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+# 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.
+
+<table><tr><td>n</td><td>k</td><td>Precision score</td></tr><tr><td>100 100</td><td>2 5</td><td>0.972 ± 0.004 0.952 ± 0.007</td></tr><tr><td>100</td><td>10</td><td>0.943 ± 0.006</td></tr><tr><td>100</td><td>20</td><td>0.955 ± 0.008</td></tr><tr><td>500</td><td>20</td><td>0.936 ± 0.007</td></tr><tr><td>1000 1000</td><td>20 50</td><td>0.933 ± 0.006</td></tr></table>
+
+![](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]

parse/train/5GihaaZKL4/5GihaaZKL4_content_list.json

@@ -0,0 +1,1272 @@
+[
+    {
+        "type": "text",
+        "text": "Pairwise Adjusted Mutual Information ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            292,
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+            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. ",
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+    {
+        "type": "text",
+        "text": "13 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            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. ",
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+        "type": "text",
+        "text": "41 2 Adjusted mutual information ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
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+        "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": [
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+    {
+        "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,
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+        "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,
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+        "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,
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+        ],
+        "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": [
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+    {
+        "type": "text",
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+        "bbox": [
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+        "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": [
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+        "page_idx": 1
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+    {
+        "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. ",
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+    {
+        "type": "text",
+        "text": "57 We have the equivalent definition: ",
+        "bbox": [
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+        "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",
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+    {
+        "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": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "60 Proposition 1. We have for any random variables $X$ and $Y$ : ",
+        "bbox": [
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+            626,
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+            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",
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+    },
+    {
+        "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": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "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,
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+        "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": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "67 By (1), we get: ",
+        "bbox": [
+            148,
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+        ],
+        "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,
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+        ],
+        "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. ",
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
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+        "bbox": [
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+    },
+    {
+        "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": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Pairwise adjusted mutual information. We define the pairwise adjusted mutual information as: ",
+        "bbox": [
+            166,
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+        ],
+        "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,
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+        ],
+        "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": [
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+        "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",
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+        "type": "text",
+        "text": "85 Moreover, $\\Delta _ { \\mathrm { p } } I ( X , Y ) = 0$ whenever $X$ or $Y$ is constant or equal to some permutation of $\\{ 1 , \\ldots , n \\}$ ",
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+    {
+        "type": "text",
+        "text": "Pairwise adjusted entropy. We also define the pairwise adjusted entropy as: ",
+        "bbox": [
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+        ],
+        "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",
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+    {
+        "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 $\\{ 1 , \\ldots , n \\}$ . ",
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+    {
+        "type": "text",
+        "text": "88 4 Application to clustering ",
+        "text_level": 1,
+        "bbox": [
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+        "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. ",
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+    {
+        "type": "text",
+        "text": "105 It is known that [Vinh et al., 2010]: ",
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+    },
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+        "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",
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+        "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 $n$ . ",
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+        "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). ",
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+    {
+        "type": "text",
+        "text": "114 Since $q ( A ) = s ( A , A )$ , it follows from (6) that: ",
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+    {
+        "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": [
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+        ],
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+    },
+    {
+        "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: ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "121 Theorem 1. We have for any clusterings $A , B$ ",
+        "bbox": [
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+        "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": [
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+    },
+    {
+        "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. ",
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+    },
+    {
+        "type": "text",
+        "text": "126 Corollary 1. We have for any clusterings $A , B$ : ",
+        "bbox": [
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+        ],
+        "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": [
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+    {
+        "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": [
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+        "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",
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+        "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 $n$ . ",
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+        "type": "text",
+        "text": "133 5 Experiments ",
+        "text_level": 1,
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+        "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: ",
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+        "img_path": "images/6c42fe6957f17783f8c1554e8baaab5de933abc19aa87ac9cef5f942232debb2.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of metrics on synthetic data $\\mathit { n } = 1 0 0 $ ). "
+        ],
+        "image_footnote": [],
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+    {
+        "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",
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+        "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. ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>n</td><td>k</td><td>Precision score</td></tr><tr><td>100 100</td><td>2 5</td><td>0.972 ± 0.004 0.952 ± 0.007</td></tr><tr><td>100</td><td>10</td><td>0.943 ± 0.006</td></tr><tr><td>100</td><td>20</td><td>0.955 ± 0.008</td></tr><tr><td>500</td><td>20</td><td>0.936 ± 0.007</td></tr><tr><td>1000 1000</td><td>20 50</td><td>0.933 ± 0.006</td></tr></table>",
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+        "image_caption": [
+            "Table 1: Precision score (mean $\\pm$ standard deviation) ",
+            "Figure 2: Computation time with respect to $n$ (mean $\\pm$ standard deviation). "
+        ],
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+        "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: ",
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+        "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 ",
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+        "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. ",
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+        "image_caption": [
+            "Figure 3: Comparison of metrics on the Gagolewski benchmark. "
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+        "img_path": "images/3056fac0045be17a9b229a2afba177861eb5726b163d2534a53a3d5166a34841.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of metrics on OpenML datasets. "
+        ],
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+        "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. ",
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+    },
+    {
+        "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]. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. For all authors... ",
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+    {
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+        "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] ",
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+        "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. ",
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+        "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] ",
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