgebras of $\\mathbb { R } ^ { p }$ 27 \nD.7 Proof of the main equivariant approximation theorem 29 \nD.8 Practical reductions 30 \nD.9 Reductions for GNNs 33 ", + "bbox": [ + 196, + 537, + 826, + 743 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "E Proofs for expressiveness of GNNs 34 ", + "text_level": 1, + "bbox": [ + 173, + 771, + 823, + 786 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "E.1 Expressivity of GNN layers . 34 \nE.2 Approximation theorems for GNNs 35 ", + "bbox": [ + 196, + 794, + 826, + 834 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "F Extension to graphs of varying sizes 37 ", + "text_level": 1, + "bbox": [ + 171, + 862, + 823, + 877 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "F.1 Extension to disconnected input spaces . . 37 \nF.2 Approximation theorem with varying graph size . . . 39 ", + "bbox": [ + 194, + 885, + 826, + 925 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 178, + 102, + 421, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 DETAILS ON THE EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 174, + 133, + 496, + 147 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We consider a $2 { \\mathrm { - } } \\mathrm { F G N N } _ { E }$ and train it to solve random planted problem instances of the QAP. Given a pair of graphs $G _ { 1 } , G _ { 2 }$ with $n$ nodes each, we consider the siamese $2 { \\mathrm { - } } \\mathrm { F G N N } _ { E }$ encoder producing embeddings $E _ { 1 } , E _ { 2 } \\in \\mathbb { R } ^ { n \\times k }$ . Those embeddings are used to predict a matching as follows: we first compute the outer product $E _ { 1 } E _ { 2 } ^ { T }$ , then we take a softmax along each row and use standard cross-entropy loss to predict the corresponding permutation index. We used $2 { \\mathrm { - } } \\mathrm { F G N N } _ { E }$ with 2 layers, each MLP having depth 3 and hidden states of size 64. We trained for 25 epochs with batches of size 32, a learning rate of 1e-4 and Adam optimizer. The PyTorch code is available in the supplementary material. ", + "bbox": [ + 173, + 159, + 825, + 271 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For each experiment, the dataset was made of 20000 graphs for the train set, 1000 for the validation set and 1000 for the test set. For the experiment with Erdos–Rényi random graphs, we consider ˝ $G _ { 1 }$ to be a random Erdos–Rényi graph with edge density˝ $p _ { e } = 0 . 2$ and $n = 5 0$ vertices. The graph $G _ { 2 }$ is a small perturbation of $G _ { 1 }$ according to the following error model considered in Feizi et al. (2016): ", + "bbox": [ + 174, + 277, + 825, + 333 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d3edbe612f8ab83349ead7085b01a5b048f3115354ca1005a051648b2a2935c3.jpg", + "text": "$$\nG _ { 2 } = G _ { 1 } \\odot ( 1 - Q ) + ( 1 - G _ { 1 } ) \\odot Q ^ { \\prime } ,\n$$", + "text_format": "latex", + "bbox": [ + 351, + 339, + 612, + 357 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $Q$ and $Q ^ { \\prime }$ are Erdos–Rényi random graphs with edge density ˝ $p _ { 1 }$ and $p _ { 2 } = p _ { 1 } p _ { e } / ( 1 - p _ { e } )$ respectively, so that $G _ { 2 }$ has the same expected degree as $G _ { 1 }$ . The noise level is the parameter $p _ { 1 }$ . For regular graphs, we followed the same experimental setup but now $G _ { 1 }$ is a random regular graph with degree $d = 1 0$ . Regular graphs are interesting example as they tend to be considered harder to align due to their more symmetric structure. ", + "bbox": [ + 174, + 362, + 825, + 433 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 EXPERIMENTAL RESULTS ON GRAPHS OF VARYING SIZE ", + "text_level": 1, + "bbox": [ + 174, + 450, + 602, + 463 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/68e538d254f564e7cec0985a6c326e6ed430e58c68d1e39f3b9e6b48208c6669.jpg", + "image_caption": [ + "Figure 2: Fraction of matched nodes for pairs of correlated graphs (with edge density 0.2) as a function of the noise, see Section A.1 for details. " + ], + "image_footnote": [], + "bbox": [ + 269, + 479, + 720, + 786 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We tested our models on dataset of graphs of varying size, as this setting is also encompassed by our theory. ", + "bbox": [ + 176, + 845, + 821, + 875 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "However, contrary to message-passing GNN, GNN based on tensors do not work well with batches of graphs of varying size. Previous implementations, such as the one of Maron et al. (2019a), group the graphs in the dataset by size, enabling the GNN to only deal with batches of graphs on the same size. ", + "bbox": [ + 176, + 881, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/030b6bc0ced87c1a9d34f138c1bc1289d003a176c8c11c4142fe35d0382d1684.jpg", + "image_caption": [ + "Figure 3: Each line corresponds to a model trained at a given noise level and shows its accuracy across all noise levels. " + ], + "image_footnote": [], + "bbox": [ + 241, + 103, + 767, + 527 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Instead, we use masking, which is a standard practice in recurrent neural networks. A batch of $b$ tensors of sizes $n _ { 1 } \\times n _ { 1 } , n _ { 2 } \\times n _ { 2 } , . . . , n _ { b } \\times n _ { b }$ is represented as a tensor $b \\times n _ { m a x } \\times n _ { m a x }$ where $n _ { m a x } = \\mathrm { m a x } _ { i = 1 , \\dots , b } n _ { i }$ . A mask is created at initialization and is used to ensure that the operations on the full tensor translates to valid operations on each of the individual tensor. ", + "bbox": [ + 174, + 608, + 825, + 665 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We implemented this functionnality as a class MaskedTensors. Thanks to the newest improvements of PyTorch (Paszke et al., 2019), MaskedTensors act as a subclass of fundamental Tensor class. Thus they almost seamlessly integrate into standard PyTorch code. We refer the reader to the code for more details: https://github.com/mlelarge/graph_neural_net ", + "bbox": [ + 174, + 671, + 825, + 728 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Results of our architecture and implementation with graphs of varying size are shown below on Figure 2. The only difference with the setting described above is that the number of nodes is now random. The number of vertices of a graph is indeed chosen randomly according to binomial distribution of parameters $n = 5 0$ and $p _ { n } = 0 . 9$ . ", + "bbox": [ + 174, + 734, + 825, + 791 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.3 GENERALIZATION FOR REGULAR GRAPHS ", + "text_level": 1, + "bbox": [ + 174, + 823, + 508, + 835 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We made the following experiment with the same general setting as in Section A.1 with regular graphs. We trained different models for all noise levels between 0 and 0.22 but in Figure 3, we plot the accuracy of each model across all noise levels. We observe that a majority of the models actually generalize to settings with noise level on which they were not trained. Indeed, the model trained with noise level $\\approx 0 . 1$ is performing best among all models across all noise levels! ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B WEISFEILER-LEHMAN TESTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 450, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Here we describe more precisely this hierarchy of tests, which will be used extensively to characterize the discirminatory power of classes of GNN. See Douglas (2011); Grohe (2017); Fürer (2017) for graph-theoretic introductions to these algorithms. ", + "bbox": [ + 174, + 133, + 825, + 176 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.1 WEISFEILER-LEHMAN TEST ON VERTICES ", + "text_level": 1, + "bbox": [ + 173, + 193, + 511, + 208 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We now present the initial vertex coloring algorithm. ", + "bbox": [ + 173, + 218, + 519, + 233 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Input. This algorithm takes as input a discrete graph structure $G = ( V , E ) \\in \\mathcal { G } _ { n }$ with $V = [ n ]$ , $E \\subseteq V ^ { 2 }$ and $h \\in \\mathbb { F } _ { 0 } ^ { n }$ features on the vertices. ", + "bbox": [ + 173, + 247, + 826, + 277 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Initialization. Each vertex $s \\in V$ is given a color $c _ { \\mathrm { W L } } ^ { 0 } ( G , h ) _ { s } = h _ { s }$ corresponding to its features vector. ", + "bbox": [ + 174, + 291, + 823, + 321 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Refining the coloring. The colors of the vertices are updated as follows, ", + "bbox": [ + 173, + 335, + 663, + 352 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/196640d3f4c82db6fed6cff254544afec733e8d2932d34fe5301a8d84fe638fd.jpg", + "text": "$$\nc _ { \\mathrm { W L } } ^ { t + 1 } ( G , h ) _ { s } = \\mathrm { L e x } \\left( c _ { \\mathrm { W L } } ^ { t } ( G , h ) _ { s } , \\{ \\{ c _ { \\mathrm { W L } } ^ { t } ( G , h ) ( \\tilde { s } ) : \\tilde { s } \\sim s \\} \\} \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 294, + 356, + 700, + 376 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "and the function Lex means that all occuring colors are lexicographically ordered. ", + "bbox": [ + 176, + 381, + 709, + 397 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For each graph $G \\in \\mathcal G _ { n }$ and each vector of features $h \\in \\mathbb { F } _ { 0 } ^ { n }$ , there exists a time $T ( G , h )$ from which the sequence of colorings $( c _ { \\mathrm { W L } } ^ { t } ( G , h ) ) _ { t \\geq 0 }$ is stationary. More exactly, the colorings are not refined anymore: for any $t \\geq T ( G , h ) , s , s ^ { \\prime } \\in \\bar { V }$ , ", + "bbox": [ + 174, + 402, + 825, + 446 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/02e5b1b8a4a10d9ecd1aa66ee50343e2ebc85ccdc400e83104810693345a3524.jpg", + "text": "$$\nc _ { \\mathrm { W L } } ^ { t } ( G , h ) _ { s } = c _ { \\mathrm { W L } } ^ { t } ( G , h ) _ { s ^ { \\prime } } \\longleftrightarrow c _ { \\mathrm { W L } } ^ { T ( G , h ) } ( G , h ) _ { s } = c _ { \\mathrm { W L } } ^ { T ( G , h ) } ( G , h ) _ { s ^ { \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 453, + 727, + 474 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Denote the resulting coloring cT (G,h)WL (G, h) by simply cWL(G, h). cWL is now a mapping from $\\mathcal { G } _ { n } \\times \\mathbb { F } _ { 0 } ^ { n } \\to Z ^ { n }$ for some space of colors $Z$ . ", + "bbox": [ + 174, + 481, + 823, + 512 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Invariant tests The proper Weisfeiler-Lehman test is invariant and is defined by, for $t \\geq 0$ and $( G , h ) \\in \\mathcal { G } _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }$ a graph, ", + "bbox": [ + 168, + 526, + 825, + 558 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c5cd809e0529fba5b151e9f721aba410e0a9a6119aaef283e45f040e1fd7f1f7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\operatorname { W L } _ { I } ^ { t } ( G ) = \\{ \\{ c _ { \\mathrm { W L } } ^ { t } ( G ) _ { s } : s \\in V \\} \\} } \\\\ & { \\operatorname { W L } _ { I } ( G ) = \\{ \\{ c _ { \\mathrm { W L } } ( G ) _ { s } : s \\in V \\} \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 561, + 612, + 603 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Equivariant tests For the vertex coloring algorithm, $c _ { \\mathrm { W L } }$ is already an equiavriant mapping so we define, for $t \\geq 0$ and $( G , h ) \\in \\mathcal { G } _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }$ a graph, ", + "bbox": [ + 173, + 632, + 823, + 661 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/3b816ab81504c732cdf8bf1ebdb7e2465674e8272ffca96f0366c65db8448bc1.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { W L } _ { E } ^ { t } = c _ { \\mathrm { W L } } ^ { t } } \\\\ { \\mathrm { W L } _ { E } = c _ { \\mathrm { W L } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 452, + 666, + 544, + 703 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.2 ISOMORPHISM TYPE ", + "text_level": 1, + "bbox": [ + 174, + 736, + 357, + 751 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The initialization of the higher-order variants of the Weisfeiler-Lehman test is slightly more intricate. For this we need to define the isomorphism type of a $k$ -tuple w.r.t. a graph described by a tensor $\\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ ", + "bbox": [ + 173, + 761, + 826, + 794 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A $k$ -tuple $( i _ { 1 } , \\dots , i _ { k } ) \\in [ n ] ^ { k }$ in a graph $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ and a $k$ -tuple $( j _ { 1 } , \\ldots , j _ { k } ) \\in [ n ] ^ { k }$ in a graph $H \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ are said to have the same isomoprhism type if the mapping $i _ { w } \\mapsto j _ { w }$ is a well-defined partial isormophism. Explicitly, this means that, ", + "bbox": [ + 173, + 800, + 826, + 848 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/503deb025b66cf785a5aa72ad92159a931ca86e1e494e5429fd82d683d208e26.jpg", + "text": "$$\n\\begin{array} { r l } & { \\bullet \\forall w , w ^ { \\prime } \\in [ k ] , i _ { w } = i _ { w ^ { \\prime } } \\iff j _ { w } = j _ { w ^ { \\prime } } . } \\\\ & { \\bullet \\forall w , w ^ { \\prime } \\in [ k ] , G _ { i _ { w } , i _ { w ^ { \\prime } } } = H _ { j _ { w } , j _ { w ^ { \\prime } } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 858, + 498, + 897 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Denote by $\\operatorname { i s o } ( G ) _ { i _ { 1 } , \\dots , i _ { k } }$ the isomorphism type of the $k$ -tuple $( i _ { 1 } , \\dots , i _ { k } ) \\in [ n ] ^ { k }$ in a graph $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ ", + "bbox": [ + 171, + 907, + 825, + 925 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.3 WEISFEILER-LEHMAN AND FOLKLORE WEISFEILER-LEHMAN TESTS OF ORDER $k \\geq 2$ ", + "bbox": [ + 169, + 103, + 815, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We now present the folklore version of the Weisfeiler-Lehman test of order $k$ ( $k$ -FWL), for $k \\geq 2$ , along with $k$ -WL for clarity. For both, we follow the presentation of Maron et al. (2019a) (except for the equivariant tests). ", + "bbox": [ + 173, + 128, + 826, + 172 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Input. These algorithms take as input a graph $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ which can be seen as a coloring on the pair of nodes. ", + "bbox": [ + 173, + 188, + 825, + 219 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Initialization. Each $k$ -tuple $s \\in V ^ { k }$ is given a color $c _ { k \\mathrm { - } \\mathrm { W L } } ^ { 0 } ( G ) _ { s } = c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { 0 } ( G ) _ { s }$ corresponding to its isomorphism type. ", + "bbox": [ + 173, + 233, + 823, + 263 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "$k$ -WL. The $k$ -WL test relies on the following notion of neighborhood, defined by, for any $w \\in [ k ]$ , and $s = ( i _ { 1 } , \\ldots , i _ { k } ) \\in V ^ { k }$ , ", + "bbox": [ + 171, + 277, + 825, + 309 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/b0cea0df1d335915b98d56959cb3b3f3cb82b4b9f2811b8af1a31933fe60e6a7.jpg", + "text": "$$\nN _ { w } ( s ) = \\left\\{ ( i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } ) : j \\in V \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 314, + 671, + 332 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Then, the colors of the $k$ -tuples $s \\in V ^ { k }$ are refined as follows, ", + "bbox": [ + 173, + 357, + 581, + 373 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d78ce16d6697822bcecb59fb56f7845cf7552927cc7c976861a3fe06e9dea723.jpg", + "text": "$$\nc _ { \\mathrm { W L } } ^ { t + 1 } ( G ) _ { s } = \\mathrm { L e x } \\left( c _ { k \\mathrm { - } \\mathrm { W L } } ^ { t } ( G ) _ { s } , ( C _ { 1 } ^ { t } ( s ) , \\ldots , C _ { k } ^ { t } ( s ) ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 378, + 671, + 398 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where, for $w \\in [ k ]$ , ", + "bbox": [ + 173, + 405, + 302, + 421 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/b13b09f22a571dfaf4adbbc908ea0db4dac836295029a65b6e1b46d1d0761506.jpg", + "text": "$$\nC _ { w } ^ { t } ( s ) = \\left\\{ \\left\\{ c _ { k - \\mathrm { W L } } ^ { t } ( G ) _ { \\tilde { s } } : \\tilde { s } \\in N _ { w } ( s ) \\right\\} \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 419, + 630, + 439 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For each graph $G ~ \\in ~ \\mathbb { F } _ { 0 } ^ { n }$ , there exists a time $T ( G )$ from which the sequence of colorings $( c _ { k - \\mathrm { W L } } ^ { t } ( G ) ) _ { t \\geq 0 }$ is stationary. More exactly, the colorings are not refined anymore: for any $t \\geq T ( G )$ , $s , s ^ { \\prime } \\in V ^ { k }$ , ", + "bbox": [ + 173, + 449, + 826, + 493 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/a334212128f3089c1df624da4629737d23128a36e588ac71c46637f3e01f739f.jpg", + "text": "$$\nc _ { k \\mathrm { - } \\mathrm { W L } } ^ { t } ( G ) _ { s } = c _ { k \\mathrm { - } \\mathrm { W L } } ^ { t } ( G ) _ { s ^ { \\prime } } \\iff c _ { k \\mathrm { - } \\mathrm { W L } } ^ { T ( G ) } ( G ) _ { s } = c _ { k \\mathrm { - } \\mathrm { W L } } ^ { T ( G ) } ( G ) _ { s ^ { \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 492, + 694, + 513 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Denote the resulting coloring $c _ { k \\mathrm { - W L } } ^ { T ( G ) } ( G )$ by simply $c _ { k - \\mathrm { W L } } ( G )$ ", + "bbox": [ + 173, + 517, + 573, + 537 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "$k$ -FWL. For $k$ -FWL, the corresponding notion of neighborhood is defined by, for any $j \\in V$ , and $s = ( i _ { 1 } , \\ldots , i _ { k } ) \\in V ^ { k }$ , ", + "bbox": [ + 173, + 550, + 825, + 580 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/785cd335b4b898ec5a563a304e0c5ecf8e32f85ee37d82a8b9ad7c3e0c1f0d03.jpg", + "text": "$$\nN _ { j } ^ { F } ( s ) = \\{ ( j , i _ { 2 } , \\ldots , i _ { k } ) , ( i _ { 1 } , j , i _ { 3 } , \\ldots , i _ { k } ) , \\ldots , ( i _ { 1 } , i _ { 2 } , \\ldots , i _ { k - 1 } , j ) \\}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 588, + 727, + 607 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Then, the colors of the $k$ -tuples $s \\in V ^ { k }$ are refined as follows, ", + "bbox": [ + 171, + 614, + 581, + 631 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/fd98e02787cb12432260d106cb7cd46bbe8b84c5db0e418a6a622fed5dd4ba21.jpg", + "text": "$$\nc _ { k - \\mathrm { W L } } ^ { t + 1 } ( G ) _ { s } = \\mathrm { L e x } \\left( c _ { k - \\mathrm { F W L } } ^ { t } ( G ) _ { s } , \\left\\{ \\left\\{ C _ { j } ^ { t } ( s ) : j \\in V \\right\\} \\right\\} \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 636, + 674, + 656 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where, for $j \\in V$ , ", + "bbox": [ + 173, + 661, + 292, + 678 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2e21790293a1a15becc5733bbed82f4a44ae0b6560304f7842088c4da650c8b0.jpg", + "text": "$$\nC _ { j } ^ { t } ( s ) = \\big ( c _ { k \\mathrm { \\scriptsize - F W L } } ^ { t } ( G ) _ { \\tilde { s } } : \\tilde { s } \\in N _ { j } ^ { F } ( s ) \\big ) \\ .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 683, + 624, + 704 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Like $k$ -WL, for each graph $G \\in \\mathbb { F } _ { 0 } ^ { n }$ , there exists a time $T ( G )$ from which the sequence of colorings $( c _ { k - \\mathrm { W L } } ^ { t } ( G ) ) _ { t \\geq 0 }$ is stationary. Similarly, denote the resulting coloring $c _ { k \\mathrm { - W L } } ^ { T ( G ) } ( G )$ by simply $c _ { k - \\mathrm { W L } } ( G )$ . The colors $c _ { k - \\mathrm { F W L } } ^ { t }$ and $c _ { k - \\mathrm { F W L } } ^ { t }$ at iteration $t$ define a mapping from $\\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ to space of colorings of $k$ -tuples, $Z ^ { n ^ { k } }$ for some space $Z$ . ", + "bbox": [ + 171, + 715, + 826, + 792 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Invariant tests The standard versions of the Weisfeiler-Lehman tests are invariant and can be defined by, for $t \\geq 0$ and $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ a graph, ", + "bbox": [ + 173, + 808, + 823, + 838 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/f830ed33a20c76bb884035d7c5e2cacdaba2984cb2f7d0ee4c04154544fb29fc.jpg", + "text": "$$\n\\begin{array} { r l } & { k { \\mathrm { – } } \\mathbf { W } \\mathbf { L } _ { I } ^ { t } ( G ) = \\left\\{ \\left\\{ c _ { k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } } ^ { t } ( G ) _ { s } : s \\in V ^ { k } \\right\\} \\right\\} } \\\\ & { k { \\mathrm { – } } \\mathbf { W } \\mathbf { L } _ { I } ( G ) = \\left\\{ \\left\\{ c _ { k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } } ( G ) _ { s } : s \\in V ^ { k } \\right\\} \\right\\} } \\\\ & { k { \\mathrm { – } } \\mathbf { F } \\mathbf { W } \\mathbf { L } _ { I } ^ { t } ( G ) = \\left\\{ \\left\\{ c _ { k { \\mathrm { - } } \\mathbf { F } \\mathbf { W } \\mathbf { L } } ^ { t } ( G ) _ { s } : s \\in V ^ { k } \\right\\} \\right\\} } \\\\ & { k { \\mathrm { – } } \\mathbf { F } \\mathbf { W } \\mathbf { L } _ { I } ( G ) = \\left\\{ \\left\\{ c _ { k { \\mathrm { - } } \\mathbf { F } \\mathbf { W } \\mathbf { L } } ( G ) _ { s } : s \\in V ^ { k } \\right\\} \\right\\} \\ . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 356, + 843, + 643, + 929 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Equivariant tests We now introduce the equivariant version of these tests. Many extensions are possible, we chose this one for its simplicity. For $t \\geq 0$ , $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ a graph, $i \\in V$ , ", + "bbox": [ + 171, + 103, + 823, + 136 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/f506e10903aa99cc433674367668e1e58f4e2a943fa7f0a817d60fd0939fac20.jpg", + "text": "$$\n\\begin{array} { r l } & { k \\mathrm { - } { \\mathbb { W } } \\mathrm { L } _ { E } ^ { t } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { W L } } ^ { t } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} } \\\\ & { k \\mathrm { - } { \\mathbb { W } } \\mathrm { L } _ { E } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { W L } } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} } \\\\ & { k \\mathrm { - } { \\mathrm { F W L } } _ { E } ^ { t } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { F W L } } ^ { t } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} } \\\\ & { k \\mathrm { - } { \\mathrm { F W L } } _ { E } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { F W L } } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} \\ . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 140, + 673, + 224 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C SEPARATING POWER OF GNN ", + "text_level": 1, + "bbox": [ + 174, + 238, + 455, + 256 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The goal of this section is to prove, ", + "bbox": [ + 174, + 270, + 406, + 285 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proposition 3. We have, for $k \\geq 2$ , ", + "bbox": [ + 174, + 289, + 410, + 303 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/183e7aa5f995506534ecae6f390331db71f497c37ba575f01945cb8f0553a7eb.jpg", + "text": "$$\n\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) = \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { E } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 308, + 758, + 363 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C.1 MUTLI-LINEAR PERCEPTRONS ", + "text_level": 1, + "bbox": [ + 176, + 376, + 428, + 390 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In the following we will use extensively multi-linear perceptrons (MLP) and their universality properties. Yet, for the sake of simplicity, we do not define precisely what we mean by MLP. ", + "bbox": [ + 174, + 401, + 825, + 431 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Given two finite-dimensional feature spaces $\\mathbb { F } _ { 0 }$ and $\\mathbb { F } _ { 1 }$ , we only assume we are given a class of (continuous) MLP from $\\mathbb { F } _ { 0 }$ to $\\mathbb { F } _ { 1 }$ which is large enough to be dense in $\\mathcal { C } ( \\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 } )$ . See for instance Hornik et al. (1989); Cybenko (1989) for precise conditions for MLP to be universal. ", + "bbox": [ + 173, + 436, + 825, + 479 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C.2 AUGMENTED SEPARATING POWER ", + "text_level": 1, + "bbox": [ + 176, + 496, + 452, + 510 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To factor the proof, we introduce another notion of separating power. ", + "bbox": [ + 173, + 521, + 625, + 536 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition 7. For $n , m \\geq 1$ fixed, let $X$ be a set, $\\mathbb { F }$ be a some space and $\\mathcal { F }$ be a set of functions from $X$ to $Y = \\mathbb { F } ^ { n \\times m }$ . Then, the augmented separating power of $\\mathcal { F }$ is, ", + "bbox": [ + 171, + 540, + 825, + 569 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/dc132d7ebe9d1be4d2a37dbc6f8762cabd671a3f2051247cd2f6667208f3f029.jpg", + "text": "$$\n\\rho _ { n , m } ^ { a u g m } ( \\mathcal { F } ) = \\rho ( \\{ ( x , i , j ) \\in X \\times \\{ 1 , \\ldots , n \\} \\times \\{ 1 , \\ldots , m \\} \\mapsto f ( x ) _ { i , j } : f \\in \\mathcal { F } \\} ) .\n$$", + "text_format": "latex", + "bbox": [ + 225, + 574, + 771, + 592 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Explicitly, for $x , y \\in X , i , j \\in \\{ 1 , \\dots , n \\}$ ", + "bbox": [ + 174, + 597, + 452, + 613 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/1cdacb6f2b7921bb2ef5bb501108c44647c647baa522694a35ed6ed1d6b02d7d.jpg", + "text": "$$\n( x , i , j , y , k , l ) \\in \\rho _ { n , m } ^ { a u g m } ( \\mathcal { F } ) \\iff \\forall f \\in \\mathcal { F } , f ( x ) _ { i , j } = f ( y ) _ { k , l } .\n$$", + "text_format": "latex", + "bbox": [ + 285, + 617, + 710, + 636 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Note that when $n = m = 1$ , the augmented separating power is exactly the same as the original separating power, so we identify $\\rho \\left( . \\right)$ with $\\rho _ { 1 , 1 } ^ { a u g m } ( . )$ . We also identify ρaugmn,1 (.) with ρaug1,n $\\rho _ { 1 , n } ^ { a u g m } \\left( . \\right)$ , that we denote by $\\rho _ { n } ^ { a u g m } \\left( . \\right)$ . ", + "bbox": [ + 174, + 646, + 825, + 691 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "First, it is easy to see that this notion is more precise than the separating power. ", + "bbox": [ + 174, + 696, + 692, + 712 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma 8. If $\\mathcal { F }$ and $\\mathcal { G }$ are set of functions from $X$ to $\\mathbb { F } ^ { n \\times m }$ , ", + "bbox": [ + 174, + 715, + 570, + 731 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "and, in particular, ", + "bbox": [ + 173, + 757, + 294, + 772 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d19483b756a4da63b35c5fcf0eec05346d4812069f8bb44c276223eb15cca0fa.jpg", + "text": "$$\n\\begin{array} { r l } & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { G } \\right) \\implies \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( \\mathcal { G } \\right) , } \\\\ & { } \\\\ & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) = \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { G } \\right) \\implies \\rho \\left( \\mathcal { F } \\right) = \\rho \\left( \\mathcal { G } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 733, + 656, + 790 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The interest in this notion is justified by the following lemma, which shows that this notion behaves well under composition with “reduction layers”. ", + "bbox": [ + 173, + 796, + 823, + 825 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma 9. For $n , m \\geq 1$ fixed, let $X$ be a compact topological space, $Y = \\mathbb { F } ^ { n \\times m }$ , $\\mathbb { F }$ some finitedimensional space, ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )$ , and $\\tau : X Z ^ { n \\times m } a$ function for some space $Z$ . Define ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , \\mathbb { F } ^ { n } )$ by, ", + "bbox": [ + 173, + 828, + 828, + 875 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/2a49405e4b16eb54031c89415bc020cc899ede13d403c3b341e7fb106d13a33b.jpg", + "text": "$$\n\\tilde { \\boldsymbol { x } } = \\{ x \\in X \\longmapsto ( \\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { 1 , j } ) , \\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { 2 , j } ) , \\ldots , \\sum _ { i = 1 } ^ { m } h ( f ( x ) _ { n , j } ) ) : h : \\mathbb { F } \\to \\mathbb { F } M L P , \\ f \\in \\mathcal { R } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 878, + 839, + 929 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "and $\\widetilde { \\tau } : X \\to \\widetilde { Z } ^ { n }$ by, for $x \\in X$ , ", + "bbox": [ + 173, + 102, + 388, + 118 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/c0ec6a0cee2e6ee850ba7b7e911c3b509138d92f07a66660af83d950e61207eb.jpg", + "text": "$$\n\\forall i \\in \\{ 1 , \\ldots , n \\} , \\ : \\widetilde { \\tau } ( x ) _ { i } = \\{ \\{ \\tau ( x ) _ { i , j } : 1 \\leq j \\leq m \\} \\} .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 122, + 671, + 140 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Then, ", + "bbox": [ + 173, + 143, + 212, + 159 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/4a68d4946173cfa2b07d1d8e2870444b4433798b733800a3d7c1afaea1635c94.jpg", + "text": "$$\n\\begin{array} { r l } & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) \\implies \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\tau } \\right) } \\\\ & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\supset \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) \\implies \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right) \\supset \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\tau } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 161, + 691, + 218 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Note that, in the statement, we implicitly see $\\widetilde { \\mathcal F }$ as a set of functions from $X$ to $\\mathbb { F } ^ { n \\times 1 }$ to fit the definition of augmented separating power. ", + "bbox": [ + 173, + 228, + 825, + 260 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. We show the two inclusions independently. ", + "bbox": [ + 173, + 273, + 506, + 290 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$( \\subseteqright)$ We first show that, ", + "bbox": [ + 200, + 303, + 356, + 318 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/f58392a527eb8b6babde2e263edda2f83030079b79b44fe5765c22c4cadbf292.jpg", + "text": "$$\n\\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) \\implies \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\tau } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 341, + 321, + 715, + 348 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Take $( x , i , y , k ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right)$ . This means that, for any $h : \\mathbb { F } \\to \\mathbb { F } , f \\in { \\mathcal { F } } .$ ", + "bbox": [ + 232, + 353, + 728, + 377 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/8898a2df002ffb2001e28207ffd84613c291cc5c72dc148d87515036641c02f9.jpg", + "text": "$$\n\\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { i , j } ) = \\sum _ { j = 1 } ^ { m } h ( f ( y ) _ { k , j } ) .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 381, + 640, + 425 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "By Lem. 31 and the universality of MLP, there exists a permutation $\\sigma \\in S _ { m }$ such that $( { \\dot { f } } ( x ) _ { i , \\sigma ( 1 ) } , \\dots , f ( x ) _ { i , \\sigma ( m ) } ) = ( f ( y ) _ { k , 1 } , \\dots , f ( y ) _ { k , m } )$ . By definition of the augmented separating power, this means that, for any $j \\in \\{ 1 , \\dots , m \\}$ , $( x , i , \\sigma ( j ) , y , k , j ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right)$ Hence, by assumption, for any $j ~ \\in ~ \\{ 1 , \\dots , m \\}$ , $( x , i , \\sigma ( j ) , y , k , j ) \\ \\in \\ \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) .$ , i.e. $\\tau ( x ) _ { i , \\sigma ( j ) } = \\tau ( y ) _ { k , j }$ . But this exactly means that $\\widetilde { \\tau } ( x ) _ { i } = \\widetilde { \\tau } ( y ) _ { k }$ so that $( x , i , y , k ) \\in$ $\\rho _ { n , m } ^ { a u g m } \\left( \\tilde { \\tau } \\right)$ as required. ", + "bbox": [ + 230, + 428, + 826, + 518 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$( \\supset )$ We now show the other inclusion, ", + "bbox": [ + 200, + 526, + 455, + 541 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/77b8f2d618c36a184815d8df8073aa520719642ac2c9cc1e5dfbd805214fc58b.jpg", + "text": "$$\n\\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\supset \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) \\implies \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right) \\supset \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\tau } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 545, + 718, + 571 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Take $( x , i , y , k ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\tau } \\right)$ . By definition of $\\widetilde { \\tau }$ , this means that there exists $\\sigma \\in S _ { m }$ such that $\\tau ( x ) _ { i , \\sigma ( j ) } = \\tau ( y ) _ { k , j }$ eso that $( x , i , \\sigma ( j ) , y , k , j ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right)$ . Hence, for any $f \\in { \\mathcal { F } }$ , $( f ( x ) _ { i , \\sigma ( 1 ) } , \\ldots , f ( x ) _ { i , \\sigma ( m ) } ) = ( f ( y ) _ { k , 1 } , \\ldots , f ( y ) _ { k , m } )$ , and so, for any $h : \\mathbb { F } \\mathbb { F }$ , ", + "bbox": [ + 230, + 575, + 826, + 635 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/b96465660ae5572919ebf8ce95b447e1d686503907b7ba9224a8d14e5c79f74e.jpg", + "text": "$$\n\\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { i , j } ) = \\sum _ { j = 1 } ^ { m } h ( f ( y ) _ { k , j } ) .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 632, + 640, + 675 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Therefore, $( x , i , y , k ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right)$ , which concludes the proof. ", + "bbox": [ + 230, + 678, + 656, + 703 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C.3 INITIALIZATION LAYER ", + "text_level": 1, + "bbox": [ + 174, + 746, + 380, + 761 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We use the same initialization layer as Maron et al. (2019a); Chen et al. (2020) and recall it below. The initial graph is a tensor of the form $G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ with $\\mathbb { F } _ { 0 } = \\mathbb { R } ^ { e + 2 }$ ; the last channel of $G _ { : , : , e + 1 }$ encodes the adjacency matrix of the graph and the first $e$ channels $G _ { : , : , 1 : e }$ are zero outside the diagonal and $G _ { i , i , 1 : e } \\in \\mathbb { R } ^ { e }$ is the color of vertex $v _ { i } \\in V$ . We then define $I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }$ with $\\mathbb { F } _ { 1 } = \\mathbb { R } ^ { k ^ { 2 } \\times ( e + 2 ) }$ as follows: ", + "bbox": [ + 171, + 771, + 826, + 848 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/c6183949be3a59426e1311b518bdaa184a3e6c351ed3da42868db7635a57e075.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\displaystyle { \\cal I } ^ { k } ( G ) _ { { \\bf i } , r , s , w } } } & { { = } } & { { G _ { i _ { r } , i _ { s } , w } , w \\in [ e + 1 ] , } } \\\\ { { \\displaystyle { \\cal I } ^ { k } ( G ) _ { { \\bf i } , r , s , e + 2 } } } & { { = } } & { { { \\bf 1 } ( i _ { r } = i _ { s } ) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 851, + 640, + 891 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "for $\\mathbf { i } \\in [ n ] ^ { k }$ and $r , s \\in [ k ]$ . This linear equivariant layer has the same separating power as the isormorphism type, which is defined in Section B.2. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "exists I k : Fn20 Lemma 10 ((Maron et al., 2019a, C.1)). For $I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }$ such that, $k \\geq 2$ , $p _ { 0 } \\geq 1$ , $\\mathbb { F } _ { 0 } = \\mathbb { R } ^ { p _ { 0 } }$ , $\\mathbb { F } _ { 1 } = \\mathbb { R } ^ { k ^ { 2 } \\times ( p _ { 0 } + 1 ) }$ , there ", + "bbox": [ + 169, + 102, + 825, + 136 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/67517ed0ab29cebaebac0e1affaaeac2c46c5ae6cf02b277b46d2a968f19d6fe.jpg", + "text": "$$\n\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( I ^ { k } \\right) = \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( \\mathrm { i s o } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 395, + 141, + 601, + 164 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.4 KNOWN RESULTS ABOUT THE SEPARATING POWER OF SOME GNN CLASSES ", + "text_level": 1, + "bbox": [ + 173, + 178, + 741, + 194 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "First, we need to define some classes of GNN from which both the invariant and equivaraint GNN we considered are built. See Section 3 for details about the different layers. ", + "bbox": [ + 169, + 204, + 825, + 233 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\mathbf { M G N N } _ { e m b } = \\left\\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } : F _ { t } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\right. } \\end{array}$ message passing layer, $t = 1 , \\ldots , T , T \\geq 1 \\}$ $\\mathfrak { k } \\mathrm { - } \\mathrm { L G N N } _ { e m b } = \\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n }$ linear equivariant layer, $t = 1 , \\ldots , T , T \\geq 1 \\}$ $\\begin{array} { r } { k \\mathrm { { \\mathrm { - } } F G N N } _ { e m b } = \\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } : \\mathbb { F } _ { t } ^ { n } \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { { F G L } } , t = 1 , \\dots , T , T \\geq 1 \\} . } \\end{array}$ ", + "bbox": [ + 173, + 239, + 825, + 299 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Then, the precise results from the literature can be rephrased as, ", + "bbox": [ + 173, + 313, + 593, + 327 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma 11 $\\mathrm { { X u } }$ et al. (2018),Maron et al. (2018),Maron et al. (2019a)). For $k \\geq 2$ , ", + "bbox": [ + 174, + 332, + 718, + 347 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/af6b0e7dd7cf584ab93d501e25a15ebd8072a2aeb5dd15951c0a1c6dfe410db9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\qquad \\rho _ { n } ^ { a u g m } \\left( M G N N _ { e m b } \\right) = \\rho _ { n } ^ { a u g m } \\left( c _ { W L } \\right) } \\\\ & { \\qquad \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k { - } L G N N _ { e m b } \\right) \\subseteq \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - W L } \\right) } \\\\ & { \\qquad \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k { - } F G N N _ { e m b } \\right) \\subseteq \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 352, + 638, + 415 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(8) comes from $\\mathrm { X u }$ et al. (2018, §A, $\\ S \\mathbf { B }$ ), (9) and (10) from Maron et al. (2019a, §C, $\\ S _ { \\mathbf { D } }$ ). ", + "bbox": [ + 169, + 426, + 758, + 443 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.5 BOUNDING THE SEPARATING POWER OF $k$ -FGNN ", + "text_level": 1, + "bbox": [ + 173, + 458, + 562, + 473 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We complete the results of the literature with a bound on the separating power of $k$ -FGNN. Note that the particular case of $k = 2$ is already proven in Geerts (2020a). ", + "bbox": [ + 174, + 484, + 823, + 513 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma 12. For any $k \\geq 2$ ", + "bbox": [ + 173, + 517, + 359, + 532 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/57725f634b162d617e57b58ba5d06ae5850c0e9dba7e2f2a5afb515e922db5c0.jpg", + "text": "$$\n\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k – F G N N _ { e m b } \\right) \\supset \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 539, + 642, + 559 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "so that ", + "bbox": [ + 173, + 566, + 222, + 580 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/5e0f120406178fafd424376749342ce29aeafc44c49af35d3f60da26e48c4e2a.jpg", + "text": "$$\n\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k – F G N N _ { e m b } \\right) = \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 578, + 643, + 599 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. Define, ", + "bbox": [ + 173, + 612, + 272, + 627 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/50fe17e1599d9711a3ac9ee55537127ef84363cd5694ca241e9747daa3fed6f7.jpg", + "text": "$$\nk \\mathrm { \\mathrm { - } F G N N } _ { e m b } ^ { T } = \\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { \\ F G L } , t = 1 , \\dots , T \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 254, + 633, + 741, + 654 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "the set of functions defined by exactly $T$ FGL layers. We show by induction that, for any $T \\geq 0$ ", + "bbox": [ + 169, + 659, + 803, + 675 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/e69282f24896e26212ce492dcdae1c32215d5abc7d87cabfa35c9f1146e5859f.jpg", + "text": "$$\n\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k \\mathrm { – F G N N } _ { e m b } ^ { T } \\right) \\supset \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k \\mathrm { – F W L } } ^ { T } \\right) \\ .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 681, + 648, + 704 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For $T = 0$ , this is immediate by the definition of $I ^ { k }$ in Section C.3. ", + "bbox": [ + 173, + 710, + 612, + 727 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Assume now that this inclusion holds at $T - 1 \\geq 0$ . We show that it also holds at $T$ . Take $G , G ^ { \\prime } \\in \\mathbb { F } _ { 0 } ^ { n ^ { k } }$ and $s , s ^ { \\prime } \\in [ n ] ^ { k }$ such that, ", + "bbox": [ + 173, + 734, + 823, + 763 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/63abf23cb5a9599ee9e322deba19ce61a1d21ccc7b93873ab0d3a64306ab7652.jpg", + "text": "$$\nc _ { k \\mathrm { - F W L } } ^ { T } ( G ) _ { s } = c _ { k \\mathrm { - F W L } } ^ { T } ( G ^ { \\prime } ) _ { s ^ { \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 762, + 594, + 781 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "$f \\in k \\mathrm { - F G N N } _ { e m b } ^ { T }$ ", + "bbox": [ + 173, + 786, + 490, + 803 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/2acebc813f004336842879d32b107c6074c4ee3478022efa24764667e3e3c62b.jpg", + "text": "$$\nf ( G ) _ { s } = f ( G ^ { \\prime } ) _ { s ^ { \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 808, + 562, + 825 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "But, by definition of the update rule $k$ -FWL, the equality of the colors of $s$ and $s ^ { \\prime }$ above implies that, ", + "bbox": [ + 171, + 832, + 825, + 848 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/7e3577071d779fef4105fa98a6ffbae6c674ac951496083ef2ad48c30dbfbe4e.jpg", + "text": "$$\nc _ { k \\mathrm { - F W L } } ^ { T - 1 } ( G ) _ { s } = c _ { k \\mathrm { - F W L } } ^ { T - 1 } ( G ^ { \\prime } ) _ { s ^ { \\prime } } ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 853, + 594, + 875 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and that there exists $\\sigma \\in S _ { n }$ such that, for any $j \\in [ n ]$ , ", + "bbox": [ + 173, + 878, + 531, + 895 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/6b8fbb4d8d849d050df1f83f3fb4bfaf2751a4b5e2d4f9a9832fb82f6c14927b.jpg", + "text": "$$\n\\left( c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ) _ { \\tilde { s } } : \\tilde { s } \\in N _ { j } ^ { F } ( s ) \\right) = \\left( c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ^ { \\prime } ) _ { \\tilde { s } } : \\tilde { s } \\in N _ { \\sigma ( j ) } ^ { F } ( s ^ { \\prime } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 901, + 707, + 929 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Let $s = ( i _ { 1 } , \\ldots , i _ { k } )$ and $s ^ { \\prime } = ( j _ { 1 } , \\ldots , j _ { k } )$ . Then this implies that, for any $w \\in [ k ] , j \\in [ n ]$ , ", + "bbox": [ + 169, + 102, + 769, + 119 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/d9777efdbdc2c4ab6bdd97cdd5280a348ca8bc7d6131e5d9eab7d5e35ee5b6f0.jpg", + "text": "$$\nc _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } = c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ^ { \\prime } ) _ { j _ { 1 } , \\dots , j _ { w - 1 } , \\sigma ( j ) , j _ { w + 1 } , \\dots , j _ { k } } .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 125, + 728, + 146 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We now use the induction hypothesis, i.e. that, ", + "bbox": [ + 174, + 159, + 480, + 174 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/e6059d4b4a652c053190e61c9921970f0dce91b93434bf1ac238b0a286ffe2d4.jpg", + "text": "$$\n\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k \\mathrm { – F G N N } _ { e m b } ^ { T - 1 } \\right) \\supset \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k \\mathrm { – F W L } } ^ { T - 1 } \\right) \\ .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 180, + 650, + 203 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Take any $f _ { T - 1 } \\in k – \\mathrm { F G N N } _ { e m b } ^ { T - 1 } .$ . By (11), ", + "bbox": [ + 174, + 210, + 442, + 229 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/f4b90fcbcdea6eea6a03fb753346b090c921a9c53d19e37288f41e03cc2d40cb.jpg", + "text": "$$\nf _ { T - 1 } ( G ) _ { s } = f _ { T - 1 } ( G ^ { \\prime } ) _ { s ^ { \\prime } } .\n$$", + "text_format": "latex", + "bbox": [ + 410, + 234, + 586, + 253 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "By (12), for any $w \\in [ k ] , j \\in [ n ]$ , ", + "bbox": [ + 173, + 258, + 395, + 276 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/bf077b8b292f7b94672d4077775f957278ef4390eedfbb528bdb63d4e3a34f39.jpg", + "text": "$$\n\\begin{array} { r } { f _ { T - 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } = f ^ { T - 1 } ( G ^ { \\prime } ) _ { j _ { 1 } , \\dots , j _ { w - 1 } , \\sigma ( j ) , j _ { w + 1 } , \\dots , j _ { k } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 282, + 720, + 303 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "By the definition of FGL Section 3.3, for any $F _ { T } : \\mathbb { F } _ { T } ^ { n ^ { k } } \\to \\mathbb { F } _ { T + 1 } ^ { n ^ { k } }$ FGL, $F _ { T } \\circ f _ { T - 1 } ( G ) _ { s } = F _ { T } \\circ$ $f _ { T - 1 } ( G ^ { \\prime } ) _ { s ^ { \\prime } }$ . Therefore, for any $f \\in k \\mathrm { - F G N N } _ { e m b } ^ { T }$ , ", + "bbox": [ + 173, + 309, + 825, + 345 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/89742c4e949b30efb3d33d960e6da5e7a3f507d7e6c5cff82647873961f8ba66.jpg", + "text": "$$\nf ( G ) _ { s } = f ( G ^ { \\prime } ) _ { s ^ { \\prime } } ,\n$$", + "text_format": "latex", + "bbox": [ + 436, + 352, + 560, + 369 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "which concludes the proof. ", + "bbox": [ + 174, + 376, + 352, + 391 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 409, + 312, + 422 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proposition 13. We have, for $k \\geq 2$ ", + "bbox": [ + 174, + 434, + 416, + 449 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/3cef8b694216a888c1056063bc649b98c2ab99a280bf36cffb70b41b2e353cc5.jpg", + "text": "$$\n\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( 2 \\ – L G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) \\subseteq \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( k \\ – F W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( k \\ – F W L _ { E } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 455, + 738, + 530 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. Most of the statements come from the literature or are direct consequences of the lemmas above. ", + "bbox": [ + 173, + 542, + 825, + 570 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Proof of (13). The invariant case is proven in $\\mathrm { X u }$ et al. (2018, Lem. 2, Thm. 3). The equivariant case comes from Lem. 11, Lem. 8, the fact that the layers $m _ { E } \\circ ( \\mathrm { I d } + \\lambda S ^ { 1 } )$ does not change the separating power and recalling that simply $\\mathrm { W L } _ { E } = c _ { W L }$ . \n• (14) is the exact result Chen et al. (2020, Thm. 6). \n• Proof of (15). The invariant case is exactly Maron et al. (2019a, Thm. 1). The equivariant case comes from Lem. 11, Lem. 9, Lem. 8 and the fact that the layer $m _ { E }$ does not change the separating power. \n• Proof of (16). The direct inclusion of the invariant case corresponds to Maron et al. (2019a, Thm. 2). The other cases are a consequence of Lem. 11, Lem. 9, Lem. 8 and the fact that the layers $m _ { I }$ and $m _ { E }$ does not change the separating power. ", + "bbox": [ + 215, + 587, + 826, + 757 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "D STONE-WEIERSTRASS THEOREM WITH SYMMETRIES ", + "text_level": 1, + "bbox": [ + 174, + 809, + 648, + 824 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "This section presents our extension of the Stone-Weierstrass theorem dealing with functions with symmetries. The scope of this section is not restricted to graphs or even tensors and we will deal with general spaces and general symmetries. To illustrate it, we will present applications for the PointNet architecture Qi et al. (2017). Our approximation results for GNNs (Theorems Thm. 5 and Thm. 6) are then obtained from these theoretical results applied to tensors and the symmetric group in Section D.9 ", + "bbox": [ + 174, + 839, + 825, + 922 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "As explained above, we are dealing in this section with a much larger scope than graphs and permutations. We first need to extend the notations introduced above. The notations introduced below will make this section self-contained. ", + "bbox": [ + 174, + 128, + 823, + 171 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "If $X$ is some topological space, and $F \\subseteq X$ , denote by $\\overline { F }$ its closure. ", + "bbox": [ + 174, + 176, + 625, + 194 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "If $X$ is a topological space and $Y = \\mathbb { R } ^ { p }$ some finite-dimensional space, denote by ${ \\mathcal { C } } ( X , Y )$ the set of continuous functions from $X$ to $Y$ . ", + "bbox": [ + 173, + 199, + 823, + 229 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Moreover, if $X$ is compact, we endow ${ \\mathcal { C } } ( X , Y )$ with the topology of uniform convergence, which is defined by the norm, ${ \\bar { f } } \\mapsto \\operatorname* { s u p } _ { x \\in X } \\| f ( x ) \\|$ for some norm $\\intercal \\Vert . \\Vert$ on $Y$ . ", + "bbox": [ + 174, + 234, + 821, + 265 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "If $G$ is a finite group acting on some topological space $X$ , we say that $G$ acts continuously on $X$ if, for all $g \\in G$ , $x \\mapsto g \\cdot x$ is continuous. ", + "bbox": [ + 174, + 270, + 821, + 299 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "If $G$ is a finite group acting on some compact set $X$ and some topological space $Y$ , we define the sets of equivariant and invariant continuous functions by, ", + "bbox": [ + 171, + 305, + 823, + 334 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/23c231efdfc47aa1918f0b7ba326bdb4b0a91e290573fc068e3abab5f862f233.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { C } _ { E } ( X , Y ) = \\{ f \\in \\mathcal { C } ( X , Y ) : \\forall x \\in X , \\forall g \\in G , f ( g \\cdot x ) = g \\cdot f ( x ) \\} } \\\\ & { \\mathcal { C } _ { I } ( X , Y ) = \\{ f \\in \\mathcal { C } ( X , Y ) : \\forall x \\in X , \\forall g \\in G , f ( g \\cdot x ) = f ( x ) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 340, + 728, + 378 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Note that these definitions extend Definition 1 to a general group. ", + "bbox": [ + 174, + 382, + 602, + 397 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "If $Y = \\mathbb { R } ^ { p }$ , we denote the coordinate-wise multiplication, or Hadamard product of $y , y ^ { \\prime } \\in Y$ simply by $y y ^ { \\prime } = ( y _ { 1 } y _ { 1 } ^ { \\prime } , \\dots , y _ { p } y _ { p } ^ { \\prime } ) \\in Y$ . We say that a subset $A \\subseteq \\mathbb { R } ^ { p }$ is a subalgebra of $\\mathbb { R } ^ { p }$ if it is both a linear space and stable by multiplication. ", + "bbox": [ + 176, + 404, + 823, + 446 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "This product in turn defines a product on ${ \\mathcal { C } } ( X , Y )$ with $Y = \\mathbb { R } ^ { p }$ by, for $f , g \\in { \\mathcal { C } } ( X , Y ) , f g : x \\mapsto$ $f ( x ) g ( x )$ . ", + "bbox": [ + 169, + 452, + 823, + 482 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In addition, we also extend the scalar-vector product of $Y = \\mathbb { R } ^ { p }$ to functions: if $g \\in { \\mathcal { C } } ( X , \\mathbb { R } )$ and $f \\in { \\mathcal { C } } ( X , Y )$ , their product $g f$ is the function $g f : x \\mapsto g ( x ) f ( x )$ . Given a set of scalar functions ${ \\mathcal { S } } \\subseteq { \\mathcal { C } } ( X , \\mathbb { R } )$ and a set of vector-valued functions ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )$ , the set of products of functions of these two sets will be denoted by, ", + "bbox": [ + 173, + 488, + 825, + 544 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/e9562e6635da981d0cc2cd7ded2bad8aea35fcd71f20471c69790057c6cae842.jpg", + "text": "$$\nS \\cdot { \\mathcal { F } } = \\left\\{ g f : g \\in { \\mathcal { S } } , f \\in { \\mathcal { F } } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 551, + 602, + 569 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Moreover, we denote by 1 the continuous function from some $X$ to $\\mathbb { R } ^ { p }$ defined by $x \\mapsto ( 1 , \\ldots , 1 )$ . In particular, if $f$ is a function from $X$ to $\\mathbb { R }$ , $f \\mathbf { 1 }$ denotes the function $x \\mapsto ( f ( x ) , \\ldots , f ( x ) )$ which goes from $X$ to $Y = \\mathbb { R } ^ { p }$ . Finally, we say that ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )$ with $Y$ being some $\\mathbb { R } ^ { p }$ is a subalgebra if it is a linear space which is also stable by multiplication. ", + "bbox": [ + 173, + 582, + 826, + 638 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "D.2 SEPARATING POWER ", + "text_level": 1, + "bbox": [ + 174, + 655, + 361, + 670 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We recall the definition of separating power that we introduced above: ", + "bbox": [ + 174, + 681, + 633, + 696 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Definition 14. Let $\\mathcal { F }$ be a set of functions $f$ defined on a set $X$ , where each $f$ takes its values in some $Y _ { f }$ . The equivalence relation $\\rho \\left( \\mathcal { F } \\right)$ defined by $\\mathcal { F }$ on $X$ is: for any $x , x ^ { \\prime } \\in X$ , ", + "bbox": [ + 171, + 700, + 825, + 729 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/05c44cd18e3afb1eedf76048f898065c5f4d6d174f976f67cf3ced3a9e12b10c.jpg", + "text": "$$\n( x , x ^ { \\prime } ) \\in \\rho \\left( { \\mathcal { F } } \\right) \\iff \\forall f \\in { \\mathcal { F } } , f ( x ) = f ( x ^ { \\prime } ) .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 734, + 653, + 752 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For a function $f$ , we write $\\rho \\left( f \\right)$ for $\\rho \\left( \\{ f \\} \\right)$ . ", + "bbox": [ + 176, + 766, + 465, + 782 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Separating power is stable by closure: ", + "bbox": [ + 174, + 787, + 423, + 804 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma 15. Let $X$ be a compact topological space, $Y$ be some finite-dimensional and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )$ . Then, ", + "bbox": [ + 168, + 808, + 825, + 834 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/b406b533381aac99e903f529cb6aa49c699c195d8f7cfbd064d2c96a945003b4.jpg", + "text": "$$\n\\rho \\left( \\mathcal { F } \\right) = \\rho \\left( \\overline { { \\mathcal { F } } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 447, + 832, + 549, + 852 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. As ${ \\mathcal { F } } \\subseteq { \\overline { { \\mathcal { F } } } }$ , $\\overline { { \\mathcal { F } } }$ is more separating than $\\mathcal { F }$ , i.e. $\\rho \\left( { \\overline { { \\mathcal { F } } } } \\right) \\subseteq \\rho \\left( { \\mathcal { F } } \\right)$ . ", + "bbox": [ + 173, + 866, + 625, + 883 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Conversely, take $( x , y ) \\notin \\rho \\left( { \\overline { { \\mathcal { F } } } } \\right)$ . By definition, there exists $h \\in { \\overline { { \\mathcal { F } } } }$ such that $h ( x ) \\neq h ( y )$ so that if $\\epsilon = \\| h ( x ) - h ( y ) \\|$ , $\\epsilon > 0$ (for some norm $\\left. . \\right.$ on $Y$ ). As $h \\in { \\overline { { \\mathcal { F } } } }$ , there is some $f \\in { \\mathcal { F } }$ such that ", + "bbox": [ + 173, + 890, + 821, + 925 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\operatorname* { s u p } _ { X } \\| h - f \\| \\leq \\frac { \\epsilon } { 3 } } \\end{array}$ . Therefore, by the triangular inequality, ", + "bbox": [ + 173, + 102, + 565, + 121 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/e9509da8d296de732297c71819ccda84bb5dfe891fbe9f3965fd3f7520115cdb.jpg", + "text": "$$\n= \\| h ( x ) - h ( y ) \\| \\leq \\| f ( x ) - h ( x ) \\| + \\| f ( x ) - f ( y ) \\| + \\| f ( y ) - h ( y ) \\| \\leq { \\frac { 2 } { 3 } } \\epsilon + \\| f ( x ) - f ( y ) \\| .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 128, + 826, + 160 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "It follows that $\\| f ( x ) - f ( y ) \\| \\ge \\frac { \\epsilon } { 3 } > 0$ so that $f ( x ) \\neq f ( y )$ and $( x , y ) \\notin \\rho \\left( { \\mathcal { F } } \\right)$ . ", + "bbox": [ + 174, + 167, + 699, + 186 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "D.3 APPROXIMATION THEOREMS FOR REAL-VALUED FUNCTIONS ", + "bbox": [ + 176, + 205, + 640, + 219 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We start by recalling Stone-Weierstrass theorem, see Rudin (1991, Thm. 5.7). ", + "bbox": [ + 174, + 231, + 681, + 247 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Theorem 16 (Stone-Weierstrass). Let $X$ be a compact space, and $\\mathcal { F }$ be a subalgebra of ${ \\mathcal { C } } ( X , \\mathbb { R } )$ the space of real-valued continuous functions of $X$ , which contains the constant function 1. If $\\mathcal { F }$ separates points, i.e. $\\rho \\left( { \\mathcal { F } } \\right) = \\{ \\left( x , x \\right) : x \\in X \\}$ , then $\\mathcal { F }$ is dense in ${ \\mathcal { C } } ( X , \\mathbb { R } )$ . ", + "bbox": [ + 174, + 252, + 826, + 296 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We now prove an extension of this classical result due to Timofte (2005) allowing us to deal with much smaller $\\mathcal { F }$ by dropping the requirement that $\\mathcal { F }$ separates points. ", + "bbox": [ + 173, + 308, + 821, + 337 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Corollary 17. Let $X$ be a compact space, and $\\mathcal { F }$ be a subalgebra of ${ \\mathcal { C } } ( X , \\mathbb { R } )$ the space of real-valued continuous functions of $X$ , which contains the constant function 1. Then, ", + "bbox": [ + 173, + 342, + 825, + 372 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/ffac440f44b4c99a5bbf84e632b8cf4f250897c42dc92577eb47bd89974fd770.jpg", + "text": "$$\n\\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 380, + 625, + 398 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Note that if $\\mathcal { F }$ separates points, we get back the classical result as every function satisfies $\\{ ( x , x ) :$ $x \\in X \\} \\subseteq \\rho \\left( f \\right)$ . ", + "bbox": [ + 171, + 415, + 825, + 445 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Example 18. The invariant version of PointNet is able to learn functions of the form $\\textstyle \\sum _ { i } f ( x _ { i } )$ for $f \\in { \\mathcal { C } } ( \\mathbb { R } ^ { p } , \\mathbb { R } )$ . We can apply Corollary 17 to this setting. Consider the case where $X$ is a compact subset of $\\mathbb { R } ^ { p }$ and $\\mathcal { F } = \\textstyle \\{ \\hat { x } \\mapsto g ( \\sum _ { i = 1 } ^ { n } \\bar { f } ( x _ { i } ) )$ , $f \\in \\mathcal { C } ( \\mathbf { \\breve { X } } , \\mathbb { R } ^ { h } ) , g \\in \\mathcal { C } ( \\mathbb { R } ^ { h } , \\mathbb { R } ) \\}$ . $\\mathcal { F }$ is a subalgebra of ${ \\mathcal { C } } ( X , \\mathbb { R } )$ (indeed of ${ \\mathcal { C } } _ { I } ( X , \\mathbb { R } ) ,$ ) which contains the constant function 1. Then, it easy to see that $\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , \\sigma \\star x ) , \\sigma \\in S _ { n } \\}$ , where $\\sigma \\star x$ is defined by $( \\sigma \\star x ) _ { \\sigma ( i ) } = x _ { i }$ for all $i$ (see Lem. 31 for a formal statement). ", + "bbox": [ + 173, + 449, + 825, + 535 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Now note that for a function $f \\in { \\mathcal { C } } ( X , \\mathbb { R } )$ , the condition $\\left\\{ ( x , \\sigma \\star x ) , \\sigma \\in S _ { n } \\right\\} \\subseteq \\rho \\left( f \\right)$ is equivalent to $f \\in { \\mathcal { C } } _ { I } ( X , \\mathbb { R } )$ . So that Corollary 17 implies that $\\overline { { \\mathcal { F } } } = \\mathcal { C } _ { I } ( X , \\mathbb { R } )$ which means that PointNet is universal for approximating invariant functions. This was already proved in Qi et al. (2017) . ", + "bbox": [ + 173, + 540, + 825, + 585 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We now provide a proof of Corollary 17 for completeness. ", + "bbox": [ + 173, + 598, + 555, + 613 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. The first inclusion ${ \\overline { { \\mathcal { F } } } } \\subseteq \\{ f \\in { \\mathcal { C } } ( X , \\mathbb { R } ) : \\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right) \\}$ follows from the same argument as the one given below Theorem 4 so we focus on the other one. ", + "bbox": [ + 173, + 638, + 823, + 667 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "For every $x \\in X$ , let $x _ { \\mathcal { F } }$ denote its $\\rho \\left( \\mathcal { F } \\right)$ -class. The quotient set and the canonical surjection are: $X _ { \\mathcal { F } } = X / \\rho \\left( \\mathcal { F } \\right) = \\left\\{ x _ { \\mathcal { F } } , x \\in X \\right\\}$ and $\\pi _ { \\mathcal { F } } : X \\to X _ { \\mathcal { F } }$ , $\\bar { \\pi } _ { \\mathcal { F } } ( x ) = x _ { \\mathcal { F } }$ . ", + "bbox": [ + 173, + 674, + 823, + 704 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "A function $g : X \\to \\mathbb { R }$ factorizes as $g = \\hat { g } \\circ \\pi _ { \\mathcal { F } }$ for some $\\hat { g } : X _ { \\mathcal { F } } \\mathbb { R }$ if and only if $\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( g \\right)$ . In this case $\\hat { g }$ is unique, since $\\pi _ { \\mathcal { F } }$ is a surjection. In particular. every $f \\in { \\mathcal { F } }$ factorizes uniquely as $f = \\hat { f } \\circ \\bar { \\pi } _ { \\mathcal { F } }$ , $\\hat { f } : X _ { \\mathcal { F } } \\mathbb { R }$ , and $\\hat { \\mathcal { F } } = \\{ \\hat { f } , f \\in \\mathcal { F } \\}$ clearly separates points on $X _ { \\mathcal { F } }$ . We refer to Munkres (2000, $\\ S 2 2 _ { , }$ ) for the properties of the quotient topology. In particular, by the properties of the quotient topology on $X _ { \\mathcal { F } }$ , $\\hat { \\mathcal { F } }$ is a subalgebra of $\\mathcal { C } ( X _ { \\mathcal { F } } , \\mathbb { R } )$ and $X _ { \\mathcal { F } }$ is compact. Hence, we can apply Theorem 16 to $\\hat { \\mathcal { F } }$ and $\\hat { \\mathcal { F } }$ is dense in $\\mathcal { C } ( X _ { \\mathcal { F } } , \\mathbb { R } )$ . ", + "bbox": [ + 173, + 709, + 826, + 801 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Now take $f \\in { \\mathcal { C } } ( X , \\mathbb { R } )$ with $\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right)$ and we show that $f \\in { \\overline { { \\mathcal { F } } } }$ . Again, $\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right)$ implies that $f = \\hat { f } \\circ \\pi _ { \\mathcal { F } }$ . Let $\\epsilon > 0$ . By density of $\\hat { \\mathcal F }$ , there is some $\\hat { h } \\in \\hat { \\mathcal F }$ such that $\\begin{array} { r } { \\operatorname* { s u p } _ { x _ { \\mathcal { F } } } | \\hat { h } ( x _ { \\mathcal { F } } ) ) - } \\end{array}$ $\\hat { f } ( x _ { \\mathcal { F } } ) | \\leq \\epsilon$ . But, by construction of $\\hat { F }$ , there exits $h \\in { \\mathcal { F } }$ such that $\\hat { h } \\circ \\pi _ { \\mathcal F } = h$ . Thus, ", + "bbox": [ + 173, + 808, + 825, + 858 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/5ff2d1ed6829d210316d46b247e2822e0e43df1a6397bce9833b1c37126e98de.jpg", + "text": "$$\n\\operatorname* { s u p } _ { x \\in X } \\left| h ( x ) - f ( x ) \\right| = \\operatorname* { s u p } _ { x \\in X } \\left| \\hat { h } ( \\pi _ { \\mathcal { F } } ( x ) ) - \\hat { f } ( \\pi _ { \\mathcal { F } } ( x ) ) \\right| = \\operatorname* { s u p } _ { x _ { \\mathcal { F } } \\in X _ { \\mathcal { F } } } \\left| \\hat { h } ( x _ { \\mathcal { F } } ) ) - \\hat { f } ( x _ { \\mathcal { F } } ) \\right| \\leq \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 215, + 867, + 781, + 897 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "As this holds for any $\\epsilon > 0$ , we have proven that $f \\in { \\overline { { \\mathcal { F } } } }$ . ", + "bbox": [ + 173, + 909, + 542, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We first need to extend Corollary 17 to vector-valued functions. For this, we need to have a vectorvalued version of the Stone-Weierstrass theorem and as shown by the example below additional assumptions have to be made. ", + "bbox": [ + 174, + 128, + 823, + 171 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Example 19. We consider now the equivariant version of PointNet corresponding to the particular case where $X$ is a compact subset of $( \\mathbb { R } ^ { p } ) ^ { n }$ , $Y = \\mathbb { R } ^ { n }$ and ${ \\mathcal { F } } = \\{ x \\mapsto ( f ( x _ { 1 } ) , \\ldots , f ( x _ { n } ) )$ , $f \\in$ ${ \\mathcal { C } } ( \\mathbb { R } , \\mathbb { R } ) \\}$ . Then clearly $\\mathcal { F }$ is a subalgebra of ${ \\mathcal { C } } _ { E } ( X , Y )$ containing the constant function 1 and $\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , x ) , x \\in X \\}$ . Hence if Corollary 17 would be true with vector-valued functions instead of real-valued functions, we would have that $\\mathcal { F }$ is dense in ${ \\mathcal { C } } ( X , Y )$ . But this can clearly not be true as ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )$ which is clearly not dense in ${ \\mathcal { C } } ( X , Y )$ . ", + "bbox": [ + 173, + 175, + 825, + 261 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We now present an extension of Corollary 17 also due to Timofte (2005): ", + "bbox": [ + 173, + 271, + 651, + 286 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proposition 20. Let $X$ be a compact space, $Y = \\mathbb { R } ^ { p }$ for some $p \\geq 1$ . Let ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )$ . If there exists a nonempty subset $S \\subseteq { \\mathcal { C } } ( X , \\mathbb { R } )$ such that: ", + "bbox": [ + 174, + 290, + 821, + 319 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/9bf2e3fc0ffc42174ae95b1307d557c76c96407660c9aa2316facd18ff88cdaf.jpg", + "text": "$$\nS \\cdot { \\mathcal { F } } \\subseteq { \\mathcal { F } } a n d , \\rho \\left( S \\right) \\subseteq \\rho \\left( { \\mathcal { F } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 325, + 589, + 343 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Then we have ", + "bbox": [ + 173, + 349, + 266, + 363 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/3985aa38ff42bdecea05fa73605e5ec939fbe675252c60270a27629afa252457.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } ( X , Y ) , \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , f ( x ) \\in \\overline { { \\mathcal { F } ( x ) } } \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 367, + 660, + 395 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where ${ \\mathcal { F } } ( x ) = \\{ f ( x ) , f \\in { \\mathcal { F } } \\}$ . Moreover in (18), we can replace $\\rho \\left( \\mathcal { F } \\right)$ by $\\rho \\left( S \\right)$ . ", + "bbox": [ + 174, + 400, + 707, + 417 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Note that in the particular case $Y = \\mathbb { R }$ and if $\\mathcal { F }$ is a subalgebra of ${ \\mathcal { C } } ( X , \\mathbb { R } )$ , then we can take $S = \\mathcal { F }$ in (17) and if the constant function 1 is in $\\mathcal { F }$ , then $\\mathcal { F } ( x ) = \\mathbb { R }$ , so that we recover Corollary 17. ", + "bbox": [ + 171, + 426, + 823, + 457 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Now consider the case where $\\mathcal { F }$ is a subalgebra of $\\mathcal { C } ( X , \\mathbb { R } ^ { p } )$ . We need to find a set $S \\in { \\mathcal { C } } ( X , \\mathbb { R } )$ satisfying (17) i.e. with a better separating power than $\\mathcal { F }$ but containing real-valued functions such that $s f \\in \\mathcal { F }$ for all $s \\in S$ and $f \\in { \\mathcal { F } }$ . In the sequel, we will consider the set $\\mathcal { F } _ { s c a l } =$ $\\{ f \\in { \\mathcal { C } } ( X , \\mathbb { R } ) : f { \\bf 1 } \\in { \\mathcal { F } } \\}$ . We clearly have $\\mathcal { F } _ { s c a l } \\cdot \\mathcal { F } \\subseteq \\mathcal { F }$ since $\\mathcal { F }$ is a subalgebra. Hence, in this setting, Prop. 20 can be rewritten as follows: ", + "bbox": [ + 173, + 462, + 825, + 534 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Corollary 21. Let $X$ be a compact space, $Y = \\mathbb { R } ^ { p }$ for some $p$ , $G$ be a finite group acting (continuously) on $X$ and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { I } ( X , Y )$ a (non-empty) set of invariant functions associated to $G$ . ", + "bbox": [ + 174, + 536, + 825, + 566 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Consider the following assumptions, ", + "bbox": [ + 174, + 571, + 413, + 588 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "1. $\\mathcal { F }$ is a sub-algebra of ${ \\mathcal { C } } ( X , Y )$ and the constant function 1 is in $\\mathcal { F }$ . ", + "bbox": [ + 212, + 598, + 671, + 614 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "2. The set of functions $\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )$ defined by, ", + "bbox": [ + 210, + 622, + 553, + 638 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/8a12e10f4bc61eb8fe4a5767fac0403ed142c50363ebd3d4959cfb569fb04c0e.jpg", + "text": "$$\n\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f { \\bf 1 } \\in \\mathcal { F } \\}\n$$", + "text_format": "latex", + "bbox": [ + 413, + 643, + 642, + 661 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "satisfy, ", + "bbox": [ + 232, + 667, + 279, + 683 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/7902cbbfce716d9f0a2b65aa41abfac2d75b55d6bfeb3453cd81d41988da130e.jpg", + "text": "$$\n\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 462, + 680, + 594, + 698 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "3. For any $x \\in X$ , there exists $f \\in { \\mathcal { F } }$ such that $f ( x )$ has pairwise distinct coordinates, i.e., for any indices $i , j \\in \\{ 1 , \\ldots , p \\}$ with $i \\neq j$ , $f ( x ) _ { i } \\not = f ( x ) _ { j }$ . ", + "bbox": [ + 210, + 707, + 825, + 738 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ (for the topology of uniform convergence) is, ", + "bbox": [ + 173, + 747, + 612, + 763 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/f7680e86a8d36497bd15a2a6ff6beb67c9224e77b9a2a32d2cda0a42d76c2b9e.jpg", + "text": "$$\n{ \\overline { { \\mathcal { F } } } } = \\{ f \\in { \\mathcal { C } } _ { I } ( X , Y ) : \\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right) \\} ~ .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 768, + 630, + 787 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Note that Assumptions 1 and 2 ensures that (17) is valid, while Assumption 3 ensures that $\\mathcal { F } ( \\boldsymbol { x } ) = \\mathbb { R } ^ { p }$ . Unfortunately, in the equivariant case, the condition $\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right)$ is too strong and we now explain how we will relax it. ", + "bbox": [ + 174, + 800, + 826, + 843 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For the sake of simplicity, we consider here the particular setting adapted to graphs: let $n \\geq 1$ be a fixed number (corresponding to the number of nodes), $X$ be a compact set of graphs in $\\mathbb { R } ^ { n ^ { 2 } }$ and $Y = \\mathbb { F } ^ { n }$ with $\\mathbb { F } = \\mathbb { R } ^ { p }$ for some $p \\geq 1$ . We define the action of the symmetric group $S _ { n }$ on $X$ by $( \\sigma \\star x ) _ { \\sigma ( i ) , \\sigma ( j ) } = x _ { i . j }$ and on $Y$ by $( \\sigma \\star y ) _ { \\sigma ( i ) } = y _ { i } \\in \\mathbb { R } ^ { p }$ . Hence the set of continuous equivariant functions ${ \\mathcal { C } } _ { E } ( X , Y )$ agrees with Definition 1. ", + "bbox": [ + 173, + 849, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Now consider the case where ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )$ is a subalgebra of equivariant functions. Then, $f \\in \\mathcal { F } _ { s c a l }$ needs to be invariant in order for $f \\mathbf { 1 }$ to be equivariant and hence in $\\mathcal { F }$ . As a result, we see that $\\mathcal { F } _ { s c a l }$ will not separate points of $X$ in the same orbit, i.e. $x$ and $\\sigma \\star x$ . But these points will typically be separated by $\\mathcal { F }$ , since for any $f \\in { \\mathcal { F } }$ , we have $f ( \\sigma \\star x ) = \\sigma \\star f ( x )$ which is not equal to $f ( x )$ unless $f$ is invariant. ", + "bbox": [ + 173, + 102, + 825, + 174 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We see that we need somehow to require a weaker separating power for $\\mathcal { F }$ . More formally, two isomorphic graphs will have permuted outputs through an equivariant function, but should not be considered as separated. Let $O r b ( x ) = \\{ \\bar { \\sigma } \\star x , \\sigma \\in S _ { n } \\}$ and $O r b ( y ) = \\{ \\sigma \\star y , \\sigma \\in S _ { n } \\}$ . For any equivariant function $f \\in { \\mathcal { C } } _ { E } ( X , Y )$ , for any $z \\in O r b ( x )$ , we have $f ( z ) \\in O r b ( f ( x ) )$ . Then let $\\pi : Y \\to Y / S _ { n }$ be the canonical projection $\\pi ( y ) = O r b ( y )$ . We define ", + "bbox": [ + 173, + 180, + 825, + 251 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/daeeb5dd9b853b3fb942d85b56c15bd90e1649f477120213c2c416c75e320852.jpg", + "text": "$$\n\\begin{array} { r } { ( x , x ^ { \\prime } ) \\in \\rho \\left( \\pi \\circ \\mathcal { F } \\right) \\Leftrightarrow \\forall f \\in \\mathcal { F } , O r b ( f ( x ) ) = O r b ( f ( x ^ { \\prime } ) ) } \\\\ { \\Leftrightarrow \\forall f \\in \\mathcal { F } , \\exists \\sigma \\in { \\cal S } _ { n } , f ( \\sigma \\star x ) = f ( x ^ { \\prime } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 258, + 712, + 296 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In particular, we see that if $x ^ { \\prime } \\in O r b ( x )$ then $( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } ) \\in \\rho \\left( \\pi \\circ \\mathcal { F } \\right)$ for any ${ \\mathcal { F } } \\in { \\mathcal { C } } _ { E } ( X , Y )$ . Moreover, two graphs $x$ and $x ^ { \\prime }$ are $\\rho \\left( \\pi \\circ { \\mathcal { F } } \\right)$ -distinct if there exists a function $f \\in { \\mathcal { F } }$ such that $\\forall \\sigma$ , $f ( \\sigma \\star x ) \\neq$ $f ( x ^ { \\prime } )$ , i.e. the function $f$ discriminates $O r b ( x )$ from $O r b ( x ^ { \\prime } )$ in the sense that for any $z \\in O r b ( x )$ and $z ^ { \\prime } \\in O r b ( x ^ { \\prime } )$ , we have $f ( z ) \\neq f ( z ^ { \\prime } )$ . ", + "bbox": [ + 173, + 303, + 826, + 361 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "To obtain an equivalent of Proposition 20 with ${ \\mathcal { C } } _ { E } ( X , Y )$ replacing ${ \\mathcal { C } } ( X , Y )$ , we are able to relax assumption (17) to $\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)$ . Our main general result in this direction is the following theorem (proved in Section D.7) which might be of independent interest: ", + "bbox": [ + 176, + 366, + 823, + 410 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Theorem 22. Let $X$ be a compact space, $Y = \\mathbb { R } ^ { p }$ for some $p$ , $G$ be a finite group acting (continuously) on $X$ and $Y$ and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )$ a (non-empty) set of equivariant functions. ", + "bbox": [ + 173, + 414, + 825, + 444 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Denote by $\\pi : Y \\longrightarrow Y / G$ the canonical projection on the quotient space $Y / G$ . Consider the following assumptions, ", + "bbox": [ + 173, + 449, + 823, + 479 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "1. $\\mathcal { F }$ is a sub-algebra of ${ \\mathcal { C } } ( X , Y )$ and the constant function $\\mathbf { 1 }$ is in $\\mathcal { F }$ . ", + "bbox": [ + 210, + 489, + 671, + 507 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "2. The set of functions $\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )$ defined by, ", + "bbox": [ + 210, + 515, + 553, + 532 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/03cc6447857d238a76028c740b03b7ff2d80f610017676843e7fac091e8ae257.jpg", + "text": "$$\n\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f \\mathbf { 1 } \\in \\mathcal { F } \\}\n$$", + "text_format": "latex", + "bbox": [ + 413, + 540, + 642, + 558 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "satisfy, ", + "bbox": [ + 232, + 568, + 279, + 582 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/4fba4c21ab85d643415a4560b55334ee53c658d3b0aadd634cb445d85b888309.jpg", + "text": "$$\n\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 583, + 607, + 602 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ (for the topology of uniform convergence) is, ", + "bbox": [ + 173, + 614, + 612, + 631 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/9af15362310d0e44775e80599a8a3238f592bd6bd7cab9f38c01d68fef6249e4.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 638, + 712, + 656 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where ${ \\mathcal { F } } ( x ) = \\{ f ( x ) , f \\in { \\mathcal { F } } \\}$ . Moreover, i ${ } ^ { F } I ( x ) = \\{ ( i , j ) \\in [ p ] ^ { 2 } : \\forall y \\in { \\mathcal { F } } ( x ) , y _ { i } = y _ { j } \\} ,$ , then we have: ", + "bbox": [ + 173, + 664, + 825, + 694 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/0e12632b7a3e314534e1078e28ddc30eb2a77ed83a8ef7fc25b8068b0ede250c.jpg", + "text": "$$\n{ \\mathcal { F } } ( x ) = \\left\\{ y \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I ( x ) , y _ { i } = y _ { j } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 695, + 648, + 713 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Example 23. We now demonstrate how Theorem 22 can be used to recover the universality results in Segol & Lipman (2020). In this paper, the authors study equivariant neural network architectures working with unordered sets, corresponding in our case to $X = Y = \\mathbb { R } ^ { n }$ and the group being the symmetric group $S _ { n }$ . They show that the PointNet architecture cannot approximate any (continuous) equivariant function and that adding a single so-called transmission layer is enough to make this architecture universal. ", + "bbox": [ + 173, + 719, + 825, + 804 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Indeed, PointNet can only learn maps of the form $x \\in \\mathbb { R } ^ { n } \\mapsto ( f ( x _ { 1 } ) \\dots f ( x _ { n } ) )$ , which are not universal in the class of equivariant functions, as shown by Segol & Lipman (2020, Lem. 3). Now, their transmission layer is a map of the form $x \\in \\mathbb { R } ^ { n } \\mapsto ( \\mathbf { 1 } ^ { T } \\dot { x } ) \\mathbf { 1 }$ . Therefore, in PointNetST, adding such a layer precisely adds a large class of functions to ${ \\mathcal { F } } = \\{ ( f ( x _ { 1 } , \\sum _ { i } g ( x _ { i } ) ) , \\dots , f ( x _ { n } , \\sum _ { i } g ( x _ { i } { \\bar { ) } } ) )$ , $f \\in$ $\\mathcal { C } ( \\mathbb { R } \\times \\mathbb { R } ^ { h } , \\mathbb { R } )$ , $g \\in \\mathcal { C } ( \\mathbb { R } , \\mathbb { R } ^ { h } ) , h \\geq 1 \\}$ . $\\mathcal { F }$ is still an algebra and as shown in Example 19, we have $\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , x ) , x \\in X \\}$ . Moreover, we have $\\mathcal { F } _ { s c a l } = \\mathcal { C } _ { I } ( X , \\mathbb { R } )$ by Lem. 33 in particular, we get $\\rho \\left( \\mathcal { F } _ { s c a l } \\right) = \\{ ( x , \\sigma \\star x )$ , $x \\in X \\}$ , so that we obviously have $\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)$ . In summary, Theorem 22 implies the universality of PointNetST in $\\mathcal { C } _ { E } ( X , Y )$ . ", + "bbox": [ + 173, + 809, + 825, + 925 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "D.5 A PRELIMINARY VERSION OF THE EQUIVARIANT APPROXIMATION THEOREM ", + "text_level": 1, + "bbox": [ + 173, + 103, + 746, + 118 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We start by proving a version of Theorem 22 with a slightly weaker condition: ", + "bbox": [ + 174, + 130, + 691, + 146 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proposition 24. Let $X$ be a compact space, $Y ~ = ~ \\mathbb { R } ^ { p }$ for some $p _ { ; }$ , $G$ be a finite group acting (continuously) on $X$ and $Y$ and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )$ a (non-empty) set of equivariant functions. ", + "bbox": [ + 176, + 148, + 823, + 180 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Consider the following assumptions, ", + "bbox": [ + 174, + 185, + 413, + 200 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "1. $\\mathcal { F }$ is a subalgebra ${ \\mathcal { C } } _ { E } ( X , Y )$ . ", + "bbox": [ + 212, + 212, + 428, + 228 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "2. The set of real-valued functions $\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )$ defined by, ", + "bbox": [ + 210, + 237, + 632, + 253 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/12582686c959485381f9aced585e93626563dbe9a5b653278b8db91a3b7216d4.jpg", + "text": "$$\n\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f \\mathbf { 1 } \\in \\mathcal { F } \\}\n$$", + "text_format": "latex", + "bbox": [ + 413, + 261, + 640, + 279 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "satisfies, ", + "bbox": [ + 232, + 286, + 289, + 301 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/d33ca0a723e770e49f6d1dfa964e55a278f958eae983b901a40bbf1ea9145dc2.jpg", + "text": "$$\n\\begin{array} { r } { \\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\left\\{ \\left( x , x ^ { \\prime } \\right) \\in X \\times X : \\exists g \\in G , \\left( g \\cdot x , x ^ { \\prime } \\right) \\in \\rho \\left( \\mathcal { F } \\right) \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 309, + 735, + 327 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ (for the topology of uniform convergence) is, ", + "bbox": [ + 171, + 343, + 612, + 358 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/05ec7a812381de297576716b9420b6e71e43c8146683f55dce0ab6c2214cfdc1.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\overline { { \\mathcal { F } ( x ) } } \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 364, + 715, + 393 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where ${ \\mathcal { F } } ( x ) = \\{ f ( x ) , f \\in { \\mathcal { F } } \\}$ . ", + "bbox": [ + 174, + 400, + 382, + 417 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "The proof of this theorem relies on two main ingredients. First, following the elegant idea of Maehara & Hoang (2019), we augment the input space to transform the vector-valued equivariant functions into scalar maps. Second, we apply the fine-grained approximation result Cor. 17. ", + "bbox": [ + 173, + 428, + 825, + 472 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. As uniform convergence implies point-wise convergence, the first inclusion is immediate, ", + "bbox": [ + 173, + 492, + 808, + 508 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/794a0f3fc800995c49e2b2dd3fc6908c6567345cf843ee5678134b228bcc364e.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } \\subseteq \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\overline { { \\mathcal { F } ( x ) } } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 515, + 715, + 542 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "The rest of the proof is devoted to the other direction. ", + "bbox": [ + 174, + 550, + 524, + 565 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "For convenience, denote by $\\Phi$ the family of linear forms associated to the canonical basis of $\\mathbb { R } ^ { p }$ , i.e., ", + "bbox": [ + 173, + 570, + 825, + 587 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/2d37fd88de4cf41ff9da7ce73017a1dd4c61010efa217538c346b53229c28d4d.jpg", + "text": "$$\n\\Phi = \\{ y \\mapsto y _ { i } : 1 \\leq i \\leq p \\} \\subseteq { \\mathcal { C } } ( Y , \\mathbb { R } ) .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 594, + 629, + 613 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Define our augmented input space as ${ \\tilde { X } } = X \\times \\Phi$ . As $\\Phi$ is finite and $X$ is compact, $\\tilde { X }$ is still a compact space. We now transform $\\mathcal { F }$ , a class of equivariant functions from $X$ to $Y$ , into $\\tilde { \\mathcal { F } }$ a class of maps from $\\tilde { X }$ to $\\mathbb { R }$ . Define ", + "bbox": [ + 173, + 621, + 826, + 667 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/27509e68cff172f4eb8660c327e0bde5412d66599da55a3e8b8e7e713dc0673f.jpg", + "text": "$$\n{ \\tilde { \\mathcal { F } } } = \\{ ( x , \\varphi ) \\mapsto \\varphi ( f ( x ) ) : f \\in { \\mathcal { F } } \\} \\ .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 676, + 619, + 696 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We check that $\\tilde { \\mathcal { F } }$ is indeed a subset of $\\mathcal { C } ( \\tilde { X } , \\mathbb { R } )$ . Indeed, as $\\Phi$ is finite, it is equipped with the discrete topology. Hence, each singleton $\\{ \\varphi \\}$ for $\\varphi \\in \\Phi$ is open in $\\Phi$ and it suffices to check the continuity in the first variable with $\\varphi$ fixed. But, if $f \\in { \\mathcal { F } }$ , $x \\stackrel { \\textstyle - } { \\mapsto } \\varphi ( f ( x ) )$ is continuous as a composition of continuous maps. ", + "bbox": [ + 173, + 705, + 826, + 763 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We can now apply Cor. 17 to $\\tilde { \\mathcal { F } } \\subseteq \\mathcal { C } ( \\tilde { X } , \\mathbb { R } )$ . Therefore, the closure of $\\tilde { F }$ in $\\mathcal { C } ( \\tilde { X } , \\mathbb { R } )$ is, ", + "bbox": [ + 173, + 768, + 743, + 787 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/2107fb44cd850c9d173aa7107d1e50cb5e747c605f6469ae17534e59a649cd53.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\tilde { \\mathcal { F } } } } = \\left\\{ v \\in \\mathcal { C } ( \\tilde { X } , \\mathbb { R } ) : \\rho \\left( \\tilde { \\mathcal { F } } \\right) \\subseteq \\rho \\left( v \\right) , \\forall ( x , \\varphi ) \\in \\tilde { X } , v ( x , \\varphi ) \\in \\overline { { \\tilde { \\mathcal { F } } ( x , \\varphi ) } } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 795, + 745, + 823 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We now show the equality of (19). Take $h$ in the right-hand side of (19), i.e. $h \\in { \\mathcal { C } } _ { E } ( X , Y )$ such that $\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( h \\right)$ and $h ( x ) \\in { \\overline { { \\mathcal { F } ( x ) } } }$ for all $x \\in X$ . We show that $\\tilde { h }$ , defined by ${ \\tilde { h } } : ( x , \\varphi ) \\mapsto \\varphi ( h ( x ) )$ , belongs to $\\overline { { \\tilde { F } } }$ using the result above. ", + "bbox": [ + 173, + 837, + 825, + 888 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "• As $h$ is continuous, by the same argument as above, $( x , \\varphi ) \\mapsto \\varphi ( h ( x ) )$ is continuous on $\\tilde { X }$ ", + "bbox": [ + 214, + 907, + 825, + 925 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "• We check that $\\rho \\left( \\tilde { \\mathcal { F } } \\right) \\subseteq \\rho \\left( \\tilde { h } \\right)$ ", + "bbox": [ + 218, + 101, + 441, + 126 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Take $( x , \\varphi ) , ( y , \\psi ) \\in \\tilde { X }$ such that, for all $f \\in { \\mathcal { F } }$ , ", + "bbox": [ + 230, + 132, + 553, + 150 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/64bc529b69a27cebb8112efbedaa7468f004fa7d508eb5d84669e797cc4d7a27.jpg", + "text": "$$\n\\varphi ( f ( x ) ) = \\psi ( f ( y ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 457, + 156, + 598, + 174 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "and we aim at showing that $\\varphi ( h ( x ) ) = \\psi ( h ( y ) )$ . ", + "bbox": [ + 233, + 180, + 552, + 195 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "To gain more information from (20), we apply it to functions of the form $f \\mathbf { 1 }$ with $f \\in \\mathcal { F } _ { s c a l }$ By definition of $\\Phi$ , this translates to $f ( x ) = f ( y )$ , for any $f \\in \\mathcal { F } _ { s c a l }$ . Therefore $( x , y ) \\in$ $\\rho \\left( \\mathcal { F } _ { s c a l } \\right)$ and so there exists $g \\in G$ such that $( g \\cdot { \\dot { x } } , y ) \\in \\rho \\left( { \\mathcal { F } } \\right)$ . Plugging this into (20) and using the equivariance of $f$ , ", + "bbox": [ + 230, + 202, + 825, + 258 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/033a2bd5f1112ad6911eead0dea8fa6c3f13ff16c9486825398a757c6b1aeebf.jpg", + "text": "$$\n\\forall f \\in { \\mathcal { F } } , \\varphi ( f ( x ) ) = \\psi ( g \\cdot f ( x ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 265, + 642, + 282 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "As $G$ acts continuously on $Y$ , both $\\varphi$ and $z \\mapsto \\psi ( g \\cdot z )$ are continuous and, as a consequence of the equality above, coincide on $\\overline { { \\mathcal { F } ( \\boldsymbol { x } ) } }$ . But we assumed that $h ( x ) \\in { \\overline { { \\mathcal { F } ( x ) } } }$ and therefore the equality also holds for $h$ , i.e. $\\varphi ( h ( x ) ) = \\psi ( g \\cdot h ( x ) )$ . ", + "bbox": [ + 230, + 290, + 825, + 335 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Finally, recalls that, by assumption, $\\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( h \\right)$ . Therefore $( g \\cdot x , y ) \\in \\rho \\left( \\mathcal { F } \\right)$ implies that $h ( g \\cdot x ) = h ( y )$ and, combined with the result above, $\\varphi ( h ( x ) ) = \\psi ( h ( y ) )$ . ", + "bbox": [ + 230, + 340, + 823, + 371 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "• We verify that, for $x \\in X$ , $\\varphi \\in \\Phi$ , $\\varphi ( h ( x ) )$ belongs to $\\scriptstyle { \\overline { { { \\tilde { \\mathcal { F } } } ( x ) } } }$ . Indeed, recall that $h ( x ) \\in$ $\\overline { { \\mathcal { F } ( \\boldsymbol { x } ) } }$ . Therefore, as $\\varphi$ is continuous, $\\varphi ( h ( x ) )$ is in $\\overline { { \\varphi ( \\mathcal { F } ( x ) ) } }$ which is included in $\\overline { { { \\tilde { \\mathcal { F } } } ( \\boldsymbol { x } ) } }$ . ", + "bbox": [ + 220, + 381, + 825, + 420 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "This shows that ${ \\tilde { h } } : ( x , \\varphi ) \\mapsto \\varphi ( h ( x ) )$ is in $\\overrightharpoon { \\tilde { \\mathcal { F } } }$ . Consequently, for any $\\epsilon > 0$ , there exists $f \\in { \\mathcal { F } }$ such that, ", + "bbox": [ + 173, + 434, + 823, + 467 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/708ed24e96d96e58357b2923137214da3a9064a289cbef2c414e19f0a28196c9.jpg", + "text": "$$\n\\forall x \\in X , \\forall \\varphi \\in \\Phi , | \\varphi ( h ( x ) ) - \\varphi ( f ( x ) ) | \\leq \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 465, + 650, + 483 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "If $Y = \\mathbb { R } ^ { p }$ is endowed with the infinity norms on coordinates, by definition of $\\Phi$ , this means that, ", + "bbox": [ + 174, + 486, + 807, + 501 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/3d9fc9de0ff0108475fea7d28f6b74a60c3fd339808fdb0bdbc7092a77ac33ec.jpg", + "text": "$$\n\\forall x \\in X , \\ \\| h ( x ) - f ( x ) \\| \\leq \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 507, + 599, + 526 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Remark 25. In the particular case of $G \\subseteq S _ { n }$ being a group of permutations acting on $\\mathbb { R } ^ { p }$ by, for $g \\in G , x \\in \\mathbb { R } ^ { p }$ , ", + "bbox": [ + 168, + 556, + 825, + 585 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/d523a36173a945c4e3d5fbad4635b0203faf11deb1e0fa0d240057f4c095baef.jpg", + "text": "$$\n\\forall i \\in \\{ 1 , . . . , p \\} , ( g \\cdot x ) _ { i } = x _ { g ^ { - 1 } ( i ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 584, + 619, + 604 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "the functions of $\\tilde { \\mathcal { F } }$ are indeed invariant, as shown by Maehara & Hoang (2019). For this, a left action on $\\Phi$ is defined by, for $g \\in G , \\varphi \\in \\Phi$ , ", + "bbox": [ + 169, + 609, + 825, + 638 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/ee13f6483cb00aefd95dc003fc417daa656fcbc5d6ed453ce2ed5e3b0a04ed4a.jpg", + "text": "$$\n\\forall x \\in \\mathbb { R } ^ { p } , ( g \\cdot \\varphi ) ( x ) = \\varphi ( g ^ { - 1 } \\cdot x ) .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 643, + 617, + 662 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In other words, the action of $g$ on the linear form associated to the $i ^ { t h }$ coordinate yields the linear form associated to the $g ( i ) ^ { t h }$ coordinate. One can now check that the functions $\\tilde { \\mathcal { F } }$ are invariant. ", + "bbox": [ + 173, + 669, + 825, + 700 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "D.6 CHARACTERIZING THE SUBALGEBRAS OF $\\mathbb { R } ^ { p }$ ", + "text_level": 1, + "bbox": [ + 173, + 717, + 532, + 733 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Before moving to our general result, we need to study the structure of the subalgebras of $\\mathbb { R } ^ { p }$ . For this, we will use the following simple lemma. ", + "bbox": [ + 173, + 743, + 826, + 773 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In the following lemma, $\\mathbb { R } [ X _ { 1 } , \\ldots , X _ { p } ]$ denotes the set of multivariate polynomials with $p$ indeterminates (and real coefficients). ", + "bbox": [ + 173, + 779, + 825, + 808 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma 26. Let $C \\subseteq \\mathbb { R } ^ { p }$ be a finite subset of $\\mathbb { R } ^ { p }$ . There exists $P \\in \\mathbb { R } [ X _ { 1 } , \\ldots , X _ { p } ]$ such that $P _ { \\mid C }$ the restriction of $P$ to $C$ , is an injective map. ", + "bbox": [ + 173, + 810, + 825, + 842 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. Let $x ^ { 1 } , \\ldots , x ^ { m } \\in \\mathbb { R } ^ { p }$ be distinct vectors such that $\\{ x ^ { 1 } , \\ldots , x ^ { m } \\} = C$ . Similarly to Lagrange polynomials, define, ", + "bbox": [ + 173, + 856, + 823, + 886 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/f8371e8980afafc2711ddd424c2b254491f520a2f1a0dcbb7cb5a4eb3aa1b463.jpg", + "text": "$$\nP ( X _ { 1 } , \\dots , X _ { p } ) = \\sum _ { i = 1 } ^ { m } i \\prod _ { j \\neq i } { \\frac { \\sum _ { l = 1 } ^ { p } ( X _ { l } - x _ { l } ^ { j } ) ^ { 2 } } { \\| x ^ { i } - x ^ { j } \\| _ { 2 } ^ { 2 } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 885, + 653, + 928 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "which is a well-defined multivariate polynomial. Note that, seeing $X = ( X _ { 1 } , \\ldots , X _ { p } )$ as a vector in $\\mathbb { R } ^ { p }$ , it can also be written as, ", + "bbox": [ + 171, + 102, + 823, + 132 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/71b7a3cb91ea4aceadcd115f8876316fe24d2d83c7805e7db5a7ec9bb853da21.jpg", + "text": "$$\nP ( X _ { 1 } , \\dots , X _ { p } ) = \\sum _ { i = 1 } ^ { m } i \\prod _ { j \\neq i } { \\frac { \\| X - x ^ { j } \\| _ { 2 } ^ { 2 } } { \\| x ^ { i } - x ^ { j } \\| _ { 2 } ^ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 136, + 635, + 179 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "By construction, $P ( x ^ { i } ) = i$ and therefore $P$ is an injective map on $C$ ", + "bbox": [ + 174, + 184, + 627, + 200 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lemma 27. For a subalgebra $\\mathcal { A }$ of $\\mathbb { R } ^ { p }$ , we define: ", + "bbox": [ + 173, + 207, + 508, + 222 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/92c67c82ca0b8d150e299a02adbf121f99a6519ba0b3999082238dd92333789a.jpg", + "text": "$$\n\\begin{array} { l } { J = \\left\\{ j \\in \\left\\{ 1 , \\dots , p \\right\\} : \\forall x \\in A , x _ { j } = 0 \\right\\} } \\\\ { I = \\left\\{ ( i , j ) \\in ( J ^ { C } ) ^ { 2 } : \\forall x \\in A , x _ { i } = x _ { j } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 356, + 226, + 642, + 266 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Then, we have : ", + "bbox": [ + 174, + 268, + 277, + 284 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/4337e4d6447d1bbf034a9cef86423630e4a3411c710fb76d3fc543091d4bdc30.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { A } = \\{ x \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I , x _ { i } = x _ { j } , \\forall j \\in J , x _ { j } = 0 \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 281, + 684, + 299 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. Before proving the general case, we focus on the situation where $\\mathcal { A }$ will turn out to be the whole $\\mathbb { R } ^ { p }$ . ", + "bbox": [ + 174, + 313, + 825, + 340 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Assume that the two following conditions holds, ", + "bbox": [ + 174, + 348, + 490, + 362 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/5ea3193ce50a0f2b509b532a2849b6d114051253e5947087548db89353b07d18.jpg", + "text": "$$\n\\begin{array} { l } { \\forall i \\neq j , \\exists x \\in A , x _ { i } \\neq x _ { j } } \\\\ { \\forall i , \\exists x \\in A , x _ { i } \\neq 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 410, + 366, + 586, + 404 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Our goal is to show that, under these additional assumptions, $\\ b { A } = \\mathbb { R } ^ { p }$ . We divide the proof in three parts, first we show that $\\mathbf { 1 } \\in { \\mathcal { A } }$ using (24), giving us that $\\mathcal { A }$ is closed under polynomials, then that there is $x \\in { \\mathcal { A } }$ with pairwise distinct coordinates thanks to (23) and finally that this implies that $\\mathcal { A }$ is the whole space. ", + "bbox": [ + 174, + 406, + 825, + 463 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Note that if $p = 1$ , (24) and the linear space property of $\\mathcal { A }$ immediately give the result. ", + "bbox": [ + 174, + 468, + 738, + 484 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "• Here we prove that (24) implies that $\\mathbf { 1 } \\in { \\mathcal { A } }$ . ", + "bbox": [ + 217, + 498, + 521, + 513 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "– First, we construct by induction $x \\in { \\mathcal { A } }$ such that $x _ { i } \\neq 0$ for any index $i$ . More precisely, our induction hypothesis at step $j \\in \\{ 1 , \\ldots , p \\}$ is, ", + "bbox": [ + 246, + 522, + 825, + 551 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/162404fff5d0df1e41233f017ae5e4769a242fa2fa391d06d358bf1c9e2532c7.jpg", + "text": "$$\n\\exists x \\in A , \\forall 1 \\leq i \\leq j , x _ { i } \\neq 0 .\n$$", + "text_format": "latex", + "bbox": [ + 442, + 556, + 645, + 573 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "By (24), this holds for $j = 1$ . ", + "bbox": [ + 264, + 577, + 457, + 592 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Now, assume that it holds at $j - 1$ for some $p \\ge j \\ge 2$ and take $x \\in { \\mathcal { A } }$ such that $x _ { i } \\neq 0$ for any $1 \\leq i \\leq j - 1$ and $y \\in { \\mathcal { A } }$ such that $j _ { j } \\neq 0$ by (24). ", + "bbox": [ + 264, + 593, + 823, + 621 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "By definition of $x$ , the set, ", + "bbox": [ + 264, + 622, + 439, + 635 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/49ab1ad10d515cf88d9f9a6b982c686f6ba79bf53100ebfb0a98f9ec25c6da42.jpg", + "text": "$$\n\\left\\{ \\lambda \\in \\mathbb { R } : \\exists 1 \\leq i \\leq j - 1 , \\lambda x _ { i } + y _ { i } = 0 \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 640, + 684, + 657 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "is finite. Thus, there exists $\\lambda \\in \\mathbb { R }$ such that $\\lambda x _ { i } + y _ { i } \\neq 0$ for any $1 \\leq i \\leq j - 1$ and for $i = j$ too as $x _ { j } = 0$ and $y _ { j } \\neq 0$ . As $\\mathcal { A }$ is a subalgebra, $\\lambda x + y \\in A$ and this concludes the induction step. ", + "bbox": [ + 259, + 662, + 825, + 704 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "– Let $x \\in { \\mathcal { A } }$ be the vector constructed, i.e. such that $x _ { i } \\neq 0$ for every index $i$ . We prove that $\\mathbf { 1 } \\in { \\mathcal { A } }$ by constructing 1 from $x$ . ", + "bbox": [ + 246, + 707, + 823, + 734 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Indeed, using Lagrange interpolation, take $P \\in \\mathbb { R } [ X ]$ such that $\\textstyle P ( x _ { i } ) = { \\frac { 1 } { x _ { i } } }$ for every $i$ (Note that this noes not matter if some $x _ { i }$ are equal, as the $\\frac { 1 } { x _ { i } }$ would also be the same.) ", + "bbox": [ + 261, + 734, + 823, + 750 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 263, + 752, + 821, + 767 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Finally, as $\\mathcal { A }$ is a subalgebra, $x P ( x )$ , which is to be understood coordinate-wise, is in $\\mathcal { A }$ and so is $\\mathbf { 1 } = x P ( x )$ . ", + "bbox": [ + 263, + 768, + 825, + 799 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "• We show that the previous point and (23) imply that there exists a vector in $\\mathcal { A }$ with pairwise distinct coordinates, i.e. that there exists $x \\in { \\mathcal { A } }$ such that, for any $i \\neq j$ , $x _ { i } \\neq x _ { j }$ . Using (23), for any $i < j$ , there exists $x ^ { i j } \\in { \\mathcal { A } }$ such that $x _ { i } ^ { i j } \\neq x _ { j } ^ { i j }$ . We wish to combine the family $( x ^ { i j } ) _ { i < j }$ into a single vector. ", + "bbox": [ + 220, + 806, + 825, + 869 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "For this we use Lem. 26. Seeing each collection $( x _ { k } ^ { i j } ) _ { i < j }$ as vector of $\\mathbb { R } ^ { p ( p - 1 ) / 2 }$ , we define $C = \\{ ( x _ { k } ^ { i j } ) _ { i < j } : 1 \\leq k \\leq p \\}$ , which is a finite subset (of cardinal $p$ ) of $\\mathbb { R } ^ { p ( p - 1 ) / 2 }$ . By Lem. 26, there exists $P \\in \\mathbb { R } [ \\dot { X } _ { 1 } , \\dots , X _ { p ( p - 1 ) / 2 } ]$ such that $P$ is an injective map on $C$ . ", + "bbox": [ + 230, + 876, + 825, + 926 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "As $\\mathcal { A }$ is a subalgebra and $\\mathbf { 1 } \\in { \\mathcal { A } }$ , the vector, ", + "bbox": [ + 233, + 103, + 521, + 118 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/8a423ec43f7c8b56165d79739a5fd17ad4b59ffd27935efb091e0f871ca435b7.jpg", + "text": "$$\nP \\left( ( x ^ { i j } ) _ { i < j } \\right) = \\left( \\begin{array} { c } { { P \\left( ( x _ { 1 } ^ { i j } ) _ { i < j } \\right) } } \\\\ { { \\vdots } } \\\\ { { P \\left( ( x _ { p } ^ { i j } ) _ { i < j } \\right) } } \\end{array} \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 408, + 123, + 647, + 190 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "is in $\\mathcal { A }$ too. ", + "bbox": [ + 232, + 194, + 308, + 209 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "We now check that this vector has pairwise distinct coordinates. Let $l < k$ , then $x _ { l } ^ { l k } \\neq x _ { k } ^ { l k }$ and therefore $( x _ { l } ^ { i j } ) _ { i < j } \\neq ( x _ { k } ^ { i j } ) _ { i < j }$ . By construction of $P$ , $P \\left( ( x _ { l } ^ { i j } ) _ { i < j } \\right) \\neq P \\left( ( { \\bar { x } } _ { k } ^ { i j } ) _ { i < j } \\right)$ , i.e. $P \\left( ( x ^ { i j } ) _ { i < j } \\right) _ { l } \\neq P \\left( ( x ^ { i j } ) _ { i < j } \\right) _ { k }$ . Thus, $P \\left( ( x ^ { i j } ) _ { i < j } \\right) \\in { \\mathcal { A } }$ has pairwise distinct coordinates as required. ", + "bbox": [ + 230, + 215, + 826, + 285 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "• Finally, we show that $\\ b { \\mathcal { A } } = \\mathbb { R } ^ { p }$ . This is a direct consequence of the point above and of Lagrange interpolation. Indeed, take any $y \\in \\mathbb { R } ^ { p }$ and denote by $x \\in { \\mathcal { A } }$ the vector that we just constructed with pairwise distinct coordinates. Therefore, by Lagrange interpolation, there exists $P \\in \\mathbb { R } [ X ]$ such that $P ( x _ { i } ) = y _ { i }$ for every $i \\in \\{ 1 , \\ldots , p \\}$ . As $\\mathcal { A }$ is a sublagebra and $\\mathbf { 1 } \\in { \\mathcal { A } }$ , $P ( x ) \\in { \\mathcal { A } }$ and hence $y \\in { \\mathcal { A } }$ . ", + "bbox": [ + 218, + 292, + 825, + 364 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Finally, we return to the general case. We introduce the set of indexes of $I$ and $J$ which appear in the result and use them to reduce the situation to the previous case. Define, ", + "bbox": [ + 174, + 377, + 823, + 406 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/615c88863e30d76445c12239748a02e2fbc812b000b763b7f2d5cb94742d866b.jpg", + "text": "$$\n\\begin{array} { l } { J = \\left\\{ j \\in \\left\\{ 1 , \\dots , p \\right\\} : \\forall x \\in A , x _ { j } = 0 \\right\\} } \\\\ { I = \\left\\{ ( i , j ) \\in ( J ^ { C } ) ^ { 2 } : \\forall x \\in A , x _ { i } = x _ { j } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 356, + 410, + 642, + 450 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "and denote by $\\mathcal { A } ^ { \\prime } = \\{ x \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I$ , $x _ { i } = x _ { j }$ , $\\forall j \\in J$ , $x _ { j } = 0 \\}$ , which is also a subalgebra. \nBy definition, it holds that ${ \\mathcal { A } } \\subseteq { \\mathcal { A } } ^ { \\prime }$ . ", + "bbox": [ + 173, + 454, + 825, + 484 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "By construction, $I$ is an equivalence relation on $J ^ { C }$ and denote by $J ^ { C } / I$ its equivalence classes. Let $p ^ { \\prime } = | J ^ { C } / I |$ and choose $i _ { 1 } , \\ldots , i _ { p ^ { \\prime } }$ representatives of the equivalence classes. Consider the map, ", + "bbox": [ + 171, + 489, + 825, + 520 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/97ecd28ce3499ceb0e04a23f8974bdc752aee7ad49d59f710f10913d981ef607.jpg", + "text": "$$\n\\varphi : \\left\\{ \\begin{array} { l l } { \\mathbb { R } ^ { p } \\longrightarrow \\mathbb { R } ^ { p ^ { \\prime } } } \\\\ { x \\longmapsto \\left( x _ { i _ { 1 } } , \\ldots , x _ { i _ { p ^ { \\prime } } } \\right) . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 398, + 525, + 598, + 568 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "$\\varphi$ is an algebra homomoprhism so that $\\varphi ( A )$ is a subalgebra of $\\mathbb { R } ^ { p ^ { \\prime } }$ . But, by construction of $I$ and $J$ , $\\varphi ( A )$ satisfies (23) and (24). Whence, by our result in this particular case, $\\varphi ( \\mathcal { A } ) = \\mathbb { R } ^ { p ^ { \\prime } }$ . ", + "bbox": [ + 173, + 575, + 826, + 608 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "However, ${ \\mathcal { A } } \\subseteq { \\mathcal { A } } ^ { \\prime }$ implies that $\\varphi ( \\mathcal { A } ) \\subseteq \\varphi ( \\mathcal { A } ^ { \\prime } ) \\subseteq \\mathbb { R } ^ { p ^ { \\prime } }$ . Therefore, $\\mathbb { R } ^ { p ^ { \\prime } } = \\varphi ( \\mathcal { A } ) = \\varphi ( \\mathcal { A } ^ { \\prime } )$ . But, by construction of $\\varphi$ , $\\varphi$ is actually an injective map on $\\mathcal { A } ^ { \\prime }$ . Therefore, we deduce from $\\varphi ( \\mathcal { A } ) = \\varphi ( \\mathcal { A } ^ { \\prime } )$ that $A = A ^ { \\prime }$ , concluding the proof. □ ", + "bbox": [ + 173, + 614, + 826, + 660 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "D.7 PROOF OF THE MAIN EQUIVARIANT APPROXIMATION THEOREM ", + "text_level": 1, + "bbox": [ + 174, + 675, + 656, + 690 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "We can now fully exploit the structure of subalgebra of $\\mathcal { F }$ thanks to the results above, and in particular relax the second assumption of Prop. 24 to give our main theorem 22. We first prove the following lemma. ", + "bbox": [ + 173, + 700, + 826, + 744 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Lemma 28. Under the assumptions of Thm. 22, for any $H \\subseteq G$ subgroup, for any $x , y \\in X$ ", + "bbox": [ + 168, + 747, + 781, + 763 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/b7db3ce16d5008e1f009c7818d8f281516103e840409f9608ad66a97256ec1b3.jpg", + "text": "$$\n\\forall f \\in { \\mathcal { F } } , \\ \\exists g \\in H , \\ f ( g \\cdot x ) = f ( y ) \\iff \\exists g \\in H , \\forall f \\in { \\mathcal { F } } , \\ f ( g \\cdot x ) = f ( y ) \\ .\n$$", + "text_format": "latex", + "bbox": [ + 236, + 768, + 759, + 785 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "In particular, ", + "bbox": [ + 173, + 791, + 261, + 806 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/ab22f84c31a9f4593badad02620729e5e11ee6dd1faba5d27148306d967aa618.jpg", + "text": "$$\n( x , y ) \\in \\rho \\left( \\pi \\circ { \\mathcal { F } } \\right) \\iff \\exists g \\in G , ( g \\cdot x , y ) \\in \\rho \\left( { \\mathcal { F } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 804, + 678, + 821 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. The reverse implication is immediate so we focus on the direct one and prove its contraposition, i.e., ", + "bbox": [ + 171, + 834, + 826, + 862 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/b395fbffcee7ecbc4b354d6f02d2911e9145b5fbefb83ae9bdbe81d1a8e5f9ed.jpg", + "text": "$$\n\\forall g \\in H , \\exists f \\in { \\mathcal { F } } , f ( g \\cdot x ) \\neq f ( y ) \\implies \\exists f \\in { \\mathcal { F } } , \\forall g \\in H , f ( g \\cdot x ) \\neq f ( y ) .\n$$", + "text_format": "latex", + "bbox": [ + 238, + 862, + 754, + 878 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "rove this, we take advantage of $\\mathcal { F }$ being a subalgebra and $H$ being finite. Let $H = \\{ g _ { 1 } , \\dots , g _ { h } \\}$ define, ", + "bbox": [ + 209, + 881, + 823, + 910 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/a34a4d5cbdef0f4bf83dc57f8952a85841745f08c0fae14a48ae6298660a3dbc.jpg", + "text": "$$\n{ \\mathcal { A } } = \\left\\{ ( f ( g _ { 1 } \\cdot x ) , \\ldots , f ( g _ { h } \\cdot x ) , f ( y ) ) : f \\in { \\mathcal { F } } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 907, + 660, + 925 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "As $\\mathcal { F }$ is a subalgebra of ${ \\mathcal { C } } ( X , \\mathbb { R } ^ { p } )$ , $\\mathcal { A }$ is a subalgebra of $\\mathbb { R } ^ { p ^ { \\prime } }$ with $p ^ { \\prime } = p ( h + 1 )$ . By Lem. 27, ", + "bbox": [ + 169, + 102, + 785, + 119 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/04e890a881a31113eae5d451c5a0a4718ca9f0b06807c6843664026b66fcdcd7.jpg", + "text": "$$\n\\begin{array} { r } { A = \\{ z \\in \\mathbb { R } ^ { p ^ { \\prime } } : \\forall ( i , j ) \\in I , z _ { i } = z _ { j } , \\forall j \\in J , z _ { j } = 0 \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 122, + 681, + 142 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where $I$ and $J$ can be chosen to be,1 ", + "bbox": [ + 174, + 146, + 411, + 161 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/4c3670f40a2b4811024b76646c565eee5388ad7fbec47e3932ddc11251b784d0.jpg", + "text": "$$\n\\begin{array} { r l } & { J = \\left\\{ j \\in \\{ 1 , \\dots , p \\} : \\forall z \\in A , z _ { j } = 0 \\right\\} } \\\\ & { I = \\left\\{ ( i , j ) \\in \\{ 1 , \\dots , p \\} ^ { 2 } : \\forall z \\in A , z _ { i } = z _ { j } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 164, + 656, + 204 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "As $I$ is an equivalence relation, an element of $\\mathcal { A }$ is uniquely defined by its coordinates on equivalence classes of $I$ . Therefore, one can choose $z \\in { \\mathcal { A } }$ such that $z _ { i } = z _ { j } \\iff ( i , j ) \\in I$ . By definition of $\\mathcal { A }$ , there exists $f ^ { * } \\in { \\mathcal { F } }$ such that $( f ^ { * } ( g _ { 1 } \\cdot x ) , \\ldots , f ^ { * } ( g _ { h } \\cdot x ) , ^ { ^ { \\scriptstyle * } } f ^ { * } ( y ) ) = z$ . We now check that $f ^ { * }$ is indeed appropriate. Take $l \\in \\{ 1 , \\ldots , h \\}$ , we want to show that $f ^ { * } ( g _ { l } \\cdot x ) \\neq f ^ { * } ( y )$ . By assumption, there exists $f \\in { \\mathcal { F } }$ such that $f ( g _ { l } \\cdot x ) \\neq f ( y )$ , i.e. there exists $i \\in \\{ 1 , \\ldots , p \\}$ such that $f ( g _ { l } \\cdot \\bar { x } ) _ { i } \\neq f ( y ) _ { i }$ . Therefore, $( ( l - 1 ) p + i , h p + i )$ cannot be in $I$ so that $z _ { ( l - 1 ) p + i } \\neq z _ { h p + 1 }$ , i.e. $f ^ { * } ( g _ { l } \\cdot x ) _ { i } \\neq f ^ { * } ( y ) _ { i }$ . □ ", + "bbox": [ + 173, + 205, + 826, + 306 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "We now prove our main abstract theorem. ", + "bbox": [ + 174, + 320, + 447, + 334 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Theorem 22. Let $X$ be a compact space, $Y = \\mathbb { R } ^ { p }$ for some $p , G$ be a finite group acting (continuously) on $X$ and $Y$ and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )$ a (non-empty) set of equivariant functions. ", + "bbox": [ + 173, + 337, + 823, + 366 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Denote by $\\pi : Y \\longrightarrow Y / G$ the canonical projection on the quotient space $Y / G$ . Consider the following assumptions, ", + "bbox": [ + 173, + 371, + 823, + 401 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "1. $\\mathcal { F }$ is a sub-algebra of ${ \\mathcal { C } } ( X , Y )$ and the constant function 1 is in $\\mathcal { F }$ . ", + "bbox": [ + 212, + 410, + 671, + 426 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "2. The set of functions $\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )$ defined by, ", + "bbox": [ + 210, + 434, + 553, + 450 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/aec4b63f7be76749d333d69c884dc783f9808cb3fb1837b8856ba5fa5945c340.jpg", + "text": "$$\n\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f \\mathbf { 1 } \\in \\mathcal { F } \\}\n$$", + "text_format": "latex", + "bbox": [ + 413, + 453, + 640, + 469 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "satisfy, ", + "bbox": [ + 232, + 474, + 279, + 488 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/6d2b6941bb70c4e04e2b0f4c6c74f3cb939e5a886170303da4d6fa0ce3a348cd.jpg", + "text": "$$\n\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 486, + 606, + 503 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ (for the topology of uniform convergence) is, ", + "bbox": [ + 171, + 511, + 612, + 526 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/cc66543d3e6cc2fe54822c74c03220eca072fdebb5a9c9c4b89f242633664615.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { F } = \\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 529, + 712, + 546 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where ${ \\mathcal { F } } ( x ) = \\{ f ( x ) , f \\in { \\mathcal { F } } \\}$ . Moreover, $i f I ( x ) = \\{ ( i , j ) \\in [ p ] ^ { 2 } : \\forall y \\in \\mathcal { F } ( x ) , y _ { i } = y _ { j } \\} ,$ , then we have: ", + "bbox": [ + 176, + 549, + 826, + 575 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/c8aad53115a1bdb638775a43c7e91e813bd2537293c4bbb3e3dc650edd87ccaa.jpg", + "text": "$$\n{ \\mathcal { F } } ( x ) = \\left\\{ y \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I ( x ) , y _ { i } = y _ { j } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 577, + 648, + 594 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof of Thm. 22. By Lem. 28, the second assumption of Prop. 24 is also satisfied. To get the conclusion of Thm. 22, note that $\\mathcal { F } ( x )$ is now a linear subspace of a finite-dimensional vector space and therefore it is closed. Thus, $\\overline { { \\mathcal { F } ( \\boldsymbol { x } ) } } = \\mathcal { F } ( \\boldsymbol { x } )$ which is a subalgebra. Applying Lem. 27 to $\\mathcal { F } ( x )$ and noting that, necessarily $J = \\emptyset$ as $\\mathbf { 1 } \\in { \\mathcal { F } } ( x )$ by assumption, gives the result of Thm. 22. □ ", + "bbox": [ + 173, + 607, + 826, + 667 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "D.8 PRACTICAL REDUCTIONS ", + "text_level": 1, + "bbox": [ + 176, + 683, + 395, + 696 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Though the results we proved above were formulated using classic hypotheses, such as requiring $\\mathcal { F }$ to be a subalgebra, we can give much more compact versions for our setting. We also reduce the assumption that $\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)$ to a more practical one. ", + "bbox": [ + 173, + 708, + 825, + 752 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "We start with the invariant case. ", + "bbox": [ + 174, + 757, + 383, + 772 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Corollary 29. Let $X$ be a compact space, $Y = \\mathbb { F } = \\mathbb { R } ^ { p }$ be some finite-dimensional vector space, $G$ be a finite group acting (continuously) on $X$ and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { I } ( X , Y )$ a (non-empty) set of invariant functions. ", + "bbox": [ + 173, + 775, + 826, + 818 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Assume that, for any $h \\in \\mathcal { C } ( \\mathbb { F } ^ { 2 } , \\mathbb { F } )$ and $f , g \\in { \\mathcal { F } }$ , ", + "bbox": [ + 173, + 823, + 495, + 839 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/fba3191b3745bcb49ea559ab9201e28774f6fb98ea7b5935034a8a4ed3946959.jpg", + "text": "$$\nx \\mapsto h ( f ( x ) , g ( x ) ) \\in \\mathcal { F } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 843, + 583, + 859 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ is, ", + "bbox": [ + 173, + 863, + 334, + 877 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/213c730c6ead213f127264ed917eab21831a945babe84bbae665a7b2ba6efd4d.jpg", + "text": "$$\n{ \\overline { { \\mathcal { F } } } } = \\{ f \\in { \\mathcal { C } } _ { I } ( X , Y ) : \\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right) \\} ~ .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 875, + 630, + 893 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. We wish to apply Thm. 22 but for this we need $G$ to act on $Y$ . Define a (trivial) action of $G$ on $Y$ by, ", + "bbox": [ + 169, + 102, + 823, + 132 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/876c54c0978ad3da4b5bef218e3ab2c82dd8756d68110a702bac85c6eda8cd57.jpg", + "text": "$$\n\\forall g \\in G , \\forall y \\in Y , g \\cdot y = y .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 131, + 591, + 147 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "With this action on $Y$ , $\\mathcal { C } _ { E } ( X , Y ) = \\mathcal { C } _ { I } ( X , Y )$ . Moreover, $Y / G = Y$ , $\\pi : Y \\to Y / G$ is the identity so that $\\rho \\left( \\pi \\circ { \\mathcal { F } } \\right) = \\rho \\left( { \\mathcal { F } } \\right)$ . ", + "bbox": [ + 173, + 150, + 823, + 181 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Our assumption clearly ensure that $\\mathcal { F }$ is indeed a subalgebra and contains the constant function $\\nVdash$ ", + "bbox": [ + 171, + 185, + 813, + 202 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "All that is left to show to apply Thm. 22 is that the set of functions $\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )$ defined by, ", + "bbox": [ + 173, + 207, + 802, + 223 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/a4aee7bf4e3d7ee1b9b42b7225bf97b97c7f7abc6d37497a4a94fac5521183f6.jpg", + "text": "$$\n\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f { \\bf 1 } \\in \\mathcal { F } \\}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 229, + 612, + 247 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "satisfies, ", + "bbox": [ + 173, + 253, + 232, + 268 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/9ee2e44e05c46ddd1e82de80db72593e52f35cec674fc29c5c639f6f8bee8532.jpg", + "text": "$$\n\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 267, + 565, + 285 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Take $( x , y ) \\notin \\rho \\left( { \\mathcal { F } } \\right)$ and we show that $( x , y ) \\notin \\rho \\left( \\mathcal { F } _ { s c a l } \\right)$ . Indeed, by definition there exists $f \\in { \\mathcal { F } }$ , $i \\in \\{ 1 , \\ldots , p \\}$ such that $f ( x ) _ { i } \\neq f ( y ) _ { i }$ . Let $l \\in \\mathcal { C } ( \\mathbb { F } , \\mathbb { R } )$ defined by $l ( z ) = z _ { i }$ and $h \\in { \\mathcal { C } } ( \\mathbb { F } , \\mathbb { F } )$ defined by, for $z \\in \\mathbb { F }$ , $h ( z ) = ( l ( z ) , \\ldots , l ( z ) ) = l ( z ) \\mathcal { k }$ . Then, by assumption $h \\circ f \\in { \\mathcal { F } }$ . But $h \\circ f$ is $( l \\circ f ) \\mathbf { 1 }$ with $l \\circ f \\in { \\mathcal { C } } ( X , \\mathbb { R } )$ , so that, by definition, $l \\circ f \\in \\mathcal { F } _ { s c a l }$ . Moreover, $l \\circ f ( x ) \\neq l \\circ f ( y )$ . Therefore, $( x , y ) \\notin \\rho \\left( \\mathcal { F } _ { s c a l } \\right)$ . Therefore, we can apply Thm. 22. We get that, ", + "bbox": [ + 173, + 286, + 828, + 358 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/c22a924963bfe3f0033632ac94ce7a1a79d5e21e18a0d9f63230224a231794d4.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 366, + 714, + 383 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 391, + 200, + 405 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/60314bb19560d5cb1ff7863d8d45d9323b300e62994f4df7908a62e09c5736d1.jpg", + "text": "$$\n\\mathcal { F } ( x ) = \\left\\{ y \\in \\mathbb { F } ^ { p } : \\forall ( i , j ) \\in I ( x ) , y _ { i } = y _ { j } \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 404, + 647, + 421 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "with $I ( x )$ given by, ", + "bbox": [ + 173, + 424, + 302, + 439 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/978f27184d1e0aba2e190c33a088c3e9651dba9cab7b08e8ab81918b0ed2ed37.jpg", + "text": "$$\nI ( x ) = \\{ ( i , j ) \\in \\{ 1 , \\ldots , p \\} ^ { 2 } : \\forall y \\in { \\mathcal { F } } ( x ) , y _ { i } = y _ { j } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 445, + 678, + 465 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "To conclude the proof, we now show that $I ( x ) = \\{ ( i , i ) : i \\in \\{ 1 , . . . , p \\} \\}$ , which will imply that $\\mathcal { F } ( x ) = \\mathbb { R } ^ { p } = Y$ . ", + "bbox": [ + 171, + 478, + 825, + 508 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Indeed, the constant function $z \\mapsto ( 1 , 2 , . . . , p )$ is in $\\mathcal { F }$ by assumption. Therefore, $I ( x )$ is reduced to $\\{ ( i , i ) : i \\in \\{ 1 , . . . , p \\} \\}$ . ", + "bbox": [ + 171, + 513, + 825, + 545 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "In our previous version of our approximation result for node embedding, we did not allow features in the output as it would have made the statement and the proof a bit convoluted. With this new assumption, this is much easier. ", + "bbox": [ + 173, + 559, + 825, + 602 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Corollary 30. Let $X$ be a compact space, $Y = \\mathbb { F } ^ { n }$ , with $\\mathbb { F } = \\mathbb { R } ^ { p }$ and $G = S _ { n }$ the permutation group, acting (continuously) on $X$ and acting on $\\mathbb { F } ^ { n }$ by, for $\\sigma \\in S _ { n }$ , $x \\in \\mathbb { F } ^ { n }$ , ", + "bbox": [ + 171, + 606, + 823, + 636 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/ed49a1ecff370f2b6433c694a710654149b8e628db355dbbe7d4044927b99253.jpg", + "text": "$$\n\\forall i \\in \\{ 1 , \\ldots , p \\} , ( \\sigma \\cdot x ) _ { i } = x _ { \\sigma ^ { - 1 } ( i ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 642, + 619, + 660 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Let ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , \\mathbb { F } ^ { n } )$ be a (non-empty) set of equivariant functions. ", + "bbox": [ + 171, + 674, + 609, + 690 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Consider the following assumptions, ", + "bbox": [ + 174, + 696, + 413, + 712 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "1. For any $h \\in { \\mathcal { C } } ( \\mathbb { F } ^ { 2 } , \\mathbb { F } ) , f , g \\in { \\mathcal { F } } ,$ , ", + "bbox": [ + 214, + 722, + 446, + 738 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/3a61925655e02c78331af4851a2d3764db14fe1a7491d247b013f0087c62cd39.jpg", + "text": "$$\nx \\mapsto \\left( h ( f ( x ) _ { 1 } , g ( x ) _ { 1 } ) , \\ldots , h ( f ( x ) _ { n } , g ( x ) _ { n } ) \\right) \\in { \\mathcal { F } } .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 746, + 699, + 762 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "2. If $f \\in { \\mathcal { F } }$ , ", + "bbox": [ + 212, + 775, + 295, + 791 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/29a9e5ae2854f30fe148cb7383de6d7a0c2fabe37622628c4c521f9bfd0c3ac7.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\in { \\mathcal { F } } .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 790, + 696, + 832 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ (for the topology of uniform convergence) is, ", + "bbox": [ + 173, + 842, + 612, + 857 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/aef3243baf83d19394dd7cb9157e0ca7cbef6c2ce43f5452a9c0b3d1ff4ca76f.jpg", + "text": "$$\n\\overline { { \\mathcal { F } } } = \\{ f \\in \\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) \\} \\ .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 862, + 633, + 881 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "For this we need a handy lemma, whose proof relies on a result about multi-symmetric polynomials from Maron et al. (2019a). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Lemma 31. Let $\\mathbb { F } = \\mathbb { R } ^ { p }$ be some finite-dimensional space. Take $x _ { 1 } , \\ldots , x _ { n } , y _ { 1 } , \\ldots , y _ { n } \\in \\mathbb { F }$ such that, for any $\\sigma \\in S _ { n }$ , $( x _ { 1 } , \\ldots , x _ { n } ) \\neq ( y _ { \\sigma ( 1 ) } , \\ldots , y _ { \\sigma ( n ) } )$ . Then, there exists $h \\in { \\mathcal { C } } ( \\mathbb { F } , \\mathbb { R } )$ such that, ", + "bbox": [ + 169, + 103, + 825, + 133 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/5091f441721b714fe2aafee9d3f8ee760ddbeee4b3a1429228917790a90b1955.jpg", + "text": "$$\n\\sum _ { i = 1 } ^ { n } h ( x _ { i } ) \\neq \\sum _ { i = 1 } ^ { n } h ( y _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 416, + 138, + 576, + 180 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Moreover, $h$ can be written as $h ( \\boldsymbol { x } ) = h _ { 1 } ( x _ { 1 } ) h _ { 2 } ( x _ { 2 } ) \\ldots h _ { p } ( x _ { p } )$ for any $x \\in \\mathbb { F }$ with $h _ { 1 } , \\ldots , h _ { p } \\in$ ${ \\mathcal { C } } ( \\mathbb { R } , \\mathbb { R } )$ . ", + "bbox": [ + 173, + 185, + 823, + 214 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof. By Maron et al. (2019a, Prop. 1), there exists $\\alpha _ { 1 } , \\dotsc , \\alpha _ { p }$ non-negative integers such that $\\textstyle \\sum _ { i = 1 } ^ { n } x _ { i 1 } ^ { \\alpha _ { 1 } } x _ { i 2 } ^ { \\alpha _ { 2 } } \\dotsc x _ { i p } ^ { \\alpha _ { p } } \\neq \\textstyle \\sum _ { i = 1 } ^ { n } y _ { i 1 } ^ { \\alpha _ { 1 } } y _ { i 2 } ^ { \\alpha _ { 2 } } \\dotsc x _ { i p } ^ { \\alpha _ { p } }$ . Taking $h : x \\mapsto x _ { i 1 } ^ { \\alpha _ { 1 } } x _ { i 2 } ^ { \\alpha _ { 2 } } \\ldots x _ { i p } ^ { \\alpha _ { p } }$ yields the result. □ ", + "bbox": [ + 173, + 229, + 826, + 275 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof of Cor. 30. We want to apply Thm. 22. With our first assumption, the first assumption of Thm. 22 is easily verified. ", + "bbox": [ + 173, + 287, + 825, + 318 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We now focus on the second one. As in the statement of Thm. 22, define $\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )$ by, ", + "bbox": [ + 171, + 324, + 789, + 339 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/db936cfa300ba8752143e75c741c5d3155b2ff3e6647ba7f934b28d6dfa22f35.jpg", + "text": "$$\n\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f \\mathbf { 1 } \\in \\mathcal { F } \\} \\ .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 344, + 616, + 362 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We have to show that $\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)$ . For this, take $x , y \\notin \\rho \\left( \\pi \\circ { \\mathcal { F } } \\right)$ . There exists $f \\in { \\mathcal { F } }$ such that for any $\\sigma \\in S _ { n }$ , $\\sigma \\cdot f ( x ) \\neq f ( y )$ . We have to find $l \\in \\mathcal { F } _ { s c a l }$ such that $l ( x ) \\neq l ( y )$ . In other words, from a function in $\\mathbb { F } ^ { n }$ which discriminates between $x$ and $y$ we have to build a function in $\\mathbb { R }$ . ", + "bbox": [ + 174, + 366, + 826, + 410 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "First, we exhibit a function which discriminates between $x$ and $y$ . Apply Lem. 31 to the vectors $f ( x )$ and $f ( y )$ : there exists $h _ { 0 } \\in \\mathcal { C } ( \\mathbb { F } , \\mathbb { R } )$ such that $\\textstyle \\sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( x _ { i } ) ) \\neq \\sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( y _ { i } ) )$ . ", + "bbox": [ + 171, + 416, + 825, + 446 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "To fit the assumptions, we build $h \\in { \\mathcal { C } } ( \\mathbb { F } , \\mathbb { F } )$ from $h _ { 0 }$ by $h : x \\in \\mathbb { F } \\mapsto ( h _ { 0 } ( x ) , \\ldots , h _ { 0 } ( x ) ) \\in \\mathbb { F }$ . $g \\in \\mathcal { C } ( \\mathbb { F } , \\mathbb { F } )$ $g ( z ) = ( z _ { 1 } , \\dots , z _ { 1 } )$ r any . All $z \\in \\mathbb { F }$ and e to d $l \\in \\mathcal { C } ( X , \\mathbb { R } )$ dt , for, i.e., $w \\in X$ $\\begin{array} { r } { l ( w ) = \\sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( w _ { i } ) ) } \\end{array}$ $l ( x ) \\neq l ( y )$ $l \\in \\mathcal { F } _ { s c a l }$ $l \\mathbf { 1 } \\in \\mathcal { F }$ . This is where the two assumptions we made come into play. Indeed, the first one implies that $h \\circ f \\in { \\mathcal { F } }$ and the second gives ", + "bbox": [ + 173, + 450, + 826, + 522 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/93c26c55dcf847f5ff1bc1f9e46830bd4f6e51a4f573ef9a5ba7d6c9fc2951a8.jpg", + "text": "$$\nz \\mapsto \\left( \\sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) , \\ldots , \\sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) \\right) \\in { \\mathcal { F } } .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 526, + 651, + 569 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Finally, the first assumption ensure that, ", + "bbox": [ + 173, + 574, + 437, + 588 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/8c0e28f340ef1ec79a251ba575fade2483476b5a2ad9dd0aaaaea266ffbbdca7.jpg", + "text": "$$\nz \\mapsto \\left( g \\left( \\sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) \\right) , \\ldots , g \\left( \\sum _ { i = 1 } ^ { n } h ( f ( z _ { i } ) ) \\right) \\right) \\in \\mathcal { F } .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 593, + 691, + 637 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "But this last function is none other than $l \\mathbf { 1 }$ , which shows that $l \\in \\mathcal { F } _ { s c a l }$ as required. ", + "bbox": [ + 173, + 640, + 717, + 656 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We have successfully verified the hypothesis of Thm. 22. Therefore, the closure of $\\mathcal { F }$ is, ", + "bbox": [ + 173, + 661, + 750, + 678 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/4f1810143e20d9a3f80ea8eb00577506e8bda5fb555e77c06faa48b5748bb497.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 681, + 715, + 700 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "with ", + "bbox": [ + 173, + 705, + 205, + 719 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/fccf3ad61a983a11fbb0cfa462ec71fd0cf3f852483e92d5e90589bcfc77d006.jpg", + "text": "$$\n\\mathcal { F } ( x ) = \\{ y \\in \\mathbb { F } ^ { n } : \\forall ( i , i ^ { \\prime } , j , j ^ { \\prime } ) \\in I ( x ) , y _ { i , i ^ { \\prime } } = y _ { j , j ^ { \\prime } } \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 717, + 679, + 736 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 738, + 200, + 751 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/e3f1b0a7ab53ed2d1e7925d018b4c4e6230ca1179a37788dd559f04faa1b2b44.jpg", + "text": "$$\nI ( x ) = \\{ ( i , i ^ { \\prime } , j , j ^ { \\prime } ) \\in ( \\{ 1 , \\dots , n \\} \\times \\{ 1 , \\dots , p \\} ) ^ { 2 } : \\forall y \\in \\mathcal { F } ( x ) , y _ { i , i ^ { \\prime } } = y _ { j , j ^ { \\prime } } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 236, + 755, + 759, + 775 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "To get the desired result, we need to get rid of the condition ${ \\bf \\nabla } ^ { \\bullet } f ( x ) \\in \\mathcal { F } ( x ) ^ { \\mathfrak { V } }$ in the description of $\\scriptstyle { \\overline { { \\mathcal { F } } } }$ Fix $x \\in X$ . ", + "bbox": [ + 173, + 781, + 823, + 810 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "First, we show that ", + "bbox": [ + 174, + 818, + 300, + 832 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/8b133cbccac79c2a88a0e5d7a21f1c52c09d674c696411f8fcd9425224ba7937.jpg", + "text": "$$\n{ \\mathcal { F } } ( x ) = \\left\\{ y \\in \\mathbb { F } ^ { n } : \\forall ( i , j ) \\in J ( x ) , y _ { i } = y _ { j } \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 830, + 648, + 848 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "with ", + "bbox": [ + 173, + 849, + 205, + 863 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/728de8a6686783b6ecc45b9ea61c9c273610a4faca9ab9f69006eec4f1a82405.jpg", + "text": "$$\nJ ( x ) = \\left\\{ ( i , j ) \\in \\left\\{ 1 , \\dots , n \\right\\} ^ { 2 } : y _ { i } = y _ { j } \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 361, + 861, + 635, + 880 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "(note that the equalities here are not in $\\mathbb { R }$ anymore but in $\\mathbb { F }$ ). The direct inclusion ${ \\bf \\ \" } \\subseteq { \\bf \\ \" }$ is immediate by construction of $J ( x )$ so we focus on the reverse direction. For this, we show that the 4-tuples $( i , i ^ { \\prime } , j , j ^ { \\prime } )$ of $I ( x )$ necessarily satisfy $i ^ { \\prime } = j ^ { \\prime }$ and $( i , j ) \\in J ( x )$ . ", + "bbox": [ + 173, + 882, + 825, + 925 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "First note that, by the first assumption, the vector $\\boldsymbol { y } ^ { 0 } \\in \\mathbb { F } ^ { n }$ such that $y _ { i } ^ { 0 } = ( 1 , 2 , \\ldots , p )$ for $i \\in$ $\\{ 1 , \\ldots , n \\}$ is in $\\mathcal { F } ( x )$ . Indeed, take the constant function always equal to $( 1 , 2 , \\ldots , p )$ as $h$ . Now, consider a 4-tuple $( i , i ^ { \\prime } , j , j ^ { \\prime } )$ of $I ( x )$ and we show that, actually, $( i , j ) \\in { \\dot { J } } ( x )$ and $i ^ { \\prime } = j ^ { \\prime }$ . As $y ^ { 0 }$ in $\\mathcal { F } ( x )$ , and $y _ { i , i ^ { \\prime } } ^ { 0 } = \\dot { i } ^ { \\prime }$ , $y _ { j , j ^ { \\prime } } ^ { 0 } = j ^ { \\prime }$ , $( i , i ^ { \\prime } , j , j ^ { \\prime } ) \\in I ( x )$ implies that $j ^ { \\prime } = i ^ { \\prime }$ . Consider, $k \\in \\{ 1 , \\ldots , p \\}$ . We show that, for any $y \\in { \\mathcal { F } } ( x )$ , $y _ { i , k } = y _ { j , k }$ . But such a $y$ can be written as $y = f ( x )$ for some $f \\in { \\mathcal { F } }$ . Consider, the function $h \\in { \\mathcal { C } } ( \\mathbb { F } , \\mathbb { F } )$ associated to the permutation $( i ^ { \\prime } k )$ , defined by, ", + "bbox": [ + 173, + 102, + 826, + 190 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/ad0bab6d5950b92866d7f3a18d2e689db481bf39f75ca5b3e8a1a38bccd70af4.jpg", + "text": "$$\nz \\mapsto \\left( z _ { ( i ^ { \\prime } k ) ( 1 ) } , \\dots , z _ { ( i ^ { \\prime } k ) ( n ) } \\right) = \\left( z _ { 1 } , \\dots , z _ { i ^ { \\prime } - 1 } , z _ { k } , z _ { i ^ { \\prime } + 1 } , \\dots , z _ { k - 1 } , z _ { i ^ { \\prime } } , z _ { k + 1 } , \\dots , z _ { p } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 210, + 194, + 784, + 212 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "By our first assumption, $z \\mapsto ( h ( f ( z ) _ { 1 } ) , \\ldots , h ( f ( z ) _ { n } ) ) \\in \\mathcal { F }$ so that $( h ( y _ { 1 } ) , \\dots , h ( y _ { n } ) ) \\in { \\mathcal { F } } ( x )$ . In particular, as $( i , i ^ { \\prime } , j , i ^ { \\prime } ) \\in I ( x ) , h ( y _ { i } ) _ { i ^ { \\prime } } = h ( y _ { j } ) _ { i ^ { \\prime } }$ , i.e. $y _ { i , k } = y _ { j , k }$ . Therefore, $( i , j ) \\in J ( x )$ . ", + "bbox": [ + 171, + 215, + 825, + 246 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Finally, we can conclude that ${ \\mathcal { F } } ( x ) ~ \\supset ~ \\{ y ~ \\in ~ \\mathbb { F } ^ { n } ~ : ~ \\forall ( i , j ) ~ \\in ~ J ( x )$ , $y _ { i } ~ = ~ y _ { j } \\}$ . Indeed, take $y \\in \\mathbb { F } ^ { \\cdot } : \\forall ( i , j ) \\in J ( x )$ , $y _ { i } = y _ { j }$ . We show that all the constraints of $I ( x )$ are satisfied. Indeed, take $( i , i ^ { \\prime } , j , j ^ { \\prime } ) \\in I ( x )$ . We have shown that $( i , j ) \\in J ( x )$ and $i ^ { \\prime } = j ^ { \\prime }$ so that $y _ { i } = y _ { j }$ and in particular $y _ { i , i ^ { \\prime } } = y _ { j , j ^ { \\prime } }$ . Therefore, this finishes the proof of $\\mathcal { F } ( x ) \\supset \\{ y \\in \\mathbb { F } ^ { n } : \\forall ( i , j ) \\in J ( \\bar { x } ) , y$ $y _ { i } = y _ { j } \\}$ . ", + "bbox": [ + 173, + 250, + 826, + 309 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Thus, with ", + "bbox": [ + 173, + 314, + 212, + 328 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 344, + 205, + 359 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/ab5a6ce9a4b9f6761920a85e3eb269c02d9277252ebc3984d5ec95e6f0bada51.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\mathcal F } ( x ) = \\left\\{ y \\in \\mathbb { F } ^ { n } : \\forall ( i , j ) \\in J ( x ) , y _ { i } = y _ { j } \\right\\} , } \\\\ & { } \\\\ & { J ( x ) = \\left\\{ ( i , j ) \\in \\{ 1 , \\dots , n \\} ^ { 2 } : y _ { i } = y _ { j } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 325, + 648, + 377 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "We have proven so far, that, ", + "bbox": [ + 173, + 383, + 356, + 398 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/208f78dccf9ce790ee6a4031e2924baa3b61675d03d007b229b4254ee219b051.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\boldsymbol { \\varepsilon } } } = \\{ f \\in \\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } ) : \\rho ( \\mathcal { F } ) \\subseteq \\rho ( f ) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\} \\subseteq \\{ f \\in \\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } ) : \\rho ( \\mathcal { F } ) \\subseteq \\rho ( f ) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 401, + 839, + 420 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Take $h \\in { \\mathcal { C } } _ { E } ( X , \\mathbb { F } ^ { n } )$ such that $\\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( h \\right)$ and fix $x \\in X$ . Our goal is to show that, for any $( i , j ) \\in J ( x ) , h ( x ) _ { i } = h ( x ) _ { j }$ so that $h ( x ) \\in { \\mathcal { F } } ( x )$ . But, if $( i , j ) \\in J ( x )$ , then for any $f \\in { \\mathcal { F } }$ , $f ( x ) _ { i } = f ( x ) _ { j }$ so that $( i j ) \\cdot { \\dot { f } } ( x ) = f ( x )$ , where $( i j )$ denotes the permutation which exchanges $i$ and $j$ . Moreover, as $( i j ) \\in S _ { n }$ , by equivariance, this means that $f ( ( i \\dot { \\boldsymbol { j } } ) \\cdot \\boldsymbol { x } ) = f ( \\boldsymbol { x } )$ for every $f \\in { \\mathcal { F } }$ and therefore that $( ( i j ) \\cdot x , x ) \\in \\rho \\left( { \\mathcal { F } } \\right)$ . By assumption, we infer that $( ( i j ) \\cdot x , x ) \\in \\rho \\left( h \\right)$ too, i.e. that $h ( ( i j ) \\cdot x ) = \\ddot { h } ( \\dot { x } )$ and so that $h ( \\dot { x } ) _ { i } = h ( x ) _ { j }$ by equivariance, which concludes our proof. ", + "bbox": [ + 173, + 429, + 826, + 516 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "D.9 REDUCTIONS FOR GNNS ", + "text_level": 1, + "bbox": [ + 174, + 553, + 393, + 566 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "We now present a lemma which explains how to instantiate the two corollaries above in the case of GNNs by replacing continuous functions with MLPs. ", + "bbox": [ + 174, + 578, + 825, + 607 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma 32. Fix $X$ some compact space, $n \\geq 1$ and $\\mathbb { F }$ a finite-dimensional feature space. Let $\\textstyle { \\mathcal { F } } _ { 0 } \\subseteq \\bigcup _ { h = 1 } ^ { \\infty } { \\mathcal { C } } ( X , \\mathbb { R } ^ { h } )$ be stable by concatenation and consider, ", + "bbox": [ + 174, + 609, + 828, + 638 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/f7b0ee16bd4f2c034c6ab1c9cf291dbc08b874cab7656e39204e887dee7f9387.jpg", + "text": "$$\n{ \\mathbf { \\Psi } } = \\{ x \\mapsto ( m ( f ( x ) _ { 1 } ) , \\dots , m ( f ( x ) _ { n } ) ) : f \\in \\mathcal { F } _ { 0 } \\cap \\mathcal { C } ( X , \\mathbb { R } ^ { h } ) , m : \\mathbb { R } ^ { h } \\to \\mathbb { F } M L P , h \\geq 1 \\} \\subseteq \\mathcal { C } ( X , \\mathbb { F } ) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 643, + 839, + 662 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Then, i ${ \\mathcal { \\cdot } } { \\mathcal { E } } ( { \\mathcal { F } } ) \\subseteq { \\mathcal { C } } ( X , \\mathbb { F } )$ is the set of functions obtained by replacing the MLP m in the definition of $\\mathcal { F }$ by an arbitrary continuous function, $\\mathcal { E } ( \\mathcal { F } )$ satisfies, ", + "bbox": [ + 171, + 666, + 828, + 695 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "2. $\\rho \\left( \\mathcal { F } \\right) = \\rho \\left( \\mathcal { E } ( \\mathcal { F } ) \\right)$ ", + "bbox": [ + 212, + 731, + 354, + 747 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "3. For any $h \\in { \\mathcal { C } } ( \\mathbb { F } ^ { 2 } , \\mathbb { F } ) , f , g \\in { \\mathcal { E } } ( { \\mathcal { F } } )$ ", + "bbox": [ + 210, + 753, + 467, + 770 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/eb15a5da8ff2fbe2cf24c44a199e3dcfd4baedfe674500f3d4057df6c22451a3.jpg", + "text": "$$\nx \\mapsto \\left( h ( f ( x ) _ { 1 } , g ( x ) _ { 1 } ) , \\ldots , h ( f ( x ) _ { n } , g ( x ) _ { n } ) \\right) \\in \\mathcal { E } ( \\mathcal { F } ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 775, + 710, + 791 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "4. If, for any $f \\in { \\mathcal { F } }$ , ", + "bbox": [ + 212, + 801, + 351, + 818 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/7435b0755b499d7f4fab87033a768609b6de10356efa485986436768b516192b.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\in { \\mathcal { F } } ,\n$$", + "text_format": "latex", + "bbox": [ + 361, + 821, + 694, + 863 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "then, for any $f \\in { \\mathcal { E } } ( { \\mathcal { F } } )$ , ", + "bbox": [ + 232, + 866, + 392, + 882 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/a0925a636fbbae55eea51bf7ee793f2879ca05423ab29d39deb7ca1835e031d1.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\in { \\mathcal { E } } ( { \\mathcal { F } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 887, + 705, + 928 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "5. If $\\mathcal { F }$ is equivariant w.r.t. the action described in Cor. $3 0 _ { ; }$ , so is $\\mathcal { E } ( \\mathcal { F } )$ . ", + "bbox": [ + 209, + 103, + 678, + 119 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof. Define, ", + "bbox": [ + 173, + 133, + 272, + 150 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/67b1c3980d1536d31bc737bfce00c5d939a6544d57b54fe2fb76760ad37f5ede.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { E } ( \\mathcal { F } ) = \\left\\{ x \\mapsto ( m ( f ( x ) _ { 1 } ) , \\ldots , m ( f ( x ) _ { n } ) ) : f \\in \\mathcal { F } _ { 0 } \\cap \\mathcal { C } ( X , \\mathbb { R } ^ { h } ) , m \\in \\mathcal { C } ( \\mathbb { R } ^ { h } , \\mathbb { F } ) , h \\geq 1 \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 199, + 155, + 797, + 175 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "As MLP are continuous and by the universality of MLP on a compact set (see Section C.1), ", + "bbox": [ + 173, + 181, + 772, + 196 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/e25ce24def3cd7371891a91d7c83b5536c1b2051fa37a42f6ecfedf7cc27610e.jpg", + "text": "$$\n{ \\mathcal { F } } \\subseteq { \\mathcal { E } } ( { \\mathcal { F } } ) \\subseteq { \\overline { { \\mathcal { F } } } } .\n$$", + "text_format": "latex", + "bbox": [ + 439, + 202, + 557, + 220 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "This already implies 1. and that $\\rho \\left( \\overline { { \\mathcal { F } } } \\right) \\subseteq \\rho \\left( \\mathcal { E } ( \\mathcal { F } ) \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right)$ . Using Lem. 15 yields 2. ", + "bbox": [ + 174, + 227, + 735, + 246 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "We now show 3. Take $h \\in \\mathcal { C } ( \\mathbb { F } ^ { 2 } , \\mathbb { F } ) , f , g \\in \\mathcal { F } _ { 0 } , f \\in \\mathcal { C } ( X , \\mathbb { R } ^ { h _ { f } } ) , g \\in \\mathcal { C } ( X , \\mathbb { R } ^ { h _ { g } } )$ and $m \\in \\mathcal { C } ( \\mathbb { R } ^ { h _ { f } } , \\mathbb { F } )$ $l \\in \\mathcal { C } ( \\mathbb { R } ^ { h _ { g } } , \\mathbb { F } )$ . All we have to show is that, ", + "bbox": [ + 173, + 251, + 825, + 281 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/6cc40bbe0654e6dfd10db833528b252edb66f49224ec09da9f0a61681ffc4aef.jpg", + "text": "$$\nx \\mapsto \\left( h ( m ( f ( x ) _ { 1 } ) , l ( g ( x ) _ { 1 } ) ) , \\ldots , h ( m ( f ( x ) _ { n } ) , l ( g ( x ) _ { n } ) ) \\right) \\in \\mathcal { E } ( \\mathcal { F } ) .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 289, + 725, + 306 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "But as $\\mathcal { F } _ { 0 }$ is stable by concatenation, $x \\ \\mapsto \\ ( f ( x ) , g ( x ) ) \\ \\in \\ \\mathbb { R } ^ { h _ { f } + h _ { g } } .$ is still in $\\mathcal { F } _ { 0 }$ . Moreover, $y \\in \\mathbb { R } ^ { h _ { f } + h _ { g } } \\mapsto h ( m ( \\dot { y } _ { 1 } , \\dots , y _ { h _ { f } } ) , l ( y _ { h _ { f } + 1 } , \\dots , y _ { h _ { f } + h _ { g } } )$ is also in $\\mathcal { C } ( \\mathbb { R } ^ { h _ { f } + h _ { g } } , \\mathbb { F } )$ which shows that the map above is indeed in $\\mathcal { E } ( \\mathcal { F } )$ . The last two points are immediate consequences of the definition of $\\mathcal { E } ( \\mathcal { F } )$ . □ ", + "bbox": [ + 173, + 311, + 826, + 371 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Thm. 5 and Thm. 6 are now obtained by combining Lem. 32 with Cor. 29 and Cor. 30. ", + "bbox": [ + 173, + 385, + 738, + 401 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "E PROOFS FOR EXPRESSIVENESS OF GNNS ", + "text_level": 1, + "bbox": [ + 173, + 420, + 549, + 438 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Note that in the Theorem 6, the additional stability assumption is “almost” necessary to obtain the result. Indeed, if the result holds, i.e., ", + "bbox": [ + 173, + 452, + 823, + 482 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/65feb27a5360ca5b9505f712a5230ab638f90f85f2786e6ffeac207dea2f2602.jpg", + "text": "$$\n\\overline { { \\mathcal { F } } } = \\{ f \\in \\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } ) : \\rho \\left( \\mathcal { F } _ { 0 } \\right) \\subseteq \\rho \\left( f \\right) \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 359, + 487, + 635, + 506 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "then one can show, that, if $f \\in { \\mathcal { F } }$ , then ", + "bbox": [ + 174, + 512, + 431, + 527 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/5db9806d03f0db5c8ece4df5da52a7cba452c605ff5e53fb0063c2040378889c.jpg", + "text": "$$\n{ \\tilde { f } } : x \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\in { \\overline { { \\mathcal { F } } } } .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 534, + 676, + 577 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Indeed, $f \\in { \\mathcal { F } }$ so that it has a weaker discriminating power than $\\mathcal { F }$ , so that $\\rho \\left( \\mathcal { F } \\right) = \\rho \\left( \\mathcal { F } _ { 0 } \\right) \\subseteq \\rho \\left( f \\right)$ . \nBut, by construction of $\\tilde { f } , \\rho \\left( f \\right) \\subseteq \\rho \\left( \\tilde { f } \\right)$ so that $\\rho \\left( \\mathcal { F } _ { 0 } \\right) \\subseteq \\rho \\left( \\tilde { f } \\right)$ . As $\\tilde { f }$ is also in $\\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } )$ , $\\tilde { f } \\in \\overline { { \\mathcal { F } } }$ . ", + "bbox": [ + 173, + 583, + 828, + 623 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "E.1 EXPRESSIVITY OF GNN LAYERS ", + "text_level": 1, + "bbox": [ + 174, + 637, + 442, + 652 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Lemma 33. Fix $\\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 }$ (non-trivial) finite dimensionnal vector spaces. Consider the action of $G = S _ { n }$ on $\\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k }$ defined by, ", + "bbox": [ + 168, + 664, + 828, + 695 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/3db8b4157a9232bb933d4781b0c852345ad69ea01ba0bc5f06531eb739ca18b7.jpg", + "text": "$$\n\\forall \\sigma \\in { \\mathcal { S } } _ { n } , \\forall x ^ { 0 } \\in \\mathbb { F } _ { 0 } , \\forall x \\in \\mathbb { F } ^ { n \\times k } , \\sigma \\cdot ( x ^ { 0 } , x ) = ( x ^ { 0 } , x _ { \\sigma ^ { - 1 } ( 1 ) } , \\dots , x _ { \\sigma ^ { - 1 } ( n ) } )\n$$", + "text_format": "latex", + "bbox": [ + 250, + 702, + 745, + 722 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Let $K \\subseteq \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k }$ be a compact set. Then, the set of functions from $K \\subseteq \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } t o$ $\\mathbb { F } _ { 1 }$ of the form, ", + "bbox": [ + 173, + 729, + 823, + 758 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/66ca50388388f3efbf03494f9c5fbe882ce24bec62354ec82ff3e0affd7d0b43.jpg", + "text": "$$\n( x ^ { 0 } , x ) \\longmapsto f _ { 0 } \\left( x ^ { 0 } , \\sum _ { j = 1 } ^ { n } \\prod _ { w = 1 } ^ { k } f _ { w } ( x _ { j , w } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 361, + 757, + 632, + 808 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "where $f _ { 0 } : \\mathbb { F } _ { 0 } \\times \\mathbb { R } ^ { h } \\to \\mathbb { F } _ { 1 }$ , $f _ { j } : \\mathbb { F } _ { 0 } \\mathbb { R } ^ { h }$ , $j = 1 , \\dots , k$ are multi-linear perceptrons and $h \\geq 1$ , is dense in $\\mathcal { C } _ { I } ( K , \\mathbb { F } _ { 1 } )$ . ", + "bbox": [ + 173, + 810, + 826, + 842 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof. Denote by $\\mathcal { F } \\subseteq \\mathcal { C } _ { I } ( K , \\mathbb { F } _ { 1 } )$ the set of such functions. To prove that $\\overline { { \\mathcal { F } } } = \\mathcal { C } _ { I } ( K , \\mathbb { F } _ { 1 } )$ , we first apply Thm. 5. ", + "bbox": [ + 174, + 856, + 823, + 887 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "We get, that, ", + "bbox": [ + 173, + 893, + 258, + 909 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/43581b4e42a559da9806136ab4a7dab3e223a769f5f14c388869f479eb2f8607.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 906, + 663, + 926 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "We now characterize $\\rho \\left( \\mathcal { F } \\right)$ . Actually, it is equal to $\\rho \\left( i n v \\right) = \\{ ( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \\in \\left( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } \\right) ^ { 2 }$ : $\\exists \\sigma \\in S _ { n }$ , $( x ^ { 0 } , x ) = \\sigma \\cdot ( y ^ { 0 } , y ) \\}$ . As the functions of $\\mathcal { F }$ are invariant, $\\rho \\left( i n v \\right) \\subseteq \\rho \\left( { \\mathcal { F } } \\right)$ . We now ke . I $( x _ { \\hphantom { } , \\hphantom { } } ^ { 0 } , x ) , ( y ^ { 0 } , y ) \\in \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k }$ at there does notMLP such that $\\sigma \\in S _ { n }$ uch thatso that $( x ^ { 0 } , x ) = \\sigma \\cdot ( y ^ { 0 } , y )$ $x ^ { 0 } \\neq y ^ { 0 }$ $f _ { 0 } : \\mathbb { F } _ { 0 } \\to \\mathbb { F } _ { 1 }$ $f ( x ^ { 0 } ) \\neq f ( y ^ { 0 } )$ $( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \\notin \\rho \\left( { \\mathcal F } \\right)$ . ", + "bbox": [ + 173, + 102, + 826, + 181 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Otherwise, there does not exists $\\sigma ~ \\in ~ S _ { n }$ such that $( x _ { \\sigma ^ { - 1 } ( 1 ) } , \\ldots , x _ { \\sigma ^ { - 1 } ( n ) } ) \\ = \\ ( y _ { 1 } , \\ldots , y _ { n } )$ by definition of the action of $G \\ = \\ S _ { n }$ . Now, apply Lem. 31 with $\\mathbb { F } \\mathbb { F } _ { 0 } ^ { k }$ , the universality of MLP and use the decomposition given to get $f _ { j } : \\mathbb { F } _ { 0 } \\ \\to \\ \\mathbb { R } ^ { h }$ , $j ~ = ~ 1 , \\ldots , k$ such that $\\begin{array} { r } { \\sum _ { j = 1 } ^ { n } \\prod _ { w = 1 } ^ { k } f _ { w } ( x _ { j , w } ) \\neq \\sum _ { j = 1 } ^ { n } \\prod _ { w = 1 } ^ { k } f _ { w } ( y _ { j , w } ) } \\end{array}$ . Choosing an appropriate MLP $f _ { 0 } : \\mathbb { R } ^ { h } \\mathbb { F } _ { 1 }$ yields that $( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \\notin \\rho \\left( { \\mathcal F } \\right)$ . ", + "bbox": [ + 173, + 185, + 825, + 267 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Hence, we have shown that, ", + "bbox": [ + 173, + 272, + 357, + 286 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/0f0d346eb3432095077e67ee6bb80d2e1de0673a4c8183a593b7e160e54bf160.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) : \\rho \\left( i n v \\right) \\subseteq \\rho \\left( f \\right) \\right\\} = \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 290, + 741, + 310 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "E.2 APPROXIMATION THEOREMS FOR GNNS ", + "text_level": 1, + "bbox": [ + 176, + 354, + 498, + 369 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "We now have all the tools to finally prove our main result. ", + "bbox": [ + 173, + 380, + 565, + 395 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Theorem 34. Let $K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }$ , $K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ be compact sets. For the invariant case, we have: ", + "bbox": [ + 173, + 400, + 813, + 417 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/d321ad8227bdfa3b4ec773eed58e8e815884ea4450153b593fe850d2ae4d5b8e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( 2 \\overline { { N L _ { I } } } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { 2 - L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( 2 \\overline { { N L _ { I } } } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { k - L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } L G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} \\supset \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { k - F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } F W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 422, + 812, + 503 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "For the equivariant case, we have: ", + "bbox": [ + 173, + 507, + 400, + 522 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/f4cb523c47232ca86c09150d0b92cdd3f7e011f7770e63b3c00e1603e4a6fa22.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\underline { { M G N N _ { E } } } } } & { { = } } & { { \\{ f \\in { \\mathcal C } _ { E } ( K _ { d i s c r } , \\mathbb F ^ { n } ) : \\rho \\left( 2 ^ { - } { \\cal W } L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } } \\\\ { { \\boldsymbol { \\cdot } L G N N _ { E } } } & { { = } } & { { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k { \\cal - } L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} \\supset \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k { \\cal - } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } } \\\\ { { \\boldsymbol { \\cdot } R G N N _ { E } } } & { { = } } & { { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k { \\cal - } F W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 526, + 839, + 587 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "We decompose the proof with an additional lemma. ", + "bbox": [ + 173, + 597, + 513, + 612 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma 35. Let $K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }$ , $K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ be compact sets. For the invariant case, and any $k \\geq 2$ , we have, ", + "bbox": [ + 171, + 616, + 825, + 647 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/e4ff07d94a44fc45b3aaf7898ddbb5b3e17ff7671dcb5dfa780b1301c5815387.jpg", + "text": "$$\n\\begin{array} { r l r } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( M G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\ – L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – L G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\ – F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – F G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 651, + 689, + 713 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "For the equivariant case, and any $k \\geq 2$ , we have: ", + "bbox": [ + 173, + 717, + 503, + 732 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/5751b2af1c56951e33b27575c93aca698285cda04fb2919cc06480104cc9a4c9.jpg", + "text": "$$\n\\begin{array} { r l } { \\overline { { M G N N _ { E } } } } & { = \\{ f \\in \\mathcal { C } _ { E } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( M G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\overline { { k \\ – L G N N _ { E } } } } & { = \\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\overline { { k \\ – F G N N _ { E } } } } & { = \\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – F G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 736, + 689, + 796 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof of Thm. 34. The theorem is now a direct consequence of Prop. 13 and Lem. 35. ", + "bbox": [ + 174, + 808, + 735, + 823 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "We now move to the proof of Lem. 35. ", + "bbox": [ + 174, + 837, + 428, + 852 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof of Lem. 35. First, focus on the invariant case. Let $\\mathcal { F }$ denote ${ \\bf M G N N } _ { I }$ , $k { \\mathrm { - } } \\mathrm { L G N N } _ { I }$ or $k { \\mathrm { - } } \\mathrm { F G N N } _ { I }$ and $X$ be either $K _ { d i s c r }$ or $K$ so that $X$ is compact and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { I } ( X , \\mathbb { F } )$ . Applying Thm. 5 directly gives, ", + "bbox": [ + 173, + 867, + 826, + 909 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/f4621d17f08e547b700a87d4d73141cb090be96266d397a607f5941070a72b31.jpg", + "text": "$$\n\\overline { { \\mathcal { F } } } = \\{ f \\in \\mathcal { C } _ { I } ( X , \\mathbb { F } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) \\} \\mathrm { ~ , ~ }\n$$", + "text_format": "latex", + "bbox": [ + 369, + 906, + 627, + 925 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "which is the desired result. ", + "bbox": [ + 173, + 103, + 349, + 118 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "We now move to the equivariant case. First, let us replace $\\mathbf { M G N N } _ { E }$ by another class, which is slightly simpler to analyze. Define, ", + "bbox": [ + 173, + 125, + 825, + 154 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/cdb7b1b7a683c3d5bf20105efb2ebd4861790d64f7c5b7666d7e33e648b65ff5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { M G N N } _ { E ^ { \\prime } } = \\left\\{ m _ { E } \\circ \\left( \\left( 1 - \\lambda \\right) \\mathrm { I d } + \\lambda S ^ { 1 } \\right) \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } : F _ { t } : } \\\\ & { \\qquad \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { m e s s a g e ~ p a s s i n g ~ l a y e r } , t = 1 , \\dots , T , T \\geq 1 , \\lambda \\in \\left\\{ 0 , 1 \\right\\} \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 159, + 779, + 196 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "It holds that $\\rho \\left( \\mathbf { M G N N } _ { E ^ { \\prime } } \\right) = \\rho \\left( \\mathbf { M G N N } _ { E } \\right)$ and, ", + "bbox": [ + 173, + 202, + 485, + 218 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/3cc7048c2f4307bf709f4e54969cf7354e31c6a3d50806b02d03d519df03fb39.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { M G N N } _ { E ^ { \\prime } } \\subseteq \\mathbf { \\overline { { M G N N } } } _ { E } \\subseteq \\{ f \\in \\mathcal { C } _ { E } ( K _ { d i s c r } , \\mathbb { F } ^ { n } ) : \\rho \\left( \\mathbf { M G N N } _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { = \\{ f \\in \\mathcal { C } _ { E } ( K _ { d i s c r } , \\mathbb { F } ^ { n } ) : \\rho \\left( \\mathbf { M G N N } _ { E ^ { \\prime } } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 224, + 743, + 262 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Therefore, if we show that, ", + "bbox": [ + 174, + 267, + 352, + 282 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/1b677c13d4f4ea2f842757498ab08e79abd966b38b6c545c71f6b28fb6c0662a.jpg", + "text": "$$\n\\overline { { \\mathbf { M G N N } _ { E ^ { \\prime } } } } = \\left\\{ f \\in \\mathcal { C } _ { E } \\big ( K _ { d i s c r } , \\mathbb { F } ^ { n } \\big ) : \\rho \\left( \\mathbf { M G N N } _ { E ^ { \\prime } } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 292, + 289, + 702, + 308 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "we will have the desired result. ", + "bbox": [ + 173, + 313, + 377, + 328 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Let $\\mathcal { F }$ denote $\\mathbf { M G N N } _ { E ^ { \\prime } }$ , $k { \\mathrm { - } } \\mathrm { L G N N } _ { E }$ or $k { \\mathrm { - } } \\mathrm { F G N N } _ { E }$ and $X$ be either $K _ { d i s c r }$ or $K$ so that $X$ is compact and ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , \\mathbb { F } ^ { n } )$ . We now wish to apply Thm. 6, but we need to verify the stability assumption first. Thus, we show that, for any $f \\in { \\mathcal { F } }$ , ", + "bbox": [ + 174, + 334, + 825, + 377 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/71a27fb62a260de760099003bf32148b6f3a328c03049a5a7ef2d55c6b4df36c.