Signal from noise retrieval from one and two-point Green's function - comparison

TL;DR

This paper compares one-point and two-point Green's function methods for eigen-inference, finding the one-point approach faster, more stable, and more advantageous, especially for real-valued and complex Wishart distributions.

Contribution

The study demonstrates the superiority of one-point Green's function eigen-inference over two-point methods in speed, stability, and applicability to different distributions.

Findings

One-point Green's function method is significantly faster.

Two-point Green's function method has potential instability issues.

One-point method performs better for Wishart distributions.

Abstract

We compare two methods of eigen-inference from large sets of data, based on the analysis of one-point and two-point Green's functions, respectively. Our analysis points at the superiority of eigen-inference based on one-point Green's function. First, the applied by us method based on Pad?e approximants is orders of magnitude faster comparing to the eigen-inference based on uctuations (two-point Green's functions). Second, we have identified the source of potential instability of the two-point Green's function method, as arising from the spurious zero and negative modes of the estimator for a variance operator of the certain multidimensional Gaussian distribution, inherent for the two-point Green's function eigen-inference method. Third, we have presented the cases of eigen-inference based on negative spectral moments, for strictly positive spectra. Finally, we have compared the cases of eigen-inference of real-valued and complex-valued correlated Wishart distributions, reinforcing our conclusions on an advantage of the one-point Green's function method.