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

BIGDATA：F：具有几何约束的标志流形的数据驱动优化

• 批准号: 1633830
• 负责人: Michael Kirby
• 金额: $ 60万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-15 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1633830&HistoricalAwards=false
• 关键词: BIGDATA; Data; Driven; Optimization; Flag

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.

这项研究涉及创新数学理论和算法的开发，以促进科学家、工程师和当今数据驱动社会生成的海量数据集中的知识发现。 将开发新的方法，允许以能够检测隐藏在大量信息中的相似性和差异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来提供有效的数据表示，从而扩展分析师的工具包并增强他们理解大型复杂数据集的能力。 该方法将在现实世界的数据集（例如极端天气模拟）或生物数据集（例如捕获人类对病原体感染的免疫反应的数据集）上进行验证。 正在开发的技术可以被视为几何数据学习新兴领域的一部分。该数学方法利用了格拉斯曼的几何框架，即参数化向量空间给定维度的子空间集的流形。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，这些自然变化例如源于照明或噪声的变化。 如果一个数据子空间与另一个数据子空间在某些规定的维度上相交，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。当在数学框架中表述这种类型的几何图时，会导致使用标志流形和舒伯特簇来表示和比较数据。 所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，当可行集是舒伯特簇时。该框架使我们能够提取表征模式的几何模型，并自然地基于相似性度量（子空间之间角度的函数）对大量观察结果进行比较。

项目成果

Sparse Locally Linear Embedding

• DOI: 10.1016/j.procs.2017.05.171
• 发表时间: 2017
• 作者: Lori Ziegelmeier;M. Kirby;C. Peterson

A Walk Through Spectral Bands: Using Virtual Reality to Better Visualize Hyperspectral Data

光谱带概览：利用虚拟现实更好地可视化高光谱数据

• 发表时间: 2019
• 期刊: Advances in intelligent systems and computing
• 作者: Henry Kvinge, Michael Kirby

Local eigenvalue decomposition for embedded Riemannian manifolds

• DOI: 10.1016/j.laa.2020.06.006
• 发表时间: 2020-11
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Javier Álvarez-Vizoso;M. Kirby;C. Peterson

Visualizing data sets on the Grassmannian using self-organizing mappings

• DOI: 10.1109/wsom.2017.8020003
• 发表时间: 2017-06
• 期刊: 2017 12th International Workshop on Self-Organizing Maps and Learning Vector Quantization, Clustering and Data Visualization (WSOM)
• 作者: M. Kirby;C. Peterson

Geometry of curves in Rn from the local singular value decomposition

• DOI: 10.1016/j.laa.2019.02.006
• 发表时间: 2019-06-15
• 期刊: LINEAR ALGEBRA AND ITS APPLICATIONS
• 影响因子: 1.1
• 作者: Alvarez-Vizoso, J.;Arn, Robert;Draper, Bruce
• 通讯作者: Draper, Bruce