Random Search Algorithms for the Sparse Null Vector Problem

TL;DR

This paper introduces randomized and deterministic algorithms to identify small linearly dependent column subsets in matrices, addressing an NP-complete problem with practical Monte Carlo and proof methods.

Contribution

It presents novel Monte Carlo and deterministic algorithms for the sparse null vector problem, with performance analysis and numerical validation.

Findings

Monte Carlo method finds dependent subsets with high confidence

Deterministic method proves absence of small dependent subsets

Algorithms perform effectively in numerical experiments

Abstract

We consider the following problem: Given a matrix A, find minimal subsets of columns of A with cardinality no larger than a given bound that are linear dependent or nearly so. This problem arises in various forms in optimization, electrical engineering, and statistics. In its full generality, the problem is known to be NP-complete. We present a Monte Carlo method that finds such subsets with high confidence. We also give a deterministic method that is capable of proving that no subsets of linearly dependent columns up to a certain cardinality exist. The performance of both methods is analyzed and illustrated with numerical experiments.