BIGDATA: F: Data Driven Optimization on Flag Manifolds with Geometric Constraints

这项研究涉及创新的数学理论和算法的发展，以促进在科学家、工程师和当今数据驱动的社会产生的海量数据集中发现知识。将开发新的方法，允许以一种能够检测隐藏在信息量中的相似和不同的方式对大量数据进行编码。该框架对于表征相似程度以及发现可能在数据集之间共享的特征或模式特别有用。该项目侧重于使用几何学和最优化方面的工具来提供有效的数据表示，以扩展分析人员的工具包，并提高他们理解大型和复杂数据集的能力。该方法将在极端天气模拟等现实世界数据集或捕捉人类对病原体感染的免疫反应的生物数据集上进行验证。正在开发的技术可以被视为新兴的几何数据学习领域的一部分。数学方法利用了格拉斯曼流形的几何框架，格拉斯曼流形将向量空间的给定维度的子空间集参数化。这种方法的吸引力在于，子空间作为Grassmann流形上的抽象点，是捕捉数据观测中的自然可变性的有效工具，例如，照明或噪声的变化。如果一个数据的子空间与另一个数据的子空间在规定的维度上相交，那么这些抽象点应该被认为比子空间在更少的维度上相交，或者根本不相交。这种类型的几何图像，当在数学框架中表达时，导致使用旗形和舒伯特变量来表示和比较数据。所提出的研究方案解决了几何约束下的数据驱动优化中的新问题，例如，当可行集是Schubert变种时。这个框架允许我们提取表征模式的几何模型，并自然地基于作为子空间之间角度的函数的相似性度量来比较大的观测集。

This research concerns the development of innovative mathematical theory and algorithms to facilitate knowledge discovery in the massive data sets generated by scientists, engineers and today's data driven society. New approaches will be developed that permit the encoding of large quantities of data in a way that enables the detection of similarities and differences buried in the volumes of information. The framework is especially useful for characterizing degrees of similarity, and discovering features or patterns that may be shared between data sets. The project focuses on the use of tools from geometry and optimization to provide effective data representations that expand the toolkit of analysts and enhances their capacity for understanding large and complex data sets. The methodology will be validated on real world data sets like extreme weather simulations or biological data sets such as those capturing the human immune response to infection by pathogens. The techniques being developed may be viewed as part of the emerging field of geometric data learning. The mathematical approach exploits the geometric framework of the Grassmannian, the manifold that parameterizes the set of subspaces of a given dimension of a vector space. The appeal of this approach is that subspaces, as abstract points on the Grassmann manifold, are an effective tool to capture the natural variability in data observations stemming from, for example, variations in illumination, or noise. If a subspace of data intersects another subspace of data in some prescribed number of dimensions, then these abstract points should be considered to be more related than subspaces that intersect in fewer dimensions, or not at all. This type of geometric picture, when formulated in a mathematical framework, leads to the use of flag manifolds and Schubert varieties for representing and comparing data. The proposed research program addresses new problems in data driven optimization subject to geometric constraints, for example, when the feasible set is a Schubert variety. This framework allows us to extract geometric models that characterize patterns, and leads naturally to comparisons between large sets of observations based on similarity measures which are functions of angles between subspaces.