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

Xinyu Wang; Jinbo Bi; Minghu Song

cs.LGSep 25, 2026

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

Authors: Xinyu Wang, Jinbo Bi, Minghu Song

Organizations: Department of Computer Science and Engineering, University of Connecticut Storrs, CT 06269, USA · Institute of Health and Medicine, Hefei Comprehensive National Science Center Hefei 230601, China

Abstract

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.

Figures & tables

Figure 1: Budgeted quotient-residual guidance. A frozen molecular sampler moves ambient coordinates, while many scientific objectives live on quotient features such as distances, contacts, and fixed anchors. Raw quotient directions can be dormant at sampler scale. QRG keeps the quotient-residual direction but delivers it through a sampler-relative trust budget. The right panel gives the deployed Budgeted QRG step: section and quotient-residual controls are budget-capped separately by the frozen sampler’s own motion. Deployed variants use PredNext or local-radius quotient features; rollout residuals are teachers and diagnostics, not deployment requirements.

Figure 2: Controlled quotient tasks. Dormant local guidance is almost identical to the base sampler, while trust-region delivery makes the same quotient signal active under the same denoising schedule.

Task	Method	Success $\uparrow$	Key error $\downarrow$	NFE
Gaussian 2D	Base	0.159	0.528	64
	Dormant local	0.159	0.529	64
	Budgeted residual	0.246	0.451	64
Orbit points	Base	0.117	0.406	72
	Dormant local	0.117	0.406	72
	Budgeted residual	0.310	0.299	72

Table 1: Toy-task summary. The key error is quotient MMD for Gaussian 2D and invariant template error for the geometry tasks. All rows within a task use the same denoising budget.

Task	Method	Official parts	Validity $\uparrow$	$\Delta$ val.	sec/sample	Control/base	Novelty $\uparrow$	Diversity $\uparrow$
Fragment	Base	5/5	0.815	–	16.68	0.000	1.000	0.898
	PredNext-QRG	5/5	0.829	+0.014	18.28	0.192	1.000	0.899
	Local-QRG	5/5	0.864	+0.049	19.91	0.265	1.000	0.896
Scaffold	Base	5/5	0.664	–	17.99	0.000	1.000	0.913
	PredNext-QRG	5/5	0.683	+0.019	20.36	0.194	1.000	0.911
	Local-QRG	5/5	0.707	+0.043	20.18	0.268	1.000	0.912

Table 2: Budgeted inference on frozen ligand-generation/editing tasks. Validity, runtime, and control/base ratio are official seed-0 aggregates; novelty and diversity are from the matched multi-seed molecular slice for the same task and method. Fragment and scaffold improve for both deployed variants. Linker design is near-neutral for PredNext-QRG but improves for Local-QRG , showing that the quality–runtime trade-off is task-dependent rather than uniformly monotone.

Figure 3: Official-run norm accounting for the trust-region mechanism. Left: delivered control stays well below the frozen sampler’s own step scale across official seed-0 ligand-generation/editing shards. Right: for Local-QRG , the delivered section and quotient-residual components spend a small bounded budget relative to the base sampler motion. This is a kinetic-budget diagnostic, not a measured path KL: it supports the mechanism claim that QRG delivers quotient information at sampler scale rather than applying an unbounded external optimizer.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Expanded quality–runtime trade-off from the matched multi-seed molecular slice. Fragment has the richest ablation set; scaffold and linker include base, PredNext-QRG , and Local-QRG . This figure complements the official seed-0 aggregate by showing novelty and diversity preservation across matched seeds.

Cheap surrogate	Validity $\uparrow$	Unique	Diversity	sec/sample
Sampled pairs, 16	0.888	854	0.896	17.84
Sampled pairs, 128	0.886	856	0.897	18.15
Local radius	0.902	869	0.896	16.88

Appendix

Table 3: Development-slice fragment seed-0 ablation for cheap quotient surrogates. The local-radius surrogate is not introduced as an arbitrary extra branch: on this slice it outperforms sampled-pair residual surrogates at comparable runtime while preserving diversity.

