Robust Nonnegative Sparse Recovery and the Nullspace Property of 0/1 Measurements

TL;DR

This paper establishes robust compressed sensing guarantees for nonnegative vectors using 0/1 Bernoulli measurement matrices, introducing new nullspace properties and avoiding l1-regularization, with applications in wireless network detection.

Contribution

It proves the robust nullspace property for 0/1 Bernoulli matrices and links nullspace conditions to unique, noise-robust recovery of nonnegative vectors without l1-regularization.

Findings

Random 0/1 matrices satisfy nullspace properties with high probability.

Established uniform and robust recovery guarantees for nonnegative least squares.

Demonstrated applicability to wireless network activity detection.

Abstract

We investigate recovery of nonnegative vectors from non-adaptive compressive measurements in the presence of noise of unknown power. In the absence of noise, existing results in the literature identify properties of the measurement that assure uniqueness in the non-negative orthant. By linking such uniqueness results to nullspace properties, we deduce uniform and robust compressed sensing guarantees for nonnegative least squares. No l1-regularization is required. As an important proof of principle, we establish that m x n random i.i.d. 0/1-valued Bernoulli matrices obey the required conditions with overwhelming probability provided that m=O(slog(n/s)). We achieve this by establishing the robust nullspace property for random 0/1-matrices - a novel result in its own right. Our analysis is motivated by applications in wireless network activity detection.