arxiv: 2605.10651 · v1 · submitted 2026-05-11 · 💻 cs.LG · cs.AI· stat.ML

Recognition: 2 theorem links

· Lean Theorem

A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables

Feng Xie, Hao Zhang, Ruxin Wang, Shenglan Nie, Xichen Guo, Zheng Li

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:25 UTC · model grok-4.3

classification 💻 cs.LG cs.AIstat.ML

keywords causal discoverylatent variablesdivide-and-conquerrecursive decompositionconstraint-based methodscausal structure learningcomputational efficiencysoundness and completeness

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The pith

Divide-and-conquer causal discovery can be extended to settings with latent variables via recursive decomposition and reconstruction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that divide-and-conquer strategies for learning causal structures, which have been restricted to cases with no hidden variables, can be generalized to include latent variables. It does so by recursively splitting the global task into smaller subproblems whose solutions are then combined through a reconstruction step that preserves the overall causal relations. This approach reduces the computational burden of repeated conditional independence tests in high-dimensional data. The authors prove that the framework is both sound and complete, meaning it recovers the correct structure when the subproblems are solved correctly.

Core claim

A recursive decomposition framework enables divide-and-conquer causal discovery beyond causal sufficiency by breaking the global learning task into subproblems and integrating their solutions through a principled reconstruction step to recover the global causal structure, with theoretical guarantees of soundness and completeness.

What carries the argument

The DiCoLa recursive decomposition framework, which splits the causal structure learning task into subproblems and uses a reconstruction step to assemble the global graph while accounting for latent-variable effects.

If this is right

Existing constraint-based causal discovery algorithms can be wrapped inside the framework to gain efficiency gains on large graphs.
The approach removes the need to assume causal sufficiency when applying divide-and-conquer methods.
Soundness and completeness hold recursively, so correctness at each subproblem level implies correctness of the final global structure.
Computational cost scales with the sizes of the subproblems rather than the full variable set.
The same decomposition strategy applies across multiple causal discovery algorithms without altering their core conditional independence tests.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could be adapted to settings where only partial observations are available rather than fully specified latent variables.
Repeated application of the decomposition might allow causal discovery on graphs too large for any single run of current algorithms.
If reconstruction can be made parallelizable, the method would further reduce wall-clock time on distributed systems.

Load-bearing premise

The reconstruction step must correctly combine the solutions from subproblems to recover the true global causal structure without distortion from dependencies created by latent variables.

What would settle it

A dataset with known latent variables where the framework outputs a causal graph that differs from the ground-truth structure recovered by exhaustive non-decomposed search on the same data.

Figures

Figures reproduced from arXiv: 2605.10651 by Feng Xie, Hao Zhang, Ruxin Wang, Shenglan Nie, Xichen Guo, Zheng Li.

**Figure 1.** Figure 1: The divide-and-conquer for the full variable set V. problem is solved independently using existing learners such as IC (Verma & Pearl, 1990) or PC (Spirtes & Glymour, 1991), and the resulting substructures are then merged into a global solution. By replacing exhaustive CI testing over V with localized tests on subsets Vi , these methods significantly shrink the search space and avoid redundant computation… view at source ↗

**Figure 2.** Figure 2: (a) An underlying causal structure (adapted from MILDEW network (Jensen & Jensen, 1996)), where L1 and L2 are latent variables. (b) The MAG characterizes the causal relations over the observed variables in (a). (c) The inferred PAG from observed variables. (d) The decomposition tree generated by DICOLA, where each node Gi represents a sub-problem. (e) Merging process of local skeletons. graph over the same… view at source ↗

**Figure 3.** Figure 3: Performance comparison on ER(30, 5) graphs, ER(50, 3) graphs, and the real-world ANDES network. Panels (a)–(c) correspond to these three settings, respectively. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Estimated PAG obtained by applying DICOLA+FCI to the Arabidopsis thaliana dataset. The adjacency matrix visualizes the edge marks directed from the gene labelled by the i-th row to the gene labelled by the j-th column: black squares indicate arrowheads, blue squares indicate tails, and red squares indicate circles. of latent variables by leveraging m-separation properties of maximal ancestral graphs. Build… view at source ↗

**Figure 5.** Figure 5: Performance on ER(30, 3) graphs with 3 latent variables. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: Performance on ER(30, 5) graphs with 3 latent variables. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗

**Figure 7.** Figure 7: Performance on ER(50, 3) graphs with 5 latent variables. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗

