A Cryptographic View of Regularity Lemmas: Simpler Unified Proofs and Refined Bounds

Abstract

In this work we present a short and unified proof for the Strong and Weak Regularity Lemma, based on the cryptographic technique called low-complexity approximations. In short, both problems reduce to a task of finding constructively an approximation for a certain target function under a class of distinguishers (test functions), where distinguishers are combinations of simple rectangle-indicators. In our case these approximations can be learned by a simple iterative procedure, which yields a unified and simple proof, achieving for any graph with density d and any approximation parameter \(\epsilon \) the partition size a tower of 2’s of height \(O\left( d_{}\epsilon ^{-2} \right) \) for a variant of Strong Regularity a power of 2 with exponent \(O\left( d\epsilon ^{-2} \right) \) for Weak Regularity