FAST CONVOLUTIONAL SPARSE CODING WITH ℓ0 PENALTY

Given a set of dictionary filters, the most widely used formulation of the convolutional sparse coding (CSC) problem is Convolutional BPDN (CBPDN), in which an image is represented as a sum over a set of convolutions of coefficient maps; usually, the coefficient maps are ℓ1-norm penalized in order to enforce a sparse solution. Recent theoretical results, have provided meaningful guarantees for the success of popular ℓ1-norm penalized CSC algorithms in the noiseless case. However, experimental results related to the ℓ0-norm penalized CSC case have not been addressed.In this paper we propose a two-step ℓ0-norm penalized CSC (ℓ0-CSC) algorithm, which outperforms (convergence rate, reconstruction performance and sparsity) known solutions to the ℓ0-CSC problem. Furthermore, our proposed algorithm, which is a convolutional extension of our previous work [1], originally develop for the ℓ0 regularized optimization problem, includes an escape strategy to avoid being trapped in a saddle points or in inferior local solutions, which are common in nonconvex optimization problems, such those that use the ℓ0-norm as the penalty function.