{"id":"6b28d76d-f745-4df3-9220-6cb346e37777","arxiv_id":"2505.19438","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Stochastic localization drives binary quadratic posteriors into a strong-field regime where Glauber, Metropolis-Hastings, and DULA-type samplers satisfy Poincaré inequalities with polynomial mixing time.","lead":"This paper applies stochastic localization, a diffusion-based sampling framework, to binary quadratic distributions over {−1,1}^N, and proves that the intermediate posterior distributions gain an increasingly strong external field that makes them mix quickly under standard discrete MCMC samplers.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Condition 4.1 uses a scalar norm |h|, but the proofs need coordinatewise large fields; Theorem 3.1 is also proven only in scalar form, so the high-probability large-field premise is not established.","rationale":"The reader's weakest-assumption analysis identifies exactly the gap I find most load-bearing: Condition 4.1 is stated with a scalar |h|, but the proof of Theorem 4.3 requires coordinatewise control of h_i. I agree with the reader's verdict of CONDITIONAL. The paper makes a genuine contribution: it obtains explicit large-field Poincaré coefficients for Glauber dynamics, classical Metropolis chains, gradient-informed single-site MH, and DULA using Wu's Dobrushin-spectral-radius method, and it supports the approach with reproducible-looking experiments across MIS, MaxCut, and MaxClique. Those contributions do not depend on the vector-field subtlety and can likely be salvaged by restating Condition 4.1 coordinatewise and proving a coordinatewise version of Theorem 3.1. However, the headline claim that SL yields polynomial-time mixing for general BQDs does depend on the high-probability large-field premise, and that premise is not currently proven: Theorem 3.1 is scalar, and Remark 4.2 overclaims its reach. The concrete N=2 example with h=(0,2M) shows that a norm-based Condition 4.1 can hold while the Poincaré bound (15) fails, so this is a genuine internal correctness risk rather than a stylistic issue. Because the reader already recommends CONDITIONAL and my analysis supports that assessment rather than overturning it, I recommend no change to the verdict; the paper should be asked to fix the coordinatewise condition and the vector proof before the central claim is accepted.","tokens_in":34096,"tokens_out":5556,"duration_ms":53478,"concrete_test":"Numerically test the N=2 counterexample: take β=1, W12=W21=M, h=(0,2M), and compute the exact 4×4 Glauber transition matrix for M = 8, 16, 32, 64. If the true spectral gap decays as e^{-cM} while the right-hand side of inequality (15), evaluated with |h|=2M, stays within O(M e^{-M}) of 1, then Condition 4.1 cannot mean a scalar norm and Theorem 4.3 is false as stated for general vector h. A second check is to replace scalar Theorem 3.1 with a coordinatewise statement: prove that for all i, |Y_{t,i}| ≥ ζ simultaneously with probability 1−ε, by a union bound over the N coordinates, and then verify the proof of Theorem 4.3 goes through with the coordinatewise condition |h_i| ≥ 2β sup_i Σ_{k≠i}|W_ik| for every i. If the proof of Theorem 4.3 already needs that coordinatewise condition, the paper must state it in Condition 4.1; otherwise the main theorem is unproven for general BQDs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The core theoretical claim is that after enough SL iterations, the posterior h_t is strong enough for the Poincaré inequalities in Theorems 4.3–4.9, which would give polynomial-time mixing. The load-bearing condition is Condition 4.1: |h| ≥ 2β sup_i Σ_{k≠i}|W_ik|. The proofs of Theorems 4.3, 4.5, 4.8, and 4.9 do not use a norm bound on the whole vector h; they require, for every site i, that the scalar quantity h_i enters the conditional log-probability and dominates the interaction sum. For example, in the proof of Theorem 4.3 (Appendix C.2), the authors set h0 with β sup_i Σ_{k≠i}|W_ik| ≤ h0 and bound g_1'(x) by 2/(e^{3h0/2}+e^{-3h0/2}) using |x| ≥ 3h0/2, where x = h_i − (β/2)Σ_{k≠i}W_ik x_k. This lower bound holds only if |h_i| ≥ 2h0 = 2β sup_i Σ_{k≠i}|W_ik| for every coordinate i. A bound on any single norm |h|, Euclidean or otherwise, does not imply coordinatewise