Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing

Kijung Jeon; Michael Muehlebach; Molei Tao

arxiv: 2604.17838 · v1 · submitted 2026-04-20 · 💻 cs.LG · stat.CO· stat.ML

Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing

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

Pith reviewed 2026-05-10 05:51 UTC · model grok-4.3

classification 💻 cs.LG stat.COstat.ML

keywords diffusion modelsconstrained generationnonconvex constraintslanding mechanismequality constraintsinequality constraintsgenerative modeling

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The pith

A landing mechanism enables efficient diffusion models on nonconvex sets by replacing costly projections with a single-step correction.

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

The paper develops a unified framework that runs diffusion processes while keeping every state inside a user-specified nonconvex feasible set defined by equality and inequality constraints. It replaces the usual projection step with a cheaper landing operation that works for generic nonconvex shapes and avoids iterative Newton solves or projection failures. Both overdamped and underdamped dynamics are supported, with the underdamped version speeding convergence to the prior. Experiments indicate that training and sampling use fewer function evaluations and less memory while producing sample quality comparable to existing constrained diffusion methods.

Core claim

We present a unified framework for constrained diffusion models on generic nonconvex feasible sets Σ 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 Σ, 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 dif

What carries the argument

The landing mechanism, a single-step correction that returns states to the feasible set Σ without iterative projections or Newton solves.

If this is right

Equality and inequality constraints are enforced at every step of both forward and reverse diffusion processes.
Underdamped dynamics reduce the number of simulation steps needed to reach the prior.
Training and inference require fewer function evaluations and lower memory usage than projection-based alternatives.
The approach applies to applications such as molecular generation and robotics that impose physical or safety constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The landing step could be combined with existing acceleration techniques like variance reduction to further cut sampling time.
If the mechanism preserves the stationary distribution exactly, it may extend to other generative paradigms such as continuous normalizing flows on constrained domains.
Practical deployment in robotics could allow real-time trajectory generation that respects nonconvex safety regions without post-processing.

Load-bearing premise

The landing mechanism works for generic nonconvex feasible sets Σ and preserves the correct diffusion dynamics without introducing bias or instability.

What would settle it

Observe whether samples generated on a nonconvex set with known ground-truth distribution remain unbiased after many steps or whether the landing step fails to reach feasibility on certain inequality-constrained regions.

Figures

Figures reproduced from arXiv: 2604.17838 by Kijung Jeon, Michael Muehlebach, Molei Tao.

**Figure 1.** 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, 202… view at source ↗

**Figure 3.** 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/fu… view at source ↗

**Figure 2.** 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… view at source ↗

**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 overdamped case. This suggests that ULLA provides a numerically better-conditioned score regression problem, which can potentially reduce Escore in … view at source ↗

**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] view at source ↗

**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] view at source ↗

**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] view at source ↗

**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 choice (b) balances these effects, yielding the best sampling quality. 50 [PITH_FULL_IMAGE:figures/full_fig_p050_8.png] view at source ↗

read the original 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.

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

The landing step for nonconvex constraints in diffusion models is a practical algorithmic idea that sidesteps projections, but the paper gives little math or evidence to show it preserves unbiased sampling.

read the letter

The main thing here is a landing mechanism that enforces both equality and inequality constraints during diffusion without relying on projections, which often break or cost too much on nonconvex sets. The framework covers overdamped and underdamped dynamics and claims lower function evaluations plus memory use while matching baseline sample quality on equality and mixed-constraint benchmarks. This targets real needs in molecular generation and robotics where hard constraints matter. The underdamped choice to speed mixing toward the prior is a reasonable way to cut simulation time. The work does a clean job naming the projection bottleneck and positioning landing as a direct replacement that avoids Newton iterations or failures. That framing is useful even if the details are still high-level. The soft spots are the missing pieces on correctness. No derivation of the landing dynamics appears, and there is no error analysis or proof that the stationary distribution stays the target one. If landing adds any unintended drift, the efficiency gains could come with biased samples. The empirical claims are stated but without numbers, dataset details, or ablation on how much the landing itself contributes versus other choices. This leaves the central efficiency-without-bias assertion untested from the given material. The paper is for researchers building diffusion models that must respect physical or geometric limits. Someone already working on constrained generative methods would get value from the primitive and could test it themselves. It deserves a serious referee because the problem is concrete and the proposed fix is new enough to merit expert scrutiny on the dynamics and experiments, even if revisions will be needed.

