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 $s\geq2$ 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 ...
| Authors | Haoyu Chen, Chenghui Li |
|---|---|
| Primary category | stat.ML |
| Published | 11 October 2026 |
| Last updated at source | 11 October 2026 |
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