AI 中文总结
本文提出图上的代价增强薛定谔桥可通过Feynman-Kac倾斜精确求解，无需学习控制，交替端点重缩放即可高效计算，并在蛋白质折叠和路网任务中验证了效果。
AI 中文摘要
广义薛定谔桥在图上移动两个分布之间的质量，同时对访问的状态收取代价。已有方法通过学习受控连续时间马尔可夫链的速率来处理该问题，并采用时间差分惩罚来恢复代价。状态代价作为Feynman-Kac倾斜折叠进参考过程。代价增强桥随后是相对于倾斜参考的普通桥，惩罚变得不必要。该桥通过交替进行两个端点重缩放来精确计算，每次应用稀疏矩阵指数；无需在时间上离散化或进行学习。交替过程以仅由端点耦合决定的速率收敛。对于时间平均占用率上的二次拥塞代价，围绕精确桥的阻尼最佳响应是强凸函数上的梯度下降，其残差界定了其误差。在蛋白质折叠模型上，自由能代价降低了折叠路径的预期势垒。在学习方法的路网中，精确桥的滚动输出在采样误差内匹配目标，并且在具有数百万交叉点的网络上，其内存线性增长。
英文摘要
The generalized Schrödinger bridge on a graph moves mass between two distributions while charging a cost for the states visited. It has been approached by learning the rates of a controlled continuous-time Markov chain, with a temporal-difference penalty that restores the cost. A state cost folds into the reference process as a Feynman-Kac tilt. The cost-augmented bridge is then a plain bridge against the tilted reference, and the penalty is unnecessary. The bridge is computed exactly by alternating two endpoint rescalings, each one sparse matrix-exponential application; nothing is discretized in time or learned. The alternation converges at a rate set by the endpoint coupling alone. For a quadratic congestion cost on time-averaged occupancies, damped best response around the exact bridge is gradient descent on a strongly convex function, and its residual bounds its error. On a protein-folding model, a free-energy cost lowers the expected barrier of the folding paths. On the learned approach's road network, roll-outs of the exact bridge match the target within sampling error, and on networks with millions of intersections its memory grows linearly.
