Title: Learning Convex Regularizers for Optimal Bayesian Denoising,
Abstract: We propose a data-driven algorithm for the maximum a posteriori (MAP)
estimation of stochastic processes from noisy observations. The primary
statistical properties of the sought signal is specified by the penalty
function (i.e., negative logarithm of the prior probability density function).
Our alternating direction method of multipliers (ADMM)-based approach
translates the estimation task into successive applications of the proximal
mapping of the penalty function. Capitalizing on this direct link, we define
the proximal operator as a parametric spline curve and optimize the spline
coefficients by minimizing the average reconstruction error for a given
training set. The key aspects of our learning method are that the associated
penalty function is constrained to be convex and the convergence of the ADMM
iterations is proven. As a result of these theoretical guarantees, adaptation
of the proposed framework to different levels of measurement noise is extremely
simple and does not require any retraining. We apply our method to estimation
of both sparse and non-sparse models of Lévy processes for which the
minimum mean square error (MMSE) estimators are available. We carry out a
single training session and perform comparisons at various signal-to-noise
ratio (SNR) values. Simulations illustrate that the performance of our
algorithm is practically identical to the one of the MMSE estimator
irrespective of the noise power.