Conditions for a unique non-negative solution to an underdetermined system

Authors:
Meng Wang;Ao Tang
Affiliations:
School of Electrical and Computer Engineering, Cornell University, Ithaca, NY;School of Electrical and Computer Engineering, Cornell University, Ithaca, NY
Venue:
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Year:
2009

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Abstract

This paper investigates conditions for an underdetermined linear system to have a unique nonnegative solution. A necessary condition is derived which requires the measurement matrix to have a row-span intersecting the positive orthant. For systems that satisfy this necessary condition, we provide equivalent characterizations for having a unique nonnegative solution. These conditions generalize existing ones to the cases where the measurement matrix may have different column sums. Focusing on binary measurement matrices especially ones that are adjacency matrices of expander graphs, we obtain an explicit threshold. Any nonnegative solution that is sparser than the threshold is the unique nonnegative solution. Compared with previous ones, this result is not only more general as it does not require constant degree condition, but also stronger as the threshold is larger even for cases with constant degree.