State and Dynamics Estimation with the Kalman-Langevin filter

We present a novel sampling algorithm designed for the joint estimation of the states and a graph which partially models the state dynamics within a state-space representation. In particular, the dynamics are composed by the output of a graph filter whose input is a partially known adjacency matrix. Our algorithm learns the unknown part of the adjacency matrix. Our proposed method combines the classical Kalman filter for state estimation with annealed Langevin diffusion. This allows us to approximate the maximum a posteriori of the states and dynamics given observations by sampling from the posterior distribution. We exploit the prior information embedded in a dataset of graphs of varying sizes through the utilization of graph neural networks. We empirically demonstrate that integrating a learned prior distribution into the estimation process significantly improves estimation performance.