Subspace selection for projection maximization with matroid constraints

TL;DR

This paper analyzes the performance of greedy algorithms for subspace selection problems in Hilbert spaces, providing bounds based on elemental curvatures and demonstrating optimality in orthogonal cases.

Contribution

It introduces elemental curvatures to bound the performance of heuristic algorithms for subspace selection under matroid constraints, extending understanding of their effectiveness.

Findings

Algorithms are optimal for orthogonal elements with uniform matroids.

Algorithms achieve at least 50% approximation for non-uniform matroids.

Derived bounds depend on elemental curvatures of the ground set.

Abstract

Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by the vectors in the chosen subset reaches the maximum norm. This problem is generally NP-hard, and alternative approximation algorithms such as forward regression and orthogonal matching pursuit have been proposed as heuristic approaches. In this paper, we investigate bounds on the performance of these algorithms by introducing the notions of elemental curvatures. More specifically, we derive lower bounds, as functions of these elemental curvatures, for performance of the aforementioned algorithms with respect to that of the optimal solution under uniform and non-uniform matroid constraints, respectively. We show that if the elements in the ground set are mutually orthogonal, then these algorithms are optimal when the matroid is uniform and they achieve at least $1/2$-approximations of the optimal solution when the matroid is non-uniform.