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arXiv · 2408.06146

Spectral Sparsification by Deterministic Discrepancy Walk

Lap Chi Lau·Robert Wang·Hong Zhou
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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.

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Lap Chi Lau, Robert Wang, Hong Zhou. 2024-08-12. Spectral Sparsification by Deterministic Discrepancy Walk. https://arxiv.org/abs/2408.06146

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