Optimal Design of Experiments on Riemannian Manifolds

TL;DR

This paper extends the theory of optimal experimental design to Riemannian manifolds, providing new theoretical insights and an algorithm that outperforms traditional Euclidean-based methods.

Contribution

It introduces the first theoretical results and a converging algorithm for optimal experimental design specifically on Riemannian manifolds.

Findings

D-optimal and G-optimal designs are equivalent on manifolds

The proposed algorithm converges reliably to the optimal design

Considering manifold structure improves experimental design effectiveness

Abstract

The theory of optimal design of experiments has been traditionally developed on an Euclidean space. In this paper, new theoretical results and an algorithm for finding the optimal design of an experiment located on a Riemannian manifold are provided. It is shown that analogously to the results in Euclidean spaces, D-optimal and G-optimal designs are equivalent on manifolds, and we provide a lower bound for the maximum prediction variance of the response evaluated over the manifold. In addition, a converging algorithm that finds the optimal experimental design on manifold data is proposed. Numerical experiments demonstrate the importance of considering the manifold structure in a designed experiment when present, and the superiority of the proposed algorithm.