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Regularity of Solutions to Beckmann's Parametric Optimal Transport

Hanno Gottschalk, Tobias J. Riedlinger

Abstract

Beckmann's problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann's problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann's divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential's gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact Hölder regularity of the potential, the flux and the flow generating is derived on the basis of Hölder regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (``promptable'') generative learning, we provide sufficient conditions for separate and joint Hölder continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^(k,alpha)-Hölder norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.

Regularity of Solutions to Beckmann's Parametric Optimal Transport

Abstract

Beckmann's problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann's problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann's divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential's gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact Hölder regularity of the potential, the flux and the flow generating is derived on the basis of Hölder regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (``promptable'') generative learning, we provide sufficient conditions for separate and joint Hölder continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^(k,alpha)-Hölder norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.
Paper Structure (18 sections, 13 theorems, 54 equations)

This paper contains 18 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Transport Maps and Flows
  4. Beckmann's problem
  5. Unconstrained formulation of the Beckmann problem
  6. Potential theory and Schauder estimates
  7. Regular solutions to Beckmann's problem
  8. Transport Vector Field and Transport Map Regularity
  9. Parametric Beckmann Problems
  10. Banach-Valued Hölder Functions
  11. Hölder Differentiability of the Beckmann Potential in Time
  12. Inclusion of Banach-Valued Hölder Spaces in Higher-Order Joint Hölder Spaces
  13. Regular Solutions to the Parametric Beckmann Problem
  14. Regular Transport Vector Fields for Generic Probability Paths
  15. Generalized Setting for Probability Paths
  16. ...and 3 more sections

Key Result

Theorem 3.2

Let $d>2$, $\alpha\in(0,1)$ and $\Omega$ fulfill assumption assumption:smooth-domain for $k=0$. Let $f\in C^{0,\alpha}(\Omega,\mathbb{R})$ be given and $g:\partial \Omega\to \mathbb{R}$ be in $C^{1,\alpha}(\partial\Omega,\mathbb{R})$. If $\int_\Omega f\, \mathrm{d}x+\int_{\partial\Omega} g \,\mathrm

Theorems & Definitions (25)

  • Theorem 3.2: Existence and $C^{2,\alpha}$ Schauder estimate nardi2015schauder
  • Theorem 3.3: $C^{k,\alpha}$ Schauder estimate agmon1959estimates
  • Proof 1
  • Theorem 3.4: Existence, uniqueness and regularity of solutions of the Beckmann problem
  • Proof 2
  • Theorem 3.5: Regularity of the transport vector field, flow map and flow end point
  • Proof 3
  • Proposition 4.1
  • Proof 4
  • Corollary 4.2
  • ...and 15 more