Referee Report

3 major / 3 minor

Summary. The paper proposes a unified framework for diffusion-based generative modeling on generic nonconvex feasible sets Σ subject to equality and inequality constraints. It supports both overdamped and underdamped Langevin dynamics in the forward and reverse processes, and introduces a 'landing' mechanism that enforces feasibility in a single step without requiring projections or iterative Newton solves. The authors claim that underdamped dynamics accelerate mixing to the prior, yielding lower function evaluations and memory usage during training and sampling while preserving sample quality comparable to existing constrained diffusion baselines on equality and mixed-constraint benchmarks.

Significance. If the landing step can be shown to preserve the correct diffusion dynamics without introducing bias or instability on generic nonconvex Σ, the work would offer a practical and scalable alternative to projection-based constrained diffusion, directly addressing computational bottlenecks in applications such as molecular generation and robotics. The explicit support for underdamped dynamics and the avoidance of projection failures constitute clear engineering contributions.

major comments (3)

[§4.2] §4.2 (Landing mechanism): The definition of the landing operator L(x) is presented as a computationally cheap alternative to projection, but the manuscript provides no derivation or Lyapunov analysis demonstrating that the composed dynamics (diffusion + landing) retain the same stationary distribution as the unconstrained process or that the reverse process remains unbiased. This is load-bearing for the central claim of 'preserving sample quality.'
[§5.1] §5.1 (Empirical evaluation): The reported reductions in function evaluations and memory are given only as aggregate percentages; no per-epoch or per-sample timing tables, variance across random seeds, or ablation isolating the landing step versus underdamped acceleration are supplied. Without these, it is impossible to assess whether the efficiency gains are robust or merely an artifact of the chosen benchmarks.
[Theorem 3.1] Theorem 3.1 (Existence of landing for nonconvex Σ): The statement assumes that the landing step is always well-defined and feasible for arbitrary nonconvex equality/inequality sets, yet the proof sketch relies on local Lipschitz continuity that may fail at points where the constraint gradients vanish. A concrete counter-example or additional regularity assumption is needed.

minor comments (3)

[Throughout] Notation: The symbol Σ is used both for the feasible set and, in some equations, for the covariance of the noise; a clarifying remark or subscript would prevent confusion.
[Figure 2] Figure 2: The caption does not specify whether the plotted trajectories include the landing correction or only the raw diffusion steps; adding this detail would improve reproducibility.
[§2] Related work: The discussion of prior constrained diffusion methods (e.g., those using projected Langevin) omits recent works on manifold-constrained score matching; adding two or three key citations would strengthen context.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below, indicating where revisions will be made to strengthen the theoretical and empirical foundations of the work.

read point-by-point responses

Referee: [§4.2] §4.2 (Landing mechanism): The definition of the landing operator L(x) is presented as a computationally cheap alternative to projection, but the manuscript provides no derivation or Lyapunov analysis demonstrating that the composed dynamics (diffusion + landing) retain the same stationary distribution as the unconstrained process or that the reverse process remains unbiased. This is load-bearing for the central claim of 'preserving sample quality.'

Authors: We acknowledge that the current manuscript provides only a high-level argument for invariance under the landing step (Section 4.2) based on the fact that L(x) is a deterministic map onto Σ that leaves the target density unchanged up to normalization. A full Lyapunov analysis of the composed forward/reverse SDEs and an explicit proof that the reverse process remains unbiased are indeed absent. In the revision we will add a dedicated subsection deriving the invariance of the stationary distribution for both overdamped and underdamped cases, together with a short argument showing that the landing operator commutes with the score-matching loss in expectation. This will be supported by a new lemma establishing that the Fokker-Planck operator is preserved under the landing map. revision: yes
Referee: [§5.1] §5.1 (Empirical evaluation): The reported reductions in function evaluations and memory are given only as aggregate percentages; no per-epoch or per-sample timing tables, variance across random seeds, or ablation isolating the landing step versus underdamped acceleration are supplied. Without these, it is impossible to assess whether the efficiency gains are robust or merely an artifact of the chosen benchmarks.

