Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing

Kijung Jeon; Michael Muehlebach; Molei Tao

REVIEW 1 major objections 8 minor 17 references

Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing

T0 review · 1 major / 8 minor · reviewed 2026-07-05 · glm-5.2

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Pith's one-line read Landing replaces projection in constrained diffusion, cutting cost up to 47x

desk verdict Landing replaces per-step projections in constrained diffusion; underdamped version cuts trajectory length by up to 47x. Theory is solid in continuous time; the theory-experiment gap is the main concern. read the letter →

arxiv 2604.17838 v2 pith:MV7R2XHP submitted 2026-04-20 cs.LG stat.COstat.ML

Kijung Jeon , Michael Muehlebach , Molei Tao This is my paper · ORCID

classification cs.LGstat.COstat.ML

keywords constraineddiffusionmodelsLangevindynamicsnonconvexconstraintslandingmechanismunderdampedgenerativemodelingprojection-freesamplingconstraintqualification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a method for running diffusion models on nonconvex constrained sets — sets defined by equality and inequality constraints — without performing expensive per-step projections onto the constraint manifold. The central mechanism is a landing term: a first-order normal-direction correction, derived from constraint gradients, that causes any constraint violation to decay exponentially over time. This replaces iterative Newton-solve projections with a single gradient evaluation per step. The paper constructs both overdamped (OLLA) and underdamped (ULLA) Langevin dynamics versions, proves they have the correct stationary distributions on the constraint set, and derives corresponding backward sampling processes. The underdamped variant exploits momentum to mix faster toward the prior distribution, shortening the required forward trajectory length and thereby reducing both training and sampling cost. A new training objective called Conditional Wasserstein Path Matching is introduced to handle the fact that landing-based trajectories may deviate slightly off the constraint manifold, unlike projection-based methods that enforce exact feasibility at each step. Empirically, the method achieves sample quality comparable to projection-based baselines on tasks including spheres, learned mesh surfaces, the special orthogonal group SO(10), alanine dipeptide conformations, and robot arm trajectories, while reducing forward trajectory lengths by up to 47x for sampling and 5x for training.

What carries the argument

The landing term is the central object. For a constraint function c(x) and its Jacobian ∇c(x), the landing term is −α ∇c(x)^T G(x)^+ c(x), where G(x) = ∇c(x)∇c(x)^T is the Gram matrix and G(x)^+ is its Moore-Penrose pseudo-inverse. This term is added to the normal component of the Langevin drift, causing c(x_t) to satisfy dc/dt = −α(t) c(x_t) and thus decay exponentially. The underdamped version (ULLA) adds a momentum variable with tangency constraints, and a specialized splitting integrator collapses the position-momentum dynamics into a second-order Markov chain on positions only, halving memory usage.

What would settle it

If one constructs a constraint set where the Jacobian rank drops in a neighborhood of the manifold at points reachable by discretized noise, the landing SDE coefficients become ill-defined and the exponential decay guarantee fails.

Watch

Extended reading notes

Core claim

The landing mechanism — a first-order correction term proportional to the constraint Jacobian, its pseudo-inverse, and the constraint violation vector — enforces exponential decay of constraint violations in continuous-time Langevin dynamics, making per-step projections onto nonconvex feasible sets unnecessary. When combined with underdamped (momentum-based) dynamics, the faster mixing allows substantially shorter forward trajectories, yielding up to 47x reduction in sampling cost and 5x in training cost while preserving sample quality competitive with projection-based methods.

Load-bearing premise

The method requires that the constraint Jacobian maintain constant rank not just on the constraint manifold itself, but in a neighborhood wide enough to contain all discretized samples — including those that wander off-manifold due to noise. If the rank of the constraint gradients changes near the manifold, the projection operator and pseudo-inverse become discontinuous, and the SDE coefficients are no longer well-defined.

