用经验风险最小化学习扩散路径的势函数，提升样本生成精度。
Schrödinger bridge problem via empirical risk minimization
- 将桥接问题转化为单个正势函数的非线性不动点方程求解。
- 在次高斯假设下证明了经验风险的均匀收敛性。
- 适合需要高精度生成样本的机器学习与随机控制场景。
当端点分布仅以样本形式给出时，研究薛定谔桥问题。传统方法通过对经验测度进行Sinkhorn迭代估计薛定谔势函数，并通过核光滑双变量解的导数构造时变漂移。本文提出一种学习理论路径：将薛定谔系统重写为单一正变换势函数满足的非线性不动点方程，通过在函数类上进行经验风险最小化估计该势函数。在参考核和终端密度满足次高斯假设下，建立了经验风险与其总体对应项的均匀收敛性。将学习到的势函数代入桥的随机控制表示，用于生成样本。通过数值实验展示了所提方法的有效性。
原文摘要 · Abstract (English)
We study the Schrödinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schrödinger potentials via Sinkhorn iterations on empirical measures and then construct a time-inhomogeneous drift by differentiating a kernel-smoothed dual solution. In contrast, we propose a learning-theoretic route: we rewrite the Schrödinger system in terms of a single positive transformed potential that satisfies a nonlinear fixed-point equation and estimate this potential by empirical risk minimization over a function class. We establish uniform concentration of the empirical risk around its population counterpart under sub-Gaussian assumptions on the reference kernel and terminal density. We plug the learned potential into a stochastic control representation of the bridge to generate samples. We illustrate performance of the suggested approach with numerical experiments.
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