Atlas Gaussian processes on restricted domains and point clouds

• URL: http://arxiv.org/abs/2511.15822v1
• Date: Wed, 19 Nov 2025 19:28:05 GMT
• Title: Atlas Gaussian processes on restricted domains and point clouds
• Authors: Mu Niu, Yue Zhang, Ke Ye, Pokman Cheung, Yizhu Wang, Xiaochen Yang,
• Abstract summary: In real-world applications, data often reside in restricted domains with unknown boundaries.<n>Traditional Gaussian Processes (GPs) struggle to capture the underlying geometry in such settings.
• Score: 13.895603063201127
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• The Kernel Manifold: A Geometric Approach to Gaussian Process Model Selection [0.8536502059643899]
  We present a Bayesian optimization framework built on kernel-of- Kernels geometry.<n>A multidimensional scaling (MDS) embedding maps a discrete kernel library into a continuous Euclidean manifold, enabling smooth BO.<n>We demonstrate the approach on synthetic benchmarks, real-world time-series datasets, and an additive manufacturing case study.
  arXiv  Detail & Related papers  (2026-01-08T20:52:53Z)
• Bayesian Circular Regression with von Mises Quasi-Processes [57.88921637944379]
  In this work we explore a family of expressive and interpretable distributions over circle-valued random functions.<n>For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Gibbs sampling.<n>We present experiments applying this model to the prediction of wind directions and the percentage of the running gait cycle as a function of joint angles.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction [66.21060114843202]
  We propose a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure.
  arXiv  Detail & Related papers  (2023-05-30T13:58:50Z)
• Efficient Graph Field Integrators Meet Point Clouds [59.27295475120132]
  We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds.
  arXiv  Detail & Related papers  (2023-02-02T08:33:36Z)
• A mixed-categorical correlation kernel for Gaussian process [0.0]
  We present a kernel-based approach that extends continuous exponential kernels to handle mixed-categorical variables. The proposed kernel leads to a new GP surrogate that generalizes both the continuous relaxation and the Gower distance based GP models.
  arXiv  Detail & Related papers  (2022-11-15T16:13:04Z)
• Riemannian Diffusion Schr\"odinger Bridge [56.20669989459281]
  We introduce emphRiemannian Diffusion Schr"odinger Bridge to accelerate sampling of diffusion models. We validate our proposed method on synthetic data and real Earth and climate data.
  arXiv  Detail & Related papers  (2022-07-07T00:35:04Z)
• Inducing Gaussian Process Networks [80.40892394020797]
  We propose inducing Gaussian process networks (IGN), a simple framework for simultaneously learning the feature space as well as the inducing points. The inducing points, in particular, are learned directly in the feature space, enabling a seamless representation of complex structured domains. We report on experimental results for real-world data sets showing that IGNs provide significant advances over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-04-21T05:27:09Z)
• Randomized kernels for large scale Earth observation applications [8.835750299747229]
  This paper introduces an efficient kernel method for fast statistical retrieval of bio-geo-physical parameters and image classification problems. We show that kernel regression and classification is now possible for datasets with millions of examples and high dimensionality.
  arXiv  Detail & Related papers  (2020-12-07T12:23:56Z)
• Graph Based Gaussian Processes on Restricted Domains [13.416168979487118]
  In nonparametric regression, it is common for the inputs to fall in a restricted subset of Euclidean space. We propose a new class of Graph Laplacian based GPs (GL-GPs) which learn a covariance that respects the geometry of the input domain. We provide substantial theoretical support for the GL-GP methodology, and illustrate performance gains in various applications.
  arXiv  Detail & Related papers  (2020-10-14T17:01:29Z)
• Manifold Learning via Manifold Deflation [105.7418091051558]
  dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. Many popular methods can fail dramatically, even on simple two-dimensional Manifolds. This paper presents an embedding method for a novel, incremental tangent space estimator that incorporates global structure as coordinates. Empirically, we show our algorithm recovers novel and interesting embeddings on real-world and synthetic datasets.
  arXiv  Detail & Related papers  (2020-07-07T10:04:28Z)
• Intrinsic Gaussian Processes on Manifolds and Their Accelerations by Symmetry [9.773237080061815]
  Existing methods primarily focus on low dimensional constrained domains for heat kernel estimation. Our research proposes an intrinsic approach for constructing GP on general equations. Our methodology estimates the heat kernel by simulating Brownian motion sample paths using the exponential map.
  arXiv  Detail & Related papers  (2020-06-25T09:17:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.