Jigsaw-CRL: Recovering Global Latent Causal Order from Fragmented Multi-Client Interventions

Haijie Xu; Chen Zhang

stat.MLAug 29, 2026

Jigsaw-CRL: Recovering Global Latent Causal Order from Fragmented Multi-Client Interventions

Authors: Haijie Xu, Chen Zhang

Organizations: Department of Industrial Engineering Tsinghua University Beijing 100084, China

Abstract

Causal representation learning (CRL) aims to recover latent causal variables and their structural relations from high-dimensional observations. Existing CRL methods typically assume that all environments are defined over the same latent variables, or at least share a common latent representation space. We study a fragmented multi-client setting, where multiple clients interact with the same global latent causal system but each client only accesses and intervenes on a subset of the latent variables. In this regime, marginalizing unused latent variables induces bidirected edges, so a single client no longer admits a node-wise latent causal graph, and the global latent causal order must be recovered by assembling client-specific structural fragments. We propose \textbf{Jigsaw-CRL}, a framework for recovering global latent causal order from such fragmented interventions. Under soft interventions, differences between precision matrices across environments exhibit a low-rank structure governed by latent ancestor relations. This enables recovery, for each client, of a block partition, the corresponding block-level ancestral order, and latent subspaces, and then assembly of these fragments into the global node-level latent causal order. We establish identifiability guarantees, develop practical algorithms, and validate the framework on synthetic data. Our codes are available on https://anonymous.4open.science/r/code-for-Jigsaw-CRL-7B26

Figures & tables

Panel of Figure 1 — Figure 1: Fragmented client views of latent causal systems under different multi-client settings. All nodes in the figure represent latent variables. For client $k$ , $X_{k}^{(j)}$ denotes the observed data after intervention $j$ . Purple highlights the intervened node $2$ together with the edges affected by the intervention. (a) Multi-client setting. Different clients use different subsets of latent variables from the same global causal system. Marginalizing the missing latent variables induces client-specific mixed graphs, where bidirected edges appear in addition to directed relations. In a client-specific induced graph, the intervention on node $2$ may manifest as changes in directed edges ( blue ) and bidirected edges ( red ), corresponding to $\mathbf{d}_{k}^{(j)}$ and $\mathbf{u}_{k}^{(j)}$ in Assumption 2 . (b) Multi-domain setting. Adopted in Sturma et al. (2023) , latent variables are partitioned into a shared global component (e.g., $\{1,2,3\}$ ) and client-specific local components (e.g., $\{4,5,6\}$ and $\{7,8\}$ ). No edges exist across different local components, nor between global and local components. Consequently, client-specific induced graphs do not generate bidirected edges or extra directed edges after marginalization. This can be viewed as a special case of our more general setting.

Panel of Figure 2 — Figure 2: Finite-sample performance under one graph setting A and Gaussian noise. As $n$ increases, global ancestral recovery improves while subspace and support errors decrease. The rightmost marker at $\infty$ corresponds to the population-level precision-matrix input.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

[\mathbf{R}_{k}]_{C_{\ell},J_{\ell}}=\widetilde{\mathbf{H}}_{\ell}\bigl([\mathbf{Q}_{k}]_{J_{\ell},:}\bigr)^{\dagger}

Appendix

Algorithm 4 Block-RQ decomposition

\tilde{\mathbf{M}}_{k}(\hat{C})\leftarrow\begin{bmatrix}\hat{\mathbf{\Theta}}_{k}^{(j_{1})}-\hat{\mathbf{\Theta}}_{k}^{(0)}\\ \hat{\mathbf{\Theta}}_{k}^{(j_{2})}-\hat{\mathbf{\Theta}}_{k}^{(0)}\\ \vdots\\ \hat{\mathbf{\Theta}}_{k}^{(j_{|\hat{C}|})}-\hat{\mathbf{\Theta}}_{k}^{(0)}\end{bmatrix},\qquad\hat{C}=\{j_{1},\dots,j_{|\hat{C}|}\}

Appendix

Algorithm 5 ID-Block-Ancestor-FS

\tilde{\mathbf{M}}_{k}(\tilde{C})\leftarrow\begin{bmatrix}\hat{\mathbf{\Theta}}_{k}^{(j_{1})}-\hat{\mathbf{\Theta}}_{k}^{(0)}\\ \hat{\mathbf{\Theta}}_{k}^{(j_{2})}-\hat{\mathbf{\Theta}}_{k}^{(0)}\\ \vdots\\ \hat{\mathbf{\Theta}}_{k}^{(j_{r})}-\hat{\mathbf{\Theta}}_{k}^{(0)}\end{bmatrix},\qquad\tilde{C}=\{j_{1},\dots,j_{r}\}

Appendix

Algorithm 6 ID-Block-Causal-Order-FS

Panel of Figure 6 — Figure A.1 : Finite-sample performance under one graph setting A and laplace noise. As $n$ increases, global ancestral recovery improves while subspace and support errors decrease. The rightmost marker at $\infty$ corresponds to the population-level precision-matrix input.

