Secure Linear MDS Coded Matrix Inversion

TL;DR

This paper introduces a distributed, coded computing method for approximate matrix inversion that leverages MDS codes and least squares optimization, enhancing efficiency, security, and applicability to pseudoinverse computation.

Contribution

It proposes a novel approximate matrix inversion algorithm using a black-box solver and develops a distributed framework with MDS codes for secure, efficient matrix inversion and pseudoinverse computation.

Findings

Uses least squares solver for matrix inversion without factorization.

Employs balanced Reed-Solomon codes for optimal load and communication.

Provides a secure, distributed approach applicable to pseudoinverse calculation.

Abstract

A cumbersome operation in many scientific fields, is inverting large full-rank matrices. In this paper, we propose a coded computing approach for recovering matrix inverse approximations. We first present an approximate matrix inversion algorithm which does not require a matrix factorization, but uses a black-box least squares optimization solver as a subroutine, to give an estimate of the inverse of a real full-rank matrix. We then present a distributed framework for which our algorithm can be implemented, and show how we can leverage sparsest-balanced MDS generator matrices to devise matrix inversion coded computing schemes. We focus on balanced Reed-Solomon codes, which are optimal in terms of computational load; and communication from the workers to the master server. We also discuss how our algorithms can be used to compute the pseudoinverse of a full-rank matrix, and how the communication is secured from eavesdroppers.