{"id":"ec6bbf0d-9e9f-42fc-be10-6adcca4083a2","arxiv_id":"2604.17838","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":6,"one_line_summary":"Landing-based constrained Langevin dynamics (overdamped and underdamped) replace per-step projections in diffusion models on nonconvex feasible sets, achieving comparable sample quality at up to 47x lower sampling cost.","lead":"This paper replaces expensive per-step projections in constrained diffusion models with a 'landing' mechanism that self-corrects constraint violations, and uses underdamped dynamics to cut trajectory lengths by up to 47x. A smart generalist might read it to understand how to make generative models work efficiently when outputs must satisfy physical or geometric constraints, as in robotics or molecular design.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The rCRCQ neighborhood extension is the right concern: for learned neural-network-defined manifolds, rank degeneracy of the constraint Jacobian near Σ would make Π(x) and G(x)⁺ discontinuous, breaking the SDE coefficients that OLLA/ULLA rely on. The paper assumes this away without empirical check.","rationale":"The reader identified the correct load-bearing concern. The rCRCQ neighborhood extension is the most fundamental assumption because it determines whether the SDE coefficients are well-defined — everything downstream (stationarity, CWPM bound, constraint decay) depends on it. The reader's other concerns (curvature corrections disabled, Λ_k unverified, OLLA instability) are real but secondary: they only matter if the SDE is well-posed in the first place. The paper's explicit acknowledgment of the assumption in Remark 3 is commendable, but the lack of any empirical check — especially for the learned-manifold mesh experiments where the constraint is a neural network — leaves the most critical assumption untested. The CONDITIONAL verdict is appropriate: the framework is well-motivated, the continuous-time theory is sound, and the empirical results are compelling, but the gap between the assumed regularity and the actual experimental settings should be addressed. The concrete test I propose (monitoring condition numbers along trajectories) is inexpensive and would either confirm or refute the concern definitively for the specific experiments conducted.","tokens_in":64345,"tokens_out":5710,"duration_ms":140327,"concrete_test":"For the mesh experiments (Bunny/Spot, where c(x) is a neural SDF), compute the condition number κ(G(x_k)) = σ_max/σ_min of the Gram matrix G(x_k) = ∇c(x_k)∇c(x_k)^T at every step k along all forward and backward trajectories across the five independent runs. Also compute rank(∇c(x_k)) numerically (with a tolerance, e.g., σ_min > 10⁻⁶). If any trajectory visits a point where the rank drops or κ(G) exceeds ~10⁸, the rCRCQ neighborhood assumption is violated in practice, and the discontinuity of Π(x) at such points would explain any observed instability. If no such points are found, the concern does not land for these experiments.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the most load-bearing assumption. The landing mechanism's SDE coefficients (Propositions 1–2) depend on the projection operator Π(x) = I − ∇c(x)^T G(x)^† ∇c(x) and the pseudo-inverse G(x)^† = (∇c(x)∇c(x)^T)^†. By Stewart (1969), G(x)^† is continuous at x₀ if and only if rank(∇c(x)) is constant in a neighborhood of x₀. The paper extends rCRCQ from Σ to a neighborhood containing all discretized samples (Remark 3), but provides no verification that this holds in any experiment. The concern is most acute for the mesh datasets, where c(x) = SDF_NN(x) is a learned neural network — the gradient ∇c can degenerate at points where the SDF network has poor conditioning, and the paper never checks whether discretized trajectories visit such points. For the mixed-constraint tasks (robot arm, alanine dipeptide), the active inequality set changes across strata, and rank transitions at boundary strata could cause discontinuities in Π(x) that destabilize the landing drift. The reported OLLA instability on mixed-constraint tasks (Section 5.2, Table 5) is consistent with this failure mode but is not diagnosed. If rCRCQ fails on the neighborhood, the SDE is ill-defined, the stationarity results (Theorems B.2–B.3) do not apply, and the CWPM bound (Theorems 1–2) loses its foundation because the transition kernels are no longer well-posed. This is upstream of all other concerns: the curvature-correction gap and the Λ_k verification issue only matter if the SDE coefficients are well-defined in the first place.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","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.","tokens_in":65387,"tokens_out":1067,"duration_ms":139362,"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":[{"comment":"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","section":null}],"minor_comments":[{"comment":"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.","section":null},{"comment":"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":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the rCRCQ neighborhood extension is well-founded and is the primary reason for the major revision recommendation. The concern is not merely theoretical: the OLLA instability on mixed-constraint tasks (Table 5) is a concrete empirical signal that the assumption may be violated in practice. The authors should be asked to either verify the assumption empirically or provide a more robust discussion of when it is expected to hold. The paper is otherwise technically sound and well-written, and I would expect it to be acceptable after revision."},"author_rebuttal":{"model":"glm-5.2","summary":"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.","responses":[{"response":"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_made":"no","referee_comment":"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,"}],"tokens_in":63823,"tokens_out":1094,"duration_ms":43092,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","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","headline":"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.","tokens_in":65320,"tokens_out":953,"would_cite":true,"duration_ms":97045,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Landing replaces projection in constrained diffusion, cutting cost up to 47x","keywords":["constrained diffusion models","Langevin dynamics","nonconvex constraints","landing mechanism","underdamped dynamics","generative modeling","projection-free sampling","constraint qualification"],"falsifier":"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.","tokens_in":64399,"feed_emoji":"🎯","tokens_out":1078,"duration_ms":80819,"temperature":0.7,"pith_summary":"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.","feed_headline":"Landing replaces projection in constrained diffusion, cutting cost up to 47x","feed_subtitle":"First-order normal corrections enforce nonconvex constraints in diffusion models without per-step Newton solves, while underdamped dynamics ","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Landing drops projection from nonconvex diffusion, cutting cost 47x","First-order landing enables projection-free diffusion on nonconvex sets","Constrained diffusion without projections: landing cuts cost up to 47x","Landing mechanism bypasses projections in constrained diffusion","Projection-free diffusion via landing cuts sampling cost 47x"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Landing drops projection from nonconvex diffusion, cutting cost 47x","First-order landing enables projection-free diffusion on nonconvex sets","Constrained diffusion without projections: landing cuts cost up to 47x","Landing mechanism bypasses projections in constrained diffusion","Projection-free diffusion via landing cuts sampling cost 47x"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":920,"prompt_tokens":469,"completion_tokens":451,"prompt_tokens_details":null},"tokens_in":469,"tokens_out":451,"duration_ms":10269,"temperature":1.0,"reasoning_tokens":420,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-05T15:44:10.633138+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"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.","supporting_citations":[],"review_version":2}