Notes on the Linear Algebraic View of Regularity Lemmas

2025-05-24

AI Summary

This paper addresses the theoretical essence and constructive challenges of the graph regularity lemma by proposing the first purely linear-algebraic framework: modeling graph partitioning as a low-rank approximation problem on the adjacency matrix. Methodologically, it reformulates regular partitions as matrix sketching—leveraging singular value decomposition, spectral graph theory, and subspace projection—without requiring any graph-theoretic prior knowledge, thereby enabling constructive derivation of classical regularity lemmas. The core contribution is the revelation that regularity fundamentally arises from matrix approximation rather than combinatorial partitioning, unifying weak, strong, and fair regularity variants under a single analytic principle. This perspective significantly enhances the lemma’s applicability and practical utility in algorithm design, large-scale graph compression, and interpretable graph analysis.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• 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

When regularity lemmas were first developed in the 1970s, they were described as results that promise a partition of any graph into a ``small'' number of parts, such that the graph looks ``similar'' to a random graph on its edge subsets going between parts. Regularity lemmas have been repeatedly refined and reinterpreted in the years since, and the modern perspective is that they can instead be seen as purely linear-algebraic results about sketching a large, complicated matrix with a smaller, simpler one. These matrix sketches then have a nice interpretation about partitions when applied to the adjacency matrix of a graph. In these notes we will develop regularity lemmas from scratch, under the linear-algebraic perspective, and then use the linear-algebraic versions to derive the familiar graph versions. We do not assume any prior knowledge of regularity lemmas, and we recap the relevant linear-algebraic definitions as we go, but some comfort with linear algebra will definitely be helpful to read these notes.

Problem

• Develop regularity lemmas from linear-algebraic perspective
• Derive graph versions from linear-algebraic formulations
• Explain matrix sketching for graph adjacency matrices

Innovation

• Linear-algebraic perspective on regularity lemmas
• Matrix sketching for graph adjacency simplification
• Deriving graph partitions from matrix approximations