On Low Rank Directed Acyclic Graphs and Causal Structure Learning

1 INTRODUCTION . An important goal in many sciences is to discover the underlying causal structures in various domains , both for the purpose of explaining and understanding phenomena , and for the purpose of predicting effects of interventions ( Pearl , 2009 ) . Due to the relative abundance of passively observed data as opposed to experimental data , how to learn causal structures from purely observational data has been vigorously investigated ( Peters et al. , 2017 ; Spirtes et al. , 2000 ) . In this context , causal structures are usually represented by directed acyclic graphs ( DAGs ) over a set of random variables . For this task , existing methods can be roughly categorized into two classes : constraint- and scorebased . The former use statistical tests to extract from data a number of constraints in the form of conditional ( in ) dependence and seek to identify the class of causal structures compatible with those constraints ( Meek , 1995 ; Spirtes et al. , 2000 ; Zhang , 2008 ) . The latter employ a score function to evaluate candidate causal structures relative to data and seek to locate the causal structure ( or a class of causal structures ) with the optimal score . Due to the combinatorial nature of the acyclicity constraint ( Chickering , 1996 ; He et al. , 2015 ) , most score-based methods rely on local heuristics to perform the search . A particular example is the greedy equivalence search ( GES ) algorithm ( Chickering , 2002 ) that can find an optimal solution with infinite data and proper model assumptions . Recently , Zheng et al . ( 2018 ) introduced a smooth acyclicity constraint w.r.t . graph adjacency matrix , and the task on linear data models was then formulated as a continuous optimization problem with least-squares loss . This change of perspective allows using deep learning techniques to model causal mechanisms and has already given rise to several new algorithms for causal structure learning with non-linear data , e.g. , Yu et al . ( 2019 ) ; Ng et al . ( 2019b ; a ) ; Ke et al . ( 2019 ) ; Lachapelle et al . ( 2020 ) ; Zheng et al . ( 2020 ) , among others . While these new algorithms represent the current state of the art in many settings , their performance generally degrades when the target DAG becomes large and relatively dense , as seen from the empirical results reported in the referred works and also in this paper . This issue is of course a challenge to other approaches . Ramsey et al . ( 2017 ) proposed fast GES for impressively large problems , but it works reasonably well only when the large structure is very sparse . The max-min hill-climbing ( MMHC ) ( Tsamardinos et al. , 2006 ) relies on local learning methods that often do not perform well when the target node has a large neighborhood . How to improve the performance on relatively large and dense DAGs is therefore an important question . In this work , we study the potential of exploiting a kind of low rank assumption on the DAG structure to help address this problem . The rank of a graph that concerns us is the algebraic rank of its associated weighted adjacency matrix . Similar to the role of a sparsity assumption on graph structures , we treat the low rank assumption as methodological and it is not restricted to a particular DAG learning method . However , unlike sparsity assumption , it is much less apparent when DAGs tend to be low rank and how low rank DAGs behave . Thus , besides demonstrating the utility of exploiting a low rank assumption in causal structure learning , another important goal is to improve our understanding of the low rank assumption by relating the rank of a graph to its graphical structure . Such a result also enables us to characterize the rank of a graph from several structural priors and helps to choose rank related hyperparameters for the learning algorithm . Our contributions are summarized as follows : • We show how to adapt existing causal structure learning methods to take advantage of the low rank assumption , and provide a strategy to select rank related hyperparameters utilizing the lower and upper bounds on the true rank , if they are available . • To improve our understanding of