Sparsity and ℓ_p-Restricted Isometry

05/13/2022
by   Venkatesan Guruswami, et al.
0

A matrix A is said to have the ℓ_p-Restricted Isometry Property (ℓ_p-RIP) if for all vectors x of up to some sparsity k, Ax_p is roughly proportional to x_p. It is known that with high probability, random dense m× n matrices (e.g., with i.i.d. ± 1 entries) are ℓ_2-RIP with k ≈ m/log n, and sparse random matrices are ℓ_p-RIP for p ∈ [1,2) when k, m = Θ(n). However, when m = Θ(n), sparse random matrices are known to not be ℓ_2-RIP with high probability. With this backdrop, we show that there are no sparse matrices with ± 1 entries that are ℓ_2-RIP. On the other hand, for p ≠ 2, we show that any ℓ_p-RIP matrix must be sparse. Thus, sparsity is incompatible with ℓ_2-RIP, but necessary for ℓ_p-RIP for p ≠ 2.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
08/22/2018

Improved bounds for the RIP of Subsampled Circulant matrices

In this paper, we study the restricted isometry property of partial rand...
research
01/30/2018

Surjectivity of near square random matrices

We show that a nearly square iid random integral matrix is surjective ov...
research
05/22/2020

The Average-Case Time Complexity of Certifying the Restricted Isometry Property

In compressed sensing, the restricted isometry property (RIP) on M × N s...
research
08/31/2021

ℓ_p-Spread Properties of Sparse Matrices

Random subspaces X of ℝ^n of dimension proportional to n are, with high ...
research
05/29/2018

Explicit construction of RIP matrices is Ramsey-hard

Matrices Φ∈^n× p satisfying the Restricted Isometry Property (RIP) are a...
research
07/15/2022

Approximately Hadamard matrices and Riesz bases in random frames

An n × n matrix with ± 1 entries which acts on ℝ^n as a scaled isometry ...
research
04/11/2019

Restricted Isometry Property under High Correlations

Matrices satisfying the Restricted Isometry Property (RIP) play an impor...

Please sign up or login with your details

Forgot password? Click here to reset