Reproduce “Neural Optimal Transport in Hilbert Spaces: Characterizing Spurious Solutions and Gaussian Smoothing” on OpenResearch

About this paper

Jae-Hwan Choi, Jiwoo Yoon, Dohyun Kwon, Jaewoong Choi

This research extends Semi-dual Neural Optimal Transport (SNOT) to infinite-dimensional Hilbert spaces, addressing the challenge of spurious solutions through a theoretically grounded Gaussian smoothing strategy. The resulting HiSNOT framework demonstrates robust performance and achieves competitive results on real-world time-series imputation tasks.

Key insights

• In a separable Hilbert space, a unique Monge map exists under the condition of "regular measures," and SNOT can consistently recover this map.
• Gaussian smoothing, by perturbing the source measure, can resolve ill-posedness, with smoothed Monge maps converging weakly to a true optimal plan as noise amplitude vanishes.
• The effectiveness of Gaussian smoothing is precisely characterized: the smoothed measure is regular if and only if the original measure's projection onto the kernel of the covariance operator is regular, implying noise must cover all singular directions.

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