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The matching principles behind optimal transport (OT) play an increasinglyimportant role in machine learning, a trend which can be observed when OT isused to disambiguate datasets in applications (e.g. single-cell genomics) orused to improve more complex methods (e.g. balanced attention in transformersor self-supervised learning). To scale to more challenging problems, there is agrowing consensus that OT requires solvers that can operate on millions, notthousands, of points. The low-rank optimal transport (LOT) approach advocatedin \cite{scetbon2021lowrank} holds several promises in that regard, and wasshown to complement more established entropic regularization approaches, beingable to insert itself in more complex pipelines, such as quadratic OT. LOTrestricts the search for low-cost couplings to those that have alow-nonnegative rank, yielding linear time algorithms in cases of interest.However, these promises can only be fulfilled if the LOT approach is seen as alegitimate contender to entropic regularization when compared on properties ofinterest, 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 theseareas in this paper in order to cement the impact of low-rank approaches incomputational OT.

Low-rank Optimal Transport: Approximation, Statistics and Debiasing