Neural Monge Map estimation and its applications

Published: 30 Jul 2023, Last Modified: 02 Apr 2024Accepted by TMLREveryoneRevisionsBibTeX
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Abstract: Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. Neural network-based optimal transport map solver has gained great attention in recent years. Along this line, we present a scalable algorithm for computing the neural Monge map between two probability distributions. Our algorithm is based on a weak form of the optimal transport problem, thus it only requires samples from the marginals instead of their analytic expressions, and can be applied in large-scale settings. Furthermore, using the duality gap we prove rigorously \textit{a posteriori} error analysis for the method. Our algorithm is suitable for general cost functions, compared with other existing methods for estimating Monge maps using samples, which are usually for quadratic costs. The performance of our algorithms is demonstrated through a series of experiments with both synthetic and realistic data, including text-to-image generation, class-preserving map, and image inpainting tasks.
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Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: We highlight the changes in blue. The main changes are listed below. ## Notation We have added clarification for some notations, such as footnote 3. ## Experiment We added the comparison with Perrot et al. 2016 method for text-to-image embedding mapping. Please see Figure 4. We added the comparison to quadratic cost for text-to-image embedding mapping. Please see the end of Section D. We compared the advantages and disadvantages of the stochastic map and the deterministic map. Please see the end of Section 6.2. We added the comparison to quadratic cost for the population transport example. Please see Figure 12. We quantitatively compared our method to the ground truth Monge map for three low-dimension examples. Please see Table 3. ## Theory We discussed the regularity of $T_*$ and $\psi_*$ in Remark 5.
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Assigned Action Editor: ~Makoto_Yamada3
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 807