Foundations of Schrödinger Bridges for Generative Modeling

Sophia Tang

Notation	Meaning
Basic Notation
$\text{KL}(\cdot\\|\cdot)$	Kullback-Leibler divergence
$\inf$	infimum or the largest lower bound of a set
$\sup$	supremum or the smallest upper bound of a set
$\partial_{t}$	partial derivative with respect to $t$
$d,dt,d\bm{x}$	differential, time differential, state differential

Table 1

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(\bm{x},\bm{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

\|\bm{x}_{T}-\bm{x}_{0}\|^{2}

, while the dynamic (Benamou–Brenier) formulation instead seeks a time-dependent velocity field

\bm{v}_{t}(\bm{x})

that continuously transports mass from

\pi_{0}

to

\pi_{T}

while minimizing the kinetic energy

\int_{0}^{T}\|\bm{v}_{t}(\bm{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

\bm{X}_{0}=0

over the time interval

t\in[0,1]

. Left: Sample paths of pure Brownian motion of the form

d\bm{X}_{t}=\sigma_{t}d\bm{B}_{t}

, where increments are Gaussian with variance proportional to the timestep

\bm{B}_{t+\Delta t}=\bm{B}_{t}+\sqrt{\Delta t}\bm{z}

with

\bm{z}\sim\mathcal{N}(0,\bm{I}_{d})

. Right: Sample paths of a controlled Itô process of the form

d\bm{X}_{t}=(\bm{f}(\bm{X}_{t},t)+\sigma_{t}\bm{u}(\bm{X}_{t},t))+\sigma_{t}d\bm{B}_{t}

, where we set

\bm{f}\equiv 0

and

\bm{u}(\bm{x},t):=\frac{2-\bm{x}}{T-t+\epsilon}

which pulls the process to

\bm{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

\bm{f}(\bm{X}_{t},t)

. Bottom left: trajectories under the controlled measure

\mathbb{P}^{u}

, where the dynamics include an additional control drift

\bm{u}(\bm{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.

Figure 6: Time-Reversal of Stochastic Differential Equations. Left: A forward diffusion process over

t\in[0,T]

evolves from an initial state

\bm{X}_{0}=\bm{x}_{0}

under the forward-time SDE

d\bm{X}_{t}

, producing marginal densities

p_{t}(\bm{x})

that evolve via the Fokker-Planck equation. Right: The corresponding reverse-time stochastic process generates trajectories that traverse the same sequence of marginal densities in the reversed time coordinate

s=T-t\in[0,T]

. Its drift includes an additional score correction

\nabla\log p_{T-s}(\bm{x})

that accounts for the probability flow of the forward dynamics. This reverse SDE allows trajectories to be simulated starting from the terminal distribution such that they reconstruct the initial distribution.

Figure 7: Time Reversal vs. Forward-Backward SDE. Top : Reversing an uncontrolled forward diffusion requires correcting the drift using the score function

\nabla\log p_{t}(\bm{x})

, which depends on the evolving marginal density and can lead to unstable or uncontrolled trajectories. Bottom : In the more general Schrödinger bridge setting, the forward process evolves via a control drift

\bm{u}(\bm{x},t):=\sigma_{t}\nabla_{\bm{x}}\log\varphi_{t}(\bm{x})

which contains the forward Schrödinger potential

\varphi_{t}

. This controlled forward process induces a corresponding backward SDE with the control

\sigma_{t}\nabla_{\bm{x}}\log\hat{\varphi}_{t}(\bm{x})

, containing the backward Schrödinger potential

\hat{\varphi}_{t}

. This enables accurate transport between two structured distributions with optimal bridge dynamics.

Figure 8: Iterative Markovian Fitting (IMF) vs. Iterative Proportional Fitting (IPF). While IPF alternates projections onto the marginal constraint sets

\Pi(\pi_{0},\cdot)

and

\Pi(\cdot,\pi_{T})

, intermediate iterations generally do not preserve the endpoint coupling. In contrast, IMF performs Markovian and reciprocal projections that maintain both marginal constraints

\mathbb{P}^{n}\in\Pi(\pi_{0},\pi_{T})

at every iteration, progressively refining the path measure until convergence to the optimal bridge measure

\mathbb{P}^{\star}

.

Figure 9: Gaussian Schrödinger Bridge Problem. The Gaussian SB between initial and terminal Gaussian distributions

\pi_{0}=\mathcal{N}(\mu,\Sigma)

and

\pi_{T}=\mathcal{N}(\mu^{\prime},\Sigma^{\prime})

reduces to solving an action minimization problem on the Bures–Wasserstein manifold of covariance matrices, where the covariance trajectory

(\Sigma_{t})_{t\in[0,T]}

balances a kinetic energy term describing the rate of covariance change and a potential energy term induced by the diffusion of the reference process.

