Dual Diffusion Implicit Bridge (DDIB)

• Dual Diffusion Implicit Bridge (DDIB) is a framework that maps data between distinct domains using independently trained diffusion models for unpaired translation.
• It utilizes forward diffusion to a shared latent space and reverse diffusion in the target domain, ensuring cycle consistency and linking to entropy-regularized optimal transport.
• DDIB supports diverse applications—from image-to-image translation to genomic modeling—with plug-and-play flexibility and competitive empirical performance.

The Dual Diffusion Implicit Bridge (DDIB) is a framework for mapping data between distinct domains by leveraging independently trained diffusion models. Originally introduced for unpaired image-to-image translation, DDIB generalizes to other modalities such as genomics and single-cell perturbation modeling. The method achieves cycle-consistency, flexible “plug-and-play” model composition, and provable connections to entropy-regularized optimal transport. It operates by diffusing source data into a shared latent representation and reconstructing in the target domain by reverse diffusion, bridging two distributions without paired data or joint training.

1. Mathematical Foundations and Core Workflow

Let $A$ and $B$ be two domains with respective data distributions $p_0^A$ and $p_0^B$. DDIB requires independent diffusion models DM$^A$ and DM$^B$, each trained solely on their respective domains. The core translation workflow comprises:

• Forward Diffusion in Source Domain: Start from $x_0^A \sim p_0^A$, and apply the forward SDE:

$$dx_t^A = f^A(x_t^A, t)\,dt + g^A(t)\,dW_t, \quad x_0^A\sim p_0^A$$

or in discrete form,

$$z_{t+1} = \sqrt{1-\beta_t}\,z_t + \sqrt{\beta_t}\,\varepsilon_t, \quad z_0 = x_0^A, \quad \varepsilon_t\sim\mathcal N(0,I)$$

After $T$ steps, the process yields a latent code $z_T^A$ with marginal density $p_T^A(z)$.

• Reverse Diffusion in Target Domain: Use DM$^B$, trained only on $p_0^B$, to solve the reverse SDE:

$$dx_t^B = \bigl(f^B(x_t^B, t) - g^B(t)^2\,S_\theta^B(x_t^B, t)\bigr)\,dt + g^B(t)\,d\bar W_t$$

starting at $z_T^B = z_T^A$. The denoised output $x_0^B$ is the translated image or datum.

The bridge is “implicit” since it is defined by the composition of forward and reverse flows, not by direct parameterization. Both directions (A→B and B→A) can be performed by switching roles, yielding sample-level and distribution-level cycle consistency when models are exact and discretization is ignored.

2. Connection to Optimal Transport and Schrödinger Bridge

DDIB inherits a deep connection to entropy-regularized optimal transport, specifically the Schrödinger bridge formulation. Each learned diffusion (probability flow) ODE between the data distribution and Gaussian prior encodes the unique entropic OT map (i.e., Schrödinger bridge) between its endpoints. The overall DDIB translation consists of concatenating:

• a Schrödinger bridge $p_0^A \to \mathcal N(0,I)$ (source to latent) and
• a reversed bridge $\mathcal N(0,I) \to p_0^B$ (latent to target).

This geometric structure explains the faithfulness of the translation and the observed cycle consistency. The intermediate latent (Gaussian) space is intrinsic to diffusion-based modeling, and DDIB exploits its standardized nature to “couple” the bridges without requiring paired samples (Su et al., 2022).

3. Theoretical Properties: Cycle Consistency and Latent Mismatch

DDIB’s invertibility and cycle consistency reflect the properties of the underlying PF-ODEs. In the continuous setting, mapping data A→B→A (or vice versa) recovers the original sample exactly. Discretization with methods like DDIM induces minor cycle errors, quantified empirically to be less than 0.02 relative $\ell_2$ (data std = 1) on synthetic datasets (Su et al., 2022).

A key limitation is the potential “latent mismatch gap.” The forward noising in domain A produces $p_T^A$, and denoising in B expects $p_T^B$. Unless $p_T^A \simeq p_T^B$, the initial state for the reverse SDE/SDE in B is not the model's intended prior. The mismatch $W_2(p_T^A, p_T^B)$ (Wasserstein-2 distance) bounds the translation error:

$$\bar I^B(T)\,W_2(p_T^A,p_T^B) \le W_2\bigl(p_0^B, q_0^B\bigr) \le I^B(T) W_2(p_T^A, p_T^B)$$

where $I^B(T)$ and $\bar I^B(T)$ depend on drift and score smoothness (Wang et al., 14 Nov 2025).

Therefore, high-fidelity translation requires either long diffusion time (to contract the mismatch by SDE dynamics) or explicit alignment of the latent distributions.

