Abstract
Motivated by the observation that a given signal x admits sparse representations in multiple dictionaries 𝐵𝐵𝑑𝑑 but with varying levels of sparsity across dictionaries, we propose two new algorithms for the reconstruction of (approximately) sparse signals from noisy linear measurements. Our first algorithm, Co-L1, extends the well-known lasso algorithm from the L1 regularizer𝜆𝜆𝐵𝐵𝐵𝐵1to composite regularizers of the form Σ𝑑𝑑𝜆𝜆𝑑𝑑𝐵𝐵𝑑𝑑𝑥 1 while self-adjusting the regularization weights 𝜆𝜆𝑑𝑑. Our second algorithm, Co-IRW-L1, extends the well-known iteratively reweighted L1 algorithm to the same family of composite regularizers. We provide several interpretations of both algorithms: i) majorize-minimization (MM) applied to a non-convex log-sum-type penalty, ii) MM applied to an approximate l0-type penalty, iii) MM applied to fully-Bayesian inference under a particular hierarchical prior, and iv) variational expectation - maximization (VEM) under a particular prior with deterministic unknown parameters. A detailed numerical study suggests that our proposed algorithms yield significantly improved recovery SNR when compared to their non-composite L1 and IRW-L1 counterparts.

Bio
Phil Schniter received the BS and MS degrees in EE from UIUC in 1992 and 1993, respectively, and the PhD degree in EE from Cornell in 2000. He has been with the Dept. of ECE at Ohio State since 2000,where he is now a Professor. Phil is an IEEE Fellow, and serves on the IEEE Sensor Array and Multichannel (SAM) Technical Committee, as well as the Computational Imaging Special Interest Group. His recent awards include the 2016 IEEE Signal Processing Society Best Paper Award and the 2017 Qualcomm Faculty Award.