Conditional Independent Component Analysis for Estimating Causal Structure with Latent Variables
Authors: Yewei Xia
Zhengming Chen
Haoyue Dai
Fuhong Wang
Yixin Ren
Yiqing Li
Kun Zhang
Shuigeng Zhou
Venue: ICLR 2026 Poster (2026)
Keywords: Causal Discovery
Latent Structure Learning
Conditional Independent Component Analysis
Sparsity
Primary area: causal reasoning

Abstract: Identifying latent variables and their induced causal structure is fundamental in various scientific fields. Existing approaches often rely on restrictive structural assumptions (e.g., purity assumption) and may become invalid when these assumptions are violated. We introduce Conditional Independent Component Analysis (CICA), a new principle that extracts components that are conditionally independent given latent variables. Under mild conditions, CICA can be optimized using a tractable proxy such as rank-deficiency constraints. Building on CICA, we establish an identifiability theory for linear non-Gaussian acyclic models with latent variables: solving CICA and then applying an appropriate row permutation to the sparsest CICA solution enables recovery of the causal structure. Accordingly, we propose an estimation method based on the identifiability theory and substantiate the algorithm with experiments on both synthetic and real-world datasets.

Published as a conference paper at ICLR 2026
CONDITIONAL INDEPENDENT COMPONENT ANALYSIS FOR ESTIMATING CAUSAL STRUCTURE WITH LATENT VARIABLES
Yewei Xia
Zhengming Chen
Haoyue Dai
Fuhong Wang
Yixin Ren
Yiqing LiC
Kun Zhang
Shuigeng Zhou!,*
Fudan University
Shantou University
*Carnegie Mellon University
*Guangdong University of Technology
Moohaeed bin Zayed University of Artificial Intelligence
ABSTRACT
Identifying latent variables and their induced causal structure is fundamental in various scientific fields. Existing approaches often rely on restrictive structural as- sumptions (e.g.. purity assumption) and may become invalid when these assump- tions are violated. We introduce Conditional Independent Component Analysis (CICA), new principle that extracts components that are conditionally indepen- dent given latent variables. Under mild conditions, CICA can be optimized using a tractable proxy such as rank-deficiency constraints. Building tablish an identifiability theory for linear non- Gaussian acyclic models with latent variables: solving CICA and then applying appropriate row permutation to the sparsest CICA solution enables recovery of the causal structure. Accordingly, we propose estimation method based the identifiability theory and substantiate the algorithm with experiments on both synthetic and real-world datasets.
CICA, we es-
INTRODUCTION
Understanding causal structures is essential in numerous scientific domains, such as biology (Wood- ward, 2010), psychology (Eronen, 2020), and economics (Hicks et al., 1980). To uncover the under- lying causal structures in data-driven manner, various methods have been proposed (Peters et al., 2017). Most traditional causal discovery methods rely on the causal sufficiency assumption (Spirtes et al., 2000), i.e., no latent confounders exist between any pair of observed variables. However, in many real-world applications, it is often infeasible to measure all the underlying causal variables. For example, in psychology, researchers investigate the impact of social behavior mental health, while intelligence or personality may often act as latent confounders. It is difficult to precisely measure these variables, yet ignoring such latent confounders can lead to misleading conclusions. Generally, identifying the presence of latent variables and recovering the causal structure involving both observed and latent variables remains a significant challenge.
Some approaches attempt to address the challenge by exploiting conditional independence con- straints, such as the FCI algorithm (Spirtes et al., 1995) and its variants (Colombo et al., 2012). However, their results capture only the causal relationships among observed variables. To further discover causal relationships between latent variables, additional parametric assumptions are typi- cally required. For linear Gaussian causal models, several methods leverage rank-deficiency con- straints to recover the underlying structure, including latent variables, up to the Markov equivalence class (Silva et al., 2006; Kummerfeld & Ramsey, 2016; Huang et al., 2022; Dong al., 2023). To take into account higher-order statistics, (Xie et al., 2020) develops a generalized independent noise (GIN) condition and establishes its corresponding estimation algorithm for linear non-Gaassian data. TIN (Dai et al., 2022) defines the independent linear transformation subspace and its dimension can be used further improve the identifiability of causal discovery with measurement error.
Although these methods have achieved some progress, they typically involve certain structural as- sumptions to simplify the problem. In particular, the purity assumption (Cai et al., 2019; Xie al.,
"Corresponding Authors

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X1
X3
X1
X2
(a)
(b)
Figure I: An example of a non-identifiability issue of most existing methods.
2020) rules out edges between observed variables. Violating these assumptions can lead failures in determining the true causal graph. For example, in Fig. 1, the two graphs cannot be distinguished by most existing methods. Only a few methods can theoretically distinguish these two graphs, pri- marily overcomplete ICA (OICA) (Friksson & Koivunen, 2004)-based methods and higher-order cumulant-bar baaeed methods (Schkoda et al., 2024; Chen al., 2024a). However, OICA typically relies on the expectation maximization (EM) procedure along with approximate inference, which is com- putationally prohibitive and prone to local optima (Cai et al., 2023). On the other hand, higher-order statisties can be very sensitive to outliers in the data (Hyvärinen & Oja, 2000), reliably estimating higher-order cumulants requires massive samples (Nikias & Mendel, 1993). This raises an impor- tant question: we strike better balance between identifiability and practical feasibility? Our findings indicate that this could be possible.
Concretely, by analyzing why GIN and TIN conditions fail distinguish Fig. (la) and (lb), argue that relying solely on a one-sided projection w Y. (Y, Z are two subsets of observed variables) could be restricted. Instead, two-sided projections w Y Luy Z may leave additional identifiable traces. Accordingly, we seek a unified procedure that estimates latent causal structure by searching for non-zero W1, W1,42 W2 with wi Y [.ww] Motivated by this, we introduce new principle named conditional independent component analysis (CICA), which extracts components that are conditionally independent given latent variables. Under mild conditions, CICA can be optimized using tractable proxy such as rank-deficiency constraints, which avoid involving the estimation of high-order cumulants like OICA or cumulant-based methods. Building on CICA, we establish an identifiability theory and estimation algorithm for linear non-Gaussian acyclic models with latent variables: first solve CICA, then apply an appropriate row permutation to the sparsest CICA solution to recover the underlying causal structure, yielding more general identifiability result with a tolerable computational burden.
(1) We introduce conditional independent component analysis (CICA), novel prin- ciple that extracts components conditionally independent given latent variables. CICA fundamen- tally overcomes the theoretical bottlenecks of existing methods in highly impure structures. (2) We establish an identification theory and an estimation algorithm for recovering the underlying causal structure based on CICA, where sparsity is provable property in resolving CICA's inherent indeter- minacies. (3) We validate our framework through extensive synthetic and real-world experiments.
Contributions:
2 BACKGROUND
2.1 PROBLEM SETUP
Suppose directed acyclic graph (DAG) G = (V,E), where the variables V are generated following a linear causal structural model:
V = BV + E.
Here V = X L contains observed variables X = {X,]m and latent variables L = d E are mutually independent non-Siass exogenous noises. The adjacency matrix B captures direct causal effects (B # 0 < Vú + V, E &). Assuming observed variables do not cause latent ones (BI.x = 0), can be naturally partitioned into three sub-matrices BI,L, Bx,x, and Bx,,- Specifically, BI. and Bx,x encode the direct causal effects among the latent and observed vari- ables, respectively, while BX.L captures the direct causal effects from the latent to the observed one. Through simple mathematical manipulation, V can also be expressed with the mixing matrix A = (I- B) B)- The objective of latent causal structure learning is to fully recover the underlying causal structure G from pure observational data.
linear mixture V AE
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Notations. For matrix M, denote by Ms.: the rows in M indexed by set S, and similarly by M.s the columns. In addition, let GL(m)' be the invertible matrix W E Rm Further, use Pa(V)), Ch(V)), Ane De()) as parents, children, ancestors and descendants of V,, respectively. We use LPa(S) for subset S C V to denote the set that contains all the common latent parents of any two nodes in S, excluding the variables in S. Denote equivalence relation M1 ~ M2 if and only if there exist row permutation matrix P and non-singular diagonal matrix D that makes PDM, = M2. By default, Y and Z denote two subsets of observed random variables.
2.2 PRELIMINARIES
We first briefly review the Independent Noise (IN) condition, fundamental principle that exploits asymmetry to discover causal directions in linear non-Gaussian acyclic models (LiNGAM) without latent variables, and then explain how it was extended to scenarios involving latent variables.
Definition 1 (IN condition (Shimizu et al., 2011)), Let Y be single variable and Z be set of variables. Suppose all variables follow LiNGAM. We say that (Z, Y) follows the IN condition, if and only if the residual of regressing Y on Z is statistically independent from Z.
With hidden confounders behind observed variables, independence between regressor and residual typically does not exist. However, we could still benefit from a similar idea to leverage the non- Gaussianity of exogenous noises to bring asymmetry. That is, involving other observed variables to serve proxy of latent variables and calculate ppseudo-residue the Generalized Independent Noise (GIN) condition:
Definition 2 (GIN condition (Xie et al., 2020)). Let Y and Z be two observed random vectors. Suppose that the variables follow LiNGAM. We say (Z, Y) satisfies the GIN condition, if and only if the following two conditions are satisfidd: 1) E non-zero w E TR| that solves the equation cov [, Y)u 0, and 2) Any such solution makes the linear transformation w I.
GIN condition needs to be equipped with enough pure children, which is defined as follows:
Definition 3 (Purity (Xie et al., 2024)). Let L be a set of latent variables, and S be subset of L for any Va, Vo S, and üi) SV \ }e(L)] [L. In addition, we say that a variable V. S relative to L is a pure variable if is a pure set relative to Specifically, if S Ch(L), we say that each variable V. E S is a pure child relative to L.
descendant nodes of L, i.e., S C De(L). We say S is pure set relative to L iff V. V.
In the definition of the GIN condition, the coefficient vector is only characterized by the covariance matrix, which may limit the power of non-Gaussianity, Therefore, TIN condition are proposed to bring more higher-order information.
Definition 4 (TIN condition (Dai et al., 2022)). Let Z and Y be two subsets of random variables. Denote the independent linear transformation subspace flz,y == {w E ERIY| w T I. The TIN condition of Z and Y is defined as: TIN(Z, Y) := YY| di,()) where dim(S denotes the dimension of the subspace Ω2z,,, i.e., the degree of freedom of w.
Although TIN condition generalizes GIN condition, it is fundamentally restricted by the utility of only one-sided projections w Y IL Z. As we will reveal in the next section, relying solely on such one-sided projections remains non- non- con-informative when faced with impure causal structures.
METHOD
In this section, we develop a principled framework for causal discovery in the presence of latent confounders. We first describe our motivation by analyzing why existing tools that rely on con- structing independence fail (53.1). We then formalize our proposed tool, conditional independent component analysis (CICA), and discuss its identifiability and indeterminacy (33.2), optimization criterion ($3.3). Next, we provide a comprehensive introduction to the identifiability guarantee of latent causal structure based on CICA ($3.4). Finally, we discuss the connection between CICA and independent subspace analysis (ISA) and why ISA not informative in our settings ($3.5).
the general linear group of degree
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3.1 MOTIVATION: BEYOND ONE-SIDED PROJECTIONS
As discussed, existing criteria such as GIN and TIN conditions are built on one-sided projections w I. To ensure identifiability, these methods require that latent variables L have enough pure children (Xie et al., 2024). The rationale is that pure children are mutually conditionally indepen- dent given L. With sufficient pure children, one can construct linear combination of Y to remove the dependence entirely attributable to the common ancestors L and thus induce independence.
In contrast, in Fig. la and Ib, every pair of observed variables share not only L but also E1. In this case, no one- one-sided projection of the form w Y IL Z with non-zero w can eliminate both sources of dependence simultaneously. As result, the GIN and TIN conditions fail to distinguish between the two graphs since both exhibit no non-degenerate independence pattern under one-sided projections.
This limitation highlights the insufficiency of these tools based on one-sided projections in the presence of multiple latent influences. In fact, not all constructive independence patterns can be expressed as w Y J. A natural step forward is to consider two-sided projections of the form wi Y !لW! Z. to remove the dependence from both sides. The following lemma shows that the independence patterns in the form Y are a subset of those of w Y -uy Z.
Lemma 1. Let Z and Y be two subsets of random variables. Y Y.J.J.12 .1. W1, then there must exist non-zero vector w2 makes wi Y Ilw Z.
has a non-zero solution
Essentially, the richer the independence information that a principle exploits, the stronger its identi- fication power. As shown next, Fig. la and Fig. 1b fall into different equivalent classes when using the information contained in w| [Lw Z.
Remark 1. The causal structures in Fig. Ia and Fig. lb exhibit asynmetry independence infor- mation. Specifically, non-zero solutions (w), satisfying W1,1 12 + wi 241,2Xallus X1 X2 exist in Fig. la but not in Fig. lb. Conversely, the alternative independence constraint w1,1 X2 + w1,2 X3 ll 12, X1 + 2 admits non-zero solutions in Fig. lb, but not in Fig. la.
Motivated by these asymmetries, although two causal graphs cannot be distinguished using only one-sided projection wi Y J. Z. two-sided projections Wi Y I.w.y Z can leave additional identifiable traces for the causal direction. This prompts natural question: Can we develop unified procedure that searches for non-zero Wh, W2 with w Y llwa Z to enhance identifiability?
3.2 CONDITIONAL INDEPENDENT COMPONENT ANALYSIS
A direct route construct W1 Y ILw, Z is to use overcomplete ICA (OICA), which separates more mutually independent sources from fewer observed signals. However, OICA is known to be com- putationally and statistically ineffective (Ding al., 2019). An alternative solution is to brute-force searching for each "two-sided projection" (WI, w2). However, it is difficult to guarantee that all feasible (Wi,w) have been found.
