{"id":"8096cbcc-def3-4f28-8e71-27f75c127137","arxiv_id":"2501.11743","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper proposes skew-reflected non-reversible Langevin dynamics and claims faster constrained sampling than projected Langevin Monte Carlo, with non-asymptotic bounds that are only partly justified.","lead":"This paper proposes a new Langevin sampler, SRNLMC, that adds a non-reversible drift and a skew reflection to keep samples inside a convex domain, with new convergence bounds. It claims this converges faster than the standard projected Langevin Monte Carlo, but the algorithm definition and several proof steps contain gaps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (1.9) defines a support-face map, not the oblique projection used by SRNLD; the claimed parallelism is false, so SRNLMC and its convergence analysis are not well-defined.","rationale":"I read the paper as trying to establish that skew-reflected non-reversible Langevin dynamics and its discretization converge to the constrained Gibbs distribution with accelerated rates. For that claim to hold, the map P^J_C in Eq. (1.9) must be the same object used in the continuous reflection (1.5)-(1.8): a projection along the skew normal \\nu_J that preserves the parallelism stated in the text. It is not. Equation (1.9) minimizes a linear functional over \\bar C, so it returns a supporting face or extreme point of C, not an oblique projection of the specific point being updated. The explicit two-dimensional ball example shows the parallelism assertion fails for any nonzero a, and even in the J=0, flat-face case the argmin is non-unique unless one adds an unstated tie-breaking rule. Since the discrete algorithm (1.10) is defined through this map, the algorithmic object analyzed in Theorem 2.24 and Corollary 2.25 is not the algorithm the authors describe or implement in their experiments. The secondary concerns, such as the unstated divergence-free condition on J in Lemma 2.7 and the treatment of local-time atoms in Lemma 2.20, are real but secondary; the mis-specification of P^J_C is sufficient by itself to invalidate the present version's central convergence guarantees. This agrees with the reader's weakest-assumption identification. The present version should be rejected, though the underlying idea could be salvageable with a corrected projection and a reworked analysis.","tokens_in":22033,"tokens_out":11761,"duration_ms":125210,"concrete_test":"Run the elementary two-dimensional check with C = {x : |x| \\le 1}, a = 1, and \\tilde x = (2,0). Compute P_C(\\tilde x) = (1,0), \\nu_J(P_C(\\tilde x)) = (-1,1)/\\sqrt{2}, R^J_C(\\tilde x) - P_C(\\tilde x) = (-1,1), and P^J_C(\\tilde x) = argmin_{|y|\\le 1} \\langle y, (-1,1)\\rangle = (1,-1)/\\sqrt{2}. Then verify that \\tilde x - P^J_C(\\tilde x) = (2 - 1/\\sqrt2, 1/\\sqrt2) is not a scalar multiple of (-1,1). If the intended map is instead the projection along \\nu_J, replace Eq. (1.9) accordingly and re-check Lemma 2.3 and Lemma 2.20 with the corrected P^J_C; also check that the J=0 case of the corrected map agrees with P_C on flat faces, not merely up to non-unique support points.","verdict_should_be":"REJECT","load_bearing_attack":"The central construction hinges on Eq. (1.9), which defines P^J_C(x) as the argmin over \\bar C of a linear functional in y. For a fixed \\tilde x, this is a support point of C in direction -\\nu_J(P_C(\\tilde x)); its location depends only on P_C(\\tilde x), not on where \\tilde x sits along the normal ray. This is not the skew projection along \\nu_J used in the reflected SDE (1.5) and in the skew-reflection formula (1.8). The paper's claim that \"x - P^J_C(x) is parallel to R^J_C(x) - P_C(x)\" is false in general. Example: take C the unit disk in R^2, J = [[0,a],[-a,0]], and x = (2,0). Then P_C(x) = (1,0), the inner normal is \\nu = (-1,0), and R^J_C(x) - P_C(x) = (-1,a). Equation (1.9) gives P^J_C(x) = argmin_{|y|\\le 1} \\langle y,(-1,a)\\rangle = (1,-a)/\\sqrt{1+a^2}. For a = 1, x - P^J_C(x) = (2-1/\\sqrt2, 1/\\sqrt2), which is not a scalar multiple of (-1,1). Thus the discrete update (1.10) does not implement the skew-reflected dynamics (1.5); instead it jumps to a boundary support point. Consequently Lemma 2.3, the discretization lemmas in Section 2.3, and the complexity claims in Theorem 2.24 and Corollary 2.25 analyze a different, unspecified algorithm. This is load-bearing because SRNLMC is the paper's practical method and every guarantee about it depends on P^J_C having the claimed geometric meaning.