Sparse nonnegative solution of underdetermined linear equations by linear programming

Consider an underdetermined system of linear equations y = Ax with known y and d x n matrix A. We seek the nonnegative x with the fewest nonzeros satisfying y = Ax. In general, this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. We explain this by the theory of convex polytopes. Let a(j) denote the jth column of A, 1 < or = j < or = n, let a0 = 0 and P denote the convex hull of the a(j). We say the polytope P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. We also consider weak neighborliness, where the overwhelming majority of k-sets of a(j)s not containing 0 span a face of P. This implies that most nonnegative vectors x with k nonzeros are uniquely recoverable from y = Ax by linear programming. Numerous corollaries follow by invoking neighborliness results. For example, for most large n by 2n underdetermined systems having a solution with fewer nonzeros than roughly half the number of equations, the sparsest solution can be found by linear programming.