IPGM: Inertial Proximal Gradient Method for Convolutional Dictionary Learning

Abstract

Inspired by the recent success of the proximal gradient method (PGM) and recent efforts to develop an inertial algorithm, we propose an inertial PGM (IPGM) for convolutional dictionary learning (CDL) by jointly optimizing both an ℓ2-norm data fidelity term and a sparsity term that enforces an ℓ1 penalty. Contrary to other CDL methods, in the proposed approach, the dictionary and needles are updated with an inertial force by the PGM. We obtain a novel derivative formula for the needles and dictionary with respect to the data fidelity term. At the same time, a gradient descent step is designed to add an inertial term. The proximal operation uses the thresholding operation for needles and projects the dictionary to a unit-norm sphere. We prove the convergence property of the proposed IPGM algorithm in a backtracking case. Simulation results show that the proposed IPGM achieves better performance than the PGM and slice-based methods that possess the same structure and are optimized using the alternating-direction method of multipliers (ADMM).