CAREER: Learning from Data on Structured Complexes: Products, Bundles, and Limits

人工智能（AI）在涉及文本、音频和图像形式的数据的各种任务中表现出了令人印象深刻的表现，例如人脸识别和文本摘要。然而，来自许多知识领域的数据可以在不太传统的领域中自然地定义，例如在球体（代表地球）上定义的气候数据和在道路网络上定义的交通数据。因此，在过去的几年里，人工智能已经扩展到这些设置，通常是通过将这些域表示为图形。然而，设计用于任何图形的人工智能工具必然会忽略感兴趣的特定图形的结构。例如，随时间变化的交通数据可以用结构化图表示，该图结合了底层道路网络和时间的线性演变。在这种观点的激励下，研究人员将把重点放在图的类别上，这些图的附加结构属性既具有实际意义（即，它们代表了现实世界的数据域），又具有方法上的优势（即，它们可以被利用来更好地从数据中学习）。更广泛地说，该项目旨在通过利用现实世界数据中常见的结构属性来提高人工智能的效用。这项研究将通过为高中、本科生和研究生开发课程和教学模块，纳入教育活动，并包括扩大拉丁美洲裔学生的参与活动。该项目的主要研究目标是开发一种原则性理论来处理和学习结构化（高阶）网络上定义的数据。特别是，研究人员将集中在图和（简单和细胞）复合体的三种类型的结构上。首先，将考虑产品复合体，其中数据域可以分解为两个（或更多）组成的更简单域的乘积。通过将离散霍奇理论专门应用于该数据结构，将推导出具有可转移性保证的新型神经结构。其次，将研究图束，它不是全局可分解的，而是呈现局部产品结构。在这种情况下，重点将是通过设计信号处理转换来改进数据表示，这些转换可以自然地处理bundle的不可面向特性。最后，研究人员将分析简单复合体随着节点数量增长的极限。这里的目标是利用这些限制对象的规律性来有效地设计大规模关系域的神经结构。从理论的角度来看，该项目独特地结合了图论，代数拓扑，信号处理，深度学习和线性积分算子的概念，以获得对结构化领域学习的基本理解。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。

Artificial intelligence (AI) has shown impressive performance in a variety of tasks involving data in the form of text, audio, and images, such as face recognition and text summarization. However, data from many fields of knowledge can be naturally defined over less conventional domains, such as climate data defined on spheres (representing the Earth) and traffic data defined on road networks. Consequently, over the past few years, AI has been extended to these settings, oftentimes by representing these domains as graphs. However, AI tools designed to be implemented in any graph necessarily disregard the structure of specific graphs of interest. For example, traffic data that changes over time can be represented on a structured graph that combines the underlying road network with the linear evolution of time. Motivated by this view, the investigators will focus on classes of graphs whose additional structural properties are both practically relevant (i.e., they represent real-world data domains) and methodologically advantageous (i.e., they can be exploited to better learn from data). More broadly, this project will aim to boost the utility of AI by leveraging structural properties often found in real-world data. The research will be integrated into educational activities by development of courses and teaching modules for high school, undergraduate, and graduate students and includes broadening participation activities with students of Latin American origin. The primary research goal of this project is to develop a principled theory to process and learn from data defined on structured (higher-order) networks. In particular, the investigators will focus on three types of structures for graphs and (simplicial and cell) complexes. First, product complexes will be considered, where the data domain can be decomposed as the product of two (or more) constituent simpler domains. By specializing discrete Hodge theory to this data structure, novel neural architectures with transferability guarantees will be derived. Second, graph bundles will be studied, which are not globally decomposable but present a local product-like structure. In this case, the focus will be to improve data representation by designing signal processing transformations that can naturally handle the non-orientable nature of bundles. Lastly, the researchers will analyze the limits of simplicial complexes as the number of nodes grows. The objective here is to leverage the regularity of these limiting objects to efficiently design neural architectures for large-scale relational domains. From a theoretical standpoint, this project uniquely combines concepts from graph theory, algebraic topology, signal processing, deep learning, and linear integral operators to derive a fundamental understanding of learning in structured domains.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.