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Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces

Athina Sotiropoulou, David Alvarez-Melis

TL;DR

A novel neural formulation of the Gromov-Monge (GM) problem rooted in one of its fundamental properties: invariance to strong isomorphisms is presented, yielding a simple but practical method that enjoys favorable theoretical guarantees.

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.

Strongly Isomorphic Neural Optimal Transport Across Incomparable Spaces

TL;DR

A novel neural formulation of the Gromov-Monge (GM) problem rooted in one of its fundamental properties: invariance to strong isomorphisms is presented, yielding a simple but practical method that enjoys favorable theoretical guarantees.

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.
Paper Structure (18 sections, 3 theorems, 29 equations, 8 figures, 1 algorithm)

This paper contains 18 sections, 3 theorems, 29 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Metric measure spaces and Strong Isomorphism
  4. The Gromov-Monge problem
  5. The Gromov-Wasserstein Problem
  6. GM-Optimal Map Decomposition
  7. A tripod structure of mm-spaces
  8. Optimal map decomposition
  9. Isomorphism Invariant Neural Gromov-Monge
  10. Neural Network Compositions as Universal Approximators of Transport Maps
  11. Learning GM-optimal maps
  12. Experiments
  13. Discussion
  14. Proofs
  15. Proof of Proposition \ref{['prop: gm_mn_rn_equality']}
  16. ...and 3 more sections

Key Result

Proposition 3.1

Let $\mathcal{X}_{\mu},\mathcal{Z}_{\rho}$ ,$\mathcal{Y}_{\nu} \in \mathcal{M}_p$ such that $\mathcal{X}_{\mu} \cong^{s} \mathcal{Z}_{\rho}$. Then for $p \in \left[ 1, \infty \right)$ it holds that $GM_p(\mu,\nu)=GM_p(\rho,\nu)$.

Figures (8)

  • Figure 1: A tripod structure between mm-spaces where $\mathcal{X}_{\mu} \cong^s \mathcal{Z}_{\rho}$. $\phi \in \Phi(\mu,\rho)$ represents the collection of isomorphisms, while $\mathcal{T}(\mu,\nu)$ and $\widetilde{\mathcal{T}}(\rho,\nu)$ are the sets of all transport maps between the corresponding spaces.
  • Figure : (a)
  • Figure : (a)
  • Figure : (a)
  • Figure : (b)
  • ...and 3 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 4.1