Gradient Discrete Schrodinger Bridge Matching for Peptide Generation
Abstract
The optimal control of a discrete Schrödinger bridge is a discrete gradient, a difference of vertex potentials, but recent models such as MadSBM instead learn a free control on the edges by cross-entropy. We prove that this leaves the action uncontrolled: cross-entropy identifies the control only up to a state-dependent shift (a gauge), and the action is KL(r|p) plus a gauge term the loss never constrains. We propose GDSBM, which writes the control as a difference of vertex potentials and adds the action to the loss; pinning the gauge costs no expressiveness and makes the action a property of the model, so the penalty acts on the model and not on the gauge. On peptide generation our MadSBM reimplementation has a free-gauge action of order 108 for a gauge-fixed value of $6.5$, against $1.5$ for GDSBM, with no quality loss that three seeds can detect. The gradient form also fits guidance, which becomes the addition of a potential: guided toward a net charge of +3 at pH~7, GDSBM is closer than MadSBM, over three seeds, to its exact tempered target in total variation at every guidance strength (at β=1, 81% of samples in the target window against 74%, for a target of 97%); we claim no statistical significance.
Related
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BibTeX
@misc{benabdallah2026gradient,
title = {{Gradient Discrete Schrodinger Bridge Matching for Peptide Generation}},
author = {Abdallah BenAbdallah and Mohsen Dlala},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31361},
eprint = {2610.31361},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31361},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gradient Discrete Schrodinger Bridge Matching for Peptide Generation
AU - Abdallah BenAbdallah
AU - Mohsen Dlala
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31361
UR - https://arxiv.org/abs/2610.31361
AB - The optimal control of a discrete Schrödinger bridge is a discrete gradient, a difference of vertex potentials, but recent models such as MadSBM instead learn a free control on the edges by cross-entropy. We prove that this leaves the action uncontrolled: cross-entropy identifies the control only up to a state-dependent shift (a gauge), and the action is KL(r|p) plus a gauge term the loss never constrains. We propose GDSBM, which writes the control as a difference of vertex potentials and adds the action to the loss; pinning the gauge costs no expressiveness and makes the action a property of the model, so the penalty acts on the model and not on the gauge. On peptide generation our MadSBM reimplementation has a free-gauge action of order 10^8 for a gauge-fixed value of $6.5$, against $1.5$ for GDSBM, with no quality loss that three seeds can detect. The gradient form also fits guidance, which becomes the addition of a potential: guided toward a net charge of +3 at pH~7, GDSBM is closer than MadSBM, over three seeds, to its exact tempered target in total variation at every guidance strength (at β=1, 81% of samples in the target window against 74%, for a target of 97%); we claim no statistical significance.
ER -