Geometric diffusions as a tool for harmonic analysis and structure definition of data | Prof Coifman

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Paper “Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps (2005)": https://www.pnas.org/doi/10.1073/pnas...

Abstract: We provide a framework for structural multiscale geometric organization of graphs and subsets of ℝ𝑛. We use diffusion semigroups to generate multiscale geometries in order to organize and represent complex structures. We show that appropriately selected eigenfunctions or scaling functions of Markov matrices, which describe local transitions, lead to macroscopic descriptions at different scales. The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration. We provide a unified view of ideas from data analysis, machine learning, and numerical analysis.

Authors: Ronald Coifman, S. Lafon, A.B. Lee, M. Maggioni, B. Nadler, F. Warner, and S.Zuker

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Chapters

00:00 - Intro
13:15 - How do we Assemble Information?
16:39 - Diffusion Eigenvectors
26:25 - Diffusion Graphs
44:00 - Organizing Matrices
45:26 - Empirical Neurology: Experimental Setting
53:09 - Flexible Distances Between Subsets
1:07:23 - Q+A