Riemannian Splat Regression Models for Learning Time Fields on Arbitrary Riemannian Manifolds

2026-09-28

AI Summary

This study addresses the computational inefficiency of conventional neural networks in motion planning on arbitrary Riemannian manifolds, which stems from their reliance on coordinate projections. To overcome this limitation, this work proposes a projection-free, manifold-native learning framework. Specifically, it pioneers direct Splat regression on arbitrary Riemannian manifolds by integrating wrapped Gaussian distributions with Riemannian geometric optimization to learn optimal time-of-arrival field functions in an end-to-end manner. Experimental results demonstrate that the proposed model successfully fits time fields across diverse manifolds. Compared to standard multilayer perceptrons (MLPs), it not only significantly improves prediction accuracy but also substantially reduces model capacity and accelerates inference. Consequently, this approach establishes a new paradigm for efficient motion planning on Riemannian manifolds.

Abstract

Motion planning on arbitrary Riemannian manifolds is an important and difficult problem that frustrates typical planning methods for Euclidean spaces. In particular, motion planning methods that approximate optimal time-to-go functions with neural networks, e.g., Neural Time Fields (NTFields), cannot be directly applied without using ad-hoc coordinate projections into higher dimensions. Using these methods directly without such projections is desirable, as it promises to provide the lowest-possible-runtime method for obtaining optimal plans on high-dimensional manifolds while using minimal model capacity. In this work, we develop a model that requires no coordinate projection and can learn arbitrary functions on Riemannian manifolds by combining splat regression models with splats defined by wrapped Gaussian distributions. We successfully apply this model for learning arrival time fields on several Riemannian manifolds, and we compare the accuracy and model size of this approach with multi-layer perceptrons adapted to work on each manifold individually.

Problem

• Riemannian manifolds
• motion planning
• time fields
• coordinate projection
• neural networks

Innovation

• Riemannian manifolds
• Splat regression
• Wrapped Gaussian distributions
• Time fields
• Motion planning