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Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps

Sachin Shivakumar, Georgiy A. Bondar, Gabriel Khan, Abhishek Halder

TL;DR

This work tackles certifying the regularity of Monge optimal transport maps by verifying the Ma-Trudinger-Wang (MTW) tensor and the non-negative cost curvature (NNCC) through Sum-of-Squares (SOS) programming. It shows how to transform MTW/NNCC constraints into semidefinite programs when the MTW tensor entries are rational functions of the decision variables, and it extends to an inverse problem that computes semialgebraic regions where these conditions hold. The approach is demonstrated on multiple forward and inverse problems with both rational and certain non-rational costs, recovering known results and producing new inner approximations of regularity regions. The framework broadens the applicability of OT regularity verification to a wide class of ground costs and provides a practical tool for engineers and researchers working with OT-based methods.

Abstract

For a given ground cost, approximating the Monge optimal transport map that pushes forward a given probability measure onto another has become a staple in several modern machine learning algorithms. The fourth-order Ma-Trudinger-Wang (MTW) tensor associated with this ground cost function provides a notion of curvature in optimal transport. The non-negativity of this tensor plays a crucial role for establishing continuity for the Monge optimal transport map. It is, however, generally difficult to analytically verify this condition for any given ground cost. To expand the class of cost functions for which MTW non-negativity can be verified, we propose a provably correct computational approach which provides certificates of non-negativity for the MTW tensor using Sum-of-Squares (SOS) programming. We further show that our SOS technique can also be used to compute an inner approximation of the region where MTW non-negativity holds. We apply our proposed SOS programming method to several practical ground cost functions to approximate the regions of regularity of their corresponding optimal transport maps.

Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps

TL;DR

This work tackles certifying the regularity of Monge optimal transport maps by verifying the Ma-Trudinger-Wang (MTW) tensor and the non-negative cost curvature (NNCC) through Sum-of-Squares (SOS) programming. It shows how to transform MTW/NNCC constraints into semidefinite programs when the MTW tensor entries are rational functions of the decision variables, and it extends to an inverse problem that computes semialgebraic regions where these conditions hold. The approach is demonstrated on multiple forward and inverse problems with both rational and certain non-rational costs, recovering known results and producing new inner approximations of regularity regions. The framework broadens the applicability of OT regularity verification to a wide class of ground costs and provides a practical tool for engineers and researchers working with OT-based methods.

Abstract

For a given ground cost, approximating the Monge optimal transport map that pushes forward a given probability measure onto another has become a staple in several modern machine learning algorithms. The fourth-order Ma-Trudinger-Wang (MTW) tensor associated with this ground cost function provides a notion of curvature in optimal transport. The non-negativity of this tensor plays a crucial role for establishing continuity for the Monge optimal transport map. It is, however, generally difficult to analytically verify this condition for any given ground cost. To expand the class of cost functions for which MTW non-negativity can be verified, we propose a provably correct computational approach which provides certificates of non-negativity for the MTW tensor using Sum-of-Squares (SOS) programming. We further show that our SOS technique can also be used to compute an inner approximation of the region where MTW non-negativity holds. We apply our proposed SOS programming method to several practical ground cost functions to approximate the regions of regularity of their corresponding optimal transport maps.

Paper Structure

This paper contains 27 sections, 6 theorems, 70 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and Background
  3. The regularity problem of optimal transport
  4. Sum-of-Squares programming
  5. Problem Formulation
  6. Forward problem
  7. Inverse problem
  8. Numerical Results
  9. Forward problem
  10. Inverse problem
  11. Conclusions
  12. Nonnegative Polynomials and Sum-of-Squares Programming
  13. SOS polynomial
  14. Matrix-valued SOS polynomial
  15. SOS polynomials and Archimedean semialgebraic sets
  16. ...and 12 more sections

Key Result

Theorem 5

Given the semialgebraic set (CartesianProductSemialgebraic) with a ground cost function $c: \mathcal{X}\times\mathcal{Y}\to \mathbb{R}_{\geq 0}$, let $F$ in (eq:F_matrix) be of the form $F = \frac{F_N}{F_D}\in \mathbb{R}_{N,D}[x,y]$, $N,D\in\mathbb{N}$. If there exist $s_0,s_1,\hdots,s_{\ell}\in \su then $c$ satisfies the NNCC condition on $\mathcal{X}\times \mathcal{Y}$.

Figures (5)

  • Figure 1: Inner approximation of the region where MTW tensor is $\geq 0$ for Example 3.
  • Figure 2: Inner approximation of the region where MTW tensor is $\geq 0$ for Example 4.
  • Figure 3: The Motzkin polynomial.
  • Figure 4: Comparing contour plots of various ground costs $c(x,0)$ in Sec. \ref{['sec:Numerical']} vis-à-vis the squared Euclidean cost in $n=2$ dimensions. From left to right: squared Euclidean cost, perturbed squared Euclidean cost (Examples 1 and 3), log-partition cost (Example 2), squared distance cost for a surface with positive curvature (Example 4).
  • Figure : Bisection method to estimate the largest $\varepsilon$ for which MTW(0) holds

Theorems & Definitions (20)

  • Definition 1: MTW tensor or curvature
  • Definition 2: MTW($0$), MTW($\kappa$)
  • Definition 3: Non-negative cost curvature (NNCC)
  • Definition 4: Semialgebraic set
  • Theorem 5: NNCC forward problem
  • proof
  • Theorem 6: MTW($\kappa$) forward problem
  • proof
  • Theorem 7: NNCC inverse problem
  • proof
  • ...and 10 more