Figure 10: Generalized Schrödinger Bridge Problem. The generalized SB problem extends the dynamic SB problem by introducing a mean-field interaction cost

\mathcal{I}(x,p_{t})

that depends on the evolving marginal distribution. The resulting objective balances kinetic energy

\frac{1}{2}\|\bm{u}(\bm{x},t)\|^{2}

with an interaction energy term integrated against the state density

p_{t}(x)

. The optimal dynamics transport particles between the prescribed marginals

(\pi_{0},\pi_{T})

while accounting for collective interactions encoded by

\mathcal{I}(x,p_{t})

, leading to a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system that characterizes the optimal bridge.

Figure 11: Multi-Marginal Schrödinger Bridge Problem. The multi-marginal Schrödinger bridge constructs an optimal stochastic process whose marginal distributions match a sequence of prescribed intermediate densities

\{\pi_{t_{k}}\}_{k=0}^{K}

. Starting from a reference stochastic process

\mathbb{Q}

, the optimal path measure

\mathbb{P}^{\star}

is obtained by minimizing the path-space KL divergence while matching the marginal constraints.

Figure 12: Unbalanced Schrödinger Bridge Problem. The unbalanced SB problem extends the classical formulation to settings where the endpoint marginals

\pi_{0}

and

\pi_{T}

have different total mass. In addition to the control

\bm{u}(\bm{x},t)

that transports probability, the dynamics contain a growth term

g(\bm{x},t)

which allows mass to be created or removed along trajectories, which contributes a growth penalty

\Psi(g)

in the minimization objective. The dynamics of the density evolve according to the unbalanced Fokker–Planck equation, which augments the standard continuity equation with the mass-change term

gp_{t}

.

Figure 13: Branched Schrödinger Bridge Problem. The branched Schrödinger bridge generalizes the classical formulation to transport an initial distribution

\pi_{0}

toward multiple target marginals

\{\pi_{T,k}\}_{k=1}^{K}

. A primary trajectory evolves from the source distribution and probabilistically branches into secondary branched paths, each governed by its own control and growth dynamics to reach a different terminal distribution. The objective minimizes the weighted kinetic energy of the controls across all branches, producing a stochastic branching flow that connects a single source to multiple target modes.

Figure 14: Score-Based Generative Models. Score-based generative models learn the score function

\nabla\log p_{t}(\bm{x})

of a diffusion process that progressively perturbs data with noise through a forward SDE. A neural network

\bm{s}_{\theta}(\bm{x},t)

is trained using the score matching loss to approximate this time-dependent score. Once learned, the reverse-time SDE uses the estimated score to iteratively denoise samples, transforming noise into data.

Figure 15: Likelihood Training of Forward and Backward Schrödinger Potentials. The Schrödinger bridge can be learned by parameterizing the log gradient of the forward and backward potentials

\bm{Z}^{\theta}_{t}=\nabla\log\varphi_{t}(\bm{X}_{t})

and

\widehat{\bm{Z}}_{t}^{\phi}=\nabla\log\hat{\varphi}_{t}(\bm{X}_{t})

, which define the forward and backward SDEs between endpoint distributions.

\theta

and

\phi

are trained by maximizing path likelihoods through forward and backward objective functions, each evaluated under trajectories generated by the corresponding controlled SDE. Together, the two likelihood losses couple the forward and reverse dynamics, enabling the model to learn a consistent Schrödinger bridge between the marginals.

Figure 16: Diffusion Schrödinger Bridge Matching. Illustration of the diffusion Schrödinger bridge matching (DSBM) procedure with alternating Markovian and reciprocal projections. Starting from an initial reciprocal measure

\Pi^{0}=\pi_{0,T}\mathbb{Q}_{\cdot|0,T}

, the algorithm alternates between (i) forward-time Markovian projections

\mathbb{M}^{2n+1}

which updates the parameterized forward drift

\bm{u}_{\theta}

to match

\Pi^{2n}

and (ii) reverse-time Markovian projections

\mathbb{M}^{2n+2}

which updates the parameterized reverse drift

\bm{u}_{\phi}

to match

\Pi^{2n+1}

. Each Markovian projection is followed by a reciprocal projection that constructs a mixture of reference bridges conditioned on the updated endpoint laws, denoted

\Pi^{2n+1}

and

\Pi^{2n+2}

. This alternating procedure produces a sequence of path measures that progressively reduce the KL divergence to the optimal bridge, converging to the Schrödinger bridge solution which occurs when the Markov and reciprocal projections reach equilibrium

\mathbb{M}^{\star}=\Pi^{\star}=\mathbb{P}^{\star}

.

Figure 17: Conditional Score and Flow Matching. Conditional score and flow matching learn the dynamics of stochastic bridges by conditioning trajectories on endpoint pairs

(\bm{x}_{0},\bm{x}_{T})\sim\pi_{0,T}

. The model jointly estimates the conditional control drift

\bm{u}(\bm{x},t|\bm{x}_{0},\bm{x}_{T})

and the conditional score

\nabla\log p_{t}(\bm{x}|\bm{x}_{0},\bm{x}_{T})

, which describe how trajectories evolve between the endpoints. By minimizing the combined objective, the learned dynamics produce the mixture of Brownian bridges which is equal to the Schrödinger bridge

\mathbb{P}^{\star}

given the entropic OT coupling

\pi_{0,T}^{\star}

.

