用多项式优化证明最优传输映射的光滑性条件，扩展可验证场景。
Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps
- 基于平方和编程构造MTW曲率非负性证书
- 可计算出使映射光滑的代价函数区域范围
- 适用于机器学习中常见代价函数的理论验证
给定一个基代价函数，逼近将一个概率测度推送到另一个的概率测度的Monge最优传输映射已成为现代机器学习算法中的标准工具。与该代价函数相关的四阶Ma-Trudinger-Wang（MTW）张量提供了最优传输中的曲率概念。该张量的非负性对建立Monge最优传输映射的连续性至关重要。然而，对于任意给定的代价函数，通常难以解析验证这一条件。为扩大可验证MTW非负性的代价函数类别，我们提出一种可证明正确的计算方法，利用平方和（SOS）编程为MTW张量的非负性提供证书。我们进一步表明，该SOS技术还可用于计算MTW非负性成立的内部近似区域。我们将所提出的SOS编程方法应用于多个实际代价函数，以近似其对应最优传输映射的正则性区域。
原文摘要 · Abstract (English)
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.
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