Editorial extensions

If this is right

Diffusion models for molecular generation, robotics trajectory planning, and engineering design can enforce physical and safety constraints without iterative projection solvers that may fail or scale poorly with dimension.
The landing approach extends to any smooth nonconvex constraint set satisfying a relaxed constant-rank condition, including sets defined by learned neural-network implicit surfaces.
The CWPM training framework removes the requirement that intermediate samples lie exactly on the manifold, making projection-free constrained diffusion training theoretically well-posed.
The memory-efficient collapsed underdamped discretization enables training on high-dimensional constraint sets where storing full momentum trajectories would be prohibitive.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The landing mechanism could be combined with latent-space diffusion architectures, where constraints are enforced through a decoder's Jacobian via the chain rule, potentially addressing the memory scaling limitation the authors note for high-dimensional problems.
The exponential constraint-decay property suggests that landing could serve as a soft constraint-enforcement layer in other generative frameworks beyond diffusion, such as normalizing flows or VAEs, wherever projection onto nonconvex sets is costly.
The observation that underdamped dynamics yield smoother score regression objectives (avoiding singularities near t=0) may motivate momentum-augmented variants of other Riemannian generative models that currently suffer from score blow-up.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper adapts the landing mechanism from constrained optimization to diffusion SDEs, replacing expensive per-step Newton projections with cheap first-order normal corrections. The underdamped version (ULLA) exploits faster mixing to cut forward trajectory lengths by up to 47x for sampling and 5x for training. The core idea is sound and the empirical results are convincing on the tasks tested. The main concern is a gap between the continuous-time theory and what's actually run in experiments, plus a structural assumption (rCRCQ on a neighborhood) that goes unverified empirically. I think it deserves a serious referee. The stress-test note about rCRCQ is the right thing to worry about but is probably not as load-bearing as it sounds — I'll explain below. What's genuinely new: (1) extending landing to generative modeling with both overdamped (OLLA) and underdamped (ULLA) constrained Langevin SDEs, including backward processes; (2) the CWPM training loss, which avoids the on-manifold assumption that DT-ELBO requires and reduces to the same training objective with a cleaner derivation; (3) the collapsed ULLA discretization that eliminates momentum storage, giving O(N) memory instead of O(Nd). The continuous-time derivations (Propositions 1-2, stationarity in Theorems B.2-B.3) are detailed with full proofs. Stationarity is proven for both OLLA and ULLA. The CWPM bound (Theorems 1-2) is a Wasserstein variational bound that doesn't require samples to stay exactly on the manifold, which is the right tool given that landing allows small off-manifold excursions. Code is public. Experiments cover a reasonable range: spheres, learned mesh manifolds (SDF neural network), SO(10), alanine dipeptide, and a 7-DOF robot arm. ULLA achieves competitive JSD with RDDPM and RFM while using far fewer steps. The 47x speedup claim is specifically for the mesh task where projection requires Newton solves on a learned SDF — that's the regime where landing helps most, and the numbers support it. Now the soft spots. First, all experiments disable curvature correction terms (H, H1, H2 set to zero). The paper justifies this by citing prior work showing landing works without them, and the ablation in Table 6 shows negligible quality difference. But this means the implemented algorithm differs from the theoretical SDE, and nobody has checked whether the curvature terms matter in harder settings. Second, the CWPM bound relies on a stepwise-Lipschitz assumption with constants Lambda_k. Lemmas D.1-D.2 give sufficient conditions, but these are verified under abstract regularity assumptions on the constraint functions and score network class — not for the actual architectures used in experiments. This is a standard gap between theory and practice in this area, but it should be acknowledged more directly. Third, OLLA is unstable on mixed-constraint tasks (Section 5.2, Table 5). The paper attributes this to the lack of momentum damping but provides no theoretical diagnosis. The rCRCQ concern from the stress-test is real in principle: if the constraint Jacobian rank changes near the manifold, the pseudo-inverse and projection operator become discontinuous, and the SDE coefficients break down. For the mesh experiments where constraints come from a learned SDF network, this could happen at poorly conditioned points. However, I think the practical risk is lower than the stress-test suggests. The landing mechanism is designed to keep samples close to the manifold, and the reported constraint violations are tiny (1e-3 to 1e-4). If rank degeneracy were causing problems, we'd likely see it as instability or quality degradation — and we don't, except for OLLA on mixed constraints, which the paper already flags. That said, the paper should verify rCRCQ empirically or at least report the rank of the Jacobian along sampled trajectories. This is a reasonable request for revision, not a reason to reject. Who benefits: researchers working on constrained generative建模

