# Flow Matching with Missing Data

arXiv: https://arxiv.org/abs/2607.28698 · HTML: https://mlsanity.com/paper/2607.28698/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 30 Jul 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 14 of 14)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [GAIN: Missing Data Imputation using Generative Adversarial Nets](https://mlsanity.com/paper/1806.02920/) (2018)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [HyperImpute: Generalized Iterative Imputation with Automatic Model Selection](https://mlsanity.com/paper/2206.07769/) (2022)
- [MisGAN: Learning from Incomplete Data with Generative Adversarial Networks](https://mlsanity.com/paper/1902.09599/) (2019)
- [Variational Autoencoder with Arbitrary Conditioning](https://mlsanity.com/paper/1806.02382/) (2018)
- [not-MIWAE: Deep Generative Modelling with Missing not at Random Data](https://mlsanity.com/paper/2006.12871/) (2020)
- [Generating and Imputing Tabular Data via Diffusion and Flow-based Gradient-Boosted Trees](https://mlsanity.com/paper/2309.09968/) (2023)
- [Flow Models for Arbitrary Conditional Likelihoods](https://mlsanity.com/paper/1909.06319/) (2019)
- [Score Matching With Missing Data](https://mlsanity.com/paper/2506.00557/) (2025)
- [CFMI: Flow Matching for Missing Data Imputation](https://mlsanity.com/paper/2506.09258/) (2025)

## Abstract (arXiv)

Flow matching assumes fully observed training data, which many real-world applications rarely provide. We study Missing-Data Flow Matching, which treats the missing coordinates of training samples as latent variables and averages the flow matching loss over the values they could take. We first prove that the oracle correction is exact: completing from the true pattern-conditioned posterior recovers the complete-data flow-matching objective under any missingness mechanism. Under missing completely at random (MCAR) and missing at random (MAR), this posterior equals the ordinary complete-data conditional, while under missing not at random (MNAR) it generally depends on the missingness mechanism and may require additional structure to identify. Our finite-sample analysis then answers design questions that the algorithm leaves open. Missingness transfers estimator variance rather than adding it, and for the fixed-parameter loss estimator one completion per example matches complete-data variance and is optimal under a fixed evaluation budget. A learned completion model contributes population completion bias, which we bound by its expected conditional Wasserstein distance to the true completion law. Experiments numerically validate the theoretical predictions, show that deterministic rather than frozen imputation reduces generated conditional variability, and place our method alongside strong classical and deep imputation baselines on real tabular data.