Research map / Probabilistic and scientific modeling / Variational and generative sampling
Categorical Schrödinger Bridge Matching
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Proves that discrete-time Iterative Markovian Fitting converges to the Schrödinger Bridge on finite discrete spaces and derives a practical algorithm (CSBM) that applies it to categorical data such as VQ codebooks, text tokens and colored images.
Problem
Schrödinger Bridge and unpaired translation methods are mostly developed for continuous spaces, and the discrete-time IMF procedure lacks theoretical guarantees and practical algorithms for discrete (categorical) state spaces such as VQ codebooks, text tokens and image pixels.
Approach
Shows that for a finite discrete state space and a general Markov reference process with full support, the unique process that is both Markov and reciprocal is the dynamic SB solution, which justifies D-IMF convergence; then parameterizes forward/backward transition models via reference-bridge posterior sampling with per-dimension factorization and trains them with D3PM-style losses inside a bidir
Key results
- On CelebA with N=100 and alpha=0.005, CSBM reaches FID 10.60 versus 16.86 for ASBM and 24.06 for DSBM at epsilon=1.
- On Amazon Reviews, CSBM (alpha=0.01) attains the highest BLEU of 74.8 and lowest NLL of 5.4, but its sentiment accuracy (76.5-79.3) is lower than most baselines.
- In the synthetic D-IMF test with S=50, KL between iterates and the Sinkhorn ground-truth bridge converges notably fast.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CSBM instead of ASBM or DSBM for latent-space translation | CelebA 128×128 male→female · CelebA 128×128 male→female | FID: 10.60 vs 16.86 (ASBM); 10.60 vs 24.06 (DSBM) | Helped |
| Increase stochasticity from α=0.005 to α=0.01 (ablation) | CSBM on CelebA 128×128 male→female · CelebA 128×128 male→female | FID: 10.60 -> 14.68 | Hurt |
| Increase stochasticity from α=0.005 to α=0.01 (ablation) | CSBM on CelebA 128×128 male→female · CelebA 128×128 male→female | LPIPS: 0.175 -> 0.170 | Helped |
| Use CSBM instead of other methods | Amazon Reviews negative↔positive transfer · Amazon Reviews negative↔positive | Accuracy: 79.3 vs 93.5 | Hurt |
| Use CSBM instead of other methods | Amazon Reviews negative↔positive transfer · Amazon Reviews negative↔positive | BLEU: 74.8 vs 13.7 | Helped |
| Increase the number of intermediate steps N (ablation) | CSBM on Colored MNIST · Colored MNIST classes 2↔3 | visual similarity | Hurt |
| Increase the number of intermediate steps N (ablation) | CSBM on Colored MNIST · Colored MNIST classes 2↔3 | pixelation | Helped |
| Replace Gaussian reference with uniform reference (ablation) | CSBM on Colored MNIST · Colored MNIST classes 2↔3 | color matching | Hurt |
| Replace KL loss with MSE loss (ablation) | CSBM on 2D Gaussian-to-Swiss-Roll translation · 2D Gaussian→Swiss Roll | visual sample and trajectory quality | No clear effect |
| Apply DSBM in the VQ-GAN latent space instead of CSBM (ablation) | CelebA male→female translation · CelebA VQ-GAN latent space | translation quality | Hurt |
| Increase α (ablation) | CSBM reference process · 2D categorical experiments | number of jumps | Helped |
Surprising findings
- DSBM trained directly in the VQ-GAN latent space collapsed to near-identity mappings, while CSBM worked in that space.
- A uniform reference process failed to match digit colors on Colored MNIST, showing reference choice matters.
Limitations
- Per-dimension factorization of the learned transitions ignores dependencies between features, causing pixelation in image results.
- Reference processes (uniform, Gaussian) are hand-built and alpha must be tuned per dataset; non-uniform references are harder to construct.
- Comparisons on CelebA and Amazon Reviews rely on baseline numbers taken from prior papers, and sentiment accuracy is lower than several baselines.
Experimental setup
Datasets: CelebA Colored MNIST Amazon Reviews Swiss Roll synthetic 2D
Benchmarks: CelebA 128x128 male-to-female translation Amazon Reviews negative-positive style transfer Colored MNIST 2-to-3 translation synthetic 2D Gaussian-to-Swiss Roll
Metrics: FID CMMD LPIPS accuracy NLL BLEU KL divergence to ground-truth SB
Baselines: ASBM DSBM CAAE Del.&Ret. Seq2SentiSeq BST FGIM PST SCT_1
Builds on: Iterative Markovian Fitting discrete-time IMF D3PM ASBM DSBM
Closest papers by meaning
- Discrete Diffusion Schrödinger Bridge Matching for Graph Transformation (ICLR 2025 · similarity 0.89)
- Schrödinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres (NeurIPS 2025 · similarity 0.89)
- Momentum Multi-Marginal Schrödinger Bridge Matching (NeurIPS 2025 · similarity 0.88)
- Topological Schrödinger Bridge Matching (ICLR 2025 · similarity 0.87)
- Adversarial Schrödinger Bridge Matching (NeurIPS 2024 · similarity 0.87)
- Light Schrödinger Bridge (ICLR 2024 · similarity 0.87)
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