{"id":"afae5b5d-5e1d-460f-81f9-111b2ad9df54","arxiv_id":"2508.03636","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Likelihood Matching trains diffusion models by maximizing a Gaussian quasi-likelihood of reverse transitions driven by score and Hessian estimates, with consistency and total-variation convergence guarantees.","lead":"This paper proposes a new way to train diffusion models, the AI systems that create images by removing noise, by matching the likelihood of the data along the reverse noise-removal path. It reports math guarantees and small image tests, but no code or large-scale results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 omits the Gaussian quasi-transition error; its bound contradicts a one-step reverse for non-Gaussian q0 even with exact score and Hessian.","rationale":"The reader identified the unquantified Gaussian approximation of the reverse transition as the load-bearing assumption. My analysis sharpens this: the approximation error is not merely unquantified but appears to be omitted entirely from Theorem 1. The theorem's bound vanishes with T even when the true reverse transition is non-Gaussian, and the T=1 case exposes an immediate contradiction. Since Theorem 1 is the main theoretical guarantee for the proposed sampler, this is a central flaw in the paper's argument. The empirical results are suggestive but do not validate the stated bound, and the missing term could dominate the claimed rates. The paper's core theoretical claim is therefore not established as written. I recommend REJECT rather than CONDITIONAL because the issue is a demonstrable inconsistency in a central theorem, not a missing appendix or an addressable minor assumption. A corrected version would need to add an explicit Gaussian-approximation error term to Theorem 1 and restate the main claims accordingly.","tokens_in":11108,"tokens_out":14620,"duration_ms":173924,"concrete_test":"Simulate q0 = 0.5 N(-10,0.1) + 0.5 N(10,0.1) in d=1 with β_t=1 and T ∈ {1,2,5,10,20}. Compute the exact score and Hessian of q_t analytically and run the sampler (13)-(14) with N=10^5 samples from Y_T~N(0,1). Estimate TV(q0, p̃0) via a fine histogram. If the estimated TV stays roughly constant and far above d³ log^{4.5}T/T (which tends to 0), the Gaussian-transition error is confirmed as missing from Theorem 1. Also report the TV at T=1, which should be compared directly with the claimed bound of 0.","verdict_should_be":"REJECT","load_bearing_attack":"The central sampler guarantee, Theorem 1, bounds TV(q0||p̃0) by d³ log^{4.5}T/T + √log T ε_s + (log T/√T) ε_H. With ε_s=ε_H=0, the bound tends to 0 as T→∞. But the sampler (13)-(14) replaces the true reverse density p_{t_{k-1}|t_k}, which is non-Gaussian whenever q0 is non-Gaussian, by a Gaussian with matched conditional mean and covariance, equation (9). This Gaussian-shape approximation error is not quantified in the theorem. It is also not vanishing in T: for any fixed k, the reverse transition p_{k-1|k} is independent of the horizon T, and the one-step Gaussian quasi-kernel does not become exact as T grows. Thus the leading term in (17) cannot cover the mismatch. A concrete contradiction: take q0 to be a well-separated two-component Gaussian mixture in R, set T=1, β_t=1, and use the exact score and Hessian. The sampler starts from Y_1~N(0,1) and the Gaussian quasi-kernel with matched moments is close to N(0,1), so TV(q0,p̃0) is near 1, whereas the theorem's right-hand side is d³ log^{4.5}(1)/1 = 0. Equivalently, as T→∞ the final reverse step still incurs a positive error that is absent from (17). This is an internal inconsistency in the stated theorem, not merely an unquantified approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Likelihood Matching (LM), a training objective for diffusion models that approximates the reverse-path log-likelihood by replacing the true reverse transition densities with Gaussian quasi-transition densities whose conditional means and covariances are matched via estimated score and Hessian functions. The authors derive a stochastic sampler that uses the estimated score and Hessian, prove consistency of the quasi-maximum-likelihood estimator in an oracle setting, and state a non-asymptotic total-variation bound for the sampler. Experiments on synthetic mixture models and low-resolution image datasets compare LM with score matching and study the effect of the number of transition points N and the Hessian rank r.","tokens_in":11415,"tokens_out":5861,"duration_ms":85314,"significance":"If the theoretical