O N L OW R ANK D IRECTED A CYCLIC G RAPHS AND C AUSAL S TRUCTURE L EARNING
Anonymous authors
Paper under double-blind review
A BSTRACT
Despite several important advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high dimensional settings when the graphs to be learned are not sparse.
In this paper, we propose to exploit a low rank assumption regarding the (weighted) adjacency matrix of a DAG causal model to mitigate this problem.
We demonstrate how to adapt existing methods for causal structure learning to take advantage of this assumption and establish several useful results relating interpretable graphical conditions to the low rank assumption.
In particular, we show that the maximum rank is highly related to hubs, suggesting that scale-free networks which are frequently encountered in real applications tend to be low rank.
We also provide empirical evidence for the utility of our low rank adaptations, especially on relatively large and dense graphs.
Not only do they outperform existing algorithms when the low rank condition is satisﬁed, the performance is also competitive even though the true rank may not be as low as is assumed.
I NTRODUCTION
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.
; Ng et al.
(2019b;a); Ke et al.
; Lachapelle et al.
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.
Of course this issue is a challenge to other approaches as well.
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.
Yet, we notice two works Barik & Honorio (2019); Tichavsk ` y & 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.
P RELIMINARIES 2. G RAPH T ERMINOLOGY
A graph G is deﬁned as a pair ( V , E ) , where V =
{ X 1 , X 2 , · · · , X d } is the vertex set and E ⊂ Vdenotes 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 ∈ R d × d , where W ( i, j ) is the weight of edge X i → X j and W ( i, j ) (cid:54)
=
if and only if X i → X j exists in G .
The binary adjacency matrix A ∈ { 0 , 1 } d × d is such that A ( i, j )
= 1 if X i → X j in G and A ( i, j )
=
otherwise.
The rank of a weighted graph is deﬁned as the rank of the associated weighted adjacency matrix.
2. C AUSAL S TRUCTURE L EARNING AND R ECENT G RADIENT - BASED M ETHODS
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 X i ’s to denote random variables associated with the nodes in a graph G .
Assuming G being a DAG, then the SEM is given by
[Equation]
where f i is a deterministic function and (cid:15) i ’s are jointly independent noises.
The SEM induces a marginal distribution P ( X ) over X =
[ X 1 , X 2 , · · · , X d ] 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.
They 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:
[Equation]
where h : R d → R d 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 e M 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 ) =
W T X 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.
We remark that 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).
For technical issues such as statistical consistency and parameter optimization of these methods (and also for the second class of methods described below), we refer the reader to the respective works and references therein for details.
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
[Equation]
[Equation]
[Equation]
[Equation]
, subject to trace
[Equation]
[Equation] 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.
Ke et al.
; Ng et al. (2019a); Zhu et al.
E XPLOITING L OW R ANK A SSUMPTION IN C AUSAL S TRUCTURE L EARNING
This section shows how to adapt existing gradient-based methods to take advantage of the low rank assumption, by providing a way for each class to utilize this assumption using techniques from the matrix completion literature.
We remark that our adaptations with the low rank assumption are not restricted to a particular learning algorithm; other DAG learning methods may potentially combine one of the proposed modiﬁcations for learning low rank causal graphs, too.
Matrix Factorization
Since the weighted adjacency matrix W is explicitly optimized in the ﬁrst class of methods, we can then apply the matrix factorization technique.
Speciﬁcally, with an estimate
ˆ r for the graph rank, we can factorize W as W = UV T with U, V ∈ R d × ˆ r .
Problem (1) is then to optimize U and V that minimizes the score function under the DAG constraint, and has the same solution W (obtained from the product UV T ) as the original one if ˆ r is greater than or equal to the true rank.
Furthermore, if ˆ r (cid:28) d , we have a much reduced number of parameters to optimize.
Nuclear Norm For the second class of methods, the adjacency matrix W ( θ ) is not an explicit parameter to be optimized.
In such a case, we can adopt a commonly used technique to add a nuclear norm term λ (cid:107) W ( θ ) (cid:107) ∗ , with λ > 0 being a tuning parameter, to the objective to induce low-rankness.
The optimization procedures in these recent structure learning methods can directly incorporate the two adaptations as they are all gradient-based, though some extra care needs to be taken.
Appendix C provides a detailed description of the optimization procedure and our implementation.
The second approach is also feasible for the ﬁrst class of methods, but we ﬁnd that it does not work as well as the matrix factorization approach, possibly due to the singular value decomposition to compute the (sub-)gradient w.r.t.
W at each optimization step.
An acute reader may have noticed that we assumed a proper rank estimate ˆ r or a proper penalty parameter λ .
Yet knowing exactly the rank of the graph to be learned can be difﬁcult in practice.
Similar to the sparsity assumption, one may determine the hyperparameters ˆ r and λ assisted by a validation dataset (or by cross-validation if the observed dataset is not sufﬁciently large).
Alternatively, we can try different choices of the hyperparameters and then apply traditional score-based method where the search space is restricted to the resulting DAGs.
However, since we are more concerned with relatively large and dense problems, the possible ranks may be too many to choose.
As such, a lower bound r l and an upper bound r u on the graph rank would be beneﬁcial—we need only consider ranks in [ r l , r u ] in the matrix factorization method, while the bounds are still useful by providing qualitative information for the nuclear norm approach: the lower an upper bound, the higher the tuning parameter λ should be chosen.
Moreover, a lower bound can also justify the low rank assumption, i.e., if the lower bound is high, then the low rank assumption is likely to fail to hold.
G RAPHICAL B OUNDS ON R ANKS
Obtaining exact algebraic information of a DAG such as its rank and eigenvalues may be infeasible in practice, because it may require a full knowledge of the graph to be learned.
On the other hand, we usually have access to some structural information, including graph connectivity, distributions of in-degrees and out-degrees, an estimate of number of hubs, etc.
