TY - JOUR

T1 - Mean Estimation on the Diagonal of Product Manifolds

AU - Jensen, Mathias Højgaard

AU - Sommer, Stefan

PY - 2022

Y1 - 2022

N2 - Computing sample means on Riemannian manifolds is typically computationally costly, as exemplified by computation of the Fréchet mean, which often requires finding minimizing geodesics to each data point for each step of an iterative optimization scheme. When closed-form expressions for geodesics are not available, this leads to a nested optimization problem that is costly to solve. The implied computational cost impacts applications in both geometric statistics and in geometric deep learning. The weighted diffusion mean offers an alternative to the weighted Fréchet mean. We show how the diffusion mean and the weighted diffusion mean can be estimated with a stochastic simulation scheme that does not require nested optimization. We achieve this by conditioning a Brownian motion in a product manifold to hit the diagonal at a predetermined time. We develop the theoretical foundation for the sampling-based mean estimation, we develop two simulation schemes, and we demonstrate the applicability of the method with examples of sampled means on two manifolds.

AB - Computing sample means on Riemannian manifolds is typically computationally costly, as exemplified by computation of the Fréchet mean, which often requires finding minimizing geodesics to each data point for each step of an iterative optimization scheme. When closed-form expressions for geodesics are not available, this leads to a nested optimization problem that is costly to solve. The implied computational cost impacts applications in both geometric statistics and in geometric deep learning. The weighted diffusion mean offers an alternative to the weighted Fréchet mean. We show how the diffusion mean and the weighted diffusion mean can be estimated with a stochastic simulation scheme that does not require nested optimization. We achieve this by conditioning a Brownian motion in a product manifold to hit the diagonal at a predetermined time. We develop the theoretical foundation for the sampling-based mean estimation, we develop two simulation schemes, and we demonstrate the applicability of the method with examples of sampled means on two manifolds.

U2 - 10.3390/a15030092

DO - 10.3390/a15030092

M3 - Journal article

VL - 15

SP - 1

EP - 16

JO - Algorithms

JF - Algorithms

SN - 1999-4893

IS - 3

M1 - 92

ER -