Experimental analysis of local searches for sparse reflexive generalized inverses

TL;DR

This paper experimentally analyzes local search algorithms for constructing sparse, reflexive generalized inverses of matrices, aiming to reduce computational complexity while maintaining key properties, with tests on synthetic and real-world data.

Contribution

It provides an empirical evaluation of local-search methods for sparse generalized inverses, extending previous theoretical work with practical experiments on diverse datasets.

Findings

Local-search procedures effectively produce sparse generalized inverses.

Sparsity and property satisfaction depend on matrix characteristics.

Case study demonstrates real-world applicability.

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. Irrespective of whether a given matrix is sparse, its M-P pseudoinverse can be completely dense, potentially leading to high computational burden and numerical difficulties, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but not all of them need to be satisfied for some applications. In this context, Fampa and Lee (Oper. Res. Letters, 46:605--610, 2018) and Xu, Fampa, Lee and Ponte (SIAM J. on Optimization, to appear) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy only some of the M-P properties. (Vector) 1-norm minimization is used to induce sparsity and to keep the magnitude of the entries under control, and theoretical results limit the distance between the 1-norm of the solution of the local searches and the minimum 1-norm of generalized inverses with corresponding properties. We have implemented several local-search procedures based on results presented in these two papers and make here an experimental analysis of them, considering their application to randomly generated matrices of varied dimensions, ranks, and densities. Further, we carried out a case study on a real-world data set.