On Robust Optimal Transport: Computational Complexity, Low-rank Approximation, and Barycenter Computation

We consider two robust versions of optimal transport, named $\textit{Robust Semi-constrained Optimal Transport}$ (RSOT) and $\textit{Robust Unconstrained Optimal Transport}$ (ROT), formulated by relaxing the marginal constraints with Kullback-Leibler divergence. For both problems in the discrete settings, we propose Sinkhorn-based algorithms that produce $\varepsilon$-approximations of RSOT and ROT in $\widetilde{\mathcal{O}}(\frac{n^2}{\varepsilon})$ time, where $n$ is the number of supports of the probability distributions. Furthermore, to reduce the dependency of the complexity of the Sinkhorn-based algorithms on $n$, we apply Nystr\"{o}m method to approximate the kernel matrix in both RSOT and ROT by a matrix of rank $r$ before passing it to these Sinkhorn-based algorithms. We demonstrate that these new algorithms have $\widetilde{\mathcal{O}}(n r^2 + \frac{nr}{\varepsilon})$ runtime to obtain the RSOT and ROT $\varepsilon$-approximations. Finally, we consider a barycenter problem based on RSOT, named $\textit{Robust Semi-Constrained Barycenter}$ problem (RSBP), and develop a robust iterative Bregman projection algorithm, called $\textbf{Normalized-RobustIBP}$ algorithm, to solve the RSBP in the discrete settings of probability distributions. We show that an $\varepsilon$-approximated solution of the RSBP can be achieved in $\widetilde{\mathcal{O}}(\frac{mn^2}{\varepsilon})$ time using $\textbf{Normalized-RobustIBP}$ algorithm when $m = 2$, which is better than the previous complexity $\widetilde{\mathcal{O}}(\frac{mn^2}{\varepsilon^2})$ of IBP algorithm for approximating the Wasserstein barycenter. Extensive experiments confirm our theoretical results.