Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates
This work provides a faster algorithm for spectral sparsification, which is incremental but improves computational efficiency for applications in graph theory and numerical linear algebra.
The paper tackles the problem of constructing linear-sized spectral sparsifiers more efficiently, reducing the running time from Ω(n^4) to O(n^{2+ε}) by leveraging a connection to regret minimization over density matrices and generalizing matrix multiplicative weight updates.
In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous constructions required $Ω(n^4)$ 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].