Iteratively Reweighted $\ell_1$ Approaches to Sparse Composite Regularization

Motivated by the observation that a given signal $\boldsymbol{x}$ admits sparse representations in multiple dictionaries $\boldsymbol{\Psi}d$ 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 regularizer $|\boldsymbol{\Psi x}|1$ to composite regularizers of the form $\sumd \lambdad |\boldsymbol{\Psi}d \boldsymbol{x}|1$ while self-adjusting the regularization weights $\lambdad$. 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) majorization-minimization (MM) applied to a non-convex log-sum-type penalty, ii) MM applied to an approximate $\ell0$-type penalty, iii) MM applied to Bayesian MAP 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.