CAREER: Foundations for Geometric Analysis of Noisy Data

职业：噪声数据几何分析的基础

• 批准号: 1350888
• 负责人: Jeff Phillips
• 金额: $ 52.21万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-05-15 至 2021-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1350888&HistoricalAwards=false
• 关键词: CAREER; Foundations; Geometric; Analysis; Noisy

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.

计算几何的一个重要作用是理解和形式化数据的结构。 随着数据正在成为现代科学的核心货币，这一作用变得越来越重要。 然而，许多经典计算几何本质上假设数据的所有方面都是已知的且精确的。 实践中这种情况很少见。 该项目的重点是为与噪声数据相关的经典几何设置的两个扩展奠定基础。 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，目标是形式化如何构造、近似和简洁地表示这种不确定数据上的几何查询的分布。 2. PI 将研究将统计核（例如高斯核）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、消除简并性以及隐式简化和规范算法。 此外，他将探索所得核密度估计的几何结构，以及它与数据算法和数据近似表示的关系。 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正在为这个数据挖掘项目开发一个模型课程；它侧重于数据的几何、统计和算法属性。 一套内容广泛的课程笔记正在编写中，并附有在线免费提供的讲座录像。 这个课程和项目吸引了许多跨学科和多元化的学生和观察者。 该计划是一项更大努力的一部分，旨在将计算几何的相关数据分析技术提供给更广泛的数据丰富的受众。