Advanced Models and Algorithms for Large-Scale High-Dimensional Probabilistic Graph Structure Learning

随着科学技术的快速发展，几乎所有科学和工程领域都已经并正在收集大量数据用于各种实际目的，例如用于提高人类对疾病和治疗方法的认识以挽救生命的医疗数据，来自太阳系和人类下一个太空前沿的图像数据，以及用于更好地了解社会和经济发展的社交网络数据。但要实现这些目标，必须对数据进行合理的分析，以揭示重要的内容。虽然数据分析已经存在了几个世纪，但今天的数据在数量和维度上都要大得多，而且更加复杂，这给现代数据分析带来了众所周知的挑战。通常，真实的世界数据是有噪声的，并且隐藏了固有的隐藏结构，这些隐藏结构可以由使用节点表示对象/事件并且使用边表示节点之间的关系的图来简洁地表示。现有的方法依赖于预先和事后可构建的图，表明它们无法处理当今复杂的数据。该项目旨在通过为科学家，工程师和医学专业人员开发新颖的数学模型和高效的计算工具来改变现状，他们可以使用模型和工具来挖掘隐藏的结构，以实现以前被认为是不可能的科学发现。主要研究人员将把本项目的研究活动与教学和教育相结合，并将在计算数学、数据科学和跨学科研究方面对本科生和研究生进行培训。拟议的研究将产生基于图的机器学习的先进模型和高效算法。与现有的基于图的学习方法的两个主要区别是：（1）新模型具有内置的概率组件，可以鲁棒地处理高噪声数据，以及（2）动态图结构学习组件，可以发现隐藏在真实的世界数据中的隐藏图结构，但还不够明显，无法预先或预先构建。这些模型比现有的基于图的学习方法具有更广泛的适用性，因为图结构现在是一个变量，将被优化，以便为给定的数据集产生最佳的隐藏图结构，并且算法不仅能够产生鲁棒的嵌入，同时学习隐藏图结构，而且还将通过地标和低秩矩阵近似策略对大数据进行实用化。拟议的研究将在分析高维数据集发挥至关重要作用的领域产生潜在的科学影响，例如数据可视化，发现计算生物学，大脑网络和其他领域的结构模式。该项目将在统计机器学习领域开辟一个新的研究方向。该奖项反映了NSF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。

With the rapid development of science and technology, vast amount of data has been and is being collected in nearly all fields of science and engineering for various practical purposes, such as medical data for advancing human knowledge in diseases and treatments to save lives, image data from the solar system and beyond for human's next frontier in space, and social network data for better understanding the society and economic developments. But to make these aims realities, data must be soundly analyzed to uncover what matters. While data analysis has been around for centuries, today's data is much bigger in amount and dimension and more complex, presenting notorious challenges to modern data analysis. Often real world data is noisy and conceals inherent hidden structures that can be concisely represented by graphs that use nodes for objects/events and edges for relations between nodes. Existing approaches rely on pre- and heuristically constructible graphs show their inability in handling nowadays complicated data. This project aims to change the status quo by developing novel mathematical models and efficient computational tools for scientists, engineers, and medical professionals who can use the models and tools to unearth the hidden structures to achieve scientific discoveries previously considered impossible. The principle investigators will integrate their research activities of this project with teaching and education, and will train undergraduate and graduate students in computational mathematics, data science, and interdisciplinary studies.The proposed research will result in advanced models and efficient algorithms for graph-based machine learning. Two major distinctions from existing graph-based learning methods are (1) new models have a built-in probabilistic component that can robustly deal with high noisy data, and (2) a dynamic graph structure learning component that can uncover hidden graph structures concealed in real world data and yet not obvious enough to be pre- or heuristically constructed. The models have much wider applicability than existing graph-based learning methods because graph structure is now a variable that will be optimized over so as to yield an optimal hidden graph structure for a given data set, and the algorithms not only are capable of producing robust embeddings with simultaneously learned hidden graph structures but also will be made practical for big data through landmark and low-rank matrix approximation strategies. The proposed research will have potentially high impacts scientifically in areas where analyzing high-dimensional datasets plays critically important roles, such as data visualization, discovering structural patterns in computational biology, brain networks and other areas. The project will open up a new research direction in statistical 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.