Optimal Transport with Heterogeneously Missing Data
This addresses a practical data analysis problem for researchers and practitioners dealing with incomplete datasets in optimal transport applications, representing a methodological advancement rather than a paradigm shift.
The paper tackles the optimal transport problem between empirical distributions with heterogeneously missing data under MCAR assumptions, showing that Wasserstein distances and linear Monge maps can be debiased without significant sample complexity increases, and proposing an efficient ISVT-based estimator for entropic regularized optimal transport with a hyperparameter selection strategy.
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.