Neural Optimal Transport in Hilbert Spaces: Characterizing Spurious Solutions and Gaussian Smoothing

TL;DR

This paper investigates neural optimal transport in infinite-dimensional Hilbert spaces, identifying issues with spurious solutions in non-regular settings and proposing Gaussian smoothing to ensure well-posedness and accurate distribution matching.

Contribution

It analytically characterizes spurious solutions in neural optimal transport and introduces a Gaussian smoothing method that guarantees unique solutions under regular measures.

Findings

Gaussian smoothing suppresses spurious solutions

The method recovers a unique Monge map under regular measures

Outperforms existing baselines on synthetic and time-series data

Abstract

We study Neural Optimal Transport in infinite-dimensional Hilbert spaces. In non-regular settings, Semi-dual Neural OT often generates spurious solutions that fail to accurately capture target distributions. We analytically characterize this spurious solution problem using the framework of regular measures, which generalize Lebesgue absolute continuity in finite dimensions. To resolve ill-posedness, we extend the semi-dual framework via a Gaussian smoothing strategy based on Brownian motion. Our primary theoretical contribution proves that under a regular source measure, the formulation is well-posed and recovers a unique Monge map. Furthermore, we establish a sharp characterization for the regularity of smoothed measures, proving that the success of smoothing depends strictly on the kernel of the covariance operator. Empirical results on synthetic functional data and time-series datasets demonstrate that our approach effectively suppresses spurious solutions and outperforms existing baselines.