Iteratively Reweighted $\ell_1$ Approaches to Sparse Composite Regularization

Motivated by the observation that a given signal x admits sparse representations in multiple dictionaries Ψ
_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 ∥Ψx∥
₁
to composite regularizers of the form Σ
_d
λ
_d
∥Ψ
_d
x
₁
while self-adjusting the regularization weights λd. 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: 1) majorization-minimization (MM) applied to a nonconvex log-sum-type penalty; 2) MM applied to an approximate Bo-type penalty; 3) MM applied to Bayesian MAP inference under a particular hierarchical prior; and 4) 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 noncomposite L1 and IRW-L1 counterparts.