Original: Learning Linear Non-Gaussian Polytree Models
Revised: Learning Linear Non-Gaussian Polytree Models

Original: Abstract
Revised: Abstract

Original: In the context of graphical causal discovery, we adapt the versatile framework of linear nonGaussian acyclic models (LiNGAMs) to propose new algorithms to efficiently learn graphs that are polytrees.
Revised: In the context of graphical causal discovery, we adapt the versatile framework of linear nonGaussian acyclic models (LiNGAMs) to propose new algorithms to efficiently learn graphs that are polytrees.

Original: Our approach combines the Chow–Liu algorithm, which first learns the undirected tree structure, with novel schemes to orient the edges by imposing an algebraic condition on moments of the data-generating distribution.
Revised: Our approach combines the Chow–Liu algorithm, which first learns the undirected tree structure, with novel schemes to orient the edges.

Original: This algebraic edge orientation gains significant efficiency over current algorithms that assess conditional independence.
Revised: The orientation schemes assess algebraic relations among moments of the data-generating distribution and are computationally inexpensive.

Original: We establish high-dimensional consistency results  and compare the algorithms in numerical experiments.
Revised: We establish high-dimensional consistency results for our approach and compare different algorithmic versions in numerical experiments.

Original:  INTRODUCTION
Revised:  INTRODUCTION

Original: Directed acyclic graphs (DAGs) have been extensively used in causal modeling; the nodes of the graph represent the random variables of the model while the edges represent directed causal effects from source to sink.
Revised: Directed acyclic graphs (DAGs) have been extensively used in causal modeling; the nodes of a graph represent the random variables of the model while the edges represent directed causal effects from source to sink.

Original: The causal effects of the parent nodes on the children are quantified by structural equations.
Revised: These causal effects of the parent nodes on the children are quantified by structural equations.

Original: In this paper, we  study the problem of inferring the graphical structure underlying the causal model, given only observational data [Drton and Maathuis, 2017].
Revised: In this paper, we take up this framework and study the problem of inferring the graphical structure underlying the causal model, given only observational data [Drton and Maathuis, 2017].

Original: Referred to as structural learning or causal discovery, it is an intractable problem  due to the statistical curse of dimensionality and computational issues.
Revised: Referred to as structure learning or causal discovery, it is a problem that is difficult due to the statistical curse of dimensionality and computational issues.

Original: Imposing restrictions on the random variables, graphical structure, or structural equationssimplifies the problem and contributes to a better understanding of theoverarching problemof causal discovery [Pearl et al., 2016, Peters et al., 2017].
Revised: Effective methods, thus, need to exploit restrictions on the random variables, graphical structure, or structural equations to simplify the problem [Pearl et al., 2016, Peters et al., 2017].

Original: Here, we consider a class of graphs that are trees, together with linear structural equations where the error terms are mutually independent and non-Gaussian.
Revised: Here, we consider a class of tree-structured graphs, together with linear structural equations where the error terms are mutually independent and non-Gaussian.

Original: Specifically, we work in the versatile causal discovery framework of linear non-Gaussian acyclic models (LiNGAMs)
Revised: Specifically, we work in the versatile causal discovery framework of linear non-Gaussian acyclic models (LiNGAMs)

Original: [Shimizu et al., 2006].
Revised: [Shimizu et al., 2006, Shimizu and Kano, 2008].

Original: LiNGAMs postulate linear structural equations  to describe the relation among observed variables featuring non-Gaussian noise terms.
Revised: LiNGAMs postulate linear structural equations with non-Gaussian noise terms to describe the relationships among observed variables.

Original: The non-Gaussianity assumption allows for consistent estimation of the graph encoding the model from observational data alone and for efficient structural learning algorithms [e.g., Shimizu et al., 2011, Hyvärinen and Smith, 2013, Wang and Drton, 2020].
Revised: The non-Gaussianity assumption allows for consistent estimation of the graph encoding the model from observational data alone and for efficient structure learning algorithms [e.g., Shimizu et al., 2011, Hyvärinen and Smith, 2013, Wang and Drton, 2020, Hoyer et al., 2008b].

Original: Since the complexity of the structural learning problem depends directly  the underlying graph, consistency results for causal discovery algorithms often require some restrictions on the graph.
Revised: Since the complexity of the structure learning problem depends directly on the underlying graph, consistency results for causal discovery algorithms often require some restrictions on the graph, particularly, when high-dimensional consistency results are desired.

Original: DAGs are a general framework that also control the complexity of the learning problem; in particular, the subset of DAGs whose underlying skeleton is a tree—a polytree —is the most scalable setting, offering low computational complexity whilst retaining model expressiveness [Pearl, 1988].
Revised: In this context, the subset of DAGs whose underlying skeleton is a tree—a polytree —is the most scalable setting, offering low computational complexity whilst retaining model expressiveness [Pearl, 1988].

Original: In this paper, we propose algorithms to learn a polytree underlying a LiNGAM model.
Revised: In this paper, we propose algorithms to learn a polytree underlying a LiNGAM model.

Original: Learning a polytree may be decomposed into two tasks: extracting the skeleton and determining the orientation of the edges [Rebane and Pearl, 1987, Jakobsen et al., 2021].
Revised: Learning a polytree may be decomposed into two tasks: extracting the skeleton and determining the orientation of the edges [Rebane and Pearl, 1987, Jakobsen et al., 2021].

Original: Recovering the underlying skeleton may be achieved via the Chow–Liu algorithm
Revised: Recovering the underlying skeleton may be achieved via the Chow–Liu algorithm

Original: [Chow and Liu, 1968].
Revised: [Chow and Liu, 1968].

Original: Existing methods for edge orientation entail checking conditional independence, which is usually carried out by serial hypothesis testing and hinders efficiency.
Revised: Existing methods for edge orientation entail checking conditional independence, which is usually carried out by serial hypothesis testing and impacts computational efficiency.

Original: We bypass this limitation by imposing an algebraic condition and building on recent insights concerning algebraic relations among moments to determine edge orientation [Robeva and Seby, 2021, Améndola et al., 2021].
Revised: We instead proceed by exploiting recent insights concerning algebraic relations among moments to determine edge orientation [Robeva and Seby, 2021, Améndola et al., 2021, Wiedermann, 2015, Dodge and Rousson, 2001].

Original: The result is an efficient approach that adapts a classical algorithm to recover the core causal tree structure and augments it with a novel algebraic strategy to determine edge orientation, thus fully learning the polytree from observational data alone, in a far more scalable manner than existing LiNGAM algorithms that learn more general graph structures.
Revised: The result is an efficient approach that adapts a classical algorithm to recover the core causal tree structure and augments it with a novel algebraic strategy to determine edge orientation. The proposed algorithms learn the polytree from observational data alone, in a far more scalable manner than existing LiNGAM algorithms that learn more general graph structures.

Original: The remainder of the paper is organized as follows.
Revised: The remainder of the paper is organized as follows.

Original: Section 2 sets the background and theory.
Revised: Section 2 sets the background and theory.

Original: Section 3 presents our contributions where a general population version and three cases are studied in detail.
Revised: Section 3 presents our contributions where a general population version and three algorithmic scenarios are studied in detail.

Original: Corresponding theoretical guarantees to our proposed algorithms are given in Section 4.
Revised: Corresponding theoretical guarantees for our proposed algorithms are given in Section 4.

Original: Results to numerical experiments are presented in Section 5.
Revised: Results of numerical experiments are presented in Section 5.

Original: We close with a discussion and suggestions for future research in Section 6.
Revised: We close with a discussion and suggestions for future research in Section 6.

Original: The proofs of all the results are provided in the appendix A.
Revised: The proofs of all the results are provided in the Appendix A, which is part of the supplementary material.

Revised: Appendix B in the supplementary material gives the detailed description of the sample versions of the algorithms considered in the paper.

Original:  LINEAR NON-GAUSSIAN STRUCTURAL CAUSAL MODELS
Revised:  LINEAR NON-GAUSSIAN STRUCTURAL CAUSAL MODELS

Original: A directed graph (digraph) is a pair G =
Revised: A directed graph (digraph) is a pair G =

Original: ( V, E ) , where V is the set of vertices and E ⊂ V × V is the set of directed edges.
Revised: ( V, E ) , where V is the set of vertices and E ⊂ V × V is the set of directed edges.

Original: Let V =
Revised: We let V =

Original: [ p ] := { 1 , . .
Revised: [ p ] := { 1 , . .

Original: . , p } ; an element ( i, j ) ∈ E may also be denoted by i −→ j .
Revised: . , p } . An element ( i, j ) ∈ E may also be denoted by i −→ j .

Original: A digraph G is acyclic (i.e., a DAG) if it does not contain any directed cycle: there is no sequence of vertices i 0 , . . 
Revised: A digraph G is acyclic (i.e., a DAG) if it does not contain any directed cycle: there is no sequence of vertices i 0 , . . .

Original: , i k with i j −→ i j +1 ∈ E for j = 0 , . . .,
Revised: , i k with i j −→ i j +1 ∈ E for j = 0 , . . .

Original:  k −and i 

Revised: , k − 1 and i

Original: = i k .
Revised: = i k .

Original: A path  G is a sequence of vertices i 0 , . .
Revised: A path in G is a sequence of vertices i 0 , . .

Original: , i k such that i j −→ i j +1 ∈ E or i j +1 −→ i j ∈ E for all j .
Revised: , i k such that i j −→ i j +1 ∈ E or i j +1 −→ i j ∈ E for all j .

Original: It isis directed if all the arrows point in the same direction.
Revised: It is directed if all the arrows point in the same direction.

Original: A polytree is a DAG in which there is a unique path between any two vertices.
Revised: A polytree is a DAG in which there is a unique path between any two vertices.

Original: A polytree is a directed tree if all the connecting paths are directed paths.

Original: If i −→ j ∈ E , then i is a parent of j , and j is a child of i .
Revised: If i −→ j ∈ E , then i is a parent of j , and j is a child of i .

Original: If G contains a directed path from i to j , then i is an ancestor of j and j is a descendant of i .
Revised: If G contains a directed path from i to j , then i is an ancestor of j and j is a descendant of i .

Original: The sets of parents, children, ancestors, and descendants of i ∈ V are denoted by pa( i ) , ch( i ) , an( i ) , de( i ) , respectively.
Revised: The sets of parents, children, ancestors, and descendants of i ∈ V are denoted by pa( i ) , ch( i ) , an( i ) , de( i ) , respectively.

Original: Let X =
Revised: Let X =

Original: ( X i )
Revised: ( X i )

Original: i ∈
Revised: i ∈

Original: [ p ] be a random vector indexed by the vertices of a DAG G .
Revised: [ p ] be a random vector indexed by the vertices of a DAG G .