Figure 18: Schrödinger Bridge with Adjoint Matching. Adjoint matching learns the Schrödinger bridge by alternating between optimizing the forward half-bridge through the adjoint matching loss for

\bm{u}(\bm{x},t)

and optimizing the backward half-bridge through the corrector matching loss for

\widehat{\bm{Z}}_{T}(\bm{X}_{T})

.

Figure 19: Piecewise Càdlàg Path of Continuous-Time Markov Chains. Illustration of a càdlàg stochastic trajectory, which is right-continuous with left limits. Between jump times

t_{1},\dots,t_{5}

the process evolves continuously, while discontinuities occur at the marked times. The filled markers indicate the value of the process

\bm{X}_{t}

(the right limit), while the open markers denote the left limit

\bm{X}_{t^{-}}

.

Figure 20: Continuous-Time Markov Chains (CTMC). A CTMC defines a stochastic process on a finite state space

\mathcal{X}=\{1,\dots d\}

that evolves through jumps between discrete states governed by a time-dependent transition rate matrix

\bm{Q}_{t}

. At each jump time

t_{k}

, the system transitions from the current state

\bm{X}_{t_{k}}

to a different state

\bm{y}

according to the rate

\bm{Q}_{t}(\bm{X}_{t_{k}},\bm{y})

. The evolution of the probability distribution

p_{t}(\bm{x})

satisfies the Kolmogorov forward equation, which balances incoming and outgoing probability flows induced by the transition rates.

Figure 21: Discrete Schrödinger Bridge Problem. The discrete SB problem seeks a controlled CTMC path measure

\mathbb{P}^{\star}

that is closest, in KL divergence, to a reference CTMC

\mathbb{Q}

while matching the prescribed marginal distributions

\pi_{0}

and

\pi_{T}

. Top : CTMC trajectories under the sub-optimal controlled path measure

\mathbb{P}^{u}

with generator

\bm{Q}^{u}

. Bottom : CTMC trajectories under the Schrödinger bridge path measure

\mathbb{P}^{\star}

with generator

\bm{Q}^{\star}

obtained by minimizing the KL divergence with the reference generator

\bm{Q}^{0}

while maintaining the terminal marginals.

Notation for Section 7.5 and 7.6
Notation	Meaning
$\mathbb{Q}$	reference CTMC path measure
$\bm{Q}^{0}_{t}(\bm{x},\bm{y})$	generator of the reference process $\mathbb{Q}$
$\mathbb{M}$	Markov CTMC path measure in the Markov class $\mathcal{M}$
$\Pi$	CTMC path measure in the reciprocal class $\mathcal{R}(\mathbb{Q})$
$\mathbb{M}_{t},\Pi_{t}$	marginal distributions at time $t$ under a Markov or reciprocal path measure

Table 23

Figure 22: Image to Image Translation with Schrödinger Bridges. Schrödinger bridges enable image-to-image translation by learning a stochastic process that transports samples between a structured prior distribution

\pi_{0}

(e.g., corrupted or partially observed images) and a clean target distribution

\pi_{T}

(right to left). Rather than generating images from pure Gaussian noise, the model learns the optimal stochastic interpolation conditioned on the endpoints, corresponding to the dynamic optimal transport map when diffusion is small. This yields generation trajectories that progressively restore structure, for tasks such as inpainting, deblurring, or translation, leading to more efficient and interpretable transformations than denoising from noise alone.

Figure 23: Modeling Cellular Dynamics with Schrödinger Bridges. Cellular processes such as differentiation, perturbation responses, and development can be viewed as stochastic trajectories through a high-dimensional cell state space inferred from single-cell measurements. Because single-cell sequencing provides only static snapshots of cell populations at discrete times, the underlying dynamics are not directly observed. Schrödinger bridges infer the most likely stochastic process that transports an initial cell distribution

\pi_{0}

to a later observed distribution

\pi_{T}

, yielding a probabilistic model of cell state evolution that captures both deterministic differentiation trends and stochastic biological variability.

Figure 24: Sampling the Boltzmann Distribution with Schrödinger Bridges. Boltzmann distributions

\pi_{T}(\bm{x})\propto e^{-U(\bm{x})}

arise in many scientific applications but are often difficult to sample from due to the intractable normalization constant

Z

. Schrödinger bridges address this by learning a controlled stochastic process that transports samples from an easy-to-sample prior

\pi_{0}

(e.g., a Gaussian) to the target Boltzmann distribution. The learned control drift guides trajectories so that their terminal distribution matches

\pi_{T}

, enabling efficient sampling from energy-based models.

Foundations of Schrödinger Bridges for Generative Modeling

Abstract

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