Problem
Research questions and friction points this paper is trying to address.
Schrödinger bridge
graphs
cost-augmented
continuous-time Markov chain
Feynman-Kac tilt
Cost-augmented Schrödinger bridges on graphs are exactly solvable: a Feynman-Kac tilt replaces learned control
🤖 AI Summary
This study addresses the challenge of solving cost-constrained Schrödinger bridges on graphs, which conventionally relies on complex learned controls and temporal-difference penalties. The proposed method leverages Feynman-Kac tilting to incorporate state costs into a reference process, combining alternating endpoint rescaling with continuous-time Markov chains to analytically solve the cost-augmented bridge via sparse matrix exponentiation. This work mathematically demonstrates that the problem admits an exact analytical solution without time discretization or model training. Empirical evaluations in protein folding and road network simulations show that the approach effectively reduces path barriers while matching target distributions. Furthermore, it achieves efficient convergence with memory scaling linearly relative to the number of nodes.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Schrödinger bridges
Feynman-Kac tilt
graph Markov chains
exact solvability
sparse matrix-exponential
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