CAREER: Foundations for Geometric Analysis of Noisy Data

计算几何的一个重要作用是理解和形式化数据的结构。 随着数据成为现代科学的核心货币，这一角色的重要性越来越大。 然而，许多经典的计算几何固有地假设数据的所有方面都是已知的和精确的。 在实践中，情况很少如此。 这个项目的重点是建立两个扩展的基础，经典的几何设置相关的噪音数据。 1. PI将研究点集中的位置不确定性，其中每个数据点的位置由概率分布描述。 给定这样的输入，我们的目标是形式化如何构造，近似和简洁地表示几何查询的分布在这个不确定的数据。 2. PI将研究将统计内核（例如高斯内核）应用于数据集的几何后果。 他将研究这个过程如何平滑数据，消除退化，并隐式简化和正则化算法。 此外，他还将探索由此产生的核密度估计的几何结构，以及它如何与数据的算法和数据的近似表示相关。 PI将领导围绕数据分析，算法和可视化主题开发以数据为中心的教育计划。 PI正在为这个程序开发一个关于数据挖掘的模型课程;它侧重于数据的几何，统计和算法属性。 目前正在编写一套内容广泛的课程笔记，并附有在线免费提供的讲课录像。 这个类和程序吸引了许多跨学科和不同的学生和观察员。 该计划是一个更大的努力，使相关的数据分析技术，从计算几何提供给更广泛的数据丰富的观众的一部分。

An important role of computational geometry is to understand and formalize the structure of data. And as data is becoming a central currency of modern science, this role is growing in importance. However, much of classical computational geometry inherently assumes that all aspects of data are known and precise. This is rarely the case in practice. This project focuses on building the foundations for two extensions to classic geometric settings pertinent to noisy data. 1. The PI will study locational uncertainty in point sets, where the location of each data point is described by a probability distribution. Given such an input, the goal is to formalize how to construct, approximate, and concisely represent the distribution of geometric queries on this uncertain data. 2. The PI will study the geometric consequences of applying a statistical kernel (e.g. a Gaussian kernel) to a data set. He will investigate how this process can smooth data, remove degeneracies, and implicitly simplify and regularize algorithms. Moreover, he will explore the geometric structure of the resulting kernel density estimate, and how it relates to algorithms for the data and approximate representations of the data. The PI will lead the development of a data-focused educational program around the themes of data analysis, algorithmics, and visualization. The PI is developing a model course for this program on data mining; it focuses on the geometric, statistical, and algorithmic properties of data. An extensive set of course notes is being compiled, accompanied with videotaped lectures freely available online. This class and program attract many interdisciplinary and diverse students and observers. This program is part of a larger effort to make relevant data analysis techniques from computational geometry available to a broader data-rich audience.