Latent Bridge Models: Foundations & Applications

• Latent Bridge Models are probabilistic generative models that create stochastic bridges between latent representations to connect data distributions and support tasks like image translation, metric estimation, and audio super-resolution.
• They employ techniques such as Monte Carlo simulation, diffusion processes, and neural drift prediction to efficiently estimate transition densities and optimize latent space interpolation.
• LBMs enable modular cross-domain transfer by leveraging pretrained encoders and decoders, facilitating rapid prototyping, state-of-the-art performance, and reduced retraining costs.

Latent Bridge Models (LBMs) denote a category of probabilistic and generative models that establish stochastic bridges between latent representations of data, particularly by simulating or learning transition processes or transport maps in latent spaces. LBMs subsume methods ranging from Riemannian Brownian bridges for geometric shape analysis, through latent-space translation for cross-modal generative synthesis, to conditional and bridge-matching models for efficient image and audio translation. LBMs share the defining trait of modeling continuous or stochastic interpolations in learned or structured latent spaces, leveraging geometric, statistical, or learned priors to connect distributions or configurations across tasks as diverse as image-to-image translation, metric estimation on manifolds, and cross-domain transfer in generative models.

1. Mathematical Foundations: Bridge Processes in Latent Spaces

At the core of LBMs is the mathematical concept of a stochastic bridge, which is a conditioned diffusion process constrained to reach prescribed endpoints in latent space. In geometric contexts, this is formulated as a Riemannian Brownian motion on a manifold $Q$ with endpoints set by observed data. The transition density $p_{T, \theta}(v)$ of the process, governed by high-dimensional partial differential equations, is intractable in nonlinear cases and must be estimated via Monte Carlo simulation of bridges:

$$dy_{t} = K(y_{t}, y_{t})^{kl}\Gamma(y_{t})_{kl}\,dt - \frac{y_{t} - v}{T - t}\,dt + \sqrt{K(y_{t}, y_{t})}\,dW_{t}$$

for a landmark process on manifold $Q$ (Sommer et al., 2017). Correction factors, derived from the Radon–Nikodym derivative, adjust for the attraction term forcing the bridge and are averaged to numerically estimate transition densities for likelihood maximization.

In learning-based generative settings, LBMs often reparameterize bridge processes for tractable implementation:

• In image translation, a latent bridge $z_{t}$ is constructed via a stochastic interpolation between two latent codes $z_{0}$ and $z_{1}$:

$$z_t = (1-t)z_0 + t z_1 + \sigma \sqrt{t(1-t)} \epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$

where the drift and noise schedule (with $\sigma$ controlling stochasticity) are learned, typically via a neural network (Chadebec et al., 10 Mar 2025).

• In audio super-resolution, the latent bridge includes frequency-aware weighting:

$$z_t = \left(\frac{\alpha_t \bar{\sigma}_t^2}{\sigma_1^2}\right) z_0 + \left(\frac{\sigma_t^2 \bar{\alpha}_t}{\sigma_1^2}\right) z_T + \left(\frac{\alpha_t \bar{\sigma}_t \sigma_t}{\sigma_1}\right) \epsilon$$

allowing the model to leverage structured prior information from LR inputs and bridge to HR targets (Li et al., 22 Sep 2025).

2. Inference and Estimation Algorithms

Parameter estimation in LBMs is typically performed by maximizing the likelihood of observed endpoints, requiring efficient simulation and estimation of transition densities. In geometric LBMs (Sommer et al., 2017), each iteration of the estimation algorithm involves:

• Simulating sample paths of conditioned Brownian bridges.
• Computing correction factors for the drift-induced bridge.
• Updating parameters, such as the LDDMM kernel amplitude $\alpha$ and covariance $\Sigma$ via gradient ascent on the likelihood.

In learning-based LBMs such as Latent Bridge Matching (LBM), the drift function $v_\theta(z_t, t)$ is trained to minimize:

$$\mathcal{L}_{LBM} = \mathbb{E}_{t, z_0, z_1, \epsilon}\left[ \left\|\frac{z_1 - z_t}{1-t} - v_\theta(z_t, t) \right\|^2 \right]$$

where the expectation is over random interpolation times $t$, latent codes, and noise (Chadebec et al., 10 Mar 2025).

Diffusion bridge variants (e.g., DPBridge (Ji et al., 29 Dec 2024)) derive reverse transition kernels analytically using Doob’s $h$-transform to allow maximum likelihood optimization compatible with pretrained backbones. This yields closed-form transitions for the reverse process, supporting efficient ELBO maximization and tractable training.

