Stochastic Modelling of a Collection of Correlated Sparse Signals and its Recovery via Belief Propagation Methods

The field of compressive sensing deals with the recovery of a sparse signal from a small set of measurements or linear projections of the signal. In this thesis, we introduce a stochastic framework that allows a collection of correlated sparse signals to be recovered by exploiting both intra and inter signal correlation. Our approach differs from others by not assuming that the collection of sparse signals have a common support or a common component; in some cases, this assumption does not hold true. Imagine a simplified cognitive radio problem, where users can send a single tone (sine-wave) in a finite number of frequencies; it is desired to find the used frequencies over a large area (creation of a radio map). This is a sparse problem; however, as we move spatially, the occuppied frequencies change, thus voiding the assumption of a common support/component. Our solution to multi sparse signal recovery addresses this problem, where signals that are close geographically are highly correlated and their support gradually changes as the distance between signals grow. Our approach consists of the creation of a probabilistic model that accounts for inter and intra signal correlation and then using belief propagation to calculate the posterior distribution of the signals and perform recovery.MAS