Duke Applied Mathematics

Publications [#243825] of Mauro Maggioni

Papers Published

1. Coifman, RR; Lafon, S; Lee, AB; Maggioni, M; Nadler, B; Warner, F; Zucker, SW, Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps., Proceedings of the National Academy of Sciences of the United States of America, vol. 102 no. 21 (May, 2005), pp. 7426-7431, ISSN 0027-8424 [15899970], [doi]
   (last updated on 2019/04/10)

   Abstract:
   We provide a framework for structural multiscale geometric organization of graphs and subsets of 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.