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\in { \\mathcal { F } } ,\n$$", + "text_format": "latex", + "bbox": [ + 331, + 383, + 665, + 428 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "• If $\\mathcal { F } = \\mathbf { M G N N } _ { E ^ { \\prime } }$ .Take $f \\in \\mathbf { M G N N } _ { E ^ { \\prime } }$ . $f$ is of the form, ", + "bbox": [ + 217, + 439, + 599, + 455 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/f087c9ea945b8ecba4e0b0ff11682dd7bd66084cf81166d1d055004a5088961a.jpg", + "text": "$$\nm _ { E } \\circ ( ( 1 - \\lambda ) \\operatorname { I d } + \\lambda S ^ { 1 } ) \\circ F _ { T } \\circ . . . F _ { 2 } \\circ F _ { 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 460, + 676, + 479 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "where $F _ { t } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n }$ are message passing layers, $\\mathbb { F } _ { T + 1 } = \\mathbb { F }$ and $\\lambda \\in \\{ 0 , 1 \\}$ . We need to show that there is a $\\mathrm { M G N N } _ { E ^ { \\prime } }$ which implements, ", + "bbox": [ + 230, + 486, + 825, + 515 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/4443de2bad38d106dae6dad438a93474985761077251261e0ce1c711919f43be.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) ~ .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 521, + 679, + 564 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "If $\\lambda = 1$ , $f$ is exactly the function of (26). Otherwise, if $\\lambda = 0$ , we build another $\\mathbf { M G N N } _ { E ^ { \\prime } }$ implementing this function. Denote by $F _ { T + 1 } : \\mathbb { F } ^ { n } \\mathbb { F } ^ { n }$ the (simple) message passing layer defined by $F _ { T + 1 } ( h ) _ { i } = m _ { E } ( h _ { i } )$ for $i \\in [ n ]$ and any $h \\in \\mathbb { F } ^ { n }$ . Then, the $\\mathbf { M G N N } _ { E ^ { \\prime } }$ $( 1 - \\lambda ^ { \\prime } ) \\operatorname { I d } + \\lambda ^ { \\prime } S ^ { 1 } ) \\circ F _ { T + 1 } \\circ F _ { T } \\circ . . . F _ { 2 } \\circ F _ { 1 }$ with $\\lambda ^ { \\prime } = 1$ exactly implements (26). ", + "bbox": [ + 230, + 570, + 825, + 627 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "• If $\\mathcal { F } = k \\mathrm { - } \\mathrm { L G N N } _ { E }$ . A function $f$ of this class is of the form, ", + "bbox": [ + 217, + 635, + 625, + 651 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/c62bc9ab6a30958cbde384ef5c30e3f69da7f23a51af654b1bc610bd430b388d.jpg", + "text": "$$\nm _ { E } \\circ S _ { 1 } ^ { k } \\circ F _ { T } \\circ . . . F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } ,\n$$", + "text_format": "latex", + "bbox": [ + 416, + 656, + 638, + 675 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "where $F _ { t } : \\mathbb { F } _ { t } ^ { n ^ { k } } \\to \\mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ linear graph layers and $\\mathbb { F } _ { T + 1 } = \\mathbb { F }$ . Our goal is to show that there is a GNN of $k { \\mathrm { - } } \\mathrm { L G N N } _ { E }$ ", + "bbox": [ + 232, + 684, + 825, + 714 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/d1aea1281c866138ecaf0cca02cdfc9b744a84004837a148026d16cb634c2432.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\ ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 720, + 678, + 763 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "By definition of linear graph layers, the map $F _ { T + 1 } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F } ^ { n ^ { k } }$ defined by, for $G \\in \\mathbb { F } ^ { n ^ { k } }$ , ", + "bbox": [ + 230, + 770, + 820, + 790 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/8955bfaa61d937bf7377a2c8788824578503f785e1ff05db92fde053c2fe4624.jpg", + "text": "$$\n\\forall ( i _ { 1 } , \\ldots , i _ { k } ) \\in [ n ] ^ { k } , F _ { T + 1 } ( G ) _ { i _ { 1 } , \\ldots , i _ { k } } = m _ { E } ( S _ { 1 } ^ { k } ( G ) _ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 795, + 710, + 815 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "is a linear graph layer as defined in Section 3. Now, consider, the linear graph layer, $F _ { T + 2 } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F } ^ { n ^ { k } }$ defined by, for $G \\in \\mathbb { F } ^ { n ^ { k } }$ , ", + "bbox": [ + 232, + 820, + 825, + 853 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/a1173d3b8d757b95ca597409a915ae557ba034b27cbbc59f3121aa9ad21c3482.jpg", + "text": "$$\n\\forall ( i _ { 1 } , \\dots , i _ { k } ) \\in [ n ] ^ { k } , F _ { T + 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { k } } = \\sum _ { i = 1 } ^ { n } G _ { i , i _ { 2 } , \\dots , i _ { k } } .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 859, + 714, + 901 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Then, the $k$ -LGNN $F _ { T + 2 } \\circ F _ { T + 1 } \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k }$ exactly implements (27). ", + "bbox": [ + 232, + 907, + 774, + 925 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "• If $\\mathcal { F } = k \\mathrm { - F G N N } _ { E }$ . A function $f$ of this class is of the form, ", + "bbox": [ + 217, + 102, + 625, + 118 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/16b9eee7542789dd241ec03ebd2bfa6c951e740595f3b7b3b18571fd542453ba.jpg", + "text": "$$\nm _ { E } \\circ S _ { 1 } ^ { k } \\circ F _ { T } \\circ . . . F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 123, + 638, + 142 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "where Ft : Fnkt $F _ { t } : \\mathbb { F } _ { t } ^ { n ^ { k } } \\to \\mathbb { F } _ { t + 1 } ^ { n ^ { k } }$ → Fnkt+1 a re FGL (see Section 3) and $\\mathbb { F } _ { T + 1 } = \\mathbb { F }$ . We build a GNN of which implements, ", + "bbox": [ + 232, + 150, + 823, + 181 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/1fbe907c8ac7acbb08291dd04a40575bac7f71f5a8ce5a44f5c715037404857f.jpg", + "text": "$$\nx \\mapsto \\left( \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\ldots , \\sum _ { i = 1 } ^ { n } f ( x ) _ { i } \\right) \\ ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 188, + 678, + 231 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "$w \\in [ k ]$ $H _ { w } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F } ^ { n ^ { k } }$ ", + "bbox": [ + 230, + 238, + 560, + 256 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/8fca2f1c8f5508ba1c98cfeb0c63a2d80d5a801ac142c7154640ee0754b1dd97.jpg", + "text": "$$\n\\forall G \\in \\mathbb { F } ^ { n ^ { k } } , \\forall ( i _ { 1 } , \\dots , i _ { k } ) \\in [ n ] ^ { k } , H _ { w } ( G ) _ { i _ { 1 } , \\dots , i _ { k } } = \\sum _ { j = 1 } ^ { n } G _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 262, + 787, + 305 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Then $H _ { 2 } \\circ \\cdot \\cdot \\cdot \\circ H _ { k } : \\mathbb { F } ^ { n ^ { k } } \\to \\mathbb { F } ^ { n ^ { k } }$ computes the sum of the elements of the input tensor over the last $k - 1$ dimensions like to $S _ { 1 } ^ { k } : \\mathbb { F } ^ { n ^ { k } } \\to \\mathbb { F } ^ { n }$ and $H _ { 1 } \\circ H _ { 2 } \\circ \\cdots \\circ H _ { k } : \\mathbb { F } ^ { n ^ { k } } \\to \\mathbb { F } ^ { n ^ { k } }$ computes the full sum of the elements of the input tensor like to $S ^ { k } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F }$ . Finally, consider the FGL $F _ { T + 1 } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F } ^ { n ^ { k } }$ associated to $m _ { E }$ , i.e. such that, for any $G \\in \\mathbb { F } ^ { n ^ { k } }$ , ", + "bbox": [ + 232, + 313, + 826, + 382 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/ee301528d4c7e4fb457dda4eed261959c527509bf3e1daaed5e1b40bddccc4f8.jpg", + "text": "$$\n\\forall ( i _ { 1 } , \\dots , i _ { k } ) \\in [ n ] ^ { k } , F _ { T + 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { k } } = m _ { E } ( G _ { i _ { 1 } , \\dots , i _ { k } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 388, + 715, + 407 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Now, the $k$ -FGNN ", + "bbox": [ + 232, + 412, + 357, + 428 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/ed34686b25d00bbbac03b9eee4493e9d7068c2206d93dff1dcb4389dc788f793.jpg", + "text": "$$\nH _ { 1 } \\circ H _ { 2 } \\circ \\cdot \\cdot \\cdot \\circ H _ { k } \\circ F _ { T + 1 } \\circ H _ { 2 } \\circ \\cdot \\cdot \\cdot \\circ H _ { k } \\circ F _ { T + 1 } \\circ F _ { T } \\circ . . . F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 434, + 779, + 453 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "exactly implements (28). ", + "bbox": [ + 232, + 459, + 397, + 474 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "F EXTENSION TO GRAPHS OF VARYING SIZES ", + "text_level": 1, + "bbox": [ + 173, + 525, + 565, + 540 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "F.1 EXTENSION TO DISCONNECTED INPUT SPACES ", + "text_level": 1, + "bbox": [ + 173, + 556, + 535, + 570 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Here we show that our results can be extended to graphs of varying sizes similarly to Keriven & Peyré (2019). There are two ways to do it. The first would be to directly adapt all the proofs but it would make them more cumbersome. Instead, we extend them with a simple argument presented below. As a side benefit, this general lemma makes it possible to extend almost all the approximation results for graph neural networks from the literature. ", + "bbox": [ + 173, + 582, + 825, + 651 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "The abstract setting is the following. Given a compact input space $X$ , assume that there is some finite set $A$ , and $X _ { \\alpha }$ , $\\alpha \\in A$ a family of pairwise disjoints compact sets such that $\\textstyle X = \\bigcup _ { \\alpha \\in A } X _ { \\alpha }$ Crucially, we will assume that the $X _ { \\alpha }$ are in distinct connected components. Intuitively, they do not \"touch\" each other. Similarly, assume that the output space $Y$ can be written as $\\textstyle Y = \\operatorname { L } _ { \\alpha \\in A } Y _ { \\alpha }$ , with $Y _ { \\alpha }$ , $\\alpha \\in A$ a family of (pairwise disjoints) real vector spaces. ", + "bbox": [ + 173, + 659, + 825, + 729 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Before moving to the results, we need a last definition. Informally, we need that the functions $f$ that we will consider do not change the number of nodes of their inputs. Formally, we say that $f : X \\longrightarrow Y$ is adapted if $f ( X _ { \\alpha } ) \\subseteq Y _ { \\alpha }$ for each $\\alpha \\in A$ , and denote by $\\mathcal { C } _ { a d } ( X , Y )$ the of continuous and adapted functions from $X$ to $Y$ . ", + "bbox": [ + 174, + 734, + 825, + 791 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "We can now state our lemma to adapt our results to graphs with varying node sizes. ", + "bbox": [ + 174, + 797, + 717, + 814 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Lemma 36. Let $X$ be a compact space, $Y$ be a topological space and $A$ a finite set. Assume that there exists $X _ { \\alpha }$ , $\\alpha \\in A$ a family of pairwise disjoints compact sets such that $\\textstyle X = \\bigcup _ { \\alpha \\in A } X _ { \\alpha }$ and the $X _ { \\alpha }$ ’s are in distinct connected components. Also assume that there is $Y _ { \\alpha }$ , $\\alpha \\in A$ a family of (pairwise disjoints) real vector spaces such that $\\textstyle Y = \\operatorname { L } _ { \\alpha \\in A } Y _ { \\alpha }$ . Consider ${ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { a d } ( X , Y )$ a (non-empty) set of adapted functions and, for each $\\alpha \\in A$ , define $\\mathcal { F } _ { | X _ { \\alpha } } = \\{ f _ { | X _ { \\alpha } } : f \\in \\mathcal { F } \\} \\subseteq \\mathcal { C } ( X _ { \\alpha } , Y _ { \\alpha } )$ the restriction of the functions of $\\mathcal { F }$ to $X _ { \\alpha }$ . ", + "bbox": [ + 173, + 818, + 826, + 904 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Assume that the following holds, ", + "bbox": [ + 173, + 909, + 387, + 924 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "1. Each $\\mathcal { F } _ { | X _ { \\alpha } }$ is a sub-algebra of $\\mathcal { C } ( X _ { \\alpha } , Y _ { \\alpha } )$ which contains the constant function ${ \\mathbf { 1 } } _ { Y _ { \\alpha } }$ ", + "bbox": [ + 209, + 102, + 784, + 119 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "2. There is a set of functions $\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )$ such that $\\mathcal { F } _ { s c a l } . \\mathcal { F } \\subseteq \\mathcal { F }$ and it discriminates between the $X _ { \\alpha }$ ’s, i.e. $\\textstyle \\rho ( { \\mathcal { F } } _ { s c a l } ) \\subseteq \\sqcup _ { \\alpha \\in A } X _ { \\alpha } ^ { 2 }$ . ", + "bbox": [ + 210, + 127, + 825, + 161 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Then the closure of $\\mathcal { F }$ is, ", + "bbox": [ + 173, + 170, + 336, + 185 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/15c3113b0c53c293ba047220c3308641ac5905c4bfe1553c29d47dcef9ed7ac0.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { a d } ( X , Y ) : \\forall \\alpha \\in A , f _ { | X _ { \\alpha } } \\in \\overline { { \\mathcal { F } _ { | X _ { \\alpha } } } } \\right\\} , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 191, + 661, + 212 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "for the distance on $\\mathcal { C } _ { a d } ( X , Y )$ defined by, for $f , g \\in \\mathcal { C } _ { a d } ( X , Y )$ , ", + "bbox": [ + 171, + 218, + 593, + 236 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/c56f61b78fa1259034a3d9e0cb065065593a29b6cb3223d6ce193c87dd6c6b2b.jpg", + "text": "$$\nd ( f , g ) = \\operatorname* { m a x } _ { \\alpha \\in A } \\operatorname* { s u p } _ { X _ { \\alpha } } \\| f - g \\| _ { Y _ { \\alpha } } .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 241, + 604, + 268 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Define the set of functions $S \\subseteq { \\mathcal { C } } ( X , \\mathbb { R } )$ by, ", + "bbox": [ + 173, + 286, + 513, + 303 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/be77f553a5bdc294bd9576eaa4830b41d0390d766e23dcc82d4adaf93454d48a.jpg", + "text": "$$\nS = \\left\\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f . \\mathcal { F } \\subseteq \\mathcal { F } \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 309, + 607, + 327 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "By definition, $\\mathcal { F } _ { s c a l } \\subseteq S$ . Moreover, as each $\\mathcal { F } _ { | X _ { \\alpha } }$ is a subalgebra, $S$ is also a subalgebra of ${ \\mathcal { C } } ( X , \\mathbb { R } )$ ", + "bbox": [ + 173, + 334, + 825, + 351 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "As $X$ is compact, we can apply Cor. 17 to $S$ to get that, in ${ \\mathcal { C } } ( X , \\mathbb { R } )$ , ", + "bbox": [ + 173, + 356, + 620, + 372 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/f37425b4eef4156e7d98715ba8e8920756fbc7226cc02fecd37f13807bd823f6.jpg", + "text": "$$\n{ \\overline { { S } } } = \\left\\{ f \\in { \\mathcal { C } } ( X , \\mathbb { R } ) : \\rho \\left( S \\right) \\subseteq \\rho \\left( f \\right) \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 374, + 380, + 622, + 400 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "In the first part of the proof, we check that, for each $\\alpha$ , the function $f _ { \\alpha } : X \\longrightarrow [ 0 , 1 ]$ defined by ", + "bbox": [ + 169, + 406, + 803, + 422 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/180342dd096dff111fa51d4d74927e8de3b25499f22b8ecd4bb0c04c706522c4.jpg", + "text": "$$\n\\begin{array} { r } { f _ { \\alpha | X _ { \\alpha } } = 1 } \\\\ { \\forall \\beta \\neq \\alpha , f _ { \\alpha | X _ { \\beta } } = 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 426, + 429, + 570, + 470 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "is continuous and satisfy $\\rho \\left( S \\right) \\subseteq \\rho \\left( f _ { \\alpha } \\right)$ . This will means that such functions belong to $\\overline { { \\mathcal { F } _ { s c a l } } }$ . ", + "bbox": [ + 169, + 477, + 789, + 494 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "As the $X _ { \\beta }$ ’s are in different connected components, it is enough to check its continuity on each $X _ { \\beta }$ . \nIndeed, each $f _ { \\alpha | X _ { \\beta } }$ is constant so continuous. The second fact comes from the second assumption. \nIndeed, take $( x , y ) \\in X$ such that $( x , y ) \\in \\rho \\left( S \\right)$ . As $\\mathcal { F } _ { s c a l } \\subseteq S$ , in particular $( x , y ) \\in \\rho \\left( \\mathcal { F } _ { s c a l } \\right)$ . \nBut, by assumption, $\\textstyle \\rho ( { \\mathcal { F } } _ { s c a l } ) \\subseteq \\sqcup _ { \\beta \\in A } X _ { \\beta } ^ { 2 }$ . Thus, necessarily $x$ and $y$ belong to the same $X _ { \\beta }$ so that $f _ { \\alpha } ( x ) = f _ { \\alpha } ( y )$ . ", + "bbox": [ + 173, + 500, + 828, + 575 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Therefore, we conclude that $f _ { \\alpha } \\in \\overline { { S } }$ . ", + "bbox": [ + 174, + 582, + 415, + 598 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "We now prove the announced equality. By the definition of the distance, the first inclusion, $\\overline { { \\mathcal { F } } } \\subseteq$ $\\{ f \\in \\mathcal { C } _ { a d } ( X , Y ) : f _ { | X _ { \\alpha } } \\in \\overline { { \\mathcal { F } _ { \\alpha } } } \\}$ is immediate. We focus on the other way. Take $h \\in \\mathcal { C } _ { a d } ( X , Y )$ such that $h _ { | X _ { \\alpha } } \\in \\overline { { \\mathcal { F } _ { \\alpha } } }$ for every $\\alpha \\in A$ and $\\epsilon > 0$ . By definition of $h$ , there exists, for each $\\alpha \\in A$ , $g _ { \\alpha } \\in \\mathcal { F }$ such that, ", + "bbox": [ + 173, + 604, + 826, + 662 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/2859e3dbe7165d617346b38b4980e09134067d5415d28b262b4e07416137aee9.jpg", + "text": "$$\n\\underset { X _ { \\alpha } } { \\operatorname* { s u p } } \\Vert h - g _ { \\alpha } \\Vert _ { Y _ { \\alpha } } \\leq \\epsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 424, + 662, + 570, + 689 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "As the $X _ { \\alpha }$ ’s are compact and the $g _ { \\alpha }$ ’s are continuous, $\\begin{array} { r } { \\operatorname* { m a x } _ { \\alpha \\in A } \\operatorname* { s u p } _ { X _ { \\alpha } } \\| g _ { \\alpha } \\| < + \\infty } \\end{array}$ and denote by $M > 0$ a bound on this quantity. We have shown above that each $f _ { \\alpha }$ is in $\\overline { S }$ so there exists, for each $\\alpha \\in A$ , $l _ { \\alpha } \\in S$ such that, ", + "bbox": [ + 173, + 694, + 826, + 741 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/3857bce1f5b8af129edabf9438ee6a8ff2e5deadb6332c5fb6c5beb7f08cf3ee.jpg", + "text": "$$\n\\operatorname* { s u p } _ { X } | f _ { \\alpha } - l _ { \\alpha } | \\leq \\frac { \\epsilon } { M } .\n$$", + "text_format": "latex", + "bbox": [ + 428, + 757, + 570, + 787 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "By definition of $S$ and, as $\\mathcal { F }$ is a subalgebra, $\\textstyle \\sum _ { \\alpha \\in A } l _ { \\alpha } g _ { \\alpha } \\in { \\mathcal { F } }$ and, for each $\\beta \\in A$ , ", + "bbox": [ + 173, + 799, + 723, + 818 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/ee1cc62d1feb8d94283b3b4cd6b530226ffefaba1c1c03bf219ae007483023f8.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\operatorname* { s u p } _ { X _ { \\beta } } \\| h - \\sum _ { \\alpha \\in A } l _ { \\alpha } g _ { \\alpha } \\in \\mathcal { F } \\| \\le \\operatorname* { s u p } _ { X _ { \\beta } } \\| h - g _ { \\beta } \\| + \\| g _ { \\beta } - l _ { \\beta } g _ { \\beta } \\| + \\| \\sum _ { \\alpha \\neq \\beta } l _ { \\alpha } g _ { \\alpha } \\| } } \\\\ & { } & { \\le \\epsilon + M \\times \\frac { \\epsilon } { M } + ( | A | - 1 ) M \\times \\frac { \\epsilon } { M } = ( | A | + 1 ) \\epsilon , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 824, + 759, + 905 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "which concludes the proof. ", + "bbox": [ + 173, + 909, + 352, + 924 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "We now state our theorem in the case of varying graph size like Keriven & Peyré (2019). With Lem. 36, it is indeed straightforward to extend any approximation result initially proven for a class of graphs of fixed size. However, as the complete proof would require new notations again, we only give a sketch of proof. ", + "bbox": [ + 173, + 128, + 825, + 185 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Fix $N \\geq 1$ and consider the space of graphs (described by tensors) of size less than $N$ , $\\mathbb { F } _ { 0 } ^ { \\le N ^ { 2 } } =$ $\\textstyle \\bigcup _ { n = 1 } ^ { N } \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }$ . Equip this space with the final topology or, equivalently, the graph edit distance. Then, the last thing to check to apply Lem. 36 is that the classes of GNN that we consider indeed discriminate between graphs of different sizes, which is immediate. ", + "bbox": [ + 173, + 195, + 825, + 256 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Likewise, for message passing GNN, consider $\\begin{array} { r } { \\mathcal { G } _ { \\leq N } \\times \\mathbb { F } _ { 0 } ^ { \\leq N } = \\bigcup _ { n = 1 } ^ { N } \\mathcal { G } _ { n } \\times \\mathbb { F } _ { 0 } ^ { n } } \\end{array}$ , with a similar topology or the graph edit distance. ", + "bbox": [ + 174, + 261, + 823, + 294 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Corollary 37. Let $K _ { d i s c r } \\subseteq \\mathcal { G } _ { \\leq N } \\times \\mathbb { F } _ { 0 } ^ { \\leq N }$ , $K \\subseteq \\mathbb { F } _ { 0 } ^ { \\leq N ^ { 2 } }$ be compact sets. For the invariant case, we ", + "bbox": [ + 174, + 297, + 825, + 332 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/63912604e84dba728358730e5448f7d98201bcf0ec25b0731d21df360d8d3606.jpg", + "text": "$$\n\\begin{array} { r l } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( 2 \\overline { { N L _ { I } } } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { 2 - L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( 2 \\overline { { N L _ { I } } } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { k - L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } L G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} \\supset \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { k - F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } F W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 334, + 812, + 417 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For the equivariant case, we have: ", + "bbox": [ + 173, + 421, + 400, + 436 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/1b754682727c68869ab50ecb95c7dca25cdf715332ec9a807997228809fd7995.jpg", + "text": "$$\n\\begin{array} { r c l } { \\overline { { M G N N _ { E } } } } & { = } & { \\left\\{ f \\in \\mathcal { C } _ { E } ( K _ { d i s c r } , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( 2 \\overline { { { \\cdot } } } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} } \\\\ { \\widehat { \\cdot } L G N N _ { E } } & { = } & { \\left\\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( k { - } L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} \\supset \\left\\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( k { - } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} } \\\\ { \\cdot { F G N N _ { E } } } & { = } & { \\left\\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( k { - } F W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 440, + 839, + 506 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Proof. This corollary is a direct consequence of Thm. 34 and Lem. 36, using Lem. 32 to satisfy the sub-algebra assumption of Lem. 36. □ ", + "bbox": [ + 169, + 515, + 825, + 545 + ], + "page_idx": 38 + } +] \ No newline at end of file diff --git a/parse/train/lxHgXYN4bwl/lxHgXYN4bwl_middle.json b/parse/train/lxHgXYN4bwl/lxHgXYN4bwl_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..51aab68a5f81aa94b7c44a8e85d2cec6ef11b989 --- /dev/null +++ b/parse/train/lxHgXYN4bwl/lxHgXYN4bwl_middle.json @@ -0,0 +1,159851 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 428, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 408, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 408, + 97 + ], + "score": 1.0, + "content": "EXPRESSIVE POWER OF INVARIANT AND", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 428, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 428, + 117 + ], + "score": 1.0, + "content": "EQUIVARIANT GRAPH NEURAL NETWORKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 137, + 255, + 170 + ], + "lines": [ + { + "bbox": [ + 112, + 137, + 174, + 149 + ], + "spans": [ + { + "bbox": [ + 112, + 137, + 174, + 149 + ], + "score": 1.0, + "content": "Waïss Azizian", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 148, + 256, + 160 + ], + "spans": [ + { + "bbox": [ + 112, + 148, + 256, + 160 + ], + "score": 1.0, + "content": "ENS, PSL University, Paris, France", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 113, + 160, + 234, + 171 + ], + "spans": [ + { + "bbox": [ + 113, + 160, + 234, + 171 + ], + "score": 1.0, + "content": "waiss.azizian@ens.fr", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 290, + 137, + 473, + 171 + ], + "lines": [ + { + "bbox": [ + 290, + 136, + 351, + 150 + ], + "spans": [ + { + "bbox": [ + 290, + 136, + 351, + 150 + ], + "score": 1.0, + "content": "Marc Lelarge", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 289, + 147, + 474, + 160 + ], + "spans": [ + { + "bbox": [ + 289, + 147, + 474, + 160 + ], + "score": 1.0, + "content": "INRIA & ENS, PSL University, Paris, France", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 289, + 160, + 406, + 171 + ], + "spans": [ + { + "bbox": [ + 289, + 160, + 406, + 171 + ], + "score": 1.0, + "content": "marc.lelarge@ens.fr", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 278, + 200, + 333, + 212 + ], + "lines": [ + { + "bbox": [ + 276, + 199, + 335, + 213 + ], + "spans": [ + { + "bbox": [ + 276, + 199, + 335, + 213 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 143, + 224, + 469, + 411 + ], + "lines": [ + { + "bbox": [ + 142, + 223, + 469, + 237 + ], + "spans": [ + { + "bbox": [ + 142, + 223, + 469, + 237 + ], + "score": 1.0, + "content": "Various classes of Graph Neural Networks (GNN) have been proposed and shown", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 236, + 469, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 236, + 469, + 249 + ], + "score": 1.0, + "content": "to be successful in a wide range of applications with graph structured data. 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Namely, for", + "type": "text" + }, + { + "bbox": [ + 478, + 644, + 502, + 655 + ], + "score": 0.9, + "content": "k \\geq 2", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 147, + 666 + ], + "score": 0.62, + "content": "k { \\mathrm { - W L } } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "are invariants based on the Weisfeiler-Lehman tests (described in Section 4.1). 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In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 247, + 469, + 259 + ], + "spans": [ + { + "bbox": [ + 142, + 247, + 469, + 259 + ], + "score": 1.0, + "content": "this paper, we propose a theoretical framework able to compare the expressive", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 257, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 470, + 270 + ], + "score": 1.0, + "content": "power of these GNN architectures. The current universality theorems only apply to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 268, + 470, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 470, + 281 + ], + "score": 1.0, + "content": "intractable classes of GNNs. Here, we prove the first approximation guarantees for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "score": 1.0, + "content": "practical GNNs, paving the way for a better understanding of their generalization.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "score": 1.0, + "content": "Our theoretical results are proved for invariant GNNs computing a graph embedding", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 313 + ], + "score": 1.0, + "content": "(permutation of the nodes of the input graph does not affect the output) and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "score": 1.0, + "content": "equivariant GNNs computing an embedding of the nodes (permutation of the input", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 322, + 471, + 336 + ], + "spans": [ + { + "bbox": [ + 140, + 322, + 471, + 336 + ], + "score": 1.0, + "content": "permutes the output). We show that Folklore Graph Neural Networks (FGNN),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "score": 1.0, + "content": "which are tensor based GNNs augmented with matrix multiplication are the most", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "score": 1.0, + "content": "expressive architectures proposed so far for a given tensor order. We illustrate our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 356, + 470, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 356, + 470, + 369 + ], + "score": 1.0, + "content": "results on the Quadratic Assignment Problem (a NP-Hard combinatorial problem)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 367, + 470, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 470, + 380 + ], + "score": 1.0, + "content": "by showing that FGNNs are able to learn how to solve the problem, leading to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 378, + 470, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 470, + 390 + ], + "score": 1.0, + "content": "much better average performances than existing algorithms (based on spectral,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "score": 1.0, + "content": "SDP or other GNNs architectures). On a practical side, we also implement masked", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 399, + 352, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 352, + 412 + ], + "score": 1.0, + "content": "tensors to handle batches of graphs of varying sizes.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 17, + "bbox_fs": [ + 140, + 223, + 471, + 412 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 432, + 205, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "Graph Neural Networks (GNN) are designed to deal with graph structured data. Since a graph is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "not changed by permutation of its nodes, GNNs should be either invariant if they return a result that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "must not depend on the representation of the input (typically when building a graph embedding) or", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "equivariant if the output must be permuted when the input is permuted (typically when building an", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "embedding of the nodes). More fundamentally, incorporating symmetries in machine learning is a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 471, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 471, + 524 + ], + "score": 1.0, + "content": "fundamental problem as it allows to reduce the number of degree of freedom to be learned.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 456, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "Deep learning on graphs. This paper focuses on learning deep representation of graphs with network", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "architectures, namely GNN, designed to be invariant to permutation or equivariant by permutation.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "From a practical perspective, various message passing GNNs have been proposed, see Dwivedi et al.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "(2020) for a recent survey and benchmarking on learning tasks. 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Results on the separating power of GNNs only deal with pairwise", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "comparison of graphs: we need a priori a different GNN for each pair of graphs in order to distinguish", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "them. Such results are of little help in a practical learning scenario. Our main contribution in this", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 104, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "paper overcomes this issue and we show that a single GNN can give a meaningful representation for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "all graphs. More precisely, we characterize the set of functions that can be approximated by MGNNs,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 177, + 113, + 186 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 175, + 166, + 189 + ], + "score": 1.0, + "content": "-LGNNs and", + "type": "text" + }, + { + "bbox": [ + 166, + 177, + 173, + 186 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "-FGNNs respectively. Standard Stone-Weierstrass theorem shows that if an algebra", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 107, + 187, + 116, + 198 + ], + "score": 0.7, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 186, + 317, + 200 + ], + "score": 1.0, + "content": "of real continuous functions separates points, then", + "type": "text" + }, + { + "bbox": [ + 318, + 187, + 327, + 198 + ], + "score": 0.77, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "is dense in the set of continuous function on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "a compact set. Here we extend such a theorem to general functions with symmetries and apply it to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "invariant and equivaraint functions to get our main result for GNNs. As a consequence, we show that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 502, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 113, + 230 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 219, + 491, + 233 + ], + "score": 1.0, + "content": "-FGNNs have the best approximation power among architectures dealing with tensors of order", + "type": "text" + }, + { + "bbox": [ + 491, + 221, + 497, + 230 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 219, + 502, + 233 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "Universality results for GNNs. Universal approximation theorems (similar to Cybenko (1989) for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "multi-layers perceptron) have been proved for linear GNNs in Maron et al. (2019b); Keriven & Peyré", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "(2019); Chen et al. (2019). They show that some classes of GNNs can approximate any function", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "defined on graphs. To be able to approximate any invariant function, they require the use of very", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 214, + 293 + ], + "score": 1.0, + "content": "complex networks, namely", + "type": "text" + }, + { + "bbox": [ + 215, + 281, + 222, + 291 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 281, + 280, + 293 + ], + "score": 1.0, + "content": "-LGNN where", + "type": "text" + }, + { + "bbox": [ + 280, + 281, + 287, + 291 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 281, + 372, + 293 + ], + "score": 1.0, + "content": "tends to infinity with", + "type": "text" + }, + { + "bbox": [ + 372, + 282, + 380, + 291 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "the number of nodes. Since we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 507, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 311, + 304 + ], + "score": 1.0, + "content": "prove that any invariant function less powerful than", + "type": "text" + }, + { + "bbox": [ + 312, + 292, + 342, + 303 + ], + "score": 0.9, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 291, + 466, + 304 + ], + "score": 1.0, + "content": "-WL can be approximated by a", + "type": "text" + }, + { + "bbox": [ + 466, + 292, + 473, + 302 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 291, + 507, + 304 + ], + "score": 1.0, + "content": "-FGNN,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 135, + 315 + ], + "score": 1.0, + "content": "letting", + "type": "text" + }, + { + "bbox": [ + 135, + 303, + 142, + 313 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 303, + 424, + 315 + ], + "score": 1.0, + "content": "tends to infinity directly implies universality. Universality results for", + "type": "text" + }, + { + "bbox": [ + 424, + 303, + 431, + 313 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "-FGNN is another", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 209, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 209, + 325 + ], + "score": 1.0, + "content": "contribution of our work.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 504, + 342 + ], + "score": 1.0, + "content": "Equivariant GNNs. Our second set of results extends previous analysis from invariant functions", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "to equivariant functions. There are much less results about equivariant GNNs: Keriven & Peyré", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "score": 1.0, + "content": "(2019) proves the universality of linear equivariant GNNs, and Maehara & Hoang (2019) shows the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "universality of a new class of networks they introduced. Here, we consider a natural equivariant", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 159, + 388 + ], + "score": 1.0, + "content": "extension of", + "type": "text" + }, + { + "bbox": [ + 159, + 375, + 165, + 385 + ], + "score": 0.73, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 373, + 294, + 388 + ], + "score": 1.0, + "content": "-WL and prove that equivariant", + "type": "text" + }, + { + "bbox": [ + 294, + 374, + 325, + 386 + ], + "score": 0.91, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 373, + 379, + 388 + ], + "score": 1.0, + "content": "-LGNNs and", + "type": "text" + }, + { + "bbox": [ + 380, + 375, + 386, + 384 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 373, + 506, + 388 + ], + "score": 1.0, + "content": "-FGNN can approximate any", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 335, + 398 + ], + "score": 1.0, + "content": "equivariant function less powerful than this equivariant", + "type": "text" + }, + { + "bbox": [ + 335, + 385, + 366, + 397 + ], + "score": 0.91, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 385, + 402, + 398 + ], + "score": 1.0, + "content": "-WL for", + "type": "text" + }, + { + "bbox": [ + 402, + 385, + 428, + 396 + ], + "score": 0.91, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 385, + 505, + 398 + ], + "score": 1.0, + "content": ". At this stage, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "should note that all universality results for GNNs by Maron et al. (2019b); Keriven & Peyré (2019);", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "Chen et al. (2019) are easily recovered from our main results. Also our analysis is valid for graphs of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 162, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 162, + 430 + ], + "score": 1.0, + "content": "varying sizes.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 506, + 447 + ], + "score": 1.0, + "content": "Empirical results for the Quadratic Assigment Problem (QAP). To validate our theoretical con-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "tributions, we empirically show that 2-FGNN outperforms classical MGNN. Indeed, Maron et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "(2019a) already demonstrate state of the art results for the invariant version of 2-FGNNs (for graph", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "classification or graph regression). Here we consider the graph alignment problem and show that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "the equivariant 2-FGNN is able to learn a node embedding which beats by a large margin other", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 490, + 327, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 327, + 503 + ], + "score": 1.0, + "content": "algorithms (based on spectral method, SDP or GNNs).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "Outline and contribution. After reviewing more previous works and notations in the next section,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 394, + 530 + ], + "score": 1.0, + "content": "we define the various classes of GNNs studied in this paper in Section", + "type": "text" + }, + { + "bbox": [ + 395, + 518, + 408, + 528 + ], + "score": 0.25, + "content": "3 :", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "message passing GNN,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "linear GNN and folklore GNN. Section 4 contains our main theoretical results for GNNs. First in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "Section 4.2 we describe the separating power of each GNN architecture with respect to the Weisfeiler-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "Lehman test. In Section 4.3, we give approximation guarantees for MGNNs, LGNNs and FGNNs at", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "fixed order of tensor. They cover both the invariant and equivariant cases and are our main theoretical", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "contributions. For these, we develop in Section D a fine-grained Stone-Weierstrass approximation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "theorem for vector-valued functions with symmetries. Our theorem handles both invariant and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "equivariant cases and is inspired by recent works in approximation theory. In Section 6, we illustrate", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "our theoretical results on a practical application: the graph alignment problem, a well-known NP-hard", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "score": 1.0, + "content": "problem. We highlight a previously overlooked implementation question: the handling of batches of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "graphs of varying sizes. A PyTorch implementation of the code necessary to reproduce the results is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 638, + 419, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 419, + 651 + ], + "score": 1.0, + "content": "available at https://github.com/mlelarge/graph_neural_net", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42 + }, + { + "type": "title", + "bbox": [ + 108, + 671, + 209, + 683 + ], + "lines": [ + { + "bbox": [ + 104, + 669, + 211, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 211, + 685 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "The pioneering works that applied neural networks to graphs are Gori et al. (2005) and Scarselli et al.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "(2009) that learn node representation with recurrent neural networks. More recent message passing", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "architectures make use of non-linear functions of the adjacency matrix (Kipf & Welling, 2016),", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 117 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "score": 1.0, + "content": "Approximation results for GNNs. Results on the separating power of GNNs only deal with pairwise", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "comparison of graphs: we need a priori a different GNN for each pair of graphs in order to distinguish", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "them. Such results are of little help in a practical learning scenario. Our main contribution in this", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 104, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "paper overcomes this issue and we show that a single GNN can give a meaningful representation for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "all graphs. More precisely, we characterize the set of functions that can be approximated by MGNNs,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 177, + 113, + 186 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 175, + 166, + 189 + ], + "score": 1.0, + "content": "-LGNNs and", + "type": "text" + }, + { + "bbox": [ + 166, + 177, + 173, + 186 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "-FGNNs respectively. Standard Stone-Weierstrass theorem shows that if an algebra", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 107, + 187, + 116, + 198 + ], + "score": 0.7, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 186, + 317, + 200 + ], + "score": 1.0, + "content": "of real continuous functions separates points, then", + "type": "text" + }, + { + "bbox": [ + 318, + 187, + 327, + 198 + ], + "score": 0.77, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "is dense in the set of continuous function on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "a compact set. Here we extend such a theorem to general functions with symmetries and apply it to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "invariant and equivaraint functions to get our main result for GNNs. As a consequence, we show that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 502, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 113, + 230 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 219, + 491, + 233 + ], + "score": 1.0, + "content": "-FGNNs have the best approximation power among architectures dealing with tensors of order", + "type": "text" + }, + { + "bbox": [ + 491, + 221, + 497, + 230 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 219, + 502, + 233 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 120, + 506, + 233 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "Universality results for GNNs. Universal approximation theorems (similar to Cybenko (1989) for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "multi-layers perceptron) have been proved for linear GNNs in Maron et al. (2019b); Keriven & Peyré", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "(2019); Chen et al. (2019). They show that some classes of GNNs can approximate any function", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "defined on graphs. To be able to approximate any invariant function, they require the use of very", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 214, + 293 + ], + "score": 1.0, + "content": "complex networks, namely", + "type": "text" + }, + { + "bbox": [ + 215, + 281, + 222, + 291 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 281, + 280, + 293 + ], + "score": 1.0, + "content": "-LGNN where", + "type": "text" + }, + { + "bbox": [ + 280, + 281, + 287, + 291 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 281, + 372, + 293 + ], + "score": 1.0, + "content": "tends to infinity with", + "type": "text" + }, + { + "bbox": [ + 372, + 282, + 380, + 291 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "the number of nodes. Since we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 507, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 311, + 304 + ], + "score": 1.0, + "content": "prove that any invariant function less powerful than", + "type": "text" + }, + { + "bbox": [ + 312, + 292, + 342, + 303 + ], + "score": 0.9, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 291, + 466, + 304 + ], + "score": 1.0, + "content": "-WL can be approximated by a", + "type": "text" + }, + { + "bbox": [ + 466, + 292, + 473, + 302 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 291, + 507, + 304 + ], + "score": 1.0, + "content": "-FGNN,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 135, + 315 + ], + "score": 1.0, + "content": "letting", + "type": "text" + }, + { + "bbox": [ + 135, + 303, + 142, + 313 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 303, + 424, + 315 + ], + "score": 1.0, + "content": "tends to infinity directly implies universality. Universality results for", + "type": "text" + }, + { + "bbox": [ + 424, + 303, + 431, + 313 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "-FGNN is another", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 209, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 209, + 325 + ], + "score": 1.0, + "content": "contribution of our work.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 237, + 507, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 504, + 342 + ], + "score": 1.0, + "content": "Equivariant GNNs. Our second set of results extends previous analysis from invariant functions", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "to equivariant functions. There are much less results about equivariant GNNs: Keriven & Peyré", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "score": 1.0, + "content": "(2019) proves the universality of linear equivariant GNNs, and Maehara & Hoang (2019) shows the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "universality of a new class of networks they introduced. Here, we consider a natural equivariant", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 159, + 388 + ], + "score": 1.0, + "content": "extension of", + "type": "text" + }, + { + "bbox": [ + 159, + 375, + 165, + 385 + ], + "score": 0.73, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 373, + 294, + 388 + ], + "score": 1.0, + "content": "-WL and prove that equivariant", + "type": "text" + }, + { + "bbox": [ + 294, + 374, + 325, + 386 + ], + "score": 0.91, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 373, + 379, + 388 + ], + "score": 1.0, + "content": "-LGNNs and", + "type": "text" + }, + { + "bbox": [ + 380, + 375, + 386, + 384 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 373, + 506, + 388 + ], + "score": 1.0, + "content": "-FGNN can approximate any", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 335, + 398 + ], + "score": 1.0, + "content": "equivariant function less powerful than this equivariant", + "type": "text" + }, + { + "bbox": [ + 335, + 385, + 366, + 397 + ], + "score": 0.91, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 385, + 402, + 398 + ], + "score": 1.0, + "content": "-WL for", + "type": "text" + }, + { + "bbox": [ + 402, + 385, + 428, + 396 + ], + "score": 0.91, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 385, + 505, + 398 + ], + "score": 1.0, + "content": ". At this stage, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "should note that all universality results for GNNs by Maron et al. (2019b); Keriven & Peyré (2019);", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "Chen et al. (2019) are easily recovered from our main results. Also our analysis is valid for graphs of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 162, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 162, + 430 + ], + "score": 1.0, + "content": "varying sizes.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 330, + 506, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 506, + 447 + ], + "score": 1.0, + "content": "Empirical results for the Quadratic Assigment Problem (QAP). To validate our theoretical con-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "tributions, we empirically show that 2-FGNN outperforms classical MGNN. Indeed, Maron et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "(2019a) already demonstrate state of the art results for the invariant version of 2-FGNNs (for graph", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "classification or graph regression). Here we consider the graph alignment problem and show that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "the equivariant 2-FGNN is able to learn a node embedding which beats by a large margin other", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 490, + 327, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 327, + 503 + ], + "score": 1.0, + "content": "algorithms (based on spectral method, SDP or GNNs).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 435, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "Outline and contribution. After reviewing more previous works and notations in the next section,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 394, + 530 + ], + "score": 1.0, + "content": "we define the various classes of GNNs studied in this paper in Section", + "type": "text" + }, + { + "bbox": [ + 395, + 518, + 408, + 528 + ], + "score": 0.25, + "content": "3 :", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "message passing GNN,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "linear GNN and folklore GNN. Section 4 contains our main theoretical results for GNNs. First in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "Section 4.2 we describe the separating power of each GNN architecture with respect to the Weisfeiler-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "Lehman test. In Section 4.3, we give approximation guarantees for MGNNs, LGNNs and FGNNs at", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "fixed order of tensor. They cover both the invariant and equivariant cases and are our main theoretical", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "contributions. For these, we develop in Section D a fine-grained Stone-Weierstrass approximation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "theorem for vector-valued functions with symmetries. Our theorem handles both invariant and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "equivariant cases and is inspired by recent works in approximation theory. In Section 6, we illustrate", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "our theoretical results on a practical application: the graph alignment problem, a well-known NP-hard", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "score": 1.0, + "content": "problem. We highlight a previously overlooked implementation question: the handling of batches of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "graphs of varying sizes. A PyTorch implementation of the code necessary to reproduce the results is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 638, + 419, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 419, + 651 + ], + "score": 1.0, + "content": "available at https://github.com/mlelarge/graph_neural_net", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 506, + 506, + 651 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 671, + 209, + 683 + ], + "lines": [ + { + "bbox": [ + 104, + 669, + 211, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 211, + 685 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "The pioneering works that applied neural networks to graphs are Gori et al. (2005) and Scarselli et al.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "(2009) that learn node representation with recurrent neural networks. More recent message passing", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "architectures make use of non-linear functions of the adjacency matrix (Kipf & Welling, 2016),", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "for example polynomials (Defferrard et al., 2016). For regular-grid graphs, they match classical", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "convolutional networks which by design can only approximate translation-invariant functions and", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "hence have limited expressive power. In this paper, we focus instead on more expressive architectures.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "for example polynomials (Defferrard et al., 2016). For regular-grid graphs, they match classical", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "convolutional networks which by design can only approximate translation-invariant functions and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "hence have limited expressive power. In this paper, we focus instead on more expressive architectures.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "Following the recent surge in interest in graph neural networks, some works have tried to extend the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "pioneering work of Cybenko (1989); Hornik et al. (1989) for various GNN architectures. Among the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "first ones is Scarselli et al. (2009), which studied invariant message-passing GNNs. They showed that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "such networks can approximate, in a weak sense, all functions whose discriminatory power is weaker", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "than 1-WL. Yarotsky (2018) described universal architectures which are invariant or equivariant", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "to some group action. These models rely on polynomial intermediate layers of arbitrary degrees,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "which would be prohibitive in practice. Maron et al. (2019b) leveraged classical results about the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 323, + 211 + ], + "score": 1.0, + "content": "polynomials invariant to a group action to show that", + "type": "text" + }, + { + "bbox": [ + 324, + 199, + 330, + 208 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 198, + 430, + 211 + ], + "score": 1.0, + "content": "-LGNN are universal as", + "type": "text" + }, + { + "bbox": [ + 430, + 199, + 437, + 208 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "tends to infinity", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "score": 1.0, + "content": "with the number of nodes. Keriven & Peyré (2019) derived a similar result, in the more complicated", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "equivariant case by introducing a new Stone-Weierstrass theorem. Similarly to Maron et al. (2019b),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 505, + 243 + ], + "score": 1.0, + "content": "they require the order of tensors to go to infinity. Another route towards universality is the one of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "Chen et al. (2019). In the invariant setting, they show for a class of GNN that universality is equivalent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "to being able to discriminate between (non-isomorphic) graphs. However, the only way to achieve", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "such discriminatory power is to use tensors of arbitrary high order, see also Ravanbakhsh (2020).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "Our work encompass and precise these results using high-order tensors as it yields approximation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 268, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 268, + 298 + ], + "score": 1.0, + "content": "guarantees even at fixed order of tensor.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "CPNGNN in Sato et al. (2019) and DimeNet in Klicpera et al. (2020) are message passing GNN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 327 + ], + "score": 1.0, + "content": "incorporating more information than those studied here. Partial results about their separating power", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "follows from Garg et al. (2020) which provides impossibility results to decide graph properties", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 465, + 348 + ], + "score": 1.0, + "content": "including girth, circumference, diameter, radius, conjoint cycle, total number of cycles, and", + "type": "text" + }, + { + "bbox": [ + 466, + 336, + 472, + 345 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "-cliques.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "Chen et al. (2020) studies the ability of GNNs to count graph substructures. Though our theorems are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 504, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 504, + 369 + ], + "score": 1.0, + "content": "much more general, note that their results are improved by the present work. Note also, that if the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "nodes are given distinct features, MGNNs become much more expressive Loukas (2019) but looses", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "their invariant or equivariant properties. Averaging i.e. relational pooling (RP) has been proposed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "to recover these properties Murphy et al. (2019a). However, the ideal RP, leading to a universal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 388, + 414 + ], + "score": 1.0, + "content": "approximation, cannot be used for large graphs due to its complexity of", + "type": "text" + }, + { + "bbox": [ + 388, + 401, + 421, + 414 + ], + "score": 0.95, + "content": "O ( | V | ! )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 402, + 505, + 414 + ], + "score": 1.0, + "content": ". Regarding the other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 219, + 425 + ], + "score": 1.0, + "content": "classes of RPGNN i.e. the", + "type": "text" + }, + { + "bbox": [ + 219, + 413, + 226, + 422 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "-ary pooling (Murphy et al., 2019b), we will show how our general", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "theorems in the invariant case can be applied to characterize their approximation power (see Section", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 120, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 120, + 447 + ], + "score": 1.0, + "content": "5).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 462 + ], + "score": 1.0, + "content": "Note that for neural networks on sets, the situation is a bit simpler. Efficient architectures such as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "DeepSets (Zaheer et al., 2017) or PointNet (Qi et al., 2017) have been shown to be invariant universal.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "Similar results exist in the equivariant case (Segol & Lipman, 2020; Maron et al., 2020), whose", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "proofs rely on polynomial arguments. Though this is not our main motivation, our approximation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 495, + 396, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 396, + 507 + ], + "score": 1.0, + "content": "theorems could also be applied in this context see Sections D.3 and D.4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 108, + 526, + 276, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 279, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 279, + 538 + ], + "score": 1.0, + "content": "2.1 NOTATIONS: GRAPHS AS TENSORS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 165, + 562 + ], + "score": 1.0, + "content": "We denote by", + "type": "text" + }, + { + "bbox": [ + 166, + 549, + 241, + 562 + ], + "score": 0.93, + "content": "\\mathbb { F } , \\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 / 2 } , \\mathbb { F } _ { 1 } , \\dots .", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 548, + 439, + 562 + ], + "score": 1.0, + "content": "arbitrary finite-dimensional spaces of the form", + "type": "text" + }, + { + "bbox": [ + 440, + 550, + 453, + 559 + ], + "score": 0.85, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "(for various", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 146, + 573 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 146, + 561, + 153, + 571 + ], + "score": 0.67, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 559, + 427, + 573 + ], + "score": 1.0, + "content": ") typically representing the space of features. Product of vectors in", + "type": "text" + }, + { + "bbox": [ + 428, + 560, + 440, + 570 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "always refer to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "component-wise product. There are two ways to see graphs with features. First, graphs can be seen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 582, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 182, + 598 + ], + "score": 1.0, + "content": "as tensors of order", + "type": "text" + }, + { + "bbox": [ + 183, + 584, + 190, + 594 + ], + "score": 0.43, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 582, + 194, + 598 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 194, + 582, + 231, + 595 + ], + "score": 0.89, + "content": "G \\in \\mathbb { F } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 582, + 506, + 598 + ], + "score": 1.0, + "content": ". The classical representation of a graph by its (weighted) adjacency", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 594, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 594, + 150, + 612 + ], + "score": 1.0, + "content": "matrix for", + "type": "text" + }, + { + "bbox": [ + 150, + 597, + 175, + 607 + ], + "score": 0.89, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 594, + 273, + 612 + ], + "score": 1.0, + "content": "is a tensor of order 2 in", + "type": "text" + }, + { + "bbox": [ + 273, + 595, + 290, + 608 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 594, + 506, + 612 + ], + "score": 1.0, + "content": ". 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Second, graphs can also be represented by their discrete structure", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 340, + 633 + ], + "score": 1.0, + "content": "with an additional feature vector. More exactly, denote by", + "type": "text" + }, + { + "bbox": [ + 340, + 622, + 353, + 632 + ], + "score": 0.89, + "content": "{ \\mathcal { G } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 620, + 456, + 633 + ], + "score": 1.0, + "content": "the set of discrete graphs", + "type": "text" + }, + { + "bbox": [ + 456, + 621, + 505, + 633 + ], + "score": 0.93, + "content": "G = ( V , E )", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 126, + 645 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 634, + 133, + 642 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 631, + 160, + 645 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 160, + 632, + 195, + 644 + ], + "score": 0.92, + "content": "V = [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 631, + 237, + 645 + ], + "score": 1.0, + "content": "and edges", + "type": "text" + }, + { + "bbox": [ + 238, + 632, + 273, + 643 + ], + "score": 0.92, + "content": "E \\subseteq V ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 631, + 417, + 645 + ], + "score": 1.0, + "content": "(with no weights on edges). 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The set of permutations on", + "type": "text" + }, + { + "bbox": [ + 311, + 700, + 323, + 711 + ], + "score": 0.88, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 696, + 382, + 712 + ], + "score": 1.0, + "content": "is denoted by", + "type": "text" + }, + { + "bbox": [ + 382, + 699, + 394, + 710 + ], + "score": 0.86, + "content": "S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 696, + 416, + 712 + ], + "score": 1.0, + "content": ". 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(1989) for various GNN architectures. Among the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "first ones is Scarselli et al. (2009), which studied invariant message-passing GNNs. They showed that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "such networks can approximate, in a weak sense, all functions whose discriminatory power is weaker", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "than 1-WL. Yarotsky (2018) described universal architectures which are invariant or equivariant", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "to some group action. These models rely on polynomial intermediate layers of arbitrary degrees,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "which would be prohibitive in practice. Maron et al. (2019b) leveraged classical results about the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 323, + 211 + ], + "score": 1.0, + "content": "polynomials invariant to a group action to show that", + "type": "text" + }, + { + "bbox": [ + 324, + 199, + 330, + 208 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 198, + 430, + 211 + ], + "score": 1.0, + "content": "-LGNN are universal as", + "type": "text" + }, + { + "bbox": [ + 430, + 199, + 437, + 208 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "tends to infinity", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "score": 1.0, + "content": "with the number of nodes. Keriven & Peyré (2019) derived a similar result, in the more complicated", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "equivariant case by introducing a new Stone-Weierstrass theorem. Similarly to Maron et al. (2019b),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 505, + 243 + ], + "score": 1.0, + "content": "they require the order of tensors to go to infinity. Another route towards universality is the one of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "Chen et al. (2019). In the invariant setting, they show for a class of GNN that universality is equivalent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "to being able to discriminate between (non-isomorphic) graphs. However, the only way to achieve", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "such discriminatory power is to use tensors of arbitrary high order, see also Ravanbakhsh (2020).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "Our work encompass and precise these results using high-order tensors as it yields approximation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 268, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 268, + 298 + ], + "score": 1.0, + "content": "guarantees even at fixed order of tensor.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 122, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "CPNGNN in Sato et al. (2019) and DimeNet in Klicpera et al. (2020) are message passing GNN", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 327 + ], + "score": 1.0, + "content": "incorporating more information than those studied here. Partial results about their separating power", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "follows from Garg et al. (2020) which provides impossibility results to decide graph properties", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 465, + 348 + ], + "score": 1.0, + "content": "including girth, circumference, diameter, radius, conjoint cycle, total number of cycles, and", + "type": "text" + }, + { + "bbox": [ + 466, + 336, + 472, + 345 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "-cliques.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "Chen et al. (2020) studies the ability of GNNs to count graph substructures. Though our theorems are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 504, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 504, + 369 + ], + "score": 1.0, + "content": "much more general, note that their results are improved by the present work. Note also, that if the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "nodes are given distinct features, MGNNs become much more expressive Loukas (2019) but looses", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "their invariant or equivariant properties. Averaging i.e. relational pooling (RP) has been proposed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "to recover these properties Murphy et al. (2019a). However, the ideal RP, leading to a universal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 388, + 414 + ], + "score": 1.0, + "content": "approximation, cannot be used for large graphs due to its complexity of", + "type": "text" + }, + { + "bbox": [ + 388, + 401, + 421, + 414 + ], + "score": 0.95, + "content": "O ( | V | ! )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 402, + 505, + 414 + ], + "score": 1.0, + "content": ". Regarding the other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 219, + 425 + ], + "score": 1.0, + "content": "classes of RPGNN i.e. the", + "type": "text" + }, + { + "bbox": [ + 219, + 413, + 226, + 422 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "-ary pooling (Murphy et al., 2019b), we will show how our general", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "theorems in the invariant case can be applied to characterize their approximation power (see Section", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 120, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 120, + 447 + ], + "score": 1.0, + "content": "5).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 303, + 506, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 462 + ], + "score": 1.0, + "content": "Note that for neural networks on sets, the situation is a bit simpler. Efficient architectures such as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "DeepSets (Zaheer et al., 2017) or PointNet (Qi et al., 2017) have been shown to be invariant universal.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "Similar results exist in the equivariant case (Segol & Lipman, 2020; Maron et al., 2020), whose", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "proofs rely on polynomial arguments. Though this is not our main motivation, our approximation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 495, + 396, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 396, + 507 + ], + "score": 1.0, + "content": "theorems could also be applied in this context see Sections D.3 and D.4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 452, + 506, + 507 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 526, + 276, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 279, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 279, + 538 + ], + "score": 1.0, + "content": "2.1 NOTATIONS: GRAPHS AS TENSORS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 165, + 562 + ], + "score": 1.0, + "content": "We denote by", + "type": "text" + }, + { + "bbox": [ + 166, + 549, + 241, + 562 + ], + "score": 0.93, + "content": "\\mathbb { F } , \\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 / 2 } , \\mathbb { F } _ { 1 } , \\dots .", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 548, + 439, + 562 + ], + "score": 1.0, + "content": "arbitrary finite-dimensional spaces of the form", + "type": "text" + }, + { + "bbox": [ + 440, + 550, + 453, + 559 + ], + "score": 0.85, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "(for various", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 146, + 573 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 146, + 561, + 153, + 571 + ], + "score": 0.67, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 559, + 427, + 573 + ], + "score": 1.0, + "content": ") typically representing the space of features. Product of vectors in", + "type": "text" + }, + { + "bbox": [ + 428, + 560, + 440, + 570 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "always refer to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "component-wise product. There are two ways to see graphs with features. First, graphs can be seen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 582, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 182, + 598 + ], + "score": 1.0, + "content": "as tensors of order", + "type": "text" + }, + { + "bbox": [ + 183, + 584, + 190, + 594 + ], + "score": 0.43, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 582, + 194, + 598 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 194, + 582, + 231, + 595 + ], + "score": 0.89, + "content": "G \\in \\mathbb { F } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 582, + 506, + 598 + ], + "score": 1.0, + "content": ". The classical representation of a graph by its (weighted) adjacency", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 594, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 594, + 150, + 612 + ], + "score": 1.0, + "content": "matrix for", + "type": "text" + }, + { + "bbox": [ + 150, + 597, + 175, + 607 + ], + "score": 0.89, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 594, + 273, + 612 + ], + "score": 1.0, + "content": "is a tensor of order 2 in", + "type": "text" + }, + { + "bbox": [ + 273, + 595, + 290, + 608 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 594, + 506, + 612 + ], + "score": 1.0, + "content": ". This case allows for features on edges by replacing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 607, + 507, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 124, + 620 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 607, + 145, + 624 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 145, + 608, + 162, + 620 + ], + "score": 0.89, + "content": "\\mathbb { F } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 607, + 189, + 624 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 189, + 610, + 197, + 620 + ], + "score": 0.76, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 607, + 230, + 624 + ], + "score": 1.0, + "content": "is some", + "type": "text" + }, + { + "bbox": [ + 230, + 610, + 243, + 620 + ], + "score": 0.84, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 607, + 507, + 624 + ], + "score": 1.0, + "content": ". Second, graphs can also be represented by their discrete structure", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 340, + 633 + ], + "score": 1.0, + "content": "with an additional feature vector. More exactly, denote by", + "type": "text" + }, + { + "bbox": [ + 340, + 622, + 353, + 632 + ], + "score": 0.89, + "content": "{ \\mathcal { G } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 620, + 456, + 633 + ], + "score": 1.0, + "content": "the set of discrete graphs", + "type": "text" + }, + { + "bbox": [ + 456, + 621, + 505, + 633 + ], + "score": 0.93, + "content": "G = ( V , E )", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 126, + 645 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 634, + 133, + 642 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 631, + 160, + 645 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 160, + 632, + 195, + 644 + ], + "score": 0.92, + "content": "V = [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 631, + 237, + 645 + ], + "score": 1.0, + "content": "and edges", + "type": "text" + }, + { + "bbox": [ + 238, + 632, + 273, + 643 + ], + "score": 0.92, + "content": "E \\subseteq V ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 631, + 417, + 645 + ], + "score": 1.0, + "content": "(with no weights on edges). Such a", + "type": "text" + }, + { + "bbox": [ + 417, + 632, + 449, + 643 + ], + "score": 0.92, + "content": "G \\in \\mathcal G _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "with a vector", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 641, + 340, + 657 + ], + "spans": [ + { + "bbox": [ + 107, + 643, + 141, + 653 + ], + "score": 0.91, + "content": "h ^ { 0 } \\in \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 641, + 340, + 657 + ], + "score": 1.0, + "content": "represents a graphs with features on the vertices.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 548, + 507, + 657 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 674, + 378, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 674, + 379, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 379, + 686 + ], + "score": 1.0, + "content": "2.2 DEFINITIONS: INVARIANT AND EQUIVARIANT OPERATORS", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 123, + 712 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 699, + 193, + 711 + ], + "score": 0.9, + "content": "[ n ] = \\{ 1 , \\dots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 696, + 311, + 712 + ], + "score": 1.0, + "content": ". 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We will define three different types of layers for message", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 329, + 387 + ], + "score": 1.0, + "content": "passing GNN, linear GNN and folklore GNN.The case", + "type": "text" + }, + { + "bbox": [ + 330, + 374, + 355, + 384 + ], + "score": 0.89, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "is probably the most interesting case", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "from a practical point view and corresponds to a case where a layer takes as input a graph (with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "features on nodes and edges) and produces as output a graph (with new features on nodes and edges).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 102, + 406, + 508, + 435 + ], + "spans": [ + { + "bbox": [ + 102, + 406, + 221, + 435 + ], + "score": 1.0, + "content": "For each type of GNNs, thwill share the last function:", + "type": "text" + }, + { + "bbox": [ + 222, + 418, + 290, + 429 + ], + "score": 0.93, + "content": "m _ { I } : \\mathbb { F } _ { T + 1 } \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 406, + 397, + 435 + ], + "score": 1.0, + "content": "variant and an equivaraint for the invariant case and", + "type": "text" + }, + { + "bbox": [ + 398, + 418, + 474, + 430 + ], + "score": 0.92, + "content": "m _ { E } : \\mathbb { F } _ { T + 1 } ^ { n } \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 406, + 508, + 435 + ], + "score": 1.0, + "content": "ecturesfor the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 428, + 507, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 507, + 442 + ], + "score": 1.0, + "content": "equivariant case which are continuous functions. It is typically modeled by a Multi Layer Perceptron,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "which is applied on each component for the equivariant case. In words, each network takes as input a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 450, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 132, + 468 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 451, + 169, + 465 + ], + "score": 0.92, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 450, + 389, + 468 + ], + "score": 1.0, + "content": ", produces in the invariant case a graph embedding in", + "type": "text" + }, + { + "bbox": [ + 389, + 452, + 412, + 464 + ], + "score": 0.91, + "content": "\\mathbb { F } _ { T + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 450, + 505, + 468 + ], + "score": 1.0, + "content": "and in the equivaraint", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 460, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 212, + 480 + ], + "score": 1.0, + "content": "case a node embedding in", + "type": "text" + }, + { + "bbox": [ + 212, + 464, + 235, + 476 + ], + "score": 0.91, + "content": "\\mathbb { F } _ { T + 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 460, + 460, + 480 + ], + "score": 1.0, + "content": ", then these embeddings are passed through the function", + "type": "text" + }, + { + "bbox": [ + 461, + 465, + 475, + 475 + ], + "score": 0.87, + "content": "m _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 460, + 487, + 480 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 487, + 465, + 504, + 475 + ], + "score": 0.83, + "content": "m _ { E }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 474, + 348, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 228, + 487 + ], + "score": 1.0, + "content": "respectively to get a feature in", + "type": "text" + }, + { + "bbox": [ + 229, + 476, + 236, + 484 + ], + "score": 0.83, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 474, + 248, + 487 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 248, + 475, + 261, + 484 + ], + "score": 0.89, + "content": "\\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 474, + 348, + 487 + ], + "score": 1.0, + "content": "for the learning task.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 238, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 239, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 239, + 512 + ], + "score": 1.0, + "content": "3.1 MESSAGE PASSING GNN", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 506, + 620 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 362, + 533 + ], + "score": 1.0, + "content": "Message passing GNN (MGNN) are defined for classical graphs", + "type": "text" + }, + { + "bbox": [ + 362, + 520, + 371, + 530 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "with features on the nodes. More", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 281, + 543 + ], + "score": 1.0, + "content": "exactly they take as input a discrete graph", + "type": "text" + }, + { + "bbox": [ + 282, + 531, + 357, + 542 + ], + "score": 0.89, + "content": "G = ( V , E ) \\in \\mathcal { G } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 530, + 466, + 543 + ], + "score": 1.0, + "content": "and features on the nodes", + "type": "text" + }, + { + "bbox": [ + 466, + 530, + 502, + 541 + ], + "score": 0.91, + "content": "h ^ { 0 } \\in \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 530, + 505, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 313, + 554 + ], + "score": 1.0, + "content": "MGNN are then defined inductively as 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Then an equivariant message passing GNN is simply", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 326, + 712 + ], + "score": 1.0, + "content": "obtained by the composition of message passing layers:", + "type": "text" + }, + { + "bbox": [ + 327, + 701, + 392, + 710 + ], + "score": 0.9, + "content": "F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 698, + 442, + 712 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 442, + 699, + 452, + 710 + ], + "score": 0.87, + "content": "F _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "is a message", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 243, + 722 + ], + "score": 1.0, + "content": "passing layer. Clearly since each", + "type": "text" + }, + { + "bbox": [ + 244, + 711, + 254, + 721 + ], + "score": 0.87, + "content": "F _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "is equivariant, this message passing GNN is also equivariant", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 301, + 734 + ], + "score": 1.0, + "content": "and produces features on each node in the space", + "type": "text" + }, + { + "bbox": [ + 301, + 723, + 314, + 732 + ], + "score": 0.89, + "content": "\\mathbb { F } _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 719, + 505, + 734 + ], + "score": 1.0, + "content": ". In order to obtain an invariant GNN, we apply", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 108, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 104, + 105, + 506, + 125 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 214, + 125 + ], + "score": 1.0, + "content": "Definition 1. A function", + "type": "text" + }, + { + "bbox": [ + 214, + 107, + 279, + 122 + ], + "score": 0.92, + "content": "f : \\mathbb { F } _ { 0 } ^ { n ^ { k } } \\to \\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 105, + 385, + 125 + ], + "score": 1.0, + "content": "is said to be invariant if", + "type": "text" + }, + { + "bbox": [ + 385, + 109, + 463, + 122 + ], + "score": 0.95, + "content": "f ( \\sigma \\star G ) = f ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 105, + 506, + 125 + ], + "score": 1.0, + "content": "for every", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 103, + 118, + 508, + 139 + ], + "spans": [ + { + "bbox": [ + 103, + 118, + 158, + 139 + ], + "score": 1.0, + "content": "permutation", + "type": "text" + }, + { + "bbox": [ + 159, + 123, + 191, + 135 + ], + "score": 0.91, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 118, + 236, + 139 + ], + "score": 1.0, + "content": "and every", + "type": "text" + }, + { + "bbox": [ + 236, + 122, + 274, + 136 + ], + "score": 0.92, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 118, + 328, + 139 + ], + "score": 1.0, + "content": ". A function", + "type": "text" + }, + { + "bbox": [ + 328, + 121, + 393, + 136 + ], + "score": 0.92, + "content": "f : \\mathbb { F } _ { 0 } ^ { n ^ { k } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { \\ell } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 118, + 508, + 139 + ], + "score": 1.0, + "content": "is said to be equivariant if", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 132, + 402, + 155 + ], + "spans": [ + { + "bbox": [ + 107, + 137, + 195, + 149 + ], + "score": 0.93, + "content": "f ( \\sigma \\star G ) = \\sigma \\star f ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 132, + 285, + 155 + ], + "score": 1.0, + "content": "for every permutation", + "type": "text" + }, + { + "bbox": [ + 285, + 137, + 316, + 148 + ], + "score": 0.91, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 132, + 358, + 155 + ], + "score": 1.0, + "content": "and every", + "type": "text" + }, + { + "bbox": [ + 359, + 135, + 395, + 149 + ], + "score": 0.91, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 132, + 402, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 103, + 105, + 508, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 506, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 170 + ], + "score": 1.0, + "content": "Note that composing an equivariant function with an invariant function gives an invariant function.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 103, + 164, + 508, + 187 + ], + "spans": [ + { + "bbox": [ + 103, + 164, + 123, + 187 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 171, + 150, + 182 + ], + "score": 0.89, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 164, + 323, + 187 + ], + "score": 1.0, + "content": ", we define the invariant summation layer", + "type": "text" + }, + { + "bbox": [ + 323, + 168, + 386, + 181 + ], + "score": 0.92, + "content": "S ^ { k } : \\mathbb { F } ^ { n ^ { k } } \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 164, + 400, + 187 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 401, + 169, + 489, + 185 + ], + "score": 0.93, + "content": "\\begin{array} { r } { S ^ { k } ( G ) = \\sum _ { \\mathbf i \\in [ n ] ^ { k } } G _ { \\mathbf i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 164, + 508, + 187 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 145, + 197 + ], + "score": 0.9, + "content": "G \\in \\mathbb { F } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 182, + 343, + 200 + ], + "score": 1.0, + "content": ". 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We refer to the appendix Section C.3 for a precise description of this linear equivariant function.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 261, + 504, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 261, + 199, + 279 + ], + "score": 1.0, + "content": "Note at this stage that", + "type": "text" + }, + { + "bbox": [ + 199, + 264, + 209, + 275 + ], + "score": 0.88, + "content": "I ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 261, + 276, + 279 + ], + "score": 1.0, + "content": "is given by, for", + "type": "text" + }, + { + "bbox": [ + 276, + 263, + 313, + 276 + ], + "score": 0.9, + "content": "G \\in \\mathbb { F } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 261, + 318, + 279 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 318, + 264, + 407, + 277 + ], + "score": 0.92, + "content": "I ( G ) _ { i , j } = ( G _ { i , j } , \\delta _ { i , j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 261, + 436, + 279 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 437, + 265, + 451, + 277 + ], + "score": 0.91, + "content": "\\delta _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 261, + 479, + 279 + ], + "score": 1.0, + "content": "is 0 if", + "type": "text" + }, + { + "bbox": [ + 480, + 265, + 504, + 276 + ], + "score": 0.89, + "content": "i \\neq j", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 286, + 288 + ], + "score": 1.0, + "content": "and 1 otherwise. Indeed for a pair of nodes", + "type": "text" + }, + { + "bbox": [ + 286, + 277, + 300, + 287 + ], + "score": 0.88, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "in a graph (without features), there are only three", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 286, + 445, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 187, + 300 + ], + "score": 1.0, + "content": "isomorphism types:", + "type": "text" + }, + { + "bbox": [ + 188, + 287, + 237, + 298 + ], + "score": 0.39, + "content": "i = j ; i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 286, + 255, + 300 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 287, + 276, + 299 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 286, + 323, + 300 + ], + "score": 1.0, + "content": "is an edge;", + "type": "text" + }, + { + "bbox": [ + 323, + 287, + 346, + 298 + ], + "score": 0.91, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 286, + 362, + 300 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 363, + 287, + 383, + 299 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 286, + 445, + 300 + ], + "score": 1.0, + "content": "is not an edge.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 226, + 506, + 300 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 314, + 222, + 327 + ], + "lines": [ + { + "bbox": [ + 104, + 312, + 224, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 224, + 330 + ], + "score": 1.0, + "content": "3 GNN DEFINITIONS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 506, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 507, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 507, + 352 + ], + "score": 1.0, + "content": "In this section, we define the various GNN architectures studied in this paper. In all architectures,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 100, + 346, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 100, + 347, + 297, + 371 + ], + "score": 1.0, + "content": "there is a main building block or layer mapping", + "type": "text" + }, + { + "bbox": [ + 297, + 349, + 314, + 363 + ], + "score": 0.91, + "content": "\\mathbb { F } _ { t } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 346, + 353, + 371 + ], + "score": 1.0, + "content": "k to F n kt +1", + "type": "text" + }, + { + "bbox": [ + 347, + 348, + 374, + 369 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 375, + 349, + 391, + 363 + ], + "score": 0.91, + "content": "\\mathbb { F } _ { t } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 348, + 505, + 369 + ], + "score": 1.0, + "content": "can be seen as the space for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 362, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 263, + 377 + ], + "score": 1.0, + "content": "the representation of the graph at layer", + "type": "text" + }, + { + "bbox": [ + 269, + 362, + 505, + 377 + ], + "score": 1.0, + "content": ". We will define three different types of layers for message", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 329, + 387 + ], + "score": 1.0, + "content": "passing GNN, linear GNN and folklore GNN.The case", + "type": "text" + }, + { + "bbox": [ + 330, + 374, + 355, + 384 + ], + "score": 0.89, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "is probably the most interesting case", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "from a practical point view and corresponds to a case where a layer takes as input a graph (with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "features on nodes and edges) and produces as output a graph (with new features on nodes and edges).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 102, + 406, + 508, + 435 + ], + "spans": [ + { + "bbox": [ + 102, + 406, + 221, + 435 + ], + "score": 1.0, + "content": "For each type of GNNs, thwill share the last function:", + "type": "text" + }, + { + "bbox": [ + 222, + 418, + 290, + 429 + ], + "score": 0.93, + "content": "m _ { I } : \\mathbb { F } _ { T + 1 } \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 406, + 397, + 435 + ], + "score": 1.0, + "content": "variant and an equivaraint for the invariant case and", + "type": "text" + }, + { + "bbox": [ + 398, + 418, + 474, + 430 + ], + "score": 0.92, + "content": "m _ { E } : \\mathbb { F } _ { T + 1 } ^ { n } \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 406, + 508, + 435 + ], + "score": 1.0, + "content": "ecturesfor the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 428, + 507, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 507, + 442 + ], + "score": 1.0, + "content": "equivariant case which are continuous functions. It is typically modeled by a Multi Layer Perceptron,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "which is applied on each component for the equivariant case. In words, each network takes as input a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 450, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 132, + 468 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 451, + 169, + 465 + ], + "score": 0.92, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 450, + 389, + 468 + ], + "score": 1.0, + "content": ", produces in the invariant case a graph embedding in", + "type": "text" + }, + { + "bbox": [ + 389, + 452, + 412, + 464 + ], + "score": 0.91, + "content": "\\mathbb { F } _ { T + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 450, + 505, + 468 + ], + "score": 1.0, + "content": "and in the equivaraint", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 460, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 212, + 480 + ], + "score": 1.0, + "content": "case a node embedding in", + "type": "text" + }, + { + "bbox": [ + 212, + 464, + 235, + 476 + ], + "score": 0.91, + "content": "\\mathbb { F } _ { T + 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 460, + 460, + 480 + ], + "score": 1.0, + "content": ", then these embeddings are passed through the function", + "type": "text" + }, + { + "bbox": [ + 461, + 465, + 475, + 475 + ], + "score": 0.87, + "content": "m _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 460, + 487, + 480 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 487, + 465, + 504, + 475 + ], + "score": 0.83, + "content": "m _ { E }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 474, + 348, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 228, + 487 + ], + "score": 1.0, + "content": "respectively to get a feature in", + "type": "text" + }, + { + "bbox": [ + 229, + 476, + 236, + 484 + ], + "score": 0.83, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 474, + 248, + 487 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 248, + 475, + 261, + 484 + ], + "score": 0.89, + "content": "\\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 474, + 348, + 487 + ], + "score": 1.0, + "content": "for the learning task.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5, + "bbox_fs": [ + 100, + 337, + 508, + 487 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 238, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 239, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 239, + 512 + ], + "score": 1.0, + "content": "3.1 MESSAGE PASSING GNN", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 506, + 620 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 362, + 533 + ], + "score": 1.0, + "content": "Message passing GNN (MGNN) are defined for classical graphs", + "type": "text" + }, + { + "bbox": [ + 362, + 520, + 371, + 530 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "with features on the nodes. More", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 281, + 543 + ], + "score": 1.0, + "content": "exactly they take as input a discrete graph", + "type": "text" + }, + { + "bbox": [ + 282, + 531, + 357, + 542 + ], + "score": 0.89, + "content": "G = ( V , E ) \\in \\mathcal { G } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 530, + 466, + 543 + ], + "score": 1.0, + "content": "and features on the nodes", + "type": "text" + }, + { + "bbox": [ + 466, + 530, + 502, + 541 + ], + "score": 0.91, + "content": "h ^ { 0 } \\in \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 530, + 505, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 313, + 554 + ], + "score": 1.0, + "content": "MGNN are then defined inductively as follows: let", + "type": "text" + }, + { + "bbox": [ + 313, + 542, + 346, + 554 + ], + "score": 0.91, + "content": "h _ { i } ^ { \\ell } \\in \\mathbb { F } _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 541, + 454, + 554 + ], + "score": 1.0, + "content": "denote the feature at layer", + "type": "text" + }, + { + "bbox": [ + 455, + 542, + 460, + 551 + ], + "score": 0.77, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "associated", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 549, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 103, + 549, + 150, + 574 + ], + "score": 1.0, + "content": "with node", + "type": "text" + }, + { + "bbox": [ + 150, + 558, + 154, + 567 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 549, + 244, + 574 + ], + "score": 1.0, + "content": ", the updated features", + "type": "text" 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Then an equivariant message passing GNN is simply", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 326, + 712 + ], + "score": 1.0, + "content": "obtained by the composition of message passing layers:", + "type": "text" + }, + { + "bbox": [ + 327, + 701, + 392, + 710 + ], + "score": 0.9, + "content": "F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 698, + 442, + 712 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 442, + 699, + 452, + 710 + ], + "score": 0.87, + "content": "F _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "is a message", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 243, + 722 + ], + "score": 1.0, + "content": "passing layer. Clearly since each", + "type": "text" + }, + { + "bbox": [ + 244, + 711, + 254, + 721 + ], + "score": 0.87, + "content": "F _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "is equivariant, this message passing GNN is also equivariant", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 301, + 734 + ], + "score": 1.0, + "content": "and produces features on each node in the space", + "type": "text" + }, + { + "bbox": [ + 301, + 723, + 314, + 732 + ], + "score": 0.89, + "content": "\\mathbb { F } _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 719, + 505, + 734 + ], + "score": 1.0, + "content": ". In order to obtain an invariant GNN, we apply", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 215, + 96 + ], + "score": 1.0, + "content": "an invariant function from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 216, + 83, + 268, + 95 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { T } ^ { n } \\to \\mathbb { F } _ { T + 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 268, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "on the output of an equivariant message passing GNN. In", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "practice, a symmetric function is applied on the vectors of features indexed by the nodes, typically", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 202, + 117 + ], + "score": 1.0, + "content": "the sum of the features", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 202, + 104, + 311, + 117 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i } ( F _ { T } \\circ . . . 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\\to \\mathbb { F } _ { \\ell + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 445, + 252, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 447, + 324, + 460 + ], + "score": 0.91, + "content": "f _ { k } : \\mathbb { F } _ { \\ell } \\to \\mathbb { F } _ { \\ell + 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "are learnable functions. As shown in Lem. 33", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 417, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 417, + 470 + ], + "score": 1.0, + "content": "in Appendix, FGL is an equivariant function which is indeed very expressive.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 101, + 474, + 511, + 509 + ], + "spans": [ + { + "bbox": [ + 101, + 484, + 149, + 507 + ], + "score": 1.0, + "content": "F t : F n 2t", + "type": "text" + }, + { + "bbox": [ + 107, + 487, + 178, + 502 + ], + "score": 0.93, + "content": "F _ { t } : \\mathbb { F } _ { t } ^ { n ^ { 2 } } \\mathbb { F } _ { t + 1 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 474, + 191, + 509 + ], + "score": 1.0, + "content": "phs , so", + "type": "text" + }, + { + "bbox": [ + 192, + 474, + 230, + 488 + ], + "score": 0.93, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 489, + 280, + 501 + ], + "score": 0.91, + "content": "F _ { T } \\circ \\dots F _ { 1 } \\circ F _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 474, + 463, + 509 + ], + "score": 1.0, + "content": "w define 2-FGNN by composing folklore gris an equivariant GNN producing a graph in", + "type": "text" + }, + { + "bbox": [ + 464, + 487, + 487, + 502 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { T + 1 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 474, + 511, + 509 + ], + "score": 1.0, + "content": "yers. 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As shown in Lem. 33", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 417, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 417, + 470 + ], + "score": 1.0, + "content": "in Appendix, FGL is an equivariant function which is indeed very expressive.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 445, + 505, + 470 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 101, + 474, + 511, + 509 + ], + "spans": [ + { + "bbox": [ + 101, + 484, + 149, + 507 + ], + "score": 1.0, + "content": "F t : F n 2t", + "type": "text" + }, + { + "bbox": [ + 107, + 487, + 178, + 502 + ], + "score": 0.93, + "content": "F _ { t } : \\mathbb { F } _ { t } ^ { n ^ { 2 } } \\mathbb { F } _ { t + 1 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 474, + 191, + 509 + ], + "score": 1.0, + "content": "phs , so", + "type": "text" + }, + { + "bbox": [ + 192, + 474, + 230, + 488 + ], + "score": 0.93, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 489, + 280, + 501 + ], + "score": 0.91, + "content": "F _ { T } \\circ \\dots F _ { 1 } \\circ F _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 474, + 463, + 509 + ], + "score": 1.0, + "content": "w define 2-FGNN by composing folklore gris an equivariant GNN producing a graph in", + "type": "text" + }, + { + "bbox": [ + 464, + 487, + 487, + 502 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { T + 1 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 474, + 511, + 509 + ], + "score": 1.0, + "content": "yers. 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We denote", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 119, + 207 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 194, + 165, + 207 + ], + "score": 0.91, + "content": "k { \\mathrm { - W L } } _ { I } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "the multiset of colors in the stable coloring. This is a graph invariant that is usually", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 503, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 425, + 217 + ], + "score": 1.0, + "content": "used to test if graphs are isomorphic. 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Formally, define", + "type": "text" + }, + { + "bbox": [ + 299, + 244, + 382, + 258 + ], + "score": 0.93, + "content": "k – \\mathbf { W L } _ { E } ^ { T } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 244, + 412, + 260 + ], + "score": 1.0, + "content": "by, for", + "type": "text" + }, + { + "bbox": [ + 413, + 246, + 437, + 257 + ], + "score": 0.77, + "content": "i \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 244, + 443, + 260 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 443, + 245, + 506, + 258 + ], + "score": 0.82, + "content": "k { \\mathrm { - W L } } _ { E } ^ { T } ( G ) _ { i } =", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 255, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 107, + 257, + 215, + 271 + ], + "score": 0.92, + "content": "\\left\\{ \\left\\{ c ^ { T } ( \\mathbf { i } ) : \\mathbf { i } \\in V ^ { k } , i _ { 1 } = i \\right\\} \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 255, + 287, + 272 + ], + "score": 1.0, + "content": ". 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We have, for", + "type": "text" + }, + { + "bbox": [ + 223, + 540, + 248, + 551 + ], + "score": 0.88, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 539, + 251, + 553 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 558, + 464, + 602 + ], + "lines": [ + { + "bbox": [ + 146, + 558, + 464, + 602 + ], + "spans": [ + { + "bbox": [ + 146, + 558, + 464, + 602 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) = \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { E } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "0ad3c4570af4e4e7eccbc889a029a90cb3a7dc28ac37a8b37f26d70ae2528aa1.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 146, + 558, + 464, + 572.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 146, + 572.6666666666666, + 464, + 587.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 146, + 587.3333333333333, + 464, + 601.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 613, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 423, + 627 + ], + "score": 1.0, + "content": "Only results about the invariant cases were previously known: (4) comes from", + "type": "text" + }, + { + "bbox": [ + 423, + 614, + 437, + 624 + ], + "score": 0.54, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 613, + 506, + 627 + ], + "score": 1.0, + "content": "et al. (2018), (5)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "from Maron et al. (2018) Geerts (2020a) and one inclusion of (6) comes from Maron et al. 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We denote", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 119, + 207 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 194, + 165, + 207 + ], + "score": 0.91, + "content": "k { \\mathrm { - W L } } _ { I } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "the multiset of colors in the stable coloring. This is a graph invariant that is usually", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 503, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 425, + 217 + ], + "score": 1.0, + "content": "used to test if graphs are isomorphic. 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Note that all the functions in", + "type": "text" + }, + { + "bbox": [ + 422, + 413, + 432, + 423 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 412, + 451, + 425 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 451, + 413, + 459, + 423 + ], + "score": 0.8, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "need to be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "defined on the same set but can take values in different sets. For example, we can easily see that for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 507, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 121, + 447 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 435, + 128, + 444 + ], + "score": 0.74, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 434, + 507, + 447 + ], + "score": 1.0, + "content": "-WL algorithm defined above, the equivariant version is more separating than the invariant one.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 401, + 507, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 506, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 507, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 507, + 465 + ], + "score": 1.0, + "content": "Some properties of the WL hierarchy of tests can be rephrased with the notion of separating power.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 284, + 475 + ], + "score": 1.0, + "content": "In particular, Cai et al. (1989) showed that", + "type": "text" + }, + { + "bbox": [ + 285, + 462, + 339, + 474 + ], + "score": 0.9, + "content": "( k + 1 ) \\ – \\mathbf { W } \\mathbf { L } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 462, + 472, + 475 + ], + "score": 1.0, + "content": "distinguishes strictly more than", + "type": "text" + }, + { + "bbox": [ + 473, + 463, + 502, + 474 + ], + "score": 0.81, + "content": "k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 462, + 506, + 475 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 421, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 300, + 486 + ], + "score": 1.0, + "content": "which can be rewritten simply as (for a function", + "type": "text" + }, + { + "bbox": [ + 300, + 475, + 307, + 485 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 473, + 348, + 486 + ], + "score": 1.0, + "content": ", we write", + "type": "text" + }, + { + "bbox": [ + 348, + 474, + 369, + 486 + ], + "score": 0.91, + "content": "\\rho \\left( f \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 473, + 385, + 486 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 385, + 473, + 418, + 486 + ], + "score": 0.9, + "content": "\\rho \\left( \\left\\{ f \\right\\} \\right) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 473, + 421, + 486 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 450, + 507, + 486 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 492, + 373, + 506 + ], + "lines": [ + { + "bbox": [ + 238, + 492, + 373, + 506 + ], + "spans": [ + { + "bbox": [ + 238, + 492, + 373, + 506 + ], + "score": 0.92, + "content": "\\rho \\left( { ( k + 1 ) } { - } \\mathbf { W } \\mathbf { L } _ { I } \\right) \\subsetneq \\rho \\left( k { - } \\mathbf { W } \\mathbf { L } _ { I } \\right) .", + "type": "interline_equation", + "image_path": "9baf958ba7bd01950cd74622f6d97fced58c1f937e57c1ae4de760f6a0fed1ee.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 238, + 492, + 373, + 506 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 513, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "This notion of separating power enables us to concisely summarize the current knowledge about the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 524, + 272, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 272, + 536 + ], + "score": 1.0, + "content": "discriminatory power of classes of GNN.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 513, + 505, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 250, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 251, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 222, + 553 + ], + "score": 1.0, + "content": "Proposition 3. We have, for", + "type": "text" + }, + { + "bbox": [ + 223, + 540, + 248, + 551 + ], + "score": 0.88, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 539, + 251, + 553 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 539, + 251, + 553 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 558, + 464, + 602 + ], + "lines": [ + { + "bbox": [ + 146, + 558, + 464, + 602 + ], + "spans": [ + { + "bbox": [ + 146, + 558, + 464, + 602 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) = \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { E } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "0ad3c4570af4e4e7eccbc889a029a90cb3a7dc28ac37a8b37f26d70ae2528aa1.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 146, + 558, + 464, + 572.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 146, + 572.6666666666666, + 464, + 587.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 146, + 587.3333333333333, + 464, + 601.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 613, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 423, + 627 + ], + "score": 1.0, + "content": "Only results about the invariant cases were previously known: (4) comes from", + "type": "text" + }, + { + "bbox": [ + 423, + 614, + 437, + 624 + ], + "score": 0.54, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 613, + 506, + 627 + ], + "score": 1.0, + "content": "et al. (2018), (5)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "from Maron et al. (2018) Geerts (2020a) and one inclusion of (6) comes from Maron et al. (2019a).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 634, + 349, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 229, + 648 + ], + "score": 1.0, + "content": "The equality in (6) for general", + "type": "text" + }, + { + "bbox": [ + 229, + 636, + 254, + 647 + ], + "score": 0.91, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 634, + 349, + 648 + ], + "score": 1.0, + "content": "is proved in Section C.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 613, + 506, + 648 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 158, + 665 + ], + "score": 1.0, + "content": "Note that for", + "type": "text" + }, + { + "bbox": [ + 159, + 653, + 183, + 663 + ], + "score": 0.9, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 653, + 505, + 665 + ], + "score": 1.0, + "content": ", all GNNs are dealing with tensors of order 2 i.e. with the adjacency matrix of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "graph. However, the complexities of the various layers are quite different: for the message passing", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 674, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 506, + 687 + ], + "score": 1.0, + "content": "GNN, all computations are local (scaling with the maximum degree in the graph) and can be done in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 686, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 380, + 700 + ], + "score": 1.0, + "content": "parallel; for the linear layer, there are only 15 linear functions from", + "type": "text" + }, + { + "bbox": [ + 380, + 686, + 429, + 698 + ], + "score": 0.92, + "content": "\\mathbb { R } ^ { n ^ { 2 } } \\to \\mathbb { R } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 686, + 497, + 700 + ], + "score": 1.0, + "content": "for all values of", + "type": "text" + }, + { + "bbox": [ + 497, + 690, + 504, + 698 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 466, + 711 + ], + "score": 1.0, + "content": "(Maron et al., 2018); the folklore layer involves a (dense) matrix multiplication of shape", + "type": "text" + }, + { + "bbox": [ + 466, + 700, + 491, + 709 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 698, + 506, + 711 + ], + "score": 1.0, + "content": ". If", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "2-FGNN is the most complex architecture, we see that it has the best separating power among all", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 351, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 351, + 732 + ], + "score": 1.0, + "content": "architectures proposed so far dealing with tensors of order 2.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 653, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 506, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 123, + 115 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 124, + 103, + 146, + 114 + ], + "score": 0.89, + "content": "X , Y", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 102, + 328, + 115 + ], + "score": 1.0, + "content": "finite-dimensional spaces, let us denote by", + "type": "text" + }, + { + "bbox": [ + 328, + 102, + 413, + 115 + ], + "score": 0.91, + "content": "\\mathcal { C } _ { I } ( X , Y ) , \\mathcal { C } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 102, + 506, + 115 + ], + "score": 1.0, + "content": ", , the set of invariant,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 114, + 504, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 318, + 126 + ], + "score": 1.0, + "content": "respectively equivariant, continuous functions from", + "type": "text" + }, + { + "bbox": [ + 319, + 115, + 329, + 124 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 114, + 341, + 126 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 341, + 115, + 350, + 124 + ], + "score": 0.79, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 114, + 495, + 126 + ], + "score": 1.0, + "content": ". The closure of a class of function", + "type": "text" + }, + { + "bbox": [ + 495, + 114, + 504, + 124 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 249, + 138 + ], + "score": 1.0, + "content": "for the uniform norm is denoted by", + "type": "text" + }, + { + "bbox": [ + 249, + 124, + 258, + 135 + ], + "score": 0.84, + "content": "\\overline { { \\mathcal { F } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 124, + 506, + 138 + ], + "score": 1.0, + "content": ". Our result extend easily to graphs of varying sizes but this is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 134, + 244, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 244, + 149 + ], + "score": 1.0, + "content": "deferred to Section F.2 for clarity.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 152, + 504, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 327, + 166 + ], + "score": 1.0, + "content": "The theorem below states in particular that the class", + "type": "text" + }, + { + "bbox": [ + 328, + 153, + 335, + 163 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 152, + 505, + 166 + ], + "score": 1.0, + "content": "-FGNN can approximate any continuous", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 163, + 478, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 249, + 177 + ], + "score": 1.0, + "content": "function that is less separating than", + "type": "text" + }, + { + "bbox": [ + 250, + 164, + 280, + 176 + ], + "score": 0.91, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 163, + 478, + 177 + ], + "score": 1.0, + "content": "-WL in the invariant and in the equivariant cases.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 494, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 497, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 175, + 194 + ], + "score": 1.0, + "content": "Theorem 4. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 179, + 252, + 192 + ], + "score": 0.87, + "content": "K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 176, + 255, + 194 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 177, + 294, + 192 + ], + "score": 0.9, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 176, + 497, + 194 + ], + "score": 1.0, + "content": "be compact sets. For the invariant case, we have:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 193, + 423, + 241 + ], + "lines": [ + { + "bbox": [ + 189, + 193, + 423, + 241 + ], + "spans": [ + { + "bbox": [ + 189, + 193, + 423, + 241 + ], + "score": 0.79, + "content": "\\begin{array} { r l r } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( 2 \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\mathrm { - } L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\mathrm { - } F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\end{array}", + "type": "interline_equation", + "image_path": "16f667e67876d4cc8a9da81a88fff721b73209be35db8a2b53edba1076f7a893.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 189, + 193, + 423, + 209.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 189, + 209.0, + 423, + 225.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 189, + 225.0, + 423, + 241.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 245, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 246, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 246, + 254 + ], + "score": 1.0, + "content": "For the equivariant case, we have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 253, + 514, + 301 + ], + "lines": [ + { + "bbox": [ + 111, + 253, + 514, + 301 + ], + "spans": [ + { + "bbox": [ + 111, + 253, + 514, + 301 + ], + "score": 0.87, + "content": "\\begin{array} { r c l } { \\underline { { M G N N _ { E } } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K _ { d i s c r } , \\mathbb F ^ { n } ) : \\rho \\left( 2 ^ { - } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\underline { { \\cdot } L G N N _ { E } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k \\cdot L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} \\supset \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k \\cdot W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\cdot \\overline { { F G N N _ { E } } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( ( k + 1 ) \\cdot W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}", + "type": "interline_equation", + "image_path": "e1989546bf6a79c24c53ac33c959e55f6831fa898bb4bf80d0a859a679de07db.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 111, + 253, + 514, + 269.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 269.0, + 514, + 285.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 285.0, + 514, + 301.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 506, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 204, + 320 + ], + "score": 1.0, + "content": "In the invariant case for", + "type": "text" + }, + { + "bbox": [ + 205, + 308, + 230, + 318 + ], + "score": 0.87, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 307, + 266, + 320 + ], + "score": 1.0, + "content": ",we have", + "type": "text" + }, + { + "bbox": [ + 267, + 306, + 410, + 319 + ], + "score": 0.88, + "content": "\\mathbf { M G N N } _ { I } = \\overline { { 2 \\mathrm { - } \\mathbf { L G N N } _ { I } } } \\subsetneq \\overline { { 2 \\mathrm { - } \\mathbf { F G N N } _ { I } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "where the strictness of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 312, + 331 + ], + "score": 1.0, + "content": "the last inclusion comes from (3). In other words,", + "type": "text" + }, + { + "bbox": [ + 312, + 319, + 353, + 330 + ], + "score": 0.86, + "content": "{ \\mathrm { 2 - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "has a better power of approximation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "than the other architectures working with tensors of order 2. We already knew by Proposition 3 that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 147, + 352 + ], + "score": 0.85, + "content": "{ \\mathrm { 2 - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "is the best separating architecture among those studied in this paper, dealing with tensor of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 452, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 452, + 365 + ], + "score": 1.0, + "content": "order 2 and our theorem implies that this is also the case for the approximation power.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 398, + 381 + ], + "score": 1.0, + "content": "To clarify the meaning of these statements, we explain why the inclusions", + "type": "text" + }, + { + "bbox": [ + 398, + 369, + 412, + 379 + ], + "score": 0.64, + "content": "\\mathbf { \\check { \\Psi } } \\subseteq \\mathbf { \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "are actually straightfor-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 378, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 266, + 394 + ], + "score": 1.0, + "content": "ward. For concreteness, we focus on", + "type": "text" + }, + { + "bbox": [ + 266, + 380, + 503, + 392 + ], + "score": 0.89, + "content": "\\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } } \\ \\subseteq \\ \\left\\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } \\mathrm { W L } _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 378, + 506, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 130, + 406 + ], + "score": 1.0, + "content": "Take", + "type": "text" + }, + { + "bbox": [ + 131, + 391, + 199, + 403 + ], + "score": 0.9, + "content": "h ~ \\in ~ \\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 392, + 365, + 406 + ], + "score": 1.0, + "content": ", this means that there is a sequence", + "type": "text" + }, + { + "bbox": [ + 365, + 393, + 460, + 405 + ], + "score": 0.85, + "content": "\\mathbf { G } \\mathbf { N } \\mathbf { N } _ { j } \\in k \\mathrm { \\mathrm { - } } \\mathbf { F } \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 392, + 507, + 406 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 403, + 381, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 234, + 416 + ], + "score": 0.89, + "content": "\\mathrm { s u p } _ { G \\in K } \\| h ( G ) - \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { j } ( G ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 403, + 309, + 417 + ], + "score": 1.0, + "content": "goes to zero when", + "type": "text" + }, + { + "bbox": [ + 310, + 405, + 316, + 415 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 403, + 381, + 417 + ], + "score": 1.0, + "content": "goes to infinity.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 152, + 434 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 152, + 421, + 159, + 430 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 419, + 309, + 434 + ], + "score": 1.0, + "content": "is continuous and constant on each", + "type": "text" + }, + { + "bbox": [ + 309, + 420, + 364, + 432 + ], + "score": 0.83, + "content": "\\rho \\left( k { \\mathrm { - F G N N } } _ { I } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 419, + 461, + 434 + ], + "score": 1.0, + "content": "-class. Indeed, for any", + "type": "text" + }, + { + "bbox": [ + 462, + 420, + 505, + 432 + ], + "score": 0.91, + "content": "( G , G ^ { \\prime } ) \\in", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 431, + 503, + 445 + ], + "spans": [ + { + "bbox": [ + 107, + 432, + 162, + 443 + ], + "score": 0.74, + "content": "\\rho \\left( k { \\mathrm { - } } \\mathrm { F G N N } _ { I } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 431, + 167, + 445 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 167, + 432, + 278, + 443 + ], + "score": 0.86, + "content": "\\mathbf { G N N } _ { j } ( G ) = \\mathbf { G N N } _ { j } ( G ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 431, + 308, + 445 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 308, + 432, + 503, + 443 + ], + "score": 0.84, + "content": "h ( G ) = \\mathrm { l i m } _ { i } { \\bf G } { \\bf N } { \\bf N } _ { j } ( G ) = \\mathrm { l i m } _ { j } { \\bf G } { \\bf N } { \\bf N } _ { j } ( G ^ { \\prime } ) =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 441, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 107, + 443, + 131, + 454 + ], + "score": 0.9, + "content": "h ( G ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 441, + 204, + 456 + ], + "score": 1.0, + "content": ". Hence we have", + "type": "text" + }, + { + "bbox": [ + 204, + 443, + 295, + 455 + ], + "score": 0.83, + "content": "\\rho \\left( k { \\mathrm { - F G N N } } _ { I } \\right) \\subseteq \\rho \\left( h \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 441, + 363, + 456 + ], + "score": 1.0, + "content": "and by Prop. 3,", + "type": "text" + }, + { + "bbox": [ + 363, + 443, + 502, + 455 + ], + "score": 0.83, + "content": "\\rho \\left( k { \\mathrm { - } } \\mathrm { F G N N } _ { I } \\right) = \\rho \\left( ( k + 1 ) { \\mathrm { - } } \\mathrm { W L } _ { I } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 441, + 507, + 456 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 453, + 263, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 263, + 466 + ], + "score": 1.0, + "content": "allowing us to get the inclusion above.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 263, + 482 + ], + "score": 1.0, + "content": "On the contrary, the reverse inclusions", + "type": "text" + }, + { + "bbox": [ + 263, + 471, + 281, + 481 + ], + "score": 0.56, + "content": "\\cdot \\bigcirc ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 470, + 506, + 482 + ], + "score": 1.0, + "content": "are much more intricate but they are also the most valu-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 270, + 495 + ], + "score": 1.0, + "content": "able. For instance, consider the inclusion", + "type": "text" + }, + { + "bbox": [ + 270, + 482, + 503, + 495 + ], + "score": 0.86, + "content": "\\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } } \\supset \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } \\mathrm { W L } _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 482, + 506, + 495 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 247, + 506 + ], + "score": 1.0, + "content": "If one wishes to learn a function", + "type": "text" + }, + { + "bbox": [ + 247, + 494, + 307, + 505 + ], + "score": 0.93, + "content": "h \\in { \\mathcal { C } } _ { I } ( K , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 493, + 330, + 506 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 331, + 495, + 372, + 505 + ], + "score": 0.85, + "content": "k { \\mathrm { - } } \\mathrm { F G N N } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 493, + 506, + 506 + ], + "score": 1.0, + "content": ", this function must at least be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 222, + 518 + ], + "score": 1.0, + "content": "approximable by the class of", + "type": "text" + }, + { + "bbox": [ + 222, + 505, + 263, + 515 + ], + "score": 0.76, + "content": "k { \\mathrm { - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 504, + 428, + 518 + ], + "score": 1.0, + "content": ". Our theorem precisely guarantees that if", + "type": "text" + }, + { + "bbox": [ + 428, + 505, + 435, + 514 + ], + "score": 0.86, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "is less separating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 514, + 310, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 124, + 529 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 515, + 154, + 527 + ], + "score": 0.88, + "content": "k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 514, + 263, + 529 + ], + "score": 1.0, + "content": ", it can be approximated by", + "type": "text" + }, + { + "bbox": [ + 264, + 515, + 305, + 527 + ], + "score": 0.86, + "content": "k { \\mathrm { - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 514, + 310, + 529 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 528, + 434, + 549 + ], + "lines": [ + { + "bbox": [ + 174, + 528, + 434, + 549 + ], + "spans": [ + { + "bbox": [ + 174, + 528, + 434, + 549 + ], + "score": 0.86, + "content": "\\forall \\epsilon > 0 , \\exists \\mathbf { G } \\mathbf { N } \\mathbf { N } \\in k \\mathbf { \\mathrm { \\mathrm { - } } } \\mathbf { F } \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { I } , \\mathbf { \\Pi } _ { G \\in K } \\| h ( G ) - \\mathbf { G } \\mathbf { N } \\mathbf { N } ( G ) \\| \\leq \\epsilon .", + "type": "interline_equation", + "image_path": "ff04799ea56c8034a8553f9bbd7340bb36bf1df903d57bf77f028e4df5c1159f.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 174, + 528, + 434, + 549 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "For this, we show a much more general version of the famous Stone-Weierstrass theorem (see Section", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "D) which relates the separating power with the approximation power. Following the elegant idea", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "of Maehara & Hoang (2019), we augment the input space to transform vector-valued equivariant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 581, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 506, + 597 + ], + "score": 1.0, + "content": "functions into scalar invariant maps. Then, we apply a fine-grained approximation theorem from", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "Timofte (2005).We also provide specialized versions of our abstract theorem in Section 5, which can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 477, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 477, + 618 + ], + "score": 1.0, + "content": "be easily used to determine the approximation capabilities of any deep learning architecture.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "Our theorem has also implications for universality results like Maron et al. (2019b); Keriven & Peyré", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 406, + 645 + ], + "score": 1.0, + "content": "(2019). A class of GNN is said to be universal if its closure on a compact set", + "type": "text" + }, + { + "bbox": [ + 406, + 633, + 417, + 643 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 632, + 467, + 645 + ], + "score": 1.0, + "content": "is the whole", + "type": "text" + }, + { + "bbox": [ + 467, + 633, + 505, + 645 + ], + "score": 0.93, + "content": "\\mathcal { C } _ { I } ( K , \\mathbb { F } )", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 120, + 655 + ], + "score": 1.0, + "content": "(or", + "type": "text" + }, + { + "bbox": [ + 121, + 644, + 168, + 656 + ], + "score": 0.91, + "content": "\\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { n } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 643, + 314, + 655 + ], + "score": 1.0, + "content": "). In particular, Thm. 4 implies that", + "type": "text" + }, + { + "bbox": [ + 314, + 645, + 321, + 654 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 643, + 372, + 655 + ], + "score": 1.0, + "content": "-LGNN and", + "type": "text" + }, + { + "bbox": [ + 372, + 645, + 379, + 654 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 643, + 478, + 655 + ], + "score": 1.0, + "content": "-FGNN are universal as", + "type": "text" + }, + { + "bbox": [ + 478, + 646, + 485, + 654 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 643, + 506, + 655 + ], + "score": 1.0, + "content": "-WL", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 289, + 667 + ], + "score": 1.0, + "content": "distinguishes non-isomorphic graphs of size", + "type": "text" + }, + { + "bbox": [ + 289, + 657, + 297, + 665 + ], + "score": 0.62, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". This recovers a result of Ravanbakhsh (2020) for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "LGNN. Moreover, we can leverage the extensive literature on the WL tests to give more subtle√", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 255, + 689 + ], + "score": 1.0, + "content": "results. For instance, Cai et al. (1989,", + "type": "text" + }, + { + "bbox": [ + 255, + 677, + 276, + 688 + ], + "score": 0.64, + "content": "\\ S 8 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 676, + 392, + 689 + ], + "score": 1.0, + "content": "show that, for planar graphs,", + "type": "text" + }, + { + "bbox": [ + 393, + 676, + 423, + 689 + ], + "score": 0.91, + "content": "O ( { \\sqrt { n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "-WL can distinguish", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 260, + 700 + ], + "score": 1.0, + "content": "non-isomoprhic instances. Therefore,", + "type": "text" + }, + { + "bbox": [ + 260, + 688, + 291, + 700 + ], + "score": 0.92, + "content": "O ( \\sqrt { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 687, + 334, + 700 + ], + "score": 1.0, + "content": "-LGNN or", + "type": "text" + }, + { + "bbox": [ + 335, + 687, + 365, + 700 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { O } } ( { \\sqrt { n } } ) ", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "-FGNN achieve universality in the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "particular, yet common, case of planar graphs. On a more practical side, Fürer (2010, Thm. 4.5)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "shows that the spectrum of a graph is less separating than 3-WL so that functions of the spectrum can", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 282, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 282, + 732 + ], + "score": 1.0, + "content": "actually be well approximated by 2-FGNN.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 82, + 294, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 295, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 295, + 95 + ], + "score": 1.0, + "content": "4.3 APPROXIMATION RESULTS FOR GNNS", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 506, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 123, + 115 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 124, + 103, + 146, + 114 + ], + "score": 0.89, + "content": "X , Y", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 102, + 328, + 115 + ], + "score": 1.0, + "content": "finite-dimensional spaces, let us denote by", + "type": "text" + }, + { + "bbox": [ + 328, + 102, + 413, + 115 + ], + "score": 0.91, + "content": "\\mathcal { C } _ { I } ( X , Y ) , \\mathcal { C } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 102, + 506, + 115 + ], + "score": 1.0, + "content": ", , the set of invariant,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 114, + 504, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 318, + 126 + ], + "score": 1.0, + "content": "respectively equivariant, continuous functions from", + "type": "text" + }, + { + "bbox": [ + 319, + 115, + 329, + 124 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 114, + 341, + 126 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 341, + 115, + 350, + 124 + ], + "score": 0.79, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 114, + 495, + 126 + ], + "score": 1.0, + "content": ". The closure of a class of function", + "type": "text" + }, + { + "bbox": [ + 495, + 114, + 504, + 124 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 249, + 138 + ], + "score": 1.0, + "content": "for the uniform norm is denoted by", + "type": "text" + }, + { + "bbox": [ + 249, + 124, + 258, + 135 + ], + "score": 0.84, + "content": "\\overline { { \\mathcal { F } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 124, + 506, + 138 + ], + "score": 1.0, + "content": ". Our result extend easily to graphs of varying sizes but this is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 134, + 244, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 244, + 149 + ], + "score": 1.0, + "content": "deferred to Section F.2 for clarity.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 102, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 152, + 504, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 327, + 166 + ], + "score": 1.0, + "content": "The theorem below states in particular that the class", + "type": "text" + }, + { + "bbox": [ + 328, + 153, + 335, + 163 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 152, + 505, + 166 + ], + "score": 1.0, + "content": "-FGNN can approximate any continuous", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 163, + 478, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 249, + 177 + ], + "score": 1.0, + "content": "function that is less separating than", + "type": "text" + }, + { + "bbox": [ + 250, + 164, + 280, + 176 + ], + "score": 0.91, + "content": "( k + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 163, + 478, + 177 + ], + "score": 1.0, + "content": "-WL in the invariant and in the equivariant cases.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 152, + 505, + 177 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 494, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 497, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 175, + 194 + ], + "score": 1.0, + "content": "Theorem 4. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 179, + 252, + 192 + ], + "score": 0.87, + "content": "K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 176, + 255, + 194 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 177, + 294, + 192 + ], + "score": 0.9, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 176, + 497, + 194 + ], + "score": 1.0, + "content": "be compact sets. For the invariant case, we have:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 176, + 497, + 194 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 193, + 423, + 241 + ], + "lines": [ + { + "bbox": [ + 189, + 193, + 423, + 241 + ], + "spans": [ + { + "bbox": [ + 189, + 193, + 423, + 241 + ], + "score": 0.79, + "content": "\\begin{array} { r l r } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( 2 \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\mathrm { - } L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\mathrm { - } F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\end{array}", + "type": "interline_equation", + "image_path": "16f667e67876d4cc8a9da81a88fff721b73209be35db8a2b53edba1076f7a893.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 189, + 193, + 423, + 209.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 189, + 209.0, + 423, + 225.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 189, + 225.0, + 423, + 241.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 245, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 246, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 246, + 254 + ], + "score": 1.0, + "content": "For the equivariant case, we have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 241, + 246, + 254 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 253, + 514, + 301 + ], + "lines": [ + { + "bbox": [ + 111, + 253, + 514, + 301 + ], + "spans": [ + { + "bbox": [ + 111, + 253, + 514, + 301 + ], + "score": 0.87, + "content": "\\begin{array} { r c l } { \\underline { { M G N N _ { E } } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K _ { d i s c r } , \\mathbb F ^ { n } ) : \\rho \\left( 2 ^ { - } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\underline { { \\cdot } L G N N _ { E } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k \\cdot L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} \\supset \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( k \\cdot W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\cdot \\overline { { F G N N _ { E } } } } & { = } & { \\{ f \\in { \\mathcal C } _ { E } ( K , \\mathbb F ^ { n } ) : \\rho \\left( ( k + 1 ) \\cdot W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}", + "type": "interline_equation", + "image_path": "e1989546bf6a79c24c53ac33c959e55f6831fa898bb4bf80d0a859a679de07db.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 111, + 253, + 514, + 269.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 269.0, + 514, + 285.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 285.0, + 514, + 301.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 506, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 204, + 320 + ], + "score": 1.0, + "content": "In the invariant case for", + "type": "text" + }, + { + "bbox": [ + 205, + 308, + 230, + 318 + ], + "score": 0.87, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 307, + 266, + 320 + ], + "score": 1.0, + "content": ",we have", + "type": "text" + }, + { + "bbox": [ + 267, + 306, + 410, + 319 + ], + "score": 0.88, + "content": "\\mathbf { M G N N } _ { I } = \\overline { { 2 \\mathrm { - } \\mathbf { L G N N } _ { I } } } \\subsetneq \\overline { { 2 \\mathrm { - } \\mathbf { F G N N } _ { I } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "where the strictness of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 312, + 331 + ], + "score": 1.0, + "content": "the last inclusion comes from (3). In other words,", + "type": "text" + }, + { + "bbox": [ + 312, + 319, + 353, + 330 + ], + "score": 0.86, + "content": "{ \\mathrm { 2 - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "has a better power of approximation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "than the other architectures working with tensors of order 2. We already knew by Proposition 3 that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 147, + 352 + ], + "score": 0.85, + "content": "{ \\mathrm { 2 - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "is the best separating architecture among those studied in this paper, dealing with tensor of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 452, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 452, + 365 + ], + "score": 1.0, + "content": "order 2 and our theorem implies that this is also the case for the approximation power.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 306, + 506, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 398, + 381 + ], + "score": 1.0, + "content": "To clarify the meaning of these statements, we explain why the inclusions", + "type": "text" + }, + { + "bbox": [ + 398, + 369, + 412, + 379 + ], + "score": 0.64, + "content": "\\mathbf { \\check { \\Psi } } \\subseteq \\mathbf { \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "are actually straightfor-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 378, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 266, + 394 + ], + "score": 1.0, + "content": "ward. For concreteness, we focus on", + "type": "text" + }, + { + "bbox": [ + 266, + 380, + 503, + 392 + ], + "score": 0.89, + "content": "\\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } } \\ \\subseteq \\ \\left\\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } \\mathrm { W L } _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 378, + 506, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 130, + 406 + ], + "score": 1.0, + "content": "Take", + "type": "text" + }, + { + "bbox": [ + 131, + 391, + 199, + 403 + ], + "score": 0.9, + "content": "h ~ \\in ~ \\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 392, + 365, + 406 + ], + "score": 1.0, + "content": ", this means that there is a sequence", + "type": "text" + }, + { + "bbox": [ + 365, + 393, + 460, + 405 + ], + "score": 0.85, + "content": "\\mathbf { G } \\mathbf { N } \\mathbf { N } _ { j } \\in k \\mathrm { \\mathrm { - } } \\mathbf { F } \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 392, + 507, + 406 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 403, + 381, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 234, + 416 + ], + "score": 0.89, + "content": "\\mathrm { s u p } _ { G \\in K } \\| h ( G ) - \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { j } ( G ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 403, + 309, + 417 + ], + "score": 1.0, + "content": "goes to zero when", + "type": "text" + }, + { + "bbox": [ + 310, + 405, + 316, + 415 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 403, + 381, + 417 + ], + "score": 1.0, + "content": "goes to infinity.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 367, + 507, + 417 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 152, + 434 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 152, + 421, + 159, + 430 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 419, + 309, + 434 + ], + "score": 1.0, + "content": "is continuous and constant on each", + "type": "text" + }, + { + "bbox": [ + 309, + 420, + 364, + 432 + ], + "score": 0.83, + "content": "\\rho \\left( k { \\mathrm { - F G N N } } _ { I } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 419, + 461, + 434 + ], + "score": 1.0, + "content": "-class. Indeed, for any", + "type": "text" + }, + { + "bbox": [ + 462, + 420, + 505, + 432 + ], + "score": 0.91, + "content": "( G , G ^ { \\prime } ) \\in", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 431, + 503, + 445 + ], + "spans": [ + { + "bbox": [ + 107, + 432, + 162, + 443 + ], + "score": 0.74, + "content": "\\rho \\left( k { \\mathrm { - } } \\mathrm { F G N N } _ { I } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 431, + 167, + 445 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 167, + 432, + 278, + 443 + ], + "score": 0.86, + "content": "\\mathbf { G N N } _ { j } ( G ) = \\mathbf { G N N } _ { j } ( G ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 431, + 308, + 445 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 308, + 432, + 503, + 443 + ], + "score": 0.84, + "content": "h ( G ) = \\mathrm { l i m } _ { i } { \\bf G } { \\bf N } { \\bf N } _ { j } ( G ) = \\mathrm { l i m } _ { j } { \\bf G } { \\bf N } { \\bf N } _ { j } ( G ^ { \\prime } ) =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 441, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 107, + 443, + 131, + 454 + ], + "score": 0.9, + "content": "h ( G ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 441, + 204, + 456 + ], + "score": 1.0, + "content": ". Hence we have", + "type": "text" + }, + { + "bbox": [ + 204, + 443, + 295, + 455 + ], + "score": 0.83, + "content": "\\rho \\left( k { \\mathrm { - F G N N } } _ { I } \\right) \\subseteq \\rho \\left( h \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 441, + 363, + 456 + ], + "score": 1.0, + "content": "and by Prop. 3,", + "type": "text" + }, + { + "bbox": [ + 363, + 443, + 502, + 455 + ], + "score": 0.83, + "content": "\\rho \\left( k { \\mathrm { - } } \\mathrm { F G N N } _ { I } \\right) = \\rho \\left( ( k + 1 ) { \\mathrm { - } } \\mathrm { W L } _ { I } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 441, + 507, + 456 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 453, + 263, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 263, + 466 + ], + "score": 1.0, + "content": "allowing us to get the inclusion above.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 419, + 507, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 263, + 482 + ], + "score": 1.0, + "content": "On the contrary, the reverse inclusions", + "type": "text" + }, + { + "bbox": [ + 263, + 471, + 281, + 481 + ], + "score": 0.56, + "content": "\\cdot \\bigcirc ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 470, + 506, + 482 + ], + "score": 1.0, + "content": "are much more intricate but they are also the most valu-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 270, + 495 + ], + "score": 1.0, + "content": "able. For instance, consider the inclusion", + "type": "text" + }, + { + "bbox": [ + 270, + 482, + 503, + 495 + ], + "score": 0.86, + "content": "\\overline { { k \\mathrm { - } \\mathrm { F G N N } _ { I } } } \\supset \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( ( k + 1 ) \\mathrm { - } \\mathrm { W L } _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 482, + 506, + 495 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 247, + 506 + ], + "score": 1.0, + "content": "If one wishes to learn a function", + "type": "text" + }, + { + "bbox": [ + 247, + 494, + 307, + 505 + ], + "score": 0.93, + "content": "h \\in { \\mathcal { C } } _ { I } ( K , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 493, + 330, + 506 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 331, + 495, + 372, + 505 + ], + "score": 0.85, + "content": "k { \\mathrm { - } } \\mathrm { F G N N } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 493, + 506, + 506 + ], + "score": 1.0, + "content": ", this function must at least be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 222, + 518 + ], + "score": 1.0, + "content": "approximable by the class of", + "type": "text" + }, + { + "bbox": [ + 222, + 505, + 263, + 515 + ], + "score": 0.76, + "content": "k { \\mathrm { - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 504, + 428, + 518 + ], + "score": 1.0, + "content": ". Our theorem precisely guarantees that if", + "type": "text" + }, + { + "bbox": [ + 428, + 505, + 435, + 514 + ], + "score": 0.86, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "is less separating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 514, + 310, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 124, + 529 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 515, + 154, + 527 + ], + "score": 0.88, + "content": "k { \\mathrm { - } } \\mathbf { W } \\mathbf { L } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 514, + 263, + 529 + ], + "score": 1.0, + "content": ", it can be approximated by", + "type": "text" + }, + { + "bbox": [ + 264, + 515, + 305, + 527 + ], + "score": 0.86, + "content": "k { \\mathrm { - F G N N } } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 514, + 310, + 529 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 470, + 506, + 529 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 528, + 434, + 549 + ], + "lines": [ + { + "bbox": [ + 174, + 528, + 434, + 549 + ], + "spans": [ + { + "bbox": [ + 174, + 528, + 434, + 549 + ], + "score": 0.86, + "content": "\\forall \\epsilon > 0 , \\exists \\mathbf { G } \\mathbf { N } \\mathbf { N } \\in k \\mathbf { \\mathrm { \\mathrm { - } } } \\mathbf { F } \\mathbf { G } \\mathbf { N } \\mathbf { N } _ { I } , \\mathbf { \\Pi } _ { G \\in K } \\| h ( G ) - \\mathbf { G } \\mathbf { N } \\mathbf { N } ( G ) \\| \\leq \\epsilon .", + "type": "interline_equation", + "image_path": "ff04799ea56c8034a8553f9bbd7340bb36bf1df903d57bf77f028e4df5c1159f.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 174, + 528, + 434, + 549 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "For this, we show a much more general version of the famous Stone-Weierstrass theorem (see Section", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "D) which relates the separating power with the approximation power. Following the elegant idea", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "of Maehara & Hoang (2019), we augment the input space to transform vector-valued equivariant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 581, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 506, + 597 + ], + "score": 1.0, + "content": "functions into scalar invariant maps. Then, we apply a fine-grained approximation theorem from", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "Timofte (2005).We also provide specialized versions of our abstract theorem in Section 5, which can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 477, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 477, + 618 + ], + "score": 1.0, + "content": "be easily used to determine the approximation capabilities of any deep learning architecture.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 549, + 506, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "Our theorem has also implications for universality results like Maron et al. (2019b); Keriven & Peyré", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 406, + 645 + ], + "score": 1.0, + "content": "(2019). A class of GNN is said to be universal if its closure on a compact set", + "type": "text" + }, + { + "bbox": [ + 406, + 633, + 417, + 643 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 632, + 467, + 645 + ], + "score": 1.0, + "content": "is the whole", + "type": "text" + }, + { + "bbox": [ + 467, + 633, + 505, + 645 + ], + "score": 0.93, + "content": "\\mathcal { C } _ { I } ( K , \\mathbb { F } )", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 120, + 655 + ], + "score": 1.0, + "content": "(or", + "type": "text" + }, + { + "bbox": [ + 121, + 644, + 168, + 656 + ], + "score": 0.91, + "content": "\\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { n } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 643, + 314, + 655 + ], + "score": 1.0, + "content": "). In particular, Thm. 4 implies that", + "type": "text" + }, + { + "bbox": [ + 314, + 645, + 321, + 654 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 643, + 372, + 655 + ], + "score": 1.0, + "content": "-LGNN and", + "type": "text" + }, + { + "bbox": [ + 372, + 645, + 379, + 654 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 643, + 478, + 655 + ], + "score": 1.0, + "content": "-FGNN are universal as", + "type": "text" + }, + { + "bbox": [ + 478, + 646, + 485, + 654 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 643, + 506, + 655 + ], + "score": 1.0, + "content": "-WL", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 289, + 667 + ], + "score": 1.0, + "content": "distinguishes non-isomorphic graphs of size", + "type": "text" + }, + { + "bbox": [ + 289, + 657, + 297, + 665 + ], + "score": 0.62, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". This recovers a result of Ravanbakhsh (2020) for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "LGNN. Moreover, we can leverage the extensive literature on the WL tests to give more subtle√", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 255, + 689 + ], + "score": 1.0, + "content": "results. For instance, Cai et al. (1989,", + "type": "text" + }, + { + "bbox": [ + 255, + 677, + 276, + 688 + ], + "score": 0.64, + "content": "\\ S 8 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 676, + 392, + 689 + ], + "score": 1.0, + "content": "show that, for planar graphs,", + "type": "text" + }, + { + "bbox": [ + 393, + 676, + 423, + 689 + ], + "score": 0.91, + "content": "O ( { \\sqrt { n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "-WL can distinguish", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 260, + 700 + ], + "score": 1.0, + "content": "non-isomoprhic instances. Therefore,", + "type": "text" + }, + { + "bbox": [ + 260, + 688, + 291, + 700 + ], + "score": 0.92, + "content": "O ( \\sqrt { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 687, + 334, + 700 + ], + "score": 1.0, + "content": "-LGNN or", + "type": "text" + }, + { + "bbox": [ + 335, + 687, + 365, + 700 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { O } } ( { \\sqrt { n } } ) ", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "-FGNN achieve universality in the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "particular, yet common, case of planar graphs. On a more practical side, Fürer (2010, Thm. 4.5)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "shows that the spectrum of a graph is less separating than 3-WL so that functions of the spectrum can", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 282, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 282, + 732 + ], + "score": 1.0, + "content": "actually be well approximated by 2-FGNN.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 622, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 270, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 270, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 270, + 95 + ], + "score": 1.0, + "content": "5 EXPRESSIVENESS OF GNNS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 506, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "We now state the general theorems which are our main tools in proving our approximation guarantees", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "for GNNs. Theirs proofs are deferred to Section D.9 which contains our generalization of the Stone-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 478, + 140 + ], + "score": 1.0, + "content": "Weierstrass theorem with symmetries. 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If", + "type": "text" + }, + { + "bbox": [ + 255, + 150, + 264, + 160 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 149, + 451, + 163 + ], + "score": 1.0, + "content": "is a finite group acting on some compact set", + "type": "text" + }, + { + "bbox": [ + 451, + 150, + 462, + 160 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "and some", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 477, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 178, + 174 + ], + "score": 1.0, + "content": "topological space", + "type": "text" + }, + { + "bbox": [ + 178, + 162, + 187, + 171 + ], + "score": 0.82, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 161, + 477, + 174 + ], + "score": 1.0, + "content": ", we define the sets of equivariant and invariant continuous functions by,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 176, + 446, + 206 + ], + "lines": [ + { + "bbox": [ + 165, + 176, + 446, + 206 + ], + "spans": [ + { + "bbox": [ + 165, + 176, + 446, + 206 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathcal { C } _ { E } ( X , Y ) = \\{ f \\in \\mathcal { C } ( X , Y ) : \\forall x \\in X , \\forall g \\in G , f ( g \\cdot x ) = g \\cdot f ( x ) \\} } \\\\ & { \\mathcal { C } _ { I } ( X , Y ) = \\{ f \\in \\mathcal { C } ( X , Y ) : \\forall x \\in X , \\forall g \\in G , f ( g \\cdot x ) = f ( x ) \\} } \\end{array}", + "type": "interline_equation", + "image_path": "2ac7d4cb09dc16302a9ddf1068259e35c1e0efc44f1ac821016dc286cbf91083.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 165, + 176, + 446, + 186.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 165, + 186.0, + 446, + 196.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 165, + 196.0, + 446, + 206.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 209, + 369, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 370, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 370, + 224 + ], + "score": 1.0, + "content": "Note that these definitions extend Definition 1 to a general group.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 105, + 224, + 504, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 176, + 238 + ], + "score": 1.0, + "content": "Theorem 5. