A regularized eigenmatrix method for unstructured sparse recovery

TL;DR

This paper introduces a regularized eigenmatrix approach for unstructured sparse recovery that enhances numerical stability and robustness against noise by incorporating Tikhonov regularization into the eigenmatrix method.

Contribution

It proposes a novel regularization technique for the eigenmatrix method, replacing pseudo-inverse computations with Tikhonov regularization to improve stability and performance.

Findings

Improved reconstruction accuracy in noisy conditions

Enhanced numerical stability with regularization

Effective in a wide range of kernel functions

Abstract

The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to numerical instability if the threshold tolerance is not appropriately chosen. To incorporate regularization techniques, we propose to regularize the eigenmatrix method by replacing the computation of an ill-conditioned pseudo-inverse by the solution of an ill-conditioned least square system, which can be efficiently treated by Tikhonov regularization. Extensive numerical examples confirmed the improved effectiveness of our proposed method, especially when the noise levels are relatively high.