Injective Sliced-Wasserstein embedding for weighted sets and point clouds

We present the Sliced Wasserstein Embedding x2014 a novel method to embed multisets and distributions over Rᵈ into Euclidean space. Our embedding is injective and approximately preserves the Sliced Wasserstein distance. Moreover, when restricted to multisets, it is bi-Lipschitz. We also prove that it is impossible to embed distributions over Rᵈ into a Euclidean space in a bi-Lipschitz manner, even under the assumption that their support is bounded and finite. We demonstrate empirically that our embedding offers practical advantage in learning tasks over existing methods for handling multisets.