Latent Aligned Diffusion Bridges (LADB)

• LADB is a semi-supervised framework for sample-to-sample domain translation that aligns latent endpoints using paired source data and a shared latent space.
• It leverages deterministic PF-ODE flows and a mixture coupling distribution to integrate scarce paired correspondences with unpaired target samples.
• Empirical evaluations on benchmarks like LSUN demonstrate that LADB maintains competitive generation quality, even under limited paired supervision and multi-source settings.

Latent Aligned Diffusion Bridges (LADB) is a semi-supervised framework for sample-to-sample domain translation that uses partially paired data to bridge source and target domains through a shared latent space. Its central construction couples a pretrained source-domain latent diffusion model with a target-domain Latent Aligned Diffusion Model (LADM), so that scarce source–target correspondences are converted into latent–target correspondences, while unpaired target samples are incorporated through a mixture coupling distribution. The method is designed for settings in which exhaustive paired supervision is unavailable and target-domain data are limited, and it is presented as a deterministic PF-ODE-based translation procedure with extensions to multi-source and multi-target settings (Wang et al., 10 Sep 2025).

1. Problem setting and motivation

LADB addresses sample-to-sample domain translation under partial supervision, particularly when the target domain is data-scarce and only a limited number of source–target paired correspondences are available. The motivating regime assumes the existence of a strong pretrained generative model in a source domain, such as depth or segmentation, together with a target domain such as RGB imagery, but without the resources required for exhaustive paired annotation or full target-domain diffusion retraining (Wang et al., 10 Sep 2025).

The framework is positioned between three established paradigms. First, unpaired translation methods can bridge distributions without paired supervision, but are described as lacking controllability in translation. Second, fully paired bridge methods such as DDBM require fully paired data and usually per-domain training. Third, conditional latent diffusion pipelines such as ControlNet-style conditioning require condition-specific architecture or adaptation. LADB is formulated as a middle ground: it uses partial source–target correspondences to create controllable translation while retaining access to unpaired target data (Wang et al., 10 Sep 2025).

A defining assumption is that source and target latent diffusion models share a common latent space. This shared latent domain allows a paired source sample to be mapped, through a pretrained source PF-ODE, to a latent endpoint that can be treated as aligned with its paired target sample. The result is not a direct observation-space bridge, but a latent-endpoint bridge assembled from deterministic source-side and target-side PF-ODE flows. This suggests that LADB should be understood primarily as a latent-space bridge method rather than an observation-space Schrödinger bridge.

2. Latent alignment and bridge construction

The core mechanism of LADB is the conversion of scarce paired source–target examples into latent–target supervision. For each paired sample $(x_{0,k}^{(s)}, x_{0,k}^{(t)})$, the pretrained source latent diffusion model computes a terminal latent endpoint by solving the source PF-ODE: $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$ This endpoint is then treated as the latent representative aligned with the paired target sample $x_{0,k}^{(t)}$ (Wang et al., 10 Sep 2025).

The target-side supervision is encoded by a mixture coupling distribution over target samples and latent endpoints: $q_{01}^{(t)}(x_0^{(t)},x_1^{(t)}) = \frac{1}{|K|+|L|}\cdot \Big( \sum_{k\in K}\delta_{({x}_{0,k}^{(t)}, {x}_{1,k}^{(t)})}(x_0^{(t)},x_1^{(t)}) + \sum_{l\in L} \delta_{x}_{0,l}^{(t)}(x_0^{(t)})\otimes q_1(x_1^{(t)}) \Big).$ Here the first term contains empirical paired latent–target atoms derived from partial supervision, while the second term couples unpaired target samples with independent latent prior draws. This construction is the main mathematical device by which LADB mixes paired and unpaired information (Wang et al., 10 Sep 2025).

The framework therefore does not learn a direct source–target bridge in raw data space. Instead, it concatenates two deterministic flows: a source-to-latent transport induced by the pretrained source LDM, and a latent-to-target transport learned by the target LADM. Relative to earlier diffusion bridge formulations, this places LADB closest to latent bridge methods that treat the bridge variable as an endpoint in a shared latent domain rather than as an observation-space state. A useful contrast is "Aligned Diffusion Schrödinger Bridges" (Somnath et al., 2023), which exploits aligned endpoint correspondences but operates directly in observation space and has no encoder–decoder latent bridge.

