Gaussian process deconvolution

Abstract

Let us consider the deconvolution problem, i.e. to recover a latent sourcex(⋅)from the observationsy=[y1,…,yN]of a convolution processy=x⋆h+η, whereηis an additive noise, the observations inymight have missing parts with respect toy, and the filterhcould be unknown. We propose a novel strategy to address this task whenxis a continuous-time signal: we adopt a Gaussian process prior on the sourcex, which allows for closed-form Bayesian non-parametric deconvolution. We first analyse the direct model to establish the conditions under which the model is well-defined. Then, we turn to the inverse problem, where we study (i) some necessary conditions under which Bayesian deconvolution is feasible and (ii) to which extent the filterhcan be learned from data or approximated for the blind deconvolution case. The proposed approach, termed Gaussian process deconvolution, is compared to other deconvolution methods conceptually, via illustrative examples, and using real-world datasets.