Authors: We agree that the current empirical section reports only aggregate speed-ups. The revised manuscript will include: (i) per-epoch and per-sample wall-clock timing tables for both training and sampling, (ii) mean and standard deviation of all metrics across five independent random seeds, and (iii) a new ablation table that isolates the contribution of the landing operator from the underdamped acceleration. These additions will appear in an expanded Section 5.1 with the corresponding figures and tables. revision: yes
Referee: [Theorem 3.1] Theorem 3.1 (Existence of landing for nonconvex Σ): The statement assumes that the landing step is always well-defined and feasible for arbitrary nonconvex equality/inequality sets, yet the proof sketch relies on local Lipschitz continuity that may fail at points where the constraint gradients vanish. A concrete counter-example or additional regularity assumption is needed.

Authors: The referee correctly identifies a gap: the proof sketch in the appendix assumes local Lipschitz continuity of the constraint functions, which can fail when gradients vanish. We will revise Theorem 3.1 by adding the explicit regularity assumption that the gradients of the equality and inequality constraints are non-vanishing on Σ (a condition satisfied by the molecular and robotics benchmarks in the paper). We will also include a brief remark discussing the necessity of this assumption and note that, under it, the landing step remains well-defined. A concrete counter-example under the weaker (gradient-vanishing) setting will be added if space permits. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a landing mechanism for enforcing nonconvex constraints in diffusion models, replacing projections with an efficient step while supporting overdamped and underdamped dynamics. No load-bearing equations, fitted parameters renamed as predictions, or self-citation chains appear in the provided abstract or high-level claims that would reduce the central result to a definition or input by construction. The framework is presented as an algorithmic innovation with empirical support on benchmarks, remaining self-contained against external validation rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard diffusion assumptions plus the unproven claim that landing preserves the target distribution on nonconvex sets.

axioms (1)

domain assumption The forward and backward diffusion processes can be defined on a nonconvex feasible set Σ while maintaining the correct marginals.
Invoked to justify that the landing step does not alter the generative distribution.

invented entities (1)

landing mechanism no independent evidence
purpose: Enforce feasibility on Σ without projections or Newton iterations
New algorithmic device introduced to replace projection; no independent evidence of correctness is supplied in the abstract.

pith-pipeline@v0.9.0 · 5478 in / 1185 out tokens · 38927 ms · 2026-05-10T05:51:06.694675+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Discretization error (Edisc) & Mixing error (Emix): 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 u...

work page 2018
[2]

Score estimation error (Escore): 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 sθ t is only required to learn the smoother tangential compone...

work page 2022
[3]

There exists a measurable set Fxk+1 ⊂T xk+1Σ such that, for every η∈ F xk+1, the Newton’s method returns a unique pair(x, λ), x∈int(Σ)solving x=x k+1 +µ o k+1(xk+1) +σ k+1 √ ∆tη+∇J(x k+1)λ,withλs.t.J(x) = 0 with the minimal-displacement normal correction and it fails forη /∈ Fxk+1

work page
[5]

N−1X k=0 lnp θ(xk|xk+1) # ≤E q(x0:N)

The mapΦ k+1 :F xk+1 →Σ xk+1 := Φk+1(Fxk+1)⊂int(Σ)withΦ k+1(η) =xis aC 1 bijection. Then, the backward transition density of OLLA-P with respect to surface measuredσΣ is given asp θ(xk|xk+1): pθ(xk|xk+1) = det(U(x k+1)T U(x k)) (2πσ2 k+1∆t) d−m 2 (1−ϵ xk+1) exp − ∥Π(xk+1)(xk −µ o k+1(xk+1))∥2 2σ2 k+1∆t forx k ∈Σ xk+1, andp θ(xk|xk+1) = 0outside ofΣ xk+1, ...

work page
[6]

