Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces
<|reference_start|>Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces: 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.<|reference_end|>
@article{sotiropoulou2024strongly,
    title={Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces},
    author={Athina Sotiropoulou, David Alvarez-Melis},
    journal={arXiv preprint arXiv:2407.14957},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.14957},
    primaryClass={cs.LG}
}