Covariance loss, Szemeredi regularity, and differential privacy

TL;DR

This paper introduces a novel approach using randomized rounding and Grothendieck's identity to bound covariance loss, leading to a weak Szemeredi regularity lemma for matrices and applications in differentially private data synthesis.

Contribution

It presents a new method for bounding covariance loss and deriving a regularity lemma, with implications for differential privacy and matrix analysis.

Findings

Bound on covariance loss using randomized rounding

A new weak Szemeredi regularity lemma for positive semidefinite matrices

Application to differentially private synthetic data generation

Abstract

We show how randomized rounding based on Grothendieck's identity can be used to prove a nearly tight bound on the covariance loss--the amount of covariance that is lost by taking conditional expectation. This result yields a new type of weak Szemeredi regularity lemma for positive semidefinite matrices and kernels. Moreover, it can be used to construct differentially private synthetic data.