Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering
Abstract
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≥2 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.
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BibTeX
@misc{chen2026minimax,
title = {{Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering}},
author = {Haoyu Chen and Chenghui Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15892},
eprint = {2610.15892},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15892},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering
AU - Haoyu Chen
AU - Chenghui Li
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15892
UR - https://arxiv.org/abs/2610.15892
AB - 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≥2 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.
ER -