Original: For A ⊂ [ p ] , let X A
Revised: For A ⊂ [ p ] , let X A

Original: =
Revised: =

Original: ( X i )
Revised: ( X i )

Original: i ∈ A .
Revised: i ∈ A .

Original: When
Revised: When

Original: X A is conditionally independent of X B given X C for disjoint subsets A, B, C ⊂
Revised: X A is conditionally independent of X B given X C for disjoint subsets A, B, C ⊂

Original: [ p ] , we write A ⊥⊥ B | C .
Revised: [ p ] , we write A ⊥⊥ B | C .

Original: The joint distribution of X satisfies the local Markov property with respect to G if { i } ⊥⊥ [ p ] \ (pa( i ) ∪ de( i ))
Revised: The joint distribution of X satisfies the local Markov property with respect to G if { i } ⊥⊥ [ p ] \ (pa( i ) ∪ de( i ))

Original: | pa( i ) ∀ i ∈ [ p ] .
Revised: | pa( i ) ∀ i ∈ [ p ] .

Original: The Markov equivalence class of G is the set of all DAGs that encode the same conditional independence relations, i.e., for which the set of distributions satisfying the local
Revised: The Markov equivalence class of G is the set of all DAGs that encode the same conditional independence relations, i.e., for which the set of distributions satisfying the local

Original: Markov property is the same.
Revised: Markov property is the same.

Original: See Maathuis et al.
Revised: See Maathuis et al.

Original: Chap. 1] for further details.
Revised: Chap. 1] for further details.

Original: The skeleton of a DAG is the undirected graph obtained by replacing each directed edge by an undirected edge.
Revised: The skeleton of a DAG is the undirected graph obtained by replacing each directed edge by an undirected edge.

Original: Here, edges are denoted by { i, j } ⊆ E .
Revised: Here, edges are denoted by { i, j } ⊆ E .

Original: 2.1 STRUCTURAL EQUATIONS
Revised: 2.1 STRUCTURAL EQUATIONS

Original: A structural equation model hypothesizes that every random variable in X is functionally related to its parent variables:
Revised: A structural equation model hypothesizes that every random variable in X is functionally related to its parent variables:

Original: X i =
Revised: X i =

Original: f i ( X pa( i ) , ε i ) , i ∈ V, where the ε i are independent noise terms and the f i are measurable functions.
Revised: f i ( X pa( i ) , ε i ) , i ∈ V, where the ε i are independent noise terms and the f i are measurable functions.

Original: If the f i are linear, then we obtain a linear structural equation model (LSEM).
Revised: If the f i are linear, then we obtain a linear structural equation model (LSEM).

Original: An LSEM can be written in matrix form as
Revised: An LSEM can be written in matrix form as

Original: [Equation]
Revised: [Equation]

Original: where Λ = ( λ ij ) with λ ij ̸ = 0 only if i → j ∈ E .
Revised: where Λ = ( λ ij ) with λ ij ̸ = 0 only if i → j ∈ E .

Original: An
Revised: An

Original: LSEM constrains the dependence structure on the coordinates of X , but not the mean.
Revised: LSEM constrains the dependence structure on the coordinates of X , but not the mean.

Original: Hence, when working with the LSEM, we may assume without loss of generality that E [ ε i ] = 0 , which implies E [ X i ] = 0 for all i ∈ V .
Revised: Hence, when working with the LSEM, we may assume without loss of generality that E [ ε i ] = 0 , which implies E [ X i ] = 0 for all i ∈ V .

Original: Let ε (2) =
Revised: Let ε (2) =

Original: ( E [ ε i ε j ]) ij be the covariance matrix of ε , which is a diagonal matrix by independence, and write ε (2) i :=
Revised: ( E [ ε i ε j ]) ij be the covariance matrix of ε , which is a diagonal matrix by independence, and write ε (2) i :=

Original: E [ ε 2 i ] > 0 for its i th diagonal entry.
Revised: E [ ε 2 i ] > 0 for its i th diagonal entry.

Original: The covariance matrix of X is then the positive definite matrix
Revised: The covariance matrix of X is then the positive definite matrix

Original: [Equation]
Revised: [Equation]

Revised: − 1 .

Original: 2.2 CUMULANTS IN GAUSSIAN AND NON-GAUSSIAN MODELS
Revised: 2.2 CUMULANTS IN GAUSSIAN AND NON-GAUSSIAN MODELS

Original: Cumulants can be seen as alternative representations of moments of a distribution.
Revised: Cumulants are alternative representations of moments of a distribution.

Original: Here, we formalize the definition in higher order settings and discuss their implications under Gaussian and non-Gaussian errors.
Revised: Here, we formalize the definition in higher order settings and discuss their implications under Gaussian and non-Gaussian errors.

Original: Definition 2.1.
Revised: Definition 2.1.

Original: The k th cumulant tensor of a random vector ( X 1 , . . .
Revised: The k th cumulant tensor of a random vector ( X 1 , . . .

Original: , X p ) is the k -way tensor in R p ×···× p ≡ ( R p ) k whose entry in position ( i 1 , . .
Revised: , X p ) is the k -way tensor in R p ×···× p ≡ ( R p ) k whose entry in position ( i 1 , . .

Original: , i k ) is the joint cumulant cum( X i 1 , . . .
Revised: , i k ) is the joint cumulant cum( X i 1 , . . .

Original: , X i k) :
Revised: , X i k

Revised: [Equation]

Original: = (cid:88)
Revised: := (cid:88)

Original: [Equation]
Revised: [Equation]

Original: where the sum is taken over all partitions ( A 1 , . .
Revised: where the sum is taken over all partitions ( A 1 , . .

Original: A L ) of the multiset { i 1 , . .
Revised: A L ) of the multiset { i 1 , . .

Original: , i k } .
Revised: , i k } .

Original: In our context, the variables have mean 0 , so cum( X i ) =
Revised: In our context, the variables have mean 0 , so cum( X i ) =

Original: E [ X i ] = 0 , cum( X i 1 , X i 2 ) =
Revised: E [ X i ] = 0 , cum( X i 1 , X i 2 ) =

Original: Cov
Revised: Cov

Original: [ X i 1 , X i 2 ] =
Revised: [ X i 1 , X i 2 ] =

Original: E [ X i 1 X i 2 ] .
Revised: E [ X i 1 X i 2 ] .

Original: More generally, the sum can be restricted to the partitions in which all blocks A i have at least two elements.
Revised: More generally, the sum can be restricted to the partitions in which all blocks A i have at least two elements.

Original: In particular, cum( X i 1 , X i 2 , X i 3 ) =
Revised: In particular, cum( X i 1 , X i 2 , X i 3 ) =

Original: E [ X i 1 X i 2 X i 3 ] , cum( X i 1 , X i 2 , X i 3 , X i 4 ) =E [ X i 1 X i 2 X i 3 X i 4 ] − E [ [Equation] E [ X
Revised: E [ X i 1 X i 2 X i 3 ] , cum( X i 1 , X i 2 , X i 3 , X i 4 ) =

Original: [Equation]

Original: 3 X

Original: [Equation] ] − E [ X i 1 X i 3 ] E [ X i 2 X i 4 ] − [Equation] [Equation]
Revised: E [ X i 1 X i 2 X i 3 X i 4 ] − E [ X i 1 X i 2 ] E [ X i 3 X i 4 ] − E [ X i 1 X i 3 ] E [ X i 2 X i 4 ] − E [ X i 1 X i 4 ] E

Original: X [Equation] E [ X i 2 X i 3 ] .
Revised: [ X i 2 X i 3 ] .

Original: The following powerful result dictates a simple condition for Gaussianity of X .
Revised: The following powerful result dictates a simple condition for Gaussianity of X .

Original: Theorem 2.2 (Marcinkiewicz) .
Revised: Theorem 2.2.

Original:  If there exists k such that cum( X i 1 , .., X i j ) = 0 for all j ≥ k , then k = 3 and X has a multivariate Gaussian distribution.
Revised: [Marcinkiewicz, 1939, Theorem 2] If there exists k such that cum( X i 1 , .., X i j ) = 0 for all j ≥ k , then k = 3 and X has a multivariate Gaussian distribution.

Original: Furthermore, the following results dictate when the assumptions of Theorem 2.2 are satisfied, thus giving rise to Gaussianity, especially under LSEMs.
Revised: Furthermore, the following results dictate when the assumptions of Theorem 2.2 are satisfied, thus giving rise to Gaussianity, especially under LSEMs.

Original: Lemma 2.3.
Revised: Lemma 2.3.

Original: If the variables X 1 , . ..
Revised: If the variables ε 1 , . .

Original:  , X n are independent, then cum( X i 1 , . . .
Revised: . , ε n are independent, then cum( ε i 1 , . . .

Original: , X i k )
Revised: , ε i k )

Original: = 

Revised: =

Original: unless i 1 = · · · = i k .
Revised: unless i 1 = · · · = i k .

Original: Lemma 2.4.
Revised: Lemma 2.4.

Original: Let the random vector X follow the LSEM from (2.1) with noise vector ε .
Revised: Let the random vector X follow the LSEM from (2.1) with noise vector ε .

Original: Let C ( k ) and ε ( k ) be the k th order cumulant tensors of X and ε , respectively.
Revised: Let C ( k ) and ε ( k ) be the k th order cumulant tensors of X and ε , respectively.

Original: Then
Revised: Then

Original: C ( k ) =

Original: ε ( k ) • (cid:2)

Original: ( I

Original: [Equation]

Original: [Equation]

Original: 1 (cid:3) k

Original: j

Original: [Equation]
Revised: [Equation]

Original: =

Original: [Equation]

Original: [Equation] [Equation] [Equation] is the Tucker product of ε ( k ) and k copies of ( I − Λ) − 1 .
Revised: is the Tucker product of ε ( k ) and k copies of ( I − Λ) − 1 .

Original: Notice here that C ( k ) reduces to (2.2)
Revised: Notice here that C ( k ) reduces to (2.2)

Original: when k = 2 .
Revised: when k = 2 .

Original: See Comon and Jutten [2010] and references therein for proofs of Theorem 2.2 and Lemmas 2.3 and 2.4.
Revised: See Comon and Jutten [2010] and references therein for proofs of Theorem 2.2 and Lemmas 2.3 and 2.4.

Original: The next definition introduces the cumulant model obtained from the LSEM (2.1).
Revised: The next definition introduces the cumulant model obtained from the LSEM (2.1).

Original: Definition 2.5.
Revised: Definition 2.5.

