Differentially Private Topological Data Analysis

This paper is the first to attempt differentially private (DP) topological data analysis (TDA), producing near-optimal private persistence diagrams. We analyze the sensitivity of persistence diagrams in terms of the bottleneck distance, and we show that the commonly used & Ccaron;ech complex has sensitivity that does not decrease as the sample size n increases. This makes it challenging for the persistence diagrams of & Ccaron;ech complexes to be privatized. As an alternative, we show that the persistence diagram obtained by the L 1 -distance to measure (DTM) has sensitivity O (1 /n ) . Based on the sensitivity analysis, we propose using the exponential mechanism whose utility function is defined in terms of the bottleneck distance of the L 1 -DTM persistence diagrams. We also derive upper and lower bounds of the accuracy of our privacy mechanism; the obtained bounds indicate that the privacy error of our mechanism is near-optimal. We demonstrate the performance of our privatized persistence diagrams through simulations as well as on a real data set tracking human movement.