3. Model components, objective, and algorithms

LADB consists of a pretrained source-domain latent diffusion model and a target-domain Latent Aligned Diffusion Model. The source side contains a source encoder $E_{\phi'}^{(s)}$, a source decoder $D_{\psi'}^{(s)}$, and a source score network $s_{\theta'}^{(s)}$. The target side contains a pretrained target autoencoder $(E_\phi^{(t)}, D_\psi^{(t)})$ together with a trainable target score network $s_\theta^{(t)}$. During LADB training, the source model is used to infer aligned latent endpoints, whereas the target score is trained on the mixture coupling distribution (Wang et al., 10 Sep 2025).

The target LADM is optimized with the score-matching objective

$$\min_\theta E_{\substack{t\sim U(0,1),\,(x_0^{(t)},x_1^{(t)})\sim q_{01}^{(t)},\ x_t=\alpha_tE_\phi^{(t)}(x_0^{(t)})+\sigma_t x_1^{(t)} }} \Big[ \omega(t)\big\|s_\theta^{(t)}(x_t,t) - \nabla_{x_t}\log q(x_t|E_\phi^{(t)}(x_0^{(t)}))\big\|^2 \Big].$$

The provided formulation introduces no additional explicit latent-alignment loss, cycle-consistency penalty, adversarial term, or separate bridge reconstruction loss in the main objective; the alignment is induced by the way paired latent endpoints are constructed and injected into $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$0 (Wang et al., 10 Sep 2025).

The training algorithm is correspondingly direct. First, source-to-latent correspondences are inferred from partially paired data. Second, the mixture coupling distribution is assembled from paired latent–target atoms and unpaired target–prior couplings. Third, the target score network is trained by score matching on noisy interpolants of encoded target latents and endpoint variables. The output of training is the target autoencoder together with the learned target score network (Wang et al., 10 Sep 2025).

This architecture places LADB in close conceptual proximity to latent bridge models that perform transport in a shared latent domain. "DPBridge: Latent Diffusion Bridge for Dense Prediction" (Ji et al., 2024) is a related latent bridge framework in which paired, spatially aligned endpoints define a bridge in VAE latent space, though its alignment is implicit through dense correspondence rather than through a mixture of partially supervised latent couplings. "Towards General Modality Translation with Contrastive and Predictive Latent Diffusion Bridge" (Berman et al., 23 Oct 2025) provides another nearby construction, but সেখানে latent alignment is enforced with a contrastive InfoNCE-style loss rather than through source-inferred paired latent endpoints.

4. Deterministic translation, multi-source fusion, and conditioned targets

Inference in LADB is deterministic because both translation legs are PF-ODE solves. Given a source sample $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$1, the source-domain model first produces a latent endpoint

$${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$2

after which the target-domain LADM maps this endpoint back to a target latent: $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$3 The final translated sample is obtained by decoding $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$4 with the target decoder $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$5 (Wang et al., 10 Sep 2025).

The paper states that concatenating source and target PF-ODEs yields cycle consistency on the latent level. The bridge interpretation is therefore realized by meeting at the shared latent endpoint $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$6, not by explicitly simulating a single joint source–target bridge process. This distinguishes LADB from classical paired bridge training in which a single bridge law is learned directly over endpoint pairs.

LADB also extends naturally to multi-source translation. For $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$7 source domains, each source sample is first mapped to a latent endpoint

$${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$8

and the training distribution becomes a sum of mixture components over all source domains. At inference time, multiple source endpoints can be combined through weighted averaging: $${x}_{1,k}^{(t)} = {x}_{1,k}^{(s)} = ODESolve(E_{\phi'}^{(s)}({x}_{0,k}^{(s)}), s_{\theta'}^{(s)}, 0, 1).$$9 The paper interprets this weighted average as the Fréchet mean over the shared latent space (Wang et al., 10 Sep 2025).

For multi-target or class-conditioned generation, the target score can be conditioned as $x_{0,k}^{(t)}$0, where $x_{0,k}^{(t)}$1 may be a class label or a continuous text embedding. The provided description states that concatenation or cross-attention can be used for this conditioning, and that class-conditioned style transfer is demonstrated in supplementary material (Wang et al., 10 Sep 2025).

