Monte Carlo methods for compressed sensing

In this paper we study Monte Carlo type approaches to Bayesian sparse inference under a squared error loss. This problem arises in Compressed Sensing, where sparse signals are to be estimated and where recovery performance is measured in terms of the expected sum of squared error. In this setting, it is common knowledge that the mean over the posterior is the optimal estimator. The problem is however that the posterior distribution has to be estimated, which is extremely difficult. We here contrast approaches that use a Monte Carlo estimate for the posterior mean. The randomised Iterative Hard Thresholding algorithm is compared to a new approach that is inspired by sequential importance sampling and uses a bootstrap re-sampling step based on importance weights.