Flow Matching with Missing Data
Missing-Data Flow Matching treats missing coordinates as latent variables and averages the flow matching loss over possible completions, with the correction proven exact rather than approximate Under missing completely at random with true completions, the incomplete-data objective equals the complete-data objective, shifting all difficulty to the completion model Finite-sample analysis reveals counterintuitive results: missingness transfers rather than adds estimator variance, and a single compl
Analysis
TL;DR
- Missing-Data Flow Matching treats missing coordinates as latent variables and averages the flow matching loss over possible completions, with the correction proven exact rather than approximate
- Under missing completely at random with true completions, the incomplete-data objective equals the complete-data objective, shifting all difficulty to the completion model
- Finite-sample analysis reveals counterintuitive results: missingness transfers rather than adds estimator variance, and a single completion per example already matches complete-data variance exactly
- A learned completion model introduces a single irreducible bias bounded by its expected conditional Wasserstein distance to the true completion law
- Experiments validate theory and show deterministic (rather than frozen) imputation is what collapses the generated distribution
Why It Matters
This work bridges a critical gap between flow matching theory and real-world practice, where missing data is the norm rather than the exception. By proving that the proposed correction is exact and providing finite-sample guarantees, it gives practitioners a theoretically sound foundation for training generative models on incomplete datasets without resorting to ad-hoc imputation strategies.
Technical Details
- The method formulates Missing-Data Flow Matching by treating missing coordinates as latent variables and averaging the flow matching loss over their possible values, with a proof that the correction is exact under missing completely at random (MCAR) with true completions
- Finite-sample analysis demonstrates that missingness transfers estimator variance rather than adding it, and that one completion per example achieves complete-data variance exactly, making it optimal under a fixed evaluation budget
- The irreducible bias from a learned completion model is bounded by the expected conditional Wasserstein distance between the learned completion distribution and the true completion law
- Experiments on real tabular data compare against strong classical and deep imputation baselines, with key findings that deterministic imputation collapses the generated distribution while learned (non-frozen) imputation preserves it
Industry Insight
- Practitioners working with real-world tabular data should avoid deterministic imputation pipelines when combining with flow matching, as they provably collapse distributional fidelity; learned, non-frozen completion models are essential
- The theoretical result that one completion suffices under a fixed budget can significantly reduce computational costs in training, challenging the intuition that multiple imputations are necessary
- As missing data is pervasive in healthcare, finance, and scientific domains, this framework provides a principled path toward deploying flow-based generative models in production settings where data completeness cannot be guaranteed
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