2. Data Translation: Diffusion Bridges and Stochastic Interpolants

I. Introduction

In this post, we present data translation methods grounded in diffusion bridges and stochastic interpolants. These approaches extend diffusion and flow-matching frameworks to data-to-data translation, that is, the transport between two arbitrary distributions. We assume access only to samples from the source and target distributions, \(p_0\) and \(p_1\), with no prior knowledge of their structure and without access to likelihoods or energy scores.

While many problems can be framed as data translation tasks, for instance, classification can be interpreted as transporting data from the input space to the label space, we restrict this post to translations that preserve dimensionality. Specifically, we consider mappings between data points that lie in the same space, such as image-to-image translation.

II. Motivation

Our goal is to generate samples from \(p_1\) given an initial observation \(x_0 \sim p_0\), that is, to draw from the conditional distribution \(p(x_1 \mid x_0)\). A natural approach is to train a generative model that outputs \(x_1\) from noise while being conditioned on \(x_0\). However, this strategy relies on a sufficiently expressive and robust conditioning mechanism to effectively steer the generation process. Furthermore, initializing from an uninformative prior, such as a Gaussian distribution, can be inefficient in scenarios where the source and target distributions are close in the data space, as in tasks like denoising or deblurring.

These observations motivate the development of bridges between data distributions. Early methods addressing this objective include adversarial approaches, with notable examples such as Pix2Pix (Isola et al., 2017) and CycleGAN (Zhu et al., 2017). Although successful in practice, these methods often face difficulties when modeling complex transport maps and are susceptible to training instabilities. By contrast, the strong generative performance of diffusion models provides a natural incentive to adapt them to data-to-data translation problems.

Initial attempts to employ diffusion models for data-to-data translation focused on combining two separately trained diffusion models, one for \(p_0\) and another for \(p_1\). The central idea was to identify a noise level \(t_0\) at which the forward noising processes applied to both distributions coincide. SDEdit and Dual Diffusion Implicit Bridges use two-stage procedures: first corrupt the initial data up to time \(t_0\), and then apply the reverse diffusion process learned on \(p_1\) to guide the corrupted sample toward \(p_1\) (Meng et al., 2021; Su et al., 2023). In practice, determining an appropriate value of \(t_0\) such that the noised distributions align is challenging, and the required noise level may be so large as to remove most of the informative content of the initial sample \(x_0\). This limitation underscores the necessity of end-to-end approaches that explicitly connect the source and target distributions.

More recently, successful applications of the Doob \(h\)-transform to twist diffusion processes to match the target distribution \(p_1\) at the terminal time (Zhou et al., 2024; Somnath et al., 2023) have inspired a shift toward simpler formulations of the bridge matching problem, notably through the use of stochastic interpolants.

III. Mixture of Bridges

Motivated by the success of diffusion models, the transport between an initial distribution \(p_0\) and a target distribution \(p_1\) is modeled by a stochastic differential equation (Fig. 1) that defines a probabilistic bridge between the two:

This SDE induces a probability measure over continuous paths \((x_t)_{t\in[0,1]}\), whose marginals evolve from \(p_0\) at time \(t=0\) to \(p_1\) at time \(t=1\). The drift function \(v(t,\cdot): [0,1]\times\mathbb{R}^d \to \mathbb{R}^d\) steers particles along the bridge, while the diffusion term \(\sigma\,dW_t\) introduces stochasticity.

In contrast to the forward SDEs used in diffusion models, where the drift is fixed by design, the drift in the bridge SDE is data-dependent. It must be learned so that the induced path measure has prescribed endpoint marginals, ensuring in particular that \(x_1 \sim p_1\). In practice, we parametrize the drift by a neural network \(v_\theta\).

Rather than reasoning directly in terms of drift, it is often convenient to characterize the induced path measure itself. In this section, we focus on transports that can be expressed as mixtures of bridges (Zhou et al., 2024; Tong et al., 2023). Specifically, we consider stochastic processes \(\mathbb{P}\) whose law over paths admits the decomposition

where \(\mathbb{P}_{0,1}\) denotes a joint distribution over endpoints and \(\mathbb{Q}(\cdot \mid x_0, x_1)\) is a reference bridge conditioned on \((x_0, x_1)\). Capital \(\mathbb{P}\) denotes a stochastic process (distribution over continuous paths \([0, 1] \rightarrow \mathbb{R}^d\)) of law \(p\). Intermediate marginals are denoted \(p_t\) as previously.

Here, \(\mathbb{Q}\) is a scaled Brownian motion, i.e., \(\mathbb{Q} = \sigma \mathbb{W}\) with \(\mathbb{W}\) the Wiener process and \(\sigma\) the constant diffusion rate introduced above. The global transport can be interpreted as a mixture of simple stochastic bridges, each connecting a pair of endpoints \((x_0, x_1)\), with mixing weights given by \(\mathbb{P}_{0,1}\). It can be rewritten as:

This formula casts light on the two main components of the transport, namely the conditional Brownian bridge \(\mathbb{Q}_{\mid 0, 1}(\cdot\mid x_0, x_1)\) and the coupling \(\mathbb{P}_{0, 1}\).

Conditional Brownian Bridges

Mixture of bridges uses a Brownian bridge as its reference process. Given two data points \(x_0 \sim p_0\) and \(x_1 \sim p_1\), the conditional Brownian bridge defines a stochastic path \(\mathbb{Q}_{\mid 0,1}(\cdot \mid x_0, x_1)\) that starts at \(x_0\) and ends at \(x_1\) (Fig. 2). Intermediate points along this path can be generated using a stochastic interpolant (Albergo et al., 2025; Albergo & Vanden-Eijnden, 2022):

which provides a simple and exact way to sample from the Brownian bridge at any intermediate time \(t \in [0,1]\). The diffusion rate \(\sigma\) controls the stochasticity of the conditional path. In the case where \(\sigma=0\), we retrieve flow matching’s linear interpolation.