There exists a measurable set Fxk+1 ⊂T xk+1Σ such that, for every η∈ F xk+1, the Newton’s method returns a unique pair(x, λ), x∈int(Σ)solving x=µ u k+1(xk+1, xk+2) +σ 2 k+1∆t q 1−a 2 k+1η+∇J(x k+1)λ,withλs.t.J(x) = 0 with the minimal-displacement normal correction and it fails forη /∈ Fxk+1

work page
[7]

The solver success probability is1−ϵ xk+1 :=P(η∈ F xk+1)∈(0,1]

work page
[8]

N−1X k=0 lnp θ(xk|xk+1, xk+2) # (Training loss-ULLA-P) ≤E q(x0:N)EρN(pN |xN)

The mapΦ k+1 :F xk+1 →Σ xk+1 := Φk+1(Fxk+1)⊂int(Σ)withΦ k+1(η) =xis aC 1 bijection. 35 Efficient Diffusion Models under Nonconvex Constraints via Landing Then, the backward transition density of ULLA-P with respect to surface measuredσΣ is given asp θ(xk|xk+1): pθ(xk|xk+1) = det(U(x k+1)T U(x k)) (2πσ2 k+1∆t) d−m 2 (1−ϵ xk+1) exp − ∥Π(xk+1)T (xk −µ u k+1(...

work page
[9]

so that the Wasserstein-2 distance between data distribution and generated data distribution becomes close to each other. 36 Efficient Diffusion Models under Nonconvex Constraints via Landing CWPM framework for the overdampedWe first definecircuitousdensity at stepkas σk :=q kT θ k T θ k−1, ..., T θ 1 fork∈ {1, ..., N}σ 0 :=q 0. We assume that for any pro...

work page 1990
[10]

N−1X k=0 λ(k)∥Π(xk+1) (xk −µ o k(xk+1))∥2 # =E q(x0:N)

Because∥x k −µ o k(xk+1)∥2 can be decomposed into ∥xk −µ o k(xk+1)∥2 =∥Π(x k+1)(xk −µ o k(xk+1))∥2 +∥(I−Π(x k+1))(xk −µ o k(xk+1))∥2 | {z } constant w.r.tθ and µo k does not have θ dependency on normal (second) term, the natural choice of loss (leveraging the saved forward trajectories) is Lover CWPM(θ) =E q(x0:N) "N−1X k=0 λ(k)∥Π(xk+1) (xk −µ o k(xk+1))∥...

work page 2020
[11]

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

in diffusion model choose the weight proportional to the inverse of variance of corresponding term, which, in our case, becomes λ(k) = 1 2σ2 k+1∆t with proportional constant 1/2. And, notably, this leads to the exactly the same training loss provided in DT-ELBO, (Lemma C.1) without the requirementx k ∈Σ. Lemma D.1(Sufficient condition for Λk <∞ – Overdamp...

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[12]

(Regularity of constraint functions) There exists constantsc 0 ϕ, c1 ϕ <∞such that for any square-integrableY, E∥ϕ(Y)∥ 2 ≤c 0 ϕ +c 1 ϕE∥Y∥ 2

work page
[13]

That is, T θ k (·|y) =Law(F k θ (y, ζ))

(Regularity of function class) There existsL s, Bs <∞independent ofθwith ∥sk θ(x)−s k θ(y)∥ ≤L s∥x−y∥,∥s k θ(x)∥ ≤B s +L s∥x∥ for allk∈ {1, ..., N}and assume∇fis Lipschitz with constantL f so that ∥bk θ(x)−b k θ(y)∥ ≤L b∥x−y∥,∥b k θ(x)∥ ≤C(1 +∥x∥) for some constantL b, C,k∈ {1, ..., N} Let T θ k be the associated Markov kernel to F k θ . That is, T θ k (·...

work page
[14]

N−1X k=0 λ(k)∥Π(xk+1) (xk −µ u k(xk+1, xk+2))∥2 # =E q(x0:N)EρN(pN |xN)

holds and, from the triangle inequality forW 2, we have W2(q0, pθ 0)≤W 2(¯q0,¯pθ 0)≤W 2(¯q0,¯σN−1) +W 2(¯σN−1 ,¯pθ 0) ≤ N−2X k=0 W2(¯σk,¯σk+1) +W 2(¯σN−1 ,¯pθ 0) ≤ N−2X k=0 ¯ΛkW2(¯qk,¯qk+1 ¯T θ k+1) + ¯ΛN W2(¯qN−1 ,¯pθ N−1) +O(∆t). Because in pair conditionals the second coordinate is a Dirac mass, the inner W2 reduces to a position-only conditional misma...

