Developing machine learning methods that are privacy preserving is today a
central topic of research, with huge practical impacts. Among the numerous ways
to address privacy-preserving learning, we here take the perspective of
computing the divergences between distributions under the Differential Privacy
(DP) framework -- being able to compute divergences between distributions is
pivotal for many machine learning problems, such as learning generative models
or domain adaptation problems. Instead of resorting to the popular
gradient-based sanitization method for DP, we tackle the problem at its roots
by focusing on the Sliced Wasserstein Distance and seamlessly making it
differentially private. Our main contribution is as follows: we analyze the
property of adding a Gaussian perturbation to the intrinsic randomized
mechanism of the Sliced Wasserstein Distance, and we establish the
sensitivityof the resulting differentially private mechanism. One of our
important findings is that this DP mechanism transforms the Sliced Wasserstein
distance into another distance, that we call the Smoothed Sliced Wasserstein
Distance. This new differentially private distribution distance can be plugged
into generative models and domain adaptation algorithms in a transparent way,
and we empirically show that it yields highly competitive performance compared
with gradient-based DP approaches from the literature, with almost no loss in
accuracy for the domain adaptation problems that we consider