Rank-Independent Spectral Hypergraph Sparsification via Global-Dictionary Chaining

2026-07-09

AI Summary

This work addresses the dependence on hyperedge rank in weighted hypergraph spectral sparsification by proposing a rank-independent sparsification method. By constructing a global dictionary chain and assigning weights to clique edges under effective resistance balancing, the approach ensures that all hyperedge seminorms satisfy a Lipschitz condition with respect to a unified global dictionary norm induced by normalized vertex-pair directions. This transformation recasts local rank complexity into the Gaussian width of the dictionary. The method achieves, for the first time, $\varepsilon$-spectral sparsification independent of hyperedge rank, resolving an open problem posed by Lee. It further reduces the size of the sparsifier to $O(n \log n / \varepsilon^2)$ hyperedges, significantly improving upon the sampling bounds and subsequent theoretical guarantees established in prior work presented at STOC 2023.

Abstract

We show that every weighted hypergraph on $n$ vertices admits a spectral $\varepsilon$-sparsifier with $O(n\log n/\varepsilon^2)$ hyperedges, strengthening the independent STOC 2023 works of Lee and Jambulapati--Liu--Sidford by removing their rank dependence and answering Lee's open question on whether this loss is inherent. The key idea is global-dictionary chaining: after choosing clique edge weights with balanced effective resistances, every hyperedge seminorm is Lipschitz with respect to the same global-dictionary norm generated by normalized vertex-pair directions; the local rank complexity is thereby replaced by the Gaussian width of this common dictionary. Since these STOC 2023 works have become standard analytic primitives across a broad subsequent literature on spectral hypergraph sparsification and its variants, our rank-independent theorem sharpens many later guarantees that inherit their sampling bounds.

Problem

• spectral sparsification
• hypergraph
• rank independence
• sparsifier
• Gaussian width

Innovation

• spectral sparsification
• hypergraph
• rank independence
• global-dictionary chaining
• Gaussian width