guarantees were correct, the paper would make a useful contribution by connecting diffusion-model training to quasi-likelihood estimation, by explicitly using Hessian information in both training and sampling, and by providing finite-sample TV bounds. The derivation of the conditional mean and covariance formulas in Proposition 2 and the use of external reverse-time diffusion theory are clear and standard. However, the central convergence theorem, as stated, omits a non-vanishing approximation error introduced by the Gaussian quasi-likelihood, and the consistency theorem is limited to an oracle score model. The paper also honestly acknowledges the lack of large-scale experimental validation; the experiments on MNIST and low-resolution image data are encouraging but not definitive. Overall, the methodological idea is interesting, but the main theoretical claim needs substantial revision before the paper can be accepted.","major_comments":[{"comment":"The bound in (17) does not include any term for the error caused by replacing the true reverse transition density p_{t_{k-1}|t_k} by the Gaussian quasi-density in (9). This error is not vanishing in T. For a fixed step, say from time 1 to time 0, the true reverse transition is non-Gaussian whenever q0 is non-Gaussian, and the Gaussian quasi-kernel with matched conditional mean and covariance is not equal to it, even with exact score and Hessian. Concretely, take q0 to be a well-separated two-component Gaussian mixture, set T=1 with the paper's convention t_k=k and β_t=1, and use the exact score and Hessian. The sampler (13)-(14) then applies one Gaussian quasi-kernel to Y_1 ~ N(0, I_d), producing a distribution that is not q0, so TV(q0||p̃0) is strictly positive, while the right-hand side of (17) is d^3 log^{4.5}(1)/1 = 0 when the score and Hessian errors vanish. More generally, as T grows, the final reverse step from time 1 to time 0 is independent of the horizon and its Gaussian approximation error persists, so the first term in (17) cannot account for it. The theorem as stated is therefore internally inconsistent for non-Gaussian q0. The authors need to add an explicit quasi-Gaussian approximation error term, or restrict the theorem to cases where the reverse transitions are exactly Gaussian, or otherwise explain why the missing term is covered by the existing terms.","section":"Theorem 1, Eq. (17), sampler (13)-(14)"},{"comment":"The consistency result is proved only for the oracle case where the score is known up to the true parameter θ, and it relies on the Gaussian quasi-likelihood being a valid inference objective. For non-Gaussian q0, the quasi-likelihood (10) is misspecified at the level of the full transition density, even when the conditional means and covariances are correct. Consistency of the quasi-MLE then requires additional identification and regularity conditions that are not stated in the main text and are referenced only to an Appendix B.4. The statement 'θ* is the parameter of the original data distribution q0' needs support showing that the population quasi-likelihood L(ϕ) is uniquely minimized at the true θ, and that the empirical objective (15) converges uniformly over the parameter space. Without these details, the consistency claim is not established for the actual setting of the paper.","section":"Theorem 2 and Section 3.1"},{"comment":"The text replaces E[log p_{t_N}(X_{t_N}; θ)] by a θ-free constant on the grounds that the marginal converges to N_d(0, I_d). For a finite diffusion horizon T, which is the horizon used in Theorem 1 and in the experiments, this term is not exactly θ-free; it depends on the initial distribution through q_T(·; θ). The approximation in (8) therefore introduces an additional error of order depending on how close q_T is to stationarity. This error should be made explicit and accounted for in the objective (15), or at least bounded in the theory.","section":"Equation (8) and the finite-T approximation"}],"minor_comments":[{"comment":"There are several typographical issues in the notation for transition densities, e.g., 'pt−1|t(Ytt−1|Yt; θ)' should be 'p_{t_{k-1}|t_k}(Y_{t_{k-1}}|Y_{t_k}; θ)', and 'Yt d = Xt' is typeset inconsistently.","section":"Section 3.1"},{"comment":"The caption contains the typo 'Socre Matching' instead of 'Score Matching'.","section":"Figure 3 caption"},{"comment":"The underbrace text 'converge to Nd(0,Id)' is grammatically incomplete; it should state that p_{t_N} converges to the standard normal density, and the convergence should be qualified