  As a means of improving analysis of biological shapes, we propose an
algorithm for sampling a Riemannian manifold by sequentially selecting points
with maximum uncertainty under a Gaussian process model. This greedy strategy
is known to be near-optimal in the experimental design literature, and appears
to outperform the use of user-placed landmarks in representing the geometry of
biological objects in our application. In the noiseless regime, we establish an
upper bound for the mean squared prediction error (MSPE) in terms of the number
of samples and geometric quantities of the manifold, demonstrating that the
MSPE for our proposed sequential design decays at a rate comparable to the
oracle rate achievable by any sequential or non-sequential optimal design; to
our knowledge this is the first result of this type for sequential experimental
design. The key is to link the greedy algorithm to reduced basis methods in the
context of model reduction for partial differential equations. We expect this
approach will find additional applications in other fields of research.
