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The Wasserstein distance has become increasingly important in machinelearning and deep learning. Despite its popularity, the Wasserstein distance ishard to approximate because of the curse of dimensionality. A recently proposedapproach to alleviate the curse of dimensionality is to project the sampleddata from the high dimensional probability distribution onto alower-dimensional subspace, and then compute the Wasserstein distance betweenthe projected data. However, this approach requires to solve a max-min problemover the Stiefel manifold, which is very challenging in practice. The onlyexisting work that solves this problem directly is the RGAS (RiemannianGradient Ascent with Sinkhorn Iteration) algorithm, which requires to solve anentropy-regularized optimal transport problem in each iteration, and thus canbe costly for large-scale problems. In this paper, we propose a Riemannianblock coordinate descent (RBCD) method to solve this problem, which is based ona novel reformulation of the regularized max-min problem over the Stiefelmanifold. We show that the complexity of arithmetic operations for RBCD toobtain an $\epsilon$-stationary point is $O(\epsilon^{-3})$. This significantlyimproves the corresponding complexity of RGAS, which is $O(\epsilon^{-12})$.Moreover, our RBCD has very low per-iteration complexity, and hence is suitablefor large-scale problems. Numerical results on both synthetic and real datasetsdemonstrate that our method is more efficient than existing methods, especiallywhen the number of sampled data is very large.

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