Publishing Location Dataset Differential Privately with Isotonic Regression

We consider the problem of publishing location datasets, in particular 2D spatial pointsets, in a differentially private manner. Many existing mechanisms focus on frequency counts of the points in some a priori partition of the domain that is difficult to determine. We propose an approach that adds noise directly to the point, or to a group of neighboring points. Our approach is based on the observation that, the sensitivity of sorting, as a function on sets of real numbers, can be bounded. Together with isotonic regression, the dataset can be accurately reconstructed. To extend the mechanism to higher dimension, we employ locality preserving function to map the dataset to a bounded interval. Although there are fundamental limits on the performance of locality preserving functions, fortunately, our problem only requires distance preservation in the "easier" direction, and the well-known Hilbert space-filling curve suffices to provide high accuracy. The publishing process is simple from the publisher's point of view: the publisher just needs to map the data, sort them, group them, add Laplace noise and publish the dataset. The only parameter to determine is the group size which can be chosen based on predicted generalization errors. Empirical study shows that the published dataset can also exploited to answer other queries, for example, range query and median query, accurately.