A Proximal Method for Convolutional Dictionary Learning with Convergence Property

The convolutional sparse coding (CSC) is superior in representing signals, and to obtain the best performance of CSC, the dictionary is usually learned from data. The so-called convolution dictionary learning (CDL) problem is therefore formulated for the purpose. Most of the solvers for CDL alternately update the coefficients and dictionary in an iterative manner, and as a consequence, numerous redundant iterations incur slow speed in achieving convergence. Moreover, their convergence properties can hardly be analyzed even though the $\ell_{0}$ sparse inducing function is approximated by the convex $\ell_{0}$ norm. In this article, we propose an algorithm which directly deals with the non-convex non-smooth $\ell_{0}$ constraint and provides a sound convergence property. The experimental results show that, the proposed method achieves the convergence point with less time and a smaller final function value compared to the existing convolutional dictionary learning algorithms.