Diffusion Mean Estimation on the Diagonal of Product Manifolds

Computing sample means on Riemannian manifolds is typically computationally costly. The Fr\'echet mean offers a generalization of the Euclidean mean to general metric spaces, particularly to Riemannian manifolds. Evaluating the Fr\'echet mean numerically on Riemannian manifolds requires the computation of geodesics for each sample point. When closed-form expressions do not exist for geodesics, an optimization-based approach is employed. In geometric deep-learning, particularly Riemannian convolutional neural networks, a weighted Fr\'echet mean enters each layer of the network, potentially requiring an optimization in each layer. The weighted diffusion-mean offers an alternative weighted mean sample estimator on Riemannian manifolds that do not require the computation of geodesics. Instead, we present a simulation scheme to sample guided diffusion bridges on a product manifold conditioned to intersect at a predetermined time. Such a conditioning is non-trivial since, in general, manifolds cannot be covered by a single chart. Exploiting the exponential chart, the conditioning can be made similar to that in the Euclidean setting.