A Distance Correlation-based Kernel for Nonlinear Causal Clustering in Heterogeneous Populations
Authors: Alex Markham
Moritz Grosse-Wentrup
Venue: NeurIPS 2021 Submitted (2021)
Keywords: causality
kernels
clustering
structure learning
distance correlation
heterogeneous populations

Abstract: We consider the problem of causal structure learning in the setting of heterogeneous populations, i.e., populations in which a single causal structure does not adequately represent all population members, as is common in biological and social sciences. To this end, we introduce a distance covariance-based kernel designed specifically to measure the similarity between the underlying nonlinear causal structures of different samples.This kernel enables us to perform clustering to identify the homogeneous subpopulations. Indeed, we prove the corresponding feature map is a statistically consistent estimator of nonlinear independence structure, rendering the kernel itself a statistical test for the hypothesis that sets of samples come from different generating causal structures. We can then use existing methods to learn a causal structure for each of these subpopulations. We demonstrate using our kernel for causal clustering with an application in genetics, allowing us to reason about the latent transcription factor networks regulating measured gene expression levels.


A Distance Covariance-based Kernel for Nonlinear
Causal Clustering in Heterogeneous Populations
Anonymous Author(s)
Affiliation
Address
email
Abstract
We consider the problem of causal structure learning in the setting of heterogeneous populations. i... populations in which single causal structure does adequately represent all population members, as is common in biological and social sciences. To this end, we introduce distance covariance-based kernel designed specifically to measure the similarity between the underlying nonlinear causal structures of different samples. This kernel enables us to perform clustering to identify the homogeneous subpopulations. Indeed, we prove the corresponding feature map is a statistically consistent estimator of nonlinear independence structure, rendering the kernel itself statistical test for the hypothesis that sets of samples come from different generating causal structures. We then use existing methods to learn causal structure for each of these subpopulations. We demonstrate using our kernel for causal clustering with an application in genetics, allowing us to reason about the latent transcription factor networks regulating measured gene expression levels.
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Introduction
Learning causal relationships from observational and experimental data is one of the fundamental goals of scientific research, and causal inference methods are thus used in a wide variety of fields. The resulting variety of applications nevertheless share some common difficulties, such as causal inference from complex time-series data (Eichler 2012) or the underlying causal structure being obscured by unmeasured confounders (Greenland al. 1999). Another common difficulty, especially for
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applications in the biological and social sciences, causal inference from heterogeneous populations
Xie
2013
Brand and Thomas 2013) addressing this difficulty is main motivation.
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In general terms, we understand heterogeneous population to be one whose members are not adequately described by a single model but rather better described by collection models. Within 25 our context causal structure learning, this means population is heterogeneous some samples
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26 are generated by different causal structures we call this structural heterogeneity. We note that there
27 are other kinds of heterogeneity, such as that samples generated by different joint distributions over the same causal structure, which are not the scope of this work.
29 specific example of structural heterogeneity can be found in genetics: causal methods are used to
30 learn the structure of gene regulatory networks (Emmert-Streill et al. 2012), and gene expression data
31 from a single recording or experiment may include thousands genes, many of which are involved
32 entirely different networks (Liu 2015); thus, attempting to lear a single causal structure for all of
38 the genes will obscure the fact that different sets of them have different structures.
Submitted to 35th Conference Neural Information Processing Systems (NeurIPS 2021). Do distribute.

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75 Moonsaaassinnty-www assume there are enough samples for statistical inference, as well as the usual causal Markov and faithfulness assumptions. For the first step, cluster samples together if The bulk of our work in this paper, and our main contribution, is to introduce the dependence contribution kernel, which facilitates a flexible and easily extensible approach to causal clustering: first perform clustering to identify structurally homogeneous subsets of samples, and then proceed with the actual learning task on each cluster. We prove that our kernel is a statistically consistent estimator of the similarity of the causal structures underlying different samples and can thus be used 39 to find clusters that minimize structural heterogeneity for causal structure learning tasks. Furthermore,
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40 the kernel is derived from the distance covariance (Székely et al. 2007), imbuing it with the ability to detect nonlinear dependence. It can easily be used in wide array of clustering algorithms, such as k-manss, DBSCAN, spectral clustering, or any other method that analogously makes use of a similarity (or distance) measure between samples (Filippone et al.) 2008).
