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Diffusion Wavelets for multiscale analysis on manifolds and graphs: constructions and applications

Abstract

The study of diffusion operators of manifolds, graphs and data has many applications to the analysis of the structure of the underlying space and of functions on the space. This has applications to data analysis, clustering, learning, and partial differential equations. Given a local operator T on a manifold or a graph representing local similarities, we present a general multiresolution construction for efficiently computing, representing and compressing T^t, and construct multiscale basis functions and a multiscale organization of the set. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding efficient algorithms. Applications include function approximation, denoising, and learning on data sets, value function approximation in Markov decision processes, organization of complex networks and document corpora, mesh and texture compression in 3D computer graphics.