Sinkhorn黎曼度量在有限支撑测度上的收敛性
Convergence of the Sinkhorn Riemannian metric for finitely supported measures
- Yale University（耶鲁大学）
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明了有限支撑测度在水平扰动下Sinkhorn黎曼度量及其高阶导数、曲率张量和Christoffel符号随正则化参数趋于零时的收敛性，并指出了与绝对连续情形的差异。
中文摘要 AI 辅助
Sinkhorn黎曼度量刻画了Sinkhorn散度的局部行为，Sinkhorn散度是熵正则化最优传输的一种去偏版本。尽管已知当正则化参数$\varepsilon$趋于0时，Sinkhorn散度逼近平方Wasserstein-2距离，但Sinkhorn黎曼度量在$\varepsilon\to 0$时的行为在很大程度上仍是未解问题。据我们所知，现有唯一论证是针对具有密度的测度，且仍属于形式化计算。本文中，我们证明了在有限支撑测度经历水平扰动的情形下，Sinkhorn黎曼度量在$\varepsilon\to 0$时的收敛性。在我们的假设下，我们的技术还使我们能够证明Sinkhorn黎曼度量的高阶导数的收敛性，从而确保相关黎曼曲率张量和Christoffel符号在$\varepsilon\to 0$时的收敛性。我们还与绝对连续情形进行了比较，并展示了我们的方法在具有密度的测度上失效之处。
英文摘要
The Sinkhorn Riemannian metric characterizes the local behavior of the Sinkhorn divergence, a debiased version of entropy-regularized optimal transport. Although the Sinkhorn divergence is known to approximate the squared Wasserstein-2 distance as the regularization parameter $\varepsilon$ goes to $0$, the behavior of the Sinkhorn Riemannian metric as $\varepsilon\to 0$ remains largely open. To the best of our knowledge, the only existing argument is for measures with a density and is still a formal computation. In this work, we prove the convergence of the Sinkhorn Riemannian metric as $\varepsilon\to 0$ in the case of finitely supported measures undergoing horizontal perturbations. Under our assumptions, our techniques additionally allow us to prove the convergence of higher order derivatives of the Sinkhorn Riemannian metric, thereby ensuring the convergence of the associated Riemann curvature tensor and Christoffel symbols as $\varepsilon\to 0$. We also compare with the absolutely continuous setting and show where our approach fails for measures with a density.
