On the Uniqueness of Nonnegative Sparse Solutions to Underdetermined Systems of Equations

An underdetermined linear system of equations Ax = b with non-negativity constraint x 0 is considered. It is shown that for matrices A with a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required sparsity depends on a coherence property of the matrix A. This coherence measure can be improved by applying a conditioning stage on A, thereby strengthening the claimed result. The obtained uniqueness theorem relies on an extended theoretical analysis of the 0 - 1 equivalence developed here as well, considering a matrix A with arbitrary column norms, and an arbitrary monotone element-wise concave penalty replacing the 1-norm objective function. Finally, from a numerical point of view, a greedy algorithm

• Type: Journal
• Year: 2008
• Where: TIT
• Authors: Alfred M. Bruckstein, Michael Elad, Michael Zibulevsky
• Keywords: Arbitrary Column Norms, Extended Theoretical Analysis, Sparse Solution, TIT 2008