On Sparse Reflexive Generalized Inverses

TL;DR

This paper develops methods to construct sparse reflexive generalized inverses of rank-$r$ matrices, minimizing their 1-norm and providing efficient approximations for general cases.

Contribution

It introduces a construction for sparse reflexive generalized inverses with at most $r^2$ nonzeros and algorithms to approximate minimal 1-norm inverses efficiently.

Findings

Constructed reflexive inverses with at most $r^2$ nonzeros.

Minimized 1-norm for $r=1$ and nonnegative $r=2$ matrices.

Efficiently approximated minimal 1-norm inverses within a factor of $r^2$.

Abstract

We study sparse generalized inverses $H$ of a rank-$r$ real matrix $A$. We give a construction for reflexive generalized inverses having at most $r^2$ nonzeros. For $r=1$ and for $r=2$ with $A$ nonnegative, we demonstrate how to minimize the (vector) 1-norm over reflexive generalized inverses. For general $r$, we efficiently find reflexive generalized inverses with 1-norm within approximately a factor of $r^2$ of the minimum 1-norm generalized inverse.