Exact Uncertainty Propagation via Gaussian Process Neurons

Exact marginalization in deep {Gaussian} processes (DGPs) is analytically intractable, typically requiring computationally costly sampling-based approximations. We introduce the {Gaussian} process neuron, a computational unit built on a novel {Wasserstein} exponential kernel. By defining a composable map between factorized {Gaussian} measures, this unit enables analytical uncertainty propagation; concurrently, optimizing deterministic inducing variables via maximum a posteriori estimation ensures strict probabilistic rigor. By eliminating sampling variance and the depth-scaling computational bottlenecks of Monte Carlo methods, our approach enables exact probabilistic variants of diverse architectures, including DGPs, multilayer perceptrons, and {Kolmogorov}-Arnold Networks. Empirically, these networks demonstrate superior uncertainty calibration relative to state-of-the-art approximate {DGP} frameworks, alongside high-dimensional scalability and robust compositional expressiveness.