# Long-Time Asymptotics of the Sliced-Wasserstein Flow

arXiv: https://arxiv.org/abs/2405.06313 · HTML: https://mlsanity.com/paper/2405.06313/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2024.

## Built on by (most cited first, 7 of 7)

- [On the Wasserstein Gradient Flow Interpretation of Drifting Models](https://mlsanity.com/paper/2605.05118/) (2026)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [Convergence Rates for Distribution Matching with Sliced Optimal Transport](https://mlsanity.com/paper/2602.10691/) (2026)
- [Sliced Wasserstein Steering between Gaussian Measures](https://mlsanity.com/paper/2604.22807/) (2026)
- [ReSWD: ReSTIR'd, not shaken. Combining Reservoir Sampling and Sliced Wasserstein Distance for Variance Reduction](https://mlsanity.com/paper/2510.01061/) (2025)
- [AdamFlow: Adam-based Wasserstein Gradient Flows for Surface Registration in Medical Imaging](https://mlsanity.com/paper/2604.02290/) (2026)
- [Distribution Steering via Sliced Optimal Transport Control](https://mlsanity.com/paper/2608.12828/) (2026)

## Abstract (arXiv)

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.