magnitude. If one coordinate has small h_i, the conditional distribution at that site can depend sharply on neighboring spins, the Dobrushin row sum for that site can approach 1, and the claimed spectral gap is not obtained. The same coordinatewise dependence is repeated in C.4, C.6, and C.7. Separately, Theorem 3.1 is proved in Appendix C.1 in scalar form: the proof writes P(|Yt| ≥ ζ) = P(|Bt ± α(t)/σ| ≥ ζ/σ) and estimates with the one-dimensional Gaussian CDF. For Y_t ∈ R^N and X ∈ {−1,1}^N, this does not show that all N coordinates of Y_t are simultaneously large enough to make every h_{t,i} large. Thus Remark 4.2, which uses Theorem 3.1 to justify Condition 4.1 with probability near 1, is not justified for general BQDs. Concretely, take N=2, β=1, W12=W21=M large, and h=(0,2M). Then |h| equals 2M under the Euclidean norm and the max norm, so both readings of Condition 4.1 are satisfied, but h_1=0 violates the coordinatewise condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a stochastic localization (SL) framework for sampling from binary quadratic distributions (BQDs) of the form ν(x) ∝ exp(−β/2 ⟨x,Wx⟩ + ⟨x,b⟩). The main theoretical claim is that, after sufficiently many SL iterations, the external field h_t of the SL posterior grows large, so the posterior satisfies a Poincaré inequality with high probability, which implies polynomial-time mixing for inner discrete MCMC samplers. The authors prove spectral-gap bounds for Glauber dynamics (Theorem 4.3), classical Metropolis chains (Theorem 4.5), a generic single-site Metropolis–Hastings kernel (Theorem 4.6), gradient-informed single-site MH (Theorem 4.8), and DULA (Theorem 4.9), all under a strong-field condition (Condition 4.1). They also report experiments on MIS, MaxCut, and MaxClique benchmarks showing modest consistent improvements of SL-augmented samplers over their non-SL counterparts.","tokens_in":34577,"tokens_out":7217,"duration_ms":71646,"significance":"If the main claim were established, the paper would be a valuable contribution: it would give the first general theoretical analysis of SL for discrete BQD sampling with explicit, field-dependent spectral gap bounds, and it would identify a practically relevant regime in which inner MCMC steps mix fast. The paper also ships an organized application of Wu's Dobrushin-based Poincaré framework to several discrete MCMC algorithms and provides a broad experimental comparison with code. However, the central theoretical chain contains a load-bearing gap: Condition 4.1 is stated with a scalar norm, whereas the proofs require coordinatewise large fields, and Theorem 3.1 is proved only for a scalar observation. Thus, as written, the high-probability large-field premise for the N-dimensional posterior is not established. This makes the main claim currently unsupported, though the gap appears fixable.","major_comments":[{"comment":"Condition 4.1 is stated as |h| ≥ 2β sup_i Σ_{k≠i}|W_ik| with an unspecified norm, but the proofs require a coordinatewise condition. In the proof of Theorem 4.3 (Appendix C.2), the key estimate on g_1'(x) is applied at x = −β/2 Σ_{k≠i,j} W_ik x_k − β/2 W_ij x_j + h_i; the inequality |x| ≥ 3h0/2 is obtained only if |h_i| = 2h0 for the specific coordinate i. A large Euclidean (or any single) norm of the vector h does not imply |h_i| is large for every i. The same coordinatewise dependence appears in C.4, C.6, and C.7. Therefore the Poincaré inequalities do not follow from Condition 4.1 as written. The authors should either state Condition 4.1 coordinatewise as min_i |h_i| ≥ 2β sup_i Σ_{k≠i}|W_ik|, or prove that the scalar-norm condition implies the coordinatewise lower bounds used in the proofs; the latter is generally false.","section":"Condition 4.1 and proofs of Theorems 4.3, 4.5, 4.8, 4.9 (Appendix C)"},{"comment":"Theorem 3.1 is proved only for a scalar observation. The proof writes P(|Y_t| ≥ ζ) = P(|B_t ± α(t)/σ| ≥ ζ/σ) and estimates with a one-dimensional Gaussian CDF. For X ∈ {−1,1}^N and Y_t ∈ R^N, these identities do not apply; |Y_t| is the Euclidean norm of a vector with random signs, and the argument does not show that all N coordinates of Y_t are simultaneously large. The statement actually needed for the posterior field h_t in equation (12) is a coordinatewise bound, e.g., P(min_i |Y_{t,i}| ≥ ζ) ≥ 1−ε, or at least a bound on all coordinates entering h_t. Without this, Remark 4.2's claim that Theorem 3.1 guarantees Condition 4.1 with high probability is not justified.","section":"Theorem 3.1 and Appendix C.1"},{"comment":"The paper explicitly states that \"establishing rigorous theoretical guarantees for the convergence rate of the overall SL process to the final target distribution\" remains future work. Consequently, the headline claim of \"polynomial-time mixing\" applies only to the posterior distributions used inside the SL iterations, not to the full SL algorithm that produces the final sample. This is an important scope limitation that should be stated in the abstract and introduction; as it stands, a reader could reasonably infer a stronger guarantee for the whole sampler than the paper actually proves.","section":"Section 7 (Conclusion and Discussion)"},{"comment":"Theorem 4.6 is conditional on an assumption that the transition kernel P(x^i|x) is Lipschitz with a constant C_Lip(β,h) that decreases exponentially as |h| grows, but no verifiable condition or proof is given for the proposals used in the experiments (GWG and PAS). Remark 4.7 only says this property \"can be expected\" from the warm-up cases. For the paper's claim of covering \"enormous discrete MCMC samplers,\" the relevant Lipschitz condition should be verified for the actual single-site gradient-informed proposals, or the scope should be restricted to the kernels covered by Theorems 4.8 and 4.9.","section":"Theorem 4.6 and Remark 4.7"}],"minor_comments":[{"comment":"The phrase \"covering enormous discrete MCMC samplers\" is odd; something like \"covering a broad range of discrete MCMC samplers\" would be clearer.","section":"Abstract"},{"comment":"The sentence \"It proves that as t→T_gen, Y_t/α(t)→X ∈ {−1,1}^N\" is stronger than what Theorem 3.1 states; the theorem only gives a high-probability norm lower bound for large t.","section":"Section 3, text after Theorem 3.1"},{"comment":"The notation |h| is used for a vector h without specifying the norm; the spectral-gap formulas in Section 4 (e.g., |h|/(e^{3|h|/4}+e^{-3|h|/4})) depend on which norm is intended, and this is not clarified in the main text.","section":"Condition 4.1 and Section 4 spectral-gap formulas"},{"comment":"There is a typo: \"benchamrk\" should be \"benchmark\".","section":"Appendix D.1"},{"comment":"The reference spelling is inconsistent: \"Dobruschin\" in the bibliography entries for Dobruschin (1968) and Dobrushin (1970) should be unified to \"Dobrushin\".","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is valid and lands exactly on the central claim. The gap between the scalar proof of Theorem 3.1 and the coordinatewise large-field requirement in the proofs of Theorems 4.3–4.9 is not a cosmetic issue; it is the bridge that connects the SL dynamics to the Poincaré inequalities. That said, the gap appears fixable in principle: the coordinatewise bound should follow from a union bound over coordinates when α(t)/(σ√t)→∞, and Condition 4.1 can be reformulated coordinatewise. I therefore recommend major revision rather than rejection. The empirical results are encouraging but do not compensate for the missing proof, since they compare optima rather than sampling quality and are tuned on the same benchmarks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine first step: it applies stochastic localization to general binary quadratic distributions and proves explicit strong-field Poincaré inequalities for several discrete MCMC samplers (Glauber, Metropolis, gradient-informed single-site MH, DULA). The proofs are direct applications of Wu's Dobrushin-matrix theorem, and the logistic-function contraction estimates in Appendix C are clean and mostly correct. The experimental work is also solid in scope: three QUBO problems, fourteen datasets, three samplers, and consistent (if modest) gains from the SL wrapper. Those parts deserve credit.\n\nThe problem is the load-bearing link between the main theorems and the abstract's claim of polynomial-time mixing. Condition 4.1 states |h| ≥ 2β sup_i Σ_{k≠i}|W_ik| for an unspecified norm, but every proof actually requires |h_i| ≥ 2β sup_i Σ_{k≠i}|W_ik| for each coordinate i. A norm bound does not imply coordinatewise bounds: with N=2, W_12 = M large, and h=(0,2M), both Euclidean and max norms satisfy Condition 4.1 while h_1=0 fails. The Dobrushin row sum for site 1 then is not controlled. This is not a cosmetic issue; the spectral gap estimates collapse without the coordinatewise condition.