A proximal approach to the inversion of ill-conditioned matrices

TL;DR

This paper introduces a proximal algorithm for inverting ill-conditioned matrices, leveraging a variational approach to pseudo-inverses, with convergence analysis and connections to fixed point methods.

Contribution

It presents a novel proximal algorithm for matrix inversion based on variational characterization, including convergence properties and relation to fixed point methods.

Findings

Algorithm successfully inverts ill-conditioned matrices.

Convergence of the proposed method is established.

Connection to fixed point methods is demonstrated.

Abstract

We propose a general proximal algorithm for the inversion of ill-conditioned matrices. This algorithm is based on a variational characterization of pseudo-inverses. We show that a particular instance of it (with constant regularization parameter) belongs to the class of {\sl fixed point} methods. Convergence of the algorithm is also discussed.