FRG: Collaborative Research: Mathematical and Statistical Analysis of Compressible Data on Compressive Networks

大规模高维数据集在现代社会中变得无处不在，特别是在物理、生物医学和社会应用领域。这个重点研究小组（FRG）将通过利用高维数据的可压缩特性来解决高维数据分析中出现的计算和理论基础挑战。发现这种可压缩特征是数据分析中的一个主要挑战，研究团队将使用从数据的光谱、统计和代数几何分析中得出的分层分解方法来解决这个问题。与基于插值的方法（如深度神经网络）相比，深度神经网络通常难以解释，该团队将构建最佳定义的压缩网络，专门针对此类可压缩特征进行定制。这样做将能够以固有的可解释的方式准确有效地提取和操作高维数据的稀疏表示。例如，该项目的一个重点是扩展该小组成员开发的二进制展开测试方法，这些方法在低维环境下的统计能力和计算复杂性方面都显示出前景。二元展开测试的高维泛化反过来又使其能够直接应用于基于日益复杂的数据集选择个性化医疗计划。FRG的研究人员将在数学分析、数据科学、统计学和计算等学科之间以及跨机构之间进行合作。该项目的具体目标包括利用谱理论、代数几何、能量和优化以及网络相互作用的思想推广“可压缩”特征的经典概念。这将导致对压缩网络上的可压缩高维数据集的数学和统计基础有更深的理解。利用新开发的可压缩特性，FRG团队将为大规模高维数据集设计和开发准确高效的计算工具。所有要做的工作都旨在与应用领域的科学家直接合作，以提高所提出方法的有效性。FRG研究人员还将联合指导研究生和本科生，他们将受益于跨学科培训，并获得互补和综合研究领域的各种思想和工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Large-scale high-dimensional data sets are becoming ubiquitous in modern society, particularly in the areas of physical, biomedical, and social applications. This focused research group (FRG) will address the foundational challenges, both computational and theoretical, arising in the analysis of high-dimensional data by leveraging its compressible features. Discovering such compressible features is a major challenge in data analysis, which the team of investigators will approach using hierarchical decompositions derived from spectral, statistical, and algebraic geometric analysis of data. In contrast to interpolation-based methods, such as deep neural networks which are often difficult to interpret, the group will construct optimally defined compressive networks, specifically tailored to such compressible features. Doing so will enable an accurate and efficient extraction and manipulation of sparse representations of high-dimensional data in an inherently interpretable manner. For instance, one focus of the project is to extend the binary expansion testing methods developed by members of the group, which have shown promise in both statistical power and computational complexity in low-dimensional settings. A high-dimensional generalization of binary expansion testing would, in turn, enable the direct application to selecting personalized medical treatment plans based on increasingly complex data sets. The FRG investigators will collaborate across the disciplines of mathematical analysis, data science, statistics, and computation, as well as across institutions. The specific goals of this project include generalizing classical concepts of "compressible" features using ideas from spectral theory, algebraic geometry, energy and optimization, and network interactions. This will lead to a deeper understanding of the mathematical and statistical foundations of compressible high-dimensional data sets on compressive networks. Using newly developed compressible features, the FRG team will then design and develop accurate and efficient computational tools for large-scale high-dimensional data sets. All the work to be done will be aimed at collaborating directly with application domain scientists to enhance the efficacy of the proposed methods. The FRG investigators will also jointly mentor graduate and undergraduate students, who will then have the benefits of training across disciplines and access to a variety of ideas and tools in complementary and integrative research areas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.