Asymptotically Optimal Method for Manifold Estimation Problem
Kuleshov A., Bernstein A., Yanovich Yu.
Abstracts of the 29-th European Meeting of Statisticians, 2013
Manifold learning is considered as manifold estimation problem: to estimate an unknown well-conditioned q-dimensional manifold embedded in a high-dimensional observation space given sample of n data points from the manifold. It is shown that the proposed Grassmann & Stiefel Eigenmaps algorithm estimates the manifold with a rate n to the power of −2/(q+2), where q is dimension of the manifold; this rate coincides with a minimax lower bound for Hausdorff distance between the manifold and its estimator (Genovese et al. Minimax manifold estimation. Journal of machine learning research, 13, 2012).