Sparsest Solutions of Underdetermined Linear Systems via ` q - minimization for 0 < q ≤ 1

We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal `q-quasinorm is also the sparsest one. This generalizes, and sightly improves, a similar result for the `1-norm. We then introduce a simple numerical scheme to compute solutions with minimal `q-quasinorm, and we study its convergence. Finally, we display the results of some experiments which indicate that the `q-method performs better than other available methods.