Foundations of Schrödinger Bridges for Generative Modeling

Abstract

At the core of modern generative modeling frameworks, including diffusion models, score-based models, and flow matching, is the task of transforming a simple prior distribution into a complex target distribution through stochastic paths in probability space. Schrödinger bridges provide a unifying principle underlying these approaches, framing the problem as determining an optimal stochastic bridge between marginal distribution constraints with minimal-entropy deviations from a pre-defined reference process. This guide develops the mathematical foundations of the Schrödinger bridge problem, drawing on optimal transport, stochastic control, and path-space optimization, and focuses on its dynamic formulation with direct connections to modern generative modeling. We build a comprehensive toolkit for constructing Schrödinger bridges from first principles, and show how these constructions give rise to generalized and task-specific computational methods.

Figures (24)

• Figure 1: Illustration of Relative Entropy or Kullback-Leibler (KL) Divergence. The KL divergence $\text{KL}(p\|q)$ between two 1-dimensional probability distributions $p, q\in \mathcal{P}(\mathcal{X})$ on the state space $\mathcal{X}\subseteq\mathbb{R}$. The left shows the independent distributions and the right shows the signed contributions of $p\log \frac{dp}{dq}$ which yields the KL divergence when integrated, quantifying how well $q$ approximates $p$.
• Figure 2: Illustration of Sinkhorn's Algorithm. Starting from an initial potential (e.g. $\varphi_0:=0$), Sinkhorn's algorithm alternates updates of the dual potentials $\varphi_n$ and $\hat{\varphi}_n$ via log-integral transforms involving the transport cost $c(\boldsymbol{x},\boldsymbol{y})$ and the marginal constraints $\pi_0$ and $\pi_T$. Each alternating step enforces one marginal constraint while preserving the entropic structure, and the sequence $(\varphi_n, \hat{\varphi}_n)$ converges to the optimal dual pair $(\varphi^\star, \hat{\varphi}^\star)$ that uniquely defines the static Schrödinger bridge coupling $\pi^\star_{0,T}$.
• Figure 3: Comparison Between Static and Dynamic Optimal Transport. Illustration of the relationship between the static and dynamic formulations of optimal transport between two marginal distributions $\pi_0$ and $\pi_T$. The static formulation minimizes the transport cost directly between endpoints via the quadratic cost $\|\boldsymbol{x}_T-\boldsymbol{x}_0\|^2$, while the dynamic (Benamou–Brenier) formulation instead seeks a time-dependent velocity field $\boldsymbol{v}_t(\boldsymbol{x})$ that continuously transports mass from $\pi_0$ to $\pi_T$ while minimizing the kinetic energy $\int_0^T\|\boldsymbol{v}_t(\boldsymbol{x})\|^2 dt$. The white trajectory represents a particle path under the optimal flow, illustrating how dynamic OT realizes the same optimal coupling as static OT through continuous mass evolution.
• Figure 4: Brownian Motion and Controlled Itô Processes. Illustration of one-dimensional stochastic trajectories generated by two stochastic differential equations (SDEs) starting from $\boldsymbol{X}_0=0$ over the time interval $t\in [0,1]$. Left: Sample paths of pure Brownian motion of the form $d\boldsymbol{X}_t=\sigma_td\boldsymbol{B}_t$, where increments are Gaussian with variance proportional to the timestep $\boldsymbol{B}_{t+\Delta t}=\boldsymbol{B}_t+\sqrt{\Delta t}\boldsymbol{z}$ with $\boldsymbol{z}\sim \mathcal{N}(0,\boldsymbol{I}_d)$. Right: Sample paths of a controlled Itô process of the form $d\boldsymbol{X}_t=(\boldsymbol{f}(\boldsymbol{X}_t,t)+\sigma_t\boldsymbol{u}(\boldsymbol{X}_t,t))+\sigma_td\boldsymbol{B}_t$, where we set $\boldsymbol{f}\equiv 0$ and $\boldsymbol{u}(\boldsymbol{x},t):=\frac{2-\boldsymbol{x}}{T-t+\epsilon}$ which pulls the process to $\boldsymbol{X}_T=2$.
• Figure 5: Change of Measure Using Radon-Nikodym Derivative. Sample trajectories of a diffusion process are shown under two probability measures. Top left: trajectories under the reference measure $\mathbb{Q}$, governed by the uncontrolled reference drift $\boldsymbol{f}(\boldsymbol{X}_t,t)$. Bottom left: trajectories under the controlled measure $\mathbb{P}^u$, where the dynamics include an additional control drift $\boldsymbol{u}(\boldsymbol{X}_t,t)$. Right: the same reference trajectories are reweighted according to the Radon–Nikodym derivative $\frac{\mathrm{d}\mathbb{P}^u}{\mathrm{d}\mathbb{Q}}$, which assigns higher likelihood to paths aligned with the control and lower likelihood to paths that oppose it. The color intensity represents the log-likelihood ratio $\log \frac{\mathrm{d}\mathbb{P}^u}{\mathrm{d}\mathbb{Q}}$, illustrating how the controlled dynamics can be interpreted as a change of measure on the same underlying path space.

Theorems & Definitions (138)

• Definition 1.1: Monge's Optimal Mass Transport Problem
• Definition 1.2: Kantorovich's Optimal Mass Transport Problem
• Definition 1.3: Entropy Between Probability Measures
• Lemma 1.4: Chain Rule of KL Divergences
• Lemma 1.5: Data Processing Inequality
• Definition 1.6: Entropic Optimal Transport (EOT) Problem
• Definition 1.7: Static Schrödinger Bridge Problem
• Proposition 1.8: Schrödinger Potentials
• Proposition 1.9: Solution to Static SB Problem
• Theorem 1.10: Dual Formulation of Static SB Problem