$ell_2/ell_2$ Sparse Recovery via Weighted Hypergraph Peeling

2025-10-23

AI Summary

This paper addresses the problem of efficiently recovering the optimal $k$-sparse approximation of an $n$-dimensional vector, aiming for a $(1+varepsilon)$-approximation while minimizing runtime. We introduce a novel analytical framework based on **weighted hypergraph peeling**, generalizing classical hypergraph peeling to settings where both vertices and hyperedges carry weights—thereby significantly enhancing modeling capability for non-uniform structures. Coupled with a **non-adaptive linear sketch** having $O((k/varepsilon)log n)$ rows and $O(log n)$ column sparsity, our method achieves signal recovery in $O((k/varepsilon)log n)$ time. This improves upon the previous best runtime by a $log n$ factor and attains theoretically optimal time complexity across all parameters $k$, $varepsilon$, and $n$. Notably, it is the first algorithm to break the logarithmic-factor barrier for $(1+varepsilon)$-approximate sparse recovery.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We demonstrate that the best $k$-sparse approximation of a length-$n$ vector can be recovered within a $(1+ε)$-factor approximation in $O((k/ε) log n)$ time using a non-adaptive linear sketch with $O((k/ε) log n)$ 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.

Problem

• Recover k-sparse approximations efficiently
• Improve runtime of existing sparse recovery methods
• Extend hypergraph peeling to weighted settings

Innovation

• Weighted hypergraph peeling for sparse recovery
• Non-adaptive linear sketch with logarithmic sparsity
• Optimal time complexity for approximation guarantees