Differentially Private Detection of Sparse Gaussian Mixtures
Ruizhi Zhang, Shanshan Cao · 2023
Detection of sparse signals arises in many modern applications such as signal processing, bioinformatics, finance, and disease surveillance. However, in many of these applications, the data may contain sensitive personal information, which is desirable to be protected during the data analysis. In this article, we consider the problem of ϵ-differentially private detection of the sparse Gaussian mixtures with the focus on how privacy affects the detection power. We first study the private likelihood ratio test and show it achieves vanishing error probabilities in the detectable region proposed by Ingster [1] and Donoho and Jin [2] when ϵ > n--β. Moreover, we propose an adaptive ϵ-differentially private test, which achieves vanishing error probabilities in the same detectable region when $\sqrt {\log (\log n)} \varepsilon \to \infty $. Several numerical experiments are conducted to verify our theoretical results and illustrate the performance of our proposed test.