ℓ1-Analysis minimization and generalized (co-)sparsity: When does recovery succeed?

Abstract

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the 1 -analysis basis pursuit, enabling highly accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend on the Gram matrix of the analysis operator and therefore particularly account for its mutual coherence structure. Our findings defy conventional wisdom which promotes the sparsity of analysis coefficients as the crucial quantity to study. In fact, this common paradigm breaks down completely in many situations of practical interest, for instance, when applying a redundant (multilevel) frame as analysis prior. By extensive numerical experiments, we demonstrate that, in contrast, our theoretical sampling-rate bounds reliably capture the recovery capability of various examples, such as redundant Haar wavelets systems, total variation, or random frames. The proofs of our main results build upon recent achievements in the convex geometry of data mining problems. More precisely, we establish a sophisticated upper bound on the conic Gaussian mean width that is associated with the underlying 1 -analysis polytope. Due to a novel localization argument, it turns out that the presented framework naturally extends to stable recovery, allowing us to incorporate compressible coefficient sequences as well.

Living Near the Edge: A Lower-Bound on the Phase Transition of Total Variation Minimization

“…By using a more general analysis of [27], the authors in [19] obtain an explicit formula describing the required number of measurements. Their bound depends on the coherence structure of Ω.…”

On the Error in Phase Transition Computations for Compressed Sensing

“…In [1], implicit formulas are derived for the upper-bound (5) in case of ℓ 1 and nuclear norm. Recently, an explicit upper-bound for U δ in case of ℓ 1 analysis and TV minimization is presented in [11]. The proposed bound depends on a notion called "generalized analysis sparsity" and is numerically observed to be tight for many analysis operators.…”

“…p ď n; due to the similarity of the arguments used in this paper, we include square matrices in the category of fat matrices. Tall Ω matrices cover various redundant 1 analysis operators that are common in practice; in particular, redundant wavelet frames [12], and redundant random frames (widely used as a benchmark template in [9], [11], [13], [14]). Also fat matrices include examples such as the one-dimensional finite difference operator Ω d and nonredundant random analysis operators (used in [11,Section 3.3]).…”

“…Tall Ω matrices cover various redundant 1 analysis operators that are common in practice; in particular, redundant wavelet frames [12], and redundant random frames (widely used as a benchmark template in [9], [11], [13], [14]). Also fat matrices include examples such as the one-dimensional finite difference operator Ω d and nonredundant random analysis operators (used in [11,Section 3.3]). We call a signal x P R n an analysis-sparse vector (also called as cosparse vector [9]) with respect to the analysis operator Ω P R pˆn if after applying Ω the resulting vector becomes sparse; i.e., Ωx is sparse.…”

“…We denote the support set of Ωx with S; similarly, S stands for the zero set of Ωx. The number of zeros in Ωx, i.e., |S|, is called the cosparsity of x with respect to Ω [9], [11], [14]. We should highlight that in most of the existing literature regarding the ℓ 1 -analysis problem it is assumed that Ω P R pˆn has rows in general position 2 .…”