The Whitney Reduction Network: Theory, Algorithms and Applications

研究者利用了Whitney Reduction Network，这是一种处理和分析高环境维数数据集的新工具。惠特尼定理激发了网络的架构，它包括一个“好的投影”，被设计成bilipschitz并保留了数据的微分结构，然后是一个非线性逆，从投影的选择中继承了良好的插值特性。该方法提供了数据的自然表示，即使用径向基函数（rbf）近似的函数图。基本技术可以通过多种方式实现，因此开发和评估广义算法是研究的主要焦点。研究的问题和算法包括自适应基方法、RBF中心选择、（时空）聚类方法、求解RBF权重的直接和迭代方法、带约束的RBF近似和基本算法的局部实现。科学研究在一定程度上已经成为一个数据处理问题。自然现象的复杂模型方程的高速数值模拟产生了大量的数据。理解现象意味着提取和解释数据中的信息。该项目的主要目标是开发数学工具来生成数据的有效表示，以便以原始数据所不具备的方式暴露其包含的信息。这种方法的潜在应用是广泛的和多学科的。特别是，这里进行的基础研究通过与霍尼韦尔公司和赖特帕特森空军基地的空军材料研究项目的合作直接应用。

Kirby9973303 The investigator exploits the Whitney Reduction Network, a new tool for processing and analyzing data sets of high ambient dimension. Whitney's theorem motivates the architecture of the network, which consists of a "good projection," designed to be bilipschitz and retain the differential structure of the data, followed by a nonlinear inverse with good interpolation properties inherited from the choice of projection. The approach provides a natural representation of the data as the graph of a function that is approximated using radial basis functions (RBFs). The basic technique may be implemented in a variety of ways, hence the development and evaluation of a broad class algorithms is a primary focus of the research. Problems and algorithms examined include adaptive basis methods, RBF center selection, (spatio-temporal) clustering methods, direct and iterative methods for solving for RBF weights, RBF approximation with constraints and a local implementation of the basic algorithm. Scientific research has to a certain degree become a problem in data processing. High-speed numerical simulations of intractable model equations of natural phenomena produce enormous quantities of data. Understanding the phenomena means extracting and interpretting information in the data. The main goal of the project is to develop mathematical tools to produce efficient representations of data so as to expose the information it contains in a manner that the orignal data does not. Potential applications of this methodology are widespread and multi-disciplinary. In particular, the basic research undertaken here has direct applications through collaboration with Honeywell Corporation and the Air Force materials research program at Wright Patterson Air Force Base.