论文标题
离散Schrödinger桥的渐近学通过混乱分解
Asymptotics of Discrete Schrödinger Bridges via Chaos Decomposition
论文作者
论文摘要
考虑匹配两个独立I.I.D.的问题来自两个分布的$ n $的样本$ p $和$ q $ in $ \ mathbb {r}^d $。对于任意连续的成本功能，最佳分配问题的位置是最小化总成本的匹配。相反，我们在本文中考虑了每个匹配的问题，其中gibbs的概率权重与该匹配的负总成本的指数成正比。然后将每个匹配视为与$ n $原子的联合分布，然后就上述Gibbs概率度量进行凸组合。我们表明，由Föllmer引入的变异问题解决了这种随机的关节分布，称为$ n \ rightarrow \ infty $，称为schrödinger问题。我们还分别得出了订单的前两个错误项$ n^{ - 1/2} $和$ n^{ - 1} $。这为我们提供了集成测试功能的中心限制定理，包括运输成本，以及限制高斯差异为零时高斯混乱限制。这些证明是基于一对经验分布的多项式函数对离散schrödinger桥的新型混乱分解，作为度量空间中的第一阶和二阶泰勒近似值。这是通过从U统计学的经典理论中扩展Hoffding分解来实现的。
Consider the problem of matching two independent i.i.d. samples of size $N$ from two distributions $P$ and $Q$ in $\mathbb{R}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead in this paper the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with $N$ atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show that this resulting random joint distribution converges, as $N\rightarrow \infty$, to the solution of a variational problem, introduced by Föllmer, called the Schrödinger problem. We also derive the first two error terms of orders $N^{-1/2}$ and $N^{-1}$, respectively. This gives us central limit theorems for integrated test functions, including for the cost of transport, and second order Gaussian chaos limits when the limiting Gaussian variance is zero. The proofs are based on a novel chaos decomposition of the discrete Schrödinger bridge by polynomial functions of the pair of empirical distributions as the first and second order Taylor approximations in the space of measures. This is achieved by extending the Hoeffding decomposition from the classical theory of U-statistics.
