Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates

TL;DR

This paper introduces a faster method for constructing spectral sparsifiers using a novel connection to regret minimization over density matrices, improving efficiency and providing new theoretical insights.

Contribution

It presents a nearly quadratic time algorithm for spectral sparsification by leveraging a generalized regret minimization framework over density matrices.

Findings

Constructed spectral sparsifiers in $O(n^{2+ ext{epsilon}})$ time.

Connected sparsification to regret minimization over density matrices.

Generalized matrix multiplicative weight updates for faster sparsifier construction.

Abstract

In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous constructions required $\Omega(n^4)$ running time [BSS14, Zou12], our sparsification routine can be implemented in almost-quadratic running time $O(n^{2+\varepsilon})$. The fundamental conceptual novelty of our work is the leveraging of a strong connection between sparsification and a regret minimization problem over density matrices. This connection was known to provide an interpretation of the randomized sparsifiers of Spielman and Srivastava [SS11] via the application of matrix multiplicative weight updates (MWU) [CHS11, Vis14]. In this paper, we explain how matrix MWU naturally arises as an instance of the Follow-the-Regularized-Leader framework and generalize this approach to yield a larger class of updates. This new class allows us to accelerate the construction of linear-sized spectral sparsifiers, and give novel insights on the motivation behind Batson, Spielman and Srivastava [BSS14].