Joint Nuclear and $\ell_1$ Regularization for Logistic Matrix Regression with Applications to Brain Imaging

2026-06-12

AI Summary

This study addresses the challenge of interpretable modeling between high-dimensional matrix-valued covariates and binary responses, such as identifying functional connectivity patterns associated with a family history of alcohol use disorder in neuroimaging data. The authors propose a novel logistic matrix regression method that uniquely integrates nuclear norm and ℓ₁ regularization to simultaneously induce low-rank and sparse structures in the coefficient matrix. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed, accompanied by theoretical guarantees. Numerical experiments and real neuroimaging data analysis demonstrate that the proposed approach effectively recovers biologically meaningful predictive structures and successfully identifies key brain functional connections significantly associated with a family history of alcohol use disorder.

Abstract

We introduce a new convex optimization framework for logistic scalar-on-matrix regression which incorporates nuclear and $\ell_1$ norm penalties to enforce simultaneously low-rank and sparse structures in the estimated coefficient matrix. The proposed method enables interpretable modeling of high-dimensional matrix-valued predictors in the presence of binary responses. We derive a custom algorithm based on the Alternating Direction Method of Multipliers (ADMM) to efficiently solve the resulting convex optimization problem and establish the theoretical properties of the obtained solution. Numerical experiments clearly demonstrate the effectiveness of our method in recovering meaningful predictive patterns. Finally, we apply our method to the brain imaging data to identify structures in functional brain connectivity matrices that are characteristic of subjects with a family history of alcohol use disorders (AUDs).

Problem

• logistic matrix regression
• low-rank
• sparsity
• brain imaging
• binary response

Innovation

• nuclear norm regularization
• ℓ₁ regularization
• logistic matrix regression
• ADMM algorithm
• brain connectivity