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Asymptotics of the graph Laplace operator near an isolated singularity

Susovan Pal

TL;DR

The paper investigates how the continuous graph Laplacian on a smooth manifold behaves near an isolated singularity as the bandwidth $t$ tends to zero. It identifies two regimes: (i) if the metric extends across the singularity under a controlled curvature blow-up, the Laplacian converges to the weighted Laplace–Beltrami operator with density $p$; (ii) if a locally angular conformal modification makes the metric non-extendable, curvature blows up like $\kappa(s)\sim c/s^2$, causing intrinsic and extrinsic Laplacians to blow up at rate $O(t^{-1/2})$ under mild non-degeneracy. The work provides explicit asymptotic expansions (including $t^{-1/2}$ and Taylor-type terms) and supports them with numerical simulations on punctured disks and cone-apex geometries. These results deepen understanding of graph-based approximations to Riemannian operators in the presence of singularities and have implications for data-driven manifold learning near singular structures.

Abstract

In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold $(M,g)$ admitting an isolated singularity $x$. We show that if the curvature function $κ$ doesn't grow too fast near $x$, then the graph Laplace operator at $x$ converges to the weighted Laplace-Beltrami operator as the bandwidth $t\downarrow 0.$ On the other hand, we also prove that if one locally modifies a given Riemannian metric across $x$ by a non-constant \textit{purely angular }conformal factor, then $κ$ grows too fast and the graph Laplace operator behaves like $O(\frac{1}{\sqrt{t}})$ near $x$, as $t\downarrow 0$, given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as $t\downarrow 0$ in specific cases. Numerical simulations at the end illustrate our results.

Asymptotics of the graph Laplace operator near an isolated singularity

TL;DR

The paper investigates how the continuous graph Laplacian on a smooth manifold behaves near an isolated singularity as the bandwidth tends to zero. It identifies two regimes: (i) if the metric extends across the singularity under a controlled curvature blow-up, the Laplacian converges to the weighted Laplace–Beltrami operator with density ; (ii) if a locally angular conformal modification makes the metric non-extendable, curvature blows up like , causing intrinsic and extrinsic Laplacians to blow up at rate under mild non-degeneracy. The work provides explicit asymptotic expansions (including and Taylor-type terms) and supports them with numerical simulations on punctured disks and cone-apex geometries. These results deepen understanding of graph-based approximations to Riemannian operators in the presence of singularities and have implications for data-driven manifold learning near singular structures.

Abstract

In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold admitting an isolated singularity . We show that if the curvature function doesn't grow too fast near , then the graph Laplace operator at converges to the weighted Laplace-Beltrami operator as the bandwidth On the other hand, we also prove that if one locally modifies a given Riemannian metric across by a non-constant \textit{purely angular }conformal factor, then grows too fast and the graph Laplace operator behaves like near , as , given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as in specific cases. Numerical simulations at the end illustrate our results.

Paper Structure

This paper contains 27 sections, 14 theorems, 100 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Graph Laplacian and its Continuous Limit
  3. Unnormalized graph Laplacian: discrete and continuous.
  4. Intrinsic and extrinsic unnormalized graph Laplacians:
  5. Asymptotic behavior of the graph Laplace operator for manifolds without boundary.
  6. Extension to manifolds with smooth and non-smooth boundaries.
  7. Singularities in question:isolated singularities
  8. Asymptotics of graph Laplace operator at isolated singularities
  9. Main results
  10. Organization of this paper
  11. Extending smooth structure at isolated singularities
  12. Definitions and theorems
  13. Proof of Theorem \ref{['thm:Asymptotics of Continuous Graph Laplace Operator']}
  14. Non-extension of Locally Angularly Conformal Riemannian metrics across singularity and blow-up of graph Laplacians
  15. Locally Angularly Conformal Metrics
  16. ...and 12 more sections

Key Result

Theorem 1.5

(Asymptotics of the continuous graph Laplacian near isolated singularity of a manifold) Let $M$ be a $d$-dimensional Riemannian manifold with an isolated singularity at $x.$ Case I: Assume that $M$ has an intrinsic Riemannian metric $g,$ inducing a metric (distance) $d_g.$ Case II: Assume that $M$ i

Theorems & Definitions (45)

  • Remark 1.1
  • Remark 1.2: Abuse of notation
  • Remark 1.3: Dropping the terms 'unnormalized' and focusing on the continuous graph Laplacian only
  • Remark 1.4: Identification of constants at interior points
  • Theorem 1.5
  • Remark 1.6
  • Proposition 1.7: Extendability of locally angularly conformal metrics $\Leftrightarrow$ constant angular conformal factor
  • Theorem 1.8: Locally angularly conformal metrics and controlled blow–up of curvature
  • Theorem 1.9: Taylor expansion of the intrinsic graph Laplacian near an isolated singularity for a non-constant locally angular conformal change
  • Remark 1.10: Interpretation of $L(\Theta)$
  • ...and 35 more