  In this paper, we propose a proximal splitting methodology with a non-convex
penalty function based on the heavy-tailed Cauchy distribution. We first
suggest a closed-form expression for calculating the proximal operator of the
Cauchy prior, which then makes it applicable in generic proximal splitting
algorithms. We further derive the condition required for guaranteed convergence
to a solution in optimisation problems involving the Cauchy based penalty
function. Setting the system parameters by satisfying the proposed condition
ensures convergence even though the overall cost function is non-convex when
minimisation is performed via a proximal splitting algorithm. The proposed
method based on Cauchy regularisation is evaluated by solving generic signal
processing examples, i.e. 1D signal denoising in the frequency domain, two
image reconstruction tasks including de-blurring and denoising, and error
recovery in a multiple-antenna communication system. We experimentally verify
the proposed convergence conditions for various cases, and show the
effectiveness of the proposed Cauchy based non-convex penalty function over
state-of-the-art penalty functions such as $L_1$ and total variation ($TV$)
norms.