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The rest of the paper is organized follows: We finish this section by discussing some of the most relevant related work from the causal inference and statistics literature. All of Section 2 is devoted to the theory underlying our dependence contribution kernel, including a comparison of the familiar product- moment covariance with the distance covariance (Section 2.1). defining an equivalence 48 class of causal models with a convenient representation in the kernel space (Section [.2), and the
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49 actual definition of our kernel and proofs of its relevant properties (Section 2.3). Next, in Section
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3 we demonstrate causal clustering with the kernel on a heterogeneous gene expression data set, finding structurally homogeneous clusters for which we then learn latent causal measurement models, allowing us to reason about the different transcription factor networks responsible for regulating the measured gene expression levels. Finally, we conclude in Section 4 mentioning possible future work.
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1.1
Related Work
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Causal inference in heterogeneous populations sometimes refers to data-fusion (Bareinboim and Pearl 2016). i.e., combining known homogeneous subpopulations and performing causal inference on the resulting heterogeneous population, similarly, it can refer to meta-learning using known subpopulations (Sharma et al.) 2019). Other times, it refers to estimating heterogencous treatment effects (Xie et al. 2012: Athey and Imbens] 2015). However, in our case, the subpopulations are not known and we rather consider the problem of learning which samples come from which subpopulation, and these are differentiated according to structure instead of treatment effect.
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Previous work on causal clustering has focused more on the causal modeling aspect, using stronger assumptions about the underlying structures learn more detailed models. For example, Kummerrfll et al. 2) Kummerfeld and Ramsey (2016) focus on causal clustering in measurement models, with the goal of clustering different features together to study their latent causal structure, based on tetrad constraints within the linear product-moment covariance matrix. Huang and Zhang (2019) define a class of causal models facilitating mechanism- based clustering, learning causal models both for clusters of samples as well as a shared one for all samples, assuming the underlying structures
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89 are linear non-Gaussian. Saeed et al. (2020) characterize distributions arising from mixtures of directed acyelic graph (DAG) causal models (i.e., causal models without latent selection variables),
71 trying to learn both the component DAGs and representation of how they mixed. All of these
72 approaches, like most causal inference methods, make specific (and for some applications, restrictive) assumptions about the underlying distributions or causal structures.
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74 In contrast, our method not tied to specific distributional assumptions such as linearity or
77 they (implicitly, in the kernel space) have similar nonlinear independence structures. For the second
78 step, causal structure learning, any existing method (along with its corresponding assumptions) can in
79 principle be used. In our gene expression data application (Section 3), the measurement dependence
80 inducing latent (MeDIL) causal model framework (Markham and Grosse- Wenteupnnnrpp 2020), which
81 assumes the data consists of measurement variables that are causally connected only through latent
82 variables, seems appropriate, however other applications can easily use other methods. For example, component and mixture DAGs (Saced et al. 2020) can be better learned when one first knows which samples come from which component component-clusterrig clustering with our kernel ensures samples in different

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clusters come from different DAGs, and so using their method instead of the MeDIL framework would be a natural choice for applications in which a DAG (without any latents) is more appropriate.
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Theory
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2.1
Product-moment Covariance, Distance Covariance, and Dependence Contribution
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Though there is more causal relationships than probabílistic dependence, causal inference methods based graphical models ultimately rely at least implicitly learning conditional independence
91 (CI) relations. CI relations can be estimated in many ways, with different dependence measures and tests each having their own theoretical guarantees and being better suited for distributions of
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88 various different kinds of data (e.g., categorical, discrete, or continuous) and with various kinds
94 of relationships (e.g,, linear, monotonic nonlinear, arbitrary nonlinear) and with different testing assumptions (see Tiostheim et al. 2018 for a comprehensive overview).
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A widely used measure of dependence is the product- covariance, often just called covariance, which is defined for two zero-mean random variables X1 and X₂ as the scalar value cov (xv(X,1 X2) = E|X, X2). This can be extended from a pair of random variables to every pair of variables in a random vector, thus returning a matrix instead of a scalar. The covariance matrix for a vector of zero- mean random variables X = (X1,.... Xm) can be estimated from a set E R of n samples ST S, and the j.j' value of Σx is thus the estimate côv (X,, xx))
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as Σx
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Two random variables being probabilistically independent (denoted I1.) implies that their product- moment covariance is zero, i.c., X, IL X,' = cov(X,, X,)) = 0 (importantly, the inverse of this
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does not hold). Thus, the estimated product-moment covariance can be used in statistical hypothesis testing for probabilistic independence (Wasserman] 2013] Ch. 10): X, and Xy are assumed to be independent if and only cóv(X,, xy.) is sufficiently close to O. However, this method has an important problem: the product-moment covariance is only valid test statistic against linear dependence.
