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

本研究计画的重点是可拓学在几何、偏微分方程式、电脑科学及资料处理等问题上的应用。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。为了学习数据固有的物理结构，人们寻找少量的潜变量，这些变量描述了包含或“接近”数据点的低维流形。经典的统计回归寻找数据中的线性关系，而流形学习允许可能的非线性描述。首席研究员计划使用可拓理论的技术开发实用的流形学习算法。例如，考虑用光滑凸超曲面插值一组数据的问题。鉴于所有目前可用的解决方案，这个问题需要从整个表面均匀采样的数据点，主要研究者提出了一种方法，即使没有采样数据的大片表面。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，为了研究平均曲率流的奇异解，发展一个广义的曲率概念是很重要的。另一项成就已Cheeger的理论微分度量测度空间。契格空间带有距离和体积的概念，但令人惊讶的是，它们缺乏通常定义微分所需的局部坐标图的结构。这种方法的一个缺点是其固有的局限性，一阶理论。也就是说，可以定义一阶微分算子，但是二阶算子的概念，例如拉普拉斯算子或热算子，在这个抽象层次上是没有意义的。主要研究者提出了另一种观点：假设空间被嵌入为欧几里得空间的子集。人们可以把欧几里得空间上的微分推到子集上定义一个微积分。这种简单的技术允许人们在欧几里得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，束的计算是非常不平凡的。一阶切丛可以用一个标准的爆破论证来定义，但是高阶的“paratangent bundle”很难理解。通过专注于一类明确的例子代数簇尖点奇点，主要研究者将找到新的方法来计算这些抽象空间。主要研究者将发展代数和半代数集上的除差因子的概念。

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.