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

许多现实生活中的应用都会产生一个数据集，它本身表现为某个度量空间中的点的集合。在某些情况下，人们可以将该度量空间视为高维欧几里德空间，而在某些情况下，这甚至是一个过于严格的假设。一个人面临的标准任务或挑战是找到这个点集(或其中的很大一部分)的忠实表示，作为低维欧几里德空间的子集。“忠诚”、“大”和“低”这三个词的意思显然取决于上下文，通常情况下，在这三个类别中赢得多一点就会在另两个类别中失去一些。在许多情况下，假设数据集的几何结构是有意义的。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。PI建议使用和发展调和分析和几何测度论技术来解决这些问题。尤其是定量可纠正性理论(源于调和分析的几何测量理论的一种定量方法)和扩散几何(通过扩散过程、热核特征函数等研究几何)。将会被使用。我们所处的时代，人们有能力在合理的时间内收集大量数据，但在许多情况下，我们没有能力充分利用这些数据，因为数学工具还不够发达。我们提出的研究涉及开发新的数学技术，用于从大数据集中提取信息并以更简单的形式表示，从而使其更容易使用和理解。应用包括研究图像和文件的数据库，以及复杂动力系统的建模(例如，交易数据、天气模式、分子动力学)。

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).