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Székely et al. 2007) introduce the distance covariance to remedy this problem: random variables are probabilistically independent and only if their distance covariance is zero, i.e., X, IL Xy, < (Cov(XX; Xy) = O, resulting in the estimated distance covariance being a valid test statistic against all types of dependence. The distance covariance is related to the product-moment covariance by dCov² (X,,(XX,,) Xy) cov(|X; Xx1,, |X - Xf.I) 2eov - XíI. [X,' - -X,1), where X;, and (X, X are independent and identically distributed (iid) copies of xx,) (Székely and Rizzo, 2014). The key intuition here is that the distances (e.g., XX, XÿD constitute nonlinear
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projection, so that using the linear product-momen covariance in this projected space allows for the detection of nonlinear dependence in the original space.
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Note that dCov is typically defined be a scalar value when taken between two uubitrary-dimensie random vectors, but restricted presentation of it above in terms of random variables is to make it more obviously analogous to the product-moment covariance between random variables. Thus, corresponding to εxx for random vectors, we define the following:
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Definition Let S Ra,m be a set of n samples from the vector of random variables X (X, 1x....: Xm). For each
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.1.....m. and i E {1,..., n). define the pairwise distance matrix Dj, with values given by Dj == [S:j Se.gl Now define the corresponding doubly-centered matrices Ch,p. = :J, DJ,,- DJ x' + DJ.... where putting bar over the matrix and replacing an index or with denotes taking the mean over that index. Define the matrix L E R so that each column a flattened doubly-centered distance matrix, := (veo(C"),.... vec(C )), where ve)(C") denotes "flattening" matrix O³ into column vector. Finally, the estimated distance covariance matrix over sample S is defined Åx L L.
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Analogous to Σx, the j. entry of Åx corresponds to dCoov (X,. cally equivalent to computing each pairwise distance covariance value and then manually filling in
(CCo2(XX,. (yy)-ineed it is mathemati-
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the matrix. The novelty of our Definition Tis in finding matrix of pairwise values instead of a single value for the distance covariance between random vectors, which helps provide an intuition for our next definition:
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Definition Let S E Ru. be a set of n samples from the vector random variables X (Xi...., :m)): note that we consistently use indices i, E [I....,7] n) and j.J' E fl....,my m).. Let D Ru... denote the 3-dimensional array of stacked pairwise distance matrices defined by Di.t. ر Se and use C E R.... to denote these same distance matrices after being doubly-centered, i.e., C.p' :- - - "i + D...g, where replacing an index or i' with denotes the entire sewer-dimensionall subarray over that index, and writing a bar, D, denotes taking the mean over that subarray. Then standardize the doubly-centered distances get Zi,'' D., Finally, the dependence contribution map,
IR - IR,,,, is defined as
(S Z... T(a),
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where T(a) E Rm.. is matrix of scaled critical values corresponding to a given significance level
0,
if j'
with (1) being the with zeros along the diagonal, i.e., T(a)ss' x otherwise"
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quantile of the chi-square distribution with degree of freedom.
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Notice the similarity between Definitions 2 and if we set T(a)) to be a matrix of Os and forgo standardization (i.e., use C' instead of Z). then i= (S = -x. Now, the differences: Åx is a single matrix computed over an entire set of samples, whereas is a map that projects each given sample to a new feature space; each entry of Åx is simply a distance covariance value, whereas each entry of the sum of (SS...) over i, by using standardization (using Z instead of C) and subtracting a critical value, corresponds to the result of using a distance covariance value in statistical hypothesis
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test for wewpeenddneeeideeede
Lemma 3 Let S RT.m be set of iid samples from random variables Xi...., .m with finite first moments. For a given significance level under the null hypothesis of X, -1 Xyr, , the test
reject hy if
(S. i. 0
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is statistically consistent against all types of dependence.
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Proof. This follows from (Székely and Rizzo 2009 Theorem 5 and Corollary 2) and how yp is defined to correspond to the difference between distance covariance and critical values.
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These differences between Åx and oo serve two important purposes: first, they ensure &p maps to a Hilbert space so that our Definition D is a corresponding kernel function (Schölkopf et al. 200T); and second, as the name "dependence contribution map" suggests, they ensure ((St.) is informative not just about distance covariance but about nonlinear dependence and about how the inclusion of sample S in set of samples S contributes to the dependence patterns estimated from S- this is the key intuition behind how our kernel function is used to learn structurally homogeneous sample subsets, as explicated in the following sections.
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2.2
Causal Graphs Kernel Space
156 In general, a full causal structure can only be learned with sufficient data about the effects of interventions, and thus causal structure learning from purely observational data is usually possible only up to equivalence class of causal graphs (Spirtes al. 2000 Pearl 2009). For example, the classic PC and IC algorithms, under the assumptions of no selection bias and no confounding by latent variables, do not necessarily return a fully-specified DAG but instead return a mixed graph,
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containing possibly directed and undirected edges, representing the Markov equivalence class (Spirtes)
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and Glymour 1991: Pearl and Verma 1995).
