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，以促进科学家，工程师和当今数据驱动的社会产生的海量数据集的知识发现。 将开发新的方法，允许对大量数据进行编码，以便能够发现隐藏在大量信息中的相似性和差异性。 该框架对于表征相似度以及发现数据集之间可能共享的特征或模式特别有用。该项目的重点是使用几何和优化工具，提供有效的数据表示，扩大分析人员的工具包，提高他们理解大型复杂数据集的能力。 该方法将在真实的世界数据集上得到验证，如极端天气模拟或生物数据集，如捕获人体对病原体感染的免疫反应的数据集。 正在开发的技术可以被看作是新兴的几何数据学习领域的一部分。数学方法利用几何框架的格拉斯曼，流形参数化的一组子空间的一个给定的维度的向量空间。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，是捕获数据观测中的自然变化的有效工具，例如，照明或噪声的变化。 如果一个数据子空间与另一个数据子空间在某个规定的维度上相交，那么这些抽象点应该被认为比在更少维度上相交的子空间更相关，或者根本不相关。当在数学框架中制定这种类型的几何图像时，会导致使用旗形流形和舒伯特变体来表示和比较数据。 建议的研究计划解决了新的问题，在数据驱动的优化几何约束，例如，当可行集是舒伯特品种。这个框架使我们能够提取几何模型的特征模式，并自然导致大型观测集之间的比较，基于相似性的措施，这是子空间之间的角度的函数。

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.