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 225, + 186, + 234 + ], + "score": 0.75, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 222, + 272, + 238 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 272, + 225, + 305, + 234 + ], + "score": 0.9, + "content": "\\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 222, + 474, + 238 + ], + "score": 1.0, + "content": "be some finite-dimensional vector space,", + "type": "text" + }, + { + "bbox": [ + 475, + 225, + 484, + 235 + ], + "score": 0.67, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 222, + 506, + 238 + ], + "score": 1.0, + "content": "be a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 268, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 254, + 248 + ], + "score": 1.0, + "content": "finite group acting (continuously) on", + "type": "text" + }, + { + "bbox": [ + 255, + 236, + 264, + 245 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 236, + 268, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 105, + 252, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 122, + 266 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 251, + 216, + 265 + ], + "score": 0.92, + "content": "\\textstyle { \\mathcal { F } } _ { 0 } \\subseteq \\bigcup _ { h = 1 } ^ { \\infty } { \\mathcal { C } } _ { I } ( X , \\mathbb { R } ^ { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 250, + 506, + 266 + ], + "score": 1.0, + "content": "be a non-empty set of invariant functions, stable by concatenation, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 145, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 145, + 276 + ], + "score": 1.0, + "content": "consider,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 278, + 457, + 294 + ], + "lines": [ + { + "bbox": [ + 152, + 278, + 457, + 294 + ], + "spans": [ + { + "bbox": [ + 152, + 278, + 457, + 294 + ], + "score": 0.88, + "content": "\\mathcal { F } = \\{ m \\circ f : f \\in \\mathcal { F } _ { 0 } \\cap \\mathcal { C } ( X , \\mathbb { R } ^ { h } ) , m : \\mathbb { R } ^ { h } \\to \\mathbb { F } M L P , h \\geq 1 \\} \\subseteq \\mathcal { C } ( X , \\mathbb { F } ) .", + "type": "interline_equation", + "image_path": "8d34581ed3a7abea357c426b47d31508df5c1f5fb95a6affd0874c735b33d5a8.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 152, + 278, + 457, + 294 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 206, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 207, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 185, + 318 + ], + "score": 1.0, + "content": "Then the closure of", + "type": "text" + }, + { + "bbox": [ + 185, + 305, + 194, + 314 + ], + "score": 0.8, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 302, + 207, + 318 + ], + "score": 1.0, + "content": "is,", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 320, + 387, + 334 + ], + "lines": [ + { + "bbox": [ + 224, + 320, + 387, + 334 + ], + "spans": [ + { + "bbox": [ + 224, + 320, + 387, + 334 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\{ f \\in \\mathcal { C } _ { I } ( X , \\mathbb { F } ) : \\rho \\left( \\mathcal { F } _ { 0 } \\right) \\subseteq \\rho \\left( f \\right) \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "08c9a6f5fed1d948af94274f1e35938867f3f5bbdfdea98e54744a5fd142b60a.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 224, + 320, + 387, + 334 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 269, + 358 + ], + "score": 1.0, + "content": "We can apply Theorem 5 to the class of", + "type": "text" + }, + { + "bbox": [ + 270, + 345, + 276, + 355 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "-ary relational pooling GNN introduced in Murphy et al.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 325, + 369 + ], + "score": 1.0, + "content": "(2019a). As a result, we get that this class of invariant", + "type": "text" + }, + { + "bbox": [ + 325, + 357, + 331, + 366 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "-RP GNN can approximate any continuous", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 143, + 380 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 143, + 367, + 150, + 379 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 366, + 173, + 380 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 174, + 367, + 272, + 379 + ], + "score": 0.88, + "content": "\\rho ( k - \\mathrm { R P G N N } ) \\subseteq \\rho ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 366, + 412, + 380 + ], + "score": 1.0, + "content": "but to the best of our knowledge,", + "type": "text" + }, + { + "bbox": [ + 412, + 367, + 478, + 379 + ], + "score": 0.85, + "content": "\\rho ( k - \\mathrm { R P G N N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "is not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 173, + 391 + ], + "score": 1.0, + "content": "known and only", + "type": "text" + }, + { + "bbox": [ + 174, + 379, + 304, + 390 + ], + "score": 0.87, + "content": "\\rho ( k - \\mathrm { R P G N N } ) \\subset \\rho ( 2 - W L _ { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "is proved in Murphy et al. (2019a). We now state", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 390, + 287, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 287, + 402 + ], + "score": 1.0, + "content": "our general theorem for the equivariant case:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 105, + 403, + 504, + 426 + ], + "lines": [ + { + "bbox": [ + 104, + 401, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 178, + 418 + ], + "score": 1.0, + "content": "Theorem 6. Let", + "type": "text" + }, + { + "bbox": [ + 178, + 404, + 188, + 414 + ], + "score": 0.78, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 401, + 279, + 418 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 280, + 404, + 316, + 414 + ], + "score": 0.91, + "content": "\\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 401, + 337, + 418 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 338, + 404, + 375, + 415 + ], + "score": 0.91, + "content": "G = S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 401, + 506, + 418 + ], + "score": 1.0, + "content": "the permutation group, acting", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 413, + 358, + 427 + ], + "spans": [ + { + "bbox": [ + 104, + 413, + 178, + 427 + ], + "score": 1.0, + "content": "(continuously) on", + "type": "text" + }, + { + "bbox": [ + 178, + 415, + 189, + 424 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 413, + 247, + 427 + ], + "score": 1.0, + "content": "and acting on", + "type": "text" + }, + { + "bbox": [ + 247, + 415, + 259, + 424 + ], + "score": 0.86, + "content": "\\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 413, + 289, + 427 + ], + "score": 1.0, + "content": "by, for", + "type": "text" + }, + { + "bbox": [ + 289, + 415, + 320, + 426 + ], + "score": 0.78, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 413, + 324, + 427 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 324, + 415, + 354, + 425 + ], + "score": 0.67, + "content": "x \\in \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 413, + 358, + 427 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 430, + 379, + 446 + ], + "lines": [ + { + "bbox": [ + 230, + 430, + 379, + 446 + ], + "spans": [ + { + "bbox": [ + 230, + 430, + 379, + 446 + ], + "score": 0.89, + "content": "\\forall i \\in \\{ 1 , \\ldots , p \\} , ( \\sigma \\cdot x ) _ { i } = x _ { \\sigma ^ { - 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From a practical perspective, we demonstrate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "their improved performance on the QAP with a significant gap in performances compared to other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 636, + 156, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 156, + 648 + ], + "score": 1.0, + "content": "approaches.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 663, + 218, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 662, + 220, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 220, + 678 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "This work was supported in part by the French government under management of Agence Nationale", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "de la Recherche as part of the “Investissements d’avenir” program, reference ANR19-P3IA-0001", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "(PRAIRIE 3IA Institute). M.L. thanks Google for Google Cloud Platform research credits and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 253, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 253, + 732 + ], + "score": 1.0, + "content": "NVIDIA for a NVIDIA GPU Grant.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 506, + 151 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 264, + 166 + ], + "score": 1.0, + "content": "To remedy this weakness, we consider", + "type": "text" + }, + { + "bbox": [ + 264, + 154, + 308, + 165 + ], + "score": 0.85, + "content": "2 { \\mathrm { - } } \\mathrm { F G N N } _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 154, + 506, + 166 + ], + "score": 1.0, + "content": ". 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Given", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 172, + 151 + ], + "score": 1.0, + "content": "a pair of graphs", + "type": "text" + }, + { + "bbox": [ + 172, + 138, + 202, + 149 + ], + "score": 0.9, + "content": "G _ { 1 } , G _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 137, + 223, + 151 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 224, + 140, + 231, + 147 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 137, + 382, + 151 + ], + "score": 1.0, + "content": "nodes each, we consider the siamese", + "type": "text" + }, + { + "bbox": [ + 382, + 138, + 426, + 149 + ], + "score": 0.85, + "content": "2 { \\mathrm { - } } \\mathrm { F G N N } _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "encoder producing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 146, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 159, + 162 + ], + "score": 1.0, + "content": "embeddings", + "type": "text" + }, + { + "bbox": [ + 159, + 148, + 226, + 160 + ], + "score": 0.91, + "content": "E _ { 1 } , E _ { 2 } \\in \\mathbb { R } ^ { n \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 146, + 506, + 162 + ], + "score": 1.0, + "content": ". 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We used", + "type": "text" + }, + { + "bbox": [ + 406, + 171, + 449, + 181 + ], + "score": 0.86, + "content": "2 { \\mathrm { - } } \\mathrm { F G N N } _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 169, + 507, + 184 + ], + "score": 1.0, + "content": "with 2 layers,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "each MLP having depth 3 and hidden states of size 64. We trained for 25 epochs with batches of size", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "32, a learning rate of 1e-4 and Adam optimizer. 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A mask is created at initialization and is used to ensure that the operations", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 516, + 424, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 424, + 527 + ], + "score": 1.0, + "content": "on the full tensor translates to valid operations on each of the individual tensor.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 507, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 507, + 545 + ], + "score": 1.0, + "content": "We implemented this functionnality as a class MaskedTensors. Thanks to the newest improve-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 544, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 555 + ], + "score": 1.0, + "content": "ments of PyTorch (Paszke et al., 2019), MaskedTensors act as a subclass of fundamental Tensor", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "class. Thus they almost seamlessly integrate into standard PyTorch code. 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A batch of", + "type": "text" + }, + { + "bbox": [ + 498, + 483, + 504, + 493 + ], + "score": 0.64, + "content": "b", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 171, + 507 + ], + "score": 1.0, + "content": "tensors of sizes", + "type": "text" + }, + { + "bbox": [ + 171, + 495, + 297, + 505 + ], + "score": 0.91, + "content": "n _ { 1 } \\times n _ { 1 } , n _ { 2 } \\times n _ { 2 } , . . . , n _ { b } \\times n _ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 493, + 402, + 507 + ], + "score": 1.0, + "content": "is represented as a tensor", + "type": "text" + }, + { + "bbox": [ + 402, + 494, + 477, + 505 + ], + "score": 0.91, + "content": "b \\times n _ { m a x } \\times n _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 201, + 517 + ], + "score": 0.89, + "content": "n _ { m a x } = \\mathrm { m a x } _ { i = 1 , \\dots , b } n _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 505, + 506, + 517 + ], + "score": 1.0, + "content": ". 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W L } } ^ { t } } \\\\ { \\mathrm { W L } _ { E } = c _ { \\mathrm { W L } } } \\end{array}", + "type": "interline_equation", + "image_path": "3b816ab81504c732cdf8bf1ebdb7e2465674e8272ffca96f0366c65db8448bc1.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 277, + 528, + 333, + 542.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 277, + 542.5, + 333, + 557.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 583, + 219, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 220, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 220, + 595 + ], + "score": 1.0, + "content": "B.2 ISOMORPHISM TYPE", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 506, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "The initialization of the 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\\} } \\\\ & { \\operatorname { W L } _ { I } ( G ) = \\{ \\{ c _ { \\mathrm { W L } } ( G ) _ { s } : s \\in V \\} \\} } \\end{array}", + "type": "interline_equation", + "image_path": "c5cd809e0529fba5b151e9f721aba410e0a9a6119aaef283e45f040e1fd7f1f7.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 445, + 375, + 461.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 234, + 461.5, + 375, + 478.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 504, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 325, + 515 + ], + "score": 1.0, + "content": "Equivariant tests For the vertex coloring algorithm,", + "type": "text" + }, + { + "bbox": [ + 326, + 503, + 342, + 513 + ], + "score": 0.7, + "content": "c _ { \\mathrm { W L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 499, + 505, + 515 + ], + "score": 1.0, + "content": 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We have, for", + "type": "text" + }, + { + "bbox": [ + 223, + 229, + 248, + 240 + ], + "score": 0.9, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 227, + 251, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 244, + 464, + 288 + ], + "lines": [ + { + "bbox": [ + 146, + 244, + 464, + 288 + ], + "spans": [ + { + "bbox": [ + 146, + 244, + 464, + 288 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) = \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { E } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "183e7aa5f995506534ecae6f390331db71f497c37ba575f01945cb8f0553a7eb.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 146, + 244, + 464, + 258.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 146, + 258.6666666666667, + 464, + 273.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 146, + 273.33333333333337, + 464, + 288.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 298, + 262, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 263, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 263, + 311 + ], + "score": 1.0, + "content": "C.1 MUTLI-LINEAR PERCEPTRONS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "In the following we will use extensively multi-linear perceptrons (MLP) and their universality", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 329, + 478, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 478, + 343 + ], + "score": 1.0, + "content": "properties. Yet, for the sake of simplicity, we do not define precisely what we mean by MLP.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 290, + 360 + ], + "score": 1.0, + "content": "Given two finite-dimensional feature spaces", + "type": "text" + }, + { + "bbox": [ + 290, + 347, + 302, + 358 + ], + "score": 0.88, + "content": "\\mathbb { F } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 346, + 321, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 347, + 334, + 358 + ], + "score": 0.87, + "content": "\\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 346, + 506, + 360 + ], + "score": 1.0, + "content": ", we only assume we are given a class of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 205, + 371 + ], + "score": 1.0, + "content": "(continuous) MLP from", + "type": "text" + }, + { + "bbox": [ + 205, + 358, + 218, + 369 + ], + "score": 0.88, + "content": "\\mathbb { F } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 357, + 229, + 371 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 229, + 358, + 241, + 369 + ], + "score": 0.89, + "content": "\\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 357, + 393, + 371 + ], + "score": 1.0, + "content": "which is large enough to be dense in", + "type": "text" + }, + { + "bbox": [ + 393, + 357, + 433, + 370 + ], + "score": 0.93, + "content": "\\mathcal { C } ( \\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 357, + 506, + 371 + ], + "score": 1.0, + "content": ". See for instance", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 369, + 447, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 447, + 381 + ], + "score": 1.0, + "content": "Hornik et al. (1989); Cybenko (1989) for precise conditions for MLP to be universal.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 277, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 279, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 279, + 406 + ], + "score": 1.0, + "content": "C.2 AUGMENTED SEPARATING POWER", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 383, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 384, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 384, + 428 + ], + "score": 1.0, + "content": "To factor the proof, we introduce another notion of separating power.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 105, + 428, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 179, + 441 + ], + "score": 1.0, + "content": "Definition 7. For", + "type": "text" + }, + { + "bbox": [ + 179, + 428, + 218, + 439 + ], + "score": 0.9, + "content": "n , m \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 427, + 255, + 441 + ], + "score": 1.0, + "content": "fixed, let", + "type": "text" + }, + { + "bbox": [ + 256, + 429, + 266, + 438 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 427, + 301, + 441 + ], + "score": 1.0, + "content": "be a set,", + "type": "text" + }, + { + "bbox": [ + 301, + 429, + 309, + 438 + ], + "score": 0.75, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 427, + 392, + 441 + ], + "score": 1.0, + "content": "be a some space and", + "type": "text" + }, + { + "bbox": [ + 392, + 429, + 402, + 438 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "be a set of functions from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 437, + 371, + 453 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 117, + 449 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 437, + 128, + 453 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 128, + 439, + 175, + 449 + ], + "score": 0.92, + "content": "Y = \\mathbb { F } ^ { n \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 437, + 348, + 453 + ], + "score": 1.0, + "content": ". Then, the augmented separating power of", + "type": "text" + }, + { + "bbox": [ + 348, + 440, + 357, + 449 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 437, + 371, + 453 + ], + "score": 1.0, + "content": "is,", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 455, + 472, + 469 + ], + "lines": [ + { + "bbox": [ + 138, + 455, + 472, + 469 + ], + "spans": [ + { + "bbox": [ + 138, + 455, + 472, + 469 + ], + "score": 0.84, + "content": "\\rho _ { n , m } ^ { a u g m } ( \\mathcal { F } ) = \\rho ( \\{ ( x , i , j ) \\in X \\times \\{ 1 , \\ldots , n \\} \\times \\{ 1 , \\ldots , m \\} \\mapsto f ( x ) _ { i , j } : f \\in \\mathcal { F } \\} ) .", + "type": "interline_equation", + "image_path": "dc132d7ebe9d1be4d2a37dbc6f8762cabd671a3f2051247cd2f6667208f3f029.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 138, + 455, + 472, + 469 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 277, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 275, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 164, + 488 + ], + "score": 1.0, + "content": "Explicitly, for", + "type": "text" + }, + { + "bbox": [ + 165, + 473, + 275, + 486 + ], + "score": 0.85, + "content": "x , y \\in X , i , j \\in \\{ 1 , \\dots , n \\}", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 489, + 435, + 504 + ], + "lines": [ + { + "bbox": [ + 175, + 489, + 435, + 504 + ], + "spans": [ + { + "bbox": [ + 175, + 489, + 435, + 504 + ], + "score": 0.87, + "content": "( x , i , j , y , k , l ) \\in \\rho _ { n , m } ^ { a u g m } ( \\mathcal { F } ) \\iff \\forall f \\in \\mathcal { F } , f ( x ) _ { i , j } = f ( y ) _ { k , l } .", + "type": "interline_equation", + "image_path": "1cdacb6f2b7921bb2ef5bb501108c44647c647baa522694a35ed6ed1d6b02d7d.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 175, + 489, + 435, + 504 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 172, + 524 + ], + "score": 1.0, + "content": "Note that when", + "type": "text" + }, + { + "bbox": [ + 172, + 513, + 223, + 523 + ], + "score": 0.91, + "content": "n = m = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 512, + 505, + 524 + ], + "score": 1.0, + "content": ", the augmented separating power is exactly the same as the original", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 520, + 507, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 235, + 538 + ], + "score": 1.0, + "content": "separating power, so we identify", + "type": "text" + }, + { + "bbox": [ + 235, + 523, + 254, + 536 + ], + "score": 0.9, + "content": "\\rho \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 522, + 275, + 538 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 275, + 523, + 314, + 537 + ], + "score": 0.91, + "content": "\\rho _ { 1 , 1 } ^ { a u g m } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 520, + 385, + 539 + ], + "score": 1.0, + "content": ". We also identify", + "type": "text" + }, + { + "bbox": [ + 386, + 520, + 466, + 538 + ], + "score": 1.0, + "content": "ρaugmn,1 (.) with ρaug1,n", + "type": "text" + }, + { + "bbox": [ + 445, + 523, + 484, + 537 + ], + "score": 0.92, + "content": "\\rho _ { 1 , n } ^ { a u g m } \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 521, + 507, + 537 + ], + "score": 1.0, + "content": ", that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 533, + 207, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 533, + 162, + 550 + ], + "score": 1.0, + "content": "we denote by", + "type": "text" + }, + { + "bbox": [ + 162, + 536, + 201, + 548 + ], + "score": 0.9, + "content": "\\rho _ { n } ^ { a u g m } \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 533, + 207, + 550 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 424, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 425, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 425, + 567 + ], + "score": 1.0, + "content": "First, it is easy to see that this notion is more precise than the separating power.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 349, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 350, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 163, + 580 + ], + "score": 1.0, + "content": "Lemma 8. If", + "type": "text" + }, + { + "bbox": [ + 163, + 567, + 172, + 577 + ], + "score": 0.75, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 565, + 191, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 568, + 199, + 578 + ], + "score": 0.81, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 565, + 299, + 580 + ], + "score": 1.0, + "content": "are set of functions from", + "type": "text" + }, + { + "bbox": [ + 299, + 567, + 309, + 577 + ], + "score": 0.71, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 565, + 320, + 580 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 321, + 567, + 346, + 577 + ], + "score": 0.86, + "content": "\\mathbb { F } ^ { n \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 565, + 350, + 580 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 180, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 181, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 181, + 613 + ], + "score": 1.0, + "content": "and, in particular,", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 581, + 402, + 626 + ], + "lines": [ + { + "bbox": [ + 210, + 581, + 402, + 626 + ], + "spans": [ + { + "bbox": [ + 210, + 581, + 402, + 626 + ], + "score": 0.68, + "content": "\\begin{array} { r l } & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { G } \\right) \\implies \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( \\mathcal { G } \\right) , } \\\\ & { } \\\\ & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) = \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { G } \\right) \\implies \\rho \\left( \\mathcal { F } \\right) = \\rho \\left( \\mathcal { G } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "d19483b756a4da63b35c5fcf0eec05346d4812069f8bb44c276223eb15cca0fa.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 209, + 581, + 402, + 596.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 209, + 596.0, + 402, + 611.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 209, + 611.0, + 402, + 626.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 504, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "The interest in this notion is justified by the following lemma, which shows that this notion behaves", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 642, + 300, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 300, + 655 + ], + "score": 1.0, + "content": "well under composition with “reduction layers”.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 656, + 507, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 171, + 669 + ], + "score": 1.0, + "content": "Lemma 9. For", + "type": "text" + }, + { + "bbox": [ + 171, + 657, + 211, + 668 + ], + "score": 0.89, + "content": "n , m \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 655, + 249, + 669 + ], + "score": 1.0, + "content": "fixed, let", + "type": "text" + }, + { + "bbox": [ + 250, + 657, + 260, + 666 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 655, + 396, + 669 + ], + "score": 1.0, + "content": "be a compact topological space,", + "type": "text" + }, + { + "bbox": [ + 396, + 656, + 444, + 667 + ], + "score": 0.9, + "content": "Y = \\mathbb { F } ^ { n \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 655, + 447, + 669 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 448, + 657, + 456, + 666 + ], + "score": 0.45, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "some finite-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 664, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 189, + 681 + ], + "score": 1.0, + "content": "dimensional space,", + "type": "text" + }, + { + "bbox": [ + 189, + 668, + 249, + 680 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 664, + 273, + 681 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 273, + 668, + 356, + 678 + ], + "score": 0.86, + "content": "\\tau : X Z ^ { n \\times m } a", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 664, + 460, + 681 + ], + "score": 1.0, + "content": "function for some space", + "type": "text" + }, + { + "bbox": [ + 460, + 668, + 469, + 677 + ], + "score": 0.74, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 664, + 506, + 681 + ], + "score": 1.0, + "content": ". Define", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 678, + 182, + 694 + ], + "spans": [ + { + "bbox": [ + 107, + 679, + 167, + 693 + ], + "score": 0.91, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , \\mathbb { F } ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 678, + 182, + 694 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 696, + 514, + 736 + ], + "lines": [ + { + "bbox": [ + 111, + 696, + 514, + 736 + ], + "spans": [ + { + "bbox": [ + 111, + 696, + 514, + 736 + ], + "score": 0.9, + "content": "\\tilde { \\boldsymbol { x } } = \\{ x \\in X \\longmapsto ( \\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { 1 , j } ) , \\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { 2 , j } ) , \\ldots , \\sum _ { i = 1 } ^ { m } h ( f ( x ) _ { n , j } ) ) : h : \\mathbb { F } \\to \\mathbb { F } M L P , \\ f \\in \\mathcal { R } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F }", + "type": "interline_equation", + "image_path": "2a49405e4b16eb54031c89415bc020cc899ede13d403c3b341e7fb106d13a33b.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 111, + 696, + 514, + 709.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 111, + 709.3333333333334, + 514, + 722.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 111, + 722.6666666666667, + 514, + 736.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 108 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "Equivariant tests We now introduce the equivariant version of these tests. Many extensions are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 434, + 110 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 304, + 110 + ], + "score": 1.0, + "content": "possible, we chose this one for its simplicity. For", + "type": "text" + }, + { + "bbox": [ + 304, + 95, + 327, + 107 + ], + "score": 0.83, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 92, + 331, + 110 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 331, + 93, + 367, + 108 + ], + "score": 0.87, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 92, + 403, + 110 + ], + "score": 1.0, + "content": "a graph,", + "type": "text" + }, + { + "bbox": [ + 403, + 95, + 428, + 106 + ], + "score": 0.89, + "content": "i \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 92, + 434, + 110 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 81, + 505, + 110 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 111, + 412, + 178 + ], + "lines": [ + { + "bbox": [ + 199, + 111, + 412, + 178 + ], + "spans": [ + { + "bbox": [ + 199, + 111, + 412, + 178 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { k \\mathrm { - } { \\mathbb { W } } \\mathrm { L } _ { E } ^ { t } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { W L } } ^ { t } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} } \\\\ & { k \\mathrm { - } { \\mathbb { W } } \\mathrm { L } _ { E } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { W L } } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} } \\\\ & { k \\mathrm { - } { \\mathrm { F W L } } _ { E } ^ { t } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { F W L } } ^ { t } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} } \\\\ & { k \\mathrm { - } { \\mathrm { F W L } } _ { E } ( G ) _ { i } = \\{ \\{ c _ { k - \\mathrm { F W L } } ( G ) _ { s } : s \\in V ^ { k } , s _ { 1 } = i \\} \\} \\ . } \\end{array}", + "type": "interline_equation", + "image_path": "f506e10903aa99cc433674367668e1e58f4e2a943fa7f0a817d60fd0939fac20.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 111, + 412, + 127.75 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 199, + 127.75, + 412, + 144.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 199, + 144.5, + 412, + 161.25 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 199, + 161.25, + 412, + 178.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 189, + 279, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 281, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 281, + 204 + ], + "score": 1.0, + "content": "C SEPARATING POWER OF GNN", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 249, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 250, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 250, + 228 + ], + "score": 1.0, + "content": "The goal of this section is to prove,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 213, + 250, + 228 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 229, + 251, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 251, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 222, + 243 + ], + "score": 1.0, + "content": "Proposition 3. We have, for", + "type": "text" + }, + { + "bbox": [ + 223, + 229, + 248, + 240 + ], + "score": 0.9, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 227, + 251, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 227, + 251, + 243 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 244, + 464, + 288 + ], + "lines": [ + { + "bbox": [ + 146, + 244, + 464, + 288 + ], + "spans": [ + { + "bbox": [ + 146, + 244, + 464, + 288 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) = \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( ( k + 1 ) \\ – W L _ { E } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "183e7aa5f995506534ecae6f390331db71f497c37ba575f01945cb8f0553a7eb.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 146, + 244, + 464, + 258.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 146, + 258.6666666666667, + 464, + 273.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 146, + 273.33333333333337, + 464, + 288.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 298, + 262, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 263, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 263, + 311 + ], + "score": 1.0, + "content": "C.1 MUTLI-LINEAR PERCEPTRONS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "In the following we will use extensively multi-linear perceptrons (MLP) and their universality", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 329, + 478, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 478, + 343 + ], + "score": 1.0, + "content": "properties. Yet, for the sake of simplicity, we do not define precisely what we mean by MLP.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 318, + 505, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 290, + 360 + ], + "score": 1.0, + "content": "Given two finite-dimensional feature spaces", + "type": "text" + }, + { + "bbox": [ + 290, + 347, + 302, + 358 + ], + "score": 0.88, + "content": "\\mathbb { F } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 346, + 321, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 347, + 334, + 358 + ], + "score": 0.87, + "content": "\\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 346, + 506, + 360 + ], + "score": 1.0, + "content": ", we only assume we are given a class of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 205, + 371 + ], + "score": 1.0, + "content": "(continuous) MLP from", + "type": "text" + }, + { + "bbox": [ + 205, + 358, + 218, + 369 + ], + "score": 0.88, + "content": "\\mathbb { F } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 357, + 229, + 371 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 229, + 358, + 241, + 369 + ], + "score": 0.89, + "content": "\\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 357, + 393, + 371 + ], + "score": 1.0, + "content": "which is large enough to be dense in", + "type": "text" + }, + { + "bbox": [ + 393, + 357, + 433, + 370 + ], + "score": 0.93, + "content": "\\mathcal { C } ( \\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 357, + 506, + 371 + ], + "score": 1.0, + "content": ". See for instance", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 369, + 447, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 447, + 381 + ], + "score": 1.0, + "content": "Hornik et al. (1989); Cybenko (1989) for precise conditions for MLP to be universal.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 346, + 506, + 381 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 277, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 279, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 279, + 406 + ], + "score": 1.0, + "content": "C.2 AUGMENTED SEPARATING POWER", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 383, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 384, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 384, + 428 + ], + "score": 1.0, + "content": "To factor the proof, we introduce another notion of separating power.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 411, + 384, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 428, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 179, + 441 + ], + "score": 1.0, + "content": "Definition 7. For", + "type": "text" + }, + { + "bbox": [ + 179, + 428, + 218, + 439 + ], + "score": 0.9, + "content": "n , m \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 427, + 255, + 441 + ], + "score": 1.0, + "content": "fixed, let", + "type": "text" + }, + { + "bbox": [ + 256, + 429, + 266, + 438 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 427, + 301, + 441 + ], + "score": 1.0, + "content": "be a set,", + "type": "text" + }, + { + "bbox": [ + 301, + 429, + 309, + 438 + ], + "score": 0.75, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 427, + 392, + 441 + ], + "score": 1.0, + "content": "be a some space and", + "type": "text" + }, + { + "bbox": [ + 392, + 429, + 402, + 438 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "be a set of functions from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 437, + 371, + 453 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 117, + 449 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 437, + 128, + 453 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 128, + 439, + 175, + 449 + ], + "score": 0.92, + "content": "Y = \\mathbb { F } ^ { n \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 437, + 348, + 453 + ], + "score": 1.0, + "content": ". Then, the augmented separating power of", + "type": "text" + }, + { + "bbox": [ + 348, + 440, + 357, + 449 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 437, + 371, + 453 + ], + "score": 1.0, + "content": "is,", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 427, + 505, + 453 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 455, + 472, + 469 + ], + "lines": [ + { + "bbox": [ + 138, + 455, + 472, + 469 + ], + "spans": [ + { + "bbox": [ + 138, + 455, + 472, + 469 + ], + "score": 0.84, + "content": "\\rho _ { n , m } ^ { a u g m } ( \\mathcal { F } ) = \\rho ( \\{ ( x , i , j ) \\in X \\times \\{ 1 , \\ldots , n \\} \\times \\{ 1 , \\ldots , m \\} \\mapsto f ( x ) _ { i , j } : f \\in \\mathcal { F } \\} ) .", + "type": "interline_equation", + "image_path": "dc132d7ebe9d1be4d2a37dbc6f8762cabd671a3f2051247cd2f6667208f3f029.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 138, + 455, + 472, + 469 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 277, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 275, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 164, + 488 + ], + "score": 1.0, + "content": "Explicitly, for", + "type": "text" + }, + { + "bbox": [ + 165, + 473, + 275, + 486 + ], + "score": 0.85, + "content": "x , y \\in X , i , j \\in \\{ 1 , \\dots , n \\}", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 472, + 275, + 488 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 489, + 435, + 504 + ], + "lines": [ + { + "bbox": [ + 175, + 489, + 435, + 504 + ], + "spans": [ + { + "bbox": [ + 175, + 489, + 435, + 504 + ], + "score": 0.87, + "content": "( x , i , j , y , k , l ) \\in \\rho _ { n , m } ^ { a u g m } ( \\mathcal { F } ) \\iff \\forall f \\in \\mathcal { F } , f ( x ) _ { i , j } = f ( y ) _ { k , l } .", + "type": "interline_equation", + "image_path": "1cdacb6f2b7921bb2ef5bb501108c44647c647baa522694a35ed6ed1d6b02d7d.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 175, + 489, + 435, + 504 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 172, + 524 + ], + "score": 1.0, + "content": "Note that when", + "type": "text" + }, + { + "bbox": [ + 172, + 513, + 223, + 523 + ], + "score": 0.91, + "content": "n = m = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 512, + 505, + 524 + ], + "score": 1.0, + "content": ", the augmented separating power is exactly the same as the original", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 520, + 507, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 235, + 538 + ], + "score": 1.0, + "content": "separating power, so we identify", + "type": "text" + }, + { + "bbox": [ + 235, + 523, + 254, + 536 + ], + "score": 0.9, + "content": "\\rho \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 522, + 275, + 538 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 275, + 523, + 314, + 537 + ], + "score": 0.91, + "content": "\\rho _ { 1 , 1 } ^ { a u g m } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 520, + 385, + 539 + ], + "score": 1.0, + "content": ". We also identify", + "type": "text" + }, + { + "bbox": [ + 386, + 520, + 466, + 538 + ], + "score": 1.0, + "content": "ρaugmn,1 (.) with ρaug1,n", + "type": "text" + }, + { + "bbox": [ + 445, + 523, + 484, + 537 + ], + "score": 0.92, + "content": "\\rho _ { 1 , n } ^ { a u g m } \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 521, + 507, + 537 + ], + "score": 1.0, + "content": ", that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 533, + 207, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 533, + 162, + 550 + ], + "score": 1.0, + "content": "we denote by", + "type": "text" + }, + { + "bbox": [ + 162, + 536, + 201, + 548 + ], + "score": 0.9, + "content": "\\rho _ { n } ^ { a u g m } \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 533, + 207, + 550 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 512, + 507, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 424, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 425, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 425, + 567 + ], + "score": 1.0, + "content": "First, it is easy to see that this notion is more precise than the separating power.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 550, + 425, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 349, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 350, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 163, + 580 + ], + "score": 1.0, + "content": "Lemma 8. If", + "type": "text" + }, + { + "bbox": [ + 163, + 567, + 172, + 577 + ], + "score": 0.75, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 565, + 191, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 568, + 199, + 578 + ], + "score": 0.81, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 565, + 299, + 580 + ], + "score": 1.0, + "content": "are set of functions from", + "type": "text" + }, + { + "bbox": [ + 299, + 567, + 309, + 577 + ], + "score": 0.71, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 565, + 320, + 580 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 321, + 567, + 346, + 577 + ], + "score": 0.86, + "content": "\\mathbb { F } ^ { n \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 565, + 350, + 580 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 565, + 350, + 580 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 180, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 181, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 181, + 613 + ], + "score": 1.0, + "content": "and, in particular,", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 600, + 181, + 613 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 581, + 402, + 626 + ], + "lines": [ + { + "bbox": [ + 210, + 581, + 402, + 626 + ], + "spans": [ + { + "bbox": [ + 210, + 581, + 402, + 626 + ], + "score": 0.68, + "content": "\\begin{array} { r l } & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { G } \\right) \\implies \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( \\mathcal { G } \\right) , } \\\\ & { } \\\\ & { \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) = \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { G } \\right) \\implies \\rho \\left( \\mathcal { F } \\right) = \\rho \\left( \\mathcal { G } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "d19483b756a4da63b35c5fcf0eec05346d4812069f8bb44c276223eb15cca0fa.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 209, + 581, + 402, + 596.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 209, + 596.0, + 402, + 611.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 209, + 611.0, + 402, + 626.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 504, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "The interest in this notion is justified by the following lemma, which shows that this notion behaves", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 642, + 300, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 300, + 655 + ], + "score": 1.0, + "content": "well under composition with “reduction layers”.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 631, + 505, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 656, + 507, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 171, + 669 + ], + "score": 1.0, + "content": "Lemma 9. For", + "type": "text" + }, + { + "bbox": [ + 171, + 657, + 211, + 668 + ], + "score": 0.89, + "content": "n , m \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 655, + 249, + 669 + ], + "score": 1.0, + "content": "fixed, let", + "type": "text" + }, + { + "bbox": [ + 250, + 657, + 260, + 666 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 655, + 396, + 669 + ], + "score": 1.0, + "content": "be a compact topological space,", + "type": "text" + }, + { + "bbox": [ + 396, + 656, + 444, + 667 + ], + "score": 0.9, + "content": "Y = \\mathbb { F } ^ { n \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 655, + 447, + 669 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 448, + 657, + 456, + 666 + ], + "score": 0.45, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "some finite-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 664, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 189, + 681 + ], + "score": 1.0, + "content": "dimensional space,", + "type": "text" + }, + { + "bbox": [ + 189, + 668, + 249, + 680 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 664, + 273, + 681 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 273, + 668, + 356, + 678 + ], + "score": 0.86, + "content": "\\tau : X Z ^ { n \\times m } a", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 664, + 460, + 681 + ], + "score": 1.0, + "content": "function for some space", + "type": "text" + }, + { + "bbox": [ + 460, + 668, + 469, + 677 + ], + "score": 0.74, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 664, + 506, + 681 + ], + "score": 1.0, + "content": ". 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\\mathbb { F } M L P , \\ f \\in \\mathcal { R } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } \\mathrm { ~ a n d ~ } \\mathbb { F } \\to \\mathbb { F } 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195 + ], + "score": 1.0, + "content": "as a set of functions from", + "type": "text" + }, + { + "bbox": [ + 419, + 183, + 429, + 193 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 181, + 442, + 195 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 442, + 182, + 465, + 193 + ], + "score": 0.91, + "content": "\\mathbb { F } ^ { n \\times 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "to fit the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 194, + 277, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 277, + 207 + ], + "score": 1.0, + "content": "definition of augmented separating power.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 310, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 311, + 232 + ], + "spans": [ + { + "bbox": [ 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By definition of the augmented", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 138, + 357, + 503, + 381 + ], + "spans": [ + { + "bbox": [ + 138, + 357, + 306, + 381 + ], + "score": 1.0, + "content": "separating power, this means that, for any", + "type": "text" + }, + { + "bbox": [ + 307, + 362, + 369, + 374 + ], + "score": 0.87, + "content": "j \\in \\{ 1 , \\dots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 357, + 373, + 381 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 374, + 362, + 503, + 375 + ], + "score": 0.84, + "content": "( x , i , \\sigma ( j ) , y , k , j ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right)", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 139, + 370, + 510, + 392 + ], + "spans": [ + { + "bbox": [ + 139, + 370, + 277, + 392 + ], + "score": 1.0, + "content": "Hence, by assumption, for any", + "type": "text" + }, + { + "bbox": [ + 278, + 375, + 347, + 387 + ], + "score": 0.86, + "content": "j ~ \\in ~ \\{ 1 , \\dots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 370, + 353, + 392 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 353, + 375, + 487, + 387 + ], + "score": 0.87, + "content": "( x , i , \\sigma ( j ) , y , k , j ) \\ \\in \\ \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) .", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 370, + 510, + 392 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 385, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 142, + 387, + 229, + 399 + ], + "score": 0.91, + "content": "\\tau ( x ) _ { i , \\sigma ( j ) } = \\tau ( y ) _ { k , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 385, + 353, + 401 + ], + "score": 1.0, + "content": ". 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(2019a); Chen et al. (2020) and recall it below. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 621, + 508, + 639 + ], + "spans": [ + { + "bbox": [ + 103, + 621, + 250, + 639 + ], + "score": 1.0, + "content": "initial graph is a tensor of the form", + "type": "text" + }, + { + "bbox": [ + 250, + 623, + 286, + 637 + ], + "score": 0.93, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 621, + 308, + 639 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 308, + 624, + 355, + 636 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { 0 } = \\mathbb { R } ^ { e + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 621, + 436, + 639 + ], + "score": 1.0, + "content": "; the last channel of", + "type": "text" + }, + { + "bbox": [ + 436, + 624, + 469, + 637 + ], + "score": 0.92, + "content": "G _ { : , : , e + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 621, + 508, + 639 + ], + "score": 1.0, + "content": "encodes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 294, + 649 + ], + "score": 1.0, + "content": "the adjacency matrix of the graph and the first", + "type": "text" + }, + { + "bbox": [ + 294, + 638, + 300, + 646 + ], + "score": 0.7, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 635, + 339, + 649 + ], + "score": 1.0, + "content": "channels", + "type": "text" + }, + { + "bbox": [ + 339, + 636, + 368, + 648 + ], + "score": 0.91, + "content": "G _ { : , : , 1 : e }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "are zero outside the diagonal and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 643, + 509, + 664 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 160, + 662 + ], + "score": 0.91, + "content": "G _ { i , i , 1 : e } \\in \\mathbb { R } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 643, + 244, + 664 + ], + "score": 1.0, + "content": "is the color of vertex", + "type": "text" + }, + { + "bbox": [ + 244, + 650, + 273, + 661 + ], + "score": 0.91, + "content": "v _ { i } \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 643, + 338, + 664 + ], + "score": 1.0, + "content": ". We then define", + "type": "text" + }, + { + "bbox": [ + 339, + 647, + 404, + 662 + ], + "score": 0.93, + "content": "I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 643, + 426, + 664 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 426, + 647, + 493, + 661 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { 1 } = \\mathbb { R } ^ { k ^ { 2 } \\times ( e + 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 643, + 509, + 664 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 659, + 142, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 142, + 673 + ], + "score": 1.0, + "content": "follows:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 674, + 392, + 706 + ], + "lines": [ + { + "bbox": [ + 216, + 674, + 392, + 706 + ], + "spans": [ + { + "bbox": [ + 216, + 674, + 392, + 706 + ], + "score": 0.91, + "content": "\\begin{array} { r c l } { { \\displaystyle { \\cal I } ^ { k } ( G ) _ { { \\bf i } , r , s , w } } } & { { = } } & { { G _ { i _ { r } , i _ { s } , w } , w \\in [ e + 1 ] , } } \\\\ { { \\displaystyle { \\cal I } ^ { k } ( G ) _ { { \\bf i } , r , s , e + 2 } } } & { { = } } & { { { \\bf 1 } ( i _ { r } = i _ { s } ) , } } \\end{array}", + "type": "interline_equation", + "image_path": "c6183949be3a59426e1311b518bdaa184a3e6c351ed3da42868db7635a57e075.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 216, + 674, + 392, + 706 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 121, + 723 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 709, + 158, + 722 + ], + "score": 0.94, + "content": "\\mathbf { i } \\in [ n ] ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 707, + 178, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 709, + 219, + 721 + ], + "score": 0.91, + "content": "r , s \\in [ k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 707, + 505, + 723 + ], + "score": 1.0, + "content": ". 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We show the two inclusions independently.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 215, + 311, + 232 + ] + }, + { + "type": "text", + "bbox": [ + 123, + 240, + 218, + 252 + ], + "lines": [ + { + "bbox": [ + 122, + 240, + 219, + 253 + ], + "spans": [ + { + "bbox": [ + 122, + 240, + 126, + 253 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 126, + 241, + 138, + 252 + ], + "score": 0.59, + "content": "( \\subseteqright)", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 240, + 219, + 253 + ], + "score": 1.0, + "content": "We first show that,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 122, + 240, + 219, + 253 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 255, + 438, + 276 + ], + "lines": [ + { + "bbox": [ + 209, + 255, + 438, + 276 + ], + "spans": [ + { + "bbox": [ + 209, + 255, + 438, + 276 + ], + "score": 0.9, + "content": "\\rho _ { n , m } ^ { a u g m } \\left( \\mathcal { F } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\tau \\right) \\implies \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right) \\subseteq \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\tau } \\right)", + "type": "interline_equation", + "image_path": "f58392a527eb8b6babde2e263edda2f83030079b79b44fe5765c22c4cadbf292.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 209, + 255, + 438, + 276 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 280, + 446, + 299 + ], + "lines": [ + { + "bbox": [ + 138, + 276, + 444, + 303 + ], + "spans": [ + { + "bbox": [ + 138, + 276, + 164, + 303 + ], + "score": 1.0, + "content": "Take", + "type": "text" + }, + { + "bbox": [ + 164, + 280, + 266, + 300 + ], + "score": 0.93, + "content": "( x , i , y , k ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 276, + 369, + 303 + ], + "score": 1.0, + "content": ". This means that, for any", + "type": "text" + }, + { + "bbox": [ + 369, + 283, + 444, + 295 + ], + "score": 0.56, + "content": "h : \\mathbb { F } \\to \\mathbb { F } , f \\in { \\mathcal { F } } .", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 138, + 276, + 444, + 303 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 302, + 392, + 337 + ], + "lines": [ + { + "bbox": [ + 253, + 302, + 392, + 337 + ], + "spans": [ + { + "bbox": [ + 253, + 302, + 392, + 337 + ], + "score": 0.93, + "content": "\\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { i , j } ) = \\sum _ { j = 1 } ^ { m } h ( f ( y ) _ { k , j } ) .", + "type": "interline_equation", + "image_path": "8898a2df002ffb2001e28207ffd84613c291cc5c72dc148d87515036641c02f9.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 253, + 302, + 392, + 319.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 253, + 319.5, + 392, + 337.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 339, + 506, + 411 + ], + "lines": [ + { + "bbox": [ + 142, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 142, + 339, + 427, + 351 + ], + "score": 1.0, + "content": "By Lem. 31 and the universality of MLP, there exists a permutation", + "type": "text" + }, + { + "bbox": [ + 428, + 340, + 463, + 351 + ], + "score": 0.92, + "content": "\\sigma \\in S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 349, + 507, + 365 + ], + "spans": [ + { + "bbox": [ + 142, + 350, + 365, + 363 + ], + "score": 0.88, + "content": "( { \\dot { f } } ( x ) _ { i , \\sigma ( 1 ) } , \\dots , f ( x ) _ { i , \\sigma ( m ) } ) = ( f ( y ) _ { k , 1 } , \\dots , f ( y ) _ { k , m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 349, + 507, + 365 + ], + "score": 1.0, + "content": ". 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Hence,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 140, + 479, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 140, + 479, + 175, + 495 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 176, + 482, + 206, + 493 + ], + "score": 0.83, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 479, + 212, + 495 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 212, + 482, + 434, + 495 + ], + "score": 0.85, + "content": "( f ( x ) _ { i , \\sigma ( 1 ) } , \\ldots , f ( x ) _ { i , \\sigma ( m ) } ) = ( f ( y ) _ { k , 1 } , \\ldots , f ( y ) _ { k , m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 479, + 506, + 495 + ], + "score": 1.0, + "content": ", and so, for any", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 492, + 191, + 504 + ], + "spans": [ + { + "bbox": [ + 142, + 492, + 185, + 502 + ], + "score": 0.89, + "content": "h : \\mathbb { F } \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 492, + 191, + 504 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 138, + 452, + 510, + 504 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 501, + 392, + 535 + ], + "lines": [ + { + "bbox": [ + 253, + 501, + 392, + 535 + ], + "spans": [ + { + "bbox": [ + 253, + 501, + 392, + 535 + ], + "score": 0.94, + "content": "\\sum _ { j = 1 } ^ { m } h ( f ( x ) _ { i , j } ) = \\sum _ { j = 1 } ^ { m } h ( f ( y ) _ { k , j } ) .", + "type": "interline_equation", + "image_path": "b96465660ae5572919ebf8ce95b447e1d686503907b7ba9224a8d14e5c79f74e.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 253, + 501, + 392, + 518.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 253, + 518.0, + 392, + 535.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 537, + 402, + 557 + ], + "lines": [ + { + "bbox": [ + 138, + 531, + 405, + 560 + ], + "spans": [ + { + "bbox": [ + 138, + 531, + 186, + 560 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 187, + 537, + 288, + 557 + ], + "score": 0.94, + "content": "( x , i , y , k ) \\in \\rho _ { n , m } ^ { a u g m } \\left( \\widetilde { \\mathcal { F } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 531, + 405, + 560 + ], + "score": 1.0, + "content": ", which concludes the proof.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 138, + 531, + 405, + 560 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 233, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 234, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 234, + 604 + ], + "score": 1.0, + "content": "C.3 INITIALIZATION LAYER", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 611, + 506, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "We use the same initialization layer as Maron et al. (2019a); Chen et al. (2020) and recall it below. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 621, + 508, + 639 + ], + "spans": [ + { + "bbox": [ + 103, + 621, + 250, + 639 + ], + "score": 1.0, + "content": "initial graph is a tensor of the form", + "type": "text" + }, + { + "bbox": [ + 250, + 623, + 286, + 637 + ], + "score": 0.93, + "content": "G \\in \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 621, + 308, + 639 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 308, + 624, + 355, + 636 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { 0 } = \\mathbb { R } ^ { e + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 621, + 436, + 639 + ], + "score": 1.0, + "content": "; the last channel of", + "type": "text" + }, + { + "bbox": [ + 436, + 624, + 469, + 637 + ], + "score": 0.92, + "content": "G _ { : , : , e + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 621, + 508, + 639 + ], + "score": 1.0, + "content": "encodes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 294, + 649 + ], + "score": 1.0, + "content": "the adjacency matrix of the graph and the first", + "type": "text" + }, + { + "bbox": [ + 294, + 638, + 300, + 646 + ], + "score": 0.7, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 635, + 339, + 649 + ], + "score": 1.0, + "content": "channels", + "type": "text" + }, + { + "bbox": [ + 339, + 636, + 368, + 648 + ], + "score": 0.91, + "content": "G _ { : , : , 1 : e }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "are zero outside the diagonal and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 643, + 509, + 664 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 160, + 662 + ], + "score": 0.91, + "content": "G _ { i , i , 1 : e } \\in \\mathbb { R } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 643, + 244, + 664 + ], + "score": 1.0, + "content": "is the color of vertex", + "type": "text" + }, + { + "bbox": [ + 244, + 650, + 273, + 661 + ], + "score": 0.91, + "content": "v _ { i } \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 643, + 338, + 664 + ], + "score": 1.0, + "content": ". We then define", + "type": "text" + }, + { + "bbox": [ + 339, + 647, + 404, + 662 + ], + "score": 0.93, + "content": "I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 643, + 426, + 664 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 426, + 647, + 493, + 661 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { 1 } = \\mathbb { R } ^ { k ^ { 2 } \\times ( e + 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 643, + 509, + 664 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 659, + 142, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 142, + 673 + ], + "score": 1.0, + "content": "follows:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 103, + 611, + 509, + 673 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 674, + 392, + 706 + ], + "lines": [ + { + "bbox": [ + 216, + 674, + 392, + 706 + ], + "spans": [ + { + "bbox": [ + 216, + 674, + 392, + 706 + ], + "score": 0.91, + "content": "\\begin{array} { r c l } { { \\displaystyle { \\cal I } ^ { k } ( G ) _ { { \\bf i } , r , s , w } } } & { { = } } & { { G _ { i _ { r } , i _ { s } , w } , w \\in [ e + 1 ] , } } \\\\ { { \\displaystyle { \\cal I } ^ { k } ( G ) _ { { \\bf i } , r , s , e + 2 } } } & { { = } } & { { { \\bf 1 } ( i _ { r } = i _ { s } ) , } } \\end{array}", + "type": "interline_equation", + "image_path": "c6183949be3a59426e1311b518bdaa184a3e6c351ed3da42868db7635a57e075.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 216, + 674, + 392, + 706 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 121, + 723 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 709, + 158, + 722 + ], + "score": 0.94, + "content": "\\mathbf { i } \\in [ n ] ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 707, + 178, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 709, + 219, + 721 + ], + "score": 0.91, + "content": "r , s \\in [ k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 707, + 505, + 723 + ], + "score": 1.0, + "content": ". This linear equivariant layer has the same separating power as the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 317, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 317, + 732 + ], + "score": 1.0, + "content": "isormorphism type, which is defined in Section B.2.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 707, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 108 + ], + "lines": [ + { + "bbox": [ + 102, + 78, + 506, + 114 + ], + "spans": [ + { + "bbox": [ + 102, + 89, + 172, + 114 + ], + "score": 1.0, + "content": "exists I k : Fn20", + "type": "text" + }, + { + "bbox": [ + 105, + 78, + 293, + 96 + ], + "score": 1.0, + "content": "Lemma 10 ((Maron et al., 2019a, C.1)). For", + "type": "text" + }, + { + "bbox": [ + 131, + 93, + 196, + 108 + ], + "score": 0.94, + "content": "I ^ { k } : \\mathbb { F } _ { 0 } ^ { n ^ { 2 } } \\to \\mathbb { F } _ { 1 } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 86, + 243, + 111 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + }, + { + "bbox": [ + 293, + 83, + 320, + 94 + ], + "score": 0.86, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 78, + 323, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 324, + 82, + 354, + 94 + ], + "score": 0.75, + "content": "p _ { 0 } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 78, + 358, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 358, + 82, + 400, + 94 + ], + "score": 0.79, + "content": "\\mathbb { F } _ { 0 } = \\mathbb { R } ^ { p _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 78, + 404, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 405, + 80, + 477, + 93 + ], + "score": 0.91, + "content": "\\mathbb { F } _ { 1 } = \\mathbb { R } ^ { k ^ { 2 } \\times ( p _ { 0 } + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 78, + 506, + 96 + ], + "score": 1.0, + "content": ", there", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 112, + 368, + 130 + ], + "lines": [ + { + "bbox": [ + 242, + 112, + 368, + 130 + ], + "spans": [ + { + "bbox": [ + 242, + 112, + 368, + 130 + ], + "score": 0.91, + "content": "\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( I ^ { k } \\right) = \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( \\mathrm { i s o } \\right) .", + "type": "interline_equation", + "image_path": "67517ed0ab29cebaebac0e1affaaeac2c46c5ae6cf02b277b46d2a968f19d6fe.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 242, + 112, + 368, + 130 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 141, + 454, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 454, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 454, + 154 + ], + "score": 1.0, + "content": "C.4 KNOWN RESULTS ABOUT THE SEPARATING POWER OF SOME GNN CLASSES", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 104, + 162, + 505, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 161, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 175 + ], + "score": 1.0, + "content": "First, we need to define some classes of GNN from which both the invariant and equivaraint GNN we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 173, + 394, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 394, + 186 + ], + "score": 1.0, + "content": "considered are built. See Section 3 for details about the different layers.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 111, + 188, + 502, + 207 + ], + "spans": [ + { + "bbox": [ + 111, + 191, + 320, + 204 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbf { M G N N } _ { e m b } = \\left\\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } : F _ { t } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 188, + 414, + 207 + ], + "score": 1.0, + "content": "message passing layer,", + "type": "text" + }, + { + "bbox": [ + 414, + 191, + 502, + 203 + ], + "score": 0.86, + "content": "t = 1 , \\ldots , T , T \\geq 1 \\}", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 109, + 205, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 109, + 206, + 319, + 220 + ], + "score": 0.6, + "content": "\\mathfrak { k } \\mathrm { - } \\mathrm { L G N N } _ { e m b } = \\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 205, + 416, + 222 + ], + "score": 1.0, + "content": "linear equivariant layer,", + "type": "text" + }, + { + "bbox": [ + 416, + 207, + 504, + 220 + ], + "score": 0.86, + "content": "t = 1 , \\ldots , T , T \\geq 1 \\}", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 108, + 222, + 434, + 237 + ], + "spans": [ + { + "bbox": [ + 108, + 222, + 434, + 237 + ], + "score": 0.56, + "content": "\\begin{array} { r } { k \\mathrm { { \\mathrm { - } } F G N N } _ { e m b } = \\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } : \\mathbb { F } _ { t } ^ { n } \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { { F G L } } , t = 1 , \\dots , T , T \\geq 1 \\} . } \\end{array}", + "type": "inline_equation", + "image_path": "edb596394c3bb82a9156fb58bc325f815bbc622c7c670c2eaae9b48d8d42da53.jpg" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 363, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 364, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 364, + 262 + ], + "score": 1.0, + "content": "Then, the precise results from the literature can be rephrased as,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 440, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 441, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 157, + 276 + ], + "score": 1.0, + "content": "Lemma 11", + "type": "text" + }, + { + "bbox": [ + 158, + 264, + 171, + 273 + ], + "score": 0.59, + "content": "\\mathrm { { X u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 262, + 412, + 276 + ], + "score": 1.0, + "content": "et al. (2018),Maron et al. (2018),Maron et al. (2019a)). For", + "type": "text" + }, + { + "bbox": [ + 412, + 263, + 437, + 274 + ], + "score": 0.85, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 262, + 441, + 276 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 279, + 391, + 329 + ], + "lines": [ + { + "bbox": [ + 219, + 279, + 391, + 329 + ], + "spans": [ + { + "bbox": [ + 219, + 279, + 391, + 329 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\qquad \\rho _ { n } ^ { a u g m } \\left( M G N N _ { e m b } \\right) = \\rho _ { n } ^ { a u g m } \\left( c _ { W L } \\right) } \\\\ & { \\qquad \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k { - } L G N N _ { e m b } \\right) \\subseteq \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - W L } \\right) } \\\\ & { \\qquad \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k { - } F G N N _ { e m b } \\right) \\subseteq \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "af6b0e7dd7cf584ab93d501e25a15ebd8072a2aeb5dd15951c0a1c6dfe410db9.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 219, + 279, + 391, + 295.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 219, + 295.6666666666667, + 391, + 312.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 219, + 312.33333333333337, + 391, + 329.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 338, + 464, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 465, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 169, + 352 + ], + "score": 1.0, + "content": "(8) comes from", + "type": "text" + }, + { + "bbox": [ + 170, + 339, + 184, + 349 + ], + "score": 0.47, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 338, + 252, + 352 + ], + "score": 1.0, + "content": "et al. (2018, §A,", + "type": "text" + }, + { + "bbox": [ + 252, + 339, + 265, + 350 + ], + "score": 0.52, + "content": "\\ S \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 338, + 444, + 352 + ], + "score": 1.0, + "content": "), (9) and (10) from Maron et al. (2019a, §C,", + "type": "text" + }, + { + "bbox": [ + 444, + 339, + 458, + 350 + ], + "score": 0.29, + "content": "\\ S _ { \\mathbf { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 338, + 465, + 352 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 106, + 363, + 344, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 344, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 303, + 376 + ], + "score": 1.0, + "content": "C.5 BOUNDING THE SEPARATING POWER OF", + "type": "text" + }, + { + "bbox": [ + 303, + 365, + 310, + 374 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 363, + 344, + 376 + ], + "score": 1.0, + "content": "-FGNN", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 504, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 426, + 397 + ], + "score": 1.0, + "content": "We complete the results of the literature with a bound on the separating power of", + "type": "text" + }, + { + "bbox": [ + 426, + 385, + 433, + 394 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "-FGNN. Note that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 395, + 364, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 192, + 408 + ], + "score": 1.0, + "content": "the particular case of", + "type": "text" + }, + { + "bbox": [ + 192, + 396, + 217, + 406 + ], + "score": 0.93, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 395, + 364, + 408 + ], + "score": 1.0, + "content": "is already proven in Geerts (2020a).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 220, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 217, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 192, + 424 + ], + "score": 1.0, + "content": "Lemma 12. For any", + "type": "text" + }, + { + "bbox": [ + 192, + 411, + 217, + 421 + ], + "score": 0.9, + "content": "k \\geq 2", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 427, + 393, + 443 + ], + "lines": [ + { + "bbox": [ + 216, + 427, + 393, + 443 + ], + "spans": [ + { + "bbox": [ + 216, + 427, + 393, + 443 + ], + "score": 0.88, + "content": "\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k – F G N N _ { e m b } \\right) \\supset \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) ,", + "type": "interline_equation", + "image_path": "57725f634b162d617e57b58ba5d06ae5850c0e9dba7e2f2a5afb515e922db5c0.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 216, + 427, + 393, + 443 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 136, + 460 + ], + "lines": [ + { + "bbox": [ + 104, + 447, + 137, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 137, + 461 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 458, + 394, + 475 + ], + "lines": [ + { + "bbox": [ + 216, + 458, + 394, + 475 + ], + "spans": [ + { + "bbox": [ + 216, + 458, + 394, + 475 + ], + "score": 0.88, + "content": "\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k – F G N N _ { e m b } \\right) = \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) .", + "type": "interline_equation", + "image_path": "5e0f120406178fafd424376749342ce29aeafc44c49af35d3f60da26e48c4e2a.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 216, + 458, + 394, + 475 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 167, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 169, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 169, + 500 + ], + "score": 1.0, + "content": "Proof. Define,", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 502, + 454, + 518 + ], + "lines": [ + { + "bbox": [ + 156, + 502, + 454, + 518 + ], + "spans": [ + { + "bbox": [ + 156, + 502, + 454, + 518 + ], + "score": 0.9, + "content": "k \\mathrm { \\mathrm { - } F G N N } _ { e m b } ^ { T } = \\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { \\ F G L } , t = 1 , \\dots , T \\} ,", + "type": "interline_equation", + "image_path": "50fe17e1599d9711a3ac9ee55537127ef84363cd5694ca241e9747daa3fed6f7.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 156, + 502, + 454, + 518 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 522, + 492, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 489, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 260, + 537 + ], + "score": 1.0, + "content": "the set of functions defined by exactly", + "type": "text" + }, + { + "bbox": [ + 261, + 524, + 269, + 533 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 520, + 463, + 537 + ], + "score": 1.0, + "content": "FGL layers. 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Take", + "type": "text" + }, + { + "bbox": [ + 452, + 581, + 503, + 595 + ], + "score": 0.92, + "content": "G , G ^ { \\prime } \\in \\mathbb { F } _ { 0 } ^ { n ^ { k } }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 593, + 212, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 123, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 594, + 169, + 606 + ], + "score": 0.93, + "content": "s , s ^ { \\prime } \\in [ n ] ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 593, + 212, + 606 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 604, + 364, + 619 + ], + "lines": [ + { + "bbox": [ + 246, + 604, + 364, + 619 + ], + "spans": [ + { + "bbox": [ + 246, + 604, + 364, + 619 + ], + "score": 0.89, + "content": "c _ { k \\mathrm { - F W L } } ^ { T } ( G ) _ { s } = c _ { k \\mathrm { - 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For any", + "type": "text" + }, + { + "bbox": [ + 192, + 411, + 217, + 421 + ], + "score": 0.9, + "content": "k \\geq 2", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 408, + 217, + 424 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 427, + 393, + 443 + ], + "lines": [ + { + "bbox": [ + 216, + 427, + 393, + 443 + ], + "spans": [ + { + "bbox": [ + 216, + 427, + 393, + 443 + ], + "score": 0.88, + "content": "\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k – F G N N _ { e m b } \\right) \\supset \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) ,", + "type": "interline_equation", + "image_path": "57725f634b162d617e57b58ba5d06ae5850c0e9dba7e2f2a5afb515e922db5c0.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 216, + 427, + 393, + 443 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 136, + 460 + ], + "lines": [ + { + "bbox": [ + 104, + 447, + 137, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 137, + 461 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 447, + 137, + 461 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 458, + 394, + 475 + ], + "lines": [ + { + "bbox": [ + 216, + 458, + 394, + 475 + ], + "spans": [ + { + "bbox": [ + 216, + 458, + 394, + 475 + ], + "score": 0.88, + "content": "\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k – F G N N _ { e m b } \\right) = \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k - F W L } \\right) .", + "type": "interline_equation", + "image_path": "5e0f120406178fafd424376749342ce29aeafc44c49af35d3f60da26e48c4e2a.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 216, + 458, + 394, + 475 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 167, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 169, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 169, + 500 + ], + "score": 1.0, + "content": "Proof. Define,", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 484, + 169, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 502, + 454, + 518 + ], + "lines": [ + { + "bbox": [ + 156, + 502, + 454, + 518 + ], + "spans": [ + { + "bbox": [ + 156, + 502, + 454, + 518 + ], + "score": 0.9, + "content": "k \\mathrm { \\mathrm { - } F G N N } _ { e m b } ^ { T } = \\{ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k } : \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { \\ F G L } , t = 1 , \\dots , T \\} ,", + "type": "interline_equation", + "image_path": "50fe17e1599d9711a3ac9ee55537127ef84363cd5694ca241e9747daa3fed6f7.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 156, + 502, + 454, + 518 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 522, + 492, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 489, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 260, + 537 + ], + "score": 1.0, + "content": "the set of functions defined by exactly", + "type": "text" + }, + { + "bbox": [ + 261, + 524, + 269, + 533 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 520, + 463, + 537 + ], + "score": 1.0, + "content": "FGL layers. 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Take", + "type": "text" + }, + { + "bbox": [ + 452, + 581, + 503, + 595 + ], + "score": 0.92, + "content": "G , G ^ { \\prime } \\in \\mathbb { F } _ { 0 } ^ { n ^ { k } }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 593, + 212, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 123, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 594, + 169, + 606 + ], + "score": 0.93, + "content": "s , s ^ { \\prime } \\in [ n ] ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 593, + 212, + 606 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 103, + 580, + 503, + 606 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 604, + 364, + 619 + ], + "lines": [ + { + "bbox": [ + 246, + 604, + 364, + 619 + ], + "spans": [ + { + "bbox": [ + 246, + 604, + 364, + 619 + ], + "score": 0.89, + "content": "c _ { k \\mathrm { - 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} \\mathrm { F W L } } ^ { T - 1 } ( G ) _ { \\tilde { s } } : \\tilde { s } \\in N _ { j } ^ { F } ( s ) \\right) = \\left( c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ^ { \\prime } ) _ { \\tilde { s } } : \\tilde { s } \\in N _ { \\sigma ( j ) } ^ { F } ( s ^ { \\prime } ) \\right) .", + "type": "interline_equation", + "image_path": "6b8fbb4d8d849d050df1f83f3fb4bfaf2751a4b5e2d4f9a9832fb82f6c14927b.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 178, + 714, + 433, + 736 + ], + "spans": [], + "index": 34 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 471, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 471, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 122, + 97 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 82, + 187, + 95 + ], + "score": 0.93, + "content": "s = ( i _ { 1 } , \\ldots , i _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 81, + 205, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 82, + 275, + 95 + ], + "score": 0.93, + "content": "s ^ { \\prime } = ( j _ { 1 } , \\ldots , j _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 81, + 402, + 97 + ], + "score": 1.0, + "content": ". Then this implies that, for any", + "type": "text" + }, + { + "bbox": [ + 402, + 82, + 467, + 95 + ], + "score": 0.84, + "content": "w \\in [ k ] , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 81, + 471, + 97 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 99, + 446, + 116 + ], + "lines": [ + { + "bbox": [ + 164, + 99, + 446, + 116 + ], + "spans": [ + { + "bbox": [ + 164, + 99, + 446, + 116 + ], + "score": 0.9, + "content": "c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } = c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ^ { \\prime } ) _ { j _ { 1 } , \\dots , j _ { w - 1 } , \\sigma ( j ) , j _ { w + 1 } , \\dots , j _ { k } } .", + "type": "interline_equation", + "image_path": "d9777efdbdc2c4ab6bdd97cdd5280a348ca8bc7d6131e5d9eab7d5e35ee5b6f0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 164, + 99, + 446, + 116 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 294, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 294, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 294, + 140 + ], + "score": 1.0, + "content": "We now use the induction hypothesis, i.e. that,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 143, + 398, + 161 + ], + "lines": [ + { + "bbox": [ + 213, + 143, + 398, + 161 + ], + "spans": [ + { + "bbox": [ + 213, + 143, + 398, + 161 + ], + "score": 0.92, + "content": "\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k \\mathrm { – F G N N } _ { e m b } ^ { T - 1 } \\right) \\supset \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k \\mathrm { – F W L } } ^ { T - 1 } \\right) \\ .", + "type": "interline_equation", + "image_path": "e6059d4b4a652c053190e61c9921970f0dce91b93434bf1ac238b0a286ffe2d4.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 213, + 143, + 398, + 161 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 271, + 182 + ], + "lines": [ + { + "bbox": [ + 102, + 162, + 276, + 188 + ], + "spans": [ + { + "bbox": [ + 102, + 162, + 144, + 188 + ], + "score": 1.0, + "content": "Take any", + "type": "text" + }, + { + "bbox": [ + 145, + 167, + 232, + 181 + ], + "score": 0.87, + "content": "f _ { T - 1 } \\in k – \\mathrm { F G N N } _ { e m b } ^ { T - 1 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 162, + 276, + 188 + ], + "score": 1.0, + "content": ". By (11),", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 186, + 359, + 201 + ], + "lines": [ + { + "bbox": [ + 251, + 186, + 359, + 201 + ], + "spans": [ + { + "bbox": [ + 251, + 186, + 359, + 201 + ], + "score": 0.9, + "content": "f _ { T - 1 } ( G ) _ { s } = f _ { T - 1 } ( G ^ { \\prime } ) _ { s ^ { \\prime } } .", + "type": "interline_equation", + "image_path": "f4b90fcbcdea6eea6a03fb753346b090c921a9c53d19e37288f41e03cc2d40cb.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 251, + 186, + 359, + 201 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 242, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 242, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 173, + 220 + ], + "score": 1.0, + "content": "By (12), for any", + "type": "text" + }, + { + "bbox": [ + 173, + 206, + 238, + 218 + ], + "score": 0.78, + "content": "w \\in [ k ] , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 205, + 242, + 220 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 224, + 441, + 240 + ], + "lines": [ + { + "bbox": [ + 169, + 224, + 441, + 240 + ], + "spans": [ + { + "bbox": [ + 169, + 224, + 441, + 240 + ], + "score": 0.88, + "content": "\\begin{array} { r } { f _ { T - 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } = f ^ { T - 1 } ( G ^ { \\prime } ) _ { j _ { 1 } , \\dots , j _ { w - 1 } , \\sigma ( j ) , j _ { w + 1 } , \\dots , j _ { k } } . } \\end{array}", + "type": "interline_equation", + "image_path": "bf077b8b292f7b94672d4077775f957278ef4390eedfbb528bdb63d4e3a34f39.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 169, + 224, + 441, + 240 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 100, + 238, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 100, + 238, + 297, + 269 + ], + "score": 1.0, + "content": "By the definition of FGL Section 3.3, for any", + "type": "text" + }, + { + "bbox": [ + 297, + 245, + 376, + 261 + ], + "score": 0.76, + "content": "F _ { T } : \\mathbb { F } _ { T } ^ { n ^ { k } } \\to \\mathbb { F } _ { T + 1 } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 238, + 403, + 269 + ], + "score": 1.0, + "content": "FGL,", + "type": "text" + }, + { + "bbox": [ + 403, + 248, + 505, + 261 + ], + "score": 0.83, + "content": "F _ { T } \\circ f _ { T - 1 } ( G ) _ { s } = F _ { T } \\circ", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 254, + 313, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 261, + 154, + 274 + ], + "score": 0.91, + "content": "f _ { T - 1 } ( G ^ { \\prime } ) _ { s ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 254, + 235, + 280 + ], + "score": 1.0, + "content": ". 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We have, for", + "type": "text" + }, + { + "bbox": [ + 227, + 345, + 252, + 356 + ], + "score": 0.89, + "content": "k \\geq 2", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 361, + 452, + 420 + ], + "lines": [ + { + "bbox": [ + 159, + 361, + 451, + 420 + ], + "spans": [ + { + "bbox": [ + 159, + 361, + 451, + 420 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( 2 \\ – L G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) \\subseteq \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( k \\ – F W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( k \\ – F W L _ { E } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "3cef8b694216a888c1056063bc649b98c2ab99a280bf36cffb70b41b2e353cc5.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 159, + 361, + 452, + 380.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 159, + 380.6666666666667, + 452, + 400.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 159, + 400.33333333333337, + 452, + 420.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Proof. Most of the statements come from the literature or are direct consequences of the lemmas", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 439, + 137, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 439, + 137, + 454 + ], + "score": 1.0, + "content": "above.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 132, + 465, + 506, + 600 + ], + "lines": [ + { + "bbox": [ + 133, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 133, + 465, + 338, + 478 + ], + "score": 1.0, + "content": "• Proof of (13). The invariant case is proven in", + "type": "text" + }, + { + "bbox": [ + 338, + 466, + 352, + 475 + ], + "score": 0.39, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "et al. (2018, Lem. 2, Thm. 3). The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 415, + 489 + ], + "score": 1.0, + "content": "equivariant case comes from Lem. 11, Lem. 8, the fact that the layers", + "type": "text" + }, + { + "bbox": [ + 415, + 476, + 483, + 488 + ], + "score": 0.91, + "content": "m _ { E } \\circ ( \\mathrm { I d } + \\lambda S ^ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "does", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 486, + 435, + 501 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 374, + 501 + ], + "score": 1.0, + "content": "not change the separating power and recalling that simply", + "type": "text" + }, + { + "bbox": [ + 375, + 488, + 430, + 498 + ], + "score": 0.92, + "content": "\\mathrm { W L } _ { E } = c _ { W L }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 486, + 435, + 501 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 135, + 506, + 343, + 518 + ], + "spans": [ + { + "bbox": [ + 135, + 506, + 343, + 518 + ], + "score": 1.0, + "content": "• (14) is the exact result Chen et al. (2020, Thm. 6).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 133, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 133, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "• Proof of (15). The invariant case is exactly Maron et al. (2019a, Thm. 1). The equivariant", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 140, + 535, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 140, + 535, + 419, + 550 + ], + "score": 1.0, + "content": "case comes from Lem. 11, Lem. 9, Lem. 8 and the fact that the layer", + "type": "text" + }, + { + "bbox": [ + 420, + 538, + 437, + 547 + ], + "score": 0.85, + "content": "m _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 535, + 505, + 550 + ], + "score": 1.0, + "content": "does not change", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 547, + 230, + 560 + ], + "spans": [ + { + "bbox": [ + 141, + 547, + 230, + 560 + ], + "score": 1.0, + "content": "the separating power.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 132, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 132, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "• Proof of (16). The direct inclusion of the invariant case corresponds to Maron et al. (2019a,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 141, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Thm. 2). The other cases are a consequence of Lem. 11, Lem. 9, Lem. 8 and the fact that the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 588, + 374, + 601 + ], + "spans": [ + { + "bbox": [ + 141, + 588, + 168, + 601 + ], + "score": 1.0, + "content": "layers", + "type": "text" + }, + { + "bbox": [ + 169, + 590, + 183, + 599 + ], + "score": 0.87, + "content": "m _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 588, + 201, + 601 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 590, + 218, + 599 + ], + "score": 0.87, + "content": "m _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 588, + 374, + 601 + ], + "score": 1.0, + "content": "does not change the separating power.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 641, + 397, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 398, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 398, + 655 + ], + "score": 1.0, + "content": "D STONE-WEIERSTRASS THEOREM WITH SYMMETRIES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "This section presents our extension of the Stone-Weierstrass theorem dealing with functions with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "symmetries. The scope of this section is not restricted to graphs or even tensors and we will deal with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "general spaces and general symmetries. To illustrate it, we will present applications for the PointNet", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "architecture Qi et al. (2017). Our approximation results for GNNs (Theorems Thm. 5 and Thm. 6)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "are then obtained from these theoretical results applied to tensors and the symmetric group in Section", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 124, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 124, + 732 + ], + "score": 1.0, + "content": "D.9", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 298, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 495, + 300, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 495, + 300, + 505, + 309 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 613, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 496, + 614, + 504, + 622 + ], + "spans": [ + { + "bbox": [ + 496, + 614, + 504, + 622 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 471, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 471, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 122, + 97 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 82, + 187, + 95 + ], + "score": 0.93, + "content": "s = ( i _ { 1 } , \\ldots , i _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 81, + 205, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 82, + 275, + 95 + ], + "score": 0.93, + "content": "s ^ { \\prime } = ( j _ { 1 } , \\ldots , j _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 81, + 402, + 97 + ], + "score": 1.0, + "content": ". Then this implies that, for any", + "type": "text" + }, + { + "bbox": [ + 402, + 82, + 467, + 95 + ], + "score": 0.84, + "content": "w \\in [ k ] , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 81, + 471, + 97 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 471, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 99, + 446, + 116 + ], + "lines": [ + { + "bbox": [ + 164, + 99, + 446, + 116 + ], + "spans": [ + { + "bbox": [ + 164, + 99, + 446, + 116 + ], + "score": 0.9, + "content": "c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } = c _ { k \\mathrm { - } \\mathrm { F W L } } ^ { T - 1 } ( G ^ { \\prime } ) _ { j _ { 1 } , \\dots , j _ { w - 1 } , \\sigma ( j ) , j _ { w + 1 } , \\dots , j _ { k } } .", + "type": "interline_equation", + "image_path": "d9777efdbdc2c4ab6bdd97cdd5280a348ca8bc7d6131e5d9eab7d5e35ee5b6f0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 164, + 99, + 446, + 116 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 294, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 294, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 294, + 140 + ], + "score": 1.0, + "content": "We now use the induction hypothesis, i.e. that,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 125, + 294, + 140 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 143, + 398, + 161 + ], + "lines": [ + { + "bbox": [ + 213, + 143, + 398, + 161 + ], + "spans": [ + { + "bbox": [ + 213, + 143, + 398, + 161 + ], + "score": 0.92, + "content": "\\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( k \\mathrm { – F G N N } _ { e m b } ^ { T - 1 } \\right) \\supset \\rho _ { n , n ^ { k - 1 } } ^ { a u g m } \\left( c _ { k \\mathrm { – F W L } } ^ { T - 1 } \\right) \\ .", + "type": "interline_equation", + "image_path": "e6059d4b4a652c053190e61c9921970f0dce91b93434bf1ac238b0a286ffe2d4.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 213, + 143, + 398, + 161 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 271, + 182 + ], + "lines": [ + { + "bbox": [ + 102, + 162, + 276, + 188 + ], + "spans": [ + { + "bbox": [ + 102, + 162, + 144, + 188 + ], + "score": 1.0, + "content": "Take any", + "type": "text" + }, + { + "bbox": [ + 145, + 167, + 232, + 181 + ], + "score": 0.87, + "content": "f _ { T - 1 } \\in k – \\mathrm { F G N N } _ { e m b } ^ { T - 1 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 162, + 276, + 188 + ], + "score": 1.0, + "content": ". By (11),", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 102, + 162, + 276, + 188 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 186, + 359, + 201 + ], + "lines": [ + { + "bbox": [ + 251, + 186, + 359, + 201 + ], + "spans": [ + { + "bbox": [ + 251, + 186, + 359, + 201 + ], + "score": 0.9, + "content": "f _ { T - 1 } ( G ) _ { s } = f _ { T - 1 } ( G ^ { \\prime } ) _ { s ^ { \\prime } } .", + "type": "interline_equation", + "image_path": "f4b90fcbcdea6eea6a03fb753346b090c921a9c53d19e37288f41e03cc2d40cb.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 251, + 186, + 359, + 201 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 242, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 242, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 173, + 220 + ], + "score": 1.0, + "content": "By (12), for any", + "type": "text" + }, + { + "bbox": [ + 173, + 206, + 238, + 218 + ], + "score": 0.78, + "content": "w \\in [ k ] , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 205, + 242, + 220 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 205, + 242, + 220 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 224, + 441, + 240 + ], + "lines": [ + { + "bbox": [ + 169, + 224, + 441, + 240 + ], + "spans": [ + { + "bbox": [ + 169, + 224, + 441, + 240 + ], + "score": 0.88, + "content": "\\begin{array} { r } { f _ { T - 1 } ( G ) _ { i _ { 1 } , \\dots , i _ { w - 1 } , j , i _ { w + 1 } , \\dots , i _ { k } } = f ^ { T - 1 } ( G ^ { \\prime } ) _ { j _ { 1 } , \\dots , j _ { w - 1 } , \\sigma ( j ) , j _ { w + 1 } , \\dots , j _ { k } } . } \\end{array}", + "type": "interline_equation", + "image_path": "bf077b8b292f7b94672d4077775f957278ef4390eedfbb528bdb63d4e3a34f39.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 169, + 224, + 441, + 240 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 100, + 238, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 100, + 238, + 297, + 269 + ], + "score": 1.0, + "content": "By the definition of FGL Section 3.3, for any", + "type": "text" + }, + { + "bbox": [ + 297, + 245, + 376, + 261 + ], + "score": 0.76, + "content": "F _ { T } : \\mathbb { F } _ { T } ^ { n ^ { k } } \\to \\mathbb { F } _ { T + 1 } ^ { n ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 238, + 403, + 269 + ], + "score": 1.0, + "content": "FGL,", + "type": "text" + }, + { + "bbox": [ + 403, + 248, + 505, + 261 + ], + "score": 0.83, + "content": "F _ { T } \\circ f _ { T - 1 } ( G ) _ { s } = F _ { T } \\circ", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 254, + 313, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 261, + 154, + 274 + ], + "score": 0.91, + "content": "f _ { T - 1 } ( G ^ { \\prime } ) _ { s ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 254, + 235, + 280 + ], + "score": 1.0, + "content": ". Therefore, for any", + "type": "text" + }, + { + "bbox": [ + 235, + 260, + 305, + 274 + ], + "score": 0.9, + "content": "f \\in k \\mathrm { - F G N N } _ { e m b } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 254, + 313, + 280 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 100, + 238, + 505, + 280 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 279, + 343, + 293 + ], + "lines": [ + { + "bbox": [ + 267, + 279, + 343, + 293 + ], + "spans": [ + { + "bbox": [ + 267, + 279, + 343, + 293 + ], + "score": 0.89, + "content": "f ( G ) _ { s } = f ( G ^ { \\prime } ) _ { s ^ { \\prime } } ,", + "type": "interline_equation", + "image_path": "89742c4e949b30efb3d33d960e6da5e7a3f507d7e6c5cff82647873961f8ba66.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 267, + 279, + 343, + 293 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 216, + 310 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 217, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 217, + 312 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 297, + 217, + 312 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 324, + 191, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 193, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 193, + 337 + ], + "score": 1.0, + "content": "C.6 CONCLUSION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 255, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 252, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 227, + 358 + ], + "score": 1.0, + "content": "Proposition 13. We have, for", + "type": "text" + }, + { + "bbox": [ + 227, + 345, + 252, + 356 + ], + "score": 0.89, + "content": "k \\geq 2", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 342, + 252, + 358 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 361, + 452, + 420 + ], + "lines": [ + { + "bbox": [ + 159, + 361, + 451, + 420 + ], + "spans": [ + { + "bbox": [ + 159, + 361, + 451, + 420 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\rho \\left( M G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( M G N N _ { E } \\right) = \\rho \\left( 2 \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( 2 \\ – L G N N _ { I } \\right) = \\rho \\left( 2 \\ – W L _ { I } \\right) \\qquad } & { { } } \\\\ { \\rho \\left( k \\ – L G N N _ { I } \\right) \\subseteq \\rho \\left( k \\ – W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( k \\ – W L _ { E } \\right) } \\\\ { \\rho \\left( k \\ – F G N N _ { I } \\right) = \\rho \\left( k \\ – F W L _ { I } \\right) \\qquad } & { { } \\rho \\left( k \\ – F G N N _ { E } \\right) = \\rho \\left( k \\ – F W L _ { E } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "3cef8b694216a888c1056063bc649b98c2ab99a280bf36cffb70b41b2e353cc5.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 159, + 361, + 452, + 380.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 159, + 380.6666666666667, + 452, + 400.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 159, + 400.33333333333337, + 452, + 420.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Proof. Most of the statements come from the literature or are direct consequences of the lemmas", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 439, + 137, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 439, + 137, + 454 + ], + "score": 1.0, + "content": "above.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 429, + 505, + 454 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 465, + 506, + 600 + ], + "lines": [ + { + "bbox": [ + 133, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 133, + 465, + 338, + 478 + ], + "score": 1.0, + "content": "• Proof of (13). The invariant case is proven in", + "type": "text" + }, + { + "bbox": [ + 338, + 466, + 352, + 475 + ], + "score": 0.39, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "et al. (2018, Lem. 2, Thm. 3). The", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 415, + 489 + ], + "score": 1.0, + "content": "equivariant case comes from Lem. 11, Lem. 8, the fact that the layers", + "type": "text" + }, + { + "bbox": [ + 415, + 476, + 483, + 488 + ], + "score": 0.91, + "content": "m _ { E } \\circ ( \\mathrm { I d } + \\lambda S ^ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "does", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 486, + 435, + 501 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 374, + 501 + ], + "score": 1.0, + "content": "not change the separating power and recalling that simply", + "type": "text" + }, + { + "bbox": [ + 375, + 488, + 430, + 498 + ], + "score": 0.92, + "content": "\\mathrm { W L } _ { E } = c _ { W L }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 486, + 435, + 501 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + }, + { + "bbox": [ + 135, + 506, + 343, + 518 + ], + "spans": [ + { + "bbox": [ + 135, + 506, + 343, + 518 + ], + "score": 1.0, + "content": "• (14) is the exact result Chen et al. (2020, Thm. 6).", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 133, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 133, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "• Proof of (15). The invariant case is exactly Maron et al. (2019a, Thm. 1). The equivariant", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 140, + 535, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 140, + 535, + 419, + 550 + ], + "score": 1.0, + "content": "case comes from Lem. 11, Lem. 9, Lem. 8 and the fact that the layer", + "type": "text" + }, + { + "bbox": [ + 420, + 538, + 437, + 547 + ], + "score": 0.85, + "content": "m _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 535, + 505, + 550 + ], + "score": 1.0, + "content": "does not change", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 547, + 230, + 560 + ], + "spans": [ + { + "bbox": [ + 141, + 547, + 230, + 560 + ], + "score": 1.0, + "content": "the separating power.", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 132, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "• Proof of (16). The direct inclusion of the invariant case corresponds to Maron et al. (2019a,", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 141, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Thm. 2). The other cases are a consequence of Lem. 11, Lem. 9, Lem. 8 and the fact that the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 588, + 374, + 601 + ], + "spans": [ + { + "bbox": [ + 141, + 588, + 168, + 601 + ], + "score": 1.0, + "content": "layers", + "type": "text" + }, + { + "bbox": [ + 169, + 590, + 183, + 599 + ], + "score": 0.87, + "content": "m _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 588, + 201, + 601 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 590, + 218, + 599 + ], + "score": 0.87, + "content": "m _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 588, + 374, + 601 + ], + "score": 1.0, + "content": "does not change the separating power.", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + } + ], + "index": 23.5, + "bbox_fs": [ + 132, + 465, + 507, + 601 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 641, + 397, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 398, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 398, + 655 + ], + "score": 1.0, + "content": "D STONE-WEIERSTRASS THEOREM WITH SYMMETRIES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "This section presents our extension of the Stone-Weierstrass theorem dealing with functions with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "symmetries. The scope of this section is not restricted to graphs or even tensors and we will deal with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "general spaces and general symmetries. To illustrate it, we will present applications for the PointNet", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "architecture Qi et al. (2017). Our approximation results for GNNs (Theorems Thm. 5 and Thm. 6)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "are then obtained from these theoretical results applied to tensors and the symmetric group in Section", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 124, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 124, + 732 + ], + "score": 1.0, + "content": "D.9", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 666, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 102, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 115 + ], + "score": 1.0, + "content": "As explained above, we are dealing in this section with a much larger scope than graphs and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "permutations. 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Given a set of scalar functions", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 107, + 410, + 161, + 421 + ], + "score": 0.89, + "content": "{ \\mathcal { S } } \\subseteq { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 408, + 307, + 421 + ], + "score": 1.0, + "content": "and a set of vector-valued functions", + "type": "text" + }, + { + "bbox": [ + 308, + 410, + 364, + 421 + ], + "score": 0.92, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 408, + 506, + 421 + ], + "score": 1.0, + "content": ", the set of products of functions of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 419, + 242, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 242, + 432 + ], + "score": 1.0, + "content": "these two sets will be denoted by,", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 386, + 506, + 432 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 437, + 369, + 451 + ], + "lines": [ + { + "bbox": [ + 241, + 437, + 369, + 451 + ], + "spans": [ + { + "bbox": [ + 241, + 437, + 369, + 451 + ], + "score": 0.9, + "content": "S \\cdot { \\mathcal { F } } = \\left\\{ g f : g \\in { \\mathcal { S } } , f \\in { \\mathcal { F } } \\right\\} .", + "type": "interline_equation", + "image_path": "e9562e6635da981d0cc2cd7ded2bad8aea35fcd71f20471c69790057c6cae842.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 241, + 437, + 369, + 451 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 104, + 460, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 359, + 475 + ], + "score": 1.0, + "content": "Moreover, we denote by 1 the continuous function from some", + "type": "text" + }, + { + "bbox": [ + 360, + 462, + 370, + 471 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 460, + 381, + 475 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 381, + 462, + 394, + 471 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 460, + 441, + 475 + ], + "score": 1.0, + "content": "defined by", + "type": "text" + }, + { + "bbox": [ + 441, + 461, + 503, + 473 + ], + "score": 0.9, + "content": "x \\mapsto ( 1 , \\ldots , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 460, + 507, + 475 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 169, + 486 + ], + "score": 1.0, + "content": "In particular, if", + "type": "text" + }, + { + "bbox": [ + 170, + 473, + 176, + 484 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 472, + 253, + 486 + ], + "score": 1.0, + "content": "is a function from", + "type": "text" + }, + { + "bbox": [ + 253, + 473, + 263, + 483 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 472, + 274, + 486 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 275, + 473, + 283, + 483 + ], + "score": 0.69, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 472, + 286, + 486 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 287, + 473, + 300, + 484 + ], + "score": 0.8, + "content": "f \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 472, + 385, + 486 + ], + "score": 1.0, + "content": "denotes the function", + "type": "text" + }, + { + "bbox": [ + 385, + 473, + 476, + 484 + ], + "score": 0.91, + "content": "x \\mapsto ( f ( x ) , \\ldots , f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 148, + 496 + ], + "score": 1.0, + "content": "goes from", + "type": "text" + }, + { + "bbox": [ + 149, + 484, + 158, + 493 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 484, + 169, + 496 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 170, + 484, + 204, + 493 + ], + "score": 0.89, + "content": "Y = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 484, + 286, + 496 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 186, + 556, + 195, + 565 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 555, + 282, + 568 + ], + "score": 1.0, + "content": "be a set of functions", + "type": "text" + }, + { + "bbox": [ + 282, + 555, + 289, + 567 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 555, + 358, + 568 + ], + "score": 1.0, + "content": "defined on a set", + "type": "text" + }, + { + "bbox": [ + 358, + 556, + 368, + 565 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 555, + 421, + 568 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 421, + 556, + 429, + 567 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "takes its values in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 439, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 130, + 578 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 130, + 567, + 142, + 578 + ], + "score": 0.88, + "content": "Y _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 566, + 247, + 578 + ], + "score": 1.0, + "content": ". 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The invariant version of PointNet is able to learn functions of the form", + "type": "text" + }, + { + "bbox": [ + 450, + 357, + 489, + 370 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { i } f ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 107, + 368, + 163, + 380 + ], + "score": 0.92, + "content": "f \\in { \\mathcal { C } } ( \\mathbb { R } ^ { p } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 368, + 440, + 381 + ], + "score": 1.0, + "content": ". We can apply Corollary 17 to this setting. Consider the case where", + "type": "text" + }, + { + "bbox": [ + 441, + 369, + 451, + 378 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "is a compact", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 375, + 508, + 395 + ], + "spans": [ + { + "bbox": [ + 103, + 375, + 146, + 395 + ], + "score": 1.0, + "content": "subset of", + "type": "text" + }, + { + "bbox": [ + 146, + 380, + 159, + 389 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 375, + 178, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 379, + 293, + 391 + ], + "score": 0.84, + "content": "\\mathcal { F } = \\textstyle \\{ \\hat { x } \\mapsto g ( \\sum _ { i = 1 } ^ { n } \\bar { f } ( x _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 375, + 299, + 395 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 300, + 378, + 425, + 391 + ], + "score": 0.87, + "content": "f \\in \\mathcal { C } ( \\mathbf { \\breve { X } } , \\mathbb { R } ^ { h } ) , g \\in \\mathcal { C } ( \\mathbb { R } ^ { h } , \\mathbb { R } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 375, + 430, + 395 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 430, + 379, + 440, + 389 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 375, + 508, + 395 + ], + "score": 1.0, + "content": "is a subalgebra", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 118, + 403 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 390, + 152, + 401 + ], + "score": 0.9, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 390, + 198, + 403 + ], + "score": 1.0, + "content": "(indeed of", + "type": "text" + }, + { + "bbox": [ + 198, + 391, + 238, + 402 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } _ { I } ( X , \\mathbb { R } ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 390, + 506, + 403 + ], + "score": 1.0, + "content": ") which contains the constant function 1. Then, it easy to see that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 399, + 507, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 226, + 413 + ], + "score": 0.91, + "content": "\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , \\sigma \\star x ) , \\sigma \\in S _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 399, + 256, + 416 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 256, + 403, + 279, + 411 + ], + "score": 0.87, + "content": "\\sigma \\star x", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 399, + 332, + 416 + ], + "score": 1.0, + "content": "is defined by", + "type": "text" + }, + { + "bbox": [ + 333, + 401, + 398, + 414 + ], + "score": 0.93, + "content": "( \\sigma \\star x ) _ { \\sigma ( i ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 399, + 425, + 416 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 425, + 402, + 430, + 411 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 399, + 507, + 416 + ], + "score": 1.0, + "content": "(see Lem. 31 for a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 412, + 182, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 182, + 424 + ], + "score": 1.0, + "content": "formal statement).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 222, + 442 + ], + "score": 1.0, + "content": "Now note that for a function", + "type": "text" + }, + { + "bbox": [ + 222, + 429, + 274, + 441 + ], + "score": 0.94, + "content": "f \\in { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 426, + 333, + 442 + ], + "score": 1.0, + "content": ", the condition", + "type": "text" + }, + { + "bbox": [ + 334, + 429, + 451, + 441 + ], + "score": 0.91, + "content": "\\left\\{ ( x , \\sigma \\star x ) , \\sigma \\in S _ { n } \\right\\} \\subseteq \\rho \\left( f \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 426, + 506, + 442 + ], + "score": 1.0, + "content": "is equivalent", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 117, + 454 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 441, + 175, + 453 + ], + "score": 0.93, + "content": "f \\in { \\mathcal { C } } _ { I } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 441, + 319, + 454 + ], + "score": 1.0, + "content": ". So that Corollary 17 implies that", + "type": "text" + }, + { + "bbox": [ + 319, + 441, + 380, + 453 + ], + "score": 0.92, + "content": "\\overline { { \\mathcal { F } } } = \\mathcal { C } _ { I } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "which means that PointNet is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 452, + 477, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 477, + 465 + ], + "score": 1.0, + "content": "universal for approximating invariant functions. This was already proved in Qi et al. (2017) .", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 340, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 342, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 342, + 489 + ], + "score": 1.0, + "content": "We now provide a proof of Corollary 17 for completeness.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 212, + 520 + ], + "score": 1.0, + "content": "Proof. The first inclusion", + "type": "text" + }, + { + "bbox": [ + 212, + 506, + 361, + 519 + ], + "score": 0.93, + "content": "{ \\overline { { \\mathcal { F } } } } \\subseteq \\{ f \\in { \\mathcal { C } } ( X , \\mathbb { R } ) : \\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "follows from the same argument as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 517, + 354, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 354, + 530 + ], + "score": 1.0, + "content": "the one given below Theorem 4 so we focus on the other one.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 504, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 148, + 549 + ], + "score": 1.0, + "content": "For every", + "type": "text" + }, + { + "bbox": [ + 148, + 536, + 177, + 545 + ], + "score": 0.9, + "content": "x \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 533, + 194, + 549 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 194, + 536, + 208, + 546 + ], + "score": 0.85, + "content": "x _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 533, + 251, + 549 + ], + "score": 1.0, + "content": "denote its", + "type": "text" + }, + { + "bbox": [ + 251, + 535, + 275, + 547 + ], + "score": 0.92, + "content": "\\rho \\left( \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 533, + 506, + 549 + ], + "score": 1.0, + "content": "-class. The quotient set and the canonical surjection are:", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 544, + 382, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 242, + 558 + ], + "score": 0.89, + "content": "X _ { \\mathcal { F } } = X / \\rho \\left( \\mathcal { F } \\right) = \\left\\{ x _ { \\mathcal { F } } , x \\in X \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 544, + 259, + 560 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 546, + 321, + 557 + ], + "score": 0.85, + "content": "\\pi _ { \\mathcal { F } } : X \\to X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 544, + 325, + 560 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 325, + 546, + 378, + 558 + ], + "score": 0.89, + "content": "\\bar { \\pi } _ { \\mathcal { F } } ( x ) = x _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 544, + 382, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 506, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 151, + 576 + ], + "score": 1.0, + "content": "A function", + "type": "text" + }, + { + "bbox": [ + 152, + 563, + 198, + 574 + ], + "score": 0.91, + "content": "g : X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 562, + 252, + 576 + ], + "score": 1.0, + "content": "factorizes as", + "type": "text" + }, + { + "bbox": [ + 252, + 563, + 299, + 574 + ], + "score": 0.92, + "content": "g = \\hat { g } \\circ \\pi _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 562, + 338, + 576 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 338, + 563, + 390, + 574 + ], + "score": 0.92, + "content": "\\hat { g } : X _ { \\mathcal { F } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 562, + 446, + 576 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 446, + 562, + 503, + 575 + ], + "score": 0.92, + "content": "\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( g \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 562, + 507, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 156, + 587 + ], + "score": 1.0, + "content": "In this case", + "type": "text" + }, + { + "bbox": [ + 156, + 574, + 163, + 585 + ], + "score": 0.83, + "content": "\\hat { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 573, + 231, + 587 + ], + "score": 1.0, + "content": "is unique, since", + "type": "text" + }, + { + "bbox": [ + 231, + 576, + 245, + 585 + ], + "score": 0.86, + "content": "\\pi _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 573, + 393, + 587 + ], + "score": 1.0, + "content": "is a surjection. In particular. every", + "type": "text" + }, + { + "bbox": [ + 394, + 574, + 423, + 585 + ], + "score": 0.94, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "factorizes uniquely", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 118, + 599 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 584, + 167, + 598 + ], + "score": 0.88, + "content": "f = \\hat { f } \\circ \\bar { \\pi } _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 586, + 171, + 599 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 172, + 585, + 227, + 598 + ], + "score": 0.9, + "content": "\\hat { f } : X _ { \\mathcal { F } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 586, + 249, + 599 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 249, + 585, + 322, + 599 + ], + "score": 0.89, + "content": "\\hat { \\mathcal { F } } = \\{ \\hat { f } , f \\in \\mathcal { F } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 586, + 435, + 599 + ], + "score": 1.0, + "content": "clearly separates points on", + "type": "text" + }, + { + "bbox": [ + 435, + 587, + 451, + 598 + ], + "score": 0.9, + "content": "X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 586, + 506, + 599 + ], + "score": 1.0, + "content": ". We refer to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 173, + 610 + ], + "score": 1.0, + "content": "Munkres (2000,", + "type": "text" + }, + { + "bbox": [ + 174, + 598, + 191, + 609 + ], + "score": 0.5, + "content": "\\ S 2 2 _ { , }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 596, + 506, + 610 + ], + "score": 1.0, + "content": ") for the properties of the quotient topology. In particular, by the properties of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 209, + 622 + ], + "score": 1.0, + "content": "the quotient topology on", + "type": "text" + }, + { + "bbox": [ + 209, + 609, + 225, + 621 + ], + "score": 0.82, + "content": "X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 609, + 229, + 622 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 608, + 239, + 620 + ], + "score": 0.8, + "content": "\\hat { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 609, + 315, + 622 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 315, + 610, + 356, + 622 + ], + "score": 0.93, + "content": "\\mathcal { C } ( X _ { \\mathcal { F } } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 609, + 375, + 622 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 375, + 610, + 392, + 621 + ], + "score": 0.9, + "content": "X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "is compact. 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Let", + "type": "text" + }, + { + "bbox": [ + 264, + 201, + 274, + 210 + ], + "score": 0.73, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 200, + 380, + 214 + ], + "score": 1.0, + "content": "be a compact space, and", + "type": "text" + }, + { + "bbox": [ + 380, + 201, + 389, + 210 + ], + "score": 0.8, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 200, + 470, + 214 + ], + "score": 1.0, + "content": "be a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 470, + 200, + 505, + 212 + ], + "score": 0.95, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 211, + 504, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 306, + 225 + ], + "score": 1.0, + "content": "the space of real-valued continuous functions of", + "type": "text" + }, + { + "bbox": [ + 306, + 212, + 316, + 221 + ], + "score": 0.81, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 211, + 495, + 225 + ], + "score": 1.0, + "content": ", which contains the constant function 1. If", + "type": "text" + }, + { + "bbox": [ + 495, + 213, + 504, + 221 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 222, + 415, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 191, + 235 + ], + "score": 1.0, + "content": "separates points, i.e.", + "type": "text" + }, + { + "bbox": [ + 192, + 222, + 297, + 235 + ], + "score": 0.92, + "content": "\\rho \\left( { \\mathcal { F } } \\right) = \\{ \\left( x , x \\right) : x \\in X \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 223, + 321, + 235 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 321, + 223, + 330, + 232 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 223, + 376, + 235 + ], + "score": 1.0, + "content": "is dense in", + "type": "text" + }, + { + "bbox": [ + 376, + 223, + 411, + 235 + ], + "score": 0.95, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 223, + 415, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 200, + 505, + 235 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 503, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 504, + 257 + ], + "score": 1.0, + "content": "We now prove an extension of this classical result due to Timofte (2005) allowing us to deal with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 255, + 387, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 163, + 268 + ], + "score": 1.0, + "content": "much smaller", + "type": "text" + }, + { + "bbox": [ + 163, + 256, + 173, + 266 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 255, + 307, + 268 + ], + "score": 1.0, + "content": "by dropping the requirement that", + "type": "text" + }, + { + "bbox": [ + 307, + 256, + 317, + 266 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 255, + 387, + 268 + ], + "score": 1.0, + "content": "separates points.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 106, + 245, + 504, + 268 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 181, + 284 + ], + "score": 1.0, + "content": "Corollary 17. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 272, + 192, + 282 + ], + "score": 0.73, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 271, + 290, + 284 + ], + "score": 1.0, + "content": "be a compact space, and", + "type": "text" + }, + { + "bbox": [ + 290, + 272, + 299, + 282 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 271, + 374, + 284 + ], + "score": 1.0, + "content": "be a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 374, + 272, + 408, + 284 + ], + "score": 0.92, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "the space of real-valued", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 283, + 401, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 202, + 296 + ], + "score": 1.0, + "content": "continuous functions of", + "type": "text" + }, + { + "bbox": [ + 202, + 284, + 212, + 293 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 283, + 401, + 296 + ], + "score": 1.0, + "content": ", which contains the constant function 1. Then,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 106, + 271, + 506, + 296 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 301, + 383, + 316 + ], + "lines": [ + { + "bbox": [ + 228, + 301, + 383, + 316 + ], + "spans": [ + { + "bbox": [ + 228, + 301, + 383, + 316 + ], + "score": 0.91, + "content": "\\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} .", + "type": "interline_equation", + "image_path": "ffac440f44b4c99a5bbf84e632b8cf4f250897c42dc92577eb47bd89974fd770.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 228, + 301, + 383, + 316 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 329, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 155, + 343 + ], + "score": 1.0, + "content": "Note that if", + "type": "text" + }, + { + "bbox": [ + 155, + 331, + 165, + 340 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 329, + 470, + 343 + ], + "score": 1.0, + "content": "separates points, we get back the classical result as every function satisfies", + "type": "text" + }, + { + "bbox": [ + 470, + 330, + 505, + 342 + ], + "score": 0.9, + "content": "\\{ ( x , x ) :", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 338, + 178, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 173, + 353 + ], + "score": 0.92, + "content": "x \\in X \\} \\subseteq \\rho \\left( f \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 338, + 178, + 354 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 329, + 505, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 450, + 370 + ], + "score": 1.0, + "content": "Example 18. The invariant version of PointNet is able to learn functions of the form", + "type": "text" + }, + { + "bbox": [ + 450, + 357, + 489, + 370 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { i } f ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 107, + 368, + 163, + 380 + ], + "score": 0.92, + "content": "f \\in { \\mathcal { C } } ( \\mathbb { R } ^ { p } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 368, + 440, + 381 + ], + "score": 1.0, + "content": ". We can apply Corollary 17 to this setting. Consider the case where", + "type": "text" + }, + { + "bbox": [ + 441, + 369, + 451, + 378 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "is a compact", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 375, + 508, + 395 + ], + "spans": [ + { + "bbox": [ + 103, + 375, + 146, + 395 + ], + "score": 1.0, + "content": "subset of", + "type": "text" + }, + { + "bbox": [ + 146, + 380, + 159, + 389 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 375, + 178, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 379, + 293, + 391 + ], + "score": 0.84, + "content": "\\mathcal { F } = \\textstyle \\{ \\hat { x } \\mapsto g ( \\sum _ { i = 1 } ^ { n } \\bar { f } ( x _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 375, + 299, + 395 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 300, + 378, + 425, + 391 + ], + "score": 0.87, + "content": "f \\in \\mathcal { C } ( \\mathbf { \\breve { X } } , \\mathbb { R } ^ { h } ) , g \\in \\mathcal { C } ( \\mathbb { R } ^ { h } , \\mathbb { R } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 375, + 430, + 395 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 430, + 379, + 440, + 389 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 375, + 508, + 395 + ], + "score": 1.0, + "content": "is a subalgebra", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 118, + 403 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 390, + 152, + 401 + ], + "score": 0.9, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 390, + 198, + 403 + ], + "score": 1.0, + "content": "(indeed of", + "type": "text" + }, + { + "bbox": [ + 198, + 391, + 238, + 402 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } _ { I } ( X , \\mathbb { R } ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 390, + 506, + 403 + ], + "score": 1.0, + "content": ") which contains the constant function 1. Then, it easy to see that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 399, + 507, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 226, + 413 + ], + "score": 0.91, + "content": "\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , \\sigma \\star x ) , \\sigma \\in S _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 399, + 256, + 416 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 256, + 403, + 279, + 411 + ], + "score": 0.87, + "content": "\\sigma \\star x", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 399, + 332, + 416 + ], + "score": 1.0, + "content": "is defined by", + "type": "text" + }, + { + "bbox": [ + 333, + 401, + 398, + 414 + ], + "score": 0.93, + "content": "( \\sigma \\star x ) _ { \\sigma ( i ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 399, + 425, + 416 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 425, + 402, + 430, + 411 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 399, + 507, + 416 + ], + "score": 1.0, + "content": "(see Lem. 31 for a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 412, + 182, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 182, + 424 + ], + "score": 1.0, + "content": "formal statement).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 103, + 356, + 508, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 222, + 442 + ], + "score": 1.0, + "content": "Now note that for a function", + "type": "text" + }, + { + "bbox": [ + 222, + 429, + 274, + 441 + ], + "score": 0.94, + "content": "f \\in { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 426, + 333, + 442 + ], + "score": 1.0, + "content": ", the condition", + "type": "text" + }, + { + "bbox": [ + 334, + 429, + 451, + 441 + ], + "score": 0.91, + "content": "\\left\\{ ( x , \\sigma \\star x ) , \\sigma \\in S _ { n } \\right\\} \\subseteq \\rho \\left( f \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 426, + 506, + 442 + ], + "score": 1.0, + "content": "is equivalent", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 117, + 454 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 441, + 175, + 453 + ], + "score": 0.93, + "content": "f \\in { \\mathcal { C } } _ { I } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 441, + 319, + 454 + ], + "score": 1.0, + "content": ". So that Corollary 17 implies that", + "type": "text" + }, + { + "bbox": [ + 319, + 441, + 380, + 453 + ], + "score": 0.92, + "content": "\\overline { { \\mathcal { F } } } = \\mathcal { C } _ { I } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "which means that PointNet is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 452, + 477, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 477, + 465 + ], + "score": 1.0, + "content": "universal for approximating invariant functions. This was already proved in Qi et al. (2017) .", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 426, + 506, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 340, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 342, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 342, + 489 + ], + "score": 1.0, + "content": "We now provide a proof of Corollary 17 for completeness.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 473, + 342, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 212, + 520 + ], + "score": 1.0, + "content": "Proof. The first inclusion", + "type": "text" + }, + { + "bbox": [ + 212, + 506, + 361, + 519 + ], + "score": 0.93, + "content": "{ \\overline { { \\mathcal { F } } } } \\subseteq \\{ f \\in { \\mathcal { C } } ( X , \\mathbb { R } ) : \\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "follows from the same argument as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 517, + 354, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 354, + 530 + ], + "score": 1.0, + "content": "the one given below Theorem 4 so we focus on the other one.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 505, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 504, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 148, + 549 + ], + "score": 1.0, + "content": "For every", + "type": "text" + }, + { + "bbox": [ + 148, + 536, + 177, + 545 + ], + "score": 0.9, + "content": "x \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 533, + 194, + 549 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 194, + 536, + 208, + 546 + ], + "score": 0.85, + "content": "x _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 533, + 251, + 549 + ], + "score": 1.0, + "content": "denote its", + "type": "text" + }, + { + "bbox": [ + 251, + 535, + 275, + 547 + ], + "score": 0.92, + "content": "\\rho \\left( \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 533, + 506, + 549 + ], + "score": 1.0, + "content": "-class. The quotient set and the canonical surjection are:", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 544, + 382, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 242, + 558 + ], + "score": 0.89, + "content": "X _ { \\mathcal { F } } = X / \\rho \\left( \\mathcal { F } \\right) = \\left\\{ x _ { \\mathcal { F } } , x \\in X \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 544, + 259, + 560 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 546, + 321, + 557 + ], + "score": 0.85, + "content": "\\pi _ { \\mathcal { F } } : X \\to X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 544, + 325, + 560 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 325, + 546, + 378, + 558 + ], + "score": 0.89, + "content": "\\bar { \\pi } _ { \\mathcal { F } } ( x ) = x _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 544, + 382, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 533, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 506, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 151, + 576 + ], + "score": 1.0, + "content": "A function", + "type": "text" + }, + { + "bbox": [ + 152, + 563, + 198, + 574 + ], + "score": 0.91, + "content": "g : X \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 562, + 252, + 576 + ], + "score": 1.0, + "content": "factorizes as", + "type": "text" + }, + { + "bbox": [ + 252, + 563, + 299, + 574 + ], + "score": 0.92, + "content": "g = \\hat { g } \\circ \\pi _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 562, + 338, + 576 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 338, + 563, + 390, + 574 + ], + "score": 0.92, + "content": "\\hat { g } : X _ { \\mathcal { F } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 562, + 446, + 576 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 446, + 562, + 503, + 575 + ], + "score": 0.92, + "content": "\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( g \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 562, + 507, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 156, + 587 + ], + "score": 1.0, + "content": "In this case", + "type": "text" + }, + { + "bbox": [ + 156, + 574, + 163, + 585 + ], + "score": 0.83, + "content": "\\hat { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 573, + 231, + 587 + ], + "score": 1.0, + "content": "is unique, since", + "type": "text" + }, + { + "bbox": [ + 231, + 576, + 245, + 585 + ], + "score": 0.86, + "content": "\\pi _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 573, + 393, + 587 + ], + "score": 1.0, + "content": "is a surjection. In particular. every", + "type": "text" + }, + { + "bbox": [ + 394, + 574, + 423, + 585 + ], + "score": 0.94, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "factorizes uniquely", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 118, + 599 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 584, + 167, + 598 + ], + "score": 0.88, + "content": "f = \\hat { f } \\circ \\bar { \\pi } _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 586, + 171, + 599 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 172, + 585, + 227, + 598 + ], + "score": 0.9, + "content": "\\hat { f } : X _ { \\mathcal { F } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 586, + 249, + 599 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 249, + 585, + 322, + 599 + ], + "score": 0.89, + "content": "\\hat { \\mathcal { F } } = \\{ \\hat { f } , f \\in \\mathcal { F } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 586, + 435, + 599 + ], + "score": 1.0, + "content": "clearly separates points on", + "type": "text" + }, + { + "bbox": [ + 435, + 587, + 451, + 598 + ], + "score": 0.9, + "content": "X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 586, + 506, + 599 + ], + "score": 1.0, + "content": ". We refer to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 173, + 610 + ], + "score": 1.0, + "content": "Munkres (2000,", + "type": "text" + }, + { + "bbox": [ + 174, + 598, + 191, + 609 + ], + "score": 0.5, + "content": "\\ S 2 2 _ { , }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 596, + 506, + 610 + ], + "score": 1.0, + "content": ") for the properties of the quotient topology. In particular, by the properties of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 209, + 622 + ], + "score": 1.0, + "content": "the quotient topology on", + "type": "text" + }, + { + "bbox": [ + 209, + 609, + 225, + 621 + ], + "score": 0.82, + "content": "X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 609, + 229, + 622 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 608, + 239, + 620 + ], + "score": 0.8, + "content": "\\hat { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 609, + 315, + 622 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 315, + 610, + 356, + 622 + ], + "score": 0.93, + "content": "\\mathcal { C } ( X _ { \\mathcal { F } } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 609, + 375, + 622 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 375, + 610, + 392, + 621 + ], + "score": 0.9, + "content": "X _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "is compact. Hence, we can", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 621, + 321, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 192, + 636 + ], + "score": 1.0, + "content": "apply Theorem 16 to", + "type": "text" + }, + { + "bbox": [ + 192, + 621, + 202, + 633 + ], + "score": 0.86, + "content": "\\hat { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 622, + 220, + 636 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 220, + 622, + 230, + 633 + ], + "score": 0.87, + "content": "\\hat { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 622, + 275, + 636 + ], + "score": 1.0, + "content": "is dense in", + "type": "text" + }, + { + "bbox": [ + 275, + 622, + 316, + 635 + ], + "score": 0.93, + "content": "\\mathcal { C } ( X _ { \\mathcal { F } } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 622, + 321, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 562, + 507, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 147, + 655 + ], + "score": 1.0, + "content": "Now take", + "type": "text" + }, + { + "bbox": [ + 147, + 641, + 200, + 653 + ], + "score": 0.92, + "content": "f \\in { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 639, + 222, + 655 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 222, + 641, + 279, + 653 + ], + "score": 0.93, + "content": "\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 639, + 353, + 655 + ], + "score": 1.0, + "content": "and we show that", + "type": "text" + }, + { + "bbox": [ + 353, + 640, + 380, + 653 + ], + "score": 0.91, + "content": "f \\in { \\overline { { \\mathcal { F } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 639, + 414, + 655 + ], + "score": 1.0, + "content": ". Again,", + "type": "text" + }, + { + "bbox": [ + 414, + 641, + 471, + 653 + ], + "score": 0.91, + "content": "\\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 639, + 506, + 655 + ], + "score": 1.0, + "content": "implies", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 651, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 125, + 668 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 653, + 176, + 666 + ], + "score": 0.93, + "content": "f = \\hat { f } \\circ \\pi _ { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 651, + 198, + 668 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 199, + 654, + 224, + 665 + ], + "score": 0.9, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 651, + 289, + 668 + ], + "score": 1.0, + "content": ". By density of", + "type": "text" + }, + { + "bbox": [ + 289, + 653, + 299, + 664 + ], + "score": 0.87, + "content": "\\hat { \\mathcal F }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 651, + 360, + 668 + ], + "score": 1.0, + "content": ", there is some", + "type": "text" + }, + { + "bbox": [ + 361, + 653, + 390, + 665 + ], + "score": 0.91, + "content": "\\hat { h } \\in \\hat { \\mathcal F }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 651, + 432, + 668 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 433, + 653, + 505, + 667 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { x _ { \\mathcal { F } } } | \\hat { h } ( x _ { \\mathcal { F } } ) ) - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 666, + 454, + 682 + ], + "spans": [ + { + "bbox": [ + 107, + 666, + 154, + 680 + ], + "score": 0.91, + "content": "\\hat { f } ( x _ { \\mathcal { F } } ) | \\leq \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 666, + 253, + 682 + ], + "score": 1.0, + "content": ". But, by construction of", + "type": "text" + }, + { + "bbox": [ + 253, + 667, + 262, + 678 + ], + "score": 0.87, + "content": "\\hat { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 666, + 309, + 682 + ], + "score": 1.0, + "content": ", there exits", + "type": "text" + }, + { + "bbox": [ + 309, + 668, + 337, + 678 + ], + "score": 0.9, + "content": "h \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 666, + 376, + 682 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 376, + 667, + 424, + 679 + ], + "score": 0.93, + "content": "\\hat { h } \\circ \\pi _ { \\mathcal F } = h", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 666, + 454, + 682 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 639, + 506, + 682 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 687, + 478, + 711 + ], + "lines": [ + { + "bbox": [ + 132, + 687, + 478, + 711 + ], + "spans": [ + { + "bbox": [ + 132, + 687, + 478, + 711 + ], + "score": 0.9, + "content": "\\operatorname* { s u p } _ { x \\in X } \\left| h ( x ) - f ( x ) \\right| = \\operatorname* { s u p } _ { x \\in X } \\left| \\hat { h } ( \\pi _ { \\mathcal { F } } ( x ) ) - \\hat { f } ( \\pi _ { \\mathcal { F } } ( x ) ) \\right| = \\operatorname* { s u p } _ { x _ { \\mathcal { F } } \\in X _ { \\mathcal { F } } } \\left| \\hat { h } ( x _ { \\mathcal { F } } ) ) - \\hat { f } ( x _ { \\mathcal { F } } ) \\right| \\leq \\epsilon .", + "type": "interline_equation", + "image_path": "5ff2d1ed6829d210316d46b247e2822e0e43df1a6397bce9833b1c37126e98de.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 132, + 687, + 478, + 711 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 332, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 718, + 333, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 191, + 733 + ], + "score": 1.0, + "content": "As this holds for any", + "type": "text" + }, + { + "bbox": [ + 192, + 721, + 215, + 731 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 718, + 301, + 733 + ], + "score": 1.0, + "content": ", we have proven that", + "type": "text" + }, + { + "bbox": [ + 302, + 720, + 328, + 732 + ], + "score": 0.93, + "content": "f \\in { \\overline { { \\mathcal { F } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 718, + 333, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 718, + 333, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 102, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "We first need to extend Corollary 17 to vector-valued functions. For this, we need to have a vector-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "valued version of the Stone-Weierstrass theorem and as shown by the example below additional", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 125, + 228, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 228, + 137 + ], + "score": 1.0, + "content": "assumptions have to be made.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 139, + 505, + 207 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 506, + 152 + ], + "score": 1.0, + "content": "Example 19. We consider now the equivariant version of PointNet corresponding to the particular", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 104, + 149, + 155, + 164 + ], + "score": 1.0, + "content": "case where", + "type": "text" + }, + { + "bbox": [ + 155, + 151, + 165, + 161 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 149, + 262, + 164 + ], + "score": 1.0, + "content": "is a compact subset of", + "type": "text" + }, + { + "bbox": [ + 262, + 151, + 288, + 163 + ], + "score": 0.9, + "content": "( \\mathbb { R } ^ { p } ) ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 149, + 292, + 164 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 292, + 151, + 329, + 161 + ], + "score": 0.88, + "content": "Y = \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 149, + 349, + 164 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 151, + 480, + 163 + ], + "score": 0.89, + "content": "{ \\mathcal { F } } = \\{ x \\mapsto ( f ( x _ { 1 } ) , \\ldots , f ( x _ { n } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 149, + 486, + 164 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 486, + 151, + 505, + 163 + ], + "score": 0.77, + "content": "f \\in", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 107, + 162, + 145, + 173 + ], + "score": 0.9, + "content": "{ \\mathcal { C } } ( \\mathbb { R } , \\mathbb { R } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 161, + 208, + 174 + ], + "score": 1.0, + "content": ". Then clearly", + "type": "text" + }, + { + "bbox": [ + 208, + 162, + 218, + 172 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 161, + 298, + 174 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 298, + 162, + 339, + 174 + ], + "score": 0.92, + "content": "{ \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 161, + 506, + 174 + ], + "score": 1.0, + "content": "containing the constant function 1 and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 211, + 185 + ], + "score": 0.85, + "content": "\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , x ) , x \\in X \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 173, + 506, + 186 + ], + "score": 1.0, + "content": ". Hence if Corollary 17 would be true with vector-valued functions instead", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 285, + 196 + ], + "score": 1.0, + "content": "of real-valued functions, we would have that", + "type": "text" + }, + { + "bbox": [ + 286, + 184, + 295, + 194 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 183, + 340, + 196 + ], + "score": 1.0, + "content": "is dense in", + "type": "text" + }, + { + "bbox": [ + 341, + 183, + 376, + 195 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 183, + 506, + 196 + ], + "score": 1.0, + "content": ". But this can clearly not be true", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 337, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 117, + 207 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 195, + 180, + 207 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 194, + 298, + 207 + ], + "score": 1.0, + "content": "which is clearly not dense in", + "type": "text" + }, + { + "bbox": [ + 298, + 195, + 334, + 207 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 194, + 337, + 207 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 399, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 401, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 401, + 229 + ], + "score": 1.0, + "content": "We now present an extension of Corollary 17 also due to Timofte (2005):", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 503, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 192, + 244 + ], + "score": 1.0, + "content": "Proposition 20. Let", + "type": "text" + }, + { + "bbox": [ + 192, + 231, + 202, + 240 + ], + "score": 0.77, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 230, + 289, + 244 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 289, + 231, + 324, + 241 + ], + "score": 0.9, + "content": "Y = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 230, + 364, + 244 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 364, + 231, + 389, + 242 + ], + "score": 0.89, + "content": "p \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 230, + 410, + 244 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 410, + 230, + 468, + 243 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 230, + 505, + 244 + ], + "score": 1.0, + "content": ". If there", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 241, + 306, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 207, + 253 + ], + "score": 1.0, + "content": "exists a nonempty subset", + "type": "text" + }, + { + "bbox": [ + 208, + 241, + 262, + 253 + ], + "score": 0.93, + "content": "S \\subseteq { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 241, + 306, + 253 + ], + "score": 1.0, + "content": "such that:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 258, + 361, + 272 + ], + "lines": [ + { + "bbox": [ + 230, + 258, + 361, + 272 + ], + "spans": [ + { + "bbox": [ + 230, + 258, + 361, + 272 + ], + "score": 0.9, + "content": "S \\cdot { \\mathcal { F } } \\subseteq { \\mathcal { F } } a n d , \\rho \\left( S \\right) \\subseteq \\rho \\left( { \\mathcal { F } } \\right) .", + "type": "interline_equation", + "image_path": "9bf2e3fc0ffc42174ae95b1307d557c76c96407660c9aa2316facd18ff88cdaf.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 230, + 258, + 361, + 272 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 163, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 164, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 164, + 289 + ], + "score": 1.0, + "content": "Then we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 291, + 404, + 313 + ], + "lines": [ + { + "bbox": [ + 185, + 291, + 404, + 313 + ], + "spans": [ + { + "bbox": [ + 185, + 291, + 404, + 313 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } ( X , Y ) , \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , f ( x ) \\in \\overline { { \\mathcal { F } ( x ) } } \\right\\} , } \\end{array}", + "type": "interline_equation", + "image_path": "3985aa38ff42bdecea05fa73605e5ec939fbe675252c60270a27629afa252457.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 185, + 291, + 404, + 313 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 433, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 434, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 133, + 332 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 318, + 231, + 330 + ], + "score": 0.91, + "content": "{ \\mathcal { F } } ( x ) = \\{ f ( x ) , f \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 317, + 371, + 332 + ], + "score": 1.0, + "content": ". Moreover in (18), we can replace", + "type": "text" + }, + { + "bbox": [ + 371, + 318, + 395, + 330 + ], + "score": 0.87, + "content": "\\rho \\left( \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 317, + 408, + 332 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 408, + 318, + 430, + 330 + ], + "score": 0.85, + "content": "\\rho \\left( S \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 317, + 434, + 332 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 105, + 338, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 229, + 351 + ], + "score": 1.0, + "content": "Note that in the particular case", + "type": "text" + }, + { + "bbox": [ + 229, + 339, + 259, + 349 + ], + "score": 0.91, + "content": "Y = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 338, + 285, + 351 + ], + "score": 1.0, + "content": "and if", + "type": "text" + }, + { + "bbox": [ + 286, + 340, + 295, + 349 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 338, + 367, + 351 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 368, + 339, + 402, + 351 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 338, + 474, + 351 + ], + "score": 1.0, + "content": ", then we can take", + "type": "text" + }, + { + "bbox": [ + 475, + 339, + 504, + 349 + ], + "score": 0.87, + "content": "S = \\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 350, + 484, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 274, + 362 + ], + "score": 1.0, + "content": "in (17) and if the constant function 1 is in", + "type": "text" + }, + { + "bbox": [ + 275, + 351, + 284, + 360 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 351, + 307, + 362 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 307, + 350, + 350, + 362 + ], + "score": 0.92, + "content": "\\mathcal { F } ( x ) = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 351, + 484, + 362 + ], + "score": 1.0, + "content": ", so that we recover Corollary 17.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 229, + 379 + ], + "score": 1.0, + "content": "Now consider the case where", + "type": "text" + }, + { + "bbox": [ + 230, + 368, + 239, + 377 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 367, + 315, + 379 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 316, + 367, + 355, + 379 + ], + "score": 0.92, + "content": "\\mathcal { C } ( X , \\mathbb { R } ^ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 367, + 450, + 379 + ], + "score": 1.0, + "content": ". We need to find a set", + "type": "text" + }, + { + "bbox": [ + 450, + 367, + 505, + 379 + ], + "score": 0.94, + "content": "S \\in { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 341, + 390 + ], + "score": 1.0, + "content": "satisfying (17) i.e. with a better separating power than", + "type": "text" + }, + { + "bbox": [ + 341, + 379, + 351, + 388 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "but containing real-valued functions", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 387, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 147, + 403 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 147, + 389, + 184, + 400 + ], + "score": 0.9, + "content": "s f \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 387, + 215, + 403 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 216, + 389, + 244, + 399 + ], + "score": 0.9, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 387, + 264, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 389, + 296, + 401 + ], + "score": 0.91, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 387, + 468, + 403 + ], + "score": 1.0, + "content": ". In the sequel, we will consider the set", + "type": "text" + }, + { + "bbox": [ + 468, + 389, + 505, + 401 + ], + "score": 0.89, + "content": "\\mathcal { F } _ { s c a l } =", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 107, + 400, + 210, + 412 + ], + "score": 0.91, + "content": "\\{ f \\in { \\mathcal { C } } ( X , \\mathbb { R } ) : f { \\bf 1 } \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 399, + 282, + 413 + ], + "score": 1.0, + "content": ". We clearly have", + "type": "text" + }, + { + "bbox": [ + 283, + 400, + 344, + 411 + ], + "score": 0.92, + "content": "\\mathcal { F } _ { s c a l } \\cdot \\mathcal { F } \\subseteq \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 399, + 368, + 413 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 369, + 401, + 378, + 410 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "is a subalgebra. Hence, in this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 411, + 287, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 287, + 423 + ], + "score": 1.0, + "content": "setting, Prop. 20 can be rewritten as follows:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 507, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 183, + 440 + ], + "score": 1.0, + "content": "Corollary 21. Let", + "type": "text" + }, + { + "bbox": [ + 183, + 426, + 193, + 436 + ], + "score": 0.72, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 424, + 278, + 440 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 278, + 426, + 312, + 437 + ], + "score": 0.91, + "content": "Y = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 424, + 351, + 440 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 351, + 428, + 357, + 437 + ], + "score": 0.55, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 424, + 361, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 361, + 426, + 370, + 436 + ], + "score": 0.58, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 424, + 507, + 440 + ], + "score": 1.0, + "content": "be a finite group acting (continu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 437, + 465, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 146, + 450 + ], + "score": 1.0, + "content": "ously) on", + "type": "text" + }, + { + "bbox": [ + 146, + 438, + 156, + 447 + ], + "score": 0.77, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 437, + 175, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 437, + 235, + 449 + ], + "score": 0.92, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { I } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 437, + 452, + 450 + ], + "score": 1.0, + "content": "a (non-empty) set of invariant functions associated to", + "type": "text" + }, + { + "bbox": [ + 452, + 438, + 461, + 447 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 437, + 465, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 253, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 254, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 254, + 469 + ], + "score": 1.0, + "content": "Consider the following assumptions,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 130, + 474, + 411, + 487 + ], + "lines": [ + { + "bbox": [ + 130, + 474, + 411, + 488 + ], + "spans": [ + { + "bbox": [ + 130, + 474, + 142, + 488 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 475, + 152, + 485 + ], + "score": 0.78, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 474, + 230, + 488 + ], + "score": 1.0, + "content": "is a sub-algebra of", + "type": "text" + }, + { + "bbox": [ + 231, + 475, + 266, + 487 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 474, + 398, + 488 + ], + "score": 1.0, + "content": "and the constant function 1 is in", + "type": "text" + }, + { + "bbox": [ + 399, + 476, + 407, + 485 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 474, + 411, + 488 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 129, + 493, + 339, + 506 + ], + "lines": [ + { + "bbox": [ + 128, + 491, + 339, + 508 + ], + "spans": [ + { + "bbox": [ + 128, + 491, + 223, + 508 + ], + "score": 1.0, + "content": "2. 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For any", + "type": "text" + }, + { + "bbox": [ + 175, + 562, + 202, + 572 + ], + "score": 0.89, + "content": "x \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 559, + 253, + 574 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 253, + 561, + 280, + 573 + ], + "score": 0.91, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 559, + 319, + 574 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 319, + 561, + 339, + 573 + ], + "score": 0.9, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "has pairwise distinct coordinates, i.e., for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 571, + 369, + 586 + ], + "spans": [ + { + "bbox": [ + 141, + 571, + 190, + 586 + ], + "score": 1.0, + "content": "any indices", + "type": "text" + }, + { + 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For this, we need to have a vector-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "valued version of the Stone-Weierstrass theorem and as shown by the example below additional", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 125, + 228, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 228, + 137 + ], + "score": 1.0, + "content": "assumptions have to be made.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 102, + 506, + 137 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 139, + 505, + 207 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 506, + 152 + ], + "score": 1.0, + "content": "Example 19. 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Hence if Corollary 17 would be true with vector-valued functions instead", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 285, + 196 + ], + "score": 1.0, + "content": "of real-valued functions, we would have that", + "type": "text" + }, + { + "bbox": [ + 286, + 184, + 295, + 194 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 183, + 340, + 196 + ], + "score": 1.0, + "content": "is dense in", + "type": "text" + }, + { + "bbox": [ + 341, + 183, + 376, + 195 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 183, + 506, + 196 + ], + "score": 1.0, + "content": ". But this can clearly not be true", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 337, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 117, + 207 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 195, + 180, + 207 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 194, + 298, + 207 + ], + "score": 1.0, + "content": "which is clearly not dense in", + "type": "text" + }, + { + "bbox": [ + 298, + 195, + 334, + 207 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 194, + 337, + 207 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 139, + 506, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 399, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 401, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 401, + 229 + ], + "score": 1.0, + "content": "We now present an extension of Corollary 17 also due to Timofte (2005):", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 214, + 401, + 229 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 503, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 192, + 244 + ], + "score": 1.0, + "content": "Proposition 20. Let", + "type": "text" + }, + { + "bbox": [ + 192, + 231, + 202, + 240 + ], + "score": 0.77, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 230, + 289, + 244 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 289, + 231, + 324, + 241 + ], + "score": 0.9, + "content": "Y = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 230, + 364, + 244 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 364, + 231, + 389, + 242 + ], + "score": 0.89, + "content": "p \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 230, + 410, + 244 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 410, + 230, + 468, + 243 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 230, + 505, + 244 + ], + "score": 1.0, + "content": ". If there", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 241, + 306, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 207, + 253 + ], + "score": 1.0, + "content": "exists a nonempty subset", + "type": "text" + }, + { + "bbox": [ + 208, + 241, + 262, + 253 + ], + "score": 0.93, + "content": "S \\subseteq { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 241, + 306, + 253 + ], + "score": 1.0, + "content": "such that:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 230, + 505, + 253 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 258, + 361, + 272 + ], + "lines": [ + { + "bbox": [ + 230, + 258, + 361, + 272 + ], + "spans": [ + { + "bbox": [ + 230, + 258, + 361, + 272 + ], + "score": 0.9, + "content": "S \\cdot { \\mathcal { F } } \\subseteq { \\mathcal { F } } a n d , \\rho \\left( S \\right) \\subseteq \\rho \\left( { \\mathcal { F } } \\right) .", + "type": "interline_equation", + "image_path": "9bf2e3fc0ffc42174ae95b1307d557c76c96407660c9aa2316facd18ff88cdaf.