Instead of fully separating all latent sources in OICA, we propose to factor out the shared influ- ences explicitly and only require independence conditional on a latent vector. Concretely, we seek an invertible transform W such that Z = WX has mutually independent coordinates given some latent L E IRP This approach is powerful for two reasons: 1. As we will show in Section 3.3, when p is known, this principle allows for more tractable optimization proxies, avoiding the statistical and computational burdens of OICA. 2. As we will prove in Lemma 3, any solution that satisfies this generative principle (Z, L. provably induces the two-sided projections wi Y Ilw] Z required for identifiability. We formalize this core generative principle as follows:
Assumption (Linear mixing with conditionally independent sources). Let X be an observed vari- able set with X = m. There exist constant invertible matrix A E R mm p latent variables L
with Lt. > 0, a constant matrix M E mpxxp, and noise variables E = (E,,.., Fmm" )T such that
AS,
S MLL+E,
E L.L.
(E.)m are mutually independent with finite, non-zero variances, and at most one Eg Gaussian. Eg is not a sealar multiple of the identity matrix I E pmxm
Definition (p-order Conditional Independent Component Analysis (CICA)). Let X be m observed variables. An matrix W GT.(m) is called a p-order CICA solution for X if there exists p latent
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variables L s.t., the components Z := WX (Zi...., Zm) are mutually conditionally inde- pendent given L. Besides, we introduce Punin (X) == min k k E N, k-order CICA solution of X exists} to measure the size of the minimal latent conditional set of
When p- = O. conditionally independent reduces to mutual independence of Z. and then CICA coincides with ICA, where the unmixing matrix can be identified up to permutation and scaling equivalence class (Comon, 1994). In CICA, does the introduction of latent conditional variables create new, complex mixing ambiguities? The following lemma formally establishes the results.
Lemma 2 (Identifiability and Indeterminacy of CICA). Suppose Assumption 1 holds, let X be m observed variables. there exist Pmii (X) latent variables L such that for two valid CICA solutions W1 and 22 the components of ( == W X are mutually conditionally independent given L for l E (1,2), then there uniquely exist a permutation matrix P= (for some permutation ι of [m]) and non-singular diagonal matrix D such that W2 1.
In particular, when p (the ICA case), the conclusion reduces to the classical permutation and scaling indeterminacy of ICA. Therefore, Lemma 2 tells us that CICA introduces an additional indeterminacy about the conditional set L compared ICA. In addition, based on the CICA solution, one can naturally induce two-sided projections w Y [.w] Z.
Lemma 3. Let X be observed variables, and W be p-order CICA solution of X. Let X' WWX, Y and Z are two subsets of X', then if max ([Y), [Z]] > w Y' Il w Z' has a non- zero solution (wi, for (Y'ZZ) ) where Y' = Ex.eV & + 0}, Z' are defined similarly.
w
Example 1. The following structural causal model serves as an instantiation of Fig. la, where L. FA, E2, Fa are independent non- non-Gause Gaussian variables, o, b, c, u, v are non-zero coefficients. The identity matrix I GL(3) is a 3-order CICA solution of X (the conditional set can be {L, E1, E2}). The right-hand side below shows an example of a 1-order CICA solution of X (the conditional set {L}). The existence of L leads to the absence of a O-order CICA (i.e., ICA) solution of X.
W
X1 =al + Er, {
X1
a
E1 E2
X2 bL - uXi + Ez,
1
0
X2 Xa
b
L+
X3 = cL + vX2 + Es.
U
Ex
Besides, we can construct two-sided projections W1 Y [.w! with non-zero w1, W2. based on the
CICA solution of X. Taking Y [X X1}. Z {X1} as an example, denoting X'= = WW.
then we have ax, bx, 11 Le.. - (au + b) + bx, 11 Xx3 - vX2. A non-zero solution
W1 = lI[(au + b), bj_ W2 = 1-v, 1] exists for (Y' (X,, Z' Xz)).
3.3 OPTIMIZATION CRITERION FOR CICA
Since the conditional set is latent, the definition of CICA does not specify a testable optimization ob- jective. A practical question arises: which optimization criterion should for CICA? Inspired by (Huang et al., 2022: Dong et al,, 2023), we characterize conditional independence by introducing the following rank-deficiency constraint.
Lemma 4. For an observed variable set X with [X| = m. denote p = Pmin (). Suppose m 2 2p + 2, and set X = WX, then W is a p-order CICA solution of X if and only if for every pair of disjoint coordinate subsets X,, X2 of X' with X,| = X21 = p + 1, det Ex,,x2) = 0, where Σ Cov(X') denotes the covariance matrix on X' and Xx is the (p + 1) x (p +1+) sub-matrix of Σ with rows indexed by X, and columns by X2.
In fact, here matrix with a higher-order cumulant tensor. More details are included in Appendix B.3. When Pmin (X) 1, we can use another proxy objective of CICA, equipped with weaker condition. Definition 6 (Triad constraint (Cai et al., 2019)). Define the pseudo-residun of {X,, X,) relative to XR as E(j]ik) == Cov(X;, Xx) X, Cov(C,,
2p + 2 is not strict restriction; we can relax it by replacing the covariance
Xj.. We say that the pair of variables Xx) and X₃ satisfy the Triad constraint if E(c,j)))
Lemma 5. For an observed variable set X with X = m, suppose that Punin (X) = and m 2 hold, set X & WX, then the invertible matrix W is a 1-order CICA solution of X if and only for every ordered triple (X), X3, x)) X', (X,, X and X3. satisfies the Triad constraint.
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In both Lemma 4 and 5, we assume Pmán (X) is known, then characterize Pmii (X) -order CICA using the zero-determinant and independence constraint, respectively. In our estimation algo- rithm, we can determine the value of Pmin (X) in principle, without requiring prior knowledge (see Lemma 11) I). Since both the determinant and dependence measures (e.g.. HSIC (Gretton et al., 2005)) used in Def. 6 are differentiable, these lemmas actually provide optimization criterion for CICA.
3.4 IDENTIFIABILITY OF LATENT CAUSAL STRUCTURE BASED ON CICA
In this section, we establish an identifiability theory for causal structure in the linear non- Gaussian acyclic models with latent variables. Once CICA is solved, when and how can the causal structure be recovered from the CICA solutions W First, we have the following basic assumptions.
Assumption (Rank Faithfulness Assumption (Spirtes, 2013)), We assume that the distribution P is (linearly) rank-fatte rank-faithful to DAG G if every rank constraint on sub- covariiaee matrix that holds in P is entailed by every pee-paraeeter linear structural model with a path diagram equal to G.
Assumption 2 holds generically, since the set of values of the free parameters of the SCM for which the rank not faithful is of Lebesgue measure 0 (Spirtes, 2013). Besides, recovering the latent struc- ture requires the latent variables to leave sufficient structural footprint in the observed variables, we then introduce the following standard requirement:
Condition 1. Each latent variable in G has least three neighbors and two children (which can be latent or observed).
In this section, for the sake of brevity, will primarily discuss the results under the one-factor scenario (observed variables share at most one common latent confounder). Most results can be extended into the multi-factor scenario (observed variables can share multiple latent variables) di- rectly. Since directly recovering the global causal structure is inherently challenging due to the entanglement of causal effects from latent and observed variables, decouple the objective by first recovering the causal structure among the observed variables Bx.x. As we will demonstrate, our CICA framework exactly isolates Bx.x from latent confounding. The following lemma shows that Bx,x naturally constitutes valid CICA solution when conditioning on the true latent confounders.
Lemma 6. I Bx a Pmin X) -oordr CICA solution of X with latent conditional set LPa(X).
To identify the causal structure based on CICA, we must resolve all inherent indeterminacies. In- spired by (Shimizu et al., 2006), the permutation and scaling indeterminacies can be eliminated by the acyclicity of the causal graph. Therefore, we have the following lemma.
Lemma 7. Suppose W is a Punin (X) (X)-order CICA solution of whose latent conditional set is LPa(X), then W I Bx,x and variables Bx.xfrrom W.
uniquely recover the true causal matrix among observed
Based on Lemma 6 and Lemma 7, whose latent conditional set is LPa(X). As stated in Lemma 2, CICA introduces an additional indeterminacy in the choice of the latent conditional set. If W is a Pmin (X) CICA solution of observed variables X, the conditional set does not need to coincide with the latent confounders LPa(X). Instead, it may correspond the exogenous noise of the observed variables. Therefore, to solve the indeterminacy of the latent conditional set, we must further identify the CICA solution that aligns with the ground-truth causal structure.
correctly recover Bx,x x provided that the CICA solution
As shown in Fig. 2, for X = [X1,X,, X3}. W, is a 1-order CICA solution of X given L, thus W1 I Bx.x according to Lemma In contrast, the ambiguity of the latent conditional set allows alternative solutions, such as W2, to also qualify as feasible 1-order CICA solutions of X, although without direct correspondence to Bx,x. Essentially, W2X can be interpreted as swapping the roles of L and E₁ on X. Although conditional independence is preserved after swapping the latent variables, the sparsity of the solution matrix changes. Specifically,, it becomes denser. This additional discriminative signal: the sparsest
observation highlights that sparsity can serve CICA solution better aligns with the underlying causal structure.
Lemma 8. I Bx,x E arg min [| W is a Ponin (X)-order CICA solution of X).
Lemma 8 shows that I- B- is a Pmin (X) orddr CICA solution of X with the minimum number of non-zero entries. Notably, we do not assume that real-word weal-world causal structure is maximally sparse. On

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W1
W2 2
E
X1
X3
X
X
Figure 2: An example of different 1-order CICA solutions for X. WT a 1-order CICA solution that renders X' conditionally independent given I,, while W2 renders X' conditionally independent given Ej, the exogenous noise of X1. The gray/white rectangle denotes non con-zero/zero entries.
the contrary, it can be arbitrarily dense. The minimal sparsity principle is not prior assumption or convenient choice impose on the causal structure. Instead, it is provable theoretical property that emerges from the CICA framework itself, which we then exploit for identifiability. To ensure identifiability, we seek conditions under which Bx.x x is the unique sparsest Pmin (min(XXorrdee CICA solution of X, up to some permutation and scale indeterminacies.
Condition 2. For any X; X, EX₃ X (X,) with LPa({X, X₁1) +1
Intuitively, Condition 2 ensures that every observed variable has at least one "confounded sibling" that it does not directly cause, making it possible to disentangle the latent variable's 's effect uniquely. Example 2. In the figure on the left below, since X1 is not the parent of X3. X2 and X3 are not the parents of X1, then Condition 2 holds. contrast, in the figure on the right below, X1 is both the parent of X2 and X3, thus Condition 2 does not hold.
X1
X3
X3
Figure 3: Two example graphs understand Condition
Lemma 9. If Condition holds, W E ming llo W is a Pmin (X) -ordde CICA solution of X) if and only if W Bx.x.
Under Condition 2, Lemma 9 establishes that the sparsest Pmin (o)drrder CICA solution recovers I --x up to permutation and scale indeterminacies. By Lemma the remaining gap can be eliminated by acyclicity. Consequently, Bx.x is uniquely identified, including both the causal graph among the observed variables and its edge coefficients.
Conversely, when Condition 2 does not hold, I - Bx,x is non-identifiable: there exists distinct Pmin PX) (X)-order CICA solution W' with an equal number non-zero entries. Surprisingly, although W' has different parameters from I Bx,x, x. their support matrix remains the same. Therefore, the causal structure among observed variables is identifiable, which we summarized as follows.
Theorem 1. All latent variables in LPa X) can be identified. Besides, the causal edges of LPa(X) to X and the causal edges between the observed variables are also identifiable.
When the variables form hierarchical structure and some latent variables may have no observed children, due to the linearity assumption and the transitivity of linear causal relations, we can use certain observed descendant of the latent variables to implement CICA and apply Theorem recursively. The question is, which one is suitable to serve surrogate for the latent variable?
Lemma 10. Let be a latent variable discovered in the current iteration. Denote S Ch(L). Let Se have the highest causal order in S whose index in S is k, and W be the sparsest Pmin (S)-order CICA solution of'S. S. P is the permutation matrix that makes PW have non-zero diagonal elements, simultaneously. Denote Z = PWS, then the value of Zy, can be suitable surrogate for L.
Example 3. Taking Fig. Ia as an example, denote W as the sparsest 1-order CICA solution of X = [X1.Xy Xs), P is the permutation matrix that makes PW whose diagonal elements have non-zero values, simultaneously. Let PWX. As X, is the variable that has the highest causal order in Ch( T.), then we can take the value of Zn as the surrogate L..
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Equipped with the surrogate selection strategy provided by Lemma 10, the identifiability guarantee of Theorem can be naturally generalized to a whole hierarchical structure.
Theorem 2. If Condition holds, then the underlying causal graph G is fully identifiable, including both the location of latent variables L and the recovery of causal relationships E.
Based on the identifiability guarantee, we develop an estimation algorithm named CICA-LiNGAM to recover the latent causal structure from the CICA solution. Suppose that some observed variables form a causal cluster, we can determine the value of Pmmi using the GIN condition. Here we say that an observed variable set S is causal cluster if the variables in partially share the same latent parents that satisfy S Ch(LPa(S)) or LPa(S) d-separates S and Ch(LPa(S)) S. The causal cluster serves as a basic unit that helps us quickly locate the latent variables. The following lemma states basic criterion for identifying causal clusters from active variables A (active variables contain some variables that may form causal clusters the bottom-up recursive procedure).
Lemma 11 (Identifying Causal Clusters (Xie et al., 2022)). Let A be the active variable set and S A. Then S is causal cluster with LPPaSS) = Pmii (S) 1 if: for any subset S of Y with [S| = 2. (A S, follows the GIN condition, and 2) no proper subset of S satisfies 1).
Algorithm CICA-LINGAM
Require:
Observed variables X.
Ensure:
Fully identified causal structure G' on X and discovered latent variables. Initialize active variable set A and 0.
2: while A * ( do
3: Identify causal clusters in the current active variable set A (Lemma 11).
4: Obtain the sparsest CICA solution W of each cluster (Lemma 4 or 5).
5: Find a permutation matrix P to make the diagonal elements of PW non-zero (Lemma 7),
6: Obtain causal structure within causal cluster (Theorem 1).
7: Merge clusters share the common latent parent (Proposition in Appendix B).
8: Determine whether new latent variables should be introduced (Corollary 2 in Appendix B).
9: Update the active variable set A according to Lemma 10.
10:
end while
11:
Return G.
The algorithm adopts a recursive procedure (see Algorithm 1 for pseudo code). In each iteration, performs four steps: i) identify causal clusters (line 3); infer the causal structure within each cluster based on the sparsest CICA solution (lines 4~6)); iil) merge the clusters share the common latent parent and determine how many new latent variables are required in the current iteration (lines 7~88, details see Appendix B); and iv) update the active variable set accordingly (line 9).
