Joint Estimation of Precision Matrices in Heterogeneous Populations

TL;DR

This paper presents a novel framework for estimating precision matrices across heterogeneous populations using Laplacian shrinkage, an efficient ADMM algorithm, and a hierarchical clustering-based penalty, with proven consistency and demonstrated advantages in gene expression data analysis.

Contribution

It introduces a flexible, scalable method for joint precision matrix estimation in heterogeneous settings, incorporating a Laplacian penalty and hierarchical clustering for unknown population structures.

Findings

The proposed method achieves variable selection and norm consistency.

It outperforms existing approaches in gene expression data applications.

The ADMM algorithm efficiently handles high-dimensional data.

Abstract

We introduce a general framework for estimation of inverse covariance, or precision, matrices from heterogeneous populations. The proposed framework uses a Laplacian shrinkage penalty to encourage similarity among estimates from disparate, but related, subpopulations, while allowing for differences among matrices. We propose an efficient alternating direction method of multipliers (ADMM) algorithm for parameter estimation, as well as its extension for faster computation in high dimensions by thresholding the empirical covariance matrix to identify the joint block diagonal structure in the estimated precision matrices. We establish both variable selection and norm consistency of the proposed estimator for distributions with exponential or polynomial tails. Further, to extend the applicability of the method to the settings with unknown populations structure, we propose a Laplacian penalty based on hierarchical clustering, and discuss conditions under which this data-driven choice results in consistent estimation of precision matrices in heterogenous populations. Extensive numerical studies and applications to gene expression data from subtypes of cancer with distinct clinical outcomes indicate the potential advantages of the proposed method over existing approaches.