Optimal Design of Experiments on Riemannian Manifolds

ABSTRACT

The theory of optimal design of experiments has been traditionally developed on an Euclidean space. In this article, 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. Supplementary materials for this article are available online.

KEYWORDS:

• Active learning
• High-dimensional data analysis
• Manifold learning
• Regularization

Supplementary Materials

1. Proofs.

2. Further performance evidence of ODOEM for synthetic manifold datasets.

3. Further performance on image datasets.

4. Sensitivity analysis of the regularization and range hyperparameters used by ODOEM.

5. Matlab codes and datasets (zipped file).

Additional information

Funding

The authors gratefully acknowledge NSF grant CMII 1537987.