Machine learning meets Kalman Filtering

In this work we study the problem of efficient non-parametric estimation for non-linear time-space dynamic Gaussian processes (GP). We propose a systematic and explicit procedure to address this problem by pairing GP regression with Kalman Filtering. Under a specific separability assumption of the modeling kernel and periodic sampling on a (possibly non-uniform) space-grid, we show how to build an exact finite dimensional discrete-time state-space representation for the modeled process. The major finding is that the state at instant k of the associated Kalman Filter represents a sufficient statistic to compute the minimum variance prediction of the process at instant k over any arbitrary finite subset of the space. Finally, we compare the proposed strategy with standard approaches.