Bridge matching is a framework for learning transport maps by fitting a Markov diffusion to a mixture of diffusion bridges. Given a coupling of initial and terminal distributions, bridge matching constructs reference bridges between paired endpoints and learns a neural drift that matches these bridges via regression. The learned Markov process approximates the mixture of bridges, providing a tractable generative model that can sample new trajectories without conditioning on specific endpoints.
In the deterministic limit (σ→0), bridge matching reduces to flow matching, revealing a deep connection between diffusion-based and flow-based generative models. Each DSBM iteration amounts to a bridge matching step: the Markovian projection of a reciprocal measure is computed by matching the reference bridges. This makes bridge matching the core computational primitive underlying the DSBM framework.
Algorithm
- Sample pairs (x₀, x_T) from coupling.
- Construct reference bridge Q_{|0,T} between each pair.
- Sample bridge states at random times t along each bridge trajectory.
- Regress neural drift on bridge conditional score ∇log Q_{t|0} or ∇log Q_{T|t}.
- The resulting Markov process matches the bridge mixture.
Key Properties
- Each iteration = one Markovian projection
- σ→0 recovers Flow Matching
- First iteration of DSBM-IPF recovers DDMs
- Generalizes to arbitrary couplings
- Peluchetti (2021, 2023) introduced independently
- Avoids time-discretization artifacts of DSB
Relationship to GSBM
In GSBM (Liu et al., 2024), bridge matching serves as Stage 1 of the alternating optimization for the generalized Schrödinger bridge problem. Proposition 1 of GSBM proves that once marginals are fixed, the state cost drops out of the optimization, and the explicit matching loss (bridge matching) provides the same algorithmic purpose as the implicit matching loss (entropic action matching). The key generalization is that the conditional probability paths used in the regression target are no longer Brownian bridges but instead solutions to the CondSOC problem, which accounts for via Gaussian path approximation or path integral control.