We now define set of equivalence classes for ancestral graphs (AGs), do not assume the absence of selection bias and latent confounders (Richardson et al., 2002):
causal DAGs-

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Definition 4 Consider an arbitrary ancestral graph A with the set of vertices V and edge function FM,, and denote the set of unconditional m-connection statements entailed by their corresponding unique maximal ancestral graph as M = {(i.j'): fm J' 0) S V x VA, For any ancestral graph A' such that V = V-d, define the unconditional equivalence relation denoted by 'mo'" as
A A A' if and only if M M
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This lemma has two parts: (i) the relation ~y an equivalence relation over the set of ancestral graphs A; (ii) for an arbitrary ancestral graph E A, the bidirected graph 4 (VA, E²). where F. maps all pairs (i,j") MA to the bidirected edge symbol '+>", is unique representative of the equivalence class [A].
Lemma
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Proof. For (i), recall that an equivalence relation is any relation satisfying reflexivity, symmerry, and transitivity (Devlin 2003), all of which are satisfied by ~u because of its correspondence to the relation "=' between sets. Thus, to prove (ii), it suffices to show that the map A. A//r ->
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A. [A] + 4 is injective (i.e, that it is a section) and that [s([[A]] = [A] (Mac Lane 2013). The key to the proof is the observation that 2/-A, because it contains only bidirected edges, is maximal and therefore entails exactly the unconditional m-separation statements MA,, thus by (i) we have UA ~U A or equivalently 2 E [A] or equivalently [4A1] = [A]. Let A, A'' be arbitrary AGs, and assume s([AA) = s([4"). Then by definition of we have U = 2 and by the observation above, 4 E [A'] and thus [A]] = [ making s injective. And finally, by the definition of s and also by the observation above, [s{[[])] = [ = [A]. completing the proof.
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This equivalence relation and its representatives has some important but perhaps subtle properties. First, it is different from Markov equivalence over AGs (which is characterized by partial ancestral graphs, PAGs) (Zhang] 2007) it uses only unconditional m-separation while PAGs are learned from conditional m-separation statements. Second, because all DAGs are AGs, ~C also an equivalence relation over DAGs. Third, being a representative means that every equivalence class includes exactly one fully bidirected graph (along with other equivalent AGs). Fourth, because each representative is formed by considering m-connected paths, 4A is not equivalent to what would be generated by some redge-wise" procedure, such as simply replacing every edge in a PAG/AG//DAG/Marko random field/moralized DAG with bidirected Finally, its most important property is that it facilitates Theorem g for which we first need a few more definitions.
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Definition 6 Given arbitrary ancestral graphs A, E A over the same set of vertices, define the Hamming similarity product, denoted 'e' as
A x A A and A' H.
where H = (V-, F and the function E. (j.j') = '++* if and only if E EA())))= E (.J))
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190 In words, the Hamming similarity product between two ancestral graphs returns fully bidirected graph, with edges only where the two graphs have the same edge type. Now, shifting from ancestral graphs to real- valued square matrices:
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Definition 7 Let '~o" denote the orthant equivalence relation ('orthant' is the generalization of 'quadrant" from I to arbitrarily higher dimensions) in square real matrices, i.c., for matrices
Y,Y" E R and with the element-wise function sign(Y) = [1, if Yj > or =
11, otherwise
Y Yoo Y if and only if sign(Y) = sign (Y)) 3') for all j.j'.
193 Theorem 8 Let a be the map from the set of unconditional equivalence classes over ancestral graphs 194 with m vertices, Am//ru = Umm to the set of orthant equivalence classes over the image of s, 185 i.e, m x m symmetric real matrices with positive diagonal entries, ((IR""//~ = 0m,, defined by 1, if EE (j.j') '++' or = Then is group isomorphism
196 a: 4 + O, where Ojs" -1- 11 otherwise
between (Umm, .) and (O"h (), where 'o' denotes the element-wise product.
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Proof. First, note that (U"h, is indeed a group, satisfying the three group axioms (Atin 201 the representative of its identity element is the fully connected bidirected graph over m vertices, U each element is its own inverse; and is associative. Likewise, (Omh )) is a group with identity element [1 m..",, each element its own inverse, and the associative element- wise product operator.
