发表机构
University of Washington（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究熵正则化最优传输中熵Brenier映射与Brenier映射的渐近展开，证明其差为ε乘以得分函数一半加o(ε)误差，并给出Schrödinger桥的两种扩散近似及其联系。
AI 中文摘要
考虑两个欧几里得密度 $\mu$ 和 $\nu$ 之间的Monge-Kantorovich最优传输问题，代价为二次代价。$\varepsilon$-Schrödinger桥是正则化参数为 $\varepsilon$ 的熵正则化问题的解。通过在该耦合下取第二坐标关于第一坐标的条件期望，我们得到 $\varepsilon$-熵Brenier映射。已知当 $\varepsilon$ 趋于零时，熵Brenier映射收敛到两个测度之间的二次代价最优传输映射，即Brenier映射。我们证明，在边缘分布满足一定的光滑性和对数凹性约束下，熵Brenier映射与Brenier映射之差等于 $\varepsilon$ 乘以第一边缘分布的得分函数的一半，再加上一个在 $\mathbf{L}^2(\mu)$ 中为 $o(\varepsilon)$ 的误差。该展开式不依赖于第二边缘分布 $\nu$。证明依赖于用McCann插值的噪声版本近似Schrödinger桥。在额外假设下，我们展示了通过所谓的Mirror Langevin扩散对Schrödinger桥的第二种近似，这些扩散是在由Brenier映射生成的Hessian流形上的Langevin扩散。这两种近似虽然乍看之下颇为不同，但被证明是紧密相关的。我们的证明利用了一个随机曲面和一种称为切向Markov投影的新颖随机运算。
英文摘要
Consider the Monge-Kantorovich optimal transport problem between two Euclidean densities $μ$ and $ν$ and quadratic cost. The $\varepsilon$-Schrödinger bridge is the solution to the entropic regularized problem with regularization parameter $\varepsilon$. By taking the conditional expectation of the second coordinate given the first under this coupling, we obtain the $\varepsilon$-entropic Brenier map. As $\varepsilon$ goes down to zero, it is known that the entropic Brenier map converges to the quadratic cost optimal transport map between the two measures, i.e., the Brenier map. We show that, under some smoothness and log-concavity constraints on the marginals, the difference between the entropic Brenier map and the Brenier map is equal to $\varepsilon$ times one-half of the score function of the first marginal plus an error that is $o(\varepsilon)$ in $\mathbf{L}^2(μ)$. This expansion holds irrespective of the second marginal $ν$. The proof relies on an approximation of the Schrödinger bridge by a noisy version of the McCann interpolation. Under additional assumptions we show a second approximation to the Schrödinger bridge via so-called Mirror Langevin diffusions, which are Langevin diffusions on the Hessian manifold generated by the Brenier map. These two approximations, that appear quite different at first glance, are nonetheless shown to be closely connected. Our proofs utilize a random surface and a novel stochastic operation called the tangent Markov projection.
Comments46 pages, 2 figures
