Exact Uncertainty Propagation via Gaussian Process Neurons

Abstract

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.