Original: Let G =
Revised: Let G =

Original: ( V, E ) be a DAG, and let K ≥be an integer.
Revised: ( V, E ) be a DAG, and let K ≥be an integer.

Original: The K th cumulant model of G is the set of
Revised: The K th cumulant model of G is the set of

Original: K -way tensors
Revised: K -way tensors

Original: [Equation]
Revised: [Equation]

Original: Here, R E is the set of p × p matrices with support E .
Revised: Here, R E is the set of p × p matrices with support E .

Original: Further, the cumulants up to order K defined by G are modeled by
Revised: Further, the cumulants up to order K defined by G are modeled by

Original: [Equation]
Revised: [Equation]

Original: By Theorem 2.2, all multivariate Gaussian vectors
Revised: By Theorem 2.2, all multivariate Gaussian vectors

Original: X correspond to the zero element of M ( K ) ( G ) for k ≥ 3 .
Revised: X correspond to the zero element of M ( K ) ( G ) for k ≥ 3 .

Original: When the errors in an LSEM are Gaussian, all distributional information is captured by the covariance matrix and equivalence issues arise that hinder identifiability of the full graph.
Revised: When the errors in an LSEM are Gaussian, all distributional information is captured by the covariance matrix and equivalence issues arise that hinder identifiability of the full graph.

Original: It then becomes necessary to expand to non-Gaussian settings.
Revised: It then becomes necessary to consider non-Gaussian settings.

Original: Relaxing the constraint of Gaussianity gives rise to the class of LiNGAMs where the underlying graph now becomes identifiable [Shimizu et al., 2006, 2011].
Revised: Relaxing the constraint of Gaussianity gives rise to the class of LiNGAMs where the underlying graph now becomes identifiable [Shimizu et al., 2006, 2011].

Original: We will exploit this property algorithmically and use thethe signal provided by higher cumulants; we do this by way of treks .
Revised: We will exploit this property algorithmically and use the signal provided by higher cumulants; we do this by way of treks .

Original: Definition 2.6 (Multi-Trek) .
Revised: Definition 2.6 (Multi-Trek) .

Original: A k -trek between vertices i 1 , . . .
Revised: A k -trek between vertices i 1 , . . .

Original: , i k ∈ V of a DAG G =
Revised: , i k ∈ V of a DAG G =

Original: ( V, E ) is a collection of directed paths T
Revised: ( V, E ) is a collection of directed paths T

Original: = ( P 1 , . . .
Revised: = ( P 1 , . . .

Original: , P k ) in G that share the same source and have i j as the sink of P j for all j .
Revised: , P k ) in G that share the same source and have i j as the sink of P j for all j .

Original: The common source node is the top of the trek top( T ) .
Revised: The common source node is the top of the trek top( T ) .

Original: A trek is simple if the top node  the unique node on all the paths.
Revised: A trek is simple if the top node is the unique node on all the paths.

Original: Denote the set of k -treks between i 1 , . .
Revised: We denote the set of k -treks between i 1 , . .

Original: , i k by T ( i 1 , . . 
Revised: , i k by T ( i 1 , . . .

Original: , i k ) and the set of simple treks by S ( i 1 , . .
Revised: , i k ) and the set of simple treks by S ( i 1 , . .

Original: , i k ) .
Revised: , i k ) .

Revised: See Figure 1 for an example.

Original: If P is a directed path in the DAG G =
Revised: If P is a directed path in the DAG G =

Original: ( V, E ) and Λ = ( λ ij ) ∈ R E , then λ P is the path monomial given by the product of λ ij with i → j an edge in P .
Revised: ( V, E ) and Λ = ( λ ij ) ∈ R E , then λ P = (cid:81) ( i,j ) ∈ P λ ij is a path monomial.

Original: For a k -trek T = ( P 1 , . . 
Revised: For a k -trek T = ( P 1 , . . .

Original: , P k ) , set λ T :=
Revised: , P k ) , set λ T :=

Original: λ P 1 · · · λ P k .
Revised: λ P 1 · · · λ P k .

Original: Proposition 2.7 (Multi-Trek Rule) .
Revised: Proposition 2.7 (Multi-Trek Rule) .

Original: The k th order cumulant tensor C ( k ) ( G ) of X can be expressed as
Revised: The k th order cumulant tensor C ( k ) ( G ) of X can be expressed as

Original: [Equation]
Revised: [Equation]

Original: where the sum is over all the treks T in T ( i 1 , .
Revised: where the sum is over all the treks T in T ( i 1 , .

Original: , i k ) and ε ( k ) top( T ) denotes the top( T ) diagonal entry of ε ( k ) .
Revised: , i k ) and ε ( k ) top( T ) denotes the top( T ) diagonal entry of ε ( k ) .

Original: Proposition 2.7 follows from Lemma 2.4 and expanding the entries of ( I − Λ) − 1 into sums of path monomials as in the usual trek rule for covariances [Robeva and Seby, 2021].
Revised: Proposition 2.7 follows from Lemma 2.4 and expanding the entries of ( I − Λ) − 1 into sums of path monomials as in the usual trek rule for covariances [Robeva and Seby, 2021].

Original: Corollary 2.8 (Simple Multi-Trek Rule) .
Revised: Corollary 2.8 (Simple Multi-Trek Rule) .

Original: The k th order cumulant tensor C ( k ) ( G ) of X can be expressed as
Revised: The k th order cumulant tensor C ( k ) ( G ) of X can be expressed as

Original: [Equation]
Revised: [Equation]

Original: where the sum is extended to all the simple treks S in
Revised: where the sum is extended to all the simple treks S in

Original: S ( i 1 , . . .
Revised: S ( i 1 , . . .

Original: , i k ) .
Revised: , i k ) .

Original: Corollary 2.9.
Revised: Corollary 2.9.

Original: The i th diagonal entry of C ( k ) is
Revised: The i th diagonal entry of C ( k ) is

Original: [Equation]
Revised: [Equation]

Original: 2.3 POLYTREE MODELS
Revised: 2.3 POLYTREE MODELS

Original: For general graphs, the algebraic relations among the cumulants may be far more complicated than the bivariate case (see Example A.1) and have not yet been fully characterized.
Revised: For general graphs, the algebraic relations among the cumulants may be far more complicated than the bivariate case (which is discussed in Example A.1) and have not yet been fully characterized.

Original: However, there exists a generalization ofthe rank-one constraint for polytrees, which we now discuss.
Revised: However, there exists a generalization of rank-one constraints for polytrees, which we now discuss.

Original: By consequence of there being at most one directed path between any two noes of a polytree G , there is at most one simple trek between any set of nodes i 1 , . . .
Revised: By consequence of there being at most one directed path between any two nodes of a polytree G , there is at most one simple trek between any set of nodes i 1 , . . .

Original: , i k .
Revised: , i k .

Original: The simple multi-trek rule then reduces to C ( k ) i 1 ,...,i k ( G ) =
Revised: The simple multi-trek rule then reduces to C ( k ) i 1 ,...,i k ( G ) =

Original: λ S C ( k ) top( S ) for a trek between nodes with S being the unique simple trek; denote the top of the simple trek between i 1 , . .
Revised: λ S C ( k ) top( S ) for a trek between nodes with S being the unique simple trek; denote the top of the simple trek between i 1 , . .

Original: , i k , if it exists by top( i 1 , . .
Revised: , i k , if it exists by top( i 1 , . .

Original: , i k ) .
Revised: , i k ) .

Original: Also, C ( k ) i 1 ,...,i k ( G ) =
Revised: Also, C ( k ) i 1 ,...,i k ( G ) =

Original: if there is no k -trek between the nodes.
Revised: if there is no k -trek between the nodes.

Original: For any two vertices i ̸
Revised: For any two vertices i ̸

Original: = j , let c ( i,j ) ,k m denote the k th order cumulant C ( k ) i...
Revised: = j , let c ( i,j ) ,k m denote the k th order cumulant C ( k ) i...

Original: i,j...j ( G ) , where the first m indices are equal to i and the remaining m − k equal j .
Revised: i,j...j ( G ) , where the first m indices are equal to i and the remaining m − k equal j .

Original: Proposition 2.10.
Revised: Proposition 2.10.

Original: Let e : i → j be an edge of a polytree G .
Revised: Let e : i → j be an edge of a polytree G .

Original: Then the following matrix is of rank one
Revised: Then the following matrix is of rank one

Original: [Equation]
Revised: [Equation]

Original: The first column of A e,K contains E [ X 2 i ] > 0 .
Revised: The first column of A e,K contains E [ X 2 i ] > 0 .

Original: Moreover, for every distribution induced by non-Gaussian errors, there exists k such that C ( k ) i ̸
Revised: Moreover, for every distribution induced by non-Gaussian errors, there exists k such that C ( k ) i ̸

Original: = 0 .
Revised: = 0 .

Original: Hence, at least one minor of
Revised: Hence, at least one minor of

Original: A e,K gives us an equation that is satisfied if i → j is in G , and is not satisfied in general for the graph with the edge reversed.
Revised: A e,K gives us an equation that is satisfied if i → j is in G , and is not satisfied in general for the graph with the edge reversed.

Original: This observation will provide the foundation for our learning algorithm, which we now present.
Revised: This observation will provide the foundation for our learning algorithm, which we now present.

Original:   POLYTREES FROM MOMENTS
Revised:  LEARNING NON-GAUSSIAN POLYTREES FROM MOMENTS

Original: We now present our population algorithm for learning polytrees with three versions for learning the edge orientations.
Revised: We now present our population algorithm for learning polytrees with three versions for learning the edge orientations.

Original: The first common phase is skeleton recovery.
Revised: The first common phase is skeleton recovery.

Original: 3.1 LEARNING THE SKELETON
Revised: 3.1 LEARNING THE SKELETON

Original: In its original formulation, the Chow–Liu algorithm gives the maximum likelihood tree approximation of a given discrete distribution [Chow and Liu, 1968].
Revised: In its original formulation, the Chow–Liu algorithm gives the maximum likelihood tree approximation of a given discrete distribution [Chow and Liu, 1968].

Original: The tree obtained is the maximum weight spanning tree of the complete undirected graph with edge weights w ( i, j ) , given by the mutual information between X i and X j .
Revised: The tree obtained is the maximum weight spanning tree of the complete undirected graph with edge weights w ( i, j ) , given by the mutual information between X i and X j .

Original: [Rebane and Pearl, 1987, Theo. 1] show that under a non-degeneracy assumption, the same Chow–Liu algorithm can be used to recover skeletons in the polytree setting. Their proof is based on the following property of the mutual information.
Revised: Under a non-degeneracy assumption, the same Chow–Liu algorithm can be used to recover skeletons in the polytree setting [Rebane and Pearl, 1987, Theorem 1]; the proof is based on the following property of the mutual information.