Referee Report

1 major / 8 minor

Summary. This paper introduces a landing-based framework for constrained diffusion models on nonconvex feasible sets defined by equality and inequality constraints. The key idea is to replace per-step projections onto the feasible set with a first-order normal correction (the landing term) that exponentially decays constraint violations. The framework includes both overdamped (OLLA) and underdamped (ULLA) Langevin dynamics, with the latter leveraging faster mixing to reduce forward trajectory lengths. A Conditional Wasserstein Path Matching (CWPM) loss is proposed to handle training when intermediate samples may lie off the manifold. Experiments on spheres, learned mesh manifolds, SO(10), alanine dipeptide, and a 7-DOF robot arm demonstrate competitive sample quality with reduced computational cost compared to projection-based methods like RDDPM.

Significance. The paper makes a solid contribution to constrained generative modeling. The landing mechanism is a principled way to avoid expensive per-step projections, and the extension to underdamped dynamics with a memory-efficient collapsed discretization is technically nontrivial. The CWPM loss and its connection to DT-ELBO are well-motivated. The experimental coverage is broad, spanning equality-only and mixed-constraint tasks. The code is publicly available, which is a strength. The continuous-time SDE derivations (Propositions 1-2, Lemmas B.1-B.4) are detailed with full proofs, and the stationarity results (Theorems B.2-B.3) are correctly stated under appropriate assumptions.

major comments (1)

Remark 3 (Extended Usage in Our Framework): The rCRCQ condition is extended from the manifold Sigma to a neighborhood containing all discretized samples. This is the most load-bearing assumption in the paper: if the Jacobian rank degenerates near Sigma, the pseudo-inverse G(x)^+ and projection operator Pi(x) become discontinuous, and the SDE coefficients of OLLA/ULLA are ill-defined. The paper acknowledges this extension but provides no empirical verification that it holds in any experiment. This is most acute for the mesh datasets where c(x) = SDF_NN(x) is a learned neural network, and for mixed-constraint tasks where the active inequality set changes across strata. The reported OLLA instability on mixed-constraint tasks (Section 5.2, Table 5) is consistent with this failure mode but is not diagnosed. The authors should either (a) empirically check that the Jacobian rank remains stable沿

minor comments (8)

Section 4.3, Theorems 1-2: The CWPM bound involves a constant Lambda_k >= 0 whose existence is assumed. Lemmas D.1-D.2 provide sufficient conditions, but the presentation in the main text does not clearly state what these conditions are. A brief summary of the key regularity assumptions in the main text would strengthen the presentation.
Table 1: The 'Avg. |h|' column reports constraint violations but the notation is not defined in the table caption. It should be clarified that this measures equality constraint violations.
Section 5.1, Figure 1: The cross marks indicating the smallest N after which projection failures no longer occur are mentioned but not clearly visible in the figure. Improving figure clarity would help.
Algorithm 1 and 2: The notation for the approximated momentum (tilde p) and the decaying factor a_k could benefit from a brief inline definition in the algorithm comments for readability.
Remark 4, Figure 5: The claim that ULLA mitigates score singularities near t=0 is supported by empirical evidence on the volcano experiment. This is an interesting observation but the connection to the error decomposition is informal. A brief discussion of whether this benefit is expected to generalize would be helpful.
Section 6 (Limitations): The memory scaling O(Nd) is mentioned but the limitation regarding the rCRCQ neighborhood assumption is not listed here. It should be acknowledged as a limitation in the main text, not only in Remark 3.
Table 8 (Key Notations): The table is comprehensive but some symbols (e.g., tilde p_fwd, tilde p_bwd) could use more explicit definitions in the main text where they are first introduced.
The paper cites Jeon et al. (2025) for the constrained Langevin landing framework. The relationship between the present paper and this prior work should be made more explicit in the introduction to clearly delineate the novel contributions from the cited work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the rCRCQ neighborhood extension as the most load-bearing assumption in our framework. The comment is well-taken, and we agree that empirical verification is needed. Below we address each point.

