arXiv：cs.LG（机器学习，全量分类）· Akshay Balsubramani·· 14 小时前AI 评分36
图上带代价的 Schrödinger bridge 可精确求解：Feynman-Kac 倾斜替代学习式控制
Cost-augmented Schr\"odinger bridges on graphs are exactly solvable: a Feynman-Kac tilt replaces learned control
AI 导读
图上的带代价 Schrödinger bridge 可通过 Feynman-Kac 倾斜把状态代价折入参考过程，从而化为普通 bridge，无需学习控制率或时间离散化，仅靠两次端点重缩放（各一次稀疏矩阵指数运算）交替即可精确计算，收敛速度仅由端点耦合决定。
正文
Abstract: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.
| Subjects: | Machine Learning (cs.LG) |
| MSC classes: | 60J27, 49Q22, 93E20, 65F60, 49N80 |
| Cite as: | arXiv:2610.02195 [cs.LG] |
| (or arXiv:2610.02195v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.02195 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Akshay Balsubramani [view email]
[v1]
Thu, 1 Oct 2026 17:59:39 UTC (131 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org