Almost Perfect Privacy for Additive Gaussian Privacy Filters
This work addresses privacy protection in data sharing scenarios, particularly for applications involving correlated variables, but it is incremental as it builds on existing information-theoretic frameworks.
The paper tackles the problem of maximizing mutual information about non-private data Y through an additive Gaussian channel while limiting information leakage about correlated private data X to ε bits, showing that perfect privacy (ε=0) yields zero mutual information and deriving a second-order approximation for small ε. It also formulates a privacy-utility tradeoff using estimation theory and provides explicit bounds for small ε based on this approximation.
We study the maximal mutual information about a random variable $Y$ (representing non-private information) displayed through an additive Gaussian channel when guaranteeing that only $ε$ bits of information is leaked about a random variable $X$ (representing private information) that is correlated with $Y$. Denoting this quantity by $g_ε(X,Y)$, we show that for perfect privacy, i.e., $ε=0$, one has $g_0(X,Y)=0$ for any pair of absolutely continuous random variables $(X,Y)$ and then derive a second-order approximation for $g_ε(X,Y)$ for small $ε$. This approximation is shown to be related to the strong data processing inequality for mutual information under suitable conditions on the joint distribution $P_{XY}$. Next, motivated by an operational interpretation of data privacy, we formulate the privacy-utility tradeoff in the same setup using estimation-theoretic quantities and obtain explicit bounds for this tradeoff when $ε$ is sufficiently small using the approximation formula derived for $g_ε(X,Y)$.