Blind free deconvolution over one-parameter sparse families via eigenmatrix

TL;DR

This paper introduces a novel eigenmatrix-based approach to solve nonlinear blind free deconvolution problems for sparse spectral measures within one-parameter families, transforming them into linear problems using free probability transforms.

Contribution

It develops a new eigenmatrix method leveraging free probability transforms to linearize and solve nonlinear sparse spectral deconvolution problems in a parametric setting.

Findings

Successfully applied to additive free deconvolution

Effective for multiplicative free deconvolution

Numerical results demonstrate the method's viability

Abstract

This note considers the blind free deconvolution problems of sparse spectral measures from one-parameter families. These problems pose significant challenges since they involve nonlinear sparse recovery. The main technical tool is the eigenmatrix method for solving unstructured sparse recovery problems. The key idea is to turn the nonlinear inverse problem into a linear inverse problem by leveraging the R-transform for free addition and the S-transform for free product. The resulting linear problem is solved with the eigenmatrix method tailored to the domain of the parametric family. Numerical results are provided for both the additive and multiplicative free deconvolutions.