  Several recent applications of optimal transport (OT) theory to machine
learning have relied on regularization, notably entropy and the Sinkhorn
algorithm. Because matrix-vector products are pervasive in the Sinkhorn
algorithm, several works have proposed to \textit{approximate} kernel matrices
appearing in its iterations using low-rank factors. Another route lies instead
in imposing low-rank constraints on the feasible set of couplings considered in
OT problems, with no approximations on cost nor kernel matrices. This route was
first explored by Forrow et al., 2018, who proposed an algorithm tailored for
the squared Euclidean ground cost, using a proxy objective that can be solved
through the machinery of regularized 2-Wasserstein barycenters. Building on
this, we introduce in this work a generic approach that aims at solving, in
full generality, the OT problem under low-rank constraints with arbitrary
costs. Our algorithm relies on an explicit factorization of low rank couplings
as a product of \textit{sub-coupling} factors linked by a common marginal;
similar to an NMF approach, we alternatively updates these factors. We prove
the non-asymptotic stationary convergence of this algorithm and illustrate its
efficiency on benchmark experiments.