5. Empirical evaluation

The main reported evaluation uses LSUN-Bedroom and LSUN-Churches, with depth images and segmentation masks derived from LSUN-Bedroom by open-source annotators. The depth maps are produced using LeRes and the segmentation masks using DDPM-segmentation. The paper reports experiments on randomly selected 100,000 training images, with validation performed on 1,000 samples at resolution $x_{0,k}^{(t)}$2 (Wang et al., 10 Sep 2025).

The principal task is depth-to-image translation under varying proportions of paired data: 10%, 25%, 50%, and 100%. Metrics are FID, IS, LPIPS, and MSE. The reported LADB depth-to-image results are:

Paired data	FID	IS	LPIPS	MSE
10%	33.29	2.23	0.6499	0.1191
25%	33.44	2.29	0.6375	0.1195
50%	33.78	2.35	0.6335	0.1125
100%	34.78	2.43	0.6387	0.1052

The corresponding latent DDBM baseline results are reported as 39.39/2.15/0.6145/0.1147 at 10%, 35.69/2.21/0.6585/0.1117 at 25%, 34.23/2.23/0.5947/0.1118 at 50%, and 33.72/2.38/0.6030/0.1206 at 100% for FID/IS/LPIPS/MSE, while latent DDIB performs substantially worse in FID and MSE than either paired-semi-supervised bridge model (Wang et al., 10 Sep 2025).

The paper’s interpretation is that LADB is strongest in generation quality under partial supervision, especially at 10%, 25%, and 50% pairing, while remaining competitive on fidelity-oriented metrics. A central empirical claim is robustness to pair scarcity: performance degrades less sharply than baseline methods as the proportion of paired supervision decreases. The qualitative discussion attributes this to the mixture of paired and unpaired couplings and to the use of pretrained source latent diffusion as a latent correspondence engine (Wang et al., 10 Sep 2025).

Additional experiments cover multi-source translation from depth maps and segmentation masks, as well as HED-to-image, canny-to-image, sketch-to-image, and class-conditioned style transfer in supplementary material. The provided description states that LADB preserves fine-grained detail such as sheets and curtains, while multi-source interpolation yields coherent blends of style and content (Wang et al., 10 Sep 2025).

6. Position within diffusion bridge research and limitations

LADB inherits elements from several strands of diffusion bridge research while combining them in a distinct semi-supervised latent formulation. From DDBM it inherits the idea that diffusion should directly connect paired endpoint distributions rather than only map noise to data; however, DDBM requires fully paired training data and is formulated as a bridge between paired endpoint distributions (Zhou et al., 2023). From observation-space aligned bridge methods such as SBalign it inherits the emphasis on endpoint correspondence, but unlike SBalign it pushes the bridge into a shared latent domain and uses pretrained source-domain diffusion to manufacture latent correspondences from partial supervision (Somnath et al., 2023). From latent bridge methods such as DPBridge and LDDBM it inherits the move to latent-space transport, but its alignment mechanism is neither purely structural nor contrastive; it is defined by source-inferred endpoint pairing and a mixture coupling distribution (Ji et al., 2024).

The method’s most important explicit assumption is the existence of a shared latent space between source and target LDMs. This assumption underwrites the equation $x_{0,k}^{(t)}$3 and makes latent–target supervision meaningful. A plausible implication is that failures of latent compatibility would directly impair the quality of the inferred correspondences and, consequently, the target bridge model.

The paper also makes clear that LADB remains dependent on at least some paired correspondences. It is not an unpaired method in the DDIB sense, nor a fully paired bridge in the DDBM sense. Its intended operating regime is partial supervision, with unpaired target data supplementing rather than replacing paired anchors. The provided discussion further suggests that sparse or ambiguous controls, such as segmentation masks, can make latent alignment more difficult, and that performance depends on the quality of the pretrained source LDM and the quality of the shared latent domain (Wang et al., 10 Sep 2025).

In that sense, LADB is best characterized as a deterministic latent bridge framework for semi-supervised domain translation. Its defining contribution is not merely the use of latent diffusion, but the construction of aligned latent endpoints from partially paired source data and the use of a mixture of paired and unpaired latent-target couplings to train the target-side bridge model (Wang et al., 10 Sep 2025).