Fourier Sliced-Wasserstein Embedding for Multisets and Measures
Tal Amir Nadav Dym
We present the _Fourier Sliced Wasserstein (FSW) embedding_—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 _bi-Lipschitz_ on multisets—a significant advantage over prevalent 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 input multiset size.

Furthermore, we prove that it is _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 possible.

Through numerical experiments, we demonstrate that our method yields superior multiset representations that improve performance in practical learning tasks. Specifically, we show that (a) a simple combination of the FSW embedding with an MLP achieves state-of-the-art performance in learning the (non-sliced) Wasserstein distance; and (b) replacing max-pooling with the FSW embedding makes PointNet significantly more robust to parameter reduction, with only minor performance degradation even after a 40-fold reduction.
Sliced Wasserstein distance, Euclidean embedding, multiset embedding, bi-Lipschitz, permutation invariant
A bi-Lipschitz Euclidean embedding for multisets that extends injectively to measures, based on the sliced Wasserstein distance