Gaussian Process Landmarking on Manifolds

As a means of improving analysis of biological shapes, we propose an algorithm for sampling a Riemannian manifold by greedily selecting points with maximum uncertainty under a Gaussian process model. This 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 then apply the proposed landmarking algorithm to geometric morphometrics, a branch of evolutionary biology focusing on the analysis and comparisons of anatomical shapes, and compare the automatically sampled landmarks with the "ground truth" landmarks manually placed by evolutionary anthropologists; the results suggest that Gaussian process landmarks perform equally well or better, in terms of both spatial coverage and downstream statistical analysis. We expect this approach will find additional applications in other fields of research.