Differentially private depth functions and their associated medians

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In this paper, we investigate the differentially private estimation of data depth functions and their associated medians.

• We introduce several methods for privatizing depth values at a fixed point, and show that for some depth functions, when the depth is computed at an out of sample point, privacy can be gained for free when $n\rightarrow \infty$.
• We also present a method for privately estimating the vector of sample point depth values.
• Additionally, we introduce estimation methods for depth-based medians for both depth functions with low global sensitivity and depth functions with only highly probable, low local sensitivity.