Atlas Gaussian processes on restricted domains and point clouds

2025-11-19

AI Summary

Real-world data often reside on low-dimensional, nontrivial manifolds embedded in high-dimensional spaces or within constrained domains with unknown boundaries—structures that conventional Gaussian processes (GPs) fail to capture due to their implicit Euclidean and global-coordinate assumptions. Existing manifold-aware GP approaches either presuppose a single flat latent embedding or degrade significantly under sparse and non-uniform sampling. To address this, we propose Riemannian-Corrected Atlas Gaussian Processes (RC-AGPs), which estimate the manifold’s heat kernel via an Atlas Brownian motion framework, integrate global heat-kernel priors with local RBF kernels, and perform Riemannian metric correction through manifold learning—without requiring global coordinates. RC-AGPs substantially improve heat-kernel estimation accuracy and regression performance on sparse and irregular point clouds. Extensive experiments on synthetic and real-world benchmarks demonstrate consistent superiority over state-of-the-art manifold GP methods, establishing a robust, geometry-aware paradigm for high-dimensional statistical inference on complex topological domains.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Modeling data on unknown manifold geometries with Gaussian processes
• Addressing sparse point cloud limitations in traditional kernel methods
• Improving regression accuracy on high-dimensional restricted domain data

Innovation

• Atlas Brownian Motion estimates heat kernel on unknown geometries
• Riemannian corrected kernel combines global heat kernel with local RBF
• Atlas Gaussian Processes improve manifold-based regression accuracy