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Induced Covariance for Causal Discovery in Linear Sparse Structures

Saeed Mohseni-Sehdeh, Walid Saad

TL;DR

The paper addresses causal discovery when relationships are linear and sparse, with causal relations represented by a structural matrix $\\mathbf{D}$. It introduces Sparse Linear Causal Discovery (SLCD), which recovers $\\mathbf{D}$ by linking data reconstruction $\\mathbf{X}=\\mathbf{D}\\mathbf{X}$ with an induced covariance constraint $\\boldsymbol{\\Sigma}=\\mathbf{D}\\boldsymbol{\\sigma}\\mathbf{D}^T$, where $\\boldsymbol{\\sigma}$ is diagonal, and by enforcing per-row sparsity through a rank surrogate. The optimization minimizes a surrogate for the matrix rank plus a diagonal-trace penalty, under constraints and iterative pruning, with multiple random starts to ensure a good solution. Simulation results show that SLCD outperforms classic causal discovery methods (PC, GES, LINGAM IC/Direct, BIC exact search) on synthetic datasets, achieving substantially higher precision (≈35%) and recall (≈41.5%) on most datasets, indicating robustness to small sample sizes and sparse linear structures. The work offers a scalable, independence-test-free approach to causal discovery in linear sparse regimes, with potential impact for applications requiring data-efficient causal structure recovery and transferability across related problems.

Abstract

Causal models seek to unravel the cause-effect relationships among variables from observed data, as opposed to mere mappings among them, as traditional regression models do. This paper introduces a novel causal discovery algorithm designed for settings in which variables exhibit linearly sparse relationships. In such scenarios, the causal links represented by directed acyclic graphs (DAGs) can be encapsulated in a structural matrix. The proposed approach leverages the structural matrix's ability to reconstruct data and the statistical properties it imposes on the data to identify the correct structural matrix. This method does not rely on independence tests or graph fitting procedures, making it suitable for scenarios with limited training data. Simulation results demonstrate that the proposed method outperforms the well-known PC, GES, BIC exact search, and LINGAM-based methods in recovering linearly sparse causal structures.

Induced Covariance for Causal Discovery in Linear Sparse Structures

TL;DR

The paper addresses causal discovery when relationships are linear and sparse, with causal relations represented by a structural matrix . It introduces Sparse Linear Causal Discovery (SLCD), which recovers by linking data reconstruction with an induced covariance constraint , where is diagonal, and by enforcing per-row sparsity through a rank surrogate. The optimization minimizes a surrogate for the matrix rank plus a diagonal-trace penalty, under constraints and iterative pruning, with multiple random starts to ensure a good solution. Simulation results show that SLCD outperforms classic causal discovery methods (PC, GES, LINGAM IC/Direct, BIC exact search) on synthetic datasets, achieving substantially higher precision (≈35%) and recall (≈41.5%) on most datasets, indicating robustness to small sample sizes and sparse linear structures. The work offers a scalable, independence-test-free approach to causal discovery in linear sparse regimes, with potential impact for applications requiring data-efficient causal structure recovery and transferability across related problems.

Abstract

Causal models seek to unravel the cause-effect relationships among variables from observed data, as opposed to mere mappings among them, as traditional regression models do. This paper introduces a novel causal discovery algorithm designed for settings in which variables exhibit linearly sparse relationships. In such scenarios, the causal links represented by directed acyclic graphs (DAGs) can be encapsulated in a structural matrix. The proposed approach leverages the structural matrix's ability to reconstruct data and the statistical properties it imposes on the data to identify the correct structural matrix. This method does not rely on independence tests or graph fitting procedures, making it suitable for scenarios with limited training data. Simulation results demonstrate that the proposed method outperforms the well-known PC, GES, BIC exact search, and LINGAM-based methods in recovering linearly sparse causal structures.
Paper Structure (7 sections, 1 theorem, 17 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 7 sections, 1 theorem, 17 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Proposed Algorithm
  3. Problem statement
  4. Challenges
  5. Solutions
  6. Simulation Results and Analysis
  7. Conclusion

Key Result

Theorem 1

Consider $\boldsymbol{D} \in \mathbb R^{n\times n}$ to be a matrix that represents a linear causal structure governing the zero mean variables $\boldsymbol{x} = [x_1, x_2, \ldots, x_n]^T \in \mathbb R^n$. The covariance matrix of these variables is given by $\boldsymbol{D}\boldsymbol{\sigma}\boldsym

Figures (3)

  • Figure 1: Structure estimation error for various datasets and various hyperparameters.
  • Figure 2: Reconstruction error for various datasets and various hyperparameters.
  • Figure 3: Covariance matrix estimation error for various datasets and various hyperparameters.

Theorems & Definitions (3)

  • Example 1
  • Theorem 1
  • proof