SPIM: Solving Schrödinger Bridges Via Alternating Stochastic Optimal Control

Schrödinger bridges (SBs) and stochastic optimal control (SOC) are closely related, but the two-endpoint distribution constraints of SBs make terminal-cost SOC solvers difficult to apply directly. We propose Schrödinger Potential Iteration Matching (SPIM), which solves SBs through alternating SOC that realize the fixed-point iteration of Schrödinger potentials. In each direction, the terminal cost is given by the corresponding relative endpoint energy minus the initial-time value returned by the opposite direction SOC. Under the Cole–Hopf transform, each exact SOC performs one Schrödinger-potential propagation, so a forward–backward pair completes one fixed-point iteration. Under compactness and uniform kernel positivity, exact alternating SOC inherits the Hilbert-metric contraction of the classical potential iteration, yielding geometric convergence of the SOC value functions up to additive constants. The limiting value functions determine the unique optimal generalized Schrödinger bridge. The framework supports energy-to-energy transport from relative endpoint energies without endpoint samples or normalizing constants, extends naturally to generalized Schrödinger bridges with running state costs, and allows different SOC solvers in the two directions. Experiments show that SPIM achieves competitive endpoint and interior transport accuracy across classical and generalized SB benchmarks, scales to higher-dimensional problems, and supports both running-cost and two-endpoint reward fine-tuning.