**Figure 8.** Figure 8: Performance on ER(50, 3) graphs with 7 latent variables. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗

**Figure 9.** Figure 9: Performance on MILDEW network with 3 latent variables. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗

**Figure 10.** Figure 10: Performance on BARLEY network with 4 latent variables. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗

**Figure 11.** Figure 11: Performance on ANDES network with 10 latent variables. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗

**Figure 12.** Figure 12: Performance on LINK network with 50 latent variables. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗

**Figure 13.** Figure 13: Performance on ER(40, d) graphs with 2 latent variables. Importantly, this limitation affects the efficiency gain rather than the correctness of the framework. The soundness and completeness guarantees of DICOLA remain valid whenever the guarantees of the base learner hold. To empirically examine this behavior, we conduct additional experiments using two classical constraint-based methods, FCI and RFCI, a… view at source ↗

read the original abstract

Constraint-based causal discovery is widely used for learning causal structures, but heavy reliance on conditional independence (CI) testing makes it computationally expensive in high-dimensional settings. To mitigate this limitation, many divide-and-conquer frameworks have been proposed, but most assume causal sufficiency, i.e., no latent variables. In this paper, we show that divide-and-conquer strategies can be theoretically generalized beyond causal sufficiency to settings with latent variables. Specifically, we propose a recursive decomposition framework, termed DiCoLa, that enables divide-and-conquer causal discovery in the presence of latent variables. It recursively decomposes the global learning task into smaller subproblems and integrates their solutions through a principled reconstruction step to recover the global structure. We theoretically establish the soundness and completeness of the proposed framework. Extensive experiments on synthetic data demonstrate that our approach significantly improves computational efficiency across a range of causal discovery algorithms, while experiments on a real-world dataset further illustrate its practical effectiveness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

DiCoLa extends divide-and-conquer causal discovery to latent variables through recursive splitting and reconstruction, but the reconstruction step's ability to preserve cross-partition latent effects is the part that needs the most verification.

read the letter

The main takeaway is that this work generalizes existing divide-and-conquer strategies for constraint-based causal discovery to settings with latent variables. It does so by recursively breaking the variable set into smaller subproblems, solving them separately, and then using a reconstruction step to assemble the global graph. The authors call the method DiCoLa and state that they have proven its soundness and completeness while showing efficiency gains in experiments.

Referee Report

1 major / 2 minor

Summary. The paper introduces DiCoLa, a recursive decomposition framework for constraint-based causal discovery that extends divide-and-conquer strategies to settings with latent variables. It recursively partitions the variable set into subproblems, solves them independently, and integrates the results via a reconstruction step to recover the global causal graph. The authors claim to prove the soundness and completeness of this process and report substantial computational speed-ups on synthetic data plus practical utility on a real-world dataset.

Significance. If the soundness and completeness results hold under the stated conditions, the framework would meaningfully extend scalable causal discovery to the common case of latent variables, where most existing divide-and-conquer methods fail. The empirical efficiency gains across multiple base algorithms would be a practical contribution for high-dimensional applications.

major comments (1)

[§4] §4 (Theoretical Analysis), reconstruction step: the completeness proof assumes that any latent-induced dependence spanning subproblem boundaries is correctly detected and incorporated during integration, yet the precise conditions under which the reconstruction step guarantees this (e.g., when a latent creates an untestable cross-partition dependence) are not fully articulated. This is load-bearing for the central completeness claim.

minor comments (2)

[Abstract] Abstract: the claim of 'extensive experiments' would be strengthened by reporting concrete speed-up factors or runtime ratios rather than qualitative statements.
[Notation] Notation section: the recursive decomposition operator and the interface between subproblem outputs and the reconstruction step would benefit from a small illustrative example with explicit variable partitions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address the single major comment below and will revise the paper accordingly to improve clarity.

read point-by-point responses

Referee: [§4] §4 (Theoretical Analysis), reconstruction step: the completeness proof assumes that any latent-induced dependence spanning subproblem boundaries is correctly detected and incorporated during integration, yet the precise conditions under which the reconstruction step guarantees this (e.g., when a latent creates an untestable cross-partition dependence) are not fully articulated. This is load-bearing for the central completeness claim.

