Chaining, Group Leverage Score Overestimates, and Fast Spectral Hypergraph Sparsification

We present an algorithm that given any n-vertex, m-edge, rank r hypergraph constructs a spectral sparsifier with O (n epsilon(-2) log n log r) hyperedges in nearly-linear (O) over tilde (mr) time. This improves in both size and efficiency over a line of work [Bansal-Svensson-Trevisan 2019, Kapralov-Krauthgamer-Tardos-Yoshida 2021] for which the previous best size was O (min{n epsilon(-4) log(3)n, nr(3)epsilon(-2) log n}) and runtime was (O) over tilde (mr +n(O(1))).