  The matching principles behind optimal transport (OT) play an increasingly
important role in machine learning, a trend which can be observed when OT is
used to disambiguate datasets in applications (e.g. single-cell genomics) or
used to improve more complex methods (e.g. balanced attention in transformers
or self-supervised learning). To scale to more challenging problems, there is a
growing consensus that OT requires solvers that can operate on millions, not
thousands, of points. The low-rank optimal transport (LOT) approach advocated
in \cite{scetbon2021lowrank} holds several promises in that regard, and was
shown to complement more established entropic regularization approaches, being
able to insert itself in more complex pipelines, such as quadratic OT. LOT
restricts the search for low-cost couplings to those that have a
low-nonnegative rank, yielding linear time algorithms in cases of interest.
However, these promises can only be fulfilled if the LOT approach is seen as a
legitimate contender to entropic regularization when compared on properties of
interest, where the scorecard typically includes theoretical properties
(statistical complexity and relation to other methods) or practical aspects
(debiasing, hyperparameter tuning, initialization). We target each of these
areas in this paper in order to cement the impact of low-rank approaches in
computational OT.
