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A dynamical formulation of multi-marginal optimal transport

Brendan Pass, Yair Shenfeld

TL;DR

The primal-dual dynamical formulation of the multi-marginal optimal transport problem for (semi-)convex cost functions yields a convex optimization problem, enabling the use of convex optimization tools to find quasi-Monge solutions of the static multi-marginal problem for translation-invariant costs.

Abstract

We present a primal-dual dynamical formulation of the multi-marginal optimal transport problem for (semi-)convex cost functions. Even in the two-marginal setting, this formulation applies to cost functions not covered by the classical dynamical approach of Benamou-Brenier. Our dynamical formulation yields a convex optimization problem, enabling the use of convex optimization tools to find quasi-Monge solutions of the static multi-marginal problem for translation-invariant costs. We illustrate our results numerically with proximal splitting methods.

A dynamical formulation of multi-marginal optimal transport

TL;DR

The primal-dual dynamical formulation of the multi-marginal optimal transport problem for (semi-)convex cost functions yields a convex optimization problem, enabling the use of convex optimization tools to find quasi-Monge solutions of the static multi-marginal problem for translation-invariant costs.

Abstract

We present a primal-dual dynamical formulation of the multi-marginal optimal transport problem for (semi-)convex cost functions. Even in the two-marginal setting, this formulation applies to cost functions not covered by the classical dynamical approach of Benamou-Brenier. Our dynamical formulation yields a convex optimization problem, enabling the use of convex optimization tools to find quasi-Monge solutions of the static multi-marginal problem for translation-invariant costs. We illustrate our results numerically with proximal splitting methods.

Paper Structure

This paper contains 20 sections, 11 theorems, 160 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The two-marginal optimal transport problem
  3. A dynamical formulation of multi-marginal optimal transport
  4. Convex formulations and quasi-Monge solutions
  5. Duality
  6. A dynamical formulation of multi-marginal optimal transport
  7. Main result
  8. Heuristic argument for Theorem \ref{['thm:static_dynamic_equiv']}
  9. Convexity of the dynamical cost
  10. Constructions of flows
  11. Proof of Theorem \ref{['thm:static_dynamic_equiv']}
  12. The dual dynamical multi-marginal optimal transport problem
  13. Main result
  14. Heuristic argument for Theorem \ref{['thm:dual_dynamical']}
  15. Proof of Theorem \ref{['thm:dual_dynamical']}
  16. ...and 5 more sections

Key Result

Theorem 1.6

Let $\Omega\subseteq \mathbb R^{d}$ be a compact convex set, and let $\mu_1,\ldots,\mu_{k}$ be probability measures on $\Omega$. Fix a probability measure $p$ on $\Omega^{k}$, and let $L:\mathbb R^{kd}\to \mathbb R$ be a convex function which is $p$-translation-invariant. Then, where the minimum on the right-hand side of eq:static_dynamic_equiv_informal is over $(\pi,\mathrm v)$ satisfying

Figures (2)

  • Figure 1: In our numerical experiment we solve the dynamical multi-marginal optimal transport with quadratic cost \ref{['eq:quadratic_cost_mmot_intro']} for the 3 marginals $\mu_1,\mu_2,\mu_3$.
  • Figure 2: The optimal coupling of the static multi-marginal optimal transport with quadratic cost for the marginals $\mu_1,\mu_2,\mu_3$ of Figure \ref{['fig:3marginals']} can be computed analytically in terms of transport maps between $\mu_1$ to $\mu_2$ (res. $\mu_3$), which are expressed in terms of cumulative distribution functions; see Equation \ref{['eq:analytic_transport_GS']}. In the figure we compare these exact solutions to the solutions obtained from our algorithm; see Equation \ref{['eq:transport_approx']}.

Theorems & Definitions (38)

  • Definition 1.1: $p$-translation-invariant cost functions
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 1.6: Informal; Theorem \ref{['thm:static_dynamic_equiv']}
  • Corollary 1.7: Informal; Corollary \ref{['cor:static_dynamic_equiv']}
  • Remark 1.8: Semi-convex cost functions; Corollary \ref{['cor:static_dynamic_equiv_semicvx']}
  • Example 1.9: Quadratic cost
  • Theorem 1.10: Informal; Theorem \ref{['thm:dual_dynamical']}
  • ...and 28 more