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.

翻译：最佳运输(OT)的匹配原则在最佳运输(OT)中发挥着越来越重要的作用。 在机器学习中,最优化运输(OT)的匹配原则发挥着越来越重要的作用。 当OT被用于在应用(如单细胞基因组)中调离数据集(如单细胞基因组),或用来改进更复杂的方法(如在变压器或自我监督的学习中平衡关注)时,可以看到一种趋势。 为了扩大规模到更具挑战性的问题,人们日益一致认为,OT需要能够以数百万,而不是几千个点运行的解决问题解决者,这要求以更富有挑战性的问题规模更大的问题。在\cite {scetbon2021lowrank} 中倡导的低级别最佳运输(LOT)办法在这方面有几项承诺,并表明它补充了更成熟的肿瘤正规化方法,能够将自己置在更复杂的管道中,例如变压器或自自自我插入更复杂的方法(如变压的变压器),例如变压的变压器中,LOT,将寻找低成本的更低成本的组合,在感兴趣的级别上产生线时间算算算算算算。然而。然而,但这些承诺只有在只有当LOOTLOT方法(在实际利益中,通常包括这些理的理论的特性,在SDBBA的特性的每一个的每个的特性,其中,这些理论的特性,通常的学、这些理论性、这种理论性、这些方法包括了这些理论的特性,在SDR的特性,在SDRBRB的特性,在SB的、这些理论的特性,在SDR的学方法中,通常的、这种理论的、这种理论的等的、在SDB的、这种理论的、这种方法在S的、在SB的、这些理论性、这种理论性、这些等的特性的、这种理论的、在S的、在S的特性的、在S的、这些制的、这些制的、这些等的、这些理论的等的、这些理论的、这些理论的学的特性的特性的学的、这些理论的、在S的、这些制的、这些理论的、这些理论的、这些理论的、这些制的、这些理论的、这些制的、这些制的、在S的、这些理论的、