Mathematical and computational perspectives on the Boolean and binary rank and their relation to the real rank

2026-01-20

AI Summary

This study systematically investigates the mathematical structure and computational complexity of Boolean rank and binary rank, as well as their relationship to real rank. By integrating key techniques from linear algebra, combinatorics, and graph theory—alongside tools such as fooling sets, probabilistic methods, kernelization, communication protocols, and query-to-communication lifting—the work provides a deep analysis of their interdisciplinary applications in communication complexity, parameterized algorithms, and approximate matrix factorization. The paper clarifies existing theoretical boundaries, establishes nontrivial upper and lower bounds for several matrix families, and surveys recent algorithmic advances in Boolean matrix factorization, thereby charting a clear path for future research.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Applications

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

This survey provides a comprehensive overview of the study of the binary and Boolean rank from both a mathematical and a computational perspective, with particular emphasis on their relationship to the real rank. We review the basic definitions of these rank functions and present the main alternative formulations of the binary and Boolean rank, together with their computational complexity and their deep connection to the field of communication complexity. We summarize key techniques used to establish lower and upper bounds on the binary and Boolean rank, including methods from linear algebra, combinatorics and graph theory, isolation sets, the probabilistic method, kernelization, communication protocols and the query to communication lifting technique. Furthermore, we highlight the main mathematical properties of these ranks in comparison with those of the real rank, and discuss several non-trivial bounds on the rank of specific families of matrices. Finally, we present algorithmic approaches for computing and approximating these rank functions, such as parameterized algorithms, approximation algorithms, property testing and approximate Boolean matrix factorization (BMF). Together, the results presented outline the current theoretical knowledge in this area and suggest directions for further research.

Problem

• Boolean rank
• binary rank
• real rank
• matrix rank
• communication complexity

Innovation

• Boolean rank
• binary rank
• communication complexity
• matrix factorization
• computational complexity