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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan

ICML 2025 · Learning theory and algorithms › Optimal transport · approach: Transport map estimation · 0 citations (OpenAlex)

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Analyzes when max-min semi-dual neural OT recovers the true OT map and proposes OTP, which smooths the source distribution with decreasing noise to learn the optimal transport plan, outperforming entropic and semi-dual OT baselines on synthetic and image translation tasks.

Problem

Semi-dual neural OT (max-min adversarial formulation) often converges to spurious solutions that do not push the source distribution onto the target, and cannot represent stochastic OT plans when no deterministic OT map exists.

Approach

Gives a sufficient condition (source puts no mass on sets of Hausdorff dimension at most d-1) under which the max-min SNOT solution is the true OT map, and trains OTP by smoothing the source with Gaussian or variance-preserving noise, annealing the noise level to a small minimum so the learned plan converges to the OT plan.

Key results

• On Male-to-Female 128x128, OTP reaches FID 6.38 versus 7.55 for OTM, outperforming DIOTM, DSBM and ASBM.
• On MNIST-to-CMNIST, OTP achieves FID 3.18 versus 62.4 for OTM-s, avoiding the mode collapse of the deterministic-style baseline.
• On synthetic d=16 One-to-Many, OTP reduces D_target to 0.65 versus 73.23 for OTM.

Measured effects reported in the paper

Surprising findings

• The vanilla OTM fails to cover the target measure in simple Perpendicular and One-to-Many cases even though the OT map exists in those settings.
• Adding a stochastic generator (OTM-s) on MNIST-to-CMNIST caused mode collapse to blue images rather than fixing the problem.

Limitations

• Convergence of the OTP plan holds only up to a subsequence.
• The derived condition is sufficient, not necessary, for avoiding spurious solutions.
• Generalization of the analysis to Polish spaces with general costs is left open.

Experimental setup

Datasets: MNIST CMNIST (colored MNIST, constructed) Male-to-Female Wild-to-Cat synthetic Perpendicular / One-to-Many datasets

Benchmarks: Perpendicular and One-to-Many synthetic datasets MNIST-to-CMNIST Male-to-Female 64x64 and 128x128 Wild-to-Cat 64x64

Metrics: transport cost error D_cost target distribution error D_target (W2) FID LPIPS

Baselines: OTM OTM-s NOT DIOTM DSBM ASBM

Builds on: Semi-dual Neural OT (OTM) Kantorovich duality Gradual entropic OT schedule

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