Denoising Brownian Bridge Models

• Denoising Brownian Bridge Models are generative techniques that use stochastic differential equations with explicit endpoint conditioning to reduce prior mismatches and achieve deterministic boundaries.
• Their formulation couples forward and reverse dynamics via Brownian bridges, enabling efficient noise reduction and nearly deterministic sampling through dual approximators or consistency-based training.
• Applications span speech enhancement, time series forecasting, image-to-image translation, and biomedical modeling, with empirical results showing enhanced reconstruction fidelity and reduced computational steps.

Denoising Brownian Bridge Models are a class of generative modeling techniques in which the forward and reverse dynamics are governed by Brownian bridge stochastic differential equations (SDEs). These models couple classical score-based or diffusion processes with explicit endpoint conditioning, thereby pinning the stochastic process at both the starting and ending states. This structure provides advantageous properties for denoising, conditional generation, and sequence translation, leading to improvements in both reconstruction fidelity and sampling efficiency. Brownian bridge models have been applied across a range of domains, including speech enhancement, time series forecasting, image-to-image translation, and conditional biomedical modeling.

1. Mathematical Foundation and Formulation

The prototypical denoising Brownian bridge model defines a continuous-time SDE of the form

$$dX_t = \frac{y - X_t}{1-t}\,dt + g(t)\,dW_t, \qquad X_0 = x_0,\,X_1 = y$$

where $X_t$ interpolates between the initial state $x_0$ (such as clean signal or source domain image) and endpoint $y$ (noisy mixture, target domain, or specified condition) over $t\in[0,1]$. The diffusion coefficient $g(t)$ is typically chosen to ensure that the process variance peaks at intermediate times and vanishes at $t=0$ and $t=1$, enforcing deterministic boundary conditions.

The marginal distribution induced by this SDE at time $t$ is

$$X_t \sim \mathcal{N}\bigl((1-t)x_0 + t y,\;\sigma^2(t)I\bigr)$$

with $\sigma^2(0)=\sigma^2(1)=0$ and maximum variance at mid-bridge. This construction is used in various forms, with explicit expressions for $\sigma^2(t)$ and closed-form transition kernels adopted by different works (Li et al., 2022, Lay et al., 2023, Xiao et al., 29 Dec 2025).

In discrete time, one defines a sequence of bridge states with one-step Gaussian transitions:

$$q_{BB}(x_t|x_0,y) = \mathcal{N}\bigl(x_t\,;\,(1-m_t)x_0 + m_t y,\; \delta_t I\bigr)$$

with $m_t=t/T$ for $t=0,\dots,T$ and $\delta_t$ a bridge variance schedule (Li et al., 2022, Stoyanov et al., 10 Sep 2025).

The time-reversed SDE follows from standard results in diffusion theory (Anderson, 1982):

$$dX_t = -\frac{y-X_t}{1-t}\,dt + g^2(t)\,\nabla_{X_t}\log p_t(X_t)\,dt + g(t)\,d\overline W_t$$

where $\nabla_{X_t}\log p_t(X_t)$ is the time-$t$ score function (Lay et al., 2023, Li et al., 2022, Xiao et al., 29 Dec 2025).

2. Training Objectives and Algorithmic Details

The canonical objective is a mean-squared error (MSE) regression, either for direct denoising, score matching, or self-consistency. For score-based models, the loss is typically:

$$\mathbb{E}_{x_0,y,t,\epsilon}\bigl\| m_t(y-x_0) + \sqrt{\delta_t}\,\epsilon - \epsilon_\theta(x_t,t)\bigr\|^2$$

where $x_t$ is sampled from the bridge marginal (Li et al., 2022, Stoyanov et al., 10 Sep 2025).

For deterministic or consistency-based models, the mapping $f_\theta(x_t, y, t)$ is trained to yield the initial state $x_0$ for all $t$, enforced by the self-consistency loss:

$$\mathcal L_{SEB}(\theta,\theta^-) = \mathbb E_{x_0,y,n} \bigl\| f_\theta(x^{n+1},y,t_{n+1}) - f_{\theta^-}(x^{n},y,t_{n}) \bigr\|^2$$

with $\theta^-$ an exponential moving average parameter set (Qiu et al., 2023).

In the most recent Dual-approx Bridge framework, two neural approximators are used:

• $\epsilon_\theta(X_t, t)$ for recovering $X_0$ from noisy $X_t$
• $Z_\phi(X_t, t)$ for estimating the standardized noise at each reverse step with separate MSE losses for each, supporting nearly deterministic sampling with negligible variance (Xiao et al., 29 Dec 2025).

For time series and geometric applications, context encoders or spherical U-Nets are used to accommodate structured or multi-modal covariates (Yang et al., 2024, Stoyanov et al., 10 Sep 2025).

3. Endpoint Conditioning and Prior Matching

A key distinguishing feature of denoising Brownian bridge models is explicit endpoint (boundary) conditioning. Unlike classical diffusion (DDPM/score-SDE) where the forward process terminates in an arbitrary fixed prior (often isotropic Gaussian), here both the starting and ending states are prescribed. This property has multiple empirical and theoretical consequences:

• The prior mismatch between the terminating forward distribution and the initial law for the reverse process is eliminated or greatly reduced. For example, in BBED for speech enhancement, KL$[p_T\|\delta_y] \rightarrow 0$ at the endpoint since the forward mean and variance match the noisy mixture exactly, contrasting with non-bridge SDEs that exhibit significant mean and variance discrepancies (Lay et al., 2023).
• The variance schedule vanishes at the endpoints. This ensures zero noise at the boundaries, preventing information loss and reducing reconstruction artifacts.
• The bridge kernel in BBDM and its variants yields linear interpolation in mean, which is analytically tractable and stabilizes sampling (Li et al., 2022, Xiao et al., 29 Dec 2025, Stoyanov et al., 10 Sep 2025).

