Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps
Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps
复制标题
DOI：
10.1073/pnas.0500334102
复制
发表时间：
2005-05-24
影响因子：
11.1
通讯作者：
Zucker, SW
中科院分区：
文献类型：
--
作者：
Coifman, RR;Lafon, S;Zucker, SW
我们提供了一个框架结构的多尺度几何组织的图形和子集的R-n。我们使用扩散半群生成多尺度几何，以组织和表示复杂的结构。我们表明，适当选择的特征函数或尺度函数的马尔可夫矩阵，描述局部过渡，导致在不同尺度的宏观描述。迭代或扩散马尔可夫矩阵的过程被视为牛顿范式某些方面的推广，其中系统的局部无穷小转变通过积分导致全局宏观描述。我们从数据分析、机器学习和数值分析中提供统一的观点。
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.