On the uniqueness of non-negative sparse & redundant representations

We consider an underdetermined linear system of equations Ax = b with non-negative entries in A and b, and seek a non-negative solution x. We generalize known equivalence results for the basis pursuit, for an arbitrary matrix A, and an arbitrary monotone element-wise concave penally replacing the lscr₁-norm in the objective function. This result is then used to show that if there exists a sufficiently sparse solution to Ax = b, x > 0, it is necessarily unique.