Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Nick Fischer, Vasileios Nakos

TL;DR

This work tackles fast, robust sparse recovery in the $\,\ell_2/\ell_2\,$ sense using non-adaptive linear sketches. It introduces a one-shot hashing scheme and a weighted hypergraph peeling analysis to recover a $(1+\varepsilon)$-approximation of the best $k$-term approximation in time $O\left(\frac{k}{\varepsilon}\log n\right)$ with $O\left(\frac{k}{\varepsilon}\log n\right)$ measurements and $O(\log n)$ column sparsity. The core technical contribution is the weighted hypergraph peeling framework, which extends Invertible Bloom Filter techniques to correlated edge/node weights and quantifies when heavy-edge recovery preserves most signal energy. The resulting algorithm is simple in description, practical via Count-Sketch, and provably optimal for a broad range of parameters, representing a significant speedup over prior sublinear-time sparse recovery methods while maintaining robust guarantees. The approach also offers a conceptually appealing connection between sparse recovery and hypergraph peeling, with potential practical impact in signal processing and streaming applications.

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.

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

TL;DR

This work tackles fast, robust sparse recovery in the sense using non-adaptive linear sketches. It introduces a one-shot hashing scheme and a weighted hypergraph peeling analysis to recover a -approximation of the best -term approximation in time with measurements and column sparsity. The core technical contribution is the weighted hypergraph peeling framework, which extends Invertible Bloom Filter techniques to correlated edge/node weights and quantifies when heavy-edge recovery preserves most signal energy. The resulting algorithm is simple in description, practical via Count-Sketch, and provably optimal for a broad range of parameters, representing a significant speedup over prior sublinear-time sparse recovery methods while maintaining robust guarantees. The approach also offers a conceptually appealing connection between sparse recovery and hypergraph peeling, with potential practical impact in signal processing and streaming applications.

Abstract

We demonstrate that the best -sparse approximation of a length- vector can be recovered within a -factor approximation in time using a non-adaptive linear sketch with rows and column sparsity. This improves the running time of the fastest-known sketch [Nakos, Song; STOC '19] by a factor of , 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.
Paper Structure (25 sections, 24 theorems, 57 equations, 1 figure, 1 table)

This paper contains 25 sections, 24 theorems, 57 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Algorithm Description.
  3. Technical Overview
  4. Known Approaches: Iterative and Hierarchical.
  5. New Approach: One-Shot.
  6. Inspiration: Hypergraph Peeling.
  7. Weighted Hypergraph Peeling.
  8. Problem 1: Not All Edges Are Peelable.
  9. Problem 2: Accumulated Estimation Errors.
  10. Outline.
  11. Outline.
  12. Toolkit
  13. Hypergraph Peeling
  14. Unweighted Hypergraph Peeling.
  15. Weighted Hypergraph Peeling.
  16. ...and 10 more sections

Key Result

theorem 1.1

Let $k \leq n$ be integers and let $0 < \epsilon < 1$. There is a randomized construction of a matrix $A\in\mathbb{R}^{m\times n}$ with $m=O((k/\epsilon) \log n)$ rows and $O( \log n)$ non-zero entries per column, such that given $Ax$ we can recover in time $O(m)$ a vector $x'$ satisfying with arbitrarily large constant probability. Here, $x_{-k}$ is the vector obtained from $x$ by zeroing out th

Figures (1)

  • Figure 1: Illustrates a hypertree (left) and two unicyclic hypergraphs (middle and right).

Theorems & Definitions (45)

  • theorem 1.1: Fast $\ell_2/\ell_2$ Sparse Recovery
  • lemma 3.1
  • lemma 3.2
  • proof
  • Definition 3.3: Peeling
  • lemma 3.4: Peeling with Random Vertex Weights
  • proof
  • lemma 3.5
  • proof
  • lemma 3.6: Count-Sketch CharikarCF04MintonP14
  • ...and 35 more