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

Jaemoo Choi, Jaewoong Choi, Dohyun Kwon

TL;DR: We overcome fake solutions in semi-dual Neural Optimal Transport and propose a smoothing approach for learning the Optimal Transport Plan.

Abstract: We address the convergence problem in learning the Optimal Transport (OT) map, where the OT Map refers to a map from one distribution to another while minimizing the transport cost. Semi-dual Neural OT, a widely used approach for learning OT Maps with neural networks, often generates spurious solutions that fail to transfer one distribution to another accurately. We identify a sufficient condition under which the max-min solution of Semi-dual Neural OT recovers the true OT Map. Moreover, to address cases when this sufficient condition is not satisfied, we propose a novel method, OTP, which learns both the OT Map and the Optimal Transport Plan, representing the optimal coupling between two distributions. Under sharp assumptions on the distributions, we prove that our model eliminates the spurious solution issue and correctly solves the OT problem. Our experiments show that the OTP model recovers the optimal transport map where existing methods fail and outperforms current OT-based models in image-to-image translation tasks. Notably, the OTP model can learn stochastic transport maps when deterministic OT Maps do not exist, such as one-to-many tasks like colorization.

Lay Summary:

Optimal Transport (OT) is a powerful mathematical framework for mapping one distribution to another, widely used in machine learning tasks such as image translation and domain adaptation. While neural networks have recently been applied to learn these OT maps, many popular methods often produce spurious solutions — results that satisfy the learning objectives but fail to correctly transform the source distribution into the target distribution.

We analyzed this issue and identified a mathematical condition that guarantees these neural OT methods can recover the correct OT map. However, this condition is often violated in practice. To address this, we developed a new method — called the Optimal Transport Plan (OTP) model — that reliably learns the correct transformation even when the condition does not hold.

Our approach smooths the source data distribution to avoid pathological cases, learns the OT plan from the smoothed data, and then gradually returns to the original data. This allows the model to recover both deterministic and stochastic transport plans, where previous methods fail. OTP demonstrates strong performance on synthetic and real-world tasks, providing a robust foundation for neural OT in complex settings.

Primary Area: Deep Learning->Generative Models and Autoencoders

Keywords: Optimal Transport, Spurious solution, Semi-dual formulation, Image-to-Image translation

Submission Number: 9116