Tight Bounds for Volumetric Spanners and Applications

TL;DR

This paper establishes near-optimal bounds for volumetric spanners across all ℓ_p norms, demonstrating their construction via local search and applying these results to tasks like coresets for MVEE.

Contribution

It provides almost optimal bounds for volumetric spanners in all ℓ_p norms and introduces a simple local search method for their construction.

Findings

Bounds are nearly optimal for all ℓ_p norms.

A simple local search algorithm constructs volumetric spanners.

Applications include coresets for MVEE.

Abstract

Given a set of points of interest, a volumetric spanner is a subset of the points using which all the points can be expressed using "small" coefficients (measured in an appropriate norm). Formally, given a set of vectors $X = \{v_1, v_2, \dots, v_n\}$, the goal is to find $T \subseteq [n]$ such that every $v \in X$ can be expressed as $\sum_{i\in T} \alpha_i v_i$, with $\|\alpha\|$ being small. This notion, which has also been referred to as a well-conditioned basis, has found several applications, including bandit linear optimization, determinant maximization, and matrix low rank approximation. In this paper, we give almost optimal bounds on the size of volumetric spanners for all $\ell_p$ norms, and show that they can be constructed using a simple local search procedure. We then show the applications of our result to other tasks and in particular the problem of finding coresets for the Minimum Volume Enclosing Ellipsoid (MVEE) problem.