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 230, + 258, + 361, + 272 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 163, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 164, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 164, + 289 + ], + "score": 1.0, + "content": "Then we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 276, + 164, + 289 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 291, + 404, + 313 + ], + "lines": [ + { + "bbox": [ + 185, + 291, + 404, + 313 + ], + "spans": [ + { + "bbox": [ + 185, + 291, + 404, + 313 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } ( X , Y ) , \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , f ( x ) \\in \\overline { { \\mathcal { F } ( x ) } } \\right\\} , } \\end{array}", + "type": "interline_equation", + "image_path": "3985aa38ff42bdecea05fa73605e5ec939fbe675252c60270a27629afa252457.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 185, + 291, + 404, + 313 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 433, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 434, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 133, + 332 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 318, + 231, + 330 + ], + "score": 0.91, + "content": "{ \\mathcal { F } } ( x ) = \\{ f ( x ) , f \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 317, + 371, + 332 + ], + "score": 1.0, + "content": ". Moreover in (18), we can replace", + "type": "text" + }, + { + "bbox": [ + 371, + 318, + 395, + 330 + ], + "score": 0.87, + "content": "\\rho \\left( \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 317, + 408, + 332 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 408, + 318, + 430, + 330 + ], + "score": 0.85, + "content": "\\rho \\left( S \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 317, + 434, + 332 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 317, + 434, + 332 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 338, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 229, + 351 + ], + "score": 1.0, + "content": "Note that in the particular case", + "type": "text" + }, + { + "bbox": [ + 229, + 339, + 259, + 349 + ], + "score": 0.91, + "content": "Y = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 338, + 285, + 351 + ], + "score": 1.0, + "content": "and if", + "type": "text" + }, + { + "bbox": [ + 286, + 340, + 295, + 349 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 338, + 367, + 351 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 368, + 339, + 402, + 351 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 338, + 474, + 351 + ], + "score": 1.0, + "content": ", then we can take", + "type": "text" + }, + { + "bbox": [ + 475, + 339, + 504, + 349 + ], + "score": 0.87, + "content": "S = \\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 350, + 484, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 274, + 362 + ], + "score": 1.0, + "content": "in (17) and if the constant function 1 is in", + "type": "text" + }, + { + "bbox": [ + 275, + 351, + 284, + 360 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 351, + 307, + 362 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 307, + 350, + 350, + 362 + ], + "score": 0.92, + "content": "\\mathcal { F } ( x ) = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 351, + 484, + 362 + ], + "score": 1.0, + "content": ", so that we recover Corollary 17.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 338, + 504, + 362 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 229, + 379 + ], + "score": 1.0, + "content": "Now consider the case where", + "type": "text" + }, + { + "bbox": [ + 230, + 368, + 239, + 377 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 367, + 315, + 379 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 316, + 367, + 355, + 379 + ], + "score": 0.92, + "content": "\\mathcal { C } ( X , \\mathbb { R } ^ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 367, + 450, + 379 + ], + "score": 1.0, + "content": ". We need to find a set", + "type": "text" + }, + { + "bbox": [ + 450, + 367, + 505, + 379 + ], + "score": 0.94, + "content": "S \\in { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 341, + 390 + ], + "score": 1.0, + "content": "satisfying (17) i.e. with a better separating power than", + "type": "text" + }, + { + "bbox": [ + 341, + 379, + 351, + 388 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "but containing real-valued functions", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 387, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 147, + 403 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 147, + 389, + 184, + 400 + ], + "score": 0.9, + "content": "s f \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 387, + 215, + 403 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 216, + 389, + 244, + 399 + ], + "score": 0.9, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 387, + 264, + 403 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 265, + 389, + 296, + 401 + ], + "score": 0.91, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 387, + 468, + 403 + ], + "score": 1.0, + "content": ". 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We clearly have", + "type": "text" + }, + { + "bbox": [ + 283, + 400, + 344, + 411 + ], + "score": 0.92, + "content": "\\mathcal { F } _ { s c a l } \\cdot \\mathcal { F } \\subseteq \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 399, + 368, + 413 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 369, + 401, + 378, + 410 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "is a subalgebra. Hence, in this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 411, + 287, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 287, + 423 + ], + "score": 1.0, + "content": "setting, Prop. 20 can be rewritten as follows:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 367, + 505, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 507, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 183, + 440 + ], + "score": 1.0, + "content": "Corollary 21. Let", + "type": "text" + }, + { + "bbox": [ + 183, + 426, + 193, + 436 + ], + "score": 0.72, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 424, + 278, + 440 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 278, + 426, + 312, + 437 + ], + "score": 0.91, + "content": "Y = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 424, + 351, + 440 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 351, + 428, + 357, + 437 + ], + "score": 0.55, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 424, + 361, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 361, + 426, + 370, + 436 + ], + "score": 0.58, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 424, + 507, + 440 + ], + "score": 1.0, + "content": "be a finite group acting (continu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 437, + 465, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 146, + 450 + ], + "score": 1.0, + "content": "ously) on", + "type": "text" + }, + { + "bbox": [ + 146, + 438, + 156, + 447 + ], + "score": 0.77, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 437, + 175, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 437, + 235, + 449 + ], + "score": 0.92, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { I } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 437, + 452, + 450 + ], + "score": 1.0, + "content": "a (non-empty) set of invariant functions associated to", + "type": "text" + }, + { + "bbox": [ + 452, + 438, + 461, + 447 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 437, + 465, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 424, + 507, + 450 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 253, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 254, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 254, + 469 + ], + "score": 1.0, + "content": "Consider the following assumptions,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 452, + 254, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 474, + 411, + 487 + ], + "lines": [ + { + "bbox": [ + 130, + 474, + 411, + 488 + ], + "spans": [ + { + "bbox": [ + 130, + 474, + 142, + 488 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 475, + 152, + 485 + ], + "score": 0.78, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 474, + 230, + 488 + ], + "score": 1.0, + "content": "is a sub-algebra of", + "type": "text" + }, + { + "bbox": [ + 231, + 475, + 266, + 487 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 474, + 398, + 488 + ], + "score": 1.0, + "content": "and the constant function 1 is in", + "type": "text" + }, + { + "bbox": [ + 399, + 476, + 407, + 485 + ], + "score": 0.81, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 474, + 411, + 488 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 130, + 474, + 411, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 493, + 339, + 506 + ], + "lines": [ + { + "bbox": [ + 128, + 491, + 339, + 508 + ], + "spans": [ + { + "bbox": [ + 128, + 491, + 223, + 508 + ], + "score": 1.0, + "content": "2. The set of functions", + "type": "text" + }, + { + "bbox": [ + 223, + 493, + 292, + 506 + ], + "score": 0.94, + "content": "\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 491, + 339, + 508 + ], + "score": 1.0, + "content": "defined by,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 128, + 491, + 339, + 508 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 510, + 393, + 524 + ], + "lines": [ + { + "bbox": [ + 253, + 510, + 393, + 524 + ], + "spans": [ + { + "bbox": [ + 253, + 510, + 393, + 524 + ], + "score": 0.89, + "content": "\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f { \\bf 1 } \\in \\mathcal { F } \\}", + "type": "interline_equation", + "image_path": "8a12e10f4bc61eb8fe4a5767fac0403ed142c50363ebd3d4959cfb569fb04c0e.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 253, + 510, + 393, + 524 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 529, + 171, + 541 + ], + "lines": [ + { + "bbox": [ + 140, + 528, + 172, + 543 + ], + "spans": [ + { + "bbox": [ + 140, + 528, + 172, + 543 + ], + "score": 1.0, + "content": "satisfy,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 140, + 528, + 172, + 543 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 283, + 539, + 364, + 553 + ], + "lines": [ + { + "bbox": [ + 283, + 539, + 364, + 553 + ], + "spans": [ + { + "bbox": [ + 283, + 539, + 364, + 553 + ], + "score": 0.87, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right) .", + "type": "interline_equation", + "image_path": "7902cbbfce716d9f0a2b65aa41abfac2d75b55d6bfeb3453cd81d41988da130e.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 283, + 539, + 364, + 553 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 129, + 560, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 128, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 128, + 559, + 175, + 574 + ], + "score": 1.0, + "content": "3. For any", + "type": "text" + }, + { + "bbox": [ + 175, + 562, + 202, + 572 + ], + "score": 0.89, + "content": "x \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 559, + 253, + 574 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 253, + 561, + 280, + 573 + ], + "score": 0.91, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 559, + 319, + 574 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 319, + 561, + 339, + 573 + ], + "score": 0.9, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "has pairwise distinct coordinates, i.e., for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 571, + 369, + 586 + ], + "spans": [ + { + "bbox": [ + 141, + 571, + 190, + 586 + ], + "score": 1.0, + "content": "any indices", + "type": "text" + }, + { + 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] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 225, + 609, + 386, + 624 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 506, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 455, + 647 + ], + "score": 1.0, + "content": "Note that Assumptions 1 and 2 ensures that (17) is valid, while Assumption 3 ensures that", + "type": "text" + }, + { + "bbox": [ + 456, + 635, + 503, + 647 + ], + "score": 0.92, + "content": "\\mathcal { F } ( \\boldsymbol { x } ) = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 633, + 506, + 647 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 322, + 659 + ], + "score": 1.0, + "content": "Unfortunately, in the equivariant case, the condition", + "type": "text" + }, + { + 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696, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 698, + 141, + 708 + ], + "score": 0.9, + "content": "Y = \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 696, + 163, + 711 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 164, + 698, + 197, + 708 + ], + "score": 0.91, + "content": "\\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 696, + 238, + 711 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 238, + 698, + 263, + 709 + ], + "score": 0.89, + "content": "p \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 696, + 453, + 711 + ], + "score": 1.0, + "content": ". 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But these points will", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 206, + 128 + ], + "score": 1.0, + "content": "typically be separated by", + "type": "text" + }, + { + "bbox": [ + 206, + 116, + 216, + 125 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 115, + 272, + 128 + ], + "score": 1.0, + "content": ", since for any", + "type": "text" + }, + { + "bbox": [ + 272, + 115, + 299, + 127 + ], + "score": 0.91, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 115, + 337, + 128 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 338, + 115, + 420, + 127 + ], + "score": 0.93, + "content": "f ( \\sigma \\star x ) = \\sigma \\star f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "which is not equal to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 126, + 213, + 139 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 127, + 138 + ], + "score": 0.91, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 126, + 155, + 139 + ], + "score": 1.0, + "content": "unless", + "type": "text" + }, + { + "bbox": [ + 156, + 127, + 163, + 138 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 126, + 213, + 139 + ], + "score": 1.0, + "content": "is invariant.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 406, + 156 + ], + "score": 1.0, + "content": "We see that we need somehow to require a weaker separating power for", + "type": "text" + }, + { + "bbox": [ + 407, + 144, + 416, + 154 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 142, + 506, + 156 + ], + "score": 1.0, + "content": ". More formally, two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "isomorphic graphs will have permuted outputs through an equivariant function, but should not be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 227, + 178 + ], + "score": 1.0, + "content": "considered as separated. Let", + "type": "text" + }, + { + "bbox": [ + 228, + 165, + 346, + 177 + ], + "score": 0.72, + "content": "O r b ( x ) = \\{ \\bar { \\sigma } \\star x , \\sigma \\in S _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 165, + 366, + 178 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 165, + 482, + 177 + ], + "score": 0.84, + "content": "O r b ( y ) = \\{ \\sigma \\star y , \\sigma \\in S _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 165, + 506, + 178 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 207, + 189 + ], + "score": 1.0, + "content": "any equivariant function", + "type": "text" + }, + { + "bbox": [ + 207, + 177, + 266, + 188 + ], + "score": 0.92, + "content": "f \\in { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 176, + 301, + 189 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 302, + 177, + 349, + 188 + ], + "score": 0.88, + "content": "z \\in O r b ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 176, + 389, + 189 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 390, + 177, + 465, + 188 + ], + "score": 0.91, + "content": "f ( z ) \\in O r b ( f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 176, + 506, + 189 + ], + "score": 1.0, + "content": ". 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We now demonstrate how Theorem 22 can be used to recover the universality results", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "in Segol & Lipman (2020). In this paper, the authors study equivariant neural network architectures", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 346, + 605 + ], + "score": 1.0, + "content": "working with unordered sets, corresponding in our case to", + "type": "text" + }, + { + "bbox": [ + 346, + 593, + 404, + 603 + ], + "score": 0.92, + "content": "X = Y = \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "and the group being the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 176, + 616 + ], + "score": 1.0, + "content": "symmetric group", + "type": "text" + }, + { + "bbox": [ + 176, + 604, + 188, + 614 + ], + "score": 0.87, + "content": "S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 604, + 506, + 616 + ], + "score": 1.0, + "content": ". They show that the PointNet architecture cannot approximate any (continuous)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "equivariant function and that adding a single so-called transmission layer is enough to make this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 626, + 198, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 198, + 637 + ], + "score": 1.0, + "content": "architecture universal.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 317, + 655 + ], + "score": 1.0, + "content": "Indeed, PointNet can only learn maps of the form", + "type": "text" + }, + { + "bbox": [ + 317, + 642, + 441, + 654 + ], + "score": 0.93, + "content": "x \\in \\mathbb { R } ^ { n } \\mapsto ( f ( x _ { 1 } ) \\dots f ( x _ { n } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 641, + 506, + 655 + ], + "score": 1.0, + "content": ", which are not", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "universal in the class of equivariant functions, as shown by Segol & Lipman (2020, Lem. 3). Now, their", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 664, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 262, + 676 + ], + "score": 1.0, + "content": "transmission layer is a map of the form", + "type": "text" + }, + { + "bbox": [ + 262, + 664, + 340, + 676 + ], + "score": 0.9, + "content": "x \\in \\mathbb { R } ^ { n } \\mapsto ( \\mathbf { 1 } ^ { T } \\dot { x } ) \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 664, + 506, + 676 + ], + "score": 1.0, + "content": ". Therefore, in PointNetST, adding such a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 675, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 293, + 689 + ], + "score": 1.0, + "content": "layer precisely adds a large class of functions to", + "type": "text" + }, + { + "bbox": [ + 294, + 675, + 482, + 688 + ], + "score": 0.86, + "content": "{ \\mathcal { F } } = \\{ ( f ( x _ { 1 } , \\sum _ { i } g ( x _ { i } ) ) , \\dots , f ( x _ { n } , \\sum _ { i } g ( x _ { i } { \\bar { ) } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 675, + 487, + 689 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 487, + 675, + 505, + 687 + ], + "score": 0.85, + "content": "f \\in", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 165, + 699 + ], + "score": 0.89, + "content": "\\mathcal { C } ( \\mathbb { R } \\times \\mathbb { R } ^ { h } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 686, + 170, + 700 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 170, + 687, + 260, + 699 + ], + "score": 0.9, + "content": "g \\in \\mathcal { C } ( \\mathbb { R } , \\mathbb { R } ^ { h } ) , h \\geq 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 686, + 265, + 700 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 265, + 688, + 275, + 698 + ], + "score": 0.78, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "is still an algebra and as shown in Example 19, we have", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 210, + 710 + ], + "score": 0.82, + "content": "\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , x ) , x \\in X \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 698, + 296, + 712 + ], + "score": 1.0, + "content": ". Moreover, we have", + "type": "text" + }, + { + "bbox": [ + 296, + 699, + 370, + 711 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { s c a l } = \\mathcal { C } _ { I } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "by Lem. 33 in particular, we get", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 204, + 722 + ], + "score": 0.72, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) = \\{ ( x , \\sigma \\star x )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 709, + 209, + 723 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 209, + 710, + 243, + 722 + ], + "score": 0.61, + "content": "x \\in X \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 709, + 357, + 723 + ], + "score": 1.0, + "content": ", so that we obviously have", + "type": "text" + }, + { + "bbox": [ + 357, + 710, + 447, + 722 + ], + "score": 0.9, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 709, + 507, + 723 + ], + "score": 1.0, + "content": ". In summary,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 719, + 369, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 323, + 733 + ], + "score": 1.0, + "content": "Theorem 22 implies the universality of PointNetST in", + "type": "text" + }, + { + "bbox": [ + 324, + 721, + 365, + 732 + ], + "score": 0.93, + "content": "\\mathcal { C } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 719, + 369, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5 + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 234, + 95 + ], + "score": 1.0, + "content": "Now consider the case where", + "type": "text" + }, + { + "bbox": [ + 235, + 82, + 302, + 95 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 82, + 507, + 95 + ], + "score": 1.0, + "content": "is a subalgebra of equivariant functions. Then,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 149, + 105 + ], + "score": 0.92, + "content": "f \\in \\mathcal { F } _ { s c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 92, + 286, + 107 + ], + "score": 1.0, + "content": "needs to be invariant in order for", + "type": "text" + }, + { + "bbox": [ + 287, + 95, + 300, + 105 + ], + "score": 0.87, + "content": "f \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 92, + 426, + 107 + ], + "score": 1.0, + "content": "to be equivariant and hence in", + "type": "text" + }, + { + "bbox": [ + 427, + 94, + 436, + 104 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 92, + 506, + 107 + ], + "score": 1.0, + "content": ". As a result, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 139, + 117 + ], + "score": 1.0, + "content": "see that", + "type": "text" + }, + { + "bbox": [ + 140, + 105, + 162, + 115 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { s c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 104, + 270, + 117 + ], + "score": 1.0, + "content": "will not separate points of", + "type": "text" + }, + { + "bbox": [ + 270, + 105, + 280, + 114 + ], + "score": 0.86, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 104, + 368, + 117 + ], + "score": 1.0, + "content": "in the same orbit, i.e.", + "type": "text" + }, + { + "bbox": [ + 368, + 106, + 375, + 114 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 104, + 393, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 106, + 416, + 115 + ], + "score": 0.86, + "content": "\\sigma \\star x", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ". But these points will", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 206, + 128 + ], + "score": 1.0, + "content": "typically be separated by", + "type": "text" + }, + { + "bbox": [ + 206, + 116, + 216, + 125 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 115, + 272, + 128 + ], + "score": 1.0, + "content": ", since for any", + "type": "text" + }, + { + "bbox": [ + 272, + 115, + 299, + 127 + ], + "score": 0.91, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 115, + 337, + 128 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 338, + 115, + 420, + 127 + ], + "score": 0.93, + "content": "f ( \\sigma \\star x ) = \\sigma \\star f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "which is not equal to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 126, + 213, + 139 + ], + "spans": [ + { + "bbox": [ + 107, + 127, + 127, + 138 + ], + "score": 0.91, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 126, + 155, + 139 + ], + "score": 1.0, + "content": "unless", + "type": "text" + }, + { + "bbox": [ + 156, + 127, + 163, + 138 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 126, + 213, + 139 + ], + "score": 1.0, + "content": "is invariant.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 82, + 507, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 406, + 156 + ], + "score": 1.0, + "content": "We see that we need somehow to require a weaker separating power for", + "type": "text" + }, + { + "bbox": [ + 407, + 144, + 416, + 154 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 142, + 506, + 156 + ], + "score": 1.0, + "content": ". More formally, two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "isomorphic graphs will have permuted outputs through an equivariant function, but should not be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 227, + 178 + ], + "score": 1.0, + "content": "considered as separated. Let", + "type": "text" + }, + { + "bbox": [ + 228, + 165, + 346, + 177 + ], + "score": 0.72, + "content": "O r b ( x ) = \\{ \\bar { \\sigma } \\star x , \\sigma \\in S _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 165, + 366, + 178 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 165, + 482, + 177 + ], + "score": 0.84, + "content": "O r b ( y ) = \\{ \\sigma \\star y , \\sigma \\in S _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 165, + 506, + 178 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 207, + 189 + ], + "score": 1.0, + "content": "any equivariant function", + "type": "text" + }, + { + "bbox": [ + 207, + 177, + 266, + 188 + ], + "score": 0.92, + "content": "f \\in { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 176, + 301, + 189 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 302, + 177, + 349, + 188 + ], + "score": 0.88, + "content": "z \\in O r b ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 176, + 389, + 189 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 390, + 177, + 465, + 188 + ], + "score": 0.91, + "content": "f ( z ) \\in O r b ( f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 176, + 506, + 189 + ], + "score": 1.0, + "content": ". Then let", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 186, + 391, + 200 + ], + "spans": [ + { + "bbox": [ + 107, + 187, + 168, + 199 + ], + "score": 0.93, + "content": "\\pi : Y \\to Y / S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 186, + 280, + 200 + ], + "score": 1.0, + "content": "be the canonical projection", + "type": "text" + }, + { + "bbox": [ + 281, + 187, + 344, + 199 + ], + "score": 0.92, + "content": "\\pi ( y ) = O r b ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 186, + 391, + 200 + ], + "score": 1.0, + "content": ". We define", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 142, + 506, + 200 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 205, + 436, + 235 + ], + "lines": [ + { + "bbox": [ + 174, + 205, + 436, + 235 + ], + "spans": [ + { + "bbox": [ + 174, + 205, + 436, + 235 + ], + "score": 0.9, + "content": "\\begin{array} { r } { ( x , x ^ { \\prime } ) \\in \\rho \\left( \\pi \\circ \\mathcal { F } \\right) \\Leftrightarrow \\forall f \\in \\mathcal { F } , O r b ( f ( x ) ) = O r b ( f ( x ^ { \\prime } ) ) } \\\\ { \\Leftrightarrow \\forall f \\in \\mathcal { F } , \\exists \\sigma \\in { \\cal S } _ { n } , f ( \\sigma \\star x ) = f ( x ^ { \\prime } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "daeeb5dd9b853b3fb942d85b56c15bd90e1649f477120213c2c416c75e320852.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 174, + 205, + 436, + 215.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 174, + 215.0, + 436, + 225.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 174, + 225.0, + 436, + 235.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 506, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 239, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 214, + 254 + ], + "score": 1.0, + "content": "In particular, we see that if", + "type": "text" + }, + { + "bbox": [ + 214, + 241, + 266, + 253 + ], + "score": 0.92, + "content": "x ^ { \\prime } \\in O r b ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 239, + 287, + 254 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 288, + 241, + 365, + 253 + ], + "score": 0.92, + "content": "( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } ) \\in \\rho \\left( \\pi \\circ \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 239, + 397, + 254 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 397, + 241, + 459, + 253 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\in { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 239, + 506, + 254 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 180, + 329, + 190, + 339 + ], + "score": 0.72, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 327, + 276, + 342 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 276, + 329, + 311, + 339 + ], + "score": 0.9, + "content": "Y = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 327, + 349, + 342 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 350, + 331, + 356, + 340 + ], + "score": 0.53, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 327, + 360, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 360, + 329, + 369, + 339 + ], + "score": 0.52, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "be a finite group acting (continu-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 339, + 440, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 146, + 353 + ], + "score": 1.0, + "content": "ously) on", + "type": "text" + }, + { + "bbox": [ + 146, + 340, + 156, + 350 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 339, + 174, + 353 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 340, + 184, + 350 + ], + "score": 0.72, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 339, + 203, + 353 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 340, + 266, + 352 + ], + "score": 0.95, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 339, + 440, + 353 + ], + "score": 1.0, + "content": "a (non-empty) set of equivariant functions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 327, + 506, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 504, + 380 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 151, + 370 + ], + "score": 1.0, + "content": "Denote by", + "type": "text" + }, + { + "bbox": [ + 152, + 357, + 223, + 369 + ], + "score": 0.91, + "content": "\\pi : Y \\longrightarrow Y / G", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 355, + 421, + 370 + ], + "score": 1.0, + "content": "the canonical projection on the quotient space", + "type": "text" + }, + { + "bbox": [ + 421, + 357, + 442, + 369 + ], + "score": 0.9, + "content": "Y / G", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 355, + 506, + 370 + ], + "score": 1.0, + "content": ". 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We now demonstrate how Theorem 22 can be used to recover the universality results", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "in Segol & Lipman (2020). In this paper, the authors study equivariant neural network architectures", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 346, + 605 + ], + "score": 1.0, + "content": "working with unordered sets, corresponding in our case to", + "type": "text" + }, + { + "bbox": [ + 346, + 593, + 404, + 603 + ], + "score": 0.92, + "content": "X = Y = \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "and the group being the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 176, + 616 + ], + "score": 1.0, + "content": "symmetric group", + "type": "text" + }, + { + "bbox": [ + 176, + 604, + 188, + 614 + ], + "score": 0.87, + "content": "S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 604, + 506, + 616 + ], + "score": 1.0, + "content": ". They show that the PointNet architecture cannot approximate any (continuous)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "equivariant function and that adding a single so-called transmission layer is enough to make this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 626, + 198, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 198, + 637 + ], + "score": 1.0, + "content": "architecture universal.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 571, + 506, + 637 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 317, + 655 + ], + "score": 1.0, + "content": "Indeed, PointNet can only learn maps of the form", + "type": "text" + }, + { + "bbox": [ + 317, + 642, + 441, + 654 + ], + "score": 0.93, + "content": "x \\in \\mathbb { R } ^ { n } \\mapsto ( f ( x _ { 1 } ) \\dots f ( x _ { n } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 641, + 506, + 655 + ], + "score": 1.0, + "content": ", which are not", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "universal in the class of equivariant functions, as shown by Segol & Lipman (2020, Lem. 3). Now, their", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 664, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 262, + 676 + ], + "score": 1.0, + "content": "transmission layer is a map of the form", + "type": "text" + }, + { + "bbox": [ + 262, + 664, + 340, + 676 + ], + "score": 0.9, + "content": "x \\in \\mathbb { R } ^ { n } \\mapsto ( \\mathbf { 1 } ^ { T } \\dot { x } ) \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 664, + 506, + 676 + ], + "score": 1.0, + "content": ". Therefore, in PointNetST, adding such a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 675, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 293, + 689 + ], + "score": 1.0, + "content": "layer precisely adds a large class of functions to", + "type": "text" + }, + { + "bbox": [ + 294, + 675, + 482, + 688 + ], + "score": 0.86, + "content": "{ \\mathcal { F } } = \\{ ( f ( x _ { 1 } , \\sum _ { i } g ( x _ { i } ) ) , \\dots , f ( x _ { n } , \\sum _ { i } g ( x _ { i } { \\bar { ) } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 675, + 487, + 689 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 487, + 675, + 505, + 687 + ], + "score": 0.85, + "content": "f \\in", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 165, + 699 + ], + "score": 0.89, + "content": "\\mathcal { C } ( \\mathbb { R } \\times \\mathbb { R } ^ { h } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 686, + 170, + 700 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 170, + 687, + 260, + 699 + ], + "score": 0.9, + "content": "g \\in \\mathcal { C } ( \\mathbb { R } , \\mathbb { R } ^ { h } ) , h \\geq 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 686, + 265, + 700 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 265, + 688, + 275, + 698 + ], + "score": 0.78, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "is still an algebra and as shown in Example 19, we have", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 210, + 710 + ], + "score": 0.82, + "content": "\\rho \\left( { \\mathcal { F } } \\right) = \\{ ( x , x ) , x \\in X \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 698, + 296, + 712 + ], + "score": 1.0, + "content": ". Moreover, we have", + "type": "text" + }, + { + "bbox": [ + 296, + 699, + 370, + 711 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { s c a l } = \\mathcal { C } _ { I } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "by Lem. 33 in particular, we get", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 204, + 722 + ], + "score": 0.72, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) = \\{ ( x , \\sigma \\star x )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 709, + 209, + 723 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 209, + 710, + 243, + 722 + ], + "score": 0.61, + "content": "x \\in X \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 709, + 357, + 723 + ], + "score": 1.0, + "content": ", so that we obviously have", + "type": "text" + }, + { + "bbox": [ + 357, + 710, + 447, + 722 + ], + "score": 0.9, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 709, + 507, + 723 + ], + "score": 1.0, + "content": ". In summary,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 719, + 369, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 323, + 733 + ], + "score": 1.0, + "content": "Theorem 22 implies the universality of PointNetST in", + "type": "text" + }, + { + "bbox": [ + 324, + 721, + 365, + 732 + ], + "score": 0.93, + "content": "\\mathcal { C } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 719, + 369, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 641, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 457, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 458, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 458, + 95 + ], + "score": 1.0, + "content": "D.5 A PRELIMINARY VERSION OF THE EQUIVARIANT APPROXIMATION THEOREM", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 103, + 423, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 421, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 421, + 117 + ], + "score": 1.0, + "content": "We start by proving a version of Theorem 22 with a slightly weaker condition:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 108, + 118, + 504, + 143 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 194, + 133 + ], + "score": 1.0, + "content": "Proposition 24. Let", + "type": "text" + }, + { + "bbox": [ + 195, + 120, + 205, + 129 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 117, + 297, + 133 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 298, + 119, + 337, + 130 + ], + "score": 0.9, + "content": "Y ~ = ~ \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 117, + 379, + 133 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 379, + 122, + 386, + 131 + ], + "score": 0.49, + "content": "p _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 117, + 391, + 133 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 391, + 119, + 401, + 129 + ], + "score": 0.55, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 117, + 506, + 133 + ], + "score": 1.0, + "content": "be a finite group acting", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 472, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 178, + 143 + ], + "score": 1.0, + "content": "(continuously) on", + "type": "text" + }, + { + "bbox": [ + 179, + 131, + 189, + 140 + ], + "score": 0.76, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 129, + 207, + 143 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 131, + 217, + 140 + ], + "score": 0.75, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 129, + 235, + 143 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 236, + 131, + 299, + 142 + ], + "score": 0.91, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 129, + 472, + 143 + ], + "score": 1.0, + "content": "a (non-empty) set of equivariant functions.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 147, + 253, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 254, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 254, + 162 + ], + "score": 1.0, + "content": "Consider the following assumptions,", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 130, + 168, + 262, + 181 + ], + "lines": [ + { + "bbox": [ + 129, + 167, + 261, + 183 + ], + "spans": [ + { + "bbox": [ + 129, + 167, + 142, + 183 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 169, + 152, + 179 + ], + "score": 0.77, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 167, + 217, + 183 + ], + "score": 1.0, + "content": "is a subalgebra", + "type": "text" + }, + { + "bbox": [ + 217, + 169, + 258, + 181 + ], + "score": 0.92, + "content": "{ \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 167, + 261, + 183 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 129, + 188, + 387, + 201 + ], + "lines": [ + { + "bbox": [ + 128, + 186, + 387, + 203 + ], + "spans": [ + { + "bbox": [ + 128, + 186, + 271, + 203 + ], + "score": 1.0, + "content": "2. The set of real-valued functions", + "type": "text" + }, + { + "bbox": [ + 271, + 189, + 340, + 201 + ], + "score": 0.92, + "content": "\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 186, + 387, + 203 + ], + "score": 1.0, + "content": "defined by,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 207, + 392, + 221 + ], + "lines": [ + { + "bbox": [ + 253, + 207, + 392, + 221 + ], + "spans": [ + { + "bbox": [ + 253, + 207, + 392, + 221 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f \\mathbf { 1 } \\in \\mathcal { F } \\}", + "type": "interline_equation", + "image_path": "12582686c959485381f9aced585e93626563dbe9a5b653278b8db91a3b7216d4.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 253, + 207, + 392, + 221 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 227, + 177, + 239 + ], + "lines": [ + { + "bbox": [ + 140, + 225, + 180, + 241 + ], + "spans": [ + { + "bbox": [ + 140, + 225, + 180, + 241 + ], + "score": 1.0, + "content": "satisfies,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 245, + 450, + 259 + ], + "lines": [ + { + "bbox": [ + 196, + 245, + 450, + 259 + ], + "spans": [ + { + "bbox": [ + 196, + 245, + 450, + 259 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\left\\{ \\left( x , x ^ { \\prime } \\right) \\in X \\times X : \\exists g \\in G , \\left( g \\cdot x , x ^ { \\prime } \\right) \\in \\rho \\left( \\mathcal { F } \\right) \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "d33ca0a723e770e49f6d1dfa964e55a278f958eae983b901a40bbf1ea9145dc2.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 196, + 245, + 450, + 259 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 272, + 375, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 376, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 185, + 287 + ], + "score": 1.0, + "content": "Then the closure of", + "type": "text" + }, + { + "bbox": [ + 185, + 273, + 194, + 282 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 270, + 376, + 287 + ], + "score": 1.0, + "content": "(for the topology of uniform convergence) is,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 289, + 438, + 312 + ], + "lines": [ + { + "bbox": [ + 171, + 289, + 438, + 312 + ], + "spans": [ + { + "bbox": [ + 171, + 289, + 438, + 312 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\overline { { \\mathcal { F } ( x ) } } \\right\\} , } \\end{array}", + "type": "interline_equation", + "image_path": "05ec7a812381de297576716b9420b6e71e43c8146683f55dce0ab6c2214cfdc1.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 171, + 289, + 438, + 312 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 234, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 234, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 133, + 332 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 317, + 231, + 330 + ], + "score": 0.91, + "content": "{ \\mathcal { F } } ( x ) = \\{ f ( x ) , f \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 317, + 234, + 332 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 504, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 504, + 352 + ], + "score": 1.0, + "content": "The proof of this theorem relies on two main ingredients. First, following the elegant idea of Maehara", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "& Hoang (2019), we augment the input space to transform the vector-valued equivariant functions", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 362, + 435, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 435, + 375 + ], + "score": 1.0, + "content": "into scalar maps. Second, we apply the fine-grained approximation result Cor. 17.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 495, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 496, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 496, + 404 + ], + "score": 1.0, + "content": "Proof. As uniform convergence implies point-wise convergence, the first inclusion is immediate,", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 408, + 438, + 430 + ], + "lines": [ + { + "bbox": [ + 171, + 408, + 438, + 430 + ], + "spans": [ + { + "bbox": [ + 171, + 408, + 438, + 430 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } \\subseteq \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\overline { { \\mathcal { F } ( x ) } } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "794a0f3fc800995c49e2b2dd3fc6908c6567345cf843ee5678134b228bcc364e.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 171, + 408, + 438, + 430 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 436, + 321, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 321, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 321, + 449 + ], + "score": 1.0, + "content": "The rest of the proof is devoted to the other direction.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 507, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 217, + 467 + ], + "score": 1.0, + "content": "For convenience, denote by", + "type": "text" + }, + { + "bbox": [ + 218, + 454, + 226, + 463 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 451, + 472, + 467 + ], + "score": 1.0, + "content": "the family of linear forms associated to the canonical basis of", + "type": "text" + }, + { + "bbox": [ + 473, + 454, + 485, + 463 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 451, + 507, + 467 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 471, + 385, + 486 + ], + "lines": [ + { + "bbox": [ + 225, + 471, + 385, + 486 + ], + "spans": [ + { + "bbox": [ + 225, + 471, + 385, + 486 + ], + "score": 0.93, + "content": "\\Phi = \\{ y \\mapsto y _ { i } : 1 \\leq i \\leq p \\} \\subseteq { \\mathcal { C } } ( Y , \\mathbb { R } ) .", + "type": "interline_equation", + "image_path": "2d37fd88de4cf41ff9da7ce73017a1dd4c61010efa217538c346b53229c28d4d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 225, + 471, + 385, + 486 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 506, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 261, + 506 + ], + "score": 1.0, + "content": "Define our augmented input space as", + "type": "text" + }, + { + "bbox": [ + 261, + 492, + 315, + 505 + ], + "score": 0.92, + "content": "{ \\tilde { X } } = X \\times \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 492, + 335, + 506 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 335, + 495, + 344, + 504 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 492, + 396, + 506 + ], + "score": 1.0, + "content": "is finite and", + "type": "text" + }, + { + "bbox": [ + 397, + 495, + 407, + 504 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 492, + 458, + 506 + ], + "score": 1.0, + "content": "is compact,", + "type": "text" + }, + { + "bbox": [ + 458, + 492, + 469, + 504 + ], + "score": 0.86, + "content": "\\tilde { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "is still a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 244, + 518 + ], + "score": 1.0, + "content": "compact space. 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Define", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 536, + 379, + 552 + ], + "lines": [ + { + "bbox": [ + 231, + 536, + 379, + 552 + ], + "spans": [ + { + "bbox": [ + 231, + 536, + 379, + 552 + ], + "score": 0.91, + "content": "{ \\tilde { \\mathcal { F } } } = \\{ ( x , \\varphi ) \\mapsto \\varphi ( f ( x ) ) : f \\in { \\mathcal { F } } \\} \\ .", + "type": "interline_equation", + "image_path": "27509e68cff172f4eb8660c327e0bde5412d66599da55a3e8b8e7e713dc0673f.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 231, + 536, + 379, + 552 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 559, + 506, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 164, + 573 + ], + "score": 1.0, + "content": "We check that", + "type": "text" + }, + { + "bbox": [ + 164, + 559, + 174, + 570 + ], + "score": 0.87, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 559, + 257, + 573 + ], + "score": 1.0, + "content": "is indeed a subset of", + "type": "text" + }, + { + "bbox": [ + 257, + 559, + 292, + 572 + ], + "score": 0.93, + "content": "\\mathcal { C } ( \\tilde { X } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 559, + 338, + 573 + ], + "score": 1.0, + "content": ". Indeed, as", + "type": "text" + }, + { + "bbox": [ + 339, + 561, + 347, + 570 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "is finite, it is equipped with the discrete", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 239, + 584 + ], + "score": 1.0, + "content": "topology. Hence, each singleton", + "type": "text" + }, + { + "bbox": [ + 239, + 571, + 257, + 583 + ], + "score": 0.92, + "content": "\\{ \\varphi \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 571, + 272, + 584 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 272, + 572, + 299, + 583 + ], + "score": 0.89, + "content": "\\varphi \\in \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 571, + 342, + 584 + ], + "score": 1.0, + "content": "is open in", + "type": "text" + }, + { + "bbox": [ + 343, + 572, + 351, + 581 + ], + "score": 0.86, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "and it suffices to check the continuity", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 207, + 595 + ], + "score": 1.0, + "content": "in the first variable with", + "type": "text" + }, + { + "bbox": [ + 208, + 584, + 216, + 594 + ], + "score": 0.82, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 582, + 273, + 595 + ], + "score": 1.0, + "content": "fixed. 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First, following the elegant idea of Maehara", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "& Hoang (2019), we augment the input space to transform the vector-valued equivariant functions", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 362, + 435, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 435, + 375 + ], + "score": 1.0, + "content": "into scalar maps. Second, we apply the fine-grained approximation result Cor. 17.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 340, + 506, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 495, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 496, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 496, + 404 + ], + "score": 1.0, + "content": "Proof. 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In the particular case of", + "type": "text" + }, + { + "bbox": [ + 258, + 442, + 293, + 453 + ], + "score": 0.92, + "content": "G \\subseteq S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 441, + 461, + 455 + ], + "score": 1.0, + "content": "being a group of permutations acting on", + "type": "text" + }, + { + "bbox": [ + 462, + 443, + 474, + 452 + ], + "score": 0.84, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "by, for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 172, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 167, + 465 + ], + "score": 0.59, + "content": "g \\in G , x \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 453, + 172, + 464 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 463, + 379, + 479 + ], + "lines": [ + { + "bbox": [ + 230, + 463, + 379, + 479 + ], + "spans": [ + { + "bbox": [ + 230, + 463, + 379, + 479 + ], + "score": 0.91, + "content": "\\forall i \\in \\{ 1 , . . . , p \\} , ( g \\cdot x ) _ { i } = x _ { g ^ { - 1 } ( i ) } ,", + "type": "interline_equation", + "image_path": "d523a36173a945c4e3d5fbad4635b0203faf11deb1e0fa0d240057f4c095baef.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 230, + 463, + 379, + 479 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 483, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 171, + 496 + ], + "score": 1.0, + "content": "the functions of", + "type": "text" + }, + { + "bbox": [ + 171, + 482, + 180, + 493 + ], + "score": 0.87, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "are indeed invariant, as shown by Maehara & Hoang (2019). For this, a left action", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 494, + 260, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 119, + 507 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 495, + 128, + 504 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 494, + 198, + 507 + ], + "score": 1.0, + "content": "is defined by, for", + "type": "text" + }, + { + "bbox": [ + 198, + 494, + 255, + 506 + ], + "score": 0.3, + "content": "g \\in G , \\varphi \\in \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 494, + 260, + 507 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 510, + 378, + 525 + ], + "lines": [ + { + "bbox": [ + 232, + 510, + 378, + 525 + ], + "spans": [ + { + "bbox": [ + 232, + 510, + 378, + 525 + ], + "score": 0.91, + "content": "\\forall x \\in \\mathbb { R } ^ { p } , ( g \\cdot \\varphi ) ( x ) = \\varphi ( g ^ { - 1 } \\cdot x ) .", + "type": "interline_equation", + "image_path": "ee13f6483cb00aefd95dc003fc417daa656fcbc5d6ed453ce2ed5e3b0a04ed4a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 232, + 510, + 378, + 525 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 225, + 544 + ], + "score": 1.0, + "content": "In other words, the action of", + "type": "text" + }, + { + "bbox": [ + 225, + 533, + 231, + 542 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 528, + 378, + 544 + ], + "score": 1.0, + "content": "on the linear form associated to the", + "type": "text" + }, + { + "bbox": [ + 378, + 530, + 391, + 541 + ], + "score": 0.88, + "content": "i ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 528, + 506, + 544 + ], + "score": 1.0, + "content": "coordinate yields the linear", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 541, + 489, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 196, + 556 + ], + "score": 1.0, + "content": "form associated to the", + "type": "text" + }, + { + "bbox": [ + 197, + 542, + 222, + 555 + ], + "score": 0.93, + "content": "g ( i ) ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 541, + 423, + 556 + ], + "score": 1.0, + "content": "coordinate. One can now check that the functions", + "type": "text" + }, + { + "bbox": [ + 424, + 542, + 433, + 553 + ], + "score": 0.88, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 541, + 489, + 556 + ], + "score": 1.0, + "content": "are invariant.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 106, + 568, + 326, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 324, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 311, + 581 + ], + "score": 1.0, + "content": "D.6 CHARACTERIZING THE SUBALGEBRAS OF", + "type": "text" + }, + { + "bbox": [ + 312, + 569, + 324, + 579 + ], + "score": 0.78, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 454, + 602 + ], + "score": 1.0, + "content": "Before moving to our general result, we need to study the structure of the subalgebras of", + "type": "text" + }, + { + "bbox": [ + 454, + 590, + 467, + 599 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 589, + 506, + 602 + ], + "score": 1.0, + "content": ". For this,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 600, + 271, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 271, + 613 + ], + "score": 1.0, + "content": "we will use the following simple lemma.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 203, + 629 + ], + "score": 1.0, + "content": "In the following lemma,", + "type": "text" + }, + { + "bbox": [ + 204, + 617, + 264, + 630 + ], + "score": 0.93, + "content": "\\mathbb { R } [ X _ { 1 } , \\ldots , X _ { p } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 616, + 456, + 629 + ], + "score": 1.0, + "content": "denotes the set of multivariate polynomials with", + "type": "text" + }, + { + "bbox": [ + 456, + 620, + 462, + 629 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "indetermi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 628, + 221, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 221, + 640 + ], + "score": 1.0, + "content": "nates (and real coefficients).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 642, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 502, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 175, + 658 + ], + "score": 1.0, + "content": "Lemma 26. Let", + "type": "text" + }, + { + "bbox": [ + 175, + 643, + 210, + 654 + ], + "score": 0.89, + "content": "C \\subseteq \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 640, + 293, + 658 + ], + "score": 1.0, + "content": "be a finite subset of", + "type": "text" + }, + { + "bbox": [ + 294, + 644, + 306, + 654 + ], + "score": 0.83, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 640, + 363, + 658 + ], + "score": 1.0, + "content": ". There exists", + "type": "text" + }, + { + "bbox": [ + 363, + 643, + 444, + 656 + ], + "score": 0.92, + "content": "P \\in \\mathbb { R } [ X _ { 1 } , \\ldots , X _ { p } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 640, + 486, + 658 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 487, + 644, + 502, + 657 + ], + "score": 0.88, + "content": "P _ { \\mid C }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 653, + 288, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 175, + 668 + ], + "score": 1.0, + "content": "the restriction of", + "type": "text" + }, + { + "bbox": [ + 176, + 655, + 184, + 664 + ], + "score": 0.79, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 653, + 195, + 668 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 196, + 655, + 204, + 664 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 653, + 288, + 668 + ], + "score": 1.0, + "content": ", is an injective map.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 678, + 504, + 702 + ], + "lines": [ + { + "bbox": [ + 104, + 677, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 151, + 695 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 679, + 222, + 692 + ], + "score": 0.92, + "content": "x ^ { 1 } , \\ldots , x ^ { m } \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 677, + 335, + 695 + ], + "score": 1.0, + "content": "be distinct vectors such that", + "type": "text" + }, + { + "bbox": [ + 335, + 679, + 413, + 692 + ], + "score": 0.93, + "content": "\\{ x ^ { 1 } , \\ldots , x ^ { m } \\} = C", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 677, + 506, + 695 + ], + "score": 1.0, + "content": ". 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In the particular case of", + "type": "text" + }, + { + "bbox": [ + 258, + 442, + 293, + 453 + ], + "score": 0.92, + "content": "G \\subseteq S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 441, + 461, + 455 + ], + "score": 1.0, + "content": "being a group of permutations acting on", + "type": "text" + }, + { + "bbox": [ + 462, + 443, + 474, + 452 + ], + "score": 0.84, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "by, for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 172, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 167, + 465 + ], + "score": 0.59, + "content": "g \\in G , x \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 453, + 172, + 464 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 441, + 506, + 465 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 463, + 379, + 479 + ], + "lines": [ + { + "bbox": [ + 230, + 463, + 379, + 479 + ], + "spans": [ + { + "bbox": [ + 230, + 463, + 379, + 479 + ], + "score": 0.91, + "content": "\\forall i \\in \\{ 1 , . . . , p \\} , ( g \\cdot x ) _ { i } = x _ { g ^ { - 1 } ( i ) } ,", + "type": "interline_equation", + "image_path": "d523a36173a945c4e3d5fbad4635b0203faf11deb1e0fa0d240057f4c095baef.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 230, + 463, + 379, + 479 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 483, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 171, + 496 + ], + "score": 1.0, + "content": "the functions of", + "type": "text" + }, + { + "bbox": [ + 171, + 482, + 180, + 493 + ], + "score": 0.87, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "are indeed invariant, as shown by Maehara & Hoang (2019). For this, a left action", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 494, + 260, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 119, + 507 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 495, + 128, + 504 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 494, + 198, + 507 + ], + "score": 1.0, + "content": "is defined by, for", + "type": "text" + }, + { + "bbox": [ + 198, + 494, + 255, + 506 + ], + "score": 0.3, + "content": "g \\in G , \\varphi \\in \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 494, + 260, + 507 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 482, + 505, + 507 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 510, + 378, + 525 + ], + "lines": [ + { + "bbox": [ + 232, + 510, + 378, + 525 + ], + "spans": [ + { + "bbox": [ + 232, + 510, + 378, + 525 + ], + "score": 0.91, + "content": "\\forall x \\in \\mathbb { R } ^ { p } , ( g \\cdot \\varphi ) ( x ) = \\varphi ( g ^ { - 1 } \\cdot x ) .", + "type": "interline_equation", + "image_path": "ee13f6483cb00aefd95dc003fc417daa656fcbc5d6ed453ce2ed5e3b0a04ed4a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 232, + 510, + 378, + 525 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 225, + 544 + ], + "score": 1.0, + "content": "In other words, the action of", + "type": "text" + }, + { + "bbox": [ + 225, + 533, + 231, + 542 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 528, + 378, + 544 + ], + "score": 1.0, + "content": "on the linear form associated to the", + "type": "text" + }, + { + "bbox": [ + 378, + 530, + 391, + 541 + ], + "score": 0.88, + "content": "i ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 528, + 506, + 544 + ], + "score": 1.0, + "content": "coordinate yields the linear", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 541, + 489, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 196, + 556 + ], + "score": 1.0, + "content": "form associated to the", + "type": "text" + }, + { + "bbox": [ + 197, + 542, + 222, + 555 + ], + "score": 0.93, + "content": "g ( i ) ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 541, + 423, + 556 + ], + "score": 1.0, + "content": "coordinate. One can now check that the functions", + "type": "text" + }, + { + "bbox": [ + 424, + 542, + 433, + 553 + ], + "score": 0.88, + "content": "\\tilde { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 541, + 489, + 556 + ], + "score": 1.0, + "content": "are invariant.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 528, + 506, + 556 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 568, + 326, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 324, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 311, + 581 + ], + "score": 1.0, + "content": "D.6 CHARACTERIZING THE SUBALGEBRAS OF", + "type": "text" + }, + { + "bbox": [ + 312, + 569, + 324, + 579 + ], + "score": 0.78, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 454, + 602 + ], + "score": 1.0, + "content": "Before moving to our general result, we need to study the structure of the subalgebras of", + "type": "text" + }, + { + "bbox": [ + 454, + 590, + 467, + 599 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 589, + 506, + 602 + ], + "score": 1.0, + "content": ". For this,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 600, + 271, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 271, + 613 + ], + "score": 1.0, + "content": "we will use the following simple lemma.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 589, + 506, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 203, + 629 + ], + "score": 1.0, + "content": "In the following lemma,", + "type": "text" + }, + { + "bbox": [ + 204, + 617, + 264, + 630 + ], + "score": 0.93, + "content": "\\mathbb { R } [ X _ { 1 } , \\ldots , X _ { p } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 616, + 456, + 629 + ], + "score": 1.0, + "content": "denotes the set of multivariate polynomials with", + "type": "text" + }, + { + "bbox": [ + 456, + 620, + 462, + 629 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "indetermi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 628, + 221, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 221, + 640 + ], + "score": 1.0, + "content": "nates (and real coefficients).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 616, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 642, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 502, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 175, + 658 + ], + "score": 1.0, + "content": "Lemma 26. Let", + "type": "text" + }, + { + "bbox": [ + 175, + 643, + 210, + 654 + ], + "score": 0.89, + "content": "C \\subseteq \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 640, + 293, + 658 + ], + "score": 1.0, + "content": "be a finite subset of", + "type": "text" + }, + { + "bbox": [ + 294, + 644, + 306, + 654 + ], + "score": 0.83, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 640, + 363, + 658 + ], + "score": 1.0, + "content": ". There exists", + "type": "text" + }, + { + "bbox": [ + 363, + 643, + 444, + 656 + ], + "score": 0.92, + "content": "P \\in \\mathbb { R } [ X _ { 1 } , \\ldots , X _ { p } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 640, + 486, + 658 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 487, + 644, + 502, + 657 + ], + "score": 0.88, + "content": "P _ { \\mid C }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 653, + 288, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 175, + 668 + ], + "score": 1.0, + "content": "the restriction of", + "type": "text" + }, + { + "bbox": [ + 176, + 655, + 184, + 664 + ], + "score": 0.79, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 653, + 195, + 668 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 196, + 655, + 204, + 664 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 653, + 288, + 668 + ], + "score": 1.0, + "content": ", is an injective map.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 640, + 502, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 678, + 504, + 702 + ], + "lines": [ + { + "bbox": [ + 104, + 677, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 151, + 695 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 679, + 222, + 692 + ], + "score": 0.92, + "content": "x ^ { 1 } , \\ldots , x ^ { m } \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 677, + 335, + 695 + ], + "score": 1.0, + "content": "be distinct vectors such that", + "type": "text" + }, + { + "bbox": [ + 335, + 679, + 413, + 692 + ], + "score": 0.93, + "content": "\\{ x ^ { 1 } , \\ldots , x ^ { m } \\} = C", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 677, + 506, + 695 + ], + "score": 1.0, + "content": ". Similarly to Lagrange", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 691, + 191, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 191, + 704 + ], + "score": 1.0, + "content": "polynomials, define,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 677, + 506, + 704 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 701, + 400, + 735 + ], + "lines": [ + { + "bbox": [ + 210, + 701, + 400, + 735 + ], + "spans": [ + { + "bbox": [ + 210, + 701, + 400, + 735 + ], + "score": 0.94, + "content": "P ( X _ { 1 } , \\dots , X _ { p } ) = \\sum _ { i = 1 } ^ { m } i \\prod _ { j \\neq i } { \\frac { \\sum _ { l = 1 } ^ { p } ( X _ { l } - x _ { l } ^ { j } ) ^ { 2 } } { \\| x ^ { i } - x ^ { j } \\| _ { 2 } ^ { 2 } } } ,", + "type": "interline_equation", + "image_path": "f8371e8980afafc2711ddd424c2b254491f520a2f1a0dcbb7cb5a4eb3aa1b463.jpg" + } + ] + } + ], + "index": 38.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 701, + 400, + 718.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 210, + 718.0, + 400, + 735.0 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 370, + 96 + ], + "score": 1.0, + "content": "which is a well-defined multivariate polynomial. Note that, seeing", + "type": "text" + }, + { + "bbox": [ + 370, + 82, + 448, + 95 + ], + "score": 0.92, + "content": "X = ( X _ { 1 } , \\ldots , X _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "as a vector in", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 222, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 119, + 104 + ], + "score": 0.82, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 92, + 222, + 106 + ], + "score": 1.0, + "content": ", it can also be written as,", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 108, + 389, + 142 + ], + "lines": [ + { + "bbox": [ + 222, + 108, + 389, + 142 + ], + "spans": [ + { + "bbox": [ + 222, + 108, + 389, + 142 + ], + "score": 0.94, + "content": "P ( X _ { 1 } , \\dots , X _ { p } ) = \\sum _ { i = 1 } ^ { m } i \\prod _ { j \\neq i } { \\frac { \\| X - x ^ { j } \\| _ { 2 } ^ { 2 } } { \\| x ^ { i } - x ^ { j } \\| _ { 2 } ^ { 2 } } } .", + "type": "interline_equation", + "image_path": "71b7a3cb91ea4aceadcd115f8876316fe24d2d83c7805e7db5a7ec9bb853da21.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 222, + 108, + 389, + 125.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 222, + 125.0, + 389, + 142.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 384, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 383, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 175, + 162 + ], + "score": 1.0, + "content": "By construction,", + "type": "text" + }, + { + "bbox": [ + 175, + 146, + 217, + 159 + ], + "score": 0.94, + "content": "P ( x ^ { i } ) = i", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 144, + 274, + 162 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 274, + 147, + 283, + 156 + ], + "score": 0.85, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 144, + 374, + 162 + ], + "score": 1.0, + "content": "is an injective map on", + "type": "text" + }, + { + "bbox": [ + 374, + 147, + 383, + 156 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 311, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 311, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 230, + 178 + ], + "score": 1.0, + "content": "Lemma 27. For a subalgebra", + "type": "text" + }, + { + "bbox": [ + 231, + 165, + 239, + 174 + ], + "score": 0.69, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 162, + 250, + 178 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 251, + 165, + 263, + 175 + ], + "score": 0.85, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 162, + 311, + 178 + ], + "score": 1.0, + "content": ", we define:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 179, + 393, + 211 + ], + "lines": [ + { + "bbox": [ + 218, + 179, + 393, + 211 + ], + "spans": [ + { + "bbox": [ + 218, + 179, + 393, + 211 + ], + "score": 0.9, + "content": "\\begin{array} { l } { J = \\left\\{ j \\in \\left\\{ 1 , \\dots , p \\right\\} : \\forall x \\in A , x _ { j } = 0 \\right\\} } \\\\ { I = \\left\\{ ( i , j ) \\in ( J ^ { C } ) ^ { 2 } : \\forall x \\in A , x _ { i } = x _ { j } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "92c67c82ca0b8d150e299a02adbf121f99a6519ba0b3999082238dd92333789a.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 179, + 393, + 195.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 218, + 195.0, + 393, + 211.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 170, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 172, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 172, + 226 + ], + "score": 1.0, + "content": "Then, we have :", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 223, + 419, + 237 + ], + "lines": [ + { + "bbox": [ + 191, + 223, + 419, + 237 + ], + "spans": [ + { + "bbox": [ + 191, + 223, + 419, + 237 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathcal { A } = \\{ x \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I , x _ { i } = x _ { j } , \\forall j \\in J , x _ { j } = 0 \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "4337e4d6447d1bbf034a9cef86423630e4a3411c710fb76d3fc543091d4bdc30.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 191, + 223, + 419, + 237 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 403, + 260 + ], + "score": 1.0, + "content": "Proof. Before proving the general case, we focus on the situation where", + "type": "text" + }, + { + "bbox": [ + 404, + 249, + 413, + 258 + ], + "score": 0.8, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 247, + 504, + 260 + ], + "score": 1.0, + "content": "will turn out to be the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 258, + 150, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 133, + 270 + ], + "score": 1.0, + "content": "whole", + "type": "text" + }, + { + "bbox": [ + 133, + 259, + 146, + 269 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 258, + 150, + 270 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 300, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 302, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 302, + 289 + ], + "score": 1.0, + "content": "Assume that the two following conditions holds,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 290, + 359, + 320 + ], + "lines": [ + { + "bbox": [ + 251, + 290, + 359, + 320 + ], + "spans": [ + { + "bbox": [ + 251, + 290, + 359, + 320 + ], + "score": 0.91, + "content": "\\begin{array} { l } { \\forall i \\neq j , \\exists x \\in A , x _ { i } \\neq x _ { j } } \\\\ { \\forall i , \\exists x \\in A , x _ { i } \\neq 0 } \\end{array}", + "type": "interline_equation", + "image_path": "5ea3193ce50a0f2b509b532a2849b6d114051253e5947087548db89353b07d18.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 251, + 290, + 359, + 305.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 251, + 305.0, + 359, + 320.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 352, + 334 + ], + "score": 1.0, + "content": "Our goal is to show that, under these additional assumptions,", + "type": "text" + }, + { + "bbox": [ + 352, + 323, + 386, + 333 + ], + "score": 0.9, + "content": "\\ b { A } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 322, + 505, + 334 + ], + "score": 1.0, + "content": ". We divide the proof in three", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 334, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 205, + 345 + ], + "score": 1.0, + "content": "parts, first we show that", + "type": "text" + }, + { + "bbox": [ + 205, + 334, + 233, + 344 + ], + "score": 0.9, + "content": "\\mathbf { 1 } \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 334, + 338, + 345 + ], + "score": 1.0, + "content": "using (24), giving us that", + "type": "text" + }, + { + "bbox": [ + 338, + 334, + 348, + 343 + ], + "score": 0.84, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 334, + 506, + 345 + ], + "score": 1.0, + "content": "is closed under polynomials, then that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 137, + 357 + ], + "score": 1.0, + "content": "there is", + "type": "text" + }, + { + "bbox": [ + 138, + 345, + 165, + 354 + ], + "score": 0.9, + "content": "x \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 344, + 486, + 357 + ], + "score": 1.0, + "content": "with pairwise distinct coordinates thanks to (23) and finally that this implies that", + "type": "text" + }, + { + "bbox": [ + 486, + 345, + 495, + 354 + ], + "score": 0.82, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 354, + 175, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 175, + 369 + ], + "score": 1.0, + "content": "the whole space.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 452, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 454, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 154, + 385 + ], + "score": 1.0, + "content": "Note that if", + "type": "text" + }, + { + "bbox": [ + 154, + 373, + 179, + 384 + ], + "score": 0.89, + "content": "p = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 371, + 330, + 385 + ], + "score": 1.0, + "content": ", (24) and the linear space property of", + "type": "text" + }, + { + "bbox": [ + 330, + 373, + 339, + 382 + ], + "score": 0.81, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 371, + 454, + 385 + ], + "score": 1.0, + "content": "immediately give the result.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 133, + 395, + 319, + 407 + ], + "lines": [ + { + "bbox": [ + 132, + 393, + 320, + 408 + ], + "spans": [ + { + "bbox": [ + 132, + 393, + 289, + 408 + ], + "score": 1.0, + "content": "• Here we prove that (24) implies that", + "type": "text" + }, + { + "bbox": [ + 289, + 396, + 316, + 406 + ], + "score": 0.88, + "content": "\\mathbf { 1 } \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 393, + 320, + 408 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 151, + 414, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 150, + 411, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 150, + 411, + 287, + 428 + ], + "score": 1.0, + "content": "– First, we construct by induction", + "type": "text" + }, + { + "bbox": [ + 288, + 415, + 314, + 424 + ], + "score": 0.91, + "content": "x \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 411, + 352, + 428 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 352, + 414, + 380, + 425 + ], + "score": 0.92, + "content": "x _ { i } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 411, + 434, + 428 + ], + "score": 1.0, + "content": "for any index", + "type": "text" + }, + { + "bbox": [ + 435, + 415, + 439, + 424 + ], + "score": 0.6, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 411, + 507, + 428 + ], + "score": 1.0, + "content": ". 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As", + "type": "text" + }, + { + "bbox": [ + 306, + 536, + 315, + 546 + ], + "score": 0.83, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 535, + 379, + 548 + ], + "score": 1.0, + "content": "is a subalgebra,", + "type": "text" + }, + { + "bbox": [ + 379, + 537, + 429, + 547 + ], + "score": 0.92, + "content": "\\lambda x + y \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "and this concludes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 162, + 546, + 239, + 560 + ], + "spans": [ + { + "bbox": [ + 162, + 546, + 239, + 560 + ], + "score": 1.0, + "content": "the induction step.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 151, + 560, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 151, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 151, + 558, + 177, + 572 + ], + "score": 1.0, + "content": "– Let", + "type": "text" + }, + { + "bbox": [ + 178, + 560, + 205, + 570 + ], + "score": 0.88, + "content": "x \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 558, + 363, + 572 + ], + "score": 1.0, + "content": "be the vector constructed, i.e. such that", + "type": "text" + }, + { + "bbox": [ + 363, + 560, + 392, + 571 + ], + "score": 0.92, + "content": "x _ { i } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 558, + 456, + 572 + ], + "score": 1.0, + "content": "for every index", + "type": "text" + }, + { + "bbox": [ + 456, + 561, + 460, + 569 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 558, + 505, + 572 + ], + "score": 1.0, + "content": ". 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For a subalgebra", + "type": "text" + }, + { + "bbox": [ + 231, + 165, + 239, + 174 + ], + "score": 0.69, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 162, + 250, + 178 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 251, + 165, + 263, + 175 + ], + "score": 0.85, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 162, + 311, + 178 + ], + "score": 1.0, + "content": ", we define:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 162, + 311, + 178 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 179, + 393, + 211 + ], + "lines": [ + { + "bbox": [ + 218, + 179, + 393, + 211 + ], + "spans": [ + { + "bbox": [ + 218, + 179, + 393, + 211 + ], + "score": 0.9, + "content": "\\begin{array} { l } { J = \\left\\{ j \\in \\left\\{ 1 , \\dots , p \\right\\} : \\forall x \\in A , x _ { j } = 0 \\right\\} } \\\\ { I = \\left\\{ ( i , j ) \\in ( J ^ { C } ) ^ { 2 } : \\forall x \\in A , x _ { i } = x _ { j } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "92c67c82ca0b8d150e299a02adbf121f99a6519ba0b3999082238dd92333789a.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 179, + 393, + 195.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 218, + 195.0, + 393, + 211.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 170, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 172, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 172, + 226 + ], + "score": 1.0, + "content": "Then, we have :", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 212, + 172, + 226 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 223, + 419, + 237 + ], + "lines": [ + { + "bbox": [ + 191, + 223, + 419, + 237 + ], + "spans": [ + { + "bbox": [ + 191, + 223, + 419, + 237 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathcal { A } = \\{ x \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I , x _ { i } = x _ { j } , \\forall j \\in J , x _ { j } = 0 \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "4337e4d6447d1bbf034a9cef86423630e4a3411c710fb76d3fc543091d4bdc30.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 191, + 223, + 419, + 237 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 403, + 260 + ], + "score": 1.0, + "content": "Proof. Before proving the general case, we focus on the situation where", + "type": "text" + }, + { + "bbox": [ + 404, + 249, + 413, + 258 + ], + "score": 0.8, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 247, + 504, + 260 + ], + "score": 1.0, + "content": "will turn out to be the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 258, + 150, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 133, + 270 + ], + "score": 1.0, + "content": "whole", + "type": "text" + }, + { + "bbox": [ + 133, + 259, + 146, + 269 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 258, + 150, + 270 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 247, + 504, + 270 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 300, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 302, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 302, + 289 + ], + "score": 1.0, + "content": "Assume that the two following conditions holds,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 274, + 302, + 289 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 290, + 359, + 320 + ], + "lines": [ + { + "bbox": [ + 251, + 290, + 359, + 320 + ], + "spans": [ + { + "bbox": [ + 251, + 290, + 359, + 320 + ], + "score": 0.91, + "content": "\\begin{array} { l } { \\forall i \\neq j , \\exists x \\in A , x _ { i } \\neq x _ { j } } \\\\ { \\forall i , \\exists x \\in A , x _ { i } \\neq 0 } \\end{array}", + "type": "interline_equation", + "image_path": "5ea3193ce50a0f2b509b532a2849b6d114051253e5947087548db89353b07d18.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 251, + 290, + 359, + 305.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 251, + 305.0, + 359, + 320.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 352, + 334 + ], + "score": 1.0, + "content": "Our goal is to show that, under these additional assumptions,", + "type": "text" + }, + { + "bbox": [ + 352, + 323, + 386, + 333 + ], + "score": 0.9, + "content": "\\ b { A } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 322, + 505, + 334 + ], + "score": 1.0, + "content": ". We divide the proof in three", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 334, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 205, + 345 + ], + "score": 1.0, + "content": "parts, first we show that", + "type": "text" + }, + { + "bbox": [ + 205, + 334, + 233, + 344 + ], + "score": 0.9, + "content": "\\mathbf { 1 } \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 334, + 338, + 345 + ], + "score": 1.0, + "content": "using (24), giving us that", + "type": "text" + }, + { + "bbox": [ + 338, + 334, + 348, + 343 + ], + "score": 0.84, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 334, + 506, + 345 + ], + "score": 1.0, + "content": "is closed under polynomials, then that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 137, + 357 + ], + "score": 1.0, + "content": "there is", + "type": "text" + }, + { + "bbox": [ + 138, + 345, + 165, + 354 + ], + "score": 0.9, + "content": "x \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 344, + 486, + 357 + ], + "score": 1.0, + "content": "with pairwise distinct coordinates thanks to (23) and finally that this implies that", + "type": "text" + }, + { + "bbox": [ + 486, + 345, + 495, + 354 + ], + "score": 0.82, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 354, + 175, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 175, + 369 + ], + "score": 1.0, + "content": "the whole space.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 322, + 506, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 452, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 454, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 154, + 385 + ], + "score": 1.0, + "content": "Note that if", + "type": "text" + }, + { + "bbox": [ + 154, + 373, + 179, + 384 + ], + "score": 0.89, + "content": "p = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 371, + 330, + 385 + ], + "score": 1.0, + "content": ", (24) and the linear space property of", + "type": "text" + }, + { + "bbox": [ + 330, + 373, + 339, + 382 + ], + "score": 0.81, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 371, + 454, + 385 + ], + "score": 1.0, + "content": "immediately give the result.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 371, + 454, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 395, + 319, + 407 + ], + "lines": [ + { + "bbox": [ + 132, + 393, + 320, + 408 + ], + "spans": [ + { + "bbox": [ + 132, + 393, + 289, + 408 + ], + "score": 1.0, + "content": "• Here we prove that (24) implies that", + "type": "text" + }, + { + "bbox": [ + 289, + 396, + 316, + 406 + ], + "score": 0.88, + "content": "\\mathbf { 1 } \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 393, + 320, + 408 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 132, + 393, + 320, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 151, + 414, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 150, + 411, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 150, + 411, + 287, + 428 + ], + "score": 1.0, + "content": "– First, we construct by induction", + "type": "text" + }, + { + "bbox": [ + 288, + 415, + 314, + 424 + ], + "score": 0.91, + "content": "x \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 411, + 352, + 428 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 352, + 414, + 380, + 425 + ], + "score": 0.92, + "content": "x _ { i } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 411, + 434, + 428 + ], + "score": 1.0, + "content": "for any index", + "type": "text" + }, + { + "bbox": [ + 435, + 415, + 439, + 424 + ], + "score": 0.6, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 411, + 507, + 428 + ], + "score": 1.0, + "content": ". 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By construction of", + "type": "text" + }, + { + "bbox": [ + 365, + 187, + 373, + 197 + ], + "score": 0.6, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 183, + 377, + 204 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 378, + 183, + 502, + 203 + ], + "score": 0.91, + "content": "P \\left( ( x _ { l } ^ { i j } ) _ { i < j } \\right) \\neq P \\left( ( { \\bar { x } } _ { k } ^ { i j } ) _ { i < j } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 183, + 506, + 204 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 201, + 507, + 217 + ], + "spans": [ + { + "bbox": [ + 141, + 201, + 158, + 217 + ], + "score": 1.0, + "content": "i.e.", + "type": "text" + }, + { + "bbox": [ + 158, + 201, + 285, + 216 + ], + "score": 0.92, + "content": "P \\left( ( x ^ { i j } ) _ { i < j } \\right) _ { l } \\neq P \\left( ( x ^ { i j } ) _ { i < j } \\right) _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 201, + 315, + 217 + ], + "score": 1.0, + "content": ". 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Indeed, take any", + "type": "text" + }, + { + "bbox": [ + 310, + 244, + 340, + 255 + ], + "score": 0.92, + "content": "y \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 243, + 401, + 256 + ], + "score": 1.0, + "content": "and denote by", + "type": "text" + }, + { + "bbox": [ + 402, + 244, + 429, + 254 + ], + "score": 0.92, + "content": "x \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "the vector that we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "just constructed with pairwise distinct coordinates. Therefore, by Lagrange interpolation,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 189, + 279 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 189, + 266, + 232, + 277 + ], + "score": 0.89, + "content": "P \\in \\mathbb { R } [ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 265, + 271, + 279 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 271, + 266, + 318, + 277 + ], + "score": 0.92, + "content": "P ( x _ { i } ) = y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 265, + 357, + 279 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 357, + 266, + 416, + 278 + ], + "score": 0.94, + "content": "i \\in \\{ 1 , \\ldots , p \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 265, + 434, + 279 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 434, + 266, + 443, + 276 + ], + "score": 0.81, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "is a sublagebra", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 277, + 307, + 289 + ], + "spans": [ + { + "bbox": [ + 142, + 277, + 159, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 277, + 186, + 287 + ], + "score": 0.85, + "content": "\\mathbf { 1 } \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 277, + 190, + 289 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 190, + 277, + 232, + 289 + ], + "score": 0.88, + "content": "P ( x ) \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 277, + 276, + 289 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + }, + { + "bbox": [ + 276, + 278, + 303, + 288 + ], + "score": 0.92, + "content": "y \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 277, + 307, + 289 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 504, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 392, + 312 + ], + "score": 1.0, + "content": "Finally, we return to the general case. We introduce the set of indexes of", + "type": "text" + }, + { + "bbox": [ + 392, + 300, + 399, + 309 + ], + "score": 0.82, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 298, + 416, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 417, + 300, + 424, + 309 + ], + "score": 0.85, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "which appear in the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 309, + 393, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 393, + 324 + ], + "score": 1.0, + "content": "result and use them to reduce the situation to the previous case. Define,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 325, + 393, + 357 + ], + "lines": [ + { + "bbox": [ + 218, + 325, + 393, + 357 + ], + "spans": [ + { + "bbox": [ + 218, + 325, + 393, + 357 + ], + "score": 0.91, + "content": "\\begin{array} { l } { J = \\left\\{ j \\in \\left\\{ 1 , \\dots , p \\right\\} : \\forall x \\in A , x _ { j } = 0 \\right\\} } \\\\ { I = \\left\\{ ( i , j ) \\in ( J ^ { C } ) ^ { 2 } : \\forall x \\in A , x _ { i } = x _ { j } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "615c88863e30d76445c12239748a02e2fbc812b000b763b7f2d5cb94742d866b.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 325, + 393, + 341.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 218, + 341.0, + 393, + 357.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 166, + 375 + ], + "score": 1.0, + "content": "and denote by", + "type": "text" + }, + { + "bbox": [ + 167, + 361, + 278, + 373 + ], + "score": 0.83, + "content": "\\mathcal { A } ^ { \\prime } = \\{ x \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 358, + 282, + 375 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 362, + 316, + 374 + ], + "score": 0.62, + "content": "x _ { i } = x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 358, + 321, + 375 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 322, + 361, + 352, + 373 + ], + "score": 0.44, + "content": "\\forall j \\in J", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 358, + 356, + 375 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 357, + 361, + 391, + 373 + ], + "score": 0.58, + "content": "x _ { j } = 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 358, + 507, + 375 + ], + "score": 1.0, + "content": ", which is also a subalgebra.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 372, + 250, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 213, + 384 + ], + "score": 1.0, + "content": "By definition, it holds that", + "type": "text" + }, + { + "bbox": [ + 214, + 373, + 246, + 383 + ], + "score": 0.89, + "content": "{ \\mathcal { A } } \\subseteq { \\mathcal { A } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 372, + 250, + 384 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 105, + 388, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 174, + 402 + ], + "score": 1.0, + "content": "By construction,", + "type": "text" + }, + { + "bbox": [ + 174, + 390, + 181, + 399 + ], + "score": 0.79, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 387, + 296, + 402 + ], + "score": 1.0, + "content": "is an equivalence relation on", + "type": "text" + }, + { + "bbox": [ + 297, + 388, + 311, + 399 + ], + "score": 0.89, + "content": "J ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 387, + 370, + 402 + ], + "score": 1.0, + "content": "and denote by", + "type": "text" + }, + { + "bbox": [ + 370, + 388, + 395, + 401 + ], + "score": 0.93, + "content": "J ^ { C } / I", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "its equivalence classes. Let", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 398, + 495, + 414 + ], + "spans": [ + { + "bbox": [ + 107, + 400, + 158, + 412 + ], + "score": 0.91, + "content": "p ^ { \\prime } = | J ^ { C } / I |", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 398, + 206, + 414 + ], + "score": 1.0, + "content": "and choose", + "type": "text" + }, + { + "bbox": [ + 206, + 401, + 248, + 412 + ], + "score": 0.93, + "content": "i _ { 1 } , \\ldots , i _ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 398, + 495, + 414 + ], + "score": 1.0, + "content": "representatives of the equivalence classes. Consider the map,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 416, + 366, + 450 + ], + "lines": [ + { + "bbox": [ + 244, + 416, + 366, + 450 + ], + "spans": [ + { + "bbox": [ + 244, + 416, + 366, + 450 + ], + "score": 0.93, + "content": "\\varphi : \\left\\{ \\begin{array} { l l } { \\mathbb { R } ^ { p } \\longrightarrow \\mathbb { R } ^ { p ^ { \\prime } } } \\\\ { x \\longmapsto \\left( x _ { i _ { 1 } } , \\ldots , x _ { i _ { p ^ { \\prime } } } \\right) . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "97ecd28ce3499ceb0e04a23f8974bdc752aee7ad49d59f710f10913d981ef607.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 416, + 366, + 433.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 244, + 433.0, + 366, + 450.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 482 + ], + "lines": [ + { + "bbox": [ + 107, + 455, + 507, + 470 + ], + "spans": [ + { + "bbox": [ + 107, + 460, + 114, + 469 + ], + "score": 0.77, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 455, + 261, + 470 + ], + "score": 1.0, + "content": "is an algebra homomoprhism so that", + "type": "text" + }, + { + "bbox": [ + 261, + 456, + 284, + 469 + ], + "score": 0.92, + "content": "\\varphi ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 455, + 357, + 470 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 357, + 455, + 372, + 467 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 455, + 470, + 470 + ], + "score": 1.0, + "content": ". But, by construction of", + "type": "text" + }, + { + "bbox": [ + 471, + 457, + 477, + 467 + ], + "score": 0.82, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 455, + 495, + 470 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 495, + 457, + 502, + 467 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 455, + 507, + 470 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 468, + 459, + 483 + ], + "spans": [ + { + "bbox": [ + 107, + 470, + 129, + 482 + ], + "score": 0.9, + "content": "\\varphi ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 469, + 404, + 483 + ], + "score": 1.0, + "content": "satisfies (23) and (24). Whence, by our result in this particular case,", + "type": "text" + }, + { + "bbox": [ + 404, + 468, + 455, + 482 + ], + "score": 0.93, + "content": "\\varphi ( \\mathcal { A } ) = \\mathbb { R } ^ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 469, + 459, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 148, + 502 + ], + "score": 1.0, + "content": "However,", + "type": "text" + }, + { + "bbox": [ + 148, + 488, + 182, + 500 + ], + "score": 0.9, + "content": "{ \\mathcal { A } } \\subseteq { \\mathcal { A } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 486, + 234, + 502 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 235, + 487, + 326, + 501 + ], + "score": 0.93, + "content": "\\varphi ( \\mathcal { A } ) \\subseteq \\varphi ( \\mathcal { A } ^ { \\prime } ) \\subseteq \\mathbb { R } ^ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 486, + 376, + 502 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 376, + 487, + 467, + 501 + ], + "score": 0.93, + "content": "\\mathbb { R } ^ { p ^ { \\prime } } = \\varphi ( \\mathcal { A } ) = \\varphi ( \\mathcal { A } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 486, + 506, + 502 + ], + "score": 1.0, + "content": ". But, by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 169, + 513 + ], + "score": 1.0, + "content": "construction of", + "type": "text" + }, + { + "bbox": [ + 170, + 502, + 177, + 511 + ], + "score": 0.72, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 499, + 181, + 513 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 181, + 502, + 189, + 511 + ], + "score": 0.72, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 499, + 315, + 513 + ], + "score": 1.0, + "content": "is actually an injective map on", + "type": "text" + }, + { + "bbox": [ + 315, + 500, + 327, + 510 + ], + "score": 0.84, + "content": "\\mathcal { A } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 499, + 443, + 513 + ], + "score": 1.0, + "content": ". Therefore, we deduce from", + "type": "text" + }, + { + "bbox": [ + 444, + 500, + 505, + 511 + ], + "score": 0.91, + "content": "\\varphi ( \\mathcal { A } ) = \\varphi ( \\mathcal { A } ^ { \\prime } )", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 510, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 124, + 523 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 511, + 157, + 520 + ], + "score": 0.89, + "content": "A = A ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 510, + 249, + 523 + ], + "score": 1.0, + "content": ", concluding the proof.", + "type": "text" + }, + { + "bbox": [ + 494, + 511, + 504, + 521 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 107, + 535, + 402, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 403, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 403, + 549 + ], + "score": 1.0, + "content": "D.7 PROOF OF THE MAIN EQUIVARIANT APPROXIMATION THEOREM", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 506, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 321, + 569 + ], + "score": 1.0, + "content": "We can now fully exploit the structure of subalgebra of", + "type": "text" + }, + { + "bbox": [ + 322, + 556, + 331, + 566 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "thanks to the results above, and in particular", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "relax the second assumption of Prop. 24 to give our main theorem 22. We first prove the following", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 139, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 139, + 590 + ], + "score": 1.0, + "content": "lemma.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 103, + 592, + 478, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 476, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 332, + 607 + ], + "score": 1.0, + "content": "Lemma 28. Under the assumptions of Thm. 22, for any", + "type": "text" + }, + { + "bbox": [ + 333, + 593, + 364, + 604 + ], + "score": 0.9, + "content": "H \\subseteq G", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 591, + 438, + 607 + ], + "score": 1.0, + "content": "subgroup, for any", + "type": "text" + }, + { + "bbox": [ + 438, + 593, + 476, + 604 + ], + "score": 0.89, + "content": "x , y \\in X", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 609, + 465, + 622 + ], + "lines": [ + { + "bbox": [ + 145, + 609, + 465, + 622 + ], + "spans": [ + { + "bbox": [ + 145, + 609, + 465, + 622 + ], + "score": 0.84, + "content": "\\forall f \\in { \\mathcal { F } } , \\ \\exists g \\in H , \\ f ( g \\cdot x ) = f ( y ) \\iff \\exists g \\in H , \\forall f \\in { \\mathcal { F } } , \\ f ( g \\cdot x ) = f ( y ) \\ .", + "type": "interline_equation", + "image_path": "b7db3ce16d5008e1f009c7818d8f281516103e840409f9608ad66a97256ec1b3.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 145, + 609, + 465, + 622 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 160, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 162, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 162, + 641 + ], + "score": 1.0, + "content": "In particular,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 637, + 415, + 651 + ], + "lines": [ + { + "bbox": [ + 196, + 637, + 415, + 651 + ], + "spans": [ + { + "bbox": [ + 196, + 637, + 415, + 651 + ], + "score": 0.84, + "content": "( x , y ) \\in \\rho \\left( \\pi \\circ { \\mathcal { F } } \\right) \\iff \\exists g \\in G , ( g \\cdot x , y ) \\in \\rho \\left( { \\mathcal { F } } \\right) .", + "type": "interline_equation", + "image_path": "ab22f84c31a9f4593badad02620729e5e11ee6dd1faba5d27148306d967aa618.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 196, + 637, + 415, + 651 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 661, + 506, + 683 + ], + "lines": [ + { + "bbox": [ + 104, + 659, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 506, + 676 + ], + "score": 1.0, + "content": "Proof. The reverse implication is immediate so we focus on the direct one and prove its contraposition,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 671, + 127, + 687 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 127, + 687 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 683, + 462, + 696 + ], + "lines": [ + { + "bbox": [ + 146, + 683, + 462, + 696 + ], + "spans": [ + { + "bbox": [ + 146, + 683, + 462, + 696 + ], + "score": 0.87, + "content": "\\forall g \\in H , \\exists f \\in { \\mathcal { F } } , f ( g \\cdot x ) \\neq f ( y ) \\implies \\exists f \\in { \\mathcal { F } } , \\forall g \\in H , f ( g \\cdot x ) \\neq f ( y ) .", + "type": "interline_equation", + "image_path": "b395fbffcee7ecbc4b354d6f02d2911e9145b5fbefb83ae9bdbe81d1a8e5f9ed.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 146, + 683, + 462, + 696 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 128, + 698, + 504, + 721 + ], + "lines": [ + { + "bbox": [ + 124, + 695, + 504, + 713 + ], + "spans": [ + { + "bbox": [ + 124, + 695, + 248, + 713 + ], + "score": 1.0, + "content": "rove this, we take advantage of", + "type": "text" + }, + { + "bbox": [ + 248, + 699, + 258, + 709 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 695, + 352, + 713 + ], + "score": 1.0, + "content": "being a subalgebra and", + "type": "text" + }, + { + "bbox": [ + 352, + 699, + 362, + 709 + ], + "score": 0.84, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 695, + 429, + 713 + ], + "score": 1.0, + "content": "being finite. Let", + "type": "text" + }, + { + "bbox": [ + 429, + 699, + 504, + 711 + ], + "score": 0.91, + "content": "H = \\{ g _ { 1 } , \\dots , g _ { h } \\}", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 126, + 707, + 154, + 723 + ], + "spans": [ + { + "bbox": [ + 126, + 707, + 154, + 723 + ], + "score": 1.0, + "content": "define,", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 719, + 404, + 733 + ], + "lines": [ + { + "bbox": [ + 206, + 719, + 404, + 733 + ], + "spans": [ + { + "bbox": [ + 206, + 719, + 404, + 733 + ], + "score": 0.87, + "content": "{ \\mathcal { A } } = \\left\\{ ( f ( g _ { 1 } \\cdot x ) , \\ldots , f ( g _ { h } \\cdot x ) , f ( y ) ) : f \\in { \\mathcal { F } } \\right\\} .", + "type": "interline_equation", + "image_path": "a34a4d5cbdef0f4bf83dc57f8952a85841745f08c0fae14a48ae6298660a3dbc.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 206, + 719, + 404, + 733 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "page_idx": 28, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 143, + 82, + 319, + 94 + ], + "lines": [ + { + "bbox": [ + 141, + 81, + 320, + 96 + ], + "spans": [ + { + "bbox": [ + 141, + 81, + 155, + 96 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 156, + 83, + 165, + 92 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 81, + 244, + 96 + ], + "score": 1.0, + "content": "is a subalgebra and", + "type": "text" + }, + { + "bbox": [ + 245, + 83, + 272, + 93 + ], + "score": 0.9, + "content": "\\mathbf { 1 } \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 81, + 320, + 96 + ], + "score": 1.0, + "content": ", the vector,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 141, + 81, + 320, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 98, + 396, + 151 + ], + "lines": [ + { + "bbox": [ + 250, + 98, + 396, + 151 + ], + "spans": [ + { + "bbox": [ + 250, + 98, + 396, + 151 + ], + "score": 0.94, + "content": "P \\left( ( x ^ { i j } ) _ { i < j } \\right) = \\left( \\begin{array} { c } { { P \\left( ( x _ { 1 } ^ { i j } ) _ { i < j } \\right) } } \\\\ { { \\vdots } } \\\\ { { P \\left( ( x _ { p } ^ { i j } ) _ { i < j } \\right) } } \\end{array} \\right) ,", + "type": "interline_equation", + "image_path": "8a423ec43f7c8b56165d79739a5fd17ad4b59ffd27935efb091e0f871ca435b7.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 98, + 396, + 124.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 250, + 124.5, + 396, + 151.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 154, + 189, + 166 + ], + "lines": [ + { + "bbox": [ + 140, + 152, + 191, + 168 + ], + "spans": [ + { + "bbox": [ + 140, + 152, + 162, + 168 + ], + "score": 1.0, + "content": "is in", + "type": "text" + }, + { + "bbox": [ + 162, + 155, + 171, + 165 + ], + "score": 0.78, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 152, + 191, + 168 + ], + "score": 1.0, + "content": "too.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 140, + 152, + 191, + 168 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 171, + 506, + 226 + ], + "lines": [ + { + "bbox": [ + 140, + 168, + 504, + 187 + ], + "spans": [ + { + "bbox": [ + 140, + 168, + 416, + 187 + ], + "score": 1.0, + "content": "We now check that this vector has pairwise distinct coordinates. Let", + "type": "text" + }, + { + "bbox": [ + 416, + 172, + 439, + 182 + ], + "score": 0.89, + "content": "l < k", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 168, + 462, + 187 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 463, + 171, + 504, + 184 + ], + "score": 0.91, + "content": "x _ { l } ^ { l k } \\neq x _ { k } ^ { l k }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 140, + 183, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 140, + 183, + 199, + 204 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 199, + 185, + 281, + 200 + ], + "score": 0.94, + "content": "( x _ { l } ^ { i j } ) _ { i < j } \\neq ( x _ { k } ^ { i j } ) _ { i < j }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 183, + 364, + 204 + ], + "score": 1.0, + "content": ". By construction of", + "type": "text" + }, + { + "bbox": [ + 365, + 187, + 373, + 197 + ], + "score": 0.6, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 183, + 377, + 204 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 378, + 183, + 502, + 203 + ], + "score": 0.91, + "content": "P \\left( ( x _ { l } ^ { i j } ) _ { i < j } \\right) \\neq P \\left( ( { \\bar { x } } _ { k } ^ { i j } ) _ { i < j } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 183, + 506, + 204 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 201, + 507, + 217 + ], + "spans": [ + { + "bbox": [ + 141, + 201, + 158, + 217 + ], + "score": 1.0, + "content": "i.e.", + "type": "text" + }, + { + "bbox": [ + 158, + 201, + 285, + 216 + ], + "score": 0.92, + "content": "P \\left( ( x ^ { i j } ) _ { i < j } \\right) _ { l } \\neq P \\left( ( x ^ { i j } ) _ { i < j } \\right) _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 201, + 315, + 217 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + }, + { + "bbox": [ + 315, + 201, + 388, + 216 + ], + "score": 0.93, + "content": "P \\left( ( x ^ { i j } ) _ { i < j } \\right) \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 201, + 507, + 217 + ], + "score": 1.0, + "content": "has pairwise distinct coordi-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 214, + 214, + 226 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 214, + 226 + ], + "score": 1.0, + "content": "nates as required.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 140, + 168, + 507, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 134, + 232, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 133, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 133, + 232, + 234, + 245 + ], + "score": 1.0, + "content": "• Finally, we show that", + "type": "text" + }, + { + "bbox": [ + 235, + 233, + 271, + 243 + ], + "score": 0.91, + "content": "\\ b { \\mathcal { A } } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 232, + 506, + 245 + ], + "score": 1.0, + "content": ". This is a direct consequence of the point above and of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 140, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 140, + 243, + 310, + 256 + ], + "score": 1.0, + "content": "Lagrange interpolation. Indeed, take any", + "type": "text" + }, + { + "bbox": [ + 310, + 244, + 340, + 255 + ], + "score": 0.92, + "content": "y \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 243, + 401, + 256 + ], + "score": 1.0, + "content": "and denote by", + "type": "text" + }, + { + "bbox": [ + 402, + 244, + 429, + 254 + ], + "score": 0.92, + "content": "x \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "the vector that we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "just constructed with pairwise distinct coordinates. Therefore, by Lagrange interpolation,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 189, + 279 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 189, + 266, + 232, + 277 + ], + "score": 0.89, + "content": "P \\in \\mathbb { R } [ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 265, + 271, + 279 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 271, + 266, + 318, + 277 + ], + "score": 0.92, + "content": "P ( x _ { i } ) = y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 265, + 357, + 279 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 357, + 266, + 416, + 278 + ], + "score": 0.94, + "content": "i \\in \\{ 1 , \\ldots , p \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 265, + 434, + 279 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 434, + 266, + 443, + 276 + ], + "score": 0.81, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "is a sublagebra", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 277, + 307, + 289 + ], + "spans": [ + { + "bbox": [ + 142, + 277, + 159, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 277, + 186, + 287 + ], + "score": 0.85, + "content": "\\mathbf { 1 } \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 277, + 190, + 289 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 190, + 277, + 232, + 289 + ], + "score": 0.88, + "content": "P ( x ) \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 277, + 276, + 289 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + }, + { + "bbox": [ + 276, + 278, + 303, + 288 + ], + "score": 0.92, + "content": "y \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 277, + 307, + 289 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 133, + 232, + 506, + 289 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 504, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 392, + 312 + ], + "score": 1.0, + "content": "Finally, we return to the general case. We introduce the set of indexes of", + "type": "text" + }, + { + "bbox": [ + 392, + 300, + 399, + 309 + ], + "score": 0.82, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 298, + 416, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 417, + 300, + 424, + 309 + ], + "score": 0.85, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "which appear in the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 309, + 393, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 393, + 324 + ], + "score": 1.0, + "content": "result and use them to reduce the situation to the previous case. Define,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 298, + 505, + 324 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 325, + 393, + 357 + ], + "lines": [ + { + "bbox": [ + 218, + 325, + 393, + 357 + ], + "spans": [ + { + "bbox": [ + 218, + 325, + 393, + 357 + ], + "score": 0.91, + "content": "\\begin{array} { l } { J = \\left\\{ j \\in \\left\\{ 1 , \\dots , p \\right\\} : \\forall x \\in A , x _ { j } = 0 \\right\\} } \\\\ { I = \\left\\{ ( i , j ) \\in ( J ^ { C } ) ^ { 2 } : \\forall x \\in A , x _ { i } = x _ { j } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "615c88863e30d76445c12239748a02e2fbc812b000b763b7f2d5cb94742d866b.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 325, + 393, + 341.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 218, + 341.0, + 393, + 357.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "list", + "bbox": [ + 106, + 360, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 166, + 375 + ], + "score": 1.0, + "content": "and denote by", + "type": "text" + }, + { + "bbox": [ + 167, + 361, + 278, + 373 + ], + "score": 0.83, + "content": "\\mathcal { A } ^ { \\prime } = \\{ x \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 358, + 282, + 375 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 362, + 316, + 374 + ], + "score": 0.62, + "content": "x _ { i } = x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 358, + 321, + 375 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 322, + 361, + 352, + 373 + ], + "score": 0.44, + "content": "\\forall j \\in J", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 358, + 356, + 375 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 357, + 361, + 391, + 373 + ], + "score": 0.58, + "content": "x _ { j } = 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 358, + 507, + 375 + ], + "score": 1.0, + "content": ", which is also a subalgebra.", + "type": "text" + } + ], + "index": 17, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 372, + 250, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 213, + 384 + ], + "score": 1.0, + "content": "By definition, it holds that", + "type": "text" + }, + { + "bbox": [ + 214, + 373, + 246, + 383 + ], + "score": 0.89, + "content": "{ \\mathcal { A } } \\subseteq { \\mathcal { A } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 372, + 250, + 384 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 358, + 507, + 384 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 388, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 174, + 402 + ], + "score": 1.0, + "content": "By construction,", + "type": "text" + }, + { + "bbox": [ + 174, + 390, + 181, + 399 + ], + "score": 0.79, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 387, + 296, + 402 + ], + "score": 1.0, + "content": "is an equivalence relation on", + "type": "text" + }, + { + "bbox": [ + 297, + 388, + 311, + 399 + ], + "score": 0.89, + "content": "J ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 387, + 370, + 402 + ], + "score": 1.0, + "content": "and denote by", + "type": "text" + }, + { + "bbox": [ + 370, + 388, + 395, + 401 + ], + "score": 0.93, + "content": "J ^ { C } / I", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "its equivalence classes. Let", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 398, + 495, + 414 + ], + "spans": [ + { + "bbox": [ + 107, + 400, + 158, + 412 + ], + "score": 0.91, + "content": "p ^ { \\prime } = | J ^ { C } / I |", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 398, + 206, + 414 + ], + "score": 1.0, + "content": "and choose", + "type": "text" + }, + { + "bbox": [ + 206, + 401, + 248, + 412 + ], + "score": 0.93, + "content": "i _ { 1 } , \\ldots , i _ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 398, + 495, + 414 + ], + "score": 1.0, + "content": "representatives of the equivalence classes. Consider the map,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 387, + 506, + 414 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 416, + 366, + 450 + ], + "lines": [ + { + "bbox": [ + 244, + 416, + 366, + 450 + ], + "spans": [ + { + "bbox": [ + 244, + 416, + 366, + 450 + ], + "score": 0.93, + "content": "\\varphi : \\left\\{ \\begin{array} { l l } { \\mathbb { R } ^ { p } \\longrightarrow \\mathbb { R } ^ { p ^ { \\prime } } } \\\\ { x \\longmapsto \\left( x _ { i _ { 1 } } , \\ldots , x _ { i _ { p ^ { \\prime } } } \\right) . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "97ecd28ce3499ceb0e04a23f8974bdc752aee7ad49d59f710f10913d981ef607.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 416, + 366, + 433.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 244, + 433.0, + 366, + 450.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 482 + ], + "lines": [ + { + "bbox": [ + 107, + 455, + 507, + 470 + ], + "spans": [ + { + "bbox": [ + 107, + 460, + 114, + 469 + ], + "score": 0.77, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 455, + 261, + 470 + ], + "score": 1.0, + "content": "is an algebra homomoprhism so that", + "type": "text" + }, + { + "bbox": [ + 261, + 456, + 284, + 469 + ], + "score": 0.92, + "content": "\\varphi ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 455, + 357, + 470 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 357, + 455, + 372, + 467 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 455, + 470, + 470 + ], + "score": 1.0, + "content": ". But, by construction of", + "type": "text" + }, + { + "bbox": [ + 471, + 457, + 477, + 467 + ], + "score": 0.82, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 455, + 495, + 470 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 495, + 457, + 502, + 467 + ], + "score": 0.81, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 455, + 507, + 470 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 468, + 459, + 483 + ], + "spans": [ + { + "bbox": [ + 107, + 470, + 129, + 482 + ], + "score": 0.9, + "content": "\\varphi ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 469, + 404, + 483 + ], + "score": 1.0, + "content": "satisfies (23) and (24). Whence, by our result in this particular case,", + "type": "text" + }, + { + "bbox": [ + 404, + 468, + 455, + 482 + ], + "score": 0.93, + "content": "\\varphi ( \\mathcal { A } ) = \\mathbb { R } ^ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 469, + 459, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 107, + 455, + 507, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 148, + 502 + ], + "score": 1.0, + "content": "However,", + "type": "text" + }, + { + "bbox": [ + 148, + 488, + 182, + 500 + ], + "score": 0.9, + "content": "{ \\mathcal { A } } \\subseteq { \\mathcal { A } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 486, + 234, + 502 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 235, + 487, + 326, + 501 + ], + "score": 0.93, + "content": "\\varphi ( \\mathcal { A } ) \\subseteq \\varphi ( \\mathcal { A } ^ { \\prime } ) \\subseteq \\mathbb { R } ^ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 486, + 376, + 502 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 376, + 487, + 467, + 501 + ], + "score": 0.93, + "content": "\\mathbb { R } ^ { p ^ { \\prime } } = \\varphi ( \\mathcal { A } ) = \\varphi ( \\mathcal { A } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 486, + 506, + 502 + ], + "score": 1.0, + "content": ". But, by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 169, + 513 + ], + "score": 1.0, + "content": "construction of", + "type": "text" + }, + { + "bbox": [ + 170, + 502, + 177, + 511 + ], + "score": 0.72, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 499, + 181, + 513 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 181, + 502, + 189, + 511 + ], + "score": 0.72, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 499, + 315, + 513 + ], + "score": 1.0, + "content": "is actually an injective map on", + "type": "text" + }, + { + "bbox": [ + 315, + 500, + 327, + 510 + ], + "score": 0.84, + "content": "\\mathcal { A } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 499, + 443, + 513 + ], + "score": 1.0, + "content": ". Therefore, we deduce from", + "type": "text" + }, + { + "bbox": [ + 444, + 500, + 505, + 511 + ], + "score": 0.91, + "content": "\\varphi ( \\mathcal { A } ) = \\varphi ( \\mathcal { A } ^ { \\prime } )", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 510, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 124, + 523 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 511, + 157, + 520 + ], + "score": 0.89, + "content": "A = A ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 510, + 249, + 523 + ], + "score": 1.0, + "content": ", concluding the proof.", + "type": "text" + }, + { + "bbox": [ + 494, + 511, + 504, + 521 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 486, + 506, + 523 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 535, + 402, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 403, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 403, + 549 + ], + "score": 1.0, + "content": "D.7 PROOF OF THE MAIN EQUIVARIANT APPROXIMATION THEOREM", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 506, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 321, + 569 + ], + "score": 1.0, + "content": "We can now fully exploit the structure of subalgebra of", + "type": "text" + }, + { + "bbox": [ + 322, + 556, + 331, + 566 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "thanks to the results above, and in particular", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "relax the second assumption of Prop. 24 to give our main theorem 22. We first prove the following", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 139, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 139, + 590 + ], + "score": 1.0, + "content": "lemma.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 555, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 592, + 478, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 476, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 332, + 607 + ], + "score": 1.0, + "content": "Lemma 28. Under the assumptions of Thm. 22, for any", + "type": "text" + }, + { + "bbox": [ + 333, + 593, + 364, + 604 + ], + "score": 0.9, + "content": "H \\subseteq G", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 591, + 438, + 607 + ], + "score": 1.0, + "content": "subgroup, for any", + "type": "text" + }, + { + "bbox": [ + 438, + 593, + 476, + 604 + ], + "score": 0.89, + "content": "x , y \\in X", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 591, + 476, + 607 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 609, + 465, + 622 + ], + "lines": [ + { + "bbox": [ + 145, + 609, + 465, + 622 + ], + "spans": [ + { + "bbox": [ + 145, + 609, + 465, + 622 + ], + "score": 0.84, + "content": "\\forall f \\in { \\mathcal { F } } , \\ \\exists g \\in H , \\ f ( g \\cdot x ) = f ( y ) \\iff \\exists g \\in H , \\forall f \\in { \\mathcal { F } } , \\ f ( g \\cdot x ) = f ( y ) \\ .", + "type": "interline_equation", + "image_path": "b7db3ce16d5008e1f009c7818d8f281516103e840409f9608ad66a97256ec1b3.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 145, + 609, + 465, + 622 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 160, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 162, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 162, + 641 + ], + "score": 1.0, + "content": "In particular,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 625, + 162, + 641 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 637, + 415, + 651 + ], + "lines": [ + { + "bbox": [ + 196, + 637, + 415, + 651 + ], + "spans": [ + { + "bbox": [ + 196, + 637, + 415, + 651 + ], + "score": 0.84, + "content": "( x , y ) \\in \\rho \\left( \\pi \\circ { \\mathcal { F } } \\right) \\iff \\exists g \\in G , ( g \\cdot x , y ) \\in \\rho \\left( { \\mathcal { F } } \\right) .", + "type": "interline_equation", + "image_path": "ab22f84c31a9f4593badad02620729e5e11ee6dd1faba5d27148306d967aa618.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 196, + 637, + 415, + 651 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 661, + 506, + 683 + ], + "lines": [ + { + "bbox": [ + 104, + 659, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 506, + 676 + ], + "score": 1.0, + "content": "Proof. The reverse implication is immediate so we focus on the direct one and prove its contraposition,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 671, + 127, + 687 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 127, + 687 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 659, + 506, + 687 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 683, + 462, + 696 + ], + "lines": [ + { + "bbox": [ + 146, + 683, + 462, + 696 + ], + "spans": [ + { + "bbox": [ + 146, + 683, + 462, + 696 + ], + "score": 0.87, + "content": "\\forall g \\in H , \\exists f \\in { \\mathcal { F } } , f ( g \\cdot x ) \\neq f ( y ) \\implies \\exists f \\in { \\mathcal { F } } , \\forall g \\in H , f ( g \\cdot x ) \\neq f ( y ) .", + "type": "interline_equation", + "image_path": "b395fbffcee7ecbc4b354d6f02d2911e9145b5fbefb83ae9bdbe81d1a8e5f9ed.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 146, + 683, + 462, + 696 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 128, + 698, + 504, + 721 + ], + "lines": [ + { + "bbox": [ + 124, + 695, + 504, + 713 + ], + "spans": [ + { + "bbox": [ + 124, + 695, + 248, + 713 + ], + "score": 1.0, + "content": "rove this, we take advantage of", + "type": "text" + }, + { + "bbox": [ + 248, + 699, + 258, + 709 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 695, + 352, + 713 + ], + "score": 1.0, + "content": "being a subalgebra and", + "type": "text" + }, + { + "bbox": [ + 352, + 699, + 362, + 709 + ], + "score": 0.84, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 695, + 429, + 713 + ], + "score": 1.0, + "content": "being finite. Let", + "type": "text" + }, + { + "bbox": [ + 429, + 699, + 504, + 711 + ], + "score": 0.91, + "content": "H = \\{ g _ { 1 } , \\dots , g _ { h } \\}", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 126, + 707, + 154, + 723 + ], + "spans": [ + { + "bbox": [ + 126, + 707, + 154, + 723 + ], + "score": 1.0, + "content": "define,", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 124, + 695, + 504, + 723 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 719, + 404, + 733 + ], + "lines": [ + { + "bbox": [ + 206, + 719, + 404, + 733 + ], + "spans": [ + { + "bbox": [ + 206, + 719, + 404, + 733 + ], + "score": 0.87, + "content": "{ \\mathcal { A } } = \\left\\{ ( f ( g _ { 1 } \\cdot x ) , \\ldots , f ( g _ { h } \\cdot x ) , f ( y ) ) : f \\in { \\mathcal { F } } \\right\\} .", + "type": "interline_equation", + "image_path": "a34a4d5cbdef0f4bf83dc57f8952a85841745f08c0fae14a48ae6298660a3dbc.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 206, + 719, + 404, + 733 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 481, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 482, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 120, + 96 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 120, + 83, + 129, + 92 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 80, + 203, + 96 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 203, + 82, + 242, + 95 + ], + "score": 0.92, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } ^ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 80, + 246, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 246, + 83, + 255, + 92 + ], + "score": 0.75, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 80, + 329, + 96 + ], + "score": 1.0, + "content": "is a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 329, + 81, + 344, + 92 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { p ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 80, + 366, + 96 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 366, + 82, + 424, + 95 + ], + "score": 0.93, + "content": "p ^ { \\prime } = p ( h + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 80, + 482, + 96 + ], + "score": 1.0, + "content": ". By Lem. 27,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 97, + 417, + 113 + ], + "lines": [ + { + "bbox": [ + 191, + 97, + 417, + 113 + ], + "spans": [ + { + "bbox": [ + 191, + 97, + 417, + 113 + ], + "score": 0.9, + "content": "\\begin{array} { r } { A = \\{ z \\in \\mathbb { R } ^ { p ^ { \\prime } } : \\forall ( i , j ) \\in I , z _ { i } = z _ { j } , \\forall j \\in J , z _ { j } = 0 \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "04e890a881a31113eae5d451c5a0a4718ca9f0b06807c6843664026b66fcdcd7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 191, + 97, + 417, + 113 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 116, + 252, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 252, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 133, + 129 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 118, + 139, + 127 + ], + "score": 0.81, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 115, + 157, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 117, + 165, + 127 + ], + "score": 0.82, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 115, + 252, + 129 + ], + "score": 1.0, + "content": "can be chosen to be,1", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 130, + 402, + 162 + ], + "lines": [ + { + "bbox": [ + 209, + 130, + 402, + 162 + ], + "spans": [ + { + "bbox": [ + 209, + 130, + 402, + 162 + ], + "score": 0.85, + "content": "\\begin{array} { r l } & { J = \\left\\{ j \\in \\{ 1 , \\dots , p \\} : \\forall z \\in A , z _ { j } = 0 \\right\\} } \\\\ & { I = \\left\\{ ( i , j ) \\in \\{ 1 , \\dots , p \\} ^ { 2 } : \\forall z \\in A , z _ { i } = z _ { j } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "4c3670f40a2b4811024b76646c565eee5388ad7fbec47e3932ddc11251b784d0.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 130, + 402, + 146.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 209, + 146.0, + 402, + 162.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 163, + 506, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 119, + 176 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 119, + 164, + 126, + 173 + ], + "score": 0.81, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 163, + 286, + 176 + ], + "score": 1.0, + "content": "is an equivalence relation, an element of", + "type": "text" + }, + { + "bbox": [ + 286, + 164, + 295, + 173 + ], + "score": 0.8, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "is uniquely defined by its coordinates on equivalence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 173, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 149, + 187 + ], + "score": 1.0, + "content": "classes of", + "type": "text" + }, + { + "bbox": [ + 149, + 174, + 155, + 184 + ], + "score": 0.76, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 173, + 271, + 187 + ], + "score": 1.0, + "content": ". Therefore, one can choose", + "type": "text" + }, + { + "bbox": [ + 272, + 174, + 299, + 184 + ], + "score": 0.9, + "content": "z \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 173, + 340, + 187 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 340, + 174, + 444, + 186 + ], + "score": 0.87, + "content": "z _ { i } = z _ { j } \\iff ( i , j ) \\in I", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 173, + 506, + 187 + ], + "score": 1.0, + "content": ". By definition", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 184, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 118, + 198 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 186, + 127, + 195 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 184, + 181, + 198 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 181, + 185, + 215, + 197 + ], + "score": 0.92, + "content": "f ^ { * } \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 184, + 256, + 198 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 257, + 185, + 418, + 197 + ], + "score": 0.89, + "content": "( f ^ { * } ( g _ { 1 } \\cdot x ) , \\ldots , f ^ { * } ( g _ { h } \\cdot x ) , ^ { ^ { \\scriptstyle * } } f ^ { * } ( y ) ) = z", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 184, + 506, + 198 + ], + "score": 1.0, + "content": ". We now check that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 107, + 196, + 118, + 208 + ], + "score": 0.88, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 195, + 239, + 209 + ], + "score": 1.0, + "content": "is indeed appropriate. Take", + "type": "text" + }, + { + "bbox": [ + 239, + 197, + 303, + 208 + ], + "score": 0.91, + "content": "l \\in \\{ 1 , \\ldots , h \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 195, + 401, + 209 + ], + "score": 1.0, + "content": ", we want to show that", + "type": "text" + }, + { + "bbox": [ + 401, + 196, + 483, + 208 + ], + "score": 0.92, + "content": "f ^ { * } ( g _ { l } \\cdot x ) \\neq f ^ { * } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 195, + 506, + 209 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 205, + 221 + ], + "score": 1.0, + "content": "assumption, there exists", + "type": "text" + }, + { + "bbox": [ + 205, + 207, + 232, + 218 + ], + "score": 0.91, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 207, + 272, + 221 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 273, + 207, + 339, + 219 + ], + "score": 0.91, + "content": "f ( g _ { l } \\cdot x ) \\neq f ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 207, + 406, + 221 + ], + "score": 1.0, + "content": ", i.e. there exists", + "type": "text" + }, + { + "bbox": [ + 407, + 208, + 465, + 219 + ], + "score": 0.9, + "content": "i \\in \\{ 1 , \\ldots , p \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 217, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 107, + 218, + 181, + 230 + ], + "score": 0.92, + "content": "f ( g _ { l } \\cdot \\bar { x } ) _ { i } \\neq f ( y ) _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 217, + 232, + 232 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 232, + 218, + 320, + 230 + ], + "score": 0.9, + "content": "( ( l - 1 ) p + i , h p + i )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 217, + 374, + 232 + ], + "score": 1.0, + "content": "cannot be in", + "type": "text" + }, + { + "bbox": [ + 375, + 218, + 381, + 228 + ], + "score": 0.74, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 217, + 412, + 232 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 412, + 219, + 488, + 231 + ], + "score": 0.89, + "content": "z _ { ( l - 1 ) p + i } \\neq z _ { h p + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 217, + 507, + 232 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 229, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 107, + 230, + 190, + 243 + ], + "score": 0.9, + "content": "f ^ { * } ( g _ { l } \\cdot x ) _ { i } \\neq f ^ { * } ( y ) _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 229, + 194, + 244 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 230, + 506, + 241 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 274, + 265 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 275, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 275, + 266 + ], + "score": 1.0, + "content": "We now prove our main abstract theorem.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 504, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 180, + 281 + ], + "score": 1.0, + "content": "Theorem 22. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 268, + 190, + 277 + ], + "score": 0.67, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 266, + 276, + 281 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 276, + 268, + 311, + 278 + ], + "score": 0.9, + "content": "Y = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 266, + 349, + 281 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 350, + 268, + 369, + 279 + ], + "score": 0.35, + "content": "p , G", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 266, + 506, + 281 + ], + "score": 1.0, + "content": "be a finite group acting (continu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 278, + 439, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 146, + 291 + ], + "score": 1.0, + "content": "ously) on", + "type": "text" + }, + { + "bbox": [ + 146, + 279, + 156, + 288 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 278, + 174, + 291 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 279, + 184, + 288 + ], + "score": 0.72, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 278, + 203, + 291 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 279, + 266, + 290 + ], + "score": 0.92, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 278, + 439, + 291 + ], + "score": 1.0, + "content": "a (non-empty) set of equivariant functions.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 294, + 504, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 151, + 309 + ], + "score": 1.0, + "content": "Denote by", + "type": "text" + }, + { + "bbox": [ + 152, + 295, + 223, + 307 + ], + "score": 0.91, + "content": "\\pi : Y \\longrightarrow Y / G", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 293, + 421, + 309 + ], + "score": 1.0, + "content": "the canonical projection on the quotient space", + "type": "text" + }, + { + "bbox": [ + 421, + 295, + 442, + 307 + ], + "score": 0.9, + "content": "Y / G", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 293, + 505, + 309 + ], + "score": 1.0, + "content": ". Consider the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 305, + 201, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 201, + 320 + ], + "score": 1.0, + "content": "following assumptions,", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 130, + 325, + 411, + 338 + ], + "lines": [ + { + "bbox": [ + 129, + 325, + 411, + 339 + ], + "spans": [ + { + "bbox": [ + 129, + 325, + 142, + 339 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 326, + 152, + 336 + ], + "score": 0.73, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 325, + 230, + 339 + ], + "score": 1.0, + "content": "is a sub-algebra of", + "type": "text" + }, + { + "bbox": [ + 231, + 326, + 266, + 338 + ], + "score": 0.92, + "content": "{ \\mathcal { C } } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 325, + 398, + 339 + ], + "score": 1.0, + "content": "and the constant function 1 is in", + "type": "text" + }, + { + "bbox": [ + 399, + 327, + 407, + 336 + ], + "score": 0.8, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 325, + 411, + 339 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 129, + 344, + 339, + 357 + ], + "lines": [ + { + "bbox": [ + 128, + 342, + 339, + 359 + ], + "spans": [ + { + "bbox": [ + 128, + 342, + 223, + 359 + ], + "score": 1.0, + "content": "2. 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By Lem. 28, the second assumption of Prop. 24 is also satisfied. To get the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 238, + 506 + ], + "score": 1.0, + "content": "conclusion of Thm. 22, note that", + "type": "text" + }, + { + "bbox": [ + 239, + 493, + 261, + 505 + ], + "score": 0.91, + "content": "\\mathcal { F } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "is now a linear subspace of a finite-dimensional vector space", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 239, + 519 + ], + "score": 1.0, + "content": "and therefore it is closed. Thus,", + "type": "text" + }, + { + "bbox": [ + 240, + 505, + 297, + 518 + ], + "score": 0.9, + "content": "\\overline { { \\mathcal { F } ( \\boldsymbol { x } ) } } = \\mathcal { F } ( \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 505, + 482, + 519 + ], + "score": 1.0, + "content": "which is a subalgebra. Applying Lem. 27 to", + "type": "text" + }, + { + "bbox": [ + 482, + 505, + 505, + 518 + ], + "score": 0.9, + "content": "\\mathcal { F } ( x )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 218, + 529 + ], + "score": 1.0, + "content": "and noting that, necessarily", + "type": "text" + }, + { + "bbox": [ + 218, + 517, + 244, + 527 + ], + "score": 0.91, + "content": "J = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 516, + 256, + 529 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 257, + 518, + 297, + 529 + ], + "score": 0.9, + "content": "\\mathbf { 1 } \\in { \\mathcal { F } } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 516, + 473, + 529 + ], + "score": 1.0, + "content": "by assumption, gives the result of Thm. 22.", + "type": "text" + }, + { + "bbox": [ + 493, + 516, + 505, + 528 + ], + "score": 0.985, + "content": "□", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 541, + 242, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 243, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 243, + 554 + ], + "score": 1.0, + "content": "D.8 PRACTICAL REDUCTIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 494, + 574 + ], + "score": 1.0, + "content": "Though the results we proved above were formulated using classic hypotheses, such as requiring", + "type": "text" + }, + { + "bbox": [ + 495, + 562, + 504, + 572 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "to be a subalgebra, we can give much more compact versions for our setting. We also reduce the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 583, + 359, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 172, + 597 + ], + "score": 1.0, + "content": "assumption that", + "type": "text" + }, + { + "bbox": [ + 172, + 584, + 261, + 596 + ], + "score": 0.93, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 583, + 359, + 597 + ], + "score": 1.0, + "content": "to a more practical one.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 235, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 599, + 236, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 236, + 613 + ], + "score": 1.0, + "content": "We start with the invariant case.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 184, + 628 + ], + "score": 1.0, + "content": "Corollary 29. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 615, + 194, + 624 + ], + "score": 0.63, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 613, + 281, + 628 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 281, + 615, + 336, + 624 + ], + "score": 0.91, + "content": "Y = \\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 613, + 506, + 628 + ], + "score": 1.0, + "content": "be some finite-dimensional vector space,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 624, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 116, + 636 + ], + "score": 0.58, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 624, + 288, + 639 + ], + "score": 1.0, + "content": "be a finite group acting (continuously) on", + "type": "text" + }, + { + "bbox": [ + 289, + 626, + 299, + 635 + ], + "score": 0.79, + "content": "X", + 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Moreover,", + "type": "text" + }, + { + "bbox": [ + 281, + 435, + 468, + 449 + ], + "score": 0.82, + "content": "i f I ( x ) = \\{ ( i , j ) \\in [ p ] ^ { 2 } : \\forall y \\in \\mathcal { F } ( x ) , y _ { i } = y _ { j } \\} ,", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 434, + 505, + 450 + ], + "score": 1.0, + "content": ", then we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 445, + 132, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 132, + 460 + ], + "score": 1.0, + "content": "have:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 434, + 505, + 460 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 457, + 397, + 471 + ], + "lines": [ + { + "bbox": [ + 213, + 457, + 397, + 471 + ], + "spans": [ + { + "bbox": [ + 213, + 457, + 397, + 471 + ], + "score": 0.86, + "content": "{ \\mathcal { F } } ( x ) = \\left\\{ y \\in \\mathbb { R } ^ { p } : \\forall ( i , j ) \\in I ( x ) , y _ { i } = y _ { j } \\right\\} .", + "type": "interline_equation", + "image_path": "c8aad53115a1bdb638775a43c7e91e813bd2537293c4bbb3e3dc650edd87ccaa.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 213, + 457, + 397, + 471 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 481, + 506, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "Proof of Thm. 22. By Lem. 28, the second assumption of Prop. 24 is also satisfied. To get the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 238, + 506 + ], + "score": 1.0, + "content": "conclusion of Thm. 22, note that", + "type": "text" + }, + { + "bbox": [ + 239, + 493, + 261, + 505 + ], + "score": 0.91, + "content": "\\mathcal { F } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "is now a linear subspace of a finite-dimensional vector space", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 239, + 519 + ], + "score": 1.0, + "content": "and therefore it is closed. Thus,", + "type": "text" + }, + { + "bbox": [ + 240, + 505, + 297, + 518 + ], + "score": 0.9, + "content": "\\overline { { \\mathcal { F } ( \\boldsymbol { x } ) } } = \\mathcal { F } ( \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 505, + 482, + 519 + ], + "score": 1.0, + "content": "which is a subalgebra. Applying Lem. 27 to", + "type": "text" + }, + { + "bbox": [ + 482, + 505, + 505, + 518 + ], + "score": 0.9, + "content": "\\mathcal { F } ( x )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 218, + 529 + ], + "score": 1.0, + "content": "and noting that, necessarily", + "type": "text" + }, + { + "bbox": [ + 218, + 517, + 244, + 527 + ], + "score": 0.91, + "content": "J = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 516, + 256, + 529 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 257, + 518, + 297, + 529 + ], + "score": 0.9, + "content": "\\mathbf { 1 } \\in { \\mathcal { F } } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 516, + 473, + 529 + ], + "score": 1.0, + "content": "by assumption, gives the result of Thm. 22.", + "type": "text" + }, + { + "bbox": [ + 493, + 516, + 505, + 528 + ], + "score": 0.985, + "content": "□", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 481, + 506, + 529 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 541, + 242, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 243, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 243, + 554 + ], + "score": 1.0, + "content": "D.8 PRACTICAL REDUCTIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 494, + 574 + ], + "score": 1.0, + "content": "Though the results we proved above were formulated using classic hypotheses, such as requiring", + "type": "text" + }, + { + "bbox": [ + 495, + 562, + 504, + 572 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "to be a subalgebra, we can give much more compact versions for our setting. We also reduce the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 583, + 359, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 172, + 597 + ], + "score": 1.0, + "content": "assumption that", + "type": "text" + }, + { + "bbox": [ + 172, + 584, + 261, + 596 + ], + "score": 0.93, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 583, + 359, + 597 + ], + "score": 1.0, + "content": "to a more practical one.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 561, + 505, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 235, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 599, + 236, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 236, + 613 + ], + "score": 1.0, + "content": "We start with the invariant case.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 599, + 236, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 184, + 628 + ], + "score": 1.0, + "content": "Corollary 29. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 615, + 194, + 624 + ], + "score": 0.63, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 613, + 281, + 628 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 281, + 615, + 336, + 624 + ], + "score": 0.91, + "content": "Y = \\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 613, + 506, + 628 + ], + "score": 1.0, + "content": "be some finite-dimensional vector space,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 624, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 116, + 636 + ], + "score": 0.58, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 624, + 288, + 639 + ], + "score": 1.0, + "content": "be a finite group acting (continuously) on", + "type": "text" + }, + { + "bbox": [ + 289, + 626, + 299, + 635 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 624, + 318, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 318, + 626, + 379, + 637 + ], + "score": 0.92, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { I } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 624, + 506, + 639 + ], + "score": 1.0, + "content": "a (non-empty) set of invariant", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 636, + 148, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 148, + 649 + ], + "score": 1.0, + "content": "functions.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 613, + 506, + 649 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 303, + 665 + ], + "lines": [ + { + "bbox": [ + 104, + 651, + 303, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 190, + 667 + ], + "score": 1.0, + "content": "Assume that, for any", + "type": "text" + }, + { + "bbox": [ + 191, + 653, + 244, + 666 + ], + "score": 0.94, + "content": "h \\in \\mathcal { C } ( \\mathbb { F } ^ { 2 } , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 651, + 263, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 654, + 299, + 665 + ], + "score": 0.91, + "content": "f , g \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 651, + 303, + 667 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 651, + 303, + 667 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 668, + 357, + 681 + ], + "lines": [ + { + "bbox": [ + 253, + 668, + 357, + 681 + ], + "spans": [ + { + "bbox": [ + 253, + 668, + 357, + 681 + ], + "score": 0.89, + "content": "x \\mapsto h ( f ( x ) , g ( x ) ) \\in \\mathcal { F } .", + "type": "interline_equation", + "image_path": "fba3191b3745bcb49ea559ab9201e28774f6fb98ea7b5935034a8a4ed3946959.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 253, + 668, + 357, + 681 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 684, + 205, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 207, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 185, + 697 + ], + "score": 1.0, + "content": "Then the closure of", + "type": "text" + }, + { + "bbox": [ + 185, + 684, + 194, + 694 + ], + "score": 0.8, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 682, + 207, + 697 + ], + "score": 1.0, + "content": "is,", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 682, + 207, + 697 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 693, + 386, + 708 + ], + "lines": [ + { + "bbox": [ + 225, + 693, + 386, + 708 + ], + "spans": [ + { + "bbox": [ + 225, + 693, + 386, + 708 + ], + "score": 0.91, + "content": "{ \\overline { { \\mathcal { F } } } } = \\{ f \\in { \\mathcal { C } } _ { I } ( X , Y ) : \\rho \\left( { \\mathcal { F } } \\right) \\subseteq \\rho \\left( f \\right) \\} ~ .", + "type": "interline_equation", + "image_path": "213c730c6ead213f127264ed917eab21831a945babe84bbae665a7b2ba6efd4d.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 225, + 693, + 386, + 708 + ], + "spans": [], + "index": 42 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 326, + 95 + ], + "score": 1.0, + "content": "Proof. We wish to apply Thm. 22 but for this we need", + "type": "text" + }, + { + "bbox": [ + 327, + 83, + 336, + 92 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 81, + 374, + 95 + ], + "score": 1.0, + "content": "to act on", + "type": "text" + }, + { + "bbox": [ + 374, + 83, + 383, + 92 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 81, + 495, + 95 + ], + "score": 1.0, + "content": ". Define a (trivial) action of", + "type": "text" + }, + { + "bbox": [ + 495, + 83, + 504, + 92 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 145, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 118, + 108 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 94, + 128, + 104 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 91, + 145, + 108 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 104, + 362, + 117 + ], + "lines": [ + { + "bbox": [ + 247, + 104, + 362, + 117 + ], + "spans": [ + { + "bbox": [ + 247, + 104, + 362, + 117 + ], + "score": 0.91, + "content": "\\forall g \\in G , \\forall y \\in Y , g \\cdot y = y .", + "type": "interline_equation", + "image_path": "876c54c0978ad3da4b5bef218e3ab2c82dd8756d68110a702bac85c6eda8cd57.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 247, + 104, + 362, + 117 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 185, + 134 + ], + "score": 1.0, + "content": "With this action on", + "type": "text" + }, + { + "bbox": [ + 186, + 120, + 195, + 131 + ], + "score": 0.55, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 117, + 198, + 134 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 198, + 120, + 293, + 132 + ], + "score": 0.89, + "content": "\\mathcal { C } _ { E } ( X , Y ) = \\mathcal { C } _ { I } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 117, + 341, + 134 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 342, + 120, + 383, + 132 + ], + "score": 0.7, + "content": "Y / G = Y", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 117, + 387, + 134 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 388, + 120, + 446, + 132 + ], + "score": 0.81, + "content": "\\pi : Y \\to Y / G", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 117, + 506, + 134 + ], + "score": 1.0, + "content": "is the identity", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 215, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 135, + 144 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 136, + 131, + 210, + 143 + ], + "score": 0.93, + "content": "\\rho \\left( \\pi \\circ { \\mathcal { F } } \\right) = \\rho \\left( { \\mathcal { F } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 131, + 215, + 144 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 105, + 147, + 498, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 496, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 248, + 161 + ], + "score": 1.0, + "content": "Our assumption clearly ensure that", + "type": "text" + }, + { + "bbox": [ + 248, + 149, + 258, + 158 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 146, + 487, + 161 + ], + "score": 1.0, + "content": "is indeed a subalgebra and contains the constant function", + "type": "text" + }, + { + "bbox": [ + 487, + 149, + 496, + 158 + ], + "score": 0.79, + "content": "\\nVdash", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 491, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 491, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 374, + 179 + ], + "score": 1.0, + "content": "All that is left to show to apply Thm. 22 is that the set of functions", + "type": "text" + }, + { + "bbox": [ + 374, + 165, + 443, + 177 + ], + "score": 0.94, + "content": "\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 162, + 491, + 179 + ], + "score": 1.0, + "content": "defined by,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 182, + 375, + 196 + ], + "lines": [ + { + "bbox": [ + 234, + 182, + 375, + 196 + ], + "spans": [ + { + "bbox": [ + 234, + 182, + 375, + 196 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f { \\bf 1 } \\in \\mathcal { F } \\}", + "type": "interline_equation", + "image_path": "a4aee7bf4e3d7ee1b9b42b7225bf97b97c7f7abc6d37497a4a94fac5521183f6.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 234, + 182, + 375, + 196 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 142, + 213 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 144, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 144, + 214 + ], + "score": 1.0, + "content": "satisfies,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 212, + 346, + 226 + ], + "lines": [ + { + "bbox": [ + 264, + 212, + 346, + 226 + ], + "spans": [ + { + "bbox": [ + 264, + 212, + 346, + 226 + ], + "score": 0.9, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right) .", + "type": "interline_equation", + "image_path": "9ee2e44e05c46ddd1e82de80db72593e52f35cec674fc29c5c639f6f8bee8532.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 264, + 212, + 346, + 226 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 507, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 128, + 241 + ], + "score": 1.0, + "content": "Take", + "type": "text" + }, + { + "bbox": [ + 129, + 228, + 187, + 240 + ], + "score": 0.92, + "content": "( x , y ) \\notin \\rho \\left( { \\mathcal { F } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 226, + 262, + 241 + ], + "score": 1.0, + "content": "and we show that", + "type": "text" + }, + { + "bbox": [ + 263, + 228, + 335, + 240 + ], + "score": 0.88, + "content": "( x , y ) \\notin \\rho \\left( \\mathcal { F } _ { s c a l } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 226, + 475, + 241 + ], + "score": 1.0, + "content": ". Indeed, by definition there exists", + "type": "text" + }, + { + "bbox": [ + 475, + 228, + 502, + 239 + ], + "score": 0.89, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 226, + 506, + 241 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 240, + 168, + 251 + ], + "score": 0.92, + "content": "i \\in \\{ 1 , \\ldots , p \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 238, + 210, + 252 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 210, + 239, + 271, + 250 + ], + "score": 0.9, + "content": "f ( x ) _ { i } \\neq f ( y ) _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 238, + 294, + 252 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 295, + 240, + 344, + 250 + ], + "score": 0.9, + "content": "l \\in \\mathcal { C } ( \\mathbb { F } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 238, + 393, + 252 + ], + "score": 1.0, + "content": "defined by", + "type": "text" + }, + { + "bbox": [ + 393, + 239, + 433, + 251 + ], + "score": 0.92, + "content": "l ( z ) = z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 238, + 453, + 252 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 453, + 239, + 505, + 250 + ], + "score": 0.9, + "content": "h \\in { \\mathcal { C } } ( \\mathbb { F } , \\mathbb { F } )", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 167, + 263 + ], + "score": 1.0, + "content": "defined by, for", + "type": "text" + }, + { + "bbox": [ + 167, + 250, + 191, + 261 + ], + "score": 0.87, + "content": "z \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 249, + 195, + 263 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 195, + 250, + 327, + 262 + ], + "score": 0.88, + "content": "h ( z ) = ( l ( z ) , \\ldots , l ( z ) ) = l ( z ) \\mathcal { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 249, + 417, + 263 + ], + "score": 1.0, + "content": ". Then, by assumption", + "type": "text" + }, + { + "bbox": [ + 417, + 250, + 460, + 261 + ], + "score": 0.9, + "content": "h \\circ f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 249, + 481, + 263 + ], + "score": 1.0, + "content": ". But", + "type": "text" + }, + { + "bbox": [ + 482, + 251, + 504, + 261 + ], + "score": 0.87, + "content": "h \\circ f", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 260, + 507, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 116, + 275 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 261, + 148, + 273 + ], + "score": 0.92, + "content": "( l \\circ f ) \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 260, + 170, + 275 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 170, + 262, + 235, + 273 + ], + "score": 0.9, + "content": "l \\circ f \\in { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 260, + 324, + 275 + ], + "score": 1.0, + "content": ", so that, by definition,", + "type": "text" + }, + { + "bbox": [ + 325, + 261, + 378, + 272 + ], + "score": 0.92, + "content": "l \\circ f \\in \\mathcal { F } _ { s c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 260, + 425, + 275 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 426, + 261, + 503, + 273 + ], + "score": 0.92, + "content": "l \\circ f ( x ) \\neq l \\circ f ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 260, + 507, + 275 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 417, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 151, + 285 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 151, + 272, + 223, + 284 + ], + "score": 0.89, + "content": "( x , y ) \\notin \\rho \\left( \\mathcal { F } _ { s c a l } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 270, + 417, + 285 + ], + "score": 1.0, + "content": ". Therefore, we can apply Thm. 22. We get that,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 290, + 437, + 304 + ], + "lines": [ + { + "bbox": [ + 173, + 290, + 437, + 304 + ], + "spans": [ + { + "bbox": [ + 173, + 290, + 437, + 304 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\right\\} , } \\end{array}", + "type": "interline_equation", + "image_path": "c22a924963bfe3f0033632ac94ce7a1a79d5e21e18a0d9f63230224a231794d4.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 173, + 290, + 437, + 304 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 123, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 123, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 123, + 321 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 320, + 396, + 334 + ], + "lines": [ + { + "bbox": [ + 214, + 320, + 396, + 334 + ], + "spans": [ + { + "bbox": [ + 214, + 320, + 396, + 334 + ], + "score": 0.88, + "content": "\\mathcal { F } ( x ) = \\left\\{ y \\in \\mathbb { F } ^ { p } : \\forall ( i , j ) \\in I ( x ) , y _ { i } = y _ { j } \\right\\} ,", + "type": "interline_equation", + "image_path": "60314bb19560d5cb1ff7863d8d45d9323b300e62994f4df7908a62e09c5736d1.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 214, + 320, + 396, + 334 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 185, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 186, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 126, + 351 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 336, + 146, + 348 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 334, + 186, + 351 + ], + "score": 1.0, + "content": "given by,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 353, + 415, + 369 + ], + "lines": [ + { + "bbox": [ + 196, + 353, + 415, + 369 + ], + "spans": [ + { + "bbox": [ + 196, + 353, + 415, + 369 + ], + "score": 0.88, + "content": "I ( x ) = \\{ ( i , j ) \\in \\{ 1 , \\ldots , p \\} ^ { 2 } : \\forall y \\in { \\mathcal { F } } ( x ) , y _ { i } = y _ { j } \\} .", + "type": "interline_equation", + "image_path": "978f27184d1e0aba2e190c33a088c3e9651dba9cab7b08e8ab81918b0ed2ed37.