Extending Kalman Filters for non-Gaussian Process Noise Sources via Approximate Message Passing

Estimating time-varying signals becomes particularly challenging in the face of non-Gaussian (e.g., sparse) and rapidly time-varying noise dynamics. By building upon the recent progress in the approximate message passing (AMP) realm, this paper unifies the vector AMP (VAMP) paradigm and the Kalman filter (KF) into a common message passing framework. The new algorithm - coined VAMP-KF - does not restrict the process noise dynamics to a specific structure (e.g., same support over time), thereby accounting for both uncorrelated and correlated noise. Numerical results on rapidly time-varying noise with different sparsity rates and distributions demonstrate unambiguously the effectiveness of the proposed VAMP-KF algorithm and its superiority against state-of-the-art baselines in terms of reconstruction accuracy.