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Now, show the two groups are isomorphic, it suffices to show (i) that is bijective and (ii) that for arbitrary 4' E U"", a(U) O a(ll") = a(ll e ). For (i) notice that if U * U", then there must be at least one pair of vertices j.j such that (j.3') & E (j.j') and thus clearly Oj.y' * O so a in injective. Furthermore, notice that every distinct O E 50 is the image of some graph U, so a is also surjective. For (ii), for every j.j' E [l....,m) m.. the definitions of a, O and e ensure all U'), completing the
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al4) j.y o a(l
+ E (i3i" E (1,3") =
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proof.
For causal inference, which (often, but not necessarily) amounts to taking several samples in real space and inferring a single corresponding member in the space of ancestral graphs (or, more often, its quotient set by some equivalence relation), Theorem 8 means we can compare the different graphs of different sample sets without having to first move to the ancestral graph space.
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213 Finally, notice the space of real square matrices not a typical sample space but rather precisely (a superspace of) the space that our dependence contribution map yp (Definition maps samples to- thtthss means that mapping samples with oo allows us make use of the group isomorphism. Though this already provides an intuition for why using yp would help with causal clustering, explicitly mapping each sample with it would be unnecessarily computationally expensive, and we are ultimately interested in morphisms between metric spaces (not just groups) of samples and graphs. To address
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this, we thus now move on to defining a kernel for y.
2.3
The Dependence Contribution Kernel
Definition 9 Let S, T, and y be as in Definition 2 We define the dependence contribution kernel using the Frobenius (denoted by the subscript p) inner product and norm:
Sy.) = (p(S).), ())
(I, )1pp [1 1)Ip
221 A more convenient expression for applying the kernel to a data set is obtained by first defining a 222 helper kernel, y along with vec from Definition [
7(S Sv.)) (sp(S..),, )(
((vec(Z,,) veo( (eo(Z)) -2,,TZ, ZZ,TZ, + 11T112
This allows us to write
7(S,,, Sv.
x(8,6') = 7SS,,, (S,,),
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Finally, note that x can be readily implemented on an entire set of samples, returning an entire Gram (kernel) matrix instead of a scalar value, by replacing the matrix operations above with tensor operations and specifying the correct axes along which summation occurs implementation can be found in our open source Python package at htthtps: /nan://mmms/ 1 link after-
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227 A proper distance metric can also be obtained from this kernel through function composition: arccos o K. The key idea behind the kernel is that is the cosine similarity in the space that y maps to, meaning for arbitrary sample points x, x' it evaluates to cos(f), where is the angle between <(x) and vpx'). In this space, A represents the dissimilarity of the dependence patterns underlying and z', without being biased by the possibly different magnitudes of so(x) and g(x') due to differing variances. Indeed, it can be used statistical test of whether samples come from different dependence structures and therefore causal models:
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Theorem 10 Let S E Ru.m, S' E R''s be sets of n, n' iid samples drawn respectively from the with finite first moments. Then,
random variables X = (X1,..., Xm) and X' =
n(S S4
Ej.J" E {l,..., m) such that I(X,, Xy, + I( Xy. 0).
Proof. Through Slutsky's Theorem (see Takeshi 1985 Theorem 3.2.7) and the continuous mapping theorem (see Van der Vaart 2000, Theorem 2.3), the consistency of yp (Lemma] 3 guarantees the consistency of Ki Because the numerator of x: is Frobenius inner product of yp,
n' m ΣΣΣΣ 1
Thus, in order for En, K(S.,,, S,,) < O, there must be j and j' for which ((S.))).' > 0 but ((S) )a,y' < 0 (or vice versa), and thus the hypothesis test in Lemma 3 would reject the null hypothesis that X, Il. Xy but fail to reject that X 11 Xy.
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Corollary 11 Due to the relationship between independence structure and causal structure, an immediate of result of Theorem 10 is that E,; K(S.,, S) < 0 implies X and X' have different causal structures.
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Theorem 12 Let d be the distance measure between unconditional equivalence classes of ancestral graphs over m vertices, diU = m²-I((j.j") Eu (j.j")= '+*')l For given sample sets S, S' (i.e., real n n matrices), use (S) to denote the mean of the sample in kernel space, Σ (SS...). and say S ~KK S' if and only if ((S) ~o ((S"): denote the corresponding quotient set by this equivalence class as RR.m/ ~x= k and a representative from each equivalence class as Q E [S].
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Let Ò be the distance between sets of samples in K defined as 5(, Q') = m 2n Ei, 7( Q. Let b U + K"."", b : 24 + f2 where f2 is the unique element K such that ((()) alU). Then is distance- distance-preserving map (i.e., isometry) from the metric space (Umh d) to (K 8).
