Rapid evaluation of the spectral signal detection threshold and Stieltjes transform

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) in the isotropic noise setting. However, existing methods for numerically evaluating the threshold for more general noise models remain unsatisfactory. In this work, we introduce a rapid algorithm for evaluating the spectral signal detection threshold. We consider noise matrices with a separable variance profile, 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.