Statistical Wasserstein distance with rank regularization and spiked structure

In recent years, the optimal transport (OT) model has been successfully applied to machine learning, as the Wasserstein distance in the OT model can compare the distance between probability distributions and reflects the underlying geometry. However, the computation of the OT problem suffers from the sampling noise and curse of dimensionality. The rank regularization OT model has been shown that it can reduce the sampling noise in the computation, but it does not deal with the impact of the curse of dimensionality. In this article, using the idea of principal component analysis (referred to as spiked model), we improve the rank regularization OT model to reduce the impact of data dimension and propose the rank regularization spiked model. In this model, the coupling measure is decomposed by several product measures and each marginal measure of the product measures has low-dimensional structure. The factored coupling ensures the robustness of the Wasserstein distance and can reduce the sampling noise, at the same time the marginal measures are projected to the low-dimensional subspace to resolve the high-dimensional problem. For this model, we study the estimation error of the Wasserstein distance and give two statistical convergence results.