Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions

Arthur D. Szlam; Mauro Maggioni; Ronald R. Coifman; James C. Bremer Jr.

doi:10.1117/12.616931

21 September 2005 Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions

Arthur D. Szlam, Mauro Maggioni, Ronald R. Coifman, James C. Bremer Jr.

Proceedings Volume 5914, Wavelets XI; 59141D (2005) https://doi.org/10.1117/12.616931
Event: Optics and Photonics 2005, 2005, San Diego, California, United States

Abstract

Classically, analysis on manifolds and graphs has been based on the study of the eigenfunctions of the Laplacian and its generalizations. These objects from differential geometry and analysis on manifolds have proven useful in applications to partial differential equations, and their discrete counterparts have been applied to optimization problems, learning, clustering, routing and many other algorithms.¹⁻⁷ The eigenfunctions of the Laplacian are in general global: their support often coincides with the whole manifold, and they are affected by global properties of the manifold (for example certain global topological invariants). Recently a framework for building natural multiresolution structures on manifolds and graphs was introduced, that greatly generalizes, among other things, the construction of wavelets and wavelet packets in Euclidean spaces.^8,9 This allows the study of the manifold and of functions on it at different scales, which are naturally induced by the geometry of the manifold. This construction proceeds bottom-up, from the finest scale to the coarsest scale, using powers of a diffusion operator as dilations and a numerical rank constraint to critically sample the multiresolution subspaces. In this paper we introduce a novel multiscale construction, based on a top-down recursive partitioning induced by the eigenfunctions of the Laplacian. This yields associated local cosine packets on manifolds, generalizing local cosines in Euclidean spaces.¹⁰ We discuss some of the connections with the construction of diffusion wavelets. These constructions have direct applications to the approximation, denoising, compression and learning of functions on a manifold and are promising in view of applications to problems in manifold approximation, learning, dimensionality reduction.

Citation Download Citation

Arthur D. Szlam, Mauro Maggioni, Ronald R. Coifman, and James C. Bremer Jr. "Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions", Proc. SPIE 5914, Wavelets XI, 59141D (21 September 2005); https://doi.org/10.1117/12.616931

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