Empirical Hodge Laplacians, Cohomology Ring, and Manifold Learning

TL;DR

This paper develops a method to recover the geometric and topological features of a manifold from sampled data by extending Laplacian eigenmaps to differential forms and analyzing their spectral convergence.

Contribution

It introduces a family of deformed Hodge Laplacians based on extrinsic geometry and proves their convergence to the true Laplacian, enabling topological and geometric invariants to be estimated from data.

Findings

Spectral convergence of empirical operators to the Hodge Laplacian.

Recovery of the de Rham cohomology ring from sampled data.

Estimation of the second fundamental form and curvature tensor from point clouds.

Abstract

Let $M^n$ be a compact orientable Riemannian smooth submanifold of dimension $n \ge 2$ in $\mathbf R^d$. We construct a family of deformed Hodge Laplacians $\Delta ^*_t, t \in \mathbf R_{+},$ acting on differential forms using the extrinsic geometry of $M^n$ and prove their uniform convergence to the Hodge Laplacian $\Delta^*$ as $t \to 0^+$. Given a point cloud $S_m \subset M^n$, we define symmetrized empirical operators $\Delta^*_{sym, t, S_m}$ and establish their spectral convergence in probability to $\Delta^*$, as $t \to 0^+$, under suitable scaling regimes. This extends the scalar framework of Belkin--Niyogi Laplacian Eigenmaps 2003 to differential forms. As a result, we recover the de Rham cohomology ring $H^* (M^n,\mathbf R)$ from sampled data. Additionally, we also recover the second fundamental form of $M^n$, hence the Riemannian curvature tensor, and consequently, the Pontryagin characteristic classes and numbers of $M^n$ from sampled data.