As such, this section is devoted to relating the rank of a graph to more easily interpretable graphical conditions, for the sake of a better understanding of what kinds of DAGs tend to satisfy the low rank assumption and for lower and upper bounds on the graph rank from certain structural priors.
4. P ROBLEM S ETTING
Consider a DAG G =
( V , E ) with weighted adjacency matrix W and binary adjacency matrix A .
We aim to seek upper and lower bounds on rank( W ) using only the graphical structure.
Speciﬁcally, we focus on the weighted adjacency matrices with the same binary adjacency matrix A , i.e., W A = { W ∈ R d × d ; sign( | W | ) =
A } , where sign( · ) and | · | are point-wise sign and absolute value functions, respectively.
Notice that there exist trivial upper bound d − 1 and lower bound 0 for any
DAG, but they are generally too loose for our purpose.
In the following, we investigate the maximum rank max { rank( W ); W ∈ W A } and minimum rank min { rank( W ); W ∈ W A } to ﬁnd tighter upper and lower bounds for any W ∈ W A .
Before introducing two useful graph concepts, we comment that low rank DAGs are not necessarily sparse and vice versa; see a discussion in Appendix A.
Deﬁnition 1 (Height) .
Given a DAG G =
( V , E ) and a vertex X i ∈ V , the height of X i , denoted by l ( X i ) , is deﬁned as the length of the longest directed path starting from X i .
The height of G , denoted by l ( G ) , is the length of the longest path in G .
Deﬁnition 2 (Head-tail vertex cover) .
Let G =
( V , E ) be a directed graph and H , T be two subsets of V .
( H , T ) is called a head-tail vertex cover of G if every edge in G has its head vertex in H or its tail vertex in T .
The size of a head-tail vertex cover ( H , T ) is deﬁned as | H | + | T | .
As an example, Figure 1c is a head-tail vertex cover of G in Figure 1a, where H = { X 2 , X 4 , X 8 } (red nodes) and T = { X 8 , X 9 , X 10 } (blue nodes).
The size of this vertex cover is 6 .
[Figure]
L OWER B OUNDS
We ﬁrst study lower bounds on the rank of a weighted DAG.
Deﬁne V − 1 = ∅ and V s =
{ X i ; l ( X i ) =
s } for s = 0 , 1 , . . .
, l ( G ) .
Denote by G s,s − 1 the induced subgraph of G over V s ∪ V s − 1 .
Let C ( G s,s − 1 ) be the set of non-singleton connected components of G s,s − 1 and |C ( G s,s − 1 ) | the cardinality.
We have the following lower bounds.
Theorem 1.
Let G be a DAG with binary adjacency matrix A .
Then
[Equation]
All the proofs in this paper are provided in Appendix B. Theorem 1 shows that rank( W ) is greater than or equal to the sum of the number of non-singleton connected components in each G s,s − 1 .
As
G s,s − 1 has at least one non-singleton connected component, we obtain the second inequality.
In other words, the rank of a weighted DAG is at least as high as the length of the longest directed path.
As an example, consider the graph shown in Figure 1.
One can verify that min { rank( W ); W ∈ W A } = 6 , |C ( G 1 , 0 )
| = 2 , |C ( G 2 , 1 )
| = 1 , |C ( G 3 , 2 )
| = 1 , and l ( G ) = 3 .
Thus, we have min { rank( W ); W ∈
W A } = 6 > 2 + 1 + 1 = 4 > 3 .
We remark that the bounds in Theorem 1 may be loose in some cases.
To characterize the minimum rank exactly is an on-going research problem (Hogben, 2010).
4. U PPER B OUNDS
We turn to the more important issue for our purpose, regarding upper bounds on rank( W ) .
The next theorem shows that max { rank( W ); W ∈ W A } can be characterized exactly in graphical terms.
Theorem 2.
Let G be a directed graph with binary adjacency matrix A .
Then max { rank( W ); W ∈
W A } is equal to the minimum size of the head-tail vertex cover of G , that is, max { rank( W ) ; W ∈ W A }
= min {| H | + | T | ; ( H , T ) is a head-tail vertex cover of G} .
We comment that Theorem 2 holds for all directed graphs (not only DAGs), which may be of independent interest to other applications.
A head-tail vertex cover of minimum size is called a minimum head-tail vertex cover, which in general is not unique.
For a head-tail vertex cover ( H , T ) , the vertices in H cover all the edges pointing towards these vertices while the vertices in T cover the edges pointing away.
A head-tail cover of a relatively small size then indicates the presence of hubs, that is, vertices with relatively high in-degrees or out-degrees.
Therefore, Theorem 2 suggests that the maximum rank of a weighted DAG is highly related to the presence of hubs: a DAG with many hubs tends to have low rank.
Intuitively, a hub of high in-degree (out-degree) is a common effect (cause) of a number of direct causes (effect variables), comprising many V-structures (inverted V-structures).
For example, in Figure 1a, X 8 is a hub of V-structures and X 9 is a hub of inverted V-structures.
Such features are fairly common in real graph structures.
Appendix A presents a real network, called pathﬁnder , which describes the causal relations among 109 variables (Heckerman et al., 1992) with the center node being the parent of a large number of other nodes.
The famous scale-free (SF) graphs also tend to have hubs.
A scale-free graph is one whose distribution of degree k follows a power law:
P ( k ) ∼ k − γ , where γ is the power parameter typically within [2 , 3] and P ( k ) denotes the fraction of nodes with degree k (Nikolova & Aluru, 2012).
It is observed that many real-world networks are scale-free, and some of them may be viewed as causal networks (Guelzim et al., 2002; Barabasi & Oltvai, 2004; Hartemink, 2005; Egu ´ ıluz et al., 2005; Gao & en Ren, 2013; Ramsey et al., 2017).
