BAYESIAN SPARSE SIGNAL RECOVERY By XING TAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BAYESIAN SPARSE SIGNAL RECOVERY By Xing Tan December 2009 Chair: Jian Li Major: Electrical and Computer Engineering Sparse Bayesian learning (SBL) was first proposed in the machine learning literature and later applied to sparse signal recovery. It has been shown that SBL is easy to use and can recover sparse signals more accurately than the l1-norm based optimization approaches. In this dissertation, we present several Bayesian algorithms for sparse signal recovery in different applications. Firstly, we propose a general sparse signal recovery algorithm. The computational complexities of the widely-used SBL approaches are quite high, which limit their use in large-scale problems. We propose herein an efficient Gibbs sampling approach, referred to as GS-SBL, for general sparse signal recovery problems. We show that GS-SBL provides better performance than Basis Pursuit (BP) and other SBL approaches for both smalland large-scale compressed sensing problems. For large-scale compressed sensing problems, GS-SBL can be faster than the so-called Fast Marginal Likelihood Maximization method, which is currently the fastest SBL approach among all existing SBL approaches. Secondly, we propose a belief propagation (BP) sparse Bayesian learning algorithm, referred to as the BP-SBL, to recover sparse transform coefficients in large scale compressed sensing problems. BP-SBL is based on a widely-used hierarchical Bayesian model, which is turned into a sparse factor graph so that BP can be applied to achieve computational efficiency. The computational complexity of BP-SBL is