Rectifiability of Measures in Euclidean and Metric Spaces

现代技术的一个方面是很容易收集数据。一项非常具有挑战性的任务是筛选大量的数据，以找到有意义的信息。人们希望以一种易于使用的方式来组织数据，或者至少组织其中的一部分。想象一下，这些数据都是互联网上的所有图像，而你寻找的“组织”是能够根据你选择的特定人的活动来绘制出哪些图像是你所选择的人的图像，并根据他或她从事的活动进行细分排序。以有用的方式组织大量信息，或对其进行分类并找到您关心的部分，这些都是可以转换的任务，或者与数学问题相关的任务。这项提案试图解决其中一些问题。基本问题是对以下问题的数学类比：组织数据后，我们希望得到什么样的结构？我们能期望以一种有用的方式组织多少数据？如果我们愿意在这个过程中“丢失”一些信息，答案会改变吗？我们会知道丢失的数据量吗？最后但并非最不重要的一点是，我们能否以一种实用的方式访问组织的数据或其中的重要部分？在许多应用中，人们被给予一个表示为度量空间的子集的大数据集，例如表示大维d的R^d，并寻求将该数据集的一个“大”部分“忠实地”表示为维度k“小于”d的R^k的子集。这里的“忠实地”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。到目前为止，这项任务得到了使用各种方法的计算机科学家和应用数学家的极大关注。降维的框架还包括数据压缩和数据逼近。它们在许多科学领域都有应用。几何测度论和几何函数论是尚未被充分利用的工具。一个关键点是，给定的数据集通常具有一些额外的几何结构，例如小的Hausdorff维(离散模拟)，或者接近低维流形的并。这使得人们可以使用调和分析和几何测量理论。该项目旨在研究受此启发的数学问题。要讨论的基本问题可以表述为“度量空间的一部分何时由‘标准’块的Lipschitz像组成，我们如何找到这些块？”或者“何时以低维的方式最好地描述一组点？”将使用的工具来自调和分析和几何测量理论的组合，通常被称为定量可纠正性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.