  This paper focuses on multi-block optimization problems over transport
polytopes, which underlie various applications including strongly correlated
quantum physics and machine learning. Conventional block coordinate
descent-type methods for the general multi-block problems store and operate on
the matrix variables directly, resulting in formidable expenditure for
large-scale settings. On the other hand, optimal transport problems, as a
special case, have attracted extensive attention and numerical techniques that
waive the use of the full matrices have recently emerged. However, it remains
nontrivial to apply these techniques to the multi-block, possibly nonconvex
problems with theoretical guarantees. In this work, we leverage the benefits of
both sides and develop novel sampling-based block coordinate descent-type
methods, which are equipped with either entropy regularization or
Kullback-Leibler divergence. Each iteration of these methods solves subproblems
restricted on the sampled degrees of freedom. Consequently, they involve only
sparse matrices, which amounts to considerable complexity reductions. We
explicitly characterize the sampling-induced errors and establish convergence
and asymptotic properties for the methods equipped with the entropy
regularization. Numerical experiments on typical strongly correlated electron
systems corroborate their superior scalability over the methods utilizing full
matrices. The advantage also enables the first visualization of approximate
optimal transport maps between electron positions in three-dimensional
contexts.
