Leveraging connections between diffusion-based sampling, optimal transport, and optimal stochastic control through their shared links to the Schr\"odinger bridge problem, we propose novel objective functions that can be used to transport $\nu$ to $\mu$, consequently sample from the target $\mu$, via optimally controlled dynamics. We highlight the importance of the pathwise perspective and the role various optimality conditions on the path measure can play for the design of valid training losses, the careful choice of which offer numerical advantages in practical implementation.

翻译：通过扩散采样、最优传输与最优随机控制在薛定谔桥问题上的内在联系，我们提出了新颖的目标函数，该函数能够通过最优控制动力学将分布 $\nu$ 传输至 $\mu$，从而实现对目标分布 $\mu$ 的采样。我们强调了路径视角的重要性，并阐明了路径测度上各类最优性条件对于设计有效训练损失函数的关键作用，其审慎选择可在实际实现中带来数值计算上的优势。