Atlas Gaussian processes on restricted domains and point clouds

TL;DR

This paper introduces a novel Riemannian-corrected Gaussian Process framework that estimates heat kernels on complex point clouds and improves regression accuracy on real-world and synthetic data by capturing underlying geometries.

Contribution

It develops the Atlas Brownian Motion framework for heat kernel estimation and constructs Riemannian-corrected kernels for enhanced Gaussian Process regression on manifold-structured data.

Findings

Outperforms existing methods in heat kernel estimation.

Achieves higher regression accuracy on complex datasets.

Effectively captures geometry of high-dimensional point clouds.

Abstract

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.