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Geometric Kernel Interpolation and Regression

Patrick Guidotti

TL;DR

This work unifies kernel-based interpolation and regression to extract geometric information from point clouds representing smooth hypersurfaces. By constructing kernel-defined level-set functions in ambient space, it enables stable computation of normals, tangent planes, curvatures, and differential operators such as the surface gradient and Laplace-Beltrami operator without meshing. The regularized regression perspective acts as natural denoising and ill-posedness control, with a clear link to Gaussian Process Regression via the kernel $K$ and noise level $\sigma^2$. Numerical experiments on curves and surfaces (sphere, torus, ellipsoid) demonstrate high accuracy and robustness to sampling and noise, highlighting practical relevance for meshfree geometric analysis and manifold-based PDEs.

Abstract

Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds or samples of smooth manifolds (hypersurfaces, in particular), we show how fundamental geometric quantities such as tangent plane and principal curvatures can be computed numerically using a kernel based (approximate) level set function (often a defining function) for smooth hypersurfaces. In the case of point clouds, the approach generates an interpolated hypersurface, which is an approximation of the underlying manifold when the cloud is a sample of it. It is shown how the geometric quantities obtained can be used in the numerical approximation/computation of geometric operators like the surface gradient or the Laplace-Beltrami operator in the spirit of kernel based meshfree methods. Kernel based interpolation can be extremely ill-posed, especially when using smooth kernels, and the regression approximation offers a natural regularization that proves also quite useful when dealing with geometric or functional data that are affected by errors or noise.

Geometric Kernel Interpolation and Regression

TL;DR

This work unifies kernel-based interpolation and regression to extract geometric information from point clouds representing smooth hypersurfaces. By constructing kernel-defined level-set functions in ambient space, it enables stable computation of normals, tangent planes, curvatures, and differential operators such as the surface gradient and Laplace-Beltrami operator without meshing. The regularized regression perspective acts as natural denoising and ill-posedness control, with a clear link to Gaussian Process Regression via the kernel and noise level . Numerical experiments on curves and surfaces (sphere, torus, ellipsoid) demonstrate high accuracy and robustness to sampling and noise, highlighting practical relevance for meshfree geometric analysis and manifold-based PDEs.

Abstract

Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds or samples of smooth manifolds (hypersurfaces, in particular), we show how fundamental geometric quantities such as tangent plane and principal curvatures can be computed numerically using a kernel based (approximate) level set function (often a defining function) for smooth hypersurfaces. In the case of point clouds, the approach generates an interpolated hypersurface, which is an approximation of the underlying manifold when the cloud is a sample of it. It is shown how the geometric quantities obtained can be used in the numerical approximation/computation of geometric operators like the surface gradient or the Laplace-Beltrami operator in the spirit of kernel based meshfree methods. Kernel based interpolation can be extremely ill-posed, especially when using smooth kernels, and the regression approximation offers a natural regularization that proves also quite useful when dealing with geometric or functional data that are affected by errors or noise.
Paper Structure (13 sections, 6 theorems, 73 equations, 8 figures, 6 tables)

This paper contains 13 sections, 6 theorems, 73 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Kernel Interpolation
  3. Hypersurfaces
  4. Interpolation Results
  5. Computation of Geometric Quantities
  6. Connection to Regression
  7. Numerical Experiments
  8. Curves
  9. Quadratic Surfaces
  10. Laplace-Beltrami Operator of the Sphere
  11. The Ring Torus
  12. The Gauss Curvature of an Ellipsoid
  13. The Noisy Case

Key Result

Proposition 2.3

The optimization problems opa, $\alpha>0$, and op possess a unique minimizer $u^\alpha_\mathcal{\mathbb{X},\mathbb{Y}}\in \mathcal{H}_K$. For $\alpha>0$, the minimizer $u^\alpha_{\mathbb{X},\mathbb{Y}}$ is a weak solution of the equation i.e. a solution of the equation in $\mathcal{H}_K^*$. If $\alpha=0$, then it holds that for some Lagrange multiplier $\Lambda\in \mathbb{R}^{|\mathbb{X}|}$. The

Figures (8)

  • Figure 1: Reconstruction of a closed curve from 64 of its points (in white) on the left. Reconstruction based on an incomplete subsample consisting of 24 points on the right.
  • Figure 2: Reconstruction of a triangle from 48 sample points using Gauss and Laplace kernel interpolation, left and right, respectively.
  • Figure 3: Implied curve of an exact sample consisiting of $m$ points along half of a ellipse. Left: $m=16$. Right: $m=256$.
  • Figure 4: Implied curve of an exact sample of points of a full square and one with a corner removed.
  • Figure 5: The samples obtained with a probability distribution proportional to the surface area.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • proof
  • Corollary 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • ...and 16 more