Kalman filtering beyond Gaussian innovation processes
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Estimating time-varying signals becomes particularly challenging under non Gaussian innovation processes such as sparse and rapidly time-varying noise dynamics. In this thesis, by building upon the recent progress in the approximate message passing (AMP) algorithms, the vector AMP (VAMP) algorithm is unified with the Kalman filter (KF) into a common message passing framework that we coin VAMP-KF. The advantage of VAMP-KF is that it does not restrict the innovation dynamics to have a specific structure (e.g., same support over time when the innovation is sparse), thereby accounting for uncorrelated noise dynamics without the need of explicit innovation correlation modelling. For the sake of theoretical performance prediction, we conduct a state evolution (SE) analysis of the proposed algorithm and show its consistency with the asymptotic empirical mean-squared error (MSE). Numerical results on various rapidly time-varying innovation dynamics (e.g., with different sparsity rates) demonstrate unambiguously the effectiveness of the proposed VAMP-KF algorithm and its superiority over state of-the-art algorithms both in terms of reconstruction accuracy and computational complexity.