Empirical Hodge Laplacians, Cohomology Ring, and Manifold Learning

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 $Δ^*_t, t \in \mathbf R_{+},$ acting on differential forms using the extrinsic geometry of $M^n$ and prove their uniform convergence to the Hodge Laplacian $Δ^*$ as $t \to 0^+$. Given a point cloud $S_m \subset M^n$, we define symmetrized empirical operators $Δ^*_{sym, t, S_m}$ and establish their spectral convergence in probability to $Δ^*$, 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.