Efficient Noise-Blind \(\ell_1\)-Regression of Nonnegative Compressible Signals

Abstract

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements. The general assumption is that the signal has only a few non-zero entries. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising (BPDN). If the measurement matrix has the robust null space property with respect to the \(\ell_2\)-norm, BPDN obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the \(M^+\)-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the \(\ell_1\)-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD). For these measurement matrices, we prove a uniform, stable and robust recovery guarantee. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for group testing in the recent COVID-19 crisis and why contamination of specimens may be modeled as peaky noise, which favors \(\ell_1\) based data fidelity terms.