Computational Methods for Exploring the Geometry of Large Data Sets

首席研究员和他的同事们开发了计算和理论框架来分析具有低维内在结构的大型数据集。更具体地说，它们解决了以下挑战：在存在显著异常值和噪声的情况下构建底层曲线和表面；大噪声数据集非线性嵌入技术的改进具有低维惯性流形的特殊非线性偏微分方程生成的大数据集的分析。提出的研究有几个重要的应用：图像中的定量边缘检测，通过μ子辐射检测核装置，蛋白质结合基因组区域（甚至特定位点）的鉴定，基因本体中功能域的定量探索及其与结构特性的关系。该提议的更广泛影响如下：1)数学提出了重要的应用，其中一些在上面列出。2)这些应用指导并需要一个广泛的框架来对具有内在低维几何结构的数据集进行多尺度几何分析。3)不同数学领域之间的互动，特别是计算调和分析、科学计算、统计学习、概率论和数学建模。4)多学科合作，包括应用数学家、生物学家、计算机科学家、统计学家和数学分析师。5)产业合作。(6)在有前途的数学新领域培养年轻的研究人员。

The principal investigator and his colleagues develop computational and theoretical framework to analyze large data sets with low-dimensional intrinsic structure. More specifically, they address the following challenges: Constructions of underlying curves and surfaces in the presence of significant outliers and noise; Improvement of recent nonlinear embedding techniques for large data sets with significant noise; Analysis of large data sets generated by special nonlinear partial differential equations with low-dimensional inertial manifold. There are several important applications of the proposed research: quantitative edge detection in images, detection of nuclear devices by muon radiation, identification of protein-binding genomic regions (and even specific sites), quantitative exploration of the functional domain in the gene ontology and its relation with structural properties. The broader impacts of the proposal are as follows: 1) The mathematics suggests important applications, some of them are listed above. 2) The applications guide and demand a broad framework for multiscale geometric analysis of data sets with intrinsic low-dimensional geometric structures. 3) Interaction between different areas of mathematics, in particular, computational harmonic analysis, scientific computation, statistical learning, probability and mathematical modeling. 4) Multidisciplinary collaborations, involving applied mathematicians, biologists, computer scientists, statisticians and mathematical analysts. 5) Industrial collaborations. 6) Training of young researchers in a promising new area of mathematics.