Document Understanding, Measurement, and Manipulation Using Category Theory
Jared Claypoole, Yunye Gong, Noson S. Yanofsky, Ajay Divakaran
2025-10-27
Summary
This research uses advanced math, specifically category theory, to understand and improve how we work with documents containing different types of information like text and images. It aims to automatically figure out a document's structure, summarize it effectively, and even expand upon the original content, all while making large AI models better at understanding information.
What's the problem?
Currently, it's difficult for computers to truly *understand* the structure of complex documents, especially those with multiple types of data. Existing methods for summarizing and extending documents often miss important nuances or don't guarantee the new content logically fits with the original. Also, large AI models, while powerful, can still be improved in their ability to consistently process and interpret information.
What's the solution?
The researchers treated a document as a collection of questions and answers, using category theory to mathematically represent this relationship. They then developed a way to break down the information in a document into distinct, non-overlapping parts. This allowed them to measure the amount of information and create better summarization techniques. They also tackled the problem of 'exegesis' – expanding a document in a meaningful way – and used a reinforcement learning method called RLVR to fine-tune large AI models, ensuring they follow logical rules derived from their mathematical framework.
Why it matters?
This work is important because it provides a more rigorous and mathematically sound approach to document understanding and processing. By improving summarization and content extension, it can help people quickly grasp the key ideas in complex materials and even generate new insights. Furthermore, the self-supervised learning method offers a way to make large AI models more reliable and consistent, leading to better performance in various applications like information retrieval and content creation.
Abstract
We apply category theory to extract multimodal document structure which leads us to develop information theoretic measures, content summarization and extension, and self-supervised improvement of large pretrained models. We first develop a mathematical representation of a document as a category of question-answer pairs. Second, we develop an orthogonalization procedure to divide the information contained in one or more documents into non-overlapping pieces. The structures extracted in the first and second steps lead us to develop methods to measure and enumerate the information contained in a document. We also build on those steps to develop new summarization techniques, as well as to develop a solution to a new problem viz. exegesis resulting in an extension of the original document. Our question-answer pair methodology enables a novel rate distortion analysis of summarization techniques. We implement our techniques using large pretrained models, and we propose a multimodal extension of our overall mathematical framework. Finally, we develop a novel self-supervised method using RLVR to improve large pretrained models using consistency constraints such as composability and closure under certain operations that stem naturally from our category theoretic framework.