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This paper considers the problem of recovering time-varying sparse signals from dramatically undersampled measurements. A probabilistic signal model is presented that describes two common traits of time-varying sparse signals: a support set that changes slowly over time, and amplitudes that evolve smoothly in time. An algorithm for recovering signals that exhibit these traits is then described. Built on the belief propagation framework, the algorithm leverages recently developed approximate message passing techniques to perform rapid and accurate estimation. The algorithm is capable of performing both causal tracking and non-causal smoothing to enable both online and offline processing of sparse time series, with a complexity that is linear in all problem dimensions. Simulation results illustrate the performance gains obtained through exploiting the temporal correlation of the time series relative to …