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes skew-reflected non-reversible Langevin dynamics (SRNLD), a continuous-time diffusion on a bounded convex domain with a skew reflection at the boundary, and a discrete algorithm SRNLMC obtained by a 'skew projection' step. The main claims are that the Gibbs distribution constrained to the domain is invariant, that the continuous-time process converges exponentially in total variation and 1-Wasserstein distance, that breaking reversibility gives faster convergence than the reversible reflected Langevin dynamics, and that the discretized algorithm has explicit non-asymptotic bounds and better iteration complexity than projected Langevin Monte Carlo. Numerical experiments on a truncated Gaussian and on Bayesian linear and logistic regression are reported to illustrate the acceleration.","tokens_in":22455,"tokens_out":7987,"duration_ms":83467,"significance":"If the main claims were correct, the paper would make a genuinely useful contribution: it attempts to extend the known acceleration of non-reversible Langevin samplers to constrained sampling, and it provides non-asymptotic rates in both total variation and 1-Wasserstein metrics. The topic is timely and the ambition is appropriate. However, the central geometric construction used to define the discrete algorithm is not a projection with the claimed property, and the invariance proof silently depends on a divergence-free condition on J that is not among the stated assumptions. Because the discretization error analysis, the iteration complexity, and the headline comparison with PLMC are all built on that construction, the main results as stated do not apply to a well-defined algorithm. The paper does not provide code or machine-checked proofs, and the numerical Wasserstein comparisons are presented without confidence intervals. I therefore cannot recommend acceptance or even a minor revision; the central flaw is load-bearing.","major_comments":[{"comment":"The map P^J_C defined by P^J_C(x) := argmin_{y in \\bar C} <y - x, nu_J(P_C(x))> is not the oblique projection along nu_J that the paper needs. For a fixed x, the objective is linear in y, so the argmin is a support point of \\bar C in the direction -nu_J(P_C(x)); it depends on P_C(x) only and not on the position of x along the normal ray. Consequently, the sentence after (1.9) asserting that x - P^J_C(x) is parallel to R^J_C(x) - P_C(x) is false in general. Example: let C be the unit disk in R^2, J = [[0,a],[-a,0]], and x = (2,0). Then P_C(x) = (1,0), and R^J_C(x) - P_C(x) = (I+J)(P_C(x)-x) = (-1,a). Equation (1.9) gives P^J_C(x) = argmin_{|y| <= 1} <y,(-1,a)> = (1,-a)/sqrt(1+a^2). For a = 1, x - P^J_C(x) = (2 - 1/sqrt(2), 1/sqrt(2)), which is not a scalar multiple of (-1,1). Thus the discrete update (1.10) does not implement the skew-reflected dynamics (1.5); it jumps to a boundary support point. As a result, the discretization estimates in Lemma 2.20, Corollary 2.22, Lemma 2.23, Theorem 2.24, and Corollary 2.25 analyze a different, unspecified map. This is load-bearing because SRNLMC is the practical algorithm of the paper and every guarantee about it depends on P^J_C having the claimed geometric meaning.","section":"Eq. (1.9)"},{"comment":"The invariance proof uses the identity <grad, J^top grad g> = 0 and justifies it solely by antisymmetry of J. For a state-dependent matrix field J(x), this identity holds only if J is divergence-free in the sense that the row-wise