SDDBMs: Soft Denoising Diffusion Bridge Models

2026-08-09

AI Summary

This work addresses critical limitations in existing diffusion bridge models, which suffer from terminal singularities, degenerate to Dirac measures, and exhibit ill-conditioned drift dynamics near endpoints due to hard endpoint constraints. To overcome these issues, the authors propose the Soft Denoising Diffusion Bridge Model (SDDBM), which introduces soft terminal constraints into the diffusion bridge framework for the first time. By constructing non-degenerate Gaussian marginal distributions, SDDBM enables flexible bridging paths. The method leverages Doob’s h-transform combined with Gaussian reweighting to derive closed-form forward marginals and stochastic differential equation dynamics that are independent of the initial state. Notably, several existing bridge models emerge as special cases within this unified formulation. Experimental results demonstrate that SDDBM significantly enhances numerical stability and generation quality in image restoration tasks.

Abstract

Diffusion bridge models leverage Doob's \(h\)-transform to construct stochastic transports between arbitrary endpoint distributions, and have shown strong potential in image-to-image translation and restoration. However, most existing bridge models rely on hard endpoint conditioning, which forces the terminal state to match a prescribed target exactly. This hard constraint induces terminal-boundary singularities: the terminal law collapses to a Dirac measure, and the resulting drift coefficients become ill-conditioned near the endpoint. In this paper, we propose Soft Denoising Diffusion Bridge Models (SDDBMs), a generalized framework that regularizes diffusion bridges directly at the level of their terminal constraints. Instead of imposing an exact endpoint, SDDBMs prescribe a non-degenerate Gaussian terminal marginal under the transformed path measure, with a flexible terminal center and variance. Starting from this prescribed marginal, we develop a complete closed-form construction of the soft bridge, including the Gaussian terminal reweighting and soft \(h\)-function, the induced Gaussian forward marginals and \(\mathbf{x}_0\)-free dynamics. Theoretically, SDDBMs provide a unified probabilistic perspective that encompasses existing diffusion bridge models, including DDBMs, GOUB, and UniDB, as special cases under specific parameter choices. Extensive experiments on image restoration tasks demonstrate that SDDBMs achieve improved numerical stability and superior generation quality over existing bridge-based methods.

Problem

• diffusion bridge models
• hard endpoint conditioning
• terminal-boundary singularities
• ill-conditioned drift
• Dirac measure

Innovation

• Soft Denoising Diffusion Bridge Models
• terminal constraint regularization
• non-degenerate Gaussian terminal marginal
• closed-form bridge construction
• diffusion bridges