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

William Marsh Rice University, Houston TX

Investigators

Abstract

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. 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.