DCL-DECOR: Modular Causal Discovery

• DCL-DECOR is a modular precision-based causal discovery framework that decomposes sample precision matrices to isolate both pervasive and localized latent confounders in linear Gaussian systems.
• It applies a two-stage strategy using latent-variable graphical lasso and correlated-noise DAG learning to accurately recover bow-free directed acyclic graphs.
• Empirical evaluations show improved directed-edge F1 scores and reduced structural errors compared to alternative methods under simulated mixed confounding scenarios.

DCL-DECOR refers to a modular, precision-based framework for causal discovery in the presence of mixed latent confounding—where both pervasive (broad) and localized (subset-specific) unobserved confounders affect observed data—within linear Gaussian systems. DCL-DECOR decomposes the observed sample precision matrix into a structured component and a low-rank component, isolates the impact of pervasive confounders, and recovers the directed acyclic graph (DAG) structure using correlated-noise SEM learning, followed by a reconciliation step to ensure bow-freeness. The method establishes identifiability conditions and demonstrates empirical gains in directed-edge discovery under simulated mixed confounding scenarios (Asiaee et al., 31 Dec 2025).

1. Mixed Latent Confounding in SEMs

DCL-DECOR addresses the challenge of causal discovery when observed variables $x \in \mathbb{R}^p$ are generated by a structural equation model (SEM) with both global and local latent confounders:

$x = B^\top x + \varepsilon,$

where the exogenous noise vector $\varepsilon$ is modeled as

$\varepsilon = Ww + Vv + Uu,$

with $w \sim \mathcal{N}(0,I_p)$ (idiosyncratic noise), $v \sim \mathcal{N}(0, I_{r_S})$ (localized confounders), and $u \sim \mathcal{N}(0, I_{r_L})$ (pervasive confounders). $V \in \mathbb{R}^{p \times r_S}$ is column-sparse, representing localized effects; $U \in \mathbb{R}^{p \times r_L}$ is dense, modeling widespread confounding; $W$ is diagonal. The observed covariance $\Sigma$ and precision $\Theta = \Sigma^{-1}$ are related by

$$\Sigma = T^{-\top} \Omega T^{-1}, \quad \Theta = T \Omega^{-1} T^\top,$$

where $T = I - B$ and $\Omega = \mathrm{Var}(\varepsilon)$.

Traditional DAG-learning methods misattribute latent-effects-induced correlations, leading to incorrect causal edge inference. Latent variable graphical models capture undirected structure but fail to orient edges. DCL-DECOR modularizes the problem by first deconfounding for pervasive effects, then learning the directed structure in the conditional model (Asiaee et al., 31 Dec 2025).

2. Precision Decomposition

DCL-DECOR performs a two-stage precision decomposition:

• At the noise level:
  • Uses the Sherman–Morrison–Woodbury identity to split $\Omega^{-1}$ into a structured component $S_{\varepsilon}$ (from idiosyncratic and localized confounders) and low-rank corrective term $L_{\varepsilon}$ (from pervasive confounders):

  $\Omega^{-1} = S_\varepsilon - L_\varepsilon,$

  with explicit forms: - $$S_\varepsilon = (WW^\top + VV^\top)^{-1} = D_\varepsilon - C_\varepsilon,$$ with $D_\varepsilon = (WW^\top)^{-1}$, $$C_\varepsilon = D_\varepsilon V A^{-1} V^\top D_\varepsilon$$, $A = I + V^\top D_\varepsilon V$. - $$L_\varepsilon = S_\varepsilon U (I + U^\top S_\varepsilon U)^{-1} U^\top S_\varepsilon.$$

• At the observed level:
  • By congruence transformation, the observed precision matrix splits as

  $\Theta = S_x - L_x,$

  where $S_x = T S_\varepsilon T^\top$ encodes structured (local, typically sparse) dependencies and $L_x = T L_\varepsilon T^\top$ is low-rank, rank $\leq r_L$.

• Estimation: The decomposition is recovered via convex optimization (“latent-variable graphical lasso”):

$$\min_{S,L} \left[ -\log\det(S-L) + \mathrm{tr}(\widehat{\Sigma}(S-L)) + \lambda_s R_{\mathrm{loc}}(S) + \lambda_* \mathrm{tr}(L) \right],$$

subject to $S \succ 0, L \succeq 0, S-L \succ 0$, where $R_{\mathrm{loc}}$ enforces locality/sparsity in $S$ and the nuclear norm $\mathrm{tr}(L)$ proxies low-rank (Asiaee et al., 31 Dec 2025).

