Gaussian-Smoothed Sliced Probability Divergences

Mokhtar Z. Alaya, Alain Rakotomamonjy, Maxime Berar, Gilles Gasso

Abstract: 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 computational and 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 ${\text{G}}_{\sigma}\text{SW}_p$. 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 ${\text{G}}_{\sigma}\text{SW}_p$ and prove that it converges with a rate $O(n^{-1/{2p}})$. 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.

Submission Length: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Jonathan_Ullman1

Submission Number: 2568