Many (most?) column subset selection criteria are NP hard for a few columns

TL;DR

This paper investigates various criteria for selecting a small subset of columns from a matrix, demonstrating that most are NP-hard and lack efficient approximation schemes, with implications for matrix design and optimization.

Contribution

It proves NP-hardness for multiple column subset selection criteria and derives optimal values and expressions for related decision problems.

Findings

Most column subset selection criteria are NP-hard.

Many criteria do not admit polynomial time approximation schemes.

Derived optimal values for subset selection criteria and partitioned pseudo-inverses.

Abstract

We consider a variety of criteria for selecting k representative columns from a real mxn matrix A, when sufficiently few columns are required, i.e., 1<= k<= min{rank(A), m/3}. The criteria include the following optimization problems: absolute volume and S-optimality maximization; norm, pseudo-inverse norm, and condition minimization number in the two-norm, Frobenius norm and Schatten p-norms for p>2; stable rank maximization; and the new criterion of relative volume maximization, which is inversely proportional to a power of the condition number. We show that these criteria are NP hard and many do not admit polynomial time approximation schemes (PTAS). To formulate the optimization problems as decision problems, we derive optimal values for the subset selection criteria, as well as expressions for partitioned pseudo-inverses. The results for minimization of the pseudo-inverse in the Frobenius norm are applicable to trace optimization in A-optimal design.