Matrix-Variate Gaussian Process State-Space Model with Structured Latent Dynamics

Matrix-valued time series arise in many applications with coupled row-wise, column-wise, and temporal dependencies. Probabilistic modeling of their dynamics requires a framework that preserves row–column structure while remaining tractable for high-dimensional data. In this paper, we formulate a matrix-variate Gaussian process state-space model (MV-GPSSM) with structured latent dynamics. It represents the observed sequence as low-dimensional latent matrices through a Tucker2-type decomposition, models their temporal evolution using a matrix-variate GP, and lifts the predictive distribution back to the observation space through learned row and column bases. Within this formulation, we characterize how latent interaction mechanisms induce structured matrix-valued kernels: multiplicative and additive row–column interactions lead to Kronecker-product and Kronecker-sum covariance forms, respectively. Thus, these covariance structures arise as consequences of the assumed latent interaction mechanisms rather than being prescribed directly, thereby providing a dynamical interpretation of these structured covariance forms within the model family. We further present a computational complexity analysis of inference in the proposed MV-GPSSM. Experiments on battery electric vehicle fleet data and publicly available weather data show that the MV-GPSSM maintains predictive performance comparable to that of the baseline methods in terms of both point accuracy (RMSE) and probabilistic quality (NLL evaluated using the full predictive covariance matrix), while remaining tractable at large matrix sizes where observation-space methods become computationally or memory infeasible. Controlled comparisons clarify the roles of matrix structure, latent representation, and GP-based dynamics. The learned row- and column-wise covariance structures further reveal qualitatively interpretable dependencies in vehicle behavior and weather spatiotemporal patterns.