Metrics for Probabilistic Geometries

Alessandra Tosi, Søren Hauberg, Alfredo Vellido, Neil D. Lawrence

Abstract: We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natu- ral metric given by the models. We provide the necessary algorithms to compute expected metric tensors where the distribution over mappings is given by a Gaussian process. We treat the corre- sponding latent variable model as a Riemannian manifold and we use the expectation of the met- ric under the Gaussian process prior to define in- terpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate gen- eration of new data.