  Local differential privacy (LDP), which perturbs the data of each user
locally and only sends the noisy version of her information to the aggregator,
is a popular privacy-preserving data collection mechanism. In LDP, the data
collector could obtain accurate statistics without access to original data,
thus guaranteeing privacy. However, a primary drawback of LDP is its
disappointing utility in high-dimensional space. Although various LDP schemes
have been proposed to reduce perturbation, they share the same and naive
aggregation mechanism at the side of the collector. In this paper, we first
bring forward an analytical framework to generally measure the utilities of LDP
mechanisms in high-dimensional space, which can benchmark existing and future
LDP mechanisms without conducting any experiment. Based on this, the framework
further reveals that the naive aggregation is sub-optimal in high-dimensional
space, and there is much room for improvement. Motivated by this, we present a
re-calibration protocol HDR4ME for high-dimensional mean estimation, which
improves the utilities of existing LDP mechanisms without making any change to
them. Both theoretical analysis and extensive experiments confirm the
generality and effectiveness of our framework and protocol.