Following the same logic as flow matching and diffusion models, we are interested in the conditional drift function \(v_t(\cdot \mid x_0, x_1)\), and later average conditional drifts to estimate the true drift function from the forward SDE. This conditional drift function takes the following form:

\(v_t(x_t \mid x_0, x_1) = \frac{x_1 - x_t}{1 -t}\) with \(x_t\) defined by the stochastic interpolant above.

Couplings

The choice of the coupling \(\mathbb{P}_{0,1}\) is crucial (Fig. 3). A naive approach is to sample \((x_0, x_1)\) independently from \(p(x_0)p(x_1)\), which allows transporting \(p_0\) to \(p_1\) but fails to preserve high-level information from the original data points. For aligned datasets, where pairs of corresponding points \((x_0, x_1)\) are available, (Albergo et al., 2024; Liu et al., 2023) suggest using data-dependent couplings \(p(x_0, x_1) = p(x_1 \mid x_0)p(x_0)\), treating the dataset pairs as samples from this coupling. More recently, (Theodoropoulos et al., 2025) extended diffusion bridges to scenarios where only a small fraction of aligned data is available. These methods enforce desirable transport properties by explicitly providing pairs of points that share them. For example, in denoising, \((x_0, x_1)\) can be a noisy image and its clean counterpart, both preserving the same semantic content. In super-resolution, natural pairs are high-resolution images and their downsampled versions.

However, in many practical scenarios, paired samples are not available, leading to the problem of unpaired data translation. This challenge is often addressed using the Schrödinger bridge framework (Bortoli et al., 2024; Peluchetti, 2023), which relies on optimal transport-based couplings to define plausible correspondences between distributions. We expand on this approach in the Schrödinger Diffusion Bridges section.

Learning the Drift

The drift function of the bridge, \(v(t, \cdot)\), can be learned by matching it to the conditional drift of the Brownian bridge given the endpoints \((x_0, x_1)\). A neural network \(v_\theta(t, x_t)\) is trained via a regression objective over these conditional drifts:

where \(x_t\) is sampled via the stochastic interpolant and \(v_t(x_t \mid x_0, x_1)\) is the conditional drift of the Brownian bridge. This gives:

By averaging over all endpoint pairs \((x_0, x_1)\), this objective enables the network to learn the marginal drift \(v_t(x_t)\), which reflects the behavior of the ensemble of conditional drifts. In practice, the network is trained on minibatches of data pairs. The overall training procedure is summarized in the algorithm below. Once trained, the neural network \(v_\theta\) can be used to estimate \(v_t(x_t)\) in a numerical solver for the forward SDE.

Training: Mixture of Bridges

Require: Neural network v_θ, Dataset D = {X_0, X_1}, batch size B

While Training:
    Sample batches of size B: x_0 ∈ X_0, x_1 ∈ X_1
    Sample B timesteps: t ~ U([0, 1])
    Sample B noises: ε ~ N(0, I_d)
    Sample interpolant:
        x_t = (1-t)x_0 + t x_1 + σ √(t(1-t)) ε
    Compute loss:
        L_bridge(θ) ← ‖v_θ(t, x_t, x_0) - (x_1 - x_t)/(1-t)‖²

Return v_θ

IV. Schrödinger Diffusion Bridges

The following section introduces a specific instance of mixture of bridges derived from optimal transport, providing complementary background on principled coupling strategies.

Although mixtures of bridges define a probabilistic transport between \(p_0\) and \(p_1\), they do not necessarily satisfy properties that are desirable for data-distribution transport, such as straight trajectories for efficient sampling or the preservation of high-level semantic structure. In practice, the choice of the coupling plays a central role, as it directly influences both the curvature of the transport trajectories and the semantic similarity between the original data point \(x_0\) and its transported counterpart \(x_1\).

In the unpaired data translation setting, suitable couplings cannot be inferred directly from data and must instead be constructed implicitly. Naive choices, such as independent couplings, often lead to inefficient or semantically inconsistent transports. This motivates the use of principled coupling strategies grounded in optimal transport (OT). Within this framework, the Schrödinger bridge problem provides an entropy-regularized formulation of OT on path space. Given a reference Brownian motion \(\mathbb{Q} = \sigma \mathbb{W}\), the Schrödinger bridge is defined as the stochastic process that interpolates between prescribed marginals \(q_0\) and \(q_1\) while minimizing the relative entropy with respect to \(\mathbb{Q}\):

where \(\mathbb{P}\) is the stochastic process with marginals \(p_t\) that bridges \(q_0\) and \(q_1\).

Föllmer (1988) (Föllmer, 1988) proved that the Schrödinger problem admits a unique solution \(\mathbb{P}^*\) that is written as a mixture of (scaled) Brownian bridges weighted by an Entropic Optimal Transport (EOT) plan \(\pi^*_{2\sigma^2}\):

The induced endpoint coupling \(\pi^*_{2\sigma^2}\) coincides with the entropic optimal transport plan with regularization parameter \(2\sigma^2\):

Sampling pairs \((x_0, x_1) \sim \pi^*_{2\sigma^2}\) has traditionally relied on iterative trajectory simulations combined with drift fine-tuning (Liu et al., 2023; Bortoli et al., 2024; De Bortoli et al., 2021), resulting in slow training and high computational cost. In contrast, Tong et al. (Tong et al., 2023) introduce a simulation-free approach that approximates the Schrödinger bridge by estimating the EOT plan at the minibatch level. Notably, the EOT plan between empirical distributions can be efficiently computed using the Sinkhorn algorithm on a small number of samples of the batch.

References