Asymptotically optimal method for manifold estimation problem

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). [Joint work with Alexander Kuleshov and Alexander Bernstein (IITP and PreMoLab, Moscow)]