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Pointwise Convergence Analysis for Approximations of Optimal Transport Problems with a Target Measure that Has Unbounded Support

Axel G. R. Turnquist

Abstract

We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target measure is approximated, with special attention given to a cutoff approximation in which we parametrize the approximation by a ``cutoff" radius $R$ for the target measure. We study both the convergence of the mapping and potential functions for the forward and inverse problem in many cases such as 1) the radially symmetric case with the cutoff approximation for general cost functions and 2) the non-radially symmetric case with the squared distance cost function. We derive quantitative non-asymptotic pointwise convergence rates in special cases, building on the $L^2$ convergence rates established by Delalande and Mèrigot. These results can be used, for instance, to justify the use of certain types of numerical Monge-Ampère equation solvers in computationally solving the problem.

Pointwise Convergence Analysis for Approximations of Optimal Transport Problems with a Target Measure that Has Unbounded Support

Abstract

We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target measure is approximated, with special attention given to a cutoff approximation in which we parametrize the approximation by a ``cutoff" radius for the target measure. We study both the convergence of the mapping and potential functions for the forward and inverse problem in many cases such as 1) the radially symmetric case with the cutoff approximation for general cost functions and 2) the non-radially symmetric case with the squared distance cost function. We derive quantitative non-asymptotic pointwise convergence rates in special cases, building on the convergence rates established by Delalande and Mèrigot. These results can be used, for instance, to justify the use of certain types of numerical Monge-Ampère equation solvers in computationally solving the problem.
Paper Structure (28 sections, 37 theorems, 139 equations)

This paper contains 28 sections, 37 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. The Monge Problem of Optimal Transport
  3. The Unbounded Support Problem and the Cutoff Approximation
  4. Discussion on Computational Methods and Applications
  5. Discussion on Quantitative $L^2$ Stability Bounds
  6. Outline of Manuscript
  7. Definitions and Discussion
  8. Theoretical Background
  9. Rapid Convergence for Radial Case
  10. Radial Source and Target Distributions Lead to Radial Solutions
  11. Convergence Rates of Inverse Optimal Radial Mapping $T_{\nu}$
  12. Convergence of Radial Map $T_{\mu}$
  13. Convergence for the Inverse Radial Potential Function $\phi_{\nu; R}$
  14. Convergence Rates for Radial Potential Function $\phi_{\mu; R}$
  15. Convergence Rates of Distribution Function for Certain Classes of Distributions $\mu$
  16. ...and 13 more sections

Key Result

Theorem 2

Let $\mu$ have a probability density $f$ over a compact convex set $X$ and $f$ is bounded from above and below. Let $Y$ be a bounded connected open set with a Lipschitz boundary. Then, there exists a constant $C$ depending only on $\mu$, $X$ and $Y$ such that for any probability measures $\nu_{1}$ a where $\alpha(n)$ is an exponent that depends only on the ambient dimension $n$.

Theorems & Definitions (89)

  • Definition 1: Monge problem of optimal transport
  • Theorem 2
  • Theorem 3: Optimality Conditions for Optimal Transport with Squared Distance Cost Function, see Loeper LoeperReg, for example
  • Remark 4
  • Definition 5
  • Theorem 6
  • Definition 7: Monge-Ampère equation with second boundary value condition
  • Theorem 8: Monge problem of optimal transport in $\mathbb{R}$
  • Remark 9
  • Definition 10
  • ...and 79 more