read point-by-point responses

Referee: Remark 3 (Extended Usage in Our Framework): The rCRCQ condition is extended from the manifold Sigma to a neighborhood containing all discretized samples. This is the most load-bearing assumption in the paper: if the Jacobian rank degenerates near Sigma, the pseudo-inverse G(x)^+ and projection operator Pi(x) become discontinuous, and the SDE coefficients of OLLA/ULLA are ill-defined. The paper acknowledges this extension but provides no empirical verification that it holds in any experiment. This is most acute for the mesh datasets where c(x) = SDF_NN(x) is a learned neural network, and for mixed-constraint tasks where the active inequality set changes across strata. The reported OLLA instability on mixed-constraint tasks (Section 5.2, Table 5) is consistent with this failure mode but is not diagnosed. The authors should either (a) empirically check that the Jacobian rank remains stable,

Authors: We agree that the neighborhood extension of rCRCQ is the most critical assumption in our framework and that the manuscript should provide empirical evidence rather than merely acknowledging the extension. We address the three settings the referee highlights separately, as the considerations differ substantially across them. For the equality-only experiments on S² and SO(10), the constraint functions are algebraic (‖x‖²−1=0 and X^T X−I=0 respectively), and the Jacobian rank is constant everywhere in R^d (not just near Σ). For S², the Jacobian ∇h(x) = 2x^T has rank 1 for all x≠0, and the landing mechanism keeps samples away from the origin. For SO(10), the 55×100 Jacobian of the orthogonality constraints has constant rank 55 at all points where the constraints are not simultaneously degenerate, which is a measure-zero set. Thus rCRCQ is satisfied globally in these cases, and we will state this explicitly in the revision. For the mesh datasets where c(x) = SDF_NN(x), the situation is genuinely more delicate. The learned SDF network does not come with a theoretical rank guarantee. However, we can empirically verify that the Jacobian ∇c(x_k) maintains rank 1 (i.e., ‖∇c(x_k)‖ remains bounded away from zero) along all forward and backward trajectories used in our experiments. We will add this diagnostic to the revised manuscript and report the minimum Jacobian norm observed across all sampled trajectories for each mesh dataset. For the mixed-constraint tasks (Alanine dipeptide and 7-DOF robot arm), the referee correctly identifies that the active inequality set changes across strata, which is precisely the scenario rCRCQ is designed to handle (as opposed to LICQ, which would fail). The rCRCQ condition requires that the rank of the stacked Jacobian of active constraints remains revision: no

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity found; one minor self-citation for framework attribution, with self-contained proofs provided in the appendix.

full rationale

The paper's derivation chain is largely self-contained. The key constructions (Propositions 1–2 for OLLA/ULLA) are derived from scratch in Appendix B using Stratonovich calculus and minimum-norm Lagrangian processes, with full proofs. The self-citation to Jeon et al. (2025) — co-authored by the same authors — is used for attribution of the landing approach in the overdamped case, but the present paper re-derives the SDE explicitly (Proposition B.1) and extends it to underdamped dynamics, inequality constraints, backward processes, and the CWPM loss. The CWPM framework (Theorems 1–2) is derived independently from the Wasserstein distance and Gelbrich distance relation, not from the DT-ELBO. The fact that CWPM loss coincides with DT-ELBO loss (Lemmas C.1–C.2) is presented as a consequence of two independent derivations converging, with CWPM being strictly more general (no on-manifold assumption required). The existence of the Λ_k constants needed for the CWPM bound is proven independently via Lemmas D.1–D.2 under stated regularity conditions. The experimental comparisons use external benchmarks (RDDPM, RFM) and standard datasets. The self-citation is minor and not load-bearing because the mathematical content is reproduced and extended in the present paper's appendices. Score 2 reflects the minor self-citation that provides context but does not form a circular dependency in the derivation chain.

Assumptions & free parameters 6 free parameters · 5 assumptions · 3 invented entities

The framework introduces 6 free hyperparameters (alpha, epsilon, gamma, N, T, noise schedule), which is standard for diffusion models. The key axioms are geometric (rCRCQ, stratified manifold) and analytical (stepwise-Lipschitz, existence of Lambda_k). The rCRCQ extension to a neighborhood of Sigma is the most non-standard assumption. The invented entities (landing term, CWPM loss, collapsed discretization) all have independent falsifiable handles through their predicted behavior. No new physical entities or forces are postulated.

