Simultaneous Optimization of Geodesics and Fréchet Means

TL;DR

GEORCE-FM tackles the computational bottleneck of Fréchet mean estimation on manifolds by jointly optimizing geodesics and the mean within a local chart, yielding substantial runtime gains over traditional geodesic-then-mean schemes. The method extends to Finsler geometries and enables a scalable adaptive version that uses mini-batches while preserving convergence in expectation. Theoretical guarantees include global convergence to a local minimum and local quadratic convergence, plus adaptive convergence in expectation; empirically it demonstrates improved accuracy and speed on both classical Riemannian cases (e.g., spheres and ellipsoids) and Finsler cases, as well as downstream geometric-statistics tasks. The approach provides a practical tool for geometric data analysis, enabling efficient Fréchet-mean computations in high-dimensional or large-scale settings, with explicit handling of non-Euclidean metrics and velocity-dependent energies.

Abstract

A central part of geometric statistics is to compute the Fréchet mean. This is a well-known intrinsic mean on a Riemannian manifold that minimizes the sum of squared Riemannian distances from the mean point to all other data points. The Fréchet mean is simple to define and generalizes the Euclidean mean, but for most manifolds even minimizing the Riemannian distance involves solving an optimization problem. Therefore, numerical computations of the Fréchet mean require solving an embedded optimization problem in each iteration. We introduce the GEORCE-FM algorithm to simultaneously compute the Fréchet mean and Riemannian distances in each iteration in a local chart, making it faster than previous methods. We extend the algorithm to Finsler manifolds and introduce an adaptive extension such that GEORCE-FM scales to a large number of data points. Theoretically, we show that GEORCE-FM has global convergence and local quadratic convergence and prove that the adaptive extension converges in expectation to the Fréchet mean. We further empirically demonstrate that GEORCE-FM outperforms existing baseline methods to estimate the Fréchet mean in terms of both accuracy and runtime.

Figures (19)

• Figure 1: Iterative computation for 10 data points on a circle within the convexity radius of the north pole, and hence guaranteeing the existence and uniqueness of the Fréchet mean of the data points around the north pole on a unit sphere, $\mathbb{S}^{2}$, using GEORCE-FM. After only 3 iterations the gradient of the discretized geodesic points and Fréchet mean is less than $10^{-4}$. "MOI" refers to moment of inertia given by the sum of squared distances in Eq. \ref{['eq:frechet_mean']}.
• Figure 2: The left figure shows a distribution of data points (black) on the equator, where both the north and south pole are Fréchet means (blue). The right figure shows a distribution of data points on a circle around the north pole, but where a closed set represented by the green area is removed from the unit sphere. In this case, the Fréchet mean does not exist.
• Figure 3: We compute the geodesics and Fréchet mean simultaneously in a local chart of a Riemannian manifold. Green curves are the geodesics, black points are the datapoints, while red is the indicatrix field.
• Figure 4: The application of GEORCE-FM to Information geometry for 4 different distributions equipped with the Fisher-Rao metric miyamoto2024closedformexpressionsfisherraodistance with synthetic data. From left to right we have: Gaussian distribution, Fréchet distribution, Cauchy distribution and Pareto distribution. The black outlined distributions are the data, where the blue outlined distribution is the estimated Fréchet mean using GEORCE-FM and "MOI" refers to moment of inertia given by the sum of squared distances in Eq. \ref{['eq:frechet_mean']}. The details of the manifolds and data points can be found in Appendix \ref{['ap:manifold_description']}.
• Figure 5: We consider a constant indicatrix field of centered ellipsis in the left-most figure, and the corresponding Riemannian Fréchet mean. The center figures shows displaced ellipses such that they are non-centered and the corresponding Fréchet mean as a starting point. The right-most figure shows the Finslerian Fréchet mean as an end point.

Theorems & Definitions (35)

• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4: Convergence Properties
• proof
• Proposition 5: Adaptive Convergence
• proof