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Currently known methods for this task either employ the computationallyintensive \emph{exponential mechanism} or require an access to the covariancematrix, and therefore fail to utilize potential sparsity of the data. Theproblem of designing simpler and more efficient methods for this task has beenraised as an open problem in \cite{kapralov2013differentially}.  In this paper we address this problem by employing the output perturbationmechanism. Despite being arguably the simplest and most straightforwardtechnique, it has been overlooked due to the large \emph{global sensitivity}associated with publishing the leading eigenvector. We tackle this issue byadopting a \emph{smooth sensitivity} based approach, which allows us toestablish differential privacy (in a worst-case manner) and near-optimal samplecomplexity results under eigengap assumption. We consider both the pure and theapproximate notions of differential privacy, and demonstrate a tradeoff betweenprivacy level and sample complexity. We conclude by suggesting how our resultscan be extended to related problems.

Smooth Sensitivity Based Approach for Differentially Private Principal Component Analysis