Fourier Sliced-Wasserstein Embedding for Multisets and Measures

We present the $\textit{Fourier Sliced Wasserstein (FSW) embedding}\unicode{x2014}$a novel method to embed multisets and measures over $\mathbb{R}^d$ into Euclidean space. Our proposed embedding approximately preserves the sliced Wasserstein distance on distributions, thereby yielding geometrically meaningful representations that better capture the structure of the input. Moreover, it is injective on measures and $\textit{bi-Lipschitz}$ on multisets$\unicode{x2014}$a significant advantage over prevalent embedding methods based on sum- or max-pooling, which are provably not bi-Lipschitz, and in many cases, not even injective. The required output dimension for these guarantees is near optimal: roughly $2 n d$, where $n$ is the maximal number of support points in the input. Conversely, we prove that it is $\textit{impossible}$ to embed distributions over $\mathbb{R}^d$ into Euclidean space in a bi-Lipschitz manner. Thus, the metric properties of our embedding are, in a sense, the best achievable. Through numerical experiments, we demonstrate that our method yields superior representations of input multisets and offers practical advantage for learning on multiset data. Specifically, we show that (a) the FSW embedding induces significantly lower distortion on the space of multisets, compared to the leading method for computing sliced-Wasserstein-preserving embeddings; and (b) a simple combination of the FSW embedding and an MLP achieves state-of-the-art performance in learning the (non-sliced) Wasserstein distance.