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Characterization of Trace Spaces and Differential Structures on Subsets of Euclidean Space

欧氏空间子集上迹空间和微分结构的表征

基本信息

批准号：
1700404
负责人：
Arie Israel
金额：
$ 15.3万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700404&HistoricalAwards=false
关键词：
Characterization Trace Spaces Differential Structures

项目摘要

This research project is focused on applications of extension theory to problems in geometry, partial differential equations, computer science, and data processing. In the machine learning paradigm, one typically observes a collection of data that represents the measurements of a physical process. The data may be represented in terms of a large number of variables. To learn the physical structures inherent to the data, one looks for a small number of latent variables that describe a low-dimensional manifold which contains or "passes close to" the data points. Whereas classical statistical regression looks for a linear relationship in the data, manifold learning allows for possibly non-linear descriptions. The principal investigator plans to develop practical manifold learning algorithms using techniques from extension theory. Consider for instance the problem of interpolating a set of data by a smooth, convex hypersurface. Whereas all currently available solutions to this problem require that the data points be sampled uniformly from the entire surface, the principal investigator proposes an approach which would work even when there are no sampled data on large pieces of the surface. The proposed research will introduce new connections between the fields of harmonic analysis and machine learning.The past few decades have witnessed the development of abstract notions of differentiation and curvature on nonsmooth spaces. To study singular solutions to mean curvature flow, for instance, it is important to develop a generalized notion of curvature. Another achievement has been Cheeger's theory of differentiation on metric measure spaces. Cheeger's spaces carry a notion of distance and volume, but surprisingly, they lack the structure of local coordinate charts usually required to define a differential calculus. One shortcoming of this approach is its inherent limitation to first order theories. That is, one can define a first order differential operator, but the notion of a second order operator, such as the Laplacian or heat operator, is meaningless at this level of abstraction. The principal investigator proposes an alternate perspective: Assume the space is embedded as a subset of a Euclidean space. One can take the differential calculus on the Euclidean space and push it forward to define a calculus on the subset. This simple technique allows one to define high-order tangent and cotangent bundles on subsets of Euclidean space. An issue with the approach is that the computation of the bundles is highly nontrivial. The first order tangent bundle can be defined by a standard blow-up argument, but the higher order "paratangent bundles" are difficult to understand. By focusing on a class of explicit examples of algebraic varieties with cuspidal singularities, the principal investigator will find new methods for computing with these abstract spaces. The principal investigator will develop a notion of divided difference quotients on algebraic and semialgebraic sets.

这项研究项目集中在可拓理论在几何、偏微分方程组、计算机科学和数据处理中的应用。在机器学习范式中，人们通常观察代表物理过程的测量的数据集合。数据可以用大量变量来表示。为了了解数据固有的物理结构，人们寻找少量的潜在变量，这些变量描述了包含数据点或接近数据点的低维流形。经典的统计回归在数据中寻找线性关系，而流形学习则允许可能的非线性描述。主要研究人员计划使用可拓理论中的技术开发实用的流形学习算法。例如，考虑用光滑的凸超曲面对一组数据进行内插的问题。尽管目前所有可用的解决方案都要求从整个曲面均匀地采样数据点，但主要研究人员提出了一种即使在大片曲面上没有采样数据的情况下也可以工作的方法。这项研究将在调和分析和机器学习领域之间引入新的联系。在过去的几十年里，非光滑空间上的微分和曲率的抽象概念得到了发展。例如，要研究平均曲率流的奇异解，重要的是要发展一个广义的曲率概念。另一项成就是切格关于度量空间的微分理论。契格的空间带有距离和体积的概念，但令人惊讶的是，它们缺乏定义微积分通常所需的局部坐标图的结构。这种方法的一个缺点是其对一阶理论的固有局限性。也就是说，人们可以定义一阶微分算子，但二阶算子的概念，如拉普拉斯算子或热算子，在这个抽象级别上是没有意义的。首席研究者提出了另一种观点：假设空间被嵌入为欧几里德空间的子集。我们可以把欧几里得空间上的微积分向前推，在子集上定义一个微积分。这个简单的技巧允许我们在欧几里德空间的子集上定义高阶切丛和余切丛。这种方法的一个问题是，捆绑包的计算非常重要。一阶切丛可以用一个标准的爆破引理来定义，但更高阶的“旁切丛”很难理解。通过集中于一类具有尖端奇点的代数簇的显式例子，主要研究者将找到利用这些抽象空间进行计算的新方法。主要研究者将在代数和半代数集上提出除差商的概念。