Bridging Differential Privacy and Random Triangles

2026-08-02

AI Summary

This work transcends the limitations of traditional scalar privacy loss by adopting a geometric perspective to precisely characterize the random triangular structures induced by high-dimensional noise perturbations in differential privacy. By constructing two complementary geometric representations—mapping the random triangle formed by sensitivity and noise vectors onto a simplex and a hemisphere—it establishes, for the first time, an exact geometric connection between differential privacy and probabilistic shape analysis. Through rigorous derivations of probability densities, geometric mappings, spectral shape analysis, and high-dimensional asymptotic methods, the study reformulates classical privacy loss variables, revealing an elliptical support structure on the simplex and phenomena of equatorial drift and zonal concentration on the hemisphere. These insights offer a novel geometric framework for designing privacy-preserving mechanisms.

Abstract

The classical analysis of the Gaussian mechanism in differential privacy reduces privacy loss for a pair of neighboring datasets to a scalar random variable. While this scalar characterization is sufficient for privacy accounting, each perturbation instance also induces a high-dimensional random triangle formed by the sensitivity vector and the two corresponding noise vectors. In this work, we develop two complementary geometric representations of these random triangles. The first representation maps the normalized squared edge lengths to a simplex. We derive its exact joint density, characterize its elliptical support, and reconstruct the classical privacy loss random variable from the simplex coordinates. The second representation maps the spectral shape of each normalized triangle to a hemisphere. We derive the corresponding density and coordinate mappings, recover the same privacy loss, and characterize an equatorial drift together with band concentration as the dimension increases. These results provide two exact geometric coordinate systems that complement the scalar privacy loss and connect differential privacy with the probabilistic analysis of random shapes.

Problem

• differential privacy
• random triangles
• Gaussian mechanism
• privacy loss
• geometric representation

Innovation

• differential privacy
• random triangles
• geometric representation
• privacy loss
• Gaussian mechanism