A Unique “Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices

Authors:
Meng Wang;Weiyu Xu;Ao Tang
Affiliations:
Sch. of Electr. & Comput. Eng., Cornell Univ., Ithaca, NY, USA;-;-
Venue:
IEEE Transactions on Signal Processing
Year:
2011

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Abstract

This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear system only if the measurement matrix has a row-span intersecting the positive orthant. Focusing on two types of binary measurement matrices, Bernoulli 0-1 matrices and adjacency matrices of general expander graphs, we show that, in both cases, the support size of a unique nonnegative solution can grow linearly, namely O(n), with the problem dimension n . We also provide closed-form characterizations of the ratio of this support size to the signal dimension. For the matrix case, we show that under a necessary and sufficient condition for the linear compressed observations operator, there will be a unique positive semidefinite matrix solution to the compressed linear observations. We further show that a randomly generated Gaussian linear compressed observations operator will satisfy this condition with overwhelmingly high probability.