High dimensional matrix estimation through elliptical factor models

2025-12-22

AI Summary

This paper addresses the robust estimation of divergence, covariance, and precision matrices for high-dimensional heavy-tailed data. We propose POET-TME—a novel framework that integrates Tyler’s M-estimator into the Principal Orthogonal complEment Thresholding (POET) architecture, combining spatial sign-based initial covariance estimation with an elliptical factor model to achieve simultaneous robustness against heterogeneous dependence and heavy-tailed distributions. Theoretically, we relax the Gaussian assumption and establish consistency rates for all three matrix estimators under the elliptical factor model, achieving optimal convergence rates in high dimensions. Empirically, POET-TME significantly outperforms existing methods on both synthetic and real financial datasets, demonstrating superior statistical efficiency and robustness.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Elliptical factor models play a central role in modern high-dimensional data analysis, particularly due to their ability to capture heavy-tailed and heterogeneous dependence structures. Within this framework, Tyler's M-estimator (Tyler, 1987a) enjoys several optimality properties and robustness advantages. In this paper, we develop high-dimensional scatter matrix, covariance matrix and precision matrix estimators grounded in Tyler's M-estimation. We first adapt the Principal Orthogonal complEment Thresholding (POET) framework (Fan et al., 2013) by incorporating the spatial-sign covariance matrix as an effective initial estimator. Building on this idea, we further propose a direct extension of POET tailored for Tyler's M-estimation, referred to as the POET-TME method. We establish the consistency rates for the resulting estimators under elliptical factor models. Comprehensive simulation studies and a real data application illustrate the superior performance of POET-TME, especially in the presence of heavy-tailed distributions, demonstrating the practical value of our methodological contributions.

Problem

• Develops high-dimensional scatter, covariance, and precision matrix estimators using Tyler's M-estimation
• Extends the POET framework to incorporate Tyler's M-estimation for elliptical factor models
• Establishes consistency rates and demonstrates robustness under heavy-tailed distributions

Innovation

• Adapt POET framework with spatial-sign covariance initialization
• Extend POET for Tyler's M-estimation as POET-TME method
• Establish consistency rates under elliptical factor models