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Identifiability of the Unnormalized Graph Laplace Operators

Susovan Pal

TL;DR

This work analyzes the identifiability of unnormalized graph Laplacians on compact manifolds, comparing intrinsic versus extrinsic formulations. It shows that, for the intrinsic operator with bandwidth $t>0$ and positive density $p$, the underlying Riemannian metric $g$ and the density $p$ are uniquely recoverable (and hence $(g,p)$ is identifiable) from the operator family; separately, the extrinsic operator identifies the sampling measure $m=p\,\mu_g$, and, when the metric is induced by a Euclidean embedding, the induced metric and density are also identifiable. The paper also establishes that while the extrinsic operator determines the measure, the volume form $1mu_g$ alone does not always fix the metric (with a torus counterexample), though embedding-induced metrics can be recovered in the fixed-embedding setting. In the embedding-induced regime, the extrinsic operator fully identifies the geometry: the embedding-induced metric, volume form, and density can be recovered jointly. These results clarify what geometric and probabilistic information is recoverable from graph Laplacians in manifold learning and guide inverse-problem analysis in this setting.

Abstract

In this short note, we show that the continuous intrinsic graph Laplace operator with Gaussian kernel on a compact Riemannian manifold without boundary uniquely determines both the Riemannian metric and the sampling density, provided the latter is positive. In contrast, the corresponding continuous extrinsic graph Laplace operator uniquely determines the sampling measure; moreover, when the operator is defined via an embedding into Euclidean space, it also uniquely determines the induced Riemannian metric and the sampling density.

Identifiability of the Unnormalized Graph Laplace Operators

TL;DR

This work analyzes the identifiability of unnormalized graph Laplacians on compact manifolds, comparing intrinsic versus extrinsic formulations. It shows that, for the intrinsic operator with bandwidth and positive density , the underlying Riemannian metric and the density are uniquely recoverable (and hence is identifiable) from the operator family; separately, the extrinsic operator identifies the sampling measure , and, when the metric is induced by a Euclidean embedding, the induced metric and density are also identifiable. The paper also establishes that while the extrinsic operator determines the measure, the volume form alone does not always fix the metric (with a torus counterexample), though embedding-induced metrics can be recovered in the fixed-embedding setting. In the embedding-induced regime, the extrinsic operator fully identifies the geometry: the embedding-induced metric, volume form, and density can be recovered jointly. These results clarify what geometric and probabilistic information is recoverable from graph Laplacians in manifold learning and guide inverse-problem analysis in this setting.

Abstract

In this short note, we show that the continuous intrinsic graph Laplace operator with Gaussian kernel on a compact Riemannian manifold without boundary uniquely determines both the Riemannian metric and the sampling density, provided the latter is positive. In contrast, the corresponding continuous extrinsic graph Laplace operator uniquely determines the sampling measure; moreover, when the operator is defined via an embedding into Euclidean space, it also uniquely determines the induced Riemannian metric and the sampling density.
Paper Structure (4 sections, 10 theorems, 50 equations)

This paper contains 4 sections, 10 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Identifiability of intrinsic continuous graph Laplacians
  3. On identifiability of extrinsic/ambient continuous graph Laplacians
  4. Identifiability in the case of embedding-induced Riemannian metric

Key Result

Lemma 2.1

Let $M$ be a smooth manifold and let $g_1,g_2$ be $C^2$ Riemannian metrics on $M$ with geodesic distance functions $d_{g_1},d_{g_2}$. If $d_{g_1}(x,y)=d_{g_2}(x,y)$ for all $x,y\in M$, then $g_1=g_2$.

Theorems & Definitions (22)

  • Lemma 2.1: A $C^2$ Riemannian metric is determined by its distance
  • proof
  • Theorem 2.2: Intrinsic operator identifies the Riemannian metric (for fixed bandwidth and positive density)
  • proof
  • Theorem 2.3: Intrinsic operator identifies the positive density (for fixed bandwidth and $C^2$ Riemannian metric)
  • proof
  • Theorem 2.4: Intrinsic operator identifies the pair $(g,p)$ (for fixed $t$)
  • proof
  • Theorem 3.1: Extrinsic operator identifies the sampling measure $m=p\,\mu_g$
  • proof
  • ...and 12 more