From Deterministic to Bayesian Compressive Sensing

Abstract

Compressive Sensing (CS) is a paradigm that has gained a lot of attention in the last decade. CS allows us to reconstruct sufficiently sparse or compressible signals of dimension far greater than the number of available measurements. The reconstruction phase of CS is an inverse problem and usually requires solving a convex sparsity-constrained regularisation problem. However, much of the literature in CS is based in this deterministic setting which gives little or no information of the uncertainty in reconstructions. In this thesis we will introduce how we can generalise this inverse problem as a Bayesian inference problem and show how this generalisation can provide us with a measure of uncertainty in our predictions. We will review two of the more popular Bayesian methods that are used in CS and show how these methods give overoptimistic approximations to the uncertainty in their reconstructions. We show that as we take fewer CS measurements these methods suffer from infeasible posterior distributions. We propose a new method for Bayesian CS using a Gibbs sampler and show that the method does not suffer from these infeasibilities. We compare the results of our method with the two existing methods for a simple CS problem. Our method can also be easily extended to solve more general sparsity-constrained inverse problems outside of CS.