Schrödinger bridge

The Schrödinger bridge problem seeks the path measure $P^{\ast }$ closest in KL divergence on path space to a reference diffusion process $Q$, subject to prescribed initial and terminal marginal constraints: $P_0^{\ast }={\pi }_0$ and $P_T^{\ast }={\pi }_T$. Formally, $P^{\ast }=\arg {\min }_{P:P_0={\pi }_0,P_T={\pi }_T}\mathrm{KL}(P\Vert Q)$. The SB is the unique measure that is simultaneously Markov, in the reciprocal class $\mathcal{R}(Q)$ of the reference, and satisfies the boundary conditions.

The static (coupling) version of the SB problem is equivalent to entropy-regularized optimal transport: the SB coupling solves ${\min }_{\Pi :{\Pi }_0={\pi }_0,{\Pi }_T={\pi }_T}\mathbb{E}[\Vert X_0-X_T{\Vert }^2]-2{\sigma }^{2}T\cdot H(\Pi )$. As $\sigma \to 0$, the SB converges to the classical Monge-Kantorovich OT plan. This deep connection between SB and OT, surveyed by Léonard (2014), makes SB a principled framework for learning transport maps in machine learning.

Numerically, SBs are computed via Iterative Proportional Fitting (Sinkhorn-like iterations) or the more recent Iterative Markovian Fitting. The DSB algorithm (De Bortoli et al. 2021) showed that standard score-based generative modeling is the first IPF iteration for an SB problem. DSBM (Shi et al. 2023) significantly improved SB numerics via bridge matching, recovering DDMs, Flow Matching, and Rectified Flow as special/limiting cases.

Originally posed by Erwin Schrödinger in 1931 in the context of quantum mechanics. Léonard (2014) provides the definitive modern survey, proving that the unique solution $\hat{P}$ has the product-shaped Radon-Nikodym derivative $\hat{P}=f_0(X_0)g_1(X_1)R$ (the (f,g)-transform), where $f_0,g_1$ solve the Schrödinger system. The logarithms $\phi =\log f$, $\psi =\log g$ are the Schrödinger potentials, second-order analogues of Kantorovich potentials. The time-marginal flow defines the entropic interpolation, which converges to the displacement interpolation as noise vanishes (Γ-convergence of Schrödinger to Monge-Kantorovich).

From the stochastic control perspective (Chen, Georgiou & Pavon 2014), the SB has a fluid dynamic formulation that parallels the Benamou-Brenier formula: minimize ∫∫[½||v||² + ⅛||∇ln ρ||²]ρ dtdx subject to the continuity equation and marginal constraints. The additional Fisher information term ⅛||∇ln ρ||² (the osmotic kinetic energy) is exactly what distinguishes SB from classical OT — revealing their relationship without zero-noise limits. The optimal drift decomposes as b_+^{Q*} = b_+^P + ∇ln φ where φ is space-time harmonic, which is the score correction familiar from score-based generative models. This connects to optimal transport with prior as the zero-noise limit.

Key Details

• KL minimization on path space subject to marginal constraints
• Equivalent to entropy-regularized OT in the static formulation
• Fluid dynamic formulation: OT functional + Fisher information penalty
• Optimal drift = prior drift + ∇ln φ (score correction)
• Solved via IPF or IMF
• First IPF iteration = standard SGM
• As regularization vanishes, recovers classical OT / optimal transport with prior
• Originally from Schrödinger (1932)

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