RUI: Multiscale and Modeling of Scattered Data

在许多实际应用中，人们需要研究基于光滑流形的数据建模和分析，例如球面，更一般地，度量度量空间。例如，文档分析、人脸识别、半监督学习、图像处理、星系编目、脑电势模式分析和脑肿瘤研究。提出者将继续他的工作，通过模拟神经和径向基函数(RBF)网络的近似，有时重新定义，以利用数据的几何优势，为此类数据建模。他还将继续他的多尺度工作，以分析数据。他将在这些理论的基础上开发理论结果和高效算法。像往常一样，研究结果将通过在被引用的期刊和会议记录上发表的文章，以及在口语和会议上的陈述来传播。虽然来自近似理论的经典技术需要对收集数据的地点进行明智的选择，但许多实际应用程序不允许这样的选择。这项研究的新颖性之一是处理在任意地点收集的数据。另一个新奇之处是利用全局数据，例如正交展开中的系数，来研究数据背后的函数关系的局部特征。

In many practical applications, one needs to study modeling and analysis of data based on smooth manifolds such as a sphere, and more generally, metric measure spaces. Examples include document analysis, face recognition, semi-supervised learning, image processing, cataloguing of galaxies, pattern analysis of brain potentials, and the study of brain tumors. The proposer will continue his work on approximation by analogues of neural and radial basis function (RBF) networks, sometimes redefined to take advantage of the geometry of the data, for modeling such data. He will also continue his work on multiscales for the analysis of the data. He will develop theoretical results as well as efficient algorithms based on these theories. The findings of the research will be disseminated, as usual, through articles in refereed journals and conference proceedings, as well as presentations in colloquia and conferences.While classical techniques from approximation theory require a judicious choice of the sites where the data is collected, many practical applications do not allow such a choice. One of the novelties of the research is to deal with data collected at arbitrary sites. Another novelty is to utilize global data, such as coefficients in an orthogonal expansion, to study local features of the functional relationship underlying the data.