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We study a statistical signal processing privacy problem, where an agentobserves useful data $Y$ and wants to reveal the information to a user. Sincethe useful data is correlated with the private data $X$, the agent employs aprivacy mechanism to generate data $U$ that can be released. We study theprivacy mechanism design that maximizes the revealed information about $Y$while satisfying a strong $\ell_1$-privacy criterion. When a sufficiently smallleakage is allowed, we show that the optimizer vectors of the privacy mechanismdesign problem have a specific geometry, i.e., they are perturbations of fixedvector distributions. This geometrical structure allows us to use a localapproximation of the conditional entropy. By using this approximation theoriginal optimization problem can be reduced to a linear program so that anapproximate solution for privacy mechanism can be easily obtained. The maincontribution of this work is to consider non-zero leakage with a non-invertibleleakage matrix. In an example inspired by water mark application, we firstinvestigate the accuracy of the approximation. Then, we employ differentmeasures for utility and privacy leakage to compare the privacy-utilitytrade-off using our approach with other methods. In particular, it has beenshown that by allowing small leakage, significant utility can be achieved usingour method compared to the case where no leakage is allowed.

Data Disclosure with Non-zero Leakage and Non-invertible Leakage Matrix