The Markovian projection finds the Markov diffusion process M* that minimizes reverse KL divergence KL(M||Π) to a given (possibly non-Markov) path measure Π. The key property is that it preserves all time marginals of the original measure — the marginal distribution at every time t is the same for M* and Π. The optimal Markov drift is v*(t,x) = σ_t²E[∇log Q_{T|t}(X_T|x)|X_t=x] under Π, which can be learned by regression on the conditional score of the reference bridge.
This operation is central to both bridge matching and the DSBM framework. The Markovian projection has a time-symmetry property: projecting forward and backward Markov processes yields the same result, which DSBM exploits by alternating forward-backward projections to converge to the Schrödinger bridge.
Each iteration of DSBM amounts to a single Markovian projection step. The framework generalizes Bridge Matching by allowing iterative refinement of the coupling, rather than relying on a fixed initial coupling between source and target distributions.
Key Details
- Minimizes reverse-KL to non-Markov path measure
- Preserves all time marginals
- Drift = conditional expectation of bridge score
- Time-symmetric
- Each DSBM iteration is a Markovian projection
- Generalizes Bridge Matching