Non-Convex Compressed Sensing from Noisy Measurements
A. Majumdar*, R. K. Ward
Identifiers and Pagination:Year: 2009
First Page: 40
Last Page: 44
Publisher Id: TOSIGPJ-2-40
Article History:Received Date: 19/10/2009
Revision Received Date: 03/11/2009
Acceptance Date: 03/11/2009
Electronic publication date: 16/12/2009
Collection year: 2009
open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: https://creativecommons.org/licenses/by/4.0/legalcode. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
This paper proposes solution to the following non-convex optimization problem:
min || x || subject to || y – Ax ||≤ ε
Such an optimization problem arises in a rapidly advancing branch of signal processing called ‘Compressed Sensing’ (CS). The problem of CS is to reconstruct a k-sparse vector x, from noisy measurements y = Ax +η , where A (m < n) is the measurement matrix and η is additive noise.
In general the optimization methods developed for CS minimizes a sparsity promoting lnorm (p=1) for Gaussian noise (q=2). This is restrictive for two reasons: i) theoretically it has been shown that, with positive fractional norms (0 < p < 1), the sparse vector x can be reconstructed by fewer measurements than required by lnorm; and ii) Noises other than Gaussian require the norm of the misfit (q) to be something other than 2. To address these two issues an Iterative Reweighted Least Squares based algorithm is proposed here to solve the aforesaid optimization problem.