A1078
Title: Principal component analysis and graph Laplacians in high dimensions
Authors: Martin Wahl - Bielefeld University (Germany) [presenting]
Martin Wahl - Bielefeld University (Germany)
Abstract: Given i.i.d. observations uniformly distributed on a closed manifold $M\subseteq R^p$, the spectral properties of the associated empirical graph Laplacian are studied based on a Gaussian kernel. The main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of $M$. In the analysis, the empirical graph Laplacian is connected to kernel principal component analysis and considers the heat kernel of $M$ as a reproducing kernel feature map. This leads to novel points of view and allows leveraging results for empirical covariance operators in infinite dimensions.
