Bayesian Multi-Task Compressive Sensing with Dirichlet Process Priors

Compressive sensing (CS) is an emerging field that, under appropriate conditions, can significantly reduce the number of measurements required for a given signal. Specifically, if the m-dimensional signal u is sparse in an orthonormal basis represented by the m × m matrix Ψ, then one may infer u based on n m projection measurements. If u = Ψθ, where θ are the sparse coefficients in basis Ψ, then the CS measurements are represented by v = Φθ, where v is an n-dimensional vector and Φ is an n × m projection matrix. There are several ways in which the matrix Φ may be constituted, and one typically inverts for the signal u by solving v = Φθ under the constraint that θ is sparse (with this often performed with l1 regularization). In many applications, one is interested in multiple signals {ui}i=1,M that may be measured in multiple CS-type measurements, where here each ui corresponds to a sensing “task”. It is possible to improve the CS performance (e.g., requiring fewer total CS measurements) by exploiting the statistical inter-relationships of the associated {vi}i=1,M CS measurements, jointly inverting for the M underlying signals. In this paper we propose a novel multi-task compressive sensing framework based on a Bayesian formalism, where a sparseness prior is adopted. The key challenge is that not all of the measured {vi}i=1,M are necessarily appropriate for sharing when performing inversion, and one must therefore infer what sharing of data across the M “tasks” is appropriate. Toward this end, a Dirichlet process (DP) prior is employed, which provides a principled means of inferring the appropriate sharing mechanisms (i.e., it infers how best to cluster the M CS measurements, with CS inversion effectively performed separately within each cluster). The posteriors of the sparse signals as well as the sharing mechanism are inferred among all CS tasks. A variational Bayesian (VB) inference algorithm is employed to estimate the full posterior on the model parameters, and an even more efficient simplified VB DP algorithm is also considered. Index Terms Compressive sensing (CS), Multi-task learning, Dirichlet Process Priors, Sparse Bayesian learning, Variational Bayes