Iterative Log Thresholding

Sparse reconstruction approaches using the re-weighted l1-penalty have been shown, both empirically and theoretically, to provide a significant improvement in recovering sparse signals in comparison to the l1-relaxation. However, numerical optimization of such penalties involves solving problems with l1-norms in the objective many times. Using the direct link of reweighted l1-penalties to the concave log-regularizer for sparsity, we derive a simple prox-like algorithm for the log-regularized formulation. The proximal splitting step of the algorithm has a closed form solution, and we call the algorithm 'log-thresholding' in analogy to soft thresholding for the l1-penalty. We establish convergence results, and demonstrate that log-thresholding provides more accurate sparse reconstructions compared to both soft and hard thresholding. Furthermore, the approach can be directly extended to optimization over matrices with penalty for rank (i.e. the nuclear norm penalty and its re-weigthed version), where we suggest a singular-value log-thresholding approach.