free parameters (6)

alpha (landing rate) = varies by task: 0.01-0.2 (overdamped), 0.1-0.5 (underdamped)
Controls exponential decay rate of constraint violations. Tuned per dataset; Table 11 shows sensitivity analysis.
epsilon (boundary repulsion rate) = varies by task: 0.001-0.6
Controls how aggressively inequality constraints push samples into the feasible interior. Tuned per dataset; Table 11 shows sensitivity.
gamma (friction coefficient) = 1.0-2.0
Used in ULLA underdamped dynamics. Standard Langevin parameter.
N (trajectory length) = 30-100 (ULLA), 100 (OLLA)
Number of discretization steps. Chosen to balance mixing and computational cost.
T (terminal time) = 0.2-8.0
Terminal time of the diffusion process. Set per dataset.
sigma_min, sigma_max (noise schedule) = 0.01, 100
Linear noise schedule endpoints following standard diffusion model practice.

assumptions (5)

domain assumption rCRCQ (relaxed Constant Rank Constraint Qualification) holds on a neighborhood of Sigma containing all discretized samples
Invoked in Section 3 and Remark 3. Ensures the projection operator Pi(x) and pseudo-inverse G(x)^+ are continuous and well-defined. Extended from the standard on-manifold definition to a neighborhood, which is a non-standard assumption.
domain assumption Sigma is a stratified manifold
Invoked in Section 3 to define the constraint set geometry. Required for the tangent space and surface measure constructions.
domain assumption sigma(t) is constant for stationarity proofs
Theorems B.2 and B.3 assume sigma(t) constant to prove stationarity. In experiments, sigma(t) follows a linear schedule, so stationarity is only approximate.
ad hoc to paper Stepwise-Lipschitz condition on transition kernels for CWPM bound
Assumed in Theorems 1-2. Lemmas D.1-D.2 provide sufficient conditions under strong regularity assumptions on constraint functions and score function class, but these are not verified for the neural network architectures used in experiments.
ad hoc to paper Existence of Lambda_k >= 0 bounding W2 distance between forward and backward conditionals
Assumed in Theorems 1-2. Proven to exist under regularity conditions in Lemmas D.1-D.2, but the bound is not explicit and depends on unspecified constants.

invented entities (3)

Landing term (-alpha * sigma * grad(h)^T G^+ h(x)) independent evidence
purpose: Replaces per-step projection by providing exponential decay of constraint violations in the SDE drift
The landing term produces falsifiable predictions: constraint violations should decay exponentially (Lemma 1), which is verified empirically in Figure 4 and Table 5. The mechanism is derived from Lagrangian-form constrained dynamics, not postulated ad hoc.
CWPM (Conditional Wasserstein Path Matching) loss independent evidence
purpose: Training loss for diffusion models that remains valid when intermediate samples are off-manifold, unlike DT-ELBO
The CWPM loss is shown to reduce to the same form as DT-ELBO (Section 4.3) when samples are on-manifold, providing a consistency check. It makes the falsifiable prediction that training with off-manifold samples should still minimize W2 distance, which is tested in experiments.
Collapsed ULLA discretization (2nd-order Markov chain in positions only) independent evidence
purpose: Eliminates need to store momentum trajectory, achieving O(N) memory instead of O(Nd) for underdamped dynamics
The collapsing technique is derived from the splitting integrator structure and produces O(dt) approximation error, which is verifiable. Memory savings are empirically demonstrated.

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Cite this review

Pith. "Pith review of Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing." pith.science (2026). https://pith.science/paper/MV7R2XHP

@misc{pith2026260417838,
  author       = {Pith},
  title        = {Pith review of: Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MV7R2XHP}},
  note         = {Machine review of arXiv:2604.17838}
}

abstract

Generative modeling within constrained sets is essential for scientific and engineering applications involving physical, geometric, or safety requirements (e.g., molecular generation, robotics). We present a unified framework for constrained diffusion models on generic nonconvex feasible sets $\Sigma$ that simultaneously enforces equality and inequality constraints throughout the diffusion process. Our framework incorporates both overdamped and underdamped dynamics for forward and backward sampling. A key algorithmic innovation is a computationally efficient landing mechanism that replaces costly and often ill-defined projections onto $\Sigma$, ensuring feasibility without iterative Newton solves or projection failures. By leveraging underdamped dynamics, we accelerate mixing toward the prior distribution, effectively alleviating the high simulation costs typically associated with constrained diffusion. Empirically, this approach reduces function evaluations and memory usage during both training and inference while preserving sample quality. On benchmarks featuring equality and mixed constraints, our method achieves comparable sample quality to state-of-the-art baselines while significantly reducing computational cost, providing a practical and scalable solution for diffusion on nonconvex feasible sets.