3.5 CONNECTION WITH INDEPENDENT SUBSPACE ANALYSIS
Local ISA-LiNG (Dai et al., 2024) leverages independent subspace analysis (ISA) instead of OICA for local causal discovery. Inspired by this, we then ask whether ISA remains suitable surrogate of OICA in the presence of latent confounders and what the relationship is between CICA and ISA. To answer these questions, first review the basic terminology of ISA.
Definition 7 (Irreducible). An m-dim random vector Z is irreducible if it contains lower-dim independent components. In other words, no invertible matrix W E GL(m) can decompose WZ- (Z, ZZ) into Z ZZz-
Definition 8 (ISA solution (Theis, 2006)), For m-dim random vector X, an invertible matrix W called an independent subspace analysis (ISA) solution of Y if WX [Z]...., Zg ) consists of mutually independent, irreducible random vectors Z, The corresponding partition Tw of the indices [m] is called the ISA partition associated with W.
Although ISA seeks separation "as independent as possible", the following theorem shows that ISA is actually not informative enough in the presence of latent confounders.
Theorem (Interpretations of ISA in LiNGAM model). Let the graph obtained after removing all X in G be named by G', which form several connected components of observed the outgoing edges X
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variables X.c, C ".. .x',, where k be the number of connected components in g' For an ISA
solution W, let WX (Z) ......,,.... DT)T. Then there a permutation of []] s.t. for any E [k]] EW; E GL((C:)) makes (()) = c.
Example 4, Here we present a concrete example to aid in understanding Theorem 3. After re-
moving all outgoing edges of X in & (the graph in Fig. 4a), G' (the graph in Fig. 4b) form three
connected components of observed variables, {X1]. X1} and {X X.J. Then WX =
(Z) .2.. Zgz) 3., W1, W2 W3, s.t. (I) W Zz) W2 [2.4] (3) W x [355
X4
X5
X
X
X1
X3
X
X
X
(a)
(b)
Figure 4: An example to understand the procedure of ISA in the LiNGAM model.
According to Lemma any matrix W GL((3) is an ISA solution of Fig. 1a and 1b. Therefore, they are "ISA equivalent", which we summarize in the following remark.
Remark 2. The two causal graphs in Fig. la and Fig. lb cannot be identified by ISA.
The fundamental reason why ISA fails be informative in the presence of latent confounders is that, although it seeks components that are "as independent as possible", ISA does not impose constraints within each irreducible subspace. Consequently, regardless of how variables are connected within subspace, the corresponding graphs belong to the same equivalence class under ISA. In contrast, the absence of constraints within each subspace can be addressed by CICA. For example, the sparsest -orrder CICA solution on (X2, X1} makes the edge X2 + X, identifiable.
4 EXPERIMENTS
In this section, we first present simulation studies synthetic data to demonstrate that our algo- rithm effectively identifies causal structure. We also conduct a simulation study to quantify the gap between the sparsest and the alternative CICA solution to demonstrate the stability of CICA for practical use. Due to space limitations, real-world experiments are presented in Appendix C.
Case
Case
Case 3
Case
X3
X3
Figure 5: Causal structures used in synthetic experiments.
Dataset, Baselines, Metrics. We generate data from some typical graph structures that satisfy Condition (see Fig. 5). We consider different sample sizes = (5k, 20k). The causal strengths Bij are generated uniformly from 1-2, 0.55 U [0.5, 2]. and the non-Gaussian noise terms are generated from the square of exponential distributions. In each setting, the results are obtained after averaging the values in the 10 tests. We report both the average results and standard errors. We consider the following four methods as baselines for comparing: RLCD (Dong et al., 2023), PO-LiNGAM (Jin et al., 2023), LaHME (Xie al., 2024), CDHS (Li al., 2025). To evaluate the precision of the estimated graph, we used the following three metrics as (Li & Liu, 2025). 1) Error in Latent Variables: the absolute difference between the estimated number of latent variables and the ground-truth one; 2) Correct Ordering Rate: the number of correctly estimated causal orderings divided by that of ground-truth causal orderings; 3) Fl score of causal edges.
Results and Analysis. The experimental results are summarized in Table 1. For CDHS, the algo- rithm fails in the fully impure setting as its "Homologous Surrogates" condition (Li et al., 2025) is violated. preventing any valid output. While LaHME and PO-LiNGAM key evaluation metrics, they are unable to produce correct results in fully impure scenarios (c.g., cases 2 and 4) because their clustering step fails. RLCD is inapplicable to cases 1 and 2, as its
relatively stable on

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Table 1: Comparison on synthetic data. 1 means higher is better while
means lower is better.
Grapb
Method
Error in Latent Variables
ConrectOrdeerin Rate
FI-Score
5K
10k
20k
SK
Tdk
20K
5K
TOK
20K
CDHS
0.3080.46
0,2000.40
0.4020.49
0.65:+0.45
0.80=+0.40
0.60#0,49
0.6720.45
0.50=0.40
0.0020.49
LaHME
0,00+0.00
0.10+0.30
0.00+0.00
0.50+0.00
0.45+0.15
0.50=0.00
0.67+0.00
0.60+0.20
0.67=0.00
Casc
PO-LINGAM
0.00+0.00
0.00+0.00
0.00+0.00
0.50+0.00
0.50+0.00
0.50=0.00
0.66+0.03
0.67=0.00
0.67=0.00
RLCD
1,0000000
1.0+0.00
1.00+0.00
0.00+0.00
0.00+0000
0.00+0.00
0.00+0.00
0.00+0.00
0.00=0.00
Ours
0.0050.00
0.0010.00
0.00:0.00
0.65+0.25
0.6010.35
0.75#0.25
0.75+0.25
0.67=0.38
0.77=0.46
CDHS
1.0020.00
1,0010.00
1.0010.00
0.00+0.00
0.0010.00
0.0010.00
0,0010.00
0.0020.00
0.0020.00
LaHMME
1.00+0.00
1.0+0.00
1.000.000
0.00+0.00
0.00+0.00
0.00=0.00
0.00+0.00
0.00=0.00
0.00=0.00
PO-LINGAM
1.0000.00
1.0040.00
L0050.00
0.00+0.00
0.00+0.00
0.00:0.00
0.00+0.00
0.00
0.00
RLCD
1.00+0.00
1.000.000
1.0+00000
0.00+0.00
0.00+0000
0.00+0.00
0.00+0.00
0.00+0.00
0.00=0.00
Ours
0.00:0.00
0.00:0.00
0.00:0.00
0.60:0.25
0.60:9.25
0.66:4..27
0.67:0.44
0.6720.44
0.72s0..8
CDHS
2,0000.00
190:10.0
200000.00
0.0020000
0.0000.000
0000000000
0.0000.00
0.00200000
0.0000000
LaHME
0,00+0.00
0,20+0.60
0.10+0.30
0.44+0.00
0.40+0.13
0.40=0.13
0.73+0.00
D 0.6500.22 22
0.65
Case
PO-LINGAM
0.0020.00
0.2010.60
0.4420.00
0.4420.00
0.40
0.7320.00 0.73
0.6520.22
RLCD
0.10+0.30
0.10##.30
0.00=0.00
0.60
0.60+0.25
0.58:00116
0.70=0.24
0.70 24
0.7320.08
Ours
0.00=0.00
0 0:2040.00
0.1040.00
0.66+0.118
0.61=0.31
0.61=0.31
0.78+0.31
0.72+0 35
0.78+0.31
CDTS
2.0080.00
20000.000
2.00+0.00
0.0000.00
0.00=00000
0:00=00000
0.00=0.00
0.00=0000
0.00=0.00
LaHME
0. 0,25+0.54
0.20+0.4
D. 0.1040.44
0,30+0.15
0.30+0.15
0.36+0008 08
0.56+0..28
0. 0.66+0.. 28
0.67=0.1 15
PO-LINGAM
2.00+0.00
2.00+0.00
2.00+0.00
0.00+0.00
0.00+0.00
0.00=0.00
0.00+0.00
0.00=0.00
0.00+0.00
RLCD
0.50&0.81
1.10
0.70g0.90
0.28
0.1120.17
0.20
0.13:0.20 0.30a0.22
0.23g0.20
Ours
0.25+0.54
0.20+0.40
0.10=0.44
0.52+0.27
0.52+0.27
0.68+0.39
0.68+0.43
0.68=0.43
0.74:0.40
underlying rank test requires at least four observed variables; it also struggled to resolve the causal
structure in the remaining scenarios. contrast, our proposed algorithm demonstrated optimal per- formance across all cases. It consistently identified and characterized the impure connections among the observed variables, showcasing its advantages in handling impure structures.
Robustness Verification. Besides, we quantify the gap between the sparsest and the alternative CICA solution with a simulation study. We analyze all possible causal graphs with latent variable and observed variables (X1 ...,.... X,), for n E 4,,4,5, 5, 6). To make the enumeration feasible, assume causal order X₁ + randomizing all possible causal edges between the observed variables. For each of these possible graphs, we compute the number of non-zero entries for all n + 1 possible valid CICA solutions.
Xn, and then enumerate all 2n possible graphs by
n
#graphs
Sumg (True)
Sum;
Suma
Sum3
Sum4
Sum
Sum
3
8
36
43
50
59
4
64
448
531
602
682
809
5
1024
10240
12015
13368
14857
16802
19758
6
32768
442368
513675
563799
619008
685666
769179
895521
Table Simulation results of the sparsity gap.
Table 2 collects the total sum of non-zero entries for each of the n + 1 solutions (sorted from sparsest to densest) aggregated across all enumerated graphs, which provides clear evidence for the gap between the sparsest solution and the alternative solution. For example, when n = 6, the average number of non-zero entries in I Bx.x is 13.5 (Sumg/#graphss, and it increases 176 ((Sumy-Sump \4ggaphss 16. 12% relative increase rate) per graph compared with the second sparsest CICA solution. This confirms that the sparsity gap is a generic property, and provides practical stability and robustness guarantees for CICA in finding the true causal graph.
5 CONCLUSION
In this paper, we introduce a new principle, Conditional Independent Component Analysis (CICA), which aims to extract components that are mutually independent given a certain number of latent variables. CICA naturally induces two-sided projections wi Y ".w Z, which carry richer identifi- cation signal than one-sided projections w T Y 11 Z used in GIN/TIN conditions, thus improving the identifiability in latent causal structure learning. Although CICA involves additional indeterminacy on the latent conditional set, we show that sparsity resolves this ambiguity and yields full identifi- cation of the latent variables and causal relationships. Building on our theoretical results, we derive an estimation algorithm for latent causal structure recovery. Synthetic and real-world experiments show the superiority of our methods in dealing with impure structures. One of the future research directions is extending the linear mixing assumption to the non-linear setting, existing techniques, e.g., Chen al. (2024b; 2025b), may help to mitigate this issue.
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ACKNOWLEDGMENTS
The authors thank anonymous reviewers for their helpful comments. We also acknowledge the support from National Natural Science Foundation of China (NSFC) under grant No. 62372116, 62476163, U24A20233, National Key R&D Program of China under grant No. 2025YFC3410003, NSF Award No. 2229881, Guangdong Basic and Applied Basic Research Foundation under grant number
2023B 1 515 120020, AI Institute for Societal Decision Making (AI- SDM). the National Institutes of Health (NIH) under Contract RO1HL159805. and grants from Quris AI, Florin Court Capital, MBZUAI-WE WIS Joint Program, and the Al Deira Causal Education project.
7M would like to acknowledge the support by STU Scientific Research Initiation Grant under grant No. NTF25024T. The computations in this research were mainly performed using the CFFF platform of Fudan University.
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ORGANIZATION OF APPENDICES
Definitions, Examples, and Proofs
16
A.1
Definitions
16
A.2
Examples
16
A.3 Proof
16
A.3.1 Preliminaries
16
A.3.2
Illustration of Non-identinee Non-identifiability Issue on Fig. 1a and 1b
19
A.3.3
Proof Lemma
19
A.3.4
Proof of Remark
20
A.3.5
Proof of Lemma 2
20
A.3.6
Proof of Lemma 3
A.3.7
Proof of Lemma 4
21
A.3.8
Proof of Lemma 5
22
A.3.9 Proof of Lemma 6
A.3.10 Proof of Lemma 7
23
A.3.11 Proof of Lemma 8
23
A.3.12 Proof of Lemma
26
A.3.13 Proof of Theorem
26
A.3.14 Proof of Lemma 10.
26
A.3.15 Proof of Theorem
27
A.3.16 Proof of Theorem 3
27
A.3.17 Proof of Remark 2
28
Illustrations of Algorithms
28
B.1 Merging Rules
28
B.2 Pseudo Code
29
B.3 Discussion of Optimization Criterion of CICA
30
Additional Information on Experiments
31
C.1 Computing Infrastructure
C.2 Implementation Details
31
C.3 Real-world Experiments
31
C.3.1 Teacher's Burnout Study
C.3.2 Big Five personality
32
Related Work
34
E The Use of Large Language Models (LLMss)
35
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A DEFINITIONS, EXAMPLES, AND PROOFS
A.1 DEFINITIONS
Definition 9 (Treks (Sullivant et al., 2010)). In G, a trek from X Y is an ordered pair of directed paths (P1, P2) where Pi has a sink X, P2 has a sink Y, and both P1 and P2 have the same source Z. Definition 10 (T-separation (Sullivant et al., 2010)). Let A. B, CA. and CB be four subsets of V in graph G (not necessarilly disjoint). (Ca, CB) t-separates A from B if for every trek (P1, P2) from vertex in A to vertex in B, either P1 contains vertex in CA or P2 contains vertex in CB.
Lemma 12 (Rank and T-separation (Sullivant al., 2010)). Given two sets of variables A and B from a linear model with graph G. we have rank (EA.B) = min {|CA| + |Bll : (Ca, CB) t-
separates A from B in 9}, where EA,B is the cross-covariance over A and B.
A.2 EXAMPLES
0
0
0
0
11
1
(av +b//a c/a
0
0
1
UU 1
W1 1
W2
X
Figure 6: An example of different CICA solutions for X. WN is a CICA solution that renders X' conditionally independent given L, while W2 renders X' conditionally independent given E1, the exogenous noise of X1. The gray/white rectangle denotes non non-zero/zero entries.
