Vargas Francisco, Thodoroff Pierre, Lamacraft Austen, Lawrence Neil
The Computer Laboratory, Department of Computer Science and Technology, University of Cambridge, William Gates Building, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK.
The Cavendish Laboratory, Deparment of Physics, The Old Schools, Trinity Ln, Cambridge CB2 1TN, UK.
Entropy (Basel). 2021 Aug 31;23(9):1134. doi: 10.3390/e23091134.
The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.
薛定谔桥问题（SBP）在给定先验随机演化的情况下，找出两个概率分布之间最可能的随机演化。除了在自然科学中的应用外，这类问题在机器学习中也有重要应用，如数据集对齐和假设检验。虽然这个问题背后的理论相对成熟，但估计薛定谔桥的可扩展数值方法仍是一个活跃的研究领域。我们的主要贡献是证明了解决SBP与自回归最大似然估计目标之间的等价性。这种表述规避了密度估计的许多挑战，并能直接应用成功的机器学习技术。我们提出了一种使用高斯过程估计SBP的数值方法，并在数值模拟和实验中展示了我们方法的实际应用。
