Causal Discovery for Linear Models with Latent Confounding via Multi-domain Identifiable Representation Learning

Danru Xu; Aapo Hyvärinen; Sara Magliacane

cs.LGOct 20, 2026

Causal Discovery for Linear Models with Latent Confounding via Multi-domain Identifiable Representation Learning

Authors: Danru Xu, Aapo Hyvärinen, Sara Magliacane

Organizations: University of Amsterdam · Saarland University · University of Helsinki

Abstract

Causal discovery with latent confounding is particularly challenging, but a common setting in real-world data. In this work, we focus on linear causal models with latent confounding, represented by acyclic directed mixed graphs (ADMGs). Existing methods that fully identify ADMGs typically assume non-Gaussianity or graphical constraints, such as bow-freeness. In contrast, we propose Linear ADMG Learning (LinADMG), a framework for learning ADMGs for linear causal models through identifiable representation learning. Rather than relying on non-Gaussianity or bow-freeness, LinADMG leverages multi-domain settings to identify the noise variables. We show that the identifiability of the noise variables is sufficient for recovering all direct causal relations, and that, under an additional latent-confounding no-cancellation condition, LinADMG can identify the whole ADMG. We implement LinADMG with four identifiable representation learning approaches from the literature and validate the methods on simulated and semisynthetic datasets.

Figures & tables

Figure 1: Overview of LiNGAM and LinADMG (our method) pipelines. LiNGAM applies ICA to the observed data $\mathbf{X}$ to estimate independent non-Gaussian noise variable $\widehat{\mathbf{Z}}$ , from which it infers the directed edge set $\widehat{E}_{\rightarrow}$ of a DAG. LinADMG replaces ICA with any appropriate IRL/CRL on a multi-domain dataset $\mathbf{X}^{\tau}$ with domain index $\tau$ to estimate dependent noise variable $\widehat{\mathbf{Z}}^{\tau}$ without specific parametric assumptions on the noise variables, and thereby enabling the additional recovery of bidirected edges $\widehat{E}^{\tau}_{\leftrightarrow}$ representing latent confounding.

Figure 2: Examples showing that Asm. 4.2 and bow-freeness/faithfulness capture different properties. (a)

X_{1}

and

X_{2}

share the latent parent

U_{1}

while

X_{1}\to X_{2}

, resulting a violation of bow-freeness. Nevertheless, the latent confounding is detectable by the dependency of noise variables, so Asm. 4.2 holds. (b) Conversely, bow-freeness and faithfulness do not guarantee Asm. 4.2 . Let

U_{1}

and

U_{2}

be independent Gaussian variables with equal variance. The resulting composite noise terms are

Z_{1}=U_{1}+U_{2}

and

Z_{2}=U_{1}-U_{2}

, so

\operatorname{Cov}(Z_{1},Z_{2})=0

and hence

Z_{1}\perp\!\!\!\perp Z_{2}

, despite the existence of

X_{1}

and

X_{2}

as they share latent confounders.

Figure 3: Estimation of the directed (left) and bidirected (right) edges of the ground-truth ADMG under four DGPs listed above. Bars report means over 20 random seeds, with 95% confidence intervals. Warm colors are the proposed LinADMG- methods, while hatched cold colors are baselines. The top row reports F1 score (

\uparrow

), and the bottom row reports SHD (

\downarrow

). LiNGAM does not infer bidirected edges and is therefore omitted.

SCM	Latent	Method	Directed F1	Bidirected F1
Nonlinear	$n_{1}$	-diagGMM	$.020\pm.000$	$.004\pm.000$
Nonlinear	$n_{1}$	-MuDoNLL	$.074\pm.000$	$.004\pm.000$
Nonlinear	$n_{1}$ + 50% $n_{2}$	-diagGMM	$.000\pm.000$	$.554\pm.000$
Nonlinear	$n_{1}$ + 50% $n_{2}$	-MuDoNLL	$.085\pm.000$	$.576\pm.000$
Linear	$n_{1}$ + 50% $n_{2}$	-diagGMM	$.508\pm.020$	$.800\pm.009$
Linear	$n_{1}$ + 50% $n_{2}$	-MuDoNLL	$.147\pm.031$	$.554\pm.024$

Table 1: Average F1 scores

\pm

standard deviation over 5 random seeds across different versions of causalAssembly with latent confounding with linear/nonlinear SCM and latent confounding settings.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Description
$\tau$	Domain index
$\mathbf{U}$	Unobserved variables
$\mathbf{X}$	Observed variables
$\mathbf{E}_{1}$	Exogenous variables of $\mathbf{X}$
$\mathbf{E}_{2}$	Exogenous variables of $\mathbf{U}$
$\mathbf{Z}^{\tau}$	Latent noise variables from domain $\tau$

Appendix

Table 5

Method	Hyperparameters
LinADMG-SP	lr = $10^{-4}$ ; num_steps =15000; num_initializations =1; batch_size =5000; sparse_level =0.01; aug_lag_coef =0
LinADMG-MuDoNLL	lr =0.01; num_steps =5000; num_initializations =1; scheduler = cosine ; grad_clip =1;
LinADMG-UMNI	atol_eigv =0.05; atol_ci_test =0.05; kappa =1; umni_batch_size =2048;
LinADMG-diagGMM	lr =0.001; num_steps =5000; num_initializations =1; batch_size =6144;
LiNGAM	lingam_measure = pwling ;
BANG	bang_moment_degree =3; bang_level =0.01; bang_restriction =1; bang_max_iterations =10000; bang_condition_limit = $10^{12}$ ;

Appendix

Table 2: Hyperparameters used in the experiments for all methods and baselines.

Figure 4: True vs. estimated directed-edge weights by LinADMG-MuDoNLL under multi-domain scalings DGP.

Setting	Value	LinADMG-MuDoNLL	LiNGAM	BANG	DCD	RCD
Directed edges
Dimension
$n$	5	0.927 $\pm$ 0.125	0.943 $\pm$ 0.107	0.000 $\pm$ 0.000	0.156 $\pm$ 0.139	0.197 $\pm$ 0.173
$n$	10	0.883 $\pm$ 0.196	0.815 $\pm$ 0.096	0.005 $\pm$ 0.021	0.078 $\pm$ 0.073	0.064 $\pm$ 0.083
$n$	20	0.854 $\pm$ 0.280	0.754 $\pm$ 0.069	0.005 $\pm$ 0.015	0.026 $\pm$ 0.034	0.033 $\pm$ 0.038
Number of domains

Appendix

Table 3: MD-Scalings results: average F1

\pm