# A Riemannian Block Coordinate Descent Method for Computing the Projection Robust Wasserstein Distance

@article{Huang2020ARB, title={A Riemannian Block Coordinate Descent Method for Computing the Projection Robust Wasserstein Distance}, author={Minhui Huang and Shiqian Ma and Lifeng Lai}, journal={ArXiv}, year={2020}, volume={abs/2012.05199} }

The Wasserstein distance has become increasingly important in machine learning and deep learning. Despite its popularity, the Wasserstein distance is hard to approximate because of the curse of dimensionality. A recently proposed approach to alleviate the curse of dimensionality is to project the sampled data from the high dimensional probability distribution onto a lower-dimensional subspace, and then compute the Wasserstein distance between the projected data. However, this approach requires…

## 19 Citations

### Projection Robust Wasserstein Barycenter

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This paper proposes the projection robust Wasserstein barycenter (PR WB) that has the potential to mitigate the curse of dimensionality, and a relaxed PRWB (RPRWB) model that is computationally more tractable.

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A projected Wasserstein distance is developed for the two-sample test, a fundamental problem in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution, to circumvent the curse of dimensionality.

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This is the first study of the minimax optimization over the Riemannian manifold and it is proved that the MVR-RSGDA algorithm achieves a lower sample complexity of $\tilde{O}(\kappa^{4}\epsilon^{-3})$ without large batches, which reaches near the best known sample complexity for its Euclidean counterparts.

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