Sparsest solutions of underdetermined linear systems via ℓq-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 ℓq-quasinorm is also the sparsest one. This generalizes, and slightly improves, a similar result for the ℓ1-norm. We then introduce a simple numerical scheme to compute solutions with minimal ℓq-quasinorm, and we study its convergence. Finally, we display the results of some experiments which indicate that the ℓq-method performs better than other available methods.