解决无限维空间中神经最优传输的虚假解问题，提升分布对齐精度。
Neural Optimal Transport in Hilbert Spaces: Characterizing Spurious Solutions and Gaussian Smoothing
- 基于正则测度理论分析虚假解成因，提出高斯平滑改进半对偶框架。
- 在正则源测度下保证解唯一性，且能恢复唯一的Monge映射。
- 适用于函数型数据与时间序列，特别适合对平滑性敏感的任务。
我们研究了无限维希尔伯特空间中的神经最优传输。在非正则设置下，半对偶神经OT常产生无法准确捕捉目标分布的虚假解。本文通过正则测度框架（推广有限维中的勒贝格绝对连续性）对这一问题进行解析表征。为解决不适定性，我们基于布朗运动引入高斯平滑策略，扩展半对偶框架。主要理论贡献证明：在正则源测度条件下，该形式是适定的，并可恢复唯一Monge映射。进一步建立平滑测度正则性的精确刻画，证明平滑效果严格依赖于协方差算子的核。在合成函数数据与时间序列数据上的实验表明，该方法有效抑制虚假解，显著优于现有基线。
原文摘要 · Abstract (English)
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.
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