Sampling Rates for l1-Synthesis
Year: 2020
Type: preprint
Abstract: This work investigates the problem of signal recovery from undersampled noisy sub-Gaussian measurements under the assumption of a synthesis-based sparsity model. Solving the $\ell^1$-synthesis basis pursuit allows for a simultaneous estimation of a coefficient representation as well as the sought-for signal. However, due to linear dependencies within redundant dictionary atoms it might be impossible to identify a specific representation vector, although the actual signal is still successfully recovered. The present manuscript studies both estimation problems from a non-uniform, signal-dependent perspective. By utilizing recent results on the convex geometry of linear inverse problems, the sampling rates describing the phase transitions of each formulation are identified. In both cases, they are given by the conic Gaussian mean width of an $\ell^1$-descent cone that is linearly transformed by the dictionary.
In general, this expression does not allow a simple calculation by following the polarity-based approach commonly found in the literature. Hence, two upper bounds involving the sparsity of coefficient representations are provided: The first one is based on a local condition number and the second one on a geometric analysis that makes use of the thinness of high-dimensional polyhedral cones with not too many generators. It is furthermore revealed that both recovery problems can differ dramatically with respect to robustness to measurement noise -- a fact that seems to have gone unnoticed in most of the related literature. All insights are carefully undermined by numerical simulations.
Source: HAL
Institutions: École Normale Supérieure - PSL, Département de mathématiques et applications, Laboratoire de Probabilités, Statistique et Modélisation, Centre National de la Recherche Scientifique, Université Toulouse III - Paul Sabatier, Institut National des Sciences Appliquées de Toulouse, Institut de Recherche en Informatique de Toulouse, Unité de biologie Moléculaire, Cellulaire et du Développement, Centre de Biologie Intégrative de Toulouse, Institut de Mathématiques de Toulouse
Language: English
Subfield: Signal Processing
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3016449737