IPF is an iterative algorithm for solving the Schrödinger bridge problem by alternately projecting onto marginal constraint sets. In discrete settings, it is known as the Sinkhorn algorithm. Each iteration alternates between: (1) projecting onto the set of measures with correct terminal marginal (learning backward transitions); (2) projecting onto the set with correct initial marginal (learning forward transitions). Each projection minimizes KL divergence.
The DSB algorithm approximates IPF by learning neural network drifts via regression at each half-step (mean-matching formulation). After iterations, the forward and backward drifts converge to the Schrödinger bridge. IPF convergence requires mild assumptions and achieves monotone decrease of Jeffrey’s divergence.
A limitation of IPF is that each iteration preserves only one marginal constraint at a time, and the alternating projections can accumulate errors. Iterative Markovian Fitting (IMF) addresses this by preserving both marginals throughout.
Algorithm
- Initialize with reference process
- Project onto terminal marginal constraint (learn backward drift)
- Project onto initial marginal constraint (learn forward drift)
- Repeat until convergence
Key Properties
- Continuous-state analogue of Sinkhorn
- Monotone KL decrease
- First iteration = standard SGM
- May accumulate errors across iterations
- DSB implements IPF via neural network regression
- The Sinkhorn approach was extended to the generalized Schrödinger bridge by DeepGSB (Liu et al., 2022), but this Sinkhorn-inspired method prioritizes optimality over feasibility. GSBM addresses this by using matching algorithms instead.