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Solving Monge problem by Hilbert space embeddings of probability measures

Takafumi Saito, Yumiharu Nakano

TL;DR

It is proved that the transport maps given by the proposed methods converge to optimal transport maps in the problem with $L^2$ cost, which is applicable to large-scale Monge problems.

Abstract

We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space embeddings of probability measures. We prove that the transport maps given by the proposed methods converge to optimal transport maps in the problem with $L^2$ cost. Several numerical experiments validate our methods. In particular, we show that our methods are applicable to large-scale Monge problems. This is a corrected version of the ICORES 2025 proceedings paper.

Solving Monge problem by Hilbert space embeddings of probability measures

TL;DR

It is proved that the transport maps given by the proposed methods converge to optimal transport maps in the problem with cost, which is applicable to large-scale Monge problems.

Abstract

We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space embeddings of probability measures. We prove that the transport maps given by the proposed methods converge to optimal transport maps in the problem with cost. Several numerical experiments validate our methods. In particular, we show that our methods are applicable to large-scale Monge problems. This is a corrected version of the ICORES 2025 proceedings paper.

Paper Structure

This paper contains 12 sections, 1 theorem, 27 equations, 9 figures, 4 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PENALTY METHOD
  3. Hilbert space embeddings of probability measures
  4. Proposed methods
  5. Theoretical result
  6. NUMERICAL EXPERIMENTS
  7. Interpolation of synthetic datasets
  8. From moon to circle
  9. From normal distribution to moon
  10. From normal distribution to normal ditribution
  11. Comparison with POT
  12. CONCLUSION

Key Result

Theorem 2.1

Let $c(x,y)=|x-y|^2$. Suppose that $\mu$ is absolutely continuous with respect to the Lebesgue measure and that Suppose moreover that $\gamma$ metrizes the weak topology on $\mathcal{P}(\mathbb{R}^d)$. Then, where $T^*$ is the unique optimal transport map. In particular, $\{T_n\}_{n=1}^{\infty}$ converges to $T^*$ in law under $\mu$.

Figures (9)

  • Figure 1: Initial distribution
  • Figure 2: Target distribution (blue) and generated samples (orange).
  • Figure 3: Loss curve
  • Figure 4: Initial distribution
  • Figure 5: Target distribution (blue) and generated samples (orange)
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 2.1
  • Proof 1