Representation Learning via Cauchy Convolutional Sparse Coding

In representation learning, Convolutional Sparse Coding (CSC) enables unsupervised learning of features by jointly optimising both an
$\ell _{2}$
-norm fidelity term and a sparsity enforcing penalty. This work investigates using a regularisation term derived from an assumed Cauchy prior for the coefficients of the feature maps of a CSC generative model. The sparsity penalty term resulting from this prior is solved via its proximal operator, which is then applied iteratively, element-wise, on the coefficients of the feature maps to optimise the CSC cost function. The performance of the proposed Iterative Cauchy Thresholding (ICT) algorithm in reconstructing natural images is compared against algorithms based on minimising standard penalty functions via soft and hard thresholding as well as against the Iterative Log-Thresholding (ILT) method. ICT outperforms the Iterative Hard Thresholding (IHT), Iterative Soft Thresholding (IST), and ILT algorithms in most of our reconstruction experiments across various datasets, with an average Peak Signal to Noise Ratio (PSNR) of up to 11.30 dB, 7.04 dB, and 7.74 dB over IST, IHT, and ILT respectively. The source code for the implementation of the proposed approach is publicly available at
https://github.com/p-mayo/cauchycsc