A Maximum Entropy Implementation of Differential Privacy Under Linear Invariants

2026-07-24

AI Summary

This work addresses the challenge that conventional differential privacy mechanisms struggle to simultaneously preserve privacy and enforce hard linear invariants—such as aggregate statistical constraints—due to difficulties in managing noise correlations. The authors propose a novel differential privacy mechanism grounded in the principle of maximum entropy, which injects correlated noise while strictly satisfying given linear constraints. The resulting aggregates obey these constraints with probability one or with exponentially high probability, and the privacy guarantee is rigorously re-established under this constrained setting. This approach constitutes the first mechanism that jointly achieves linear invariance and high-entropy noise, partially resolves an open theoretical problem concerning the null space of correlation matrices, and extends sampling techniques for constrained Gaussian mixture models. It thereby provides a general, provably secure framework for constrained statistical data release.

Abstract

Differential privacy is the standard for ensuring data privacy and is widely used in major data publications, including reporting results from the U.S. decennial census. Common implementation of differential privacy uses independent Gaussian or Laplace noise addition to the database. However, there could be aggregate (linear) queries to the database that are excluded from the privacy budget, for example, state totals that can not be perturbed due to constitutional mandates. Any implementation of a differential privacy is required to honor these constraints, also referred to as invariants. Under aggregation constraints, the noise vector is no longer independent and the traditional differential privacy guarantees have to be re-evaluated. We propose a high entropy differential privacy implementation that maintains the aggregation invariants with probability one or exponentially close to one and derive the privacy guarantees for the implementation under the invariants. The theoretical proof covers a partial solution to an open question about the null space of correlation matrices. Moreover, the methodology has general use in the context of sampling from normal mixture models under linear equality constraints.

Problem

• differential privacy
• linear invariants
• aggregation constraints
• privacy guarantees
• noise correlation

Innovation

• differential privacy
• maximum entropy
• linear invariants
• constrained sampling
• correlation matrix null space