Breakdown of equivalence between the minimal l1-norm solution and the sparsest solution

Finding the sparsest solution to a set of underdetermined linear equations is NP-hard in general. However, recent research has shown that for certain systems of linear equations, the sparsest solution (i.e. the solution with the smallest number of nonzeros), is also the solution with minimal l(1) norm, and so can be found by a computationally tractable method. For a given n by m matrix Phi defining a system y = Phi alpha, with n