Coifman, R. R.; Lafon, S.; Lee, A. B.; Maggioni, M.; Nadler, B.; Warner, F.; Zucker, S. W. Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. (English) Zbl 1405.42043 Proc. Natl. Acad. Sci. USA 102, No. 21, 7426-7431 (2005). Summary: We provide a framework for structural multiscale geometric organization of graphs and subsets of \(\mathbb{R}^n\). 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. Cited in 2 ReviewsCited in 165 Documents MSC: 42B35 Function spaces arising in harmonic analysis 68U10 Computing methodologies for image processing 68P05 Data structures × Cite Format Result Cite Review PDF Full Text: DOI