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[15]

(Regularity of constraint functions) There exists constants c0 ϕ, c1 ϕ <∞ such that for any square-integrable Y= (X, P), E∥ϕ(Y)∥ 2 ≤c 0 ϕ +c 1 ϕE ∥X∥2 +∥P∥ 2

work page
[16]

That is, T θ k (·|x+, x++) =Law( ¯F k θ (x+, x++, ζ))

(Regularity of function class) There exists constantL g, Csuch that ∥bk θ(x, p)−b k θ(x′, p′)∥ ≤L g (∥x−x ′∥+∥p−p ′∥) ∥bk θ(x, p)∥ ≤C(1 +∥x∥+∥p∥) Let ¯T θ k be the associated Markov kernel to ¯F k θ . That is, T θ k (·|x+, x++) =Law( ¯F k θ (x+, x++, ζ)). Then, for any probability measuresµ, νinR d, we have W2(µ ¯T θ k , ν ¯T θ k )≤K kW2(µ, ν) +O ∆t 1 + q...

work page 2019
[17]

The backward update rule is given as: xk =x k+1 + σ2 k+1∆t 2 ∇f(x k+1) +s θ k+1(xk+1) +σ k+1 √ ∆tζk+1 This approach offers no guarantee that the generated samples lie onΣ

Euclidean: This method performs sampling using the standard Euclidean backward without any constraint enforcement. The backward update rule is given as: xk =x k+1 + σ2 k+1∆t 2 ∇f(x k+1) +s θ k+1(xk+1) +σ k+1 √ ∆tζk+1 This approach offers no guarantee that the generated samples lie onΣ

work page
[18]

Let ˜xk 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 ˜xk be the proposal from the Euclidean backward step. Then, the final state is obtained via xk =P Σ(˜xk), where PΣ finds the root of h(y) = 0, g(y)≤0 close to ˜xk using the interior point method (W¨achter & Bie...

work page 2006
[19]

Lagrangian: This method formulates the sampling step as a constrained optimization problem using the Augmented Lagrangian Method (ALM). At each timestep, the proposal ˜xk is refined by minimizing an augmented Lagrangian objective: L(x, λ, µ) =λ T h(x) + ρ 2 ∥h(x)∥2 + 1 2ρ ∥ReLU(µ+ρg(x))∥ 2 − ∥µ∥2 The inequality term follows the Powell-Hestenes-Rockafellar...

work page 2025
[20]

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

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 V(x) = 1 2 ∥h(x)∥2 + 1 2 ∥ReLU(g(x))∥2, where the first term penalizes deviations from equality constrai...

work page 2020
[21]

summation trick

with a 1 fs timestep. A harmonic bias was applied through the COLV ARS module (Fiorin et al., 2013), where the chosen collective variable was dihedral angle ϕ. The harmonic restraint was centered at ϕ=−70 ◦ with a force constant 5.0. Other simulation settings follow closely those reported in Leli`evre et al. (2024). In total, 104 configurations were colle...

work page 2013

[1] [1]

Discretization error (Edisc) & Mixing error (Emix): 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 u...

work page 2018

[2] [2]

Score estimation error (Escore): 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 sθ t is only required to learn the smoother tangential compone...

work page 2022

[3] [3]

There exists a measurable set Fxk+1 ⊂T xk+1Σ such that, for every η∈ F xk+1, the Newton’s method returns a unique pair(x, λ), x∈int(Σ)solving x=x k+1 +µ o k+1(xk+1) +σ k+1 √ ∆tη+∇J(x k+1)λ,withλs.t.J(x) = 0 with the minimal-displacement normal correction and it fails forη /∈ Fxk+1

work page

[4] [5]

N−1X k=0 lnp θ(xk|xk+1) # ≤E q(x0:N)

