Over the past few years, numerous computational models have been developed to solve Optimal Transport (OT) in a stochastic setting, where distributions are represented by samples and where the goal is to find the closest map to the ground truth OT map, unknown in practical settings. So far, no quantitative criterion has yet been put forward to tune the parameters of these models and select maps that best approximate the ground truth. To perform this task, we propose to leverage the Brenier formulation of OT.Theoretically, we show that this formulation guarantees that, up to sharp a distortion parameter depending on the smoothness/strong convexity and a statistical deviation term, the selected map achieves the lowest quadratic error to the ground truth. This criterion, estimated via convex optimization, enables parameter tuning and model selection among entropic regularization of OT, input convex neural networks and smooth and strongly convex nearest-Brenier (SSNB) models.We also use this criterion to question the use of OT in Domain-Adaptation (DA). In a standard DA experiment, it enables us to identify the potential that is closest to the true OT map between the source and the target. Yet, we observe that this selected potential is far from being the one that performs best for the downstream transfer classification task.

翻译：近年来，学术界开发了大量计算模型来解决随机环境下的最优传输问题——在此类问题中，分布由样本表示，目标是寻找最接近真实OT映射（实际场景中未知）的映射。但目前尚未建立定量准则来调节这些模型的参数并选择最逼近真实OT映射的映射。为此，我们提出利用OT的Brenier公式来解决该问题。理论层面，我们证明该公式能确保所选映射达到与真实OT映射的最低二次误差，其误差边界由取决于光滑性/强凸性的尖锐形变参数与统计偏差项共同决定。该准则通过凸优化进行估计，可用于熵正则化OT、输入凸神经网络以及光滑强凸近Brenier（SSNB）模型的参数调优与模型选择。我们还进一步将该准则应用于域自适应中的OT有效性评估。在标准DA实验中，该准则能识别出源域与目标域之间最接近真实OT映射的势函数。然而我们观察到，这个被选中的最优势函数并非在下游迁移分类任务中表现最佳的映射。