Residual Deep Gaussian Processes on Manifolds

We propose practical deep Gaussian process models on Riemannian manifolds, similar in spirit to residual neural networks. With manifold-to-manifold hidden layers and an arbitrary last layer, they can model manifold-and scalar-valued functions, as well as vector fields. We target data inherently supported on manifolds, which is too complex for shallow Gaussian processes thereon. For example, while the latter perform well on high-altitude wind data, they struggle with the more intricate, nonstationary patterns at low altitudes. Our models significantly improve performance in these settings, enhancing prediction quality and uncertainty calibration, and remain robust to overfitting, reverting to shallow models when additional complexity is unneeded. We further showcase our models on Bayesian optimisation problems on manifolds, using stylised examples motivated by robotics, and obtain substantial improvements in later stages of the optimisation process. Finally, we show our models to have potential for speeding up inference for nonmanifold data, when, and if, it can be mapped to a proxy manifold well enough. Gaussian processes (GPs) are a widely adopted model class for learning functions within the Bayesian framework (Rasmussen and Williams, 2006). They offer accurate uncertainty estimates and perform well even when data is scarce. Consequently, GPs have found success in decisionmaking tasks, where well-calibrated uncertainty is key, including Bayesian optimisation (Snoek et al., 2012), active (Krause et al., 2008) and reinforcement (Kamthe and Deisenroth, 2018) learning. In recent years, substantial work went into developing the analogs of practical GP models on various non-Euclidean domains (Borovitskiy et al., 2021; 2023; 2020; Fichera et al., 2023).