A Unique "Nonnegative" Solution to an Underdetermined System: from Vectors to Matrices

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 compr...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Meng Wang, Weiyu Xu, Ao Tang
• Keywords: CORR 2010, Education, Linear Compressed Observations, Positive Semidefinite Matrix, Semidefinite Matrix Solution