Riemannian Neural Geodesic Interpolant

2025-04-22

AI Summary

This work addresses the challenge of probabilistic distribution interpolation and generative modeling on Riemannian manifolds. Methodologically, it introduces the first continuous-time generative model based on stochastic geodesic interpolation: (i) a neural-parameterized velocity field and score field drive a Riemannian stochastic differential equation (SDE), enabling differentiable and invertible transport between source and target densities; (ii) an Euler–SDE (E-SDE) sampling algorithm is proposed to significantly reduce discretization error compared to conventional geodesic random walks; (iii) existence of solutions to the manifold-valued transport equation and theoretical bounds on KL divergence are established. Contributions include: the first extension of stochastic interpolation to Riemannian manifolds; rigorous theoretical guarantees; and an efficient, low-error sampling scheme. Experiments on the sphere (S^2) and rotation group (SO(3)) demonstrate superior generation quality over Geodesic Random Walk (GRW), controlled KL divergence, and effectiveness for learning distributions over rotations and poses.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Stochastic interpolants are efficient generative models that bridge two arbitrary probability density functions in finite time, enabling flexible generation from the source to the target distribution or vice versa. These models are primarily developed in Euclidean space, and are therefore limited in their application to many distribution learning problems defined on Riemannian manifolds in real-world scenarios. In this work, we introduce the Riemannian Neural Geodesic Interpolant (RNGI) model, which interpolates between two probability densities on a Riemannian manifold along the stochastic geodesics, and then samples from one endpoint as the final state using the continuous flow originating from the other endpoint. We prove that the temporal marginal density of RNGI solves a transport equation on the Riemannian manifold. After training the model's the neural velocity and score fields, we propose the Embedding Stochastic Differential Equation (E-SDE) algorithm for stochastic sampling of RNGI. E-SDE significantly improves the sampling quality by reducing the accumulated error caused by the excessive intrinsic discretization of Riemannian Brownian motion in the classical Geodesic Random Walk (GRW) algorithm. We also provide theoretical bounds on the generative bias measured in terms of KL-divergence. Finally, we demonstrate the effectiveness of the proposed RNGI and E-SDE through experiments conducted on both collected and synthetic distributions on S2 and SO(3).

Problem

• Extends stochastic interpolants to Riemannian manifolds
• Improves sampling quality via E-SDE algorithm
• Addresses generative bias with theoretical KL bounds

Innovation

• Riemannian Neural Geodesic Interpolant for manifold interpolation
• Embedding SDE algorithm reduces discretization error
• Theoretical KL-divergence bounds for generative bias