On Perfect Privacy and Maximal Correlation
Borzoo Rassouli, Denız Gündüz · arXiv (Cornell University) · 2017
The problem of private data disclosure is studied from an information theoretic perspective. Considering a pair of correlated random variables $(X,Y)$, where $Y$ denotes the observed data while $X$ denotes the private latent variables, the following problem is addressed: What is the maximum information that can be revealed about $Y$, while disclosing no information about $X$? Assuming that a Markov kernel maps $Y$ to the revealed information $U$, it is shown that the maximum mutual information between $Y$ and $U$, i.e., $I(Y;U)$, can be obtained as the solution of a standard linear program, when $X$ and $U$ are required to be independent, called \textit{perfect privacy}. This solution is shown to be greater than or equal to the \textit{non-private information about $X$ carried by $Y$.} Maximal information disclosure under perfect privacy is is shown to be the solution of a linear program also when the utility is measured by the reduction in the mean square error, $\mathbb{E}[(Y-U)^2]$, or the probability of error, $\mbox{Pr}\{Y eq U\}$. For jointly Gaussian $(X,Y)$, it is shown that perfect privacy is not possible if the kernel is applied to only $Y$; whereas perfect privacy can be achieved if the mapping is from both $X$ and $Y$; that is, if the private latent variables can also be observed at the encoder. Next, measuring the utility and privacy by $I(Y;U)$ and $I(X;U)$, respectively, the slope of the optimal utility-privacy trade-off curve is studied when $I(X;U)=0$. Finally, through a similar but independent analysis, an alternative characterization of the maximal correlation between two random variables is provided.