We provide a framework for structural multiscale geometric organization of graphs and subsets of ‫ޒ‬ 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.T he geometric organization of graphs and data sets in ‫ޒ‬ n is a central problem in statistical data analysis. In the continuous Euclidean setting, tools from harmonic analysis, such as Fourier decompositions, wavelets, and spectral analysis of pseudodifferential operators, have proven highly successful in many areas such as compression, denoising, and density estimation (1, 2). In this paper, we extend multiscale harmonic analysis to discrete graphs and subsets of ‫ޒ‬ n . We use diffusion semigroups to define and generate multiscale geometries of complex structures. This framework generalizes some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration, the global functions being characterized by differential equations. We show that appropriately selected eigenfunctions of Markov matrices (describing local transitions, or affinities in the system) lead to macroscopic representations at different scales. In particular, the top eigenfunctions permit a low-dimensional geometric embedding of the set into ‫ޒ‬ k , with k Ͻ Ͻ n, so that the ordinary Euclidean distance in the embedding space measures intrinsic diffusion metrics on the data. Many of these ideas appear in a variety of contexts of data analysis, such as spectral graph theory, manifold learning, nonlinear principal components, and kernel methods. We augment these approaches by showing that the diffusion distance is a key intrinsic geometric quantity linking spectral theory of the Markov process, Laplace operators, or kernels, to the corresponding geometry and density of the data. This opens the door to the application of methods from numerical analysis and signal processing to the analysis of functions and transformations of the data. Diffusion MapsThe problem of finding meaningful structures and geometric descriptions of a data set X is often tied to that of dimensionality reduction. Among the different techniques developed, particular attention has been paid to kernel methods (3). Their nonlinearity as well as their locality-preserving property are generally viewed as a major advantage over classical methods like principal component analysis and classical multidimensional scaling. Several other methods to achieve dimensional reduction have also eme...

In the companion article, a framework for structural multiscale geometric organization of subsets of ‫ޒ‬ 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.I n the companion article (1), it is shown that the eigenfunctions of a diffusion operator, A, can be used to perform global analysis of the set and of functions on a set. Here, we present a construction of a multiresolution analysis of functions on the set related to the diffusion operator A. This allows one to perform a local analysis at different diffusion scales. This is motivated by the fact that in many situations one is interested not in the data themselves but in functions on the data, and in general these functions exhibit different behaviors at different scales. This is the case in many problems in learning, in analysis on graphs, in dynamical systems, etc. The analysis through the eigenfunctions of Laplacian considered in the companion article (1) are global and are affected by global characteristics of the space. It can be thought of as global Fourier analysis. The multiscale analysis proposed here is in the spirit of wavelet analysis.We refer the reader to (2-4) for further details and applications of this construction, as well as a discussion of the many relationships between this work and the work of many other researchers in several branches of mathematics and applied mathematics. Here, we would like to at least mention the relationship with fast multiple methods (5, 6), algebraic multigrid (7), and lifting (8, 9). Multiscale Analysis of DiffusionConstruction of the Multiresolution Analysis. Suppose we are given a self-adjoint diffusion operator A as in ref. 1 acting on L 2 of a metric measure space (X, d, ). We interpret A as a dilation operator and use it to define a multiresolution analysis. It is natural to discretize the semigroup {A t } tՆ0 of the powers of A at a logarithmic scale, for example at the times t j ϭ 1 ϩ 2 ϩ 2 2 ϩ · · · ϩ 2 j ϭ 2 jϩ1 Ϫ 1.[1]For a fixed ʦ (0,1), we define the approximation spaces bywhere the i s are the eigenvectors of A, ordered by decreasing eigenvalue. We will denote by P j the orthogonal projection onto V j . The set of subspaces {V j } jʦ‫ޚ‬ is a multiresolution analysis in the sense that it satisfies the following properties:We can also define the detail subspaces W j as the orthogonal complement of V j in V jϩ1 , so that we have the familiar ...

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