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.

中文翻译：

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Tracy-Widom 分布的异构革兰氏矩阵在信号检测中的应用

检测被高维噪声破坏的信号数量是信号处理和统计中的一个基本问题。本文重点关注高维噪声具有未知复杂异质方差结构的一般设置。我们提出了一种利用数据矩阵的边缘奇异值（即最大的几个奇异值）的顺序测试。它还自然地导致信号数量的一致顺序测试估计。我们用 Tracy-Widom 分布来描述检验统计量的渐近分布。该测试被证明是准确的，并且在理论上和数值上都具有对抗替代方案的全部力量。