Metrics for Probabilistic Geometries

2014/11/27 by Alessandra Tosi, Tosi, Alessandra, Søren Hauberg +5 · 9 citations
Computer Science · Engineering · Mathematics · #3D Shape Modeling and Analysis #FOS: Computer and information sciences #Gaussian Processes and Bayesian Inference #Human Motion and Animation #Machine Learning (cs.LG) #Machine Learning (stat.ML) #cs.LG #stat.ML

paper · pdf · doi:10.48550/arxiv.1411.7432

UAI 2014

arxiv created 2014/11/27 · openalex publication_date 2014/11/27 · arxiv updated 2014/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Citations

• A BVP solver based on residual control and the Maltab PSE

Cited by

• Likely Interpolants of Generative Models
• Geodesic Calculus on Implicitly Defined Latent Manifolds
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• Pulling back information geometry
• Expected path length on random manifolds
• Connecting Neural Models Latent Geometries with Relative Geodesic Representations
• Bayesian Quadrature on Riemannian Data Manifolds
• Geometrically Enriched Latent Spaces
• Intrinsic-Hybrid Latent Diffusion Models for Generative Modeling on Unknown Manifolds

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