Lossy Compression with Privacy Constraints: Optimality of Polar Codes

A lossy source coding problem with privacy constraint is studied in which two correlated discrete sources $X$ and $Y$ are compressed into a reconstruction $\hat{X}$ with some prescribed distortion $D$. In addition, a privacy constraint is specified as the equivocation between the lossy reconstruction $\hat{X}$ and $Y$. This models the situation where a certain amount of source information from one user is provided as utility (given by the fidelity of its reconstruction) to another user or the public, while some other correlated part of the source information $Y$ must be kept private. In this work, we show that polar codes are able, possibly with the aid of time sharing, to achieve any point in the optimal rate-distortion-equivocation region identified by Yamamoto, thus providing a constructive scheme that obtains the optimal tradeoff between utility and privacy in this framework.