  When collecting information, local differential privacy (LDP) relieves the
concern of privacy leakage from users' perspective, as user's private
information is randomized before sent to the aggregator. We study the problem
of recovering the distribution over a numerical domain while satisfying LDP.
While one can discretize a numerical domain and then apply the protocols
developed for categorical domains, we show that taking advantage of the
numerical nature of the domain results in better trade-off of privacy and
utility. We introduce a new reporting mechanism, called the square wave SW
mechanism, which exploits the numerical nature in reporting. We also develop an
Expectation Maximization with Smoothing (EMS) algorithm, which is applied to
aggregated histograms from the SW mechanism to estimate the original
distributions. Extensive experiments demonstrate that our proposed approach, SW
with EMS, consistently outperforms other methods in a variety of utility
metrics.
