Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces

• URL: http://arxiv.org/abs/2407.14957v1
• Date: Sat, 20 Jul 2024 18:27:11 GMT
• Title: Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces
• Authors: Athina Sotiropoulou, David Alvarez-Melis,
• Abstract summary: We present a novel neural formulation of the Gromov-Monge problem rooted in one of its fundamental properties. We operationalize this property by decomposing the learnable OT map into two components. Our framework provides a promising approach to learn OT maps across diverse spaces.
• Score: 7.535219325248997
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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