Convex Formulation for Regularized Estimation of Structural Equation Models

TL;DR

This paper introduces a convex, regularized estimation approach for structural equation models, enabling efficient causal inference and low-rank solutions, demonstrated on climate and neuroimaging data.

Contribution

It proposes a convex formulation for SEM with regularization, relaxing nonlinear constraints to improve scalability and interpretability in causal modeling.

Findings

Convex formulation enables scalable SEM estimation.

Regularization reveals low-rank, interpretable solutions.

Method successfully applied to climate and fMRI data.

Abstract

Path analysis is a model class of structural equation modeling (SEM), which it describes causal relations among measured variables in the form of a multiple linear regression. This paper presents two estimation formulations, one each for confirmatory and exploratory SEM, where a zero pattern of the estimated path coefficient matrix can explain a causality structure of the variables. The original nonlinear equality constraints of the model parameters were relaxed to an inequality, allowing the transformation of the original problem into a convex framework. A regularized estimation formulation was then proposed for exploratory SEM using an l1-type penalty of the path coefficient matrix. Under a condition on problem parameters, our optimal solution is low rank and provides a useful solution to the original problem. Proximal algorithms were applied to solve our convex programs in a large-scale setting. The performance of this approach was demonstrated in both simulated and real data sets, and in comparison with an existing method. When applied to two real application results (learning causality among climate variables in Thailand and examining connectivity differences in autism patients using fMRI time series from ABIDE data sets) the findings could explain known relationships among environmental variables and discern known and new brain connectivity differences, respectively.