On the Error of Estimating the Sparsest Solution of Underdetermined Linear Systems

Let A be an n × 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 ŝ as an estimation of s₀, and suppose that ŝ 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 ||ŝ - 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 ||ŝ - s₀ ||₂, ŝ 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.