Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces

Athina Sotiropoulou, David Alvarez-Melis
Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM), PMLR 251:381-393, 2024.

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 \citet{gap_monge} 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v251-sotiropoulou24a, title = {Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces}, author = {Sotiropoulou, Athina and Alvarez-Melis, David}, booktitle = {Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM)}, pages = {381--393}, year = {2024}, editor = {Vadgama, Sharvaree and Bekkers, Erik and Pouplin, Alison and Kaba, Sekou-Oumar and Walters, Robin and Lawrence, Hannah and Emerson, Tegan and Kvinge, Henry and Tomczak, Jakub and Jegelka, Stephanie}, volume = {251}, series = {Proceedings of Machine Learning Research}, month = {29 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v251/main/assets/sotiropoulou24a/sotiropoulou24a.pdf}, url = {https://proceedings.mlr.press/v251/sotiropoulou24a.html}, 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 \citet{gap_monge} 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.} }
Endnote
%0 Conference Paper %T Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces %A Athina Sotiropoulou %A David Alvarez-Melis %B Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM) %C Proceedings of Machine Learning Research %D 2024 %E Sharvaree Vadgama %E Erik Bekkers %E Alison Pouplin %E Sekou-Oumar Kaba %E Robin Walters %E Hannah Lawrence %E Tegan Emerson %E Henry Kvinge %E Jakub Tomczak %E Stephanie Jegelka %F pmlr-v251-sotiropoulou24a %I PMLR %P 381--393 %U https://proceedings.mlr.press/v251/sotiropoulou24a.html %V 251 %X 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 \citet{gap_monge} 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.
APA
Sotiropoulou, A. & Alvarez-Melis, D.. (2024). Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces. Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM), in Proceedings of Machine Learning Research 251:381-393 Available from https://proceedings.mlr.press/v251/sotiropoulou24a.html.

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