Spectral Sparsification by Deterministic Discrepancy Walk

TL;DR

This paper extends the deterministic discrepancy walk framework to matrix discrepancy, providing a unified, simpler, and deterministic approach to spectral sparsification problems with improved results.

Contribution

It generalizes the discrepancy walk framework from vector to matrix discrepancy and applies it to various spectral sparsification problems, simplifying analysis and improving results.

Findings

Unified approach for spectral sparsification problems

Simpler deterministic proofs for matrix discrepancy

Improved spectral sparsification results

Abstract

Spectral sparsification and discrepancy minimization are two well-studied areas that are closely related. Building on recent connections between these two areas, we generalize the "deterministic discrepancy walk" framework by Pesenti and Vladu [SODA~23] for vector discrepancy to matrix discrepancy, and use it to give a simpler proof of the matrix partial coloring theorem of Reis and Rothvoss [SODA~20]. Moreover, we show that this matrix discrepancy framework provides a unified approach for various spectral sparsification problems, from stronger notions including unit-circle approximation and singular-value approximation to weaker notions including graphical spectral sketching and effective resistance sparsification. In all of these applications, our framework produces improved results with a simpler and deterministic analysis.