Spectral estimation of weighted Laplace operators
A talk in the Oberseminar Probability Theory and Mathematical Statistics series by
Vincent Divol from ENSAE Paris
| Abstract: | Spectral representations of datasets are routinely used in data science (e.g. for dimension reduction through diffusion maps, for clustering with spectral clustering, or as a frame in a regression task). These spectral representations are computed using the transition matrix of a random walk on the dataset, known as the Laplacian matrix. As the number of observations approaches infinity, this matrix is known to converge to a continuous diffusion operator, depending on the density generating the observations. Recently, several studies have focused on the exact convergence rate of the matrix's spectrum to the spectrum of the limiting Laplace operator, with various proposed rates. In this work, we address this question from a minimax perspective by providing estimators (not based on a Laplacian matrix) for the eigenfunctions and eigenvalues of a weighted Laplace operator. We establish tight minimax rates of convergence and demonstrate that, surprisingly, asymptotically normal estimators exist for eigenvalue estimation, provided the underlying density is sufficiently smooth. Joint work with Clément Berenfeld and Yann Chaubet Within the CRC this talk is associated to the project(s): B10 |