On the error of estimating the sparsest solution of underdetermined linear systems

TL;DR

This paper derives bounds on the estimation error when approximating the sparsest solution of underdetermined linear systems, considering nearly sparse solutions, noise, and the effect of sparsity level on recovery accuracy.

Contribution

It introduces new bounds based on minimal singular values of submatrices of A, including tight and probabilistic bounds, to analyze approximate and noisy sparse recovery.

Findings

Bound on estimation error without knowing the true solution

Tight bounds enable probabilistic analysis for random dictionaries

Error bounds degrade as sparsity level increases

Abstract

Let A be an n by m matrix with m>n, and suppose that the underdetermined linear system As=x admits a sparse solution s0 for which ||s0||_0 < 1/2 spark(A). Such a sparse solution is unique due to a well-known uniqueness theorem. Suppose now that we have somehow a solution s_hat as an estimation of s0, and suppose that s_hat is only `approximately sparse', that is, many of its components are very small and nearly zero, but not mathematically equal to zero. Is such a solution necessarily close to the true sparsest solution? More generally, is it possible to construct an upper bound on the estimation error ||s_hat-s0||_2 without knowing s0? The answer is positive, and in this paper we construct such a bound based on minimal singular values of submatrices of A. We will also state a tight bound, which is more complicated, but besides being tight, enables us to study the case of random dictionaries and obtain probabilistic upper bounds. We will also study the noisy case, that is, where x=As+n. Moreover, we will see that where ||s0||_0 grows, to obtain a predetermined guaranty on the maximum of ||s_hat-s0||_2, s_hat is needed to be sparse with a better approximation. This can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where ||s0||_0 grows.