Tracy-Widom Distribution for Heterogeneous Gram Matrices With Applications in Signal Detection
Tracy-Widom Distribution for Heterogeneous Gram Matrices With Applications in Signal Detection
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DOI：
10.1109/tit.2022.3176784
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发表时间：
2020-08
影响因子：
2.5
通讯作者：
Xiucai Ding;F. Yang
中科院分区：
文献类型：
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作者：
Xiucai Ding;F. Yang
高维噪声污染信号的检测是信号处理和统计学中的一个基本问题。本文研究了高维噪声具有未知的复杂异质方差结构的一般情况。我们提出了一个顺序测试，利用边缘奇异值（即，最大的几个奇异值）。它也自然地导致信号数量的一致的顺序测试估计。我们描述的Tracy-Widom分布的检验统计量的渐近分布。该测试被证明是准确的，并有充分的权力对替代品，无论是理论上和数值。理论分析依赖于建立Tracy-Widom定律的一大类Gram型随机矩阵的非零均值和完全任意的方差轮廓，这可能是独立的利益。
Detection of the number of signals corrupted by high-dimensional noise is a fundamental problem in signal processing and statistics. This paper focuses on a general setting where the high-dimensional noise has an unknown complicated heterogeneous variance structure. We propose a sequential test which utilizes the edge singular values (i.e., the largest few singular values) of the data matrix. It also naturally leads to a consistent sequential testing estimate of the number of signals. We describe the asymptotic distribution of the test statistic in terms of the Tracy-Widom distribution. The test is shown to be accurate and have full power against the alternative, both theoretically and numerically. The theoretical analysis relies on establishing the Tracy-Widom law for a large class of Gram type random matrices with non-zero means and completely arbitrary variance profiles, which can be of independent interest.