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Generative Conditional Distributions by Neural (Entropic) Optimal Transport
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Proposes GENTLE, a neural generative model that learns conditional distributions by minimax training of a generative inverse-CDF network and a semi-dual entropic OT potential network, with a Lipschitz-style OT regularizer across covariate neighbors.
Problem
Learning generative models of conditional distributions Y|X=x from small, unbalanced samples where many covariate values have only one or few responses, causing overfitting and mode collapse.
Approach
Parameterize the conditional inverse CDF with a network T(x,U), fit it with a CDF-based fitness term, and regularize smoothness across covariates by penalizing semi-dual entropic OT distances between MST-linked covariate pairs, optimized with a doubly-smoothed gradient descent-ascent.
Key results
- On LDW-CPS, GENTLE reaches WD 3767.62 and KS 0.48, versus 4970.66 / 0.56 for CDSB and 13371.99 / 0.88 for CWGAN.
- On ECM, GENTLE reaches WD 0.97 and KS 0.21, versus 1.09 / 0.55 for MGAN and 4.42 / 0.94 for CWGAN.
- Removing the OT regularizer raises LDW-CPS WD to 13625.83, showing the regularizer matters most on unbalanced covariates.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use GENTLE instead of CWGAN | conditional distribution generation · LDW-CPS test | Wasserstein distance: 13371.99 -> 3767.62 | Helped |
| use GENTLE instead of CWGAN | conditional distribution generation · ECM test | KS statistic: 0.94 -> 0.21 | Helped |
| use GENTLE instead of MGAN | conditional distribution generation · LDW-CPS test | Wasserstein distance: 4096.75 -> 3767.62 | Helped |
| use GENTLE instead of MGAN | conditional distribution generation · ECM test | Wasserstein distance: 1.09 -> 0.97 | Helped |
| use GENTLE instead of WGAN-GP | conditional distribution generation · LDW-CPS test | KS statistic: 0.99 -> 0.48 | Helped |
| use GENTLE instead of CDSB | conditional distribution generation · ECM test | Wasserstein distance: 2.04 -> 0.97 | Helped |
| add the regularization term (ablation) | GENTLE · LDW-CPS test | Wasserstein distance: 13625.83 -> 3767.62 | Helped |
| add the regularization term (ablation) | GENTLE · LDW-CPS test | KS statistic: 0.57 -> 0.48 | Helped |
| add the regularization term (ablation) | GENTLE · ECM test | Wasserstein distance: 1.13 -> 0.97 | Helped |
| add the regularization term (ablation) | GENTLE · ECM test | KS statistic: 0.29 -> 0.21 | Helped |
| add the smoothing term (ablation) | GENTLE · LDW-CPS test | Wasserstein distance: 6075.02 -> 3767.62 | Helped |
| add the smoothing term (ablation) | GENTLE · LDW-CPS test | KS statistic: 0.41 -> 0.48 | Hurt |
Surprising findings
- Removing the smoothing term gives a lower KS (0.41) than the full GENTLE model on LDW-CPS, though with much worse WD.
- MGAN achieves MSE and R^2 comparable to GENTLE on LDW-CPS despite collapsing toward the mean in distribution metrics.
Limitations
- Restricted to one-dimensional responses; the paper says extension beyond 1D is future work.
- Regularizer requires a valid edge set (at most one partner per covariate), built as an MST over training covariates.
- Hyperparameters (KDE bandwidth, epsilon, lambda, r1, r2) are selected by grid search on a validation set.
Experimental setup
Datasets: LDW-CPS ECM (Esophageal Cancer Markov Chain Simulation Model)
Benchmarks: LDW-CPS ECM
Metrics: Wasserstein distance Kolmogorov-Smirnov statistic MSE R^2
Baselines: CWGAN WGAN-GP MGAN CDSB
Builds on: Wasserstein GAN neural optimal transport Sinkhorn algorithm noise outsourcing lemma doubly-smoothed gradient descent-ascent
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