Accurate detection of signal components is a frequently-encountered challenge in statistical applications with a low signal-to-noise ratio. This problem is particularly challenging in settings with heteroscedastic noise. In certain signal-plus-noise models of data, such as the classical spiked covariance model and its variants, there are closed formulas for the spectral signal detection threshold (the largest sample eigenvalue attributable solely to noise) for isotropic noise in the limit of infinitely large data matrices. However, more general noise models currently lack provably fast and accurate methods for numerically evaluating the threshold. In this work, we introduce a rapid algorithm for evaluating the spectral signal detection threshold in the limit of infinitely large data matrices. We consider noise matrices with a separable variance profile (whose variance matrix is rank 1), as these arise often in applications. The solution is based on nested applications of Newton’s method. We also devise a new algorithm for evaluating the Stieltjes transform of the spectral distribution at real values exceeding the threshold. The Stieltjes transform on this domain is known to be a key quantity in parameter estimation for spectral denoising methods. The correctness of both algorithms is proven from a detailed analysis of the master equations characterizing the Stieltjes transform, and their performance is demonstrated in numerical experiments.

信号分量的准确检测是低信噪比统计应用中经常遇到的挑战。这个问题在具有异方差噪声的环境中尤其具有挑战性。在某些数据的信号加噪声模型中，例如经典的尖峰协方差模型及其变体，对于无限大范围内的各向同性噪声的频谱信号检测阈值（仅可归因于噪声的最大样本特征值）存在闭合公式大数据矩阵。然而，更通用的噪声模型目前缺乏可证明快速和准确的数值评估阈值的方法。在这项工作中，我们引入了一种快速算法，用于在无限大数据矩阵的限制下评估光谱信号检测阈值。我们考虑具有可分离方差分布的噪声矩阵（其方差矩阵为 1 级），因为这些在应用程序中经常出现。该解决方案基于牛顿方法的嵌套应用。我们还设计了一种新算法，用于评估超出阈值的实际值时光谱分布的 Stieltjes 变换。已知该域上的 Stieltjes 变换是频谱去噪方法参数估计中的关键量。通过对表征 Stieltjes 变换的主方程的详细分析，证明了两种算法的正确性，并在数值实验中证明了它们的性能。我们还设计了一种新算法，用于评估超出阈值的实际值时光谱分布的 Stieltjes 变换。已知该域上的 Stieltjes 变换是光谱去噪方法参数估计中的关键量。通过对表征 Stieltjes 变换的主方程的详细分析，证明了两种算法的正确性，并在数值实验中证明了它们的性能。我们还设计了一种新算法，用于评估超出阈值的实际值时光谱分布的 Stieltjes 变换。已知该域上的 Stieltjes 变换是光谱去噪方法参数估计中的关键量。通过对表征 Stieltjes 变换的主方程的详细分析，证明了两种算法的正确性，并在数值实验中证明了它们的性能。