A Differential Privacy Mechanism Design Under Matrix-Valued Query
This work addresses a specific bottleneck in differential privacy for matrix-valued data, offering an incremental improvement over existing methods.
The paper tackles the problem of designing differential privacy mechanisms for matrix-valued queries, proposing a Matrix-Variate Gaussian (MVG) mechanism that adds matrix-valued noise to exploit structural characteristics, and demonstrates that it outperforms four previous state-of-the-art approaches while providing utility comparable to non-private baselines.
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 $(ε,δ)$-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.