COVARIANCE LOSS, SZEMEREDI REGULARITY, AND DIFFERENTIAL PRIVACY

Area of Science:

• Mathematics
• Computer Science
• Statistics

Background:

• Covariance loss quantifies information reduction in conditional expectation.
• Randomized rounding is a technique used in approximation algorithms.
• Szemeredi regularity lemmas are fundamental in graph theory and combinatorics.

Purpose of the Study:

• To establish a nearly tight bound on covariance loss using randomized rounding.
• To introduce a novel weak Szemeredi regularity lemma for positive semidefinite matrices and kernels.
• To explore the application of this method in constructing differentially private synthetic data.

Main Methods:

• Application of randomized rounding based on Grothendieck's identity.
• Development of a new weak Szemeredi regularity lemma.
• Utilizing the lemma for synthetic data generation.

Main Results:

• A nearly tight bound on covariance loss was proven.
• A new type of weak Szemeredi regularity lemma for positive semidefinite matrices and kernels was established.
• The method was shown to be applicable for creating differentially private synthetic data.

Conclusions:

• Randomized rounding provides a powerful tool for analyzing covariance loss.
• The developed regularity lemma offers new theoretical insights.
• The approach facilitates the creation of privacy-preserving synthetic datasets.