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 196, + 353, + 415, + 369 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 379, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 279, + 393 + ], + "score": 1.0, + "content": "To conclude the proof, we now show that", + "type": "text" + }, + { + "bbox": [ + 279, + 379, + 410, + 392 + ], + "score": 0.93, + "content": "I ( x ) = \\{ ( i , i ) : i \\in \\{ 1 , . . . , p \\} \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 378, + 506, + 393 + ], + "score": 1.0, + "content": ", which will imply that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 390, + 181, + 403 + ], + "spans": [ + { + "bbox": [ + 107, + 391, + 176, + 403 + ], + "score": 0.91, + "content": "\\mathcal { F } ( x ) = \\mathbb { R } ^ { p } = Y", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 390, + 181, + 403 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 105, + 407, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 223, + 421 + ], + "score": 1.0, + "content": "Indeed, the constant function", + "type": "text" + }, + { + "bbox": [ + 223, + 408, + 294, + 420 + ], + "score": 0.93, + "content": "z \\mapsto ( 1 , 2 , . . . , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 406, + 314, + 421 + ], + "score": 1.0, + "content": "is in", + "type": "text" + }, + { + "bbox": [ + 315, + 408, + 324, + 418 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 406, + 431, + 421 + ], + "score": 1.0, + "content": "by assumption. 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With this new", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 235, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 235, + 477 + ], + "score": 1.0, + "content": "assumption, this is much easier.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 105, + 480, + 504, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 185, + 494 + ], + "score": 1.0, + "content": "Corollary 30. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 481, + 195, + 491 + ], + "score": 0.66, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 479, + 284, + 494 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 284, + 481, + 320, + 491 + ], + "score": 0.91, + "content": "Y = \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 479, + 344, + 494 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 345, + 481, + 379, + 491 + ], + "score": 0.91, + "content": "\\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 479, + 399, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 400, + 481, + 435, + 492 + ], + "score": 0.91, + "content": "G = S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 479, + 506, + 494 + ], + "score": 1.0, + "content": "the permutation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 491, + 415, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 235, + 504 + ], + "score": 1.0, + "content": "group, acting (continuously) on", + "type": "text" + }, + { + "bbox": [ + 235, + 492, + 245, + 501 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 491, + 303, + 504 + ], + "score": 1.0, + "content": "and acting on", + "type": "text" + }, + { + "bbox": [ + 304, + 492, + 316, + 501 + ], + "score": 0.86, + "content": "\\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 491, + 345, + 504 + ], + "score": 1.0, + "content": "by, for", + "type": "text" + }, + { + "bbox": [ + 345, + 492, + 376, + 502 + ], + "score": 0.79, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 491, + 380, + 504 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 380, + 492, + 410, + 502 + ], + "score": 0.84, + "content": "x \\in \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 491, + 415, + 504 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 509, + 379, + 523 + ], + "lines": [ + { + "bbox": [ + 230, + 509, + 379, + 523 + ], + "spans": [ + { + "bbox": [ + 230, + 509, + 379, + 523 + ], + "score": 0.9, + "content": "\\forall i \\in \\{ 1 , \\ldots , p \\} , ( \\sigma \\cdot x ) _ { i } = x _ { \\sigma ^ { - 1 } ( i ) } ,", + "type": "interline_equation", + "image_path": "ed49a1ecff370f2b6433c694a710654149b8e628db355dbbe7d4044927b99253.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 230, + 509, + 379, + 523 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 534, + 373, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 373, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 122, + 549 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 535, + 188, + 547 + ], + "score": 0.92, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , \\mathbb { F } ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 534, + 373, + 549 + ], + "score": 1.0, + "content": "be a (non-empty) set of equivariant functions.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 253, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 254, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 254, + 566 + ], + "score": 1.0, + "content": "Consider the following assumptions,", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 131, + 572, + 273, + 585 + ], + "lines": [ + { + "bbox": [ + 129, + 571, + 273, + 587 + ], + "spans": [ + { + "bbox": [ + 129, + 571, + 175, + 587 + ], + "score": 1.0, + "content": "1. 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(2019a).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 30, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 326, + 95 + ], + "score": 1.0, + "content": "Proof. We wish to apply Thm. 22 but for this we need", + "type": "text" + }, + { + "bbox": [ + 327, + 83, + 336, + 92 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 81, + 374, + 95 + ], + "score": 1.0, + "content": "to act on", + "type": "text" + }, + { + "bbox": [ + 374, + 83, + 383, + 92 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 81, + 495, + 95 + ], + "score": 1.0, + "content": ". Define a (trivial) action of", + "type": "text" + }, + { + "bbox": [ + 495, + 83, + 504, + 92 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 145, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 118, + 108 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 94, + 128, + 104 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 91, + 145, + 108 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 81, + 504, + 108 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 104, + 362, + 117 + ], + "lines": [ + { + "bbox": [ + 247, + 104, + 362, + 117 + ], + "spans": [ + { + "bbox": [ + 247, + 104, + 362, + 117 + ], + "score": 0.91, + "content": "\\forall g \\in G , \\forall y \\in Y , g \\cdot y = y .", + "type": "interline_equation", + "image_path": "876c54c0978ad3da4b5bef218e3ab2c82dd8756d68110a702bac85c6eda8cd57.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 247, + 104, + 362, + 117 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 185, + 134 + ], + "score": 1.0, + "content": "With this action on", + "type": "text" + }, + { + "bbox": [ + 186, + 120, + 195, + 131 + ], + "score": 0.55, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 117, + 198, + 134 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 198, + 120, + 293, + 132 + ], + "score": 0.89, + "content": "\\mathcal { C } _ { E } ( X , Y ) = \\mathcal { C } _ { I } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 117, + 341, + 134 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 342, + 120, + 383, + 132 + ], + "score": 0.7, + "content": "Y / G = Y", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 117, + 387, + 134 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 388, + 120, + 446, + 132 + ], + "score": 0.81, + "content": "\\pi : Y \\to Y / G", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 117, + 506, + 134 + ], + "score": 1.0, + "content": "is the identity", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 215, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 135, + 144 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 136, + 131, + 210, + 143 + ], + "score": 0.93, + "content": "\\rho \\left( \\pi \\circ { \\mathcal { F } } \\right) = \\rho \\left( { \\mathcal { F } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 131, + 215, + 144 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 117, + 506, + 144 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 147, + 498, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 496, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 248, + 161 + ], + "score": 1.0, + "content": "Our assumption clearly ensure that", + "type": "text" + }, + { + "bbox": [ + 248, + 149, + 258, + 158 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 146, + 487, + 161 + ], + "score": 1.0, + "content": "is indeed a subalgebra and contains the constant function", + "type": "text" + }, + { + "bbox": [ + 487, + 149, + 496, + 158 + ], + "score": 0.79, + "content": "\\nVdash", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 146, + 496, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 491, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 491, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 374, + 179 + ], + "score": 1.0, + "content": "All that is left to show to apply Thm. 22 is that the set of functions", + "type": "text" + }, + { + "bbox": [ + 374, + 165, + 443, + 177 + ], + "score": 0.94, + "content": "\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 162, + 491, + 179 + ], + "score": 1.0, + "content": "defined by,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 162, + 491, + 179 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 182, + 375, + 196 + ], + "lines": [ + { + "bbox": [ + 234, + 182, + 375, + 196 + ], + "spans": [ + { + "bbox": [ + 234, + 182, + 375, + 196 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f { \\bf 1 } \\in \\mathcal { F } \\}", + "type": "interline_equation", + "image_path": "a4aee7bf4e3d7ee1b9b42b7225bf97b97c7f7abc6d37497a4a94fac5521183f6.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 234, + 182, + 375, + 196 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 142, + 213 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 144, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 144, + 214 + ], + "score": 1.0, + "content": "satisfies,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 199, + 144, + 214 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 212, + 346, + 226 + ], + "lines": [ + { + "bbox": [ + 264, + 212, + 346, + 226 + ], + "spans": [ + { + "bbox": [ + 264, + 212, + 346, + 226 + ], + "score": 0.9, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right) .", + "type": "interline_equation", + "image_path": "9ee2e44e05c46ddd1e82de80db72593e52f35cec674fc29c5c639f6f8bee8532.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 264, + 212, + 346, + 226 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 507, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 128, + 241 + ], + "score": 1.0, + "content": "Take", + "type": "text" + }, + { + "bbox": [ + 129, + 228, + 187, + 240 + ], + "score": 0.92, + "content": "( x , y ) \\notin \\rho \\left( { \\mathcal { F } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 226, + 262, + 241 + ], + "score": 1.0, + "content": "and we show that", + "type": "text" + }, + { + "bbox": [ + 263, + 228, + 335, + 240 + ], + "score": 0.88, + "content": "( x , y ) \\notin \\rho \\left( \\mathcal { F } _ { s c a l } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 226, + 475, + 241 + ], + "score": 1.0, + "content": ". Indeed, by definition there exists", + "type": "text" + }, + { + "bbox": [ + 475, + 228, + 502, + 239 + ], + "score": 0.89, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 226, + 506, + 241 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 107, + 240, + 168, + 251 + ], + "score": 0.92, + "content": "i \\in \\{ 1 , \\ldots , p \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 238, + 210, + 252 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 210, + 239, + 271, + 250 + ], + "score": 0.9, + "content": "f ( x ) _ { i } \\neq f ( y ) _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 238, + 294, + 252 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 295, + 240, + 344, + 250 + ], + "score": 0.9, + "content": "l \\in \\mathcal { C } ( \\mathbb { F } , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 238, + 393, + 252 + ], + "score": 1.0, + "content": "defined by", + "type": "text" + }, + { + "bbox": [ + 393, + 239, + 433, + 251 + ], + "score": 0.92, + "content": "l ( z ) = z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 238, + 453, + 252 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 453, + 239, + 505, + 250 + ], + "score": 0.9, + "content": "h \\in { \\mathcal { C } } ( \\mathbb { F } , \\mathbb { F } )", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 167, + 263 + ], + "score": 1.0, + "content": "defined by, for", + "type": "text" + }, + { + "bbox": [ + 167, + 250, + 191, + 261 + ], + "score": 0.87, + "content": "z \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 249, + 195, + 263 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 195, + 250, + 327, + 262 + ], + "score": 0.88, + "content": "h ( z ) = ( l ( z ) , \\ldots , l ( z ) ) = l ( z ) \\mathcal { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 249, + 417, + 263 + ], + "score": 1.0, + "content": ". 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But", + "type": "text" + }, + { + "bbox": [ + 482, + 251, + 504, + 261 + ], + "score": 0.87, + "content": "h \\circ f", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 260, + 507, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 116, + 275 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 116, + 261, + 148, + 273 + ], + "score": 0.92, + "content": "( l \\circ f ) \\mathbf { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 260, + 170, + 275 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 170, + 262, + 235, + 273 + ], + "score": 0.9, + "content": "l \\circ f \\in { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 260, + 324, + 275 + ], + "score": 1.0, + "content": ", so that, by definition,", + "type": "text" + }, + { + "bbox": [ + 325, + 261, + 378, + 272 + ], + "score": 0.92, + "content": "l \\circ f \\in \\mathcal { F } _ { s c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 260, + 425, + 275 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 426, + 261, + 503, + 273 + ], + "score": 0.92, + "content": "l \\circ f ( x ) \\neq l \\circ f ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 260, + 507, + 275 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 417, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 151, + 285 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 151, + 272, + 223, + 284 + ], + "score": 0.89, + "content": "( x , y ) \\notin \\rho \\left( \\mathcal { F } _ { s c a l } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 270, + 417, + 285 + ], + "score": 1.0, + "content": ". Therefore, we can apply Thm. 22. We get that,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 226, + 507, + 285 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 290, + 437, + 304 + ], + "lines": [ + { + "bbox": [ + 173, + 290, + 437, + 304 + ], + "spans": [ + { + "bbox": [ + 173, + 290, + 437, + 304 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , Y ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\right\\} , } \\end{array}", + "type": "interline_equation", + "image_path": "c22a924963bfe3f0033632ac94ce7a1a79d5e21e18a0d9f63230224a231794d4.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 173, + 290, + 437, + 304 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 123, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 123, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 123, + 321 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 310, + 123, + 321 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 320, + 396, + 334 + ], + "lines": [ + { + "bbox": [ + 214, + 320, + 396, + 334 + ], + "spans": [ + { + "bbox": [ + 214, + 320, + 396, + 334 + ], + "score": 0.88, + "content": "\\mathcal { F } ( x ) = \\left\\{ y \\in \\mathbb { F } ^ { p } : \\forall ( i , j ) \\in I ( x ) , y _ { i } = y _ { j } \\right\\} ,", + "type": "interline_equation", + "image_path": "60314bb19560d5cb1ff7863d8d45d9323b300e62994f4df7908a62e09c5736d1.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 214, + 320, + 396, + 334 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 185, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 186, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 126, + 351 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 336, + 146, + 348 + ], + "score": 0.92, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 334, + 186, + 351 + ], + "score": 1.0, + "content": "given by,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 334, + 186, + 351 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 353, + 415, + 369 + ], + "lines": [ + { + "bbox": [ + 196, + 353, + 415, + 369 + ], + "spans": [ + { + "bbox": [ + 196, + 353, + 415, + 369 + ], + "score": 0.88, + "content": "I ( x ) = \\{ ( i , j ) \\in \\{ 1 , \\ldots , p \\} ^ { 2 } : \\forall y \\in { \\mathcal { F } } ( x ) , y _ { i } = y _ { j } \\} .", + "type": "interline_equation", + "image_path": "978f27184d1e0aba2e190c33a088c3e9651dba9cab7b08e8ab81918b0ed2ed37.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 196, + 353, + 415, + 369 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 379, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 279, + 393 + ], + "score": 1.0, + "content": "To conclude the proof, we now show that", + "type": "text" + }, + { + "bbox": [ + 279, + 379, + 410, + 392 + ], + "score": 0.93, + "content": "I ( x ) = \\{ ( i , i ) : i \\in \\{ 1 , . . . , p \\} \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 378, + 506, + 393 + ], + "score": 1.0, + "content": ", which will imply that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 390, + 181, + 403 + ], + "spans": [ + { + "bbox": [ + 107, + 391, + 176, + 403 + ], + "score": 0.91, + "content": "\\mathcal { F } ( x ) = \\mathbb { R } ^ { p } = Y", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 390, + 181, + 403 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 378, + 506, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 407, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 223, + 421 + ], + "score": 1.0, + "content": "Indeed, the constant function", + "type": "text" + }, + { + "bbox": [ + 223, + 408, + 294, + 420 + ], + "score": 0.93, + "content": "z \\mapsto ( 1 , 2 , . . . , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 406, + 314, + 421 + ], + "score": 1.0, + "content": "is in", + "type": "text" + }, + { + "bbox": [ + 315, + 408, + 324, + 418 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 406, + 431, + 421 + ], + "score": 1.0, + "content": "by assumption. 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With this new", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 235, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 235, + 477 + ], + "score": 1.0, + "content": "assumption, this is much easier.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 443, + 505, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 480, + 504, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 185, + 494 + ], + "score": 1.0, + "content": "Corollary 30. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 481, + 195, + 491 + ], + "score": 0.66, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 479, + 284, + 494 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 284, + 481, + 320, + 491 + ], + "score": 0.91, + "content": "Y = \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 479, + 344, + 494 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 345, + 481, + 379, + 491 + ], + "score": 0.91, + "content": "\\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 479, + 399, + 494 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 400, + 481, + 435, + 492 + ], + "score": 0.91, + "content": "G = S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 479, + 506, + 494 + ], + "score": 1.0, + "content": "the permutation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 491, + 415, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 235, + 504 + ], + "score": 1.0, + "content": "group, acting (continuously) on", + "type": "text" + }, + { + "bbox": [ + 235, + 492, + 245, + 501 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 491, + 303, + 504 + ], + "score": 1.0, + "content": "and acting on", + "type": "text" + }, + { + "bbox": [ + 304, + 492, + 316, + 501 + ], + "score": 0.86, + "content": "\\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 491, + 345, + 504 + ], + "score": 1.0, + "content": "by, for", + "type": "text" + }, + { + "bbox": [ + 345, + 492, + 376, + 502 + ], + "score": 0.79, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 491, + 380, + 504 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 380, + 492, + 410, + 502 + ], + "score": 0.84, + "content": "x \\in \\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 491, + 415, + 504 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 479, + 506, + 504 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 509, + 379, + 523 + ], + "lines": [ + { + "bbox": [ + 230, + 509, + 379, + 523 + ], + "spans": [ + { + "bbox": [ + 230, + 509, + 379, + 523 + ], + "score": 0.9, + "content": "\\forall i \\in \\{ 1 , \\ldots , p \\} , ( \\sigma \\cdot x ) _ { i } = x _ { \\sigma ^ { - 1 } ( i ) } ,", + "type": "interline_equation", + "image_path": "ed49a1ecff370f2b6433c694a710654149b8e628db355dbbe7d4044927b99253.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 230, + 509, + 379, + 523 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 534, + 373, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 373, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 122, + 549 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 535, + 188, + 547 + ], + "score": 0.92, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { E } ( X , \\mathbb { F } ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 534, + 373, + 549 + ], + "score": 1.0, + "content": "be a (non-empty) set of equivariant functions.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 534, + 373, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 253, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 254, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 254, + 566 + ], + "score": 1.0, + "content": "Consider the following assumptions,", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 550, + 254, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 572, + 273, + 585 + ], + "lines": [ + { + "bbox": [ + 129, + 571, + 273, + 587 + ], + "spans": [ + { + "bbox": [ + 129, + 571, + 175, + 587 + ], + "score": 1.0, + "content": "1. 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Take", + "type": "text" + }, + { + "bbox": [ + 372, + 83, + 483, + 94 + ], + "score": 0.89, + "content": "x _ { 1 } , \\ldots , x _ { n } , y _ { 1 } , \\ldots , y _ { n } \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 80, + 506, + 97 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 500, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 158, + 108 + ], + "score": 1.0, + "content": "that, for any", + "type": "text" + }, + { + "bbox": [ + 158, + 94, + 189, + 105 + ], + "score": 0.86, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 92, + 192, + 108 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 193, + 93, + 329, + 107 + ], + "score": 0.89, + "content": "( x _ { 1 } , \\ldots , x _ { n } ) \\neq ( y _ { \\sigma ( 1 ) } , \\ldots , y _ { \\sigma ( n ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 92, + 406, + 108 + ], + "score": 1.0, + "content": ". 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With our first assumption, the first assumption of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 240, + 213, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 213, + 253 + ], + "score": 1.0, + "content": "Thm. 22 is easily verified.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 105, + 257, + 483, + 269 + ], + "lines": [ + { + "bbox": [ + 104, + 254, + 484, + 272 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 398, + 272 + ], + "score": 1.0, + "content": "We now focus on the second one. As in the statement of Thm. 22, define", + "type": "text" + }, + { + "bbox": [ + 398, + 257, + 468, + 270 + ], + "score": 0.93, + "content": "\\mathcal { F } _ { s c a l } \\subseteq \\mathcal { C } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 254, + 484, + 272 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 273, + 377, + 287 + ], + "lines": [ + { + "bbox": [ + 232, + 273, + 377, + 287 + ], + "spans": [ + { + "bbox": [ + 232, + 273, + 377, + 287 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { s c a l } = \\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f \\mathbf { 1 } \\in \\mathcal { F } \\} \\ .", + "type": "interline_equation", + "image_path": "db936cfa300ba8752143e75c741c5d3155b2ff3e6647ba7f934b28d6dfa22f35.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 232, + 273, + 377, + 287 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 506, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 197, + 304 + ], + "score": 1.0, + "content": "We have to show that", + "type": "text" + }, + { + "bbox": [ + 197, + 291, + 286, + 303 + ], + "score": 0.91, + "content": "\\rho \\left( \\mathcal { F } _ { s c a l } \\right) \\subseteq \\rho \\left( \\pi \\circ \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 290, + 349, + 304 + ], + "score": 1.0, + "content": ". 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There exists", + "type": "text" + }, + { + "bbox": [ + 475, + 291, + 504, + 303 + ], + "score": 0.9, + "content": "f \\in { \\mathcal { F } }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 174, + 315 + ], + "score": 1.0, + "content": "such that for any", + "type": "text" + }, + { + "bbox": [ + 174, + 303, + 205, + 313 + ], + "score": 0.88, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 302, + 209, + 315 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 210, + 303, + 275, + 314 + ], + "score": 0.89, + "content": "\\sigma \\cdot f ( x ) \\neq f ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 302, + 343, + 315 + ], + "score": 1.0, + "content": ". We have to find", + "type": "text" + }, + { + "bbox": [ + 343, + 303, + 381, + 313 + ], + "score": 0.91, + "content": "l \\in \\mathcal { F } _ { s c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 302, + 420, + 315 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 421, + 302, + 467, + 314 + ], + "score": 0.92, + "content": "l ( x ) \\neq l ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 302, + 506, + 315 + ], + "score": 1.0, + "content": ". In other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 313, + 507, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 209, + 325 + ], + "score": 1.0, + "content": "words, from a function in", + "type": "text" + }, + { + "bbox": [ + 210, + 314, + 222, + 323 + ], + "score": 0.87, + "content": "\\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 313, + 341, + 325 + ], + "score": 1.0, + "content": "which discriminates between", + "type": "text" + }, + { + "bbox": [ + 342, + 316, + 348, + 323 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 313, + 366, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 315, + 373, + 325 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 313, + 495, + 325 + ], + "score": 1.0, + "content": "we have to build a function in", + "type": "text" + }, + { + "bbox": [ + 495, + 314, + 503, + 323 + ], + "score": 0.83, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 313, + 507, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 330, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 331, + 344 + ], + "score": 1.0, + "content": "First, we exhibit a function which discriminates between", + "type": "text" + }, + { + "bbox": [ + 331, + 333, + 338, + 340 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 328, + 356, + 344 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 333, + 363, + 341 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 328, + 484, + 344 + ], + "score": 1.0, + "content": ". 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All", + "type": "text" + }, + { + "bbox": [ + 340, + 369, + 368, + 379 + ], + "score": 0.9, + "content": "z \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 367, + 388, + 397 + ], + "score": 1.0, + "content": "and e to d", + "type": "text" + }, + { + "bbox": [ + 388, + 370, + 440, + 381 + ], + "score": 0.91, + "content": "l \\in \\mathcal { C } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 367, + 447, + 397 + ], + "score": 1.0, + "content": "dt", + "type": "text" + }, + { + "bbox": [ + 485, + 367, + 510, + 397 + ], + "score": 1.0, + "content": ", for, i.e.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 380, + 485, + 392 + ], + "spans": [ + { + "bbox": [ + 107, + 381, + 136, + 390 + ], + "score": 0.82, + "content": "w \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 380, + 241, + 392 + ], + "score": 0.88, + "content": "\\begin{array} { r } { l ( w ) = \\sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( w _ { i } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 380, + 318, + 392 + ], + "score": 0.9, + "content": "l ( x ) \\neq l ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 380, + 485, + 391 + ], + "score": 0.91, + "content": "l \\in \\mathcal { F } _ { s c a l }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 389, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 107, + 391, + 137, + 401 + ], + "score": 0.87, + "content": "l \\mathbf { 1 } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 389, + 506, + 404 + ], + "score": 1.0, + "content": ". 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Therefore, the closure of", + "type": "text" + }, + { + "bbox": [ + 438, + 525, + 447, + 535 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 524, + 460, + 538 + ], + "score": 1.0, + "content": "is,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 540, + 438, + 555 + ], + "lines": [ + { + "bbox": [ + 171, + 540, + 438, + 555 + ], + "spans": [ + { + "bbox": [ + 171, + 540, + 438, + 555 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { E } ( X , \\mathbb { F } ^ { n } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) , \\forall x \\in X , f ( x ) \\in \\mathcal { F } ( x ) \\right\\} , } \\end{array}", + "type": "interline_equation", + "image_path": "4f1810143e20d9a3f80ea8eb00577506e8bda5fb555e77c06faa48b5748bb497.jpg" + } + ] + } + ], + 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Let", + "type": "text" + }, + { + "bbox": [ + 175, + 83, + 208, + 92 + ], + "score": 0.9, + "content": "\\mathbb { F } = \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 80, + 372, + 97 + ], + "score": 1.0, + "content": "be some finite-dimensional space. Take", + "type": "text" + }, + { + "bbox": [ + 372, + 83, + 483, + 94 + ], + "score": 0.89, + "content": "x _ { 1 } , \\ldots , x _ { n } , y _ { 1 } , \\ldots , y _ { n } \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 80, + 506, + 97 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 500, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 158, + 108 + ], + "score": 1.0, + "content": "that, for any", + "type": "text" + }, + { + "bbox": [ + 158, + 94, + 189, + 105 + ], + "score": 0.86, + "content": "\\sigma \\in S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 92, + 192, + 108 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 193, + 93, + 329, + 107 + ], + "score": 0.89, + "content": "( x _ { 1 } , \\ldots , x _ { n } ) \\neq ( y _ { \\sigma ( 1 ) } , \\ldots , y _ { \\sigma ( n ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 92, + 406, + 108 + ], + "score": 1.0, + "content": ". 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Taking", + "type": "text" + }, + { + "bbox": [ + 333, + 194, + 434, + 208 + ], + "score": 0.9, + "content": "h : x \\mapsto x _ { i 1 } ^ { \\alpha _ { 1 } } x _ { i 2 } ^ { \\alpha _ { 2 } } \\ldots x _ { i p } ^ { \\alpha _ { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 192, + 506, + 206 + ], + "score": 1.0, + "content": "yields the result.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 494, + 206, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 494, + 206, + 506, + 216 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 181, + 506, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 228, + 505, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "Proof of Cor. 30. We want to apply Thm. 22. 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We have to find", + "type": "text" + }, + { + "bbox": [ + 343, + 303, + 381, + 313 + ], + "score": 0.91, + "content": "l \\in \\mathcal { F } _ { s c a l }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 302, + 420, + 315 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 421, + 302, + 467, + 314 + ], + "score": 0.92, + "content": "l ( x ) \\neq l ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 302, + 506, + 315 + ], + "score": 1.0, + "content": ". In other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 313, + 507, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 209, + 325 + ], + "score": 1.0, + "content": "words, from a function in", + "type": "text" + }, + { + "bbox": [ + 210, + 314, + 222, + 323 + ], + "score": 0.87, + "content": "\\mathbb { F } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 313, + 341, + 325 + ], + "score": 1.0, + "content": "which discriminates between", + "type": "text" + }, + { + "bbox": [ + 342, + 316, + 348, + 323 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 313, + 366, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 315, + 373, + 325 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 313, + 495, + 325 + ], + "score": 1.0, + "content": "we have to build a function in", + "type": "text" + }, + { + "bbox": [ + 495, + 314, + 503, + 323 + ], + "score": 0.83, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 313, + 507, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 290, + 507, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 330, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 331, + 344 + ], + "score": 1.0, + "content": "First, we exhibit a function which discriminates between", + "type": "text" + }, + { + "bbox": [ + 331, + 333, + 338, + 340 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 328, + 356, + 344 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 333, + 363, + 341 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 328, + 484, + 344 + ], + "score": 1.0, + "content": ". 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All", + "type": "text" + }, + { + "bbox": [ + 340, + 369, + 368, + 379 + ], + "score": 0.9, + "content": "z \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 367, + 388, + 397 + ], + "score": 1.0, + "content": "and e to d", + "type": "text" + }, + { + "bbox": [ + 388, + 370, + 440, + 381 + ], + "score": 0.91, + "content": "l \\in \\mathcal { C } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 367, + 447, + 397 + ], + "score": 1.0, + "content": "dt", + "type": "text" + }, + { + "bbox": [ + 485, + 367, + 510, + 397 + ], + "score": 1.0, + "content": ", for, i.e.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 380, + 485, + 392 + ], + "spans": [ + { + "bbox": [ + 107, + 381, + 136, + 390 + ], + "score": 0.82, + "content": "w \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 380, + 241, + 392 + ], + "score": 0.88, + "content": "\\begin{array} { r } { l ( w ) = \\sum _ { i = 1 } ^ { n } h _ { 0 } ( f ( w _ { i } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 380, + 318, + 392 + ], + "score": 0.9, + "content": "l ( x ) \\neq l ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 380, + 485, + 391 + ], + "score": 0.91, + "content": "l \\in \\mathcal { F } _ { s c a l }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 389, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 107, + 391, + 137, + 401 + ], + "score": 0.87, + "content": "l \\mathbf { 1 } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 389, + 506, + 404 + ], + "score": 1.0, + "content": ". This is where the two assumptions we made come into play. 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As", + "type": "text" + }, + { + "bbox": [ + 493, + 104, + 504, + 116 + ], + "score": 0.86, + "content": "y ^ { 0 }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 508, + 131 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 117, + 131 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 115, + 139, + 127 + ], + "score": 0.92, + "content": "\\mathcal { F } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 113, + 160, + 131 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 160, + 115, + 197, + 129 + ], + "score": 0.78, + "content": "y _ { i , i ^ { \\prime } } ^ { 0 } = \\dot { i } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 113, + 200, + 131 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 201, + 115, + 240, + 128 + ], + "score": 0.59, + "content": "y _ { j , j ^ { \\prime } } ^ { 0 } = j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 113, + 244, + 131 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 244, + 115, + 318, + 127 + ], + "score": 0.91, + "content": "( i , i ^ { \\prime } , j , j ^ { \\prime } ) \\in I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 113, + 369, + 131 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 369, + 116, + 397, + 127 + ], + "score": 0.9, + "content": "j ^ { \\prime } = i ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 113, + 442, + 131 + ], + "score": 1.0, + "content": ". Consider,", + "type": "text" + }, + { + "bbox": [ + 442, + 116, + 503, + 127 + ], + "score": 0.92, + "content": "k \\in \\{ 1 , \\ldots , p \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 113, + 508, + 131 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 199, + 142 + ], + "score": 1.0, + "content": "We show that, for any", + "type": "text" + }, + { + "bbox": [ + 199, + 129, + 241, + 140 + ], + "score": 0.86, + "content": "y \\in { \\mathcal { F } } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 126, + 245, + 142 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 245, + 129, + 291, + 140 + ], + "score": 0.84, + "content": "y _ { i , k } = y _ { j , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 126, + 343, + 142 + ], + "score": 1.0, + "content": ". 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In", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 479, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 160, + 196 + ], + "score": 1.0, + "content": "particular, as", + "type": "text" + }, + { + "bbox": [ + 160, + 183, + 307, + 195 + ], + "score": 0.79, + "content": "( i , i ^ { \\prime } , j , i ^ { \\prime } ) \\in I ( x ) , h ( y _ { i } ) _ { i ^ { \\prime } } = h ( y _ { j } ) _ { i ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 181, + 326, + 196 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 327, + 184, + 371, + 195 + ], + "score": 0.89, + "content": "y _ { i , k } = y _ { j , k }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 181, + 420, + 196 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 421, + 183, + 473, + 194 + ], + "score": 0.91, + "content": "( i , j ) \\in J ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 181, + 479, + 196 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 506, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 235, + 212 + ], + "score": 1.0, + "content": "Finally, we can conclude that", + "type": "text" + }, + { + "bbox": [ + 235, + 199, + 394, + 212 + ], + "score": 0.89, + "content": "{ \\mathcal { F } } ( x ) ~ \\supset ~ \\{ y ~ \\in ~ \\mathbb { F } ^ { n } ~ : ~ \\forall ( i , j ) ~ \\in ~ J ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 198, + 400, + 212 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 400, + 199, + 442, + 212 + ], + "score": 0.88, + "content": "y _ { i } ~ = ~ y _ { j } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 198, + 506, + 212 + ], + "score": 1.0, + "content": ". Indeed, take", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 203, + 222 + ], + "score": 0.8, + "content": "y \\in \\mathbb { F } ^ { \\cdot } : \\forall ( i , j ) \\in J ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 209, + 208, + 225 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 208, + 212, + 240, + 222 + ], + "score": 0.57, + "content": "y _ { i } = y _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 209, + 382, + 225 + ], + "score": 1.0, + "content": ". We show that all the constraints of", + "type": "text" + }, + { + "bbox": [ + 382, + 211, + 402, + 222 + ], + "score": 0.91, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 209, + 506, + 225 + ], + "score": 1.0, + "content": "are satisfied. 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Fix", + "type": "text" + }, + { + "bbox": [ + 177, + 484, + 187, + 493 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 482, + 281, + 497 + ], + "score": 1.0, + "content": "some compact space,", + "type": "text" + }, + { + "bbox": [ + 281, + 484, + 311, + 495 + ], + "score": 0.89, + "content": "n \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 482, + 331, + 497 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 331, + 484, + 339, + 493 + ], + "score": 0.6, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 482, + 506, + 497 + ], + "score": 1.0, + "content": "a finite-dimensional feature space. 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Indeed, take", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 203, + 222 + ], + "score": 0.8, + "content": "y \\in \\mathbb { F } ^ { \\cdot } : \\forall ( i , j ) \\in J ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 209, + 208, + 225 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 208, + 212, + 240, + 222 + ], + "score": 0.57, + "content": "y _ { i } = y _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 209, + 382, + 225 + ], + "score": 1.0, + "content": ". We show that all the constraints of", + "type": "text" + }, + { + "bbox": [ + 382, + 211, + 402, + 222 + ], + "score": 0.91, + "content": "I ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 209, + 506, + 225 + ], + "score": 1.0, + "content": "are satisfied. 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By assumption, we infer that", + "type": "text" + }, + { + "bbox": [ + 390, + 386, + 472, + 398 + ], + "score": 0.89, + "content": "( ( i j ) \\cdot x , x ) \\in \\rho \\left( h \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 384, + 507, + 399 + ], + "score": 1.0, + "content": "too, i.e.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 490, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 124, + 409 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 396, + 202, + 408 + ], + "score": 0.92, + "content": "h ( ( i j ) \\cdot x ) = \\ddot { h } ( \\dot { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 396, + 249, + 409 + ], + "score": 1.0, + "content": "and so that", + "type": "text" + }, + { + "bbox": [ + 250, + 397, + 309, + 408 + ], + "score": 0.93, + "content": "h ( \\dot { x } ) _ { i } = h ( x ) _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 396, + 490, + 409 + ], + "score": 1.0, + "content": "by equivariance, which concludes our proof.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 340, + 507, + 409 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 438, + 241, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 242, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 242, + 451 + ], + "score": 1.0, + "content": "D.9 REDUCTIONS FOR GNNS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "We now present a lemma which explains how to instantiate the two corollaries above in the case of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 470, + 321, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 321, + 481 + ], + "score": 1.0, + "content": "GNNs by replacing continuous functions with MLPs.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 106, + 458, + 506, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 507, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 177, + 497 + ], + "score": 1.0, + "content": "Lemma 32. 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Let", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 493, + 363, + 509 + ], + "spans": [ + { + "bbox": [ + 107, + 494, + 197, + 507 + ], + "score": 0.92, + "content": "\\textstyle { \\mathcal { F } } _ { 0 } \\subseteq \\bigcup _ { h = 1 } ^ { \\infty } { \\mathcal { C } } ( X , \\mathbb { R } ^ { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 493, + 363, + 509 + ], + "score": 1.0, + "content": "be stable by concatenation and consider,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 482, + 506, + 509 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 510, + 514, + 525 + ], + "lines": [ + { + "bbox": [ + 111, + 510, + 514, + 525 + ], + "spans": [ + { + "bbox": [ + 111, + 510, + 514, + 525 + ], + "score": 0.83, + "content": "{ \\mathbf { \\Psi } } = \\{ x \\mapsto ( m ( f ( x ) _ { 1 } ) , \\dots , m ( f ( x ) _ { n } ) ) : f \\in \\mathcal { F } _ { 0 } \\cap \\mathcal { C } ( X , \\mathbb { R } ^ { h } ) , m : \\mathbb { R } ^ { h } \\to \\mathbb { F } M L P , h \\geq 1 \\} \\subseteq \\mathcal { C } ( X , \\mathbb { F } ) .", + "type": "interline_equation", + "image_path": "f7b0ee16bd4f2c034c6ab1c9cf291dbc08b874cab7656e39204e887dee7f9387.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 111, + 510, + 514, + 525 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 528, + 507, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 136, + 542 + ], + "score": 1.0, + "content": "Then, i", + "type": "text" + }, + { + "bbox": [ + 137, + 528, + 208, + 540 + ], + "score": 0.9, + "content": "{ \\mathcal { \\cdot } } { \\mathcal { E } } ( { \\mathcal { F } } ) \\subseteq { \\mathcal { C } } ( X , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 526, + 506, + 542 + ], + "score": 1.0, + "content": "is the set of functions obtained by replacing the MLP m in the definition of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 538, + 325, + 552 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 116, + 549 + ], + "score": 0.78, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 538, + 264, + 552 + ], + "score": 1.0, + "content": "by an arbitrary continuous function,", + "type": "text" + }, + { + "bbox": [ + 264, + 539, + 287, + 551 + ], + "score": 0.92, + "content": "\\mathcal { E } ( \\mathcal { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 538, + 325, + 552 + ], + "score": 1.0, + "content": "satisfies,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 526, + 506, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 559, + 188, + 573 + ], + "lines": [], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 133, + 559, + 188, + 573 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 130, + 579, + 217, + 592 + ], + "lines": [ + { + "bbox": [ + 129, + 578, + 216, + 594 + ], + "spans": [ + { + "bbox": [ + 129, + 578, + 142, + 594 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 579, + 216, + 592 + ], + "score": 0.89, + "content": "\\rho \\left( \\mathcal { F } \\right) = \\rho \\left( \\mathcal { E } ( \\mathcal { F } ) \\right)", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 129, + 578, + 216, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 597, + 286, + 610 + ], + "lines": [ + { + "bbox": [ + 128, + 595, + 284, + 613 + ], + "spans": [ + { + "bbox": [ + 128, + 595, + 175, + 613 + ], + "score": 1.0, + "content": "3. 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Define,", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 123, + 488, + 139 + ], + "lines": [ + { + "bbox": [ + 122, + 123, + 488, + 139 + ], + "spans": [ + { + "bbox": [ + 122, + 123, + 488, + 139 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { E } ( \\mathcal { F } ) = \\left\\{ x \\mapsto ( m ( f ( x ) _ { 1 } ) , \\ldots , m ( f ( x ) _ { n } ) ) : f \\in \\mathcal { F } _ { 0 } \\cap \\mathcal { C } ( X , \\mathbb { R } ^ { h } ) , m \\in \\mathcal { C } ( \\mathbb { R } ^ { h } , \\mathbb { F } ) , h \\geq 1 \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "67b1c3980d1536d31bc737bfce00c5d939a6544d57b54fe2fb76760ad37f5ede.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 122, + 123, + 488, + 139 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 473, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 473, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 473, + 158 + ], + "score": 1.0, + "content": "As MLP are continuous and by the universality of MLP on a compact set (see Section C.1),", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 160, + 341, + 175 + ], + "lines": [ + { + "bbox": [ + 269, + 160, + 341, + 175 + ], + "spans": [ + { + "bbox": [ + 269, + 160, + 341, + 175 + ], + "score": 0.89, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { E } } ( { \\mathcal { F } } ) \\subseteq { \\overline { { \\mathcal { F } } } } .", + "type": "interline_equation", + "image_path": "e25ce24def3cd7371891a91d7c83b5536c1b2051fa37a42f6ecfedf7cc27610e.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 269, + 160, + 341, + 175 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 450, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 451, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 237, + 195 + ], + "score": 1.0, + "content": "This already implies 1. and that", + "type": "text" + }, + { + "bbox": [ + 237, + 181, + 349, + 195 + ], + "score": 0.92, + "content": "\\rho \\left( \\overline { { \\mathcal { F } } } \\right) \\subseteq \\rho \\left( \\mathcal { E } ( \\mathcal { F } ) \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 180, + 451, + 195 + ], + "score": 1.0, + "content": ". 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Take", + "type": "text" + }, + { + "bbox": [ + 193, + 200, + 422, + 213 + ], + "score": 0.62, + "content": "h \\in \\mathcal { C } ( \\mathbb { F } ^ { 2 } , \\mathbb { F } ) , f , g \\in \\mathcal { F } _ { 0 } , f \\in \\mathcal { C } ( X , \\mathbb { R } ^ { h _ { f } } ) , g \\in \\mathcal { C } ( X , \\mathbb { R } ^ { h _ { g } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 199, + 439, + 215 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 200, + 502, + 213 + ], + "score": 0.93, + "content": "m \\in \\mathcal { C } ( \\mathbb { R } ^ { h _ { f } } , \\mathbb { F } )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 210, + 282, + 225 + ], + "spans": [ + { + "bbox": [ + 107, + 211, + 163, + 224 + ], + "score": 0.93, + "content": "l \\in \\mathcal { C } ( \\mathbb { R } ^ { h _ { g } } , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 210, + 282, + 225 + ], + "score": 1.0, + "content": ". 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Fix", + "type": "text" + }, + { + "bbox": [ + 177, + 527, + 204, + 538 + ], + "score": 0.89, + "content": "\\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 524, + 506, + 540 + ], + "score": 1.0, + "content": "(non-trivial) finite dimensionnal vector spaces. Consider the action of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 535, + 248, + 553 + ], + "spans": [ + { + "bbox": [ + 107, + 538, + 140, + 550 + ], + "score": 0.91, + "content": "G = S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 535, + 154, + 553 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 154, + 538, + 200, + 551 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 535, + 248, + 553 + ], + "score": 1.0, + "content": "defined by,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 556, + 456, + 572 + ], + "lines": [ + { + "bbox": [ + 153, + 556, + 456, + 572 + ], + "spans": [ + { + "bbox": [ + 153, + 556, + 456, + 572 + ], + "score": 0.89, + "content": "\\forall \\sigma \\in { \\mathcal { S } } _ { n } , \\forall x ^ { 0 } \\in \\mathbb { F } _ { 0 } , \\forall x \\in \\mathbb { F } ^ { n \\times k } , \\sigma \\cdot ( x ^ { 0 } , x ) = ( x ^ { 0 } , x _ { \\sigma ^ { - 1 } ( 1 ) } , \\dots , x _ { \\sigma ^ { - 1 } ( n ) } )", + "type": "interline_equation", + "image_path": "3db8b4157a9232bb933d4781b0c852345ad69ea01ba0bc5f06531eb739ca18b7.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 153, + 556, + 456, + 572 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 504, + 601 + ], + "lines": [ + { + "bbox": [ + 101, + 574, + 509, + 598 + ], + "spans": [ + { + "bbox": [ + 101, + 574, + 122, + 598 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 578, + 190, + 592 + ], + "score": 0.93, + "content": "K \\subseteq \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 574, + 387, + 598 + ], + "score": 1.0, + "content": "be a compact set. Then, the set of functions from", + "type": "text" + }, + { + "bbox": [ + 387, + 578, + 465, + 592 + ], + "score": 0.9, + "content": "K \\subseteq \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } t o", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 574, + 466, + 598 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 467, + 579, + 479, + 591 + ], + "score": 0.7, + "content": "\\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 574, + 509, + 598 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 588, + 132, + 605 + ], + "spans": [ + { + "bbox": [ + 103, + 588, + 132, + 605 + ], + "score": 1.0, + "content": "form,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 600, + 387, + 640 + ], + "lines": [ + { + "bbox": [ + 221, + 600, + 387, + 640 + ], + "spans": [ + { + "bbox": [ + 221, + 600, + 387, + 640 + ], + "score": 0.95, + "content": "( x ^ { 0 } , x ) \\longmapsto f _ { 0 } \\left( x ^ { 0 } , \\sum _ { j = 1 } ^ { n } \\prod _ { w = 1 } ^ { k } f _ { w } ( x _ { j , w } ) \\right)", + "type": "interline_equation", + "image_path": "66ca50388388f3efbf03494f9c5fbe882ce24bec62354ec82ff3e0affd7d0b43.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 221, + 600, + 387, + 620.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 221, + 620.0, + 387, + 640.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 642, + 506, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 641, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 133, + 657 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 642, + 214, + 654 + ], + "score": 0.87, + "content": "f _ { 0 } : \\mathbb { F } _ { 0 } \\times \\mathbb { R } ^ { h } \\to \\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 641, + 218, + 657 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 218, + 642, + 274, + 656 + ], + "score": 0.83, + "content": "f _ { j } : \\mathbb { F } _ { 0 } \\mathbb { R } ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 641, + 278, + 657 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 279, + 643, + 330, + 655 + ], + "score": 0.87, + "content": "j = 1 , \\dots , k", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 641, + 466, + 657 + ], + "score": 1.0, + "content": "are multi-linear perceptrons and", + "type": "text" + }, + { + "bbox": [ + 467, + 644, + 491, + 654 + ], + "score": 0.9, + "content": "h \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 641, + 506, + 657 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 654, + 187, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 142, + 666 + ], + "score": 1.0, + "content": "dense in", + "type": "text" + }, + { + "bbox": [ + 142, + 655, + 183, + 666 + ], + "score": 0.91, + "content": "\\mathcal { C } _ { I } ( K , \\mathbb { F } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 654, + 187, + 666 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 504, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 181, + 693 + ], + "score": 1.0, + "content": "Proof. Denote by", + "type": "text" + }, + { + "bbox": [ + 181, + 680, + 244, + 692 + ], + "score": 0.93, + "content": "\\mathcal { F } \\subseteq \\mathcal { C } _ { I } ( K , \\mathbb { F } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 677, + 405, + 693 + ], + "score": 1.0, + "content": "the set of such functions. To prove that", + "type": "text" + }, + { + "bbox": [ + 406, + 679, + 469, + 692 + ], + "score": 0.93, + "content": "\\overline { { \\mathcal { F } } } = \\mathcal { C } _ { I } ( K , \\mathbb { F } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 677, + 506, + 693 + ], + "score": 1.0, + "content": ", we first", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 690, + 166, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 166, + 703 + ], + "score": 1.0, + "content": "apply Thm. 5.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 708, + 158, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 159, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 159, + 721 + ], + "score": 1.0, + "content": "We get, that,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 718, + 406, + 734 + ], + "lines": [ + { + "bbox": [ + 204, + 718, + 406, + 734 + ], + "spans": [ + { + "bbox": [ + 204, + 718, + 406, + 734 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) : \\rho \\left( \\mathcal { F } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "43581b4e42a559da9806136ab4a7dab3e223a769f5f14c388869f479eb2f8607.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 204, + 718, + 406, + 734 + ], + "spans": [], + "index": 36 + } + ] + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 128, + 82, + 415, + 95 + ], + "lines": [ + { + "bbox": [ + 129, + 81, + 415, + 96 + ], + "spans": [ + { + "bbox": [ + 129, + 81, + 151, + 96 + ], + "score": 1.0, + "content": "5. If", + "type": "text" + }, + { + "bbox": [ + 151, + 83, + 161, + 92 + ], + "score": 0.73, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 81, + 353, + 96 + ], + "score": 1.0, + "content": "is equivariant w.r.t. the action described in Cor.", + "type": "text" + }, + { + "bbox": [ + 353, + 83, + 365, + 93 + ], + "score": 0.31, + "content": "3 0 _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 81, + 389, + 96 + ], + "score": 1.0, + "content": ", so is", + "type": "text" + }, + { + "bbox": [ + 389, + 82, + 412, + 95 + ], + "score": 0.91, + "content": "\\mathcal { E } ( \\mathcal { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 81, + 415, + 96 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 129, + 81, + 415, + 96 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 167, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 169, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 169, + 120 + ], + "score": 1.0, + "content": "Proof. Define,", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 105, + 169, + 120 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 123, + 488, + 139 + ], + "lines": [ + { + "bbox": [ + 122, + 123, + 488, + 139 + ], + "spans": [ + { + "bbox": [ + 122, + 123, + 488, + 139 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { E } ( \\mathcal { F } ) = \\left\\{ x \\mapsto ( m ( f ( x ) _ { 1 } ) , \\ldots , m ( f ( x ) _ { n } ) ) : f \\in \\mathcal { F } _ { 0 } \\cap \\mathcal { C } ( X , \\mathbb { R } ^ { h } ) , m \\in \\mathcal { C } ( \\mathbb { R } ^ { h } , \\mathbb { F } ) , h \\geq 1 \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "67b1c3980d1536d31bc737bfce00c5d939a6544d57b54fe2fb76760ad37f5ede.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 122, + 123, + 488, + 139 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 473, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 473, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 473, + 158 + ], + "score": 1.0, + "content": "As MLP are continuous and by the universality of MLP on a compact set (see Section C.1),", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 143, + 473, + 158 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 160, + 341, + 175 + ], + "lines": [ + { + "bbox": [ + 269, + 160, + 341, + 175 + ], + "spans": [ + { + "bbox": [ + 269, + 160, + 341, + 175 + ], + "score": 0.89, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { E } } ( { \\mathcal { F } } ) \\subseteq { \\overline { { \\mathcal { F } } } } .", + "type": "interline_equation", + "image_path": "e25ce24def3cd7371891a91d7c83b5536c1b2051fa37a42f6ecfedf7cc27610e.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 269, + 160, + 341, + 175 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 450, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 451, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 237, + 195 + ], + "score": 1.0, + "content": "This already implies 1. and that", + "type": "text" + }, + { + "bbox": [ + 237, + 181, + 349, + 195 + ], + "score": 0.92, + "content": "\\rho \\left( \\overline { { \\mathcal { F } } } \\right) \\subseteq \\rho \\left( \\mathcal { E } ( \\mathcal { F } ) \\right) \\subseteq \\rho \\left( \\mathcal { F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 180, + 451, + 195 + ], + "score": 1.0, + "content": ". Using Lem. 15 yields 2.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 180, + 451, + 195 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 502, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 193, + 215 + ], + "score": 1.0, + "content": "We now show 3. Take", + "type": "text" + }, + { + "bbox": [ + 193, + 200, + 422, + 213 + ], + "score": 0.62, + "content": "h \\in \\mathcal { C } ( \\mathbb { F } ^ { 2 } , \\mathbb { F } ) , f , g \\in \\mathcal { F } _ { 0 } , f \\in \\mathcal { C } ( X , \\mathbb { R } ^ { h _ { f } } ) , g \\in \\mathcal { C } ( X , \\mathbb { R } ^ { h _ { g } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 199, + 439, + 215 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 200, + 502, + 213 + ], + "score": 0.93, + "content": "m \\in \\mathcal { C } ( \\mathbb { R } ^ { h _ { f } } , \\mathbb { F } )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 210, + 282, + 225 + ], + "spans": [ + { + "bbox": [ + 107, + 211, + 163, + 224 + ], + "score": 0.93, + "content": "l \\in \\mathcal { C } ( \\mathbb { R } ^ { h _ { g } } , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 210, + 282, + 225 + ], + "score": 1.0, + "content": ". All we have to show is that,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 199, + 502, + 225 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 229, + 444, + 243 + ], + "lines": [ + { + "bbox": [ + 165, + 229, + 444, + 243 + ], + "spans": [ + { + "bbox": [ + 165, + 229, + 444, + 243 + ], + "score": 0.86, + "content": "x \\mapsto \\left( h ( m ( f ( x ) _ { 1 } ) , l ( g ( x ) _ { 1 } ) ) , \\ldots , h ( m ( f ( x ) _ { n } ) , l ( g ( x ) _ { n } ) ) \\right) \\in \\mathcal { E } ( \\mathcal { F } ) .", + "type": "interline_equation", + "image_path": "6cc40bbe0654e6dfd10db833528b252edb66f49224ec09da9f0a61681ffc4aef.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 165, + 229, + 444, + 243 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 506, + 294 + ], + "lines": [ + { + "bbox": [ + 104, + 244, + 508, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 244, + 137, + 262 + ], + "score": 1.0, + "content": "But as", + "type": "text" + }, + { + "bbox": [ + 137, + 249, + 150, + 259 + ], + "score": 0.86, + "content": "\\mathcal { F } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 244, + 269, + 262 + ], + "score": 1.0, + "content": "is stable by concatenation,", + "type": "text" + }, + { + "bbox": [ + 269, + 247, + 396, + 260 + ], + "score": 0.9, + "content": "x \\ \\mapsto \\ ( f ( x ) , g ( x ) ) \\ \\in \\ \\mathbb { R } ^ { h _ { f } + h _ { g } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 244, + 440, + 262 + ], + "score": 1.0, + "content": "is still in", + "type": "text" + }, + { + "bbox": [ + 440, + 248, + 453, + 259 + ], + "score": 0.88, + "content": "\\mathcal { F } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 244, + 508, + 262 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 255, + 508, + 274 + ], + "spans": [ + { + "bbox": [ + 107, + 259, + 336, + 271 + ], + "score": 0.88, + "content": "y \\in \\mathbb { R } ^ { h _ { f } + h _ { g } } \\mapsto h ( m ( \\dot { y } _ { 1 } , \\dots , y _ { h _ { f } } ) , l ( y _ { h _ { f } + 1 } , \\dots , y _ { h _ { f } + h _ { g } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 255, + 375, + 274 + ], + "score": 1.0, + "content": "is also in", + "type": "text" + }, + { + "bbox": [ + 376, + 258, + 432, + 271 + ], + "score": 0.93, + "content": "\\mathcal { C } ( \\mathbb { R } ^ { h _ { f } + h _ { g } } , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 255, + 508, + 274 + ], + "score": 1.0, + "content": "which shows that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 270, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 215, + 284 + ], + "score": 1.0, + "content": "the map above is indeed in", + "type": "text" + }, + { + "bbox": [ + 216, + 272, + 238, + 283 + ], + "score": 0.91, + "content": "\\mathcal { E } ( \\mathcal { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 270, + 506, + 284 + ], + "score": 1.0, + "content": ". 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Fix", + "type": "text" + }, + { + "bbox": [ + 177, + 527, + 204, + 538 + ], + "score": 0.89, + "content": "\\mathbb { F } _ { 0 } , \\mathbb { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 524, + 506, + 540 + ], + "score": 1.0, + "content": "(non-trivial) finite dimensionnal vector spaces. Consider the action of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 535, + 248, + 553 + ], + "spans": [ + { + "bbox": [ + 107, + 538, + 140, + 550 + ], + "score": 0.91, + "content": "G = S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 535, + 154, + 553 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 154, + 538, + 200, + 551 + ], + "score": 0.92, + "content": "\\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 535, + 248, + 553 + ], + "score": 1.0, + "content": "defined by,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 524, + 506, + 553 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 556, + 456, + 572 + ], + "lines": [ + { + "bbox": [ + 153, + 556, + 456, + 572 + ], + "spans": [ + { + "bbox": [ + 153, + 556, + 456, + 572 + ], + "score": 0.89, + "content": "\\forall \\sigma \\in { \\mathcal { S } } _ { n } , \\forall x ^ { 0 } \\in \\mathbb { F } _ { 0 } , \\forall x \\in \\mathbb { F } ^ { n \\times k } , \\sigma \\cdot ( x ^ { 0 } , x ) = ( x ^ { 0 } , x _ { \\sigma ^ { - 1 } ( 1 ) } , \\dots , x _ { \\sigma ^ { - 1 } ( n ) } )", + "type": "interline_equation", + "image_path": "3db8b4157a9232bb933d4781b0c852345ad69ea01ba0bc5f06531eb739ca18b7.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 153, + 556, + 456, + 572 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 504, + 601 + ], + "lines": [ + { + "bbox": [ + 101, + 574, + 509, + 598 + ], + "spans": [ + { + "bbox": [ + 101, + 574, + 122, + 598 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 578, + 190, + 592 + ], + "score": 0.93, + "content": "K \\subseteq \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 574, + 387, + 598 + ], + "score": 1.0, + "content": "be a compact set. 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Denote by", + "type": "text" + }, + { + "bbox": [ + 181, + 680, + 244, + 692 + ], + "score": 0.93, + "content": "\\mathcal { F } \\subseteq \\mathcal { C } _ { I } ( K , \\mathbb { F } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 677, + 405, + 693 + ], + "score": 1.0, + "content": "the set of such functions. 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Now, apply Lem. 31 with", + "type": "text" + }, + { + "bbox": [ + 390, + 160, + 432, + 172 + ], + "score": 0.89, + "content": "\\mathbb { F } \\mathbb { F } _ { 0 } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 159, + 506, + 174 + ], + "score": 1.0, + "content": ", the universality", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 170, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 104, + 170, + 321, + 186 + ], + "score": 1.0, + "content": "of MLP and use the decomposition given to get", + "type": "text" + }, + { + "bbox": [ + 321, + 172, + 394, + 184 + ], + "score": 0.85, + "content": "f _ { j } : \\mathbb { F } _ { 0 } \\ \\to \\ \\mathbb { R } ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 170, + 400, + 186 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 401, + 172, + 461, + 183 + ], + "score": 0.91, + "content": "j ~ = ~ 1 , \\ldots , k", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 170, + 506, + 186 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 305, + 199 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { j = 1 } ^ { n } \\prod _ { w = 1 } ^ { k } f _ { w } ( x _ { j , w } ) \\neq \\sum _ { j = 1 } ^ { n } \\prod _ { w = 1 } ^ { k } f _ { w } ( y _ { j , w } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 177, + 441, + 204 + ], + "score": 1.0, + "content": ". Choosing an appropriate MLP", + "type": "text" + }, + { + "bbox": [ + 442, + 184, + 504, + 197 + ], + "score": 0.91, + "content": "f _ { 0 } : \\mathbb { R } ^ { h } \\mathbb { F } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 258, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 150, + 212 + ], + "score": 1.0, + "content": "yields that", + "type": "text" + }, + { + "bbox": [ + 150, + 199, + 253, + 211 + ], + "score": 0.89, + "content": "( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \\notin \\rho \\left( { \\mathcal F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 198, + 258, + 212 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 219, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 220, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 220, + 229 + ], + "score": 1.0, + "content": "Hence, we have shown that,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 230, + 454, + 246 + ], + "lines": [ + { + "bbox": [ + 156, + 230, + 454, + 246 + ], + "spans": [ + { + "bbox": [ + 156, + 230, + 454, + 246 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) : \\rho \\left( i n v \\right) \\subseteq \\rho \\left( f \\right) \\right\\} = \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "0f0d346eb3432095077e67ee6bb80d2e1de0673a4c8183a593b7e160e54bf160.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 156, + 230, + 454, + 246 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 281, + 305, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 306, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 306, + 294 + ], + "score": 1.0, + "content": "E.2 APPROXIMATION THEOREMS FOR GNNS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 301, + 346, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 339, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 339, + 314 + ], + "score": 1.0, + "content": "We now have all the tools to finally prove our main result.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 498, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 501, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 180, + 332 + ], + "score": 1.0, + "content": "Theorem 34. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 318, + 256, + 330 + ], + "score": 0.88, + "content": "K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 316, + 260, + 332 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 261, + 316, + 299, + 331 + ], + "score": 0.9, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 316, + 501, + 332 + ], + "score": 1.0, + "content": "be compact sets. 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Let", + "type": "text" + }, + { + "bbox": [ + 176, + 490, + 255, + 502 + ], + "score": 0.88, + "content": "K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 487, + 259, + 505 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 488, + 299, + 502 + ], + "score": 0.89, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 487, + 506, + 505 + ], + "score": 1.0, + "content": "be compact sets. 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Choosing an appropriate MLP", + "type": "text" + }, + { + "bbox": [ + 442, + 184, + 504, + 197 + ], + "score": 0.91, + "content": "f _ { 0 } : \\mathbb { R } ^ { h } \\mathbb { F } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 258, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 150, + 212 + ], + "score": 1.0, + "content": "yields that", + "type": "text" + }, + { + "bbox": [ + 150, + 199, + 253, + 211 + ], + "score": 0.89, + "content": "( ( x ^ { 0 } , x ) , ( y ^ { 0 } , y ) ) \\notin \\rho \\left( { \\mathcal F } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 198, + 258, + 212 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 145, + 506, + 212 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 219, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 220, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 220, + 229 + ], + "score": 1.0, + "content": "Hence, we have shown that,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 214, + 220, + 229 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 230, + 454, + 246 + ], + "lines": [ + { + "bbox": [ + 156, + 230, + 454, + 246 + ], + "spans": [ + { + "bbox": [ + 156, + 230, + 454, + 246 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\overline { { \\mathcal { F } } } = \\left\\{ f \\in \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) : \\rho \\left( i n v \\right) \\subseteq \\rho \\left( f \\right) \\right\\} = \\mathcal { C } _ { I } ( \\mathbb { F } _ { 0 } \\times \\mathbb { F } _ { 0 } ^ { n \\times k } , \\mathbb { F } _ { 1 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "0f0d346eb3432095077e67ee6bb80d2e1de0673a4c8183a593b7e160e54bf160.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 156, + 230, + 454, + 246 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 281, + 305, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 306, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 306, + 294 + ], + "score": 1.0, + "content": "E.2 APPROXIMATION THEOREMS FOR GNNS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 301, + 346, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 339, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 339, + 314 + ], + "score": 1.0, + "content": "We now have all the tools to finally prove our main result.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 300, + 339, + 314 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 498, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 501, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 180, + 332 + ], + "score": 1.0, + "content": "Theorem 34. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 318, + 256, + 330 + ], + "score": 0.88, + "content": "K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 316, + 260, + 332 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 261, + 316, + 299, + 331 + ], + "score": 0.9, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 316, + 501, + 332 + ], + "score": 1.0, + "content": "be compact sets. 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Let", + "type": "text" + }, + { + "bbox": [ + 176, + 490, + 255, + 502 + ], + "score": 0.88, + "content": "K _ { d i s c r } \\subseteq \\mathcal G _ { n } \\times \\mathbb { F } _ { 0 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 487, + 259, + 505 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 488, + 299, + 502 + ], + "score": 0.89, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { n ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 487, + 506, + 505 + ], + "score": 1.0, + "content": "be compact sets. For the invariant case, and any", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 499, + 173, + 514 + ], + "spans": [ + { + "bbox": [ + 107, + 501, + 131, + 512 + ], + "score": 0.9, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 499, + 173, + 514 + ], + "score": 1.0, + "content": ", we have,", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 487, + 506, + 514 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 516, + 422, + 565 + ], + "lines": [ + { + "bbox": [ + 190, + 516, + 422, + 565 + ], + "spans": [ + { + "bbox": [ + 190, + 516, + 422, + 565 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( M G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\ – L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – L G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\\\ & { \\overline { { k \\ – F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – F G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } & \\end{array}", + "type": "interline_equation", + "image_path": "e4ff07d94a44fc45b3aaf7898ddbb5b3e17ff7671dcb5dfa780b1301c5815387.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 190, + 516, + 422, + 532.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 190, + 532.3333333333334, + 422, + 548.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 190, + 548.6666666666667, + 422, + 565.0000000000001 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 308, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 309, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 242, + 581 + ], + "score": 1.0, + "content": "For the equivariant case, and any", + "type": "text" + }, + { + "bbox": [ + 243, + 568, + 267, + 579 + ], + "score": 0.9, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 568, + 309, + 581 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 568, + 309, + 581 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 583, + 422, + 631 + ], + "lines": [ + { + "bbox": [ + 176, + 583, + 422, + 631 + ], + "spans": [ + { + "bbox": [ + 176, + 583, + 422, + 631 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\overline { { M G N N _ { E } } } } & { = \\{ f \\in \\mathcal { C } _ { E } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( M G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\overline { { k \\ – L G N N _ { E } } } } & { = \\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ { \\overline { { k \\ – F G N N _ { E } } } } & { = \\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ) : \\rho \\left( k \\ – F G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}", + "type": "interline_equation", + "image_path": "5751b2af1c56951e33b27575c93aca698285cda04fb2919cc06480104cc9a4c9.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 176, + 583, + 422, + 599.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 176, + 599.0, + 422, + 615.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 176, + 615.0, + 422, + 631.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 450, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 451, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 451, + 653 + ], + "score": 1.0, + "content": "Proof of Thm. 34. The theorem is now a direct consequence of Prop. 13 and Lem. 35.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 639, + 451, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 663, + 262, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 263, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 263, + 676 + ], + "score": 1.0, + "content": "We now move to the proof of Lem. 35.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 663, + 263, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 506, + 720 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 329, + 699 + ], + "score": 1.0, + "content": "Proof of Lem. 35. First, focus on the invariant case. Let", + "type": "text" + }, + { + "bbox": [ + 329, + 688, + 338, + 698 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 687, + 367, + 699 + ], + "score": 1.0, + "content": "denote", + "type": "text" + }, + { + "bbox": [ + 368, + 688, + 404, + 699 + ], + "score": 0.77, + "content": "{ \\bf M G N N } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 687, + 408, + 699 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 408, + 688, + 450, + 699 + ], + "score": 0.72, + "content": "k { \\mathrm { - } } \\mathrm { L G N N } _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 687, + 462, + 699 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 462, + 687, + 504, + 699 + ], + "score": 0.8, + "content": "k { \\mathrm { - } } \\mathrm { F G N N } _ { I }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 124, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 699, + 134, + 709 + ], + "score": 0.83, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 698, + 174, + 712 + ], + "score": 1.0, + "content": "be either", + "type": "text" + }, + { + "bbox": [ + 174, + 699, + 202, + 710 + ], + "score": 0.92, + "content": "K _ { d i s c r }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 698, + 215, + 712 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 216, + 699, + 226, + 709 + ], + "score": 0.86, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 698, + 257, + 712 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 257, + 699, + 267, + 708 + ], + "score": 0.86, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 698, + 333, + 712 + ], + "score": 1.0, + "content": "is compact and", + "type": "text" + }, + { + "bbox": [ + 333, + 699, + 393, + 711 + ], + "score": 0.93, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { I } ( X , \\mathbb { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". 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First, let us replace", + "type": "text" + }, + { + "bbox": [ + 332, + 100, + 370, + 111 + ], + "score": 0.83, + "content": "\\mathbf { M G N N } _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 97, + 505, + 113 + ], + "score": 1.0, + "content": "by another class, which is slightly", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 110, + 217, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 217, + 123 + ], + "score": 1.0, + "content": "simpler to analyze. Define,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 126, + 477, + 156 + ], + "lines": [ + { + "bbox": [ + 132, + 126, + 477, + 156 + ], + "spans": [ + { + "bbox": [ + 132, + 126, + 477, + 156 + ], + "score": 0.68, + "content": "\\begin{array} { r l } & { \\mathbf { M G N N } _ { E ^ { \\prime } } = \\left\\{ m _ { E } \\circ \\left( \\left( 1 - \\lambda \\right) \\mathrm { I d } + \\lambda S ^ { 1 } \\right) \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } : F _ { t } : } \\\\ & { \\qquad \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { m e s s a g e ~ p a s s i n g ~ l a y e r } , t = 1 , \\dots , T , T \\geq 1 , \\lambda \\in \\left\\{ 0 , 1 \\right\\} \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "cdb7b1b7a683c3d5bf20105efb2ebd4861790d64f7c5b7666d7e33e648b65ff5.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + 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Define,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 97, + 505, + 123 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 126, + 477, + 156 + ], + "lines": [ + { + "bbox": [ + 132, + 126, + 477, + 156 + ], + "spans": [ + { + "bbox": [ + 132, + 126, + 477, + 156 + ], + "score": 0.68, + "content": "\\begin{array} { r l } & { \\mathbf { M G N N } _ { E ^ { \\prime } } = \\left\\{ m _ { E } \\circ \\left( \\left( 1 - \\lambda \\right) \\mathrm { I d } + \\lambda S ^ { 1 } \\right) \\circ F _ { T } \\circ \\dots F _ { 2 } \\circ F _ { 1 } : F _ { t } : } \\\\ & { \\qquad \\mathbb { F } _ { t } ^ { n } \\to \\mathbb { F } _ { t + 1 } ^ { n } \\mathrm { m e s s a g e ~ p a s s i n g ~ l a y e r } , t = 1 , \\dots , T , T \\geq 1 , \\lambda \\in \\left\\{ 0 , 1 \\right\\} \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "cdb7b1b7a683c3d5bf20105efb2ebd4861790d64f7c5b7666d7e33e648b65ff5.jpg" + } + ] + } + ], 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There are two ways to do it. The first would be to directly adapt all the proofs but it", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "would make them more cumbersome. Instead, we extend them with a simple argument presented", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "below. 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Similarly, assume that the output space", + "type": "text" + }, + { + "bbox": [ + 341, + 555, + 350, + 565 + ], + "score": 0.73, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 552, + 420, + 570 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + }, + { + "bbox": [ + 421, + 555, + 481, + 568 + ], + "score": 0.93, + "content": "\\textstyle Y = \\operatorname { L } _ { \\alpha \\in A } Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 552, + 506, + 570 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 565, + 353, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 119, + 577 + ], + "score": 0.8, + "content": "Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 565, + 123, + 579 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 123, + 566, + 150, + 577 + ], + "score": 0.85, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 565, + 353, + 579 + ], + "score": 1.0, + "content": "a family of (pairwise disjoints) real vector spaces.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 496, + 595 + ], + "score": 1.0, + "content": "Before moving to the results, we need a last definition. Informally, we need that the functions", + "type": "text" + }, + { + "bbox": [ + 497, + 583, + 504, + 595 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "that we will consider do not change the number of nodes of their inputs. Formally, we say that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 160, + 616 + ], + "score": 0.91, + "content": "f : X \\longrightarrow Y", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 605, + 213, + 617 + ], + "score": 1.0, + "content": "is adapted if", + "type": "text" + }, + { + "bbox": [ + 214, + 605, + 266, + 616 + ], + "score": 0.92, + "content": "f ( X _ { \\alpha } ) \\subseteq Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 605, + 302, + 617 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 302, + 605, + 329, + 615 + ], + "score": 0.89, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 605, + 389, + 617 + ], + "score": 1.0, + "content": ", and denote by", + "type": "text" + }, + { + "bbox": [ + 390, + 605, + 434, + 617 + ], + "score": 0.95, + "content": "\\mathcal { C } _ { a d } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "the of continuous", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 616, + 253, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 218, + 627 + ], + "score": 1.0, + "content": "and adapted functions from", + "type": "text" + }, + { + "bbox": [ + 218, + 617, + 228, + 626 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 616, + 240, + 627 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 240, + 617, + 249, + 626 + ], + "score": 0.8, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 616, + 253, + 627 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 439, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 441, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 441, + 646 + ], + "score": 1.0, + "content": "We can now state our lemma to adapt our results to graphs with varying node sizes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 506, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 175, + 659 + ], + "score": 1.0, + "content": "Lemma 36. Let", + "type": "text" + }, + { + "bbox": [ + 175, + 648, + 185, + 658 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 648, + 271, + 659 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 272, + 648, + 281, + 658 + ], + "score": 0.66, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 648, + 395, + 659 + ], + "score": 1.0, + "content": "be a topological space and", + "type": "text" + }, + { + "bbox": [ + 395, + 648, + 404, + 658 + ], + "score": 0.6, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 648, + 505, + 659 + ], + "score": 1.0, + "content": "a finite set. Assume that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 656, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 153, + 673 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 153, + 659, + 168, + 670 + ], + "score": 0.86, + "content": "X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 656, + 171, + 673 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 172, + 659, + 199, + 669 + ], + "score": 0.89, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 656, + 408, + 673 + ], + "score": 1.0, + "content": "a family of pairwise disjoints compact sets such that", + "type": "text" + }, + { + "bbox": [ + 408, + 659, + 472, + 671 + ], + "score": 0.91, + "content": "\\textstyle X = \\bigcup _ { \\alpha \\in A } X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 656, + 506, + 673 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 122, + 681 + ], + "score": 0.87, + "content": "X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 669, + 378, + 682 + ], + "score": 1.0, + "content": "’s are in distinct connected components. Also assume that there is", + "type": "text" + }, + { + "bbox": [ + 378, + 670, + 390, + 681 + ], + "score": 0.77, + "content": "Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 669, + 394, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 671, + 421, + 680 + ], + "score": 0.84, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "a family of (pairwise", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 680, + 507, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 255, + 695 + ], + "score": 1.0, + "content": "disjoints) real vector spaces such that", + "type": "text" + }, + { + "bbox": [ + 255, + 681, + 315, + 693 + ], + "score": 0.91, + "content": "\\textstyle Y = \\operatorname { L } _ { \\alpha \\in A } Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 680, + 358, + 695 + ], + "score": 1.0, + "content": ". Consider", + "type": "text" + }, + { + "bbox": [ + 359, + 681, + 424, + 693 + ], + "score": 0.89, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { a d } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 680, + 507, + 695 + ], + "score": 1.0, + "content": "a (non-empty) set of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 691, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 233, + 707 + ], + "score": 1.0, + "content": "adapted functions and, for each", + "type": "text" + }, + { + "bbox": [ + 233, + 693, + 260, + 703 + ], + "score": 0.88, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 691, + 290, + 707 + ], + "score": 1.0, + "content": ", define", + "type": "text" + }, + { + "bbox": [ + 290, + 693, + 447, + 706 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { | X _ { \\alpha } } = \\{ f _ { | X _ { \\alpha } } : f \\in \\mathcal { F } \\} \\subseteq \\mathcal { C } ( X _ { \\alpha } , Y _ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 691, + 506, + 707 + ], + "score": 1.0, + "content": "the restriction", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 702, + 221, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 181, + 717 + ], + "score": 1.0, + "content": "of the functions of", + "type": "text" + }, + { + "bbox": [ + 181, + 704, + 190, + 713 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 702, + 201, + 717 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 202, + 704, + 216, + 715 + ], + "score": 0.89, + "content": "X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 702, + 221, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 237, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 719, + 237, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 719, + 237, + 734 + ], + "score": 1.0, + "content": "Assume that the following holds,", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "page_idx": 36, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "37", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 388, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 496, + 389, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 496, + 389, + 505, + 399 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 133, + 81, + 383, + 94 + ], + "lines": [ + { + "bbox": [ + 133, + 81, + 384, + 96 + ], + "spans": [ + { + "bbox": [ + 133, + 81, + 151, + 96 + ], + "score": 1.0, + "content": "• If", + "type": "text" + }, + { + "bbox": [ + 152, + 82, + 217, + 93 + ], + "score": 0.72, + "content": "\\mathcal { F } = k \\mathrm { - F G N N } _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 81, + 266, + 96 + ], + "score": 1.0, + "content": ". A function", + "type": "text" + }, + { + "bbox": [ + 267, + 83, + 274, + 94 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 81, + 384, + 96 + ], + "score": 1.0, + "content": "of this class is of the form,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 133, + 81, + 384, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 98, + 391, + 113 + ], + "lines": [ + { + "bbox": [ + 254, + 98, + 391, + 113 + ], + "spans": [ + { + "bbox": [ + 254, + 98, + 391, + 113 + ], + "score": 0.91, + "content": "m _ { E } \\circ S _ { 1 } ^ { k } \\circ F _ { T } \\circ . . . 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F _ { 2 } \\circ F _ { 1 } \\circ I ^ { k }", + "type": "interline_equation", + "image_path": "ed34686b25d00bbbac03b9eee4493e9d7068c2206d93dff1dcb4389dc788f793.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 170, + 344, + 477, + 359 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 364, + 243, + 376 + ], + "lines": [ + { + "bbox": [ + 142, + 363, + 244, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 363, + 244, + 378 + ], + "score": 1.0, + "content": "exactly implements (28).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 142, + 363, + 244, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 416, + 346, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 347, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 347, + 430 + ], + "score": 1.0, + "content": "F EXTENSION TO GRAPHS OF VARYING SIZES", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 106, + 441, + 328, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 329, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 329, + 453 + ], + "score": 1.0, + "content": "F.1 EXTENSION TO DISCONNECTED INPUT SPACES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "Here we show that our results can be extended to graphs of varying sizes similarly to Keriven &", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "Peyré (2019). There are two ways to do it. The first would be to directly adapt all the proofs but it", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "would make them more cumbersome. Instead, we extend them with a simple argument presented", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "below. As a side benefit, this general lemma makes it possible to extend almost all the approximation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 505, + 318, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 318, + 517 + ], + "score": 1.0, + "content": "results for graph neural networks from the literature.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 461, + 506, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 382, + 535 + ], + "score": 1.0, + "content": "The abstract setting is the following. 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Informally, we need that the functions", + "type": "text" + }, + { + "bbox": [ + 497, + 583, + 504, + 595 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "that we will consider do not change the number of nodes of their inputs. Formally, we say that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 160, + 616 + ], + "score": 0.91, + "content": "f : X \\longrightarrow Y", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 605, + 213, + 617 + ], + "score": 1.0, + "content": "is adapted if", + "type": "text" + }, + { + "bbox": [ + 214, + 605, + 266, + 616 + ], + "score": 0.92, + "content": "f ( X _ { \\alpha } ) \\subseteq Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 605, + 302, + 617 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 302, + 605, + 329, + 615 + ], + "score": 0.89, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 605, + 389, + 617 + ], + "score": 1.0, + "content": ", and denote by", + "type": "text" + }, + { + "bbox": [ + 390, + 605, + 434, + 617 + ], + "score": 0.95, + "content": "\\mathcal { C } _ { a d } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "the of continuous", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 616, + 253, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 218, + 627 + ], + "score": 1.0, + "content": "and adapted functions from", + "type": "text" + }, + { + "bbox": [ + 218, + 617, + 228, + 626 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 616, + 240, + 627 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 240, + 617, + 249, + 626 + ], + "score": 0.8, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 616, + 253, + 627 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 583, + 506, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 439, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 441, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 441, + 646 + ], + "score": 1.0, + "content": "We can now state our lemma to adapt our results to graphs with varying node sizes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 631, + 441, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 506, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 175, + 659 + ], + "score": 1.0, + "content": "Lemma 36. Let", + "type": "text" + }, + { + "bbox": [ + 175, + 648, + 185, + 658 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 648, + 271, + 659 + ], + "score": 1.0, + "content": "be a compact space,", + "type": "text" + }, + { + "bbox": [ + 272, + 648, + 281, + 658 + ], + "score": 0.66, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 648, + 395, + 659 + ], + "score": 1.0, + "content": "be a topological space and", + "type": "text" + }, + { + "bbox": [ + 395, + 648, + 404, + 658 + ], + "score": 0.6, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 648, + 505, + 659 + ], + "score": 1.0, + "content": "a finite set. Assume that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 656, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 153, + 673 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 153, + 659, + 168, + 670 + ], + "score": 0.86, + "content": "X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 656, + 171, + 673 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 172, + 659, + 199, + 669 + ], + "score": 0.89, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 656, + 408, + 673 + ], + "score": 1.0, + "content": "a family of pairwise disjoints compact sets such that", + "type": "text" + }, + { + "bbox": [ + 408, + 659, + 472, + 671 + ], + "score": 0.91, + "content": "\\textstyle X = \\bigcup _ { \\alpha \\in A } X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 656, + 506, + 673 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 122, + 681 + ], + "score": 0.87, + "content": "X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 669, + 378, + 682 + ], + "score": 1.0, + "content": "’s are in distinct connected components. Also assume that there is", + "type": "text" + }, + { + "bbox": [ + 378, + 670, + 390, + 681 + ], + "score": 0.77, + "content": "Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 669, + 394, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 671, + 421, + 680 + ], + "score": 0.84, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "a family of (pairwise", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 680, + 507, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 255, + 695 + ], + "score": 1.0, + "content": "disjoints) real vector spaces such that", + "type": "text" + }, + { + "bbox": [ + 255, + 681, + 315, + 693 + ], + "score": 0.91, + "content": "\\textstyle Y = \\operatorname { L } _ { \\alpha \\in A } Y _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 680, + 358, + 695 + ], + "score": 1.0, + "content": ". Consider", + "type": "text" + }, + { + "bbox": [ + 359, + 681, + 424, + 693 + ], + "score": 0.89, + "content": "{ \\mathcal { F } } \\subseteq { \\mathcal { C } } _ { a d } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 680, + 507, + 695 + ], + "score": 1.0, + "content": "a (non-empty) set of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 691, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 233, + 707 + ], + "score": 1.0, + "content": "adapted functions and, for each", + "type": "text" + }, + { + "bbox": [ + 233, + 693, + 260, + 703 + ], + "score": 0.88, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 691, + 290, + 707 + ], + "score": 1.0, + "content": ", define", + "type": "text" + }, + { + "bbox": [ + 290, + 693, + 447, + 706 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { | X _ { \\alpha } } = \\{ f _ { | X _ { \\alpha } } : f \\in \\mathcal { F } \\} \\subseteq \\mathcal { C } ( X _ { \\alpha } , Y _ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 691, + 506, + 707 + ], + "score": 1.0, + "content": "the restriction", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 702, + 221, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 181, + 717 + ], + "score": 1.0, + "content": "of the functions of", + "type": "text" + }, + { + "bbox": [ + 181, + 704, + 190, + 713 + ], + "score": 0.79, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 702, + 201, + 717 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 202, + 704, + 216, + 715 + ], + "score": 0.89, + "content": "X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 702, + 221, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 648, + 507, + 717 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 237, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 719, + 237, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 719, + 237, + 734 + ], + "score": 1.0, + "content": "Assume that the following holds,", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 719, + 237, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 128, + 81, + 480, + 95 + ], + "lines": [ + { + "bbox": [ + 128, + 79, + 477, + 98 + ], + "spans": [ + { + "bbox": [ + 128, + 79, + 165, + 98 + ], + "score": 1.0, + "content": "1. 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Define the set of functions", + "type": "text" + }, + { + "bbox": [ + 244, + 228, + 298, + 240 + ], + "score": 0.94, + "content": "S \\subseteq { \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 225, + 315, + 242 + ], + "score": 1.0, + "content": "by,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 245, + 372, + 259 + ], + "lines": [ + { + "bbox": [ + 238, + 245, + 372, + 259 + ], + "spans": [ + { + "bbox": [ + 238, + 245, + 372, + 259 + ], + "score": 0.91, + "content": "S = \\left\\{ f \\in \\mathcal { C } ( X , \\mathbb { R } ) : f . \\mathcal { F } \\subseteq \\mathcal { F } \\right\\} .", + "type": "interline_equation", + "image_path": "be77f553a5bdc294bd9576eaa4830b41d0390d766e23dcc82d4adaf93454d48a.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 238, + 245, + 372, + 259 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 503, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 163, + 279 + ], + "score": 1.0, + "content": "By definition,", + "type": "text" + }, + { + "bbox": [ + 163, + 266, + 205, + 277 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { s c a l } \\subseteq S", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 263, + 283, + 279 + ], + "score": 1.0, + "content": ". Moreover, as each", + "type": "text" + }, + { + "bbox": [ + 284, + 266, + 306, + 279 + ], + "score": 0.93, + "content": "\\mathcal { F } _ { | X _ { \\alpha } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 263, + 370, + 279 + ], + "score": 1.0, + "content": "is a subalgebra,", + "type": "text" + }, + { + "bbox": [ + 370, + 266, + 378, + 275 + ], + "score": 0.85, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 263, + 468, + 279 + ], + "score": 1.0, + "content": "is also a subalgebra of", + "type": "text" + }, + { + "bbox": [ + 469, + 265, + 503, + 277 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 380, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 380, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 119, + 297 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 120, + 284, + 130, + 293 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 282, + 276, + 297 + ], + "score": 1.0, + "content": "is compact, we can apply Cor. 17 to", + "type": "text" + }, + { + "bbox": [ + 276, + 284, + 284, + 293 + ], + "score": 0.84, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 282, + 340, + 297 + ], + "score": 1.0, + "content": "to get that, in", + "type": "text" + }, + { + "bbox": [ + 340, + 283, + 375, + 295 + ], + "score": 0.93, + "content": "{ \\mathcal { C } } ( X , \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 282, + 380, + 297 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 301, + 381, + 317 + ], + "lines": [ + { + "bbox": [ + 229, + 301, + 381, + 317 + ], + "spans": [ + { + "bbox": [ + 229, + 301, + 381, + 317 + ], + "score": 0.9, + "content": "{ \\overline { { S } } } = \\left\\{ f \\in { \\mathcal { C } } ( X , \\mathbb { R } ) : \\rho \\left( S \\right) \\subseteq \\rho \\left( f \\right) \\right\\} .", + "type": "interline_equation", + "image_path": "f37425b4eef4156e7d98715ba8e8920756fbc7226cc02fecd37f13807bd823f6.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 229, + 301, + 381, + 317 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 322, + 492, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 492, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 314, + 337 + ], + "score": 1.0, + "content": "In the first part of the proof, we check that, for each", + "type": "text" + }, + { + "bbox": [ + 315, + 325, + 322, + 333 + ], + "score": 0.81, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 320, + 376, + 337 + ], + "score": 1.0, + "content": ", the function", + "type": "text" + }, + { + "bbox": [ + 377, + 323, + 447, + 335 + ], + "score": 0.91, + "content": "f _ { \\alpha } : X \\longrightarrow [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 320, + 492, + 337 + ], + "score": 1.0, + "content": "defined by", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 340, + 349, + 373 + ], + "lines": [ + { + "bbox": [ + 261, + 340, + 349, + 373 + ], + "spans": [ + { + "bbox": [ + 261, + 340, + 349, + 373 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f _ { \\alpha | X _ { \\alpha } } = 1 } \\\\ { \\forall \\beta \\neq \\alpha , f _ { \\alpha | X _ { \\beta } } = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "180342dd096dff111fa51d4d74927e8de3b25499f22b8ecd4bb0c04c706522c4.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 261, + 340, + 349, + 356.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 261, + 356.5, + 349, + 373.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 378, + 483, + 392 + ], + "lines": [ + { + "bbox": [ + 104, + 377, + 484, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 377, + 207, + 394 + ], + "score": 1.0, + "content": "is continuous and satisfy", + "type": "text" + }, + { + "bbox": [ + 208, + 379, + 268, + 392 + ], + "score": 0.92, + "content": "\\rho \\left( S \\right) \\subseteq \\rho \\left( f _ { \\alpha } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 377, + 456, + 394 + ], + "score": 1.0, + "content": ". This will means that such functions belong to", + "type": "text" + }, + { + "bbox": [ + 456, + 379, + 479, + 390 + ], + "score": 0.91, + "content": "\\overline { { \\mathcal { F } _ { s c a l } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 377, + 484, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 507, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 507, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 134, + 409 + ], + "score": 1.0, + "content": "As the", + "type": "text" + }, + { + "bbox": [ + 135, + 397, + 149, + 408 + ], + "score": 0.92, + "content": "X _ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 395, + 488, + 409 + ], + "score": 1.0, + "content": "’s are in different connected components, it is enough to check its continuity on each", + "type": "text" + }, + { + "bbox": [ + 488, + 397, + 502, + 408 + ], + "score": 0.89, + "content": "X _ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 395, + 507, + 409 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 406, + 508, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 160, + 421 + ], + "score": 1.0, + "content": "Indeed, each", + "type": "text" + }, + { + "bbox": [ + 160, + 408, + 185, + 420 + ], + "score": 0.91, + "content": "f _ { \\alpha | X _ { \\beta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 406, + 508, + 421 + ], + "score": 1.0, + "content": "is constant so continuous. The second fact comes from the second assumption.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 418, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 160, + 434 + ], + "score": 1.0, + "content": "Indeed, take", + "type": "text" + }, + { + "bbox": [ + 160, + 420, + 207, + 432 + ], + "score": 0.9, + "content": "( x , y ) \\in X", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 418, + 248, + 434 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 249, + 420, + 307, + 432 + ], + "score": 0.92, + "content": "( x , y ) \\in \\rho \\left( S \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 418, + 327, + 434 + ], + "score": 1.0, + "content": ". As", + "type": "text" + }, + { + "bbox": [ + 327, + 420, + 371, + 432 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { s c a l } \\subseteq S", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 418, + 429, + 434 + ], + "score": 1.0, + "content": ", in particular", + "type": "text" + }, + { + "bbox": [ + 429, + 420, + 503, + 432 + ], + "score": 0.91, + "content": "( x , y ) \\in \\rho \\left( \\mathcal { F } _ { s c a l } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 418, + 506, + 434 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 429, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 187, + 446 + ], + "score": 1.0, + "content": "But, by assumption,", + "type": "text" + }, + { + "bbox": [ + 187, + 431, + 279, + 445 + ], + "score": 0.93, + "content": "\\textstyle \\rho ( { \\mathcal { F } } _ { s c a l } ) \\subseteq \\sqcup _ { \\beta \\in A } X _ { \\beta } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 429, + 352, + 446 + ], + "score": 1.0, + "content": ". Thus, necessarily", + "type": "text" + }, + { + "bbox": [ + 353, + 434, + 360, + 441 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 429, + 377, + 446 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 433, + 384, + 443 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 429, + 460, + 446 + ], + "score": 1.0, + "content": "belong to the same", + "type": "text" + }, + { + "bbox": [ + 461, + 432, + 475, + 443 + ], + "score": 0.89, + "content": "X _ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 429, + 507, + 446 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 443, + 173, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 168, + 456 + ], + "score": 0.91, + "content": "f _ { \\alpha } ( x ) = f _ { \\alpha } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 443, + 173, + 457 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 254, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 255, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 220, + 474 + ], + "score": 1.0, + "content": "Therefore, we conclude that", + "type": "text" + }, + { + "bbox": [ + 221, + 461, + 251, + 474 + ], + "score": 0.92, + "content": "f _ { \\alpha } \\in \\overline { { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 461, + 255, + 474 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 506, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 483, + 491 + ], + "score": 1.0, + "content": "We now prove the announced equality. By the definition of the distance, the first inclusion,", + "type": "text" + }, + { + "bbox": [ + 483, + 478, + 505, + 491 + ], + "score": 0.88, + "content": "\\overline { { \\mathcal { F } } } \\subseteq", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 107, + 489, + 233, + 503 + ], + "score": 0.9, + "content": "\\{ f \\in \\mathcal { C } _ { a d } ( X , Y ) : f _ { | X _ { \\alpha } } \\in \\overline { { \\mathcal { F } _ { \\alpha } } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 489, + 421, + 504 + ], + "score": 1.0, + "content": "is immediate. We focus on the other way. Take", + "type": "text" + }, + { + "bbox": [ + 421, + 490, + 483, + 502 + ], + "score": 0.93, + "content": "h \\in \\mathcal { C } _ { a d } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 502, + 504, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 124, + 517 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 503, + 170, + 516 + ], + "score": 0.91, + "content": "h _ { | X _ { \\alpha } } \\in \\overline { { \\mathcal { F } _ { \\alpha } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 502, + 209, + 517 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 209, + 504, + 236, + 514 + ], + "score": 0.9, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 502, + 254, + 517 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 504, + 278, + 514 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 502, + 347, + 517 + ], + "score": 1.0, + "content": ". By definition of", + "type": "text" + }, + { + "bbox": [ + 347, + 504, + 353, + 513 + ], + "score": 0.8, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 502, + 441, + 517 + ], + "score": 1.0, + "content": ", there exists, for each", + "type": "text" + }, + { + "bbox": [ + 441, + 504, + 468, + 514 + ], + "score": 0.83, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 502, + 471, + 517 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 472, + 503, + 504, + 515 + ], + "score": 0.79, + "content": "g _ { \\alpha } \\in \\mathcal { F }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 513, + 149, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 149, + 527 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 525, + 349, + 546 + ], + "lines": [ + { + "bbox": [ + 260, + 525, + 349, + 546 + ], + "spans": [ + { + "bbox": [ + 260, + 525, + 349, + 546 + ], + "score": 0.91, + "content": "\\underset { X _ { \\alpha } } { \\operatorname* { s u p } } \\Vert h - g _ { \\alpha } \\Vert _ { Y _ { \\alpha } } \\leq \\epsilon ,", + "type": "interline_equation", + "image_path": "2859e3dbe7165d617346b38b4980e09134067d5415d28b262b4e07416137aee9.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 260, + 525, + 349, + 546 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 506, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 135, + 565 + ], + "score": 1.0, + "content": "As the", + "type": "text" + }, + { + "bbox": [ + 135, + 551, + 150, + 562 + ], + "score": 0.9, + "content": "X _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 550, + 241, + 565 + ], + "score": 1.0, + "content": "’s are compact and the", + "type": "text" + }, + { + "bbox": [ + 242, + 552, + 253, + 563 + ], + "score": 0.86, + "content": "g _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 550, + 326, + 565 + ], + "score": 1.0, + "content": "’s are continuous,", + "type": "text" + }, + { + "bbox": [ + 326, + 551, + 445, + 564 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { \\alpha \\in A } \\operatorname* { s u p } _ { X _ { \\alpha } } \\| g _ { \\alpha } \\| < + \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 550, + 506, + 565 + ], + "score": 1.0, + "content": "and denote by", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 107, + 564, + 137, + 574 + ], + "score": 0.89, + "content": "M > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 563, + 367, + 577 + ], + "score": 1.0, + "content": "a bound on this quantity. 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l _ { \\alpha } | \\leq \\frac { \\epsilon } { M } .", + "type": "interline_equation", + "image_path": "3857bce1f5b8af129edabf9438ee6a8ff2e5deadb6332c5fb6c5beb7f08cf3ee.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 262, + 600, + 349, + 624 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 443, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 443, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 172, + 649 + ], + "score": 1.0, + "content": "By definition of", + "type": "text" + }, + { + "bbox": [ + 172, + 635, + 180, + 645 + ], + "score": 0.85, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 633, + 211, + 649 + ], + "score": 1.0, + "content": "and, as", + "type": "text" + }, + { + "bbox": [ + 212, + 635, + 221, + 645 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 633, + 286, + 649 + ], + "score": 1.0, + "content": "is a subalgebra,", + "type": "text" + }, + { + "bbox": [ + 286, + 634, + 356, + 648 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { \\alpha \\in A } l _ { \\alpha } g _ { \\alpha } \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 633, + 411, + 649 + ], + "score": 1.0, + "content": "and, for each", + "type": "text" + }, + { + "bbox": [ + 412, + 635, + 438, + 646 + ], + "score": 0.91, + "content": "\\beta \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 633, + 443, + 649 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 653, + 465, + 717 + ], + "lines": [ + { + "bbox": [ + 146, + 653, + 465, + 717 + ], + "spans": [ + { + "bbox": [ + 146, + 653, + 465, + 717 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { { \\operatorname* { s u p } _ { X _ { \\beta } } \\| h - \\sum _ { \\alpha \\in A } l _ { \\alpha } g _ { \\alpha } \\in \\mathcal { F } \\| \\le \\operatorname* { s u p } _ { X _ { \\beta } } \\| h - g _ { \\beta } \\| + \\| g _ { \\beta } - l _ { \\beta } g _ { \\beta } \\| + \\| \\sum _ { \\alpha \\neq \\beta } l _ { \\alpha } g _ { \\alpha } \\| } } \\\\ & { } & { \\le \\epsilon + M \\times \\frac { \\epsilon } { M } + ( | A | - 1 ) M \\times \\frac { \\epsilon } { M } = ( | A | + 1 ) \\epsilon , } \\end{array}", + "type": "interline_equation", + "image_path": "ee1cc62d1feb8d94283b3b4cd6b530226ffefaba1c1c03bf219ae007483023f8.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 146, + 653, + 465, + 674.3333333333334 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 146, + 674.3333333333334, + 465, + 695.6666666666667 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 146, + 695.6666666666667, + 465, + 717.0000000000001 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 216, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 216, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 216, + 733 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + } + ], + "page_idx": 37, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "38", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 721, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 722, + 504, + 731 + ], + "spans": [ + { + "bbox": [ + 496, + 722, + 504, + 731 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 128, + 81, + 480, + 95 + ], + "lines": [ + { + "bbox": [ + 128, + 79, + 477, + 98 + ], + "spans": [ + { + "bbox": [ + 128, + 79, + 165, + 98 + ], + "score": 1.0, + "content": "1. 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This will means that such functions belong to", + "type": "text" + }, + { + "bbox": [ + 456, + 379, + 479, + 390 + ], + "score": 0.91, + "content": "\\overline { { \\mathcal { F } _ { s c a l } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 377, + 484, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 377, + 484, + 394 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 396, + 507, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 507, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 134, + 409 + ], + "score": 1.0, + "content": "As the", + "type": "text" + }, + { + "bbox": [ + 135, + 397, + 149, + 408 + ], + "score": 0.92, + "content": "X _ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 395, + 488, + 409 + ], + "score": 1.0, + "content": "’s are in different connected components, it is enough to check its continuity on each", + "type": "text" + }, + { + "bbox": [ + 488, + 397, + 502, + 408 + ], + "score": 0.89, + "content": "X _ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 395, + 507, + 409 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 406, + 508, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 160, + 421 + ], + "score": 1.0, + "content": "Indeed, each", + "type": "text" + }, + { + "bbox": [ + 160, + 408, + 185, + 420 + ], + "score": 0.91, + "content": "f _ { \\alpha | X _ { \\beta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 406, + 508, + 421 + ], + "score": 1.0, + "content": "is constant so continuous. 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As", + "type": "text" + }, + { + "bbox": [ + 327, + 420, + 371, + 432 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { s c a l } \\subseteq S", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 418, + 429, + 434 + ], + "score": 1.0, + "content": ", in particular", + "type": "text" + }, + { + "bbox": [ + 429, + 420, + 503, + 432 + ], + "score": 0.91, + "content": "( x , y ) \\in \\rho \\left( \\mathcal { F } _ { s c a l } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 418, + 506, + 434 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 429, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 187, + 446 + ], + "score": 1.0, + "content": "But, by assumption,", + "type": "text" + }, + { + "bbox": [ + 187, + 431, + 279, + 445 + ], + "score": 0.93, + "content": "\\textstyle \\rho ( { \\mathcal { F } } _ { s c a l } ) \\subseteq \\sqcup _ { \\beta \\in A } X _ { \\beta } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 429, + 352, + 446 + ], + "score": 1.0, + "content": ". 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Take", + "type": "text" + }, + { + "bbox": [ + 421, + 490, + 483, + 502 + ], + "score": 0.93, + "content": "h \\in \\mathcal { C } _ { a d } ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 502, + 504, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 124, + 517 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 503, + 170, + 516 + ], + "score": 0.91, + "content": "h _ { | X _ { \\alpha } } \\in \\overline { { \\mathcal { F } _ { \\alpha } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 502, + 209, + 517 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 209, + 504, + 236, + 514 + ], + "score": 0.9, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 502, + 254, + 517 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 504, + 278, + 514 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 502, + 347, + 517 + ], + "score": 1.0, + "content": ". 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We have shown above that each", + "type": "text" + }, + { + "bbox": [ + 368, + 565, + 379, + 576 + ], + "score": 0.88, + "content": "f _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 563, + 400, + 577 + ], + "score": 1.0, + "content": "is in", + "type": "text" + }, + { + "bbox": [ + 400, + 563, + 408, + 574 + ], + "score": 0.83, + "content": "\\overline { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 563, + 505, + 577 + ], + "score": 1.0, + "content": "so there exists, for each", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 576, + 209, + 587 + ], + "spans": [ + { + "bbox": [ + 107, + 576, + 134, + 586 + ], + "score": 0.78, + "content": "\\alpha \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 576, + 137, + 587 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 137, + 576, + 166, + 586 + ], + "score": 0.76, + "content": "l _ { \\alpha } \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 576, + 209, + 587 + ], + "score": 1.0, + "content": "such that,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 550, + 506, + 587 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 600, + 349, + 624 + ], + "lines": [ + { + "bbox": [ + 262, + 600, + 349, + 624 + ], + "spans": [ + { + "bbox": [ + 262, + 600, + 349, + 624 + ], + "score": 0.92, + "content": "\\operatorname* { s u p } _ { X } | f _ { \\alpha } - 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\\sum _ { \\alpha \\in A } l _ { \\alpha } g _ { \\alpha } \\in \\mathcal { F } \\| \\le \\operatorname* { s u p } _ { X _ { \\beta } } \\| h - g _ { \\beta } \\| + \\| g _ { \\beta } - l _ { \\beta } g _ { \\beta } \\| + \\| \\sum _ { \\alpha \\neq \\beta } l _ { \\alpha } g _ { \\alpha } \\| } } \\\\ & { } & { \\le \\epsilon + M \\times \\frac { \\epsilon } { M } + ( | A | - 1 ) M \\times \\frac { \\epsilon } { M } = ( | A | + 1 ) \\epsilon , } \\end{array}", + "type": "interline_equation", + "image_path": "ee1cc62d1feb8d94283b3b4cd6b530226ffefaba1c1c03bf219ae007483023f8.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 146, + 653, + 465, + 674.3333333333334 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 146, + 674.3333333333334, + 465, + 695.6666666666667 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 146, + 695.6666666666667, + 465, + 717.0000000000001 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 216, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 216, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 216, + 733 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 720, + 216, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "We now state our theorem in the case of varying graph size like Keriven & Peyré (2019). 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Equip this space with the final topology or, equivalently, the graph edit distance. Then, the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "last thing to check to apply Lem. 36 is that the classes of GNN that we consider indeed discriminate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 191, + 326, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 326, + 203 + ], + "score": 1.0, + "content": "between graphs of different sizes, which is immediate.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 504, + 233 + ], + "lines": [ + { + "bbox": [ + 104, + 205, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 288, + 226 + ], + "score": 1.0, + "content": "Likewise, for message passing GNN, consider", + "type": "text" + }, + { + "bbox": [ + 288, + 208, + 409, + 223 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { G } _ { \\leq N } \\times \\mathbb { F } _ { 0 } ^ { \\leq N } = \\bigcup _ { n = 1 } ^ { N } \\mathcal { G } _ { n } \\times \\mathbb { F } _ { 0 } ^ { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 205, + 506, + 226 + ], + "score": 1.0, + "content": ", with a similar topology", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 221, + 212, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 212, + 233 + ], + "score": 1.0, + "content": "or the graph edit distance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 101, + 232, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 101, + 232, + 183, + 259 + ], + "score": 1.0, + "content": "Corollary 37. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 237, + 276, + 252 + ], + "score": 0.9, + "content": "K _ { d i s c r } \\subseteq \\mathcal { G } _ { \\leq N } \\times \\mathbb { F } _ { 0 } ^ { \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 232, + 280, + 259 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 280, + 236, + 327, + 252 + ], + "score": 0.9, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { \\leq N ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 232, + 506, + 259 + ], + "score": 1.0, + "content": "be compact sets. 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This corollary is a direct consequence of Thm. 34 and Lem. 36, using Lem. 32 to satisfy the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 253, + 432 + ], + "score": 1.0, + "content": "sub-algebra assumption of Lem. 36.", + "type": "text" + }, + { + "bbox": [ + 495, + 421, + 505, + 430 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + } + ], + "page_idx": 38, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "39", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 110, + 83, + 366, + 93 + ], + "lines": [ + { + "bbox": [ + 108, + 82, + 368, + 95 + ], + "spans": [ + { + "bbox": [ + 108, + 82, + 368, + 95 + ], + "score": 1.0, + "content": "F.2 APPROXIMATION THEOREM WITH VARYING GRAPH SIZE", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "We now state our theorem in the case of varying graph size like Keriven & Peyré (2019). 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Then, the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "last thing to check to apply Lem. 36 is that the classes of GNN that we consider indeed discriminate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 191, + 326, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 326, + 203 + ], + "score": 1.0, + "content": "between graphs of different sizes, which is immediate.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 100, + 150, + 505, + 203 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 504, + 233 + ], + "lines": [ + { + "bbox": [ + 104, + 205, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 288, + 226 + ], + "score": 1.0, + "content": "Likewise, for message passing GNN, consider", + "type": "text" + }, + { + "bbox": [ + 288, + 208, + 409, + 223 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { G } _ { \\leq N } \\times \\mathbb { F } _ { 0 } ^ { \\leq N } = \\bigcup _ { n = 1 } ^ { N } \\mathcal { G } _ { n } \\times \\mathbb { F } _ { 0 } ^ { n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 205, + 506, + 226 + ], + "score": 1.0, + "content": ", with a similar topology", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 221, + 212, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 212, + 233 + ], + "score": 1.0, + "content": "or the graph edit distance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 205, + 506, + 233 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 101, + 232, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 101, + 232, + 183, + 259 + ], + "score": 1.0, + "content": "Corollary 37. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 237, + 276, + 252 + ], + "score": 0.9, + "content": "K _ { d i s c r } \\subseteq \\mathcal { G } _ { \\leq N } \\times \\mathbb { F } _ { 0 } ^ { \\leq N }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 232, + 280, + 259 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 280, + 236, + 327, + 252 + ], + "score": 0.9, + "content": "K \\subseteq \\mathbb { F } _ { 0 } ^ { \\leq N ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 232, + 506, + 259 + ], + "score": 1.0, + "content": "be compact sets. For the invariant case, we", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 101, + 232, + 506, + 259 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 265, + 497, + 331 + ], + "lines": [ + { + "bbox": [ + 113, + 265, + 497, + 331 + ], + "spans": [ + { + "bbox": [ + 113, + 265, + 497, + 331 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\overline { { M G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K _ { d i s c r } , \\mathbb { F } ) : \\rho \\left( 2 \\overline { { N L _ { I } } } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { 2 - L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( 2 \\overline { { N L _ { I } } } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { k - L G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } L G N N _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} \\supset \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\\\ & { \\overline { { k - F G N N _ { I } } } = \\{ f \\in \\mathcal { C } _ { I } ( K , \\mathbb { F } ) : \\rho \\left( k { - } F W L _ { I } \\right) \\subseteq \\rho \\left( f \\right) \\} } \\end{array}", + "type": "interline_equation", + "image_path": "63912604e84dba728358730e5448f7d98201bcf0ec25b0731d21df360d8d3606.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 113, + 265, + 497, + 287.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 113, + 287.0, + 497, + 309.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 113, + 309.0, + 497, + 331.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 245, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 246, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 246, + 347 + ], + "score": 1.0, + "content": "For the equivariant case, we have:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 334, + 246, + 347 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 349, + 514, + 401 + ], + "lines": [ + { + "bbox": [ + 111, + 349, + 514, + 401 + ], + "spans": [ + { + "bbox": [ + 111, + 349, + 514, + 401 + ], + "score": 0.89, + "content": "\\begin{array} { r c l } { \\overline { { M G N N _ { E } } } } & { = } & { \\left\\{ f \\in \\mathcal { C } _ { E } ( K _ { d i s c r } , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( 2 \\overline { { { \\cdot } } } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} } \\\\ { \\widehat { \\cdot } L G N N _ { E } } & { = } & { \\left\\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( k { - } L G N N _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} \\supset \\left\\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( k { - } W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} } \\\\ { \\cdot { F G N N _ { E } } } & { = } & { \\left\\{ f \\in \\mathcal { C } _ { E } ( K , \\mathbb { F } ^ { \\leq N } ) : \\rho \\left( k { - } F W L _ { E } \\right) \\subseteq \\rho \\left( f \\right) \\right\\} } \\end{array}", + "type": "interline_equation", + "image_path": "1b754682727c68869ab50ecb95c7dca25cdf715332ec9a807997228809fd7995.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 111, + 349, + 514, + 366.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 111, + 366.3333333333333, + 514, + 383.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 111, + 383.66666666666663, + 514, + 400.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 408, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "score": 1.0, + "content": "Proof. 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MethodsAP25AP50AP100AP150Recall@500MPJPE[mm]Time[ms]
VoxelPose 40]84.096.497.597.898.117.8320
MvP (Ours)92.396.697.597.798.215.8170
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MethodsShelfCampus
Actor 1Actor 2Actor3AverageActor 1Actor 2Actor3 Average
Belagiannis et al. [2]75.369.787.677.593.575.784.484.5
Ershadi et al. [9]93.375.994.888.094.292.984.690.6
Dong et al. [6]98.894.197.896.997.693.398.096.3
VoxelPose 40]99.394.197.697.097.693.898.896.7
MvP (Ours)99.395.197.897.498.294.197.496.6
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QueryAP25 AP100 MPJPE
Per-joint67.484.741.2
Hier.82.593.219.5
Hier.+ad.92.397.515.8
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Cam.AP25 AP100MPJPE
14.761.0 93.8
237.793.0 34.8
371.895.1 21.1
484.196.7 19.3
592.397.5 15.8
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Dec.AP25 AP100MPJPE
26.392.5 49.6
363.495.6 22.8
486.896.8 17.5
591.897.6 16.2
692.397.5 15.8
792.097.5 15.9
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KAP25 AP100MPJPE
188.696.3 18.2
289.397.5 17.4
492.397.7 15.8
884.491.1 20.3
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Thr. AP25 AP100 MPJPE
0.093.198.5 16.3
0.192.3 97.515.8
0.291.1 96.215.5
0.489.2 93.715.0
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RConv AP25 AP100 MPJPE
w/92.3 97.515.8
w/o87.5 96.217.4
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Allen School of Computer Science & Engineering +University of Washington +{antoineb,omerlevy,ahai,fox,yejin}@cs.washington.edu +‡School of Science, Technology, Engineering & Mathematics +University of Washington - Bothell +{corin123}@uw.edu + +# ABSTRACT + +Understanding procedural language requires anticipating the causal effects of actions, even when they are not explicitly stated. In this work, we introduce Neural Process Networks to understand procedural text through (neural) simulation of action dynamics. Our model complements existing memory architectures with dynamic entity tracking by explicitly modeling actions as state transformers. The model updates the states of the entities by executing learned action operators. Empirical results demonstrate that our proposed model can reason about the unstated causal effects of actions, allowing it to provide more accurate contextual information for understanding and generating procedural text, all while offering more interpretable internal representations than existing alternatives. + +# 1 INTRODUCTION + +Understanding procedural text such as instructions or stories requires anticipating the implicit causal effects of actions on entities. For example, given instructions such as “add blueberries to the muffin mix, then bake for one half hour,” an intelligent agent must be able to anticipate a number of entailed facts (e.g., the blueberries are now in the oven; their “temperature” will increase). While this common sense reasoning is trivial for humans, most natural language understanding algorithms do not have the capacity to reason about causal effects not mentioned directly in the surface strings (Levy et al., 2015; Jia & Liang, 2017; Lucy & Gauthier, 2017). + +In this paper, we introduce Neural Process Networks, a procedural language understanding system that tracks common sense attributes through neural simulation of action dynamics. Our network models interpretation of natural language instructions as a process of actions and their cumulative effects on entities. More concretely, reading one sentence at a time, our model attentively selects what actions to execute on which entities, and remembers the state changes induced with a recurrent memory structure. In Figure 1, for example, our model indexes the “tomato” embedding, selects the “wash” and “cut” functions and performs a computation that changes the “tomato” embedding so that it can reason about attributes such as its “SHAPE” and “CLEANLINESS”. + +![](images/ea83afe00b344242d8798385881abffd1d2f37254531c7595434b55036942bed.jpg) +Figure 1: The process is a narrative of entity state changes induced by actions. In each sentence, these state changes are induced by simulated actions and must be remembered. + +Our model contributes to a recent line of research that aims to model aspects of world state changes, such as language models and machine readers with explicit entity representations (Henaff et al., 2016; Yang et al., 2016; Ji et al., 2017), as well as other more general purpose memory network variants (Weston et al., 2014; Sukhbaatar et al., 2015; Hill et al., 2015; Seo et al., 2016). This worldcentric modeling of procedural language (i.e., understanding by simulation) abstracts away from the surface strings, complementing text-centric modeling of language, which focuses on syntactic and semantic labeling of surface words (i.e., understanding by labeling). + +Unlike previous approaches, however, our model also learns explicit action representations as functional operators (See Figure 1). While representations of action semantics could be acquired through an embodied agent that can see and interact with the world (Oh et al., 2015), we propose to learn these representations from text. In particular, we require the model to be able to explain the causal effects of actions by predicting natural language attributes about entities such as “LOCATION” and “TEMPERATURE”. The model adjusts its representations of actions based on errors it makes in predicting the resultant state changes to attributes. This textual simulation allows us to model aspects of action causality that are not readily available in existing simulation environments. Indeed, most virtual environments offer limited aspects of the world – with a primary focus on spatial relations (Oh et al., 2015; Chiappa et al., 2017; Wahlstrom et al., 2015). They leave out various other dimensions of the world states that are implied by diverse everyday actions such as “dissolve” (change of “COMPOSITION”) and “wash” (change of “CLEANLINESS”). + +Empirical results demonstrate that parametrizing explicit action embeddings provides an inductive bias that allows the neural process network to learn more informative context representations for understanding and generating natural language procedural text. In addition, our model offers more interpretable internal representations and can reason about the unstated causal effects of actions explained through natural language descriptors. Finally, we include a new dataset with fine-grained annotations on state changes, to be shared publicly, to encourage future research in this direction. + +# 2 NEURAL PROCESS NETWORK + +The neural process network is an interpreter that reads in natural language sentences, one at a time, and simulates the process of actions being applied to relevant entities through learned representations of actions and entities. + +# 2.1 OVERVIEW AND NOTATION + +The main component of the neural process network is the simulation module (§2.5), a recurrent unit whose internals simulate the effects of actions being applied to entities. A set of $V$ actions is known a priori and an embedding is initialized for each one, $\mathcal { F } = \{ f _ { 1 } , . . . f _ { V } \}$ . Similarly, a set of $I$ entities is known and an embedding is initialized for each one: $\mathcal { E } = \{ e _ { 1 } , . . . e _ { I } \}$ . Each $e _ { i }$ can be considered to encode information about state attributes of that entity, which can be extracted by a set of state predictors (§2.6). As the model reads text, it “applies” action embeddings to the entity vectors, thereby changing the state information encoded about the entities. + +For any document $d$ , an initial list of entities $I _ { d }$ is known and $\mathcal { E } _ { d } = \{ e _ { i } | i \in I _ { d } \} \subset \mathcal { E }$ entity state embeddings are initialized. As the neural process network reads a sentence from the document, it selects a subset of both $\mathcal { F } \left( \ S 2 . 3 \right)$ and $\mathcal { E } _ { d }$ $( \ S 2 . 4 )$ based on the actions performed and entities affected in the sentence. The entity state embeddings are changed by the action and the new embeddings are used to predict end states for a set of state changes (§2.6). The prediction error for end states is backpropagated to the action embeddings, learning action representations that model the simulation of desired causal effects on entities. This process is broken down into five modules below. Unless explicitly defined, all $W$ and $b$ variables are parametrized linear projections and biases. We use the notation $\{ e _ { i } \} _ { t }$ when referring to the values of the entity embeddings before processing sentence $s _ { t }$ . + +# 2.2 SENTENCE ENCODER + +Given a sentence $s _ { t }$ , a Gated Recurrent Unit (Cho et al., 2014) encodes each word and outputs its last hidden vector as a sentence encoding $h _ { t }$ (Sutskever et al., 2014). + +# 2.3 ACTION SELECTOR + +Given $h _ { t }$ from the sentence encoder, the action selector (bottom left in Fig. 2) contextually determines which action(s) from $\mathcal { F }$ to execute. For example, if the input sentence is “wash and cut + +![](images/11046f3337ccdae5eb2a85134800ce6ee2fb6e8d92d2d1d3ef9346c712c08f29.jpg) +Figure 2: Model Summary. The sentence encoder converts a sentence to a vector representation, $h _ { t }$ . The action selector and entity selector use the vector representation to choose the actions that are applied and the entities that are acted upon in the sentence. The simulation module indexes the action and entity state embeddings, and applies the transformation to the entities. The state predictors predict the new state of the entities if a state change has occurred. Equation references are provided in parentheses. + +beets”, both $f _ { w a s h }$ and $f _ { c u t }$ must be selected. To account for multiple actions, we make a soft selection over $\mathcal { F }$ , yielding a weighted sum of the selected action embeddings $\bar { f } _ { t }$ : + +$$ +\begin{array} { l } { { \displaystyle w _ { p } = \mathbf { M L P } \big ( h _ { t } \big ) } } \\ { { \displaystyle \bar { w } _ { p } = \frac { w _ { p } } { \sum _ { j = 1 } ^ { V } w _ { p _ { j } } } } } \\ { { \displaystyle \bar { f } _ { t } = \bar { w } _ { p } ^ { \top } \mathcal { F } } } \end{array} +$$ + +where MLP is a parametrized feed-forward network with a sigmoid activation and $w _ { p } \in \mathbb { R } ^ { V }$ is the attention distribution over $V$ possible actions (§3.1). We compose the action embedding by taking the weighted average of the selected actions. + +# 2.4 ENTITY SELECTOR + +Sentence Attention Given $h _ { t }$ from the sentence encoder, the entity selector chooses relevant entities using a soft attention mechanism: + +$$ +\begin{array} { r l } & { \tilde { h } _ { t } = \mathrm { R e L U } ( W _ { 1 } h _ { t } + b _ { 1 } ) } \\ & { d _ { i } = \sigma ( e _ { i _ { 0 } } ^ { \top } W _ { 2 } [ \tilde { h } _ { t } ; w _ { p } ] ) } \end{array} +$$ + +where $W _ { 2 }$ is a bilinear mapping, $e _ { i _ { 0 } }$ is a unique key for each entity (§2.5), and $d _ { i }$ is the attention weight for entity embedding $e _ { i }$ . For example, in “wash and cut beets and carrots”, the model should select $e _ { b e e t }$ and ecarrot. + +Recurrent Attention While sentence attention would suffice if entities were always explicitly mentioned, natural language often elides arguments or uses referent pronouns. As such, the module must be able to consider entities mentioned in previous sentences. Using $\tilde { h } _ { t }$ , the model computes a soft choice over whether to choose affected entities from this step’s attention $d _ { i }$ or the previous step’s attention distribution. + +$$ +\begin{array} { r } { c = s o f t m a x ( W _ { 3 } \tilde { h } _ { t } + b _ { 3 } ) } \\ { a _ { i _ { t } } = c _ { 1 } d _ { i } + c _ { 2 } a _ { i _ { t - 1 } } + c _ { 3 } \mathbf { 0 } } \end{array} +$$ + +where $c \in \mathbb { R } ^ { 3 }$ is the choice distribution, $a _ { i _ { t - 1 } }$ is the previous sentence’s attention weight for each entity, $a _ { i _ { t } }$ is the final attention for each entity, and 0 is a vector of zeroes (providing the option to not change any entity). Prior entity attentions can propagate forward for multiple steps. + +# 2.5 SIMULATION MODULE + +Entity Memory A unique state embedding $e _ { i }$ is initialized for every entity $i$ in the document. A unique key to index each embedding $e _ { i _ { 0 } }$ is set as the initial value of the embedding (Henaff et al., 2016; Miller et al., 2016). After the model reads $s _ { t }$ , it modifies $\{ e _ { i } \} _ { t }$ to reflect changes influenced by actions. At every time step, the entity memory receives the attention weights from the entity selector, normalizes them and computes a weighted average of the relevant entity state embeddings: + +$$ +\alpha _ { i } = \frac { a _ { i } } { \sum _ { j = 1 } ^ { I _ { d } } a _ { j } } +$$ + +$$ +\bar { e } _ { t } = \sum _ { j = 1 } ^ { I _ { d } } \alpha _ { i } e _ { i } +$$ + +Applicator Given the action summary embedding $\bar { f } _ { t }$ and the entity summary embedding $\bar { e _ { t } }$ , the applicator (middle right in Fig. 2) applies the selected actions to the selected entities, and outputs the new proposal entity embedding $k _ { t }$ . + +$$ +k _ { t } = \mathrm { R e L U } ( \bar { f } _ { t } W _ { 4 } \bar { e } _ { t } + b _ { 4 } ) +$$ + +where $W _ { 4 }$ is a third order tensor projection. The vector $k _ { t }$ is the new representation of the entity $\bar { e } _ { t }$ after the applicator simulates the action being applied to it. + +Entity Updater The entity updater interpolates the new proposal entity embedding $k _ { t }$ and the set of current entity embeddings $\{ e _ { i } \} _ { t }$ : + +$$ +e _ { i _ { t + 1 } } = a _ { i _ { t } } k _ { t } + ( 1 - a _ { i _ { t } } ) e _ { i _ { t } } +$$ + +yielding an updated set of entity embeddings $\{ e _ { i } \} _ { t + 1 }$ . Each embedding is updated proportional to its entity’s unnormalized attention $a _ { i }$ , allowing the model to completely overwrite the state embedding for any entity. For example, in the sentence “mix the flour and water,” the embeddings for $e _ { f l o u r }$ and $e _ { w a t e r }$ must both be overwritten by $k _ { t }$ because they no longer exist outside of this new composition. + +# 2.6 STATE PREDICTORS + +Given the new proposal entity embedding $k _ { t }$ , the state predictor (bottom right in Fig. 2) predicts changes to the resulting entity embedding $k _ { t }$ along the following six dimensions: location, cookedness, temperature, composition, shape, and cleanliness. Discrete multi-class classifiers, one for each dimension, take in $k _ { t }$ and predict a unique end state for their corresponding state change type: + +$$ +P ( Y _ { s } | k _ { t } ) = s o f t m a x ( W _ { s } k _ { t } + b _ { s } ) +$$ + +For location changes, which require contextual information to predict the end state, $k _ { t }$ is concatenated with the original sentence representation $h _ { t }$ to predict the final state. + +# 3 TRAINING + +# 3.1 STATE CHANGE KNOWLEDGE + +In this work we focus on physical action verbs in cooking recipes. We manually collect a set of 384 actions such as cut, bake, boil, arrange, and place, organizing their causal effects along the following predefined dimensions: LOCATION, COOKEDNESS, TEMPERATURE, SHAPE, CLEANLINESS and COMPOSITION. The textual simulation operated by the model induces state changes along these dimensions by applying actions functions from the above set of 384. For example, cut entails a change in SHAPE, while bake entails a change in TEMPERATURE, COOKEDNESS, and even LOCATION. We annotate the state changes each action induces, as well as the end state of the action, using Amazon Mechanical Turk. The set of possible end states for a state change can range from 2 for binary state changes to more than 200 (See Appendix C for details). Table 1 provides examples of annotations in this action lexicon. + +Table 1: Example actions, the state changes they induce, and the possible end states + +
ActionState Change TypesEnd States
braiseCOOKEDNESS;TEMPERATURECOOKED;HOTCOLDMOLDEDCLEANCOMPOSEDCOLD;REFRIGERATORSEPARATED
chillTEMPERATURE
kneadSHAPECLEANLINESS
wash
dissolveCOMPOSITION
refrigerateTEMPERATURE;LOCATION
sliceSHAPE
+ +# 3.2 DATASET + +For learning and evaluation, we use a subset of the Now You’re Cooking dataset (Kiddon et al., 2016). We chose 65816 recipes for training, 175 recipes for development, and 700 recipes for testing. For the development and test sets, crowdsourced workers densely annotate actions, entities and state changes that occur in each sentence so that we can tune hyperparameters and evaluate on gold evaluation sets. Annotation details are provided in Appendix C.3. + +# 3.3 COMPONENT-WISE TRAINING + +The neural process network is trained by jointly optimizing multiple losses for the action selector, entity selector, and state change predictors. Importantly, our training scheme uses weak supervision because dense annotations are prohibitively expensive to acquire at a very large scale. Thus, we heuristically extract verb mentions from each recipe step and assign a state change label based on the state changes induced by that action (§3.1). Entities are extracted similarly based on string matching between the instructions and the ingredient list. We use the following losses for training: + +Action Selection Loss Using noisy supervision, the action selector is trained to minimize the cross-entropy loss for each possible action, allowing multiple actions to be chosen at each step if multiple actions are mentioned in a sentence. The MLP in the action selector (Eq. 1) is pretrained. + +Entity Selection Loss Similarly, to train the attentive entity selector, we minimize the binary cross-entropy loss of predicting whether each entity is affected in the sentence. + +State Change Loss For each state change predictor, we minimize the negative loglikelihood of predicting the correct end state for each state change. + +Coverage Loss An underlying assumption in many narratives is that all entities that are mentioned should be important to the narrative. We add a loss term that penalizes narratives whose combined attention weights for each entity does not sum to more than 1. + +$$ +\mathcal { L } _ { c o v e r } = - \frac { 1 } { I _ { d } } \sum _ { i = 1 } ^ { I _ { d } } \log \sum _ { t = 1 } ^ { S } a _ { i _ { t } } +$$ + +where $a _ { i _ { t } }$ is the attention weight for a particular entity at sentence $t$ and $I _ { d }$ is the number of entities in a document. $\textstyle \sum _ { t = 1 } ^ { S } a _ { i _ { t } }$ is upper bounded by 1. This is similar to the coverage penalty used in neural machine translation (Tu et al., 2016). + +# 4 EXPERIMENTAL SETUP + +We evaluate our model on a set of intrinsic tasks centered around tracking entities and state changes in recipes to show that the model can simulate preliminary dynamics of the recipe task. Additionally, we provide a qualitative analysis of the internal components of our model. Finally, we evaluate the quality of the states encoded by our model on the extrinsic task of generating future steps in a recipe. + +# 4.1 INTRINSIC EVALUATION - TRACKING + +In the tracking task, we evaluate the model’s ability to identify which entities are selected and what changes have been made to them in every step. We break the tracking task into two separate evaluations, entity selection and end state prediction, and also investigate whether the model learns internal representations that approximate recipe dynamics. + +Table 2: Results for entity selection and state change selection + +
ModelEntity SelectionState Change
F1URCRF1Acc
2-layer LSTM Entity Recognizer50.9874.0313.3311
Adapted Gated Recurrent Unit Adapted Recurrent Entity Network - Recurrent Attention (Eq. 3)45.9467.697.7441.1652.69
48.5771.889.8742.3253.47
48.9172.3212.6742.1450.48
- Coverage Loss (Eq. 9) - Action Connections (Eq. 2)55.1873.9820.2344.4455.20
54.8573.5420.0344.0554.81
54.9173.8720.3044.2855.00
- Action Selector Pretraining + Pretrained Action Embeddings - Action Embedding Updates55.1674.0220.3244.0255.03
53.7970.7718.6044.2755.02
FullModel55.3974.8820.4544.6555.07
+ +Metrics In the entity selection test, we report the F1 score of choosing the correct entities in any step. A selected entity is defined as one whose attention weight $a _ { i }$ is greater than $50 \%$ $( \ S 2 . 4 )$ . Because entities may be harder to predict when they have been combined with other entities (e.g., the mixture may have a new name), we also report the recall for selecting combined (CR) and uncombined (UR) entities. In the end state prediction test, we report how often the model correctly predicts the state change performed in a recipe step and the resultant end state. This score is then scaled by the accuracy of predicting which entities were changed in that same step. We report the average F1 and accuracy across the six state change types. + +Baselines We compare our models against two baselines. First, we built a GRU model that is trained to predict entities and state changes independently. This can be viewed as a bare minimum network with no action representations or recurrent entity memory. The second baseline is a Recurrent Entity Network (Henaff et al., 2016) with changes to fit our task. First, the model can tie memory cells to a subset of the full list of entities so that it only considers entities that are present in a particular recipe. Second, the entity distribution for writing to the memory cells is re-used when we query the memory cells. The normalized weighted average of the entity cells is used as the input to the state predictors. The unnormalized attention when writing to each cell is used to predict selected entities. Both baselines are trained with entity selection and state change losses (§3.3). + +Ablations We report results on six ablations. First, we remove the recurrent attention (Eq. 3). The model only predicts entities using the current encoder hidden state. In the second ablation, the model is trained with no coverage penalty (Eq. 9). The third ablation prunes the connection from the action selector $w _ { p }$ to the entity selector (Eq. 2). We also explore not pretraining the action selector. Finally, we look at two ablations where we intialize the action embeddings with vectors from a skipgram model. In the first, the model operates normally, and in the second, we do not allow gradients to backpropagate to the action embeddings, updating only the mapping tensor $W _ { 4 }$ instead (Eq. 6). + +# 4.2 EXTRINSIC EVALUATION - GENERATION + +The generation task tests whether our system can produce the next step in a recipe based on the previous steps that have been performed. The model is provided all of the previous steps as context. + +Metrics We report the combined BLEU score and ROUGE score of the generated sequence relative to the reference sequence. Each candidate sequence has one reference sentence. Both metrics are computed at the corpus-level. Also reported are “VF1”, the F1 score for the overlap of the actions performed in the reference sequence and the verbs mentioned in the generated sequence, and “SF1”, the F1 score for the overlap of end states annotated in the reference sequence and predicted by the generated sequences. End states for the generated sequences are extracted using the lexicon from Section 3.1 based on the actions performed in the sentence. + +Setup To apply our model to the task of recipe step generation, we input the context sentences through the neural process network and record the entity state vectors once the entire context has been read (§2.5). These vectors can be viewed as a snapshot of the current state of the entities once the preceding context has been simulated inside the neural process network. We encode these vectors using a bidirectional GRU (Cho et al., 2014) and take the final time step hidden state $\mathbb { e } _ { I }$ . A different GRU encodes the context words in the same way (yielding $\mathbb { h } _ { T }$ ) and the first hidden state input to the decoder is computed using the projection function: + +Table 3: Examples of the model selecting entities for sentence $s _ { t }$ . The previous sentences are provided as context in cases where they are relevant. + +
GoodSt-1StselectedcorrectAdd tomato paste,broth, garlic,chili powder,cumin,chile peppers,and water.Bring to boil, then turn very low,cover and simmer until meat is tender.meat,garlic, chili powder, tomato paste,cumin,chiles,beef broth,waterSame + [oil]
GoodSt-4St-3St-2St-1StselectedcorrectStir in oats, sugar, flour, corn syrup,milk, vanilla extract, and salt.Mix well.Drop by measuring teaspoonfuls onto cookie sheets.Bake 5-7 minutes.Let cool.oats,sugar, flour, corn syrup,milk,vanilla extract, saltoats,sugar, flour, corn syrup,milk,vanilla extract, salt
GoodSt-1StselectedcorrectIn a large saucepan over low heat, melt marshmallows.Add sprinkles,cereal,and raisins,stir until well coated.marshmallows,cereal,raisinsmarshmallows,cereal, raisins,sprinkles
BadSt-3St-2St-1StselectedcorrectLadle the barbecue sauce around the crust and spread.Add mozzarella, yellow cheddar,and monterey jack cheese.Next,add onion mixture and sliced chicken breast .Top pizza with jalapeno peppers.jalapenoscrust,sauce,mozzarella,cheddar,monterey jack,white onion,chicken, jalapenos
BadSt-2St-1StselectedcorrectCombine 1 cup flour, salt,and 1 tbsp sugar.Cut in butter until mixture is crumbly, then sprinkle with vinegar .Gather dough into a ball and press into bottom of 9 inch springform pan.butter, vinegarflour, salt,sugar, butter, vinegar
+ +$$ +\tilde { h } _ { 0 } = W _ { 5 } ( \Subset _ { I } \circ \Join _ { T } ) +$$ + +where $\circ$ is the Hadamard product between the two encoder outputs. All models are trained by minimizing the negative loglikelihood of predicting the next word for the full sequence. Implementation details can be found in Appendix A. + +Baselines For the generation task, we use three baselines: a seq2seq model with no attention (Sutskever et al., 2014), an attentive seq2seq model (Bahdanau et al., 2014), and a similar variant as our NPN generator, except where the entity states have been computed by the Recurrent Entity Network (EntNet) baseline $( \ S 4 . 1 )$ . Implementation details for baselines can be found in Appendix B. + +# 5 EXPERIMENTAL RESULTS + +# 5.1 INTRINSIC EVALUATIONS + +Entity Selection As shown in Table 8, our full model outperforms all baselines at selecting entities, with an F1 score of $5 5 . 3 9 \%$ . The ablation study shows that the recurrent attention, coverage loss, action connections and action selector pretraining improve performance. Our success at predicting entities extends to both uncomposed entities, which are still in their raw forms (e.g., melt the butter butter), and composed entities, in which all of the entities that make up a composition must be selected. For example, in a Cooking lasagna recipe, if the final step involves baking the prepared lasagna, the model must select all the entities that make up the lasagna (e.g., lasagna sheets, beef, tomato sauce). In Table 3, we provide examples of our model’s ability to handle complex cases such as compositional entities (Ex. 1, 3), and elided arguments over long time windows (Ex. 2). We also provide examples where the model fails to select the correct entities because it does not identify the mapping between a reference construct such as “pizza” (Ex. 4) or “dough” (Ex. 5) and the set of entities that composes it, showcasing the difficulty of selecting the full set for a composed entity. + +Table 4: Most similar actions based on cosine similarity of action embeddings + +
ActionNearestNeighbor Actions
cut boilslice,split, snap,slash,carve,slit,chop cook,microwave,fry,steam,simmer
addsprinkle,mix,reduce,splash,stir,dust
washrinse,scrub,refresh,soak,wipe,scale
mashspread, puree,squeeze,liquefy, blend
place rinseease,put,lace,arrange,leave
wash,refresh,soak,wipe,scrub,clean
warmreheat, ignite,heat,light,crisp,preheat
steam
microwave,crisp,boil, parboil,heat
sprinkletop,pat,add,dip,salt, season
greasecoat,rub,dribble,spray,smear, line
+ +![](images/7d3dd2e0d40faf65ae5f5c239788d29171cd4654468f9e42a1ba283ad355c5c2.jpg) +Figure 3: Change in cosine similarity of entity state embeddings + +Table 5: Generation Results + +
ModelBLEUBLEUROUGE-LRVF1
Vanilla Seq2SeqAttentive Seq2SeqEntNet Generator2.812.832.3033.0033.1832.7116.1716.9717.5340.2141.4342.43
NPN Generator3.7435.6420.1243.40
+ +State Change Tracking In Table 8, we show that our full model outperforms competitive baselines such as Recurrent Entity Networks (Henaff et al., 2016) and jointly trained GRUs. While the ablation without the coverage loss shows higher accuracy, we attribute this to the fact that it predicts a smaller number of total state changes. Interestingly, initializing action embeddings with skipgram vectors and locking their values shows relatively high performance, indicating the potential gains in using powerful pretrained representations to represent actions. + +Action Embeddings In our model, each action is assigned its own embedding, but many actions induce similar changes in the physical world (e.g.,“cut” and “slice”). After training, we compute the pairwise cosine similarity between each pair of action embeddings. In Table 4, we see that actions that perform similar functions are neighbors in embedding space, indicating the model has captured certain semantic properties of these actions. Learning action representations through the state changes they induce has allowed the model to cluster actions by their transformation functions. + +Entity Compositions When individual entities are combined into new constructs, our model averages their state embeddings (Eq. 5), applies an action embedding to them (Eq. 6), and writes them to memory (Eq. 7). The state embeddings of entities that are combined should be overwritten by the same new embedding. In Figure 3, we present the percentage increase in cosine similarity for state embeddings of entities that are combined in a sentence (blue) as opposed to the percentage increase for those that are not (red bars). While the soft attention mechanism for entity selection allows similarities to leak between entity embeddings, our system is generally able to model the compositionality patterns that result from entities being combined into new constructs. + +# 5.2 EXTRINSIC EVALUATIONS + +Recipe Step Generation Our results in Table 5 indicate that sequences generated using the neural process network entity states as additional input yield higher scores than competitive baselines. The entity states allow the model to predict next steps conditioned on a representation of the world being simulated by the neural process network. Additionally, the higher VF1 and SF1 scores indicate that the model is indeed using the extra information to better predict the actions that should follow the context provided. Example generations for each baselines from the dev set are provided in Table 6, showing that the NPN generator can use information about ingredient states to reason about the most likely next step. The first and second examples are interesting as it shows that the NPN-aware model has learned to condition on entity state – knowing that raw butter will likely be melted or that a cooked flan must be refrigerated. The third example is also interesting because the model learns that cooked vegetables such as squash will sometimes be drained, even if it is not relevant to this recipe because the squash is steamed. The seq2seq and EntNet baselines, meanwhile, output reasonable sentences given the immediate context, but do not exhibit understanding of global patterns. + +Table 6: Examples of the model generating sentences compared to baselines. The context and reference are provided first, followed by our model’s generation and then the baseline generations + +
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPreheat oven to 425 degrees.Melt butter in saucepan and mix in bourbon, thyme,pepper, and salt.Melt butter in skillet.Lightly grease 4 x 8 baking pan with sunflower oil.Combine all ingredients and mix well.In a large bowl, combine flour,baking powder,baking soda, salt,and pepper.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPour egg mixture over caramelized sugar in cake pan. Place cake pan in large shallowbaking dish.Bake for 55 minutes or until knife inserted into flan comes out clean.Cover and chill at least 8 hours.Refrigerate until ready to use.Serve at room temperature.Store in an airtight container.Store in an airtight container.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetCut squash into large pieces and steam. Remove cooked squash from shells;Measure 4 cups pulp and reserve remainder for another dish.Drain.Mash pulp with a fork.Set aside.Set aside.
