# Budgeted Quotient-Residual Guidance for Frozen Pocket-Conditioned Molecular Diffusion

arXiv: https://arxiv.org/abs/2609.31222 · HTML: https://mlsanity.com/paper/2609.31222/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 45 of its 56 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 30 of 45)

- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [GuacaMol: Benchmarking Models for De Novo Molecular Design](https://mlsanity.com/paper/1811.09621/) (2018)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Molecular Sets (MOSES): A Benchmarking Platform for Molecular Generation Models](https://mlsanity.com/paper/1811.12823/) (2018)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)

## Abstract (arXiv)

Pocket-conditioned molecular diffusion updates ambient atom coordinates, but many lead-optimization objectives are expressed on quotient features such as distances, contacts, and anchored substructures. We introduce budgeted quotient-residual guidance (QRG), an inference-time correction that makes these quotient objectives active without retraining the molecular generator. QRG lifts quotient covectors to metric-horizontal ambient directions and delivers them through a trust budget set by the frozen sampler's own step norm: quotient geometry chooses the direction, while sampler motion bounds the scale. We derive the horizontal lift, closed-form sampler-budget update, KL/kinetic interpretation around a frozen reverse step, equivariance conditions, and a product-budget split for budget-capped section and residual controls. Controlled quotient tasks confirm that sampler-relative delivery activates signals that raw local quotient gradients leave dormant. On frozen TargetDiff backbones, official seed-0 CBGBench ligand-generation/editing sweeps show practical quality-runtime gains: Local-QRG improves validity from 0.815 to 0.864 on fragment growing, 0.664 to 0.707 on scaffold hopping, and 0.681 to 0.712 on linker design, while PredNext-QRG improves fragment/scaffold and remains near-neutral on linker. Novelty remains 1.000 and diversity is preserved in the matched multi-seed molecular slice, giving task-dependent improvements without sampler retraining or backbone modification. Overall, QRG provides a lightweight route to quotient-aware inference for frozen molecular samplers with explicit runtime accounting.