divergence of J vanishes on each column. The paper first introduces the condition 'grad . J = 0' in Remarks 2.8 and 2.9, but it is not stated in Assumption 2.2 or anywhere in the standing assumptions. Without that condition, grad . (J^top grad g) = (div J) . grad g + sum_{i,j} J_{ji} partial_{ij} g, and the first term does not vanish. Thus Theorem 2.6, Lemma 2.7, and Equation (2.11) require an additional hypothesis on J. Since the continuous-time analysis and the definition of the spectral gap lambda_J rest on the invariant measure being pi, this is a load-bearing gap in the paper's main theorem.","section":"Theorem 2.6 and Remarks 2.8-2.9"},{"comment":"The comparison with PLMC in Remark 2.26 uses only the inequality rho_J >= rho_0 from Theorem 2.11. This inequality is not strict for a general J, so the complexity bound in Corollary 2.25 does not imply that SRNLMC has a strictly lower iteration complexity than PLMC; it could be the same if rho_J = rho_0. Moreover, complexity is expressed through the unknown spectral gap rho_J, and the paper explicitly says after Theorem 2.11 that no explicit bound on rho_J is available. The quadratic example in Proposition 2.16 does give an explicit rate for constant H and J, but it does not cover the general non-convex setting of Theorem 2.24. Therefore the assertion that breaking reversibility yields better convergence guarantees than the reversible PLMC is not established by the stated results.","section":"Theorem 2.11 and Remark 2.26"}],"minor_comments":[{"comment":"There is a typo 'the soluiton is unique'; also the proof jumps from the definition of nu_J^s to existence without verifying the conditions of the cited Tanaka/Lions-Sznitman theorems in the text.","section":"Section 2.1, Lemma 2.3 proof"},{"comment":"The expression '(Xt - eX ^top_t )' contains a typo and should be '(Xt - eXt)^top'.","section":"Equation (2.29)"},{"comment":"The notation for the log factors is inconsistent: Lemma 2.19 has (log(T/eta))^{1/2}, while Lemma 2.20 and Corollary 2.21 use (log(T/eta))^{1/4}; please state the exact intermediate inequalities.","section":"Lemma 2.19 and Corollary 2.21"},{"comment":"The 1-Wasserstein distances are computed separately for each dimension, not in the full R^3 metric; the text should say so explicitly because this weakens the visual comparison.","section":"Section 3.1"},{"comment":"The Wasserstein convergence plots and accuracy plots are shown without error bars or multiple seeds; please report standard errors or confidence intervals so the empirical acceleration can be assessed.","section":"Numerical experiments"},{"comment":"The regularity argument invokes [Lie90] for oblique parabolic equations, but the boundary condition (2.3) involves the state-dependent skew normal nu_J; please justify that the cited result applies to this nonsymmetric, state-dependent boundary condition.","section":"Lemma 2.13"}],"recommendation":"reject","confidential_remarks":"To the editor: the manuscript's central algorithm is not well defined because Eq. (1.9) defines a support map rather than the oblique projection used in the SDE, and the invariance proof hides a divergence-free assumption. These are not presentation-level issues; fixing them would change the algorithm and require reworking the entire discretization analysis. A major revision cannot credibly repair the paper within its current scope, so I recommend rejection. I note also that the claimed acceleration over PLMC is not implied by the stated non-strict inequality on spectral gaps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The idea of adding a non-reversible drift to reflected Langevin dynamics on a bounded convex domain is worth taking seriously. The continuous-time SRNLD construction, with its skew unit normal and the local-time estimate in Lemma 2.17, is a legitimate technical contribution. The paper also honestly extends the HHMS05 spectral comparison to the constrained setting and gives non-asymptotic bounds in both TV and 1-Wasserstein distance. The numerical experiments are standard but do show the claimed acceleration in test cases.