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

@article{citeulike:1176894, abstract = {In the companion article, a framework for structural multiscale geometric organization of subsets of [IMG]f1.gif" BORDER="0"> 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. 10.1073/pnas.0500896102}, author = {Coifman, R. R. and Lafon, S. and Lee, A. B. and Maggioni, M. and Nadler, B. and Warner, F. and Zucker, S. W. }, citeulike-article-id = {1176894}, journal = {PNAS}, keywords = {analysis, diffusion, harmonic, multiscale}, month = {May}, number = {21}, pages = {7432--7437}, posted-at = {2007-03-20 03:22:59}, priority = {0}, title = {Geometric diffusions as a tool for harmonic analysis and structure definition of data: Multiscale methods}, url = {http://www.pnas.org/cgi/content/abstract/102/21/7432}, volume = {102}, year = {2005} }

TY - JOUR ID - citeulike:1176894 L3 - citeulike-article-id:1176894 TI - Geometric diffusions as a tool for harmonic analysis and structure definition of data: Multiscale methods JF - PNAS VL - 102 IS - 21 SP - 7432 EP - 7437 N2 - In the companion article, a framework for structural multiscale geometric organization of subsets of [IMG]f1.gif" BORDER="0"> 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. 10.1073/pnas.0500896102 KW - analysis KW - diffusion KW - harmonic KW - multiscale AU - Coifman, RR AU - Lafon, S AU - Lee, AB AU - Maggioni, M AU - Nadler, B AU - Warner, F AU - Zucker, SW PY - 2005/05/24/ UR - http://www.pnas.org/cgi/content/abstract/102/21/7432 ER -