Collaborative Research: Integrating Algebraic Topology, Graph Theory, and Multiscale Analysis for Learning Complex and Diverse Datasets

尽管机器学习和深度学习在过去十年中取得了巨大的成就，但结构复杂和多样化的数据仍然存在挑战。例如，用于药物设计的数据库中的单个数据点可能具有数万个内部自由度，而这样的数据库可能具有数万个这样的数据点。这种结构复杂性的特征是对深度学习方法的主要挑战。此外，不同的数据通常来自于一个巨大空间的稀疏采样，这种稀疏性主要是由于实验数据采集的成本和时间限制。该项目将通过融合和整合几个子领域的数学技术，包括代数拓扑、谱图理论和多尺度分析，来解决复杂和多样化数据集的挑战。开发的方法将适用于数据表示、先进的机器学习方法和深度学习算法，并将实现到社区可用的软件包中。该项目将培训研究生和本科生，并吸引数据科学研究中代表性不足的群体。该项目将开发新颖的基于拓扑和图论的方法，以彻底改变当前数据分析的实践，并处理结构复杂数据和多样化数据的挑战。首先，研究人员将发展持久组合图论，作为同时进行拓扑数据分析和光谱数据分析的统一范式。特别是，他们将开发系统的、可扩展的、精确的、持久的组合图表示来提取丰富的拓扑和光谱信息。其次，研究人员将开发多尺度图模型，以创建一系列嵌套子流形，以处理来自大空间中稀疏采样数据点的各种数据。这些方法将与先进的机器学习和深度学习算法相结合，用于复杂和多样化的数据集。第三，所提出的方法将广泛应用于数据科学的案例研究。用户友好的软件包和在线服务器将使用并行和GPU架构为没有受过数学或机器学习正规训练的研究人员开发。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。

Despite the tremendous accomplishments of machine learning and deep learning in the past decade, challenges remain for structurally complex and diverse data. For example, a single data point in a database used for drug design might have tens of thousands of internal degrees of freedom, and such a database may have tens of thousands of such data points. This feature of structural complexity is a major challenge to deep learning methods. Moreover, diverse data typically originate from sparse sampling of a huge space, and this sparsity is due, in particular, to the cost and time constraints in experimental data acquisition. This project will address the challenges of complex and diverse datasets with ideas that blend and integrate mathematical techniques from several subfields including algebraic topology, spectral graph theory and multiscale analysis. The methods developed will apply to data representation, advanced machine learning methods, and deep learning algorithms, and will be implemented into software packages available to the community. This project will train graduate and undergraduate students and engage underrepresented groups in data science research. This project will develop novel topology and graph theory-based approaches to revolutionize the current practice in data analysis and to deal with the challenge of structurally complex data and diverse data. First, the investigators will develop persistent combinatorial graph theory as a unified paradigm for simultaneous topological data analysis and spectral data analysis. In particular, they will develop systematic, scalable, accurate persistent combinatorial graph representations to extract rich topological and spectral information. Secondly, the investigators will develop multiscale graph models to create a family of nested submanifolds to handle the diverse data originated from sparsely sampled data points in a huge space. These methods will be integrated with advanced machine learning and deep learning algorithms for complex and diverse datasets. Thirdly, the proposed methods will be applied to a wide range of case studies in data science. User-friendly software packages and online servers will be developed using parallel and GPU architectures for researchers who are not formally trained in mathematics or machine learning.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.