Panel of Figure 7 — Figure A.2 : Finite-sample performance under one graph setting B and gaussian noise. As $n$ increases, global ancestral recovery improves while subspace and support errors decrease. The rightmost marker at $\infty$ corresponds to the population-level precision-matrix input.

Panel of Figure 8 — Figure A.3 : Finite-sample performance under one graph setting B and laplace noise. As $n$ increases, global ancestral recovery improves while subspace and support errors decrease. The rightmost marker at $\infty$ corresponds to the population-level precision-matrix input.

Panel of Figure 9 — Figure A.4 : Finite-sample performance under one graph setting C and gaussian noise. As $n$ increases, global ancestral recovery improves while subspace and support errors decrease. The rightmost marker at $\infty$ corresponds to the population-level precision-matrix input.

Panel of Figure 10 — Figure A.5 : Finite-sample performance under one graph setting C and laplace noise. As $n$ increases, global ancestral recovery improves while subspace and support errors decrease. The rightmost marker at $\infty$ corresponds to the population-level precision-matrix input.

$p$	Metric	$10^{4}$	$5\!\times\!10^{4}$	$10^{5}$	$5\!\times\!10^{5}$	$10^{6}$	$\infty$
10	$E_{1}$	0.19	0.45	0.56	0.58	0.81	1.00
	$E_{2}$	0.32	0.30	0.24	0.14	0.00	0.00
	$E_{3}$	0.57	0.39	0.25	0.18	0.07	0.00
	Runtime (s)	–	–	–	–	–	0.17
20	$E_{1}$	0.17	0.49	0.50	0.62	0.85	0.93
	$E_{2}$	0.45	0.28	0.27	0.18	0.10	0.00

Appendix

Table 1: Sensitivity to the global latent dimension

p

(

K=15

q=0.6

). Runtime is measured at

n=\infty

$K$	Metric	$10^{4}$	$5\!\times\!10^{4}$	$10^{5}$	$5\!\times\!10^{5}$	$10^{6}$	$\infty$
5	$E_{1}$	0.13	0.18	0.19	0.24	0.32	0.34
	$E_{2}$	0.42	0.34	0.22	0.18	0.10	0.00
	$E_{3}$	0.58	0.48	0.34	0.11	0.00	0.00
	Runtime (s)	–	–	–	–	–	0.25
10	$E_{1}$	0.15	0.44	0.46	0.55	0.76	0.81
	$E_{2}$	0.36	0.28	0.27	0.14	0.10	0.00

Appendix

Table 2: Sensitivity to the number of clients

K

(

p=20

q=0.6

). Runtime is measured at

n=\infty

$q$	Metric	$10^{4}$	$5\!\times\!10^{4}$	$10^{5}$	$5\!\times\!10^{5}$	$10^{6}$	$\infty$
0.5	$E_{1}$	0.12	0.46	0.46	0.57	0.75	0.82
	$E_{2}$	0.55	0.36	0.29	0.15	0.08	0.00
	$E_{3}$	0.47	0.39	0.18	0.10	0.08	0.00
	Runtime (s)	–	–	–	–	–	0.53
0.6	$E_{1}$	0.17	0.49	0.50	0.62	0.85	0.93
	$E_{2}$	0.45	0.28	0.27	0.18	0.10	0.00

Appendix

Table 3: Sensitivity to the client coverage probability

q

(

p=20

K=15

). Runtime is measured at

n=\infty

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Causal representation learning aims to recover the latent causal variables and their causal relations, typically represented by directed acyclic graphs (DAGs), from low-level observations such as image pixels. A prevailing line of research exploits multiple environments, which assume how data distributions change, including single-node interventions, coupled interventions, or hard interventions, or parametric constraints on the mixing function or the latent causal model, such as linearity. Despite the novelty and elegance of the results, they are often violated in real problems. Accordingly, we formalize a set of desiderata for causal representation learning that applies to a broader class of environments, referred to as general environments. Interestingly, we show that one can fully recover the latent DAG and identify the latent variables up to minor indeterminacies under a nonparametric mixing function and nonlinear latent causal models, such as additive (Gaussian) noise models or heteroscedastic noise models, by properly leveraging sufficient change conditions on the causal mechanisms up to third-order derivatives. These represent, to our knowledge, the first results to fully recover the latent DAG from general environments under nonparametric mixing. Notably, our results match or improve upon many existing works, but require less restrictive assumptions about changing environments.

Abstract

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