3. Correlated-Noise DAG Learning on the Structured Component

Once pervasive confounders have been partialled out, the conditional covariance $\Sigma_{\mathrm{cond}} = S_x^{-1}$ captures only local structure and localized confounders. DCL-DECOR employs a correlated-noise DAG learner (DECOR-GL):

• Objective:

$$\min_{B, S_\varepsilon \succ 0} \,\, \mathcal{L}(B, S_\varepsilon; \widehat{\Sigma}_{\mathrm{cond}})$$

$$\mathcal{L} = \mathrm{tr}[\widehat{\Sigma}_{\mathrm{cond}} T S_\varepsilon T^\top] - \log\det S_\varepsilon + \lambda_B \|B\|_1 + \lambda_S \|S_\varepsilon\|_{1,\mathrm{off}} + \rho h(B)$$

where $h(B) = \mathrm{tr}(e^{B \circ B}) - p$ enforces acyclicity, $T = I - B$.

• Optimization: Alternates between graph-step (proximal gradient on $B$) and noise-step (graphical lasso on $S_\varepsilon$). After convergence, hard-thresholding is used to obtain a sparse graph (Asiaee et al., 31 Dec 2025).

4. Bow-Freeness Reconciliation

Due to unidentifiability in linear Gaussian ADMGs, "bows"—simultaneous presence of a directed edge and residual correlation between the same variable pair—cannot be inferred. DCL-DECOR enforces bow-freeness post hoc:

• For each pair with both a directed edge ($B_{ij} \neq 0$ or $B_{ji} \neq 0$) and bidirected residual entry ($\Gamma_{ij} \neq 0$ in $\Gamma = S_\varepsilon^{-1}$), retain the element with larger normalized magnitude; set the other to zero.
• The bow constant $c$ (typically $c=1$) calibrates the comparison.

This ensures the final output is a bow-free mixed graph (Asiaee et al., 31 Dec 2025).

5. Identifiability and Theoretical Guarantees

DCL-DECOR’s structure–low-rank decomposition is uniquely identifiable under the transversality condition (tangent cones at $(S_x, L_x)$ of the structured and low-rank varieties intersect only at the origin) and standard incoherence requirements. Given convergence of the decomposition solver and stability in DAG recovery, the causal target is characterized as the minimal bow-free equivalence class

$$\mathcal{E}_{\mathrm{bow}}^{\min}(\Sigma_{\mathrm{cond}}) = \arg\min \{\|B\|_0 + \|\Gamma_{\mathrm{off}}\|_0 : (B,\Gamma) \text{ bow-free}, \Sigma(B,\Gamma) = \Sigma_{\mathrm{cond}} \}.$$

End-to-end consistency is established: under appropriate conditions, the DCL-DECOR output converges to an element in this equivalence class (Asiaee et al., 31 Dec 2025).

6. Algorithmic Workflow

The DCL-DECOR pipeline follows:

1. Compute the empirical covariance $\widehat{\Sigma}$ of observed data.
2. Solve the structured–low-rank split via convex optimization (latent-variable graphical lasso), yielding $(\widehat{S}_x, \widehat{L}_x)$.
3. Invert $\widehat{S}_x$ (using sparse Cholesky) to obtain $\widehat{\Sigma}_{\mathrm{cond}}$.
4. Apply correlated-noise DAG learning (DECOR-GL) to $\widehat{\Sigma}_{\mathrm{cond}}$, alternating optimization over $B$, $S_\varepsilon$ until convergence.
5. Threshold $B$ and $\Gamma = S_\varepsilon^{-1}$; enforce bow-freeness using the bow reconciliation rule.
6. Output: final bow-free estimate $(\widehat{B}, \widehat{\Gamma}_\varepsilon)$ (Asiaee et al., 31 Dec 2025).

7. Empirical Evaluation

Extensive synthetic experiments evaluate DCL-DECOR’s performance. Scenarios with $p = 40$ variables, $n = 600$ samples, and both pervasive ($r_L$ up to 5) and localized ($r_S=15,$ support size 6) confounders were considered. DCL-DECOR consistently outperformed alternative methods (DECOR-GL, DeCAMFounder, NOTEARS, GOLEM, GES, LiNGAM) in directed-edge $F_1$ score and structural Hamming distance, especially as the strength and rank of pervasive confounding increased. For example, with $q_P=3$, $U_d=1.0$, mean $F_1$ was $0.431$ for DCL-DECOR versus $0.266$ for DECOR-GL; mean SHD $55.0$ versus $76.2$. The results demonstrate that decomposing precision matrices to account for mixed confounding and imposing bow-freeness are critical for accurate causal discovery in this regime (Asiaee et al., 31 Dec 2025).