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

许多实际应用产生的数据集表现为某个度量空间中的点的集合。在某些情况下，人们可以把这个度规空间当作高维欧几里德空间，在某些情况下，甚至这个假设也太严格了。人们面临的一个标准任务或挑战是找到这个点集合（或它的很大一部分）作为低维欧几里得空间子集的忠实表示。“忠实”、“大”和“低”这三个词的含义显然取决于上下文，通常情况下，在这三个类别中赢得多一点，在其他两个类别中就会损失。在许多情况下，假设数据集的几何形状是有意义的。到目前为止，这项任务已经引起了计算机科学家和应用数学家的广泛关注，他们使用了各种各样的方法。PI建议使用和发展谐波分析和几何测量理论技术来解决这些问题。特别是，定量可纠偏性理论（源于谐波分析的几何测量理论的定量方法）和扩散几何（通过扩散过程，热核特征函数等研究几何）将被使用。我们正处于一个有能力在合理的时间内收集大量数据的时代，但在很多情况下，由于数学工具不够发达，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，用于从大型数据集中提取信息，并以更简单的形式表示它，从而使其更容易处理和理解。应用包括图像和文档数据库的研究，以及复杂动态系统的建模（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).