Iterative Cauchy Thresholding: Regularisation with a heavy-tailed prior

In the machine learning era, sparsity continues to attract significant interest due to the benefits it provides to learning models. Algorithms aiming to optimise the $\ell_0$- and $\ell_1$-norm are the common choices to achieve sparsity. In this work, an alternative algorithm is proposed, which is derived based on the assumption of a Cauchy distribution characterising the coefficients in sparse domains. The Cauchy distribution is known to be able to capture heavy-tails in the data, which are linked to sparse processes. We begin by deriving the Cauchy proximal operator and subsequently propose an algorithm for optimising a cost function which includes a Cauchy penalty term. We have coined our contribution as Iterative Cauchy Thresholding (ICT). Results indicate that sparser solutions can be achieved using ICT in conjunction with a fixed over-complete discrete cosine transform dictionary under a sparse coding methodology.