+ +# 6 RELATED WORK + +Recent studies in machine comprehension have used a neural memory component to store a running representation of processed text (Weston et al., 2014; Sukhbaatar et al., 2015; Hill et al., 2015; Seo et al., 2016). While these approaches map text to memory vectors using standard neural encoder approaches, our model, in contrast, directly interprets text in terms of the effects actions induce in entities, providing an inductive bias for learning how to represent stored memories. More recent work in machine comprehension also sought to couple the memory representation with tracking entity states (Henaff et al., 2016). Our work seeks to provide a relatively more structured representation of domain-specific action knowledge to provide an inductive bias to the reasoning process. + +Neural Programmers (Neelakantan et al., 2015; 2016) have also used functions to simulate reasoning, by building a model to select rows in a database and applying operation on those selected rows. While their work explicitly defined the effect of a number of operations for those rows, we provide a framework for learning representations for a more expansive set of actions, allowing the model to learn representations for how actions change the state space. + +Works on instructional language studied the task of building discrete graph representations of recipes using probabilistic models (Kiddon et al., 2015; Mori et al., 2014; 2012). We propose a complementary new model by integrating action and entity relations into the neural network architecture and also address the additional challenge of tracking the state changes of the entities. + +Additional work in tracking states with visual or multimodal context has focused on 1) building graph representations for how entities change in goal-oriented domains (Gao et al., 2016; Liu et al., 2016; Si et al., 2011) or 2) tracking visual state changes based on decisions taken by agents in environment simulators such as videos or games (Chiappa et al., 2017; Wahlstrom et al., 2015; Oh et al., 2015). Our work, in contrast, models state changes in embedding space using only text-based signals to map real-world actions to algebraic transformations. + +# 7 CONCLUSION + +We introduced the Neural Process Network for modeling a process of actions and their causal effects on entities by learning action transformations that change entity state representations. The model maintains a recurrent memory structure to track entity states and is trained to predict the state changes that entities undergo. Empirical results demonstrate that our model can learn the causal effects of action semantics in the cooking domain and track the dynamic state changes of entities, showing advantages over competitive baselines. + +# ACKNOWLEDGMENTS + +We thank Yonatan Bisk, Peter Clark, Bhavana Dalvi, Niket Tandon, and Yuyin Sun for helpful discussions at various stages of this work. This research was supported in part by NSF (IIS-1524371, IIS-1714566, NRI-1525251), DARPA under the CwC program through the ARO (W911NF-15-1- 0543), Samsung Research, and gifts by Google and Facebook. + +# REFERENCES + +Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In ICLR 2015, 2014. + +Silvia Chiappa, Sebastien Racani ´ ere, Daan Wierstra, and Shakir Mohamed. Recurrent environment \` simulators. CoRR, abs/1704.02254, 2017. + +Kyunghyun Cho, Bart van Merrienboer, Caglar Gulcehre, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. In Conference on Empirical Methods in Natural Language Processing (EMNLP 2014), 2014. + +Qiaozi Gao, Malcolm Doering, Shaohua Yang, and Joyce Y Chai. Physical causality of action verbs in grounded language understanding. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (ACL), volume 1, pp. 1814–1824, 2016. + +Mikael Henaff, Jason Weston, Arthur Szlam, Antoine Bordes, and Yann LeCun. Tracking the world state with recurrent entity networks. arXiv preprint arXiv:1612.03969, 2016. + +Felix Hill, Antoine Bordes, Sumit Chopra, and Jason Weston. The goldilocks principle: Reading children’s books with explicit memory representations. arXiv preprint arXiv:1511.02301, 2015. + +Yangfeng Ji, Chenhao Tan, Sebastian Martschat, Yejin Choi, and Noah A Smith. Dynamic entity representations in neural language models. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 1831–1840, 2017. + +Robin Jia and Percy Liang. Adversarial examples for evaluating reading comprehension systems. In Empirical Methods in Natural Language Processing (EMNLP), 2017. + +Chloe Kiddon, Ganesa Thandavam Ponnuraj, Luke Zettlemoyer, and Yejin Choi. Mise en place: ´ Unsupervised interpretation of instructional recipes. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pp. 982–992, 2015. + +Chloe Kiddon, Luke Zettlemoyer, and Yejin Choi. Globally coherent text generation with neu-´ ral checklist models. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing (EMNLP), 2016. + +Jin-Hwa Kim, Kyoung Woon On, Jeonghee Kim, JungWoo Ha, and Byoung-Tak Zhang. Hadamard product for low-rank bilinear pooling. CoRR, abs/1610.04325, 2016. + +Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. + +Omer Levy, Steffen Remus, Christian Biemann, and Ido Dagan. Do supervised distributional methods really learn lexical inference relations? In HLT-NAACL, 2015. + +Changsong Liu, Shaohua Yang, Sari Saba-Sadiya, Nishant Shukla, Yunzhong He, Song-Chun Zhu, and Joyce Y Chai. Jointly learning grounded task structures from language instruction and visual demonstration. In Conference on Empirical Methods in Natural Language Processing (EMNLP), 2016. + +Li Lucy and Jon Gauthier. Are distributional representations ready for the real world? evaluating word vectors for grounded perceptual meaning. CoRR, abs/1705.11168, 2017. + +Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013a. + +Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 26, pp. 3111–3119, 2013b. + +Alexander Miller, Adam Fisch, Jesse Dodge, Amir-Hossein Karimi, Antoine Bordes, and Jason Weston. Key-value memory networks for directly reading documents. arXiv preprint arXiv:1606.03126, 2016. +Shinsuke Mori, Tetsuro Sasada, Yoko Yamakata, and Koichiro Yoshino. A machine learning approach to recipe text processing. In Proceedings of the 1st Cooking with Computer Workshop, pp. 29–34, 2012. +Shinsuke Mori, Hirokuni Maeta, Yoko Yamakata, and Tetsuro Sasada. Flow graph corpus from recipe texts. In LREC, pp. 2370–2377, 2014. +Arvind Neelakantan, Quoc V. Le, and Ilya Sutskever. Neural programmer: Inducing latent programs with gradient descent. CoRR, abs/1511.04834, 2015. +Arvind Neelakantan, Quoc V. Le, Mart´ın Abadi, Andrew McCallum, and Dario Amodei. Learning a natural language interface with neural programmer. CoRR, abs/1611.08945, 2016. +Junhyuk Oh, Xiaoxiao Guo, Honglak Lee, Richard L. Lewis, and Satinder P. Singh. Actionconditional video prediction using deep networks in atari games. In NIPS, 2015. +Minjoon Seo, Sewon Min, Ali Farhadi, and Hannaneh Hajishirzi. Query-reduction networks for question answering. arXiv preprint arXiv:1606.04582, 2016. +Zhangzhang Si, Mingtao Pei, Benjamin Yao, and Song-Chun Zhu. Unsupervised learning of event and-or grammar and semantics from video. In Computer Vision (ICCV), 2011 IEEE International Conference on, pp. 41–48. IEEE, 2011. +Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929–1958, 2014. +Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015. +Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pp. 3104–3112, 2014. +Zhaopeng Tu, Zhengdong Lu, Yang Liu, Xiaohua Liu, and Hang Li. Modeling coverage for neural machine translation. In ACL, 2016. +Niklas Wahlstrom, Thomas B. Schon, and Marc Peter Deisenroth. From pixels to torques: Policy ¨ learning with deep dynamical models. CoRR, abs/1502.02251, 2015. +Jason Weston, Sumit Chopra, and Antoine Bordes. Memory networks. arXiv preprint arXiv:1410.3916, 2014. +Zichao Yang, Phil Blunsom, Chris Dyer, and Wang Ling. Reference-aware language models. CoRR, abs/1611.01628, 2016. URL http://arxiv.org/abs/1611.01628. + +# A TRAINING DETAILS OF OUR FULL MODEL AND ABLATIONS + +# A.1 TRACKING MODELS + +The hidden size of the instruction encoder is 100, the embedding sizes of action functions and entities are 30. We use dropout with a rate of 0.3 before any non-recurrent fully connected layers Srivastava et al. (2014). We use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over three epochs. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use two instruction encoders, one for the entity selector, and one for the action selector. Word embeddings and entity embeddings are initialized with skipgram embeddings (Mikolov et al., 2013a;b) using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words, and 2996 for entities. Gradients with respect to the coverage loss (Eq. 9) are only backpropagated in steps where no entity is annotated as being selected. To account for the false negatives in the training data due to the heuristic generation of the labels, gradients with respect to the entity selection loss are zeroed when no entity label is present. + +# A.2 GENERATION MODEL + +The hidden size of the context encoder is 200. The hidden size of the state vector encoder is 200. State vectors have dimensionality 30 (the same as in the neural process network). Dropout of 0.3 is used during training in the decoder. The context and state representations are projected jointly using an element-wise product followed by a linear projection Kim et al. (2016). Both encoders and the decoder are single layer. The learning rate is 0.0003 initially and is halved every 5 epochs. The model is trained with the Adam optimizer. + +# B TRAINING DETAILS OF BASELINES + +# B.1 TRACKING BASELINES + +Joint Gated Recurrent Unit The hidden state of the GRU is 100. We use a dropout with a rate of 0.3 before any non-recurrent fully connected layers. We use the Adam optimizer with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over a single epoch. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use encoders, one for the entity selector, and one for the state change predictors. Word embeddings are initialized with skipgram embeddings using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words. + +Recurrent Entity Networks Memory cells are tied to the entities in the document. For a recipe with 12 ingredients, 12 entity cells are initialized. All hyperparameters are the same as the in the bAbI task from Henaff et al. (2016). The learning rate start at 0.01 and is halved every 25 epochs. Entity cells and word embeddings are 100 dimensional. The encoder is a multiplicative mask initialized the same as in Henaff et al. (2016). Intermediate supervision from the weak labels is provided to help predict entities. A separate encoder is used for computing the attention over memory cells and the content to write to the memory. Dropout of 0.3 is used in the encoders. The batch size is 64. We use a vocabulary size of 7358 for words, and 2996 for entities. + +# B.2 GENERATION BASELINES + +Seq2seq The encoder and decoder are both single-layer GRUs with hidden size 200. We use dropout with probability 0.3 in the decoder. We train with the Adam optimizer starting with a learning rate 0.0003 that is halved every 5 epochs. The encoder is bidirectional. The model is trained to minimize the negative loglikelihood of predicting the next word. + +Attentive Seq2seq The encoder is the same as in the seq2seq baseline. A multiplicative attention between the decoder hidden state and the context vectors is used to compute the attention over the context at every decoder time step. The model is trained with the same learning rate, learning schedule and loss function as the seq2seq baseline. + +EntNet Generator The model is trained in the same way as the NPN generator model in Appendix A.2 except that the state representations used as input are produced from by EntNet baseline described in Section 4.1 and Appendix B.1. + +# C ANNOTATIONS + +# C.1 ANNOTATING STATE CHANGES + +We provide workers with a verb, its definition, an illustrative image of the action, and a set of sentences where the verb is mentioned. Workers are provided a checklist of the six state change types and instructed to identify which of them the verb causes. They are free to identify multiple changes. Seven workers annotate each verb and we assign a state change based on majority vote. Of the set of 384 verbs extracted, only 342 have a state change type identified with them. Of those, 74 entail multiple state change types. + +# C.2 ANNOTATING END STATES + +We give workers a verb, a state change type, and an example with the verb and ask them to provide an end state for the ingredient the verb is applied to in the example. We then use the answers to manually aggregate a set of end states for each state change type. These end states are used as labels when the model predicting state changes. For example, a LOCATION change might lead to an end state of “pan,” “pot”, or “oven.” End states for each state change type are provided in Table 7. + +Table 7: End states for each state change type + +
State Change TypeEnd States
Temperature Composition Cleanliness Cookedness Shape Locationhot; cold; room composed; not composed clean; dirty; dry cooked; raw molded; hit; deformed; separated pan,pot, cupboard, screen, scale,
+ +# C.3 ANNOTATING DEVELOPMENT AND TEST SETS + +Annotators are instructed to note any entities that undergo one of the six state changes in each step, as well as to identify new combinations of ingredients that are created. For example, the sentence “Cut the tomatoes and add to the onions” would involve a SHAPE change for the tomatoes and a combination created from the “tomatoes” and “onions”. In a separate task, three workers are asked to identify the actions performed in every sentence of the development and test set recipes. If an action receives a majority vote that it is performed, it is included in the annotations. + +D ADDITIONAL RESULTS D.1 REMOVING TRAINING DATA + +Table 8: Results for entity selection and state change selection on the development set when randomly dropping a percentage of the training labels + +
ModelEntity Selection State Change
F1URCRF1Acc
25% training data kept54.3475.7121.172.5250.23
50% training data kept55.1276.0419.0536.3454.66
75% training data kept56.6476.0321.0048.8657.62
100% training data56.8474.9821.1450.5657.87
\ No newline at end of file diff --git a/parse/train/rJYFzMZC-/rJYFzMZC-_content_list.json b/parse/train/rJYFzMZC-/rJYFzMZC-_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5c0a821ab5b90db8c8a78a4995f2ead70872b6b2 --- /dev/null +++ b/parse/train/rJYFzMZC-/rJYFzMZC-_content_list.json @@ -0,0 +1,1692 @@ +[ + { + "type": "text", + "text": "SIMULATING ACTION DYNAMICS WITH NEURAL PROCESS NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 643, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Antoine Bosselut†, Omer Levy†, Ari Holtzman†, Corin Ennis‡, Dieter Fox† & Yejin Choi† ", + "bbox": [ + 184, + 169, + 802, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "†Paul G. Allen School of Computer Science & Engineering \nUniversity of Washington \n{antoineb,omerlevy,ahai,fox,yejin}@cs.washington.edu \n‡School of Science, Technology, Engineering & Mathematics \nUniversity of Washington - Bothell \n{corin123}@uw.edu ", + "bbox": [ + 184, + 185, + 691, + 270 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 306, + 544, + 321 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Understanding procedural language requires anticipating the causal effects of actions, even when they are not explicitly stated. In this work, we introduce Neural Process Networks to understand procedural text through (neural) simulation of action dynamics. Our model complements existing memory architectures with dynamic entity tracking by explicitly modeling actions as state transformers. The model updates the states of the entities by executing learned action operators. Empirical results demonstrate that our proposed model can reason about the unstated causal effects of actions, allowing it to provide more accurate contextual information for understanding and generating procedural text, all while offering more interpretable internal representations than existing alternatives. ", + "bbox": [ + 232, + 338, + 764, + 477 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 502, + 336, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Understanding procedural text such as instructions or stories requires anticipating the implicit causal effects of actions on entities. For example, given instructions such as “add blueberries to the muffin mix, then bake for one half hour,” an intelligent agent must be able to anticipate a number of entailed facts (e.g., the blueberries are now in the oven; their “temperature” will increase). While this common sense reasoning is trivial for humans, most natural language understanding algorithms do not have the capacity to reason about causal effects not mentioned directly in the surface strings (Levy et al., 2015; Jia & Liang, 2017; Lucy & Gauthier, 2017). ", + "bbox": [ + 173, + 535, + 825, + 632 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we introduce Neural Process Networks, a procedural language understanding system that tracks common sense attributes through neural simulation of action dynamics. Our network models interpretation of natural language instructions as a process of actions and their cumulative effects on entities. More concretely, reading one sentence at a time, our model attentively selects what actions to execute on which entities, and remembers the state changes induced with a recurrent memory structure. In Figure 1, for example, our model indexes the “tomato” embedding, selects the “wash” and “cut” functions and performs a computation that changes the “tomato” embedding so that it can reason about attributes such as its “SHAPE” and “CLEANLINESS”. ", + "bbox": [ + 174, + 638, + 495, + 875 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/ea83afe00b344242d8798385881abffd1d2f37254531c7595434b55036942bed.jpg", + "image_caption": [ + "Figure 1: The process is a narrative of entity state changes induced by actions. In each sentence, these state changes are induced by simulated actions and must be remembered. " + ], + "image_footnote": [], + "bbox": [ + 511, + 648, + 820, + 775 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our model contributes to a recent line of research that aims to model aspects of world state changes, such as language models and machine readers with explicit entity representations (Henaff et al., 2016; Yang et al., 2016; Ji et al., 2017), as well as other more general purpose memory network variants (Weston et al., 2014; Sukhbaatar et al., 2015; Hill et al., 2015; Seo et al., 2016). This worldcentric modeling of procedural language (i.e., understanding by simulation) abstracts away from the surface strings, complementing text-centric modeling of language, which focuses on syntactic and semantic labeling of surface words (i.e., understanding by labeling). ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 104, + 823, + 161 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Unlike previous approaches, however, our model also learns explicit action representations as functional operators (See Figure 1). While representations of action semantics could be acquired through an embodied agent that can see and interact with the world (Oh et al., 2015), we propose to learn these representations from text. In particular, we require the model to be able to explain the causal effects of actions by predicting natural language attributes about entities such as “LOCATION” and “TEMPERATURE”. The model adjusts its representations of actions based on errors it makes in predicting the resultant state changes to attributes. This textual simulation allows us to model aspects of action causality that are not readily available in existing simulation environments. Indeed, most virtual environments offer limited aspects of the world – with a primary focus on spatial relations (Oh et al., 2015; Chiappa et al., 2017; Wahlstrom et al., 2015). They leave out various other dimensions of the world states that are implied by diverse everyday actions such as “dissolve” (change of “COMPOSITION”) and “wash” (change of “CLEANLINESS”). ", + "bbox": [ + 174, + 167, + 825, + 334 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Empirical results demonstrate that parametrizing explicit action embeddings provides an inductive bias that allows the neural process network to learn more informative context representations for understanding and generating natural language procedural text. In addition, our model offers more interpretable internal representations and can reason about the unstated causal effects of actions explained through natural language descriptors. Finally, we include a new dataset with fine-grained annotations on state changes, to be shared publicly, to encourage future research in this direction. ", + "bbox": [ + 174, + 342, + 825, + 424 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 NEURAL PROCESS NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 445, + 449, + 462 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The neural process network is an interpreter that reads in natural language sentences, one at a time, and simulates the process of actions being applied to relevant entities through learned representations of actions and entities. ", + "bbox": [ + 176, + 479, + 823, + 521 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 OVERVIEW AND NOTATION ", + "text_level": 1, + "bbox": [ + 176, + 535, + 401, + 550 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main component of the neural process network is the simulation module (§2.5), a recurrent unit whose internals simulate the effects of actions being applied to entities. A set of $V$ actions is known a priori and an embedding is initialized for each one, $\\mathcal { F } = \\{ f _ { 1 } , . . . f _ { V } \\}$ . Similarly, a set of $I$ entities is known and an embedding is initialized for each one: $\\mathcal { E } = \\{ e _ { 1 } , . . . e _ { I } \\}$ . Each $e _ { i }$ can be considered to encode information about state attributes of that entity, which can be extracted by a set of state predictors (§2.6). As the model reads text, it “applies” action embeddings to the entity vectors, thereby changing the state information encoded about the entities. ", + "bbox": [ + 174, + 563, + 825, + 660 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "For any document $d$ , an initial list of entities $I _ { d }$ is known and $\\mathcal { E } _ { d } = \\{ e _ { i } | i \\in I _ { d } \\} \\subset \\mathcal { E }$ entity state embeddings are initialized. As the neural process network reads a sentence from the document, it selects a subset of both $\\mathcal { F } \\left( \\ S 2 . 3 \\right)$ and $\\mathcal { E } _ { d }$ $( \\ S 2 . 4 )$ based on the actions performed and entities affected in the sentence. The entity state embeddings are changed by the action and the new embeddings are used to predict end states for a set of state changes (§2.6). The prediction error for end states is backpropagated to the action embeddings, learning action representations that model the simulation of desired causal effects on entities. This process is broken down into five modules below. Unless explicitly defined, all $W$ and $b$ variables are parametrized linear projections and biases. We use the notation $\\{ e _ { i } \\} _ { t }$ when referring to the values of the entity embeddings before processing sentence $s _ { t }$ . ", + "bbox": [ + 173, + 666, + 825, + 791 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 SENTENCE ENCODER ", + "text_level": 1, + "bbox": [ + 176, + 803, + 362, + 818 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given a sentence $s _ { t }$ , a Gated Recurrent Unit (Cho et al., 2014) encodes each word and outputs its last hidden vector as a sentence encoding $h _ { t }$ (Sutskever et al., 2014). ", + "bbox": [ + 174, + 829, + 823, + 858 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.3 ACTION SELECTOR ", + "text_level": 1, + "bbox": [ + 176, + 868, + 348, + 883 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given $h _ { t }$ from the sentence encoder, the action selector (bottom left in Fig. 2) contextually determines which action(s) from $\\mathcal { F }$ to execute. For example, if the input sentence is “wash and cut ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/11046f3337ccdae5eb2a85134800ce6ee2fb6e8d92d2d1d3ef9346c712c08f29.jpg", + "image_caption": [ + "Figure 2: Model Summary. The sentence encoder converts a sentence to a vector representation, $h _ { t }$ . The action selector and entity selector use the vector representation to choose the actions that are applied and the entities that are acted upon in the sentence. The simulation module indexes the action and entity state embeddings, and applies the transformation to the entities. The state predictors predict the new state of the entities if a state change has occurred. Equation references are provided in parentheses. " + ], + "image_footnote": [], + "bbox": [ + 204, + 101, + 787, + 411 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "beets”, both $f _ { w a s h }$ and $f _ { c u t }$ must be selected. To account for multiple actions, we make a soft selection over $\\mathcal { F }$ , yielding a weighted sum of the selected action embeddings $\\bar { f } _ { t }$ : ", + "bbox": [ + 173, + 511, + 821, + 540 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d3ee757336701155a43e5a49e62228e879a98b54e5ac102b0e448e27fb6428cc.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle w _ { p } = \\mathbf { M L P } \\big ( h _ { t } \\big ) } } \\\\ { { \\displaystyle \\bar { w } _ { p } = \\frac { w _ { p } } { \\sum _ { j = 1 } ^ { V } w _ { p _ { j } } } } } \\\\ { { \\displaystyle \\bar { f } _ { t } = \\bar { w } _ { p } ^ { \\top } \\mathcal { F } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 437, + 540, + 557, + 616 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where MLP is a parametrized feed-forward network with a sigmoid activation and $w _ { p } \\in \\mathbb { R } ^ { V }$ is the attention distribution over $V$ possible actions (§3.1). We compose the action embedding by taking the weighted average of the selected actions. ", + "bbox": [ + 174, + 618, + 825, + 660 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.4 ENTITY SELECTOR ", + "text_level": 1, + "bbox": [ + 174, + 676, + 346, + 690 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Sentence Attention Given $h _ { t }$ from the sentence encoder, the entity selector chooses relevant entities using a soft attention mechanism: ", + "bbox": [ + 176, + 702, + 823, + 729 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/60ce406e4ca6640c1854f90296811c5503f219490c3c30a660449781617d8052.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { h } _ { t } = \\mathrm { R e L U } ( W _ { 1 } h _ { t } + b _ { 1 } ) } \\\\ & { d _ { i } = \\sigma ( e _ { i _ { 0 } } ^ { \\top } W _ { 2 } [ \\tilde { h } _ { t } ; w _ { p } ] ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 415, + 732, + 581, + 775 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $W _ { 2 }$ is a bilinear mapping, $e _ { i _ { 0 } }$ is a unique key for each entity (§2.5), and $d _ { i }$ is the attention weight for entity embedding $e _ { i }$ . For example, in “wash and cut beets and carrots”, the model should select $e _ { b e e t }$ and ecarrot. ", + "bbox": [ + 174, + 775, + 825, + 818 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Recurrent Attention While sentence attention would suffice if entities were always explicitly mentioned, natural language often elides arguments or uses referent pronouns. As such, the module must be able to consider entities mentioned in previous sentences. Using $\\tilde { h } _ { t }$ , the model computes a soft choice over whether to choose affected entities from this step’s attention $d _ { i }$ or the previous step’s attention distribution. ", + "bbox": [ + 173, + 821, + 825, + 892 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/809e14240baae8481ab6678e7fbb4b93ad05bf3bf29313ca8c8d9391c1384c93.jpg", + "text": "$$\n\\begin{array} { r } { c = s o f t m a x ( W _ { 3 } \\tilde { h } _ { t } + b _ { 3 } ) } \\\\ { a _ { i _ { t } } = c _ { 1 } d _ { i } + c _ { 2 } a _ { i _ { t - 1 } } + c _ { 3 } \\mathbf { 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 406, + 890, + 593, + 930 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $c \\in \\mathbb { R } ^ { 3 }$ is the choice distribution, $a _ { i _ { t - 1 } }$ is the previous sentence’s attention weight for each entity, $a _ { i _ { t } }$ is the final attention for each entity, and 0 is a vector of zeroes (providing the option to not change any entity). Prior entity attentions can propagate forward for multiple steps. ", + "bbox": [ + 174, + 104, + 825, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.5 SIMULATION MODULE ", + "text_level": 1, + "bbox": [ + 174, + 164, + 372, + 179 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Entity Memory A unique state embedding $e _ { i }$ is initialized for every entity $i$ in the document. A unique key to index each embedding $e _ { i _ { 0 } }$ is set as the initial value of the embedding (Henaff et al., 2016; Miller et al., 2016). After the model reads $s _ { t }$ , it modifies $\\{ e _ { i } \\} _ { t }$ to reflect changes influenced by actions. At every time step, the entity memory receives the attention weights from the entity selector, normalizes them and computes a weighted average of the relevant entity state embeddings: ", + "bbox": [ + 174, + 190, + 825, + 261 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/dfb2b6fe3a723a736218fb350bea287b156648b8d247a517376fd73b08a8a4f5.jpg", + "text": "$$\n\\alpha _ { i } = \\frac { a _ { i } } { \\sum _ { j = 1 } ^ { I _ { d } } a _ { j } }\n$$", + "text_format": "latex", + "bbox": [ + 284, + 272, + 390, + 308 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/593860c8a158df665aead8506ea5217074e7384bc812ff50209b0a2879a70840.jpg", + "text": "$$\n\\bar { e } _ { t } = \\sum _ { j = 1 } ^ { I _ { d } } \\alpha _ { i } e _ { i }\n$$", + "text_format": "latex", + "bbox": [ + 612, + 267, + 707, + 311 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Applicator Given the action summary embedding $\\bar { f } _ { t }$ and the entity summary embedding $\\bar { e _ { t } }$ , the applicator (middle right in Fig. 2) applies the selected actions to the selected entities, and outputs the new proposal entity embedding $k _ { t }$ . ", + "bbox": [ + 173, + 327, + 825, + 369 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/346f7d67e8cac541fc410c1301f49116e6e2e08cf0cfbb5dbcc5471b6d6d3a17.jpg", + "text": "$$\nk _ { t } = \\mathrm { R e L U } ( \\bar { f } _ { t } W _ { 4 } \\bar { e } _ { t } + b _ { 4 } )\n$$", + "text_format": "latex", + "bbox": [ + 410, + 376, + 588, + 393 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $W _ { 4 }$ is a third order tensor projection. The vector $k _ { t }$ is the new representation of the entity $\\bar { e } _ { t }$ after the applicator simulates the action being applied to it. ", + "bbox": [ + 173, + 401, + 821, + 430 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Entity Updater The entity updater interpolates the new proposal entity embedding $k _ { t }$ and the set of current entity embeddings $\\{ e _ { i } \\} _ { t }$ : ", + "bbox": [ + 171, + 434, + 823, + 463 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1e369b0dc94112425385833f02da302dbf65de7979234edc8b624c3f1916fbb1.jpg", + "text": "$$\ne _ { i _ { t + 1 } } = a _ { i _ { t } } k _ { t } + ( 1 - a _ { i _ { t } } ) e _ { i _ { t } }\n$$", + "text_format": "latex", + "bbox": [ + 403, + 470, + 593, + 488 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "yielding an updated set of entity embeddings $\\{ e _ { i } \\} _ { t + 1 }$ . Each embedding is updated proportional to its entity’s unnormalized attention $a _ { i }$ , allowing the model to completely overwrite the state embedding for any entity. For example, in the sentence “mix the flour and water,” the embeddings for $e _ { f l o u r }$ and $e _ { w a t e r }$ must both be overwritten by $k _ { t }$ because they no longer exist outside of this new composition. ", + "bbox": [ + 174, + 494, + 825, + 551 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.6 STATE PREDICTORS ", + "text_level": 1, + "bbox": [ + 176, + 568, + 352, + 582 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given the new proposal entity embedding $k _ { t }$ , the state predictor (bottom right in Fig. 2) predicts changes to the resulting entity embedding $k _ { t }$ along the following six dimensions: location, cookedness, temperature, composition, shape, and cleanliness. Discrete multi-class classifiers, one for each dimension, take in $k _ { t }$ and predict a unique end state for their corresponding state change type: ", + "bbox": [ + 173, + 593, + 825, + 650 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b26ddd4f9eb46bcc4c4435b260bdb7f30438307da730a1555ba3404f23e9ae6d.jpg", + "text": "$$\nP ( Y _ { s } | k _ { t } ) = s o f t m a x ( W _ { s } k _ { t } + b _ { s } )\n$$", + "text_format": "latex", + "bbox": [ + 382, + 657, + 616, + 674 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For location changes, which require contextual information to predict the end state, $k _ { t }$ is concatenated with the original sentence representation $h _ { t }$ to predict the final state. ", + "bbox": [ + 173, + 681, + 821, + 710 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 727, + 292, + 742 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 STATE CHANGE KNOWLEDGE ", + "text_level": 1, + "bbox": [ + 176, + 758, + 419, + 773 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this work we focus on physical action verbs in cooking recipes. We manually collect a set of 384 actions such as cut, bake, boil, arrange, and place, organizing their causal effects along the following predefined dimensions: LOCATION, COOKEDNESS, TEMPERATURE, SHAPE, CLEANLINESS and COMPOSITION. The textual simulation operated by the model induces state changes along these dimensions by applying actions functions from the above set of 384. For example, cut entails a change in SHAPE, while bake entails a change in TEMPERATURE, COOKEDNESS, and even LOCATION. We annotate the state changes each action induces, as well as the end state of the action, using Amazon Mechanical Turk. The set of possible end states for a state change can range from 2 for binary state changes to more than 200 (See Appendix C for details). Table 1 provides examples of annotations in this action lexicon. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/a7c065bb6cd8e80813300b1626a454595b2cfdab5ce724bc644f0601ad8c6afa.jpg", + "table_caption": [ + "Table 1: Example actions, the state changes they induce, and the possible end states " + ], + "table_footnote": [], + "table_body": "
ActionState Change TypesEnd States
braiseCOOKEDNESS;TEMPERATURECOOKED;HOTCOLDMOLDEDCLEANCOMPOSEDCOLD;REFRIGERATORSEPARATED
chillTEMPERATURE
kneadSHAPECLEANLINESS
wash
dissolveCOMPOSITION
refrigerateTEMPERATURE;LOCATION
sliceSHAPE
", + "bbox": [ + 272, + 99, + 720, + 208 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 DATASET ", + "text_level": 1, + "bbox": [ + 174, + 257, + 279, + 272 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For learning and evaluation, we use a subset of the Now You’re Cooking dataset (Kiddon et al., 2016). We chose 65816 recipes for training, 175 recipes for development, and 700 recipes for testing. For the development and test sets, crowdsourced workers densely annotate actions, entities and state changes that occur in each sentence so that we can tune hyperparameters and evaluate on gold evaluation sets. Annotation details are provided in Appendix C.3. ", + "bbox": [ + 174, + 284, + 825, + 353 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 COMPONENT-WISE TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 367, + 419, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The neural process network is trained by jointly optimizing multiple losses for the action selector, entity selector, and state change predictors. Importantly, our training scheme uses weak supervision because dense annotations are prohibitively expensive to acquire at a very large scale. Thus, we heuristically extract verb mentions from each recipe step and assign a state change label based on the state changes induced by that action (§3.1). Entities are extracted similarly based on string matching between the instructions and the ingredient list. We use the following losses for training: ", + "bbox": [ + 174, + 392, + 825, + 477 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Action Selection Loss Using noisy supervision, the action selector is trained to minimize the cross-entropy loss for each possible action, allowing multiple actions to be chosen at each step if multiple actions are mentioned in a sentence. The MLP in the action selector (Eq. 1) is pretrained. ", + "bbox": [ + 176, + 481, + 825, + 523 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Entity Selection Loss Similarly, to train the attentive entity selector, we minimize the binary cross-entropy loss of predicting whether each entity is affected in the sentence. ", + "bbox": [ + 173, + 529, + 821, + 556 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "State Change Loss For each state change predictor, we minimize the negative loglikelihood of predicting the correct end state for each state change. ", + "bbox": [ + 176, + 560, + 823, + 589 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Coverage Loss An underlying assumption in many narratives is that all entities that are mentioned should be important to the narrative. We add a loss term that penalizes narratives whose combined attention weights for each entity does not sum to more than 1. ", + "bbox": [ + 174, + 594, + 825, + 636 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c7a9994e4b0354440095398942defe6adef1dc35873358208b358ab4fdf6c107.jpg", + "text": "$$\n\\mathcal { L } _ { c o v e r } = - \\frac { 1 } { I _ { d } } \\sum _ { i = 1 } ^ { I _ { d } } \\log \\sum _ { t = 1 } ^ { S } a _ { i _ { t } }\n$$", + "text_format": "latex", + "bbox": [ + 398, + 642, + 598, + 686 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $a _ { i _ { t } }$ is the attention weight for a particular entity at sentence $t$ and $I _ { d }$ is the number of entities in a document. $\\textstyle \\sum _ { t = 1 } ^ { S } a _ { i _ { t } }$ is upper bounded by 1. This is similar to the coverage penalty used in neural machine translation (Tu et al., 2016). ", + "bbox": [ + 174, + 699, + 825, + 744 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 765, + 398, + 781 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We evaluate our model on a set of intrinsic tasks centered around tracking entities and state changes in recipes to show that the model can simulate preliminary dynamics of the recipe task. Additionally, we provide a qualitative analysis of the internal components of our model. Finally, we evaluate the quality of the states encoded by our model on the extrinsic task of generating future steps in a recipe. ", + "bbox": [ + 174, + 796, + 825, + 853 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 INTRINSIC EVALUATION - TRACKING ", + "text_level": 1, + "bbox": [ + 176, + 869, + 472, + 883 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the tracking task, we evaluate the model’s ability to identify which entities are selected and what changes have been made to them in every step. We break the tracking task into two separate evaluations, entity selection and end state prediction, and also investigate whether the model learns internal representations that approximate recipe dynamics. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/1e596fb9114761de312e6312db961f042ca0f8d3f30b762a408e48bbf13cacaf.jpg", + "table_caption": [ + "Table 2: Results for entity selection and state change selection " + ], + "table_footnote": [], + "table_body": "
ModelEntity SelectionState Change
F1URCRF1Acc
2-layer LSTM Entity Recognizer50.9874.0313.3311
Adapted Gated Recurrent Unit Adapted Recurrent Entity Network - Recurrent Attention (Eq. 3)45.9467.697.7441.1652.69
48.5771.889.8742.3253.47
48.9172.3212.6742.1450.48
- Coverage Loss (Eq. 9) - Action Connections (Eq. 2)55.1873.9820.2344.4455.20
54.8573.5420.0344.0554.81
54.9173.8720.3044.2855.00
- Action Selector Pretraining + Pretrained Action Embeddings - Action Embedding Updates55.1674.0220.3244.0255.03
53.7970.7718.6044.2755.02
FullModel55.3974.8820.4544.6555.07
", + "bbox": [ + 232, + 99, + 766, + 275 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 325, + 823, + 353 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Metrics In the entity selection test, we report the F1 score of choosing the correct entities in any step. A selected entity is defined as one whose attention weight $a _ { i }$ is greater than $50 \\%$ $( \\ S 2 . 4 )$ . Because entities may be harder to predict when they have been combined with other entities (e.g., the mixture may have a new name), we also report the recall for selecting combined (CR) and uncombined (UR) entities. In the end state prediction test, we report how often the model correctly predicts the state change performed in a recipe step and the resultant end state. This score is then scaled by the accuracy of predicting which entities were changed in that same step. We report the average F1 and accuracy across the six state change types. ", + "bbox": [ + 173, + 358, + 825, + 469 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines We compare our models against two baselines. First, we built a GRU model that is trained to predict entities and state changes independently. This can be viewed as a bare minimum network with no action representations or recurrent entity memory. The second baseline is a Recurrent Entity Network (Henaff et al., 2016) with changes to fit our task. First, the model can tie memory cells to a subset of the full list of entities so that it only considers entities that are present in a particular recipe. Second, the entity distribution for writing to the memory cells is re-used when we query the memory cells. The normalized weighted average of the entity cells is used as the input to the state predictors. The unnormalized attention when writing to each cell is used to predict selected entities. Both baselines are trained with entity selection and state change losses (§3.3). ", + "bbox": [ + 173, + 474, + 825, + 599 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Ablations We report results on six ablations. First, we remove the recurrent attention (Eq. 3). The model only predicts entities using the current encoder hidden state. In the second ablation, the model is trained with no coverage penalty (Eq. 9). The third ablation prunes the connection from the action selector $w _ { p }$ to the entity selector (Eq. 2). We also explore not pretraining the action selector. Finally, we look at two ablations where we intialize the action embeddings with vectors from a skipgram model. In the first, the model operates normally, and in the second, we do not allow gradients to backpropagate to the action embeddings, updating only the mapping tensor $W _ { 4 }$ instead (Eq. 6). ", + "bbox": [ + 174, + 604, + 825, + 703 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 EXTRINSIC EVALUATION - GENERATION ", + "text_level": 1, + "bbox": [ + 176, + 719, + 493, + 734 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The generation task tests whether our system can produce the next step in a recipe based on the previous steps that have been performed. The model is provided all of the previous steps as context. ", + "bbox": [ + 176, + 746, + 823, + 773 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Metrics We report the combined BLEU score and ROUGE score of the generated sequence relative to the reference sequence. Each candidate sequence has one reference sentence. Both metrics are computed at the corpus-level. Also reported are “VF1”, the F1 score for the overlap of the actions performed in the reference sequence and the verbs mentioned in the generated sequence, and “SF1”, the F1 score for the overlap of end states annotated in the reference sequence and predicted by the generated sequences. End states for the generated sequences are extracted using the lexicon from Section 3.1 based on the actions performed in the sentence. ", + "bbox": [ + 174, + 792, + 823, + 890 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Setup To apply our model to the task of recipe step generation, we input the context sentences through the neural process network and record the entity state vectors once the entire context has been read (§2.5). These vectors can be viewed as a snapshot of the current state of the entities once the preceding context has been simulated inside the neural process network. We encode these vectors using a bidirectional GRU (Cho et al., 2014) and take the final time step hidden state $\\mathbb { e } _ { I }$ . A different GRU encodes the context words in the same way (yielding $\\mathbb { h } _ { T }$ ) and the first hidden state input to the decoder is computed using the projection function: ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/8c5b1defdf590d6b21c7ca6d95dad5df2a53465ef67a22faffadcb383958becb.jpg", + "table_caption": [ + "Table 3: Examples of the model selecting entities for sentence $s _ { t }$ . The previous sentences are provided as context in cases where they are relevant. " + ], + "table_footnote": [], + "table_body": "
GoodSt-1StselectedcorrectAdd tomato paste,broth, garlic,chili powder,cumin,chile peppers,and water.Bring to boil, then turn very low,cover and simmer until meat is tender.meat,garlic, chili powder, tomato paste,cumin,chiles,beef broth,waterSame + [oil]
GoodSt-4St-3St-2St-1StselectedcorrectStir in oats, sugar, flour, corn syrup,milk, vanilla extract, and salt.Mix well.Drop by measuring teaspoonfuls onto cookie sheets.Bake 5-7 minutes.Let cool.oats,sugar, flour, corn syrup,milk,vanilla extract, saltoats,sugar, flour, corn syrup,milk,vanilla extract, salt
GoodSt-1StselectedcorrectIn a large saucepan over low heat, melt marshmallows.Add sprinkles,cereal,and raisins,stir until well coated.marshmallows,cereal,raisinsmarshmallows,cereal, raisins,sprinkles
BadSt-3St-2St-1StselectedcorrectLadle the barbecue sauce around the crust and spread.Add mozzarella, yellow cheddar,and monterey jack cheese.Next,add onion mixture and sliced chicken breast .Top pizza with jalapeno peppers.jalapenoscrust,sauce,mozzarella,cheddar,monterey jack,white onion,chicken, jalapenos
BadSt-2St-1StselectedcorrectCombine 1 cup flour, salt,and 1 tbsp sugar.Cut in butter until mixture is crumbly, then sprinkle with vinegar .Gather dough into a ball and press into bottom of 9 inch springform pan.butter, vinegarflour, salt,sugar, butter, vinegar
", + "bbox": [ + 187, + 101, + 810, + 446 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 515, + 825, + 585 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/6f299b808f9e52043586a07ebd2c9d3a92114b1f8f9167d426b1e6630f15360e.jpg", + "text": "$$\n\\tilde { h } _ { 0 } = W _ { 5 } ( \\Subset _ { I } \\circ \\Join _ { T } )\n$$", + "text_format": "latex", + "bbox": [ + 434, + 602, + 563, + 622 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\circ$ is the Hadamard product between the two encoder outputs. All models are trained by minimizing the negative loglikelihood of predicting the next word for the full sequence. Implementation details can be found in Appendix A. ", + "bbox": [ + 174, + 632, + 825, + 674 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines For the generation task, we use three baselines: a seq2seq model with no attention (Sutskever et al., 2014), an attentive seq2seq model (Bahdanau et al., 2014), and a similar variant as our NPN generator, except where the entity states have been computed by the Recurrent Entity Network (EntNet) baseline $( \\ S 4 . 1 )$ . Implementation details for baselines can be found in Appendix B. ", + "bbox": [ + 174, + 679, + 823, + 734 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 755, + 418, + 771 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 INTRINSIC EVALUATIONS ", + "text_level": 1, + "bbox": [ + 174, + 786, + 392, + 800 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Entity Selection As shown in Table 8, our full model outperforms all baselines at selecting entities, with an F1 score of $5 5 . 3 9 \\%$ . The ablation study shows that the recurrent attention, coverage loss, action connections and action selector pretraining improve performance. Our success at predicting entities extends to both uncomposed entities, which are still in their raw forms (e.g., melt the butter butter), and composed entities, in which all of the entities that make up a composition must be selected. For example, in a Cooking lasagna recipe, if the final step involves baking the prepared lasagna, the model must select all the entities that make up the lasagna (e.g., lasagna sheets, beef, tomato sauce). In Table 3, we provide examples of our model’s ability to handle complex cases such as compositional entities (Ex. 1, 3), and elided arguments over long time windows (Ex. 2). We also provide examples where the model fails to select the correct entities because it does not identify the mapping between a reference construct such as “pizza” (Ex. 4) or “dough” (Ex. 5) and the set of entities that composes it, showcasing the difficulty of selecting the full set for a composed entity. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/9e5de9b4391228ff4564923e7f8a213de1be9c7578d0535d7773d8900f4d8738.jpg", + "table_caption": [ + "Table 4: Most similar actions based on cosine similarity of action embeddings " + ], + "table_footnote": [], + "table_body": "
ActionNearestNeighbor Actions
cut boilslice,split, snap,slash,carve,slit,chop cook,microwave,fry,steam,simmer
addsprinkle,mix,reduce,splash,stir,dust
washrinse,scrub,refresh,soak,wipe,scale
mashspread, puree,squeeze,liquefy, blend
place rinseease,put,lace,arrange,leave
wash,refresh,soak,wipe,scrub,clean
warmreheat, ignite,heat,light,crisp,preheat
steam
microwave,crisp,boil, parboil,heat
sprinkletop,pat,add,dip,salt, season
greasecoat,rub,dribble,spray,smear, line
", + "bbox": [ + 179, + 119, + 496, + 277 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/7d3dd2e0d40faf65ae5f5c239788d29171cd4654468f9e42a1ba283ad355c5c2.jpg", + "image_caption": [ + "Figure 3: Change in cosine similarity of entity state embeddings " + ], + "image_footnote": [], + "bbox": [ + 526, + 104, + 833, + 273 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/c22fac19765a68784c96a30aae8849865e186a561f4ccd4b28946ed693ea6a5b.jpg", + "table_caption": [ + "Table 5: Generation Results " + ], + "table_footnote": [], + "table_body": "
ModelBLEUBLEUROUGE-LRVF1
Vanilla Seq2SeqAttentive Seq2SeqEntNet Generator2.812.832.3033.0033.1832.7116.1716.9717.5340.2141.4342.43
NPN Generator3.7435.6420.1243.40
", + "bbox": [ + 287, + 325, + 710, + 404 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 449, + 825, + 506 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "State Change Tracking In Table 8, we show that our full model outperforms competitive baselines such as Recurrent Entity Networks (Henaff et al., 2016) and jointly trained GRUs. While the ablation without the coverage loss shows higher accuracy, we attribute this to the fact that it predicts a smaller number of total state changes. Interestingly, initializing action embeddings with skipgram vectors and locking their values shows relatively high performance, indicating the potential gains in using powerful pretrained representations to represent actions. ", + "bbox": [ + 174, + 510, + 825, + 593 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Action Embeddings In our model, each action is assigned its own embedding, but many actions induce similar changes in the physical world (e.g.,“cut” and “slice”). After training, we compute the pairwise cosine similarity between each pair of action embeddings. In Table 4, we see that actions that perform similar functions are neighbors in embedding space, indicating the model has captured certain semantic properties of these actions. Learning action representations through the state changes they induce has allowed the model to cluster actions by their transformation functions. ", + "bbox": [ + 174, + 598, + 825, + 681 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Entity Compositions When individual entities are combined into new constructs, our model averages their state embeddings (Eq. 5), applies an action embedding to them (Eq. 6), and writes them to memory (Eq. 7). The state embeddings of entities that are combined should be overwritten by the same new embedding. In Figure 3, we present the percentage increase in cosine similarity for state embeddings of entities that are combined in a sentence (blue) as opposed to the percentage increase for those that are not (red bars). While the soft attention mechanism for entity selection allows similarities to leak between entity embeddings, our system is generally able to model the compositionality patterns that result from entities being combined into new constructs. ", + "bbox": [ + 174, + 685, + 825, + 797 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 EXTRINSIC EVALUATIONS ", + "text_level": 1, + "bbox": [ + 176, + 814, + 397, + 828 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recipe Step Generation Our results in Table 5 indicate that sequences generated using the neural process network entity states as additional input yield higher scores than competitive baselines. The entity states allow the model to predict next steps conditioned on a representation of the world being simulated by the neural process network. Additionally, the higher VF1 and SF1 scores indicate that the model is indeed using the extra information to better predict the actions that should follow the context provided. Example generations for each baselines from the dev set are provided in Table 6, showing that the NPN generator can use information about ingredient states to reason about the most likely next step. The first and second examples are interesting as it shows that the NPN-aware model has learned to condition on entity state – knowing that raw butter will likely be melted or that a cooked flan must be refrigerated. The third example is also interesting because the model learns that cooked vegetables such as squash will sometimes be drained, even if it is not relevant to this recipe because the squash is steamed. The seq2seq and EntNet baselines, meanwhile, output reasonable sentences given the immediate context, but do not exhibit understanding of global patterns. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/843410b03c74e76b2d9c424b4dcc95dce8401733e1cd3252326097e814a50b57.jpg", + "table_caption": [ + "Table 6: Examples of the model generating sentences compared to baselines. The context and reference are provided first, followed by our model’s generation and then the baseline generations " + ], + "table_footnote": [], + "table_body": "
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPreheat oven to 425 degrees.Melt butter in saucepan and mix in bourbon, thyme,pepper, and salt.Melt butter in skillet.Lightly grease 4 x 8 baking pan with sunflower oil.Combine all ingredients and mix well.In a large bowl, combine flour,baking powder,baking soda, salt,and pepper.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPour egg mixture over caramelized sugar in cake pan. Place cake pan in large shallowbaking dish.Bake for 55 minutes or until knife inserted into flan comes out clean.Cover and chill at least 8 hours.Refrigerate until ready to use.Serve at room temperature.Store in an airtight container.Store in an airtight container.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetCut squash into large pieces and steam. Remove cooked squash from shells;Measure 4 cups pulp and reserve remainder for another dish.Drain.Mash pulp with a fork.Set aside.Set aside.
", + "bbox": [ + 173, + 103, + 831, + 352 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 428, + 825, + 526 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 549, + 344, + 565 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Recent studies in machine comprehension have used a neural memory component to store a running representation of processed text (Weston et al., 2014; Sukhbaatar et al., 2015; Hill et al., 2015; Seo et al., 2016). While these approaches map text to memory vectors using standard neural encoder approaches, our model, in contrast, directly interprets text in terms of the effects actions induce in entities, providing an inductive bias for learning how to represent stored memories. More recent work in machine comprehension also sought to couple the memory representation with tracking entity states (Henaff et al., 2016). Our work seeks to provide a relatively more structured representation of domain-specific action knowledge to provide an inductive bias to the reasoning process. ", + "bbox": [ + 174, + 582, + 825, + 694 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Neural Programmers (Neelakantan et al., 2015; 2016) have also used functions to simulate reasoning, by building a model to select rows in a database and applying operation on those selected rows. While their work explicitly defined the effect of a number of operations for those rows, we provide a framework for learning representations for a more expansive set of actions, allowing the model to learn representations for how actions change the state space. ", + "bbox": [ + 174, + 700, + 823, + 770 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Works on instructional language studied the task of building discrete graph representations of recipes using probabilistic models (Kiddon et al., 2015; Mori et al., 2014; 2012). We propose a complementary new model by integrating action and entity relations into the neural network architecture and also address the additional challenge of tracking the state changes of the entities. ", + "bbox": [ + 174, + 776, + 823, + 833 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Additional work in tracking states with visual or multimodal context has focused on 1) building graph representations for how entities change in goal-oriented domains (Gao et al., 2016; Liu et al., 2016; Si et al., 2011) or 2) tracking visual state changes based on decisions taken by agents in environment simulators such as videos or games (Chiappa et al., 2017; Wahlstrom et al., 2015; Oh et al., 2015). Our work, in contrast, models state changes in embedding space using only text-based signals to map real-world actions to algebraic transformations. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 318, + 119 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We introduced the Neural Process Network for modeling a process of actions and their causal effects on entities by learning action transformations that change entity state representations. The model maintains a recurrent memory structure to track entity states and is trained to predict the state changes that entities undergo. Empirical results demonstrate that our model can learn the causal effects of action semantics in the cooking domain and track the dynamic state changes of entities, showing advantages over competitive baselines. ", + "bbox": [ + 174, + 131, + 825, + 214 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 236, + 356, + 251 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Yonatan Bisk, Peter Clark, Bhavana Dalvi, Niket Tandon, and Yuyin Sun for helpful discussions at various stages of this work. This research was supported in part by NSF (IIS-1524371, IIS-1714566, NRI-1525251), DARPA under the CwC program through the ARO (W911NF-15-1- 0543), Samsung Research, and gifts by Google and Facebook. ", + "bbox": [ + 174, + 267, + 825, + 321 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 287, + 119 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In ICLR 2015, 2014. 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", + "bbox": [ + 169, + 98, + 826, + 738 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A TRAINING DETAILS OF OUR FULL MODEL AND ABLATIONS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 705, + 119 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 TRACKING MODELS ", + "text_level": 1, + "bbox": [ + 176, + 133, + 357, + 148 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The hidden size of the instruction encoder is 100, the embedding sizes of action functions and entities are 30. We use dropout with a rate of 0.3 before any non-recurrent fully connected layers Srivastava et al. (2014). We use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over three epochs. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use two instruction encoders, one for the entity selector, and one for the action selector. Word embeddings and entity embeddings are initialized with skipgram embeddings (Mikolov et al., 2013a;b) using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words, and 2996 for entities. Gradients with respect to the coverage loss (Eq. 9) are only backpropagated in steps where no entity is annotated as being selected. To account for the false negatives in the training data due to the heuristic generation of the labels, gradients with respect to the entity selection loss are zeroed when no entity label is present. ", + "bbox": [ + 174, + 160, + 825, + 327 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 GENERATION MODEL ", + "text_level": 1, + "bbox": [ + 176, + 343, + 367, + 358 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The hidden size of the context encoder is 200. The hidden size of the state vector encoder is 200. State vectors have dimensionality 30 (the same as in the neural process network). Dropout of 0.3 is used during training in the decoder. The context and state representations are projected jointly using an element-wise product followed by a linear projection Kim et al. (2016). Both encoders and the decoder are single layer. The learning rate is 0.0003 initially and is halved every 5 epochs. The model is trained with the Adam optimizer. ", + "bbox": [ + 174, + 369, + 825, + 453 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B TRAINING DETAILS OF BASELINES ", + "text_level": 1, + "bbox": [ + 174, + 473, + 500, + 489 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.1 TRACKING BASELINES ", + "text_level": 1, + "bbox": [ + 176, + 503, + 375, + 518 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Joint Gated Recurrent Unit The hidden state of the GRU is 100. We use a dropout with a rate of 0.3 before any non-recurrent fully connected layers. We use the Adam optimizer with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over a single epoch. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use encoders, one for the entity selector, and one for the state change predictors. Word embeddings are initialized with skipgram embeddings using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words. ", + "bbox": [ + 174, + 530, + 825, + 627 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Recurrent Entity Networks Memory cells are tied to the entities in the document. For a recipe with 12 ingredients, 12 entity cells are initialized. All hyperparameters are the same as the in the bAbI task from Henaff et al. (2016). The learning rate start at 0.01 and is halved every 25 epochs. Entity cells and word embeddings are 100 dimensional. The encoder is a multiplicative mask initialized the same as in Henaff et al. (2016). Intermediate supervision from the weak labels is provided to help predict entities. A separate encoder is used for computing the attention over memory cells and the content to write to the memory. Dropout of 0.3 is used in the encoders. The batch size is 64. We use a vocabulary size of 7358 for words, and 2996 for entities. ", + "bbox": [ + 174, + 642, + 825, + 753 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.2 GENERATION BASELINES ", + "text_level": 1, + "bbox": [ + 176, + 770, + 393, + 785 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Seq2seq The encoder and decoder are both single-layer GRUs with hidden size 200. We use dropout with probability 0.3 in the decoder. We train with the Adam optimizer starting with a learning rate 0.0003 that is halved every 5 epochs. The encoder is bidirectional. The model is trained to minimize the negative loglikelihood of predicting the next word. ", + "bbox": [ + 174, + 796, + 823, + 852 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Attentive Seq2seq The encoder is the same as in the seq2seq baseline. A multiplicative attention between the decoder hidden state and the context vectors is used to compute the attention over the context at every decoder time step. The model is trained with the same learning rate, learning schedule and loss function as the seq2seq baseline. ", + "bbox": [ + 176, + 867, + 823, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "EntNet Generator The model is trained in the same way as the NPN generator model in Appendix A.2 except that the state representations used as input are produced from by EntNet baseline described in Section 4.1 and Appendix B.1. ", + "bbox": [ + 174, + 104, + 823, + 147 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C ANNOTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 167, + 331, + 183 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.1 ANNOTATING STATE CHANGES ", + "text_level": 1, + "bbox": [ + 176, + 198, + 431, + 213 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We provide workers with a verb, its definition, an illustrative image of the action, and a set of sentences where the verb is mentioned. Workers are provided a checklist of the six state change types and instructed to identify which of them the verb causes. They are free to identify multiple changes. Seven workers annotate each verb and we assign a state change based on majority vote. Of the set of 384 verbs extracted, only 342 have a state change type identified with them. Of those, 74 entail multiple state change types. ", + "bbox": [ + 174, + 224, + 825, + 309 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.2 ANNOTATING END STATES ", + "text_level": 1, + "bbox": [ + 176, + 325, + 401, + 339 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We give workers a verb, a state change type, and an example with the verb and ask them to provide an end state for the ingredient the verb is applied to in the example. We then use the answers to manually aggregate a set of end states for each state change type. These end states are used as labels when the model predicting state changes. For example, a LOCATION change might lead to an end state of “pan,” “pot”, or “oven.” End states for each state change type are provided in Table 7. ", + "bbox": [ + 173, + 351, + 826, + 421 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/dd18fa443a20db6dcbd11f7367458abd0dd1e585b5c29e43a3e6e878562ffebd.jpg", + "table_caption": [ + "Table 7: End states for each state change type " + ], + "table_footnote": [], + "table_body": "
State Change TypeEnd States
Temperature Composition Cleanliness Cookedness Shape Locationhot; cold; room composed; not composed clean; dirty; dry cooked; raw molded; hit; deformed; separated pan,pot, cupboard, screen, scale,
", + "bbox": [ + 308, + 435, + 687, + 550 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.3 ANNOTATING DEVELOPMENT AND TEST SETS ", + "text_level": 1, + "bbox": [ + 174, + 597, + 537, + 612 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Annotators are instructed to note any entities that undergo one of the six state changes in each step, as well as to identify new combinations of ingredients that are created. For example, the sentence “Cut the tomatoes and add to the onions” would involve a SHAPE change for the tomatoes and a combination created from the “tomatoes” and “onions”. In a separate task, three workers are asked to identify the actions performed in every sentence of the development and test set recipes. If an action receives a majority vote that it is performed, it is included in the annotations. ", + "bbox": [ + 173, + 623, + 825, + 708 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D ADDITIONAL RESULTS D.1 REMOVING TRAINING DATA ", + "bbox": [ + 173, + 727, + 413, + 775 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/66fa690166b0e4d1edda0a8cd6657e3dc7165a05f999185c81957b0169c7a4ba.jpg", + "table_caption": [ + "Table 8: Results for entity selection and state change selection on the development set when randomly dropping a percentage of the training labels " + ], + "table_footnote": [], + "table_body": "
ModelEntity Selection State Change
F1URCRF1Acc
25% training data kept54.3475.7121.172.5250.23
50% training data kept55.1276.0419.0536.3454.66
75% training data kept56.6476.0321.0048.8657.62
100% training data56.8474.9821.1450.5657.87
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While representations of action semantics could be acquired through", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "an embodied agent that can see and interact with the world (Oh et al., 2015), we propose to learn", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "these representations from text. 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While representations of action semantics could be acquired through", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "an embodied agent that can see and interact with the world (Oh et al., 2015), we propose to learn", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "these representations from text. 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chillTEMPERATURE
kneadSHAPECLEANLINESS
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dissolveCOMPOSITION
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Annotation details are provided in Appendix C.3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 226, + 505, + 281 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 291, + 257, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 259, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 259, + 304 + ], + "score": 1.0, + "content": "3.3 COMPONENT-WISE TRAINING", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 311, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "The neural process network is trained by jointly optimizing multiple losses for the action selector,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "entity selector, and state change predictors. 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Entities are extracted similarly based on string", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 365, + 502, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 502, + 381 + ], + "score": 1.0, + "content": "matching between the instructions and the ingredient list. 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This is similar to the coverage penalty used in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 579, + 284, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 284, + 590 + ], + "score": 1.0, + "content": "neural machine translation (Tu et al., 2016).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 554, + 506, + 590 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 244, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 245, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 245, + 620 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "We evaluate our model on a set of intrinsic tasks centered around tracking entities and state changes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "in recipes to show that the model can simulate preliminary dynamics of the recipe task. Additionally,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "we provide a qualitative analysis of the internal components of our model. Finally, we evaluate the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "quality of the states encoded by our model on the extrinsic task of generating future steps in a recipe.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 630, + 505, + 678 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 289, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 290, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 290, + 702 + ], + "score": 1.0, + "content": "4.1 INTRINSIC EVALUATION - TRACKING", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In the tracking task, we evaluate the model’s ability to identify which entities are selected and what", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "changes have been made to them in every step. We break the tracking task into two separate evalua-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "tions, entity selection and end state prediction, and also investigate whether the model learns internal", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 269, + 309, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 309, + 282 + ], + "score": 1.0, + "content": "representations that approximate recipe dynamics.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 142, + 79, + 469, + 218 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 142, + 79, + 469, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 79, + 469, + 218 + ], + "spans": [ + { + "bbox": [ + 142, + 79, + 469, + 218 + ], + "score": 0.983, + "html": "
ModelEntity SelectionState Change
F1URCRF1Acc
2-layer LSTM Entity Recognizer50.9874.0313.3311
Adapted Gated Recurrent Unit Adapted Recurrent Entity Network - Recurrent Attention (Eq. 3)45.9467.697.7441.1652.69
48.5771.889.8742.3253.47
48.9172.3212.6742.1450.48
- Coverage Loss (Eq. 9) - Action Connections (Eq. 2)55.1873.9820.2344.4455.20
54.8573.5420.0344.0554.81
54.9173.8720.3044.2855.00
- Action Selector Pretraining + Pretrained Action Embeddings - Action Embedding Updates55.1674.0220.3244.0255.03
53.7970.7718.6044.2755.02
FullModel55.3974.8820.4544.6555.07
", + "type": "table", + "image_path": "1e596fb9114761de312e6312db961f042ca0f8d3f30b762a408e48bbf13cacaf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 142, + 79, + 469, + 125.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 142, + 125.33333333333334, + 469, + 171.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 142, + 171.66666666666669, + 469, + 218.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 180, + 225, + 430, + 236 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 180, + 223, + 431, + 238 + ], + "spans": [ + { + "bbox": [ + 180, + 223, + 431, + 238 + ], + "score": 1.0, + "content": "Table 2: Results for entity selection and state change selection", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 504, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "tions, entity selection and end state prediction, and also investigate whether the model learns internal", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 269, + 309, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 309, + 282 + ], + "score": 1.0, + "content": "representations that approximate recipe dynamics.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "score": 1.0, + "content": "Metrics In the entity selection test, we report the F1 score of choosing the correct entities in any", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 378, + 308 + ], + "score": 1.0, + "content": "step. A selected entity is defined as one whose attention weight", + "type": "text" + }, + { + "bbox": [ + 379, + 297, + 389, + 306 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 295, + 453, + 308 + ], + "score": 1.0, + "content": "is greater than", + "type": "text" + }, + { + "bbox": [ + 454, + 295, + 474, + 306 + ], + "score": 0.83, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 295, + 501, + 307 + ], + "score": 0.28, + "content": "( \\ S 2 . 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 295, + 505, + 308 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 305, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 320 + ], + "score": 1.0, + "content": "Because entities may be harder to predict when they have been combined with other entities (e.g.,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "the mixture may have a new name), we also report the recall for selecting combined (CR) and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "uncombined (UR) entities. In the end state prediction test, we report how often the model correctly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "predicts the state change performed in a recipe step and the resultant end state. This score is then", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "scaled by the accuracy of predicting which entities were changed in that same step. We report the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 361, + 341, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 341, + 374 + ], + "score": 1.0, + "content": "average F1 and accuracy across the six state change types.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "score": 1.0, + "content": "Baselines We compare our models against two baselines. First, we built a GRU model that is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "trained to predict entities and state changes independently. This can be viewed as a bare minimum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "network with no action representations or recurrent entity memory. The second baseline is a Re-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "current Entity Network (Henaff et al., 2016) with changes to fit our task. First, the model can tie", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "memory cells to a subset of the full list of entities so that it only considers entities that are present in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "a particular recipe. Second, the entity distribution for writing to the memory cells is re-used when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "we query the memory cells. The normalized weighted average of the entity cells is used as the in-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "put to the state predictors. The unnormalized attention when writing to each cell is used to predict", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 464, + 487, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 487, + 477 + ], + "score": 1.0, + "content": "selected entities. Both baselines are trained with entity selection and state change losses (§3.3).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "score": 1.0, + "content": "Ablations We report results on six ablations. First, we remove the recurrent attention (Eq. 3). The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "model only predicts entities using the current encoder hidden state. In the second ablation, the model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "is trained with no coverage penalty (Eq. 9). The third ablation prunes the connection from the action", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 513, + 504, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 140, + 525 + ], + "score": 1.0, + "content": "selector", + "type": "text" + }, + { + "bbox": [ + 140, + 514, + 153, + 525 + ], + "score": 0.89, + "content": "w _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 513, + 504, + 525 + ], + "score": 1.0, + "content": "to the entity selector (Eq. 2). We also explore not pretraining the action selector. Finally,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "we look at two ablations where we intialize the action embeddings with vectors from a skipgram", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "model. In the first, the model operates normally, and in the second, we do not allow gradients to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 545, + 488, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 408, + 558 + ], + "score": 1.0, + "content": "backpropagate to the action embeddings, updating only the mapping tensor", + "type": "text" + }, + { + "bbox": [ + 408, + 546, + 424, + 556 + ], + "score": 0.89, + "content": "W _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 545, + 488, + 558 + ], + "score": 1.0, + "content": "instead (Eq. 6).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 302, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 303, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 303, + 583 + ], + "score": 1.0, + "content": "4.2 EXTRINSIC EVALUATION - GENERATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 591, + 504, + 613 + ], + "lines": [ + { + "bbox": [ + 107, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 107, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "The generation task tests whether our system can produce the next step in a recipe based on the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "previous steps that have been performed. The model is provided all of the previous steps as context.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 504, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "Metrics We report the combined BLEU score and ROUGE score of the generated sequence rela-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "tive to the reference sequence. Each candidate sequence has one reference sentence. Both metrics", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "are computed at the corpus-level. 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ModelEntity SelectionState Change
F1URCRF1Acc
2-layer LSTM Entity Recognizer50.9874.0313.3311
Adapted Gated Recurrent Unit Adapted Recurrent Entity Network - Recurrent Attention (Eq. 3)45.9467.697.7441.1652.69
48.5771.889.8742.3253.47
48.9172.3212.6742.1450.48
- Coverage Loss (Eq. 9) - Action Connections (Eq. 2)55.1873.9820.2344.4455.20
54.8573.5420.0344.0554.81
54.9173.8720.3044.2855.00
- Action Selector Pretraining + Pretrained Action Embeddings - Action Embedding Updates55.1674.0220.3244.0255.03
53.7970.7718.6044.2755.02
FullModel55.3974.8820.4544.6555.07
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A selected entity is defined as one whose attention weight", + "type": "text" + }, + { + "bbox": [ + 379, + 297, + 389, + 306 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 295, + 453, + 308 + ], + "score": 1.0, + "content": "is greater than", + "type": "text" + }, + { + "bbox": [ + 454, + 295, + 474, + 306 + ], + "score": 0.83, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 295, + 501, + 307 + ], + "score": 0.28, + "content": "( \\ S 2 . 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 295, + 505, + 308 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 305, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 320 + ], + "score": 1.0, + "content": "Because entities may be harder to predict when they have been combined with other entities (e.g.,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "the mixture may have a new name), we also report the recall for selecting combined (CR) and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "uncombined (UR) entities. In the end state prediction test, we report how often the model correctly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "predicts the state change performed in a recipe step and the resultant end state. This score is then", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "scaled by the accuracy of predicting which entities were changed in that same step. We report the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 361, + 341, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 341, + 374 + ], + "score": 1.0, + "content": "average F1 and accuracy across the six state change types.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 283, + 506, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "score": 1.0, + "content": "Baselines We compare our models against two baselines. First, we built a GRU model that is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 399 + ], + "score": 1.0, + "content": "trained to predict entities and state changes independently. This can be viewed as a bare minimum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "network with no action representations or recurrent entity memory. The second baseline is a Re-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "current Entity Network (Henaff et al., 2016) with changes to fit our task. First, the model can tie", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "memory cells to a subset of the full list of entities so that it only considers entities that are present in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "a particular recipe. Second, the entity distribution for writing to the memory cells is re-used when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "we query the memory cells. The normalized weighted average of the entity cells is used as the in-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "put to the state predictors. The unnormalized attention when writing to each cell is used to predict", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 464, + 487, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 487, + 477 + ], + "score": 1.0, + "content": "selected entities. Both baselines are trained with entity selection and state change losses (§3.3).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 377, + 506, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "score": 1.0, + "content": "Ablations We report results on six ablations. First, we remove the recurrent attention (Eq. 3). The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "model only predicts entities using the current encoder hidden state. In the second ablation, the model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "is trained with no coverage penalty (Eq. 9). The third ablation prunes the connection from the action", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 513, + 504, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 140, + 525 + ], + "score": 1.0, + "content": "selector", + "type": "text" + }, + { + "bbox": [ + 140, + 514, + 153, + 525 + ], + "score": 0.89, + "content": "w _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 513, + 504, + 525 + ], + "score": 1.0, + "content": "to the entity selector (Eq. 2). We also explore not pretraining the action selector. Finally,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "we look at two ablations where we intialize the action embeddings with vectors from a skipgram", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "model. In the first, the model operates normally, and in the second, we do not allow gradients to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 545, + 488, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 408, + 558 + ], + "score": 1.0, + "content": "backpropagate to the action embeddings, updating only the mapping tensor", + "type": "text" + }, + { + "bbox": [ + 408, + 546, + 424, + 556 + ], + "score": 0.89, + "content": "W _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 545, + 488, + 558 + ], + "score": 1.0, + "content": "instead (Eq. 6).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 480, + 506, + 558 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 302, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 303, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 303, + 583 + ], + "score": 1.0, + "content": "4.2 EXTRINSIC EVALUATION - GENERATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 591, + 504, + 613 + ], + "lines": [ + { + "bbox": [ + 107, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 107, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "The generation task tests whether our system can produce the next step in a recipe based on the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "previous steps that have been performed. The model is provided all of the previous steps as context.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 592, + 506, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 504, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "Metrics We report the combined BLEU score and ROUGE score of the generated sequence rela-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "tive to the reference sequence. Each candidate sequence has one reference sentence. Both metrics", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "are computed at the corpus-level. Also reported are “VF1”, the F1 score for the overlap of the ac-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 662, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 673 + ], + "score": 1.0, + "content": "tions performed in the reference sequence and the verbs mentioned in the generated sequence, and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 670, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 686 + ], + "score": 1.0, + "content": "“SF1”, the F1 score for the overlap of end states annotated in the reference sequence and predicted", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 684, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 696 + ], + "score": 1.0, + "content": "by the generated sequences. End states for the generated sequences are extracted using the lexicon", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 694, + 366, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 366, + 706 + ], + "score": 1.0, + "content": "from Section 3.1 based on the actions performed in the sentence.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 629, + 506, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Setup To apply our model to the task of recipe step generation, we input the context sentences", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "through the neural process network and record the entity state vectors once the entire context has", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "been read (§2.5). 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GoodSt-1StselectedcorrectAdd tomato paste,broth, garlic,chili powder,cumin,chile peppers,and water.Bring to boil, then turn very low,cover and simmer until meat is tender.meat,garlic, chili powder, tomato paste,cumin,chiles,beef broth,waterSame + [oil]
GoodSt-4St-3St-2St-1StselectedcorrectStir in oats, sugar, flour, corn syrup,milk, vanilla extract, and salt.Mix well.Drop by measuring teaspoonfuls onto cookie sheets.Bake 5-7 minutes.Let cool.oats,sugar, flour, corn syrup,milk,vanilla extract, saltoats,sugar, flour, corn syrup,milk,vanilla extract, salt
GoodSt-1StselectedcorrectIn a large saucepan over low heat, melt marshmallows.Add sprinkles,cereal,and raisins,stir until well coated.marshmallows,cereal,raisinsmarshmallows,cereal, raisins,sprinkles
BadSt-3St-2St-1StselectedcorrectLadle the barbecue sauce around the crust and spread.Add mozzarella, yellow cheddar,and monterey jack cheese.Next,add onion mixture and sliced chicken breast .Top pizza with jalapeno peppers.jalapenoscrust,sauce,mozzarella,cheddar,monterey jack,white onion,chicken, jalapenos
BadSt-2St-1StselectedcorrectCombine 1 cup flour, salt,and 1 tbsp sugar.Cut in butter until mixture is crumbly, then sprinkle with vinegar .Gather dough into a ball and press into bottom of 9 inch springform pan.butter, vinegarflour, salt,sugar, butter, vinegar
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The previous sentences are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 376, + 318, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 318, + 389 + ], + "score": 1.0, + "content": "provided as context in cases where they are relevant.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "been read (§2.5). These vectors can be viewed as a snapshot of the current state of the entities", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "once the preceding context has been simulated inside the neural process network. 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GoodSt-1StselectedcorrectAdd tomato paste,broth, garlic,chili powder,cumin,chile peppers,and water.Bring to boil, then turn very low,cover and simmer until meat is tender.meat,garlic, chili powder, tomato paste,cumin,chiles,beef broth,waterSame + [oil]
GoodSt-4St-3St-2St-1StselectedcorrectStir in oats, sugar, flour, corn syrup,milk, vanilla extract, and salt.Mix well.Drop by measuring teaspoonfuls onto cookie sheets.Bake 5-7 minutes.Let cool.oats,sugar, flour, corn syrup,milk,vanilla extract, saltoats,sugar, flour, corn syrup,milk,vanilla extract, salt
GoodSt-1StselectedcorrectIn a large saucepan over low heat, melt marshmallows.Add sprinkles,cereal,and raisins,stir until well coated.marshmallows,cereal,raisinsmarshmallows,cereal, raisins,sprinkles
BadSt-3St-2St-1StselectedcorrectLadle the barbecue sauce around the crust and spread.Add mozzarella, yellow cheddar,and monterey jack cheese.Next,add onion mixture and sliced chicken breast .Top pizza with jalapeno peppers.jalapenoscrust,sauce,mozzarella,cheddar,monterey jack,white onion,chicken, jalapenos
BadSt-2St-1StselectedcorrectCombine 1 cup flour, salt,and 1 tbsp sugar.Cut in butter until mixture is crumbly, then sprinkle with vinegar .Gather dough into a ball and press into bottom of 9 inch springform pan.butter, vinegarflour, salt,sugar, butter, vinegar
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ActionNearestNeighbor Actions
cut boilslice,split, snap,slash,carve,slit,chop cook,microwave,fry,steam,simmer
addsprinkle,mix,reduce,splash,stir,dust
washrinse,scrub,refresh,soak,wipe,scale
mashspread, puree,squeeze,liquefy, blend
place rinseease,put,lace,arrange,leave
wash,refresh,soak,wipe,scrub,clean
warmreheat, ignite,heat,light,crisp,preheat
steam
microwave,crisp,boil, parboil,heat
sprinkletop,pat,add,dip,salt, season
greasecoat,rub,dribble,spray,smear, line
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ModelBLEUBLEUROUGE-LRVF1
Vanilla Seq2SeqAttentive Seq2SeqEntNet Generator2.812.832.3033.0033.1832.7116.1716.9717.5340.2141.4342.43
NPN Generator3.7435.6420.1243.40
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We also", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "provide examples where the model fails to select the correct entities because it does not identify the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 380, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 390 + ], + "score": 1.0, + "content": "mapping between a reference construct such as “pizza” (Ex. 4) or “dough” (Ex. 5) and the set of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 389, + 493, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 493, + 402 + ], + "score": 1.0, + "content": "entities that composes it, showcasing the difficulty of selecting the full set for a composed entity.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "State Change Tracking In Table 8, we show that our full model outperforms competitive base-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "lines such as Recurrent Entity Networks (Henaff et al., 2016) and jointly trained GRUs. 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After training, we compute", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "the pairwise cosine similarity between each pair of action embeddings. In Table 4, we see that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "actions that perform similar functions are neighbors in embedding space, indicating the model has", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "captured certain semantic properties of these actions. Learning action representations through the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "state changes they induce has allowed the model to cluster actions by their transformation functions.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "Entity Compositions When individual entities are combined into new constructs, our model av-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "erages their state embeddings (Eq. 5), applies an action embedding to them (Eq. 6), and writes them", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "to memory (Eq. 7). 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ActionNearestNeighbor Actions
cut boilslice,split, snap,slash,carve,slit,chop cook,microwave,fry,steam,simmer
addsprinkle,mix,reduce,splash,stir,dust
washrinse,scrub,refresh,soak,wipe,scale
mashspread, puree,squeeze,liquefy, blend
place rinseease,put,lace,arrange,leave
wash,refresh,soak,wipe,scrub,clean
warmreheat, ignite,heat,light,crisp,preheat
steam
microwave,crisp,boil, parboil,heat
sprinkletop,pat,add,dip,salt, season
greasecoat,rub,dribble,spray,smear, line
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ModelBLEUBLEUROUGE-LRVF1
Vanilla Seq2SeqAttentive Seq2SeqEntNet Generator2.812.832.3033.0033.1832.7116.1716.9717.5340.2141.4342.43
NPN Generator3.7435.6420.1243.40
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While the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "ablation without the coverage loss shows higher accuracy, we attribute this to the fact that it predicts", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "a smaller number of total state changes. Interestingly, initializing action embeddings with skipgram", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "vectors and locking their values shows relatively high performance, indicating the potential gains in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 459, + 357, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 357, + 472 + ], + "score": 1.0, + "content": "using powerful pretrained representations to represent actions.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 405, + 505, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "Action Embeddings In our model, each action is assigned its own embedding, but many actions", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "induce similar changes in the physical world (e.g.,“cut” and “slice”). After training, we compute", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "the pairwise cosine similarity between each pair of action embeddings. In Table 4, we see that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "actions that perform similar functions are neighbors in embedding space, indicating the model has", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "captured certain semantic properties of these actions. Learning action representations through the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "state changes they induce has allowed the model to cluster actions by their transformation functions.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 474, + 506, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "Entity Compositions When individual entities are combined into new constructs, our model av-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "erages their state embeddings (Eq. 5), applies an action embedding to them (Eq. 6), and writes them", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "to memory (Eq. 7). The state embeddings of entities that are combined should be overwritten by", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "the same new embedding. In Figure 3, we present the percentage increase in cosine similarity for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "state embeddings of entities that are combined in a sentence (blue) as opposed to the percentage", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "increase for those that are not (red bars). While the soft attention mechanism for entity selection", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "allows similarities to leak between entity embeddings, our system is generally able to model the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 621, + 454, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 454, + 632 + ], + "score": 1.0, + "content": "compositionality patterns that result from entities being combined into new constructs.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 543, + 505, + 632 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 645, + 243, + 656 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 243, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 243, + 658 + ], + "score": 1.0, + "content": "5.2 EXTRINSIC EVALUATIONS", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Recipe Step Generation Our results in Table 5 indicate that sequences generated using the neural", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "process network entity states as additional input yield higher scores than competitive baselines. The", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "entity states allow the model to predict next steps conditioned on a representation of the world being", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "simulated by the neural process network. Additionally, the higher VF1 and SF1 scores indicate that", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "the model is indeed using the extra information to better predict the actions that should follow the", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "context provided. Example generations for each baselines from the dev set are provided in Table 6,", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "showing that the NPN generator can use information about ingredient states to reason about the", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "most likely next step. The first and second examples are interesting as it shows that the NPN-aware", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "model has learned to condition on entity state – knowing that raw butter will likely be melted or that a", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "cooked flan must be refrigerated. 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ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPreheat oven to 425 degrees.Melt butter in saucepan and mix in bourbon, thyme,pepper, and salt.Melt butter in skillet.Lightly grease 4 x 8 baking pan with sunflower oil.Combine all ingredients and mix well.In a large bowl, combine flour,baking powder,baking soda, salt,and pepper.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPour egg mixture over caramelized sugar in cake pan. Place cake pan in large shallowbaking dish.Bake for 55 minutes or until knife inserted into flan comes out clean.Cover and chill at least 8 hours.Refrigerate until ready to use.Serve at room temperature.Store in an airtight container.Store in an airtight container.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetCut squash into large pieces and steam. Remove cooked squash from shells;Measure 4 cups pulp and reserve remainder for another dish.Drain.Mash pulp with a fork.Set aside.Set aside.
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The context and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 302, + 498, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 498, + 315 + ], + "score": 1.0, + "content": "reference are provided first, followed by our model’s generation and then the baseline generations", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "showing that the NPN generator can use information about ingredient states to reason about the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "most likely next step. The first and second examples are interesting as it shows that the NPN-aware", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "model has learned to condition on entity state – knowing that raw butter will likely be melted or that a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "cooked flan must be refrigerated. The third example is also interesting because the model learns that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "cooked vegetables such as squash will sometimes be drained, even if it is not relevant to this recipe", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "because the squash is steamed. The seq2seq and EntNet baselines, meanwhile, output reasonable", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 406, + 471, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 471, + 418 + ], + "score": 1.0, + "content": "sentences given the immediate context, but do not exhibit understanding of global patterns.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 435, + 211, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 213, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 213, + 450 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "Recent studies in machine comprehension have used a neural memory component to store a running", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "representation of processed text (Weston et al., 2014; Sukhbaatar et al., 2015; Hill et al., 2015; Seo", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "et al., 2016). While these approaches map text to memory vectors using standard neural encoder", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "approaches, our model, in contrast, directly interprets text in terms of the effects actions induce in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "entities, providing an inductive bias for learning how to represent stored memories. 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ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPreheat oven to 425 degrees.Melt butter in saucepan and mix in bourbon, thyme,pepper, and salt.Melt butter in skillet.Lightly grease 4 x 8 baking pan with sunflower oil.Combine all ingredients and mix well.In a large bowl, combine flour,baking powder,baking soda, salt,and pepper.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPour egg mixture over caramelized sugar in cake pan. Place cake pan in large shallowbaking dish.Bake for 55 minutes or until knife inserted into flan comes out clean.Cover and chill at least 8 hours.Refrigerate until ready to use.Serve at room temperature.Store in an airtight container.Store in an airtight container.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetCut squash into large pieces and steam. Remove cooked squash from shells;Measure 4 cups pulp and reserve remainder for another dish.Drain.Mash pulp with a fork.Set aside.Set aside.
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Temperature Composition Cleanliness Cookedness Shape Locationhot; cold; room composed; not composed clean; dirty; dry cooked; raw molded; hit; deformed; separated pan,pot, cupboard, screen, scale,
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We then use the answers to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "manually aggregate a set of end states for each state change type. These end states are used as labels", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "when the model predicting state changes. 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State Change TypeEnd States
Temperature Composition Cleanliness Cookedness Shape Locationhot; cold; room composed; not composed clean; dirty; dry cooked; raw molded; hit; deformed; separated pan,pot, cupboard, screen, scale,
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ModelEntity Selection State Change
F1URCRF1Acc
25% training data kept54.3475.7121.172.5250.23
50% training data kept55.1276.0419.0536.3454.66
75% training data kept56.6476.0321.0048.8657.62
100% training data56.8474.9821.1450.5657.87
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ModelEntity Selection State Change
F1URCRF1Acc
25% training data kept54.3475.7121.172.5250.23
50% training data kept55.1276.0419.0536.3454.66
75% training data kept56.6476.0321.0048.8657.62
100% training data56.8474.9821.1450.5657.87
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ModelEntity SelectionState Change
F1URCRF1Acc
2-layer LSTM Entity Recognizer50.9874.0313.3311
Adapted Gated Recurrent Unit Adapted Recurrent Entity Network - Recurrent Attention (Eq. 3)45.9467.697.7441.1652.69
48.5771.889.8742.3253.47
48.9172.3212.6742.1450.48
- Coverage Loss (Eq. 9) - Action Connections (Eq. 2)55.1873.9820.2344.4455.20
54.8573.5420.0344.0554.81
54.9173.8720.3044.2855.00
- Action Selector Pretraining + Pretrained Action Embeddings - Action Embedding Updates55.1674.0220.3244.0255.03
53.7970.7718.6044.2755.02
FullModel55.3974.8820.4544.6555.07
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GoodSt-1StselectedcorrectAdd tomato paste,broth, garlic,chili powder,cumin,chile peppers,and water.Bring to boil, then turn very low,cover and simmer until meat is tender.meat,garlic, chili powder, tomato paste,cumin,chiles,beef broth,waterSame + [oil]
GoodSt-4St-3St-2St-1StselectedcorrectStir in oats, sugar, flour, corn syrup,milk, vanilla extract, and salt.Mix well.Drop by measuring teaspoonfuls onto cookie sheets.Bake 5-7 minutes.Let cool.oats,sugar, flour, corn syrup,milk,vanilla extract, saltoats,sugar, flour, corn syrup,milk,vanilla extract, salt
GoodSt-1StselectedcorrectIn a large saucepan over low heat, melt marshmallows.Add sprinkles,cereal,and raisins,stir until well coated.marshmallows,cereal,raisinsmarshmallows,cereal, raisins,sprinkles
BadSt-3St-2St-1StselectedcorrectLadle the barbecue sauce around the crust and spread.Add mozzarella, yellow cheddar,and monterey jack cheese.Next,add onion mixture and sliced chicken breast .Top pizza with jalapeno peppers.jalapenoscrust,sauce,mozzarella,cheddar,monterey jack,white onion,chicken, jalapenos
BadSt-2St-1StselectedcorrectCombine 1 cup flour, salt,and 1 tbsp sugar.Cut in butter until mixture is crumbly, then sprinkle with vinegar .Gather dough into a ball and press into bottom of 9 inch springform pan.butter, vinegarflour, salt,sugar, butter, vinegar
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ModelBLEUBLEUROUGE-LRVF1
Vanilla Seq2SeqAttentive Seq2SeqEntNet Generator2.812.832.3033.0033.1832.7116.1716.9717.5340.2141.4342.43
NPN Generator3.7435.6420.1243.40
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ActionNearestNeighbor Actions
cut boilslice,split, snap,slash,carve,slit,chop cook,microwave,fry,steam,simmer
addsprinkle,mix,reduce,splash,stir,dust
washrinse,scrub,refresh,soak,wipe,scale
mashspread, puree,squeeze,liquefy, blend
place rinseease,put,lace,arrange,leave
wash,refresh,soak,wipe,scrub,clean
warmreheat, ignite,heat,light,crisp,preheat
steam
microwave,crisp,boil, parboil,heat
sprinkletop,pat,add,dip,salt, season
greasecoat,rub,dribble,spray,smear, line
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ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPreheat oven to 425 degrees.Melt butter in saucepan and mix in bourbon, thyme,pepper, and salt.Melt butter in skillet.Lightly grease 4 x 8 baking pan with sunflower oil.Combine all ingredients and mix well.In a large bowl, combine flour,baking powder,baking soda, salt,and pepper.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetPour egg mixture over caramelized sugar in cake pan. Place cake pan in large shallowbaking dish.Bake for 55 minutes or until knife inserted into flan comes out clean.Cover and chill at least 8 hours.Refrigerate until ready to use.Serve at room temperature.Store in an airtight container.Store in an airtight container.
ContextReferenceNPNSeq2seqAttentive Seq2seqEntNetCut squash into large pieces and steam. Remove cooked squash from shells;Measure 4 cups pulp and reserve remainder for another dish.Drain.Mash pulp with a fork.Set aside.Set aside.
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ModelEntity Selection State Change
F1URCRF1Acc
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50% training data kept55.1276.0419.0536.3454.66
75% training data kept56.6476.0321.0048.8657.62
100% training data56.8474.9821.1450.5657.87
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Both in graph kernels and graph neural networks, one of the implicit assumptions of successful state-of-the-art models was that incorporating graph isomorphism features into the architecture leads to better empirical performance. However, as we discover in this work, commonly used data sets for graph classification have repeating instances which cause the problem of isomorphism bias, i.e. artificially increasing the accuracy of the models by memorizing target information from the training set. The problem does not vanish even in consideration of node labels. This prevents fair competition of the algorithms and raises a question of the validity of the obtained results. First, we characterize this effect theoretically by reducing the graph classification to a weighted classification problem and estimating the corresponding generalization gap. Then we analyze 54 data sets, previously extensively used for graph-related tasks, on the existence of isomorphism bias, give a set of recommendations to machine learning practitioners to properly set up their models, and open source new data sets for the future experiments. + +# 1 INTRODUCTION + +Recently there has been an increasing interest in the development of machine learning models that operate on graph structured data. Such models have found applications in chemoinformatics (Ralaivola et al. (2005); Rupp & Schneider (2010); Ferre et al. (2017)) and bioinformatics (Borg- ´ wardt et al. (2005); Kundu et al. (2013)), neuroscience (Sharaev et al. (2018); Jie et al. (2016); Wang et al. (2016)), computer vision (Stumm et al. (2016)) and system security (Li et al. (2016)), natural language processing (Glavas &ˇ Snajder (2013)), and others (Kriege et al. (2019); Nikolentzos ˇ et al. (2019)). One of the popular tasks that encompasses these applications is graph classification problem for which many graph kernels and graph neural networks have been developed. + +One of the implicit assumptions that many practitioners adhere to is that models that can distinguish isomorphic instances from non-isomorphic ones possess higher expressiveness in classification problem and hence much efforts have been devoted to incorporate efficient graph isomorphism methods into the classification models. As the problem of computing complete graph invariant is $\mathbb { G } \mathbb { I }$ -hard (Gartner et al. (2003)), for which no known polynomial-time algorithm exists, other heuris- ¨ tics have been proposed as a proxy for deciding whether two graphs are isomorphic. Indeed, from the early days topological descriptors such Wiener index (Wiener (1947a;b)) attempted to find a single number that uniquely identifies a graph. Later, graph kernels that model pairwise similarities between graphs utilized theoretical developments in graph isomorphism literature. For example, graphlet kernel (Shervashidze et al. (2009)) is based on the Kelly conjecture (see also Kelly (1957)), anonymous walk kernel (Ivanov & Burnaev (2018)) derives insights from the reconstruction properties of anonymous experiments (see also Micali & Allen Zhu (2016)), and WL kernel (Shervashidze et al. (2011a)) is based on an efficient graph isomorphism algorithm. For sufficiently large $k$ , $k$ - dimensional WL algorithm includes all combinatorial properties of a graph (Cai et al. (1992a)), so one may hope its power is enough for the data set at hand. Since only for $k = \Omega ( n )$ WL algorithm is guaranteed to distinguish all graphs (for which the running time becomes exponential; see also Furer (2017)), in the general case WL algorithm can be used only as a strong baseline for graph ¨ isomorphism. In similar fashion, graph neural networks exploit graph isomorphism algorithms and + +have been shown to be as powerful as $\mathbf { k }$ -dimensional WL algorithm (see for example Maron et al. +(2019); Xu et al. (2018); Morris et al. (2019)). + +Experimental evaluation reveals that models based on the theoretical constructions with high combinatorial power such as WL algorithm performs better than the models without them such as Vertex histogram kernel (Vishwanathan et al. (2010)) on a commonly used data sets. This could add additional bias to results of comparison of classification algorithms since the models could simply apply a graph isomorphism method (or an efficient approximation) to determine a target label at the inference time. However, purely judging on the accuracy of the algorithms in such cases would imply an unfair comparison between the methods as it does not measure correctly generalization ability of the models on the new test instances. As we discover, indeed many of the data sets used in graph classification have isomorphic instances so much that in some of them the fraction of the unique non-repeating graphs is as low as $20 \%$ of the total size. This challenges previous experimental results and requires understanding of how influential isomorphic instances on the final performance of the models. Our contributions are: + +• We analyze the quality of 54 graph data sets which are used ubiquitously in graph classification comparison. Our findings suggest that in the most of the data sets there are isomorphic graphs and their proportion varies from as much as $100 \%$ to $0 \%$ . Surprisingly, we also found that there are isomorphic instances that have different target labels suggesting they are not suitable for learning a classifier at all. We investigate the causes of isomorphic graphs and show that node and edge labels are important to identify isomorphic graphs. Other causes include numerical attributes of nodes and edges as well as the sizes of the data set. We express an upper bound for the generalization gap through the Radamacher complexity of a classifier and the number of isomorphic graphs in a data set. This bound presents theoretical evidence on how weightning of each graph in the training influences classification accuracy. +We evaluate a classification model’s performance on isomorphic instances and show that even strong models do not achieve optimal accuracy even if the instances have been seen at the training time. Hence we show a model-agnostic way to artificially increase performance on several widely used data sets. We open-source new cleaned data sets that contain only non-isomorphic instances with no noisy target labels. We give a set of recommendations regarding applying new models that work with graph structured data. + +# 2 RELATED WORK + +Measuring quality of data sets. A similar issue of duplicates instances in commonly used data sets was recently discovered in computer vision domain. Recht et al. (2019); Barz & Denzler (2019); Birodkar et al. (2019) discover that image data sets CIFAR and ImageNet contain at least $10 \%$ of the duplicate images in the test test invalidating previous performance and questioning generalization abilities of previous successful architectures. In particular, evaluating the models in new test sets shows a drop of accuracy by as much as $15 \%$ (Recht et al., 2019), which is explained by models’ incapability to generalize to unseen slightly ”harder” instances than in the original test sets. In graph domain, a fresh look into understanding of expressiveness of graph kernels and the quality of data sets has been considered in Kriege et al. (2019), where an extensive comparison of existing graph kernels is done and a few insights about models’ behavior are suggested. In contrast, we conduct a broader study of isomorphism metrics, revealing all isomorphism pairs in proposed 54 data sets, and propose new cleaned data. Additionally we also consider graph neural network performance and argue that current data sets present isomorphism bias which can artificially boost evaluation metrics in a model-agnostic way. + +Explaining performance of graph models. Graph kernels (Kriege et al. (2019)) and graph neural networks (Wu et al. (2019)) are two competing paradigms for designing graph representations and solving graph classification and have significantly advanced empirical results due to more efficient algorithms, incorporating graph invariance into the models, and end-to-end training. Several papers have tried to justify performance of different families of methods by studying different statistical properties. For example, in Ying et al. (2019) by maximizing mutual information between explanation variables and predicted label distribution, the model is trained to return a small subgraph and the graph-specific attributes that are the most influential on the decision made by a GNN, which allows inspection of single- and multi-level predictions in an agnostic manner for GNNs. In another work (Scarselli et al. (2018)), the VC dimension of GNNs models has been shown to grow as $\mathcal { O } ( p ^ { 4 } N ^ { 2 } )$ , where $p$ is the number of network parameters and $N$ is the number of nodes in a graph, which is comparable to RNN models. Furthermore, stability and generalization properties of convolutional GNNs have been shown to depend on the largest eigenvalue of the graph filter and therefore are attained for properly normalized graph convolutions such as symmetric normalized graph Laplacian (Verma & Zhang (2019)). Finally, expressivity of graph kernels has been studied from statistical learning theory (Oneto et al. (2017b)) and property testing (Kriege et al. (2018b)), showing that graph kernels can capture certain graph properties such as planarity, girth, and chromatic number (Johansson et al. (2014)). Our approach is complementary to all of the above as we analyze if the data sets used in experiments have any effect on the final performance. + +# 3 PRELIMINARIES + +In this work we analyze 54 graph data sets from Kersting et al. (2016) that are commonly used in graph classification task. Examples of popular graph data sets are presented in Table 1 and statistics of all 54 data sets can be found in Table 5, see Section A in the appendix. All data sets represent a collection of graphs and accompanying categorical label for each graph in the data sets. Some data sets also include node and/or edge labels that graph classification methods can use to improve the scoring. Most of the data sets come either from biological domain or from social network domain. Biological data sets such as MUTAG, ENZYMES, PROTEINS are graphs that represent small or large molecules, where edges of the graphs are chemical bonds or spatial proximity between different atoms. Graph labels in these cases encode different properties like toxicity. In social data sets such as IMDB-BINARY, REDDIT-MULTI-5K, COLLAB the nodes represent people and edges are relationships in movies, discussion threads, or citation network respectively. Labels in these cases denote the type of interaction like the genre of the movie/thread or a research subfield. For completeness we also include synthetic data sets SYNTHETIC (Morris et al. (2016)) that have continuous attributes and computer vision data sets MSRC (Neumann et al. (2016)), where images are encoded as graphs. The origin of all data sets can be found in the Table 5. + +Table 1: Example of graph data sets. $N$ is the number of graphs, $C$ is the number of different classes. Avg. Nodes and Avg. Edges is the average number of nodes and edges. N.L. and E.L indicate if the graphs in a data set contain node or edge labels. + +
data setTypeNCAvg. NodesAvg. EdgesN.L.E.L.
