Identifiability of the Unnormalized Graph Laplace Operators

TL;DR

This paper demonstrates that certain types of graph Laplace operators uniquely determine the underlying geometric and sampling properties of a manifold, with distinctions based on whether the operator is intrinsic or extrinsic.

Contribution

It establishes the identifiability of the Riemannian metric and sampling density from continuous graph Laplace operators, clarifying what geometric information can be recovered.

Findings

Intrinsic operator determines metric and density.

Extrinsic operator determines sampling measure.

Embedding-based operator recovers metric and density.

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.