Accurate detection of signal components is a frequently-encountered challenge in statistical applications with 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 one), 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.

翻译：对信号元件的精确检测是信号到噪音比率低的统计应用中经常遇到的挑战。 这个问题在信息到音量比率低的统计应用中尤其具有挑战性。 在使用超强噪音的环境下, 这个问题特别具有挑战性。 在一些信号加噪音的数据模型中, 如古典的加热共变异模型及其变异模型, 光谱信号检测阈值( 最大的样本比值仅因噪音) 是无限大的数据矩阵限制范围内的异调噪音的最大样本。 然而, 更普遍的噪音模型目前缺乏可辨别的快速和准确的数值评估阈值的方法。 在这项工作中, 我们采用快速的算法来评估无穷大数据矩阵限内的光谱信号检测阈值。 我们考虑的是具有分立差异性特征的矩阵( 其差异矩阵为一级 ), 其解决方案以牛顿方法的嵌套应用为基础。 我们还设计了一种新的算法, 用于评价真实值超过阈值的光谱分布变异的转换方法。 Stieltjejels 和这个域的变形模型的变异性模型分析中, 其变的精确性模型的模型被证实成为一个关键数的模型。