解决特征缺失不均的最优传输问题，提升距离估计精度。
Optimal Transport with Heterogeneously Missing Data
- 假设数据完全随机缺失，提出无偏化方法修正分布距离
- 用迭代奇异值阈值法高效稳定估算正则化最优传输
- 无需验证集选择超参数，适合矩阵补全等场景
研究在数据完全随机缺失（MCAR）条件下，两个经验分布间的最优传输问题。允许不同特征及两分布间缺失概率异质。首先证明：基于经验高斯分布的Wasserstein距离与任意分布间的线性Monge映射可无偏估计，且样本复杂度基本不变。其次，提出使用迭代奇异值阈值法（ISVT）高效一致地估计熵正则化最优传输。设计一种无需验证集的超参数选择策略，利用对Bures-Wasserstein距离的估计，该方法对一般矩阵补全问题亦具独立价值。最后，在多种数值实验中验证了方法的有效性。
原文摘要 · Abstract (English)
We consider the problem of solving the optimal transport problem between two empirical distributions with missing values. Our main assumption is that the data is missing completely at random (MCAR), but we allow for heterogeneous missingness probabilities across features and across the two distributions. As a first contribution, we show that the Wasserstein distance between empirical Gaussian distributions and linear Monge maps between arbitrary distributions can be debiased without significantly affecting the sample complexity. Secondly, we show that entropic regularized optimal transport can be estimated efficiently and consistently using iterative singular value thresholding (ISVT). We propose a validation set-free hyperparameter selection strategy for ISVT that leverages our estimator of the Bures-Wasserstein distance, which could be of independent interest in general matrix completion problems. Finally, we validate our findings on a wide range of numerical applications.
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