Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices

02/09/2018
by   Ahmed Elzanaty, et al.
0

One of the key issues in the acquisition of sparse data by means of compressed sensing (CS) is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the required number of measurements for sparse recovery. In this paper we provide a new approach for the analysis of the restricted isometry constant (RIC) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distributions of the extreme eigenvalues for Wishart matrices. First, we derive the probability that the restricted isometry property is satisfied for a given sufficient recovery condition on the RIC, and propose a probabilistic framework to study both the symmetric and asymmetric RICs. Then, we analyze the recovery of compressible signals in noise through the statistical characterization of stability and robustness. The presented framework determines limits on various sparse recovery algorithms for finite size problems. In particular, it provides a tight lower bound on the maximum sparsity order of the acquired data allowing signal recovery with a given target probability. Also, we derive simple approximations for the RICs based on the Tracy-Widom distribution.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
11/18/2019

Deterministic partial binary circulant compressed sensing matrices

Compressed sensing (CS) is a signal acquisition paradigm to simultaneous...
research
01/01/2018

Statistical and Computational Limits for Sparse Matrix Detection

This paper investigates the fundamental limits for detecting a high-dime...
research
11/10/2017

A Theoretical Analysis of Sparse Recovery Stability of Dantzig Selector and LASSO

Dantzig selector (DS) and LASSO problems have attracted plenty of attent...
research
04/09/2020

Sparse recovery of noisy data and Grothendieck inequality

We present a detailed analysis of the unconstrained ℓ_1-method LASSO for...
research
05/20/2008

High-dimensional subset recovery in noise: Sparsified measurements without loss of statistical efficiency

We consider the problem of estimating the support of a vector β^* ∈R^p b...
research
01/18/2019

The Restricted Isometry Property of Block Diagonal Matrices for Group-Sparse Signal Recovery

Group-sparsity is a common low-complexity signal model with widespread a...
research
04/09/2020

Sparse recovery of noisy data using the Lasso method

We present a detailed analysis of the unconstrained ℓ_1-method Lasso met...

Please sign up or login with your details

Forgot password? Click here to reset