\n\nThe problem is that the discrete algorithm SRNLMC, which is the practical method, is not actually defined by the paper. Equation (1.9) defines P^J_C(x) as the argmin of a linear functional over the domain. For a fixed direction, that argmin lands on a support face of the boundary and depends only on the projection P_C(x), not on where x sits along the normal ray. It is not the oblique projection along the skew normal used in the SDE (1.5) and the reflection formula (1.8). The paper's claim that x - P^J_C(x) is parallel to R^J_C(x) - P_C(x) is false in general; the unit disk example in the stress-test note makes this concrete. As a result, the update in (1.10) does not implement the skew-reflected dynamics, and all of Section 2.3 analyzes a different, unspecified algorithm. This is load-bearing.\n\nThere is a second, smaller issue: the invariance proof in Theorem 2.6 silently uses div J = 0. The paper never states this as an assumption, and the assertion that antisymmetry alone makes the divergence term vanish is incorrect. The authors do mention div J = 0 later in Remark 2.8, but it is not in Assumption 2.2, and the proof as written is not valid without it.\n\nOn the positive side, the continuous-time analysis is mostly sound in spirit. The spectral gap comparison λ_J ≥ λ_0 is a known argument, and the quadratic case in Proposition 2.16 is a clean extension. The local-time bound is new and likely useful even if the current discretization proof does not go through.\n\nThis seems salvageable. A corrected skew projection—defined so that x - P^J_C(x) is parallel to R^J_C(x) - P_C(x)—would restore the link between the SDE and the algorithm. But as submitted, the practical algorithm and its guarantees are not well defined.\n\nI would send this to peer review, with referees who know oblique reflection and Langevin discretization. It is the kind of paper where the core idea is promising but the details need real scrutiny. For my own work, I would not cite it in its current form.","headline":"The paper has a genuinely new idea for constrained sampling, but the central discrete algorithm is mis-specified: Eq. (1.9) defines a support map, not the skew projection the rest of the paper relies on.","tokens_in":22959,"tokens_out":4368,"would_cite":false,"duration_ms":45938,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J60","60H10","65C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding a non-reversible drift to a reflected Langevin diffusion on a convex domain preserves the constrained Gibbs target and provably accelerates convergence to it.","keywords":["constrained sampling","non-reversible Langevin dynamics","skew reflection","projected Langevin Monte Carlo","1-Wasserstein distance","total variation","spectral gap","Skorokhod problem"],"falsifier":"Take $C=[-1,1]^2$, a fixed nonzero anti-symmetric $2\\times 2$ matrix $J$, and a point $x$ outside $C$ near a corner; compute the skew-projection $P^J_C$ defined in equation (1.9) and test whether $x-P^J_C(x)$ is parallel to $R^J_C(x)-P_C(x)$. A single counterexample would show that the skew-projection is not the oblique reflection that the SDE analysis requires.","tokens_in":21840,"feed_emoji":"⚡","tokens_out":12737,"duration_ms":121826,"temperature":0.7,"pith_summary":"Constrained sampling asks for random draws from a target distribution $\\pi(x)\\propto e^{-f(x)}$ living on a bounded convex set. This paper proposes a way to speed that up: add a non-reversible (anti-symmetric) drift $J$ to the usual Langevin dynamics and reflect the process off the boundary along a skew direction tied to $J$. The authors prove the resulting continuous-time process still has $\\pi$ as its stationary distribution, and that it approaches $\\pi$ at an exponential rate at least as fast as the reversible reflected process, with strict acceleration exhibited in explicit quadratic examples. They also analyze the