A.3 PROOF
A.3.1 PRELIMINARIES
Lemma 13 -SitovicS Theorem (Darmois, 1953; Skitovich, 1954)). Given n independent scalar random variables XI...., X., that are not necessarily identically distributed. Consider two linear statistics L1- = Σ a, L2 = Σ B;X,, where ag, 3r are constant coefficients. L1 and L2 are independent if and only if the random variables f which a, F ffoloo a normal distribution. Lemma 14 (Graphical implication of TIN (Dai et al., 2022)). Let Z, Y be two subsets of variables, we have:
TIN(Z, Y)= = [|S] S is a vertex cut from Ane(Z) to Y).
In a linear non- non-Gaussian system, the Darmois-Skitovich theorem (Darmois, 1953) plays a key role in determining the independence of two linear statisties. It tells that two linear combinations of independent non- Gaus-Gassin variables are independent they do not share any non-Gaussian com- ponent. As w X is linear combination of independent noises of V, characterizing all possible independence that can be constructed from observational data requires understanding which noise combinations can be represented by w T X. To this end, we introduce a new definition that describes the noise combinations attainable through linear combinations of observed variables.
Definition 11 (Constructible Noise Combination). A noise combination Z C E, which consists of some independent noises of variables in V. The noise combination Z is constructible by some observed variables X if there exists a coefficient vector w such that w T " is a linear combination of the noise variables in Z with non-zero coefficients, i.e., w X Ez + 0). In other words, w X contains and only contains noise variables in Z.
Example 5. In the figure below, L is the latent confounder of two observed variables X and Y. We have 0. {FL. Fxx), ) (FI. Fxy } Fxx Fy) and (F, Fxx, Fxy) are constructible while the other noise combinations are not.
Definition 12 (Bottleneck). Let J, K. and B be three subsets of V that are not necessarily disjoint. We say that is bottleneck from J to K if. for every j E and every l E K. each directed path from j to k includes some b E B.
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E
x
1
0
Ex
v
0.3 +
B 1
Ey
0
E1
[xx]
Ey
(EL, Ex
{EL, Ey}
{Ex, Ey
E,, Ex Ey
X
X
x
Table 3: All constructive noise combinations of the graph above.
Definition (Latest Minimal bottleneck (LM bottleneck)). Let J. K and B be three subsets of that not necessarily disjoint. We say that bottleneck B from J to K called minimal if every bottleneck B' from J to K has B'| 2 [B.. Furthermore, B is the (topologically) latest minimal bottleneck (LM bottleneck) from J K if for every minimal bottleneck B' from J K. B is the bottleneck from B' to K.
X2
LA
X1
Example In the figure above, {L3} is a minimal bottleneck from {L,, L2} to X. More precisely, it is also the corresponding LM bottleneck. {La, Ls} is minimal bottleneck from {L1, Lo} to X but it is not the corresponding LM bottleneck. Instead, it should be {La, Ls).
Definition 14. We define the LM cottleneck-dominatee set of B with respect to K as the set of all nodes in V such that B is the LM bottleneck from the node to K. Formally,
DB, - == {v VBB is the LM bottleneck from v to K)
(4)
This is the maximal set of nodes for which B serves as bottleneck toward K. Lemma 15. Let J, K C V that are not necessarily disjoint. The LM bottleneck from J to K always exists and is unique.
Proof. Build the standard vertex-splitting network G' (V', E') with capacities as follows. For each v E V, create two nodes vー v= and add unit-capacity edge v' - vt. For each - E E, add an infinite-capacity edge =t - v-. Add source a and sink 1; for each j E add an infinite-capacity edge +j and for each k K add infinite infinite-cpaccity capacity edge k+ t.
Then for any C V, B is bottleneck from J to K + C(B) = {v - ut u E B) is s-t cut in G". Moreover, the capacity of CBB) equals [B). Indeed, every path k+ in G' that necessarily traverses the split edge x + x for each visited x; cutting precisely the split edges in C(B) blocks all lifted s-t paths iff every ~s path in G meets B. Since only split edges have finite capacity, the cut capacity is [B].
k in G
lifts to a path s ~+ j ~4
Therefore, minimal bottleneck (of smallest cardinality) exists because it corresponds to minimum 8-f cut the finite network G'.
Let be any maximum flow G' and let Ry be the residual network. Define
Ty = (I V can reach in Ry }.
Sy:=
Standard max-flow theory implies that (S,,Ty) is minimum s-t cut, and that T'y is inclusion- maximal among the sink sides of all minimum cuts (the "closest-to-f" minimum cut); in particular, Ty is unique. For completeness: if (S', T²) is any minimum cut, then edges from T" to S' carry residual capacity and edges from S' to T are saturated; hence every node reachable from in R must lie in T', so T' C T
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Map the t-closest minimum cut back to vertex set:
B == {oEV e- E and at ETy)-
By construction, C(B*) the cut (S), henee B is a minimal bottleneck.
Let B' be any other minimal bottleneck, and let (5', (") be its corresponding minimum cut in G'. From the previous paragraph T' C Ty (equivalently S C S'). Take any path b' ~ k in G with V' E B' and l E K; its lift in G' goes from 1/- E S 2 S to A+ E Tr hence must cross the cut (S;:7) through some split edge r > e+ with e E B Therefore every V/ ~ k path passes through B" i... B is a bottleneck from B' to K. Since B' was an arbitrary minimal bottleneck, B" is the latest minimal (LM) bottleneck.
If B is another LM bottleneck with minimum cut (S, (), then by the same argument its sink side T must contain the sink side of every minimum cut, hence T = T'y by the maximality/uniquee TG. Thus B B In summary, the LM bottleneck from J to K exists and is unique.
of
Lemma 16. A variable set Vo S V is an LM bottleneck from some variable set Vs to X if and only if V itself is the LM bottleneckfrow Vn to X.
Proof. If V6 is the LM bottleneck from some V, X, then V& is the LM bottleneck from Vo to X. Since V,, is a bottleneck from V, to X, every V, ~o X path meets Vo. Consequently Vo is trivially a bottleneck from V to X (every E Vi-X path contains E Va at its first node).
We show that V, is minimal for the pair (Vo, X). Assume, for contradiction, that there exists a bottleneck C from V to X with |C| < Vol. Then for any s B V, and x E X. each s path first hits V2 and, from that hit, must pass C (because C meets every V5 ~+ X path). Hence C is also a bottleneck from V, to X, contradicting the minimality of VB for (V,,X)
It remains to verify the latest property for X). Let C be any minimal bottleneck from V to X. We claim that every C- X path meets Vo. Indeed, otherwise there would exist c C and X with a path c z avoiding V,, Concatenate a path - with s E V. whose internal nodes avoid V, (which exists because Va is minimal for (V., v, X); otherwise e would redundant in C), and then follow the e~ x path; this would give an V X path avoiding V,, contradicting that V, is bottleneck from V, to X. Thus Vo is a bottleneck from C to X; since C was arbitrary minimal for (V,,X), X), Vo is the LM bottleneck from Vo to X.
Vf, is the LM bottleneck from V, to X, then V6 is the LM bottleneck from some V, to X. Take V, , : V.. By assumption, V, is (latest) minimal bottleneck for (V,, b, X); in particular it is bottleneck from V. to X and, for every minimal bottleneck C from V. to X, every C ~ X path meets Vb. Hence V, is the LM bottleneck from V. X.
Theorem 4 (Graphical criteria of the constructible noise combination). Any noise combination constructible by X if and only if (i) Tт C V s.t. T is the LM bottleneck of T to X in G. (ii) VV; E V, a, = > T is bottleneck from V: to X in G.
Proof. Constructibility = (i)-(ii, Assume a is constructible, let S = {i V : a, # 0} be the support of a. By Lemma 15, the LM bottleneck T from S to X exists and is unique; by Lemma 16, T is also the LM bottleneck from T to X. This gives (i).
remains show (ii). i E V. If a; O. then T* is a bottleneck from V; X. Suppose the contrary that there exists a directed path P i ~4 x avoiding T* (with æ E X). Since T is the LM bottleneck from X, it is, by definition, the bottleneck from every minimal bottleneck for (S, X) to X; in particular, P can be concatenated with an S ~ i path that avoids T up to i (otherwise i would be separated from S by T and as would inherit nonzero contribution through i's first hit in T*). Consequently there exists at least directed path from S to that avoids T and can be continued by P. contradicting that T intercepts all S ~* X paths. Hence every i ~ path hits T ie., T is bottleneck from Vi to X.
If a4 + 0, then T is not a bottleneck from V3 to X. If every i x path met T then any w whose latent terms have been canceled via constraints indexed by T would give v; 0 (all contributions must pass through T* and are nullified), contradicting a; * Thus has a path to some x e X that avoids T
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Combining the two implications yields (ii) with T = T
(+) (i) (ii = constructibility. Assume (i) hold for some T S V. Let
S {i e V a #0 fieV:
T not a bottleneck from V; to X }.
By (i) and Lemma 16, T is the LM bottleneck from T to X and, therefore, from S to X as well (latest with respect to any minimal bottleneck for (S, X)).
Consider the vertex-splittnng network G' used in Lemma 15. Let (Ty be the unique t-closest minimum cut in G' (induced by any maximum flow); it induces T by T = {u E V v E Sy, e* E T. Choose [ distinct nodes {I ................................ [ S X reached by the [T] vertex disjoit paths guaranteed by Menger's theorem from T to X (tightness of the cut). Define supported on (zi,...,,Ip } as the unique solution to the linear system that zeroes the contributions flowing through T (the (T) x [T] system is non-singular because the T ~~ {xr) paths are vertex-disjoint). Then 1) for any such that T is a bottleneck from V; to X, every i ~ X path must traverse some t e T, hence its contribution to v4 is canceled by construction; thus v, = 0. 2) for any such that T is not a bottleneck from V; to X. there exists path P : ~e that avoids T. Since our constraints only cancel flows that pass through T, the term corresponding to P survives so v4 + O.
Finally, impose additional linear constraints (orthogonality) w to remove latent terms (these con- straints are independent of the T-cancellation because the latter acts only on flows that cross T), only eliminate TT directions associated with the cut while retaining
which is always possible degrees of freedom on X. Thus a is constructible by X.
Corollary 1. Any noise combination a is constructible by X X if and only if TT C V s.s. T is the LM bottleneck of T to X in G. (ii) VV; E V, a; = + T is a bottleneck from V; to X in G.
A.3.2 ILLUSTRATION OF NON-IDENTIHABILITY ISSUE ON FIG. IA AND 1B
{
{X2]
{X3}
{X1,
X2)
{X1,
X32,
{X1, X2, X3}
Z
X1
1
2
2
2
(X2)
1
2
2
2
{X3)
1
2
2
2
Table 4: TIN value of different Y and Z of Fig. la
Y
}
(X2)
(X3)
{X1, X2)
{X1, Xz}
(X2, X3}
(X1,X2, X3}
Z
1
2
2
2
2
(2
1
2
2
2
X3)
1
1
2
2
2
Table 5: TIN value of different Y and Z of Fig. 1b
Proof. We use G1 and G2 to represent the causal graph in Fig. 1a, and Fig. 1b, respectively. By some simple calculations, we can find that both G1 and G2 have no rank-deficiency constraints. Thus, for each pair of (Z, Y), rankg, (Ez,y) Table 4 and 5, 9, and G2 have the same TIN value for each (Z, Y). As GIN(Z, Y) is satisfied if and only if TZ Y) = rank(2z y) < Y (Dai et al., 2022), whether the GIN condition is satisfied for a certain pair (Z, Y) keeps the same in G1 and G2.
min(Z, Y) = rankg, (Ez,y). In addition, we can see in
A.3.3 PROOF OF LEMMA
Proof. NS(Z) Anc(Z), By Theorem 4, Anc(Z) is constructible. Therefore, according to the definition of constructible noise combination, we can always find a non-zero coefficient W2 such that NS Z)= Ane(Z). Since wil Y II Z, naturally obtain w| Y Il u1 Z.
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A.3.4 PROOF OF REMARK
Proof. By Theorem 4, can enumerate all constructive noise combinations by finding all LM bottlenecks. All LM bottlenecks can be identified by testing for Lemma 16. All constructive noise combinations by {2), {X,, X3) and (X2, X3} in Fig. la are shown Tab. 6. All constructive noise combinations by {_1 X2), {X1. N3) and {X2, X3 in Fig. 1b are shown in Tab. 7.
Z (21,
0 EL F1, E2
{x1}
{x2]
F1,, F2
FL F2
[E Ei
(X1, X3)
{
{
{x3}
{Er, Ej, B2, Ezj
Ei, E2, Ea}
{E E2, Ez}
EE Ei, Ez}
Z = [X2, X3
{
(22)
[xat
E E1. E2, E3
En, Ez, Es}
E E3
LE. Ei. E2}
Table 6: All constructive noise combinations by (XI,,X2); {-1, and {33 in Fig. la. Each constructive noise combination is shown together with its corresponding LM bottleneck in pairwise manner.
X2)
[T.}
{X1]
{X2]
[E Eg. E2, Eg)
F1 Eg, Fej
F2 Ez, Eig)
F2 E1.
Eg}
Z = {3)
{
{x
[xat
Er, Ei, Ez}
Ei, Ea}
Ei, Ea}
Er, Es
Z = [x2, Xa}
{
{x2)
{Xa}
EL, E1, E2, E3
E1. E2, E3}
EE. E1. Es}
EE E2
Table 7: All constructive noise combinations by {X1, X2), (X1, N3} and (X2,Xy, in Fig. 1b. Each constructive noise combination is shown together with its corresponding LM bottleneck in pairwise manner.
construct NS(w 2) (F,, F2) and NS(w; Y) = [E1, E3} with non-zero W1 E R2. In contrast, in Tab. 7, each pair of constructive noise combinations by Z and Y has shared noise components, thus cannot be independent. The conclusion for Z = {X1,Xs X3} and Y = (X2, X3} can be analyzed similarly.