Table: Comparison of Terminal Distributions

Model	Terminal Mean	Terminal Variance	Prior Mismatch
BBED	Matches noisy $y$	$=0$	Negligible
OUVE SDE	$e^{-\gamma T}S + \ldots$	$>0$	Significant (KL≫0)

(Lay et al., 2023)

4. Sampling and Inference Algorithms

Sampling in denoising Brownian bridge models proceeds in reverse time, typically from the endpoint $y$ or a predicted future/prior. The main classes of algorithms are:

• Stochastic reverse SDE (Euler–Maruyama with score approximation) (Lay et al., 2023, Li et al., 2022).
• Probability-flow ODEs for deterministic sampling, where the reverse-time SDE noise is eliminated (Qiu et al., 2023, Xiao et al., 29 Dec 2025).
• One-step or few-shot deterministic mappings with a learned consistency or dual-approximator network (Qiu et al., 2023, Xiao et al., 29 Dec 2025).
• Structured context encoding in the denoiser for high-dimensional or structured data, such as the CoS-UNet for spherical mesh signals (Stoyanov et al., 10 Sep 2025).

Pseudocode typically involves iteratively updating $x_{t-1}$ from $x_t$ via closed-form Gaussian transitions, with neural or analytic estimation of the noise or score term.

Empirically, bridge models require fewer reverse steps than standard diffusion methods, supporting high-quality reconstruction with reduced computational cost. In SE-Bridge, single-step inference achieves competitive performance versus hundreds of diffusion steps (Qiu et al., 2023). BBED achieves state-of-the-art enhancement using only half the number of reverse steps compared to variance-exploding SDE baselines (Lay et al., 2023).

5. Applications Across Domains

Denoising Brownian bridge models have demonstrated substantial impact in several areas:

• Speech Enhancement: BBED and SE-Bridge models utilize the Brownian bridge structure for endpoint consistency between clean and noisy audio signals, reducing prior mismatch and improving objective metrics such as POLQA, PESQ, ESTOI, and SI-SDR (Lay et al., 2023, Qiu et al., 2023).
• Time Series Forecasting: S$^2$DBM leverages the Brownian bridge to "pin down" both ends of a forecast, reducing variance and outperforming non-autoregressive diffusion baselines on standard benchmarks (Yang et al., 2024).
• Image-to-Image Translation: BBDM and Dual-approx Bridge implement image translation as a latent-space bridge, yielding high-fidelity, low-variance outputs and competitive scores for FID, LPIPS, and PSNR on standard vision datasets (Li et al., 2022, Xiao et al., 29 Dec 2025).
• Geometric/Biomedical Modeling: SBDM enables vertex-wise forecasting of cortical thickness on the sphere via a Brownian bridge, supporting factual and counterfactual scenario generation in neuroimaging (Stoyanov et al., 10 Sep 2025).

6. Empirical Results and Comparative Performance

Consistent empirical gains have been demonstrated:

• Speech (WSJ0-CHiME3): BBED with 30 steps: POLQA=4.01, PESQ=3.08, ESTOI=0.94, SI-SDR=19.26 dB, surpassing variance-exploding SDE baselines on all counts (Lay et al., 2023).
• Time Series: S$^2$DBM achieves the best point-forecast MSE/MAE in 21/56 settings, with deterministic sampling (variance $s=0$) eliminating mid-trajectory oscillations (Yang et al., 2024).
• Images: Dual-approx Bridge: Cityscapes FID=48.70, PSNR=15.70, SSIM=53.26%, with minimal output variance and enhanced fidelity relative to both stochastic and deterministic comparators (Xiao et al., 29 Dec 2025). BBDM achieves similar improvements and flexible diversity control via the $s$ parameter (Li et al., 2022).
• Cortical Thickness: SBDM attains lower mean absolute error in longitudinal surface predictions than DDPM and deterministic baselines, while maintaining endpoint accuracy (Stoyanov et al., 10 Sep 2025).

A salient property in all applications is the ability to achieve stable, low-variance, and high-fidelity outputs, with controllable stochasticity through the bridge variance or auxiliary networks.

7. Theoretical and Practical Implications

The adoption of Brownian bridge dynamics corrects the prior mismatch present in standard diffusion generative models. Endpoint conditioning enforces both boundary integrity and improved sample quality in tasks with known input-output (or past-future) correspondences. The variance structure suppresses the information loss or excessive smoothing seen in models where the forward process terminates in a pure noise prior.

Practical benefits include:

• Elimination of extra stiffness or diversity hyperparameters (e.g., the $\gamma$ of OUVE SDEs)
• Dramatic reduction in the number of required sampling steps
• Straightforward integration of prior information and covariates at both endpoints
• Enhanced sample fidelity absent of stochastic artifacts common in unconstrained diffusion samplers

The theoretical guarantee that the reverse bridge inverts the forward process in law (with exact scores) establishes a clear blueprint for constructing statistically consistent denoisers across a multitude of conditional or paired-sample learning problems (Li et al., 2022, Lay et al., 2023, Stoyanov et al., 10 Sep 2025, Xiao et al., 29 Dec 2025, Yang et al., 2024).