Multitask compressive sensing

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 u∈RℝM to a desired signal u∈RℝM with N≪ M; this is performed under the assumption that uis sparse in the basis represented by the matrix Ψ∈RℝM×M. It has been demonstrated that with appropriate design of the compressive measurements used to define v, the decompressive mapping v⇁ umay be performed with error ∥u-u∥22 having asymptotic properties analogous to those of the best transform-coding algorithm applied in the basis Ψ. The mapping v⇁u constitutes an inverse problem, often solved using ℓ1 regularization or related techniques. In most previous research, if L〉 sets of compressive measurements vii=1,L are performed, each of the associated uii=1,L are recovered one at a time, independently. In many applications the "tasks"defined by the mappings vi⇁ ui 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 ⇁uifor 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 of hyperparameters 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. © 2008 IEEE.