Rolled Gaussian process models for curves on manifolds

• URL: http://arxiv.org/abs/2503.21980v1
• Date: Thu, 27 Mar 2025 20:52:18 GMT
• Title: Rolled Gaussian process models for curves on manifolds
• Authors: Simon Preston, Karthik Bharath, Pablo Lopez-Custodio, Alfred Kume,
• Abstract summary: rolling operation induces a local isometry between the sphere and the plane so that the two curves uniquely determine each other.<n>We use rolling to construct an analogue of a Gaussian process on a manifold starting from a Euclidean Gaussian process.<n>The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold.
• Score: 3.499870393443268
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Given a planar curve, imagine rolling a sphere along that curve without slipping or twisting, and by this means tracing out a curve on the sphere. It is well known that such a rolling operation induces a local isometry between the sphere and the plane so that the two curves uniquely determine each other, and moreover, the operation extends to a general class of manifolds in any dimension. We use rolling to construct an analogue of a Gaussian process on a manifold starting from a Euclidean Gaussian process. The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold. We illustrate with examples on the unit sphere, symmetric positive-definite matrices, and with a robotics application involving 3D orientations.

Related papers

• 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)
• Implicit Manifold Gaussian Process Regression [49.0787777751317]
  Gaussian process regression is widely used to provide well-calibrated uncertainty estimates. It struggles with high-dimensional data because of the implicit low-dimensional manifold upon which the data actually lies. In this paper we propose a technique capable of inferring implicit structure directly from data (labeled and unlabeled) in a fully differentiable way.
  arXiv  Detail & Related papers  (2023-10-30T09:52:48Z)
• Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds [5.0401589279256065]
  We show how the geometry of a manifold impacts the distance field, and exploit the geodesic flow to obtain globally length-minimising curves directly. This work opens opportunities for statistics and reduced-order modelling on differentiable manifold.
  arXiv  Detail & Related papers  (2023-10-09T21:11:13Z)
• The Fisher-Rao geometry of CES distributions [50.50897590847961]
  The Fisher-Rao information geometry allows for leveraging tools from differential geometry. We will present some practical uses of these geometric tools in the framework of elliptical distributions.
  arXiv  Detail & Related papers  (2023-10-02T09:23:32Z)
• Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics [5.582101184758529]
  This article presents a novel approach to construct Intrinsic Gaussian Processes for regression on unknown manifold with probabilistic metrics in point clouds. The geometry of manifold is in general different from the usual Euclidean geometry. 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.
  arXiv  Detail & Related papers  (2023-01-16T17:42:40Z)
• Shape And Structure Preserving Differential Privacy [70.08490462870144]
  We show how the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism. We also show how using the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism.
  arXiv  Detail & Related papers  (2022-09-21T18:14:38Z)
• Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Equivariant Projected Kernels [108.60991563944351]
  We present a recipe for constructing gauge equivariant kernels, which induce vector-valued Gaussian processes coherent with geometry. We extend standard Gaussian process training methods, such as variational inference, to this setting.
  arXiv  Detail & Related papers  (2021-10-27T13:31:10Z)
• Curvatures of Stiefel manifolds with deformation metrics [0.0]
  We compute curvatures of a family of tractable metrics on Stiefel manifold. The metrics could be identified with the Cheeger deformation metrics.
  arXiv  Detail & Related papers  (2021-05-05T02:13:38Z)
• Disentangling by Subspace Diffusion [72.1895236605335]
  We show that fully unsupervised factorization of a data manifold is possible if the true metric of the manifold is known. Our work reduces the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.
  arXiv  Detail & Related papers  (2020-06-23T13:33:19Z)

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.