Task	Method	$n$	Mean $\Delta$	Median $\Delta$	95% bootstrap CI	Wins/losses/ties
Fragment	PredNext-QRG	61	+0.013	+0.010	$[-0.006,0.032]$	32/27/2
Fragment	Local-QRG	61	+0.049	+0.030	$[0.019,0.078]$	42/14/5
Scaffold	PredNext-QRG	64	+0.019	+0.040	$[-0.029,0.061]$	44/17/3
Scaffold	Local-QRG	64	+0.043	+0.060	$[-0.016,0.099]$	43/19/2
Linker	PredNext-QRG	43	-0.001	+0.020	$[-0.037,0.033]$	22/19/2
Linker	Local-QRG	43	+0.030	+0.040	$[-0.012,0.073]$	25/14/4

Appendix

Table 4: Paired official seed-0 target-level validity deltas. Each row compares a guided method to the unguided sampler on the same targets; bootstrap intervals resample targets. These statistics support the aggregate table while showing target heterogeneity.

Task	Method	$n$	Mean $\Delta$	Median	P10	P25	P75	P90	Min/Max
Fragment	PredNext-QRG	61	+0.013	+0.010	-0.060	-0.020	+0.050	+0.100	-0.160/+0.210
Fragment	Local-QRG	61	+0.049	+0.030	-0.060	+0.000	+0.100	+0.190	-0.280/+0.400
Scaffold	PredNext-QRG	64	+0.019	+0.040	-0.097	-0.010	+0.103	+0.167	-0.860/+0.330
Scaffold	Local-QRG	64	+0.043	+0.060	-0.246	-0.023	+0.190	+0.271	-0.860/+0.520
Linker	PredNext-QRG	43	-0.001	+0.020	-0.158	-0.035	+0.060	+0.130	-0.340/+0.270
Linker	Local-QRG	43	+0.030	+0.040	-0.148	-0.050	+0.130	+0.156	-0.260/+0.440

Appendix

Table 5: Distribution of per-target official seed-0 validity deltas versus the unguided sampler. Each target contributes one target-level validity difference for the same task and method. The quantiles show that aggregate gains coexist with heterogeneous pockets, including a small number of large negative and large positive target-level shifts.

Fragment variant	Seeds	Validity $\uparrow$	Runtime	Control/base
Base	3	$0.842$	$1.00\times$	0.000
Section-only	1	$0.896$	$1.07\times$	–
Sham-QRG	3	$0.888$	$1.22\times$	0.255
PredNext-QRG	3	$0.896$	$1.10\times$	0.193
Local-QRG	3	$\mathbf{0.905}$	$1.22\times$	0.271
Rollout-QRG Teacher	3	$0.898$	$2.15\times$	0.253

Appendix

Table 6: Fragment ablation frontier. Rollout-QRG Teacher is useful as a diagnostic upper-cost point but is not the deployed method. The comparison separates the core trust-region delivery rule from rollout-heavy or unstructured guidance controls.

Figure 5: Side-chain decoration exposes a task-dependent budget mismatch. Default ligand-generation budgets reduce side-chain validity relative to the unguided sampler. A sparse low-budget rescue nearly closes the gap, but does not beat unguided in the full sweep.

Task / method	Paired targets	Mean $\Delta$ Vina $\downarrow$	Better targets
Fragment / PredNext-QRG	30	$0.061\;[-0.049,0.174]$	14/30
Fragment / Local-QRG	30	$-0.101\;[-0.189,-0.023]$	17/30
Scaffold / PredNext-QRG	44	$-0.132\;[-0.234,-0.035]$	28/44
Scaffold / Local-QRG	44	$\mathbf{-0.285}\;[-0.397,-0.178]$	35/44
Linker / PredNext-QRG	42	$-0.001\;[-0.087,0.081]$	20/42
Linker / Local-QRG	42	$0.038\;[-0.055,0.130]$	17/42

Appendix

Table 7: Corrected raw-pocket docking diagnostic. Values are paired mean differences versus the base sampler with bootstrap intervals. Negative means the guided variant has better raw AutoDock Vina score. The signal is clearest on scaffold hopping and mixed on the other tasks.

