Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces
| Autor: | Sotiropoulou, Athina, Alvarez-Melis, David |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | Optimal Transport (OT) has recently emerged as a powerful framework for learning minimal-displacement maps between distributions. The predominant approach involves a neural parametrization of the Monge formulation of OT, typically assuming the same space for both distributions. However, the setting across ``incomparable spaces'' (e.g., of different dimensionality), corresponding to the Gromov- Wasserstein distance, remains underexplored, with existing methods often imposing restrictive assumptions on the cost function. In this paper, we present a novel neural formulation of the Gromov-Monge (GM) problem rooted in one of its fundamental properties: invariance to strong isomorphisms. We operationalize this property by decomposing the learnable OT map into two components: (i) an approximate strong isomorphism between the source distribution and an intermediate reference distribution, and (ii) a GM-optimal map between this reference and the target distribution. Our formulation leverages and extends the Monge gap regularizer of Uscidda & Cuturi (2023) to eliminate the need for complex architectural requirements of other neural OT methods, yielding a simple but practical method that enjoys favorable theoretical guarantees. Our preliminary empirical results show that our framework provides a promising approach to learn OT maps across diverse spaces. Comment: ICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling |
| Databáze: | arXiv |
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