# Atlas Gaussian processes on restricted domains and point clouds

arXiv: https://arxiv.org/abs/2511.15822 · HTML: https://mlsanity.com/paper/2511.15822/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 19 Nov 2025.

## Builds on (most cited first, 13 of 13)

- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Inducing Point Allocation for Sparse Gaussian Processes in High-Throughput Bayesian Optimisation](https://mlsanity.com/paper/2301.10123/) (2023)
- [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)
- [Adaptive Inducing Points Selection For Gaussian Processes](https://mlsanity.com/paper/2107.10066/) (2021)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics](https://mlsanity.com/paper/2301.06533/) (2023)
- [Atlas Generative Models and Geodesic Interpolation](https://mlsanity.com/paper/2102.00264/) (2021)
- [Revisiting Active Sets for Gaussian Process Decoders](https://mlsanity.com/paper/2209.04636/) (2022)
- [Scalable Bayesian inference for heat kernel Gaussian processes on manifolds](https://mlsanity.com/paper/2405.13342/) (2024)
- [Atlas flow : compatible local structures on the manifold](https://mlsanity.com/paper/2210.14149/) (2022)

## Abstract (arXiv)

In real-world applications, data often reside in restricted domains with unknown boundaries, or as high-dimensional point clouds lying on a lower-dimensional, nontrivial, unknown manifold. Traditional Gaussian Processes (GPs) struggle to capture the underlying geometry in such settings. Some existing methods assume a flat space embedded in a point cloud, which can be represented by a single latent chart (latent space), while others exhibit weak performance when the point cloud is sparse or irregularly sampled. The goal of this work is to address these challenges. The main contributions are twofold: (1) We establish the Atlas Brownian Motion (BM) framework for estimating the heat kernel on point clouds with unknown geometries and nontrivial topological structures; (2) Instead of directly using the heat kernel estimates, we construct a Riemannian corrected kernel by combining the global heat kernel with local RBF kernel and leading to the formulation of Riemannian-corrected Atlas Gaussian Processes (RC-AGPs). The resulting RC-AGPs are applied to regression tasks across synthetic and real-world datasets. These examples demonstrate that our method outperforms existing approaches in both heat kernel estimation and regression accuracy. It improves statistical inference by effectively bridging the gap between complex, high-dimensional observations and manifold-based inferences.