Abstract
In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements with the general assumption that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the ℓ2-norm, basis pursuit denoising 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 ℓ1-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD), which is free of tuning parameters. 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 viral detection in the recent COVID-19 crisis.
Highlights
Since it has been realized that many signals admit a sparse representation in some frames, the question arose whether or not such signals can be recovered from less samples than the dimension of the domain by utilizing the low dimensional structure of the signal
The basis pursuit denoising requires an upper bound on the norm of the noise ([3], Theorem 4.22), the least shrinkage and selection operator an estimate on the l1-norm of the signal ([4], Theorem 11.1) and the Lagrangian version of least shrinkage and selection operator allegedly needs to be tuned to the order of the the noise level ([4], Theorem 11.1)
Our results state that if the M+ criterion is fulfilled, the basis pursuit denoising can be replaced by the tuning-less non-negative least residual (NNLR) for non-negative signals
Summary
Since it has been realized that many signals admit a sparse representation in some frames, the question arose whether or not such signals can be recovered from less samples than the dimension of the domain by utilizing the low dimensional structure of the signal. Other thresholding based decoders like sparse matching pursuit and expander matching pursuit have the same limitations as the expander iterative hard thresholding [6] If these side information is not known a priori, many decoders yield either no recovery guarantees or, in their imperfectly tuned versions, yield sub-optimal estimation errors ([3], Theorem 11.12). The most prominent achievement for that is the non-negative least squares (NNLS) [7,8,9,10,11] It is completely tuning free [12] and in [13, 14] it was proven that it achieves robust recovery guarantees if the measurement matrix consists of independent biased sub-Gaussian random variables
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