Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schr\"odinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

翻译：近期，一系列论文提出了基于深度学习的方法，通过受控扩散过程从非归一化目标密度中进行采样。在本工作中，我们将这些方法识别为薛定谔桥问题的特例，即寻找给定先验分布与指定目标之间最可能的随机演化过程。我们进一步通过引入基于时间反演扩散过程路径空间散度的变分公式来推广这一框架。这一抽象视角导出了可通过梯度算法优化的实用损失函数，并将先前目标函数作为特例包含其中。同时，它允许我们考虑除已知存在模式坍缩问题的反向KL散度之外的其他散度。特别地，我们提出了一种名为对数方差损失的散度，该散度具有优异的数值性质，并在所有考虑的方法中显著提升了性能。