# Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics

arXiv: https://arxiv.org/abs/2301.06533 · HTML: https://mlsanity.com/paper/2301.06533/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 31 references that are in the map.
- First posted on arXiv 16 Jan 2023.

## Builds on (most cited first, 9 of 9)

- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Metrics for Probabilistic Geometries](https://mlsanity.com/paper/1411.7432/) (2014)
- [Fast and Robust Shortest Paths on Manifolds Learned from Data](https://mlsanity.com/paper/1901.07229/) (2019)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Extrinsic Gaussian processes for regression and classification on manifolds](https://mlsanity.com/paper/1706.08757/) (2017)
- [Intrinsic Gaussian processes on complex constrained domains](https://mlsanity.com/paper/1801.01061/) (2018)

## Built on by (most cited first, 6 of 6)

- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [Diffusion Processes on Implicit Manifolds](https://mlsanity.com/paper/2604.07213/) (2026)
- [Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimation](https://mlsanity.com/paper/2410.16698/) (2024)
- [Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable](https://mlsanity.com/paper/2411.18989/) (2024)
- [Atlas Gaussian processes on restricted domains and point clouds](https://mlsanity.com/paper/2511.15822/) (2025)

## Abstract (arXiv)

This article presents a novel approach to construct Intrinsic Gaussian Processes for regression on unknown manifolds with probabilistic metrics (GPUM) in point clouds. In many real world applications, one often encounters high dimensional data (e.g. point cloud data) centred around some lower dimensional unknown manifolds. The geometry of manifold is in general different from the usual Euclidean geometry. Naively applying traditional smoothing methods such as Euclidean Gaussian Processes (GPs) to manifold valued data and so ignoring the geometry of the space can potentially lead to highly misleading predictions and inferences. A manifold embedded in a high dimensional Euclidean space can be well described by a probabilistic mapping function and the corresponding latent space. We investigate the geometrical structure of the unknown manifolds using the Bayesian Gaussian Processes latent variable models(BGPLVM) and Riemannian geometry. The distribution of the metric tensor is learned using BGPLVM. The boundary of the resulting manifold is defined based on the uncertainty quantification of the mapping. We use the the probabilistic metric tensor to simulate Brownian Motion paths on the unknown manifold. The heat kernel is estimated as the transition density of Brownian Motion and used as the covariance functions of GPUM. The applications of GPUM are illustrated in the simulation studies on the Swiss roll, high dimensional real datasets of WiFi signals and image data examples. Its performance is compared with the Graph Laplacian GP, Graph Matern GP and Euclidean GP.