X2); From Tab. 6, when Z X2),
(X2, X3),
A.3.5 PROOF OF LEMMA 2
Lemma 2 (Identifiability and indeterminacy of CICA). Suppose Assumption 1 holds, let X be m observed variables. there exist Pmin (X) latent variables L such that for two valid CICA solutions Wi and W2, the components of (k) == W are mutually conditionally independent given L for k (1,2), then there uniquely exist a permutation matrix P., (for some permutation π of [ml) and non-singular diagonal matrix D such that W = P.DW,.
Proof. Under Assumption there exist an invertible E mm a latent vector L E RP, matrix M E gmxp, and noise E (Ei,.... Em)) with mutually independent coordinates. E.lL L, finite non-zero variances, and with at most one Gaussian coordinate, such that X AS and S ML+ E. For (1,2} write Z W.X and set B W (hence Z (b) B,
For every
[L [L= B +E) ( + B.E.
Thus, for each k the coordinates of (k) are mutually independent given L if and only if the coordi- nates of E, are mutually independent deterministic shift M& does not affect independence). In particular, we have an ICA model with independent sources E and mixing matrices B
(+) Necessity. Assume (i) holds: there exists a specific latent vector such that (*)) has mutu- ally independent coordinates conditional on L for 1 1,2. By the reduction above, both B, E and
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ByE have mutually independent coordinates. Since E has mutually independent coordinates with most one Gaussian, the standard ICA identifiability theory (Comon, 1994) tells that the only invert- ible linear maps sending E to vector with independent coordinates are generalized permutation matrices. That is, the invertible matrices B₁ and B2 must both be of the form B = P, and B2 22D2, where PA: permutation matrices and D non-singular diagonal matrices. Con- sequently, we can express B2 in terms of B1 as B2 = P2D2(P
"B, = P2D2D, P Bj. follows that Pa is permutation matrix, D is a non- singular diagonal matrix, and Bg P DB1 Multiplying on the right by A (recall B = Wx yields We 1.1.
TPT Let P = P2P1 and D D,D2D, PT.
(+) Sufficiency. Assume (ii) holds: W2 D, with Px permutation matrix and D non-singular diagonal matrix. Let L be any latent vector for which W1 is p-order CICA solution (which exists by assumption that W1 a CICA solution). Then for every given condition L =
2 - f] W {L E} P, [L= {} P, DZ
{ = 2}. ()=
Since permutation and nonzero per-coordinate scaling preserve mutual independence of coordinates, the coordinates of 2 are mutually independent given whenever those of Z are. Hence (i) holds. Therefore, the two statements equivalent.
A.3.6 PROOF OF LEMMA 3
Lemma 3. Let X be m observed variables, and W be a p-order CICA solution of X. Let X WWX,Y Y and are two subsets of X', then if max ([Y), [Z]] > p. w Y ww has a non- zero solution (w1,W2) for (Y', ZZ), where Y' [Xa] E EXLY WE, # 0}. Z' are defined similarly.
Proof. Y WY. == Wz, Z' Y and Z are conditional independent given p latent variables. Since max{ YY) | > p. without losing generality, we assume Y > p. Then we can find non-zero W1 that w1 Y Y_ Z. By Lemma there exist a non-zero W2 that makes w Y Ilwa Z. Thus,
wi Wy, Y' W. W Wz.Z''
A.3.7 PROOF OF LEMMA
Lemma 4. For an observed variable set X with X = m. denote p = Pmin (X). Suppose m
2p + 2, and set X' == WX, then W is a p-order CICA solution of X if and only if for every pair of disjoint coordinate subsets X1.X2 of X' with X1 X2| = + 1, det (et(Ex,,x, x2) = 0, where = Cov(X') denotes the covariance matrix on X' and Σx, X2 is the (p+ 1) x (p 1) sub-matrix of E with rows indexed by X1 and columns by X2.
Proof. (=) Necessity. If W is p-order CICA solution, there exist a p- p-dimensional latent vector L and independent noises E (E1...., Em) such that Z, = a. L + E, .,..,, m. Hence,
Σ = AΣLA (diag( Var(E.)).
mankSpp
diagonal
For disjoint X,, X2 the diagonal term vanishes, so Σx,x, Ax, ELAX, has rank at most p. Therefore det(('x,.)) xx) = O for every such pair.
(+) Sufficiency. Assume that for every disjoint X,, X2 of size p 1, rank(Ex,,xx S p (equiv- alently, all (m-noss vanish). By the trek separation theorem, for each such pair there exists t-separating pair (L1, L2) with [L,] + L2 S p that t-separates X, from X2. Since p = Pmiu (X), no separator of size < works uniformly: hence the minimum size is exactly for all these pairs.
Consider some (XA: XB) with Xa 0 X
and both | and X&l equals p + let (L1, L2) We claim that (L, L2) t-separates every
be a minimal f-separator for this pair, so |L, + L2 other disjoint (Xc, Xp) with - p + and is minimal for that pair as well. Suppose (L,, L2) does not (-separate X from Xp. Then there exists a trek from some E Xc to d E X avoiding while | = [ = + 1, Menger's 's theorem for treks imply that there p vertex- vertex-disjoint treks connecting X {a} to X \ {b) for some
L1 U L2. Because [L + L2
XA. b X all avoiding L, U Lg. Together with the trek e ~ d (also avoiding L U L2) we obtain p+ vertex- disjoint treks between the modified sets X', (X, (a)) (c) and X = (Xn (b)) U (d).
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hence rank( .x²) 2 p + 1. contradicting our hypothesis. Thus (L1, L2) t-separates every such pair. If for some (X X there were smaller separator (I₃ L2) with [L]| LL < p. then rank(Xx;;Xo) S [[] + [L < p. again contradicting the assumption that all those ranks equal p by minimality of Pmin (X). Hence the same (L,, L2) is minimal E-separator (of size p) for every such pair. We fix this global separator (L,, L2).
Let i A j be nodes outside L1 U L2. If there existed a trek from to avoiding L, L2, then by the same replacement argument as above we could build p+ vertex-disjoint treks between some disjoint (pubsssubsetss, forcing cross-rank p+ 1, which a contradiction. Therefore, every trek from i to meets L U L2, i.c. all covariance between distinct observed coordinates flows through L2). Equivalently, Σ admits decomposition diagonal D collecting variances.
AEL T +
with rank(AALAT)) S p and
Under Assumption I, noises are mutually independent and independent of the latents. The di- agonal D found in Step 2 implies that each observed coordinate has unique private noise, and distinct coordinates share no private noise. Therefore, for some p-vector L, Z; = a L + E, = (E1,..., Fm) mutually independent, E L. Thus, the coordinates of Z are mutually independent given L, i.c., W a p-order CICA solution.
Combining both directions proves the equivalence.
A.3.8 PROOF OF LEMMA 5
(X) and m 2 3 hold, set X A WX, then the invertible matrix W is l-onder CICA solution of X if and only for every ordered triple (X;,X) x1) of X', Xy) and X satisfies the Triad constraint.
Lemma 5. For an observed variable set X with [X| = suppose that Pmin
Proof. (=) Necessity. If W is 1-order CICA solution of X. then there exists some latent L that have X + + FA with F- (F1,... F. mutually independent and F 11 L. For i $ i k Cov(X), X() = my Var(L) and hence the pseudo-residual of {y, relative to X₁ as
E(g)y == Cov(X,, X - Cov(X;, X
mm Var(L) (L) (m - mx, m. Var (L) (m, m,E,), which depends only on (Fe, Fry) and is independent of X = m. + Fs.. Thus the Triad constraint holds for all distinct triples.
(+=) Sufficiency. Assume the Triad constraint holds for every distinct (i.j,
Suppose, for the sake of contradiction, that there exist two variables X and X that share least two independent components. Since the Triad constraint E(s,jkk) 1l. X holds for any j $ {i, k). X must also contain the components shared by X and X Consequently, every pair of variables in X' shares least two identical independent components. Let us denote two of these shared components as L1 and L2. Each X can then be expressed as X m,L1 + L2 + É,, where É, L1, and L are mutually independent (note that the residual terms E, may be dependent across different variables). Varying the triplet (i,j, k), we have:
E(s.jaa= == Cov(X", X X Cov(X,, X"j
(m Var Var(L, + ū. Var L2)) (m, + n. + E,) n, (mans Var(T.) म Var (2)) i To2
FFg)
(Aym; n, ) Lar(L2)) (m,,. -, m. Va ....
The independence condition F((.)1R) I1. X establishes that the coefficients of Ta and T.2 must vanish, yielding i.m, n,m, 0 for any pair (i,j). This implies that the vectors (m, ū.) are collinear, meaning Ar äg
Am (assuming non-zero denominators). This directly contradicts the rank faithfulness assumption. Therefore, any two variables X and X share at most one component.
th 182
Next, suppose that two variables X and X are independent. As E(g,jK) IL X holds for any j & ki, it follows that X3 11. X(. By extension, the variables in X' are pairwise independent.
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According the definition of CICA solution, this implies Pmin (X) which contradicts the prerequisite Puii (X) = 1. Consequently, any pair of variables in X' must be dependent.
From the above, any pair of variables must be dependent but can share at most component; thus, they must share exactly one independent component. Let us denote the single independent component shared by X and X as L. Following the previous logic, all other variables must also contain L. Because any pair of variables in X' is dependent, we can write X; = m, + Fs, where E = (Ei..... Em) are mutually independent and E IT L. Therefore, by definition, W is 1-order CICA solution for X.
Combining both directions proves the claim.
A.3.9 PROOF OF LEMMA 6
Lemma 6. I - Bx.x is a pmin (X)-order CICA solution X with latent conditional set LPa(X).
Bx.x. the proof of Theorem 3, we prove that x deletes all the outgoing edges from X graphically. Therefore, xx.xX is conditional in- dependent given LPa(X). Given Condition I holds, Pmin (X) LPa(X)] Thus, I Bx,x x is Pmin (o)-r CICA solution of with latent conditional set LPa.X).
Proof. In the setting our paper, Ax,x
A.3.10 PROOF OF LEMMA 7
Lemma 7. Suppose W is Pmii -ororr CICA solution of X whose latent conditional set is LPa(X), then W - Bx.x and we can uniquely recover the true causal matrix among observed variables Bx.x from W.
Proof. By Lemma 2 and Lemma 6, find a permutation matrix P and non-singular diagonal matrix that makes W PD(I Bx.x). Subsequent proofs can be analogized to Lemma in (Shimizu al., 2006).
A.3.11 PROOF OF LEMMA
Lemma 8. I Bx.x mig Wilo
W a Pmin Pmin (X)er CICA solution of X).
Proof. For a vector R" denote
X
X T AE T( -IIBxxx)--B where a E Rm++d
Since Axxx (I - Bxxx)= is non-singular matrix, let us denote the row indices corresponding to X as a* for convenience. We then have A = ax which yields
w = A xx (xxxx T Tax x==aa ax Bxxxx
Defining T similarly, we obtain
at A = (I- B
In summary, we can represent w and at as linear combinations of x
L
A L.X
L
AL A T x.
x
(5)
x
A x.x
A xx
Here, since focus mainly on the sparsity of W (i.c.. ||W]))) rather than its specific values, we use O to represent value of O, and x to represent a nonzero value, following (Ghassami et al., 2020).
For W
x denoting its corresponding noise coefficients of observed variables as ax
[oxx11, ax,2 ... 0x,m), we have ax.; -0.-... X, O, ,,. and an, = " x]
times
(m times
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Any other feasible solution, aside from A x. corresponds to choosing d variables from the d m independent noises.
W is I I-order CICA solution whose latent conditional set I.Pa(X) X), we can find permutation matrix P and non-singular diagonal matrix D such that W X Because and D do not alter the sparsity pattern of W, we can analyze W A X.X directly for convenience.
For the j-th row of W, there is exactly one in the j-th column of the coefficient matrix a (specifically at ), Because W acts as the unmixing matrix where W,:- = W3 we have
W (a.))) T( -I-Bxxx.
For any / E [m].
W Ja
Σ
a x
2a (T. -B,,) = =0ag - -By,).
(6)
X Case (i): If = (1, * 0), then Wij = JJ = x.
Case (ii): If X, E Pa(X)) (By, + 0), then W j.t
-a j, Bj = x.
Case (iii): X, does not fall into either of the two cases above, then Wj. In summary,
1I WE [m), W #0 (X))
Now, if W is a 1-order CICA solution whose latent conditional set not LPa(X), then X' conditionally independent given a latent variable other than L.. Without loss of generality, assume X' is conditionally independent given the exogenous noise of Xg, denoted Ex. Therefore, F2 NS(X)) for any j [m]] In other words, the column vector xx m has a in its k-th position. Furthermore, exactly one column x has
the j-th position for j E [m] {k} On the other hand, we have a = [0....00 0 X, 0. 0.....0 Essentially, this scenario exchanges the positions
.0
(k- 1i-mmes (m-4)-times
of Eg and Eg compared to W A X.X.
For example, Equation (8) presents an example of the matrix x when A:
X
1x
x
1000
0 X
x
X O
x
0
X
(8)
Based on this structure of x we can now check the sparsity of each row W.
For the k-th row of W, we have exactly one x in axx, located in the k-th row (i.e., X k. In addition,
a. Therefore, the support of a is exactly the same in the scenario W Ax
For the j-th row of W with + k, we have two x in a..j" located in the j-th and k-th rows (i.e., ax and a ) respectively. Since W,. = (a [[I- T(I
Bx,x), for any E [m].
Wi Σ
a
x (Li - BBa) + ax (Jr, (Ta, BR.)). X sy (I,,t - B.,)) = ju
(9)
Case (i): lf = (1i + 0, Ik.j 0), then
W i.j و
x (1ا5 -B,y) + akj
x (Te. BR.s) = X - -ox B
a
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On the other hand, we have the following.
a A B )xx)
A B )ays x
+ (L,, - - Dak
(AL.s AL Jais (AL. Ax,
=0
If W J.j - O, given that ox and ax. are non-zero, the following system of equations must have non-zero solution 21
axx j.g 22 ok.
