Multiscale Geometric Dictionaries for Point-cloud Data

We develop a novel geometric multiresolution analysis for analyzing intrinsically low dimensional point clouds in high-dimensional spaces, modeled as samples from a d-dimensional set M (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, images, and gene arrays. In this paper we construct data-dependent multiscale dictionaries that aim at efficient encoding and manipulating of the data. Unlike existing constructions, our construction is fast, and so are the algorithms that map data points to dictionary coefficients and vice versa. In addition, data points have a guaranteed sparsity in terms of the dictionary.