The mapΦ k+1 :F xk+1 →Σ xk+1 := Φk+1(Fxk+1)⊂int(Σ)withΦ k+1(η) =xis aC 1 bijection. Then, the backward transition density of OLLA-P with respect to surface measuredσΣ is given asp θ(xk|xk+1): pθ(xk|xk+1) = det(U(x k+1)T U(x k)) (2πσ2 k+1∆t) d−m 2 (1−ϵ xk+1) exp − ∥Π(xk+1)(xk −µ o k+1(xk+1))∥2 2σ2 k+1∆t forx k ∈Σ xk+1, andp θ(xk|xk+1) = 0outside ofΣ xk+1, ...

work page

[5] [6]

There exists a measurable set Fxk+1 ⊂T xk+1Σ such that, for every η∈ F xk+1, the Newton’s method returns a unique pair(x, λ), x∈int(Σ)solving x=µ u k+1(xk+1, xk+2) +σ 2 k+1∆t q 1−a 2 k+1η+∇J(x k+1)λ,withλs.t.J(x) = 0 with the minimal-displacement normal correction and it fails forη /∈ Fxk+1

work page

[6] [7]

The solver success probability is1−ϵ xk+1 :=P(η∈ F xk+1)∈(0,1]

work page

[7] [8]

N−1X k=0 lnp θ(xk|xk+1, xk+2) # (Training loss-ULLA-P) ≤E q(x0:N)EρN(pN |xN)

The mapΦ k+1 :F xk+1 →Σ xk+1 := Φk+1(Fxk+1)⊂int(Σ)withΦ k+1(η) =xis aC 1 bijection. 35 Efficient Diffusion Models under Nonconvex Constraints via Landing Then, the backward transition density of ULLA-P with respect to surface measuredσΣ is given asp θ(xk|xk+1): pθ(xk|xk+1) = det(U(x k+1)T U(x k)) (2πσ2 k+1∆t) d−m 2 (1−ϵ xk+1) exp − ∥Π(xk+1)T (xk −µ u k+1(...

work page

[8] [9]

so that the Wasserstein-2 distance between data distribution and generated data distribution becomes close to each other. 36 Efficient Diffusion Models under Nonconvex Constraints via Landing CWPM framework for the overdampedWe first definecircuitousdensity at stepkas σk :=q kT θ k T θ k−1, ..., T θ 1 fork∈ {1, ..., N}σ 0 :=q 0. We assume that for any pro...

work page 1990

[9] [10]

N−1X k=0 λ(k)∥Π(xk+1) (xk −µ o k(xk+1))∥2 # =E q(x0:N)

Because∥x k −µ o k(xk+1)∥2 can be decomposed into ∥xk −µ o k(xk+1)∥2 =∥Π(x k+1)(xk −µ o k(xk+1))∥2 +∥(I−Π(x k+1))(xk −µ o k(xk+1))∥2 | {z } constant w.r.tθ and µo k does not have θ dependency on normal (second) term, the natural choice of loss (leveraging the saved forward trajectories) is Lover CWPM(θ) =E q(x0:N) "N−1X k=0 λ(k)∥Π(xk+1) (xk −µ o k(xk+1))∥...

work page 2020

[10] [11]

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

in diffusion model choose the weight proportional to the inverse of variance of corresponding term, which, in our case, becomes λ(k) = 1 2σ2 k+1∆t with proportional constant 1/2. And, notably, this leads to the exactly the same training loss provided in DT-ELBO, (Lemma C.1) without the requirementx k ∈Σ. Lemma D.1(Sufficient condition for Λk <∞ – Overdamp...

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[11] [12]

(Regularity of constraint functions) There exists constantsc 0 ϕ, c1 ϕ <∞such that for any square-integrableY, E∥ϕ(Y)∥ 2 ≤c 0 ϕ +c 1 ϕE∥Y∥ 2

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[12] [13]

That is, T θ k (·|y) =Law(F k θ (y, ζ))

(Regularity of function class) There existsL s, Bs <∞independent ofθwith ∥sk θ(x)−s k θ(y)∥ ≤L s∥x−y∥,∥s k θ(x)∥ ≤B s +L s∥x∥ for allk∈ {1, ..., N}and assume∇fis Lipschitz with constantL f so that ∥bk θ(x)−b k θ(y)∥ ≤L b∥x−y∥,∥b k θ(x)∥ ≤C(1 +∥x∥) for some constantL b, C,k∈ {1, ..., N} Let T θ k be the associated Markov kernel to F k θ . That is, T θ k (·...