4. Empirical and Practical Considerations

The DDIB paradigm enables several practical advantages:

• Unpaired Translation & Privacy: Source and target models never require joint training or access to both datasets, supporting data privacy and decentralized workflows (Su et al., 2022).
• Plug-and-Play Flexibility: Arbitrary pre-trained DMs can be “bridged” for new translation pairs.
• Image-to-Image and Genomic Applications: DDIB has been applied to unpaired image-to-image translation (including high-resolution and class-conditional ImageNet), color transfer, and unpaired multi-perturbation single-cell response estimation (Chi et al., 26 Jun 2025).
• Experimental Results: Empirical benchmarks demonstrate near-zero cycle errors on synthetic data, competitive image translation metrics, and plausible faithfulness of content.

However, the need for diffusion ODE solving incurs higher computational cost—typically $O(N\cdot C)$ per image, where $N$ is the number of steps and $C$ is the cost per network evaluation.

5. Addressing Latent Distribution Mismatch: The OT-ALD Extension

A core challenge for DDIB is the latent mismatch between $p_T^A$ and $p_T^B$ after finite diffusion. The Optimal-Transport Aligning Latent Distributions (OT-ALD) method addresses this by computing an explicit optimal transport (OT) map $M_{ot} = \nabla u_T^{A \to B}$ to push $p_T^A$ onto $p_T^B$, with quadratic cost:

$$M_{ot} = \arg\min_{M: M_\# p_T^A = p_T^B} \mathbb{E}_{z \sim p_T^A} \|z - M(z)\|^2$$

This map is constructed using a semi-discrete Brenier approach, with the Brenier potential $$u_h(x) = \max_{1 \leq i \leq m} \langle x, y_i^B \rangle + h_i$$ where $\{y_i^B\}_{i=1}^m$ are samples from $p_T^B$ and $h \in \mathbb R^m$ is optimized to ensure mass balance.

In translation, one applies the sequence:

1. Run forward DM$^A$ to obtain $z_T^A$,
2. Map $z_T^A$ via $M_{ot}$,
3. Start reverse DM$^B$ from the OT-aligned $z_T^{B'} = M_{ot}(z_T^A)$.

Error analysis confirms that with exact score models and accurate OT solve, the output mismatch vanishes in $W_2$:

$$W_2(p_0^B, q_0^B) \leq \sqrt{2T}\,(\mathcal J_{SM}^B)^{1/2}$$

with no latent mismatch residual (Wang et al., 14 Nov 2025).

6. Experimental Results and Applications

Benchmarking on AFHQ (Cat→Dog, Wild→Dog), Summer2Winter, and CelebA-HQ (Male→Female), DDIB (with OT-ALD) achieves:

• Sampling efficiency: 20.29% faster than ILVR, enabled by shorter diffusion schedules due to OT alignment.
• FID reductions (lower is better): OT-ALD is, on average, 2.6 points lower in FID than the top-performing baseline (e.g., Cat→Dog FID: 44.3 vs. 47.9 for DMT).
• Cycle consistency: Empirical faithfulness of round-trip mapping, e.g., mean pixel MSE below 0.02 on synthetic data.
• Visual preservation: Structure and fine details of input images are better preserved, especially under reduced $T$.
• Flexibility: Supports arbitrary pairs of off-the-shelf DMs.

In single-cell domain translation, DDIB enables inference of perturbed states given only unpaired control data and vice versa. Enhancements such as integration with gene regulatory networks and masking for silent genes further tailor DDIB to biological applications (Chi et al., 26 Jun 2025).

Dataset/Task	FID (OT-ALD)	FID (Best Baseline)	Efficiency Gain (%)
Cat→Dog (AFHQ)	44.3	47.9 (DMT)	20.29
Summer2Winter	52.9	54.6	20.29
Male→Female (CelebA)	25.2	29.0	20.29

7. Limitations and Theoretical Insights

DDIB inherits both strengths and limitations of decoupled diffusion modeling:

• Exact cycle consistency and OT optimality hold only in the continuous, idealized setting. Discretization, score model approximation, and stochasticity introduce errors that are generally controlled and empirically negligible.
• ODE-based sampling is slower than feedforward generative adversarial networks, though solver improvements (Heun, DPM-Solver) ameliorate this.
• Each domain requires a separate model or a sufficiently expressive conditional DM.
• In settings with highly non-overlapping domains, the entropy-regularized nature of the Schrödinger bridge may smooth over sharp domain features.

In summary, the Dual Diffusion Implicit Bridge framework provides a modular, theoretically-grounded solution to unpaired domain translation, synthesizing advances in diffusion-based generative modeling and optimal transport. The introduction of OT-ALD resolves translation inefficiency and latent mismatch, establishing new empirical and theoretical benchmarks in both image and non-image domains (Wang et al., 14 Nov 2025, Su et al., 2022, Chi et al., 26 Jun 2025).