Metrics for Probabilistic Geometries

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.

UAI 2014

References in corpus (2)

• Variational Gaussian Process Dynamical Systems
• Geodesics of Random Riemannian Metrics

Cited by in corpus (13)

• Only Bayes should learn a manifold (on the estimation of differential geometric structure from data)
• Expected path length on random manifolds
• Variational Autoencoders with Riemannian Brownian Motion Priors
• Learning Flat Latent Manifolds with VAEs
• Isometric Gaussian Process Latent Variable Model for Dissimilarity Data
• A prior-based approximate latent Riemannian metric
• Geometrically Enriched Latent Spaces
• Fast Approximate Geodesics for Deep Generative Models
• Fast Adaptation with Linearized Neural Networks
• Pulling back information geometry
• Identifying latent distances with Finslerian geometry
• Riemannian Laplace approximations for Bayesian neural networks
• Bayesian Quadrature on Riemannian Data Manifolds