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.

翻译：本文研究当端点分布仅通过样本可获得时的薛定谔桥问题。经典计算方法通过经验测度上的Sinkhorn迭代估计薛定谔势，然后通过对核平滑对偶解求导构建时变漂移项。与之相对，我们提出一种学习理论路径：将薛定谔系统重写为满足非线性不动点方程的单正变换势形式，并通过函数类上的经验风险最小化估计该势函数。在参考核与终端密度满足次高斯假设的条件下，我们建立了经验风险围绕其总体对应值的一致集中性。将学习得到的势函数代入桥过程的随机控制表示中以生成样本。我们通过数值实验展示了所提方法的性能表现。