low rank DAGs , we establish some lower bounds on the rank of a DAG in terms of simple graphical conditions , which imply necessary conditions for DAGs to be low rank . • We also show that the maximum possible rank of weighted adjacency matrices associated with a directed graph is highly related to hubs in the graph , which suggests that scale-free networks tend to be low rank . From this result , we derive several graphical conditions to bound the rank of a DAG from above , providing simple sufficient conditions for low rank . • Empirically , we demonstrate that the low rank adaptations are indeed useful . Not only do they outperform the original algorithms when the low rank condition is satisfied , the performance is also very competitive even when the true rank is not as low as is assumed . Related Work The low rank assumption is frequently adopted in graph-based applications ( Smith et al. , 2012 ; Zhou et al. , 2013 ; Yao & Kwok , 2016 ; Frot et al. , 2019 ) , matrix completion and factorization ( Recht , 2011 ; Koltchinskii et al. , 2011 ; Cao et al. , 2015 ; Davenport & Romberg , 2016 ) , network sciences ( Hsieh et al. , 2012 ; Huang et al. , 2013 ; Zhang et al. , 2017 ) and so on , but to our best knowledge , has not been used on the DAG structures in the context of learning causal DAGs . We notice two works Barik & Honorio ( 2019 ) ; Tichavskỳ & Vomlel ( 2018 ) that assume low rank conditional probability tables in learning Bayesian networks , which are different from ours . Also related are existing works that studied the rank of real weighted matrices described by a given simple directed/undirected graph . However , most works only considered the zero-nonzero pattern of off-diagonal entries ( see , e.g. , Fallat & Hogben ( 2007 ) ; Hogben ( 2010 ) ; Mitchell et al . ( 2010 ) ) , whereas we also take into account the diagonal entries . This difference is crucial : if one only considers the off-diagonal entries , then the maximum rank over all possible weighted matrices is trivial and is always equal to the number of vertices . Consequently , many works focus on the minimum rank of a given graph , but to characterize exactly the minimum rank remains open , except for some special graph structures like trees ( Hogben , 2010 ) . Apart from these works , Edmonds ( 1967 ) studied algebraically the maximum rank for matrices with a common zero-nonzero pattern . In Section 4 , we use this result to relate the maximum possible rank to a more interpretable graphical condition , which further implies several structural conditions of DAGs that may be easier to obtain in practice . 2 PRELIMINARIES . 2.1 GRAPH TERMINOLOGY . A graph G is defined as a pair ( V , E ) , where V = { X1 , X2 , · · · , Xd } is the vertex set and E ⊂ V2 denotes the edge set . We are particularly interested in directed ( acyclic ) graphs in the context of causal structure learning . For any S ⊂ V , we use pa ( S , G ) , ch ( S , G ) , and adj ( S , G ) to denote the union of all parents , children , and adjacent vertices of the nodes of S in G , respectively . A graph is called weighted if every edge in the graph is associated with a non-zero value . We will work with weighted graphs and treat unweighted graphs as a special case where the edge weights are set to 1 . Weighted graphs can be treated algebraically via weighted adjacency matrices . Specifically , the weighted adjacency matrix of a weighted graph G is a matrix W ∈ Rd×d , where W ( i , j ) is the weight of edge Xi → Xj and W ( i , j ) 6= 0 if and only if Xi → Xj exists in G. The binary adjacency matrix A ∈ { 0 , 1 } d×d is such that A ( i , j ) = 1 if Xi → Xj in G and A ( i , j ) = 0 otherwise . The rank of a weighted graph is defined as the rank of the associated weighted adjacency matrix . 