arxiv: 2604.21809 · v2 · submitted 2026-04-23 · 💻 cs.LG · cs.AI· q-bio.QM· stat.ML

Recognition: 2 theorem links

· Lean Theorem

Quotient-Space Diffusion Models

Yixian Xu , Yusong Wang , Shengjie Luo , Kaiyuan Gao , Tianyu He , Di He , Chang Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-15 07:07 UTC · model grok-4.3

classification 💻 cs.LG cs.AIq-bio.QMstat.ML

keywords diffusion modelsquotient spacesymmetry handlingmolecular generationSE(3)equivariant modelsgenerative models

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The pith

Quotient-space diffusion models generate correct symmetric distributions by allowing arbitrary outputs within equivalence classes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard diffusion models for data with symmetries such as molecular structures under rotations and translations must either enforce equivariance which complicates learning or use alignments that can distort the target distribution. This paper proposes quotient-space diffusion models that operate on the space where symmetric equivalents are identified as single points. This construction lets the model produce outputs that move freely inside equivalence classes without affecting the final distribution. If the framework holds, training becomes simpler while still guaranteeing that the generated samples follow the exact symmetric distribution desired. The method is tested on generating 3D molecular structures and shows better results than previous approaches for both small molecules and proteins.

Core claim

By viewing the intrinsic generation process on the quotient space, the exact construction that removes symmetry redundancy, the framework simplifies learning by allowing model output to have an arbitrary intra-equivalence-class movement, while generating the correct symmetric target distribution with guarantee.

What carries the argument

The quotient space construction that removes symmetry redundancy and supports a lifting operation to recover the original space distribution.

Load-bearing premise

The quotient-space construction exactly removes symmetry redundancy while permitting a well-defined lifting operation that preserves the target distribution without additional assumptions on the data manifold or the diffusion process.

What would settle it

Train the model on symmetric data such as molecular conformations and lift the outputs to the original space; if the empirical distribution of the lifted samples deviates from the target symmetric distribution under group transformations, the guarantee fails.

Figures

Figures reproduced from arXiv: 2604.21809 by Chang Liu, Di He, Kaiyuan Gao, Shengjie Luo, Tianyu He, Yixian Xu, Yusong Wang.

**Figure 2.** Figure 2: Illustration of the relation between the total space M and the quotient space Q and the correspondence of tangent vectors among them. See Appx. D.1 for formal definitions of the concepts and the proof. Thm. 1 shows that the projected process on the quotient space is indeed a diffusion process, and running the original process followed by projection gives the same process by running Eq. (7) on the quotien… view at source ↗

**Figure 3.** Figure 3: Illustration of denoising-model learning target using conventional training and using [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Training loss vs. training epochs. We find that our training is stable in practice. [PITH_FULL_IMAGE:figures/full_fig_p035_4.png] view at source ↗

**Figure 5.** Figure 5: Training and sampling convergence speed comparison on GEOM-DRUGS. [PITH_FULL_IMAGE:figures/full_fig_p037_5.png] view at source ↗

read the original abstract

Diffusion-based generative models have reformed generative AI, and also enabled new capabilities in the science domain, e.g., fast generation of 3D structures of molecules. In such tasks, there is often a symmetry in the system, identifying elements that can be converted by certain transformations as equivalent. Equivariant diffusion models guarantee a symmetric distribution, but miss the opportunity to make learning easier, while alignment-based simplification attempts fail to preserve the target distribution. In this work, we develop quotient-space diffusion models, a principled generative framework to fully handle and leverage symmetry. By viewing the intrinsic generation process on the quotient space, the exact construction that removes symmetry redundancy, the framework simplifies learning by allowing model output to have an arbitrary intra-equivalence-class movement, while generating the correct symmetric target distribution with guarantee. We instantiate the framework for molecular structure generation which follows $\mathrm{SE}(3)$ (rigid-body movement) symmetry. It improves the performance over equivariant diffusion models and outperforms alignment-based methods universally for small molecules and proteins, representing a new framework that surpasses previous symmetry treatments in generative models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Quotient-space diffusion moves the whole process off the original manifold to remove symmetry redundancy at the source, which is a real shift from equivariant networks or post-hoc alignment.

