We provide a framework for structural multiscale geometric organization of subsets
(data) in Euclidean spaces and on graphs. 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 (describing 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. These methods provide a unified augmented view of data analysis machine learning numerical analysis.
In particular we obtain fast order NlogN algorithms for homogenization of heterogeneous structures as well as for data representation.
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