Properties of Discrete Sliced Wasserstein Losses
share
The Sliced Wasserstein (SW) distance has become a popular alternative to the Wasserstein distance for comparing probability measures. Widespread applications include image processing, domain adaptation and generative modelling, where it is common to optimise some parameters in order to minimise SW, which serves as a loss function between discrete probability measures (since measures admitting densities are numerically unattainable). All these optimisation problems bear the same sub-problem, which is minimising the Sliced Wasserstein energy. In this paper we study the properties of ℰ: Y ⟼SW_2^2(γ_Y, γ_Z), i.e. the SW distance between two uniform discrete measures with the same amount of points as a function of the support Y ∈ℝ^n × d of one of the measures. We investigate the regularity and optimisation properties of this energy, as well as its Monte-Carlo approximation ℰ_p (estimating the expectation in SW using only p samples) and show convergence results on the critical points of ℰ_p to those of ℰ, as well as an almost-sure uniform convergence. Finally, we show that in a certain sense, Stochastic Gradient Descent methods minimising ℰ and ℰ_p converge towards (Clarke) critical points of these energies.
READ FULL TEXT