Original: Proposition 3.1.
Revised: Proposition 3.1.

Original: If the polytree that defines the model contains the subgraph i → j → l or i ←− j → l , then min { I ( X i , X j ) , I ( X j , X l ) } > I ( X i , X l ) , where I ( · , · ) is the mutual information.
Revised: If the polytree that defines the model contains the subgraph i → j → l or i ←− j → l , then min { I ( X i , X j ) , I ( X j , X l ) } > I ( X i , X l ) , where I ( · , · ) is the mutual information.

Original: When working with an LSEM, a stronger result justifies the use of the absolute value of the correlation coefficient instead of the mutual information.
Revised: When working with an LSEM, a stronger result justifies the use of the absolute value of the correlation coefficient instead of the mutual information.

Original: Lemma 3.2 (Wright’s Formula, [Wright, 1960]) .
Revised: Lemma 3.2 (Wright’s Formula, [Wright, 1960]) .

Original: In the
Revised: In the

Original: LSEM defined by a polytree, the correlation ρ i,j =
Revised: LSEM defined by a polytree, the correlation ρ i,j =

Original: Corr [ X i , X j ] satisfies | ρ i,j | = (cid:40)(cid:89)| ρ e | ,
Revised: Corr [ X i , X j ] satisfies | ρ [Equation] [Equation] = (cid:40)(cid:89)

Revised: [Equation]

Original:  T ( i, j ) ̸ = ∅ , 0 , otherwise , (3.1) where the product is taken over the edges of the unique trek from i to j , and ρ e denotes the correlation between the random variables indexed by the endpoints of the edge e .
Revised: e | , T ( i, j ) ̸ = [Equation] , 0 , otherwise , (3.1) where the product is taken over the edges of the unique trek from i to j , and ρ e denotes the correlation between the random variables indexed by the endpoints of the edge e .

Original: Definition 3.3.
Revised: Definition 3.3.

Original: Let R =
Revised: Let R =

Original: ( ρ i,j ) be the correlation matrix of a random vector X .
Revised: ( ρ i,j ) be the correlation matrix of a random vector X .

Original: The Chow–Liu tree M ( R ) is the (undirected) maximum weight spanning tree over [ p ] , with weights given by | ρ i,j | .
Revised: The Chow–Liu tree M ( R ) is the (undirected) maximum weight spanning tree over [ p ] , with weights given by | ρ i,j | .

Original: Kruskal’s algorithm may be applied to compute the Chow–
Revised: Kruskal’s algorithm may be applied to compute the Chow–

Original: Liu tree Kruskal [1956].
Revised: Liu tree [Kruskal, 1956].

Original: Proposition 3.4.
Revised: Proposition 3.4.

Original: Let R =
Revised: Let R =

Original: ( ρ i,j ) be the correlation matrix of a random vector X =
Revised: ( ρ i,j ) be the correlation matrix of a random vector X =

Original: ( X 1 , . . .
Revised: ( X 1 , . . .

Original: , X p ) that follows the
Revised: , X p ) that follows the

Original: LSEM given by a polytree G .
Revised: LSEM given by a polytree G .

Original: If 0 < | ρ i,j | < 1 for every e : i → j ∈ E , then M ( R ) equals the skeleton of G .
Revised: If 0 < | ρ i,j | < 1 for every e : i → j ∈ E , then M ( R ) equals the skeleton of G .

Original: The assumption | ρ i,j | < 1 holds for all random vectors with positive definite covariance matrix.
Revised: The assumption | ρ i,j | < 1 holds for all random vectors with positive definite covariance matrix.

Original: Moreover, in a polytree model, | ρ i,j | > 0 for an edge ( i, j ) if λ ij ̸ = 0 .
Revised: Moreover, in a polytree model, | ρ i,j | > 0 for an edge ( i, j ) if λ ij ̸ = 0 .

Original: 3.2 LEARNING ORIENTATIONS
Revised: 3.2 LEARNING ORIENTATIONS

Original: We now present three ways to orient the edges in the estimated skeleton.
Revised: We now present three ways to orient the edges in the estimated skeleton.

Original: The three resulting orientation algorithms areall based on Proposition 2.10 and the following result.
Revised: The three resulting orientation algorithms are based on Proposition 2.10 and the following result.

Original: Theorem 3.5.
Revised: Theorem 3.5.

Original: Consider the LSEM given the polytree G , and let e : i → j be an edge of G .
Revised: Consider the LSEM given the polytree G , and let e : i → j be an edge of G .

Original: Then
Revised: Then

Original: (i) rank( A i → j,K )
Revised: (i) rank( A i → j,K )

Original: = 1 , (ii) rank( A j → i,K )
Revised: = 1 , (ii) rank( A j → i,K )

Original: = 2 ,for
Revised: = 2 ,

Original:  generic edge coefficients and error cumulants up to order K .
Revised: for generic edge coefficients and error cumulants up to order K .

Original: Proposition 3.6.
Revised: Proposition 3.6.

Original: Suppose the skeleton of the polytree G contains the subgraph i − j − l with ρ i,j , ρ j,l ̸ = 0 .
Revised: Suppose the skeleton of the polytree G contains the subgraph i − j − l with ρ i,j , ρ j,l ̸ = 0 .

Original: Then the corresponding subgraph of G is i → j ←− l iff ρ i,l = 0 .
Revised: Then the corresponding subgraph of G is i → j ←− l iff ρ i,l = 0 .

Original: We now present PairwiseOrientation_Pop ; Algorithm 1.
Revised: We now present PairwiseOrientation_Pop ; Algorithm 1.

Original: This algorithm takes as input the list of unoriented edges and the parameter K ≥ 3 , which defines the highest order cumulant used in A i → j,K .
Revised: This algorithm takes as input the list of unoriented edges and the parameter K ≥ 3 , which defines the highest order cumulant used in A i → j,K .

Original: It orients each edge separately by checking whether the rank of A i → j,K is 1 or not.
Revised: It orients each edge separately by checking whether the rank of A i → j,K is 1 or not.

Original:  do 
Revised: Algorithm 1 PairwiseOrientation_Pop ( E, K ) 1: O ← ∅ 2: for { i, j } ∈ E do 3: if rank( A i → j,K ) = 1 then 4: O ← O ∪ { i → j } 5: else 6:

Original: : else return O
Revised: O ← O ∪ { j → i } return O

Original: Our second algorithm PTO_Pop , Algorithm 2, proceeds recursively.
Revised: Our second algorithm TPO_Pop , Algorithm 2, proceeds recursively.

Original: At each step, it takesas input the order K , a list of already oriented edges O , a list of still unoriented edges E , and, possibly, an oriented edge o .
Revised: At each step, it takes the order K , a list of already oriented edges O , a list of still unoriented edges E , and, possibly, an oriented edge o , as inputs.

Original: Here t ( o ) is the target/sink of the edge and E ∩ t ( o ) is the (possibly empty) set of unoriented edges containing t ( o ) .
Revised: Here t ( o ) is the target/sink of the edge and E ∩ t ( o ) is the (possibly empty) set of unoriented edges containing t ( o ) .

Original: The procedure checks if there are unoriented edges, and if so, it searches for triplets of the form i →
Revised: The procedure checks if there are unoriented edges, and if so, it searches for triplets of the form i →

Original: j − k , where the oriented edge o = i
Revised: j − k , where the oriented edge o = i

Original: → j can come either from the previous call of the procedure or from checking the rank of A i
Revised: → j can come either from the previous call of the procedure or from checking the rank of A i

Original: → j,K .
Revised: → j,K .

Original: For such a triplet, the method checks if ρ i,k = 0 , orienting the other edge according to the result.
Revised: For such a triplet, the method determines whether ρ i,k = 0 , orienting the other edge according to the result.

Original: The algorithm is initialized with O =
Revised: The algorithm is initialized with O =

Original: o = ∅ and the full list of undirected edges, E .
Revised: o = ∅ and the full list of undirected edges, E .

Original: Our third proposed algorithm TPO_Pop , Algorithm 3, can be seen as a direct extension of learning completed partially directed graphs (CPDAG)—a mixed graph that encodes the causal information common to all the members of a Markov equivalence class.

Original: Here, we first compute the CPDAG as in [Rebane and Pearl, 1987], then we orient all remaining undirected edges by checking the rank of A i → j,K .

Original: This ensures that no other unshielded colliders appear.

Original: The following example compares our three algorithms. 1: if E ̸ = ∅ then 2: if o = ∅ then 3: ( i, j ) ← E [1] 4: if rank( A i → j,K ) = 1 then 5: o ← ( i → j ) 6:
Revised: 1: if E ̸ = ∅ then 2: if o = ∅ then 3: { i, j } ← E [1] 4: if rank( A i → j,K ) = 1 then 5: o ← ( i → j ) 6:

Original: O ← O ∪ { o } 7: else 8: o ← ( j → i ) 9:
Revised: O ← O ∪ { o } 7: else 8: o ← ( j → i ) 9:

Original: O ← O ∪ { o } 10:
Revised: O ← O ∪ { o } 10:

Original: E o ← E ∩ t ( o ) 11: if E o ̸ = ∅ then 12:
Revised: E o ← E ∩ t ( o ) 11: if E o ̸ = ∅ then 12:

Original: E ← E \ E o 13: for t ( o ) − k ∈ E o do 14: if ρ s ( o ) ,k = 0 then 15:
Revised: E ← E \ E o 13: for t ( o ) − k ∈ E o do 14: if ρ s ( o ) ,k = 0 then 15:

Original: O ← O ∪ { k → t ( o ) } 16: else 17:
Revised: O ← O ∪ { k → t ( o ) } 16: else 17:

Original: O ← O ∪ { t ( o ) → w } 18: o ← ( t ( o ) → w ) 19:
Revised: O ← O ∪ { t ( o ) → w } 18: o ← ( t ( o ) → w ) 19:

Original: O, E ← PTO _ Pop ( E, K, O, o ) 20:
Revised: O, E ← TPO _ Pop ( E, K, O, o ) 20:

Original: O, E ← PTO _ Pop ( E, K, O, ∅ ) return O,E
Revised: O, E ← TPO _ Pop ( E, K, O, ∅ ) return O,E

Original: Algorithm 1: O ← ∅ 2: for i − j − k ∈ E do 3: if ρ i,k = 0 then 4: E ← E \ { ( i, j ) , ( j, k ) } 5:
Revised: Our third proposed algorithm PTO_Pop , Algorithm 3, can be seen as a direct extension of learning completed partially directed graphs (CPDAG)—a mixed graph that encodes the causal information common to all the members of a Markov equivalence class.

