Notes on the Linear Algebraic View of Regularity Lemmas

TL;DR

This paper presents a linear-algebraic perspective on regularity lemmas, developing the theory from scratch and deriving classical graph results through matrix sketching techniques.

Contribution

It introduces a linear-algebraic framework for regularity lemmas, providing a new foundational approach and accessible derivations of traditional graph regularity results.

Findings

Linear-algebraic approach to regularity lemmas

Derivation of graph regularity results from matrix sketches

Accessible development without prior regularity lemma knowledge

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.