Diffusion Schrödinger Bridge (DSB)

Updated 5 December 2025

Diffusion Schrödinger Bridge (DSB) is a framework that steers a reference diffusion process to match prescribed distributions by minimizing relative entropy.
It generalizes classical optimal transport and denoising diffusion models, enabling robust generative modeling and inference across diverse domains.
Practical implementations use IPF-like algorithms with neural parameterization to iteratively update forward and backward SDEs, reducing endpoint errors.

A Diffusion Schrödinger Bridge (DSB) is a stochastic-process-based framework that solves the problem of probabilistically matching two distributions in finite time by minimally steering a reference diffusion process—such as Brownian motion—between prescribed start and end marginals. The DSB framework constitutes the dynamic, entropy-regularized generalization of optimal transport (OT) and underpins a range of modern generative modeling and inference methodologies. It generalizes both classical denoising diffusion models and entropic optimal transport, provides a principled link to stochastic control, and supports rich algorithmic variants for continuous, discrete, unbalanced, and manifold-valued domains.

1. Mathematical Foundations and Core Formulation

Consider two probability distributions $\pi_0$ and $\pi_1$ on $\mathbb{R}^d$ at times $t=0$ and $t=1$ . The DSB problem is to find a path measure $\mathbb{P}$ over trajectories $(X_t)_{t\in[0,1]}$ , absolutely continuous with respect to a reference diffusion process $\mathbb{Q}$ (e.g., Brownian motion or an SDE with given drift), that transports $\pi_0$ to $\pi_1$ while minimizing the relative entropy (KL divergence) to $\mathbb{Q}$ : $\mathbb{P}^{\rm SB} = \arg\min_{\mathbb{P} \in \mathcal{P}(C)} \ \mathrm{KL}(\mathbb{P} \Vert \mathbb{Q}) \quad \text{s.t.} \quad \mathbb{P}(X_0 \in dx) = \pi_0(dx),\ \mathbb{P}(X_1 \in dx) = \pi_1(dx)$ where $C = C([0,1], \mathbb{R}^d)$ (Kim, 27 Mar 2025, Tang et al., 21 Mar 2024, Bortoli et al., 2021).

Equivalently, the DSB can be formulated as a stochastic optimal control problem: minimize the expected control energy needed to steer the reference SDE to the desired endpoint distributions,

$dx_t = f(x_t, t)\,dt + u_t(x_t)\,dt + \sigma(t)\,dW_t$

with cost

$\E_{\mathbb{P}}\left[\frac{1}{2} \int_0^1 \|u_t(x_t)\|^2 dt\right],$

s.t. $x_0 \sim \pi_0$ , $x_1 \sim \pi_1$ . The optimal control $u_t^*$ is realized as a gradient of certain potentials and relates to score functions (Kim, 27 Mar 2025, Jiang et al., 18 Sep 2024, Zhu et al., 9 Jun 2025).

This setting generalizes classical static OT and, in the zero-noise limit, recovers the Benamou–Brenier dynamic OT (Shi et al., 2023). DSB also connects directly to entropy-regularized OT through the static coupling of endpoints via the reference path measure (Kim, 27 Mar 2025).

2. Algorithmic Principles and Neural Parameterization

Practical DSB solvers exploit the structure of IPF (iterative proportional fitting) or IMF (iterative Markovian fitting). The standard numerical procedure alternates between:

Reciprocal/bridge projection: Fix the joint endpoints, reconstruct paths via the reference bridges.
Markovian/diffusion projection: Given the path marginals, fit a Markov SDE minimizing KL divergence subject to the required marginals.

In the discrete setting, this involves alternately updating forward and backward SDEs (or Markov chains) with learned, time-dependent drift terms (Bortoli et al., 2021, Tang et al., 21 Mar 2024, Kim, 27 Mar 2025). For continuous state spaces, the drift at each iteration is parameterized via neural score models or mean prediction networks, typically U-Nets or MLPs, depending on the data domain (Shi et al., 2023, Jiang et al., 18 Sep 2024). The main regression objectives are denoising-score-matching for score estimation and mean-matching for transition means.

Bidirectional alternation and IPF-like updates ensure monotonic reduction in endpoint-matching error and overall KL divergence (Kim, 27 Mar 2025, Bortoli et al., 2021). In practice, many state-of-the-art approaches (e.g., DSBM, IPMF) utilize minibatch Sinkhorn coupling, Sinkhorn flows, or stochastic approximation to enforce endpoint constraints efficiently (Shi et al., 2023, Kholkin et al., 3 Oct 2024).