MUTAGMolecular188217.9319.79++
ENZYMESMolecular600632.6362.14+二
PROTEINSMolecular1113239.0672.82+-
IMDB-BINARYSocial1000219.7796.53二二
REDDIT-MULTI-5KSocial49995508.52594.87二二
COLLABSocial5000374.492457.78--
SYNTHETICSynthetic3002100196二二
SynthieSynthetic400495172.93-二
MSRC_21CVision2092040.2896.6+二
MSRC_9Vision221840.5897.94+1
+ +Graph isomorphism. Isomorphism between two graphs $G _ { 1 } = ( V _ { 1 } , E _ { 1 } )$ and $G _ { 2 } = ( V _ { 2 } , E _ { 2 } )$ is a bijective function $\phi : V _ { 1 } \mapsto V _ { 2 }$ such that any edge $( u , v ) \in E _ { 1 }$ if and only if $( \phi ( u ) , \phi ( v ) ) \in E _ { 2 }$ . Graph isomorphism problem asks if such function exists for given two graphs $G _ { 1 }$ and $G _ { 2 }$ . We denote isomorphic graphs as $G _ { 1 } \cong G _ { 2 }$ . The problem has efficient algorithms in $\mathbb { P }$ for certain classes of graphs such as planar or bounded-degree graphs (Hopcroft $\&$ Wong (1974); Luks (1980)), but in the general case admits only quasi-polynomial algorithm (Babai (2015)). In practice many GI solvers are based on individualization-refinement paradigm (Mckay & Piperno (2014)), which for each graph iteratively updates a permutation of the nodes such that the resulted permutations of two graphs are identical if an only if they are isomorphic. Importantly, while finding such canonical permutation of a graph is at least as hard as solving GI problem, state-of-the-art solvers tackle majority of pairs of graphs efficiently, only taking exponential time on the specific hard instances of graphs that possess highly symmetrical structures (Cai et al. (1992b)). + +# 4 IDENTIFYING ISOMORPHISM IN DATA SETS + +To distinguish between different isomorphic graphs inside a data set we use the notion of graph orbits: + +Definition 4.1 (Graph orbit). Let $\mathcal { D } = \{ G _ { i } , y _ { i } \} _ { i = 1 } ^ { N }$ be a data set of graphs and target labels. For a graph $G _ { i }$ let a set $o _ { i } = \{ G _ { k } \}$ be a set of all isomorphic graphs in $\mathcal { D }$ to $G _ { i }$ , including $G _ { i }$ . We call $o _ { i }$ the orbit of graph $G _ { i }$ in $\mathcal { D }$ . The cardinality of the orbit is called orbit size. An orbit with size one is called trivial. + +In a data set with no isomorphic graphs, the number of orbits equals to the number of graphs in a data set, $N$ . Hence, the more orbits in a data set, the ”cleaner” it is. Note however that the distribution of orbit sizes in two different data sets can vary even if they have the same number of orbits. Therefore, we look at additional metrics that describe the data set. + +• $I$ , aggregated number of graphs that belong to an orbit of size greater than one, i.e. those graphs that isomorphic counterparts in a data set; +• $I , \%$ , proportion of isomorphic graphs to the total data set size, i.e. $\textstyle { \frac { I } { N } }$ ; +• $I P , \%$ , proportion of isomorphic pairs to the total number of graph pairs in a data set ( $\frac { N ( N - 1 ) } { 2 } )$ . + +If we consider target labels of graphs in a data set $\mathcal { D } = \{ G _ { i } , y _ { i } \} _ { i = 1 } ^ { N }$ we can also measure agreement between the labels of two isomorphic data set. If $G _ { 1 } \cong G _ { 2 }$ and $y _ { 1 } ~ \ne ~ y _ { 2 }$ , then we call graphs mismatched. Note that if there is more than one target label in an orbit $o$ , then all graphs in this orbit are mismatched. To obtain isomorphic graphs, we run nauty algorithm (Mckay & Piperno, 2014) on all possible pairs of graphs in a data set. We substantially reduce the number of calls between the graphs by verifying that a pair has the same number of nodes and edges before the call. + +The metrics are presented in Table 2 for top-10 data sets and in Table 6 (see the appendix) for all data sets. The graphs in Table 2 are sorted by the proportion of isomorphic graphs $I \%$ . The results for the first Top-10 data sets are somewhat surprising: almost all graphs in the selected data sets have other isomorphic graphs. If we look at all data sets in Table 6, we see that the proportion of isomorphic graphs in the data sets varies from $100 \%$ to $0 \%$ . However, more than $80 \%$ of the analyzed data sets have at least $10 \%$ of the graphs in a non-trivial orbit. + +Table 2: Isomorphic metrics for Top-10 data sets based on the proportion of isomorphic graphs $I \%$ $I P \%$ is the proportion of isomorphic pairs of graphs, Mismatched $\%$ is the proportion of mismatched labels. + +
data setSize, NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
+ +Another surprising observation is that the proportion of mismatched graphs is significant, ranging from $100 \%$ to $0 \%$ . This clearly indicates that such graphs are not suitable for graph classification and require additional information to distinguish the models. We analyze the reasons for this in the next section. + +Also, the distribution of orbit sizes can vary significantly across the data sets. In Figure 1 we plot a distribution of orbit sizes for several examples of data sets (and distributions for other data sets can be found in Appendix C). For example, for IMDB-BINARY data set the number of orbits of small sizes, e.g. two or three, goes to 100, which indicate prevalence of pairs of isomorphic graphs that are non-isomorphic to the rest. However, for Letter-med data set there are many orbits of sizes more than 100, while small orbits are not that common. In this case, the graphs in this data set are equivalent to a lot of other graphs, which may have a substantial effect on the corresponding metrics. While the orbit distribution changes from one data set to another, it is clear that in many situations there are isomorphic graphs that can affect training procedure by effectively increasing the weights for the corresponding graphs, change performance on the test by validating on the already seen instances, and by confusing the model by utilizing different target labels for topologically-equivalent graphs. We analyze the reasons for it further. + +![](images/f6a14c15fa6baf9143bba7daa52c62de70ad1b25b506e43721c025cb6a81eba4.jpg) +Figure 1: Examples of distributions of orbit sizes without considering labels. + +# 5 EXPLAINING ISOMORPHISM + +Meta-information about graphs. In addition to the topology of a graph, many data sets also include meta information about nodes and/or edges. Out of 54 analyzed data sets there are 40 that additionally include node features and 25 that include edge features. For example, in Synthetic data set all graphs are topologically identical but the nodes are endowed with normally distributed scalar attributes and in DHFR MD edges are attributed with distances and labeled according to a chemical bond type. Alternatively, some graphs can have parallel edges which is equivalent to have a corresponding weight on the edges. Thus some data sets include node/edge categorical features (labels) and numerical features (attributes), which leads to better distinction between the graphs and therefore their corresponding labels. + +To see this, we rerun our previous analysis but now include the node labels, if any, when computing isomorphism between graphs. Consider a tuple $( G , l )$ , where $G$ is a graph and $l : V ( G ) \mapsto$ $\{ 1 , 2 , \ldots , k \}$ is a $k$ -labeling of $G$ . In this case of node label-preserving graph isomorphism from graph $( G _ { 1 } , l _ { 1 } )$ to graph $\left( { { G } _ { 2 } } , { { l } _ { 2 } } \right)$ we seek an isomorphism function $\phi : { \bar { V } } ( { \bar { G } } _ { 1 } { \bar { ) } } \mapsto V ( { \bar { G _ { 2 } } } )$ such that $l _ { 1 } ( v ) = l _ { 2 } ( \phi ( v ) )$ . + +Tables 3 and 7 (see the appendix) show the number of isomorphic graphs after considering node labels. While for the first six data sets the proportion of isomorphic graphs has not changed much, it is clearly the case for the remaining data sets. In particular, almost $90 \%$ of the analyzed data sets include less than $20 \%$ of isomorphic graphs. Also, the number of mismatched graphs significantly decreases after considering node labels. For example, for MUTAG data set the proportion of isomorphic graphs went down from $4 2 . 0 2 \%$ to $1 9 . 1 5 \%$ and the proportion of mismatched graphs from $6 . 9 1 \%$ to $0 \%$ . + +Table 3: Isomorphic metrics with node labels for Top-10 data sets based on the proportion of isomorphic graphs $\bar { I \% }$ . $I P \%$ is the proportion of isomorphic pairs of graphs, Mismatched $\%$ is the proportion of mismatched labels. + +
data setSize, NNum.orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
DHFR_MD3932539299.756.8794.91
COX2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
BZR_MD3062230399.027.1695.75
MUTAG188173619.150.140
PTC_FM349225415.470.0810.89
PTC_MM336225014.880.077.74
DHFR756399812.960.043.97
+ +Likewise, the orbit size distribution also changes significantly after considering node labels. Figure 2 shows a changed distribution of orbits with and without considering node labels. For majority of data sets large orbits vanish and the number of small orbits is substantially decreased in label-preserving graph isomorphism setting. This indicates one of the reasons for presence of many isomorphic graphs in the data sets, which implies that including node/edge labels/attributes can be important for graph classification models. + +![](images/d7e9dbdde3017047ec255b42935b6e9cc68b9f8d7a5f186905404b51e128b64f.jpg) +Figure 2: Examples of distributions of orbit sizes with node labels. + +Sizes of the data sets. Another reason for having isomorphism in a data set is the sizes of graphs, which could be too small on average to lead to a diversity in a data set. In general, the number of non-isomorphic graphs with $n$ vertices and $m$ edges can be computed using Polya enumeration theory and grows very fast. For example, for a graph with 15 nodes and 15 edges, there are 2,632,420 non-isomorphic graphs. Nevertheless, specifics of the origin of the data set may affect possible configurations that graphs have (e.g. structure of chemical compounds in COX2 MD or ego-networks for actors in IMDB-BINARY) and thus smaller graphs may tend to be close to isomorphic structures. On the other hand, all five data sets with the average number of nodes greater than 100 have very low or zero proportion of isomorphic graphs. Hence, the average size of the graphs directly impacts the possible structure of the data set and thus data sets with larger graphs tend to be more diverse. We next analyze the consequences of the isomorphic graphs on classification methods. + +# 6 ISOMORPHIC GRAPHS AND WEIGHTED CLASSIFICATION + +We denote by + +• $\mathcal { F } \subseteq Y ^ { X }$ a class of binary classifiers with an input space $X$ and an output space $Y =$ $\{ - 1 , + 1 \}$ , +• $\mathbb { P }$ a distribution on $X \times Y$ , +• $\pi$ a prior probability of a positive class, i.e. $\mathbb { P } = \pi \mathbb { P } _ { x | y = + 1 } + ( 1 - \pi ) \mathbb { P } _ { x | y = - 1 } ,$ , +• $\mathcal { D } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ a training sample, $x _ { i } \in X$ , $y _ { i } \in Y$ . In our case $x _ { i } \in \mathcal D$ are graphs $G _ { i }$ , see notations in section 4. + +We consider a zero-one loss function $l ( \hat { y } , y ) = \mathbb { I } _ { \hat { y } \neq y }$ . Let us assume that classifiers from $\mathcal { F }$ can detect which graphs in $\mathcal { D }$ are isomorphic. E.g. classifiers based on Weisfeiler-Lehman graph kernels (see Shervashidze et al. (2011a)) are capable to do it for majority of graphs. In such case the empirical risk $\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } l ( f ( x ) , y ) = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } l ( f ( x _ { j } ) , y _ { j } ) } \end{array}$ reduces to $\begin{array} { r } { \frac { 1 } { N } \sum _ { j \in J } u _ { j } l ( f ( x _ { j } ) , y _ { j } ) } \end{array}$ , where $J$ is an index set of non-isomorphic graphs, $u _ { j } ~ \geq ~ 1$ is equal to the number of graphs in the initial sample , isomorphic to the graph $x _ { j }$ (we count $x _ { j }$ as well). + +Thus under such assumptions the graph classification problem with the training data set, containing isomorphic graphs, can be interpreted as a classification problem with a weighted loss. Let us introduce general notations for this problem. We define some (fixed) measurable weighting function $u : ( X \times Y ) \to ( 0 , + \infty )$ . Then the theoretical risk is equal to $\mathbb { E } _ { \mathbb { P } } l ( f ( x ) , y )$ and the weighted empirical risk is equal to $\begin{array} { r } { \mathbb { E } _ { \mathcal { D } } u ( x , y ) l ( f ( x ) , y ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } u ( x _ { i } , y _ { i } ) l ( f ( x _ { i } ) , y _ { i } ) } \end{array}$ ). We would like to derive an upper bound for the excess risk $\begin{array} { r } { \operatorname* { s u p } _ { f \in \bar { \mathcal { F } } } \left( \mathbb { E } _ { \mathbb { P } } l ( f ( x ) , y ) - \mathbb { E } _ { \mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \right) } \end{array}$ , i.e. we would like to quantify an upper bound for a generalization gap. We optimize the weighted empirical risk when training a classifier and measure its accuracy using non-weighted theoretical risk. + +There are some results about classification performance with a weighted loss. E.g. in (Dupret & Koda, 2001) a bayesian framework for imbalanced classification with a weighted risk is proposed. Scott (2012) investigated the calibration of asymmetric surrogate losses. Natarajan et al. (2018) considered the case of cost-sensitive learning with noisy labels. However, to the best of our knowledge, there is no studied upper bound for the excess risk with explicit dependence on the class imbalance and the weighting scheme that quantifies the influence on the overall classification performance. We show this result next. + +To derive explicit expressions we use some additional modeling asumption, namely, we consider $u ( x , y ) = ( \bar { 1 } + g _ { + } ( w \bar { ) } ) \mathbb { I } _ { \{ y = + 1 \} } + ( 1 + g _ { - } ( w ) ) \mathbb { I } _ { \{ y = - 1 \} }$ for some non-negative weighting functions $g _ { + } ( w )$ and $g _ { - } ( w )$ of the weight value $w \geq 0$ . E.g. we can use $g _ { + } ( w ) = w$ and $g _ { - } ( w ) = 1 / w$ . + +Theorem 6.1. With probability $1 - \delta , \delta > 0$ for $\mathcal { D } \sim \mathbb { P } ^ { N }$ the excess risk is upper bounded by + +$$ +\begin{array} { r l r } { { \operatorname* { s u p } _ { f \in \mathcal { F } } \Big ( \mathbb { E } _ { \mathbb { P } } l ( f ( x ) , y ) - \mathbb { E } _ { \mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \Big ) \leq 3 ( g _ { + } ( w ) \pi + g _ { - } ( w ) ( 1 - \pi ) ) + } } \\ & { } & { + \mathcal { R } _ { N } ( \mathcal { F } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \sqrt { ( \log \delta ^ { - 1 } ) / ( 2 N ) } , } \end{array} +$$ + +where $\mathcal { R } _ { N } ( \mathcal { F } )$ is a Rademacher complexity of the function class $\mathcal { F }$ . + +Let us note that the Rademacher complexity of the function class $\mathcal { F }$ , defined by a graph kernel, was studied e.g. in Oneto et al. (2018); Oneto et al. (2017a). + +From equation 1 it follows that by tuning the weight parameter $w$ we can make the upper bound tighter, namely collecting the terms with $w$ in the RHS of equation 1 we solve + +$$ +g _ { + } ( w ) \left( 3 \pi + \sqrt { ( \log \delta ^ { - 1 } ) / ( 2 N ) } \right) + g _ { - } ( w ) \left( 3 ( 1 - \pi ) + \sqrt { ( \log \delta ^ { - 1 } ) / ( 2 N ) } \right) \to \operatorname* { m i n } _ { w } . +$$ + +In case we set $g _ { + } ( w ) = w$ and $g _ { - } ( w ) = 1 / w$ , the optimal weight $\begin{array} { r } { w _ { o p t } = \sqrt { \frac { 3 ( 1 - \pi ) + \alpha _ { N } } { 3 \pi + \alpha _ { N } } } \approx \sqrt { \frac { 1 - \pi } { \pi } } , } \end{array}$ , where αN = $\begin{array} { r } { \alpha _ { N } = \sqrt { \frac { \log \delta ^ { - 1 } } { 2 N } } \approx 0 } \end{array}$ for $N \gg 1$ . For such optimal $w _ { o p t }$ the RHS of equation 1 has the form + +$$ +6 \sqrt { \pi ( 1 - \pi ) } + \mathcal { R } _ { N } ( \mathcal { F } ) + \alpha _ { N } \left( 2 + [ \pi ( 1 - \pi ) ] ^ { - 1 / 2 } \right) . +$$ + +Thus we get theoretical evidence on how the weighting influences the classification accuracy: e.g. in the imbalanced case (when $\pi \approx 0$ or $\pi \approx 1$ ) selecting the weight optimally we reduce the generalization gap almost to zero for $N \gg 1$ ; at the same time, not optimal weight can lead to overfitting. + +As we already discussed, under some mild modeling assumptions the graph classification problem with isomorphic graphs in the training data set can be interpreted as the classification problem with a weighted loss. Therefore the obtained estimate provides additional evidence on a negative effect of the isomorphic graphs when solving the graph classification problems: the presence of isomorphic graphs in the training data set could have the same negative effect as not optimal weight value for the classification with a weighted loss function. + +# 7 INFLUENCE OF ISOMORPHISM BIAS: EMPIRICAL RESULTS + +To understand the impact of isomorphic graphs in the data set on the final metric we consider separately the results on two subparts of the data set. In particular, let $Y _ { t r a i n }$ and $Y _ { t e s t }$ be train and test splits of a data set. Let $H _ { i } \in Y _ { t e s t }$ be a graph such that there exists an isomorphic graph $G _ { i } \in Y _ { t r a i n }$ in the train data set. Let $\{ Y _ { i s o } \}$ be a set of all such graphs $H _ { i }$ for which there exists an isomorphic graph $G _ { i }$ in $Y _ { t r a i n }$ . Note that the graphs in $\{ Y _ { i s o } \}$ are not necessarily isomorphic. We denote by $Y _ { n e w }$ the test graphs that do not have isomorphic copies in the train data set, i.e. $Y _ { n e w } = Y _ { t e s t } \backslash Y _ { i s o }$ . If we want to test generalization of classification models, we need to test it on new instances of the data sets and therefore at least consider $Y _ { n e w }$ instead of $Y _ { t e s t }$ . One question regarding the performance of the models on this new test set $Y _ { n e w }$ is whether the performance on it will be lower than on the original test set $Y _ { t e s t }$ . As we show below answer to this question solely depends on the accuracy of the model on isomorphic instances $Y _ { i s o }$ . + +Consider a graph classification model that is evaluated on normalized accuracy over a data set $Y$ + +$$ +\operatorname { a c c } ( Y ) = { \frac { \sum _ { G \in Y } \operatorname { a c c } ( G _ { i } ) } { | Y | } } , +$$ + +where $\operatorname { a c c } ( G )$ equals to one if the model predicts the label of $G _ { i }$ correctly, and zero otherwise. If $| Y | = 0$ , then we consider $\operatorname { a c c } ( Y ) = 0$ . We can see that the accuracy on the test data set can be written as the sum of two terms: + +$$ +\begin{array} { r l } & { \qquad \displaystyle \sum _ { G \subset ( Y _ { t e s t } ) } \operatorname { a c c } ( G ) } \\ & { \qquad \quad = \frac { | Y _ { n e w } | } { | Y _ { t e s t } | } = \frac { \displaystyle \sum _ { G \in Y _ { n e w } } \operatorname { a c c } ( G ) + \sum _ { G \in Y _ { i s o } } \operatorname { a c c } ( G ) } { \displaystyle | Y _ { t e s t } | } = } \\ & { \qquad = \frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \mathrm { a c c } ( Y _ { n e w } ) + \frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \mathrm { a c c } ( Y _ { i s o } ) . } \end{array} +$$ + +Equation 3 decomposes accuracy on the original data set as the weighted sum of two accuracies on the set of the new test instances $Y _ { n e w }$ and a set of the instances $Y _ { i s o }$ already appeared in the train set and therefore available to the model. We call the term $\operatorname { a c c } ( Y _ { i s o } )$ as isomorphism bias, which corresponds to the accuracy of the model on the isomorphic test instances. As we will see next, the accuracy of the model on the new set $Y _ { n e w }$ will be less if only if the model performs better on the isomorphic set $Y _ { i s o }$ .1 + +$\mathbf { C l a i m 7 . 1 }$ . Let $Y _ { t e s t } = Y _ { n e w } \cup Y _ { i s o }$ , $Y _ { n e w } \cap Y _ { i s o } = \emptyset$ and $Y _ { i s o } \neq \emptyset$ , where $Y _ { i s o } \subset Y _ { t r a i n }$ . Then for any classification model accuracy on the new test instances $Y _ { n e w }$ is smaller than on the test set $Y _ { t e s t }$ if and only if it is smaller than accuracy on the isomorphic test instances $Y _ { i s o }$ , i.e.: + +$$ +\operatorname { a c c } ( Y _ { t e s t } ) > \operatorname { a c c } ( Y _ { n e w } ) \iff \operatorname { a c c } ( Y _ { i s o } ) > \operatorname { a c c } ( Y _ { n e w } ) +$$ + +The equation 4 gives a definite answer with the possible performance of the model on a new test set. If the model performs well on isomorphic instances $Y _ { i s o }$ , then it will falsely increase performance on $Y _ { t e s t }$ in comparison to $Y _ { n e w }$ . Conversely, if the model performs poorly on the instances that appeared in the training set, then removing them from the test set and evaluating the model purely on $Y _ { n e w }$ will demonstrate higher accuracy. There are two reasons for the model to misclassify isomorphic instances $Y _ { i s o }$ : (i) the instances contain target labels that are different than those that it has seen, as we show in Table 2 the percentage of mismatched labels can be high in some data sets; or (ii) the model is not expressive enough to map the structure of the graphs to the target label correctly. + +Table 4: Mean classification accuracy for test sets $Y _ { t e s t }$ and $Y _ { i s o }$ (in brackets) in 10-fold crossvalidation. Top-1 result in bold. + +
MUTAGIMDB-BIMDB-MCOX2AIDSPROTEINS
NN0.829 (0.840)0.737 (0.733)0.501 (0.488)0.82 (0.872)0.996 (0.998)0.737 (0.834)
NN-PH0.867 (1.000)0.756 (1.000)0.522 (1.000)0.838 (1.000)0.996 (1.000)0.742 (1.000)
NN-P0.856 (0.847)0.737 (0.731)0.499 (0.486)0.795 (0.83)0.996 (0.999)0.729 (0.709)
WL0.862 (0.867)0.734 (0.990)0.502 (0.953)0.800 (0.974)0.993 (0.999)0.747 (0.950)
WL-PH0.907 (1.000)0.736 (1.000)0.504 (1.000)0.810 (1.000)0.994 (1.000)0.749 (1.000)
WL-P0.870 (0.838)0.724 (0.715)0.495 (0.487)0.794 (0.844)0.994 (0.999)0.740 (0.742)
V0.836 (0.902)0.707(0.820)0.503 (0.732)0.7811(0.966)0.994 (0.997)0.726 (0.946)
V-PH0.859 (1.000)0.750 (1.000)0.517 (1.000)0.794(1.000)0.996 (1.000)0.729 (1.000)
V-P0.827 (0.844)0.724 (0.728)0.496 (0.481)0.768 (0.852)0.996 (0.999)0.719 (0.741)
+ +Crucially, while $Y _ { n e w }$ tests generalization capabilities of the models, on $Y _ { i s o }$ the models can explicitly or implicitly memorize the right labels from the training. We describe a model-agnostic way to guarantee increase of classification performance if $| Y _ { i s o } | \ne 0$ . + +Let $G \cong { \widehat { G } }$ such that $G \in Y _ { i s o }$ and $\widehat { G } \in Y _ { t r a i n }$ . Note that there can be multiple isomorphic graphs $\{ \widehat { G } _ { i } \} \subset Y _ { t r a i n }$ . If for any $G \in Y _ { i s o }$ all target labels of the orbit of $G$ are the same we call the set $Y _ { i s o }$ as homogeneous. Consider a classification model $\mathcal { M }$ that maps each graph $G$ to its label $l ( G )$ . We define a peering model $\widehat { \mathcal { M } }$ such that for each $G \in Y _ { i s o }$ it outputs the target label $l ( \widehat { G } )$ . Then the accuracy of the model $\widehat { \mathcal { M } }$ is at least as the accuracy of the original model $\mathcal { M }$ . + +Claim 7.2. Let $Y _ { t e s t } = Y _ { n e w } \cup Y _ { i s o }$ , $Y _ { n e w } \cap Y _ { i s o } = \emptyset$ . If $Y _ { i s o }$ is homogeneous, then the accuracy on $Y _ { t e s t }$ of a classification model $\mathcal { M }$ is at most as the accuracy of its peering model $\hat { \mathcal { M } }$ , i.e.: + +$$ +\operatorname { a c c } _ { \mathcal { M } } ( Y _ { t e s t } ) \leq \operatorname { a c c } _ { \widehat { \mathcal { M } } } ( Y _ { t e s t } ) . +$$ + +Claim 7.2 establishes a way to increase performance only for homogeneous $Y _ { i s o }$ . If there are noisy labels in the training set and hence the set is not homogeneous, the model cannot guarantee the right target label for these instances. Nonetheless, one can select a heuristic such as majority vote among the training isomorphic instances to select a proper label at the testing time. + +In experiments, we compare neural network model (NN) $\mathrm { { X u } }$ et al., 2018) with graph kernels, Weisfeiler-Lehman (WL) (Shervashidze et al., 2011b) and vertex histogram $( \mathbf { V } )$ (Sugiyama & Borgwardt, 2015). For each model we consider two modifications: one for peering model on homogeneous $Y _ { i s o }$ (e.g. NN-PH) and one for peering model on all $Y _ { i s o }$ (e.g. NN-P). We show accuracy on $Y _ { t e s t }$ and on $Y _ { i s o }$ (in brackets) in Table 4. Experimentation details can be found in Appendix G. + +From Table 4 we can conclude that peering model on homogeneous data is always the top performer. This is aligned with the result of Claim 7.2, which guarantees that $a c c ( Y _ { i s o } ) ~ = ~ 1$ , but it is an interesting observation if we compare it to the peering model on all isomorphic instances $Y _ { i s o }$ (-P models). Moreover, the latter model often loses even to the original model, where no information from the train set is explicitly taken into the test set. This can be explained by the noisy target labels in the orbits of isomorphic graphs, as can be seen both from the statistics for these datasets (Table 6) and accuracy measured just on isomorphic instances $Y _ { i s o }$ . These results show that due to the presence of isomorphism bias performance of any classification model can be overestimated by as much as $5 \%$ of accuracy on these datasets and hence future comparison of classification models should be estimated on $Y _ { n e w }$ instead. These observations conforms with our theoretical findings and conclusions in Section 6. + +# 7.1 GENERAL RECOMMENDATIONS + +In order to avoid measuring performance over the wrong test sets, we provide a set of recommendations that will guarantee measuring the right metrics for the models. + +• We open-source new, ”clean” data sets that do not include isomorphic instances that are in Table 8. To tackle this problem in the future, we propose to use clean versions of the data set for which isomorphism bias vanishes. For each data set we consider the found graph orbits and keep only one graph from each orbit if and only if the graphs in the orbit have the same label. If the orbit contains more than one label, a classification model can do little to predict a correct label at the inference time and hence we remove such orbit completely. In this case, for a new data set $Y _ { i s o } = \emptyset$ and hence it prevents the models to implicitly memorize the labels from the training set. We consider the data set orbits that do not account for neither node nor edge labels because the remaining graphs are not isomorphic based purely on graph topology. Incorporating node and edge features into the models may be necessary to distinguish the graphs. As we have seen, just using node labels can reduce the number of isomorphic graphs significantly and many data sets provide additional information to distinguish the models at full scope. +Verification of the models on bigger graphs in general is more challenging due to the sheer number of non-isomorphic graphs. For example, data sets related to REDDIT or DD include a number of big graphs for classification. + +# 8 CONCLUSION + +In this work we study isomorphism bias of the classification models in graph structured data that originates from substantial amount of isomorphic graphs in the data sets. We analyzed 54 graph data sets and provide the reasons for it as well as a set of rules to avoid unfair comparison of the models. 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A comprehensive survey on graph neural networks. arXiv preprint arXiv:1901.00596, 2019. + +Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018. + +Pinar Yanardag and SVN Vishwanathan. Deep graph kernels. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1365–1374. ACM, 2015. + +Rex Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec. Gnn explainer: A tool for post-hoc explanation of graph neural networks. arXiv preprint arXiv:1903.03894, 2019. + +# A STATISTICS FOR ORIGINAL DATA SETS + +Table 5: All original graph data sets. $N$ is the number of graphs, $C$ is the number of different classes. Avg. Nodes and Avg. Edges is the average number of nodes and edges. N.L. and E.L indicate if the graphs in a data set contain node or edge labels. + +
data setTypeNCAvg.NodesAvg.EdgesN.L.E.L.Source
FIRSTMM_DBMolecular41111377.273074.1+1Neumann et al. (2016)
OHSUMolecular79282.01199.66+1Pan (2018)
KKIMolecular83226.9648.42+1Pan (2018)
Peking_1Molecular85239.3177.35+1Pan (2018)
MUTAGMolecular188217.9319.79++Kriege&Mutzel (2012)
MSRC_21CVision2092040.2896.6+1Neumann et al. (2016)
MSRC_9Vision221840.5897.94+1Neumann et al. (2016)
CuneiformMolecular2673021.2744.8++Kriege et al. (2018a)
SYNTHETICSynthetic300210019611Feragen et al. (2013)
COX2_MDMolecular303226.28335.12++Kriege&Mutzel (2012)
BZR_MDMolecular306221.3225.06++Kriege&Mutzel(2012)
PTC_MMMolecular336213.9714.32++Kriege&Mutzel (2012)
PTC_MRMolecular344214.2914.69++Kriege&Mutzel (2012)
PTC_FMMolecular349214.1114.48++Kriege&Mutzel(2012)
PTC_FRMolecular351214.5615++Kriege&Mutzel(2012)
DHFR_MDMolecular393223.87283.01++Kriege&Mutzel (2012)
SynthieSynthetic400495172.9311Morris et al. (2016)
BZRMolecular405235.7538.36+1Sutherland et al. (2003)
ER_MDMolecular446221.33234.85++Kriege&Mutzel (2012)
COX2Molecular467241.2243.45+1Sutherland etal. (2003)
DHFRMolecular467242.4344.54+-Sutherland et al. (2003)
MSRC_21Vision5632077.52198.32+1Neumann et al. (2016)
ENZYMESMolecular600632.6362.14+-Borgwardt et al. (2005)
IMDB-BINARYSocial1000219.7796.53-1Yanardag&Vishwanathan (2015)
PROTEINSMolecular1113239.0672.82+1Borgwardt etal. (2005)
DDMolecular11782284.32715.66+1Shervashidze et al. (2011a)
IMDB-MULTISocial150031365.94-1Yanardag&Vishwanathan (2015)
AIDSMolecular2000215.6916.2++Riesen&Bunke (2008)
REDDIT-BINARYSocial20002429.63497.75-1Yanardag&Vishwanathan (2015)
Letter-highMolecular2250154.674.5-1Riesen&Bunke (2008)
Letter-lowMolecular2250154.683.13-1Riesen&Bunke (2008)
Letter-medMolecular2250154.674.5-1Riesen&Bunke (2008)
FingerprintMolecular280045.424.42-1Riesen&Bunke(2008)
COIL-DELMolecular390010021.5454.24-+Riesen&Bunke (2008)
COIL-RAGMolecular39001003.013.02-1Riesen&Bunke (2008)
NCI1Molecular4110229.8732.3+1Shervashidze et al. (2011a)
NCI109Molecular4127229.6832.13+1Shervashidze et al. (2011a)
FRANKENSTEINMolecular4337216.917.88-1Orsini et al. (2015)
MutagenicityMolecular4337230.3230.77++Riesen&Bunke (2008)
REDDIT-MULTI-5KSocial49995508.52594.87-1Yanardag&Vishwanathan (2015)
COLLABSocial5000374.492457.78-1Yanardag&Vishwanathan(2015)
Tox21_AREMolecular7167216.2816.52++Challenge (2014)
Tox21_aromataseMolecular7226217.517.79++Challenge (2014)
Tox21_MMPMolecular7320217.4917.83++Challenge (2014)
Tox21_ERMolecular7697217.5817.94++Challenge (2014)
Tox21_HSEMolecular8150216.7217.04++Challenge (2014)
Tox21_AHRMolecular8169218.0918.5++Challenge (2014)
Tox21_PPAR-gammaMolecular8184217.2317.55++Challenge (2014)
Tox21_AR-LBDMolecular8599217.7718.16++Challenge (2014)
Tox21_p53Molecular8634217.7918.19++Challenge (2014)
Tox21ER_LBDMolecular8753218.0618.47++Challenge (2014)
Tox21_ATAD5Molecular9091217.8918.3++Challenge (2014)
Tox21_ARMolecular9362218.3918.84++Challenge (2014)
REDDIT-MULTI-12KSocial1192911391.41456.89-1Yanardag& Vishwanathan (2015)
+ +# B ISOMORPHISM METRICS FOR ALL DATA SETS + +Table 6: Isomorphic metrics for all data sets. Sorting is based on the proportion of isomorphic graphs $I \%$ . Num. orbits is the number of non-trivial orbits. $I P \%$ is the proportion of isomorphic pairs of graphs, Mismatched $\%$ is the proportion of mismatched labels. + +
data setSize,NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
Letter-high225094220097.783.6795.91
IMDB-MULTI1500100121280.86.3974.67
Tox21_ATAD590911461616767.840.099.15
Tox21PPAR-gamma81841265551367.360.17.77
Tox21_AR93621519629567.240.088
Tox21_p5386341345580067.180.0911.28
Tox21_AR-LBD85991354576667.050.096.88
Tox21_MMP73201138487566.60.118.76
Tox21_HSE81501218542566.560.118.02
Tox21ERLBD87531375579166.160.0912.41
Tox21_ER76971203507865.970.0927.32
Tox21_AHR81691299537765.820.0915.61
Tox21_aromatase72261084472765.420.15.07
Tox21_ARE71671047468265.330.1126.45
AIDS2000371125962.950.130.35
COX24677628360.60.620.56
IMDB-BINARY100011757957.90.6731.8
FRANKENSTEIN4337574223051.420.0930.87
MUTAG188317942.020.496.91
BZR4054316540.740.68.89
PTC_MM3364213239.290.4623.21
PTC_MR3444012536.340.4125
PTC_FM3493912435.530.3923.5
DHFR7568925033.070.149.13
PTC_FR3513611633.050.3720.51
Mutagenicity4337397127429.380.0313.1
COLLAB5000158107721.540.116.68
COIL-DEL390015579620.410.0618.56
PROTEINS11133515113.570.19.07
NCI1411022552312.730.011.7
NCI109412722251912.580.011.7
ENZYMES6006101.6700
REDDIT-BINARY2000340.200
REDDIT-MULTI-12K119298170.1400.04
FIRSTMM_DB4110000
OHSU790000
KKI830000
Peking_18510000
MSRC_21C20910000
MSRC.92210000
Synthie40010000
MSRC_2156310000
DD117810000
REDDIT-MULTI-5K499910000
+ +In the plots 3, 4, 5 the sizes of orbits are presented for each data set. Empty plots correspond to data sets with no isomorphic graphs. Plots with just wo-labels correspond to cases when there are no node labels available for the graphs in a data set. + +![](images/0fde0b75916ba7570274a87b8ac0c5c557c2f78e60370bcab5c1031d2cd9cb89.jpg) +Figure 3: Distribution of orbits sizes. wo-labels correspond to isomorphism without considering the labels. node-labels correspond to isomorphism that considers node labels. Part-1. + +![](images/2a469f104aace5fa5c90a6e844e1d29a83b4449bea7160fea0e8ddc0b0df7ab9.jpg) +Figure 4: Distribution of orbits sizes. Part-2. + +![](images/83854384c25f71c93f0795e1a4a52c8322953ac8c4bc31056ed2200bd095b89d.jpg) +Figure 5: Distribution of orbits sizes. Part-3. + +# D ISOMORPHISM METRICS FOR DATA SETS WITH NODE LABELS + +Table 7: Isomorphic metrics for data sets with node labels. Sorting is based on the proportion of isomorphic graphs $I \%$ . Num. orbits is the number of non-trivial orbits. $I P \%$ is the proportion of isomorphic pairs of graphs, Mismatched $\%$ is the proportion of mismatched labels. Table does not include data sets with no node labels. + +
data setSize, NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
DHFR_MD3932539299.756.8794.91
COX2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
BZR_MD3062230399.027.1695.75
MUTAG188173619.150.140
PTC_FM349225415.470.0810.89
PTC_MM336225014.880.077.74
DHFR756399812.960.043.97
PTC_FR351204312.250.056.27
PTC_MR344194111.920.056.4
Tox21_ARE716722882011.440.013.91
Tox21_HSE815025091911.280.012.25
Tox21_aromatase722622380511.140.010.77
Tox21-p53863425894610.960.011.83
Tox21ER769723184010.910.012.4
COX2467255010.710.031.07
Tox21_PPAR-gamma818422786910.620.010.76
Tox21_MMP732020477010.520.012.19
Tox21_ERLBD875325591310.430.011.28
Tox21_AR936226596510.310.010.68
Tox21_ATAD5909125592410.160.011.04
Tox211_AR-LBD859923887110.130.010.7
BZR40516409.880.060.99
Tox21_AHR81692247959.730.012.07
PROTEINS111321746.650.032.61
AIDS200022542.700
Mutagenicity433731751.7300.92
NCI141109170.4100.05
ENZYMES600220.3300
NCI10941277120.2900.05
FIRSTMM_DB4110000
OHSU7910000
KKI8310000
Peking_18510000
MSRC_21C20910000
MSRC_922110000
MSRC_2156310000
DD117810000
+ +# E PROOF OF CLAIM 7.1 + +Proof. From the equation 3 we have: + +$$ +\begin{array} { l } { \displaystyle \csc ( Y _ { t e s t } ) = \frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \mathrm { a c c } ( Y _ { n e w } ) + \frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \mathrm { a c c } ( Y _ { i s o } ) > \operatorname { a c c } ( Y _ { i s o } ) \Rightarrow } \\ { \displaystyle \frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \mathrm { a c c } ( Y _ { n e w } ) > ( 1 - \frac { | Y _ { i s o } | } { | Y _ { t e s t } | } ) \mathrm { a c c } ( Y _ { i s o } ) \Rightarrow } \\ { \displaystyle \operatorname { a c c } ( Y _ { i s o } ) > \operatorname { a c c } ( Y _ { n e w } ) } \end{array} +$$ + +# F PROOF OF CLAIM 7.2 + +Proof. From the definition of the peering model we have: + +$$ +\begin{array} { l } { \operatorname { a c c } _ { \mathcal { M } } ( Y _ { n e w } ) = \operatorname { a c c } _ { \widehat { \mathcal { M } } } ( Y _ { n e w } ) } \\ { \operatorname { a c c } _ { \mathcal { M } } ( Y _ { i s o } ) \leq \operatorname { a c c } _ { \widehat { \mathcal { M } } } ( Y _ { i s o } ) = 1 } \end{array} +$$ + +Substituting these into the equation 3 we have: + +$$ +\operatorname { a c c } _ { \mathcal { M } } ( Y _ { t e s t } ) \leq \operatorname { a c c } _ { \widehat { \mathcal { M } } } ( Y _ { t e s t } ) . +$$ + +# G EXPERIMENTATION DETAILS + +NN model is from $\mathrm { X u }$ et al. (2018) and evaluate it on the data sets from PyTorch-Geometric (Fey & Lenssen, 2019). For each data set we perform 10-fold cross-validation such that each fold is evaluated on $10 \%$ of hold-out instances $Y _ { t e s t }$ . For each fold we train the model for 350 epochs selecting the final model with the best performance on the validation set ( $20 \%$ from hold-out trained split) across all epochs. Additionally we found that for small data set performance during the first epochs can be unstable on the validation set and thus we select our model only after the first 50 epochs. The final model is evaluated on the test instances and corresponds to NN in the experiments. Peering models NN-PH and NN-P are obtained from NN by replicating the target labels for homogeneous $Y _ { i s o }$ and non-homogeneous $Y _ { i s o }$ respectively. Weisfeiler-Lehman and Vertex histogram kernels are taken from the code2 of Sugiyama & Borgwardt (2015). We selected the height of subtree $h = 5$ for WL kernel. We train an SVM model selecting $C$ parameter from the range [0.001, 0.01, 0.1, 1, 10]. + +# H NEW DATA SETS + +Table 8: Statistics for new clean data sets. Retention is the proportion of graphs remaining after cleaning procedure. Min. Class and Max. Class are minimum and maximum number of graphs in a class. Sorted by retention. + +
Data setSizeRetention, %Avg.NodesAvg.EdgesClassesMin. ClassMax. Class
SYNTHETIC0000000
Cuneiform0000000
COIL-RAG130.335.779.62715
Letter-low120.537.175.42513
COX2_MD30.9930.33467212
DHFR_MD41.0225.25366.25213
Letter-med291.296.835.668110
Fingerprint511.8214.4513.12640
BZR_MD61.9614.17130.17215
Letter-high602.677.037.237132
ER_MD143.1418.07227.362113
IMDB-MULTI32121.422.35249.46385144
Tox21_ARE330246.0720.9621.8626022700
Tox21_ER356046.2522.2423.2924193141
Tox21_MMP340546.5222.423.4326182787
Tox21_HSE381446.821.2922.2822073607
Tox21-p53408847.3522.6223.7222913797
Tox21_PPAR-gamma387747.3721.9122.9121203757
Tox21_ATAD5431247.4322.5223.6121684144
Tox21_AR-LBD413448.0822.4123.4821503984
Tox21_AR450648.1322.9324.0521894317
Tox21_AHR393548.1722.8823.9824903445
Tox21ERLBD422448.2622.6923.821934031
Tox21_aromatase352448.7722.2423.2222343290
IMDB-BINARY49349.324.08221.962232261
COX223750.7542.1444.43268169
AIDS111055.518.2219.12310800
FRANKENSTEIN244856.4420.7822.35210201428
PTC_MM22667.2617.0417.74277149
BZR27668.1536.2538.81272204
PTC_MR23568.3117.2317.97296139
PTC_FM24269.3416.9617.66285157
MUTAG13571.8118.8520.8424293
PTC_FR25372.0817.1117.86286167
DHFR57876.4643.3745.532205373
Mutagenicity333576.932.9634.16214841851
COIL-DEL313380.3325.0564.2698139
COLLAB406481.2876.944667.9237702289
PROTEINS97587.643.4181.042343632
NCI1378592.0929.8432.37217812004
NCI109380192.129.6632.22218012000
ENZYMES59599.1732.4862.17698100
REDDIT-BINARY199899.9430.04996.4829981000
REDDIT-MULTI-12K1191799.9391.79914.68115132586
FIRSTMM_DB411001377.273073.931126
OHSU7910082.01199.6623544
KKI8310026.9648.4223746
Peking_18510039.3177.3523649
MSRC_21C20910040.2896.617129
MSRC_922110040.5897.9481930
Synthie40010091.6202.18490110
MSRC_2156310077.52198.32201034
DD1178100284.32715.662487691
REDDIT-MULTI-5K4999100508.511189.7559991000
+ +# I PROOF OF THEOREM 6.1 + +Proof. Let us prove Theorem 6.1. We denote by ${ \mathcal { L } } = \{ ( x , y ) \to L ( f ( x ) , y ) : f \in { \mathcal { F } } \}$ a composite loss class. For any $L \in { \mathcal { L } }$ we get that + +$$ +\begin{array} { r } { \mathbb { E } _ { \mathbb { P } } L - \mathbb { E } _ { \mathcal { D } } u L = \mathbb { E } _ { \mathbb { P } } L - \mathbb { E } _ { \mathbb { P } } u L + \mathbb { E } _ { \mathbb { P } } u L - \mathbb { E } _ { \mathcal { D } } u L \leq } \\ { \leq \mathbb { E } _ { \mathbb { P } } \lvert ( 1 - u ) L \rvert + ( \mathbb { E } _ { \mathbb { P } } u L - \mathbb { E } _ { \mathcal { D } } u L ) . } \end{array} +$$ + +Since any $L \in { \mathcal { L } }$ is bounded from above by 1 for the first term in equation 5 we obtain + +$$ +\begin{array} { r } { { { \mathbb E } } _ { \mathbb P } | ( 1 - u ) L | \le { { \mathbb E } } _ { \mathbb P } g _ { + } ( w ) { { \mathbb I } } _ { \{ y = + 1 \} } + { { \mathbb E } } _ { \mathbb P } g _ { - } ( w ) { { \mathbb I } } _ { \{ y = - 1 \} } = g _ { + } ( w ) \pi + g _ { - } ( w ) ( 1 - \pi ) . } \end{array} +$$ + +Thanks to McDiarmid’d concentration inequality Mohri et al. (2012), applied to the function class ${ \mathcal { L } } _ { u } = \{ u L : L \in { \mathcal { L } } \}$ , with probability $1 - \delta , \delta > 0$ for $\mathcal { D } \sim \mathbb { P } ^ { N }$ we get the upper bound on the excess risk + +$$ +\begin{array} { r l } & { \underset { L \in \mathcal { L } } { \operatorname* { s u p } } ( \mathbb { E } _ { \mathbb { P } } u L - \mathbb { E } _ { \mathcal { D } } u L ) \leq 2 \mathcal { R } _ { N } ( \mathcal { L } _ { u } ) + \operatorname* { m a x } [ ( 1 + g _ { + } ( w ) ) , ( 1 + g _ { - } ( w ) ) ] \sqrt { ( \log \delta ^ { - 1 } ) / ( 2 N ) } \leq } \\ & { \qquad \leq 2 \mathcal { R } _ { N } ( \mathcal { L } _ { u } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \sqrt { ( \log \delta ^ { - 1 } ) / ( 2 N ) } . } \end{array} +$$ + +Let us find a relation between $\mathcal { R } _ { N } ( \mathcal { L } _ { u } )$ and $\mathcal { R } _ { N } ( \mathcal { L } )$ . Here we denote by $z _ { i }$ a pair $z _ { i } = ( x _ { i } , y _ { i } )$ . By the definition (see Mohri et al. (2012)) the empirical Rademacher complexity + +$$ +\begin{array} { l } { \displaystyle \hat { \mathfrak { z } } _ { \mathcal { P } } ( \mathcal { L } _ { u } ) = \displaystyle \frac { 1 } { N } \mathbb { E } _ { \sigma } \displaystyle \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i = 1 } ^ { N } \sigma _ { i } u ( z _ { i } ) L ( z _ { i } ) \leq } \\ { \displaystyle \leq \frac { 1 } { N } \mathbb { E } _ { \sigma } \displaystyle \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i = 1 } ^ { N } \sigma _ { i } L ( z _ { i } ) + \frac { g _ { + } ( w ) } { N } \mathbb { E } _ { \sigma } \displaystyle \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i : y _ { i } = + 1 } \sigma _ { i } L ( z _ { i } ) + \frac { g _ { - } ( w ) } { N } \mathbb { E } _ { \sigma } \displaystyle \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i \geq y _ { i } = - 1 } \sigma _ { i } L ( z _ { i } ) \leq } \\ { \displaystyle \leq \widehat { \mathcal { R } } _ { \mathcal { D } } ( \mathcal { L } ) + \frac { g _ { + } ( w ) } { N } \mathbb { E } _ { \sigma } \displaystyle \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i \geq y _ { i } = + 1 } \sigma _ { i } L ( z _ { i } ) + \frac { g _ { - } ( w ) } { N } \mathbb { E } _ { \sigma } \displaystyle \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i \in \mathcal { L } _ { i } : y _ { i } = - 1 } \sigma _ { i } L ( z _ { i } ) . } \end{array} +$$ + +Since we use the zero-one loss, then + +$$ +\mathbb { E } _ { \sigma } \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i : y _ { i } = + 1 } \sigma _ { i } L ( z _ { i } ) \leq \# \{ i : y _ { i } = + 1 \} , \ \mathbb { E } _ { \sigma } \operatorname* { s u p } _ { L \in \mathcal { L } _ { u } } \sum _ { i : y _ { i } = - 1 } \sigma _ { i } L ( z _ { i } ) \leq \# \{ i : y _ { i } = - 1 \} . +$$ + +The Rademacher complexity + +$$ +\begin{array} { r l r } & { } & { \mathcal { R } _ { N } ( \mathcal { L } _ { u } ) = \mathbb { E } _ { \mathcal { D } \sim \mathbb { P } ^ { N } } \hat { \mathcal { R } } _ { \mathcal { D } } ( \mathcal { L } _ { u } ) \leq \mathbb { E } _ { \mathcal { D } \sim \mathbb { P } ^ { N } } \Big [ \hat { \mathcal { R } } _ { \mathcal { D } } ( \mathcal { L } ) + \frac { g _ { + } ( w ) } { N } \# \{ i : y _ { i } = + 1 \} + } \\ & { } & { \qquad + \frac { g _ { - } ( w ) } { N } \# \{ i : y _ { i } = - 1 \} \Big ] = \mathcal { R } _ { N } ( \mathcal { L } ) + g _ { + } ( w ) \pi + g _ { - } ( w ) ( 1 - \pi ) . } \end{array} +$$ + +Using the fact that $\begin{array} { r } { \mathcal { R } _ { N } ( \mathcal { L } ) = \frac { 1 } { 2 } \mathcal { R } _ { N } ( \mathcal { F } ) } \end{array}$ , substituting inequalities 6, 7 and 8 into equation 5, we get that + +$$ +\begin{array} { r l r } { { \operatorname* { s u p } _ { f \in \mathcal { F } } \Big ( \mathbb { E } _ { \mathbb { P } } l ( f ( x ) , y ) - \mathbb { E } _ { \mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \Big ) \leq 3 ( g _ { + } ( w ) \pi + g _ { - } ( w ) ( 1 - \pi ) ) + } } \\ & { } & { + \mathcal { R } _ { N } ( \mathcal { F } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \sqrt { ( \log \delta ^ { - 1 } ) / ( 2 N ) } . } \end{array} +$$ \ No newline at end of file diff --git a/parse/train/rJlUhhVYvS/rJlUhhVYvS_content_list.json b/parse/train/rJlUhhVYvS/rJlUhhVYvS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..71b78773c68fbc8db2351543b040840b53ef692b --- /dev/null +++ b/parse/train/rJlUhhVYvS/rJlUhhVYvS_content_list.json @@ -0,0 +1,2019 @@ +[ + { + "type": "text", + "text": "UNDERSTANDING ISOMORPHISM BIAS IN GRAPH DATA SETS ", + "text_level": 1, + "bbox": [ + 176, + 99, + 823, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years there has been a rapid increase in classification methods on graph structured data. Both in graph kernels and graph neural networks, one of the implicit assumptions of successful state-of-the-art models was that incorporating graph isomorphism features into the architecture leads to better empirical performance. However, as we discover in this work, commonly used data sets for graph classification have repeating instances which cause the problem of isomorphism bias, i.e. artificially increasing the accuracy of the models by memorizing target information from the training set. The problem does not vanish even in consideration of node labels. This prevents fair competition of the algorithms and raises a question of the validity of the obtained results. First, we characterize this effect theoretically by reducing the graph classification to a weighted classification problem and estimating the corresponding generalization gap. Then we analyze 54 data sets, previously extensively used for graph-related tasks, on the existence of isomorphism bias, give a set of recommendations to machine learning practitioners to properly set up their models, and open source new data sets for the future experiments. ", + "bbox": [ + 233, + 268, + 764, + 491 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 521, + 336, + 536 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently there has been an increasing interest in the development of machine learning models that operate on graph structured data. Such models have found applications in chemoinformatics (Ralaivola et al. (2005); Rupp & Schneider (2010); Ferre et al. (2017)) and bioinformatics (Borg- ´ wardt et al. (2005); Kundu et al. (2013)), neuroscience (Sharaev et al. (2018); Jie et al. (2016); Wang et al. (2016)), computer vision (Stumm et al. (2016)) and system security (Li et al. (2016)), natural language processing (Glavas &ˇ Snajder (2013)), and others (Kriege et al. (2019); Nikolentzos ˇ et al. (2019)). One of the popular tasks that encompasses these applications is graph classification problem for which many graph kernels and graph neural networks have been developed. ", + "bbox": [ + 173, + 554, + 825, + 667 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One of the implicit assumptions that many practitioners adhere to is that models that can distinguish isomorphic instances from non-isomorphic ones possess higher expressiveness in classification problem and hence much efforts have been devoted to incorporate efficient graph isomorphism methods into the classification models. As the problem of computing complete graph invariant is $\\mathbb { G } \\mathbb { I }$ -hard (Gartner et al. (2003)), for which no known polynomial-time algorithm exists, other heuris- ¨ tics have been proposed as a proxy for deciding whether two graphs are isomorphic. Indeed, from the early days topological descriptors such Wiener index (Wiener (1947a;b)) attempted to find a single number that uniquely identifies a graph. Later, graph kernels that model pairwise similarities between graphs utilized theoretical developments in graph isomorphism literature. For example, graphlet kernel (Shervashidze et al. (2009)) is based on the Kelly conjecture (see also Kelly (1957)), anonymous walk kernel (Ivanov & Burnaev (2018)) derives insights from the reconstruction properties of anonymous experiments (see also Micali & Allen Zhu (2016)), and WL kernel (Shervashidze et al. (2011a)) is based on an efficient graph isomorphism algorithm. For sufficiently large $k$ , $k$ - dimensional WL algorithm includes all combinatorial properties of a graph (Cai et al. (1992a)), so one may hope its power is enough for the data set at hand. Since only for $k = \\Omega ( n )$ WL algorithm is guaranteed to distinguish all graphs (for which the running time becomes exponential; see also Furer (2017)), in the general case WL algorithm can be used only as a strong baseline for graph ¨ isomorphism. In similar fashion, graph neural networks exploit graph isomorphism algorithms and ", + "bbox": [ + 174, + 674, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "have been shown to be as powerful as $\\mathbf { k }$ -dimensional WL algorithm (see for example Maron et al. \n(2019); Xu et al. (2018); Morris et al. (2019)). ", + "bbox": [ + 171, + 103, + 821, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Experimental evaluation reveals that models based on the theoretical constructions with high combinatorial power such as WL algorithm performs better than the models without them such as Vertex histogram kernel (Vishwanathan et al. (2010)) on a commonly used data sets. This could add additional bias to results of comparison of classification algorithms since the models could simply apply a graph isomorphism method (or an efficient approximation) to determine a target label at the inference time. However, purely judging on the accuracy of the algorithms in such cases would imply an unfair comparison between the methods as it does not measure correctly generalization ability of the models on the new test instances. As we discover, indeed many of the data sets used in graph classification have isomorphic instances so much that in some of them the fraction of the unique non-repeating graphs is as low as $20 \\%$ of the total size. This challenges previous experimental results and requires understanding of how influential isomorphic instances on the final performance of the models. Our contributions are: ", + "bbox": [ + 174, + 138, + 825, + 304 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We analyze the quality of 54 graph data sets which are used ubiquitously in graph classification comparison. Our findings suggest that in the most of the data sets there are isomorphic graphs and their proportion varies from as much as $100 \\%$ to $0 \\%$ . Surprisingly, we also found that there are isomorphic instances that have different target labels suggesting they are not suitable for learning a classifier at all. We investigate the causes of isomorphic graphs and show that node and edge labels are important to identify isomorphic graphs. Other causes include numerical attributes of nodes and edges as well as the sizes of the data set. We express an upper bound for the generalization gap through the Radamacher complexity of a classifier and the number of isomorphic graphs in a data set. This bound presents theoretical evidence on how weightning of each graph in the training influences classification accuracy. \nWe evaluate a classification model’s performance on isomorphic instances and show that even strong models do not achieve optimal accuracy even if the instances have been seen at the training time. Hence we show a model-agnostic way to artificially increase performance on several widely used data sets. We open-source new cleaned data sets that contain only non-isomorphic instances with no noisy target labels. We give a set of recommendations regarding applying new models that work with graph structured data. ", + "bbox": [ + 215, + 318, + 825, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 621, + 341, + 637 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Measuring quality of data sets. A similar issue of duplicates instances in commonly used data sets was recently discovered in computer vision domain. Recht et al. (2019); Barz & Denzler (2019); Birodkar et al. (2019) discover that image data sets CIFAR and ImageNet contain at least $10 \\%$ of the duplicate images in the test test invalidating previous performance and questioning generalization abilities of previous successful architectures. In particular, evaluating the models in new test sets shows a drop of accuracy by as much as $15 \\%$ (Recht et al., 2019), which is explained by models’ incapability to generalize to unseen slightly ”harder” instances than in the original test sets. In graph domain, a fresh look into understanding of expressiveness of graph kernels and the quality of data sets has been considered in Kriege et al. (2019), where an extensive comparison of existing graph kernels is done and a few insights about models’ behavior are suggested. In contrast, we conduct a broader study of isomorphism metrics, revealing all isomorphism pairs in proposed 54 data sets, and propose new cleaned data. Additionally we also consider graph neural network performance and argue that current data sets present isomorphism bias which can artificially boost evaluation metrics in a model-agnostic way. ", + "bbox": [ + 174, + 652, + 825, + 847 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Explaining performance of graph models. Graph kernels (Kriege et al. (2019)) and graph neural networks (Wu et al. (2019)) are two competing paradigms for designing graph representations and solving graph classification and have significantly advanced empirical results due to more efficient algorithms, incorporating graph invariance into the models, and end-to-end training. Several papers have tried to justify performance of different families of methods by studying different statistical properties. For example, in Ying et al. (2019) by maximizing mutual information between explanation variables and predicted label distribution, the model is trained to return a small subgraph and the graph-specific attributes that are the most influential on the decision made by a GNN, which allows inspection of single- and multi-level predictions in an agnostic manner for GNNs. In another work (Scarselli et al. (2018)), the VC dimension of GNNs models has been shown to grow as $\\mathcal { O } ( p ^ { 4 } N ^ { 2 } )$ , where $p$ is the number of network parameters and $N$ is the number of nodes in a graph, which is comparable to RNN models. Furthermore, stability and generalization properties of convolutional GNNs have been shown to depend on the largest eigenvalue of the graph filter and therefore are attained for properly normalized graph convolutions such as symmetric normalized graph Laplacian (Verma & Zhang (2019)). Finally, expressivity of graph kernels has been studied from statistical learning theory (Oneto et al. (2017b)) and property testing (Kriege et al. (2018b)), showing that graph kernels can capture certain graph properties such as planarity, girth, and chromatic number (Johansson et al. (2014)). Our approach is complementary to all of the above as we analyze if the data sets used in experiments have any effect on the final performance. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 319, + 338, + 334 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work we analyze 54 graph data sets from Kersting et al. (2016) that are commonly used in graph classification task. Examples of popular graph data sets are presented in Table 1 and statistics of all 54 data sets can be found in Table 5, see Section A in the appendix. All data sets represent a collection of graphs and accompanying categorical label for each graph in the data sets. Some data sets also include node and/or edge labels that graph classification methods can use to improve the scoring. Most of the data sets come either from biological domain or from social network domain. Biological data sets such as MUTAG, ENZYMES, PROTEINS are graphs that represent small or large molecules, where edges of the graphs are chemical bonds or spatial proximity between different atoms. Graph labels in these cases encode different properties like toxicity. In social data sets such as IMDB-BINARY, REDDIT-MULTI-5K, COLLAB the nodes represent people and edges are relationships in movies, discussion threads, or citation network respectively. Labels in these cases denote the type of interaction like the genre of the movie/thread or a research subfield. For completeness we also include synthetic data sets SYNTHETIC (Morris et al. (2016)) that have continuous attributes and computer vision data sets MSRC (Neumann et al. (2016)), where images are encoded as graphs. The origin of all data sets can be found in the Table 5. ", + "bbox": [ + 174, + 349, + 825, + 559 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/3c2d392dca821a4f0d78c724c4d5f335e45effbf6691590c4c2f069946484b16.jpg", + "table_caption": [ + "Table 1: Example of graph data sets. $N$ is the number of graphs, $C$ is the number of different classes. Avg. Nodes and Avg. Edges is the average number of nodes and edges. N.L. and E.L indicate if the graphs in a data set contain node or edge labels. " + ], + "table_footnote": [], + "table_body": "
data setTypeNCAvg. NodesAvg. EdgesN.L.E.L.
MUTAGMolecular188217.9319.79++
ENZYMESMolecular600632.6362.14+二
PROTEINSMolecular1113239.0672.82+-
IMDB-BINARYSocial1000219.7796.53二二
REDDIT-MULTI-5KSocial49995508.52594.87二二
COLLABSocial5000374.492457.78--
SYNTHETICSynthetic3002100196二二
SynthieSynthetic400495172.93-二
MSRC_21CVision2092040.2896.6+二
MSRC_9Vision221840.5897.94+1
", + "bbox": [ + 183, + 631, + 813, + 791 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Graph isomorphism. Isomorphism between two graphs $G _ { 1 } = ( V _ { 1 } , E _ { 1 } )$ and $G _ { 2 } = ( V _ { 2 } , E _ { 2 } )$ is a bijective function $\\phi : V _ { 1 } \\mapsto V _ { 2 }$ such that any edge $( u , v ) \\in E _ { 1 }$ if and only if $( \\phi ( u ) , \\phi ( v ) ) \\in E _ { 2 }$ . Graph isomorphism problem asks if such function exists for given two graphs $G _ { 1 }$ and $G _ { 2 }$ . We denote isomorphic graphs as $G _ { 1 } \\cong G _ { 2 }$ . The problem has efficient algorithms in $\\mathbb { P }$ for certain classes of graphs such as planar or bounded-degree graphs (Hopcroft $\\&$ Wong (1974); Luks (1980)), but in the general case admits only quasi-polynomial algorithm (Babai (2015)). In practice many GI solvers are based on individualization-refinement paradigm (Mckay & Piperno (2014)), which for each graph iteratively updates a permutation of the nodes such that the resulted permutations of two graphs are identical if an only if they are isomorphic. Importantly, while finding such canonical permutation of a graph is at least as hard as solving GI problem, state-of-the-art solvers tackle majority of pairs of graphs efficiently, only taking exponential time on the specific hard instances of graphs that possess highly symmetrical structures (Cai et al. (1992b)). ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 IDENTIFYING ISOMORPHISM IN DATA SETS ", + "text_level": 1, + "bbox": [ + 174, + 180, + 560, + 195 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To distinguish between different isomorphic graphs inside a data set we use the notion of graph orbits: ", + "bbox": [ + 173, + 209, + 825, + 238 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 4.1 (Graph orbit). Let $\\mathcal { D } = \\{ G _ { i } , y _ { i } \\} _ { i = 1 } ^ { N }$ be a data set of graphs and target labels. For a graph $G _ { i }$ let a set $o _ { i } = \\{ G _ { k } \\}$ be a set of all isomorphic graphs in $\\mathcal { D }$ to $G _ { i }$ , including $G _ { i }$ . We call $o _ { i }$ the orbit of graph $G _ { i }$ in $\\mathcal { D }$ . The cardinality of the orbit is called orbit size. An orbit with size one is called trivial. ", + "bbox": [ + 174, + 239, + 825, + 296 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In a data set with no isomorphic graphs, the number of orbits equals to the number of graphs in a data set, $N$ . Hence, the more orbits in a data set, the ”cleaner” it is. Note however that the distribution of orbit sizes in two different data sets can vary even if they have the same number of orbits. Therefore, we look at additional metrics that describe the data set. ", + "bbox": [ + 174, + 306, + 825, + 363 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• $I$ , aggregated number of graphs that belong to an orbit of size greater than one, i.e. those graphs that isomorphic counterparts in a data set; \n• $I , \\%$ , proportion of isomorphic graphs to the total data set size, i.e. $\\textstyle { \\frac { I } { N } }$ ; \n• $I P , \\%$ , proportion of isomorphic pairs to the total number of graph pairs in a data set ( $\\frac { N ( N - 1 ) } { 2 } )$ . ", + "bbox": [ + 215, + 373, + 825, + 455 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "If we consider target labels of graphs in a data set $\\mathcal { D } = \\{ G _ { i } , y _ { i } \\} _ { i = 1 } ^ { N }$ we can also measure agreement between the labels of two isomorphic data set. If $G _ { 1 } \\cong G _ { 2 }$ and $y _ { 1 } ~ \\ne ~ y _ { 2 }$ , then we call graphs mismatched. Note that if there is more than one target label in an orbit $o$ , then all graphs in this orbit are mismatched. To obtain isomorphic graphs, we run nauty algorithm (Mckay & Piperno, 2014) on all possible pairs of graphs in a data set. We substantially reduce the number of calls between the graphs by verifying that a pair has the same number of nodes and edges before the call. ", + "bbox": [ + 174, + 467, + 825, + 551 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The metrics are presented in Table 2 for top-10 data sets and in Table 6 (see the appendix) for all data sets. The graphs in Table 2 are sorted by the proportion of isomorphic graphs $I \\%$ . The results for the first Top-10 data sets are somewhat surprising: almost all graphs in the selected data sets have other isomorphic graphs. If we look at all data sets in Table 6, we see that the proportion of isomorphic graphs in the data sets varies from $100 \\%$ to $0 \\%$ . However, more than $80 \\%$ of the analyzed data sets have at least $10 \\%$ of the graphs in a non-trivial orbit. ", + "bbox": [ + 174, + 558, + 825, + 642 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/a1599045762f9b49ec73a18a7b7256607709a6320b194d51681cde5de152ccae.jpg", + "table_caption": [ + "Table 2: Isomorphic metrics for Top-10 data sets based on the proportion of isomorphic graphs $I \\%$ $I P \\%$ is the proportion of isomorphic pairs of graphs, Mismatched $\\%$ is the proportion of mismatched labels. " + ], + "table_footnote": [], + "table_body": "
data setSize, NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
", + "bbox": [ + 184, + 710, + 812, + 875 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Another surprising observation is that the proportion of mismatched graphs is significant, ranging from $100 \\%$ to $0 \\%$ . This clearly indicates that such graphs are not suitable for graph classification and require additional information to distinguish the models. We analyze the reasons for this in the next section. ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Also, the distribution of orbit sizes can vary significantly across the data sets. In Figure 1 we plot a distribution of orbit sizes for several examples of data sets (and distributions for other data sets can be found in Appendix C). For example, for IMDB-BINARY data set the number of orbits of small sizes, e.g. two or three, goes to 100, which indicate prevalence of pairs of isomorphic graphs that are non-isomorphic to the rest. However, for Letter-med data set there are many orbits of sizes more than 100, while small orbits are not that common. In this case, the graphs in this data set are equivalent to a lot of other graphs, which may have a substantial effect on the corresponding metrics. While the orbit distribution changes from one data set to another, it is clear that in many situations there are isomorphic graphs that can affect training procedure by effectively increasing the weights for the corresponding graphs, change performance on the test by validating on the already seen instances, and by confusing the model by utilizing different target labels for topologically-equivalent graphs. We analyze the reasons for it further. ", + "bbox": [ + 174, + 138, + 825, + 305 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/f6a14c15fa6baf9143bba7daa52c62de70ad1b25b506e43721c025cb6a81eba4.jpg", + "image_caption": [ + "Figure 1: Examples of distributions of orbit sizes without considering labels. " + ], + "image_footnote": [], + "bbox": [ + 173, + 320, + 825, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EXPLAINING ISOMORPHISM ", + "text_level": 1, + "bbox": [ + 176, + 585, + 436, + 601 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Meta-information about graphs. In addition to the topology of a graph, many data sets also include meta information about nodes and/or edges. Out of 54 analyzed data sets there are 40 that additionally include node features and 25 that include edge features. For example, in Synthetic data set all graphs are topologically identical but the nodes are endowed with normally distributed scalar attributes and in DHFR MD edges are attributed with distances and labeled according to a chemical bond type. Alternatively, some graphs can have parallel edges which is equivalent to have a corresponding weight on the edges. Thus some data sets include node/edge categorical features (labels) and numerical features (attributes), which leads to better distinction between the graphs and therefore their corresponding labels. ", + "bbox": [ + 173, + 617, + 825, + 742 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To see this, we rerun our previous analysis but now include the node labels, if any, when computing isomorphism between graphs. Consider a tuple $( G , l )$ , where $G$ is a graph and $l : V ( G ) \\mapsto$ $\\{ 1 , 2 , \\ldots , k \\}$ is a $k$ -labeling of $G$ . In this case of node label-preserving graph isomorphism from graph $( G _ { 1 } , l _ { 1 } )$ to graph $\\left( { { G } _ { 2 } } , { { l } _ { 2 } } \\right)$ we seek an isomorphism function $\\phi : { \\bar { V } } ( { \\bar { G } } _ { 1 } { \\bar { ) } } \\mapsto V ( { \\bar { G _ { 2 } } } )$ such that $l _ { 1 } ( v ) = l _ { 2 } ( \\phi ( v ) )$ . ", + "bbox": [ + 173, + 750, + 825, + 820 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Tables 3 and 7 (see the appendix) show the number of isomorphic graphs after considering node labels. While for the first six data sets the proportion of isomorphic graphs has not changed much, it is clearly the case for the remaining data sets. In particular, almost $90 \\%$ of the analyzed data sets include less than $20 \\%$ of isomorphic graphs. Also, the number of mismatched graphs significantly decreases after considering node labels. For example, for MUTAG data set the proportion of isomorphic graphs went down from $4 2 . 0 2 \\%$ to $1 9 . 1 5 \\%$ and the proportion of mismatched graphs from $6 . 9 1 \\%$ to $0 \\%$ . ", + "bbox": [ + 173, + 825, + 825, + 922 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/6f10881c4d8957a21ac0d9793ad38064624d70e3e3da0459fbf8487846535fe9.jpg", + "table_caption": [ + "Table 3: Isomorphic metrics with node labels for Top-10 data sets based on the proportion of isomorphic graphs $\\bar { I \\% }$ . $I P \\%$ is the proportion of isomorphic pairs of graphs, Mismatched $\\%$ is the proportion of mismatched labels. " + ], + "table_footnote": [], + "table_body": "
data setSize, NNum.orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
DHFR_MD3932539299.756.8794.91
COX2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
BZR_MD3062230399.027.1695.75
MUTAG188173619.150.140
PTC_FM349225415.470.0810.89
PTC_MM336225014.880.077.74
DHFR756399812.960.043.97
", + "bbox": [ + 184, + 160, + 812, + 325 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Likewise, the orbit size distribution also changes significantly after considering node labels. Figure 2 shows a changed distribution of orbits with and without considering node labels. For majority of data sets large orbits vanish and the number of small orbits is substantially decreased in label-preserving graph isomorphism setting. This indicates one of the reasons for presence of many isomorphic graphs in the data sets, which implies that including node/edge labels/attributes can be important for graph classification models. ", + "bbox": [ + 173, + 358, + 825, + 443 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/d7e9dbdde3017047ec255b42935b6e9cc68b9f8d7a5f186905404b51e128b64f.jpg", + "image_caption": [ + "Figure 2: Examples of distributions of orbit sizes with node labels. " + ], + "image_footnote": [], + "bbox": [ + 173, + 455, + 825, + 654 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Sizes of the data sets. Another reason for having isomorphism in a data set is the sizes of graphs, which could be too small on average to lead to a diversity in a data set. In general, the number of non-isomorphic graphs with $n$ vertices and $m$ edges can be computed using Polya enumeration theory and grows very fast. For example, for a graph with 15 nodes and 15 edges, there are 2,632,420 non-isomorphic graphs. Nevertheless, specifics of the origin of the data set may affect possible configurations that graphs have (e.g. structure of chemical compounds in COX2 MD or ego-networks for actors in IMDB-BINARY) and thus smaller graphs may tend to be close to isomorphic structures. On the other hand, all five data sets with the average number of nodes greater than 100 have very low or zero proportion of isomorphic graphs. Hence, the average size of the graphs directly impacts the possible structure of the data set and thus data sets with larger graphs tend to be more diverse. We next analyze the consequences of the isomorphic graphs on classification methods. ", + "bbox": [ + 173, + 704, + 826, + 857 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 ISOMORPHIC GRAPHS AND WEIGHTED CLASSIFICATION", + "text_level": 1, + "bbox": [ + 173, + 878, + 666, + 893 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We denote by ", + "bbox": [ + 176, + 909, + 264, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• $\\mathcal { F } \\subseteq Y ^ { X }$ a class of binary classifiers with an input space $X$ and an output space $Y =$ $\\{ - 1 , + 1 \\}$ , \n• $\\mathbb { P }$ a distribution on $X \\times Y$ , \n• $\\pi$ a prior probability of a positive class, i.e. $\\mathbb { P } = \\pi \\mathbb { P } _ { x | y = + 1 } + ( 1 - \\pi ) \\mathbb { P } _ { x | y = - 1 } ,$ , \n• $\\mathcal { D } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N }$ a training sample, $x _ { i } \\in X$ , $y _ { i } \\in Y$ . In our case $x _ { i } \\in \\mathcal D$ are graphs $G _ { i }$ , see notations in section 4. ", + "bbox": [ + 215, + 102, + 825, + 204 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We consider a zero-one loss function $l ( \\hat { y } , y ) = \\mathbb { I } _ { \\hat { y } \\neq y }$ . Let us assume that classifiers from $\\mathcal { F }$ can detect which graphs in $\\mathcal { D }$ are isomorphic. E.g. classifiers based on Weisfeiler-Lehman graph kernels (see Shervashidze et al. (2011a)) are capable to do it for majority of graphs. In such case the empirical risk $\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } l ( f ( x ) , y ) = \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } l ( f ( x _ { j } ) , y _ { j } ) } \\end{array}$ reduces to $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { j \\in J } u _ { j } l ( f ( x _ { j } ) , y _ { j } ) } \\end{array}$ , where $J$ is an index set of non-isomorphic graphs, $u _ { j } ~ \\geq ~ 1$ is equal to the number of graphs in the initial sample , isomorphic to the graph $x _ { j }$ (we count $x _ { j }$ as well). ", + "bbox": [ + 173, + 213, + 825, + 303 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Thus under such assumptions the graph classification problem with the training data set, containing isomorphic graphs, can be interpreted as a classification problem with a weighted loss. Let us introduce general notations for this problem. We define some (fixed) measurable weighting function $u : ( X \\times Y ) \\to ( 0 , + \\infty )$ . Then the theoretical risk is equal to $\\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y )$ and the weighted empirical risk is equal to $\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } u ( x _ { i } , y _ { i } ) l ( f ( x _ { i } ) , y _ { i } ) } \\end{array}$ ). We would like to derive an upper bound for the excess risk $\\begin{array} { r } { \\operatorname* { s u p } _ { f \\in \\bar { \\mathcal { F } } } \\left( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\right) } \\end{array}$ , i.e. we would like to quantify an upper bound for a generalization gap. We optimize the weighted empirical risk when training a classifier and measure its accuracy using non-weighted theoretical risk. ", + "bbox": [ + 173, + 308, + 825, + 424 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "There are some results about classification performance with a weighted loss. E.g. in (Dupret & Koda, 2001) a bayesian framework for imbalanced classification with a weighted risk is proposed. Scott (2012) investigated the calibration of asymmetric surrogate losses. Natarajan et al. (2018) considered the case of cost-sensitive learning with noisy labels. However, to the best of our knowledge, there is no studied upper bound for the excess risk with explicit dependence on the class imbalance and the weighting scheme that quantifies the influence on the overall classification performance. We show this result next. ", + "bbox": [ + 173, + 429, + 825, + 527 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To derive explicit expressions we use some additional modeling asumption, namely, we consider $u ( x , y ) = ( \\bar { 1 } + g _ { + } ( w \\bar { ) } ) \\mathbb { I } _ { \\{ y = + 1 \\} } + ( 1 + g _ { - } ( w ) ) \\mathbb { I } _ { \\{ y = - 1 \\} }$ for some non-negative weighting functions $g _ { + } ( w )$ and $g _ { - } ( w )$ of the weight value $w \\geq 0$ . E.g. we can use $g _ { + } ( w ) = w$ and $g _ { - } ( w ) = 1 / w$ . ", + "bbox": [ + 174, + 534, + 825, + 579 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 6.1. With probability $1 - \\delta , \\delta > 0$ for $\\mathcal { D } \\sim \\mathbb { P } ^ { N }$ the excess risk is upper bounded by ", + "bbox": [ + 171, + 580, + 790, + 597 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/044123505e40689970d447aa8afe372670a9e344d6d6977add6ffe008cd79c1e.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\Big ( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\Big ) \\leq 3 ( g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) ) + } } \\\\ & { } & { + \\mathcal { R } _ { N } ( \\mathcal { F } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 599, + 753, + 655 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\mathcal { R } _ { N } ( \\mathcal { F } )$ is a Rademacher complexity of the function class $\\mathcal { F }$ . ", + "bbox": [ + 174, + 656, + 617, + 672 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Let us note that the Rademacher complexity of the function class $\\mathcal { F }$ , defined by a graph kernel, was studied e.g. in Oneto et al. (2018); Oneto et al. (2017a). ", + "bbox": [ + 173, + 681, + 823, + 712 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "From equation 1 it follows that by tuning the weight parameter $w$ we can make the upper bound tighter, namely collecting the terms with $w$ in the RHS of equation 1 we solve ", + "bbox": [ + 173, + 718, + 825, + 746 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/c45347036d971045198d5e4b93bc91153bd961611ed7b214a65fac22a3d2f38a.jpg", + "text": "$$\ng _ { + } ( w ) \\left( 3 \\pi + \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } \\right) + g _ { - } ( w ) \\left( 3 ( 1 - \\pi ) + \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } \\right) \\to \\operatorname* { m i n } _ { w } .\n$$", + "text_format": "latex", + "bbox": [ + 209, + 750, + 785, + 777 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In case we set $g _ { + } ( w ) = w$ and $g _ { - } ( w ) = 1 / w$ , the optimal weight $\\begin{array} { r } { w _ { o p t } = \\sqrt { \\frac { 3 ( 1 - \\pi ) + \\alpha _ { N } } { 3 \\pi + \\alpha _ { N } } } \\approx \\sqrt { \\frac { 1 - \\pi } { \\pi } } , } \\end{array}$ , where αN = $\\begin{array} { r } { \\alpha _ { N } = \\sqrt { \\frac { \\log \\delta ^ { - 1 } } { 2 N } } \\approx 0 } \\end{array}$ for $N \\gg 1$ . For such optimal $w _ { o p t }$ the RHS of equation 1 has the form ", + "bbox": [ + 173, + 789, + 825, + 839 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/d29d159b24e367c77f972f95f29f21df69b18312b2bcf85065681fdd5026d15e.jpg", + "text": "$$\n6 \\sqrt { \\pi ( 1 - \\pi ) } + \\mathcal { R } _ { N } ( \\mathcal { F } ) + \\alpha _ { N } \\left( 2 + [ \\pi ( 1 - \\pi ) ] ^ { - 1 / 2 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 843, + 678, + 871 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Thus we get theoretical evidence on how the weighting influences the classification accuracy: e.g. in the imbalanced case (when $\\pi \\approx 0$ or $\\pi \\approx 1$ ) selecting the weight optimally we reduce the generalization gap almost to zero for $N \\gg 1$ ; at the same time, not optimal weight can lead to overfitting. ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As we already discussed, under some mild modeling assumptions the graph classification problem with isomorphic graphs in the training data set can be interpreted as the classification problem with a weighted loss. Therefore the obtained estimate provides additional evidence on a negative effect of the isomorphic graphs when solving the graph classification problems: the presence of isomorphic graphs in the training data set could have the same negative effect as not optimal weight value for the classification with a weighted loss function. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 INFLUENCE OF ISOMORPHISM BIAS: EMPIRICAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 208, + 679, + 223 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To understand the impact of isomorphic graphs in the data set on the final metric we consider separately the results on two subparts of the data set. In particular, let $Y _ { t r a i n }$ and $Y _ { t e s t }$ be train and test splits of a data set. Let $H _ { i } \\in Y _ { t e s t }$ be a graph such that there exists an isomorphic graph $G _ { i } \\in Y _ { t r a i n }$ in the train data set. Let $\\{ Y _ { i s o } \\}$ be a set of all such graphs $H _ { i }$ for which there exists an isomorphic graph $G _ { i }$ in $Y _ { t r a i n }$ . Note that the graphs in $\\{ Y _ { i s o } \\}$ are not necessarily isomorphic. We denote by $Y _ { n e w }$ the test graphs that do not have isomorphic copies in the train data set, i.e. $Y _ { n e w } = Y _ { t e s t } \\backslash Y _ { i s o }$ . If we want to test generalization of classification models, we need to test it on new instances of the data sets and therefore at least consider $Y _ { n e w }$ instead of $Y _ { t e s t }$ . One question regarding the performance of the models on this new test set $Y _ { n e w }$ is whether the performance on it will be lower than on the original test set $Y _ { t e s t }$ . As we show below answer to this question solely depends on the accuracy of the model on isomorphic instances $Y _ { i s o }$ . ", + "bbox": [ + 173, + 238, + 825, + 392 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Consider a graph classification model that is evaluated on normalized accuracy over a data set $Y$ ", + "bbox": [ + 173, + 398, + 807, + 412 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/463bfb84bf61a927250c828511d9ac31915defaea6f86cc570b88f4c5d505c97.jpg", + "text": "$$\n\\operatorname { a c c } ( Y ) = { \\frac { \\sum _ { G \\in Y } \\operatorname { a c c } ( G _ { i } ) } { | Y | } } ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 417, + 581, + 462 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\operatorname { a c c } ( G )$ equals to one if the model predicts the label of $G _ { i }$ correctly, and zero otherwise. If $| Y | = 0$ , then we consider $\\operatorname { a c c } ( Y ) = 0$ . We can see that the accuracy on the test data set can be written as the sum of two terms: ", + "bbox": [ + 174, + 468, + 825, + 511 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/9fd7fdc4b883a86a24095df6ff685452c5c55b7e7713190fd8c78b1f14fc1841.jpg", + "text": "$$\n\\begin{array} { r l } & { \\qquad \\displaystyle \\sum _ { G \\subset ( Y _ { t e s t } ) } \\operatorname { a c c } ( G ) } \\\\ & { \\qquad \\quad = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } = \\frac { \\displaystyle \\sum _ { G \\in Y _ { n e w } } \\operatorname { a c c } ( G ) + \\sum _ { G \\in Y _ { i s o } } \\operatorname { a c c } ( G ) } { \\displaystyle | Y _ { t e s t } | } = } \\\\ & { \\qquad = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) + \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { i s o } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 534, + 723, + 616 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Equation 3 decomposes accuracy on the original data set as the weighted sum of two accuracies on the set of the new test instances $Y _ { n e w }$ and a set of the instances $Y _ { i s o }$ already appeared in the train set and therefore available to the model. We call the term $\\operatorname { a c c } ( Y _ { i s o } )$ as isomorphism bias, which corresponds to the accuracy of the model on the isomorphic test instances. As we will see next, the accuracy of the model on the new set $Y _ { n e w }$ will be less if only if the model performs better on the isomorphic set $Y _ { i s o }$ .1 ", + "bbox": [ + 173, + 626, + 825, + 710 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$\\mathbf { C l a i m 7 . 1 }$ . Let $Y _ { t e s t } = Y _ { n e w } \\cup Y _ { i s o }$ , $Y _ { n e w } \\cap Y _ { i s o } = \\emptyset$ and $Y _ { i s o } \\neq \\emptyset$ , where $Y _ { i s o } \\subset Y _ { t r a i n }$ . Then for any classification model accuracy on the new test instances $Y _ { n e w }$ is smaller than on the test set $Y _ { t e s t }$ if and only if it is smaller than accuracy on the isomorphic test instances $Y _ { i s o }$ , i.e.: ", + "bbox": [ + 174, + 713, + 825, + 756 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/1322af5fb329ca44f6fc2ccc030ade725302a3dbd9a39707197683bcdcb389f4.jpg", + "text": "$$\n\\operatorname { a c c } ( Y _ { t e s t } ) > \\operatorname { a c c } ( Y _ { n e w } ) \\iff \\operatorname { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { n e w } )\n$$", + "text_format": "latex", + "bbox": [ + 316, + 775, + 679, + 792 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The equation 4 gives a definite answer with the possible performance of the model on a new test set. If the model performs well on isomorphic instances $Y _ { i s o }$ , then it will falsely increase performance on $Y _ { t e s t }$ in comparison to $Y _ { n e w }$ . Conversely, if the model performs poorly on the instances that appeared in the training set, then removing them from the test set and evaluating the model purely on $Y _ { n e w }$ will demonstrate higher accuracy. There are two reasons for the model to misclassify isomorphic instances $Y _ { i s o }$ : (i) the instances contain target labels that are different than those that it has seen, as we show in Table 2 the percentage of mismatched labels can be high in some data sets; or (ii) the model is not expressive enough to map the structure of the graphs to the target label correctly. ", + "bbox": [ + 173, + 801, + 825, + 900 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/b1893314e6d7c75eed3cdb185d3155327912c0bc4b4e8526795cb1ad88cad6e2.jpg", + "table_caption": [ + "Table 4: Mean classification accuracy for test sets $Y _ { t e s t }$ and $Y _ { i s o }$ (in brackets) in 10-fold crossvalidation. Top-1 result in bold. " + ], + "table_footnote": [], + "table_body": "
MUTAGIMDB-BIMDB-MCOX2AIDSPROTEINS
NN0.829 (0.840)0.737 (0.733)0.501 (0.488)0.82 (0.872)0.996 (0.998)0.737 (0.834)
NN-PH0.867 (1.000)0.756 (1.000)0.522 (1.000)0.838 (1.000)0.996 (1.000)0.742 (1.000)
NN-P0.856 (0.847)0.737 (0.731)0.499 (0.486)0.795 (0.83)0.996 (0.999)0.729 (0.709)
WL0.862 (0.867)0.734 (0.990)0.502 (0.953)0.800 (0.974)0.993 (0.999)0.747 (0.950)
WL-PH0.907 (1.000)0.736 (1.000)0.504 (1.000)0.810 (1.000)0.994 (1.000)0.749 (1.000)
WL-P0.870 (0.838)0.724 (0.715)0.495 (0.487)0.794 (0.844)0.994 (0.999)0.740 (0.742)
V0.836 (0.902)0.707(0.820)0.503 (0.732)0.7811(0.966)0.994 (0.997)0.726 (0.946)
V-PH0.859 (1.000)0.750 (1.000)0.517 (1.000)0.794(1.000)0.996 (1.000)0.729 (1.000)
V-P0.827 (0.844)0.724 (0.728)0.496 (0.481)0.768 (0.852)0.996 (0.999)0.719 (0.741)
", + "bbox": [ + 176, + 156, + 825, + 294 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 324, + 823, + 352 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Crucially, while $Y _ { n e w }$ tests generalization capabilities of the models, on $Y _ { i s o }$ the models can explicitly or implicitly memorize the right labels from the training. We describe a model-agnostic way to guarantee increase of classification performance if $| Y _ { i s o } | \\ne 0$ . ", + "bbox": [ + 174, + 359, + 825, + 402 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Let $G \\cong { \\widehat { G } }$ such that $G \\in Y _ { i s o }$ and $\\widehat { G } \\in Y _ { t r a i n }$ . Note that there can be multiple isomorphic graphs $\\{ \\widehat { G } _ { i } \\} \\subset Y _ { t r a i n }$ . If for any $G \\in Y _ { i s o }$ all target labels of the orbit of $G$ are the same we call the set $Y _ { i s o }$ as homogeneous. Consider a classification model $\\mathcal { M }$ that maps each graph $G$ to its label $l ( G )$ . We define a peering model $\\widehat { \\mathcal { M } }$ such that for each $G \\in Y _ { i s o }$ it outputs the target label $l ( \\widehat { G } )$ . Then the accuracy of the model $\\widehat { \\mathcal { M } }$ is at least as the accuracy of the original model $\\mathcal { M }$ . ", + "bbox": [ + 173, + 409, + 825, + 491 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Claim 7.2. Let $Y _ { t e s t } = Y _ { n e w } \\cup Y _ { i s o }$ , $Y _ { n e w } \\cap Y _ { i s o } = \\emptyset$ . If $Y _ { i s o }$ is homogeneous, then the accuracy on $Y _ { t e s t }$ of a classification model $\\mathcal { M }$ is at most as the accuracy of its peering model $\\hat { \\mathcal { M } }$ , i.e.: ", + "bbox": [ + 173, + 492, + 823, + 525 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/907812d78c25ff43ac90709105e7b3ae11a87ba09ace62fdd34dc8cf6538e009.jpg", + "text": "$$\n\\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { t e s t } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { t e s t } ) .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 530, + 596, + 549 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Claim 7.2 establishes a way to increase performance only for homogeneous $Y _ { i s o }$ . If there are noisy labels in the training set and hence the set is not homogeneous, the model cannot guarantee the right target label for these instances. Nonetheless, one can select a heuristic such as majority vote among the training isomorphic instances to select a proper label at the testing time. ", + "bbox": [ + 173, + 559, + 825, + 616 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In experiments, we compare neural network model (NN) $\\mathrm { { X u } }$ et al., 2018) with graph kernels, Weisfeiler-Lehman (WL) (Shervashidze et al., 2011b) and vertex histogram $( \\mathbf { V } )$ (Sugiyama & Borgwardt, 2015). For each model we consider two modifications: one for peering model on homogeneous $Y _ { i s o }$ (e.g. NN-PH) and one for peering model on all $Y _ { i s o }$ (e.g. NN-P). We show accuracy on $Y _ { t e s t }$ and on $Y _ { i s o }$ (in brackets) in Table 4. Experimentation details can be found in Appendix G. ", + "bbox": [ + 174, + 622, + 825, + 693 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "From Table 4 we can conclude that peering model on homogeneous data is always the top performer. This is aligned with the result of Claim 7.2, which guarantees that $a c c ( Y _ { i s o } ) ~ = ~ 1$ , but it is an interesting observation if we compare it to the peering model on all isomorphic instances $Y _ { i s o }$ (-P models). Moreover, the latter model often loses even to the original model, where no information from the train set is explicitly taken into the test set. This can be explained by the noisy target labels in the orbits of isomorphic graphs, as can be seen both from the statistics for these datasets (Table 6) and accuracy measured just on isomorphic instances $Y _ { i s o }$ . These results show that due to the presence of isomorphism bias performance of any classification model can be overestimated by as much as $5 \\%$ of accuracy on these datasets and hence future comparison of classification models should be estimated on $Y _ { n e w }$ instead. These observations conforms with our theoretical findings and conclusions in Section 6. ", + "bbox": [ + 173, + 699, + 825, + 853 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7.1 GENERAL RECOMMENDATIONS ", + "text_level": 1, + "bbox": [ + 176, + 869, + 429, + 883 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In order to avoid measuring performance over the wrong test sets, we provide a set of recommendations that will guarantee measuring the right metrics for the models. ", + "bbox": [ + 174, + 896, + 821, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• We open-source new, ”clean” data sets that do not include isomorphic instances that are in Table 8. To tackle this problem in the future, we propose to use clean versions of the data set for which isomorphism bias vanishes. For each data set we consider the found graph orbits and keep only one graph from each orbit if and only if the graphs in the orbit have the same label. If the orbit contains more than one label, a classification model can do little to predict a correct label at the inference time and hence we remove such orbit completely. In this case, for a new data set $Y _ { i s o } = \\emptyset$ and hence it prevents the models to implicitly memorize the labels from the training set. We consider the data set orbits that do not account for neither node nor edge labels because the remaining graphs are not isomorphic based purely on graph topology. Incorporating node and edge features into the models may be necessary to distinguish the graphs. As we have seen, just using node labels can reduce the number of isomorphic graphs significantly and many data sets provide additional information to distinguish the models at full scope. \nVerification of the models on bigger graphs in general is more challenging due to the sheer number of non-isomorphic graphs. For example, data sets related to REDDIT or DD include a number of big graphs for classification. ", + "bbox": [ + 215, + 104, + 825, + 352 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 373, + 318, + 388 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this work we study isomorphism bias of the classification models in graph structured data that originates from substantial amount of isomorphic graphs in the data sets. We analyzed 54 graph data sets and provide the reasons for it as well as a set of rules to avoid unfair comparison of the models. We theoretically characterized the influence of isomorphism bias on the graph classification performance by providing an upper bound on the generalization gap. We showed that in the current data sets any model can memorize the correct answers from the training set and we open-source new clean data sets where such problems do not appear. 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", + "bbox": [ + 173, + 784, + 825, + 827 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Rex Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec. Gnn explainer: A tool for post-hoc explanation of graph neural networks. arXiv preprint arXiv:1903.03894, 2019. ", + "bbox": [ + 173, + 835, + 825, + 864 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A STATISTICS FOR ORIGINAL DATA SETS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 526, + 118 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/49fb3fe2c55dd7d765490dd414c9561eb3ab6e6ebb2d5590ce43c38393ff054e.jpg", + "table_caption": [ + "Table 5: All original graph data sets. $N$ is the number of graphs, $C$ is the number of different classes. Avg. Nodes and Avg. Edges is the average number of nodes and edges. N.L. and E.L indicate if the graphs in a data set contain node or edge labels. " + ], + "table_footnote": [], + "table_body": "
data setTypeNCAvg.NodesAvg.EdgesN.L.E.L.Source
FIRSTMM_DBMolecular41111377.273074.1+1Neumann et al. (2016)
OHSUMolecular79282.01199.66+1Pan (2018)
KKIMolecular83226.9648.42+1Pan (2018)
Peking_1Molecular85239.3177.35+1Pan (2018)
MUTAGMolecular188217.9319.79++Kriege&Mutzel (2012)
MSRC_21CVision2092040.2896.6+1Neumann et al. (2016)
MSRC_9Vision221840.5897.94+1Neumann et al. (2016)
CuneiformMolecular2673021.2744.8++Kriege et al. (2018a)
SYNTHETICSynthetic300210019611Feragen et al. (2013)
COX2_MDMolecular303226.28335.12++Kriege&Mutzel (2012)
BZR_MDMolecular306221.3225.06++Kriege&Mutzel(2012)
PTC_MMMolecular336213.9714.32++Kriege&Mutzel (2012)
PTC_MRMolecular344214.2914.69++Kriege&Mutzel (2012)
PTC_FMMolecular349214.1114.48++Kriege&Mutzel(2012)
PTC_FRMolecular351214.5615++Kriege&Mutzel(2012)
DHFR_MDMolecular393223.87283.01++Kriege&Mutzel (2012)
SynthieSynthetic400495172.9311Morris et al. (2016)
BZRMolecular405235.7538.36+1Sutherland et al. (2003)
ER_MDMolecular446221.33234.85++Kriege&Mutzel (2012)
COX2Molecular467241.2243.45+1Sutherland etal. (2003)
DHFRMolecular467242.4344.54+-Sutherland et al. (2003)
MSRC_21Vision5632077.52198.32+1Neumann et al. (2016)
ENZYMESMolecular600632.6362.14+-Borgwardt et al. (2005)
IMDB-BINARYSocial1000219.7796.53-1Yanardag&Vishwanathan (2015)
PROTEINSMolecular1113239.0672.82+1Borgwardt etal. (2005)
DDMolecular11782284.32715.66+1Shervashidze et al. (2011a)
IMDB-MULTISocial150031365.94-1Yanardag&Vishwanathan (2015)
AIDSMolecular2000215.6916.2++Riesen&Bunke (2008)
REDDIT-BINARYSocial20002429.63497.75-1Yanardag&Vishwanathan (2015)
Letter-highMolecular2250154.674.5-1Riesen&Bunke (2008)
Letter-lowMolecular2250154.683.13-1Riesen&Bunke (2008)
Letter-medMolecular2250154.674.5-1Riesen&Bunke (2008)
FingerprintMolecular280045.424.42-1Riesen&Bunke(2008)
COIL-DELMolecular390010021.5454.24-+Riesen&Bunke (2008)
COIL-RAGMolecular39001003.013.02-1Riesen&Bunke (2008)
NCI1Molecular4110229.8732.3+1Shervashidze et al. (2011a)
NCI109Molecular4127229.6832.13+1Shervashidze et al. (2011a)
FRANKENSTEINMolecular4337216.917.88-1Orsini et al. (2015)
MutagenicityMolecular4337230.3230.77++Riesen&Bunke (2008)
REDDIT-MULTI-5KSocial49995508.52594.87-1Yanardag&Vishwanathan (2015)
COLLABSocial5000374.492457.78-1Yanardag&Vishwanathan(2015)
Tox21_AREMolecular7167216.2816.52++Challenge (2014)
Tox21_aromataseMolecular7226217.517.79++Challenge (2014)
Tox21_MMPMolecular7320217.4917.83++Challenge (2014)
Tox21_ERMolecular7697217.5817.94++Challenge (2014)
Tox21_HSEMolecular8150216.7217.04++Challenge (2014)
Tox21_AHRMolecular8169218.0918.5++Challenge (2014)
Tox21_PPAR-gammaMolecular8184217.2317.55++Challenge (2014)
Tox21_AR-LBDMolecular8599217.7718.16++Challenge (2014)
Tox21_p53Molecular8634217.7918.19++Challenge (2014)
Tox21ER_LBDMolecular8753218.0618.47++Challenge (2014)
Tox21_ATAD5Molecular9091217.8918.3++Challenge (2014)
Tox21_ARMolecular9362218.3918.84++Challenge (2014)
REDDIT-MULTI-12KSocial1192911391.41456.89-1Yanardag& Vishwanathan (2015)
", + "bbox": [ + 176, + 194, + 823, + 770 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B ISOMORPHISM METRICS FOR ALL DATA SETS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 580, + 118 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/c43d0475870160d6966561be084b69e9f359ec7b05bc7d0451b8bdb193ec3282.jpg", + "table_caption": [ + "Table 6: Isomorphic metrics for all data sets. Sorting is based on the proportion of isomorphic graphs $I \\%$ . Num. orbits is the number of non-trivial orbits. $I P \\%$ is the proportion of isomorphic pairs of graphs, Mismatched $\\%$ is the proportion of mismatched labels. " + ], + "table_footnote": [], + "table_body": "
data setSize,NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
Letter-high225094220097.783.6795.91
IMDB-MULTI1500100121280.86.3974.67
Tox21_ATAD590911461616767.840.099.15
Tox21PPAR-gamma81841265551367.360.17.77
Tox21_AR93621519629567.240.088
Tox21_p5386341345580067.180.0911.28
Tox21_AR-LBD85991354576667.050.096.88
Tox21_MMP73201138487566.60.118.76
Tox21_HSE81501218542566.560.118.02
Tox21ERLBD87531375579166.160.0912.41
Tox21_ER76971203507865.970.0927.32
Tox21_AHR81691299537765.820.0915.61
Tox21_aromatase72261084472765.420.15.07
Tox21_ARE71671047468265.330.1126.45
AIDS2000371125962.950.130.35
COX24677628360.60.620.56
IMDB-BINARY100011757957.90.6731.8
FRANKENSTEIN4337574223051.420.0930.87
MUTAG188317942.020.496.91
BZR4054316540.740.68.89
PTC_MM3364213239.290.4623.21
PTC_MR3444012536.340.4125
PTC_FM3493912435.530.3923.5
DHFR7568925033.070.149.13
PTC_FR3513611633.050.3720.51
Mutagenicity4337397127429.380.0313.1
COLLAB5000158107721.540.116.68
COIL-DEL390015579620.410.0618.56
PROTEINS11133515113.570.19.07
NCI1411022552312.730.011.7
NCI109412722251912.580.011.7
ENZYMES6006101.6700
REDDIT-BINARY2000340.200
REDDIT-MULTI-12K119298170.1400.04
FIRSTMM_DB4110000
OHSU790000
KKI830000
Peking_18510000
MSRC_21C20910000
MSRC.92210000
Synthie40010000
MSRC_2156310000
DD117810000
REDDIT-MULTI-5K499910000
", + "bbox": [ + 207, + 212, + 794, + 900 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In the plots 3, 4, 5 the sizes of orbits are presented for each data set. Empty plots correspond to data sets with no isomorphic graphs. Plots with just wo-labels correspond to cases when there are no node labels available for the graphs in a data set. ", + "bbox": [ + 174, + 133, + 825, + 176 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/0fde0b75916ba7570274a87b8ac0c5c557c2f78e60370bcab5c1031d2cd9cb89.jpg", + "image_caption": [ + "Figure 3: Distribution of orbits sizes. wo-labels correspond to isomorphism without considering the labels. node-labels correspond to isomorphism that considers node labels. Part-1. " + ], + "image_footnote": [], + "bbox": [ + 200, + 188, + 797, + 869 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/2a469f104aace5fa5c90a6e844e1d29a83b4449bea7160fea0e8ddc0b0df7ab9.jpg", + "image_caption": [ + "Figure 4: Distribution of orbits sizes. Part-2. " + ], + "image_footnote": [], + "bbox": [ + 174, + 133, + 825, + 856 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/83854384c25f71c93f0795e1a4a52c8322953ac8c4bc31056ed2200bd095b89d.jpg", + "image_caption": [ + "Figure 5: Distribution of orbits sizes. Part-3. " + ], + "image_footnote": [], + "bbox": [ + 171, + 133, + 820, + 857 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D ISOMORPHISM METRICS FOR DATA SETS WITH NODE LABELS ", + "text_level": 1, + "bbox": [ + 173, + 103, + 712, + 118 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/e1e36d59f014554de5bea50ed59f8cb0d3d6137ec4db67e56c44dec8c8f8c30c.jpg", + "table_caption": [ + "Table 7: Isomorphic metrics for data sets with node labels. Sorting is based on the proportion of isomorphic graphs $I \\%$ . Num. orbits is the number of non-trivial orbits. $I P \\%$ is the proportion of isomorphic pairs of graphs, Mismatched $\\%$ is the proportion of mismatched labels. Table does not include data sets with no node labels. " + ], + "table_footnote": [], + "table_body": "
data setSize, NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
DHFR_MD3932539299.756.8794.91
COX2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
BZR_MD3062230399.027.1695.75
MUTAG188173619.150.140
PTC_FM349225415.470.0810.89
PTC_MM336225014.880.077.74
DHFR756399812.960.043.97
PTC_FR351204312.250.056.27
PTC_MR344194111.920.056.4
Tox21_ARE716722882011.440.013.91
Tox21_HSE815025091911.280.012.25
Tox21_aromatase722622380511.140.010.77
Tox21-p53863425894610.960.011.83
Tox21ER769723184010.910.012.4
COX2467255010.710.031.07
Tox21_PPAR-gamma818422786910.620.010.76
Tox21_MMP732020477010.520.012.19
Tox21_ERLBD875325591310.430.011.28
Tox21_AR936226596510.310.010.68
Tox21_ATAD5909125592410.160.011.04
Tox211_AR-LBD859923887110.130.010.7
BZR40516409.880.060.99
Tox21_AHR81692247959.730.012.07
PROTEINS111321746.650.032.61
AIDS200022542.700
Mutagenicity433731751.7300.92
NCI141109170.4100.05
ENZYMES600220.3300
NCI10941277120.2900.05
FIRSTMM_DB4110000
OHSU7910000
KKI8310000
Peking_18510000
MSRC_21C20910000
MSRC_922110000
MSRC_2156310000
DD117810000
", + "bbox": [ + 176, + 218, + 825, + 790 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E PROOF OF CLAIM 7.1 ", + "text_level": 1, + "bbox": [ + 174, + 101, + 387, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. From the equation 3 we have: ", + "bbox": [ + 174, + 133, + 418, + 148 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/4402a3a1043de6381f652a329101219c19c1de54a287521d9bf9ead50204d107.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\csc ( Y _ { t e s t } ) = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) + \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { i s o } ) \\Rightarrow } \\\\ { \\displaystyle \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) > ( 1 - \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } ) \\mathrm { a c c } ( Y _ { i s o } ) \\Rightarrow } \\\\ { \\displaystyle \\operatorname { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { n e w } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 277, + 155, + 718, + 247 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "F PROOF OF CLAIM 7.2 ", + "text_level": 1, + "bbox": [ + 174, + 287, + 387, + 304 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. From the definition of the peering model we have: ", + "bbox": [ + 173, + 319, + 552, + 334 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/c8be42f8ee557dd1f8e61b0b3e954d2d2b176960ba63099e7f99911a14946b5f.jpg", + "text": "$$\n\\begin{array} { l } { \\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { n e w } ) = \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { n e w } ) } \\\\ { \\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { i s o } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { i s o } ) = 1 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 392, + 340, + 606, + 382 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Substituting these into the equation 3 we have: ", + "bbox": [ + 174, + 387, + 480, + 401 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ddd2cf0817fbfc278e4769d342721cfdc23901a475b28ff99d329f8d1408ad5a.jpg", + "text": "$$\n\\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { t e s t } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { t e s t } ) .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 407, + 598, + 428 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "G EXPERIMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 467, + 446, + 483 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "NN model is from $\\mathrm { X u }$ et al. (2018) and evaluate it on the data sets from PyTorch-Geometric (Fey & Lenssen, 2019). For each data set we perform 10-fold cross-validation such that each fold is evaluated on $10 \\%$ of hold-out instances $Y _ { t e s t }$ . For each fold we train the model for 350 epochs selecting the final model with the best performance on the validation set ( $20 \\%$ from hold-out trained split) across all epochs. Additionally we found that for small data set performance during the first epochs can be unstable on the validation set and thus we select our model only after the first 50 epochs. The final model is evaluated on the test instances and corresponds to NN in the experiments. Peering models NN-PH and NN-P are obtained from NN by replicating the target labels for homogeneous $Y _ { i s o }$ and non-homogeneous $Y _ { i s o }$ respectively. Weisfeiler-Lehman and Vertex histogram kernels are taken from the code2 of Sugiyama & Borgwardt (2015). We selected the height of subtree $h = 5$ for WL kernel. We train an SVM model selecting $C$ parameter from the range [0.001, 0.01, 0.1, 1, 10]. ", + "bbox": [ + 173, + 497, + 825, + 651 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "H NEW DATA SETS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 346, + 118 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/aef6de398781b89fd754fe0c01d255fcfa101b2205f747d30b377656bea678c4.jpg", + "table_caption": [ + "Table 8: Statistics for new clean data sets. Retention is the proportion of graphs remaining after cleaning procedure. Min. Class and Max. Class are minimum and maximum number of graphs in a class. Sorted by retention. " + ], + "table_footnote": [], + "table_body": "
Data setSizeRetention, %Avg.NodesAvg.EdgesClassesMin. ClassMax. Class
SYNTHETIC0000000
Cuneiform0000000
COIL-RAG130.335.779.62715
Letter-low120.537.175.42513
COX2_MD30.9930.33467212
DHFR_MD41.0225.25366.25213
Letter-med291.296.835.668110
Fingerprint511.8214.4513.12640
BZR_MD61.9614.17130.17215
Letter-high602.677.037.237132
ER_MD143.1418.07227.362113
IMDB-MULTI32121.422.35249.46385144
Tox21_ARE330246.0720.9621.8626022700
Tox21_ER356046.2522.2423.2924193141
Tox21_MMP340546.5222.423.4326182787
Tox21_HSE381446.821.2922.2822073607
Tox21-p53408847.3522.6223.7222913797
Tox21_PPAR-gamma387747.3721.9122.9121203757
Tox21_ATAD5431247.4322.5223.6121684144
Tox21_AR-LBD413448.0822.4123.4821503984
Tox21_AR450648.1322.9324.0521894317
Tox21_AHR393548.1722.8823.9824903445
Tox21ERLBD422448.2622.6923.821934031
Tox21_aromatase352448.7722.2423.2222343290
IMDB-BINARY49349.324.08221.962232261
COX223750.7542.1444.43268169
AIDS111055.518.2219.12310800
FRANKENSTEIN244856.4420.7822.35210201428
PTC_MM22667.2617.0417.74277149
BZR27668.1536.2538.81272204
PTC_MR23568.3117.2317.97296139
PTC_FM24269.3416.9617.66285157
MUTAG13571.8118.8520.8424293
PTC_FR25372.0817.1117.86286167
DHFR57876.4643.3745.532205373
Mutagenicity333576.932.9634.16214841851
COIL-DEL313380.3325.0564.2698139
COLLAB406481.2876.944667.9237702289
PROTEINS97587.643.4181.042343632
NCI1378592.0929.8432.37217812004
NCI109380192.129.6632.22218012000
ENZYMES59599.1732.4862.17698100
REDDIT-BINARY199899.9430.04996.4829981000
REDDIT-MULTI-12K1191799.9391.79914.68115132586
FIRSTMM_DB411001377.273073.931126
OHSU7910082.01199.6623544
KKI8310026.9648.4223746
Peking_18510039.3177.3523649
MSRC_21C20910040.2896.617129
MSRC_922110040.5897.9481930
Synthie40010091.6202.18490110
MSRC_2156310077.52198.32201034
DD1178100284.32715.662487691
REDDIT-MULTI-5K4999100508.511189.7559991000
", + "bbox": [ + 178, + 204, + 825, + 861 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "I PROOF OF THEOREM 6.1 ", + "text_level": 1, + "bbox": [ + 174, + 102, + 406, + 118 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. Let us prove Theorem 6.1. We denote by ${ \\mathcal { L } } = \\{ ( x , y ) \\to L ( f ( x ) , y ) : f \\in { \\mathcal { F } } \\}$ a composite loss class. For any $L \\in { \\mathcal { L } }$ we get that ", + "bbox": [ + 173, + 132, + 823, + 162 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/11e7b64037a3708fadef6349f9ae48405e23717790621e622f87c44a0f2de068.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\mathbb { P } } L - \\mathbb { E } _ { \\mathcal { D } } u L = \\mathbb { E } _ { \\mathbb { P } } L - \\mathbb { E } _ { \\mathbb { P } } u L + \\mathbb { E } _ { \\mathbb { P } } u L - \\mathbb { E } _ { \\mathcal { D } } u L \\leq } \\\\ { \\leq \\mathbb { E } _ { \\mathbb { P } } \\lvert ( 1 - u ) L \\rvert + ( \\mathbb { E } _ { \\mathbb { P } } u L - \\mathbb { E } _ { \\mathcal { D } } u L ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 167, + 673, + 205 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Since any $L \\in { \\mathcal { L } }$ is bounded from above by 1 for the first term in equation 5 we obtain ", + "bbox": [ + 173, + 217, + 741, + 233 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/3101761b14a2f4da5813ee55b81c800921d633c5870ade783cb52764fe971ca8.jpg", + "text": "$$\n\\begin{array} { r } { { { \\mathbb E } } _ { \\mathbb P } | ( 1 - u ) L | \\le { { \\mathbb E } } _ { \\mathbb P } g _ { + } ( w ) { { \\mathbb I } } _ { \\{ y = + 1 \\} } + { { \\mathbb E } } _ { \\mathbb P } g _ { - } ( w ) { { \\mathbb I } } _ { \\{ y = - 1 \\} } = g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 217, + 239, + 779, + 258 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Thanks to McDiarmid’d concentration inequality Mohri et al. (2012), applied to the function class ${ \\mathcal { L } } _ { u } = \\{ u L : L \\in { \\mathcal { L } } \\}$ , with probability $1 - \\delta , \\delta > 0$ for $\\mathcal { D } \\sim \\mathbb { P } ^ { N }$ we get the upper bound on the excess risk ", + "bbox": [ + 174, + 270, + 825, + 313 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/57d7b4a98f4fd6c66f5987469f6084c49a205d98b72bbac66a02dcd108af3017.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { L \\in \\mathcal { L } } { \\operatorname* { s u p } } ( \\mathbb { E } _ { \\mathbb { P } } u L - \\mathbb { E } _ { \\mathcal { D } } u L ) \\leq 2 \\mathcal { R } _ { N } ( \\mathcal { L } _ { u } ) + \\operatorname* { m a x } [ ( 1 + g _ { + } ( w ) ) , ( 1 + g _ { - } ( w ) ) ] \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } \\leq } \\\\ & { \\qquad \\leq 2 \\mathcal { R } _ { N } ( \\mathcal { L } _ { u } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 315, + 802, + 369 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Let us find a relation between $\\mathcal { R } _ { N } ( \\mathcal { L } _ { u } )$ and $\\mathcal { R } _ { N } ( \\mathcal { L } )$ . Here we denote by $z _ { i }$ a pair $z _ { i } = ( x _ { i } , y _ { i } )$ . By the definition (see Mohri et al. (2012)) the empirical Rademacher complexity ", + "bbox": [ + 171, + 380, + 825, + 410 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/4697b3fca3bc7af60a5ce737bfcd952fce97337ac0162197567907001fa81edd.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\hat { \\mathfrak { z } } _ { \\mathcal { P } } ( \\mathcal { L } _ { u } ) = \\displaystyle \\frac { 1 } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i = 1 } ^ { N } \\sigma _ { i } u ( z _ { i } ) L ( z _ { i } ) \\leq } \\\\ { \\displaystyle \\leq \\frac { 1 } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i = 1 } ^ { N } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { + } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { - } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\geq y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) \\leq } \\\\ { \\displaystyle \\leq \\widehat { \\mathcal { R } } _ { \\mathcal { D } } ( \\mathcal { L } ) + \\frac { g _ { + } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\geq y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { - } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\in \\mathcal { L } _ { i } : y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 415, + 830, + 547 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Since we use the zero-one loss, then ", + "bbox": [ + 174, + 553, + 413, + 566 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/cd09b9406c6ab503ef1e9ad760493a4dcbfc3cbe36e0c09dde31e54768466184.jpg", + "text": "$$\n\\mathbb { E } _ { \\sigma } \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) \\leq \\# \\{ i : y _ { i } = + 1 \\} , \\ \\mathbb { E } _ { \\sigma } \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) \\leq \\# \\{ i : y _ { i } = - 1 \\} .\n$$", + "text_format": "latex", + "bbox": [ + 199, + 573, + 795, + 607 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The Rademacher complexity ", + "bbox": [ + 174, + 613, + 364, + 628 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/948c0948403923c4352c413ed56f426b51107edd87ffcf20c4aea2a386880044.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\mathcal { R } _ { N } ( \\mathcal { L } _ { u } ) = \\mathbb { E } _ { \\mathcal { D } \\sim \\mathbb { P } ^ { N } } \\hat { \\mathcal { R } } _ { \\mathcal { D } } ( \\mathcal { L } _ { u } ) \\leq \\mathbb { E } _ { \\mathcal { D } \\sim \\mathbb { P } ^ { N } } \\Big [ \\hat { \\mathcal { R } } _ { \\mathcal { D } } ( \\mathcal { L } ) + \\frac { g _ { + } ( w ) } { N } \\# \\{ i : y _ { i } = + 1 \\} + } \\\\ & { } & { \\qquad + \\frac { g _ { - } ( w ) } { N } \\# \\{ i : y _ { i } = - 1 \\} \\Big ] = \\mathcal { R } _ { N } ( \\mathcal { L } ) + g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 635, + 753, + 699 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Using the fact that $\\begin{array} { r } { \\mathcal { R } _ { N } ( \\mathcal { L } ) = \\frac { 1 } { 2 } \\mathcal { R } _ { N } ( \\mathcal { F } ) } \\end{array}$ , substituting inequalities 6, 7 and 8 into equation 5, we get that ", + "bbox": [ + 171, + 712, + 826, + 741 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/3073a61d208e71204de32dc227c6de2bff90fa4a17ed6656a80132ae19108735.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\Big ( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\Big ) \\leq 3 ( g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) ) + } } \\\\ & { } & { + \\mathcal { R } _ { N } ( \\mathcal { F } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 744, + 754, + 801 + ], + "page_idx": 21 + } +] \ No newline at end of file diff --git a/parse/train/rJlUhhVYvS/rJlUhhVYvS_middle.json b/parse/train/rJlUhhVYvS/rJlUhhVYvS_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..8c01e792611e0ce0a54f18c2744d168834b9f2b1 --- /dev/null +++ b/parse/train/rJlUhhVYvS/rJlUhhVYvS_middle.json @@ -0,0 +1,46029 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "UNDERSTANDING ISOMORPHISM BIAS IN GRAPH", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 192, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 192, + 118 + ], + "score": 1.0, + "content": "DATA SETS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 201 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 201 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 213, + 468, + 389 + ], + "lines": [ + { + "bbox": [ + 141, + 212, + 469, + 225 + ], + "spans": [ + { + "bbox": [ + 141, + 212, + 469, + 225 + ], + "score": 1.0, + "content": "In recent years there has been a rapid increase in classification methods on graph", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 224, + 469, + 236 + ], + "spans": [ + { + "bbox": [ + 142, + 224, + 469, + 236 + ], + "score": 1.0, + "content": "structured data. 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The problem does not vanish even in consider-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "score": 1.0, + "content": "ation of node labels. This prevents fair competition of the algorithms and raises", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 324 + ], + "score": 1.0, + "content": "a question of the validity of the obtained results. 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Then we analyze", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 470, + 357 + ], + "score": 1.0, + "content": "54 data sets, previously extensively used for graph-related tasks, on the existence", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 470, + 368 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 470, + 368 + ], + "score": 1.0, + "content": "of isomorphism bias, give a set of recommendations to machine learning prac-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 367, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 367, + 469, + 378 + ], + "score": 1.0, + "content": "titioners to properly set up their models, and open source new data sets for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 378, + 222, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 222, + 389 + ], + "score": 1.0, + "content": "future experiments.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 206, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 208, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 208, + 428 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "Recently there has been an increasing interest in the development of machine learning models", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "that operate on graph structured data. Such models have found applications in chemoinformatics", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 460, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 475 + ], + "score": 1.0, + "content": "(Ralaivola et al. (2005); Rupp & Schneider (2010); Ferre et al. (2017)) and bioinformatics (Borg- ´", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 471, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 487 + ], + "score": 1.0, + "content": "wardt et al. (2005); Kundu et al. (2013)), neuroscience (Sharaev et al. (2018); Jie et al. (2016); Wang", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 483, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 504, + 495 + ], + "score": 1.0, + "content": "et al. (2016)), computer vision (Stumm et al. (2016)) and system security (Li et al. (2016)), natu-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "ral language processing (Glavas &ˇ Snajder (2013)), and others (Kriege et al. (2019); Nikolentzos ˇ", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "et al. (2019)). One of the popular tasks that encompasses these applications is graph classification", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 517, + 460, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 460, + 530 + ], + "score": 1.0, + "content": "problem for which many graph kernels and graph neural networks have been developed.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "One of the implicit assumptions that many practitioners adhere to is that models that can distin-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "guish isomorphic instances from non-isomorphic ones possess higher expressiveness in classifica-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "tion problem and hence much efforts have been devoted to incorporate efficient graph isomorphism", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "methods into the classification models. 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Both in graph kernels and graph neural networks, one of the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 140, + 233, + 470, + 249 + ], + "spans": [ + { + "bbox": [ + 140, + 233, + 470, + 249 + ], + "score": 1.0, + "content": "implicit assumptions of successful state-of-the-art models was that incorporating", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 246, + 469, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 469, + 258 + ], + "score": 1.0, + "content": "graph isomorphism features into the architecture leads to better empirical perfor-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 469, + 269 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 469, + 269 + ], + "score": 1.0, + "content": "mance. However, as we discover in this work, commonly used data sets for graph", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 280 + ], + "score": 1.0, + "content": "classification have repeating instances which cause the problem of isomorphism", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "score": 1.0, + "content": "bias, i.e. artificially increasing the accuracy of the models by memorizing target", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 470, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 302 + ], + "score": 1.0, + "content": "information from the training set. The problem does not vanish even in consider-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 470, + 312 + ], + "score": 1.0, + "content": "ation of node labels. This prevents fair competition of the algorithms and raises", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 324 + ], + "score": 1.0, + "content": "a question of the validity of the obtained results. First, we characterize this ef-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 335 + ], + "score": 1.0, + "content": "fect theoretically by reducing the graph classification to a weighted classification", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 470, + 346 + ], + "score": 1.0, + "content": "problem and estimating the corresponding generalization gap. Then we analyze", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 470, + 357 + ], + "score": 1.0, + "content": "54 data sets, previously extensively used for graph-related tasks, on the existence", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 470, + 368 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 470, + 368 + ], + "score": 1.0, + "content": "of isomorphism bias, give a set of recommendations to machine learning prac-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 367, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 367, + 469, + 378 + ], + "score": 1.0, + "content": "titioners to properly set up their models, and open source new data sets for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 378, + 222, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 222, + 389 + ], + "score": 1.0, + "content": "future experiments.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 140, + 212, + 470, + 389 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 206, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 208, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 208, + 428 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "Recently there has been an increasing interest in the development of machine learning models", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "that operate on graph structured data. Such models have found applications in chemoinformatics", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 460, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 475 + ], + "score": 1.0, + "content": "(Ralaivola et al. (2005); Rupp & Schneider (2010); Ferre et al. (2017)) and bioinformatics (Borg- ´", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 471, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 487 + ], + "score": 1.0, + "content": "wardt et al. (2005); Kundu et al. (2013)), neuroscience (Sharaev et al. (2018); Jie et al. (2016); Wang", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 483, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 504, + 495 + ], + "score": 1.0, + "content": "et al. (2016)), computer vision (Stumm et al. (2016)) and system security (Li et al. (2016)), natu-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "ral language processing (Glavas &ˇ Snajder (2013)), and others (Kriege et al. (2019); Nikolentzos ˇ", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "et al. (2019)). One of the popular tasks that encompasses these applications is graph classification", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 517, + 460, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 460, + 530 + ], + "score": 1.0, + "content": "problem for which many graph kernels and graph neural networks have been developed.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 439, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "One of the implicit assumptions that many practitioners adhere to is that models that can distin-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "guish isomorphic instances from non-isomorphic ones possess higher expressiveness in classifica-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "tion problem and hence much efforts have been devoted to incorporate efficient graph isomorphism", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "methods into the classification models. As the problem of computing complete graph invariant is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 119, + 588 + ], + "score": 0.72, + "content": "\\mathbb { G } \\mathbb { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "-hard (Gartner et al. (2003)), for which no known polynomial-time algorithm exists, other heuris- ¨", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "tics have been proposed as a proxy for deciding whether two graphs are isomorphic. Indeed, from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "score": 1.0, + "content": "the early days topological descriptors such Wiener index (Wiener (1947a;b)) attempted to find a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "single number that uniquely identifies a graph. Later, graph kernels that model pairwise similarities", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "between graphs utilized theoretical developments in graph isomorphism literature. 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Since only for", + "type": "text" + }, + { + "bbox": [ + 403, + 687, + 444, + 700 + ], + "score": 0.93, + "content": "k = \\Omega ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "WL algorithm", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "is guaranteed to distinguish all graphs (for which the running time becomes exponential; see also", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Furer (2017)), in the general case WL algorithm can be used only as a strong baseline for graph ¨", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "isomorphism. 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(2019)).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Experimental evaluation reveals that models based on the theoretical constructions with high com-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "score": 1.0, + "content": "binatorial power such as WL algorithm performs better than the models without them such as Vertex", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "histogram kernel (Vishwanathan et al. (2010)) on a commonly used data sets. This could add addi-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "tional bias to results of comparison of classification algorithms since the models could simply apply", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "a graph isomorphism method (or an efficient approximation) to determine a target label at the infer-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "ence time. However, purely judging on the accuracy of the algorithms in such cases would imply", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "an unfair comparison between the methods as it does not measure correctly generalization ability of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "the models on the new test instances. As we discover, indeed many of the data sets used in graph", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "classification have isomorphic instances so much that in some of them the fraction of the unique", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 243, + 222 + ], + "score": 1.0, + "content": "non-repeating graphs is as low as", + "type": "text" + }, + { + "bbox": [ + 244, + 209, + 263, + 220 + ], + "score": 0.86, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "of the total size. This challenges previous experimental re-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "sults and requires understanding of how influential isomorphic instances on the final performance of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 246, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 246, + 243 + ], + "score": 1.0, + "content": "the models. Our contributions are:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 132, + 252, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 133, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 133, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "• We analyze the quality of 54 graph data sets which are used ubiquitously in graph classifica-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 142, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "tion comparison. Our findings suggest that in the most of the data sets there are isomorphic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 357, + 286 + ], + "score": 1.0, + "content": "graphs and their proportion varies from as much as", + "type": "text" + }, + { + "bbox": [ + 357, + 275, + 381, + 285 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 275, + 394, + 286 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 394, + 275, + 409, + 285 + ], + "score": 0.85, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 275, + 505, + 286 + ], + "score": 1.0, + "content": ". Surprisingly, we also", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 285, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 504, + 298 + ], + "score": 1.0, + "content": "found that there are isomorphic instances that have different target labels suggesting they", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 297, + 324, + 308 + ], + "spans": [ + { + "bbox": [ + 142, + 297, + 324, + 308 + ], + "score": 1.0, + "content": "are not suitable for learning a classifier at all.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "We investigate the causes of isomorphic graphs and show that node and edge labels are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "important to identify isomorphic graphs. Other causes include numerical attributes of nodes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 334, + 322, + 344 + ], + "spans": [ + { + "bbox": [ + 142, + 334, + 322, + 344 + ], + "score": 1.0, + "content": "and edges as well as the sizes of the data set.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 140, + 347, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 140, + 347, + 504, + 360 + ], + "score": 1.0, + "content": "We express an upper bound for the generalization gap through the Radamacher complexity", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 358, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 504, + 371 + ], + "score": 1.0, + "content": "of a classifier and the number of isomorphic graphs in a data set. This bound presents the-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 142, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "oretical evidence on how weightning of each graph in the training influences classification", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 381, + 182, + 393 + ], + "spans": [ + { + "bbox": [ + 141, + 381, + 182, + 393 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 137, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 137, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "We evaluate a classification model’s performance on isomorphic instances and show that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 407, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 141, + 407, + 505, + 418 + ], + "score": 1.0, + "content": "even strong models do not achieve optimal accuracy even if the instances have been seen at", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "the training time. Hence we show a model-agnostic way to artificially increase performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 429, + 275, + 441 + ], + "spans": [ + { + "bbox": [ + 141, + 429, + 275, + 441 + ], + "score": 1.0, + "content": "on several widely used data sets.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 139, + 444, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 139, + 444, + 505, + 455 + ], + "score": 1.0, + "content": "We open-source new cleaned data sets that contain only non-isomorphic instances with no", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "noisy target labels. We give a set of recommendations regarding applying new models that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 465, + 274, + 477 + ], + "spans": [ + { + "bbox": [ + 142, + 465, + 274, + 477 + ], + "score": 1.0, + "content": "work with graph structured data.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 209, + 505 + ], + "lines": [ + { + "bbox": [ + 104, + 491, + 210, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 210, + 507 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "Measuring quality of data sets. A similar issue of duplicates instances in commonly used data sets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "was recently discovered in computer vision domain. Recht et al. (2019); Barz & Denzler (2019);", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 459, + 552 + ], + "score": 1.0, + "content": "Birodkar et al. (2019) discover that image data sets CIFAR and ImageNet contain at least", + "type": "text" + }, + { + "bbox": [ + 460, + 540, + 479, + 550 + ], + "score": 0.88, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "duplicate images in the test test invalidating previous performance and questioning generalization", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "abilities of previous successful architectures. 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In graph", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "domain, a fresh look into understanding of expressiveness of graph kernels and the quality of data", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "sets has been considered in Kriege et al. (2019), where an extensive comparison of existing graph", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 617, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 628 + ], + "score": 1.0, + "content": "kernels is done and a few insights about models’ behavior are suggested. In contrast, we conduct a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "broader study of isomorphism metrics, revealing all isomorphism pairs in proposed 54 data sets, and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "propose new cleaned data. 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(2019)) and graph neural", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "networks (Wu et al. (2019)) are two competing paradigms for designing graph representations and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "solving graph classification and have significantly advanced empirical results due to more efficient", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "algorithms, incorporating graph invariance into the models, and end-to-end training. Several papers", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "have tried to justify performance of different families of methods by studying different statistical", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 105, + 82, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 262, + 95 + ], + "score": 1.0, + "content": "have been shown to be as powerful as", + "type": "text" + }, + { + "bbox": [ + 262, + 83, + 269, + 93 + ], + "score": 0.49, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "-dimensional WL algorithm (see for example Maron et al.", + "type": "text" + } + ], + "index": 0, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 93, + 293, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 293, + 106 + ], + "score": 1.0, + "content": "(2019); Xu et al. 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(2010)) on a commonly used data sets. This could add addi-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "tional bias to results of comparison of classification algorithms since the models could simply apply", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "a graph isomorphism method (or an efficient approximation) to determine a target label at the infer-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "ence time. However, purely judging on the accuracy of the algorithms in such cases would imply", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "an unfair comparison between the methods as it does not measure correctly generalization ability of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "the models on the new test instances. 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This challenges previous experimental re-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "sults and requires understanding of how influential isomorphic instances on the final performance of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 246, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 246, + 243 + ], + "score": 1.0, + "content": "the models. Our contributions are:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 110, + 506, + 243 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 252, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 133, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 133, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "• We analyze the quality of 54 graph data sets which are used ubiquitously in graph classifica-", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 142, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "tion comparison. Our findings suggest that in the most of the data sets there are isomorphic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 357, + 286 + ], + "score": 1.0, + "content": "graphs and their proportion varies from as much as", + "type": "text" + }, + { + "bbox": [ + 357, + 275, + 381, + 285 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 275, + 394, + 286 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 394, + 275, + 409, + 285 + ], + "score": 0.85, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 275, + 505, + 286 + ], + "score": 1.0, + "content": ". Surprisingly, we also", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 285, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 504, + 298 + ], + "score": 1.0, + "content": "found that there are isomorphic instances that have different target labels suggesting they", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 297, + 324, + 308 + ], + "spans": [ + { + "bbox": [ + 142, + 297, + 324, + 308 + ], + "score": 1.0, + "content": "are not suitable for learning a classifier at all.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 140, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "We investigate the causes of isomorphic graphs and show that node and edge labels are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "important to identify isomorphic graphs. Other causes include numerical attributes of nodes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 334, + 322, + 344 + ], + "spans": [ + { + "bbox": [ + 142, + 334, + 322, + 344 + ], + "score": 1.0, + "content": "and edges as well as the sizes of the data set.", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 347, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 140, + 347, + 504, + 360 + ], + "score": 1.0, + "content": "We express an upper bound for the generalization gap through the Radamacher complexity", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 358, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 504, + 371 + ], + "score": 1.0, + "content": "of a classifier and the number of isomorphic graphs in a data set. This bound presents the-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 142, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "oretical evidence on how weightning of each graph in the training influences classification", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 381, + 182, + 393 + ], + "spans": [ + { + "bbox": [ + 141, + 381, + 182, + 393 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 137, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 137, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "We evaluate a classification model’s performance on isomorphic instances and show that", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 407, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 141, + 407, + 505, + 418 + ], + "score": 1.0, + "content": "even strong models do not achieve optimal accuracy even if the instances have been seen at", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "the training time. Hence we show a model-agnostic way to artificially increase performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 429, + 275, + 441 + ], + "spans": [ + { + "bbox": [ + 141, + 429, + 275, + 441 + ], + "score": 1.0, + "content": "on several widely used data sets.", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 444, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 139, + 444, + 505, + 455 + ], + "score": 1.0, + "content": "We open-source new cleaned data sets that contain only non-isomorphic instances with no", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "noisy target labels. We give a set of recommendations regarding applying new models that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 465, + 274, + 477 + ], + "spans": [ + { + "bbox": [ + 142, + 465, + 274, + 477 + ], + "score": 1.0, + "content": "work with graph structured data.", + "type": "text" + } + ], + "index": 32, + "is_list_end_line": true + } + ], + "index": 23, + "bbox_fs": [ + 133, + 253, + 506, + 477 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 209, + 505 + ], + "lines": [ + { + "bbox": [ + 104, + 491, + 210, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 210, + 507 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "Measuring quality of data sets. A similar issue of duplicates instances in commonly used data sets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "was recently discovered in computer vision domain. Recht et al. (2019); Barz & Denzler (2019);", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 459, + 552 + ], + "score": 1.0, + "content": "Birodkar et al. (2019) discover that image data sets CIFAR and ImageNet contain at least", + "type": "text" + }, + { + "bbox": [ + 460, + 540, + 479, + 550 + ], + "score": 0.88, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "duplicate images in the test test invalidating previous performance and questioning generalization", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "abilities of previous successful architectures. In particular, evaluating the models in new test sets", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 273, + 584 + ], + "score": 1.0, + "content": "shows a drop of accuracy by as much as", + "type": "text" + }, + { + "bbox": [ + 274, + 572, + 293, + 583 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "(Recht et al., 2019), which is explained by models’", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 582, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 504, + 595 + ], + "score": 1.0, + "content": "incapability to generalize to unseen slightly ”harder” instances than in the original test sets. In graph", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "domain, a fresh look into understanding of expressiveness of graph kernels and the quality of data", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "sets has been considered in Kriege et al. (2019), where an extensive comparison of existing graph", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 617, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 628 + ], + "score": 1.0, + "content": "kernels is done and a few insights about models’ behavior are suggested. In contrast, we conduct a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "broader study of isomorphism metrics, revealing all isomorphism pairs in proposed 54 data sets, and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "propose new cleaned data. Additionally we also consider graph neural network performance and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "argue that current data sets present isomorphism bias which can artificially boost evaluation metrics", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 659, + 209, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 209, + 673 + ], + "score": 1.0, + "content": "in a model-agnostic way.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 518, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Explaining performance of graph models. Graph kernels (Kriege et al. (2019)) and graph neural", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "networks (Wu et al. (2019)) are two competing paradigms for designing graph representations and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "solving graph classification and have significantly advanced empirical results due to more efficient", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "algorithms, incorporating graph invariance into the models, and end-to-end training. Several papers", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "have tried to justify performance of different families of methods by studying different statistical", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "properties. For example, in Ying et al. (2019) by maximizing mutual information between explana-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "tion variables and predicted label distribution, the model is trained to return a small subgraph and the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "graph-specific attributes that are the most influential on the decision made by a GNN, which allows", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "inspection of single- and multi-level predictions in an agnostic manner for GNNs. In another work", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 461, + 140 + ], + "score": 1.0, + "content": "(Scarselli et al. 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Furthermore, stability and generalization properties of convolutional", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "GNNs have been shown to depend on the largest eigenvalue of the graph filter and therefore are", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "attained for properly normalized graph convolutions such as symmetric normalized graph Laplacian", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "(Verma & Zhang (2019)). Finally, expressivity of graph kernels has been studied from statistical", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "learning theory (Oneto et al. (2017b)) and property testing (Kriege et al. (2018b)), showing that", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "graph kernels can capture certain graph properties such as planarity, girth, and chromatic number", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "(Johansson et al. (2014)). Our approach is complementary to all of the above as we analyze if the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 390, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 390, + 239 + ], + "score": 1.0, + "content": "data sets used in experiments have any effect on the final performance.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 676, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "properties. For example, in Ying et al. (2019) by maximizing mutual information between explana-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "tion variables and predicted label distribution, the model is trained to return a small subgraph and the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "graph-specific attributes that are the most influential on the decision made by a GNN, which allows", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "inspection of single- and multi-level predictions in an agnostic manner for GNNs. 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(2018)), the VC dimension of GNNs models has been shown to grow as", + "type": "text" + }, + { + "bbox": [ + 461, + 126, + 501, + 138 + ], + "score": 0.92, + "content": "\\mathcal { O } ( p ^ { 4 } N ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 124, + 506, + 140 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 133, + 150 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 139, + 141, + 149 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 137, + 312, + 150 + ], + "score": 1.0, + "content": "is the number of network parameters and", + "type": "text" + }, + { + "bbox": [ + 313, + 138, + 323, + 147 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "is the number of nodes in a graph, which is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "comparable to RNN models. 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Isomorphism between two graphs", + "type": "text" + }, + { + "bbox": [ + 340, + 644, + 403, + 656 + ], + "score": 0.93, + "content": "G _ { 1 } = ( V _ { 1 } , E _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 643, + 423, + 657 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 644, + 486, + 655 + ], + "score": 0.91, + "content": "G _ { 2 } = ( V _ { 2 } , E _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 180, + 668 + ], + "score": 1.0, + "content": "bijective function", + "type": "text" + }, + { + "bbox": [ + 180, + 655, + 235, + 667 + ], + "score": 0.93, + "content": "\\phi : V _ { 1 } \\mapsto V _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 654, + 316, + 668 + ], + "score": 1.0, + "content": "such that any edge", + "type": "text" + }, + { + "bbox": [ + 316, + 655, + 365, + 667 + ], + "score": 0.92, + "content": "( u , v ) \\in E _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 654, + 424, + 668 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 424, + 655, + 501, + 667 + ], + "score": 0.88, + "content": "( \\phi ( u ) , \\phi ( v ) ) \\in E _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 654, + 506, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 433, + 679 + ], + "score": 1.0, + "content": "Graph isomorphism problem asks if such function exists for given two graphs", + "type": "text" + }, + { + "bbox": [ + 434, + 667, + 447, + 677 + ], + "score": 0.87, + "content": "G _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 665, + 468, + 679 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 468, + 667, + 481, + 677 + ], + "score": 0.86, + "content": "G _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 665, + 505, + 679 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 222, + 689 + ], + "score": 1.0, + "content": "denote isomorphic graphs as", + "type": "text" + }, + { + "bbox": [ + 222, + 677, + 261, + 688 + ], + "score": 0.92, + "content": "G _ { 1 } \\cong G _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 677, + 423, + 689 + ], + "score": 1.0, + "content": ". The problem has efficient algorithms in", + "type": "text" + }, + { + "bbox": [ + 423, + 677, + 431, + 687 + ], + "score": 0.74, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "for certain classes", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 360, + 701 + ], + "score": 1.0, + "content": "of graphs such as planar or bounded-degree graphs (Hopcroft", + "type": "text" + }, + { + "bbox": [ + 361, + 688, + 370, + 698 + ], + "score": 0.36, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "Wong (1974); Luks (1980)), but", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "in the general case admits only quasi-polynomial algorithm (Babai (2015)). In practice many GI", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "solvers are based on individualization-refinement paradigm (Mckay & Piperno (2014)), which for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "each graph iteratively updates a permutation of the nodes such that the resulted permutations of two", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "graphs are identical if an only if they are isomorphic. Importantly, while finding such canonical", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "permutation of a graph is at least as hard as solving GI problem, state-of-the-art solvers tackle", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "majority of pairs of graphs efficiently, only taking exponential time on the specific hard instances of", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 388, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 388, + 128 + ], + "score": 1.0, + "content": "graphs that possess highly symmetrical structures (Cai et al. (1992b)).", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "graphs are identical if an only if they are isomorphic. 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(1992b)).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 143, + 343, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 343, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 343, + 156 + ], + "score": 1.0, + "content": "4 IDENTIFYING ISOMORPHISM IN DATA SETS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 166, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 104, + 165, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 165, + 505, + 180 + ], + "score": 1.0, + "content": "To distinguish between different isomorphic graphs inside a data set we use the notion of graph", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 175, + 136, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 136, + 191 + ], + "score": 1.0, + "content": "orbits:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 505, + 235 + ], + "lines": [ + { + "bbox": [ + 104, + 188, + 508, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 188, + 244, + 205 + ], + "score": 1.0, + "content": "Definition 4.1 (Graph orbit). Let", + "type": "text" + }, + { + "bbox": [ + 244, + 190, + 314, + 203 + ], + "score": 0.93, + "content": "\\mathcal { D } = \\{ G _ { i } , y _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 188, + 508, + 205 + ], + "score": 1.0, + "content": "be a data set of graphs and target labels. For a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 202, + 504, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 131, + 215 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 202, + 144, + 213 + ], + "score": 0.91, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 202, + 178, + 215 + ], + "score": 1.0, + "content": "let a set", + "type": "text" + }, + { + "bbox": [ + 178, + 202, + 224, + 214 + ], + "score": 0.92, + "content": "o _ { i } = \\{ G _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 202, + 367, + 215 + ], + "score": 1.0, + "content": "be a set of all isomorphic graphs in", + "type": "text" + }, + { + "bbox": [ + 368, + 203, + 378, + 212 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 202, + 388, + 215 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 389, + 202, + 401, + 213 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 202, + 445, + 215 + ], + "score": 1.0, + "content": ", including", + "type": "text" + }, + { + "bbox": [ + 445, + 202, + 457, + 213 + ], + "score": 0.87, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 202, + 494, + 215 + ], + "score": 1.0, + "content": ". We call", + "type": "text" + }, + { + "bbox": [ + 495, + 203, + 504, + 213 + ], + "score": 0.81, + "content": "o _ { i }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 179, + 225 + ], + "score": 1.0, + "content": "the orbit of graph", + "type": "text" + }, + { + "bbox": [ + 179, + 214, + 191, + 224 + ], + "score": 0.87, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 213, + 203, + 225 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 203, + 214, + 212, + 223 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 213, + 506, + 225 + ], + "score": 1.0, + "content": ". The cardinality of the orbit is called orbit size. An orbit with size one is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 224, + 163, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 163, + 235 + ], + "score": 1.0, + "content": "called trivial.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "In a data set with no isomorphic graphs, the number of orbits equals to the number of graphs in a data", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 122, + 266 + ], + "score": 1.0, + "content": "set,", + "type": "text" + }, + { + "bbox": [ + 123, + 255, + 133, + 264 + ], + "score": 0.7, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 254, + 506, + 266 + ], + "score": 1.0, + "content": ". Hence, the more orbits in a data set, the ”cleaner” it is. Note however that the distribution of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 266, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 277 + ], + "score": 1.0, + "content": "orbit sizes in two different data sets can vary even if they have the same number of orbits. Therefore,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 327, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 327, + 288 + ], + "score": 1.0, + "content": "we look at additional metrics that describe the data set.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 132, + 296, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 132, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 132, + 296, + 142, + 308 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 297, + 149, + 306 + ], + "score": 0.56, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 296, + 505, + 308 + ], + "score": 1.0, + "content": ", aggregated number of graphs that belong to an orbit of size greater than one, i.e. those", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 307, + 340, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 340, + 320 + ], + "score": 1.0, + "content": "graphs that isomorphic counterparts in a data set;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 131, + 319, + 426, + 336 + ], + "spans": [ + { + "bbox": [ + 131, + 319, + 142, + 336 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 321, + 161, + 333 + ], + "score": 0.87, + "content": "I , \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 319, + 410, + 336 + ], + "score": 1.0, + "content": ", proportion of isomorphic graphs to the total data set size, i.e.", + "type": "text" + }, + { + "bbox": [ + 411, + 320, + 421, + 335 + ], + "score": 0.88, + "content": "\\textstyle { \\frac { I } { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 319, + 426, + 336 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 132, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 132, + 334, + 142, + 348 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 335, + 168, + 346 + ], + "score": 0.87, + "content": "I P , \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 334, + 505, + 348 + ], + "score": 1.0, + "content": ", proportion of isomorphic pairs to the total number of graph pairs in a data set", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 139, + 341, + 189, + 363 + ], + "spans": [ + { + "bbox": [ + 139, + 341, + 146, + 363 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 146, + 347, + 181, + 362 + ], + "score": 0.91, + "content": "\\frac { N ( N - 1 ) } { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 341, + 189, + 363 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 102, + 365, + 509, + 388 + ], + "spans": [ + { + "bbox": [ + 102, + 365, + 306, + 388 + ], + "score": 1.0, + "content": "If we consider target labels of graphs in a data set", + "type": "text" + }, + { + "bbox": [ + 306, + 370, + 375, + 382 + ], + "score": 0.93, + "content": "\\mathcal { D } = \\{ G _ { i } , y _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 365, + 509, + 388 + ], + "score": 1.0, + "content": "we can also measure agreement", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 315, + 395 + ], + "score": 1.0, + "content": "between the labels of two isomorphic data set. If", + "type": "text" + }, + { + "bbox": [ + 316, + 382, + 359, + 393 + ], + "score": 0.91, + "content": "G _ { 1 } \\cong G _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 380, + 379, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 379, + 382, + 416, + 393 + ], + "score": 0.91, + "content": "y _ { 1 } ~ \\ne ~ y _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 380, + 505, + 395 + ], + "score": 1.0, + "content": ", then we call graphs", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 387, + 405 + ], + "score": 1.0, + "content": "mismatched. Note that if there is more than one target label in an orbit", + "type": "text" + }, + { + "bbox": [ + 387, + 395, + 393, + 402 + ], + "score": 0.76, + "content": "o", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 392, + 506, + 405 + ], + "score": 1.0, + "content": ", then all graphs in this orbit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "are mismatched. To obtain isomorphic graphs, we run nauty algorithm (Mckay & Piperno, 2014) on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "all possible pairs of graphs in a data set. We substantially reduce the number of calls between the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 456, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 456, + 438 + ], + "score": 1.0, + "content": "graphs by verifying that a pair has the same number of nodes and edges before the call.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "The metrics are presented in Table 2 for top-10 data sets and in Table 6 (see the appendix) for all data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 412, + 465 + ], + "score": 1.0, + "content": "sets. The graphs in Table 2 are sorted by the proportion of isomorphic graphs", + "type": "text" + }, + { + "bbox": [ + 412, + 453, + 426, + 464 + ], + "score": 0.88, + "content": "I \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 453, + 505, + 465 + ], + "score": 1.0, + "content": ". The results for the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "first Top-10 data sets are somewhat surprising: almost all graphs in the selected data sets have other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "isomorphic graphs. If we look at all data sets in Table 6, we see that the proportion of isomorphic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 245, + 499 + ], + "score": 1.0, + "content": "graphs in the data sets varies from", + "type": "text" + }, + { + "bbox": [ + 245, + 486, + 270, + 497 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 486, + 281, + 499 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 281, + 486, + 295, + 496 + ], + "score": 0.86, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 486, + 383, + 499 + ], + "score": 1.0, + "content": ". However, more than", + "type": "text" + }, + { + "bbox": [ + 384, + 486, + 403, + 497 + ], + "score": 0.84, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "of the analyzed data sets", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 322, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 159, + 510 + ], + "score": 1.0, + "content": "have at least", + "type": "text" + }, + { + "bbox": [ + 160, + 497, + 179, + 508 + ], + "score": 0.75, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 497, + 322, + 510 + ], + "score": 1.0, + "content": "of the graphs in a non-trivial orbit.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "table", + "bbox": [ + 113, + 563, + 497, + 693 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 527, + 503, + 560 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 526, + 501, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 486, + 540 + ], + "score": 1.0, + "content": "Table 2: Isomorphic metrics for Top-10 data sets based on the proportion of isomorphic graphs", + "type": "text" + }, + { + "bbox": [ + 487, + 527, + 501, + 538 + ], + "score": 0.87, + "content": "I \\%", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 538, + 129, + 549 + ], + "score": 0.89, + "content": "I P \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 537, + 367, + 551 + ], + "score": 1.0, + "content": "is the proportion of isomorphic pairs of graphs, Mismatched", + "type": "text" + }, + { + "bbox": [ + 367, + 538, + 377, + 549 + ], + "score": 0.86, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "is the proportion of mismatched", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 136, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 136, + 562 + ], + "score": 1.0, + "content": "labels.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "table_body", + "bbox": [ + 113, + 563, + 497, + 693 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 563, + 497, + 693 + ], + "spans": [ + { + "bbox": [ + 113, + 563, + 497, + 693 + ], + "score": 0.985, + "html": "
