# Augmented Sliced Wasserstein Distances

arXiv: https://arxiv.org/abs/2006.08812 · HTML: https://mlsanity.com/paper/2006.08812/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2020.

## Built on by (most cited first, 14 of 14)

- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution](https://mlsanity.com/paper/2204.01188/) (2022)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Amortized Projection Optimization for Sliced Wasserstein Generative Models](https://mlsanity.com/paper/2203.13417/) (2022)
- [SAN: Inducing Metrizability of GAN with Discriminative Normalized Linear Layer](https://mlsanity.com/paper/2301.12811/) (2023)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Differentiable Generalized Sliced Wasserstein Plans](https://mlsanity.com/paper/2505.22049/) (2025)
- [Efficient Transferable Optimal Transport via Min-Sliced Transport Plans](https://mlsanity.com/paper/2511.19741/) (2025)
- [Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices](https://mlsanity.com/paper/2411.10651/) (2024)
- [Constrained Sliced Wasserstein Embedding](https://mlsanity.com/paper/2506.02203/) (2025)
- [Fast Approximation of the Generalized Sliced-Wasserstein Distance](https://mlsanity.com/paper/2210.10268/) (2022)
- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [Relation-Aware Slicing in Cross-Domain Alignment](https://mlsanity.com/paper/2507.13194/) (2025)
- [SONA: Learning Conditional, Unconditional, and Mismatching-Aware Discriminator](https://mlsanity.com/paper/2510.04576/) (2025)

## Abstract (arXiv)

While theoretically appealing, the application of the Wasserstein distance to large-scale machine learning problems has been hampered by its prohibitive computational cost. The sliced Wasserstein distance and its variants improve the computational efficiency through the random projection, yet they suffer from low accuracy if the number of projections is not sufficiently large, because the majority of projections result in trivially small values. In this work, we propose a new family of distance metrics, called augmented sliced Wasserstein distances (ASWDs), constructed by first mapping samples to higher-dimensional hypersurfaces parameterized by neural networks. It is derived from a key observation that (random) linear projections of samples residing on these hypersurfaces would translate to much more flexible nonlinear projections in the original sample space, so they can capture complex structures of the data distribution. We show that the hypersurfaces can be optimized by gradient ascent efficiently. We provide the condition under which the ASWD is a valid metric and show that this can be obtained by an injective neural network architecture. Numerical results demonstrate that the ASWD significantly outperforms other Wasserstein variants for both synthetic and real-world problems.