Structured Diffusion Bridges: Inductive Bias for Denoising Diffusion Bridges

Modality translation is inherently under-constrained, as multiple cross-modal mappings may yield the same marginals. Recent work has shown that diffusion bridges are effective for this task. However, most existing approaches rely on fully paired datasets, thereby imposing a single data-driven constraint. We propose a diffusion-bridge framework that characterizes the space of admissible solutions and restricts it via alignment constraints, treating paired supervision as an optional heuristic rather than a prerequisite. We validate our method on synthetic and real modality translation benchmarks across unpaired, semi-paired, and paired regimes, showing consistent performance across supervision levels. Notably, $\mathbf{\text{it}}\mathbf{\text{achieves}}\mathbf{\text{near}}\mathbf{\text{fully-paired}}\mathbf{\text{quality}}\mathbf{\text{with}}\mathbf{\text{a}}\mathbf{\text{substantial}}\mathbf{\text{relaxation}}\mathbf{\text{in}}\mathbf{\text{pairing}}\mathbf{\text{requirements,}}\mathbf{\text{and}}\mathbf{\text{remaining}}\mathbf{\text{applicable}}\mathbf{\text{in}}\mathbf{\text{the}}\mathbf{\text{unpaired}}\mathbf{\text{regime}}$. These results highlight diffusion bridges as a flexible foundation for modality translation beyond fully paired data.