Probabilistic Riemannian submanifold learning with wrapped Gaussian process latent variable models
This work addresses the issue of probabilistic modeling for data with inherent geometric constraints, such as in applications with spatial limitations, by enabling more accurate inference on Riemannian submanifolds.
The authors tackled the problem of latent variable models ignoring known spatial constraints and Euclidean geometry in data, which leads to assigning probability to impossible data points and suboptimal similarity measures. They proposed the wrapped Gaussian process latent variable model (WGPLVM), which restricts values to a given Riemannian manifold, improving performance on tasks like encoding, visualization, and uncertainty quantification across diverse datasets.
Latent variable models (LVMs) learn probabilistic models of data manifolds lying in an \emph{ambient} Euclidean space. In a number of applications, a priori known spatial constraints can shrink the ambient space into a considerably smaller manifold. Additionally, in these applications the Euclidean geometry might induce a suboptimal similarity measure, which could be improved by choosing a different metric. Euclidean models ignore such information and assign probability mass to data points that can never appear as data, and vastly different likelihoods to points that are similar under the desired metric. We propose the wrapped Gaussian process latent variable model (WGPLVM), that extends Gaussian process latent variable models to take values strictly on a given ambient Riemannian manifold, making the model blind to impossible data points. This allows non-linear, probabilistic inference of low-dimensional Riemannian submanifolds from data. Our evaluation on diverse datasets show that we improve performance on several tasks, including encoding, visualization and uncertainty quantification.