Metrics for Probabilistic Geometries
This addresses the problem of improving data generation quality in probabilistic dimensionality reduction for researchers in machine learning and statistics, though it appears incremental as it builds on existing Riemannian geometry and Gaussian process frameworks.
The paper tackles the problem of measuring distances in probabilistic generative dimensionality reduction models by defining a distribution over natural metrics using Riemannian geometry and Gaussian processes, showing that distances respecting the expected metric lead to more appropriate data generation.
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.