RUI: Localized function approximation based on spectral and scattered data on manifolds

该项目将研究者在环面、单位区间和球面上的局部逼近的工作扩展到数据相关流形的情况。目前关于数据依赖流形的大部分工作都集中在理解数据背后的几何结构上。研究者已经开始开发不依赖于几何结构的工具来近似定义在数据相关流形上的函数。该项目将数据依赖流形的理论提升到了已知流形的完成水平。近似是普遍定义的，可以基于谱数据或非局部分散数据，即无法控制目标函数采样点位置的数据。在这个全局定义的近似中，误差在流形上的不同点上根据目标函数在这些点附近的平滑度进行最佳调整。具体而言，本项目包括(1)在某些图上的近似理论的发展，(2)在目标函数的光滑性方面完整地描述了流形上径向基函数网络的类似物的近似性质，以及(3)研究了求解流形上伪微分方程的谱和伪谱方法的局部近似性质。本研究的一个组成部分是开发有效的算法来实现和测试理论。许多现代应用程序涉及回答基于非结构化数据集的查询。例如，基于一个由手写数字组成的数据集，作为图像扫描，人们想要确定一个新的未见过的图像代表哪个数字（如果有的话）。在过去的几年中，为了在这些数据集上施加几何结构，已经提出了新的和强大的工具。在这个项目中，研究者的目标是超越几何考虑，发展一种理论，专注于将特定查询建模为函数近似。该理论有助于开发有效的算法并对其进行测试。潜在的应用领域包括晶体学、地球物理学、生物数学、半监督学习、文件分析、人脸识别、高光谱图像处理和星系编目。

MhaskarDMS-0908037 The project extends the investigator's work on localized approximation on the torus, unit interval, and sphere to the case of data-dependent manifolds. Much of the current work on data-dependent manifolds is focussed on gaining an understanding of the geometry underlying the data. The investigator has started to develop tools that don't rely on geometric structure to approximate functions defined on data-dependent manifolds. This project brings the theory for data-dependent manifolds to the level of completion achieved in the case of the known manifolds. The approximations are defined universally, and can be based on either spectral or nonlocal, scattered data, i.e., data where one has no control on the location of the points where the target function is sampled. The error in this globally defined approximation adjusts itself optimally at different points on the manifold according to the smoothness of the target function in the vicinity of those points. In particular, the project includes (1) the development of approximation theory on certain graphs, (2) a complete characterization of the approximation properties of analogues of radial basis function networks on the manifolds in terms of the smoothness of the target functions, and (3) a study of the local approximation properties of spectral and pseudo-spectral methods to solve pseudo-differential equations on the manifold. An integral part of this research is to develop efficient algorithms to implement and test the theory. Many modern applications involve answering queries based on unstructured data sets. For example, based on a data set consisting of hand-written digits, scanned as images, one wants to determine which digit a new unseen image represents, if any. In the past few years, new and powerful tools have been proposed in order to impose a geometrical structure on such data sets. In this project the investigator aims to go beyond geometrical considerations, developing a theory that focusses on modeling of specific queries as function approximation. The theory helps to develop efficient algorithms and test them. Potential areas of applications include crystallography, geophysics, biomathematics, semi-supervised learning, document analysis, face recognition, hyperspectral image processing, and cataloguing of galaxies.