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.

翻译：本文针对高维场景下矩阵值数据的均值假设检验问题展开研究。我们探讨了最初由Cragg（1997）提出的最小差异检验，该检验是针对低维矩阵的秩检验方法。我们评估了当矩阵维度与样本量成比例增长时该检验的性能，并指出了当矩阵维度显著超过样本量时其存在的局限性。为应对这些挑战，我们提出了一种专为高维矩阵秩检验设计的新检验统计量。通过分析该统计量的理论版本，我们揭示了其理论特性。此外，我们开发了一种构建奇异向量稀疏奇异值分解（SVD）估计量的新方法，并对其理论性质进行了全面考察。利用稀疏SVD估计量，我们探究了所提出统计量的样本版本性质。本文最后通过模拟研究和两个涉及监控视频数据的案例研究，验证了所提出方法的实际效用。