read the letter

The core move is to run the forward and reverse diffusion on the quotient space itself rather than on the original data manifold. This lets the model output any point inside an equivalence class without penalty, and the lift is supposed to recover the exact symmetric target distribution. That construction is new in the diffusion literature and goes beyond forcing equivariance into the network or trying to align samples after generation. They instantiate it for SE(3) on molecular point clouds and report consistent gains over both equivariant diffusion baselines and alignment methods on small molecules and proteins. The empirical pattern is straightforward: better validity and coverage without extra symmetry-enforcing tricks. That part is useful and worth citing if the numbers hold up under closer inspection. The soft spot is the measure-preservation claim. The guarantee assumes the group action is free and proper and that the noise kernel commutes with the projection in the right way. Real molecules have identical atoms and symmetric substructures, which create stabilizers, so the action is not free. The abstract states the guarantee without visible caveats, but the stress-test concern about non-free cases and invariant kernels is not obviously resolved from the high-level description. If the full derivation handles these cases explicitly, the framework is solid; if it relies on generic positions, the claim is narrower than stated. This paper is for people who build generative models for symmetric scientific data and want a cleaner way to remove redundancy. A reader who already works with equivariant diffusion or alignment heuristics will see the difference immediately and can judge whether the extra machinery is worth the implementation cost. The idea is coherent on its own terms and the experiments are not obviously cherry-picked, so it deserves a serious referee even if the central guarantee needs tightening.

Referee Report

2 major / 2 minor

Summary. The paper introduces quotient-space diffusion models, a generative framework that operates intrinsically on the quotient space induced by a symmetry group (instantiated for SE(3) on molecules). By removing redundancy via the quotient construction, the model is permitted arbitrary intra-equivalence-class outputs while the lifting operation is claimed to recover the exact symmetric target distribution. Empirical results on small molecules and proteins show gains over both equivariant diffusion baselines and alignment-based simplifications.

Significance. If the measure-preserving guarantee holds without additional manifold or kernel assumptions, the approach offers a principled alternative to equivariant networks that reduces learning difficulty while retaining distribution correctness. The reported universal outperformance on molecular tasks would position the framework as a new standard for symmetry handling in scientific generative models.

major comments (2)

[§3] The central guarantee (abstract and §3) that arbitrary intra-class model outputs on the quotient still yield the exact symmetric target distribution after lifting requires the projection/lifting pair to commute with the diffusion process in a measure-preserving way. The manuscript treats freeness/properness of the SE(3) action and invariance of the forward kernel as automatic once the quotient is formed, but provides no explicit verification or counter-example analysis for cases with stabilizers (identical atoms, symmetric substructures) common in molecular data.
[Theorem 1] Theorem 1 (or equivalent statement of the guarantee) is stated at a high level without the full derivation of how the reverse process recovers the original measure rather than a quotient-induced one. The stress-test concern on non-free actions is not addressed by any explicit assumption check or empirical diagnostic in the experiments.

minor comments (2)

[§2] Notation for the quotient map and lifting operator is introduced without a dedicated table or diagram clarifying the relationship to the original data manifold.
[§4] Experimental details on how the SE(3) quotient is discretized or approximated for proteins are brief; a supplementary note on numerical stability of the lifting step would help reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We address the two major comments point by point below, outlining the revisions that will be incorporated to strengthen the presentation of the measure-preserving guarantee and the supporting analysis.

read point-by-point responses

Referee: [§3] The central guarantee (abstract and §3) that arbitrary intra-class model outputs on the quotient still yield the exact symmetric target distribution after lifting requires the projection/lifting pair to commute with the diffusion process in a measure-preserving way. The manuscript treats freeness/properness of the SE(3) action and invariance of the forward kernel as automatic once the quotient is formed, but provides no explicit verification or counter-example analysis for cases with stabilizers (identical atoms, symmetric substructures) common in molecular data.

Authors: We agree that the manuscript would benefit from an explicit discussion of the freeness and properness assumptions. While the SE(3) action is free for generic molecular configurations with distinct atom types, stabilizers can arise in symmetric substructures. In the revised version we will add a dedicated paragraph in §3 that states the precise conditions for the action to be free and proper, verifies that the forward kernel remains invariant under these conditions, and provides a short counter-example analysis for a non-free case (e.g., a molecule with identical atoms). This will make the commuting property and measure preservation explicit rather than implicit. revision: yes
Referee: [Theorem 1] Theorem 1 (or equivalent statement of the guarantee) is stated at a high level without the full derivation of how the reverse process recovers the original measure rather than a quotient-induced one. The stress-test concern on non-free actions is not addressed by any explicit assumption check or empirical diagnostic in the experiments.

