Optimal transport (OT) has profoundly impacted machine learning by providing theoretical and computational tools to realign datasets. In this context, given two large point clouds of sizes $n$ and $m$ in $\mathbb{R}^d$, entropic OT (EOT) solvers have emerged as the most reliable tool to either solve the Kantorovich problem and output a $n\times m$ coupling matrix, or to solve the Monge problem and learn a vector-valued push-forward map. While the robustness of EOT couplings/maps makes them a go-to choice in practical applications, EOT solvers remain difficult to tune because of a small but influential set of hyperparameters, notably the omnipresent entropic regularization strength $\varepsilon$. Setting $\varepsilon$ can be difficult, as it simultaneously impacts various performance metrics, such as compute speed, statistical performance, generalization, and bias. In this work, we propose a new class of EOT solvers (ProgOT), that can estimate both plans and transport maps. We take advantage of several opportunities to optimize the computation of EOT solutions by dividing mass displacement using a time discretization, borrowing inspiration from dynamic OT formulations, and conquering each of these steps using EOT with properly scheduled parameters. We provide experimental evidence demonstrating that ProgOT is a faster and more robust alternative to standard solvers when computing couplings at large scales, even outperforming neural network-based approaches. We also prove statistical consistency of our approach for estimating optimal transport maps.

翻译：最优传输（OT）通过提供理论及计算工具以重对齐数据集，对机器学习产生了深远影响。在此背景下，给定 $\mathbb{R}^d$ 空间中规模分别为 $n$ 和 $m$ 的两个大型点云，熵最优传输（EOT）求解器已成为最可靠的工具：既可求解康托罗维奇问题并输出 $n\times m$ 耦合矩阵，亦可求解蒙日问题并学习向量值前推映射。尽管EOT耦合/映射的鲁棒性使其成为实际应用中的首选，但EOT求解器因一组数量少但影响大的超参数（尤其是无处不在的熵正则化强度 $\varepsilon$）而仍难以调节。$\varepsilon$ 的设置可能较为困难，因其会同时影响多种性能指标，如计算速度、统计性能、泛化能力与偏差。本工作中，我们提出一类新型EOT求解器（ProgOT），能够同时估计传输方案与传输映射。我们通过时间离散化分割质量位移，借鉴动态OT公式的灵感，并利用参数经恰当调度的EOT攻克每个步骤，从而优化EOT解的计算过程。实验证据表明，在大规模计算耦合时，ProgOT相较于标准求解器具有更快的速度与更强的鲁棒性，甚至优于基于神经网络的方法。我们同时证明了该方法在估计最优传输映射方面的统计一致性。