Covariance Estimation for Matrix-valued Data

• 成果类型: Article
• 署名作者: Zhang, Yichi; Shen, Weining; Kong, Dehan
• 署名单位: North Carolina State University; University of California System; University of California Irvine; University of Toronto
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2022.2068419
• 发表日期: 2023
• 页码: 2620-2631
• 关键词: high-dimensional covariance; Optimal Rates; product; models; bounds

摘要

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 sim ulations and real applications from a gridded temperature anomalies dataset and an S&P 500 stock data analysis. Supplementary materials for this article are available online.