21
B 0
(10)
(AL.) Ax (A AL Bx. x,,)) (.)2) 0
This requires the determinant of the coefficient matrix to be zero:
(AL, x
LX =0
Here, AL, AX, measures the total causal effect of L on Xş without passing through other x x,) Bk, measures the total causal effect of on X₃ that does not pass through \ (X,) and ends with X, - X₃- Therefore, this sum implies the causal effect of L X₂ is given all observed variables other than X, and Xg. In other words, X X \ {.x,, X and Rank(2_L.x, (x),..,)) = O. However, this rank constraint is not generic and violates the rank faithfulness assumption. Therefore, we must have Wj.i at a contradiction.
observed variables, while (AI.)
j. o O, arriving
Case (ii): Ift k (I,,k I * 0), then
X Wi,k = " BBr)) + a (Ir.k BB,K) = -a xx Bj, + 0x.5
Similar to Case (i), if W j.k = 0, we have
(AL, ALxx
(AL. AL,
which means the causal effect of L on X3 is zero given all observed variables other than X, and X₁. This implies Rank(_L.x, Xxxn((X.xx)xxxxxxxxxx (xx,,,,) = 0. As this violates the rank faithfulness assumption,
we have W, j.k *
Case (iii): If X, E (X (Bj,, , * 0), then W jit x ox BR Similar to Case (i), if W J, O, we have
(A AX Bjr = (AL, - Bx, B
This implies Rank( (LL,x,) (X,, XXX sumption. Therefore, W W,t # 0.
which violates the rank faithfulness
x Case (iv): If X2, E Pa(X.) \ {X,} } (B + 0) then W = -xx By, at. BR Similar to Case (iii), if W, j.t O, we obtain
(A A B BX B (AL, ALx B
Thus, we can prove W,, j.r # 0 because Rank XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX faithfulness assumption.
violates the rank
Case (v): X, does not fall into any of the four cases above, then W,, =
Therefore, from Case (i)~(v), we have
E [m] \\C 8 { W 0 X (X,, Xx) (P) Ua(())) Combined with the scenario t = k; in summary, we have
(11)
Vt E [m], W * 0 = X E (X,, X&) UPa(XX))))))))ppp))))))pDp( Pa( ( Since (X, PU UUA,
(12)
we have | Axxxl S || Therefore,
- Bx E aig (1 [W]]o W is Pinin X -order CICA solution of X).
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A.3.12 PROOF OF LEMMA
Lemma 9. If Condition 2 holds, W arg W| W a pmii (orde CICA solution of } if and only if W ~ -x,x.
Proof. During the proof in Lemma 8 we obtain the following results. If W is a 1-order CICA solution whose latent conditional set is LPa(X), it follows that for any row index j.
vt E [m], Wj j.t *
= X; E (X,) UP)))))
If W is a I-order CICA solution whose latent conditional set is the exogenous noise of Xg, instead of LP)(X), follows that for any row index j.
E E [m]], W + = X, E (x,, Xx) (P(((X) U
If X& Pa(X)) and Pa( (x)) 0. we have
(X,, X U Pa(X,) U (X, = (X3} (X UP,()),
Thus, W jat has the exact same sparsity pattern in two secnario. If Condition holds, then there does not hold,
exists a Xj such that the constraint X5 Pa(X;) and Pa(Xa)
{X3} Pa(X)) & (Xy, X UP)(X)) U Pa(X;),
then the CICA solution whose latent conditional set is LPa(X) has strictly small number of non- zero entries.
A.3.13 PROOF OF THEOREM
Theorem 1. All latent variables in LPa(X) can be identified. Besides, the causal edges of LPa(X) X and the causal edges between the observed variables also identifiable.
Proof. By Lemma 8, Condition is satisfied, we can identify - Bx.x by adding sparsity constraints and induce the causal structure. On the other hand, if Condition 2 is not satisfied, I Bx.x not identifiable. That is, we can find another p-order CICA solution W with the same number of non-zero entries
Bx Review the results obtained in the proof of Lemma 8,
E E [m], W + 0 - X₂ E (X,, X U Pa(X)) Pa(Xx).
If X E Pa(X,) and (xx)))== 0.
X&) U Pa(X)) (X3
(X,
then Wej has the exact same sparsity pattern. If the constraint X₁ E Pa(X,) and Pa(X₂) holds for every X, (Condition does not hold), then the whole W exist exactly the same sparsity pattern. In other words, although W' has different parameters with I - Bx.x, their support matrix remains the same. Therefore, both cases, the causal structure among observed variables Bx.x within causal cluster is identifiable. Given Condition I holds, Pmin (X) LPa(XX), thus we can identify each latent variable in LPa(X). Putting these partial results together, all latent variables in LPa(X), the causal edges of LPa(X) to X and the causal edges between the observed variables can be identified.
A.3.14 PROOF OF LEMMA 10
Lemma 10. Let L be latent variable discovered the current iteration. Denote S Ch(L). Let Sy have the highest causal order in S whose index in S is k, and W be the sparsest Pmii (S) -order CICA solution of P is the permutation matrix that makes PW have zon-zero diagonal elements, simultaneously, Denote Z = PWS, then the value of Zu can be suitable surrogate for L.
Proof. By Lemma 8, if Condition is satisfied, we can identify Bs.s by adding sparsity con- straints and induce the causal structure. Then PW deletes all outgoing edges from S and makes Zy a pure child of L. As shown in (Xie et al., 2024), it can be a suitable surrogate for L. On the other identifiable. Review the results obtained in the
hand, if Condition 2 is satisfied, I Bs,s proof of Lemma 8
1T E [m], W ja # X (X,, Xx) U Pa(x).)
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If Xx Pa(X,) and Pa(X2) = 0. (x,,
(X3)
thus Wa, has the exactly same sparsity pattern. the constraint X Pa(X,) and Pa(X;) holds for every X, (Condition 2 does not hold), then the whole W exist exactly same sparsity pattern. In other words, although W has different parameters with I- Bs.s. their support matrix remains the same. Essentially, W'S can be interpreted swapping the roles of L and Eg on I- Ss.sS. Although L not contained in the latent conditional set, still included in Zx- Therefore, in both cases, Zy can be suitable surrogate for L.
A.3.15 PROOF OF THEOREM 2
Theorem 2. If Condition 1 holds, then the underlying causal graph G is fully identifiable, including both the location of latent variables and the recovery of causal relationships E.
Proof. Denote Dis( V.) the length of the longest direct path from to X. AX; Dis(X). O. We collect Y [Vi] Dis(V.) z k). The proof is based on mathematical induction:
(1) Base: for k 1, we use Theorem to identify the common latent parents of observed variables and related causal edges. In other words, we can correctly identify the induced sub- sub-graph of G with nodes in Y1.
(2) Induction: assume we have correctly identified the induced sub-graph of G with nodes in (Vi) Dis S k), then using Lemma 10 find the suitable surrogate for latent variables in Y YK-1, we can continue to use Theorem I to local the latent variables in Yk41 and related causal edges, which concludes the induction.
Therefore, the underlying causal graph G' is fully identifiable, including both latent variables and their causal relationships.
A.3.16 PROOF OF THEOREM
Theorem 3. Let the graph obtained after removing all the outgoing edges of X in & be named by G' which form several connected components of observed variables X X',, X where be the number of connected components in g' For ISA solution W, let WX (Z] ..... Z T. Then there is a permutation æ of [k] s.t. for any E [k.. 3W; GL((C;)) makes Zz(i) = W,X(,
Proof. Based on the Schur complement, we have
Ax = - Bx.x) - Bx. -L.L))'BL.. BL.
(13)
Denote Z = =xx,XX X. Then we have NS(Z.) = [E has a directed path to X; whose intermedi- ate nodes, if exist, are all latent nodes). The reasons are as follows.
X
2:
(A
[
Bx.x) BX.L -
(14)
X;
(B.,+ EB,, + B,, BL,1) BL,,),,
X,EX((X.)
Considering all directed paths into X,, we categorize them into different groups according to topologically last observed nodes before X; this path. For example, if there is path P. : X, - L1 > Xx + L2 + X4. we put this path into the group corresponding to XR. named [[ there no observed nodes before X; in this path, we put this path in the group corresponding to 0, named G[0). total, there are X groups: Ux,Ex)(x,) [[Xx] Ug
X, is a cumulative sum of all directed paths into X,. The contribution of each directed path this sum is the noise of the start point times the path coefficient. Obviously, any path will be placed
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in the group UxLexx(x.) 9[Xa] U G[O.. Then, consider what the subtrahend in the last line of Equ. (14) denotes. B denotes the direct causal effect from X, to X,, BI BI.L))'BL. denotes the indirect causal effect from X, to X; through latent variables. Consequently, (B., + BI - BI.,I)==B,,.. J includes all causal effects in X; from Ane(X;) whose last observed node before X; in the causal path is x. This term is exactly the sum of causal effects on X; by paths in g[X,]. As consequence, Z, equals the sum of causal effects on X, by the paths in G[ That is, those directed paths whose intermediate nodes are all latent.
Therefore, Ax,x deletes all the outgoing edges from X and forms several connected components which correspond to the subspace in ISA's definition. Since ISA does not pose any constraints within a subspace, any invertible matrix valid. Since ISA exists block permutation indeterminacy (Theis, 2006), then makes (i)
conclude that there is permutation π of [k] s.t. for any i [k), E GL([C.)) W.X.
A.3.17 PROOF REMARK 2
Remark 2. The two causal graphs in Fig. la and Fig. 1b cannot be identified by ISA.
Proof. For the causal graphs in Fig. la and 1b, after removing all the outgoing edges of X, X1, X2,X3 are still connected due to the existence of L. According to Theorem 3, V W E GL((3
an ISA solution in both causal graphs. Consequently, the causal graphs in Fig. 1a and Fig. 1b cannot be identified by ISA.
L
B ILLUSTRATIONS OF ALGORIHTMS
B.1 MERGING RULES
Proposition 1 (Merging Rules). Let A be the active variable set and C₁ and C2 be two causal clusters. C1 and C2 share the common latent parent, if one of the following rules holds.
RI. 1)Cr and C2 are both pure clusters, and for any subsel C S C,UC2 with |C] = 2, (A follows the GIN condition.
R2. 1) One of the clusters a pure cluster and the other not, e.g. C1 pure and C2 is impure, and 2) V V; E Ci and V V₁ E C2. (AMC2, ViJ, VV, V23) follows the GIN condition.
R3. 1) C1 and C2 both are impure clusters, and 2) for V Ō S C Up with |Õ] 2, (AVO) U
C2), C" follows the GIN condition.
Otherwise, C1 and O2 do not share the common latent parent.
Proposition 1 establishes set of GIN-based criteria for determining whether two identified causal clusters (C, and C2) originate from the same latent parent node during each iteration. This propo- sition defines three mutually exclusive merging rules (R1, R2, R3) based on the purity attribute of clusters (pure or impure). This proposition is crucial for the algorithm's 's bottom-up reconstruction of the latent structure. During causal discovery, the algorithm first identifies local causal clusters, but these clusters may merely represent different observed subsets of the same latent variable. Through Proposition refmerge, the algorithm can reaggregate these dispersed observational evidence clusters, thereby correctly inferring the number and coverage of latent variables and avoiding the erroneous splitting of a single latent variable into multiple ones.
Corollary 2. Let L₁ be latent variable that was introduced in previous iterations, C2 be new cluster, and be the active variable set in the current iteration. Suppose cluster C₁ was a subset of Ch(Li) found in previous iterations. Then C₁ and C2 share the common latent parent L if setting A A J C1 L1 be the active set, one of the three rules in Proposition 2 holds. Otherwise, C1 and C2 do not share the common latent parent.
Corollary is a natural extension of Proposition 1 during the recursive process, designed to address the connection between newly discovered causal clusters and known latent variables. Corollary prevents the algorithm from "over-gends everrgenerattig" latent variables. Without this check, the algorithm might create a new latent variable node for every occurrence of the same latent factor.
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B.2
PSEUDO CODE
Algorithm/ CICA-LINGAM
Require: Observed variables X.
Ensure: Fully identified causal structure G.
1: Initialize active variable set
and = Ø.
2: while A # 0 do
3: C < Cllustessl A); (see Algorithm 3)
4: G f- SparseCICA(C) G); (see Algorithm 4)
5:
G +
(C, A, G). (see Algorithm 5)
6:
UpdateActtveDatttAA. A, G). (see Algorithm 6)
7: end while
8: Return G.
Algorithm 3 Finding Causal Clusters
Require: Active variable set A.
Ensure: The set causal clusters C. I: Initialize C 0 and the group size GrLen 2;
2: while |A| 2 GrLen + do
3: repeat
4: Select subset Y from A such that Y = GrLen;
5: if (A Y. Y) follows GIN condition for vÝ e Y such that Y 2 then
6: Add into C;
7: end if
8: until All subsets with group length GrLen A have been selected;
9: A = A C; GrLen GrLen
10: end while
11: Return C:
Algorithm constitutes the first step of each iteration in the CICA-LiNGAM algorithm. It employs incremental subset search strategy, executing four sequential steps in each iteration: i) Enumer- ating candidate subsets of a specified size from the current active variable set (Line 4); ii) Verifying whether specific independence structure exists between the subset and its complement using the GIN condition to determine if it constitutes a causal cluster (Line 5); iii) Confirming qualified sub- sets as causal clusters and removing them from the active (Line 6); iv) Gradually increasing the subset size to continue searching for larger clusters among the remaining variables (Line 9).
Algorithm 4 Sparse CICA
Require: The set of causal clusters C, and partial graph G.
Ensure: Updated partial graph G.
1: for each C; E C' do
2: W <- sparsest CICA solution on C;;
3: P + the permutation matrix that makes diag(PW) non-zero simultaneously;
4:
W divide each row of PW by its corresponding diagonal element;
5: Compute estimate B using B = I I:W;
6: Update := GU [j
B j.j # 0):
7: end for
8: Return G;;
Algorithm 4 constitutes the second and most crucial step in each iteration of the CICA-LiNGAM algorithm. It performs structural estimation within each identified causal cluster, executing four sequential steps per iteration: i) Solving for the sparsest CICA solution on the current causal cluster (Line 2); ii) Finding a permutation matrix to eliminate permutation ambiguity and ensure nonzero diagonal elements (Line 3); iii) Eliminate scaling uncertainty through normalization compute
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the corresponding adjacency matrix (Lines 4-5); iv) Update the local causal graph structure based on nonzero connection coefficients (Line 6).
Algorithm Determine Latent Variables
Require: A cluster set C, active variable set A, and partial graph G.
Ensure: Updated partial graph G.
