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

TL;DR

This paper explores conditions for the uniqueness of nonnegative vector and positive semidefinite matrix solutions to underdetermined linear systems, with specific focus on binary and Gaussian measurement matrices, revealing linear support growth and probabilistic guarantees.

Contribution

It provides new characterizations of uniqueness conditions for both vector and matrix solutions, especially for binary and Gaussian measurement matrices, with explicit support size bounds.

Findings

Support size of unique solutions can grow linearly with problem dimension.

Binary measurement matrices like Bernoulli and expander graphs satisfy these conditions.

Gaussian measurement operators satisfy the conditions with high probability.

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.