LONG-MEMORY GAUSSIAN PROCESSES FOR STATE- SPACE INFERENCE AND REPRESENTATION LEARNING

The Mat´ern covariance kernel is a popular choice for Gaussian process (GP) based modeling of temporal dependencies in time-series data. For half-integer smoothness, the Mat´ern kernel admits a finite-dimensional state-space representation, reducing GP inference from cubic to linear complexity in the time series length. This has made Mat´ern attractive both for direct sequential modeling and for representing structured latent priors in variational autoencoders. However, Mat´ern kernels decay exponentially with lag and therefore cannot represent persistent, polynomially decaying long-range dependencies. To address this, we introduce the GAMMA covariance kernel family, obtained by gamma-mixing of half-integer Mat´ern components. The resulting kernels separate local regularity from long-range memory, allowing near-origin smoothness and asymptotic dependence to be controlled independently. For scalable inference, we approximate the continuous mixture using generalized Gauss–Laguerre quadrature, yielding a finite weighted sum of Markovian components and linear inference complexity in time series length. Building on this construction, we further propose GGPVAE, which places GAMMA priors over latent trajectories. We also theoretically prove the convergence of the proposed approximations. Our experimental results show that GAMMA and GGPVAE capture long-range structure, improve prediction and reconstruction across long missing intervals, and retain scalable inference.