Empirically, the ranks of scale-free graphs are relatively low, especially in comparison to Erd¨os-R´enyi (ER) random graphs (Mihail & Papadimitriou, 2002).
Figure 2 provides a simulated example where γ is chosen from { 2 , 3 } and each reported value is over 100 random runs.
As graph becomes denser, the graph rank also increases.
However, for scale-free graphs with a relatively large γ , the increase of their ranks is much slower than that of Erd¨os-R´enyi graphs; indeed, their ranks tend to stay fairly low even when the graph degree is large.
Theorem 2 can also be used to generate a low rank graph, or more precisely, a random DAG with a given rank r and a properly speciﬁed graph degree.
Here we brieﬂy describe the idea and leave the detailed algorithm to Appendix C.1:
ﬁrst generate a graph with r edges and rank r ; a random edge is sampled without replacement and would be added to the graph, if adding this edge does not increase the size of the minimum head-tail vertex cover; repeat the previous step until the pre-speciﬁed degree is reached or no edge could be added to the graph; ﬁnally, assign the edge weights randomly according to a continuous distribution and the weighted graph will have rank r with high probability.
The next two theorems report some looser but simpler upper bounds on rank( W ) .
Theorem 3.
Let G be a DAG with binary adjacency matrix A , and denote the set of vertices with at least one parent by V ch and those with at least one child by V pa .
Then we have
[Equation]
Since V ch and V pa are the non-root and the non-leaf vertices, respectively, the ﬁrst two inequalities of (4) indicate that the maximum rank is bounded from above by the number of non-root vertices and also by the number of non-leaf vertices.
The last inequality of (4) is a generalization of the ﬁrst two, which implies that the rank is likely to be low if most vertices have the same height.
Theorem 4.
Let G be a DAG with binary adjacency matrix A .
Denote by skeleton(A) and moral(A) the binary adjacency matrices of the skeleton and moral graph of G , respectively.
Then we have
[Equation]
The skeleton of a DAG is the undirected graph obtained by removing all the arrowheads, and the moral graph is the undirected graph where two vertices are adjacent if they are adjacent or if they share a common child in the DAG.
This result is useful when the skeleton or the moral graph can be accurately estimated and the corresponding rank is low.
In practice, we may use all available structural priors to obtain upper bounds on the underlying rank and choose the lowest one as our estimate.
E XPERIMENTS
This section reports empirical results of the low rank adaptations of existing methods, compared with their original versions.
We choose NOTEARS (Zheng et al., 2020) for linear SEMs by adopting the matrix factorization approach, denoted as NOTEARS-low-rank, and use the nuclear norm approach in combination with GraN-DAG (Lachapelle et al., 2020) for a non-linear data model.
Again we remark that the two methods are only demonstrations of the utility of low rank assumption, which can be potentially combined with other methods as well.
For more information, we also include several benchmark methods: fast GES (Ramsey et al., 2017), PC (Spirtes et al., 2000), MMHC (Tsamardinos et al., 2006),
ICA-LiNGAM
(Shimizu et al., 2006) speciﬁcally designed with non-Gaussian noises, for linear SEMs; and DAG-GNN (Yu et al., 2019), NOTEARS-MLP (Zheng et al., 2020), and CAM (B¨uhlmann et al., 2014) for the non-linear case.
Their implementations are described in Appendix C.
We consider randomly sampled DAGs with speciﬁed ranks (the generating procedure was described in Section 4.3 and is given as Algorithm 1 in Appendix C.1), scale-free graphs, and a real network
[Figure]
structure.
For linear SEMs, the weights are uniformly sampled from [ − 2 , − 0 .
5] ∪ [0 . 5 , 2] and the noises are either standard Gaussian or standard exponential.
For non-linear SEMs, we use additive
Gaussian noise model with functions sampled from Gaussian processes with RBF kernel of bandwidth one.
These data models are known to be identiﬁable (Shimizu et al., 2006; Peters & B ¨ uhlmann, 2013;
Peters et al., 2014).
From each SEM, we then generate n = 3 , 000 observations.
We repeat ten times over different seeds for each experiment setting.
Detailed information about the setup can be found in
Appendix C.3.
Below we mainly report structural Hamming distance (SHD) which takes into account both false positives and false negatives, and a smaller SHD indicates a better estimate.
5. L INEAR SEM S WITH R ANK -S PECIFIED G RAPHS
We ﬁrst consider linear SEMs on rank-speciﬁed graphs, with number of nodes d ∈ { 100 , 300 } , rank r = (cid:100) 0 .
1 d (cid:101) , and degree k ∈ { 2 , 4 , 6 , 8 } .
The true rank is assumed to be known and is used as the rank parameter ˆ r in NOTEARS-low-rank.
For a better visualization, Figure 3 only reports the average
SHDs, while the true positive rate, false discovery rate, and running time are left to Appendix D. We also show the results after using the interquartile range rule to remove outlier SHDs.
We observe that the low rank assumption can greatly improve the performance of NOTEARS, reducing the SHDs by at least a half.
For this data model, the fast GES has much higher SHDs (see also Appendix D).
PC is too slow (for example, it did not ﬁnish in 16 hours for a dataset with 100 nodes and degree 6 ), because some nodes may have a high in-degree.
For the same reason, the skeleton may not be well estimated by MMHC; its performance is slightly worse than the fast GES and is not reported.
Appendix D.1 studies graphs with higher ranks, where it is observed that the advantage of NOTEARSlow-rank over NOTEARS decreases when the rank of the underlying DAG increases.
Nevertheless,
NOTEARS-low-rank is still competitive when the true rank is (cid:100) d/ 2 (cid:101) and the factorized matrix has the same number of parameters as NOTEARS.
