# Optimal Transport with Heterogeneously Missing Data

arXiv: https://arxiv.org/abs/2505.17291 · HTML: https://mlsanity.com/paper/2505.17291/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 55 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 23 of 23)

- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares](https://mlsanity.com/paper/1410.2596/) (2014)
- [Missing Data Imputation using Optimal Transport](https://mlsanity.com/paper/2002.03860/) (2020)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Generalization Bounds and Representation Learning for Estimation of Potential Outcomes and Causal Effects](https://mlsanity.com/paper/2001.07426/) (2020)
- [What's a good imputation to predict with missing values?](https://mlsanity.com/paper/2106.00311/) (2021)
- [On the consistency of supervised learning with missing values](https://mlsanity.com/paper/1902.06931/) (2019)
- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [Benchmarking missing-values approaches for predictive models on health databases](https://mlsanity.com/paper/2202.10580/) (2022)
- [Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation](https://mlsanity.com/paper/1905.10155/) (2019)
- [High-dimensional principal component analysis with heterogeneous missingness](https://mlsanity.com/paper/1906.12125/) (2019)
- [The statistical effect of entropic regularization in optimal transportation](https://mlsanity.com/paper/2006.05199/) (2020)
- [Conformalized matrix completion](https://mlsanity.com/paper/2305.10637/) (2023)
- [Imputation for prediction: beware of diminishing returns](https://mlsanity.com/paper/2407.19804/) (2024)
- [Collective Matrix Completion](https://mlsanity.com/paper/1807.09010/) (2018)
- [Random features models: a way to study the success of naive imputation](https://mlsanity.com/paper/2402.03839/) (2024)
- [High-dimensional logistic regression with missing data: Imputation, regularization, and universality](https://mlsanity.com/paper/2410.01093/) (2024)
- [Estimation beyond Missing (Completely) at Random](https://mlsanity.com/paper/2410.10704/) (2024)
- [Domain adaptation using optimal transport for invariant learning using histopathology datasets](https://mlsanity.com/paper/2303.02241/) (2023)
- [Hypothesis Transfer Learning with Surrogate Classification Losses: Generalization Bounds through Algorithmic Stability](https://mlsanity.com/paper/2305.19694/) (2023)
- [Parametric MMD Estimation with Missing Values: Robustness to Missingness and Data Model Misspecification](https://mlsanity.com/paper/2503.00448/) (2025)
- [Entropic Optimal Transport in Random Graphs](https://mlsanity.com/paper/2201.03949/) (2022)
- [Online Matrix Completion Through Nuclear Norm Regularisation](https://mlsanity.com/paper/1401.2451/) (2014)

## Abstract (arXiv)

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.