$\ell_2/\ell_2$ Sparse Recovery via Weighted Hypergraph Peeling

TL;DR

This paper introduces a new weighted hypergraph peeling technique to improve sparse recovery algorithms, achieving faster approximation within optimal bounds using a simple, practical linear sketching method.

Contribution

It presents a novel weighted hypergraph peeling approach that enhances sparse recovery efficiency and extends existing hypergraph peeling techniques to weighted, correlated settings.

Findings

Achieves $(1+psilon)$-approximate $k$-sparse recovery in $O((k/psilon) \, ext{log} n)$ time.

Reduces the running time of previous methods by a factor of $ ext{log} n$ and is optimal for many parameters.

Provides a simple, practical algorithm with a new analytical technique called weighted hypergraph peeling.

Abstract

We demonstrate that the best $k$-sparse approximation of a length-$n$ vector can be recovered within a $(1+\epsilon)$-factor approximation in $O((k/\epsilon) \log n)$ time using a non-adaptive linear sketch with $O((k/\epsilon) \log n)$ rows and $O(\log n)$ column sparsity. This improves the running time of the fastest-known sketch [Nakos, Song; STOC '19] by a factor of $\log n$, and is optimal for a wide range of parameters. Our algorithm is simple and likely to be practical, with the analysis built on a new technique we call weighted hypergraph peeling. Our method naturally extends known hypergraph peeling processes (as in the analysis of Invertible Bloom Filters) to a setting where edges and nodes have (possibly correlated) weights.