Sparse Non-Negative Recovery from Biased Subgaussian Measurements using NNLS

TL;DR

This paper demonstrates that biased subgaussian measurement matrices enable reliable sparse non-negative vector recovery via NNLS without tuning parameters, leveraging the nullspace property for strong guarantees.

Contribution

It establishes that biased subgaussian matrices satisfy the nullspace property with high probability, ensuring effective sparse recovery using NNLS without parameter tuning.

Findings

NSP holds with high probability for biased subgaussian matrices

Bias in the sensing matrix enhances NNLS auto-regularization

Recovery guarantees are independent of the bias magnitude

Abstract

We investigate non-negative least squares (NNLS) for the recovery of sparse non-negative vectors from noisy linear and biased measurements. We build upon recent results from [1] showing that for matrices whose row-span intersects the positive orthant, the nullspace property (NSP) implies compressed sensing recovery guarantees for NNLS. Such results are as good as for $\ell_1$-regularized estimators but do not require tuning parameters that depend on the noise level. A bias in the sensing matrix improves this auto-regularization feature of NNLS and the NSP then determines the sparse recovery performance only. We show that NSP holds with high probability for biased subgaussian matrices and its quality is independent of the bias.