We propose a procedure for estimating the Schr\"odinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schr\"odinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport.

翻译：本文提出了一种估计两个概率分布间薛定谔桥的算法。与现有方法不同，本方法无需迭代模拟前向与反向扩散过程，也无需训练神经网络来拟合未知漂移项。我们证明，通过求解源样本与目标样本间静态熵最优输运问题所获得的势函数，经适当修正后可产生一个自然的插件估计量，该估计量能定义两测度间桥梁的时变漂移项。在最小假设条件下，我们证明所提出的方法（称为Sinkhorn桥）能以收敛速率可证明地估计薛定谔桥，该速率取决于目标测度的本征维度。本方法融合了采样理论、理论熵最优输运与统计熵最优输运领域的研究成果。