Neural Optimal Transport in Hilbert Spaces: Characterizing Spurious Solutions and Gaussian Smoothing
This addresses a theoretical and practical limitation in optimal transport methods for functional data, though it appears incremental as an extension of existing neural OT frameworks.
The paper tackles the problem of spurious solutions in Neural Optimal Transport in infinite-dimensional Hilbert spaces, showing that Semi-dual Neural OT often fails to accurately capture target distributions in non-regular settings. The authors resolve this by extending the framework with Gaussian smoothing, proving well-posedness and unique Monge map recovery under regular source measures, and empirically demonstrating effective suppression of spurious solutions and outperformance of baselines on synthetic functional data and time-series datasets.
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.