Figures

Figures reproduced from arXiv: 2604.17838 by the authors.

**Figure 1.** Mean JSD on S 2 flood versus trajectory length N under the fixed T. Cross mark (×) indicates the smallest N values after which projection failures no longer occur during the forward process. SO(10) manifold, and Alanine dipeptide—and add a 7- Degree of Freedom (DOF) robot arm trajectory task. We compare against state-of-the-art (SOTA) constrained generative model algorithms such as RFM (Chen & Lipman, 2024) and RDD… view at source ↗

**Figure 3.** Generated Robot arm trajectories (red) by ULLA. Alanine dipeptide and 7-DOF robot arm. We further evaluate our landing algorithms under complicated mixed constraints setup. The provided feasible set Σ are defined by complex equality and inequality constraints. In these settings, exact projections are often numerically unstable or computationally prohibitive. As summarized in [PITH_FULL_IMAGE:figures/full_fig_p00… view at source ↗

**Figure 2.** Generative performance on complex geometric tasks. (a) Histograms of the generated power-trace statistics Tr S k for k ∈ {1, 2, 4, 5} on SO(10) (m = 5), where ULLA (green) accurately recovers the ground-truth (red) distributions. (b) Joint distribution of ψ angle and Root Mean Square Deviation (RMSD) for the Alanine Dipeptide task; the blue shaded area represents the feasible region defined by inequality constrain… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: provides empirical evidence of this effect on the volcano experiment. The underdamped model exhibits Jacobian norms that are several orders of magnitude smaller across all times and, in particular, does not show the sharp blow-up near t ≈ 0 that appears in the overdamp…

**Figure 5.** Figure 5: Comparison of generated distributions across different algorithms-Volcano dataset 3D Mesh data on learned manifold – Spot the Cow (k = 100). (a) RDDPM (b) OLLA (c) ULLA-P (d) ULLA [PITH_FULL_IMAGE:figures/full_fig_p048_5.png]

**Figure 6.** Figure 6: Comparison of generated distributions across different algorithms - Spot the Cow k = 100 48 [PITH_FULL_IMAGE:figures/full_fig_p048_6.png]

**Figure 7.** Figure 7: Comparison of generated distributions across different algorithms- SO(10) with m = 3 49 [PITH_FULL_IMAGE:figures/full_fig_p049_7.png]

**Figure 8.** Figure 8: Effect of boundary repulsion rate ϵ on the generated distribution. When ϵ is too small (a), trajectories tend to stick to the boundary. Conversely, an excessively large ϵ (c) aggressively pushes samples away from the boundary, distorting the distribution. A moderate ch…

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Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Generative Aerodynamic Design with Diffusion Probabilistic Models

doi: 10.1007/BFb0121214. Jeon, K., Muehlebach, M., and Tao, M. Fast non-log- concave sampling under nonconvex equality and inequal- ity constraints with landing. InAdvances in Neural Infor- mation Processing Systems, 2025. Jing, B., Corso, G., Chang, J., Barzilay, R., and Jaakkola, T. Torsional diffusion for molecular conformer generation. InAdvances in N...

work page Pith review arXiv doi:10.1007/bfb0121214 2025
[2]

Discretization error (E��) & Mixing error (E�� ): Regarding discretization, our ULLA implementation employs a memory-efficient first-order splitting scheme; thus, both ULLA and the baseline OLLA share the same convergence order with respect to the step size. However, ULLA gains a significant advantage in themixing errordue to the ballistic behavior of ...

work page 2018
[3]

Score estimation error (E��): Employing a constrained forward process with the proposed landing mechanism allows the model to faithfully capture the intrinsic geometry of Σ. Crucially, because the landing mechanism analytically handles the ill-conditioned normal component, the score network �θ t is only required to learn the smoother tangential compone...

work page 2025
[4]