Original: O ← O ∪ { i → j, k → j } 6: for i → j ∈ O do 7: for j − l ∈ E do 8: E ← E \ { ( j, l ) } 9: O ← O \ { j → l } 10: for ( i, j ) ∈ E do 11: if rank( A i → j,K ) = 1 then 12:
Revised: Here, we first compute the CPDAG following Rebane and Pearl [1987], then we orient all remaining undirected edges by considering the rank of A i → j,K .

Original: O ← O ∪ { i → j } 13: for j − l ∈ E do 14: E ← E \ { ( j, l ) } 15: O ← O \ { j → l } 16: else 17:
Revised: This ensures that no other unshielded colliders appear.

Original: O ← O ∪ { j → i } 18: for i − l ∈ E do 19: E ← E \ { ( i, l ) } 20: O ← O \ { i → l } return O
Revised: The following example compares our three algorithms.

Original: Example 3.7.
Revised: Example 3.7.

Original: Consider the graph G with 1 −→ 2 −→ 3 ←− 4 .
Revised: Consider the graph G with 1 −→ 2 −→ 3 ←− 4 .

Original: With the skeleton 1 − 2 − 3 − 4 inferred, the algorithm
Revised: With the skeleton 1 − 2 − 3 − 4 inferred, the algorithm

Original: PairwiseOrientation_Pop sequentially computes the rank of
Revised: PairwiseOrientation_Pop sequentially computes the rank of

Original: A i → j,K in the order of all edges and orients them according to the results.
Revised: A i → j,K in the order of all edges and orients them according to the results.

Original: PTO_Pop orients 1 − 2 using the rank condition and then checks if ρ 1 , 3 = 0 .
Revised: TPO_Pop orients 1 − 2 using the rank condition and then checks if ρ 1 , 3 = 0 .

Original: Since this is not the case, it orients 2 − 3 using the rank condition and then 3 −checking that ρ 2 , 4 = 0 .
Revised: Since this is not the case, it orients 2 − 3 using the rank condition and then 3 −checking that ρ 2 , 4 = 0 .

Original: Finally, TPO_Pop first computes ρ 1 , 3 and ρ 2 , 4 .
Revised: Finally, PTO_Pop first computes ρ 1 , 3 and ρ 2 , 4 .

Original: Since ρ 2 , 4 = 0 , it orients 2 − 3 − 4 , and then orients 1 − 2 with the rank condition.
Revised: Since ρ 2 , 4 = 0 , it orients 2 − 3 − 4 , and then orients 1 − 2 with the rank condition.

Original: Theorem 3.8.
Revised: Theorem 3.8.

Original: The three versions of the algorithm are correct for generic edge coefficients and cumulants up to order
Revised: The three versions of the algorithm are correct for generic edge coefficients and cumulants up to order

Original: K .
Revised: K .

Original: LEARNING NON-GAUSSIAN POLYTREES FROM DATA
Revised: Algorithm 3 PTO_ 1: O ← ∅ 2: for i − j − k ∈ E do 3: if ρ i,k = 0 then 4: E ← E \ {{ i, j } , { j, k }} 5: O ← O ∪ { i → j, k → j } 6: for i → j ∈ O do 7: for j − l ∈ E do 8:

Revised: E ← E \ { ( j, l ) } 9:

Revised: O ← O \ { j → l } 10: for { i, j } ∈ E do 11: if rank( A i → j,K ) = 1 then 12:

Revised: O ← O ∪ { i → j } 13: for j − l ∈ E do 14:

Revised: E ← E \ { ( j, l ) } 15:

Revised: O ← O \ { j → l } 16: else 17:

Revised: O ← O ∪ { j → i } 18: for i − l ∈ E do 19:

Revised: E ← E \ { ( i, l ) } 20:

Revised: O ← O \ { i → l } return O

Revised:  LEARNING NON-GAUSSIAN POLYTREES FROM DATA

Original: We now consider the empirical versions of our algorithms, which now learn a polytree from a dataset consisting of n i.i.d.
Revised: We now consider the empirical versions of our algorithms, which now learn a polytree from a dataset consisting of n i.i.d.

Original: random vectors.
Revised: random vectors.

Original: The algorithms then process the sample correlations ˆ ρ i,j and sample cumulants ˆ c ( i,j ) ,k m .
Revised: The algorithms then process the sample correlations ˆ ρ i,j and sample cumulants ˆ c ( i,j ) ,k m .

Original: Let ˆΣ i,j be the unbiased sample covariances.
Revised: Let ˆΣ i,j be the unbiased sample covariances.

Original: Then ˆ ρ i,j = Σ i,j / (cid:112) Σ i,i Σ j,j .
Revised: Then ˆ ρ i,j = ˆΣ i,j / (cid:113) ˆΣ i,i ˆΣ j,j .

Original: Generalizing sample covariances, we take the sample cumulants
Revised: Generalizing sample covariances, we take the sample cumulants

Original: ˆ c ( i,j ) ,k m to be the k -statistics thatunbiasedly estimate c ( i,j ) ,k m  [McCullagh, 1987, §4.2].
Revised: ˆ c ( i,j ) ,k m to be the k -statistics that estimate c ( i,j ) ,k m in an unbiased manner [McCullagh, 1987, §4.2].

Original: We provide consistency results in a high dimensional setting where the size of the polytree grows at a faster rate than the sample size, subject to log-concavity of the variables.
Revised: We provide consistency results in a high dimensional setting where the size of the polytree grows at a faster rate than the sample size, subject to log-concavity of the variables.

Original: Specifically, we assume the errors ε i and thus also the observation vector X are log-concave distributed.
Revised: Specifically, we assume the errors ε i and thus also the observation vector X are log-concave distributed.

Original: This setting allows for the following corollary that builds on the concentration inequality given in Lemma B.3 of Lin et al.
Revised: This setting allows for the following corollary that builds on the concentration inequality given in Lemma B.3 of Lin et al.

Original: Corollary 4.1.
Revised: Corollary 4.1.

Original: Let K ∈ N and suppose that all moments up to order 2 K of the random vector X are bounded in magnitude by a constant M K > 0 .
Revised: Let K ∈ N and suppose that all moments up to order 2 K of the random vector X are bounded in magnitude by a constant M K > 0 .

Revised: There exists a constant

Original: For any k ≤ K , if ˆ c ( i,j ) ,k m [Equation]
Revised: L > 0 such that for any k ≤ K , if ˆ c ( i,j ) ,k m =

Original: c ( i,j ) ,k m + ϵ ( i,j ) ,k m is the k -statistic for a sample of size n ,then for every δ > 0 where[Equation]
Revised: c ( i,j ) ,k m + ϵ ( i,j ) ,k m is the k -statistic for a sample of size n , for every δ > 0 where

Original:  we have 
Revised: LK 2 √ M K (cid:18) δ √ n e (cid:19) 1 K > 2 , we have P [ | ϵ ( i,j ) ,k m | > δ ] ≤ exp (cid:26) −

Original: [Equation]
Revised: [Equation]

Original: 4.1 LEARNING THE SKELETON CONSISTENTLY
Revised: 4.1 LEARNING THE SKELETON CONSISTENTLY

Original: Let ρ min and ρ max be the respective minimum and maximum of the absolute edge correlations in the set {| ρ i,j | : i → j ∈ E } with 0 < ρ min , ρ max < 1 .
Revised: Let ρ min and ρ max be the respective minimum and maximum of the absolute edge correlations in the set {| ρ i,j | : i → j ∈ E } with 0 < ρ min , ρ max < 1 .

Original: We will use the following lemma on the correctness of the Chow–Liu tree M ( R n ) computed from the sample correlation matrix
Revised: We will use the following lemma on the correctness of the Chow–Liu tree M ( R n ) computed from the sample correlation matrix

Original: R n =
Revised: R n =

Original: (ˆ ρ i,j ) , together with Lemma 7 from Harris and Drton [2013], both restated below.
Revised: (ˆ ρ i,j ) , together with Lemma 7 from Harris and Drton [2013], both restated below.

Original: Lemma 4.2.
Revised: Lemma 4.2.

Original: Let γ = ρ min (1 − ρ max ) / 2 .
Revised: Let γ = ρ min (1 − ρ max ) / 2 .

Original: Then the event
Revised: Then the event

Original: F := (cid:84) {| ˆ
Revised: F := (cid:84) {| ˆ

Original: ρ
Revised: ρ

Original: i,j − ρ i,j | ≤ γ } satisfies F ⊂ {M ( ˆ R n ) =
Revised: i,j − ρ i,j | ≤ γ } satisfies F ⊂ {M ( ˆ R n ) =

Original: S ( G ) } .
Revised: S ( G ) } .

Original: Lemma 4.3.
Revised: Lemma 4.3.

Original: If A, B are 2 × 2 symmetric matrices, with A positive definite, a 1 , 1 , a 2 , 2 ≥ 1 , and || A − B || ∞ <
Revised: If A, B are 2 × 2 symmetric matrices, with A positive definite, a 1 , 1 , a 2 , 2 ≥ 1 , and || A − B || ∞ <

Original: δ , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a 1 , 2 √ [Equation] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [Equation]
Revised: δ , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a 1 , 2 √ a 1 , 1 a 2 , 2 − b 1 , 2 (cid:112) b 1 , 1 b 2 ,(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < 2 δ 1 − δ .

Original: We now have the following consistency result for the Chow–
Revised: We now have the following consistency result for the Chow–

Original: Liu tree M ( R n ) .
Revised: Liu tree M ( R n ) .

Original: Theorem 4.4.
Revised: Theorem 4.4.

Original: Let λ :=
Revised: Let λ :=

Original: min { min i Σ
Revised: min { min i Σ

Original: i,i , 1 } and let γ and
Revised: i,i , 1 } and let γ and

Original: M 2 be defined as in Lemma 4.2 and Corollary 4.1 respectively, then
Revised: M 2 be defined as in Lemma 4.2 and Corollary 4.1 respectively. Then

Revised: P ( M ( R n ) = S ( G )) ≥ 1 − 3 p ( p − 1) exp (cid:40)

Original: [Equation]
Revised: [Equation]

Revised: M

Revised: [Equation]

Revised: (cid:18)

Original: for all n > e
Revised: [Equation] n [Equation] +

Original: [Equation]
Revised: [Equation]

Original: {+(cid:19) 12 (cid:41) , for all n > e 2 (2+ λ ) 2 (4 L 2 √ M 2 ) 
+}
Revised: {+(cid:19) 12 (cid:41) , for all n > e 2 (2+ λ ) 2 (4 L 2 √ M 2 ) 
+}

Revised: λ 2 γ.

