复杂数据的拓扑与几何结构

Traditional mathematical or statistical models for data are based on the idea of approximation theory. The simplest examples are the linear regression models. These ideas are effective for studying the local structure of data sets, but are powerless for studying the global structure of data sets. To study their global structure, one has to resort to topological and geometric tools..The focus of this project is to systematically study the topological and geometric models of large and complex data sets, as well as the algorithms associated with these models. Through focused seminars, small scale meetings and discussion groups, we will gain a much deeper and broad understanding of these models. We will focus in particular on the topological structure of complex networks that arise from data sets. This series of activities will help to bring together industry and academics, to study these problems of common interest. One of the.driving problems for this project is the structure of the Twitter and Sina tweets network. This project will not only benefit the mathematics community but will also have some influence in information industry.

传统的数据模型，如统计模型，大都是基于逼近论的想法的。最简单的例子就是线性回归模型。这种想法对研究数据集的局部性态比较有效，但难以反映数据集的整体性态。要有效地研究数据集的整体性态，就必须采用拓扑或几何的方法。.本项目的重点是系统地研究大型复杂数据集的拓扑和几何模型，以及相关算法。我们将通过举办小型研讨会，讲习班等形式，对数据的拓扑和几何模型作全面和深刻的考察，尤其是对由数据产生出的复杂网络的拓扑结构作深入的研究。通过这些研讨会，我们将把学术界和企业界的相关人员联合起来，共同商讨这些问题。驱动这项研究的具体问题之一是新浪微博网络以及Twitter网络的拓扑结构。这项研究的受益面不仅仅是在数学界，它将会对信息产业产生一定的影响。

在本项目的支持下，我们面向应用问题中的数据分析，发展了拓扑和几何模型及相关算法。具体地，我们针对网络数据分析发展了基于离散 Morse 理论的关键点(Critical node)分析和多层次分解，并且在蛋白质折叠和社交网络分析中取得应用; 另外我们针对网络众包偏好/排序实验数据，发展了基于Hodge 理论的鲁棒排序方法，并且在计算机视觉相对属性学习、多媒体主管感受质量评价中获得应用。