Gaussian-Smoothed Sliced Probability Divergences
Mokhtar Z. Alaya Alain Rakotomamonjy Maxime Berar Gilles Gasso
Gaussian smoothed sliced Wasserstein distance has been recently introduced
for comparing probability distributions, while preserving privacy on the data.
It has been shown that it provides performances similar to its non-smoothed
(non-private) counterpart. However, the computationaland statistical properties
of such a metric have not yet been well-established. This work investigates the
theoretical properties of this distance as well as those of generalized
versions denoted as Gaussian-smoothed sliced divergences. We first show that
smoothing and slicing preserve the metric property and the weak topology. To
study the sample complexity of such divergences, we then introduce
$\hat{\hat\mu}_{n}$ the double empirical distribution for the
smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a
double sampling process: one from sampling according to the origin distribution
$\mu$ and the second according to the convolution of the projection of $\mu$ on
the unit sphere and the Gaussian smoothing. We particularly focus on the
Gaussian smoothed sliced Wasserstein distance and prove that it converges with
a rate $O(n^{-1/2})$. We also derive other properties, including continuity, of
different divergences with respect to the smoothing parameter. We support our
theoretical findings with empirical studies in the context of
privacy-preserving domain adaptation.