MULTISCALE GEOMETRIC DICTIONARIES FOR POINT-CLOUD DATA

We develop a novel geometric multiresolution analysis for analyzing intrinsically low-dimensional point clouds in h ig dimensional spaces, modeled as samples from a d-dimensional setM (in particular, a manifold) embedded in R, in the regime d ≪ D. This type of situation has been recognized as important in various applications, such as the analysis of sounds, ima ges, and gene arrays. In this paper we construct data-dependent m ultiscale dictionaries that aim at efficient encoding and mani pulating of the data. Unlike existing constructions, our constru ction is fast, and so are the algorithms that map data points to dict ionary coefficients and vice versa. In addition, data points ha ve a guaranteed sparsity in terms of the dictionary. Keywords— Data Sets. Point Clouds. Wavelets. Dictionary Learning. Multiscale Analysis. Sparse Approximation.