Metrics for Probabilistic Geometries

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Alessandra Tosi, University of Oxford
Søren Hauberg
Alfredo Vellido
Neil D. Lawrence, University of Sheffield
in Uncertainty in Artificial Intelligence 30, pp 800-808

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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 natural 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 corresponding latent variable model as a Riemannian manifold and we use the expectation of the metric under the Gaussian process prior to define interpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate generation of new data.