Geometric diffusions as a tool for harmonic analysis and structure definition of data: Multiscale methods

In the companion article, a framework for structural multiscale geometric organization of subsets of R-n and of graphs was introduced. Here, diffusion semigroups are used to generate multiscale analyses in order to organize and represent complex structures. We emphasize the multiscale nature of these problems and build scaling functions of Markov matrices (describing local transitions) that 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. This article deals with the construction of fast-order N algorithms for data representation and for homogenization of heterogeneous structures.