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

本研究关注创新数学理论和算法的发展，以促进科学家、工程师和当今数据驱动社会产生的海量数据集中的知识发现。将开发新的方法，允许对大量数据进行编码，从而能够发现隐藏在大量信息中的相似性和差异性。该框架对于描述相似度和发现数据集之间可能共享的特征或模式特别有用。该项目侧重于几何和优化工具的使用，以提供有效的数据表示，扩展分析人员的工具包，提高他们理解大型复杂数据集的能力。该方法将在真实世界的数据集（如极端天气模拟）或生物数据集（如捕捉人类对病原体感染的免疫反应）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的几何框架，即参数化向量空间中给定维数的子空间集合的流形。这种方法的吸引力在于子空间，作为格拉斯曼流形上的抽象点，是捕获数据观测中自然变化的有效工具，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某些规定的维数上相交，那么这些抽象点应该被认为比在更少维数上相交的子空间更相关，或者根本不相关。这种类型的几何图形，当在数学框架中形成时，导致使用标志流形和舒伯特变量来表示和比较数据。提出的研究计划解决了受几何约束的数据驱动优化中的新问题，例如，当可行集是Schubert变体时。该框架允许我们提取表征模式的几何模型，并自然地导致基于相似性度量的大型观察集之间的比较，相似性度量是子空间之间角度的函数。

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.