Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering

Key points

• Let $d$ denote the largest intrinsic dimension of the manifold components and $s\geq2$ the smoothness of the sampling densities.1
• 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.1
• 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.1
• These results characterize the information-theoretic difficulty of multi-manifold spectral estimation and quantify the additional cost of learning without oracle component information.1

Sources (1)

• [1]Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 11, 08:14 PM

  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.

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