Authors: We thank the referee for highlighting this point. The completeness of DiCoLa follows from the fact that the reconstruction step augments the subproblem solutions with additional CI tests performed on the full observed dataset between variables from different partitions. Under the standard assumptions (Markov condition, faithfulness, and no selection bias), any latent-induced dependence that is testable via d-separation on the observed variables will be detected by these cross-partition tests and incorporated into the global PAG. For untestable cases (e.g., certain inducing paths that do not yield observable conditional independences), the framework correctly outputs the corresponding equivalence class without claiming to resolve them, consistent with the completeness of the base algorithm (such as FCI) applied to subproblems. We agree that these conditions are currently stated implicitly rather than via an explicit lemma or corollary. In the revised manuscript we will add a dedicated proposition in §4 that enumerates the detectability conditions, with a brief discussion of untestable cross-boundary cases and how they are represented in the output. This clarification strengthens the presentation of the central result without altering its validity. revision: yes

Circularity Check

0 steps flagged

No circularity: soundness and completeness established independently of inputs.

full rationale

The abstract and description present DiCoLa as a recursive decomposition into subproblems followed by a reconstruction step, with soundness and completeness asserted via separate theoretical arguments rather than by re-deriving the global structure from the decomposition itself. No equations appear that equate a claimed result to a fitted parameter or self-referential definition (e.g., no 'prediction' that is the input by construction). Self-citations, if present, are not load-bearing for the core claim; the framework is offered as an algorithmic generalization whose validity rests on external causal assumptions and proof structure, not tautology. This matches the expected non-finding for a proposal whose central contribution is the decomposition strategy itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the domain assumption that causal structures admit recursive decomposition whose subproblem solutions can be reconstructed without loss of information about latent effects. No free parameters or new entities are mentioned.

axioms (1)

domain assumption Causal structures with latent variables admit recursive decomposition into subproblems whose solutions can be reconstructed to recover the global structure.
This is the key generalization beyond causal sufficiency asserted in the abstract.

pith-pipeline@v0.9.0 · 5474 in / 1259 out tokens · 73868 ms · 2026-05-12T04:25:39.408373+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We propose a recursive decomposition framework, termed DiCoLa, that enables divide-and-conquer causal discovery in the presence of latent variables. It recursively decomposes the global learning task into smaller subproblems and integrates their solutions through a principled reconstruction step
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Theorem 3 … the local skeleton LK can be constructed by merging LA∪C and LB∪C … remove the edge ⟨X,Y⟩ from LK if it is absent from either LA∪C or LB∪C

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

F., Tsamardinos, I., and Statnikov, A

Aliferis, C. F., Tsamardinos, I., and Statnikov, A. Hiton: a novel markov blanket algorithm for optimal variable se- lection. InAMIA annual symposium proceedings, volume 2003, pp

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Sada: A general framework to support robust causation discovery with theoretical guarantee.arXiv preprint arXiv:1707.01283, 2017a

Cai, R., Zhang, Z., and Hao, Z. Sada: A general framework to support robust causation discovery with theoretical guarantee.arXiv preprint arXiv:1707.01283, 2017a. Cai, R., Zhang, Z., Hao, Z., and Winslett, M. Sophisticated merging over random partitions: a scalable and robust causal discovery approach.IEEE Transactions on Neu- ral Networks and Learning Sy...

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A versatile causal discovery framework to allow causally-related hidden variables

Dong, X., Huang, B., Ng, I., Song, X., Zheng, Y ., Jin, S., Legaspi, R., Spirtes, P., and Zhang, K. A versatile causal discovery framework to allow causally-related hidden variables. InInternational Conference on Learning Rep- resentations, volume 2024, pp. 43084–43118,

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[4]

If there exists a directed path from X to Y , then X is anancestorof Y and Y is adescendantof X

of three pairwise disjoint subsets whose union is V. If there exists a directed path from X to Y , then X is anancestorof Y and Y is adescendantof X. For a vertex X in a MAG M, let An(X,M) denote the set of ancestors of X in M, and define An+(X,M) = An(X,M)∪ {X} . For a vertex set X, we similarly denote its set of ancestors byAn(X,M), and defineAn +(X,M) ...

work page 2000
[5]

Markov blanket

The formal definition of a Markov blanket is given below. Definition 5(Markov Blanket).The Markov blanket of a variable X relative to a set of variables V, denoted by MBV(X), is the minimal subset of V\ {X} such that X is conditionally independent of all remaining variables given MBV(X), formally, X⊥ ⊥V|MB V(X),∀V∈V\ MBV(X)∪ {X} . Under the causal Faithfu...

work page 1988
[6]

By Property 2, collider-connectedness between X and Y is equivalent to Y belonging to the Markov blanket ofXinM

