# Hierarchical Sliced Wasserstein Distance

@article{Nguyen2022HierarchicalSW, title={Hierarchical Sliced Wasserstein Distance}, author={Khai Nguyen and Tongzheng Ren and Huy Nguyen and Litu Rout and Tan Minh Nguyen and Nhat Ho}, journal={ArXiv}, year={2022}, volume={abs/2209.13570} }

Sliced Wasserstein (SW) distance has been widely used in diﬀerent application scenarios since it can be scaled to a large number of supports without suﬀering from the curse of dimensionality. The value of sliced Wasserstein distance is the average of transportation cost between one-dimensional representations (projections) of original measures that are obtained by Radon Transform (RT). Despite its eﬃciency in the number of supports, estimating the sliced Wasserstein requires a relatively large…

## 3 Citations

### Improving Generative Flow Networks with Path Regularization

- Computer ScienceArXiv
- 2022

A novel path regularization method based on optimal transport theory that places prior constraints on the underlying structure of the G FlowNets to help the GFlowNets better discover the latentructure of the target distribution or enhance its ability to explore the environment in the context of active learning.

## References

SHOWING 1-10 OF 72 REFERENCES

### Augmented Sliced Wasserstein Distances

- Computer ScienceICLR
- 2022

This work proposes 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, and provides the condition under which the ASWD is a valid metric and shows it can be obtained by an injective neural network architecture.

### Distributional Sliced-Wasserstein and Applications to Generative Modeling

- Computer ScienceICLR
- 2021

This paper proposes a novel distance that finds optimal penalized probability measure over the slices, named Distributional Sliced-Wasserstein distance (DSWD), and shows that the DSWD is a generalization of both SWD and Max-SWD, and the proposed distance could be found by searching for the push-forward measure over a set of measures satisfying some certain constraints.

### Generalized Sliced Wasserstein Distances

- Computer ScienceNeurIPS
- 2019

The generalized Radon transform is utilized to define a new family of distances for probability measures, which are called generalized sliced-Wasserstein (GSW) distances, and it is shown that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max- GSW) distance.

### Sliced Wasserstein Generative Models

- Computer Science2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
- 2019

This paper proposes to approximate SWDs with a small number of parameterized orthogonal projections in an end-to-end deep learning fashion and designs two types of differentiable SWD blocks to equip modern generative frameworks---Auto-Encoders and Generative Adversarial Networks.

### Max-Sliced Wasserstein Distance and Its Use for GANs

- Computer Science2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
- 2019

This work demonstrates that the recently proposed sliced Wasserstein distance trains GANs on high-dimensional images up to a resolution of 256x256 easily and develops the max-sliced Wasserenstein distance, which enjoys compelling sample complexity while reducing projection complexity, albeit necessitating a max estimation.

### Sliced Gromov-Wasserstein

- Computer ScienceNeurIPS
- 2019

A novel OT discrepancy is defined that can deal with large scale distributions via a slicing approach and is demonstrated to have ability to tackle similar problems as GW while being several order of magnitudes faster to compute.

### Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution

- Computer ScienceArXiv
- 2022

Convolution sliced Wasserstein (CSW) is derived via incorporating stride, dilation, and non-linear activation function into the convolution operators and is demonstrated to have favorable performance in comparing probability measures over images and in training deep generative modeling on images.

### Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections

- Computer ScienceNeurIPS
- 2021

This work adopts a new perspective to approximate SW by making use of the concentration of measure phenomenon, and develops a simple deterministic approximation that is both accurate and easy to use compared to the usual Monte Carlo approximation.

### Sliced-Wasserstein Gradient Flows

- Computer ScienceArXiv
- 2021

It is argued that this method is more flexible than JKO-ICNN, since SW enjoys a closedform differentiable approximation and the density at each step can be parameterized by any generative model which alleviates the computational burden and makes it tractable in higher dimensions.

### Sliced Wasserstein Variational Inference

- Computer ScienceArXiv
- 2022

This work proposes a new variational inference method by minimizing sliced Wasserstein distance–a valid metric arising from optimal transport and does not require a tractable density function of variational distributions so that approximating families can be amortized by generators like neural networks.