We also conduct an empirical analysis with different sample sizes in Appendix D.2, which shows that NOTEARS-low-rank performs reasonably well when the sample size is small and tends to have a better performance with a larger number of samples.
Due to space limit, please ﬁnd further details in the appendix.
5. L INEAR SEM S WITH S CALE -F REE G RAPHS
We next consider scale-free graphs with d = 100 nodes, average degree k = 6 , and power γ = 2 . 5 .
For this experiment, the minimum, maximum, and mean ranks of generated graphs are 14 , 24 , and 18 . 7 , respectively.
Here we choose the rank parameter ˆ r from { 20 , 30 , 40 } for
NOTEARS-low-rank.
As seen from Figure 4, NOTEARSlow-rank with rank parameter ˆ r = 20 performs the best, even though there are graphs with ranks greater than 20 .
5. S ENSITIVITY OF R ANK P ARAMETERS AND V ALIDATION
So far we have assumed that the true rank or an accurate estimate is known.
In this experiment, we conduct an empirical analysis with different rank parameters for linear Gaussian data model on rank-speciﬁed graphs with 100 nodes, degree 8 , and rank 10 .
We include the validation based approach where 2 , 000 samples are chosen as training dataset and the rest for validation, and also use the derived lower and upper bounds in Theorems 1 and 3 to select 7 evenly distributed rank parameters, assuming that the corresponding structural priors are available.
As seen from Figure 5,
NOTEARS-low-rank performs the best when the rank parameter is identical to the true rank, while the rank parameter chosen by validation has almost the same performance.
Compared with NOTEARS on the same datasets (cf.
Figure 3), the low rank version performs well across a range of rank parameters.
5. N ON -L INEAR SEM S
For non-linear data models, we pick rank-speciﬁed graphs with 50 nodes, rank 5 , and degree k ∈ { 2 , 4 , 6 , 8 } .
To our knowledge, the selected benchmark methods CAM, NOTEARS-MLP, and
GraN-DAG are state-of-the-art methods on this data model.
As a demonstration of the low rank assumption, we apply the nuclear norm approach to GraN-DAG and choose from { 0 . 3 , 0 . 5 , 1 . 0 } as penalty weights.
For validation, we use the same splitting ratio as in Section 5.3 and consider more penalty weights from 0 .
1 to 2 , where the one with lowest validation error is chosen as ﬁnal estimate.
Figure 6 (and Appendix D.5 with a more detailed result) shows that adding a nuclear norm can improve the performance of GraN-DAG across a large range of weights when the graph is relatively dense.
For degree 8 , the low rank version with validation achieves average SHD 77 .
4 , while the
SHDs of CAM, NOTEARS-MLP, and original GraN-DAG are 131 .
9 , 119 . 4 , and 109 . 4 , respectively.
5. R EAL N ETWORK
We apply the proposed method to the arth150 gene network, which is a DAG containing 107 genes and 150 edges.
Its maximum rank is 40 .
Since the real dataset has onlysamples, we instead use simulated data from linear Gaussian SEMs.
We pick ˆ r from { 36 , 40 , 44 } and also use validation to select the rank parameter.
We include NOTEARS with a sparsity inducing term, named NOTEARS-L1.
Here the (cid:96) 1 penalty weight is chosen from { 0 . 05 , 0 . 1 , 0 . 2 } and we treat this method favorably by reporting the lowest SHD.
The mean and median SHDs are shown in Figure 7.
The experimental result demonstrates again the utility of the low rank assumption, even when the true rank is not as low as is assumed.
C ONCLUDING R EMARKS
This paper studies the potential of low rank assumption in causal structure learning.
Empirically, we show that the low rank adaptations perform noticeably better than existing algorithms when the low rank condition is satisﬁed, and also deliver competitive performances when the rank is not as low as is assumed.
Theoretically, we provide an improved understanding of what kinds of graphs tend to be low rank and a possibility to obtain bounds on the underlying rank from several structural priors.
We treat the present work as our ﬁrst step to incorporate low-rankness into causal DAG learning.
A future direction is to approximate a high rank DAG with a low rank one.
While there is a rich literature on low rank approximations of matrices, it is non-trivial to us to conclude under what conditions such an approximation is guaranteed to be effective to learn causal DAGs.
Another direction is to investigate if a low rank DAG model implies any interesting and useful behavior in the data.
Appendix A E XAMPLES AND D ISCUSSIONS
We provide more examples and discussions in this section.
Minimum rank of the graph in Figure
We ﬁrst show that the minimum rank of the DAG structure in Figure 1 is 6 .
It is clear that the 6 -th to 10 -th rows of A are always linearly independent, so it sufﬁces to show that the 11 -th row is linearly independent of the 6 -th to 10 -th rows.
To see this, notice that if the 11 -th row is a linear combination of the 6 -th to 10 -th rows, then A (11 , 1) would be non-zero, which is a contradiction.
The pathﬁnder and arth150 networks Figure 8 visualizes the pathﬁnder and arth150 networks that are mentioned in Sections 4.3 and 5, respectively.
Both networks can be found at http: //www.bnlearn.com/bnrepository .
As one can see, these two networks contain hubs: the center note in the pathﬁnder network has a large number of children, while the arth150 network contains many ‘small’ hubs, each of which has 5 ∼ 10 children.
We also notice that nearly all the hubs in the two networks have high out-degrees.
Sparse DAGs and low rank DAGs
A sparse DAG does not necessarily indicate a low rank DAG, and vice versa.
For example, a directed linear graph with d vertices has only d − 1 edges, i.e. X 1 → X 2 → · · · → X d , while the rank of its binary adjacency matrix is d − 1 .
According to Theorems 1 and 2, the maximum and minimum ranks of a directed linear graph are equal to its number of edges.
Thus, directed linear graphs are sparse but have high ranks.
On the other hand, for some non-sparse graphs, we can assign the edge weights so that the resulting graphs have low ranks.
