In this work we are interested in the problem of reconstructing time-varying signals for which the support is assumed to be sparse. For a single time instance it is possible to reconstruct the original signal efficiently by employing a suitable algorithm for sparse signal recovery, given the sparsity level of the signal. In the case of time-varying sparse signals the sparsity level is not necessarily known a-priori. Furthermore conventional tracking by Kalman filtering fails to promote sparsity. Instead, a hierarchical Bayesian model is used in the tracking process which succeeds in modelling sparsity. One theorem is provided that extends previous work by providing some more general results. A second theorem gives the conditions under which all sparse signals are recovered exactly. It is demonstrated that the proposed method succeeds in recovering timevarying sparse signals with greater accuracy than the classic Kalman filter approach.
In this paper we present a set of theoretical results regarding inference algorithms for hierarchical Bayesian networks. More specifically we focus on a specific type of networks which result in highly sparse models for the input. Bayesian inference in these networks usually is based on optimising a non-convex cost function of the model parameters. We extend previous work done in this field by providing some global performance guarantees regarding this cost function. This is the starting point for redesigning the aforementioned algorithms by employing results from well known sparse reconstruction techniques. This contribution comes in the form of three theorems. The end result is a new view of the Bayesian sparse reconstruction problem.
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