Solving Sparse Linear Inverse Problems: Analysis of Reweighted l1 and l2 Methods

A variety of practical methods have recently been introduced for finding maximally sparse representations from overcomplete dictionaries, a central computational task in compressed sensing and source localization applications as well as numerous others. Many of the underlying algorithms rely on iterative reweighting schemes that produce more focal estimates as optimization progresses. Two such variants are iterative reweighted `1 and `2 minimization; however, some properties related to convergence and sparse estimation, as well as possible generalizations, are still not clearly understood or fully exploited. In this paper, we make the distinction between separable and nonseparable iterative reweighting algorithms. The vast majority of existing methods are separable, meaning the weighting of a given coefficient at each iteration is only a function of that individual coefficient from the previous iteration (as opposed to dependency on all coefficients). We examine two such separable reweighting schemes: an `2 method from Chartand and Yin (2008) and an `1 approach from Cand`es et al. (2008), elaborating on convergence results and explicit connections between them. We then explore an interesting non-separable variant that can be implemented via either `2 or `1 reweighting and show several desirable properties relevant to sparse recovery. For the former, we show a direct connection with Chartrand and Yin’s approach. For the latter, we demonstrate two desirable properties: (i) each iteration can only improve the sparsity and (ii), for any dictionary and sparsity profile, there will always exist cases where non-separable `1 reweighting improves over standard `1 minimization.