The GeometricKernels Package: Heat and Matérn Kernels for Geometric Learning on Manifolds, Meshes, and Graphs

• URL: http://arxiv.org/abs/2407.08086v1
• Date: Wed, 10 Jul 2024 23:09:23 GMT
• Title: The GeometricKernels Package: Heat and Matérn Kernels for Geometric Learning on Manifolds, Meshes, and Graphs
• Authors: Peter Mostowsky, Vincent Dutordoir, Iskander Azangulov, Noémie Jaquier, Michael John Hutchinson, Aditya Ravuri, Leonel Rozo, Alexander Terenin, Viacheslav Borovitskiy,
• Abstract summary: We present a package which implements the geometric analogs of classical Euclidean squared exponential - also known as heat - and Mat'ern kernels. Our implementation supports automatic differentiation in every major current framework simultaneously via a backend-agnostic design.
• Score: 40.35022755872059
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Kernels are a fundamental technical primitive in machine learning. In recent years, kernel-based methods such as Gaussian processes are becoming increasingly important in applications where quantifying uncertainty is of key interest. In settings that involve structured data defined on graphs, meshes, manifolds, or other related spaces, defining kernels with good uncertainty-quantification behavior, and computing their value numerically, is less straightforward than in the Euclidean setting. To address this difficulty, we present GeometricKernels, a software package which implements the geometric analogs of classical Euclidean squared exponential - also known as heat - and Mat\'ern kernels, which are widely-used in settings where uncertainty is of key interest. As a byproduct, we obtain the ability to compute Fourier-feature-type expansions, which are widely used in their own right, on a wide set of geometric spaces. Our implementation supports automatic differentiation in every major current framework simultaneously via a backend-agnostic design. In this companion paper to the package and its documentation, we outline the capabilities of the package and present an illustrated example of its interface. We also include a brief overview of the theory the package is built upon and provide some historic context in the appendix.

Related papers

• Unified Native Spaces in Kernel Methods [0.0]
  This paper unifies a wealth of well-known kernels into a single class that encompasses them as special cases. As a by-product, we infer the Sobolev spaces that are associated with existing classes of kernels.
  arXiv  Detail & Related papers  (2025-01-03T14:17:41Z)
• Higher-order topological kernels via quantum computation [68.8204255655161]
  Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. We propose a quantum approach to defining Betti kernels, which is based on constructing Betti curves with increasing order.
  arXiv  Detail & Related papers  (2023-07-14T14:48:52Z)
• Interpolation with the polynomial kernels [5.8720142291102135]
  kernels are widely used in machine learning and they are one of the default choices to develop kernel-based regression models. They are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. This paper is devoted to establish some initial results for the study of these kernels, and their related algorithms.
  arXiv  Detail & Related papers  (2022-12-15T08:30:23Z)
• Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case [43.877478563933316]
  In to symmetries is one of the most fundamental forms of prior information one can consider. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces.
  arXiv  Detail & Related papers  (2022-08-31T16:40:40Z)
• Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat\'ern Kernels [64.62221198500467]
  We show how to implement geometry-aware kernels for Bayesian optimization. This technique can be used for control parameter tuning, parametric policy adaptation, and structure design in robotics.
  arXiv  Detail & Related papers  (2021-11-02T09:47:22Z)
• 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)
• Hermitian Symmetric Spaces for Graph Embeddings [0.0]
  We learn continuous representations of graphs in spaces of symmetric matrices over C. These spaces offer a rich geometry that simultaneously admits hyperbolic and Euclidean subspaces. The proposed models are able to automatically adapt to very dissimilar arrangements without any apriori estimates of graph features.
  arXiv  Detail & Related papers  (2021-05-11T18:14:52Z)
• Bayesian Quadrature on Riemannian Data Manifolds [79.71142807798284]
  A principled way to model nonlinear geometric structure inherent in data is provided. However, these operations are typically computationally demanding. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws. We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations.
  arXiv  Detail & Related papers  (2021-02-12T17:38:04Z)
• Advanced Stationary and Non-Stationary Kernel Designs for Domain-Aware Gaussian Processes [0.0]
  We propose advanced kernel designs that only allow for functions with certain desirable characteristics to be elements of the reproducing kernel Hilbert space (RKHS) We will show the impact of advanced kernel designs on Gaussian processes using several synthetic and two scientific data sets.
  arXiv  Detail & Related papers  (2021-02-05T22:07:56Z)
• Mat\'ern Gaussian Processes on Graphs [67.13902825728718]
  We leverage the partial differential equation characterization of Mat'ern Gaussian processes to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Euclidian analogs. This enables graph Mat'ern Gaussian processes to be employed in mini-batch and non-conjugate settings.
  arXiv  Detail & Related papers  (2020-10-29T13:08:07Z)

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.