The Sparsest Solution of Underdetermined Linear System by ` q minimization for 0 < q ≤ 1

We study tne `q approximation of the sparsest solution of underdetermined linear systems. Mainly we present a condition on the matrix associated with an underdetermined linear system under which the solution of `q minimization is the sparsest solution of the system. Our condition generalizes a similar condition in [Candés, Romberg and Tao’06] ensuring that the solution of the `1 minimization is the sparsest solution. Our condition in `1 case slightly improves the similar condition. We present a numerical method to compute the `q minimization. Our numerical experiments show that the `q method is better than many existing methods, to find the sparsest solution.