Multitask compressive sensing

• Authors: Shihao Ji;David Dunson;Lawrence Carin
• Venue: IEEE Transactions on Signal Processing
• Year: 2009

Compressive sensing (CS) is a framework whereby one performs N nonadaptive measurements to constitute a vector v ∈ RN, with v used to recover an approximation û ∈ RM to a desired signal u ∈ RM, with N ≪ M; this is performed under the assumption that u. is sparse in the basis represented by the matrix. Ψ ∈ RM × M. It has been demonstrated that with appropriate design of the compressive measurements used to define v, the decompressive mapping v → û may be performed with error ||u - û||22 having asymptotic properties analogous to those of the best adaptive transform-coding algorithm applied in the basis Ψ. The mapping v → û constitutes an inverse problem, often solved using l1 regularization or related techniques. In most previous research, if L 1 sets of compressive measurements {vi}i=1, L are performed, each of the associated {ûi}i=1, L are recovered one at a time, independently. In many applications the L "tasks" defined by the mappings vi → ûi are not statistically independent, and it may be possible to improve the performance of the inversion if statistical interrelationships are exploited. In this paper, we address this problem within a multitask learning setting, wherein the mapping vi → ûi for each task corresponds to inferring the parameters (here, wavelet coefficients) associated with the desired signal ui, and a shared prior is placed across all of the L tasks. Under this hierarchical Bayesian modeling, data from all L tasks contribute toward inferring a posterior on the hyperparameters, and once the shared prior is thereby inferred, the data from each of the L individual tasks is then employed to estimate the task-dependent wavelet coefficients. An empirical Bayesian procedure for the estimation ofhyperparameters is considered; two fast inference algorithms extending the relevance vector machine (RVM) are developed. Example results on several data sets demonstrate the effectiveness and robustness of the proposed algorithms.