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.

翻译：实验的最佳设计理论传统上是在欧几里得空间上发展起来的。在本文中,提供了新的理论结果和一种算法,以寻找位于里伊曼多元上的实验的最佳设计。它表明,与欧几里德空间的结果相类似,D-最佳和G-最佳设计在多元上相当于多元,我们为所评价的反应的最大预测差异提供了一个较低的界限。此外,还提出了一种趋同算法,它发现对多种数据的最佳实验设计。数字实验表明,在设计实验中考虑多元结构的重要性,以及拟议的算法的优越性。