  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.
