Rectifiability of Measures in Euclidean and Metric Spaces

欧几里得和度量空间中测度的可修正性

基本信息

批准号：
1763973
负责人：
Raanan Schul
金额：
$ 18万
依托单位：
SUNY at Stony Brook
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1763973&HistoricalAwards=false
关键词：
Rectifiability Measures Euclidean Metric Spaces

项目摘要

One aspect of modern technology is that it is easy to collect data. A very challenging task is to sift through a large collection of data in order to find meaningful information. One would like to organize the data, or at least part of it, in such a way that it is easy to use. Imagine the data as being all images on the internet, and the "organization" that you seek is being able to map out which images are those of a specific person of your choosing, sub-ordered according to the activities in which he or she is engaged. Organizing large amounts of information in a useful way, or sorting through it and finding pieces you care about, are tasks that can be transformed, or related to, mathematical questions. This proposal attempts to address some of these questions. Basic questions are mathematical analogues of the following: What kind of structure can we hope to get after organizing the data? How much of the data can we expect to organize in a useful way? Do answers change if we are willing to "lose" some information in the process? Will we know the amount data lost? And, last but not least, can we, in a practical way, access the organized data or a significant part of it?In many applications one is given a large data set represented as a subset of a metric space, such as R^d for large dimension d, and seeks to `faithfully' represent a `large' portion of this data set as a subset of R^k for dimension k much `smaller' than d. `Faithfully' here, means that one can still perform the same data mining tasks on the image of the data portion. This task has thus far yielded much attention from computer scientists and applied mathematicians using a wide range of approaches. The framework of dimensionality reduction also includes data compression and data approximation. These have applications in many areas of science. Geometric Measure Theory and Geometric Function Theory are tools whose use in this matter has not been fully exploited. A key point is that often the given data set has some additional geometric structure, for example small Hausdorff dimension (a discrete analogue), or being close to a union of low dimensional manifold. This allows one to use harmonic analysis and geometric measure theory. The project aims at studying mathematical questions motivated by this. Basic questions to be discusses can be phrased as "When is part of a metric measure space composed of Lipschitz images of `standard' pieces and how do we find these pieces?" or "When is a collection of points best described in a low-dimensional way?". The tools to be used come from a combination of Harmonic Analysis and Geometric measure theory, which is usually referred to as quantitative rectifiability.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

现代技术的一个方面是很容易收集数据。一个非常具有挑战性的任务是筛选大量数据，以找到有意义的信息。人们希望以易于使用的方式组织数据或至少一部分数据。想象一下，数据是互联网上的所有图像，您寻求的“组织”能够绘制出哪些图像是您选择的特定人的图像，该图像是根据他或她参与的活动进行的。以有用的方式组织大量信息，或者通过它进行分类并找到您关心的部分，是可以转换或与数学问题相关的任务。该建议试图解决其中一些问题。基本问题是以下数学类似物：组织数据后我们希望获得什么样的结构？我们可以期望以有用的方式组织多少数据？如果我们愿意在此过程中“丢失”一些信息，答案会改变吗？我们会知道数据损失的数量吗？而且，最后但并非最不重要的一点是，我们可以以实际的方式访问有组织的数据或其中的重要部分吗？在许多应用程序中，给出了一个大的数据集，代表了一个指标空间的子集，例如对于大维度D的r^d，并寻求“忠实地”“忠实地”该数据的“大型”部分，该数据集比r^k for Dimension k小d d d d小。 ``忠实''在这里，意味着人们仍然可以在数据部分的图像上执行相同的数据挖掘任务。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。降低维度的框架还包括数据压缩和数据近似。这些在许多科学领域都有应用。几何测量理论和几何函数理论是在此问题中使用的工具，尚未得到充分利用。一个关键点是，给定的数据集通常具有一些附加的几何结构，例如小的hausdorff尺寸（一个离散的模拟），或接近低维歧管的结合。这允许人们使用谐波分析和几何测量理论。该项目旨在研究以此激励的数学问题。要讨论的基本问题可以用作“何时是由``标准''片的Lipschitz图像组成的度量测量空间的一部分，我们如何找到这些作品？”或“何时以低维方式描述最佳点的集合？”。要使用的工具来自谐波分析和几何措施理论的结合，该理论通常称为定量可重新讨论性。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的评估来审查标准的评估值得支持的。