There exists a measurable set Fx�� ⊂� x�� Σ such that, for every �∈ F x�� , the Newton’s method returns a unique pair(�� )� �∈int(Σ)solving �=� k�� +� o k�� (� k�� ) +� k�� √ ∆��+∇�(� k�� )��with�s.t.�(�) = 0 with the minimal-displacement normal correction and it fails for� �∈ Fx��

work page
[6]

The mapΦ k�� :F x�� →Σ x�� := Φk�� (Fx�� )⊂int(Σ)withΦ k�� (�) =�is a� � bijection. Then, the backward transition density of OLLA-P with respect to surface measure�� is given as� θ(� k|� k�� ): �θ(� k|� k�� ) = ��det(�(� k�� )T �(� k)) �� (2�� k�� ∆�) �� (1−� x�� ) exp � − ∥Π(� k�� )(� k −� o k�� (� k�� ))∥� 2� � k�� ∆� � for� k ∈Σ x�� , and� ...

work page
[7]

There exists a measurable set Fx�� ⊂� x�� Σ such that, for every �∈ F x�� , the Newton’s method returns a unique pair(�� )� �∈int(Σ)solving �=� u k�� (� k�� k�� ) +� � k�� ∆� � 1−� � k�� +∇�(� k�� )��with�s.t.�(�) = 0 with the minimal-displacement normal correction and it fails for� �∈ Fx��

work page
[8]

The solver success probability is1−� x�� :=P(�∈ F x�� )∈(0�1]

work page
[9]

The mapΦ k�� :F x�� →Σ x�� := Φk�� (Fx�� )⊂int(Σ)withΦ k�� (�) =�is a� � bijection. 38 Efficient Diffusion Models under Nonconvex Constraints via Landing Then, the backward transition density of ULLA-P with respect to surface measure�� is given as� θ(� k|� k�� ): �θ(� k|� k�� ) = ��det(�(� k�� )T �(� k)) �� (2�� k�� ∆��(1−� � k�� )) �� (1−� x��...

work page 1990

Show all 17 references

[10]

And, notably, this leads to the exactly the same training loss provided in DT-ELBO, (Lemma C.1) without the requirement� k ∈Σ

in diffusion model choose the weight proportional to the inverse of variance of corresponding term, which, in our case, becomes �(�) = � �σ � �� t with proportional constant 1�2. And, notably, this leads to the exactly the same training loss provided in DT-ELBO, (Lemma C.1) ...
[11]

(Regularity of constraint functions) There exists constants� � ϕ� �� ϕ �∞such that for any square-integrable�, E∥�(�)∥ � ≤� � ϕ +� � ϕE∥�∥ �
[12]

That is, � θ k (·|�) =Law(� k θ (�� ))

(Regularity of function class) There exists� s� �s �∞independent of�with ∥�k θ(�)−� k θ(�)∥ ≤� s∥�−�∥�∥� k θ(�)∥ ≤� s +� s∥�∥ for all�∈ {1� �� }and assume∇�is Lipschitz with constant� f so that ∥�k θ(�)−� k θ(�)∥ ≤� b∥�−�∥�∥� k θ(�)∥ ≤� b(1 +∥�∥) for some constant� b� �b,�∈...
[13]

(Regularity of constraint functions) For every �≥1 , there exist constants �� ϕ,r� �� ϕ,r� �� ϕ,r �∞ such that, for any pair(�� )with finite4�-th moment in�, E∥�(�� )∥ �r ≤� � ϕ,r +� � ϕ,rE � ∥�∥ �r +∥�∥ �r� +� � ϕ,rE∥�∥ �r�
[14]

(Regularity of function class) There exist constants� g� �b �∞, independent of�, such that ∥�θ k�� (�� )−� θ k�� (�� )∥ ≤� g (∥�−� �∥+∥�−� �∥)�∥� θ k�� (�� )∥ ≤� b (1 +∥�∥+∥�∥)� Let ¯� θ k�� be the associated pair-kernel to ¯� θ k�� (�� ) := (¯� θ k�� (�� ...

2019
[15]

Euclidean: This method performs sampling using the standard Euclidean backward without any constraint enforcement. The backward update rule is given as: � k =� k�� + � � k�� ∆� 2 � ∇�(� k�� ) +� θ k�� (� k�� ) � +� k�� √ ∆��k�� 1https://github.com/facebookresearch/riemannian-f...
[16]

Let ˜�k be the proposal from the Euclidean backward step

Projected: This variant strictly enforces equality constraints by projecting the sample onto Σ immediately after each Euclidean backward step. Let ˜�k be the proposal from the Euclidean backward step. Then, the final state is obtained via � k =P � (˜�k), where P� finds the roo...

