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

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

TL;DR

This work analytically characterize Semi-dual Neural OT using the framework of regular measures, which generalize Lebesgue absolute continuity in finite dimensions and establishes a sharp characterization for the regularity of smoothed measures, proving that the success of smoothing depends strictly on the kernel of the covariance operator.

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.

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

TL;DR

This work analytically characterize Semi-dual Neural OT using the framework of regular measures, which generalize Lebesgue absolute continuity in finite dimensions and establishes a sharp characterization for the regularity of smoothed measures, proving that the success of smoothing depends strictly on the kernel of the covariance operator.

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.
Paper Structure (64 sections, 14 theorems, 162 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 64 sections, 14 theorems, 162 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Neural Optimal Transport in Hilbert Spaces
  3. Ill-posedness and Gaussian Smoothing.
  4. Background
  5. Optimal Transport
  6. Neural Optimal Transport
  7. Spurious Solutions in SNOT
  8. Analytical Results for Semi-dual Neural OT in Hilbert spaces
  9. General OT on Hilbert Spaces
  10. Regular Measure on Hilbert Spaces
  11. OT on Hilbert Spaces
  12. Connections to SNOT on Hilbert Spaces
  13. Method
  14. Gaussian Smoothing in Hilbert Spaces
  15. Convergence of Smoothed Monge Maps
  16. ...and 49 more sections

Key Result

Theorem 3.2

Consider the quadratic cost $c(x,y)=\frac{1}{2}\| x-y \|_{H}^2$ along with measures $\mu \in \mathcal{P}_2^r(H)$ and $\nu \in \mathcal{P}_2(H)$. Let $V^\star$ be a Kantorovich potential. Then, the following hold:

Figures (3)

  • Figure 1: Comparison of transport maps learned by vanilla HiSNOT (top row) and HiSNOT with annealed smoothing (bottom row). The vanilla model exhibits spurious solutions in singular cases (Perpendicular, One-to-Many, Grid), while our proposed smoothing correctly recovers the optimal transport plan.
  • Figure 2: Empirical validation of Theorem \ref{['25.12.09.14.29']}. Left: Appropriate smoothing covers singular directions, leading to a unique Monge map. Right: Inappropriate smoothing fails to cover singular directions, resulting in spurious solutions.
  • Figure 3: Illustration of optimal transport maps in Example 1

Theorems & Definitions (31)

  • Definition 3.1
  • Theorem 3.2: Characterization of Monge Map and Consistency of HiSNOT
  • Definition 4.1: Gaussian Measures
  • Theorem 4.2
  • Theorem 4.3
  • Proposition 4.4
  • Definition 2.1
  • Definition 2.2: Gâteaux Differential
  • Proposition 3.1: Monge and its dual problems in Hilbert space
  • Lemma 3.2
  • ...and 21 more