Abstract
This paper motivates the use of random-bridges — stochastic processes conditioned to take target distributions at fixed timepoints — in the realm of generative modelling. Herein, random-bridges can act as stochastic transports between two probability distributions when appropriately initialized, and can display either Markovian or non-Markovian, and either continuous, discontinuous or hybrid patterns depending on the driving process. We show how one can start from general probabilistic statements and then branch out into specific representations for learning and simulation algorithms in terms of information processing. Our empirical results, built on Gaussian random bridges, produce high-quality samples in significantly fewer steps compared to traditional approaches, while achieving competitive Frechet inception distance scores. Our analysis provides evidence that the proposed framework is computationally cheap and suitable for high-speed generation tasks.
Summary
This paper introduces the formal mathematical framework of (Φ, Ψ)-bridges for generative modelling. A random bridge is a càdlàg stochastic process constrained to take a target distribution at a fixed terminal time, acting as a stochastic transport between two distributions. The framework is deliberately abstract — imposing no model-specific assumptions beyond right-continuity with left limits — so it can encompass Markovian or non-Markovian, continuous (Brownian), discontinuous (Poisson, gamma), or hybrid dynamics depending on the driving process.
For the Gaussian case, the bridge admits a closed-form anticipative representation decomposing into a signal component (the target) and a noise process, echoing information-based models from stochastic filtering. The conditional expectation E[Y | ξ_t] — the L² best-estimate — becomes the central object for both training and sampling. Training amounts to learning this conditional expectation via MSE loss, while simulation uses Euler-Maruyama discretization of the bridge SDE in a single forward pass, without any backward-time denoising procedure. The information-theoretic analysis proves that Shannon entropy of the posterior is a supermartingale converging to zero.
Experiments on MNIST and CIFAR-10 demonstrate dramatically better FID at very low step counts (2-10 steps) compared to both DDPM and improved DDPM baselines, though at 1000 steps the improved DDPM surpasses the bridge model. This positions random bridges as particularly suitable for high-speed generation tasks.
Key Contributions
- General mathematical definition of (Φ, Ψ)-bridges as stochastic transports, subsuming diffusion bridges as a special case
- Single-directional generative framework eliminating the bi-directional noising-denoising protocol of DDPMs
- Anticipative and non-anticipative bridge representations with closed-form conditional distributions
- Simple MSE training objective on E[Y | ξ_t] with analytically tractable conditional sampling
- Dramatic low-step sampling efficiency: competitive FID with 2-10 steps
- Lévy random bridge extensions to gamma, stable-1/2, and jump-diffusion processes
- Shannon entropy supermartingale property formalizing information gain along the bridge
Methodology
The framework builds on a probability space with right-continuous complete filtration. For generative modelling, a Φ-initialized random bridge to Ψ is constructed with driving process {Z_t} = {σW_t}. Training minimizes MSE between neural network output f(ξ_t, t; θ) and target y, sampling ξ_t from known Gaussian conditionals. Simulation proceeds via Euler-Maruyama discretization of dξ_t = (Ŷ_t - ξ_t)/(T-t) dt + σdW_t. Architecture uses a modified UNet (64 channels, 2 residual blocks, 4 attention heads) trained for 40k steps.
Key Findings
- 2-step MNIST FID: Bridge 61.9 vs Improved DDPM 299.2 (~5x better)
- 10-step MNIST FID: Bridge 19.3 vs Improved DDPM 136.9 (~7x better)
- At 1000 steps, Improved DDPM surpasses Bridge model
- Shannon entropy of target posterior is provably non-increasing, reaching zero at terminal time
- The Doob h-transform naturally arises in the bridge transition probabilities
Important References
- Lévy Random Bridges and the Modelling of Financial Information — Original Lévy random bridge theory
- Score-Based Generative Modeling through Stochastic Differential Equations — Improved DDPM baseline
- Denoising Diffusion Probabilistic Models — Primary DDPM baseline