Differentially Private Precision Matrix Estimation
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Differentially Private Precision Matrix Estimation Wen Qing SU
Xiao GUO
School of Mathematics, Northwest University, Xi’an 710127, P. R. China E-mail : [email protected] [email protected]
Hai ZHANG1) School of Mathematics, Northwest University, Xi’an 710127, P. R. China and Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, P. R. China and Key Laboratory of Advanced Theory and Application in Statistics and Data Science (East China Normal University), Ministry of Education, Shanghai, 200062, P. R. China E-mail : [email protected] Abstract In this paper, we study the problem of precision matrix estimation when the dataset contains sensitive information. In the differential privacy framework, we develop a differentially private ridge estimator by perturbing the sample covariance matrix. Then we develop a differentially private graphical lasso estimator by using the alternating direction method of multipliers (ADMM) algorithm. Furthermore, we prove theoretical results showing that the differentially private ridge estimator for the precision matrix is consistent under fixed-dimension asymptotic, and establish a convergence rate of differentially private graphical lasso estimator in the Frobenius norm as both data dimension p and sample size n are allowed to grow. The empirical results that show the utility of the proposed methods are also provided. Keywords
Differential privacy, graphical model, ADMM algorithm
MR(2010) Subject Classification
1
62F30
Introduction
Precision matrix plays a fundamental role in many statistical inference problems. For example, in discriminant analysis, the precision matrix needs to be estimated to compute the classification rules [24]. In graphical models, the structure exploration of Gaussian graphical model is equivalent to recovering the support of the precision matrix [20]. Moreover, the precision matrix is useful for a wide range of applications including portfolio optimization, genomics and single processing, among many others. Therefore, it is of great importance to estimate the precision matrix. Received September 4, 2019, revised January 18, 2020, accepted April 8, 2020 Supported by National Natural Science Foundation of China (Grant Nos. 11571011 and U1811461), the Open Research Fund of KLATASDS-MOE, the Fundamental Research Funds for the Central Universities 1) Corresponding author
Su W. Q. et al.
1108
Given a p × n data matrix X, where each column is sampled from a p-dimensional Gaussian distribution with mean zero and covariance matrix Σ0 . A natural way to estimate precision matrix Θ0 = Σ−1 0 is via maximum likelihood approach. Under the Gaussian setting, the negative log-likelihood takes the form − log |Θ| + tr(Σn Θ), (1.1) where Σn = XX T /n is the sample covariance matrix and tr(·) denotes the trace of the matrix. Then minimizing (1.1) with respect to Θ yields the maximum likelihood estimate of the precision matrix
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