with respect to T.","section":"Equation (7) and Proposition 1"},{"comment":"The main text refers to Appendix B for the proofs of Proposition 2, Theorem 1, and Theorem 2, and to Appendix C for experimental details, but the appendices are not included in the submitted version. The proofs and implementation details need to be available for verification.","section":"Appendix"},{"comment":"The sampler equations use both 'Y_{t-1}' and 't' in a way that is ambiguous with the continuous-time index; it would be clearer to write the discrete index explicitly, e.g., 'Y_{k-1}' for k = T, ..., 1.","section":"Notation in Eq. (13)-(14)"}],"recommendation":"major_revision","confidential_remarks":"The central issue is that Theorem 1's stated bound is false for non-Gaussian targets because the Gaussian quasi-transition approximation error is omitted. This is a load-bearing error: the headline convergence guarantee cannot stand without a correction. The correction is conceptually straightforward — add an explicit approximation-error term — but it will weaken the claimed result and may require reworking the framing of the paper. The oracle-only consistency theorem is also less convincing than the abstract suggests. I believe the paper is worth revising rather than rejecting outright, because the likelihood-matching idea and the moment-matching derivation are potentially useful, but the authors need to either prove a bound that includes the missing error or state the theorem under assumptions that make the Gaussian approximation exact."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the Likelihood Matching objective is a real addition: it assembles known ingredients into a quasi-likelihood with score and Hessian, and the conditional moment formulas in Proposition 2 are cleanly derived and reduce correctly to the Gaussian bridge case. Second, the sampler guarantee in Theorem 1 is not correct as stated: it drops the Gaussian quasi-transition error, which does not vanish as T grows. The stress-test note is right. For any non-Gaussian q0, the exact reverse transition is non-Gaussian, and the sampler replaces it with a Gaussian of matched mean and covariance. With exact score and Hessian and T large, the theorem's right-hand side goes to zero, but the one-step Gaussian mismatch remains. A well-separated two-component Gaussian mixture with T=1 makes this concrete: the bound in (17) is 0, but TV(q0||p̃0) is near 1. So the theorem overclaims.\n\nWhere the paper succeeds: the quasi-likelihood framing is more principled than the score-matching upper bound, the Hessian-aware sampler is a natural extension, and the mixture experiments show LM beats score matching for parameter recovery. The consistency theorem is honest about being oracle-only, though it doesn't cover the neural setting that matters. The low-rank Hessian approximation and the SMW-based implementation are useful practical details.\n\nSoft spots: the missing Gaussian error is load-bearing, not a nitpick. The proofs are in an appendix not present in this arXiv version, which makes verification impossible. No code or data are released. Experiments on MNIST/CIFAR are suggestive but small-scale, and the FID comparisons are not against strong baselines. Assumption 4 is mild but deserves a check.\n\nBottom line: this deserves a serious referee because the core idea is worth taking up, but the current Theorem 1 should be corrected or explicitly restricted to a bound that excludes the Gaussian approximation term. I wouldn't cite it as is. For a reading group, maybe, if people are working on diffusion objectives.","headline":"Genuinely new quasi-likelihood objective with Hessian-aware sampling, but Theorem 1's sampler bound omits a non-vanishing Gaussian approximation error.","tokens_in":11938,"tokens_out":3782,"would_cite":false,"duration_ms":45744,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Training diffusion models by directly maximizing an approximate data likelihood, using matched Gaussian reverse transitions, is consistent and yields explicit sampling error bounds.","keywords":["likelihood matching","diffusion models","quasi-maximum likelihood estimation","score matching","Hessian estimation","reverse-time stochastic differential equations","total variation convergence","stochastic sampler"],"falsifier":"Direct calculation on a distribution with known non-Gaussian reverse transitions, such as a well-separated two-component Gaussian mixture in dimension 2 or 10, would settle the matter: compute the exact negative log-likelihood $-E[\\log q_0(X_0;\\theta)]$, the LM objective (15), and the total-variation distance of samples from the sampler (13) as $T$ grows with oracle score and Hessian. If the gap between the LM objective and the true negative log-likelihood persists at a fixed positive level as $T \\to \\infty$, the Gaussian quasi-likelihood approximation is not harmless.","tokens_in":10876,"feed_emoji":"🎯","tokens_out":9870,"duration_ms":94529,"temperature":0.7,"pith_summary":"This paper proposes training diffusion models by directly maximizing an approximate likelihood of the data distribution rather than by score matching alone. The authors show that the data log-likelihood is equivalent, up to constants, to a sum of log-transition densities along the reverse diffusion path, then approximate each intractable reverse transition by a Gaussian whose mean and covariance are matched to the true conditional moments. Because those moments depend on both the score and the Hessian of the perturbed densities, the method learns both functions and uses both in a new stochastic sampler. The paper proves that the resulting quasi-maximum-likelihood estimator is consistent in the oracle-score setting and gives a non-asymptotic total-variation bound for the sampler in terms of dimension, diffusion steps, and score and Hessian estimation errors. A sympathetic reader would care because it turns likelihood maximization, the statistically principled objective, into a computable training loss for diffusion models.","feed_headline":"Diffusion training targets the true likelihood, not just an upper bound","feed_subtitle":"Matched Gaussian reverse transitions make maximum likelihood a computable loss for diffusion models.","key_machinery":"The central machinery is the reverse-path likelihood identity of Proposition 1 combined with the Gaussian quasi-likelihood of equation (9). The identity rewrites the expected log-likelihood of the initial data density as an expected sum of log-transition densities of the reverse process, so maximizing data likelihood is equivalent to maximizing reverse path likelihood up to constants. The quasi-likelihood replaces each intractable reverse transition density $p_{t_{k-1}|t_k}$ by a Gaussian $\\varphi_d(y;\\mu_{t_{k-1}|t_k},\\Sigma_{t_{k-1}|t_k})$ whose conditional mean and covariance are the true ones from Proposition 2: $\\mu_{s|t} = (x_t + \\sigma^2_{t|s}\\nabla\\log q_t(x_t;\\theta))/m_{t|s}$ and $\\Sigma_{s|t} = (\\sigma^2_{t|s}/m^2_{t|s})(I_d + \\sigma^2_{t|s}\\nabla^2\\log q_t(x_t;\\theta))$. This machinery turns maximum likelihood estimation into a tractable objective depending on the score and Hessian, and the same matched moments define the stochastic sampler (13)-(14).","core_discovery":"The paper's central claim is that minimizing the Likelihood Matching objective in equation (15) is an operational way to maximize the log-likelihood of the target data distribution. Proposition 1 gives the identity $-E[\\log q_0(X_0;\\theta)] \\approx -E[\\sum_{k=1}^N \\log p_{t_{k-1}|t_k}(X_{t_{k-1}}|X_{t_k};\\theta)] + C_T$, expressing the data negative log-likelihood as an expected sum of reverse transition log-densities along the forward sample paths. Since the true reverse transition densities are non-Gaussian and unavailable, Proposition 2 derives their conditional mean and covariance, showing they depend on the score $\\nabla \\log q_t$ and the Hessian $\\nabla^2 \\log q_t$, and equation (9) replaces each transition density with a Gaussian having those matched moments. The paper parameterizes the score and Hessian by a common parameter $\\phi$, minimizes the resulting quasi-likelihood (15), and samples by (13)-(14) using both learned moments. The result is a training objective that contains score matching as a component but adds covariance matching and likelihood weighting, with consistency of the quasi-MLE (Theorem 2) and the non-asymptotic total-variation bound (Theorem 1) as the theoretical payoffs.","pith_inferences":["The paper does not quantify the Gaussian approximation error; a direct extension would be to bound the gap between the LM objective and the true negative log-likelihood for non-Gaussian data, which would tell practitioners when the method is safe.","A testable prediction of Theorem 1 is that lowering Hessian error by increasing the Hessian rank $r$ pays off less than lowering score error, because the Hessian error term enters at rate $\\log T/\\sqrt{T}$ while the score error term enters at $\\sqrt{\\log T}$.","Because the reverse-path identity (7) holds for arbitrary