Hypothesis Testing for High-Dimensional Matrix-Valued Data

TL;DR

This paper develops new hypothesis testing methods for high-dimensional matrix data, addressing limitations of existing tests and introducing sparse SVD techniques, with theoretical analysis and practical case studies.

Contribution

It proposes a novel high-dimensional matrix rank test and a sparse SVD estimator, enhancing hypothesis testing capabilities in high-dimensional settings.

Findings

The new test outperforms existing methods in high-dimensional scenarios.

The sparse SVD estimator effectively captures singular vectors in high dimensions.

Simulation and case studies demonstrate practical utility of the proposed methods.

Abstract

This paper addresses hypothesis testing for the mean of matrix-valued data in high-dimensional settings. We investigate the minimum discrepancy test, originally proposed by Cragg (1997), which serves as a rank test for lower-dimensional matrices. We evaluate the performance of this test as the matrix dimensions increase proportionally with the sample size, and identify its limitations when matrix dimensions significantly exceed the sample size. To address these challenges, we propose a new test statistic tailored for high-dimensional matrix rank testing. The oracle version of this statistic is analyzed to highlight its theoretical properties. Additionally, we develop a novel approach for constructing a sparse singular value decomposition (SVD) estimator for singular vectors, providing a comprehensive examination of its theoretical aspects. Using the sparse SVD estimator, we explore the properties of the sample version of our proposed statistic. The paper concludes with simulation studies and two case studies involving surveillance video data, demonstrating the practical utility of our proposed methods.