Markovian Gaussian Process: A Universal State-Space Representation for Stationary Temporal Gaussian Process

Gaussian Processes (GPs) and Linear Dynamical Systems (LDSs) are essential time series and dynamic system modeling tools. GPs can handle complex, nonlinear dynamics but are computationally demanding, while LDSs offer efficient computation but lack the expressive power of GPs. To combine their benefits, we introduce a universal method that allows an LDS to mirror stationary temporal GPs. This state-space representation, known as the Markovian Gaussian Process (Markovian GP), leverages the flexibility of kernel functions while maintaining efficient linear computation. Unlike existing GP-LDS conversion methods, which require separability for most multi-output kernels, our approach works universally for single- and multi-output stationary temporal kernels. We evaluate our method by computing covariance, performing regression tasks, and applying it to a neuroscience application, demonstrating that our method provides an accurate state-space representation for stationary temporal GPs.