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Proof. Notice that (Umha)) is indeed a metric space (Choudhary 1993 Ch. 2): d(U, 2²) = iff Ul is the empty graph, which happens iff 24'; the symmetry of d follows from the symmetry and for subadditivity of d, observe that for vertices j.j in arbitrary 2-vertex graphs U, U, , we have either d(U, ) = 2, in which case d(((,(() + d(ld', ,4") = 4, or we have d(U) ) 0, in which case d(" + d(l'", (2") is either 0 or 4-in both cases d((), S (() + d(''' 2T"): this easily extends to graphs of arbitrary numbers of vertices. Likewise, (K",.T, 6) is a metric space:
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u
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5(Q, Q') =0 > 1
2n² Eie +(Q.,., Q.,) = m² - p(Q);,)' = ({Q)io': for all j. so iff Q = Q': symmetry and subadditivity of follow from the symmetry and subadditivity of y.
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Finally, to show b is an isometry, must show (i) that it is bijective and (ii) that for all U. E U u' 8(b(1), b((()). For (i, observe that by the group isomorphism a and definition of b, we have U + Lt' = alU) * a(ll') = Q + Q' = b(l)) * b(ll) and so b is injective. Also observe that because K is exactly the set of representatives of orthant equivalence classes of sample sets in kernel space, then for every Q E K, there exists a 21 such that b(24) = Q, and so b is surjective.
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For (ii), isomorphism and the relation between element-wise product and Frobenius inner product allow us write d(((,t') m² - Ejg((0 @ O')ss' m² (O, O')r. Substituting O O' with their corresponding 22, and because the Frobenius inner product is a sesquilinear form, we can write d(,) = m2 n² Ei, (),()),)) ()p,, which by Definition 10 finally gives us that d(, = 652 R¹) completing the proof.
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In less formal terms, Theorem 12 shows how the space of unconditional equivalence classes of ancestral graph corresponds to the space of real matrices, which is a common space for samples to lie
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in. More specifically, it shows how the structure defined by distances between graphs is the same as the structure defined by distances between sets of samples and how this sample distance related to our kernel K. Note that this is much stronger than Theorem 10 not only can K tell us that two sets of samples come from different causal models, it gives a measure of just how different the causal models statements.
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are, in terms of their differing unconditional nonlinear independeencies/mmsspprre
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To summarize, we began by defining (Definition 2) which maps a given data set into a new higher-dimensional feature space. This feature space corresponds to a space of causal graphical models, such that samples which are similar in the new feature space must come from similar causal models (Theorem ( Our main contribution then is to propose the dependence contribution kernel (Definition 9 This kernel is guaranteed not only to tell us that two sets of samples come from different causal models (Theorem 10 and Corollary [T but furthermore exactly how different the causal models are (Theorem 12). all without the computational expense of explicitly projecting samples or learning causal models. Thus, is well-suited for addressing the causal clustering problem and ensures that resulting clusters will be structurally homogencous so that subsequent causal structure learning will be more informative.
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Application
We use kernel k-means with our dependence contribution kernel to cluster gene expression data set and then use the measurement dependence inducing latent (MeDIL) causal model framework for structure learning within each cluster (Markham and Grosse Wentrup 2020). The goal of causal clustering here is to reason about the different latent transcription factor (TF) networks governing gene expression (see Verny et al. 2017, Hackett et al., 2020 for other latent causal model approaches to learning TF networks). The original data set comes from Iyer (1999) and can be found genome- www stanford.........o edu/ wwwteeeattaa//eeeeeeeeeeeeaaaaet httm////daa/tgte/ttttttaaaaaaaaa txt with subsequent analysis
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by Dhillon et al. (2003 2004). All of the code for our analysis is open source and available at link. afverr
httt::://non-amonyy/
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The data consists of the measured gene expression levels of 517 different genes from human fibroblast cells in response to serum exposure, measured 11 different time points, i.e., there are 517 samples and 11 different features. In genetics applications, it is not unusual to consider genes to be samples and expression (over time) be cictures-inded the three previous analyses of this data all have this approach-and the intuition is simply that we wish to cluster genes based on patterns in their expression levels over time, in order to identify subsets of genes that are controlled by the same gene regulatory network. Also notice that such data exemplifies the structurally heterogeneous populations discussed in Section [ different genes can of course be regulated by different TFs, and so we can better represent the data by first clustering it into subpopulations that are more homogeneous and then performing causal structure learning on each subpopulation.
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For clustering, we used h = 6, which we found by looking at both the Variance Ratio Criterion Calinski and Harabasz 1974) and the Silhouette Coefficients (Rousseeuw 1987). computed with the
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seikit-learn machine learning toolbox (Pedregosa et al. 2011). We implemented (unweighted) kernel
306 k- k-mans ourselves, using the pseudocode given by Dhillon et al. (2004), with initial mean points
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drawn uniformly at random from the sample set, and with significance level = 0.1 for the kernel parameter T(x).. We then used the MeDIL (Markham et al. 2020) package to learn the dependence
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structure and latent causal models for each cluster.
