Machine Learning-Assisted High-Dimensional Matrix Estimation

arXiv stat.ML / 3/31/2026

💬 OpinionIdeas & Deep AnalysisModels & Research

共有:

Key Points

The paper addresses computational challenges in estimating high-dimensional matrices such as covariance and precision matrices, moving beyond prior work that focused mainly on statistical properties like consistency and sparsity.
It proposes a machine learning–assisted optimization approach by starting with Linearized ADMM (LADMM) and introducing learnable parameters that replace/proxy proximal operators via neural networks within the iterative scheme.
The authors provide theoretical guarantees, including convergence of standard LADMM and convergence, rate, and monotonicity for the reparameterized (learnable) LADMM variant.
They claim the reparameterized LADMM achieves a faster convergence rate and that the methodology can be applied to both covariance and precision matrix estimation.
Experiments compare the proposed method against multiple classical optimization baselines across different matrix structures and dimensionalities to demonstrate improved accuracy and faster convergence.

Abstract

Efficient estimation of high-dimensional matrices-including covariance and precision matrices-is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators (e.g., consistency and sparsity), while largely overlooking the computational challenges inherent in high-dimensional settings. Motivated by recent advances in learning-based optimization method-which integrate data-driven structures with classical optimization algorithms-we explore high-dimensional matrix estimation assisted by machine learning. Specifically, for the optimization problem of high-dimensional matrix estimation, we first present a solution procedure based on the Linearized Alternating Direction Method of Multipliers (LADMM). We then introduce learnable parameters and model the proximal operators in the iterative scheme with neural networks, thereby improving estimation accuracy and accelerating convergence. Theoretically, we first prove the convergence of LADMM, and then establish the convergence, convergence rate, and monotonicity of its reparameterized counterpart; importantly, we show that the reparameterized LADMM enjoys a faster convergence rate. Notably, the proposed reparameterization theory and methodology are applicable to the estimation of both high-dimensional covariance and precision matrices. We validate the effectiveness of our method by comparing it with several classical optimization algorithms across different structures and dimensions of high-dimensional matrices.