On computing sparse generalized inverses

TL;DR

This paper explores methods for computing sparse generalized inverses, particularly focusing on 2,1-norm minimization, which yields solutions satisfying key properties of the Moore-Penrose pseudoinverse for efficient least-squares computations.

Contribution

It introduces 2,1-norm minimization formulations for row-sparse generalized inverses that satisfy important Moore-Penrose properties, enabling efficient computation.

Findings

2,1-norm minimization produces generalized inverses with key Moore-Penrose properties.

Proposed formulations can be solved efficiently using mathematical optimization.

Numerical comparisons show advantages over other sparsity-inducing methods.

Abstract

The well-known M-P (Moore-Penrose) pseudoinverse is used in several linear-algebra applications; for example, to compute least-squares solutions of inconsistent systems of linear equations. It is uniquely characterized by four properties, but not all of them need to be satisfied for some applications. For computational reasons, it is convenient then, to construct sparse block-structured matrices satisfying relevant properties of the M-P pseudoinverse for specific applications. (Vector) 1-norm minimization has been used to induce sparsity in this context. Aiming at row-sparse generalized inverses motivated by the least-squares application, we consider 2,1-norm minimization (and generalizations). In particular, we show that a 2,1-norm minimizing generalized inverse satisfies two additional M-P pseudoinverse properties, including the one needed for computing least-squares solutions. We present mathematical-optimization formulations related to finding row-sparse generalized inverses that can be solved very efficiently, and compare their solutions numerically to generalized inverses constructed by other methodologies, also aiming at sparsity and row-sparse structure.