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...

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We present a novel approach to solving the trajectory plan ning problem (TPP) in time-varying environments. The es sence of our approach lies in a heuristic but natural decom position of TPP into two subproblems: (1) planning a path to avoid collision with static obstacles and (2) planning the velocity along the path to avoid collision with moving obsta cles. We call thefirst subproblem the path planning problem (PPP) and the second the velocity planning problem (VPP). Thus, our decomposition is summarized by the equation TPP => PPP + VPP. The symbol => indicates that the de composition holds under certain assumptions, e.g., when obstacles are moving independently of ( i.e., not tracking ) the robot. Furthermore, we pose the VPP in path-time space, where time is explicitly represented as an extra dimension, and reduce it to a graph search in this space. In fact, VPP is transformed to a two-dimensional PPP in path-time space with some additional constraints. Algorithms are then pre sented to solve the VPP with different optimality criteria: minimum length in path-time space, and minimum time.