A simple example would be a fully connected directed balanced bipartite graph, as shown in Figure 9.
The deﬁnition of bipartite graphs can be found in Appendix B.1.
A bipartite graph is called balanced if its two parts contain the same number of vertices.
The rank of a fully connected balanced bipartite graph with d vertices is 1 if all the edge weights are the same (e.g., the binary adjacency matrix), but the number of edges is d 2 / 4 .
[Figure]
We also notice that there exist some connections between the maximum rank and the graph degree, or more precisely, the total number of edges in the graph, according to Theorem 2.
Intuitively, if the graph is dense, then we need more vertices to cover all the edges.
Thus, the size of the minimum head-tail vertex cover should be large.
Explicitly providing a formula to characterize these two graph parameters is an interesting problem, which will be explored in the future.
B P ROOFS
In this section, we present proofs for the theorems given in the main content.
B. P RELIMINARIES
A bipartite graph is a graph whose vertex set V can be partitioned into two disjoint subsets V 0 and
V 1 , such that the vertices within each subset are not adjacent to one another.
V 0 and V 1 are called the parts of the graph.
A matching of a graph is a subset of its edges where no two of them share a common endpoint.
A vertex cover of a graph is a subset of the vertex set where every edge in the graph has at least one endpoint in the subset.
The size of a matching (vertex cover) is the number of edges (vertices) in the matching (vertex cover).
A maximum matching of a graph is a matching of the largest possible size and a minimum vertex cover is a vertex cover of the smallest possible size.
An important result about bipartite graphs is K ¨ onig’s theorem (D ´ enes, 1931), which states that the size of a minimum vertex cover is equal to the size of a maximum matching in a bipartite graph.
Based on the heights of vertices in V , we can deﬁne a weak ordering among the vertices: X i (cid:31) X j if and only if l ( X i ) >
l ( X j ) , and X i ∼ X j if and only if l ( X i ) =
l ( X j ) .
Given this weak ordering, we can group the vertices by their heights, and the resulting graph shows a hierarchical structure; see Figure 1 in the main text for an example.
This hierarchical representation has some simple and nice properties.
Let V s =
{ X i ; l ( X i ) =
s } , s = 0 , 1 , . . .
, l ( G ) , and let V − 1 = ∅ .
We have: (1) for any given s ∈ { 0 , 1 , . . .
, l ( G ) } and two distinct vertices X 1 , X 2 ∈ V s , X 1 and X 2 are not adjacent, and (2) for any given s ∈ { 1 , 2 , . . .
, l ( G ) } and X i ∈
V s , there is at least one vertex in V s − 1 which is a child of X i .
If we denote the induced subgraph of G over V s ∪ V s − 1 by G s,s − 1 , then G s,s −is a bipartite graph with V s and V s − 1 as parts, and singletons in G s,s − 1 (i.e., vertices that are not endpoints of any edge) only appear in V s − 1 .
For ease of presentation, we occasionally use index i to represent variable X i in the following sections.
B. P ROOF OF T HEOREM
Proof.
Let G =
( V , E ) .
Consider an equivalence relation, denoted by ∼ , among vertices in V deﬁned as follows: for any X i , X j ∈ V , X i ∼ X j if and only if l ( X i )
=
l ( X j ) and X i and X j are connected.
Here, connected means that there is a path between X i and X j .
Below we use C ( X i ) to denote the equivalence class containing X i .
Next, we deﬁne a weak ordering π on V / ∼ , i.e., the equivalence classes induced by ∼ , by letting C ( X i ) (cid:23) π C ( X j ) if and only if l ( X i ) ≥ l ( X j ) .
Then, we extend (cid:23) π to a total ordering ρ on V / ∼ .
The ordering ρ also induces a weak ordering (denoted by ¯ ρ ) on V : X i (cid:23) ¯ ρ X j if and only if C ( X i ) (cid:31) ρ C ( X j ) .
Finally, we extend ¯ ρ to a total ordering γ on
V .
It can be veriﬁed that γ is a topological ordering of G , that is, if we relabel the vertices according to γ , then X i ∈ pa( X j , G ) if and only if i > j and X i and X j are adjacent, and the adjacency matrix of G becomes lower triangular.
Assume that the vertices of G are relabeled according to γ and we will consider the binary adjacency matrix A of the resulting graph throughout the rest of this proof.
Note that relabelling is equivalent to applying a permutation onto the adjacency matrix, which does not change the rank.
Let V
= { 1 , 2 , . . .
, k 1 − 1 } for some k 1 ≥ 2 .
Then the k 1 -th row of A , denoted by A ( k 1 , · ) , is the ﬁrst non-zero row vector of A .
Letting S
= { A ( k 1 , · ) } , then S contains a subset of linearly independent vector(s) of the ﬁrst k 1 rows of A .
Suppose that we have visited the ﬁrst m rows of A and S
=
{ A ( k 1 , · ) , A ( k 2 , · ) , . . .
, A ( k t , · ) } contains a subset of linearly independent vector(s) of the ﬁrst m rows of A , where k 1 ≤ m < d .
If X m +1 (cid:28) X k t , then we add A ( m +1 , · ) to S ; otherwise, we keep S unchanged.
We claim that the vectors in S are still linearly independent after the above step.
Clearly, if we do not add any new vector, then S contains only linearly independent vectors.
To show the other case, note that if l ( X m +1 ) >
l ( X k t ) ≥ · · · ≥ l ( X k 1 ) , then there is an index i ∈
V l ( X m +1 ) −such that A ( m + 1 , i ) (cid:54) = 0 , by the deﬁnition of height.
Since l ( X m +1 )
>
l ( X k t ) , we have l ( X k t ) ≤ l ( X m +1 ) − 1 and thus A ( k j , i ) = 0 for all j = 1 , 2 , . . .
, t .
