Composable Core-sets for Determinant Maximization Problems via Spectral Spanners

TL;DR

This paper introduces spectral spanners as a spectral generalization of graph spanners, enabling the construction of nearly optimal composable core-sets for determinant maximization and other spectral problems, with broad applications in distributed computing.

Contribution

It defines spectral spanners, proves their near-optimal size, and applies them to create almost optimal composable core-sets for spectral optimization problems like determinant maximization.

Findings

Existence of spectral spanners of size O(d) for any set of vectors.

Construction of almost optimal composable core-sets for determinant maximization.

Spectral analogue of the greedy algorithm for graph spanners.

Abstract

We study a spectral generalization of classical combinatorial graph spanners to the spectral setting. Given a set of vectors $V\subseteq \Re^d$, we say a set $U\subseteq V$ is an $\alpha$-spectral spanner if for all $v\in V$ there is a probability distribution $\mu_v$ supported on $U$ such that $$vv^\intercal \preceq \alpha\cdot\mathbb{E}_{u\sim\mu_v} uu^\intercal.$$ We show that any set $V$ has an $\tilde{O}(d)$-spectral spanner of size $\tilde{O}(d)$ and this bound is almost optimal in the worst case. We use spectral spanners to study composable core-sets for spectral problems. We show that for many objective functions one can use a spectral spanner, independent of the underlying functions, as a core-set and obtain almost optimal composable core-sets. For example, for the determinant maximization problem we obtain an $\tilde{O}(k)^k$-composable core-set and we show that this is almost optimal in the worst case. Our algorithm is a spectral analogue of the classical greedy algorithm for finding (combinatorial) spanners in graphs. We expect that our spanners find many other applications in distributed or parallel models of computation. Our proof is spectral. As a side result of our techniques, we show that the rank of diagonally dominant lower-triangular matrices are robust under `small perturbations' which could be of independent interests.