Automatic calculation of step size and inertia parameter for convolutional dictionary learning

Abstract

The convergent methods available for convolutional dictionary learning (CDL) are the proximal gradient method (PGM) and the inertial proximal gradient method (IPGM). However, it is not trivial and heuristic for IPGM to set the step size and inertia parameter, and IPGM produces local minima and oscillating solutions. To address these issues, in this paper, we introduce a dry friction, which has an oscillation-alleviating property. Specifically, the proposed IPGM with dry friction (IPGM-DF) generates the composite proximal mappings, whose construction and optimization solver are two challenges. An auxiliary function is designed to construct the composite proximal mappings, whose optimization problem is solved by the alternating direction method of multipliers. Fortunately, IPGM-DF obtains the formulas of the step size and inertia parameter. The finite convergence of IPGM-DF is proved. Experimental results of image reconstruction, separation, and fusion demonstrate the superiority of IPGM-DF over IPGM and the state-of-the-art methods. For image reconstruction, the objective function value of IPGM-DF is reduced by about 38.708% than that of IPGM. Throughout alleviating oscillations, IPGM-DF obtains a much lower value of the objective function than IPGM, which indicates that IPGM-DF jumps out of the local minima of IPGM. In addition, the average PSNR of IPGM-DF is about 3.5 dB larger than that of IPGM and the state-of-the-art methods. The code is available.