Original: 4.2 LEARNING ORIENTATIONS CONSISTENTLY
Revised: 4.2 LEARNING ORIENTATIONS CONSISTENTLY

Original: For every edge e = { i, j } in the skeleton S ( G ) , let v r ( e ) ,
Revised: For every edge e = { i, j } in the skeleton S ( G ) , let v r ( e ) ,

Original: v w ( e ) ∈ R B
Revised: v w ( e ) ∈ R B

Original: ( K ) be the vectors containing the minors of A ( r ( e )) ,K and A ( w ( e )) ,K involving the first column.
Revised: ( K ) be the vectors containing the minors of A ( r ( e )) ,K and A ( w ( e )) ,K involving the first column.

Original: Here, r ( e ) and w ( e ) are the correct and incorrect orientations of e in G , respectively.
Revised: Here, r ( e ) and w ( e ) are the correct and incorrect orientations of e in G , respectively.

Original: Let B ( K ) =
Revised: Let B ( K ) =

Original: K ( K − 1) / 2 − 1 be the size of the vectors.
Revised: K ( K − 1) / 2 −be the size of the vectors.

Original: We assume ∃ δ > 0 such that || v w ( e ) ||
Revised: We assume that there exists δ > 0 such that || v w ( e ) ||

Original: >
Revised: >

Original: δ for all e ∈ S ( G ) , where || · || is the 2-norm.
Revised: δ for all e ∈ S ( G ) , where || · || is the 2-norm.

Original: Let M K , L , and ϵ ( i,j ) ,k m be defined as in Corollary 4.1.
Revised: Let M K , L , and ϵ ( i,j ) ,k m be defined as in Corollary 4.1.

Original: Moreover, let c be the vector containing all the cumulants c ( i,j ) ,k m such that edge { i, j } ∈ S ( G ) and 0 ≤ m ≤ k ≤ K .
Revised: Moreover, let c be the vector containing all the cumulants c ( i,j ) ,k m such that edge { i, j } ∈ S ( G ) and 0 ≤ m ≤ k ≤ K .

Original: Write ˆ c n for the vector containing the sample versions of these cumulants.
Revised: Write ˆ c n for the vector containing the sample versions of these cumulants.

Original: Finally, let ϵ n be the corresponding error vector tracking the differences between the true and sample cumulants.
Revised: Finally, let ϵ n be the corresponding error vector tracking the differences between the true and sample cumulants.

Original: Lemma 4.5.
Revised: Lemma 4.5.

Original: If f is the difference of two monomials of degree 2 in the variables c , then
Revised: If f is the difference of two monomials of degree 2 in the variables c , then

Original: [Equation]
Revised: [Equation]

Original: For use with data, the proposed algorithms in Section 3 must be modified to allow for sampling variability.
Revised: For use with data, the proposed algorithms in Section 3 must be modified to allow for sampling variability.

Original: In particular, instead of checking whether or not rank( A i → j,K ) = 1 , we check ||
Revised: In particular, instead of assessing whether or not rank( A i → j,K ) = 1 , we check ||

Original: ˆ v i →
Revised: ˆ v i →

Original: j (( i, j ))
Revised: j ( { i, j } )

Original: || <
Revised: || <

Original: || ˆ v j → i (( i, j )) || instead.
Revised: || ˆ v j → i ( { i, j } ) || instead.

Revised: Here, ˆ v is the sample analogue of v , computed using sample moments.

Original: Similarly, for the independence test, we check whether or not the absolute  correlation is below a threshold ρ thr ; Lemma 4. clarifies the possible choices of the threshold.
Revised: Similarly, for the independence test (vanishing of correlation), we check whether or not the absolute sample correlation is below a threshold ρ θ ; Lemma 4.8 clarifies the possible choices of the threshold.

Original: The  sample versions of the algorithms are given in Appendix B.
Revised: The resulting sample versions of the algorithms are given in Appendix B.

Original: Let A POn ( E, K ) be the output of Algorithm 4 applied to a sample of size n and let E S ( G ) be the edge set of the true skeleton of G .
Revised: Let A POn ( E, K ) be the output of Algorithm 1 applied to a sample of size n and let E S ( G ) be the edge set of the true skeleton of G .

Original: Then we have the following consistency result.
Revised: Then we have the following consistency result.

Original: Lemma 4.6.
Revised: Lemma 4.6.

Original: Let δ ′ :
Revised: Let δ ′ :

Original: = min { δ 4 M K √ B
Revised: = min { [Equation]

Original: ( K ) , √ δ 4 √ 4 B ( K ) } .

Original: Then
Revised: Then

Original: P (

Original: [Equation]
Revised: [Equation]

Original: POn (
Revised: for all n > e

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation] = G ) [Equation] for all n
Revised: LK 2 √ M K ) 2 K δ

Original: [Equation]
Revised: [Equation]

Original: Theorem 4.7.
Revised: Theorem 4.7.

Original: Suppose the data are an n -sample drawn from a distribution in the LSEM given by a polytree G .
Revised: Suppose the data are an n -sample drawn from a distribution in the LSEM given by a polytree G .

Original: Let ˆ
Revised: Let ˆ

Original: G be the polytree obtained by applying Algorithm 4 to the (undirected) edge set of the Chow–Liu tree M ( R n ) .
Revised: G be the polytree obtained by applying Algorithm 1 to the (undirected) edge set of the Chow–Liu tree M ( R n ) .

Original: Then ˆ
Revised: Then ˆ

Original: G =
Revised: G =

Original: G with probability greater than
Revised: G with probability greater than

Original: [Equation]
Revised: [Equation]

Original: for all n > max 
Revised: for all n > max (cid:110) e

Original: [Equation]
Revised: [Equation]

Original: with constants defined in Lemma 4.6 and Theorem 4.4.
Revised: with constants defined in Lemma 4.6 and Theorem 4.4.

Original: Lemma 4.8.
Revised: Lemma 4.8.

Original: Let ˜ γ = min { ρ min / 3 , (1 − ρ max ) / 2 } ρ min , and let λ and M 2 be as in Theorem 4.4.
Revised: Let ˜ γ = min { ρ min / 3 , (1 − ρ max ) / 2 } ρ min , and let λ and M 2 be as in Theorem 4.4.

Original: If ˜ γ < ρ thr < ρ 2min − ˜ γ , then the probability that all independence tests carried out by Algorithm 5 yield correct decisions is bounded from below by
Revised: If ˜ γ < ρ θ < ρ 2min − ˜ γ , then the probability that all independence tests carried out by Algorithm 3 yield correct decisions is bounded from below by

Original: [Equation]
Revised: [Equation]

Original: The same statement holds for Algorithm 6.
Revised: The same statement holds for Algorithm 2.

Original: Theorem 4.9.
Revised: Theorem 4.9.

Original: Suppose the data are an n -sample drawn from a distribution in the LSEM given by a polytree G .
Revised: Suppose the data are an n -sample drawn from a distribution in the LSEM given by a polytree G .

Original: Let ˆ
Revised: Let ˆ

Original: G be the polytree obtained by applying Algorithm 5 orto the (undirected) edge set of the Chow–Liu tree M ( R n ) .
Revised: G be the polytree obtained by applying Algorithm 3 or 2 to the (undirected) edge set of the Chow–Liu tree M ( R n ) .

Original: If the threshold satisfies ˜ γ < ρ thr < ρ 2 min − ˜ γ , then there exists α ∗ < p − 1 such that ˆ G
Revised: If the threshold satisfies ˜ γ < ρ θ < ρ 2min − ˜ γ , then there exists α ∗ < p − 1 such that ˆ G

Original: =
Revised: =

Original: G with probability greater than
Revised: G with probability greater than

Original: [Equation]
Revised: [Equation]

Original: n
Revised: for all [Equation] > max

Original: [Equation]
Revised: [Equation]

Revised: Computational Complexity.

Revised: The complexity of the three algorithms is dominated by the O ( p 2 log ( p )) cost of the

Revised: Kruskal algorithm which computes the Chow–Liu tree, see [Cormen et al., 2001, Chapter VI].

Revised: In terms of the edge orientation, Algorithms 1 have linear computational complexity both in p and n which is independent of the structure of the graph, while Algorithms 3 and 2 may entail a cost that is quadratic in p in the worst case scenario, e.g., a star tree with all the edges outgoing from the center.

Original:  NUMERICAL EXPERIMENTS
Revised:  NUMERICAL EXPERIMENTS

Original: We assess and compare the accuracy of our three proposed algorithms on synthetic data, simulated as follows:
Revised: We assess and compare the accuracy of our three proposed algorithms on synthetic data, simulated as follows:

Original: For any fixed choice of n , p , and error distribution, we first generate a random undirected tree with p nodes using randomly generated Prüfer sequences [Prüfer, 1918] and then independently orient each edge.
Revised: For any fixed choice of n , p , and error distribution, we first generate a random undirected tree with p nodes using randomly generated Prüfer sequences [Prüfer, 1918] and then independently orient each edge.

Original: Next, we draw n samples for every node from the error distribution and uniformly draw the coefficients λ ij from the interval ( − 1 , − 0 . 3) ∪ (0 . 3 , 1) .
Revised: Next, we draw n samples for every node from the error distribution and uniformly draw the coefficients λ ij from the interval ( − 1 , − 0 . 3) ∪ (0 . 3 , 1) .

Original: Finally, we multiply the matrix of sampled errors by the matrix ( I − Λ) − 1 to obtain samples corresponding to the
Revised: Finally, we multiply the matrix of sampled errors by the matrix ( I − Λ) − 1 to obtain samples corresponding to the

Original: LSEM defined by the generated polytree.
Revised: LSEM defined by the generated polytree.

Original: The performance is measured by the structural Hamming distance, which is the number of incorrectly included edges plus the number of falsely omitted edges plus the number of incorrectly oriented edges divided by 2( p − 1) .
Revised: The performance is measured by the structural Hamming distance, which is the number of incorrectly included edges plus the number of incorrectly omitted edges, plus the number of incorrectly oriented edges, divided by 2( p − 1) .

Original: Small distance indicates improved performance.
Revised: Small distance indicates improved performance.

Original: We show the results in three settings: (i) low dimensional, with25 ≤ p ≤and 1 ≤ n/p ≤ 300 ; (ii) high dimensional, with 500 ≤ p ≤ 3000 and 0 .
Revised: We show the results in three settings: (i) low dimensional, with p ≤ 200 and 1 ≤ n/p ≤ 100 ; (ii) high dimensional, with 1500 ≤ p ≤ 3000 and 0 .