1: C t Merge clusters from C according to Rules R1 and R2 of Proposition 1;
2: for each C2 E C do
3: if I, and C satisfy R3 of Corollary then
4:
G (T, Vi V, & C:):
5: else
6:
Introduce a new latent variable La to L; 7: G (U + Vi Vi E:J;;
8: end if
9: end for
10: Return G;
Algorithm 5 constitutes the third step of each iteration in the CICA-LiNGAM algorithm. It aims to resolve the connection between latent variables and observed clusters, executing three sequential steps: i) Merge causal clusters potentially sharing the same latent parent node based on Proposi- tion (Line 1); ii) Determine whether the current cluster shares parent node with latent variables discovered in previous iterations (Line 3); iii) Based on the determination, either establish connec- tions between existing latent variables and clusters, or introduce new latent variables and construct causal edges pointing to variables within the cluster (Lines 4-7).
Algorithm Update Active Data
Require: Current active variable set A, partial graph G. Ensure: Updated active variable set A.
1: if no new latent variable introduced G then
2: A+
3: else
4:
for each latent variable T4 E G do
5: Initialize the value of I, according to Lemma 10;
6: Add I. into A and delete Ch(T.) from A;
7: end for
8:
end if
Algorithm 6 maintains the set of active variables for the next recursive round. It sequentially cutes three steps: i) Check whether new latent variables were introduced in the current iteration; not, terminate the algorithm (Lines 1-2); ii) If new latent variables exist, initialize their values using the observed child node with the highest causal order as a proxy according to Lemma 10 (Line 5); iii) Add the latent variable the active set while removing its child already explained by latent variables, thereby achieving bottom-up recovery of the causal structure (line 6).
B.3 DISCUSSION OF OPTIMIZATION CRITERION OF CICA
Definition 15 (Cumulant (Brillinger, 2001)). Let = (X1, X2...., .,. Xx) ) be a random vector of length n. The k-th order cumulant tensor of is defined as a n x X n (l times) table, C(k) whose entry at position (i,, 1x) is
(k) de = -xum(X,,,..... ).) = Σ (D....Ds)
(-1)h-1(h -1)IE
II X,
E
II X jED
JED;
where the is taken all partitions (D), ....... Di) of the set (i1,...,ik).
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A p-dimensional shared subspace leaves a low-rank fingerprint not only in covariance but also in higher- cumulants. In the covariance view, identifiability comes from the fact that cross- covariance blocks live in space of rank at most p; equivalently, all (p 1)-minors vanish. The same logic transfers to cumulants: when we form cumulant matrices by linearly contracting the fourth-order cumulant tensor, the contribution of the shared factors still spans most p independent directions. Hence, these cumulant blocks also satisfy a rank deficiency property.
This viewpoint treats cumulants as providing additional low-rank views of the same latent struc- ture. Because there are many ways to contract a cumulant tensor, we obtaín many rank constraints without needing two large disjoint coordinate subsets, which loosens the requirement on m. At the same time, the framework strictly contains the second-order case: if we "degrade" the cumulant to order two, we recover the original covariance criterion. In short, moving from covariance to cumu- lants preserves the rank-deficiency principle while supplying more constraints and thereby stronger identifiability with fewer observed variables.
C ADDITIONAL INFORMATION ON EXPERIMENTS
C.I COMPUTING INFRASTRUCTURE
The computing devices and platforms are listed as follows.
OS: Microsoft Windows 11.
CPU: AMD Ryzen 7 4800H with Radeon Graphics, 2900 Mhz.
Memory: 16G.
Python 3.8.18.
C.2 IMPLEMENTATION DETAILS
We use the solution of ICA as an initialization for CICA optimization. LiNGIC (Li al., 2026) serves the dependence measure for testing Triad constraints. Besides, the L1 norm of the demixing matrix is adopted as a convex relaxation of the LO norm to encourage a sparsest solution.
C.3 REAL-WORLD EXPERIMENTS
C.3.1 TEACHER'S BURNOUT STUDY
Barbara Byrne conducted study to investigate the impact of organizational (role ambiguity, role conflict, work overload, classroom climate, decision making, superior support, peer support) and personality (self-esteem, external locus of control) on three facets (emotional exhaustion, deperson- alization, personal accomplishment) of burnout in full-time elementary teachers (Byrne, 6). The data set consists of 32 observed variables with 599 samples. The details of latent factors and their indicators are shown in Table (See Chapter 6, Page 191 in (Byrne, 2016) for more details). As in practice, the ground-truth latent structure is usually hard to know, here use the hypothesized model given in (Byrne, 2016) as a reference.
Locating latent variables. We run our algorithm with the prior knowledge that the underlying graph contains only the one-factor cluster. The final output of the measurement model is shown above. Here we rename the name of the latent variables in RLCD's output for easier comparison. Compare the reference model given in (Bymee, 2016), method merges DM and SS into one latent factor and keeps other clusters correctly identified. Notice that (Dong et al., 2023) arises more errors in clustering step (L,, L2, La). possible reason is that Ln only have two measurement variables and are incapable of correctly locating by their method. These results further verify the efficacy of our algorithm. Besides, the structural model learning results (causal graph latent variables) of our method and RLCD are:
Inferring latent variable structure. The Fl score of our results is 0.522. In contrast, RLCD obtains 0.364. In the output results of the RLCD, most of the edges connected to RC are incorrect.
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Latent Factors
Children (Indicators)
Role Ambiguity (RA)
RA1, RA2
Emotional Exhaustion (EE)
Eb1, EE2, EES
Depersonalization (DP)
DP,
Role Conflict (RC)
RC1. RC2, WO,, WO
Self-Esteem (SE)
SE,, SE,,,,,,S
Personal Accomplishment (PA)
PAI, PA3
Peer Support (PS)
PSt, PS₂
Classroom (CC)
CC₁. CC2, CC3, CC2
Decision Making (DM)
DM,, 1M2
Superior Support (SS)
SS,
External Locus of Control (ELC)
ELC1, ELC2, ELC's, ELCA, ELCs
Table 8: The latent factors and their indicators in teacher's burnout study.
Ours
RLCD
L
]
L1 RA1, RA2, RC1, EE
12 ~
EE F.F2, F.FaT
L ~
EE FFSs)
La
DPs)
V
L (DP1. DP2}
L4 |RC1, RC2, WOi,WOzf
V
LA (RC2, WO,,WOzf
Lg
SE,,
L6 N. ISE,,SEs, SEx)
L6 PAA,,AA,,PA PAs}
V
L6 ~ IPA,,PAs, PAs)
L1 CC1,,C2,, CCa, CCa)
LT [CC1, (CC,,CC2, CC2, CCa)
SS,, L DM,, DM2, S8, SSat
Ls (DM,, DM2, SS2
Lg BBLC,,ECC, ELCa, ELC,, ELC,)
La M (ELCI, ELC2, ELCa, ELC,, ELC,}
Table 9: The measurement model results of our method and RLCD (Dong al., 2023).
The possible reason is that some latent factors can not be discovered correctly, which further causes some unobserved confounding between latent variables. Note that previous method can not identify SE + ELC' in principle, as they form impure structure on latent variables. By solving CICA on SE and ELC using their observed descendants, our method can recover the causal direction SE + ELC, which supports the necessitate of introducing two-sided projection.
C.3.2 BIG FIVE PERSONALITY
Dataset Description. The Big Five personality dataset is rooted in the Five-Factor Model (FFM), a seminal theoretical framework in personality psychology to characterize individual personal- ity differences, proposed by American psychologists Paul Costa and Robert McCrae (Costa & McCrae, 1992). This dataset encompasses five core personality dimensions, namely Openness, Conscientiousness, Extraversion, Agreeableness, and Neuroticism, abbreviated as the O-C-E-A-N model. Each dimension is operationally measured by 10 psychometric items, which are designed to capture the nuanced traits underlying each factor. For example, the Openness dimension includes items like "I am intrigued by abstract ideas", while the Conscientiousness dimension features items such as "I am diligent fulfilling responsibilities..
The data were collected via the online interactive personality testing platform hosted on https: / /apenpsyycooeet ricommetrriii..ooo org, a widely recognized and ethically compliant public data acqui- sition channel in psychological research. The survey implementation adhered to established ethical norms in empirical psychology, including informed consent and anonymous participation. After data cleaning and validation, the final dataset utilized in this study comprises approximately 20,000 valid samples, covering 50 psychological measurement indicators (10 items per dimension across the five factors). Prior subsequent analyses, performed standardization on the data to ensure each variable follows a distribution with a mean of 0 and a variance 1.
Measurement Model Learning. To determine the causal structure in the Big Five personality data, we first employed the GIN algorithm (Xie et al., 2022) to construct measurement model. The core objective was to identify observed items that highly correspond to each personality dimension. During the clustering process, some items may reflect multiple personality dimensions: for instance,
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Ours
RLCD
HA + PA
RA - DM/SS
EE - SE
SE + DP
SE ELC
V
SE PA
DM/SS + SE RC + DP
DP PA
DP + CC
CC + EE
V
RC + DP
ELC + PA
X
RC - SE
ELC DP
RC ELC
RC -
RC - RA
EE + ELC
Table 10: The structural model results of our method and RLCD (Dong et al., 2023).
item Og ("I spend time reflecting on things") has dual connotations. On one hand, it reflects in-depth thinking about abstract and complex issues, which is consistent with the cognitive exploration traits of Openness; on the other hand, it involves reviewing and being prudent about one's own behaviors and tasks, aligning with the rigorous and self-disciplined traits of Conscientiousness. For item A ("I make people feel at ease."), on one hand, the sense of interpersonal security brought by empathy and friendliness is in line with the cognitive exploration traits of Agrecableness; from the perspective of Extraversion, the enthusiasm and talkativeness of extroverts can easily alleviate awkwardness. Such variables cannot correspond specific cluster and are therefore not included in the output of the measurement model. After screening via the GIN algorithm, the final output of the measurement model is follows:
Openness:
On,, On) Ls00,, Conseientiousness: (C C2, Ca, C4, Cs, Co C7, Cs Cg Cio):
Extraversion: Ls{E1, E,, Es, Es, E7, Es Eg E
Agrecableness: Lo(A, A2, A3, As, As, As, A7, As, A9);
Neuroticism: [, N2, N2, Nr, No. No. Nr, No. No, N101.
The measurement model reveals that the latent variables L4, Lis Las and L, serve as unitary rep- resentations for Conscientiousness, Extraversion, Agreeableness, and Neuroticism, respectively, ex- plaining the shared variance their corresponding item responses. contrast, the Openness mension exhibits
granular internal structure, decomposing into three distinct sub-clusters: L1, L2, and L3. These sub-clusters correspond to the core components of "Cognitive exploration", namely abstract reasoning, creative imagination, and mnggustiic-ognitiin complexity.
Causal Analysis Within Clusters. After obtaining the measurement model, we further applied our algorithm to uncover causal relationships within the clusters. We found several new conclusions that were not revealed by (Dong et al., 2023).
(i) Openness: In the Openness dimension, "difficulty in understanding" is the direct cause of "lack of interest" (O2 + O4). When a person repeatedly fails understand abstract content, it will di- rectly weaken their willingness to explore this field, whereas if they can understand it easily, they will be more likely to develop interest. Imagination is the core source of creative output: on one hand, "vivid imagination" will directly give rise to "excellent and unique ideas" (O3 + Os), and conversely, lack of imagination will directly restrict the quality of ideas; on the other hand, the breadth of imagination also directly determines the quantity of ideas, and "vivid imagination" will be transformed into "a constant stream of ideas" (O3 -> O10). In addition, vocabulary reserve the foundation of the complexity of language expression: "a rich vocabulary" will directly endow people with the ability to use complex and rare words (O, + Os), while poor vocabulary cannot support the of difficult words.
(ii) Conscientiousness:: In the Conscientiousness dimension, The intrinsic core trait "liking or- der" directly drives individuals to maintain the orderly state of life and work through the behavior of "following a schedule" (C, - C)); while the behavioral tendency of "paying attention to details" directly translates into the specific manifestation of "being exacting in work" (Ca C1o) high
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sensitivity to details directly acts on the control of omissions in work, thereby presenting rigorous work state.
In the Extraversion dimension, one hand, the intrinsic mindset of "feeling comfortable around people" serves as the core prerequisite for active social interaction - if an individual feels at in crowds, this mindset will directly prompt them to initiate conversations actively (E3 + Eg), and at the same time, it will directly drive them to interact with multiple people in social scenarios such parties (E3 + E7); on the other hand, the core tendency of "not liking to draw attention to oneself" is the direct trigger for social avoidance behaviors the aversion to others' attention will directly guide the individual choose a low key position "keeping in the background" (Es + E1), and this sense of aversion will also directly suppress their desire to express themselves in front of strangers (Fis + F1o).
(iii) Extraversion:
(iv) Agreeableness: In the Agreeableness dimension, in which A4 ("I sympathize with others' feel- ings.") plays a key mediating role: "feeling others' emotions" is the prerequisite for generating "sympathizing with others' feelings" (Ag + Aa) only by accurately capturing others' emotional states can one further put oneself in others' shoes and generate emotional resonance, while the in- ability to perceive emotions will directly lead to lack of empathy. On this basis, "sympathizing with others' feelings", as a mediating variable, becomes the direct driving force for altruistic behavior deep resonance with others' feelings will directly prompt individuals to take time out for others (A4 + As); conversely, if such empathy (A4) is lacking, even if one can perceive others' emotions, it will directly reduce the willingness engage in the altruistic behavior of active companionship.
(v) Neuroticism: In the Neuroticism dimension, on one hand, the core trait of "changing mood a lot" directly externalized the specific manifestation of "having frequent mood swings" (N7 + Ng); on the other hand, the emotional tendency of "getting stressed out easily" exerts a direct impact through the accumulation of sustained states (N1 + NN1) - being in a stressed state for a long time will directly lead to the continuous superposition of negative emotions, which in turn gives rise to the emotional outcome of "often feeling blue".
Structural Model Learning. Following the learning of the measurement model and cluster causal analysis, further recovered the causal structure among latent variables. While some of our find- ings are generally consistent with (Dong et al., 2023), we present here only the newly discovered structural learning results.
Causal relationship: {L1 - L2, + La, La + Lo, Lo + Ls, La + L1 - Ls, L7 - Ls}.
