Ensemble Kalman Filtering-Aided Variational Inference for Gaussian Process State-Space Models

Gaussian process state-space models (GPSSMs) are a flexible and principled approach for modeling dynamical systems. However, existing variational learning and inference methods for GPSSMs often necessitate optimizing a substantial number of variational distribution parameters, leading to inadequate performance and efficiency. To overcome this issue, we propose incorporating the ensemble Kalman filter (EnKF), a well-established model-based filtering technique, into the variational inference framework to approximate the posterior distribution of latent states. This utilization of EnKF can effectively exploit the dependencies between latent states and GP dynamics, while eliminating the need for parameterizing the variational distribution, thereby significantly reducing the number of variational parameters. Moreover, we show that our proposed algorithm allows straightforward evaluation of an approximated evidence lower bound (ELBO) in variational inference via simply summating multiple terms with readily available closed-form solutions. Leveraging automatic differentiation tools, we hence can maximize the ELBO and train the GPSSM efficiently. We also extend the proposed algorithm to an online setting and provide detailed algorithmic analyses and insights. Extensive evaluation on diverse real and synthetic datasets demonstrates the superiority of our EnKF-aided variational inference algorithms in terms of learning and inference performance compared to existing methods.