Long-Time Asymptotics of the Sliced-Wasserstein Flow

The sliced-Wasserstein flow is an evolution equation where a probability density evolves in time, advected by a velocity field computed as the average among directions in the unit sphere of the optimal transport displacements from its 1D projections to the projections of a fixed target measure. This flow happens to be the gradient flow in the usual Wasserstein space of the squared sliced-Wasserstein distance to the target. We consider the question whether in long-time the flow converges to the target (providing a positive result when the target is Gaussian) and the question of the long-time limit of the flow map obtained by following the trajectories of each particle. We prove that this limit is in general not the optimal transport map from the starting measure to the target. Both questions come from the folklore about sliced-Wasserstein and had never been properly treated.