Metrics for Probabilistic Geometries

TL;DR

This paper explores the Riemannian geometric structure of probabilistic generative models, defining a distribution over metrics and developing algorithms to compute expected metric tensors for improved data generation.

Contribution

It introduces a method to treat probabilistic generative models as Riemannian manifolds using Gaussian processes to define and compute expected metrics.

Findings

Expected metric tensors enable more accurate data interpolation.

Distances respecting the expected metric improve data generation quality.

The approach provides a geometric framework for probabilistic models.

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.