Authors: The high-level statement of Theorem 1 follows from the standard theory of quotient manifolds and the fact that the projection is a Riemannian submersion; the reverse process on the quotient therefore lifts to the correct measure on the original space. We acknowledge that the manuscript would be clearer with the intermediate steps written out. In the revision we will expand the proof of Theorem 1 (and its supporting lemmas) in the appendix, explicitly deriving the recovery of the original measure. We will also add an empirical diagnostic in the experiments section that stress-tests the framework on synthetic data containing known stabilizers, reporting both distribution correctness and any degradation when the freeness assumption is violated. revision: yes

Circularity Check

0 steps flagged

Quotient-space construction is a self-contained redefinition with no load-bearing reductions

full rationale

The paper defines the quotient-space diffusion process directly as the intrinsic generation mechanism that removes symmetry redundancy by construction. The guarantee that arbitrary intra-equivalence-class outputs lift to the correct symmetric target distribution follows from the mathematical properties of the quotient projection and lifting operations as stated in the framework. No equations or claims reduce to fitted parameters renamed as predictions, no uniqueness theorems are imported from self-citations, and no ansatzes are smuggled via prior work. The derivation chain is independent of external benchmarks or self-referential fits; the central result is the new construction itself rather than a statistical forcing from data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from group theory and diffusion models plus the existence of a well-defined quotient and lifting map for the symmetry group.

axioms (2)

domain assumption The symmetry group (SE(3)) acts properly on the data space so that a well-defined quotient space exists.
Invoked to justify operating directly on the quotient space for molecular structures.
domain assumption The diffusion process on the quotient space can be lifted back to the original space while preserving the target distribution.
Core to the guarantee that arbitrary intra-class outputs still yield the correct symmetric distribution.

pith-pipeline@v0.9.0 · 5505 in / 1295 out tokens · 41205 ms · 2026-05-15T07:07:41.965802+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 … projected process {y_t := π(x_t)} … d y_t = ((π_* b_t)(y_t) − σ_t²/2 h(y_t)) dt + … (quotient SDE)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

P_x(v) := v_H … horizontal projection … removes total rigid-body angular momentum (Eq. 10)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Emiel Hoogeboom, V´ıctor Garcia Satorras, Cl´ement Vignac, and Max Welling

PMLR, 2022a. Emiel Hoogeboom, V´ıctor Garcia Satorras, Cl´ement Vignac, and Max Welling. Equivariant diffu- sion for molecule generation in 3D. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato (eds.),Proceedings of the 39th International Conference on Machine Learning, volume 162 ofProceedings of Machine Learn...

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12 Published as a conference paper at ICLR 2026 Elton P Hsu.Stochastic analysis on manifolds

PMLR, 17–23 Jul 2022b. 12 Published as a conference paper at ICLR 2026 Elton P Hsu.Stochastic analysis on manifolds. Number 38. American Mathematical Soc., 2002. Chenqing Hua, Sitao Luan, Minkai Xu, Zhitao Ying, Jie Fu, Stefano Ermon, and Doina Precup. Mudiff: Unified diffusion for complete molecule generation. InLearning on Graphs Conference, pp. 33–1. P...

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[3]

representation

doi: 10.1126/science.adv9817. URLhttps://www.science.org/doi/abs/10. 1126/science.adv9817. Xin Li, Wenqing Chu, Ye Wu, Weihang Yuan, Fanglong Liu, Qi Zhang, Fu Li, Haocheng Feng, Errui Ding, and Jingdong Wang. VideoGen: A reference–guided latent diffusion approach for high definition text-to-video generation.arXiv preprint arXiv:2309.00398, 2023. URLhttps...

work page doi:10.1126/science.adv9817 2023
[4]

dg1 ∧ · · · ∧dg G =∂t=0 log p det(Gx t ) Z G Volx 0(g) =V x 0 ∂t=0 log p det(Gx t ), whereV x t := R G Volx t (g)is the volume of the equivalence classπ(x)induced fromΦ x t . For the time-dependent embeddingΦ x t (g), we consider a special type that is induced from an left- invariant vector fieldvon the total spaceM: Φx t (g) := Expg·x(tvg·x), fortaround ...

work page 2026
[5]

Therefore, the mentioned expression indeed holds: ˜h(x) =− 1 2 ∇x log det(Gx).(30) (c)Finally, we derive the expression for the specific case ofG= SO(3)

=⟨∇ x log det(Gx 0),v x⟩x. Therefore, the mentioned expression indeed holds: ˜h(x) =− 1 2 ∇x log det(Gx).(30) (c)Finally, we derive the expression for the specific case ofG= SO(3). Its embedding intoM throughx∈ Mis given byΦ x(g) :=g·x=gx, where in the last expression,gis meant to be its3×3rotation-matrix form, andgx:= (g⃗ x (n))n denotes a set of matrix-...

work page 2026