By definition of the augmented graph, Y is adjacent to X in (M)a if and only if X and Y are collider-connected in M. By Property 2, collider-connectedness between X and Y is equivalent to Y belonging to the Markov blanket ofXinM. The claim therefore follows. Lemma 8.(Mokhtarian et al., 2025; Richardson & Spirtes,

work page 2025
[7]

For conditional independence testing, we apply the Fisher Z-transformation (Fisher,

Subsequently, to identify vertex separators within the UIG, we utilize the junction tree routines provided by theNetworkXlibrary (Hagberg et al., 2008). For conditional independence testing, we apply the Fisher Z-transformation (Fisher,

work page 2008
[8]

Our source code is included in the supplementary materials

with a significance level of α= 0.01 across all methods. Our source code is included in the supplementary materials. All experimental results are summarized in Table 2, with corresponding figures presented below. Marginalizing latent variables may substantially densify the induced MAGs relative to the underlying generating DAGs (Richardson & Spirtes, 2002...

work page 2002
[9]

Table 2.Summary of experimental settings and corresponding results

More details of those real-world structures can be found athttps://www.bnlearn.com/bnrepository/. Table 2.Summary of experimental settings and corresponding results. Random Structures Underlying Structure|V| |L|Avg. Degree in DAGs Avg. Degree in Induced MAGs Corresponding Results ER(30,3)graphs 30 3 3.00 3.81 Figure 5 ER(30,5)graphs 30 3 5.00 7.28 Figure ...

work page 2016
[10]

and HITON-MB (Aliferis et al., 2003). 21 A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables 2000 3000 4000 3 4 5 Number of CI Tests ×103 FCI 2000 3000 4000 2.5 3.0 ×103 RFCI 2000 3000 4000 5 6 ×103 FCI+ 2000 3000 4000 6 8 ×103 L-MARVEL 2000 3000 4000 1.0 1.2 1.4 ×103 ICD 2000 3000 4000 0.2 0.4 0.6 ln(Time...

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[11]

graphs with 3 latent variables. 22 A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables 2000 3000 4000 0.7 0.8 0.9 1.0 Number of CI Tests ×104 FCI 2000 3000 4000 5 6 7 ×103 RFCI 2000 3000 4000 1.2 1.4 ×104 FCI+ 2000 3000 4000 2.5 5.0 7.5 ×105 L-MARVEL 2000 3000 4000 2.50 2.75 3.00 3.25 ×103 ICD 2000 3000 40...

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graphs with 3 latent variables. 23 A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables 2000 3000 4000 1.0 1.5 2.0 Number of CI Tests ×104 FCI 2000 3000 4000 6 7 8 ×103 RFCI 2000 3000 4000 1.0 1.5 ×104 FCI+ 2000 3000 4000 2 3 4 ×104 L-MARVEL 2000 3000 4000 3 4 ×103 ICD 2000 3000 4000 1.2 1.4 1.6 ln(Time) 20...

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[13]

graphs with 5 latent variables. 24 A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables 2000 3000 4000 1 2 3 4 Number of CI Tests ×104 FCI 2000 3000 4000 6 8 ×103 RFCI 2000 3000 4000 0.5 1.0 1.5 ×104 FCI+ 2000 3000 4000 0.5 1.0 ×105 L-MARVEL 2000 3000 4000 2 3 ×103 ICD 2000 3000 4000 0 1 2 ln(Time) 2000 300...

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graphs with 7 latent variables. 25 A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables 2000 3000 4000 3 4 5 6 Number of CI Tests ×103 FCI 2000 3000 4000 3 4 ×103 RFCI 2000 3000 4000 0.6 0.8 1.0 ×104 FCI+ 2000 3000 4000 0.6 0.8 1.0 ×103 L-MARVEL 2000 3000 4000 2.0 2.5 3.0 ×103 ICD 2000 3000 4000 −1.0 −0.8 −...

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Notably, marginalizing latent variables can make the induced MAGs noticeably denser than the generating DAGs (Richardson & Spirtes, 2002); the average degrees of the induced MAGs are reported in Table

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[16]

These results confirm that DICOLAis most beneficial on sparse or decomposable graphs and gradually approaches the behavior of the base learner on denser graphs

Similarly, when using RFCI as the base learner, the reduction decreases from 69% to 2.7%. These results confirm that DICOLAis most beneficial on sparse or decomposable graphs and gradually approaches the behavior of the base learner on denser graphs. D.2. Relation to Local Causal Structure Learning-Based Methods We further discuss the relation between DIC...

work page 2000