Diffusion Geometries and Harmonic Analysis of Large Data Sets
Our goal is to describe a geometric analytic method to organize and map discrete subsets (data) of high dimensional Euclidean space, as well as complex graphs and submanifolds. Diffusion geometry relates spectral Analysis and multiscale geometric analysis of functions and operators on the data. It provides a setting for Harmonic Analysis, and leads to a comprehensive synthesis of a variety of methods in data processing such as clustering and regression.
These ideas augment and synthesize a rich literature in data analysis including Laplacean eigenmaps, spectral methods for clustering on graphs, kernel pca, isomap etc, to which we add concepts and tools from signal processing and Fourier analysis such as multiscale analysis, low pass filtering and wavelets. In particular we show that the diffusion map obtained by considering the top few Eigenfunctions of the diffusion process provide an embedding into low dimensions so that the usual extrinsic Euclidean (cord) metric in the embedding space measures the intrinsic diffusion distance in the data.
We apply these ideas to a the analysis of imagery of biological spectra of tissue, character recognition, molecular simulations, network analysis etc.
The first lecture will provide application examples and overview.
The second lecture will address analytical and geometric issues including the relations between intrinsic and extrinsic geometries of subsets in Euclidean space.
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