Evaluation asset	Completed rows	Generated samples	Role in the paper
Official ligand-generation/editing sweeps	9	50,400	Main validity, runtime, and control-budget evidence for fragment growing, scaffold hopping, and linker design in table 2 .
Official section-only controls	3	16,800	Residual-off controls used to separate section delivery from quotient-residual guidance.
Official side-chain sweep	4	25,600	Out-of-family budget-transfer stress test; not counted as a ligand-generation win claim.
Matched multi-seed molecular slice	40	–	Novelty/diversity preservation, development ablations, rollout-teacher diagnostics, and sham-control comparisons.
Target-balanced docking diagnostic	6	–	Supporting raw-pocket AutoDock Vina diagnostic with paired target comparisons; not a binding-affinity claim.
Official full aggregate total	16	92,800	Completed official seed-0 aggregate rows across all four molecular tasks and four reported method families.

Appendix

Table 8: Completed evaluation inventory. The table records what was completed and how each asset is used in the paper. It is an accounting table, not an additional performance ranking.

Task	Method	Generated	sec/sample	Approx. GPU h	Control/base
Fragment	Base	6,100	16.68	28.3	0.000
Fragment	PredNext-QRG	6,100	18.28	31.0	0.192
Fragment	Local-QRG	6,100	19.91	33.7	0.265
Scaffold	Base	6,400	17.99	32.0	0.000
Scaffold	PredNext-QRG	6,400	20.36	36.2	0.194
Scaffold	Local-QRG	6,400	20.18	35.9	0.268

Appendix

Table 9: Compute accounting for the official seed-0 ligand-generation/editing rows that carry the main molecular claim. Approximate GPU hours are computed from generated samples times aggregate wall-clock seconds/sample divided by 3600, and therefore include denoising, guidance, postprocessing, and file writing as recorded by the runner.

Method family	Guidance schedule	Guidance deliveries/sample	Denoiser-query interpretation
Base	Frozen reverse sampler only	0	No guidance branch and no extra denoiser rollout.
Section-only	Section branch, residual budget zero	500	Tests delivery without quotient-residual correction.
PredNext-QRG	Predicted-next residual every two denoising steps	250	Reuses the sampler’s predicted next state; deployed point avoids rollout-teacher queries.
Local-QRG	Local-radius quotient surrogate with trust-region delivery	500	Uses a cheap local surrogate rather than querying a rollout teacher.
Rollout-QRG Teacher	One shared rollout before residual evaluation	500	Diagnostic high-cost teacher/oracle point, not the deployed method.

Appendix

Table 10: Denoiser-query and guidance-call accounting. The table separates deployed QRG variants from rollout-teacher diagnostics and explains why the reported runtime overhead is modest relative to a rollout-heavy inference scheme.

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Molecular Generation Diffusion Models 3D Molecular Generation Permutation-Invariant Learning

Open paper AskPDF arXiv

Permutation invariance is fundamental in molecular point-cloud generation, yet most diffusion models enforce it indirectly via permutation-equivariant networks on an ordered space. We propose to model diffusion directly on the quotient manifold

\tilde{\calX}=\sR^{d\times N}/S_N

, where all atom permutations are identified. We show that the heat kernel on

\tilde{\calX}

admits an explicit expression as a sum of Euclidean heat kernels over permutations, which clarifies how diffusion on the quotient differs from ordered-particle diffusion. Training requires a permutation-symmetrized score involving an intractable sum over

S_N

; we derive an expectation form over a posterior on permutations and approximate it using MCMC in permutation space. We evaluate on unconditional 3D molecule generation on QM9 under the EQGAT-Diff protocol, using SemlaFlow-style backbone and treating all variables continuously. The results demonstrate that quotient-based permutation symmetrization is practical and yields competitive generation quality with improved efficiency.

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