Original: 5 ≤ n/p ≤ 1 ; and (iii) a large scale setting with 10000 ≤ p ≤ 20000 and n/p = 0 .
Revised: 5 ≤ n/p ≤ 1 ; and (iii) a large scale setting with 10000 ≤ p ≤ 20000 and n/p = 0 .

Original: We set up experiments with errors drawn from the gamma and uniform distributions; the results are displayed in Figures 2, 3, and 4.
Revised: We set up experiments with errors drawn from the gamma and uniform distributions; the results are displayed in Figure 2.

Original: For  choice of threshold required in Algorithms 5 and 6, we evaluate the algorithms on a grid of thresholds and report the value corresponding to the best result.
Revised: For the choice of threshold required in Algorithms 3 and 2, we evaluate the algorithms on a grid of thresholds and report the value corresponding to the best result.

Original: Gamma Distribution.
Revised: Gamma Distribution.

Original: Errors were drawn from Γ( α, β ) ; the shape α and the scale β parameters are uniformly drawn from (0 . 5 , 5) .
Revised: Errors were drawn from Γ( α, β ) ; the shape α and the scale β parameters are uniformly drawn from (0 . 5 , 5) .

Original: Since Γ( α, β ) is asymmetric, we tested the algorithms with K = 3 .
Revised: Since Γ( α, β ) is asymmetric, we tested the algorithms with K = 3 .

Original: The experimental results are coherent with our developed theory: for all three algorithms, the distance between the true and learned trees converges to 0 as the sample size and/or the dimension of the tree increases.
Revised: The experimental results are coherent with our developed theory: for all three algorithms, the distance between the true and learned trees converges to 0 as the sample size and/or the dimension of the tree increases.

Original: We observe that Algorithm 4 performs better both in mean accuracy and variance, despite heavily relying on higher moments which is statistically disadvantageous.
Revised: We observe that Algorithmperforms better both in mean accuracy and variance, despite heavily relying on higher moments which is statistically disadvantageous.

Original: The improved performance is potentially due to the tuning-free nature of this algorithm compared to the other where tuning of the threshold parameter is required. Additionally, and opposed to the latter two algorithms, Algorithm 4 avoids any potential error propagation since the edges are oriented independently.
Revised: The improved performance is potentially due to the fact that Algorithm 1 avoids any potential error propagation since the edges are oriented independently.

Original: Uniform Distribution.
Revised: Uniform Distribution.

Original: Errors were drawn from U ( a, b ) , with the parameter a uniformly drawn from ( − 10 , − 1) and b uniformly drawn from (1 , 10) .
Revised: Errors were drawn from U ( a, b ) , with the parameter a uniformly drawn from ( − 10 , − 1) and b uniformly drawn from (1 , 10) .

Original: Here, the uniform distribution is symmetric so the 3rd order cumulants are 0; we thus tested the algorithms with K = 4 .
Revised: Here, the uniform distribution is symmetric so the third order cumulants are 0; we thus tested the algorithms with K = 4 .

Original: The experimental results here are also consistent with our developed theory.
Revised: The experimental results here are also consistent with our developed theory.

Original: We also notice that overall, the experiments with uniform errors outperform those with gamma errors, which may be due to the greater higher order moments associated with the gamma distribution, which tend to increase the variance of the sample cumulants in Corollary 4.1.
Revised: We also notice that overall, the experiments with uniform errors outperform those with gamma errors, which may be due to the greater higher order moments associated with the gamma distribution, which tend to increase the variance of the sample cumulants in Corollary 4.1.

Revised: The code to reproduce the experiments is available at https://github.com/danieletramontano/LiNGAM-PolytreeLearning.

Original:  CONCLUSION
Revised:  CONCLUSION

Original: In this paper, we proposed three algorithmic schemes that learn linear non-Gaussian polytree models by using the Chow–Liu algorithm to infer the graph skeleton and then  applying different approaches to orient edges leveraging non-Gaussianity and/or marginal uncorrelatedness.
Revised: In this paper, we proposed three algorithms that learn linear non-Gaussian polytree models first using the Chow–Liu algorithm to infer the graph skeleton, and then subsequently applying different approaches to orient edges leveraging non-Gaussianity and marginal uncorrelatedness.

Original: The algorithms differ from one another in how much information is taken from correlations versus from higher moments.
Revised: The algorithms differ from one another in how much information is taken from correlations versus from higher moments.

Original: The numerical experiments show that the algorithmsare effective also  in very high-dimensional problems.
Revised: The numerical experiments show that the algorithms also perform well in very high-dimensional problems.

Revised: These results indicate that our approach may be applicable in preliminary data analyses towards the aim of understanding dependence structures in data, particularly since the polytree setting allows for richer dependence and causal structures than other tree-based models [e.g., Edwards et al., 2010].

Original: Our work motivates the following questions for future research:
Revised: Our work motivates the following questions for future research:

Revised: How to avoid Chow–Liu?

Revised: As pointed out above the main computational burden comes from the computation of the

Original: How to best handle Gaussian random variables when learning a polytree?
Revised: Chow–Liu tree. Another shortcoming of the Chow–Liu algorithm is that it requires the whole covariance matrix to be computed and stored beforehand, making it impractical for very large graphs. A solution to this problem that leverages on algebraic relations has been proposed by Lugosi et al. [2021] for undirected trees.

Revised: A possible extension of this

Revised: [Figure]

Revised: approach to polytrees would be of interest.

Revised: How to best handle Gaussian random variables when learning a polytree?

Original: In some settings we may encounter the situation that some but not all errors are non-Gaussian; see
Revised: In some settings we may encounter the situation that some but not all errors are non-Gaussian; see

Original: Hoyer et al.
Revised: Hoyer et al.

Original: [2008] for a characterization of equivalence of graphs in this case.
Revised: [2008a] for a characterization of equivalence of graphs in this case.

Original: An interesting problem is then to determine how the respective performance of our algorithms is affected bysuch partial Gaussianity, and provide modifications that effectively learn a polytree equivalence class.
Revised: An interesting problem is then to determine how the respective performance of our algorithms is affected by partial Gaussianity and provide modifications that effectively learn a polytree equivalence class.

Original: As an illustration, Figure 1 shows that Algorithm 4 achieves 70% accuracy when a random choice of half of the random variables are allowed to be Gaussian.
Revised: As an illustration, Figure 3 shows that Algorithm 1 achieves 70% accuracy when a random choice of half of the random variables are allowed to be Gaussian.

Original: Which tree structures are the most difficult to learn?
Revised: Which tree structures are the most difficult to learn?

Original: Tan et al.
Revised: Tan et al.

Original: [2009] show that for undirected Gaussian tree models, the star and the chain represent the most difficult and the easiest trees to learn, respectively.
Revised: [2009] show that for undirected Gaussian tree models, the star and the chain represent the most difficult and the easiest trees to learn, respectively.

Original: The difficulty is due to the correlation decay.
Revised: The difficulty is due to the correlation decay.

Original: An interesting question to pursue is what the polytree analogues for the most difficult and easiest trees to learn would be.
Revised: An interesting question to pursue is what the polytree analogues for the most difficult and easiest trees to learn would be.

Original: What happens when the graph is not a tree?
Revised: What happens when the graph is not a tree?

Original: Boix-Adserà et al.
Revised: Boix-Adserà et al.

Original: [2021] prove a weakness result of the Chow–Liu algorithm under model misspecification for the Ising model and adapt it to achieve a form of optimality.
Revised: [2021] prove a weakness result of the Chow–Liu algorithm under model misspecification for the Ising model and adapt it to achieve a form of optimality.

Original: It would be of interest to describe a similar optimality criterion in the LiNGAM setting and investigate how our algorithm performs under these terms.[Figure] [Figure] [Figure]
Revised: It would be of interest to describe a similar optimality criterion in the LiNGAM setting and investigate how our algorithm performs under these terms.

Original: A

Original: A.1 SECTION

Original: Example A.1. Assume we have two random variables Xand X 2 , coming from an unknown xSEM, and we wish to find the DAG G that generates the model. Assume further that X 1 and X 2 are correlated, leaving us with the two options
Revised: Acknowledgements

Original: G 1 = ([2] , { 1 → 2 } ) and G 2 = ([2] , { 2 → 1 } ) . Under the model given by G 1 , we have X 1 = ε 1 and X 2 = λ 12 X 1 + ε 2 , and the covariance matrix from (2.2) becomes [Equation]
Revised: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 883818).

Original: Observe that any positive definite 2 × 2 matrix Σ can be written in this way; set ε (2) =

Original: Σ 11 > 0 , λ 12 = Σ 21 / Σ 11 and ε (2)2 = Σ 22 − Σ 212 / Σ 11 > 0 .

Original: By symmetry, the model given by G 2 also allows its covariance matrices to be any positive definite 2 × 2 matrix.

Original: Hence, the sets of covariance matrices resulting from G 1 versus G 2 are the same, and nothing can be said about the graph on the basis of covariances alone.

Original: Proof of Corollary 2.8.

Original: We say that a k -trek T =

Original: ( P 1 , . . .

Original: , P k ) factorizes through k -trek S =

Original: ( Q 1 , . . .

Original: , Q k ) if Q j ⊂ P j for all j .

Original: Indeed, it is easy to see that λ T = λ S λ T − S , where T − S

Original: =

Original: ( P 1 − Q 1 , . . .

Original: P k − Q k ) ∈ T S =

Original: T (top( S ) , . .

Original: , top( S )) with P

Original: j − Q j being the directed path from top( T ) to top( S ) that remains after removing the edges in Q j from P j .

Original: For the sake of readability, when the considered set of vertices is clear from the context, we denote the set of treks (and simple treks) by only T (and S ) .

Original: Now note that every trek factorizes along one and only one simple trek.

Original: Hence, we can partition T = ˙

Original: (cid:83) S ∈S T S , according to the simple trek through which the factorization occurs.

Original: The expression in (2.4) may thus be rewritten as

Original: [Equation]

Original: By the multi-trek rule, the term in parentheses is C ( k ) top( S ) ( G ) .

Original: Proof of Corollary 2.9.

Original: Since the graph is acyclic, the only trek in T ( i, .., i ) that has i as top is the trivial trek, from which come the ε ( k ) i term.

Original: All the other treks in T ( i, .., i ) factorize through a set of distinct parents of i , so we can write the sum in 2.4 as

Original: [Equation]

Original: and the term inside the internal parenthesis is C ( k ) p 1 ,...,p k ( G ) .

Original: Proof of Proposition 2.10.

Original: By the simple multi-trek rule for polytrees c e,km = λ k − m ij

Original: C ( k ) i .

Original: Therefore, c e,km = λ ij c e,km + so that the second row of A e,K equals λ ij times the first row.

Original: A.2 SECTION

Original: Proof of Proposition 3.4.