time grids, the same quasi-likelihood construction could be used with non-Gaussian transition approximations, such as mixtures, without changing the overall maximum-likelihood framing."],"forward_implications":["The LM objective (15) is an approximation of the data log-likelihood rather than an upper bound, so training on it aims directly at the quantity statistics prefers to maximize.","With exact score and Hessian, the sampler's total variation error decays as $d^3\\log^{4.5} T/T$ plus score and Hessian error terms, so the discretization error can be driven down by increasing diffusion steps.","In the oracle-score setting, the quasi-MLE is consistent: as $n$ and $T$ grow, the estimated parameters converge to the true parameters of the data distribution.","Because the sampler (13) uses the learned covariance as well as the score, it can produce coherent samples with fewer reverse steps than a score-only DDPM sampler in the MNIST experiments.","The low-rank Hessian parameterization with the Sherman-Morrison-Woodbury formula keeps the per-step cost of evaluating the matched covariance manageable, making the likelihood objective computationally feasible."],"supporting_citations":[{"why":"Establishes the reverse-time diffusion SDE whose marginals match the forward process, the foundation for the path-likelihood equivalence.","marker":"(Anderson, 1982)"},{"why":"Provides the time-reversal likelihood equivalence between forward and reverse joint densities used in Proposition 1.","marker":"(Haussmann and Pardoux, 1986)"},{"why":"Introduces quasi-likelihood estimation, the principle behind replacing intractable transition densities with a Gaussian with matched moments.","marker":"(Wedderburn, 1974)"},{"why":"Defines score-based generative modeling through SDEs and the score matching objective that Likelihood Matching extends.","marker":"(Song et al., 2021c)"},{"why":"Shows score matching only minimizes an upper bound on negative log-likelihood, the gap Likelihood Matching targets.","marker":"(Song et al., 2021b)"},{"why":"Supplies the DDPM forward discretization and sampling baseline against which the proposed sampler is framed.","marker":"(Ho et al., 2020)"},{"why":"Provides the low-rank parameterization of the Hessian used to make high-dimensional likelihood matching tractable.","marker":"Meng et al. (2021)"},{"why":"Provides the noise schedule and non-asymptotic convergence analysis techniques that Theorem 1 builds on.","marker":"Li et al. (2023)"}],"fun_headline_variants":["Diffusion likelihood matched via reverse transitions","Training diffusion with Gaussian matched moments","Quasi-likelihood for diffusion model training","Matching score and Hessian for diffusion likelihood","Diffusion models trained to true likelihood"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that each non-Gaussian reverse transition density of the true diffusion process can be replaced by a Gaussian with the same conditional mean and covariance without losing information that matters for the final sample distribution, and that the resulting approximation error stays small enough that the Likelihood Matching objective remains a faithful surrogate for the data log-likelihood.","fun_headline_variants_meta":{"raw":{"variants":["Diffusion likelihood matched via reverse transitions","Training diffusion with Gaussian matched moments","Quasi-likelihood for diffusion model training","Matching score and Hessian for diffusion likelihood","Diffusion models trained to true likelihood"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000732,"raw_usage":{"total_tokens":3292,"prompt_tokens":978,"completion_tokens":2314,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":2250}},"tokens_in":594,"tokens_out":2314,"duration_ms":20522,"temperature":1.0,"reasoning_tokens":2250,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:18:52.932056+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Direct calculation on a distribution with known non-Gaussian reverse transitions, such as a well-separated two-component Gaussian mixture in dimension 2 or 10, would settle the matter: compute the exact negative log-likelihood $-E[\\log q_0(X_0;\\theta)]$, the LM objective (15), and the total-variation distance of samples from the sampler (13) as $T$ grows with oracle score and Hessian. If the gap between the LM objective and the true negative log-likelihood persists at a fixed positive level as $T \\to \\infty$, the Gaussian quasi-likelihood approximation is not harmless.","supporting_citations":[],"review_version":1}