Covariance Estimation for Matrix-valued Data

TL;DR

This paper introduces a distribution-free, regularized covariance estimation framework for high-dimensional matrix data, leveraging separability and bandable structures, with proven optimal convergence rates and robustness to heavy tails.

Contribution

It proposes a novel, distribution-free approach for estimating covariance matrices of matrix-valued data under separability and bandability, with theoretical guarantees and robust variants.

Findings

Establishes convergence rates and minimax optimality of the estimators.

Demonstrates superior finite-sample performance through simulations.

Validates methods on temperature and stock market datasets.

Abstract

Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.