A Differential Privacy Mechanism Design Under Matrix-Valued Query

TL;DR

This paper introduces the Matrix-Variate Gaussian (MVG) mechanism for differential privacy tailored to matrix-valued queries, exploiting matrix structure for improved privacy-utility trade-offs.

Contribution

It proposes the MVG mechanism using matrix-variate Gaussian noise, preserving differential privacy and leveraging matrix structure, with the concept of directional noise to enhance utility.

Findings

MVG mechanism outperforms four state-of-the-art methods.

MVG provides utility comparable to non-private baseline.

Directional noise reduces utility loss.

Abstract

Traditionally, differential privacy mechanism design has been tailored for a scalar-valued query function. Although many mechanisms such as the Laplace and Gaussian mechanisms can be extended to a matrix-valued query function by adding i.i.d. noise to each element of the matrix, this method is often sub-optimal as it forfeits an opportunity to exploit the structural characteristics typically associated with matrix analysis. In this work, we consider the design of differential privacy mechanism specifically for a matrix-valued query function. The proposed solution is to utilize a matrix-variate noise, as opposed to the traditional scalar-valued noise. Particularly, we propose a novel differential privacy mechanism called the Matrix-Variate Gaussian (MVG) mechanism, which adds a matrix-valued noise drawn from a matrix-variate Gaussian distribution. We prove that the MVG mechanism preserves $(\epsilon,\delta)$-differential privacy, and show that it allows the structural characteristics of the matrix-valued query function to naturally be exploited. Furthermore, due to the multi-dimensional nature of the MVG mechanism and the matrix-valued query, we introduce the concept of directional noise, which can be utilized to mitigate the impact the noise has on the utility of the query. Finally, we demonstrate the performance of the MVG mechanism and the advantages of directional noise using three matrix-valued queries on three privacy-sensitive datasets. We find that the MVG mechanism notably outperforms four previous state-of-the-art approaches, and provides comparable utility to the non-private baseline. Our work thus presents a promising prospect for both future research and implementation of differential privacy for matrix-valued query functions.