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Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs

Alessandro Scagliotti, Sara Farinelli

TL;DR

The paper addresses recovering the $W_2$-optimal transport map $T$ between absolutely continuous measures by flows of linear-control ODEs with time-dependent, finite-dimensional controls. It proves that, under regularity and Lie-algebra strong approximating conditions, $T$ is isotopic to the identity and lies in the $C^0_c$-closure of the linear-control flows, enabling approximation by such flows; it extends this to data-driven settings via a $\Gamma$-convergence framework that relates discrete OT couplings to continuous OT maps. A Pontryagin Maximum Principle-based numerical scheme is developed to minimize the discrete OT-oriented functional $\mathcal{F}^{N,\beta}$, yielding a computable normalizing flow that approximates $T$ in $L^2_\mu$ as $\beta\to 0$. The approach is validated in a two-dimensional experiment, illustrating practical recovery of the target transport and connecting optimal transport with normalizing-flow-style neural ODEs under rigorous guarantees.

Abstract

In this paper, we consider the problem of recovering the $W_2$-optimal transport map T between absolutely continuous measures $μ,ν\in\mathcal{P}(\mathbb{R}^n)$ as the flow of a linear-control neural ODE, where the control depends only on the time variable and takes values in a finite-dimensional space. We first show that, under suitable assumptions on $μ,ν$ and on the controlled vector fields governing the neural ODE, the optimal transport map is contained in the $C^0_c$-closure of the flows generated by the system. Then, we tackle the problem under the assumption that only discrete approximations of $μ_N,ν_N$ of the original measures $μ,ν$ are available: we formulate approximated optimal control problems, and we show that their solutions give flows that approximate the original optimal transport map $T$. In the framework of generative models, the approximating flow constructed here can be seen as a `Normalizing Flow', which usually refers to the task of providing invertible transport maps between probability measures by means of deep neural networks. We propose an iterative numerical scheme based on the Pontryagin Maximum Principle for the resolution of the optimal control problem, resulting in a method for the practical computation of the approximated optimal transport map, and we test it on a two-dimensional example.

Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs

TL;DR

The paper addresses recovering the -optimal transport map between absolutely continuous measures by flows of linear-control ODEs with time-dependent, finite-dimensional controls. It proves that, under regularity and Lie-algebra strong approximating conditions, is isotopic to the identity and lies in the -closure of the linear-control flows, enabling approximation by such flows; it extends this to data-driven settings via a -convergence framework that relates discrete OT couplings to continuous OT maps. A Pontryagin Maximum Principle-based numerical scheme is developed to minimize the discrete OT-oriented functional , yielding a computable normalizing flow that approximates in as . The approach is validated in a two-dimensional experiment, illustrating practical recovery of the target transport and connecting optimal transport with normalizing-flow-style neural ODEs under rigorous guarantees.

Abstract

In this paper, we consider the problem of recovering the -optimal transport map T between absolutely continuous measures as the flow of a linear-control neural ODE, where the control depends only on the time variable and takes values in a finite-dimensional space. We first show that, under suitable assumptions on and on the controlled vector fields governing the neural ODE, the optimal transport map is contained in the -closure of the flows generated by the system. Then, we tackle the problem under the assumption that only discrete approximations of of the original measures are available: we formulate approximated optimal control problems, and we show that their solutions give flows that approximate the original optimal transport map . In the framework of generative models, the approximating flow constructed here can be seen as a `Normalizing Flow', which usually refers to the task of providing invertible transport maps between probability measures by means of deep neural networks. We propose an iterative numerical scheme based on the Pontryagin Maximum Principle for the resolution of the optimal control problem, resulting in a method for the practical computation of the approximated optimal transport map, and we test it on a two-dimensional example.
Paper Structure (14 sections, 16 theorems, 100 equations, 1 figure, 1 algorithm)

This paper contains 14 sections, 16 theorems, 100 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Preliminaries and Notations
  2. Preliminaries on Optimal Transport
  3. Preliminaries on linear-control systems
  4. Approximability of the optimal transport map
  5. Optimal control problems and $\Gamma$-convergence
  6. Existence of minimizers
  7. $\Gamma$-convergence result
  8. Optimal transport map approximation
  9. Numerical approximation of the optimal transport map
  10. Preliminary optimal transport problem
  11. Pontryagin Maximum Principle
  12. Algorithm description
  13. A numerical experiment
  14. Proofs of Section \ref{['subsec:lin-ctrl']}

Key Result

Lemma 1.1

Let $(\mu_N)_{N\geq 1}$ be a sequence of probability measures of $\mathbb{R}^n$ and $\mu_{\infty}\in\mathcal{P}(\mathbb{R}^n)$ such that $\mu_N\rightharpoonup^*\mu_{\infty}$ as $N\to +\infty$. Let $T:\mathbb{R}^n\to\mathbb{R}^{n'}$ be a continuous map. Then $T_{\sharp}\mu_N\rightharpoonup^*T_{\sharp

Figures (1)

  • Figure 1: Approximation of the optimal transport map using samplings of the transported measure.

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Lemma 1.1
  • proof
  • Definition 3
  • Proposition 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • ...and 40 more