Therefore, A ( m + 1 , · ) cannot be linearly represented by { A ( k j , · ); j = 1 , 2 , . . .
, t } and the vectors in S are linearly independent.
On the other hand, if l ( X m +1 )
=
l ( X k t ) , then the deﬁnition of the equivalence relation ∼ implies that X m +1 and X k t are disconnected, which means that X m +1 and X k t do not share a common child in V l ( X m +1 ) − 1 .
Consequently, there is an index i ∈
V l ( X m +1 ) − 1 such that A ( m + 1 , i ) (cid:54) =
but A ( k t , i ) = 0 .
Similarly, we can show that A ( k j , i ) =
for all j = 1 , 2 , . . .
, t .
Thus, the vectors in S are still linearly independent.
After visiting all the rows in A , the number of vectors in S is equal to (cid:80) l ( G ) s =1 |C ( G s,s − 1 ) | based on the deﬁnition of ∼ .
The second inequality can be shown by noting that C ( G s,s − 1 ) has at least one elements.
The proof is complete.
B. P ROOF OF T HEOREM
Proof.
Denote the directed graph by G =
( V , E ) .
Edmonds (1967, Theorem 1) showed that max { rank( W ); W ∈ W A } is equal to the maximum number of nonzero entries of A , no two of which lie in a common row or column.
Therefore, it sufﬁces to show that the latter quantity is equal to the size of the minimum head-tail vertex cover.
Let V (cid:48)
=
V (cid:48)
∪ V (cid:48) 1 , where V (cid:48)
= V × { 0 } = { ( X i , 0); X i ∈ V } and V (cid:48)
= V × { 1 } = { ( X i , 1); X i ∈ V } .
Now deﬁne a bipartite graph B =
( V (cid:48) , E (cid:48) )
where E (cid:48)
= { ( X i , 0) →
( X j , 1); ( X i , X j ) ∈ E } .
Denote by M a set of nonzero entries of A so that no two entries lie in the same row or column.
Notice that M can be viewed as an edge set and no two edges in M share a common endpoint.
Thus, M is a matching of B .
Conversely, it can be shown by similar arguments that any matching of B corresponds to a set of nonzero entries of A , no two of which lie in a common row or column.
Therefore, max { rank( W ) , W ∈ W A } equals the size of the maximum matching of B , and further the size of the minimum vertex cover of B according to K¨onig’s theorem.
Note that any vertex cover of B can be equivalently transformed to a head-tail vertex cover of G , by letting H and T be the subsets of the vertex cover containing all variables in V (cid:48)
and of the vertex cover containing all variables in V (cid:48) 1 , respectively.
Thus, max { rank( W ) , W ∈ W A } is equal to the size of the minimum head-tail vertex cover.
B. P ROOF OF T HEOREM
Proof.
We start with the ﬁrst inequality in Equation (4).
Let h 1 , . .
, h p denote the heights where | V s | < | ch( V s ) | , and t 1 , . . .
, t q the height where | V s | > | ch( V s ) | .
Let H = ∪ pi =1 V h
i
and
T = ∪ qi =1 V t i .
It is straightforward to see that ( H , T ) is a head-tail vertex cover.
Thus, Equation (4) holds according to Theorem 2.
The second inequality can be shown similarly and its proof is omitted.
For the third inequality, let m = argmax {|
V s | : 0 ≤ s ≤ l ( G ) } , and deﬁne H
= ∪ i>m V i and
T = ∪ i<m V i .
Then ( H , T ) is also a head-tail vertex cover and the third inequality follows from
Theorem 2, too.
B. P ROOF OF T HEOREM
Proof.
Notice that Theorem 2 holds for all directed graphs.
This theorem then follows by treating the skeleton and the moral graph as directed graphs with loops, i.e., an undirected edge
X i − X j is treated as two directed edges X i → X j and X j → X i .
C I MPLEMENTATION D ETAILS
In this section, we present an algorithm to generate a random DAG with a given rank, a low rank version of NOTEARS and GraN-DAG, and also a description of our experimental settings.
C. G ENERATING R ANDOM DAG S
In Section 4.3, we brieﬂy discuss the idea of generating a random DAG with a given rank.
We now describe the detailed procedure in Algorithm 1.
In particular, we aim to generate a random DAG with d nodes, average degree k , and rank r .
The ﬁrst part of Algorithm 1 after initialization is to sample a number N , representing the total number of edges, from a binomial distribution B ( d ( d − 1) / 2 , p )
Require:
Number of nodes d , average degree k , and rank r .
Ensure: A randomly sampled DAG with the number of nodes d average degree k and rank r . 5: return FAIL 6: end if 7: Sample indices d them M descending order. do 9:
Sample Add edge 11: end 12: while R (cid:54) =
M do 13:
Sample 14: if adding ( i, j ) to M does not change the size of the minimum head-tail vertex cover of M then 15: Add ( i, j to M
16: end if 17: end while 18: if | M | <
N then 19: return FAIL 20: end if 21: return M where p = k/ ( d − 1) . If N < r , Algorithm 1 would return FAIL since a graph with N < r edges could never have rank r .
Otherwise, Algorithm 1 samples an initial graph with r edges and rank r , by choosing r edges such that no two of them share the same head points or the same tail points, i.e., each row and each column of the corresponding adjacency matrix have at most one non-zero entry.
Then, Algorithm 1 sequentially samples an edge from R containing all possible edges and checks whether adding this edge to the graph changes the size of the minimum head-tail vertex cover.
If not, the edge will be added to the graph; otherwise, it will be removed from R .
This is because if a graph G is a super-graph of another graph H , then the size of the minimum head-tail cover of G is no less than that of H .
We repeat the above sampling procedure until there is no edge in R or the number of edges in the resulting graph reaches N .
If the latter happens, the algorithm will return the generated graph; otherwise, it will return FAIL.
