Subset selection in sparse matrices

TL;DR

This paper explores conditions under which subset selection in sparse matrices can be solved efficiently, leveraging discrete geometry to identify polynomial-time solvable cases despite the general NP-hardness.

Contribution

It introduces sparsity conditions on data matrices that enable polynomial-time solutions for subset selection, expanding the understanding of tractable cases.

Findings

Certain sparsity conditions lead to polynomial-time algorithms

Subset selection remains NP-hard in general

Discrete geometry tools facilitate new solution approaches

Abstract

In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using mainly tools from discrete geometry, we show that some sparsity conditions on the original data matrix allow us to solve the problem in polynomial time.