Multitask Compressive Sensing

Compressive sensing (CS) is a framework whereby one performs N nonadaptive measurements to constitute a vector v isin R N used to recover an approximation u isin R M desired signal u isin R M with N MtimesM. It has been demonstrated that with appropriate design of the compressive measurements used to define v, the decompressive mapping vrarru may be performed with error ||u-u|| 2 2 having asymptotic properties analogous to those of the best adaptive transform-coding algorithm applied in the basis Psi. The mapping vrarru constitutes an inverse problem, often solved using l 1 regularization or related techniques. In most previous research, if L > 1 sets of compressive measurements v i i=1, L are performed, each of the associated u i i=1, L are recovered one at a time, independently. In many applications the L ldquotasksrdquo defined by the mappings v i rarru i 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 vrarru for each task corresponds to inferring the parameters (here, wavelet coefficients) associated with the desired signal v i, 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.