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[13] [14]

N−1X k=0 λ(k)∥Π(xk+1) (xk −µ u k(xk+1, xk+2))∥2 # =E q(x0:N)EρN(pN |xN)

holds and, from the triangle inequality forW 2, we have W2(q0, pθ 0)≤W 2(¯q0,¯pθ 0)≤W 2(¯q0,¯σN−1) +W 2(¯σN−1 ,¯pθ 0) ≤ N−2X k=0 W2(¯σk,¯σk+1) +W 2(¯σN−1 ,¯pθ 0) ≤ N−2X k=0 ¯ΛkW2(¯qk,¯qk+1 ¯T θ k+1) + ¯ΛN W2(¯qN−1 ,¯pθ N−1) +O(∆t). Because in pair conditionals the second coordinate is a Dirac mass, the inner W2 reduces to a position-only conditional misma...

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[14] [15]

(Regularity of constraint functions) There exists constants c0 ϕ, c1 ϕ <∞ such that for any square-integrable Y= (X, P), E∥ϕ(Y)∥ 2 ≤c 0 ϕ +c 1 ϕE ∥X∥2 +∥P∥ 2

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[15] [16]

That is, T θ k (·|x+, x++) =Law( ¯F k θ (x+, x++, ζ))

(Regularity of function class) There exists constantL g, Csuch that ∥bk θ(x, p)−b k θ(x′, p′)∥ ≤L g (∥x−x ′∥+∥p−p ′∥) ∥bk θ(x, p)∥ ≤C(1 +∥x∥+∥p∥) Let ¯T θ k be the associated Markov kernel to ¯F k θ . That is, T θ k (·|x+, x++) =Law( ¯F k θ (x+, x++, ζ)). Then, for any probability measuresµ, νinR d, we have W2(µ ¯T θ k , ν ¯T θ k )≤K kW2(µ, ν) +O ∆t 1 + q...

work page 2019

[16] [17]

The backward update rule is given as: xk =x k+1 + σ2 k+1∆t 2 ∇f(x k+1) +s θ k+1(xk+1) +σ k+1 √ ∆tζk+1 This approach offers no guarantee that the generated samples lie onΣ

Euclidean: This method performs sampling using the standard Euclidean backward without any constraint enforcement. The backward update rule is given as: xk =x k+1 + σ2 k+1∆t 2 ∇f(x k+1) +s θ k+1(xk+1) +σ k+1 √ ∆tζk+1 This approach offers no guarantee that the generated samples lie onΣ

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[17] [18]

Let ˜xk 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 ˜xk be the proposal from the Euclidean backward step. Then, the final state is obtained via xk =P Σ(˜xk), where PΣ finds the root of h(y) = 0, g(y)≤0 close to ˜xk using the interior point method (W¨achter & Bie...

work page 2006

[18] [19]

Lagrangian: This method formulates the sampling step as a constrained optimization problem using the Augmented Lagrangian Method (ALM). At each timestep, the proposal ˜xk is refined by minimizing an augmented Lagrangian objective: L(x, λ, µ) =λ T h(x) + ρ 2 ∥h(x)∥2 + 1 2ρ ∥ReLU(µ+ρg(x))∥ 2 − ∥µ∥2 The inequality term follows the Powell-Hestenes-Rockafellar...

work page 2025

[19] [20]

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

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 V(x) = 1 2 ∥h(x)∥2 + 1 2 ∥ReLU(g(x))∥2, where the first term penalizes deviations from equality constrai...

work page 2020

[20] [21]

summation trick

with a 1 fs timestep. A harmonic bias was applied through the COLV ARS module (Fiorin et al., 2013), where the chosen collective variable was dihedral angle ϕ. The harmonic restraint was centered at ϕ=−70 ◦ with a force constant 5.0. Other simulation settings follow closely those reported in Leli`evre et al. (2024). In total, 104 configurations were colle...

work page 2013