Mauro Maggioni, Departments of Mathematics and Computer Science, Duke University

Harmonic and multiscale analysis of and on data in high-dimensions

Friday November 2nd, 4pm, Phillips 332
(refreshments served in Phillips 330 starting at 3:30)

Abstract: In many applications one is faced with the task of analyzing large amounts of data, typically embedded in high-dimensional space, but with a lower effective dimensionality, due to physical or statistical constraints. We are interested in studying the geometry of such data sets, modeled as noisy manifolds or graphs, in particular in estimating its intrinsic dimensionality and finding intrinsic coordinate systems on the data. We discuss recent results in these directions, where eigenfunctions of a Laplacian on the data or the associated heat kernel can be used to introduce coordinates with provable guarantees on their bi-Lipschitz distortion. We also discuss ways of studying, fitting, denoising and regularizing functions defined on the data, by using Fourier or a wavelet-like multiscale analysis on the data. We present toy applications to nonlinear image denoising, semisupervised learning on a family of benchmark datasets, and Markov decision processes.