Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Signal Representations with Minimum l∞ -Norm

Maximum (or ℓ∞ ) norm minimization subject to an underdetermined system of linear equations finds use in a large number of practical applications, such as vector quantization, peak-to-average power ratio (PAPR) (or “crest factor”) reduction in wireless communication systems, approximate neighbor search, robotics, and control. In this paper, we analyze the fundamental properties of signal representations with minimum ℓ∞ -norm. In particular, we develop bounds on the maximum magnitude of such representations using the uncertainty principle (UP) introduced by Lyubarskii and Vershynin, 2010, and we characterize the limits of ℓ∞ -norm-based PAPR reduction. Our results show that matrices satisfying the UP, such as randomly subsampled Fourier or i.i.d. Gaussian matrices, enable the efficient computation of so-called democratic representations, which have both provably small ℓ∞ -norm and low PAPR