  We study a statistical signal processing privacy problem, where an agent
observes useful data $Y$ and wants to reveal the information to a user. Since
the useful data is correlated with the private data $X$, the agent employs a
privacy mechanism to generate data $U$ that can be released. We study the
privacy mechanism design that maximizes the revealed information about $Y$
while satisfying a strong $\ell_1$-privacy criterion. When a sufficiently small
leakage is allowed, we show that the optimizer vectors of the privacy mechanism
design problem have a specific geometry, i.e., they are perturbations of fixed
vector distributions. This geometrical structure allows us to use a local
approximation of the conditional entropy. By using this approximation the
original optimization problem can be reduced to a linear program so that an
approximate solution for privacy mechanism can be easily obtained. The main
contribution of this work is to consider non-zero leakage with a non-invertible
leakage matrix. In an example inspired by water mark application, we first
investigate the accuracy of the approximation. Then, we employ different
measures for utility and privacy leakage to compare the privacy-utility
trade-off using our approach with other methods. In particular, it has been
shown that by allowing small leakage, significant utility can be achieved using
our method compared to the case where no leakage is allowed.
