论文标题
具有信号检测应用的异质革兰氏矩阵的Tracy-Widom分布
Tracy-Widom distribution for heterogeneous Gram matrices with applications in signal detection
论文作者
论文摘要
检测高维噪声损坏的信号数量是信号处理和统计数据中的一个基本问题。本文着重于高维噪声具有未知复杂的异质方差结构的一般环境。我们提出了一个顺序测试，该测试利用了数据矩阵的边缘单数值（即最大的少数单数值）。它也自然会导致信号数量的一致顺序测试估计。我们描述了测试统计量的渐近分布，该分布在tracy-widom分布方面。该测试在理论上和数字上都具有准确性，并且对替代方案具有全功率。理论分析依赖于建立具有非零均值和完全任意差异概况的大型革兰氏型随机矩阵的Tracy-Widom定律，这可能具有独立的兴趣。
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.
