Hierarchical bayesian models for sparse signal recovery and sampling

Abstract

This thesis builds upon the problem of sparse signal recovery from the Bayesian standpoint. The advantages of employing Bayesian models are underscored, with the most important being the ease at which a model can be expanded or altered; leading to a fresh class of algorithms. The thesis fills out several gaps between sparse recovery algorithms and sparse Bayesian models; firstly the lack of global performance guarantees for the latter and secondly what the signifying differences are between the two. These questions are answered by providing; a refined theoretical analysis and a new class of algorithms that combines the benefits from classic recovery algorithms and sparse Bayesian modelling. The said Bayesian techniques find application in tracking dynamic sparse signals, something impossible under the Kalman filter approach.

Another innovation of this thesis are Bayesian models for signals whose components are known a priori to exhibit a certain statistical trend. These situations require that the model enforces a given statistical bias on the solutions. Existing Bayesian models can cope with this input, but the algorithms to carry out the task are computationally expensive. Several ways are proposed to remedy the associated problems while still attaining some form of optimality. The proposed framework finds application in multipath channel estimation with some very promising results.

Not far from the same area lies that of Approximate Message Passing. This includes extremely low-complexity algorithms for sparse recovery with a powerful analysis framework. Some results are derived, regarding the differences between these approximate methods and the aforementioned models. This can be seen as preliminary work for future research.

Finally, the thesis presents a hardware implementation of a wideband spectrum analyser based on sparse recovery methods. The hardware consists of a Field-Programmable Gate Array coupled with an Analogue to Digital Converter. Some critical results are drawn, regarding the gains and viability of such methods.