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).

Este artículo explora los viajes en el tiempo y sus implicaciones.

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