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Satisfying the Restricted Isometry Property with the Optimal Number of Rows and Slightly Less Randomness

Shravas Rao

TL;DR

This paper addresses constructing RIP matrices with the optimal row count while minimizing randomness. It introduces a generalized Hanson-Wright bound for $\varepsilon$-biased, almost $\ell$-wise independent distributions and uses it to show that a matrix $\Phi$ drawn from such a distribution yields $Q = O(k \log(N/k))$ rows and only $O(k\log(k)\log(N/k))$ random bits. The approach leverages a Johnson-Lindenstrauss-type projection framework to bound quadratic forms $\|\Phi x\|_2^2$ for all $k$-sparse $x$. The result improves randomness efficiency among optimal-row RIP constructions and advances deterministic/rand-bit-efficient designs in compressed sensing.

Abstract

A matrix $Φ\in \mathbb{R}^{Q \times N}$ satisfies the restricted isometry property if $\|Φx\|_2^2$ is approximately equal to $\|x\|_2^2$ for all $k$-sparse vectors $x$. We give a construction of RIP matrices with the optimal $Q = O(k \log(N/k))$ rows using $O(k\log(N/k)\log(k))$ bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to $ε$-biased distributions.

Satisfying the Restricted Isometry Property with the Optimal Number of Rows and Slightly Less Randomness

TL;DR

This paper addresses constructing RIP matrices with the optimal row count while minimizing randomness. It introduces a generalized Hanson-Wright bound for -biased, almost -wise independent distributions and uses it to show that a matrix drawn from such a distribution yields rows and only random bits. The approach leverages a Johnson-Lindenstrauss-type projection framework to bound quadratic forms for all -sparse . The result improves randomness efficiency among optimal-row RIP constructions and advances deterministic/rand-bit-efficient designs in compressed sensing.

Abstract

A matrix satisfies the restricted isometry property if is approximately equal to for all -sparse vectors . We give a construction of RIP matrices with the optimal rows using bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to -biased distributions.
Paper Structure (3 sections, 3 theorems, 14 equations)

This paper contains 3 sections, 3 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Theorem

Key Result

Theorem 1.1

There exists a distribution of $Q \times N$ matrices for $Q = O(k\log(N/k)\eta^{-2})$ that can be sampled efficiently using $O(k\log(k)\log(N/k))$ bits of randomness such that a sample $\Phi$ from this distribution satisfies the $(k, \eta)$-restricted isometry property with high probability.

Theorems & Definitions (6)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}