(i) (L, T2, In - Tz)): In the Openness dimension, "Abstract cognitive ability and interest orientation(TT] [,)" serve as the prerequisite for fostering "creative potential (I22)² and 'ccompleitty of language expression (Lz)". Only by overcoming difficulties in understanding abstract concepts and maintaining interest in them can one provide cognitive support for the operation of imagination and the accumulation of vocabulary. On this basis, L1 directly drives the manifestation of L2 and Ly: strong abstract cognitive ability translates into rich imagination and excellent creative output, while a positive orientation toward abstract thinking enhances the depth of vocabulary reserves and the ability to use complex words; conversely, deficiencies in L1 regarding abstract cognition will directly restrict the development of creativity and the complexity of language expression.
(ii) (T, + Lg)): "Emotional Instability (I,)** exerts a negative regulatory effect "social partic- ipation tendency (Lo)". Emotional fluctuations and feelings of anxiety directly suppress people's ' desire to interact, thereby leading to social avoidance behaviors such staying in the background and being quiet around strangers.
D RELATED WORK
Existing methods for handling causal discovery in the presence of latent confounders can be catego- rized into the following folds. Here list the papers focusing linear continuous variables,
Conditional independence constraints- et al., 2012; Akbari et al., 2021; Triantafillou & Tsamardinos, 2015) uses conditional independence tests to infer causal graphs. The core idea is to find patterns of conditional independence among variables to reveal the underlying causal structure. By testing for independence among observed
This line of work (Spirtes et al., 2000; Colombo
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variables, these methods discover the causal skeleton and orient some of the edges. These approaches can handle both linear and nonlinear causal relationships.
Rank sarciency-bassed This line of work (Silva et al., 2002; 2006; Kummerfeld & Ramsey, 2016; Huang et al., 2022; Li et al., 2024) uses rank constraints of covariance matrices to locate latent variables and infer the causal skeleton. The core idea is that in linear causal models, the covariance matrix or its submatrices exhibit specific rank properties. By analyzing these rank deficiencies, is possible to reveal the connection patterns between latent and observed variables.
This line of work proposes to identify the causal structure of latent variables by decomposing the covariance or precision matrix into matrices with specific structures, such as low-rank and sparse. Specifically, the low-rank matrix captures the causal relationships from latent variables observed variables, while the sparse matrix represents the direct causal relationships among observed variables. Representative works include (Chandrasekaran et al., 2011; 2012; Anandkumar al., 2013; Frot al., 2019).
Matrix decomposition-
Overcomplete independent component analysis (OICA)-based. This line of work leverages Overcomplete Independent Component Analysis (OICA) to handle problems with latent variables. OICA allows more source signals than observed signals, and thus can be used to learn the causal structure with latent variables. Related work in this area includes (Shimizu et al., 2009; Entner & Hoyer, 2010; Adams et al., 2021; Dai et al., 2026).
Generalized independent noise (GiN)-based. This line of work (Cai et al., 2019; Xie et al., 2020; Dai et al., 2022; Xie et al., 2023; Chen et al., 2022; 2023; Jin et al., 2023; Li et al., 2024; Xie et al., 2024) extends the independent noise condition to handle scenarios with latent variables. The core idea is that, for non- non-Gaussian linear causal mechanisms, higher-order statistics be leveraged to bring asymmetry and identify causal directions.
Higher-order cumulant-based. This line of work (Cai et al., 2023; Chen al., 2024a; Schkoda et al., 2024; Chen et al., 2025a) leverages higher-order cumulants to identify the causal structure when latent variables are present. For non-Gaussian distributions, cumulants can capture richer structural information than covariance alone. These studies show that the cumulant tensors of ob- served variables have specific rank constraints that can reveal the causal skeleton of latent variables.
Score-based.
This line of work (Agrawal et al., 2023; Ng et al., 2024) frames the learning of causal structure as search problem, aiming to find the graph structure that best fits the data. They typically define a scoring function to measure a graph's 's goodness of fit, then use search algorithms (like hill- climbing or greedy search) to find the sighiig--coring graph.
E THE USE OF LARGE LANGUAGE MODELS (LLMS)
We used ChatGPT refine writing only. The prompt was: "I am preparing a paper for submission to an international conference and would like your help to check for any grammatical issues and refine wording or sentence structure where necessary to ensure conciseness and precision." " Edits were applied paragraph-by-parageee and all outputs were verified and revised by the authors; no scientific content, analyses, or references were generated by the tool.
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Reviews:
[{"review_id": "t2rgNXwcUL", "reviewer": "Reviewer_479s", "summary": "The paper tackles causal discovery with latent confounders in settings where the standard “pure measurement” assumption breaks down—i.e., observed variables may also be causally linked, making many existing methods unreliable. The authors introduce Conditional Independent Component Analysis (CICA), which seeks two-sided linear projections whose outputs are conditionally independent given the latent factors. They show that CICA admits a characterization through rank-deficiency constraints and that, among all CICA-consistent solutions, the one aligned with the true causal model is the sparsest. Building on this, they design an algorithm that recursively recovers a latent-augmented DAG under mild structural assumptions. On synthetic data with non-pure, inter-connected observed variables, the method outperforms recent latent-causal baselines.", "questions": "* How robust is CICA to departures from the linear-mixing-with-independent-noise assumption? In particular, if the observables are produced by mildly nonlinear transformations of the latents, do the proposed rank-deficiency constraints still identify meaningful structure, or do they degenerate to spurious CICA solutions?\n* The method largely assumes that p (the number of conditioning latents) is known in advance. How should p be chosen in practice from data? Is there a concrete, data-driven procedure—e.g., progressively increasing p while monitoring rank deficiency or CI test statistics—to determine an appropriate order?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The motivation is clear: the paper targets exactly the regime where measurement purity is violated because observed variables also have edges, a setting in which many latent-causal methods break down.\n* The theoretical contribution is novel and well argued: introducing two-sided conditional independence provides the additional degrees of freedom needed to distinguish structures that are otherwise observationally equivalent, and the insight that the “correct” CICA solution is the sparsest one is both elegant and convincing.\n* Experiments on several deliberately hard, impure scenarios demonstrate consistent advantages over existing baselines.", "weaknesses": "* The main limitation—admittedly a common one in this area—is that all experiments are synthetic and closely matched to the paper’s own generative assumptions. There is no semi-real or real benchmark (e.g., latent-variable causal discovery datasets, or at least mixed real+synthetic scenarios) to demonstrate robustness under model misspecification.\n* Several recent works with very similar problem settings are not discussed. For instance, the ICLR’24 paper “Causal Structure Recovery with Latent Variables under Milder Distributional and Graphical Assumptions” is highly relevant and should be compared.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tyoMuItcck", "reviewer": "Reviewer_K2L3", "summary": "This paper introduces Conditional Independent Component Analysis (CICA), a novel method for identifying latent variables and causal structure in linear non-Gaussian acyclic models. Unlike existing approaches that require \"purity\" assumptions (where latent variables must have enough pure children), CICA uses two-sided projections to extract components that are conditionally independent given latent variables. The authors prove that under some conditions, the sparsest CICA solution combined with appropriate row permutation recovers the full causal structure and demonstrate their approach on synthetic data and a personality psychology dataset. The method offers tractable optimization through rank-deficiency constraints rather than higher-order statistics, achieving superior performance particularly in \"fully impure\" scenarios where existing methods struggle.", "questions": "* Is there any concrete, non-theoretical evidence, that extending the method to two-sided projections helps solve relevant problems for which previous approaches fail?\n* How does your method relate to Causal Component Analysis [1]?\n\nReferences:\n\n[1] Wendong, Liang, et al. Causal component analysis. NeurIPS 2023", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "* The paper presents a reasonable extension of prior work by generalizing from one-sided to two-sided projections in the ICA framework, addressing the one-sided limitations.\n* The theoretical analysis and identifiability results are rigorous as far as I can tell, with clear conditions under which full causal structure recovery is guaranteed.\n* Unlike overcomplete ICA methods that require EM algorithms and higher-order cumulants, CICA provides tractable optimization criteria through rank-deficiency constraints, making the approach computationally feasible while maintaining theoretical guarantees.\n* While the material is mathematically dense, the theoretical developments are presented clearly with good use of examples.", "weaknesses": "* The paper seems to be mostly driven by theoretical feasibility rather than a practical necessity of an improved approach. It is unclear to me whether the setting and the proposed method have any practical relevance. The assumptions in Equation (1), linearity and non-Gaussianity seem prohibitive for most problems. While those are common assumptions in the Causality literature, the authors would have to provide strong evidence that their method is useful in relevant problems despite those strong assumptions.\n* The real-world experiment on personality data fails to validate the method convincingly. The discovered causal relationships lack corroboration with psychological literature and rely solely on intuitive plausibility (e.g., \"difficulty understanding causes lack of interest\"), with many arrows equally plausible in reverse. The authors provide no justification for why personality data would exhibit the specific dependency structures that necessitate their two-sided projection approach over existing methods like Dong et al. Critical technical assumptions (linearity, non-Gaussianity) are never verified for the data, and the fundamental assumption that latent personality factors follow a DAG structure is also not well-justified. The real-world results appear to be post-hoc interpretations rather than meaningful validation.\n\n**Minor:**\n\n* L121: LiNGAM isn't defined.\n* Figure 2 contains little relevant information. That space might be better spent making a visual overview of the method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "M53OqbHtCv", "reviewer": "Reviewer_Lp3A", "summary": "This paper proposes Conditional Independent Component Analysis (CICA), a new statistical causal discovery method that can handle latent variables. Traditional causal discovery methods involving latent variables often require a strong assumption that each latent variable has at least one pure child node to ensure identifiability.\nIn contrast, the proposed approach proves that identifiability can be achieved without the pure-child assumption, by jointly adjusting the projections of two sets of observed variables to find directions that are conditionally independent given the latent variables, together with a sparsity condition on the causal structure.\n\nBuilding on this theoretical result, the authors introduce the CICA-LiNGAM algorithm, which implements the framework in practice.\nThey also show that, unlike Independent Subspace Analysis (ISA), which assumes independence only across subspaces, CICA can explicitly model conditional autonomy concerning latent structures.\n\nExperiments on simulated data demonstrate that the proposed method estimates causal structures more accurately than existing statistical causal discovery approaches that account for latent variables.", "questions": "Could you discuss more clearly how much denser alternative feasible solutions might differ from the sparsest one, and, if possible, include an evaluation experiment to illustrate this?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The main strength of this paper is that it removes the pure-child assumption, which is required by most existing statistical causal discovery methods dealing with latent variables, through a natural and well-motivated reformulation. This represents a significant contribution to the causal discovery research community. The ability to handle latent variables more naturally greatly expands the practical applicability of causal discovery to real-world problems where unobserved factors cannot be eliminated.\n\nThe theoretical development is carefully and clearly presented, making the reasoning easy to follow.\n\nIn addition, the paper uses clear illustrative examples that help readers understand the key ideas, and the simulation experiments convincingly demonstrate that the proposed method outperforms conventional approaches.", "weaknesses": "While the paper makes a strong theoretical contribution by proving that the sparsest solution is unique and identifiable, it does not sufficiently discuss how much denser alternative feasible solutions could differ from the sparsest one. This issue is important for understanding the algorithm’s stability and robustness. Although it is plausible that such alternative solutions would have similar structures due to the algorithmic design, explicitly discussing this point, and, if possible, providing empirical evaluation, would greatly strengthen the paper’s practical credibility.\n\nIn addition, the minimal sparsity principle, the assumption that the true causal structure is the sparsest among all feasible ones, is theoretically reasonable in the context of identifiability proofs. However, this assumption appears to be motivated mainly by the need for identifiability rather than by empirical or theoretical evidence that real causal systems are indeed maximally sparse. In this sense, the preference for sparsity may be viewed more as a convenient modeling choice than as a natural property of causal mechanisms. A clearer discussion of this limitation, or of the conditions under which sparsity aligns with real-world causal structures, would further improve the interpretability and credibility of the work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qC3PcA5xkh", "reviewer": "Reviewer_8zWU", "summary": "This paper proposes a new algorithm for identifying latent variables and causal structures in linear causal models, especially for models with unknown latent variables. A new CICA principle is proposed as a key tool for identification, which utilizes two-sided projections instead of one-sided projections, thus allowing for wider identifiable cases. Experiments on synthetic data show that the proposed algorithm performs better in both variable and structure identification, compared with several baseline methods.", "questions": "1.\tSince the proposed CICA principle is one of the major contributions, why not compare your method with GIN and TIN algorithms in experiments?\n2.\tIn Fig. 2, the CICA is given in a form conditioned on L. However, in the rest of this paper, it is more common that the independence is presented without conditioning on L. Which is right?\n3.\tIn Eqn. 1, the variable V=AE is introduced as part of problem setup. However, I find V is seldomly mentioned in the rest parts. Is V necessary here? Or it should be mentioned separately somewhere else.\n4.\tThe solution in line 234 seems to be incorrect, see if it should be $\\omega_1=[-(au+b),a], \\omega_2=[-v, 1]$.\n5.\tIn Condition 2, what does $LPa(X_i)$ mean? According to your definition in line 106, LPa takes a subset S as input and returns common latent parents of any *two* nodes in S. However, $X_i$ is a single node here.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1.\tA novel identifiability principle is proposed (CICA), allowing for identification of some cases that is originally unidentifiable. I think this is the major merit of this paper.\n2.\tA complete identification algorithm is presented, achieving better results on both latent variable identification and structure discovery than several baselines.\n3.\tExamples are provided for better illustration of certain Lemmas or Conditions.", "weaknesses": "1.\tThe major weakness lies in unsatisfying experimental results. In Tab. 1, causal discovery metrics (COR, F1-Score) are quite low for the proposed methods. If I understand it correctly, COR < 0.7 or F1-Score < 0.8 indicate a structure identification result that is far from optimal, > 0.9 results are expected for a theoretically-supported algorithm. Meanwhile, error in latent variables is supposed to decrease while sample size grows. However, the proposed method does not exhibit such monotonicity, making it questionable for the stability of proposed algorithm.\n2.\tAnother weakness is about the clarity. In the main algorithm (Alg. 1), only a rough process is given, and no implementation details is provided in this manuscript. Some subprocesses is based on Lemma or Theorem which does not contain a constructive solution itself. I suggest the authors provide details for each subprocess in Appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)