Harmonic Analysis and Faithful Data Representations. Multiscale Analysis and Diffusion Processes

谐波分析和忠实的数据表示。

• 批准号: 0800837
• 负责人: Raanan Schul
• 金额: $ 8.58万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-15 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800837&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Faithful; Data; Representations.

Many real life applications yield a data set which manifests itself as a collection of points in some metric space. In some cases one may take this metric space as a high dimensional Euclidean space, and in some cases even this is too restrictive an assumption. A standard task or challenge which one faces is finding a faithful representation of this collection of points (or a big portion of it) as a subset of a low dimensional Euclidean space. The meaning of the words faithful, big, and low is clearly context dependent, and it is often the case that winning a little more in one of these three categories yields a loss in the other two. In many cases it makes sense to assume something about the geometry of the data set. This task has thus far yielded much attention from computer scientists and applied mathematicians using a wide range of approaches. The PI proposes to use and develop harmonic analysis and geometric measure theory techniques to attack these problems. In particular, the theory of quantitative rectifiability (a quantitative approach to geometric measure theory stemming from harmonic analysis) and diffusion geometry (the study of geometry through diffusion processes, heat kernels eigenfunctions etc.) will be used. We are at an age where one has the ability to collect large amounts of data in a reasonable time, but in many cases we do not have the ability to make full use of this data since the mathematical tools are not sufficiently well developed. The research we propose involves the development of new mathematical techniques for extracting information from large data sets and representing it in simpler form, thus making it easier to work with and understand. Applications include the study of databases of images and documents, and the modeling of complex dynamical systems (e.g. transaction data, weather patterns, molecular dynamics).

许多现实生活中的应用程序产生的数据集表现为某些度量空间中的点集合。 在某些情况下，人们可能将此度量空间视为高维欧几里得空间，在某些情况下，即使这太过限制了一个假设。 一个人面临的标准任务或挑战是找到该点（或其中很大一部分）作为低维欧几里得空间的子集的忠实表示。忠实，大和低的一词的含义显然取决于上下文，而在这三个类别中，赢得更多的胜利通常会在另外两个类别中产生损失。 在许多情况下，要假设数据集的几何形状是有意义的。 迄今为止，这项任务已经通过广泛的方法引起了计算机科学家和应用数学家的广泛关注。 PI建议使用和开发谐波分析和几何测量理论技术来攻击这些问题。特别是，将使用定量重新讨论性的理论（一种源自谐波分析的几何方法理论的定量方法）和扩散几何学（通过扩散过程，热核特征函数等对几何研究的研究）。我们正处于一个年龄的时代，可以在合理的时间内收集大量数据，但是在许多情况下，由于数学工具的发展不够充分，我们没有能力充分利用这些数据。我们提出的研究涉及开发新的数学技术，以从大型数据集中提取信息并以更简单的形式表示信息，从而使其更容易使用和理解。 应用包括图像和文档数据库的研究以及复杂动力学系统的建模（例如，交易数据，天气模式，分子动力学）。