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，以分析具有低维内在结构的大数据集。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潜在的曲线和曲面；改进了最近针对具有显著噪声的大数据集的非线性嵌入技术；分析由具有低维惯性流形的特殊非线性偏微分方程组生成的大数据集。提出的研究有几个重要的应用：图像的定量边缘检测，通过Muon辐射检测核装置，识别蛋白质结合的基因组区域(甚至特定位置)，定量探索基因本体论中的功能域及其与结构特性的关系。这项提议的更广泛的影响如下：1)数学表明了一些重要的应用，上面列出了一些应用。2)这些应用指导并要求对具有内在低维几何结构的数据集进行多尺度几何分析的广泛框架。3)不同数学领域之间的相互作用，特别是计算调和分析、科学计算、统计学习、概率和数学建模。4)多学科协作，涉及应用数学家、生物学家、计算机科学家、统计学家和数学分析师。5)产业协作。6)在一个很有前途的数学新领域培训年轻的研究人员。

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.