A Non-Negative and Sparse Enough Solution of an Underdetermined Linear System of Equations is Unique
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: A Non-Negative and Sparse Enough Solution of an Underdetermined Linear System of Equations is Unique
Abstract : In this paper we consider an underdetermined linear system of equations Ax = b with non-negative entries of A and b, and the solution x being also required to be nonnegative. We show that if there exists a sufficiently sparse solution to this problem, it is necessarily unique. Furthermore, we present a greedy algorithm – a variant of the matching pursuit – that is guaranteed to find this sparse solution. We also extend the existing theoretical analysis of the basis pursuit problem, i.e. min ?x?1 s.t. Ax = b, by studying conditions for perfect recovery of sparse enough solutions. Considering a matrix A with arbitrary column norms, and an arbitrary monotone element-wise concave penalty replacing the ?1-norm objective function, we generalize known equivalence results. Beyond the desirable generalization that this result introduces, we show how it is exploited to lead to the above-mentioned uniqueness claim. 1
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