Geometric Harmonic Analysis

扩散(或推断)几何，为组织海量数字数据集提供了工具。与微积分一样，多尺度分析方法通过结合“无穷小”(线性)模型来建立对象之间的全局推理关系。对这种结构的调和分析导致了多尺度文件夹构建范式，从而产生了用于海量复杂数据的函数回归和分析的强大工具。这些几何方法在经典的微分几何中提供了新的见解，使得能够显式地嵌入定理和黎曼流形的坐标系。多尺度分析方法为看似非结构化的数据库带来了系统的分析工具，并使本体论和数据诱导语言的自动生成成为可能。这些方法广泛应用于医学诊断、心理问卷的组织和分析以及从数据挖掘到机器视觉的机器学习的各个方面。

Diffusion (or inference) geometries , provide tools for organization ofmassive digital data sets. Like differential calculus, they are used tobuild global inference relations between objects by combining ``infinitesimal" (linear) models. Harmonic analysis on such structuresleads to multiscale folder building paradigms leading to powerful toolsfor functional regression and analysis of massive complex data.These geometric methods provide new insights in classical differentialgeometry enabling explicit embedding theorems and coordinate systems forRiemannian manifolds.The multiscale analytic methods lead to a systematic analysis tool forseemingly unstructured data bases, and enables the automatic generationof Ontologies and data induced languages. These methods are broadlyapplicable in medical diagnostics, in the organization and analysis ofpsychological questionnaires, as well as in all aspects of machinelearning from data mining to machine vision.