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Figure I shows an example of our results for three of the six gene clusters: Figure la shows their distance covariance heatmaps and estimated nonlinear dependence structure with significance level 0.1 (so the axes are the 11 different features, i.e. the time, in hours, at which gene expression level was measured), while Figure lb shows their corresponding causal structures, with measurement variables Mo for each of the features and learned latent variables L for different posited TFs.
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The results show a clear difference in causal structure for the different clusters and allow us to reason about the latent TFs regulating genes in different clusters: notice that the latents in cluster K1 each cause only two or three measurement variables that tend to be close together- e.g., L1 causes Mj and M2. indicating the TF corresponding to L2 "short-acting", only affecting gene expression
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from 30 minutes (M1) to hour (M2) after serum exposure; in contrast, the latents in cluster K3
each cause between two and seven measurement variables that tend to be more spread out-
L1 causes Mj and My, indicating the corresponding TF is more complicated, "long long-acting" but not
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Figure 1: Results of dependence contribution kernel clustering with significance level a 0.1.
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continuously so, affecting gene expression 30 minutes (M1) and 12 hours (M7) after serum exposure, but independently of gene expression the time between.
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Our results especially noteworthy compared what happens if one ignores the heterogeneity of the data and Iearns causal structure for the entire data set without first clustering with kernel: in that case, all of the measurement variables are dependent, with single latent causing all of them, and no meaningful conclusions can be drawn about how unmeasured transcription factors regulate measured gene expression, i.c., the heterogeneity obscures the underlying causal structures.
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Discussion
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We address the problem of causal clustering- -hat is, finding the different causal structures underlying structurally heterogeneous data Our main contribution develop the dependence contribution kernel and prove its suitability for the causal clustering task. This allows to first use the kernel with existing clustering methods, such kernel k-means DBSCAN, to identify homogencous subpopulations. Then we use existing causal structure learning methods on each subpopulation. The kernel guarantees that each subpopulation is more structurally homogeneous and therefore the resulting causal structures better capture the causal structures within the data than if a single model were learned for the entire heterogeneous population.
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Furthermore, prove several interesting theoretical properties of kernel, including (i) that it can be used as a statistical test for the hypothesis that two sets of samples come from different causal structures, as well as (ii) how it induces metric space that is isometric to the one defined by Hamming distance between ancestral graphs, i.e., comparing sets of samples with our kernel is equivalent to first estimating the causal graphs of the different sets and then comparing those graphs.
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Beyond the practical applications of our kernel, as shown by our application reasoning about latent transcription factor networks that regulate gene expression, this work also draws from and suggests further fruitful connections between variety of fields, including causal inference, kernel methods, and algebraic statistics.
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Checklist
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the pa- per's contributions and scope? [Yes] The first, fifth, and sixth sentences are covered thoroughly in Sections 3 and while the rest are covered thoroughly in Section 2
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(b) Did you describe the limitations of your work? [Yes] In Section 1.1 and throughout
Section 2
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(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our work has no direct potential negative societal impact-just the same indirect potential most theoretical work has
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [es]]
2. If you including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Yes, general assumptions in Section 1.1 as well as more specific assumptions within the statement of each relevant theorem/lemma/etc.
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(b) Did you include complete proofs of all theoretical results? [Yes] Proofs follow each Theorem and Lemma in the text
3. If you experiments...
(a) Did you include the code, data, and instructions needed reproduce the main ex- perimental results (either in the supplemental material or as URL)? [Yes] Though the linked pages contain identifying information, so we've ve included only placeholder 'https://non htap::::/nonn-nnnmmouusss - link. after reviex" links
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
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(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [N/A] We didn't 't run experiment multiple times but rather analyzed real data set
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(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)? [No] It runs in just few seconds, even on an old, underpowered laptop
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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(b) Did you mention the license of the assets? [Yes] We mentioned that it's all open source; details can be found in their respective repos/documentation
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(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]
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(d) Did you discuss whether and how consent was obtained from people whose data you'l 're using/curating? [Yes] The data is publicly available
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(e) Did you discuss whether the data you are using/curating contains personally identi information or offensive content? [N/A] It's gene expression data, so neither of these are an issue
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5. If you used crowdsourcing or conducted research with human subjects...