The theoretic basis of Algorithm 1 is Theorem 2.
Note that the algorithm may not return a valid graph if the desired number N of edges cannot be reached.
This could happen if the input rank is too low while the input average degree is too high.
With our experiment settings, we ﬁnd it rare for
Algorithm 1 to fail to return a desired graph.
C. O PTIMIZATION
For this part, we consider a dataset consisting of n i.i.d.
observations from P ( X ) and consequently the expectations in Problems (1) and (2) are replaced by empirical means.
Denote the design matrix by X ∈ R n × d , where each row of X corresponds to an observation and each column represents a variable.
Here we use NOTEARS (Zheng et al., 2018) and Gran-DAG (Lachapelle et al., 2020) from each class of methods as examples and will describe their low rank versions in the following.
Other gradient-based methods and their optimization procedures can be similarly modiﬁed to incorporate the low rank assumption.
Require: Design matrix X , starting point ( U 0 , V 0 , α 0 ) , rate c ∈ (0 , 1) , tolerance (cid:15) > 0 , and threshold w > 0 .
Ensure: Locally optimal parameter W ∗ .
1: for t = 1 , 2 , . . . do 2: (Solve primal) U t +1 , V t +1 ← arg min U,V L ρ ( U, V, α t ) with ρ such that g ( U t +1 V
Tt +1 )
< cg
( U t V Tt ) .
3: (Dual ascent)
α t +1 ← α t
+ ρg ( U t +1 V Tt +1 ) . 4: if g ( U t +1 V Tt +1 ) < (cid:15) then 5: Set U ∗ =
U t +1 and V ∗ =
V t +1 . 6: break 7: end if 8: end for 9: (Thresholding) Set W ∗ = U ∗ V ∗ T ◦ 1 ( | U ∗ V ∗ T | > w ) . 10: return W ∗
C.2. NOTEARS WITH LOW RANK ASSUMPTION
Following Section 3, the optimization problem in our work can be written as
[Equation] (5) where U, V ∈ R d × ˆ r and ◦ is the point-wise product.
The constraint in Problem (5) holds if and only if UV T is a weighted adjacency matrix of a DAG.
This problem can then be solved by standard numeric optimization methods such as the augmented Lagrangian method (Bertsekas, 1999).
In particular, the augmented Lagrangian is given by
[Equation]
where g ( UV T ) := trace (cid:16) e UV T ◦ UV T (cid:17) − d , α is the Lagrange multiplier, and ρ > 0 is the penalty parameter.
The optimization procedure is summarized in Algorithm 2, similar to Zheng et al.
Algorithm 1).
Notice that here we do not include the (cid:96) 1 penalty term (except for the last experiment in Section 5.5), for the following reasons: (1) the thresholding procedure can also control false discoveries; (2) we consider relatively sufﬁcient data for the experiments and NOTEARS with thresholding has been shown in Zheng et al.
(2018) to perform consistently well even when the graph is sparse; (3) we are more concerned with relatively large and dense graphs, so a sparsity assumption may be harmful, as shown also by Zheng et al.
(2018); (4) the (cid:96) 1 penalty term requires a tuning parameter, which itself is not easy to choose.
Zheng et al.
(2018) used L-BFGS to solve the unconstrained subproblem in Step 2.
We alternatively use the Newton conjugate gradient method that is written in C. Empirically, these two optimizers behave similarly in terms of the estimate performance, while the latter can run much faster thanks to its C implementation.
The DAG constraint may not be satisﬁed exactly using iterative numeric methods, so it is a common practice to pick a small tolerance, followed by a thresholding procedure on the estimated entries to obtain exact DAGs.
In our implementation, we choose U 0 and V 0 to be the ﬁrst ˆ r columns of the d × d identity matrices.
Other parameter choices are: α
= 0 , c
= 0 .
25 , (cid:15) =
10 − 6 , and w = 0 .
3 , similar to those used in related methods on the same datasets (e.g., Zheng et al.
Yu et al.
Zhu et al.
The chosen threshold w = 0 .
3 works well in our experiments and in the experiments of related works that use the same data model.
In case the thresholded matrix is not a DAG, one may further increase the threshold until the resulting matrix corresponds to a DAG.
After obtaining W ∗ , we add an additional pruning step: we use linear regression to reﬁt the dataset based on the structure indicated by W ∗ and then apply another thresholding (with w = 0 . 3 ) to the reﬁtted weighted adjacency matrix.
Both the Newton conjugate gradient optimizer and the pruning technique are also applied to NOTEARS, which not only accelerate the optimization but also improve its performance by obtaining a much lower SHD, particularly for large and dense graphs.
See
Appendix D.3 for an empirical comparison.
C.2. G RA N-DAG WITH LOW RANK ASSUMPTION
We next consider a low rank version of GraN-DAG.
The optimization problem can be written as
[Equation]
subject to trace (cid:16)
[Equation]
6) where X ( l ) i is the l -th sample of variable X i and pa( X i , W ( θ ))
( l ) means the l -th sample of X
i ’s parents indicated by the adjacency matrix W ( θ ) .
Here, θ denotes the parameters of neural networks and W ( θ ) with non-negative entries is obtained from the neural network path products.
Problem (6) can be solved similarly using augmented Lagrangian.
The procedure is similar to
Algorithm 2 and is the same to that used by GraN-DAG, with slight modiﬁcations: (1) the subproblem in Step 2 is approximately solved using ﬁrst-order methods; (2) the thresholding at Step 9 is replaced by a variable selection method proposed by B¨uhlmann et al.
The same variable selection or pruning method is adopted by two other benchmark methods CAM and NOTEARS-MLP in our experiment.
Please refer to Lachapelle et al.
(2020) and B ¨ uhlmann et al.
(2014) for further details.
