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.
In recent work, REF suggested perceiving the latent space of Gaussian process latent variable models as a Riemannian manifold, where the distance between two data points is given as the shortest path along the data manifold.
Metrics for Probabilistic Geometries
ArXiv
ArXiv
Mathematics Computer Science