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Causal Discovery with Mixed Latent Confounding via Precision Decomposition

Amir Asiaee, Samhita Pal, James O'quinn, James P. Long

TL;DR

This work tackles causal discovery in linear Gaussian SEMs under mixed latent confounding, consisting of pervasive low-rank and sparse localized confounders. It introduces DCL-DECOR, a three-stage pipeline that (i) decomposes the observed precision into a structured component and a low-rank part, (ii) inverts the structured component to form a pervasive-adjusted covariance, and (iii) applies a correlated-noise DAG learner (DECOR-GL) with bow-free reconciliation to recover a minimal bow-free ADMG. The paper provides identifiability and stability guarantees for the precision decomposition and the conditional covariance, and it shows that the end-to-end approach consistently recovers a representative bow-free causal target under mild assumptions. Synthetic experiments demonstrate robust improvements in directed-edge recovery over applying standard DAG learners directly to confounded data, particularly as pervasive confounding strengthens or local confounding increases. Overall, the work offers a modular, principled path to causal discovery in the Gaussian, linear setting with mixed latent confounding, combining precision-domain deconfounding with state-of-the-art DAG learning.

Abstract

We study causal discovery from observational data in linear Gaussian systems affected by \emph{mixed latent confounding}, where some unobserved factors act broadly across many variables while others influence only small subsets. This setting is common in practice and poses a challenge for existing methods: differentiable and score-based DAG learners can misinterpret global latent effects as causal edges, while latent-variable graphical models recover only undirected structure. We propose \textsc{DCL-DECOR}, a modular, precision-led pipeline that separates these roles. The method first isolates pervasive latent effects by decomposing the observed precision matrix into a structured component and a low-rank component. The structured component corresponds to the conditional distribution after accounting for pervasive confounders and retains only local dependence induced by the causal graph and localized confounding. A correlated-noise DAG learner is then applied to this deconfounded representation to recover directed edges while modeling remaining structured error correlations, followed by a simple reconciliation step to enforce bow-freeness. We provide identifiability results that characterize the recoverable causal target under mixed confounding and show how the overall problem reduces to well-studied subproblems with modular guarantees. Synthetic experiments that vary the strength and dimensionality of pervasive confounding demonstrate consistent improvements in directed edge recovery over applying correlated-noise DAG learning directly to the confounded data.

Causal Discovery with Mixed Latent Confounding via Precision Decomposition

TL;DR

This work tackles causal discovery in linear Gaussian SEMs under mixed latent confounding, consisting of pervasive low-rank and sparse localized confounders. It introduces DCL-DECOR, a three-stage pipeline that (i) decomposes the observed precision into a structured component and a low-rank part, (ii) inverts the structured component to form a pervasive-adjusted covariance, and (iii) applies a correlated-noise DAG learner (DECOR-GL) with bow-free reconciliation to recover a minimal bow-free ADMG. The paper provides identifiability and stability guarantees for the precision decomposition and the conditional covariance, and it shows that the end-to-end approach consistently recovers a representative bow-free causal target under mild assumptions. Synthetic experiments demonstrate robust improvements in directed-edge recovery over applying standard DAG learners directly to confounded data, particularly as pervasive confounding strengthens or local confounding increases. Overall, the work offers a modular, principled path to causal discovery in the Gaussian, linear setting with mixed latent confounding, combining precision-domain deconfounding with state-of-the-art DAG learning.

Abstract

We study causal discovery from observational data in linear Gaussian systems affected by \emph{mixed latent confounding}, where some unobserved factors act broadly across many variables while others influence only small subsets. This setting is common in practice and poses a challenge for existing methods: differentiable and score-based DAG learners can misinterpret global latent effects as causal edges, while latent-variable graphical models recover only undirected structure. We propose \textsc{DCL-DECOR}, a modular, precision-led pipeline that separates these roles. The method first isolates pervasive latent effects by decomposing the observed precision matrix into a structured component and a low-rank component. The structured component corresponds to the conditional distribution after accounting for pervasive confounders and retains only local dependence induced by the causal graph and localized confounding. A correlated-noise DAG learner is then applied to this deconfounded representation to recover directed edges while modeling remaining structured error correlations, followed by a simple reconciliation step to enforce bow-freeness. We provide identifiability results that characterize the recoverable causal target under mixed confounding and show how the overall problem reduces to well-studied subproblems with modular guarantees. Synthetic experiments that vary the strength and dimensionality of pervasive confounding demonstrate consistent improvements in directed edge recovery over applying correlated-noise DAG learning directly to the confounded data.
Paper Structure (63 sections, 8 theorems, 43 equations, 1 figure)

This paper contains 63 sections, 8 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Why mixed confounding is hard.
  3. Our view: deconfound in precision, learn in data.
  4. A structural condition that pays off.
  5. Contributions.
  6. Positioning with prior art.
  7. Related Work
  8. Latent variables and mixed graphs.
  9. Continuous optimization for DAG learning.
  10. Differentiable discovery under latent confounding.
  11. Low-rank + sparse decompositions for latent structure (undirected).
  12. Non-Gaussian and nonlinear identifiability.
  13. Position of the present work.
  14. Problem Setup
  15. Subscript convention (noise vs. observed level).
  16. ...and 48 more sections

Key Result

Proposition 2

Assume the model in §sec:setup with $\mathbf{T}=\mathbf{I}-\mathbf{B}$ unit-diagonal and triangular under some causal order, and define $\mathbf{D}_\varepsilon,\mathbf{C}_\varepsilon,\mathbf{S}_\varepsilon,\mathbf{L}_\varepsilon$ as above.

Figures (1)

  • Figure 1: Mixed-confounding synthetic experiments. (a) Mean $\Delta$F1 relative to DECOR-GL across pervasive rank ($q_P$) and strength ($U_d$). (b--c) With pervasive confounding fixed, performance as localized confounding density ($L_d$) increases (mean over 10 replicates).

Theorems & Definitions (10)

  • Remark 1: Beyond sparsity: other local structure classes
  • Proposition 2: Low-rankness of $\mathbf{L}_\varepsilon$ and locality of $\mathbf{S}_\varepsilon$
  • Proposition 3: Conditional precision
  • Remark 4: Connection to NOTEARS
  • Lemma 5: Inversion perturbation bound
  • Theorem 6: Modular reduction
  • Theorem 7: End-to-end consistency, informal
  • Proposition 8: Low-rankness of $\mathbf{L}_\varepsilon$ and locality of $\mathbf{S}_\varepsilon$
  • Proposition 9: Controlled leakage under relaxed orthogonality
  • Proposition 10: Structure preservation under $\mathbf{T}$-congruence