3. Monte Carlo and Sequential Sampling Procedures

Monte Carlo techniques are central for simulating bridge processes when analytic densities are unavailable. In Riemannian settings, sample paths of guided Brownian bridges are simulated, and correction factor expectations are computed via averaging (Sommer et al., 2017). In particle filter bridge interpolation (Lindhe et al., 2021), stochastic interpolation paths are constructed in latent space using sequential Monte Carlo (SMC):

• At each time step, a set of particles propagates according to the Gaussian bridge proposal.
• Weights derived from a trained discriminator $f(z)$ (identifying high data-density regions) adjust the sampling measure, favoring trajectories that remain near the encoded data.
• Resampling increases adherence to the high-density manifold and introduces stochastic variability in interpolation.

This SMC-with-discriminator procedure ensures LBMs provide interpolation paths that stay closely aligned with the data manifold, enhancing sample quality and diversity.

4. Modularity, Shared Latent Spaces, and Cross-Modal Transfer

Bridge models facilitate modular compositionality by bridging pretrained generative models via shared latent space encoders and decoders. In Latent Translation (Tian et al., 2019), cross-modal transfer between domains (e.g., image $\rightarrow$ audio), or even VAE $\rightarrow$ GAN, is achieved via a domain-conditioned bridging VAE:

• Each domain’s pretrained model encodes to a latent code.
• A shared autoencoder maps these codes to a shared latent $z'$, allowing translation and alignment of semantic attributes.
• Sliced Wasserstein Distance and classification losses enforce overlap and alignment, yielding high transfer accuracy and low Fréchet Inception Distance (FID) in qualitative and quantitative benchmarks.

This modular structure decouples expensive base model retraining from the bridging process, enabling rapid prototyping and functional recombination of pretrained generative architectures, with speedups up to $200\times$ over monolithic retraining.

5. Extensions: Audio Super-Resolution, Dense Prediction, and Cascade Strategies

LBMs have been extended to structured prediction and signal processing tasks by formulating the translation or upsampling task as latent-to-latent bridging:

• Audio super-resolution (Li et al., 22 Sep 2025): AudioLBM compresses waveforms to latent space and learns bridges from LR to HR latent codes, leveraging explicit frequency-aware conditioning and cascade strategies (multi-stage LBMs with filtering and blurring augmentation) to support any-to-48kHz SR and seamless upsampling to 192kHz with state-of-the-art results across diverse benchmarks.
• Dense prediction (Ji et al., 29 Dec 2024): DPBridge introduces tractable reverse transition kernels and distribution-aligned normalization to exploit the visual priors of pretrained diffusion backbones. The model consistently achieves superior AbsRel and $\delta_1$ metrics in tasks such as monocular depth estimation across NYUv2, KITTI, ScanNet, ETH3D, and DIODE, with efficient inference and preserved fine spatial detail.

These extensions highlight the adaptability of the bridge paradigm to scenarios where informative priors, structural alignment, and data-efficient translation are critical.

6. Relation to Optimal Transport, Schrödinger Bridges, and Diffusion

Latent Schrödinger Bridges reformulate unpaired translation as entropy-regularized optimal transport between distributions in latent space, minimizing transport cost by solving controlled stochastic differential equations (SDEs):

$dx_t = u(x_t, t)dt + \sqrt{\tau}dw_t,$

with $x_0 \sim P_0$, $x_1 \sim P_1$ (Kim et al., 22 Nov 2024). The probability flow ODE admits a decomposition:

$$v(x_t, t) = ((1/2 - t)\sqrt{\tau})/\sqrt{t(1-t)} \cdot \text{NoisePredictor}(x_t) + \text{TargetPredictor}(x_t) - \text{SourcePredictor}(x_t)$$

LBMs leverage pre-trained Stable Diffusion backbones for efficient latent transport and prompt-level conditioning, matching SNR and noise statistics for plug-and-play translation. This framework enables unsupervised translation with far fewer function evaluations than conventional diffusion models.

7. Implementation, Performance, and Applications

Recent LBMs such as LBM (Chadebec et al., 10 Mar 2025) provide open-source implementations. Core steps include:

• Latent encoding via a pretrained VAE.
• Stochastic or deterministic bridge construction.
• Drift prediction via a neural network (optionally conditioned for controllable tasks).
• One-step inference leveraging learned transport maps.

LBMs have demonstrated versatility across domains:

• Fast image-to-image translation (e.g., object removal, relighting, geometric prediction), achieving state-of-the-art performance with single-step inference.
• Audio upsampling with frequency-aware and cascaded bridges, attaining unprecedented 192kHz super-resolution.
• Dense prediction tasks, surpassing prior generative methods in spatial fidelity and sample efficiency.

Their modular design, compatibility with pretrained backbones, and effective exploitation of prior knowledge and rich latent representations afford robust, scalable, and generalizable generative modeling paradigms.

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In summary, Latent Bridge Models unify geometric, probabilistic, and generative frameworks for connecting latent representations, supporting a wide spectrum of tasks requiring efficient, high-fidelity translation, interpolation, and upsampling. They integrate advanced statistical estimation, optimal transport, and efficient bridge simulation strategies, forming a foundational methodology in contemporary generative modeling and representation learning.