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Abstract:
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We examine the use of elliptical distributions for obtaining differentially private statistical estimates in a locally convex vector space, which includes most finite and infinite dimensional linear spaces encountered in statistics. We show that many properties established for Gaussian distributions extend readily to more general elliptical families. However, we also highlight some key breakdowns as one moves to infinite dimensional spaces. In particular, while elliptical distributions can achieve epsilon privacy in finite dimensions, we show that only a weaker form of privacy can be achieved in infinite dimensions. In finite dimensions, we also show that the privacy parameter is tied to a relatively simple one dimensional optimization problem so that the tools can be readily implemented in practice.
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