# Minimal-Action Discrete Schr"odinger Bridge Matching for Peptide Sequence Design

arXiv: https://arxiv.org/abs/2601.22408 · HTML: https://mlsanity.com/paper/2601.22408/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 53 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 30 of 31)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [ProGen2: Exploring the Boundaries of Protein Language Models](https://mlsanity.com/paper/2206.13517/) (2022)
- [Protein Design with Guided Discrete Diffusion](https://mlsanity.com/paper/2305.20009/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [A Reparameterized Discrete Diffusion Model for Text Generation](https://mlsanity.com/paper/2302.05737/) (2023)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Unlocking Guidance for Discrete State-Space Diffusion and Flow Models](https://mlsanity.com/paper/2406.01572/) (2024)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Parallel Sampling of Diffusion Models](https://mlsanity.com/paper/2305.16317/) (2023)
- [Diffusion Language Models Are Versatile Protein Learners](https://mlsanity.com/paper/2402.18567/) (2024)
- [Path Planning for Masked Diffusion Model Sampling](https://mlsanity.com/paper/2502.03540/) (2025)
- [Accelerating Diffusion Sampling with Optimized Time Steps](https://mlsanity.com/paper/2402.17376/) (2024)
- [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023)
- [PepTune: De Novo Generation of Therapeutic Peptides with Multi-Objective-Guided Discrete Diffusion](https://mlsanity.com/paper/2412.17780/) (2024)
- [Tilt Matching for Scalable Sampling and Fine-Tuning](https://mlsanity.com/paper/2512.21829/) (2025)
- [Discrete Diffusion Schr"odinger Bridge Matching for Graph Transformation](https://mlsanity.com/paper/2410.01500/) (2024)
- [Learning to design protein-protein interactions with enhanced generalization](https://mlsanity.com/paper/2310.18515/) (2023)
- [Multi-Objective-Guided Discrete Flow Matching for Controllable Biological Sequence Design](https://mlsanity.com/paper/2505.07086/) (2025)
- [TR2-D2: Tree Search Guided Trajectory-Aware Fine-Tuning for Discrete Diffusion](https://mlsanity.com/paper/2509.25171/) (2025)
- [AReUReDi: Annealed Rectified Updates for Refining Discrete Flows with Multi-Objective Guidance](https://mlsanity.com/paper/2510.00352/) (2025)
- [Exponential convergence rate for Iterative Markovian Fitting](https://mlsanity.com/paper/2508.02770/) (2025)

## Abstract (arXiv)

Generative modeling of peptide sequences requires navigating a discrete and highly constrained space in which many intermediate states are chemically implausible or unstable. Existing discrete diffusion and flow-based methods rely on reversing fixed corruption processes or following prescribed probability paths, which can force generation through low-likelihood regions and require countless sampling steps. We introduce Minimal-action discrete Schr"odinger Bridge Matching (MadSBM), a rate-based generative framework for peptide design that formulates generation as a controlled continuous-time Markov process on the amino-acid edit graph. To yield probability trajectories that remain near high-likelihood sequence neighborhoods throughout generation, MadSBM 1) defines generation relative to a biologically informed reference process derived from pre-trained protein language model logits and 2) learns a time-dependent control field that biases transition rates to produce low-action transport paths from a masked prior to the data distribution. We finally introduce guidance to the MadSBM sampling procedure towards a specific functional objective, expanding the design space of therapeutic peptides; to our knowledge, this represents the first-ever application of discrete classifier guidance to Schr"odinger bridge-based generative models.