C. E XPERIMENT S ETUP
In our experiments, we consider three data models: linear Gaussian SEMs, linear non-Gaussian
SEMs (linear exponential SEMs), and non-linear SEMs (Gaussian processes).
Given a randomly generated DAG G , the associated SEM is generated as follows:
Linear Gaussian A linear Gaussian SEM is given by
[Equation]
where pa( X i , G ) denotes
X i ’s parents in G and (cid:15)
i ’s are jointly independent standard Gaussian noises.
In our experiments, the weights W ( i, j ) ’s are uniformly sampled from [ − 2 , − 0 .
5] ∪ [0 . 5 , 2] .
Linear Exponential A linear exponential SEM is also generated according to Equation (7), where (cid:15) i ’s are replaced by jointly independent Exp(1) random variables.
The weights W ( i, j ) ’s are sampled from [ − 2 , − 0 .
5] ∪ [0 . 5 , 2] uniformly, too.
Gaussian Processes We consider the following additive noise model:
[Equation]
where (cid:15) i ’s are jointly independent standard Gaussian noises and f i ’s are functions sampled from
Gaussian processes with RBF kernel of bandwidth one.
We sample 3 , 000 observations according the SEM.
The reported results of each setting are summarized over 10 repetitions with different seeds.
The experiments are run on a Linux workstation with 16-core Intel Xeon 3.20GHz CPU and 128GB RAM.
C. B ENCHMARK M ETHODS
Existing causal structure learning methods used in our experiments all have available implementations, as listed below:
• GES and PC: an implementation of both methods is available through the py-causal package at https://github.com/bd2kccd/py-causal .
We note that, the implementation of py-causal package is based on the CMU TETRAD project, in which the version of GES is indeed the fast GES algorithm proposed by Ramsey et al.
• MMHC (Tsamardinos et al., 2006): an implementation is available in the bnlearn package at https://CRAN.R-project.org/package=bnlearn .
• CAM (Peters et al., 2014): its codes are available through the CRAN R package repository at https://cran.r-project.org/web/packages/CAM .
• NOTEARS
(Zheng et al., 2018) and NOTEARS-MLP (Zheng et al., 2020): codes are available at the ﬁrst author’s github repository https://github.com/xunzheng/ notears .
• GraN-DAG (Lachapelle et al., 2020): an implementation is available at the ﬁrst author’s github repository https://github.com/kurowasan/GraN-DAG .
Note that for graphs of 50 nodes or more, GraN-DAG performs a preliminary neighborhood selection step to avoid overﬁtting.
• DAG-GNN (Yu et al., 2019): the codes are available at the ﬁrst author’s github repository https://github.com/fishmoon1234/DAG-GNN .
In the experiments, we mostly use default hyperparameters unless otherwise stated.
D A DDITIONAL E XPERIMENTAL R ESULTS D. L INEAR SEM S WITH H IGHER R ANKS
This experiment considers graphs of higher ranks.
We use rank-speciﬁed random graphs with d = 100 nodes and rank r ∈ { 30 , 35 , 40 , 45 , 50 } on linear Gaussian SEMs.
The results are shown in Figures 10a and 10b with degrees 2 and 8 , respectively.
We observe that when the rank of the underlying graph becomes higher, the advantage of NOTEARS-low-rank over NOTEARS decreases.
Nonetheless, NOTEARS-low-rank with rank r = 50 is still comparable to NOTEARS, and has a lower average SHD after removing outlier SHDs using the interquartile range rule.
[Figure]
D. NOTEARS-L OW -R ANK WITH D IFFERENT S AMPLE S IZES
We next empirically study the consistency of NOTEARSlow-rank.
Again, we use rank-speciﬁed random graphs (sampled according to Algorithm 1) with d = 100 nodes, degree k = 8 , rank r = 10 , and linear Gaussian SEMs.
We also assume that the true rank is known.
We ﬁx the rank parameter ˆ r = 10 and use different sample sizes ranging from 200 to 5 , 000 .
From Figure 11, NOTEARSlow-rank performs reasonably well when the sample size is small and tends to have a better performance with a larger number of samples.
D. F URTHER P RUNING
We compare the empirical results before and after applying the additional pruning technique described in Appendix C.2.
The graphs are rank-speciﬁed with d ∈ { 100 , 300 } nodes, rank r = (cid:100) 0 .
1 d (cid:101) , and degree k ∈ { 2 , 4 , 6 , 8 } .
We again use linear Gaussian data model with equal noise variances to generate the datasets.
The average SHDs are reported in Figure 12.
We see that applying an additional pruning step indeed improves the ﬁnal performance of both NOTEARS and NOTEARS-low-rank, especially on relatively large and dense graphs. [Figure] [Figure]
D ETAILED E MPIRICAL R ESULTS FOR E XPERIMENT 1 WITH L INEAR G AUSSIAN SEM S
Table 1 reports detailed results including true positive rates (TPRs), false discovery rates (FDRs), structural Hamming distances (SHDs)
[Table]
and running time on rank-speciﬁed graphs with linear Gaussian data model.
Here the true rank is assumed to be known and is used as the rank parameter in NOTEARS- [Table] low-rank.
We also test (fast) GES, MMHC [Table] and PC.
However, PC is too slow since some nodes may have a high in-degree (i.e., hubs) in large, dense, and low rank graphs.
For the same reason, the skeleton may not be correctly estimated by MMHC, which has a similar performance to that of GES.
Therefore, we only include the results of GES for comparison.
We treat GES favorably by regarding undirected edges as true positives if the true graph has a directed edge in place of the undirected ones.
D. D ETAILED R ESULTS FOR E XPERIMENT 4 WITH N ON -L INEAR SEM S
Table 2 reports the detailed SHDs for each method in Section 5.4.
We also mark in bold the best results from methods with or without low rank modiﬁcations.
[Table]