CAREER: Learning of graph diffusion and transport from high dimensional data with low-dimensional structures

基于图的方法是大数据分析中的关键工具，因为它们具有强大的建模能力，可以在科学和工业的各个领域进行数据建模。对于高维数据，可以从数据云构建亲和图，并且图几何将恢复数据的隐式低维结构。因此，基于图的方法有可能克服维数灾难，并为预测和生成学习任务提供无分布方法。该项目的总体目标是开发一个基于图的数据分析的理论和计算框架，通过利用数据中的底层低维几何结构来克服高维数据的维数灾难。数学结果可以应用于数据可视化和降维，生成模型，一般的无监督学习，以及广泛的真实的应用，从单细胞测序到传感器网络。该项目将提供适合研究生和本科生的研究机会和项目，该项目的结果将产生教学材料，以纳入本科和研究生阶段的数据科学课程。该项目旨在开发理论和计算工具，用于对高维数据进行有效和准确的基于图形的分析，以捕获数据中固有的低维非线性结构。研究工作包括四个集成的主题：（1）图扩散的学习与理论保证，（2）基于图的数据分析的鲁棒图亲和力，（3）基于图的高维内在最优传输的学习，和（4）梯度流的图数据生成模型。使用应用调和分析和高维概率的工具，该项目将解决该领域的几个开放性问题。在理论方面，该项目将把隐式低维结构建模为位于或接近嵌入高维空间的隐藏流形的数据，并分析图算子在大样本限制下的收敛性。在实践方面，该项目将开发采样和计算复杂性仅取决于内在数据维度的算法。数学发现将提供计算工具，以分析数据在真实的世界中的应用，包括生物医学和网络数据。这个奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Graph-based methods are pivotal tools in big data analysis due to their powerful ability to model data in various fields of science and industry. For high-dimensional data, an affinity graph can be constructed from the data cloud and the graph geometry will recover the implicit low-dimensional structure of the data. Therefore, a graph-based approach has the potential to overcome the curse of dimensionality and provide distribution-free methods for predictive and generative learning tasks. The overarching goal of this project is to develop a theoretical and computational framework for graph-based data analysis that overcomes the curse of dimensionality of high dimensional data by leveraging the underlying low-dimensional geometric structure in the data. The mathematical results can be applied to data visualization and dimension reduction, generative models, general unsupervised learning, and a wide range of real applications, ranging from single-cell sequencing to sensor networks. The project will provide research opportunities and projects that are suitable for graduate and undergraduate students, and results of the project will produce pedagogical materials to be incorporated into data science courses at the undergraduate and graduate levels. The project aims to develop theoretical and computational tools for efficient and accurate graph-based analysis of high-dimensional data that captures the intrinsically low-dimensional, non-linear structures in the data. The research work consists of four integrated topics: (1) learning of graph diffusion with a theoretical guarantee, (2) robust graph affinity for graph-based data analysis, (3) graph-based learning of intrinsic optimal transport in high dimension, and (4) generative model of graph data by gradient flow. Using tools from applied harmonic analysis and high dimensional probability, the project will address several open questions in the field. On the theoretical side, the project will model the implicit low-dimensional structure as data lying on or near hidden manifolds embedded in the high-dimensional space and analyze the convergence of the graph operators in the limit of large samples. On the practical side, the project will develop algorithms with sampling and computational complexities only depending on the intrinsic data dimensionality. The mathematical findings will provide computational tools to analyze data in real world applications, including biomedical and network data.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.