Sparsification of sums with respect to convex cones

TL;DR

This paper generalizes spectral sparsification to convex cones, establishing bounds on the minimal size of approximate sums, and explores implications for conic optimization.

Contribution

It introduces the sparsification function for convex cones and provides bounds for cones with specific barrier properties, extending spectral sparsification theory.

Findings

Bounded the sparsification function for cones with self-concordant barriers.

Extended spectral sparsification bounds from positive semidefinite matrices to general convex cones.

Analyzed interactions of sparsification with convex geometric operations.

Abstract

Let $x_1,x_2,\ldots,x_m$ be elements of a convex cone $K$ such that their sum, $e$, is in the relative interior of $K$. An $\epsilon$-sparsification of the sum involves taking a subset of the $x_i$ and reweighting them by positive scalars, so that the resulting sum is $\epsilon$-close to $e$, where error is measured in a relative sense with respect to the order induced by $K$. This generalizes the influential spectral sparsification model for sums of positive semidefinite matrices. This paper introduces and studies the sparsification function of a convex cone, which measures, in the worst case over all possible sums from the cone, the smallest size of an $\epsilon$-sparsifier. The linear-sized spectral sparsification theorem of Batson, Spielman, and Srivastava can be viewed as a bound on the sparsification function of the cone of positive semidefinite matrices. This result is generalized to a family of convex cones (including all hyperbolicity cones) that admit a $\nu$-logarithmically homogeneous self-concordant barrier with certain additional properties. For these cones, the sparsification function is bounded above by $\lceil4\nu/\epsilon^2\rceil$. For general convex cones that only admit an ordinary $\nu$-logarithmically homogeneous self-concordant barrier, the sparsification function is bounded above by $\lceil(4\nu/\epsilon)^2\rceil$. Furthermore, the paper explores how sparsification functions interact with various convex geometric operations (such as conic lifts), and describes implications of sparsification with respect to cones for certain conic optimization problems.