Original: As noted, M ( R ) may be computed using Kruskal’s algorithm, which considers all edges in decreasing order of their weights and adds them to the spanning tree if their presence does not create a (undirected) cycle.

Original: Let S ( G ) be the skeleton of G .

Original: For a contradiction, assume that M ( R ) ̸ = S ( G ) .

Original: Since both graphs are trees, we have L :=

Original: M ( R ) \ S ( G ) ̸ = ∅ .

Original: Take ˜ e

Original: = { i, j } = arg max e ∈ L | ρ e | .

Original: Then | ρ ˜ e | ̸ = 0 and, thus, T ( i, j ) ̸ = ∅ .

Original: The unique trek in G that connects i and j must contain an edge e that is not in M ( R ) ; otherwise we would have two paths between i and j in M ( R ) which cannot occur as M ( R ) is a tree.

Original: Moreover, Wright’s formula from Lemma 3.2 and the assumption made on the correlation coefficients imply that | ρ ˜ e | < | ρ e | .

Original: But then e would appear before ˜ e in Kruskal’s algorithm and e would be added to M ( R ) since all the edges in M ( R ) with a weight higher than the weight of ˜ e are correctly classified and so the presence of e could not create a loop.

Original: We have arrived at a contradiction.

Original: Proof of Theorem 3.5.

Original: The first claim is a restatement of

Original: Proposition 2.10.

Original: To prove the second claim, we need to show that for generic error cumulants, at least one of the 2 × 2 subdeterminants of A j → i,K is nonzero.

Original: Since these minors are polynomials in the cumulants, it is enough to prove that they are not identically zero; see, e.g., the lemma in Okamoto [1973].

Original: By the simple multi-trek rule (Corollary 2.8), c j → i, 2 i = Σ

Original: i,j = λ i,j Σ i,i , c j → i,k m = λ mi,j C ( k ) i ∀ m < k ≤ K.

Original: By Corollary 2.9, c j → i, 2= Σ j,j = (cid:88) p,q ∈ pa( i ) λ p,j λ q,j Σ p,q

Original: [Equation]

Original: because in a polytree any two distinct parents p and q have

Original: T ( p, q ) = ∅ and, thus, Σ p,q = 0 by Lemma 3.2.

Original: Consider now a minor involving the first column together with any other column with m < k .

Original: This minor is

Original: [Equation]

Original: The term in parentheses is always positive, while the front factor is nonzero provided λ i,j ̸ = 0 and the error cumulants are chosen such that C ( k ) i ̸

Original: [-= 
-]
Revised: [-= 
-]

Original: (e.g., take any distribution with ε ( k ) k = 0 , for every k ̸

Original: = j , and ε ( k ) j ̸ = 0 ).

Original: Proof of Proposition 3.6.

Original: The claim follows from

Original: Lemma 3.2 as i → j ←− l is the only case in which there are no treks between i, l .

Original: Proof.

Original: From Theorem 3.5 we derive the correctness of the rank condition and thus of the entire algorithm PairwiseOrientation_Pop as well as the relevant parts of the other two algorithms.

Original: Proposition 3.6 yields the correctness of the remaining parts of the other two algorithms.

Original: A.3 SECTION

Original: Lemma A.2.

Original: Consider a degree k polynomial f ( X ) =

Original: f ( X 1 , . . .

Original: , X m ) , where X 1 , . .

Original: , X m are possibly dependent random variables with log-concave joint distribution on R m .

Original: Then exists a constant L > 0 such that for all δ with

Original: L (cid:32)

Original: [Equation]

Original: we have

Original: [Equation]

Original: Proof of Corollary 4.1.

Original: The results follows by bounding the variance of ˆ c ( i,j ) ,k m .

Original: To this end, we may express the variance as a linear combination of products of moments based on the definition of cumulants.

Original: We may then bound each product of moments by a power of M K and note that no weight may exceed ( K − 1)! .

Original: Applying Stirling’s approximation for the factorial and bounding the Bell numbers that count the number of summands gives the result.

Original: Proof of Theorem 4.4. Let F be theevent defined in Lemma 4.2. Then
Revised: DT’s PhD scholarship is funded by the

Original: [Equation]

Original: where the last inequality comes from the union bound.

Original: Correlations are scale-invariant, thus the value of ρ i,j and the distribution of ˆ

Original: ρ i,j do not change under rescaling of the observed variables to ˜ X

Original: i

Original: =

Original: X i /λ .

Original: Now, we apply

Original: Lemma 4.3 with A = ˜Σ and B = ˜Σ ∗ as the new population and sample covariance matrices, respectively.

Original: Let || ˜

Original: [Equation]

Original: |

Original: [Equation] = || ˜Σ − ˜Σ ∗ || ∞ . Then || ˜ ϵ || ∞ ≤ || ϵ
Revised: IGSSE/TUM-GS via a Technical University of Munich–

Original: ( i,j ) , 2 || ∞ λ , where we indicate with ϵ ( i,j ) , 2 the vector containing the errors

Original: ϵ ( i,j ) , 2 m , for m = 0 , 1 , 2 .

Original: This allows us to write

Original: [Equation]

Original: where the last inequality comes from a union bound and

Original: Corollary 4.1 with K = 2 .

Original: Proof of Lemma 4.5.

Original: By Taylor expansion, we have

Original: [Equation]

Original: where H ( f ) | c is the Hessian matrix of f computed in c .

Original: For the special quadratic polynomial f , the Hessian is constant with entries ± 1 , and the gradient contains entries of c , possibly negated.

Original: The result follows by triangle inequality.

Original: Proof of Theorem 4.4.

Original: The event of correctly reconstructing the polytree G is the intersection of the two events of correctly recovering of the skeleton and correctly orienting all the edges.

Original: The probability of this intersection is bounded from below by the sum of the probabilities of the two events minus one.

Original: Theorem 4.4 and Lemma 4.6 imply the result.

Original: Proof of Lemma 4.6.

Original: Given that we start with the correct skeleton, a mistake in the orientation appears if and only if there is an edge, e in the skeleton for which || ˆ v r ( e ) || ≥

Original: ||

Original: ˆ v w ( e ) | , so the union bound leads to the following lower bound

Original: [Equation]

Original: Now we need an upper bound on P ( || ˆ v r ( e )

Original: || ≥

Original: || ˆ v w ( e ) || )

Original: , adding and subtracting || v w ( e ) || on the right hand side of the inequality, and using || v w ( e ) || > δ , led to

Original: P ( || ˆ v r ( e ) || ≥

Original: ||

Original: ˆ v w

Original: [Equation]

Original: e ) || )

Original: [Equation]

Original: Now, using that v r ( e ) = 0 and applying Lemma 4.

Original: [Equation]

Original: we obtain that

Original: [Equation]

Original: where v i is one of the components of the vector.

Original: Finally

Original: [Equation]

Original: we obtain that

Original: [Equation]

Original: With δ ′ :

Original: = min { δ 4 M K √ B

Original: ( K ) ,

Original: [Equation]

Original: [Equation]

Original: K ) } we can bound the right hand side of the last inequality by

Original: [Equation]

Original: whenever 2 LK

Original: [Equation]

Original: 1 K > 2 holds, and this concludes the proof.

Original: Proof of Lemma 4.8.

Original: It suffices to show that under the event ˜

Original: F = (cid:84) {| ˆ ρ

Original: i,j − ρ i,j | < ˜ γ } all independence tests give the correct decision.

Original: This claim was proven in Theorem 3.5 of [Lou et al., 2021]. The rest of the proof is the same as for Theorem 4.4. Proof of Theorem 4.9.
Revised: Imperial College London Joint Academy of Doctoral Studies (JADS) award (2021 cohort, PIs Drton/Monod).

Original: Having ˜ γ ≤ γ from Lemma 4.2 and

Original: Lemma 4.8, the probability that the skeleton is recovered correctly and that all the independence tests give the right answer is bounded from below by the probability of the event considered in Lemma 4.2.

Original: Hence, it only remains to bound the probability that all cumulant tests correctly orient their respective edge.

Original: This can be done as in Lemma 4.6.

Original: Finally, carrying over the arguments from the proof of Theorem 4. gives the result.

Original: Notice that in the case of Algorithm 5, the constant α ∗ < p − 1 − CP , where CP is the number of directed edges in the CPAG associated to G .

Original: B SAMPLE VERSION OF THE ALGORITHMS

Original: Algorithm

Original: 1: O ← ∅ 2: for e = ( i,

Original: j ) ∈

Original: E do 3: if || ˆ v i →

Original: j (( i, j ))

Original: || <

Original: || ˆ v j → i (( i, j ))

Original: || then 4:

Original: O ← O ∪ { i → j } 5: else 6:

Original: O ← O ∪ { j → i } return O

Original: Algorithm 5 TPO ( E, K, ρ thr ) 1: O ← ∅ 2: for i − j − k ∈ E do 3: if | ρ i,k | < ρ thr then 4:

Original: E ← E \ { ( i, j ) , ( j, k ) } 5:

Original: O ← O ∪ { i → j, k → j } 6: for i → j ∈ O do 7: for j − l ∈ E do 8:

Original: E ← E \ { j − l } 9:

Original: O ← O \ { j → l } 10: for ( i, j ) ∈

Original: E do 11: if rank( A i → j,K ) = 1 then 12:

Original: O ← O ∪ { i → j } 13: for j − l ∈ E do 14:

Original: E ← E \ { j − l } 15:

Original: O ← O \ { j → l } 16: else 17:

Original: O ← O ∪ { j → i } 18: for i − l ∈ E do 19:

Original: E ← E \ { i − l } 20:

Original: O ← O \ { i → l } return O

Original: Algorithm 6 PTO ( E, K, O, o, ρ thr ) 1: if E ̸ = ∅ then 2: if o = ∅ then 3: ( i, j ) ← E [1] 4: if rank( A i → j,K ) = 1 then 5: o ← ( i → j ) 6:

Original: O ← O ∪ { o } 7: else 8: o ← ( j → i ) 9:

Original: O ← O ∪ { o } 10:

Original: E o ← E ∩ t ( o ) 11: if E o ̸ = ∅ then 12:

Original: E ← E \ E o 13: for t ( o ) − k ∈ E o do 14: if | ρ s ( o ) ,k | < ρ thr then 15:

Original: O ← O ∪ { k → t ( o ) } 16: else 17:

Original: O ← O ∪ { t ( o ) → w } 18: o ← ( t ( o ) → w ) 19:

Original: O, E ← PTO _ Pop ( E, K, O, o, ρ thr ) 20:

Original: O, E ← PTO _ Pop ( E, K, O, ∅ , ρ thr ) return O,E