2.2 CAUSAL STRUCTURE LEARNING AND RECENT GRADIENT-BASED METHODS . A commonly used model in causal structure learning is the structural equation model ( SEM ) that describes data generating procedure . In a slight abuse of notation , we also use Xi ’ s to denote random variables associated with the nodes in a graph G. Assuming G being a DAG , then the SEM is given by Xi = fi ( pa ( Xi , G ) , i ) , i = 1 , 2 , . . . , d , where fi is a deterministic function and i ’ s are jointly independent noises . The SEM induces a marginal distribution P ( X ) over X = [ X1 , X2 , · · · , Xd ] T , and G and P ( X ) are said to form a causal Bayesian network ( Pearl , 2009 ; Spirtes et al. , 2000 ) . The problem of causal structure learning is to infer the underlying causal DAG G based on the marginal distribution P ( X ) , or more practically , an empirical version consisting of a number of i.i.d . observations from P ( X ) . We next briefly review recently developed gradient-based methods that rely on a smooth characterization of acyclicity of directed graphs . These methods aim to find a DAG that optimizes a score function and can be categorized into two classes . The first class of methods explicitly associates the target causal model with a weighted adjacency matrix W and then estimate W by solving optimization problems in the following form : min W , φ EX∼P ( X ) S ( X , h ( X ; W , φ ) ) , subject to trace ( eW◦W ) − d = 0 , ( 1 ) where h : Rd → Rd is a model function parameterized by W ( and other possible parameter φ ) that aims to reconstruct X , S ( · , · ) denotes a score function between the true and reconstructed variables , notation ◦ denotes the element-wise product , and eM is the matrix exponential of a square matrix M . The constraint was proposed by Zheng et al . ( 2018 ) , which is smooth and holds if and only if W indicates a DAG . Methods in this class include : NOTEARS ( Zheng et al. , 2018 ) , which targets linear models , with h ( X ; W , φ ) = WTX and S ( · , · ) being the Frobenius norm or equivalently the least-squares loss ; and DAG-GNN ( Yu et al. , 2019 ) and the graph autoencoder approach ( Ng et al. , 2019b ) , where neural networks are used for the function h with φ being the weights of neural networks , and the score function can be chosen as the evidence lower bound ( Kingma & Welling , 2013 ) . A sparsity inducing term may be further added when the causal graph is assumed to be sparse . These objectives are equivalent to or are variants of some well studied score functions like the penalized maximum likelihood ( Chickering , 2002 ; Van de Geer et al. , 2013 ; Loh & Bühlmann , 2014 ) . The second class uses certain functions , with parameter θ , to construct a weighted adjacency matrix W ( θ ) ( or a binary one A ( θ ) ) to represent the causal structure . These methods can be summarized as min θ , φ EX∼P ( X ) S ( X , h ( X ; W ( θ ) , φ ) ) , subject to trace ( eW ( θ ) ◦W ( θ ) ) − d = 0 . ( 2 ) For example , GraN-DAG ( Lachapelle et al. , 2020 ) and NOTEARS-MLP ( Zheng et al. , 2020 ) respectively use neural network path products and partial derivatives between variables to construct W ( θ ) . The binary matrix A ( θ ) can be obtained by sampling according to some distributions with learnable parameters , as used by Kalainathan et al . ( 2018 ) ; Ke et al . ( 2019 ) ; Ng et al . ( 2019a ) ; Zhu et al . ( 2020 ) . Before ending this section , we remark that while the gradient-based methods intend to learn a causal DAG , the learned DAG may not be identical to the underlying one for general SEMs due to the Markov equivalence ( Spirtes et al. , 2000 ; Peters et al. , 2017 ) . For such cases , one may convert the obtained DAG to its corresponding Completed Partially Directed Acyclic Graph ( CPDAG ) as the estimate . Nevertheless , if the SEM is identifiable and a proper score function is used , then the exact solution to the optimization problem is consistent , i.e. , same as the true graph with probability 1 ; see , e.g. , Shimizu et al . ( 2006 ) ; Peters & Bühlmann ( 2013 ) ; Peters et al . ( 2014 ) ; Zhang & Hyvärinen ( 2009 ) . For further details and other technical issues like parameter optimization of the gradient-based methods , we refer the reader to the cited works and references therein .

Summary: This paper attempts to exploit the low-rankness of the adjacency matrix of the DAG in Bayesian network structure learning. The overall framework is similar to NOTEARS, except that the adjacency matrix W is decomposed into low rank components W = UV'. To justify the approach, the paper also includes lower and upper bounds of the rank of DAGs, albeit mostly theoretical and not applicable to real experiments.