\n\nSecond, Theorem 3.1 is proved only for a scalar observation Y_t. In the actual BQD setting Y_t ∈ R^N, and the proof's one-dimensional Gaussian CDF argument does not show that all N coordinates of the posterior field are simultaneously large. Remark 4.2 therefore overreaches. Third, Theorem 4.6's Lipschitz assumption is left unverified; Remark 4.7's \"can be expected\" is not a proof. Fourth, Section 7 explicitly says there is no end-to-end convergence guarantee for the full SL algorithm, so the phrase \"polynomial-time mixing\" should be scoped to the inner chain at late times, under an unproven condition.\n\nThese are addressable gaps, not contradictions. The conditional theorems are plausible, and the authors are honest about the missing overall-SL guarantee. But the paper as written does not support the abstract's strong claim. A serious referee should ask for either a correct coordinatewise condition plus a vector version of Theorem 3.1, or a honest rescoping of the guarantees. I would send it to review—the idea is worth refereeing—but I would not cite the current version as a reliable result.","headline":"Conditional Poincaré bounds for SL posteriors on BQDs are real, but the headline polynomial-time claim rests on a coordinatewise strong-field condition the paper never establishes.","tokens_in":35221,"tokens_out":2915,"would_cite":false,"duration_ms":30287,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J22","60J27","82B20","65C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"After enough stochastic-localization steps the posterior's external field grows without bound, so Glauber, Metropolis, and Langevin samplers on binary quadratic distributions satisfy Poincaré inequalities and mix in polynomial time, the…","keywords":["stochastic localization","binary quadratic distributions","Glauber dynamics","Metropolis-Hastings","Poincaré inequality","spectral gap","QUBO","MCMC"],"falsifier":"Run the SL observation process on an explicit $N \\ge 2$ BQD, record the trajectory of $h_t$, and check at the time $T$ promised by Theorem 3.1 whether $\\min_i |h_{t,i}| \\ge 2\\beta \\sup_i \\sum_{k\\ne i}|W_{ik}|$ on typical runs. If the scalar magnitude $|h_t|$ is large while some coordinate stays below threshold, the stated assumptions do not apply; a numerical spectral-gap estimate of Glauber dynamics at that field configuration would then show whether the Poincaré conclusion survives anyway.","tokens_in":33837,"feed_emoji":"🎲","tokens_out":15255,"duration_ms":123892,"temperature":0.7,"pith_summary":"Stochastic localization (SL) solves hard sampling problems by observing a copy of the target through noise and then sampling the sequence of posterior distributions; in continuous spaces this approach is established, and this paper aims to carry it over to binary quadratic distributions (BQDs), the Gibbs measures behind Ising models and QUBO optimization. The central assertion is that after enough SL iterations the posterior's external field becomes effectively infinite, and in that large-field regime the posterior is easy for standard discrete MCMC samplers. To back this up, the paper proves Poincaré inequalities, with explicit spectral gap formulas, for Glauber dynamics, classical Metropolis chains, gradient-informed single-site Metropolis-Hastings, and the discrete unadjusted Langevin algorithm. The paper concludes that SL turns an off-the-shelf discrete sampler into a polynomially mixing sampler even when the original BQD has no fast mixing. Experiments on maximum independent set, maximum cut, and maximum clique instances show the SL versions consistently match or outperform their base samplers at equal MCMC step budgets.","feed_headline":"Stochastic localization makes discrete samplers mix fast","feed_subtitle":"As the posterior field grows without bound, Glauber, Metropolis, and Langevin samplers mix in polynomial time.","key_machinery":"The machinery is the SL posterior together with the Dobrushin interdependence matrix. The posterior $\\nu_{\\beta,h}$ is the original Gibbs measure with an added external field $h_t$; because $x^2=1$ on the hypercube, the Gaussian observation contributes only this linear field. The proofs bound each entry $c_{ij}$ of the transition kernel's Dobrushin interdependence matrix by a small multiple of $\\beta|W_{ij}|$ divided by a growing denominator such as $e^{3|h|/4}+e^{-3|h|/4}$, and then use Wu's theorem (Lemma B.2) to convert the spectral radius bound $\\mathrm{r}_{\\mathrm{sp}}(C)<1$ into a Poincaré inequality with the quoted gap. This is the step that turns a large external field into a quantitative mixing guarantee, since a spectral gap implies polynomial mixing time by the standard Levin-Peres result.","core_discovery":"The paper's central claim is that stochastic localization removes the discrete-sampling bottleneck for general binary quadratic distributions. Writing the SL posterior as $\\nu_{\\beta,h}(x) \\propto \\exp(-\\tfrac{\\beta}{2}\\langle x, W x\\rangle + \\langle x, h\\rangle)$ with $h = b + \\alpha(t)Y_t/(\\sigma^2 t)$, the quadratic term from the Gaussian likelihood collapses to a constant because $x^2=1$ on $\\{-1,1\\}^N$, so SL acts entirely through a growing linear tilt. Theorems 4.3 through 4.9 show that when this tilt satisfies the strong-field condition $|h| \\ge 2\\beta \\sup_i \\sum_{k\\ne i}|W_{ik}|$, each of the four discrete MCMC families obeys a Poincaré inequality whose spectral gap increases with $|h|$ and tends to $1$; Theorem 3.1 is used to establish that $|Y_t|$ grows without bound with probability close to $1$, so the condition holds at late SL times. The conclusion is polynomial-time mixing for the inner sampler and exponential concentration for the Monte Carlo estimate of the posterior mean.","pith_inferences":["The paper proves mixing of the inner posterior sampler, not an end-to-end total-variation guarantee for the full SL algorithm; combining these Poincaré inequalities with discretization and posterior-estimation error would complete the overall convergence story, a direction the paper itself leaves open.","A testable implication of the coordinatewise-field concern is that SL's gain should be smallest on instances where some spin's local field stays near zero through most of the schedule; varying the row norms of $W$ in QUBO benchmarks could probe this prediction.","The same large-field mechanism suggests a natural extension to $k$-state and higher-order interaction models: wherever the likelihood tilt dominates the interaction term, the posterior becomes nearly independent, so the core argument may survive without the specific $x^2=1$ collapse."],"forward_implications":["If the central claim holds, the inner MCMC chain at late SL iterations mixes polynomially, so an off-the-shelf Glauber, Metropolis, GWG, PAS, or DULA sampler can be wrapped by SL without model-specific posterior estimators.","The explicit spectral gap formulas give finite-$N$ control on the Poincaré constant: the gap approaches $1$ as the field grows, so the number of inner samples needed for a fixed error tolerance is polynomial in $N$ and in the inverse tolerance.","The guarantee is stated for arbitrary $W$ and $b$, so SL should help BQDs that do not mix fast on their own; the only requirement is that the SL schedule drive the field past the threshold.","The Chernoff-type bounds in Corollary 4.4 and Appendix C.8 mean the posterior-mean estimate used to advance the SDE concentrates exponentially with the number of MCMC samples, which is what makes the full SL loop practically stable.","The empirical QUBO results on maximum independent set, max-cut, and max-clique show the SL variants at least match the base samplers at equal MCMC budgets, with total added cost only $O(TN)$ on top of the $O(MN^2)$ MCMC cost."],"supporting_citations":[{"why":"Supplies the observation process $Y_t=tX+B_t$ and the conditional-distribution dynamics on which the SL posterior is built.","marker":"El Alaoui & Montanari (2022)"},{"why":"Generalizes SL to an $\\alpha(t)$ schedule and iterative posterior sampling, and provides the GEOM schedule used in experiments.","marker":"Grenioux et al. (2024)"},{"why":"Gives the theorem that the conditional law of $X$ given $Y_t$ follows the localization SDE used in Section 2.3.","marker":"Liptser & Shiryaev (2013)"},{"why":"Provides the key lemma that a spectral radius below 1 for the Dobrushin matrix yields a Poincaré inequality, the template for Theorems 4.3-4.9.","marker":"Wu (2006)"},{"why":"Defines the interdependence matrix whose row sums the paper bounds under the large-field condition.","marker":"Dobrushin (1970)"},{"why":"States that a spectral gap implies polynomial mixing time, converting the Poincaré inequalities into the headline guarantee.","marker":"Levin & Peres (2017)"},{"why":"Defines the GWG sampler, whose single-site gradient-informed kernel is analyzed in Theorem 4.8.","marker":"Grathwohl et al. (2021)"},{"why":"Defines the PAS sampler, also covered by the single-site Metropolis-Hastings analysis.","marker":"Sun et al. (2021)"},{"why":"Defines DULA and DMALA, analyzed in Theorem 4.9.","marker":"Zhang et al. (2022)"},{"why":"Supplies the DISCS benchmark tasks and baselines for the empirical QUBO evaluation.","marker":"Goshvadi et al. (2024)"}],"fun_headline_variants":["Stochastic localization breaks discrete sampling bottleneck","Provable fast mixing for binary quadratic samplers","Polynomial-time mixing for discrete MCMC via localization","Localization guarantee: discrete samplers mix in polynomial time","Stochastic localization tames hard binary sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every individual binary variable ends up with a large tilt, not merely that the overall tilt magnitude is large; the paper's guarantee that this happens is proved in Appendix C.1 through a one-dimensional observation calculation, while the Poincaré theorems need the field to be large coordinatewise.","fun_headline_variants_meta":{"raw":{"variants":["Stochastic localization breaks discrete sampling bottleneck","Provable fast mixing for binary quadratic samplers","Polynomial-time mixing for discrete MCMC via localization","Localization guarantee: discrete samplers mix in polynomial time","Stochastic localization tames hard binary sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1412,"prompt_tokens":999,"completion_tokens":413,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":340}},"tokens_in":615,"tokens_out":413,"duration_ms":3908,"temperature":1.0,"reasoning_tokens":340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:14:54.986131+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the SL observation process on an explicit $N \\ge 2$ BQD, record the trajectory of $h_t$, and check at the time $T$ promised by Theorem 3.1 whether $\\min_i |h_{t,i}| \\ge 2\\beta \\sup_i \\sum_{k\\ne i}|W_{ik}|$ on typical runs. If the scalar magnitude $|h_t|$ is large while some coordinate stays below threshold, the stated assumptions do not apply; a numerical spectral-gap estimate of Glauber dynamics at that field configuration would then show whether the Poincaré conclusion survives anyway.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Generalizes SL to an $\\alpha(t)$ schedule and iterative posterior sampling, and provides the GEOM schedule used in experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the theorem that the conditional law of $X$ given $Y_t$ follows the localization SDE used in Section 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the interdependence matrix whose row sums the paper bounds under the large-field condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States that a spectral gap implies polynomial mixing time, converting the Poincaré inequalities into the headline guarantee."},{"cited_title":"Oops i took a gradient: Scalable sampling for discrete distributions","cited_arxiv_id":null,"evidence_quote":"Defines the GWG sampler, whose single-site gradient-informed kernel is analyzed in Theorem 4.8."},{"cited_title":"Path auxiliary proposal for mcmc in discrete space","cited_arxiv_id":null,"evidence_quote":"Defines the PAS sampler, also covered by the single-site Metropolis-Hastings analysis."},{"cited_title":"A langevin-like sampler for discrete distributions","cited_arxiv_id":null,"evidence_quote":"Defines DULA and DMALA, analyzed in Theorem 4.9."},{"cited_title":"Discs: a benchmark for discrete sampling","cited_arxiv_id":null,"evidence_quote":"Supplies the DISCS benchmark tasks and baselines for the empirical QUBO evaluation."}],"review_version":1}