We discuss techniques for studying geometric properties of point clouds that are embedded in high dimensional spaces but are intrinsically low-dimensional sets. Such point clouds arise when studying certain data sets such as databases of images, gene arrays, text documents, and certain dynamical systems. We discuss how the analysis of geometric properties can yield insights into properties of the data, and how certain multiscale geometric techniques may be employed to perform tasks such as estimating the intrinsic dimension of the data, constructing nonlinear models for approximating the point cloud, as well as generating data-dependent dictionaries which are hierarchically organized in multiscale data structures and yield fast nonlinear transforms of point clouds.
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