Proximal gradient nonconvex optimization algorithm for the slice-based [formula omitted]-constrained convolutional dictionary learning

Abstract

Convolutional dictionary learning (CDL) aims to learn a structured local convolutional dictionary and the sparse coefficient maps from the signals of various of interest, achieving better results than traditional patch-based dictionary learning in signal processing applications. Currently, most CDL methods are used to solve the ℓ1-constrained convex CDL problem using the patch-based, Fourier-based, or slice-based representation, which may lead to non-sparse coefficient maps, thus degrading the performance of applications. The slice-based representation via local processing has shown more beneficial than the Fourier-based representation for convex CDL problems. It can allow a different number of nonzeros for each spatial location of the signal according to the local complexity. However, it has not been used for the nonconvex CDL problem. In this paper, a slice-based ℓ0-constrained nonconvex CDL problem is proposed. It is the slice-based counterpart of the Fourier-based ℓ0-constrained nonconvex CDL problems and the nonconvex extension of the slice-based ℓ1-constrained convex CDL problems. A novel method named proximal gradient nonconvex optimization algorithm (PGNOA) is introduced. We prove that PGNOA can converge to a critical point. Extensive experiments are carried out on the benchmark data, and the results show that PGNOA is superior to the existing slice-based convex CDL methods and the Fourier-based nonconvex CDL methods in terms of the objective function value. The dictionaries learned from PGNOA are applied to image inpainting and image separation tasks. The experimental results demonstrate that PGNOA can perform better than other CDL methods.