Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps.

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.

[1]  Leon W. Cohen,et al.  Conference Board of the Mathematical Sciences , 1963 .

[2]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[3]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[4]  Yair Weiss,et al.  Segmentation using eigenvectors: a unifying view , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[5]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[6]  C. W. Gear,et al.  'Coarse' integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods , 2002 .

[7]  Mads Nielsen,et al.  Proceedings of the 7th European Conference on Computer Vision-Part III , 2002 .

[8]  C. W. Gear,et al.  Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis , 2003 .

[9]  Hongyuan Zha,et al.  Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment , 2002, ArXiv.

[10]  D. Donoho,et al.  Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Mikhail Belkin,et al.  Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , 2003, Neural Computation.

[12]  Eric R. Ziegel,et al.  The Elements of Statistical Learning , 2003, Technometrics.

[13]  Bernhard Schölkopf,et al.  A kernel view of the dimensionality reduction of manifolds , 2004, ICML.

[14]  Ann B. Lee,et al.  Geometric diffusions as a tool for harmonic analysis and structure definition of data: multiscale methods. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[15]  B. Nadler,et al.  Diffusion maps, spectral clustering and reaction coordinates of dynamical systems , 2005, math/0503445.