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(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
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13


Reviews:
[{"review_id": "pazqK-4bmNX", "reviewer": "Reviewer_TQkh", "summary": "Paper proposes a new distance covariance based kernel for causal clustering of heterogenous population data. It presents theoretical results  on how kernel it can be used as a two-sample statistical test for causal data and induces a metric space on ancestral graphs. Empirical results show leaned causal graphs for gene expression data.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Idea of a distance covariance kernel for two-sample statistical testing of causal data is novel and interesting and is able to check for non-linear dependencies which follows from properties of distance covariance. Hilbert Schmidt Independence criterion (HSIC) which is an equivalent criterion of distance covariance could also used to defined a similar kernel on causal structures but not addressed on the paper.\n\nResults on two-sample statistical testing and the induced metrics space on ancestral graphs provide a strong theoretical justification for the proposed kernel. \n\nEmpirical results on learned causal structures from gene expression data show the learned latent causal models but they still seems qualitative and verification on simulation data would be valuable here. Paper is well written and provides theoretically strong justification of the work which would be interested to the community.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Paper describes limitations of the causal structures learned and states assumptions clearly. Societal impact is not direct as it is mostly a theoretical work.", "review_text": "", "extra_scores": {"ethical_concerns": "No specific ethical concerns.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "34-3pAupOUh", "reviewer": "Reviewer_jXQL", "summary": "Authors proposed the dependence contribution kernel to resolve the homogeneous clusters in the projected feature space. This kernel is able to cluster samples based on difference in nonlinear independence structures.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The manuscript, especially the theory section, is well organized. Comparably the application section needs to be further polished.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "(1)\tIt would be helpful to perform a simulation study to demonstrate the power of the distance covariance-based kernel. The dataset (Iyer, 1999) is quite outdated as there are only 517 genes available. Nowadays it’s very common to have over 10,000 genes from the experiment. I’m curious how the kernel behaves when the number of genes scales up. The other possible source of heterogenous data is the whole-blood/PBMC gene expression as blood is a heterogenous collection of different blood cells. Maybe this kernel is helpful to discover the cell markers for different cell population which is an important question in biology.\n\n(2)\tAuthors should at least include one baseline model/kernel, e.g., plain k-means with Euclidean distance to highlight the novelty of the distance covariance-based kernels.\n\n(3)\tThere is ground truth based on biological knowledge regarding TF regulation. It’s impossible to make sense of the results without knowing what members are in each TF cluster. The other thing authors can try is to perform gene set enrichment analysis to make sense of the TF clusters.\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1"}, "extra_text": {}}, {"review_id": "2R3aOZRqZ4S", "reviewer": "Reviewer_vQoB", "summary": "This paper focuses on the problem of causal structural learning from heterogeneous populations, where the label or index of the population is unknown. The main contribution of this paper is on developing a distance covariance-based kernel to measure the similarity between the underlying nonlinear causal structures of different samples, so that we can first perform clustering to identify the homogeneous subpopulations and then use existing methods to learn a causal structure for each of these subpopulations. The proposed method is tested on genetic data.\n", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This paper studies an interesting problem—how to do causal structural learning when the data may be from different populations, without knowing their labels. However, overall, the writing needs to be polished to improve the readability, including the definitions and theorems. \n\nMy main concern of the proposed kernel metric is as follows. In Theorem 10 and Corollary 11, the authors show that if the kernel metric is negative, then X and X’ have different causal structures. The problem is that the theorem does not show the if and if only property; when the kernel metric is positive or zero, how should the conclusion be? Without the if and only if property, I do not think it can be applied to do clustering. Another issue is that Theorem 10 requires to know which n samples are in X and which n’ samples are in X’, and Theorem 10 holds asymptotically; but at the beginning of clustering, we need to start from n=n’=1, where the asymptotical result does not seem applicable.\n\nRegarding the experiments, it would be more convincing if the authors can perform some simulation studies, since we can know the ground truth and estimate the accuracy of the proposed method. In addition, it is important to compare the results with different sample sizes and graph structures. For example, when some populations have very few samples, then how is the accuracy?\n\nUpdate:\n\nThe author's response does explain some of my concerns about the if and only if property. However, I still have concerns about the writing and the experimental results. Also, the authors did not compare with any other methods.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a distance covariance-based kernel designed specifically for measuring the similarity between the underlying nonlinear causal structures of different samples, in order to deal with the problem of causal structural learning from heterogeneous populations without known labels, in which a single causal structure does not adequately describe all population members. According to the reviews, the paper would greatly benefit from simulation studies to demonstrate the power of the distance covariance-based kernel, and the authors should at least include one baseline model/kernel, e.g., plain k-means with Euclidean distance, to highlight the novelty of the distance covariance-based kernels.
Decision: Reject