data setSize, NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
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Letter-low225032224599.788.7296.22
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", + "type": "table", + "image_path": "a1599045762f9b49ec73a18a7b7256607709a6320b194d51681cde5de152ccae.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 113, + 563, + 497, + 606.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 113, + 606.3333333333334, + 497, + 649.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 113, + 649.6666666666667, + 497, + 693.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Another surprising observation is that the proportion of mismatched graphs is significant, ranging", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 129, + 733 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 721, + 154, + 731 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 720, + 165, + 733 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 166, + 721, + 180, + 731 + ], + "score": 0.86, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ". This clearly indicates that such graphs are not suitable for graph classification", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 505, + 128 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 143, + 343, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 343, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 343, + 156 + ], + "score": 1.0, + "content": "4 IDENTIFYING ISOMORPHISM IN DATA SETS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 166, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 104, + 165, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 165, + 505, + 180 + ], + "score": 1.0, + "content": "To distinguish between different isomorphic graphs inside a data set we use the notion of graph", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 175, + 136, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 136, + 191 + ], + "score": 1.0, + "content": "orbits:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 165, + 505, + 191 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 505, + 235 + ], + "lines": [ + { + "bbox": [ + 104, + 188, + 508, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 188, + 244, + 205 + ], + "score": 1.0, + "content": "Definition 4.1 (Graph orbit). Let", + "type": "text" + }, + { + "bbox": [ + 244, + 190, + 314, + 203 + ], + "score": 0.93, + "content": "\\mathcal { D } = \\{ G _ { i } , y _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 188, + 508, + 205 + ], + "score": 1.0, + "content": "be a data set of graphs and target labels. For a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 202, + 504, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 131, + 215 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 202, + 144, + 213 + ], + "score": 0.91, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 202, + 178, + 215 + ], + "score": 1.0, + "content": "let a set", + "type": "text" + }, + { + "bbox": [ + 178, + 202, + 224, + 214 + ], + "score": 0.92, + "content": "o _ { i } = \\{ G _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 202, + 367, + 215 + ], + "score": 1.0, + "content": "be a set of all isomorphic graphs in", + "type": "text" + }, + { + "bbox": [ + 368, + 203, + 378, + 212 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 202, + 388, + 215 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 389, + 202, + 401, + 213 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 202, + 445, + 215 + ], + "score": 1.0, + "content": ", including", + "type": "text" + }, + { + "bbox": [ + 445, + 202, + 457, + 213 + ], + "score": 0.87, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 202, + 494, + 215 + ], + "score": 1.0, + "content": ". We call", + "type": "text" + }, + { + "bbox": [ + 495, + 203, + 504, + 213 + ], + "score": 0.81, + "content": "o _ { i }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 179, + 225 + ], + "score": 1.0, + "content": "the orbit of graph", + "type": "text" + }, + { + "bbox": [ + 179, + 214, + 191, + 224 + ], + "score": 0.87, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 213, + 203, + 225 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 203, + 214, + 212, + 223 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 213, + 506, + 225 + ], + "score": 1.0, + "content": ". The cardinality of the orbit is called orbit size. An orbit with size one is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 224, + 163, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 163, + 235 + ], + "score": 1.0, + "content": "called trivial.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 188, + 508, + 235 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "In a data set with no isomorphic graphs, the number of orbits equals to the number of graphs in a data", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 122, + 266 + ], + "score": 1.0, + "content": "set,", + "type": "text" + }, + { + "bbox": [ + 123, + 255, + 133, + 264 + ], + "score": 0.7, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 254, + 506, + 266 + ], + "score": 1.0, + "content": ". Hence, the more orbits in a data set, the ”cleaner” it is. Note however that the distribution of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 266, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 277 + ], + "score": 1.0, + "content": "orbit sizes in two different data sets can vary even if they have the same number of orbits. Therefore,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 327, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 327, + 288 + ], + "score": 1.0, + "content": "we look at additional metrics that describe the data set.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 243, + 506, + 288 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 296, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 132, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 132, + 296, + 142, + 308 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 297, + 149, + 306 + ], + "score": 0.56, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 296, + 505, + 308 + ], + "score": 1.0, + "content": ", aggregated number of graphs that belong to an orbit of size greater than one, i.e. those", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 307, + 340, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 340, + 320 + ], + "score": 1.0, + "content": "graphs that isomorphic counterparts in a data set;", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 319, + 426, + 336 + ], + "spans": [ + { + "bbox": [ + 131, + 319, + 142, + 336 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 321, + 161, + 333 + ], + "score": 0.87, + "content": "I , \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 319, + 410, + 336 + ], + "score": 1.0, + "content": ", proportion of isomorphic graphs to the total data set size, i.e.", + "type": "text" + }, + { + "bbox": [ + 411, + 320, + 421, + 335 + ], + "score": 0.88, + "content": "\\textstyle { \\frac { I } { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 319, + 426, + 336 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 132, + 334, + 142, + 348 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 335, + 168, + 346 + ], + "score": 0.87, + "content": "I P , \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 334, + 505, + 348 + ], + "score": 1.0, + "content": ", proportion of isomorphic pairs to the total number of graph pairs in a data set", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 139, + 341, + 189, + 363 + ], + "spans": [ + { + "bbox": [ + 139, + 341, + 146, + 363 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 146, + 347, + 181, + 362 + ], + "score": 0.91, + "content": "\\frac { N ( N - 1 ) } { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 341, + 189, + 363 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + } + ], + "index": 17, + "bbox_fs": [ + 131, + 296, + 505, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 102, + 365, + 509, + 388 + ], + "spans": [ + { + "bbox": [ + 102, + 365, + 306, + 388 + ], + "score": 1.0, + "content": "If we consider target labels of graphs in a data set", + "type": "text" + }, + { + "bbox": [ + 306, + 370, + 375, + 382 + ], + "score": 0.93, + "content": "\\mathcal { D } = \\{ G _ { i } , y _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 365, + 509, + 388 + ], + "score": 1.0, + "content": "we can also measure agreement", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 315, + 395 + ], + "score": 1.0, + "content": "between the labels of two isomorphic data set. If", + "type": "text" + }, + { + "bbox": [ + 316, + 382, + 359, + 393 + ], + "score": 0.91, + "content": "G _ { 1 } \\cong G _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 380, + 379, + 395 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 379, + 382, + 416, + 393 + ], + "score": 0.91, + "content": "y _ { 1 } ~ \\ne ~ y _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 380, + 505, + 395 + ], + "score": 1.0, + "content": ", then we call graphs", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 387, + 405 + ], + "score": 1.0, + "content": "mismatched. Note that if there is more than one target label in an orbit", + "type": "text" + }, + { + "bbox": [ + 387, + 395, + 393, + 402 + ], + "score": 0.76, + "content": "o", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 392, + 506, + 405 + ], + "score": 1.0, + "content": ", then all graphs in this orbit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "are mismatched. To obtain isomorphic graphs, we run nauty algorithm (Mckay & Piperno, 2014) on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "all possible pairs of graphs in a data set. We substantially reduce the number of calls between the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 456, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 456, + 438 + ], + "score": 1.0, + "content": "graphs by verifying that a pair has the same number of nodes and edges before the call.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 102, + 365, + 509, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "The metrics are presented in Table 2 for top-10 data sets and in Table 6 (see the appendix) for all data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 412, + 465 + ], + "score": 1.0, + "content": "sets. The graphs in Table 2 are sorted by the proportion of isomorphic graphs", + "type": "text" + }, + { + "bbox": [ + 412, + 453, + 426, + 464 + ], + "score": 0.88, + "content": "I \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 453, + 505, + 465 + ], + "score": 1.0, + "content": ". The results for the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "first Top-10 data sets are somewhat surprising: almost all graphs in the selected data sets have other", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "isomorphic graphs. If we look at all data sets in Table 6, we see that the proportion of isomorphic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 245, + 499 + ], + "score": 1.0, + "content": "graphs in the data sets varies from", + "type": "text" + }, + { + "bbox": [ + 245, + 486, + 270, + 497 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 486, + 281, + 499 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 281, + 486, + 295, + 496 + ], + "score": 0.86, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 486, + 383, + 499 + ], + "score": 1.0, + "content": ". However, more than", + "type": "text" + }, + { + "bbox": [ + 384, + 486, + 403, + 497 + ], + "score": 0.84, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "of the analyzed data sets", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 322, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 159, + 510 + ], + "score": 1.0, + "content": "have at least", + "type": "text" + }, + { + "bbox": [ + 160, + 497, + 179, + 508 + ], + "score": 0.75, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 497, + 322, + 510 + ], + "score": 1.0, + "content": "of the graphs in a non-trivial orbit.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 442, + 506, + 510 + ] + }, + { + "type": "table", + "bbox": [ + 113, + 563, + 497, + 693 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 527, + 503, + 560 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 526, + 501, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 486, + 540 + ], + "score": 1.0, + "content": "Table 2: Isomorphic metrics for Top-10 data sets based on the proportion of isomorphic graphs", + "type": "text" + }, + { + "bbox": [ + 487, + 527, + 501, + 538 + ], + "score": 0.87, + "content": "I \\%", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 538, + 129, + 549 + ], + "score": 0.89, + "content": "I P \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 537, + 367, + 551 + ], + "score": 1.0, + "content": "is the proportion of isomorphic pairs of graphs, Mismatched", + "type": "text" + }, + { + "bbox": [ + 367, + 538, + 377, + 549 + ], + "score": 0.86, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "is the proportion of mismatched", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 136, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 136, + 562 + ], + "score": 1.0, + "content": "labels.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "table_body", + "bbox": [ + 113, + 563, + 497, + 693 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 563, + 497, + 693 + ], + "spans": [ + { + "bbox": [ + 113, + 563, + 497, + 693 + ], + "score": 0.985, + "html": "
data setSize, NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
", + "type": "table", + "image_path": "a1599045762f9b49ec73a18a7b7256607709a6320b194d51681cde5de152ccae.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 113, + 563, + 497, + 606.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 113, + 606.3333333333334, + 497, + 649.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 113, + 649.6666666666667, + 497, + 693.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Another surprising observation is that the proportion of mismatched graphs is significant, ranging", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 129, + 733 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 721, + 154, + 731 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 720, + 165, + 733 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 166, + 721, + 180, + 731 + ], + "score": 0.86, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ". This clearly indicates that such graphs are not suitable for graph classification", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "and require additional information to distinguish the models. We analyze the reasons for this in the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 159, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 159, + 105 + ], + "score": 1.0, + "content": "next section.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "and require additional information to distinguish the models. We analyze the reasons for this in the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 159, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 159, + 105 + ], + "score": 1.0, + "content": "next section.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "Also, the distribution of orbit sizes can vary significantly across the data sets. In Figure 1 we plot a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "score": 1.0, + "content": "distribution of orbit sizes for several examples of data sets (and distributions for other data sets can", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "be found in Appendix C). For example, for IMDB-BINARY data set the number of orbits of small", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "sizes, e.g. two or three, goes to 100, which indicate prevalence of pairs of isomorphic graphs that are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "score": 1.0, + "content": "non-isomorphic to the rest. However, for Letter-med data set there are many orbits of sizes more than", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "100, while small orbits are not that common. In this case, the graphs in this data set are equivalent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "to a lot of other graphs, which may have a substantial effect on the corresponding metrics. While", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "the orbit distribution changes from one data set to another, it is clear that in many situations there", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 504, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 504, + 211 + ], + "score": 1.0, + "content": "are isomorphic graphs that can affect training procedure by effectively increasing the weights for the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "corresponding graphs, change performance on the test by validating on the already seen instances,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 504, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 504, + 233 + ], + "score": 1.0, + "content": "and by confusing the model by utilizing different target labels for topologically-equivalent graphs.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 256, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 256, + 243 + ], + "score": 1.0, + "content": "We analyze the reasons for it further.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7.5 + }, + { + "type": "image", + "bbox": [ + 106, + 254, + 505, + 411 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 254, + 505, + 411 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 254, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 411 + ], + "score": 0.966, + "type": "image", + "image_path": "f6a14c15fa6baf9143bba7daa52c62de70ad1b25b506e43721c025cb6a81eba4.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 106, + 254, + 505, + 306.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 106, + 306.3333333333333, + 505, + 358.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 358.66666666666663, + 505, + 410.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 151, + 427, + 459, + 439 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 426, + 459, + 440 + ], + "spans": [ + { + "bbox": [ + 151, + 426, + 459, + 440 + ], + "score": 1.0, + "content": "Figure 1: Examples of distributions of orbit sizes without considering labels.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + } + ], + "index": 16.0 + }, + { + "type": "title", + "bbox": [ + 108, + 464, + 267, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 269, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 269, + 477 + ], + "score": 1.0, + "content": "5 EXPLAINING ISOMORPHISM", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "score": 1.0, + "content": "Meta-information about graphs. In addition to the topology of a graph, many data sets also in-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "clude meta information about nodes and/or edges. Out of 54 analyzed data sets there are 40 that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "additionally include node features and 25 that include edge features. For example, in Synthetic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "data set all graphs are topologically identical but the nodes are endowed with normally distributed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "scalar attributes and in DHFR MD edges are attributed with distances and labeled according to a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "chemical bond type. Alternatively, some graphs can have parallel edges which is equivalent to have", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "a corresponding weight on the edges. Thus some data sets include node/edge categorical features", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "(labels) and numerical features (attributes), which leads to better distinction between the graphs and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 578, + 253, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 253, + 590 + ], + "score": 1.0, + "content": "therefore their corresponding labels.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 504, + 605 + ], + "score": 1.0, + "content": "To see this, we rerun our previous analysis but now include the node labels, if any, when comput-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 320, + 617 + ], + "score": 1.0, + "content": "ing isomorphism between graphs. Consider a tuple", + "type": "text" + }, + { + "bbox": [ + 320, + 605, + 344, + 616 + ], + "score": 0.84, + "content": "( G , l )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 605, + 377, + 617 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 377, + 605, + 387, + 615 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 605, + 450, + 617 + ], + "score": 1.0, + "content": "is a graph and", + "type": "text" + }, + { + "bbox": [ + 450, + 605, + 505, + 617 + ], + "score": 0.91, + "content": "l : V ( G ) \\mapsto", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 107, + 616, + 160, + 628 + ], + "score": 0.91, + "content": "\\{ 1 , 2 , \\ldots , k \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 615, + 179, + 630 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 179, + 617, + 186, + 626 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 615, + 234, + 630 + ], + "score": 1.0, + "content": "-labeling of", + "type": "text" + }, + { + "bbox": [ + 235, + 617, + 244, + 626 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 615, + 506, + 630 + ], + "score": 1.0, + "content": ". In this case of node label-preserving graph isomorphism from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 132, + 640 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 628, + 165, + 638 + ], + "score": 0.86, + "content": "( G _ { 1 } , l _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 626, + 203, + 640 + ], + "score": 1.0, + "content": "to graph", + "type": "text" + }, + { + "bbox": [ + 203, + 627, + 235, + 639 + ], + "score": 0.93, + "content": "\\left( { { G } _ { 2 } } , { { l } _ { 2 } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 626, + 375, + 640 + ], + "score": 1.0, + "content": "we seek an isomorphism function", + "type": "text" + }, + { + "bbox": [ + 376, + 627, + 465, + 639 + ], + "score": 0.93, + "content": "\\phi : { \\bar { V } } ( { \\bar { G } } _ { 1 } { \\bar { ) } } \\mapsto V ( { \\bar { G _ { 2 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 636, + 180, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 175, + 650 + ], + "score": 0.92, + "content": "l _ { 1 } ( v ) = l _ { 2 } ( \\phi ( v ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 636, + 180, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "Tables 3 and 7 (see the appendix) show the number of isomorphic graphs after considering node", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "labels. While for the first six data sets the proportion of isomorphic graphs has not changed much,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 383, + 689 + ], + "score": 1.0, + "content": "it is clearly the case for the remaining data sets. In particular, almost", + "type": "text" + }, + { + "bbox": [ + 383, + 677, + 403, + 687 + ], + "score": 0.82, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "of the analyzed data sets", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 176, + 700 + ], + "score": 1.0, + "content": "include less than", + "type": "text" + }, + { + "bbox": [ + 177, + 688, + 197, + 699 + ], + "score": 0.85, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of isomorphic graphs. Also, the number of mismatched graphs significantly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "decreases after considering node labels. For example, for MUTAG data set the proportion of iso-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 239, + 722 + ], + "score": 1.0, + "content": "morphic graphs went down from", + "type": "text" + }, + { + "bbox": [ + 240, + 710, + 272, + 720 + ], + "score": 0.88, + "content": "4 2 . 0 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 710, + 283, + 722 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 284, + 710, + 315, + 720 + ], + "score": 0.88, + "content": "1 9 . 1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "and the proportion of mismatched graphs from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 163, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 134, + 731 + ], + "score": 0.87, + "content": "6 . 9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 721, + 145, + 731 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 145, + 721, + 159, + 731 + ], + "score": 0.86, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 721, + 163, + 731 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 105 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "Also, the distribution of orbit sizes can vary significantly across the data sets. In Figure 1 we plot a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 133 + ], + "score": 1.0, + "content": "distribution of orbit sizes for several examples of data sets (and distributions for other data sets can", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "be found in Appendix C). For example, for IMDB-BINARY data set the number of orbits of small", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "sizes, e.g. two or three, goes to 100, which indicate prevalence of pairs of isomorphic graphs that are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 504, + 166 + ], + "score": 1.0, + "content": "non-isomorphic to the rest. However, for Letter-med data set there are many orbits of sizes more than", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "100, while small orbits are not that common. In this case, the graphs in this data set are equivalent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "to a lot of other graphs, which may have a substantial effect on the corresponding metrics. While", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "the orbit distribution changes from one data set to another, it is clear that in many situations there", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 504, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 504, + 211 + ], + "score": 1.0, + "content": "are isomorphic graphs that can affect training procedure by effectively increasing the weights for the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "corresponding graphs, change performance on the test by validating on the already seen instances,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 504, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 504, + 233 + ], + "score": 1.0, + "content": "and by confusing the model by utilizing different target labels for topologically-equivalent graphs.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 256, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 256, + 243 + ], + "score": 1.0, + "content": "We analyze the reasons for it further.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 110, + 506, + 243 + ] + }, + { + "type": "image", + "bbox": [ + 106, + 254, + 505, + 411 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 254, + 505, + 411 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 254, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 411 + ], + "score": 0.966, + "type": "image", + "image_path": "f6a14c15fa6baf9143bba7daa52c62de70ad1b25b506e43721c025cb6a81eba4.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 106, + 254, + 505, + 306.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 106, + 306.3333333333333, + 505, + 358.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 358.66666666666663, + 505, + 410.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 151, + 427, + 459, + 439 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 426, + 459, + 440 + ], + "spans": [ + { + "bbox": [ + 151, + 426, + 459, + 440 + ], + "score": 1.0, + "content": "Figure 1: Examples of distributions of orbit sizes without considering labels.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + } + ], + "index": 16.0 + }, + { + "type": "title", + "bbox": [ + 108, + 464, + 267, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 269, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 269, + 477 + ], + "score": 1.0, + "content": "5 EXPLAINING ISOMORPHISM", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 504, + 501 + ], + "score": 1.0, + "content": "Meta-information about graphs. In addition to the topology of a graph, many data sets also in-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "clude meta information about nodes and/or edges. Out of 54 analyzed data sets there are 40 that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "additionally include node features and 25 that include edge features. For example, in Synthetic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "data set all graphs are topologically identical but the nodes are endowed with normally distributed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "scalar attributes and in DHFR MD edges are attributed with distances and labeled according to a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "chemical bond type. Alternatively, some graphs can have parallel edges which is equivalent to have", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "a corresponding weight on the edges. Thus some data sets include node/edge categorical features", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "(labels) and numerical features (attributes), which leads to better distinction between the graphs and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 578, + 253, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 253, + 590 + ], + "score": 1.0, + "content": "therefore their corresponding labels.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 489, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 504, + 605 + ], + "score": 1.0, + "content": "To see this, we rerun our previous analysis but now include the node labels, if any, when comput-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 320, + 617 + ], + "score": 1.0, + "content": "ing isomorphism between graphs. Consider a tuple", + "type": "text" + }, + { + "bbox": [ + 320, + 605, + 344, + 616 + ], + "score": 0.84, + "content": "( G , l )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 605, + 377, + 617 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 377, + 605, + 387, + 615 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 605, + 450, + 617 + ], + "score": 1.0, + "content": "is a graph and", + "type": "text" + }, + { + "bbox": [ + 450, + 605, + 505, + 617 + ], + "score": 0.91, + "content": "l : V ( G ) \\mapsto", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 107, + 616, + 160, + 628 + ], + "score": 0.91, + "content": "\\{ 1 , 2 , \\ldots , k \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 615, + 179, + 630 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 179, + 617, + 186, + 626 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 615, + 234, + 630 + ], + "score": 1.0, + "content": "-labeling of", + "type": "text" + }, + { + "bbox": [ + 235, + 617, + 244, + 626 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 615, + 506, + 630 + ], + "score": 1.0, + "content": ". In this case of node label-preserving graph isomorphism from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 132, + 640 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 628, + 165, + 638 + ], + "score": 0.86, + "content": "( G _ { 1 } , l _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 626, + 203, + 640 + ], + "score": 1.0, + "content": "to graph", + "type": "text" + }, + { + "bbox": [ + 203, + 627, + 235, + 639 + ], + "score": 0.93, + "content": "\\left( { { G } _ { 2 } } , { { l } _ { 2 } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 626, + 375, + 640 + ], + "score": 1.0, + "content": "we seek an isomorphism function", + "type": "text" + }, + { + "bbox": [ + 376, + 627, + 465, + 639 + ], + "score": 0.93, + "content": "\\phi : { \\bar { V } } ( { \\bar { G } } _ { 1 } { \\bar { ) } } \\mapsto V ( { \\bar { G _ { 2 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 636, + 180, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 175, + 650 + ], + "score": 0.92, + "content": "l _ { 1 } ( v ) = l _ { 2 } ( \\phi ( v ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 636, + 180, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 594, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "Tables 3 and 7 (see the appendix) show the number of isomorphic graphs after considering node", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "labels. While for the first six data sets the proportion of isomorphic graphs has not changed much,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 383, + 689 + ], + "score": 1.0, + "content": "it is clearly the case for the remaining data sets. In particular, almost", + "type": "text" + }, + { + "bbox": [ + 383, + 677, + 403, + 687 + ], + "score": 0.82, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "of the analyzed data sets", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 176, + 700 + ], + "score": 1.0, + "content": "include less than", + "type": "text" + }, + { + "bbox": [ + 177, + 688, + 197, + 699 + ], + "score": 0.85, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of isomorphic graphs. Also, the number of mismatched graphs significantly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "decreases after considering node labels. 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DHFR_MD3932539299.756.8794.91
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Figure 2", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "shows a changed distribution of orbits with and without considering node labels. For majority of data", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 305, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 320 + ], + "score": 1.0, + "content": "sets large orbits vanish and the number of small orbits is substantially decreased in label-preserving", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "graph isomorphism setting. This indicates one of the reasons for presence of many isomorphic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 329, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 340 + ], + "score": 1.0, + "content": "graphs in the data sets, which implies that including node/edge labels/attributes can be important for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 340, + 220, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 220, + 351 + ], + "score": 1.0, + "content": "graph classification models.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "image", + "bbox": [ + 106, + 361, + 505, + 518 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 361, + 505, + 518 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 518 + ], + "score": 0.968, + "type": "image", + "image_path": "d7e9dbdde3017047ec255b42935b6e9cc68b9f8d7a5f186905404b51e128b64f.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 106, + 361, + 505, + 413.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 106, + 413.3333333333333, + 505, + 465.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 106, + 465.66666666666663, + 505, + 518.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 171, + 534, + 439, + 546 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 171, + 533, + 440, + 547 + ], + "spans": [ + { + "bbox": [ + 171, + 533, + 440, + 547 + ], + "score": 1.0, + "content": "Figure 2: Examples of distributions of orbit sizes with node labels.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + } + ], + "index": 14.0 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 506, + 679 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "Sizes of the data sets. 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Nevertheless, specifics of the origin of the data set may affect possible con-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "figurations that graphs have (e.g. structure of chemical compounds in COX2 MD or ego-networks", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "for actors in IMDB-BINARY) and thus smaller graphs may tend to be close to isomorphic structures.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "On the other hand, all five data sets with the average number of nodes greater than 100 have very", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "low or zero proportion of isomorphic graphs. Hence, the average size of the graphs directly impacts", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "the possible structure of the data set and thus data sets with larger graphs tend to be more diverse.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 668, + 454, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 454, + 680 + ], + "score": 1.0, + "content": "We next analyze the consequences of the isomorphic graphs on classification methods.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 106, + 696, + 408, + 708 + ], + "lines": [ + { + "bbox": [ + 107, + 696, + 408, + 708 + ], + "spans": [ + { + "bbox": [ + 107, + 696, + 115, + 707 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 124, + 696, + 408, + 708 + ], + "score": 1.0, + "content": "ISOMORPHIC GRAPHS AND WEIGHTED CLASSIFICATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 720, + 162, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 718, + 164, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 718, + 164, + 734 + ], + "score": 1.0, + "content": "We denote by", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 113, + 127, + 497, + 258 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 89, + 505, + 122 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "score": 1.0, + "content": "Table 3: Isomorphic metrics with node labels for Top-10 data sets based on the proportion of iso-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 172, + 113 + ], + "score": 1.0, + "content": "morphic graphs", + "type": "text" + }, + { + "bbox": [ + 173, + 100, + 187, + 111 + ], + "score": 0.86, + "content": "\\bar { I \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 100, + 194, + 113 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 194, + 100, + 217, + 111 + ], + "score": 0.89, + "content": "I P \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 100, + 468, + 113 + ], + "score": 1.0, + "content": "is the proportion of isomorphic pairs of graphs, Mismatched", + "type": "text" + }, + { + "bbox": [ + 468, + 100, + 479, + 111 + ], + "score": 0.81, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 242, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 242, + 123 + ], + "score": 1.0, + "content": "proportion of mismatched labels.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 113, + 127, + 497, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 127, + 497, + 258 + ], + "spans": [ + { + "bbox": [ + 113, + 127, + 497, + 258 + ], + "score": 0.985, + "html": "
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In general, the number of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 221, + 592 + ], + "score": 1.0, + "content": "non-isomorphic graphs with", + "type": "text" + }, + { + "bbox": [ + 221, + 582, + 228, + 590 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 580, + 279, + 592 + ], + "score": 1.0, + "content": "vertices and", + "type": "text" + }, + { + "bbox": [ + 280, + 581, + 290, + 590 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "edges can be computed using Polya enumeration the-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "ory and grows very fast. For example, for a graph with 15 nodes and 15 edges, there are 2,632,420", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "non-isomorphic graphs. Nevertheless, specifics of the origin of the data set may affect possible con-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "figurations that graphs have (e.g. structure of chemical compounds in COX2 MD or ego-networks", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "for actors in IMDB-BINARY) and thus smaller graphs may tend to be close to isomorphic structures.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 648 + ], + "score": 1.0, + "content": "On the other hand, all five data sets with the average number of nodes greater than 100 have very", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "low or zero proportion of isomorphic graphs. Hence, the average size of the graphs directly impacts", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "the possible structure of the data set and thus data sets with larger graphs tend to be more diverse.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 668, + 454, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 454, + 680 + ], + "score": 1.0, + "content": "We next analyze the consequences of the isomorphic graphs on classification methods.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 557, + 506, + 680 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 696, + 408, + 708 + ], + "lines": [ + { + "bbox": [ + 107, + 696, + 408, + 708 + ], + "spans": [ + { + "bbox": [ + 107, + 696, + 115, + 707 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 124, + 696, + 408, + 708 + ], + "score": 1.0, + "content": "ISOMORPHIC GRAPHS AND WEIGHTED CLASSIFICATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 720, + 162, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 718, + 164, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 718, + 164, + 734 + ], + "score": 1.0, + "content": "We denote by", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 718, + 164, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 132, + 81, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 130, + 79, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 130, + 79, + 142, + 96 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 81, + 184, + 93 + ], + "score": 0.89, + "content": "\\mathcal { F } \\subseteq Y ^ { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 79, + 384, + 96 + ], + "score": 1.0, + "content": "a class of binary classifiers with an input space", + "type": "text" + }, + { + "bbox": [ + 384, + 83, + 395, + 92 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 79, + 482, + 96 + ], + "score": 1.0, + "content": "and an output space", + "type": "text" + }, + { + "bbox": [ + 482, + 83, + 505, + 94 + ], + "score": 0.83, + "content": "Y =", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 92, + 188, + 107 + ], + "spans": [ + { + "bbox": [ + 142, + 94, + 183, + 106 + ], + "score": 0.89, + "content": "\\{ - 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In our case", + "type": "text" + }, + { + "bbox": [ + 412, + 140, + 443, + 150 + ], + "score": 0.92, + "content": "x _ { i } \\in \\mathcal D", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 137, + 489, + 153 + ], + "score": 1.0, + "content": "are graphs", + "type": "text" + }, + { + "bbox": [ + 489, + 139, + 501, + 150 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 137, + 506, + 153 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 151, + 248, + 161 + ], + "spans": [ + { + "bbox": [ + 141, + 151, + 248, + 161 + ], + "score": 1.0, + "content": "see notations in section 4.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 169, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 253, + 183 + ], + "score": 1.0, + "content": "We consider a zero-one loss function", + "type": "text" + }, + { + "bbox": [ + 253, + 170, + 312, + 182 + ], + "score": 0.93, + "content": "l ( \\hat { y } , y ) = \\mathbb { I } _ { \\hat { y } \\neq y }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 169, + 453, + 183 + ], + "score": 1.0, + "content": ". Let us assume that classifiers from", + "type": "text" + }, + { + "bbox": [ + 453, + 171, + 463, + 180 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "can detect", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 173, + 194 + ], + "score": 1.0, + "content": "which graphs in", + "type": "text" + }, + { + "bbox": [ + 173, + 182, + 183, + 191 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "are isomorphic. E.g. classifiers based on Weisfeiler-Lehman graph kernels (see", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "Shervashidze et al. (2011a)) are capable to do it for majority of graphs. In such case the empirical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 101, + 197, + 509, + 224 + ], + "spans": [ + { + "bbox": [ + 101, + 197, + 124, + 224 + ], + "score": 1.0, + "content": "risk", + "type": "text" + }, + { + "bbox": [ + 124, + 203, + 277, + 218 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } l ( f ( x ) , y ) = \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } l ( f ( x _ { j } ) , y _ { j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 197, + 323, + 224 + ], + "score": 1.0, + "content": "reduces to", + "type": "text" + }, + { + "bbox": [ + 323, + 204, + 420, + 219 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { j \\in J } u _ { j } l ( f ( x _ { j } ) , y _ { j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 197, + 450, + 224 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 451, + 205, + 459, + 215 + ], + "score": 0.82, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 197, + 509, + 224 + ], + "score": 1.0, + "content": "is an index", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 232, + 230 + ], + "score": 1.0, + "content": "set of non-isomorphic graphs,", + "type": "text" + }, + { + "bbox": [ + 232, + 218, + 266, + 229 + ], + "score": 0.9, + "content": "u _ { j } ~ \\geq ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 217, + 492, + 230 + ], + "score": 1.0, + "content": "is equal to the number of graphs in the initial sample", + "type": "text" + }, + { + "bbox": [ + 501, + 217, + 506, + 230 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 307, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 203, + 241 + ], + "score": 1.0, + "content": "isomorphic to the graph", + "type": "text" + }, + { + "bbox": [ + 204, + 230, + 215, + 240 + ], + "score": 0.87, + "content": "x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 227, + 258, + 241 + ], + "score": 1.0, + "content": "(we count", + "type": "text" + }, + { + "bbox": [ + 258, + 230, + 270, + 240 + ], + "score": 0.85, + "content": "x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 227, + 307, + 241 + ], + "score": 1.0, + "content": "as well).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Thus under such assumptions the graph classification problem with the training data set, contain-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "ing isomorphic graphs, can be interpreted as a classification problem with a weighted loss. Let us", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "score": 1.0, + "content": "introduce general notations for this problem. We define some (fixed) measurable weighting func-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 124, + 290 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 125, + 278, + 229, + 290 + ], + "score": 0.93, + "content": "u : ( X \\times Y ) \\to ( 0 , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 278, + 379, + 290 + ], + "score": 1.0, + "content": ". Then the theoretical risk is equal to", + "type": "text" + }, + { + "bbox": [ + 380, + 278, + 432, + 290 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "and the weighted", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 284, + 509, + 309 + ], + "spans": [ + { + "bbox": [ + 102, + 284, + 210, + 309 + ], + "score": 1.0, + "content": "empirical risk is equal to", + "type": "text" + }, + { + "bbox": [ + 210, + 289, + 425, + 303 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } u ( x _ { i } , y _ { i } ) l ( f ( x _ { i } ) , y _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 284, + 509, + 309 + ], + "score": 1.0, + "content": "). We would like to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 301, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 279, + 317 + ], + "score": 1.0, + "content": "derive an upper bound for the excess risk", + "type": "text" + }, + { + "bbox": [ + 279, + 303, + 468, + 315 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { f \\in \\bar { \\mathcal { F } } } \\left( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 301, + 507, + 317 + ], + "score": 1.0, + "content": ", i.e. we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "would like to quantify an upper bound for a generalization gap. We optimize the weighted empirical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 474, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 474, + 337 + ], + "score": 1.0, + "content": "risk when training a classifier and measure its accuracy using non-weighted theoretical risk.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "There are some results about classification performance with a weighted loss. E.g. in (Dupret &", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "Koda, 2001) a bayesian framework for imbalanced classification with a weighted risk is proposed.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "Scott (2012) investigated the calibration of asymmetric surrogate losses. Natarajan et al. (2018) con-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "sidered the case of cost-sensitive learning with noisy labels. However, to the best of our knowledge,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "there is no studied upper bound for the excess risk with explicit dependence on the class imbalance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "and the weighting scheme that quantifies the influence on the overall classification performance. We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 407, + 192, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 192, + 418 + ], + "score": 1.0, + "content": "show this result next.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "To derive explicit expressions we use some additional modeling asumption, namely, we consider", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 330, + 447 + ], + "score": 0.9, + "content": "u ( x , y ) = ( \\bar { 1 } + g _ { + } ( w \\bar { ) } ) \\mathbb { I } _ { \\{ y = + 1 \\} } + ( 1 + g _ { - } ( w ) ) \\mathbb { I } _ { \\{ y = - 1 \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "for some non-negative weighting functions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 487, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 134, + 459 + ], + "score": 0.91, + "content": "g _ { + } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 446, + 152, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 447, + 180, + 459 + ], + "score": 0.9, + "content": "g _ { - } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 446, + 260, + 460 + ], + "score": 1.0, + "content": "of the weight value", + "type": "text" + }, + { + "bbox": [ + 261, + 448, + 287, + 457 + ], + "score": 0.88, + "content": "w \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 446, + 357, + 460 + ], + "score": 1.0, + "content": ". E.g. we can use", + "type": "text" + }, + { + "bbox": [ + 357, + 446, + 406, + 459 + ], + "score": 0.93, + "content": "g _ { + } ( w ) = w", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 446, + 423, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 424, + 446, + 482, + 459 + ], + "score": 0.93, + "content": "g _ { - } ( w ) = 1 / w", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 446, + 487, + 460 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 460, + 484, + 473 + ], + "lines": [ + { + "bbox": [ + 104, + 457, + 485, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 236, + 476 + ], + "score": 1.0, + "content": "Theorem 6.1. With probability", + "type": "text" + }, + { + "bbox": [ + 236, + 461, + 287, + 472 + ], + "score": 0.55, + "content": "1 - \\delta , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 457, + 302, + 476 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 302, + 460, + 338, + 471 + ], + "score": 0.92, + "content": "\\mathcal { D } \\sim \\mathbb { P } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 457, + 485, + 476 + ], + "score": 1.0, + "content": "the excess risk is upper bounded by", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 475, + 461, + 519 + ], + "lines": [ + { + "bbox": [ + 149, + 475, + 461, + 519 + ], + "spans": [ + { + "bbox": [ + 149, + 475, + 461, + 519 + ], + "score": 0.89, + "content": "\\begin{array} { r l r } { { \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\Big ( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\Big ) \\leq 3 ( g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) ) + } } \\\\ & { } & { + \\mathcal { R } _ { N } ( \\mathcal { F } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } , } \\end{array}", + "type": "interline_equation", + "image_path": "044123505e40689970d447aa8afe372670a9e344d6d6977add6ffe008cd79c1e.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 149, + 475, + 461, + 489.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 149, + 489.6666666666667, + 461, + 504.33333333333337 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 149, + 504.33333333333337, + 461, + 519.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 378, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 379, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 133, + 533 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 520, + 166, + 533 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { N } ( \\mathcal { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 520, + 366, + 533 + ], + "score": 1.0, + "content": "is a Rademacher complexity of the function class", + "type": "text" + }, + { + "bbox": [ + 367, + 521, + 376, + 531 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 520, + 379, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 504, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 368, + 554 + ], + "score": 1.0, + "content": "Let us note that the Rademacher complexity of the function class", + "type": "text" + }, + { + "bbox": [ + 368, + 541, + 377, + 551 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 539, + 505, + 554 + ], + "score": 1.0, + "content": ", defined by a graph kernel, was", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 552, + 329, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 329, + 563 + ], + "score": 1.0, + "content": "studied e.g. in Oneto et al. 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Let us assume that classifiers from", + "type": "text" + }, + { + "bbox": [ + 453, + 171, + 463, + 180 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "can detect", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 173, + 194 + ], + "score": 1.0, + "content": "which graphs in", + "type": "text" + }, + { + "bbox": [ + 173, + 182, + 183, + 191 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "are isomorphic. E.g. classifiers based on Weisfeiler-Lehman graph kernels (see", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "Shervashidze et al. (2011a)) are capable to do it for majority of graphs. In such case the empirical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 101, + 197, + 509, + 224 + ], + "spans": [ + { + "bbox": [ + 101, + 197, + 124, + 224 + ], + "score": 1.0, + "content": "risk", + "type": "text" + }, + { + "bbox": [ + 124, + 203, + 277, + 218 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } l ( f ( x ) , y ) = \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } l ( f ( x _ { j } ) , y _ { j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 197, + 323, + 224 + ], + "score": 1.0, + "content": "reduces to", + "type": "text" + }, + { + "bbox": [ + 323, + 204, + 420, + 219 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { j \\in J } u _ { j } l ( f ( x _ { j } ) , y _ { j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 197, + 450, + 224 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 451, + 205, + 459, + 215 + ], + "score": 0.82, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 197, + 509, + 224 + ], + "score": 1.0, + "content": "is an index", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 232, + 230 + ], + "score": 1.0, + "content": "set of non-isomorphic graphs,", + "type": "text" + }, + { + "bbox": [ + 232, + 218, + 266, + 229 + ], + "score": 0.9, + "content": "u _ { j } ~ \\geq ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 217, + 492, + 230 + ], + "score": 1.0, + "content": "is equal to the number of graphs in the initial sample", + "type": "text" + }, + { + "bbox": [ + 501, + 217, + 506, + 230 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 307, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 203, + 241 + ], + "score": 1.0, + "content": "isomorphic to the graph", + "type": "text" + }, + { + "bbox": [ + 204, + 230, + 215, + 240 + ], + "score": 0.87, + "content": "x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 227, + 258, + 241 + ], + "score": 1.0, + "content": "(we count", + "type": "text" + }, + { + "bbox": [ + 258, + 230, + 270, + 240 + ], + "score": 0.85, + "content": "x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 227, + 307, + 241 + ], + "score": 1.0, + "content": "as well).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 101, + 169, + 509, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Thus under such assumptions the graph classification problem with the training data set, contain-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "ing isomorphic graphs, can be interpreted as a classification problem with a weighted loss. Let us", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 504, + 279 + ], + "score": 1.0, + "content": "introduce general notations for this problem. We define some (fixed) measurable weighting func-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 124, + 290 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 125, + 278, + 229, + 290 + ], + "score": 0.93, + "content": "u : ( X \\times Y ) \\to ( 0 , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 278, + 379, + 290 + ], + "score": 1.0, + "content": ". Then the theoretical risk is equal to", + "type": "text" + }, + { + "bbox": [ + 380, + 278, + 432, + 290 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "and the weighted", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 284, + 509, + 309 + ], + "spans": [ + { + "bbox": [ + 102, + 284, + 210, + 309 + ], + "score": 1.0, + "content": "empirical risk is equal to", + "type": "text" + }, + { + "bbox": [ + 210, + 289, + 425, + 303 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } u ( x _ { i } , y _ { i } ) l ( f ( x _ { i } ) , y _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 284, + 509, + 309 + ], + "score": 1.0, + "content": "). We would like to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 301, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 279, + 317 + ], + "score": 1.0, + "content": "derive an upper bound for the excess risk", + "type": "text" + }, + { + "bbox": [ + 279, + 303, + 468, + 315 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { f \\in \\bar { \\mathcal { F } } } \\left( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 301, + 507, + 317 + ], + "score": 1.0, + "content": ", i.e. we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "would like to quantify an upper bound for a generalization gap. We optimize the weighted empirical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 474, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 474, + 337 + ], + "score": 1.0, + "content": "risk when training a classifier and measure its accuracy using non-weighted theoretical risk.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 102, + 245, + 509, + 337 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "There are some results about classification performance with a weighted loss. E.g. in (Dupret &", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "Koda, 2001) a bayesian framework for imbalanced classification with a weighted risk is proposed.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "Scott (2012) investigated the calibration of asymmetric surrogate losses. Natarajan et al. (2018) con-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "sidered the case of cost-sensitive learning with noisy labels. However, to the best of our knowledge,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "there is no studied upper bound for the excess risk with explicit dependence on the class imbalance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "and the weighting scheme that quantifies the influence on the overall classification performance. We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 407, + 192, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 192, + 418 + ], + "score": 1.0, + "content": "show this result next.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 341, + 505, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "To derive explicit expressions we use some additional modeling asumption, namely, we consider", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 330, + 447 + ], + "score": 0.9, + "content": "u ( x , y ) = ( \\bar { 1 } + g _ { + } ( w \\bar { ) } ) \\mathbb { I } _ { \\{ y = + 1 \\} } + ( 1 + g _ { - } ( w ) ) \\mathbb { I } _ { \\{ y = - 1 \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "for some non-negative weighting functions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 487, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 134, + 459 + ], + "score": 0.91, + "content": "g _ { + } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 446, + 152, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 447, + 180, + 459 + ], + "score": 0.9, + "content": "g _ { - } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 446, + 260, + 460 + ], + "score": 1.0, + "content": "of the weight value", + "type": "text" + }, + { + "bbox": [ + 261, + 448, + 287, + 457 + ], + "score": 0.88, + "content": "w \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 446, + 357, + 460 + ], + "score": 1.0, + "content": ". E.g. we can use", + "type": "text" + }, + { + "bbox": [ + 357, + 446, + 406, + 459 + ], + "score": 0.93, + "content": "g _ { + } ( w ) = w", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 446, + 423, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 424, + 446, + 482, + 459 + ], + "score": 0.93, + "content": "g _ { - } ( w ) = 1 / w", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 446, + 487, + 460 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 423, + 505, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 460, + 484, + 473 + ], + "lines": [ + { + "bbox": [ + 104, + 457, + 485, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 236, + 476 + ], + "score": 1.0, + "content": "Theorem 6.1. With probability", + "type": "text" + }, + { + "bbox": [ + 236, + 461, + 287, + 472 + ], + "score": 0.55, + "content": "1 - \\delta , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 457, + 302, + 476 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 302, + 460, + 338, + 471 + ], + "score": 0.92, + "content": "\\mathcal { D } \\sim \\mathbb { P } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 457, + 485, + 476 + ], + "score": 1.0, + "content": "the excess risk is upper bounded by", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 457, + 485, + 476 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 475, + 461, + 519 + ], + "lines": [ + { + "bbox": [ + 149, + 475, + 461, + 519 + ], + "spans": [ + { + "bbox": [ + 149, + 475, + 461, + 519 + ], + "score": 0.89, + "content": "\\begin{array} { r l r } { { \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\Big ( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\Big ) \\leq 3 ( g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) ) + } } \\\\ & { } & { + \\mathcal { R } _ { N } ( \\mathcal { F } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } , } \\end{array}", + "type": "interline_equation", + "image_path": "044123505e40689970d447aa8afe372670a9e344d6d6977add6ffe008cd79c1e.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 149, + 475, + 461, + 489.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 149, + 489.6666666666667, + 461, + 504.33333333333337 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 149, + 504.33333333333337, + 461, + 519.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 378, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 379, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 133, + 533 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 520, + 166, + 533 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { N } ( \\mathcal { F } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 520, + 366, + 533 + ], + "score": 1.0, + "content": "is a Rademacher complexity of the function class", + "type": "text" + }, + { + "bbox": [ + 367, + 521, + 376, + 531 + ], + "score": 0.85, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 520, + 379, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 520, + 379, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 504, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 368, + 554 + ], + "score": 1.0, + "content": "Let us note that the Rademacher complexity of the function class", + "type": "text" + }, + { + "bbox": [ + 368, + 541, + 377, + 551 + ], + "score": 0.83, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 539, + 505, + 554 + ], + "score": 1.0, + "content": ", defined by a graph kernel, was", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 552, + 329, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 329, + 563 + ], + "score": 1.0, + "content": "studied e.g. in Oneto et al. (2018); Oneto et al. 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For such optimal", + "type": "text" + }, + { + "bbox": [ + 338, + 652, + 357, + 663 + ], + "score": 0.89, + "content": "w _ { o p t }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 646, + 503, + 666 + ], + "score": 1.0, + "content": "the RHS of equation 1 has the form", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 626, + 503, + 666 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 668, + 415, + 690 + ], + "lines": [ + { + "bbox": [ + 195, + 668, + 415, + 690 + ], + "spans": [ + { + "bbox": [ + 195, + 668, + 415, + 690 + ], + "score": 0.92, + "content": "6 \\sqrt { \\pi ( 1 - \\pi ) } + \\mathcal { R } _ { N } ( \\mathcal { F } ) + \\alpha _ { N } \\left( 2 + [ \\pi ( 1 - \\pi ) ] ^ { - 1 / 2 } \\right) .", + "type": "interline_equation", + "image_path": "d29d159b24e367c77f972f95f29f21df69b18312b2bcf85065681fdd5026d15e.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 195, + 668, + 415, + 690 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 713 + ], + "score": 1.0, + "content": "Thus we get theoretical evidence on how the weighting influences the classification accuracy: e.g.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 227, + 722 + ], + "score": 1.0, + "content": "in the imbalanced case (when", + "type": "text" + }, + { + "bbox": [ + 227, + 710, + 253, + 720 + ], + "score": 0.88, + "content": "\\pi \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 709, + 265, + 722 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 266, + 710, + 291, + 720 + ], + "score": 0.87, + "content": "\\pi \\approx 1", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ") selecting the weight optimally we reduce the gener-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 232, + 733 + ], + "score": 1.0, + "content": "alization gap almost to zero for", + "type": "text" + }, + { + "bbox": [ + 232, + 721, + 263, + 731 + ], + "score": 0.87, + "content": "N \\gg 1", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "; at the same time, not optimal weight can lead to overfitting.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 696, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "As we already discussed, under some mild modeling assumptions the graph classification problem", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "with isomorphic graphs in the training data set can be interpreted as the classification problem with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "a weighted loss. Therefore the obtained estimate provides additional evidence on a negative effect of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the isomorphic graphs when solving the graph classification problems: the presence of isomorphic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "graphs in the training data set could have the same negative effect as not optimal weight value for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 298, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 298, + 150 + ], + "score": 1.0, + "content": "the classification with a weighted loss function.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 165, + 416, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 418, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 418, + 180 + ], + "score": 1.0, + "content": "7 INFLUENCE OF ISOMORPHISM BIAS: EMPIRICAL RESULTS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 203 + ], + "score": 1.0, + "content": "To understand the impact of isomorphic graphs in the data set on the final metric we consider sepa-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 370, + 213 + ], + "score": 1.0, + "content": "rately the results on two subparts of the data set. In particular, let", + "type": "text" + }, + { + "bbox": [ + 371, + 201, + 397, + 212 + ], + "score": 0.92, + "content": "Y _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 200, + 415, + 213 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 416, + 201, + 437, + 212 + ], + "score": 0.91, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "be train and test", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 209, + 504, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 198, + 226 + ], + "score": 1.0, + "content": "splits of a data set. Let", + "type": "text" + }, + { + "bbox": [ + 198, + 212, + 242, + 223 + ], + "score": 0.92, + "content": "H _ { i } \\in Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 209, + 454, + 226 + ], + "score": 1.0, + "content": "be a graph such that there exists an isomorphic graph", + "type": "text" + }, + { + "bbox": [ + 454, + 212, + 504, + 223 + ], + "score": 0.91, + "content": "G _ { i } \\in Y _ { t r a i n }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 221, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 205, + 236 + ], + "score": 1.0, + "content": "in the train data set. Let", + "type": "text" + }, + { + "bbox": [ + 206, + 223, + 233, + 235 + ], + "score": 0.92, + "content": "\\{ Y _ { i s o } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 221, + 342, + 236 + ], + "score": 1.0, + "content": "be a set of all such graphs", + "type": "text" + }, + { + "bbox": [ + 342, + 223, + 355, + 234 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 221, + 506, + 236 + ], + "score": 1.0, + "content": "for which there exists an isomorphic", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 132, + 246 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 234, + 145, + 244 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 233, + 157, + 246 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 157, + 234, + 183, + 245 + ], + "score": 0.91, + "content": "Y _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 233, + 286, + 246 + ], + "score": 1.0, + "content": ". Note that the graphs in", + "type": "text" + }, + { + "bbox": [ + 287, + 234, + 314, + 245 + ], + "score": 0.94, + "content": "\\{ Y _ { i s o } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "are not necessarily isomorphic. We denote by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 128, + 256 + ], + "score": 0.9, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 243, + 422, + 259 + ], + "score": 1.0, + "content": "the test graphs that do not have isomorphic copies in the train data set, i.e.", + "type": "text" + }, + { + "bbox": [ + 423, + 244, + 501, + 257 + ], + "score": 0.92, + "content": "Y _ { n e w } = Y _ { t e s t } \\backslash Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 243, + 505, + 259 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "If we want to test generalization of classification models, we need to test it on new instances of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 268, + 279 + ], + "score": 1.0, + "content": "data sets and therefore at least consider", + "type": "text" + }, + { + "bbox": [ + 269, + 267, + 291, + 277 + ], + "score": 0.89, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 266, + 335, + 279 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 335, + 267, + 356, + 277 + ], + "score": 0.91, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 266, + 505, + 279 + ], + "score": 1.0, + "content": ". One question regarding the perfor-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 276, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 267, + 291 + ], + "score": 1.0, + "content": "mance of the models on this new test set", + "type": "text" + }, + { + "bbox": [ + 267, + 278, + 289, + 289 + ], + "score": 0.9, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 276, + 505, + 291 + ], + "score": 1.0, + "content": "is whether the performance on it will be lower than on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 286, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 183, + 302 + ], + "score": 1.0, + "content": "the original test set", + "type": "text" + }, + { + "bbox": [ + 184, + 288, + 205, + 299 + ], + "score": 0.9, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 286, + 505, + 302 + ], + "score": 1.0, + "content": ". As we show below answer to this question solely depends on the accuracy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 297, + 280, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 258, + 313 + ], + "score": 1.0, + "content": "of the model on isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 258, + 299, + 276, + 311 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 297, + 280, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 494, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 492, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 482, + 329 + ], + "score": 1.0, + "content": "Consider a graph classification model that is evaluated on normalized accuracy over a data set", + "type": "text" + }, + { + "bbox": [ + 483, + 316, + 492, + 326 + ], + "score": 0.81, + "content": "Y", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 331, + 356, + 366 + ], + "lines": [ + { + "bbox": [ + 254, + 331, + 356, + 366 + ], + "spans": [ + { + "bbox": [ + 254, + 331, + 356, + 366 + ], + "score": 0.94, + "content": "\\operatorname { a c c } ( Y ) = { \\frac { \\sum _ { G \\in Y } \\operatorname { a c c } ( G _ { i } ) } { | Y | } } ,", + "type": "interline_equation", + "image_path": "463bfb84bf61a927250c828511d9ac31915defaea6f86cc570b88f4c5d505c97.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 254, + 331, + 356, + 348.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 254, + 348.5, + 356, + 366.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 133, + 383 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 371, + 164, + 383 + ], + "score": 0.65, + "content": "\\operatorname { a c c } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 371, + 358, + 383 + ], + "score": 1.0, + "content": "equals to one if the model predicts the label of", + "type": "text" + }, + { + "bbox": [ + 358, + 372, + 370, + 382 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 371, + 506, + 383 + ], + "score": 1.0, + "content": "correctly, and zero otherwise. If", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 107, + 382, + 142, + 394 + ], + "score": 0.91, + "content": "| Y | = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 381, + 218, + 394 + ], + "score": 1.0, + "content": ", then we consider", + "type": "text" + }, + { + "bbox": [ + 218, + 382, + 269, + 394 + ], + "score": 0.92, + "content": "\\operatorname { a c c } ( Y ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 381, + 505, + 394 + ], + "score": 1.0, + "content": ". We can see that the accuracy on the test data set can be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 394, + 237, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 237, + 405 + ], + "score": 1.0, + "content": "written as the sum of two terms:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 423, + 443, + 488 + ], + "lines": [ + { + "bbox": [ + 168, + 423, + 443, + 488 + ], + "spans": [ + { + "bbox": [ + 168, + 423, + 443, + 488 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\qquad \\displaystyle \\sum _ { G \\subset ( Y _ { t e s t } ) } \\operatorname { a c c } ( G ) } \\\\ & { \\qquad \\quad = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } = \\frac { \\displaystyle \\sum _ { G \\in Y _ { n e w } } \\operatorname { a c c } ( G ) + \\sum _ { G \\in Y _ { i s o } } \\operatorname { a c c } ( G ) } { \\displaystyle | Y _ { t e s t } | } = } \\\\ & { \\qquad = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) + \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { i s o } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "9fd7fdc4b883a86a24095df6ff685452c5c55b7e7713190fd8c78b1f14fc1841.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 168, + 423, + 443, + 444.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 168, + 444.6666666666667, + 443, + 466.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 168, + 466.33333333333337, + 443, + 488.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "Equation 3 decomposes accuracy on the original data set as the weighted sum of two accuracies on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 506, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 238, + 521 + ], + "score": 1.0, + "content": "the set of the new test instances", + "type": "text" + }, + { + "bbox": [ + 238, + 508, + 260, + 519 + ], + "score": 0.91, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 506, + 367, + 521 + ], + "score": 1.0, + "content": "and a set of the instances", + "type": "text" + }, + { + "bbox": [ + 367, + 507, + 385, + 518 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 506, + 506, + 521 + ], + "score": 1.0, + "content": "already appeared in the train", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 348, + 531 + ], + "score": 1.0, + "content": "set and therefore available to the model. We call the term", + "type": "text" + }, + { + "bbox": [ + 348, + 519, + 387, + 531 + ], + "score": 0.75, + "content": "\\operatorname { a c c } ( Y _ { i s o } )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "as isomorphism bias, which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "corresponds to the accuracy of the model on the isomorphic test instances. As we will see next, the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 259, + 552 + ], + "score": 1.0, + "content": "accuracy of the model on the new set", + "type": "text" + }, + { + "bbox": [ + 259, + 541, + 282, + 552 + ], + "score": 0.91, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "will be less if only if the model performs better on the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 191, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 167, + 565 + ], + "score": 1.0, + "content": "isomorphic set", + "type": "text" + }, + { + "bbox": [ + 167, + 551, + 185, + 563 + ], + "score": 0.89, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 551, + 191, + 565 + ], + "score": 1.0, + "content": ".1", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 149, + 577 + ], + "score": 0.41, + "content": "\\mathbf { C l a i m 7 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 564, + 171, + 579 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 171, + 566, + 255, + 577 + ], + "score": 0.89, + "content": "Y _ { t e s t } = Y _ { n e w } \\cup Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 564, + 259, + 579 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 566, + 331, + 577 + ], + "score": 0.91, + "content": "Y _ { n e w } \\cap Y _ { i s o } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 564, + 349, + 579 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 566, + 389, + 577 + ], + "score": 0.91, + "content": "Y _ { i s o } \\neq \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 564, + 420, + 579 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 420, + 566, + 477, + 577 + ], + "score": 0.92, + "content": "Y _ { i s o } \\subset Y _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 564, + 506, + 579 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 361, + 590 + ], + "score": 1.0, + "content": "for any classification model accuracy on the new test instances", + "type": "text" + }, + { + "bbox": [ + 362, + 578, + 384, + 588 + ], + "score": 0.9, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "is smaller than on the test set", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 587, + 459, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 127, + 599 + ], + "score": 0.9, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 587, + 419, + 601 + ], + "score": 1.0, + "content": "if and only if it is smaller than accuracy on the isomorphic test instances", + "type": "text" + }, + { + "bbox": [ + 419, + 588, + 437, + 599 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 587, + 459, + 601 + ], + "score": 1.0, + "content": ", i.e.:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 614, + 416, + 628 + ], + "lines": [ + { + "bbox": [ + 194, + 614, + 416, + 628 + ], + "spans": [ + { + "bbox": [ + 194, + 614, + 416, + 628 + ], + "score": 0.88, + "content": "\\operatorname { a c c } ( Y _ { t e s t } ) > \\operatorname { a c c } ( Y _ { n e w } ) \\iff \\operatorname { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { n e w } )", + "type": "interline_equation", + "image_path": "1322af5fb329ca44f6fc2ccc030ade725302a3dbd9a39707197683bcdcb389f4.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 194, + 614, + 416, + 628 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "The equation 4 gives a definite answer with the possible performance of the model on a new test set.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 317, + 660 + ], + "score": 1.0, + "content": "If the model performs well on isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 318, + 647, + 336, + 658 + ], + "score": 0.91, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 646, + 506, + 660 + ], + "score": 1.0, + "content": ", then it will falsely increase performance", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 119, + 672 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 658, + 141, + 669 + ], + "score": 0.91, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 658, + 215, + 672 + ], + "score": 1.0, + "content": "in comparison to", + "type": "text" + }, + { + "bbox": [ + 216, + 658, + 237, + 669 + ], + "score": 0.93, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 658, + 506, + 672 + ], + "score": 1.0, + "content": ". Conversely, if the model performs poorly on the instances that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "score": 1.0, + "content": "appeared in the training set, then removing them from the test set and evaluating the model purely", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 120, + 694 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 680, + 142, + 691 + ], + "score": 0.9, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "will demonstrate higher accuracy. There are two reasons for the model to misclassify", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 194, + 704 + ], + "score": 1.0, + "content": "isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 194, + 691, + 212, + 702 + ], + "score": 0.89, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 690, + 506, + 704 + ], + "score": 1.0, + "content": ": (i) the instances contain target labels that are different than those that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 701, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 701, + 505, + 714 + ], + "score": 1.0, + "content": "it has seen, as we show in Table 2 the percentage of mismatched labels can be high in some data", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 380, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 380, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 380, + 734 + ], + "score": 1.0, + "content": "1We provide the proof of Claim 7.1 and Claim 7.2 in Appendix E and F.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "As we already discussed, under some mild modeling assumptions the graph classification problem", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "with isomorphic graphs in the training data set can be interpreted as the classification problem with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "a weighted loss. Therefore the obtained estimate provides additional evidence on a negative effect of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the isomorphic graphs when solving the graph classification problems: the presence of isomorphic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "graphs in the training data set could have the same negative effect as not optimal weight value for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 298, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 298, + 150 + ], + "score": 1.0, + "content": "the classification with a weighted loss function.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 82, + 506, + 150 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 165, + 416, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 418, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 418, + 180 + ], + "score": 1.0, + "content": "7 INFLUENCE OF ISOMORPHISM BIAS: EMPIRICAL RESULTS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 203 + ], + "score": 1.0, + "content": "To understand the impact of isomorphic graphs in the data set on the final metric we consider sepa-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 370, + 213 + ], + "score": 1.0, + "content": "rately the results on two subparts of the data set. 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Let", + "type": "text" + }, + { + "bbox": [ + 198, + 212, + 242, + 223 + ], + "score": 0.92, + "content": "H _ { i } \\in Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 209, + 454, + 226 + ], + "score": 1.0, + "content": "be a graph such that there exists an isomorphic graph", + "type": "text" + }, + { + "bbox": [ + 454, + 212, + 504, + 223 + ], + "score": 0.91, + "content": "G _ { i } \\in Y _ { t r a i n }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 221, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 205, + 236 + ], + "score": 1.0, + "content": "in the train data set. Let", + "type": "text" + }, + { + "bbox": [ + 206, + 223, + 233, + 235 + ], + "score": 0.92, + "content": "\\{ Y _ { i s o } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 221, + 342, + 236 + ], + "score": 1.0, + "content": "be a set of all such graphs", + "type": "text" + }, + { + "bbox": [ + 342, + 223, + 355, + 234 + ], + "score": 0.89, + "content": "H _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 221, + 506, + 236 + ], + "score": 1.0, + "content": "for which there exists an isomorphic", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 132, + 246 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 132, + 234, + 145, + 244 + ], + "score": 0.88, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 233, + 157, + 246 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 157, + 234, + 183, + 245 + ], + "score": 0.91, + "content": "Y _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 233, + 286, + 246 + ], + "score": 1.0, + "content": ". Note that the graphs in", + "type": "text" + }, + { + "bbox": [ + 287, + 234, + 314, + 245 + ], + "score": 0.94, + "content": "\\{ Y _ { i s o } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "are not necessarily isomorphic. We denote by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 128, + 256 + ], + "score": 0.9, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 243, + 422, + 259 + ], + "score": 1.0, + "content": "the test graphs that do not have isomorphic copies in the train data set, i.e.", + "type": "text" + }, + { + "bbox": [ + 423, + 244, + 501, + 257 + ], + "score": 0.92, + "content": "Y _ { n e w } = Y _ { t e s t } \\backslash Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 243, + 505, + 259 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "If we want to test generalization of classification models, we need to test it on new instances of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 268, + 279 + ], + "score": 1.0, + "content": "data sets and therefore at least consider", + "type": "text" + }, + { + "bbox": [ + 269, + 267, + 291, + 277 + ], + "score": 0.89, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 266, + 335, + 279 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 335, + 267, + 356, + 277 + ], + "score": 0.91, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 266, + 505, + 279 + ], + "score": 1.0, + "content": ". One question regarding the perfor-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 276, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 267, + 291 + ], + "score": 1.0, + "content": "mance of the models on this new test set", + "type": "text" + }, + { + "bbox": [ + 267, + 278, + 289, + 289 + ], + "score": 0.9, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 276, + 505, + 291 + ], + "score": 1.0, + "content": "is whether the performance on it will be lower than on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 286, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 183, + 302 + ], + "score": 1.0, + "content": "the original test set", + "type": "text" + }, + { + "bbox": [ + 184, + 288, + 205, + 299 + ], + "score": 0.9, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 286, + 505, + 302 + ], + "score": 1.0, + "content": ". As we show below answer to this question solely depends on the accuracy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 297, + 280, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 258, + 313 + ], + "score": 1.0, + "content": "of the model on isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 258, + 299, + 276, + 311 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 297, + 280, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 188, + 506, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 494, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 492, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 482, + 329 + ], + "score": 1.0, + "content": "Consider a graph classification model that is evaluated on normalized accuracy over a data set", + "type": "text" + }, + { + "bbox": [ + 483, + 316, + 492, + 326 + ], + "score": 0.81, + "content": "Y", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 315, + 492, + 329 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 331, + 356, + 366 + ], + "lines": [ + { + "bbox": [ + 254, + 331, + 356, + 366 + ], + "spans": [ + { + "bbox": [ + 254, + 331, + 356, + 366 + ], + "score": 0.94, + "content": "\\operatorname { a c c } ( Y ) = { \\frac { \\sum _ { G \\in Y } \\operatorname { a c c } ( G _ { i } ) } { | Y | } } ,", + "type": "interline_equation", + "image_path": "463bfb84bf61a927250c828511d9ac31915defaea6f86cc570b88f4c5d505c97.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 254, + 331, + 356, + 348.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 254, + 348.5, + 356, + 366.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 133, + 383 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 371, + 164, + 383 + ], + "score": 0.65, + "content": "\\operatorname { a c c } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 371, + 358, + 383 + ], + "score": 1.0, + "content": "equals to one if the model predicts the label of", + "type": "text" + }, + { + "bbox": [ + 358, + 372, + 370, + 382 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 371, + 506, + 383 + ], + "score": 1.0, + "content": "correctly, and zero otherwise. If", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 107, + 382, + 142, + 394 + ], + "score": 0.91, + "content": "| Y | = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 381, + 218, + 394 + ], + "score": 1.0, + "content": ", then we consider", + "type": "text" + }, + { + "bbox": [ + 218, + 382, + 269, + 394 + ], + "score": 0.92, + "content": "\\operatorname { a c c } ( Y ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 381, + 505, + 394 + ], + "score": 1.0, + "content": ". We can see that the accuracy on the test data set can be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 394, + 237, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 237, + 405 + ], + "score": 1.0, + "content": "written as the sum of two terms:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 371, + 506, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 423, + 443, + 488 + ], + "lines": [ + { + "bbox": [ + 168, + 423, + 443, + 488 + ], + "spans": [ + { + "bbox": [ + 168, + 423, + 443, + 488 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\qquad \\displaystyle \\sum _ { G \\subset ( Y _ { t e s t } ) } \\operatorname { a c c } ( G ) } \\\\ & { \\qquad \\quad = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } = \\frac { \\displaystyle \\sum _ { G \\in Y _ { n e w } } \\operatorname { a c c } ( G ) + \\sum _ { G \\in Y _ { i s o } } \\operatorname { a c c } ( G ) } { \\displaystyle | Y _ { t e s t } | } = } \\\\ & { \\qquad = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) + \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { i s o } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "9fd7fdc4b883a86a24095df6ff685452c5c55b7e7713190fd8c78b1f14fc1841.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 168, + 423, + 443, + 444.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 168, + 444.6666666666667, + 443, + 466.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 168, + 466.33333333333337, + 443, + 488.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "Equation 3 decomposes accuracy on the original data set as the weighted sum of two accuracies on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 506, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 238, + 521 + ], + "score": 1.0, + "content": "the set of the new test instances", + "type": "text" + }, + { + "bbox": [ + 238, + 508, + 260, + 519 + ], + "score": 0.91, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 506, + 367, + 521 + ], + "score": 1.0, + "content": "and a set of the instances", + "type": "text" + }, + { + "bbox": [ + 367, + 507, + 385, + 518 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 506, + 506, + 521 + ], + "score": 1.0, + "content": "already appeared in the train", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 348, + 531 + ], + "score": 1.0, + "content": "set and therefore available to the model. We call the term", + "type": "text" + }, + { + "bbox": [ + 348, + 519, + 387, + 531 + ], + "score": 0.75, + "content": "\\operatorname { a c c } ( Y _ { i s o } )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "as isomorphism bias, which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "corresponds to the accuracy of the model on the isomorphic test instances. 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MUTAGIMDB-BIMDB-MCOX2AIDSPROTEINS
NN0.829 (0.840)0.737 (0.733)0.501 (0.488)0.82 (0.872)0.996 (0.998)0.737 (0.834)
NN-PH0.867 (1.000)0.756 (1.000)0.522 (1.000)0.838 (1.000)0.996 (1.000)0.742 (1.000)
NN-P0.856 (0.847)0.737 (0.731)0.499 (0.486)0.795 (0.83)0.996 (0.999)0.729 (0.709)
WL0.862 (0.867)0.734 (0.990)0.502 (0.953)0.800 (0.974)0.993 (0.999)0.747 (0.950)
WL-PH0.907 (1.000)0.736 (1.000)0.504 (1.000)0.810 (1.000)0.994 (1.000)0.749 (1.000)
WL-P0.870 (0.838)0.724 (0.715)0.495 (0.487)0.794 (0.844)0.994 (0.999)0.740 (0.742)
V0.836 (0.902)0.707(0.820)0.503 (0.732)0.7811(0.966)0.994 (0.997)0.726 (0.946)
V-PH0.859 (1.000)0.750 (1.000)0.517 (1.000)0.794(1.000)0.996 (1.000)0.729 (1.000)
V-P0.827 (0.844)0.724 (0.728)0.496 (0.481)0.768 (0.852)0.996 (0.999)0.719 (0.741)
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Let", + "type": "text" + }, + { + "bbox": [ + 171, + 391, + 254, + 403 + ], + "score": 0.88, + "content": "Y _ { t e s t } = Y _ { n e w } \\cup Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 389, + 258, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 258, + 391, + 330, + 403 + ], + "score": 0.9, + "content": "Y _ { n e w } \\cap Y _ { i s o } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 389, + 343, + 406 + ], + "score": 1.0, + "content": ". 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If there are noisy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "labels in the training set and hence the set is not homogeneous, the model cannot guarantee the right", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "target label for these instances. Nonetheless, one can select a heuristic such as majority vote among", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 477, + 410, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 410, + 489 + ], + "score": 1.0, + "content": "the training isomorphic instances to select a proper label at the testing time.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 351, + 506 + ], + "score": 1.0, + "content": "In experiments, we compare neural network model (NN)", + "type": "text" + }, + { + "bbox": [ + 352, + 495, + 366, + 505 + ], + "score": 0.53, + "content": "\\mathrm { { X u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "et al., 2018) with graph kernels,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 503, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 408, + 519 + ], + "score": 1.0, + "content": "Weisfeiler-Lehman (WL) (Shervashidze et al., 2011b) and vertex histogram", + "type": "text" + }, + { + "bbox": [ + 409, + 505, + 423, + 516 + ], + "score": 0.48, + "content": "( \\mathbf { V } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 503, + 505, + 519 + ], + "score": 1.0, + "content": "(Sugiyama & Borg-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "wardt, 2015). For each model we consider two modifications: one for peering model on homoge-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 525, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 132, + 541 + ], + "score": 1.0, + "content": "neous", + "type": "text" + }, + { + "bbox": [ + 132, + 527, + 150, + 538 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 525, + 343, + 541 + ], + "score": 1.0, + "content": "(e.g. NN-PH) and one for peering model on all", + "type": "text" + }, + { + "bbox": [ + 343, + 527, + 361, + 538 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 525, + 506, + 541 + ], + "score": 1.0, + "content": "(e.g. NN-P). We show accuracy on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 537, + 491, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 128, + 549 + ], + "score": 0.9, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 537, + 158, + 551 + ], + "score": 1.0, + "content": "and on", + "type": "text" + }, + { + "bbox": [ + 158, + 538, + 176, + 549 + ], + "score": 0.91, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 537, + 491, + 551 + ], + "score": 1.0, + "content": "(in brackets) in Table 4. Experimentation details can be found in Appendix G.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 104, + 553, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 505, + 568 + ], + "score": 1.0, + "content": "From Table 4 we can conclude that peering model on homogeneous data is always the top performer.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 388, + 578 + ], + "score": 1.0, + "content": "This is aligned with the result of Claim 7.2, which guarantees that", + "type": "text" + }, + { + "bbox": [ + 389, + 565, + 451, + 578 + ], + "score": 0.93, + "content": "a c c ( Y _ { i s o } ) ~ = ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 565, + 506, + 578 + ], + "score": 1.0, + "content": ", but it is an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 470, + 590 + ], + "score": 1.0, + "content": "interesting observation if we compare it to the peering model on all isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 470, + 577, + 489, + 588 + ], + "score": 0.89, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "(-P", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "models). Moreover, the latter model often loses even to the original model, where no information", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "from the train set is explicitly taken into the test set. This can be explained by the noisy target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "labels in the orbits of isomorphic graphs, as can be seen both from the statistics for these datasets", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 358, + 633 + ], + "score": 1.0, + "content": "(Table 6) and accuracy measured just on isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 358, + 621, + 376, + 631 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 620, + 506, + 633 + ], + "score": 1.0, + "content": ". These results show that due to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "the presence of isomorphism bias performance of any classification model can be overestimated by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 153, + 654 + ], + "score": 1.0, + "content": "as much as", + "type": "text" + }, + { + "bbox": [ + 153, + 642, + 168, + 653 + ], + "score": 0.87, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "of accuracy on these datasets and hence future comparison of classification models", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 199, + 666 + ], + "score": 1.0, + "content": "should be estimated on", + "type": "text" + }, + { + "bbox": [ + 199, + 653, + 221, + 664 + ], + "score": 0.92, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "instead. These observations conforms with our theoretical findings and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 665, + 208, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 208, + 676 + ], + "score": 1.0, + "content": "conclusions in Section 6.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 263, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 265, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 265, + 703 + ], + "score": 1.0, + "content": "7.1 GENERAL RECOMMENDATIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 710, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "score": 1.0, + "content": "In order to avoid measuring performance over the wrong test sets, we provide a set of recommenda-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 721, + 379, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 379, + 732 + ], + "score": 1.0, + "content": "tions that will guarantee measuring the right metrics for the models.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 124, + 505, + 233 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 89, + 504, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 317, + 102 + ], + "score": 1.0, + "content": "Table 4: Mean classification accuracy for test sets", + "type": "text" + }, + { + "bbox": [ + 317, + 90, + 339, + 101 + ], + "score": 0.91, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 88, + 359, + 102 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 90, + 377, + 101 + ], + "score": 0.89, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 88, + 505, + 102 + ], + "score": 1.0, + "content": "(in brackets) in 10-fold cross-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 235, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 235, + 113 + ], + "score": 1.0, + "content": "validation. 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MUTAGIMDB-BIMDB-MCOX2AIDSPROTEINS
NN0.829 (0.840)0.737 (0.733)0.501 (0.488)0.82 (0.872)0.996 (0.998)0.737 (0.834)
NN-PH0.867 (1.000)0.756 (1.000)0.522 (1.000)0.838 (1.000)0.996 (1.000)0.742 (1.000)
NN-P0.856 (0.847)0.737 (0.731)0.499 (0.486)0.795 (0.83)0.996 (0.999)0.729 (0.709)
WL0.862 (0.867)0.734 (0.990)0.502 (0.953)0.800 (0.974)0.993 (0.999)0.747 (0.950)
WL-PH0.907 (1.000)0.736 (1.000)0.504 (1.000)0.810 (1.000)0.994 (1.000)0.749 (1.000)
WL-P0.870 (0.838)0.724 (0.715)0.495 (0.487)0.794 (0.844)0.994 (0.999)0.740 (0.742)
V0.836 (0.902)0.707(0.820)0.503 (0.732)0.7811(0.966)0.994 (0.997)0.726 (0.946)
V-PH0.859 (1.000)0.750 (1.000)0.517 (1.000)0.794(1.000)0.996 (1.000)0.729 (1.000)
V-P0.827 (0.844)0.724 (0.728)0.496 (0.481)0.768 (0.852)0.996 (0.999)0.719 (0.741)
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Consider a classification model", + "type": "text" + }, + { + "bbox": [ + 325, + 350, + 338, + 360 + ], + "score": 0.84, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 349, + 426, + 363 + ], + "score": 1.0, + "content": "that maps each graph", + "type": "text" + }, + { + "bbox": [ + 427, + 351, + 436, + 360 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 349, + 481, + 363 + ], + "score": 1.0, + "content": "to its label", + "type": "text" + }, + { + "bbox": [ + 482, + 350, + 501, + 362 + ], + "score": 0.88, + "content": "l ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 349, + 505, + 363 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 216, + 376 + ], + "score": 1.0, + "content": "We define a peering model", + "type": "text" + }, + { + "bbox": [ + 216, + 361, + 229, + 373 + ], + "score": 0.88, + "content": "\\widehat { \\mathcal { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 362, + 303, + 376 + ], + "score": 1.0, + "content": "such that for each", + "type": "text" + }, + { + "bbox": [ + 303, + 363, + 341, + 375 + ], + "score": 0.92, + "content": "G \\in Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 362, + 443, + 376 + ], + "score": 1.0, + "content": "it outputs the target label", + "type": "text" + }, + { + "bbox": [ + 443, + 361, + 463, + 375 + ], + "score": 0.91, + "content": "l ( \\widehat { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 362, + 505, + 376 + ], + "score": 1.0, + "content": ". Then the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 374, + 417, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 197, + 390 + ], + "score": 1.0, + "content": "accuracy of the model", + "type": "text" + }, + { + "bbox": [ + 198, + 374, + 210, + 387 + ], + "score": 0.88, + "content": "\\widehat { \\mathcal { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 376, + 401, + 390 + ], + "score": 1.0, + "content": "is at least as the accuracy of the original model", + "type": "text" + }, + { + "bbox": [ + 401, + 377, + 414, + 387 + ], + "score": 0.81, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 376, + 417, + 390 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 322, + 507, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 504, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 171, + 406 + ], + "score": 1.0, + "content": "Claim 7.2. Let", + "type": "text" + }, + { + "bbox": [ + 171, + 391, + 254, + 403 + ], + "score": 0.88, + "content": "Y _ { t e s t } = Y _ { n e w } \\cup Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 389, + 258, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 258, + 391, + 330, + 403 + ], + "score": 0.9, + "content": "Y _ { n e w } \\cap Y _ { i s o } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 389, + 343, + 406 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 344, + 391, + 362, + 403 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 389, + 505, + 406 + ], + "score": 1.0, + "content": "is homogeneous, then the accuracy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 402, + 476, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 119, + 417 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 404, + 140, + 415 + ], + "score": 0.9, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 402, + 241, + 417 + ], + "score": 1.0, + "content": "of a classification model", + "type": "text" + }, + { + "bbox": [ + 241, + 405, + 254, + 414 + ], + "score": 0.84, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 402, + 442, + 417 + ], + "score": 1.0, + "content": "is at most as the accuracy of its peering model", + "type": "text" + }, + { + "bbox": [ + 442, + 402, + 455, + 414 + ], + "score": 0.85, + "content": "\\hat { \\mathcal { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 402, + 476, + 417 + ], + "score": 1.0, + "content": ", i.e.:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 389, + 505, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 420, + 365, + 435 + ], + "lines": [ + { + "bbox": [ + 245, + 420, + 365, + 435 + ], + "spans": [ + { + "bbox": [ + 245, + 420, + 365, + 435 + ], + "score": 0.91, + "content": "\\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { t e s t } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { t e s t } ) .", + "type": "interline_equation", + "image_path": "907812d78c25ff43ac90709105e7b3ae11a87ba09ace62fdd34dc8cf6538e009.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 245, + 420, + 365, + 435 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 411, + 456 + ], + "score": 1.0, + "content": "Claim 7.2 establishes a way to increase performance only for homogeneous", + "type": "text" + }, + { + "bbox": [ + 412, + 444, + 429, + 455 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 443, + 505, + 456 + ], + "score": 1.0, + "content": ". If there are noisy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "labels in the training set and hence the set is not homogeneous, the model cannot guarantee the right", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "target label for these instances. Nonetheless, one can select a heuristic such as majority vote among", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 477, + 410, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 410, + 489 + ], + "score": 1.0, + "content": "the training isomorphic instances to select a proper label at the testing time.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 443, + 505, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 351, + 506 + ], + "score": 1.0, + "content": "In experiments, we compare neural network model (NN)", + "type": "text" + }, + { + "bbox": [ + 352, + 495, + 366, + 505 + ], + "score": 0.53, + "content": "\\mathrm { { X u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "et al., 2018) with graph kernels,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 503, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 408, + 519 + ], + "score": 1.0, + "content": "Weisfeiler-Lehman (WL) (Shervashidze et al., 2011b) and vertex histogram", + "type": "text" + }, + { + "bbox": [ + 409, + 505, + 423, + 516 + ], + "score": 0.48, + "content": "( \\mathbf { V } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 503, + 505, + 519 + ], + "score": 1.0, + "content": "(Sugiyama & Borg-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "wardt, 2015). For each model we consider two modifications: one for peering model on homoge-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 525, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 132, + 541 + ], + "score": 1.0, + "content": "neous", + "type": "text" + }, + { + "bbox": [ + 132, + 527, + 150, + 538 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 525, + 343, + 541 + ], + "score": 1.0, + "content": "(e.g. NN-PH) and one for peering model on all", + "type": "text" + }, + { + "bbox": [ + 343, + 527, + 361, + 538 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 525, + 506, + 541 + ], + "score": 1.0, + "content": "(e.g. NN-P). We show accuracy on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 537, + 491, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 128, + 549 + ], + "score": 0.9, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 537, + 158, + 551 + ], + "score": 1.0, + "content": "and on", + "type": "text" + }, + { + "bbox": [ + 158, + 538, + 176, + 549 + ], + "score": 0.91, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 537, + 491, + 551 + ], + "score": 1.0, + "content": "(in brackets) in Table 4. Experimentation details can be found in Appendix G.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 493, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 104, + 553, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 505, + 568 + ], + "score": 1.0, + "content": "From Table 4 we can conclude that peering model on homogeneous data is always the top performer.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 388, + 578 + ], + "score": 1.0, + "content": "This is aligned with the result of Claim 7.2, which guarantees that", + "type": "text" + }, + { + "bbox": [ + 389, + 565, + 451, + 578 + ], + "score": 0.93, + "content": "a c c ( Y _ { i s o } ) ~ = ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 565, + 506, + 578 + ], + "score": 1.0, + "content": ", but it is an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 470, + 590 + ], + "score": 1.0, + "content": "interesting observation if we compare it to the peering model on all isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 470, + 577, + 489, + 588 + ], + "score": 0.89, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "(-P", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "models). Moreover, the latter model often loses even to the original model, where no information", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "from the train set is explicitly taken into the test set. This can be explained by the noisy target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "labels in the orbits of isomorphic graphs, as can be seen both from the statistics for these datasets", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 358, + 633 + ], + "score": 1.0, + "content": "(Table 6) and accuracy measured just on isomorphic instances", + "type": "text" + }, + { + "bbox": [ + 358, + 621, + 376, + 631 + ], + "score": 0.9, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 620, + 506, + 633 + ], + "score": 1.0, + "content": ". These results show that due to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "the presence of isomorphism bias performance of any classification model can be overestimated by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 153, + 654 + ], + "score": 1.0, + "content": "as much as", + "type": "text" + }, + { + "bbox": [ + 153, + 642, + 168, + 653 + ], + "score": 0.87, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "of accuracy on these datasets and hence future comparison of classification models", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 199, + 666 + ], + "score": 1.0, + "content": "should be estimated on", + "type": "text" + }, + { + "bbox": [ + 199, + 653, + 221, + 664 + ], + "score": 0.92, + "content": "Y _ { n e w }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "instead. These observations conforms with our theoretical findings and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 665, + 208, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 208, + 676 + ], + "score": 1.0, + "content": "conclusions in Section 6.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 553, + 506, + 676 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 263, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 265, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 265, + 703 + ], + "score": 1.0, + "content": "7.1 GENERAL RECOMMENDATIONS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 710, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "score": 1.0, + "content": "In order to avoid measuring performance over the wrong test sets, we provide a set of recommenda-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 721, + 379, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 379, + 732 + ], + "score": 1.0, + "content": "tions that will guarantee measuring the right metrics for the models.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 106, + 710, + 504, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 132, + 83, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 132, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 132, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "• We open-source new, ”clean” data sets that do not include isomorphic instances that are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "in Table 8. To tackle this problem in the future, we propose to use clean versions of the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 142, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 142, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "data set for which isomorphism bias vanishes. For each data set we consider the found", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "graph orbits and keep only one graph from each orbit if and only if the graphs in the orbit", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 141, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "have the same label. 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data setTypeNCAvg.NodesAvg.EdgesN.L.E.L.Source
FIRSTMM_DBMolecular41111377.273074.1+1Neumann et al. (2016)
OHSUMolecular79282.01199.66+1Pan (2018)
KKIMolecular83226.9648.42+1Pan (2018)
Peking_1Molecular85239.3177.35+1Pan (2018)
MUTAGMolecular188217.9319.79++Kriege&Mutzel (2012)
MSRC_21CVision2092040.2896.6+1Neumann et al. (2016)
MSRC_9Vision221840.5897.94+1Neumann et al. (2016)
CuneiformMolecular2673021.2744.8++Kriege et al. (2018a)
SYNTHETICSynthetic300210019611Feragen et al. (2013)
COX2_MDMolecular303226.28335.12++Kriege&Mutzel (2012)
BZR_MDMolecular306221.3225.06++Kriege&Mutzel(2012)
PTC_MMMolecular336213.9714.32++Kriege&Mutzel (2012)
PTC_MRMolecular344214.2914.69++Kriege&Mutzel (2012)
PTC_FMMolecular349214.1114.48++Kriege&Mutzel(2012)
PTC_FRMolecular351214.5615++Kriege&Mutzel(2012)
DHFR_MDMolecular393223.87283.01++Kriege&Mutzel (2012)
SynthieSynthetic400495172.9311Morris et al. (2016)
BZRMolecular405235.7538.36+1Sutherland et al. (2003)
ER_MDMolecular446221.33234.85++Kriege&Mutzel (2012)
COX2Molecular467241.2243.45+1Sutherland etal. (2003)
DHFRMolecular467242.4344.54+-Sutherland et al. (2003)
MSRC_21Vision5632077.52198.32+1Neumann et al. (2016)
ENZYMESMolecular600632.6362.14+-Borgwardt et al. (2005)
IMDB-BINARYSocial1000219.7796.53-1Yanardag&Vishwanathan (2015)
PROTEINSMolecular1113239.0672.82+1Borgwardt etal. (2005)
DDMolecular11782284.32715.66+1Shervashidze et al. (2011a)
IMDB-MULTISocial150031365.94-1Yanardag&Vishwanathan (2015)
AIDSMolecular2000215.6916.2++Riesen&Bunke (2008)
REDDIT-BINARYSocial20002429.63497.75-1Yanardag&Vishwanathan (2015)
Letter-highMolecular2250154.674.5-1Riesen&Bunke (2008)
Letter-lowMolecular2250154.683.13-1Riesen&Bunke (2008)
Letter-medMolecular2250154.674.5-1Riesen&Bunke (2008)
FingerprintMolecular280045.424.42-1Riesen&Bunke(2008)
COIL-DELMolecular390010021.5454.24-+Riesen&Bunke (2008)
COIL-RAGMolecular39001003.013.02-1Riesen&Bunke (2008)
NCI1Molecular4110229.8732.3+1Shervashidze et al. (2011a)
NCI109Molecular4127229.6832.13+1Shervashidze et al. (2011a)
FRANKENSTEINMolecular4337216.917.88-1Orsini et al. (2015)
MutagenicityMolecular4337230.3230.77++Riesen&Bunke (2008)
REDDIT-MULTI-5KSocial49995508.52594.87-1Yanardag&Vishwanathan (2015)
COLLABSocial5000374.492457.78-1Yanardag&Vishwanathan(2015)
Tox21_AREMolecular7167216.2816.52++Challenge (2014)
Tox21_aromataseMolecular7226217.517.79++Challenge (2014)
Tox21_MMPMolecular7320217.4917.83++Challenge (2014)
Tox21_ERMolecular7697217.5817.94++Challenge (2014)
Tox21_HSEMolecular8150216.7217.04++Challenge (2014)
Tox21_AHRMolecular8169218.0918.5++Challenge (2014)
Tox21_PPAR-gammaMolecular8184217.2317.55++Challenge (2014)
Tox21_AR-LBDMolecular8599217.7718.16++Challenge (2014)
Tox21_p53Molecular8634217.7918.19++Challenge (2014)
Tox21ER_LBDMolecular8753218.0618.47++Challenge (2014)
Tox21_ATAD5Molecular9091217.8918.3++Challenge (2014)
Tox21_ARMolecular9362218.3918.84++Challenge (2014)
REDDIT-MULTI-12KSocial1192911391.41456.89-1Yanardag& Vishwanathan (2015)
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data setTypeNCAvg.NodesAvg.EdgesN.L.E.L.Source
FIRSTMM_DBMolecular41111377.273074.1+1Neumann et al. (2016)
OHSUMolecular79282.01199.66+1Pan (2018)
KKIMolecular83226.9648.42+1Pan (2018)
Peking_1Molecular85239.3177.35+1Pan (2018)
MUTAGMolecular188217.9319.79++Kriege&Mutzel (2012)
MSRC_21CVision2092040.2896.6+1Neumann et al. (2016)
MSRC_9Vision221840.5897.94+1Neumann et al. (2016)
CuneiformMolecular2673021.2744.8++Kriege et al. (2018a)
SYNTHETICSynthetic300210019611Feragen et al. (2013)
COX2_MDMolecular303226.28335.12++Kriege&Mutzel (2012)
BZR_MDMolecular306221.3225.06++Kriege&Mutzel(2012)
PTC_MMMolecular336213.9714.32++Kriege&Mutzel (2012)
PTC_MRMolecular344214.2914.69++Kriege&Mutzel (2012)
PTC_FMMolecular349214.1114.48++Kriege&Mutzel(2012)
PTC_FRMolecular351214.5615++Kriege&Mutzel(2012)
DHFR_MDMolecular393223.87283.01++Kriege&Mutzel (2012)
SynthieSynthetic400495172.9311Morris et al. (2016)
BZRMolecular405235.7538.36+1Sutherland et al. (2003)
ER_MDMolecular446221.33234.85++Kriege&Mutzel (2012)
COX2Molecular467241.2243.45+1Sutherland etal. (2003)
DHFRMolecular467242.4344.54+-Sutherland et al. (2003)
MSRC_21Vision5632077.52198.32+1Neumann et al. (2016)
ENZYMESMolecular600632.6362.14+-Borgwardt et al. (2005)
IMDB-BINARYSocial1000219.7796.53-1Yanardag&Vishwanathan (2015)
PROTEINSMolecular1113239.0672.82+1Borgwardt etal. (2005)
DDMolecular11782284.32715.66+1Shervashidze et al. (2011a)
IMDB-MULTISocial150031365.94-1Yanardag&Vishwanathan (2015)
AIDSMolecular2000215.6916.2++Riesen&Bunke (2008)
REDDIT-BINARYSocial20002429.63497.75-1Yanardag&Vishwanathan (2015)
Letter-highMolecular2250154.674.5-1Riesen&Bunke (2008)
Letter-lowMolecular2250154.683.13-1Riesen&Bunke (2008)
Letter-medMolecular2250154.674.5-1Riesen&Bunke (2008)
FingerprintMolecular280045.424.42-1Riesen&Bunke(2008)
COIL-DELMolecular390010021.5454.24-+Riesen&Bunke (2008)
COIL-RAGMolecular39001003.013.02-1Riesen&Bunke (2008)
NCI1Molecular4110229.8732.3+1Shervashidze et al. (2011a)
NCI109Molecular4127229.6832.13+1Shervashidze et al. (2011a)
FRANKENSTEINMolecular4337216.917.88-1Orsini et al. (2015)
MutagenicityMolecular4337230.3230.77++Riesen&Bunke (2008)
REDDIT-MULTI-5KSocial49995508.52594.87-1Yanardag&Vishwanathan (2015)
COLLABSocial5000374.492457.78-1Yanardag&Vishwanathan(2015)
Tox21_AREMolecular7167216.2816.52++Challenge (2014)
Tox21_aromataseMolecular7226217.517.79++Challenge (2014)
Tox21_MMPMolecular7320217.4917.83++Challenge (2014)
Tox21_ERMolecular7697217.5817.94++Challenge (2014)
Tox21_HSEMolecular8150216.7217.04++Challenge (2014)
Tox21_AHRMolecular8169218.0918.5++Challenge (2014)
Tox21_PPAR-gammaMolecular8184217.2317.55++Challenge (2014)
Tox21_AR-LBDMolecular8599217.7718.16++Challenge (2014)
Tox21_p53Molecular8634217.7918.19++Challenge (2014)
Tox21ER_LBDMolecular8753218.0618.47++Challenge (2014)
Tox21_ATAD5Molecular9091217.8918.3++Challenge (2014)
Tox21_ARMolecular9362218.3918.84++Challenge (2014)
REDDIT-MULTI-12KSocial1192911391.41456.89-1Yanardag& Vishwanathan (2015)
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data setSize,NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
Letter-high225094220097.783.6795.91
IMDB-MULTI1500100121280.86.3974.67
Tox21_ATAD590911461616767.840.099.15
Tox21PPAR-gamma81841265551367.360.17.77
Tox21_AR93621519629567.240.088
Tox21_p5386341345580067.180.0911.28
Tox21_AR-LBD85991354576667.050.096.88
Tox21_MMP73201138487566.60.118.76
Tox21_HSE81501218542566.560.118.02
Tox21ERLBD87531375579166.160.0912.41
Tox21_ER76971203507865.970.0927.32
Tox21_AHR81691299537765.820.0915.61
Tox21_aromatase72261084472765.420.15.07
Tox21_ARE71671047468265.330.1126.45
AIDS2000371125962.950.130.35
COX24677628360.60.620.56
IMDB-BINARY100011757957.90.6731.8
FRANKENSTEIN4337574223051.420.0930.87
MUTAG188317942.020.496.91
BZR4054316540.740.68.89
PTC_MM3364213239.290.4623.21
PTC_MR3444012536.340.4125
PTC_FM3493912435.530.3923.5
DHFR7568925033.070.149.13
PTC_FR3513611633.050.3720.51
Mutagenicity4337397127429.380.0313.1
COLLAB5000158107721.540.116.68
COIL-DEL390015579620.410.0618.56
PROTEINS11133515113.570.19.07
NCI1411022552312.730.011.7
NCI109412722251912.580.011.7
ENZYMES6006101.6700
REDDIT-BINARY2000340.200
REDDIT-MULTI-12K119298170.1400.04
FIRSTMM_DB4110000
OHSU790000
KKI830000
Peking_18510000
MSRC_21C20910000
MSRC.92210000
Synthie40010000
MSRC_2156310000
DD117810000
REDDIT-MULTI-5K499910000
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data setSize,NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
Letter-high225094220097.783.6795.91
IMDB-MULTI1500100121280.86.3974.67
Tox21_ATAD590911461616767.840.099.15
Tox21PPAR-gamma81841265551367.360.17.77
Tox21_AR93621519629567.240.088
Tox21_p5386341345580067.180.0911.28
Tox21_AR-LBD85991354576667.050.096.88
Tox21_MMP73201138487566.60.118.76
Tox21_HSE81501218542566.560.118.02
Tox21ERLBD87531375579166.160.0912.41
Tox21_ER76971203507865.970.0927.32
Tox21_AHR81691299537765.820.0915.61
Tox21_aromatase72261084472765.420.15.07
Tox21_ARE71671047468265.330.1126.45
AIDS2000371125962.950.130.35
COX24677628360.60.620.56
IMDB-BINARY100011757957.90.6731.8
FRANKENSTEIN4337574223051.420.0930.87
MUTAG188317942.020.496.91
BZR4054316540.740.68.89
PTC_MM3364213239.290.4623.21
PTC_MR3444012536.340.4125
PTC_FM3493912435.530.3923.5
DHFR7568925033.070.149.13
PTC_FR3513611633.050.3720.51
Mutagenicity4337397127429.380.0313.1
COLLAB5000158107721.540.116.68
COIL-DEL390015579620.410.0618.56
PROTEINS11133515113.570.19.07
NCI1411022552312.730.011.7
NCI109412722251912.580.011.7
ENZYMES6006101.6700
REDDIT-BINARY2000340.200
REDDIT-MULTI-12K119298170.1400.04
FIRSTMM_DB4110000
OHSU790000
KKI830000
Peking_18510000
MSRC_21C20910000
MSRC.92210000
Synthie40010000
MSRC_2156310000
DD117810000
REDDIT-MULTI-5K499910000
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From the equation 3 we have:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 123, + 440, + 196 + ], + "lines": [ + { + "bbox": [ + 171, + 123, + 440, + 196 + ], + "spans": [ + { + "bbox": [ + 171, + 123, + 440, + 196 + ], + "score": 0.58, + "content": "\\begin{array} { l } { \\displaystyle \\csc ( Y _ { t e s t } ) = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) + \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { i s o } ) \\Rightarrow } \\\\ { \\displaystyle \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) > ( 1 - \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } ) \\mathrm { a c c } ( Y _ { i s o } ) \\Rightarrow } \\\\ { \\displaystyle \\operatorname { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { n e w } ) } \\end{array}", + "type": "interline_equation", + "image_path": "4402a3a1043de6381f652a329101219c19c1de54a287521d9bf9ead50204d107.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 170, + 123, + 440, + 147.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 170, + 147.33333333333334, + 440, + 171.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 170, + 171.66666666666669, + 440, + 196.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 228, + 237, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 237, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 237, + 243 + ], + "score": 1.0, + "content": "F PROOF OF CLAIM 7.2", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 338, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 339, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 339, + 267 + ], + "score": 1.0, + "content": "Proof. From the definition of the peering model we have:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 270, + 371, + 303 + ], + "lines": [ + { + "bbox": [ + 240, + 270, + 371, + 303 + ], + "spans": [ + { + "bbox": [ + 240, + 270, + 371, + 303 + ], + "score": 0.85, + "content": "\\begin{array} { l } { \\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { n e w } ) = \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { n e w } ) } \\\\ { \\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { i s o } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { i s o } ) = 1 } \\end{array}", + "type": "interline_equation", + "image_path": "c8be42f8ee557dd1f8e61b0b3e954d2d2b176960ba63099e7f99911a14946b5f.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 270, + 371, + 286.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 240, + 286.5, + 371, + 303.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 294, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 295, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 295, + 320 + ], + "score": 1.0, + "content": "Substituting these into the equation 3 we have:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 323, + 366, + 339 + ], + "lines": [ + { + "bbox": [ + 245, + 323, + 366, + 339 + ], + "spans": [ + { + "bbox": [ + 245, + 323, + 366, + 339 + ], + "score": 0.89, + "content": "\\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { t e s t } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { t e s t } ) .", + "type": "interline_equation", + "image_path": "ddd2cf0817fbfc278e4769d342721cfdc23901a475b28ff99d329f8d1408ad5a.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 245, + 323, + 366, + 339 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 273, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 275, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 275, + 384 + ], + "score": 1.0, + "content": "G EXPERIMENTATION DETAILS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 181, + 407 + ], + "score": 1.0, + "content": "NN model is from", + "type": "text" + }, + { + "bbox": [ + 181, + 396, + 195, + 406 + ], + "score": 0.28, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "et al. (2018) and evaluate it on the data sets from PyTorch-Geometric (Fey &", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "score": 1.0, + "content": "Lenssen, 2019). For each data set we perform 10-fold cross-validation such that each fold is evalu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 416, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 416, + 138, + 431 + ], + "score": 1.0, + "content": "ated on", + "type": "text" + }, + { + "bbox": [ + 138, + 417, + 158, + 428 + ], + "score": 0.9, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 416, + 245, + 431 + ], + "score": 1.0, + "content": "of hold-out instances", + "type": "text" + }, + { + "bbox": [ + 246, + 417, + 267, + 429 + ], + "score": 0.9, + "content": "Y _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 416, + 506, + 431 + ], + "score": 1.0, + "content": ". For each fold we train the model for 350 epochs selecting", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 370, + 441 + ], + "score": 1.0, + "content": "the final model with the best performance on the validation set (", + "type": "text" + }, + { + "bbox": [ + 370, + 428, + 389, + 439 + ], + "score": 0.85, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "from hold-out trained split)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "across all epochs. 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From the equation 3 we have:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 105, + 257, + 119 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 123, + 440, + 196 + ], + "lines": [ + { + "bbox": [ + 171, + 123, + 440, + 196 + ], + "spans": [ + { + "bbox": [ + 171, + 123, + 440, + 196 + ], + "score": 0.58, + "content": "\\begin{array} { l } { \\displaystyle \\csc ( Y _ { t e s t } ) = \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) + \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { i s o } ) \\Rightarrow } \\\\ { \\displaystyle \\frac { | Y _ { n e w } | } { | Y _ { t e s t } | } \\mathrm { a c c } ( Y _ { n e w } ) > ( 1 - \\frac { | Y _ { i s o } | } { | Y _ { t e s t } | } ) \\mathrm { a c c } ( Y _ { i s o } ) \\Rightarrow } \\\\ { \\displaystyle \\operatorname { a c c } ( Y _ { i s o } ) > \\operatorname { a c c } ( Y _ { n e w } ) } \\end{array}", + "type": "interline_equation", + "image_path": "4402a3a1043de6381f652a329101219c19c1de54a287521d9bf9ead50204d107.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 170, + 123, + 440, + 147.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 170, + 147.33333333333334, + 440, + 171.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 170, + 171.66666666666669, + 440, + 196.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 228, + 237, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 237, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 237, + 243 + ], + "score": 1.0, + "content": "F PROOF OF CLAIM 7.2", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 338, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 339, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 339, + 267 + ], + "score": 1.0, + "content": "Proof. From the definition of the peering model we have:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 252, + 339, + 267 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 270, + 371, + 303 + ], + "lines": [ + { + "bbox": [ + 240, + 270, + 371, + 303 + ], + "spans": [ + { + "bbox": [ + 240, + 270, + 371, + 303 + ], + "score": 0.85, + "content": "\\begin{array} { l } { \\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { n e w } ) = \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { n e w } ) } \\\\ { \\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { i s o } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { i s o } ) = 1 } \\end{array}", + "type": "interline_equation", + "image_path": "c8be42f8ee557dd1f8e61b0b3e954d2d2b176960ba63099e7f99911a14946b5f.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 270, + 371, + 286.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 240, + 286.5, + 371, + 303.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 294, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 295, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 295, + 320 + ], + "score": 1.0, + "content": "Substituting these into the equation 3 we have:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 306, + 295, + 320 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 323, + 366, + 339 + ], + "lines": [ + { + "bbox": [ + 245, + 323, + 366, + 339 + ], + "spans": [ + { + "bbox": [ + 245, + 323, + 366, + 339 + ], + "score": 0.89, + "content": "\\operatorname { a c c } _ { \\mathcal { M } } ( Y _ { t e s t } ) \\leq \\operatorname { a c c } _ { \\widehat { \\mathcal { M } } } ( Y _ { t e s t } ) .", + "type": "interline_equation", + "image_path": "ddd2cf0817fbfc278e4769d342721cfdc23901a475b28ff99d329f8d1408ad5a.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 245, + 323, + 366, + 339 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 273, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 275, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 275, + 384 + ], + "score": 1.0, + "content": "G EXPERIMENTATION DETAILS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 181, + 407 + ], + "score": 1.0, + "content": "NN model is from", + "type": "text" + }, + { + "bbox": [ + 181, + 396, + 195, + 406 + ], + "score": 0.28, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "et al. (2018) and evaluate it on the data sets from PyTorch-Geometric (Fey &", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "score": 1.0, + "content": "Lenssen, 2019). 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For each fold we train the model for 350 epochs selecting", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 370, + 441 + ], + "score": 1.0, + "content": "the final model with the best performance on the validation set (", + "type": "text" + }, + { + "bbox": [ + 370, + 428, + 389, + 439 + ], + "score": 0.85, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "from hold-out trained split)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "across all epochs. Additionally we found that for small data set performance during the first epochs", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "can be unstable on the validation set and thus we select our model only after the first 50 epochs. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "score": 1.0, + "content": "final model is evaluated on the test instances and corresponds to NN in the experiments. Peering", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 506, + 486 + ], + "score": 1.0, + "content": "models NN-PH and NN-P are obtained from NN by replicating the target labels for homogeneous", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 124, + 494 + ], + "score": 0.89, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 482, + 219, + 496 + ], + "score": 1.0, + "content": "and non-homogeneous", + "type": "text" + }, + { + "bbox": [ + 219, + 483, + 237, + 494 + ], + "score": 0.91, + "content": "Y _ { i s o }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "respectively. 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Data setSizeRetention, %Avg.NodesAvg.EdgesClassesMin. ClassMax. Class
SYNTHETIC0000000
Cuneiform0000000
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COX2_MD30.9930.33467212
DHFR_MD41.0225.25366.25213
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Fingerprint511.8214.4513.12640
BZR_MD61.9614.17130.17215
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ER_MD143.1418.07227.362113
IMDB-MULTI32121.422.35249.46385144
Tox21_ARE330246.0720.9621.8626022700
Tox21_ER356046.2522.2423.2924193141
Tox21_MMP340546.5222.423.4326182787
Tox21_HSE381446.821.2922.2822073607
Tox21-p53408847.3522.6223.7222913797
Tox21_PPAR-gamma387747.3721.9122.9121203757
Tox21_ATAD5431247.4322.5223.6121684144
Tox21_AR-LBD413448.0822.4123.4821503984
Tox21_AR450648.1322.9324.0521894317
Tox21_AHR393548.1722.8823.9824903445
Tox21ERLBD422448.2622.6923.821934031
Tox21_aromatase352448.7722.2423.2222343290
IMDB-BINARY49349.324.08221.962232261
COX223750.7542.1444.43268169
AIDS111055.518.2219.12310800
FRANKENSTEIN244856.4420.7822.35210201428
PTC_MM22667.2617.0417.74277149
BZR27668.1536.2538.81272204
PTC_MR23568.3117.2317.97296139
PTC_FM24269.3416.9617.66285157
MUTAG13571.8118.8520.8424293
PTC_FR25372.0817.1117.86286167
DHFR57876.4643.3745.532205373
Mutagenicity333576.932.9634.16214841851
COIL-DEL313380.3325.0564.2698139
COLLAB406481.2876.944667.9237702289
PROTEINS97587.643.4181.042343632
NCI1378592.0929.8432.37217812004
NCI109380192.129.6632.22218012000
ENZYMES59599.1732.4862.17698100
REDDIT-BINARY199899.9430.04996.4829981000
REDDIT-MULTI-12K1191799.9391.79914.68115132586
FIRSTMM_DB411001377.273073.931126
OHSU7910082.01199.6623544
KKI8310026.9648.4223746
Peking_18510039.3177.3523649
MSRC_21C20910040.2896.617129
MSRC_922110040.5897.9481930
Synthie40010091.6202.18490110
MSRC_2156310077.52198.32201034
DD1178100284.32715.662487691
REDDIT-MULTI-5K4999100508.511189.7559991000
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Let us prove Theorem 6.1. We denote by", + "type": "text" + }, + { + "bbox": [ + 302, + 106, + 453, + 119 + ], + "score": 0.92, + "content": "{ \\mathcal { L } } = \\{ ( x , y ) \\to L ( f ( x ) , y ) : f \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 104, + 505, + 120 + ], + "score": 1.0, + "content": "a composite", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 258, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 182, + 129 + ], + "score": 1.0, + "content": "loss class. For any", + "type": "text" + }, + { + "bbox": [ + 183, + 118, + 209, + 127 + ], + "score": 0.91, + "content": "L \\in { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 117, + 258, + 129 + ], + "score": 1.0, + "content": "we get that", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 133, + 412, + 163 + ], + "lines": [ + { + "bbox": [ + 199, + 133, + 412, + 163 + ], + "spans": [ + { + "bbox": [ + 199, + 133, + 412, + 163 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mathbb { P } } L - \\mathbb { E } _ { \\mathcal { D } } u L = \\mathbb { E } _ { \\mathbb { P } } L - \\mathbb { E } _ { \\mathbb { P } } u L + \\mathbb { E } _ { \\mathbb { P } } u L - \\mathbb { E } _ { \\mathcal { D } } u L \\leq } \\\\ { \\leq \\mathbb { E } _ { \\mathbb { P } } \\lvert ( 1 - u ) L \\rvert + ( \\mathbb { E } _ { \\mathbb { P } } u L - \\mathbb { E } _ { \\mathcal { D } } u L ) . } \\end{array}", + "type": "interline_equation", + "image_path": "11e7b64037a3708fadef6349f9ae48405e23717790621e622f87c44a0f2de068.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 133, + 412, + 148.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 199, + 148.0, + 412, + 163.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 454, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 454, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 148, + 186 + ], + "score": 1.0, + "content": "Since any", + "type": "text" + }, + { + "bbox": [ + 148, + 173, + 175, + 183 + ], + "score": 0.9, + "content": "L \\in { \\mathcal { L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 171, + 454, + 186 + ], + "score": 1.0, + "content": "is bounded from above by 1 for the first term in equation 5 we obtain", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 190, + 477, + 205 + ], + "lines": [ + { + "bbox": [ + 133, + 190, + 477, + 205 + ], + "spans": [ + { + "bbox": [ + 133, + 190, + 477, + 205 + ], + "score": 0.9, + "content": "\\begin{array} { r } { { { \\mathbb E } } _ { \\mathbb P } | ( 1 - u ) L | \\le { { \\mathbb E } } _ { \\mathbb P } g _ { + } ( w ) { { \\mathbb I } } _ { \\{ y = + 1 \\} } + { { \\mathbb E } } _ { \\mathbb P } g _ { - } ( w ) { { \\mathbb I } } _ { \\{ y = - 1 \\} } = g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) . } \\end{array}", + "type": "interline_equation", + "image_path": "3101761b14a2f4da5813ee55b81c800921d633c5870ade783cb52764fe971ca8.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 133, + 190, + 477, + 205 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "Thanks to McDiarmid’d concentration inequality Mohri et al. (2012), applied to the function class", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 196, + 238 + ], + "score": 0.91, + "content": "{ \\mathcal { L } } _ { u } = \\{ u L : L \\in { \\mathcal { L } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 225, + 268, + 239 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 269, + 226, + 322, + 237 + ], + "score": 0.5, + "content": "1 - \\delta , \\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 225, + 339, + 239 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 340, + 225, + 377, + 236 + ], + "score": 0.91, + "content": "\\mathcal { D } \\sim \\mathbb { P } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "we get the upper bound on the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 237, + 153, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 153, + 248 + ], + "score": 1.0, + "content": "excess risk", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 250, + 491, + 293 + ], + "lines": [ + { + "bbox": [ + 118, + 250, + 491, + 293 + ], + "spans": [ + { + "bbox": [ + 118, + 250, + 491, + 293 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\underset { L \\in \\mathcal { L } } { \\operatorname* { s u p } } ( \\mathbb { E } _ { \\mathbb { P } } u L - \\mathbb { E } _ { \\mathcal { D } } u L ) \\leq 2 \\mathcal { R } _ { N } ( \\mathcal { L } _ { u } ) + \\operatorname* { m a x } [ ( 1 + g _ { + } ( w ) ) , ( 1 + g _ { - } ( w ) ) ] \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } \\leq } \\\\ & { \\qquad \\leq 2 \\mathcal { R } _ { N } ( \\mathcal { L } _ { u } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "57d7b4a98f4fd6c66f5987469f6084c49a205d98b72bbac66a02dcd108af3017.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 118, + 250, + 491, + 264.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 118, + 264.3333333333333, + 491, + 278.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 118, + 278.66666666666663, + 491, + 292.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 301, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 228, + 315 + ], + "score": 1.0, + "content": "Let us find a relation between", + "type": "text" + }, + { + "bbox": [ + 228, + 302, + 265, + 314 + ], + "score": 0.92, + "content": "\\mathcal { R } _ { N } ( \\mathcal { L } _ { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 301, + 283, + 315 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 284, + 302, + 315, + 314 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { N } ( \\mathcal { L } )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 301, + 398, + 315 + ], + "score": 1.0, + "content": ". Here we denote by", + "type": "text" + }, + { + "bbox": [ + 398, + 304, + 407, + 313 + ], + "score": 0.84, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 301, + 434, + 315 + ], + "score": 1.0, + "content": "a pair", + "type": "text" + }, + { + "bbox": [ + 434, + 302, + 486, + 314 + ], + "score": 0.93, + "content": "z _ { i } = ( x _ { i } , y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 301, + 505, + 315 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 312, + 416, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 416, + 326 + ], + "score": 1.0, + "content": "the definition (see Mohri et al. (2012)) the empirical Rademacher complexity", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 329, + 508, + 434 + ], + "lines": [ + { + "bbox": [ + 111, + 329, + 508, + 434 + ], + "spans": [ + { + "bbox": [ + 111, + 329, + 508, + 434 + ], + "score": 0.92, + "content": "\\begin{array} { l } { \\displaystyle \\hat { \\mathfrak { z } } _ { \\mathcal { P } } ( \\mathcal { L } _ { u } ) = \\displaystyle \\frac { 1 } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i = 1 } ^ { N } \\sigma _ { i } u ( z _ { i } ) L ( z _ { i } ) \\leq } \\\\ { \\displaystyle \\leq \\frac { 1 } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i = 1 } ^ { N } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { + } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { - } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\geq y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) \\leq } \\\\ { \\displaystyle \\leq \\widehat { \\mathcal { R } } _ { \\mathcal { D } } ( \\mathcal { L } ) + \\frac { g _ { + } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\geq y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { - } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\in \\mathcal { L } _ { i } : y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "4697b3fca3bc7af60a5ce737bfcd952fce97337ac0162197567907001fa81edd.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 111, + 329, + 508, + 364.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 111, + 364.0, + 508, + 399.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 111, + 399.0, + 508, + 434.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 253, + 449 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 253, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 253, + 450 + ], + "score": 1.0, + "content": "Since we use the zero-one loss, then", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 454, + 487, + 481 + ], + "lines": [ + { + "bbox": [ + 122, + 454, + 487, + 481 + ], + "spans": [ + { + "bbox": [ + 122, + 454, + 487, + 481 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\sigma } \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) \\leq \\# \\{ i : y _ { i } = + 1 \\} , \\ \\mathbb { E } _ { \\sigma } \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) \\leq \\# \\{ i : y _ { i } = - 1 \\} .", + "type": "interline_equation", + "image_path": "cd09b9406c6ab503ef1e9ad760493a4dcbfc3cbe36e0c09dde31e54768466184.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 122, + 454, + 487, + 481 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 223, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 223, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 223, + 500 + ], + "score": 1.0, + "content": "The Rademacher complexity", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 503, + 461, + 554 + ], + "lines": [ + { + "bbox": [ + 150, + 503, + 461, + 554 + ], + "spans": [ + { + "bbox": [ + 150, + 503, + 461, + 554 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } & { } & { \\mathcal { R } _ { N } ( \\mathcal { L } _ { u } ) = \\mathbb { E } _ { \\mathcal { D } \\sim \\mathbb { P } ^ { N } } \\hat { \\mathcal { R } } _ { \\mathcal { D } } ( \\mathcal { L } _ { u } ) \\leq \\mathbb { E } _ { \\mathcal { D } \\sim \\mathbb { P } ^ { N } } \\Big [ \\hat { \\mathcal { R } } _ { \\mathcal { D } } ( \\mathcal { L } ) + \\frac { g _ { + } ( w ) } { N } \\# \\{ i : y _ { i } = + 1 \\} + } \\\\ & { } & { \\qquad + \\frac { g _ { - } ( w ) } { N } \\# \\{ i : y _ { i } = - 1 \\} \\Big ] = \\mathcal { R } _ { N } ( \\mathcal { L } ) + g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) . } \\end{array}", + "type": "interline_equation", + "image_path": "948c0948403923c4352c413ed56f426b51107edd87ffcf20c4aea2a386880044.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 150, + 503, + 461, + 520.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 150, + 520.0, + 461, + 537.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 150, + 537.0, + 461, + 554.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 564, + 506, + 587 + ], + "lines": [ + { + "bbox": [ + 104, + 562, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 104, + 562, + 183, + 579 + ], + "score": 1.0, + "content": "Using the fact that", + "type": "text" + }, + { + "bbox": [ + 183, + 564, + 267, + 578 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\mathcal { R } _ { N } ( \\mathcal { L } ) = \\frac { 1 } { 2 } \\mathcal { R } _ { N } ( \\mathcal { F } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 562, + 506, + 579 + ], + "score": 1.0, + "content": ", substituting inequalities 6, 7 and 8 into equation 5, we get", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 574, + 126, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 126, + 588 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 590, + 462, + 635 + ], + "lines": [ + { + "bbox": [ + 149, + 590, + 462, + 635 + ], + "spans": [ + { + "bbox": [ + 149, + 590, + 462, + 635 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } { { \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\Big ( \\mathbb { E } _ { \\mathbb { P } } l ( f ( x ) , y ) - \\mathbb { E } _ { \\mathcal { D } } u ( x , y ) l ( f ( x ) , y ) \\Big ) \\leq 3 ( g _ { + } ( w ) \\pi + g _ { - } ( w ) ( 1 - \\pi ) ) + } } \\\\ & { } & { + \\mathcal { R } _ { N } ( \\mathcal { F } ) + ( 2 + g _ { + } ( w ) + g _ { - } ( w ) ) \\sqrt { ( \\log \\delta ^ { - 1 } ) / ( 2 N ) } . } \\end{array}", + "type": "interline_equation", + "image_path": "3073a61d208e71204de32dc227c6de2bff90fa4a17ed6656a80132ae19108735.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 149, + 590, + 462, + 605.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 149, + 605.0, + 462, + 620.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 149, + 620.0, + 462, + 635.0 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 309, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 638, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 496, + 640, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 496, + 640, + 505, + 649 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 249, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 250, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 119, + 93 + ], + "score": 1.0, + "content": "I", + "type": "text" + }, + { + "bbox": [ + 122, + 81, + 250, + 93 + ], + "score": 1.0, + "content": "PROOF OF THEOREM 6.1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 504, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 302, + 120 + ], + "score": 1.0, + "content": "Proof. Let us prove Theorem 6.1. We denote by", + "type": "text" + }, + { + "bbox": [ + 302, + 106, + 453, + 119 + ], + "score": 0.92, + "content": "{ \\mathcal { L } } = \\{ ( x , y ) \\to L ( f ( x ) , y ) : f \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 104, + 505, + 120 + ], + "score": 1.0, + "content": "a composite", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 258, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 182, + 129 + ], + "score": 1.0, + "content": "loss class. 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{ \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { - } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\geq y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) \\leq } \\\\ { \\displaystyle \\leq \\widehat { \\mathcal { R } } _ { \\mathcal { D } } ( \\mathcal { L } ) + \\frac { g _ { + } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\geq y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) + \\frac { g _ { - } ( w ) } { N } \\mathbb { E } _ { \\sigma } \\displaystyle \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i \\in \\mathcal { L } _ { i } : y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) . } \\end{array}", + "type": "interline_equation", + "image_path": 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"\\mathbb { E } _ { \\sigma } \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = + 1 } \\sigma _ { i } L ( z _ { i } ) \\leq \\# \\{ i : y _ { i } = + 1 \\} , \\ \\mathbb { E } _ { \\sigma } \\operatorname* { s u p } _ { L \\in \\mathcal { L } _ { u } } \\sum _ { i : y _ { i } = - 1 } \\sigma _ { i } L ( z _ { i } ) \\leq \\# \\{ i : y _ { i } = - 1 \\} .", + "type": "interline_equation", + "image_path": "cd09b9406c6ab503ef1e9ad760493a4dcbfc3cbe36e0c09dde31e54768466184.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 122, + 454, + 487, + 481 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 223, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 223, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 223, + 500 + ], + "score": 1.0, + "content": "The Rademacher complexity", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 484, + 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MUTAGIMDB-BIMDB-MCOX2AIDSPROTEINS
NN0.829 (0.840)0.737 (0.733)0.501 (0.488)0.82 (0.872)0.996 (0.998)0.737 (0.834)
NN-PH0.867 (1.000)0.756 (1.000)0.522 (1.000)0.838 (1.000)0.996 (1.000)0.742 (1.000)
NN-P0.856 (0.847)0.737 (0.731)0.499 (0.486)0.795 (0.83)0.996 (0.999)0.729 (0.709)
WL0.862 (0.867)0.734 (0.990)0.502 (0.953)0.800 (0.974)0.993 (0.999)0.747 (0.950)
WL-PH0.907 (1.000)0.736 (1.000)0.504 (1.000)0.810 (1.000)0.994 (1.000)0.749 (1.000)
WL-P0.870 (0.838)0.724 (0.715)0.495 (0.487)0.794 (0.844)0.994 (0.999)0.740 (0.742)
V0.836 (0.902)0.707(0.820)0.503 (0.732)0.7811(0.966)0.994 (0.997)0.726 (0.946)
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data setTypeNCAvg.NodesAvg.EdgesN.L.E.L.Source
FIRSTMM_DBMolecular41111377.273074.1+1Neumann et al. (2016)
OHSUMolecular79282.01199.66+1Pan (2018)
KKIMolecular83226.9648.42+1Pan (2018)
Peking_1Molecular85239.3177.35+1Pan (2018)
MUTAGMolecular188217.9319.79++Kriege&Mutzel (2012)
MSRC_21CVision2092040.2896.6+1Neumann et al. (2016)
MSRC_9Vision221840.5897.94+1Neumann et al. (2016)
CuneiformMolecular2673021.2744.8++Kriege et al. (2018a)
SYNTHETICSynthetic300210019611Feragen et al. (2013)
COX2_MDMolecular303226.28335.12++Kriege&Mutzel (2012)
BZR_MDMolecular306221.3225.06++Kriege&Mutzel(2012)
PTC_MMMolecular336213.9714.32++Kriege&Mutzel (2012)
PTC_MRMolecular344214.2914.69++Kriege&Mutzel (2012)
PTC_FMMolecular349214.1114.48++Kriege&Mutzel(2012)
PTC_FRMolecular351214.5615++Kriege&Mutzel(2012)
DHFR_MDMolecular393223.87283.01++Kriege&Mutzel (2012)
SynthieSynthetic400495172.9311Morris et al. (2016)
BZRMolecular405235.7538.36+1Sutherland et al. (2003)
ER_MDMolecular446221.33234.85++Kriege&Mutzel (2012)
COX2Molecular467241.2243.45+1Sutherland etal. (2003)
DHFRMolecular467242.4344.54+-Sutherland et al. (2003)
MSRC_21Vision5632077.52198.32+1Neumann et al. (2016)
ENZYMESMolecular600632.6362.14+-Borgwardt et al. (2005)
IMDB-BINARYSocial1000219.7796.53-1Yanardag&Vishwanathan (2015)
PROTEINSMolecular1113239.0672.82+1Borgwardt etal. (2005)
DDMolecular11782284.32715.66+1Shervashidze et al. (2011a)
IMDB-MULTISocial150031365.94-1Yanardag&Vishwanathan (2015)
AIDSMolecular2000215.6916.2++Riesen&Bunke (2008)
REDDIT-BINARYSocial20002429.63497.75-1Yanardag&Vishwanathan (2015)
Letter-highMolecular2250154.674.5-1Riesen&Bunke (2008)
Letter-lowMolecular2250154.683.13-1Riesen&Bunke (2008)
Letter-medMolecular2250154.674.5-1Riesen&Bunke (2008)
FingerprintMolecular280045.424.42-1Riesen&Bunke(2008)
COIL-DELMolecular390010021.5454.24-+Riesen&Bunke (2008)
COIL-RAGMolecular39001003.013.02-1Riesen&Bunke (2008)
NCI1Molecular4110229.8732.3+1Shervashidze et al. (2011a)
NCI109Molecular4127229.6832.13+1Shervashidze et al. (2011a)
FRANKENSTEINMolecular4337216.917.88-1Orsini et al. (2015)
MutagenicityMolecular4337230.3230.77++Riesen&Bunke (2008)
REDDIT-MULTI-5KSocial49995508.52594.87-1Yanardag&Vishwanathan (2015)
COLLABSocial5000374.492457.78-1Yanardag&Vishwanathan(2015)
Tox21_AREMolecular7167216.2816.52++Challenge (2014)
Tox21_aromataseMolecular7226217.517.79++Challenge (2014)
Tox21_MMPMolecular7320217.4917.83++Challenge (2014)
Tox21_ERMolecular7697217.5817.94++Challenge (2014)
Tox21_HSEMolecular8150216.7217.04++Challenge (2014)
Tox21_AHRMolecular8169218.0918.5++Challenge (2014)
Tox21_PPAR-gammaMolecular8184217.2317.55++Challenge (2014)
Tox21_AR-LBDMolecular8599217.7718.16++Challenge (2014)
Tox21_p53Molecular8634217.7918.19++Challenge (2014)
Tox21ER_LBDMolecular8753218.0618.47++Challenge (2014)
Tox21_ATAD5Molecular9091217.8918.3++Challenge (2014)
Tox21_ARMolecular9362218.3918.84++Challenge (2014)
REDDIT-MULTI-12KSocial1192911391.41456.89-1Yanardag& Vishwanathan (2015)
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data setSize,NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
Letter-low225032224599.788.7296.22
DHFR_MD3932539299.756.8794.91
COIL-RAG390020389099.7425.2299.31
C0X2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
Fingerprint280069277499.0716.8689.29
BZR_MD3062230399.027.1695.75
Letter-med225039222698.938.0592.93
Letter-high225094220097.783.6795.91
IMDB-MULTI1500100121280.86.3974.67
Tox21_ATAD590911461616767.840.099.15
Tox21PPAR-gamma81841265551367.360.17.77
Tox21_AR93621519629567.240.088
Tox21_p5386341345580067.180.0911.28
Tox21_AR-LBD85991354576667.050.096.88
Tox21_MMP73201138487566.60.118.76
Tox21_HSE81501218542566.560.118.02
Tox21ERLBD87531375579166.160.0912.41
Tox21_ER76971203507865.970.0927.32
Tox21_AHR81691299537765.820.0915.61
Tox21_aromatase72261084472765.420.15.07
Tox21_ARE71671047468265.330.1126.45
AIDS2000371125962.950.130.35
COX24677628360.60.620.56
IMDB-BINARY100011757957.90.6731.8
FRANKENSTEIN4337574223051.420.0930.87
MUTAG188317942.020.496.91
BZR4054316540.740.68.89
PTC_MM3364213239.290.4623.21
PTC_MR3444012536.340.4125
PTC_FM3493912435.530.3923.5
DHFR7568925033.070.149.13
PTC_FR3513611633.050.3720.51
Mutagenicity4337397127429.380.0313.1
COLLAB5000158107721.540.116.68
COIL-DEL390015579620.410.0618.56
PROTEINS11133515113.570.19.07
NCI1411022552312.730.011.7
NCI109412722251912.580.011.7
ENZYMES6006101.6700
REDDIT-BINARY2000340.200
REDDIT-MULTI-12K119298170.1400.04
FIRSTMM_DB4110000
OHSU790000
KKI830000
Peking_18510000
MSRC_21C20910000
MSRC.92210000
Synthie40010000
MSRC_2156310000
DD117810000
REDDIT-MULTI-5K499910000
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data setSize, NNum. orbitsIso. graphs, I1%IP%Mismatched %
SYNTHETIC3002300100100100
Cuneiform267826710020.46100
DHFR_MD3932539299.756.8794.91
COX2_MD3031330199.3411.8398.68
ER_MD4463144299.15.5782.74
BZR_MD3062230399.027.1695.75
MUTAG188173619.150.140
PTC_FM349225415.470.0810.89
PTC_MM336225014.880.077.74
DHFR756399812.960.043.97
PTC_FR351204312.250.056.27
PTC_MR344194111.920.056.4
Tox21_ARE716722882011.440.013.91
Tox21_HSE815025091911.280.012.25
Tox21_aromatase722622380511.140.010.77
Tox21-p53863425894610.960.011.83
Tox21ER769723184010.910.012.4
COX2467255010.710.031.07
Tox21_PPAR-gamma818422786910.620.010.76
Tox21_MMP732020477010.520.012.19
Tox21_ERLBD875325591310.430.011.28
Tox21_AR936226596510.310.010.68
Tox21_ATAD5909125592410.160.011.04
Tox211_AR-LBD859923887110.130.010.7
BZR40516409.880.060.99
Tox21_AHR81692247959.730.012.07
PROTEINS111321746.650.032.61
AIDS200022542.700
Mutagenicity433731751.7300.92
NCI141109170.4100.05
ENZYMES600220.3300
NCI10941277120.2900.05
FIRSTMM_DB4110000
OHSU7910000
KKI8310000
Peking_18510000
MSRC_21C20910000
MSRC_922110000
MSRC_2156310000
DD117810000
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Data setSizeRetention, %Avg.NodesAvg.EdgesClassesMin. ClassMax. Class
SYNTHETIC0000000
Cuneiform0000000
COIL-RAG130.335.779.62715
Letter-low120.537.175.42513
COX2_MD30.9930.33467212
DHFR_MD41.0225.25366.25213
Letter-med291.296.835.668110
Fingerprint511.8214.4513.12640
BZR_MD61.9614.17130.17215
Letter-high602.677.037.237132
ER_MD143.1418.07227.362113
IMDB-MULTI32121.422.35249.46385144
Tox21_ARE330246.0720.9621.8626022700
Tox21_ER356046.2522.2423.2924193141
Tox21_MMP340546.5222.423.4326182787
Tox21_HSE381446.821.2922.2822073607
Tox21-p53408847.3522.6223.7222913797
Tox21_PPAR-gamma387747.3721.9122.9121203757
Tox21_ATAD5431247.4322.5223.6121684144
Tox21_AR-LBD413448.0822.4123.4821503984
Tox21_AR450648.1322.9324.0521894317
Tox21_AHR393548.1722.8823.9824903445
Tox21ERLBD422448.2622.6923.821934031
Tox21_aromatase352448.7722.2423.2222343290
IMDB-BINARY49349.324.08221.962232261
COX223750.7542.1444.43268169
AIDS111055.518.2219.12310800
FRANKENSTEIN244856.4420.7822.35210201428
PTC_MM22667.2617.0417.74277149
BZR27668.1536.2538.81272204
PTC_MR23568.3117.2317.97296139
PTC_FM24269.3416.9617.66285157
MUTAG13571.8118.8520.8424293
PTC_FR25372.0817.1117.86286167
DHFR57876.4643.3745.532205373
Mutagenicity333576.932.9634.16214841851
COIL-DEL313380.3325.0564.2698139
COLLAB406481.2876.944667.9237702289
PROTEINS97587.643.4181.042343632
NCI1378592.0929.8432.37217812004
NCI109380192.129.6632.22218012000
ENZYMES59599.1732.4862.17698100
REDDIT-BINARY199899.9430.04996.4829981000
REDDIT-MULTI-12K1191799.9391.79914.68115132586
FIRSTMM_DB411001377.273073.931126
OHSU7910082.01199.6623544
KKI8310026.9648.4223746
Peking_18510039.3177.3523649
MSRC_21C20910040.2896.617129
MSRC_922110040.5897.9481930
Synthie40010091.6202.18490110
MSRC_2156310077.52198.32201034
DD1178100284.32715.662487691
REDDIT-MULTI-5K4999100508.511189.7559991000
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