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Manifolds with kinks and the asymptotic behavior of the graph Laplacian operator with Gaussian kernel

Susovan Pal, David Tewodrose

TL;DR

This work extends the fundamental link between graph Laplacians and Laplace–Beltrami operators to manifolds with kinks, comprising interiors, boundaries, corners, and cusps. By introducing and exploiting inward tangent sectors and Bouligand cones, the authors derive exact small-bandwidth expansions for intrinsic and extrinsic Gaussian graph operators, showing leading behavior along inward directions and boundary-corrective terms. They establish concentration results under alpha-subexponential tails, providing concrete sample-size–bandwidth tradeoffs for convergence to first-order differential operators with Neumann-like boundary interpretations. The framework unifies interior and singular-point behavior, offers refined error bounds, and is validated through numerical experiments on balls, cubes, and cusps. The results pave the way for principled spectral analysis and Neumann-type boundary conditions for graphs built from data on kinked manifolds.

Abstract

We introduce manifolds with kinks, a class of manifolds with possibly singular boundary that notably contains manifolds with smooth boundary and corners. We derive the asymptotic behavior of the Graph Laplace operator with Gaussian kernel and its deterministic limit on these spaces as bandwidth goes to zero. We show that this asymptotic behavior is determined by the inward sector of the tangent space and, as special cases, we derive its behavior near interior and singular points. Lastly, we show the validity of our theoretical results using numerical simulation.

Manifolds with kinks and the asymptotic behavior of the graph Laplacian operator with Gaussian kernel

TL;DR

This work extends the fundamental link between graph Laplacians and Laplace–Beltrami operators to manifolds with kinks, comprising interiors, boundaries, corners, and cusps. By introducing and exploiting inward tangent sectors and Bouligand cones, the authors derive exact small-bandwidth expansions for intrinsic and extrinsic Gaussian graph operators, showing leading behavior along inward directions and boundary-corrective terms. They establish concentration results under alpha-subexponential tails, providing concrete sample-size–bandwidth tradeoffs for convergence to first-order differential operators with Neumann-like boundary interpretations. The framework unifies interior and singular-point behavior, offers refined error bounds, and is validated through numerical experiments on balls, cubes, and cusps. The results pave the way for principled spectral analysis and Neumann-type boundary conditions for graphs built from data on kinked manifolds.

Abstract

We introduce manifolds with kinks, a class of manifolds with possibly singular boundary that notably contains manifolds with smooth boundary and corners. We derive the asymptotic behavior of the Graph Laplace operator with Gaussian kernel and its deterministic limit on these spaces as bandwidth goes to zero. We show that this asymptotic behavior is determined by the inward sector of the tangent space and, as special cases, we derive its behavior near interior and singular points. Lastly, we show the validity of our theoretical results using numerical simulation.

Paper Structure

This paper contains 38 sections, 33 theorems, 223 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Euclidean models
  3. Boundary points
  4. Diffeomorphisms and inward sectors
  5. Boundary cones of open Euclidean subsets
  6. Non-fluctuating boundary and local blow-ups
  7. Manifolds with kinks
  8. Topological manifolds with kinks
  9. Differentiable manifolds with kinks
  10. Tangent space and inward sector
  11. Tangent space
  12. Inward sector
  13. Strictly inward sector
  14. Riemannian manifolds with kinks
  15. Tangent bundle
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $M$ be a smooth $d$-dimensional manifold with kinks endowed with a $\mathcal{C}^2$ Riemannian metric $g$, and let $x \in M$ be either an interior point, an LCDD border point, or a cusp. Consider $\eta \in (0,1/2)$. Then the intrinsic Gaussian operator associated with a density $p \in \mathcal{C} as $t \downarrow 0$, for any $f \in \mathcal{C}^3(M) \cap L^1(M,p\,\mathrm{vol}_g)$, where $\mathrm

Figures (15)

  • Figure 1: $0_2$ is not a $\mathcal{C}^0$ boundary point of $\Omega$
  • Figure 2: Two diffeomorphic open sets whose boundaries are not diffeomorphic
  • Figure 3:
  • Figure 4:
  • Figure 6:
  • ...and 10 more figures

Theorems & Definitions (108)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • ...and 98 more