discrete algorithm and give a 1-Wasserstein convergence guarantee whose iteration complexity improves on projected Langevin Monte Carlo when reversibility is broken. If correct, this is a quantitative acceleration mechanism for constrained sampling in Bayesian and machine-learning settings.","feed_headline":"Non-reversible drift speeds constrained sampling","feed_subtitle":"Skew-reflected Langevin dynamics beat reversible projected Monte Carlo in convergence complexity.","key_machinery":"The load-bearing mechanism is the skew projection $P^J_C(x)=\\operatorname{argmin}_{y\\in\\overline{C}}\\langle y-x,\\nu_J(P_C(x))\\rangle$ with skew normal $\\nu_J=(I+J)\\nu/\\sqrt{\\lVert\\nu\\rVert^2+\\lVert J\\nu\\rVert^2}$, combined with the anti-symmetric drift $J$ inside the domain. This projection returns updates to $C$ along a direction rotated by $J$, so the reflection term $\\nu_J(X_t)L(dt)$ enforces the constraint while leaving the constrained Gibbs measure $\\pi\\propto e^{-f}$ invariant: in the generator $Lg=-\\langle\\nabla g,(I+J)\\nabla f\\rangle+\\Delta g$ subject to $\\nabla g\\cdot\\nu_J=0$, the $J$ contribution cancels by anti-symmetry and $\\nabla\\cdot J=0$. The proof of acceleration then runs through the spectral-gap comparison $\\lambda_J\\ge\\lambda_0$ and, in the quadratic case, a synchronous coupling under the weighted norm $\\lVert\\cdot\\rVert_{(I+J)^{-1}}$ that yields the rate $\\lambda/C_{(I+J)^{-1}}$.","core_discovery":"The central discovery is that non-reversibility and reflecting constraints are compatible. Define the skew normal $\\nu_J=(I+J)\\nu/\\sqrt{\\lVert \\nu \\rVert^2+\\lVert J\\nu \\rVert^2}$ and the skew projection $P^J_C(x)=\\operatorname{argmin}_{y\\in\\overline{C}}\\langle y-x,\\nu_J(P_C(x))\\rangle$; then the SDE $dX_t=-(I+J(X_t))\\nabla f(X_t)dt+\\sqrt{2}\\,dW_t+\\nu_J(X_t)L(dt)$ admits $\\pi\\propto e^{-f}$ on $C$ as its invariant measure. The generator is $Lg=-\\langle\\nabla g,(I+J)\\nabla f\\rangle+\\Delta g$ with Neumann boundary condition $\\nabla g\\cdot \\nu_J=0$, and the added $J$ term contributes nothing to the invariant measure because $J$ is anti-symmetric and divergence-free. The paper proves exponential convergence in total variation and 1-Wasserstein distance with rate $\\rho_J\\ge\\lambda_J\\ge\\lambda_0=\\rho_0$, where $\\lambda_0$ is the spectral gap of the reversible dynamics, and for quadratic $f$ obtains the explicit weighted-norm rate $\\lambda/C_{(I+J)^{-1}}$ with $C_{(I+J)^{-1}}\\le 1$. For the discretized algorithm SRNLMC it bounds the 1-Wasserstein error by three terms (continuous convergence, discretization error controlled by a local-time estimate, and a Girsanov change-of-measure error) and derives an iteration complexity of $\\widetilde O(\\log(1/\\rho_J)d^{10}/(\\rho_J^7\\varepsilon^8))$, which is better than the reversible projected Langevin Monte Carlo baseline.","pith_inferences":["Inference: the achievable acceleration depends on the choice of $J$; choosing $J$ to maximize $\\rho_J-\\rho_0$ (or to minimize $C_{(I+J)^{-1}}$ in the quadratic case) is a design problem the paper leaves open, and the experiments tune the single parameter $a$ by hand.","Inference: the skew-projection construction should transfer to stochastic-gradient samplers; the paper tests the stochastic variant SRNSGLD empirically but gives no discretization theory for noisy gradients.","Inference: the method's benefit should be largest when the reversible spectral gap $\\rho_0$ is small, because $J$ opens an additional drift channel for mixing; quantifying how $\\rho_J-\\rho_0$ grows with a given $J$ for nontrivial domains is a natural next step."],"forward_implications":["Non-reversibility can be introduced in constrained Langevin sampling without changing the target: the constrained Gibbs measure $\\pi\\propto e^{-f}$ remains invariant for SRNLD for any bounded anti-symmetric Lipschitz $J$.","Continuous-time SRNLD converges to $\\pi$ in total variation and 1-Wasserstein distance at an exponential rate at