Extreme Points of the Local Differential Privacy Polytope

TL;DR

This paper characterizes the extreme points of the local differential privacy polytope, providing insights into the structure of privacy-preserving mechanisms and reducing complexity in their analysis.

Contribution

It offers a complete characterization of extreme points for matrices with specific numbers of non-zero columns, advancing understanding of local differential privacy mechanisms.

Findings

Characterization of extreme points for matrices with 1, 2, or n non-zero columns

Invariance properties and constraint reduction of the polytope

Structural insights into locally differentially private mechanisms

Abstract

We study the convex polytope of n x n stochastic matrices that define locally differentially private mechanisms. We first present invariance properties of the polytope and results reducing the number of constraints needed to define it. Our main results concern the extreme points of the polytope. In particular, we completely characterise these for matrices with 1, 2 or n non-zero columns.