Residual Deep Gaussian Processes on Manifolds

TL;DR

This paper introduces residual deep Gaussian process models on Riemannian manifolds that effectively model complex manifold-valued data, outperforming shallow models in prediction, uncertainty calibration, and Bayesian optimization tasks.

Contribution

It presents a novel deep Gaussian process framework on manifolds with residual connections, capable of modeling complex data and improving performance over traditional shallow Gaussian processes.

Findings

Significant performance improvements on manifold-valued data.

Enhanced uncertainty calibration and robustness to overfitting.

Better results in Bayesian optimization on manifolds.

Abstract

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 non-manifold data, when, and if, it can be mapped to a proxy manifold well enough.