  Spike and Slab priors have been of much recent interest in signal processing
as a means of inducing sparsity in Bayesian inference. Applications domains
that benefit from the use of these priors include sparse recovery, regression
and classification. It is well-known that solving for the sparse coefficient
vector to maximize these priors results in a hard non-convex and mixed integer
programming problem. Most existing solutions to this optimization problem
either involve simplifying assumptions/relaxations or are computationally
expensive. We propose a new greedy and adaptive matching pursuit (AMP)
algorithm to directly solve this hard problem. Essentially, in each step of the
algorithm, the set of active elements would be updated by either adding or
removing one index, whichever results in better improvement. In addition, the
intermediate steps of the algorithm are calculated via an inexpensive Cholesky
decomposition which makes the algorithm much faster. Results on simulated data
sets as well as real-world image recovery challenges confirm the benefits of
the proposed AMP, particularly in providing a superior cost-quality trade-off
over existing alternatives.
