Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering

We study eigenpair estimation for the tensorized, density-weighted Laplace--Beltrami operator from n independent data points sampled from a multi-manifold mixture distribution where manifolds may intersect. Let d denote the largest intrinsic dimension of the manifold components and s at least2 the smoothness of the sampling densities. For simple, nontrivial eigenpairs, we establish the minimax rate n-s/(d+2s) for approximating the eigenvalue and eigenfunction of the tensorized Laplace--Beltrami operator, and show that a continuum plug-in estimator based on kernel density estimation attains the same rate when the supporting manifolds are known. We further analyze a manifold-agnostic graph Laplacian estimator and prove that its eigenpairs converge to those of the tensorized Laplace--Beltrami operator at rate n-1/(2.5d-1), up to logarithmic factors, improving upon previously known spectral convergence rates for multi-manifold learning. Numerical experiments on synthetic multi-manifold data corroborate the rates guaranteed by our theorems. These results characterize the information-theoretic difficulty of multi-manifold spectral estimation and quantify the additional cost of learning without oracle component information.