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

2026-02-15

AI Summary

This work addresses the issue of spurious solutions in neural optimal transport within infinite-dimensional Hilbert spaces, which often impedes accurate approximation of the target distribution. The study provides the first rigorous characterization of the underlying cause and introduces an analytical framework grounded in regularity measure theory. To resolve ill-posedness, the authors propose a Gaussian smoothing mechanism that leverages the kernel of the covariance operator—implemented via Brownian motion—to smooth the input distribution. This regularization ensures well-posedness of the semi-dual formulation of neural optimal transport and recovers the unique Monge map. Empirical evaluations on both synthetic functions and time-series data demonstrate that the method effectively suppresses spurious solutions and significantly outperforms existing baselines.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Neural Optimal Transport
• spurious solutions
• Hilbert spaces
• regular measures
• Gaussian smoothing

Innovation

• Neural Optimal Transport
• Hilbert Spaces
• Spurious Solutions
• Gaussian Smoothing
• Monge Map