Maximum Likelihood Estimation of Riemannian Metrics from Euclidean Data

Euclidean data often exhibit a nonlinear behavior, which may be modeled by assuming the data is distributed near a nonlinear submanifold in the data space. One approach to find such a manifold is to estimate a Riemannian metric that locally models the given data. Data distributions with respect to this metric will then tend to follow the nonlinear structure of the data. In practice, the learned metric rely on parameters that are hand-tuned for a given task. We propose to estimate such parameters by maximizing the data likelihood under the assumed distribution. This is complicated by two issues: (1) a change of parameters imply a change of measure such that different likelihoods are incomparable; (2) some choice of parameters renders the numerical calculation of distances and geodesics unstable such that likelihoods cannot be evaluated. As a practical solution, we propose to (1) re-normalize likelihoods with respect to the usual Lebesgue measure of the data space, and (2) to bound the likelihood when its exact value is unattainable. We provide practical algorithms for these ideas and illustrate their use on synthetic data, images of digits and faces, as well as signals extracted from EEG scalp measurements.