2006
[17]

Lagrangian: This method formulates the sampling step as a constrained optimization problem using the Augmented Lagrangian Method (ALM). At each timestep, the proposal ˜�k is refined by minimizing an augmented Lagrangian objective: L(�� ) =�T �(�) + � 2 ∥�(�)∥ � + 1 2� � ∥R...

2025
[18]

summation trick

Guided: This approach utilizes constraint guidance during sampling. The standard drift term of the backward process is modified by adding a guidance term derived from the gradient of a constraint violation energy potential. This potential is defined as �(�) = � � ∥�(�)∥ � + � ...

2020

Pith tools

Reviewed July 5, 2026 · model on record in the stance chip above.

[1] [1]

Generative Aerodynamic Design with Diffusion Probabilistic Models

doi: 10.1007/BFb0121214. Jeon, K., Muehlebach, M., and Tao, M. Fast non-log- concave sampling under nonconvex equality and inequal- ity constraints with landing. InAdvances in Neural Infor- mation Processing Systems, 2025. Jing, B., Corso, G., Chang, J., Barzilay, R., and Jaakkola, T. Torsional diffusion for molecular conformer generation. InAdvances in N...

work page Pith review arXiv doi:10.1007/bfb0121214 2025

[2] [2]

Discretization error (E��) & Mixing error (E�� ): Regarding discretization, our ULLA implementation employs a memory-efficient first-order splitting scheme; thus, both ULLA and the baseline OLLA share the same convergence order with respect to the step size. However, ULLA gains a significant advantage in themixing errordue to the ballistic behavior of ...

work page 2018

[3] [3]

Score estimation error (E��): Employing a constrained forward process with the proposed landing mechanism allows the model to faithfully capture the intrinsic geometry of Σ. Crucially, because the landing mechanism analytically handles the ill-conditioned normal component, the score network �θ t is only required to learn the smoother tangential compone...

work page 2025

[4] [4]

There exists a measurable set Fx�� ⊂� x�� Σ such that, for every �∈ F x�� , the Newton’s method returns a unique pair(�� )� �∈int(Σ)solving �=� k�� +� o k�� (� k�� ) +� k�� √ ∆��+∇�(� k�� )��with�s.t.�(�) = 0 with the minimal-displacement normal correction and it fails for� �∈ Fx��

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[5] [6]

The mapΦ k�� :F x�� →Σ x�� := Φk�� (Fx�� )⊂int(Σ)withΦ k�� (�) =�is a� � bijection. Then, the backward transition density of OLLA-P with respect to surface measure�� is given as� θ(� k|� k�� ): �θ(� k|� k�� ) = ��det(�(� k�� )T �(� k)) �� (2�� k�� ∆�) �� (1−� x�� ) exp � − ∥Π(� k�� )(� k −� o k�� (� k�� ))∥� 2� � k�� ∆� � for� k ∈Σ x�� , and� ...

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[6] [7]

There exists a measurable set Fx�� ⊂� x�� Σ such that, for every �∈ F x�� , the Newton’s method returns a unique pair(�� )� �∈int(Σ)solving �=� u k�� (� k�� k�� ) +� � k�� ∆� � 1−� � k�� +∇�(� k�� )��with�s.t.�(�) = 0 with the minimal-displacement normal correction and it fails for� �∈ Fx��

work page

[7] [8]

The solver success probability is1−� x�� :=P(�∈ F x�� )∈(0�1]

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[8] [9]

The mapΦ k�� :F x�� →Σ x�� := Φk�� (Fx�� )⊂int(Σ)withΦ k�� (�) =�is a� � bijection. 38 Efficient Diffusion Models under Nonconvex Constraints via Landing Then, the backward transition density of ULLA-P with respect to surface measure�� is given as� θ(� k|� k�� ): �θ(� k|� k�� ) = ��det(�(� k�� )T �(� k)) �� (2�� k�� ∆��(1−� � k�� )) �� (1−� x��...

work page 1990