least as fast as the reversible reflected Langevin diffusion, under only a spectral-gap assumption on the reversible dynamics rather than a log-concavity assumption.","For quadratic targets the convergence rate is explicit, $\\lambda/C_{(I+J)^{-1}}$ with $C_{(I+J)^{-1}}\\le 1$, so a nonzero anti-symmetric $J$ strictly accelerates mixing in that case.","SRNLMC reaches $\\varepsilon$ accuracy in 1-Wasserstein distance in $\\widetilde O(\\log(1/\\rho_J)d^{10}/(\\rho_J^7\\varepsilon^8))$ iterations, improving on the reversible PLMC complexity because $\\rho_J\\ge\\rho_0$.","The discretization guarantee is made quantitative by a new bound on the boundary local time, $E[L(t)]\\le C_L+C_L(1+d)t$, which controls the error introduced at reflections."],"supporting_citations":[{"why":"Defines the reflecting-boundary Skorokhod problem in convex regions, which the paper invokes to establish existence of SRNLD.","marker":"[Tan79]"},{"why":"Supplies the oblique-reflection SDE framework used to model the skew-reflected dynamics.","marker":"[LS84]"},{"why":"States the equivalence between skew reflection and projection parallel to the skew normal vector, justifying the skew-projection step.","marker":"[BGT04]"},{"why":"Introduces projected Langevin Monte Carlo and its finite-time analysis, the reversible baseline whose guarantees SRNLMC is compared with.","marker":"[BEL15]"},{"why":"Provides the log-concave constrained-sampling analysis and the reflection machinery whose discretization argument is adapted here.","marker":"[BEL18]"},{"why":"Proves that non-reversible diffusions mix faster in the unconstrained case and supplies the spectral-gap comparison $\\lambda_J \\ge \\lambda_0$.","marker":"[HHMS05]"},{"why":"Gives Hölder regularity for oblique parabolic boundary-value problems, used to turn the $L^2$ spectral gap into total-variation convergence.","marker":"[Lie90]"},{"why":"Bounds 1-Wasserstein distance by total-variation distance on bounded domains, used to convert the TV estimates into Wasserstein guarantees.","marker":"[GS02]"}],"fun_headline_variants":["Non-reversible dynamics accelerate constrained sampling","Skew-reflected Langevin: faster constrained sampling","Non-reversibility boosts constrained sampling speed","Faster constrained sampling without reversibility","Non-reversible Langevin outpaces reversible for constraints"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the skew-projection $P^J_C$ acting as a genuine oblique projection onto $C$ along the skew normal, namely on the parallelity $x-P^J_C(x)\\parallel R^J_C(x)-P_C(x)$; as written, $P^J_C$ is the argmin of a linear functional over $C$, which lands on an extreme boundary point and does not generally satisfy that relation, so the reflected dynamics and the convergence bounds built on that geometry are not well-defined if the relation fails.","fun_headline_variants_meta":{"raw":{"variants":["Non-reversible dynamics accelerate constrained sampling","Skew-reflected Langevin: faster constrained sampling","Non-reversibility boosts constrained sampling speed","Faster constrained sampling without reversibility","Non-reversible Langevin outpaces reversible for constraints"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1582,"prompt_tokens":1074,"completion_tokens":508,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":690,"completion_tokens_details":{"reasoning_tokens":440}},"tokens_in":690,"tokens_out":508,"duration_ms":5085,"temperature":1.0,"reasoning_tokens":440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:57:33.184422+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $C=[-1,1]^2$, a fixed nonzero anti-symmetric $2\\times 2$ matrix $J$, and a point $x$ outside $C$ near a corner; compute the skew-projection $P^J_C$ defined in equation (1.9) and test whether $x-P^J_C(x)$ is parallel to $R^J_C(x)-P_C(x)$. A single counterexample would show that the skew-projection is not the oblique reflection that the SDE analysis requires.","supporting_citations":[],"review_version":1}