{"id":"989a38fc-7bca-49e1-a97f-cdac8ff68b56","arxiv_id":"2505.23652","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"Score estimation in diffusion models is reformulated as solving linear systems in an eigenbasis of a backward Kolmogorov operator, avoiding neural network training and forward SDE simulation.","lead":"This paper proposes a way to train diffusion models without neural networks or forward simulations, by expanding the score in eigenfunctions of a base distribution and solving linear systems. If it works, it could make diffusion-style generation much faster for scientific distributions like molecular systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central high-dimensional claims rest on an unverified closure assumption: for the numerically computed mean-field eigenbasis, expansions (14) are not exact, and no error control is provided for the truncation used in (15)-(16).","rationale":"I read the paper first and foremost as claiming an optimization-free and forward-SDE-free score estimator valid for general base distributions, with the high-dimensional experiments as evidence. The linchpin of that algorithm is the ability to compute A(t) and b(t) in (15)-(16) from initial integrals only, via the eigenfunction closure (14). That closure is exact for the Hermite and Fourier examples, but the main experiments deliberately use a mean-field approximated basis precisely because the data are not Gaussian or uniform. For that basis the eigenfunctions are numerical, and the required expansions of products, derivatives, and the potential gradient are necessarily truncated. The paper contains no residual estimate or convergence analysis for this truncation, and Appendix A does not supply one. This is not a disagreement with consensus; it is an internal gap between the algorithm's advertised generality and its analysis. It directly affects all reported non-trivial results: the high-dimensional double-well experiment, both Ginzburg-Landau experiments, and the MNIST experiment. The reader's weakest_assumption identifies exactly this closure issue, and I agree that it is load-bearing. I do not, however, base the rejection on the alleged false inequality in Theorem 3. The displayed bound in the theorem's proof replaces an integral against ρ_t by the same integral against ρ_0; this is not justified as written, but since ρ_t = Cρ ρ∞(1+δ P_t u) and |P_t u|∞ ≤ ||u||∞ ≤ 1, one has ρ_t ≤ (1+δ)/(1-δ) ρ_0 pointwise. Inserting that constant repairs the proof while preserving the δ^2/|λ_1| form of the bound. Thus the theoretical analysis is salvageable in spirit, but the algorithmic closure problem is not addressed or bounded anywhere in the manuscript. Given that the main experimental findings rely on that uncontrolled approximation, the reader's REJECT verdict is unchanged.","tokens_in":25565,"tokens_out":10891,"duration_ms":99814,"concrete_test":"Reproduce the 32-dimensional Ginzburg-Landau experiment (λ_GL=0.05) with the mean-field basis and n=10, and compare A(t), b(t) from (15)-(16) against direct Monte Carlo evaluation using Euler-Maruyama samples from the forward SDE at a few representative times (e.g., t=0.1, 0.5, 1.0). Re-solve the linear system with the directly estimated matrices and recompute the relative second-moment error in Table 1; if it changes materially, the closure assumption dominates. As a cheaper analytic check, compute the L2(ρ∞) relative projection error of f_k f_{k'} onto the span of the first 2n one-dimensional eigenfunctions of each marginal; if this error is not near machine precision, expansion (14) is inexact and an explicit error bound is missing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's forward-SDE-free computation of A(t) and b(t) relies on expanding f_k f_{k'}, ∂_{xi}f_k, and ∂_{xi}V in a finite eigenbasis S' via (14). For the Hermite and Fourier bases this closure is exact with |S'|=2|S| (Section 3.1.3). In the main high-dimensional experiments, however, the basis is the mean-field approximated eigenbasis of Sections 3.2.1-3.2.2, obtained by solving one-dimensional eigenvalue problems for the maximum-entropy marginals. Those eigenfunctions are not polynomials or trigonometric functions, and products, derivatives, and the potential gradient are not finite linear combinations of finitely many eigenfunctions. Appendix A simply postulates expansions such as ∂_{xi}V(x) = Σ_l q_l^{(i)} ϕ_l^{(i)}(xi) and ϕ_{n_i}^{(i)'} = Σ_l v_l^{(n_i)} ϕ_l^{(i)} (A2) without quantifying the projection error. Consequently, equations (15)-(16) compute approximate coefficients whose error is uncontrolled. Theorem 3 is also silent here: it assumes exact eigenfunctions of the base density and does not bound this truncation. The Ginzburg-Landau second-moment errors in Table 1 and the MNIST samples therefore do not substantiate the method in the non-Hermite, non-Fourier setting. A secondary gap appears in the proof of Theorem 3 when ρ_t is replaced by ρ_0 under the integral without the factor (1+δ)/(1-δ), but that gap is repairable and does not change the δ→0 scaling.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a diffusion-model score estimation method that avoids neural-network optimization and forward-SDE simulation. The score is expanded in eigenfunctions of the backward Kolmogorov operator of a user-chosen base distribution, and the variational score-matching objective is reduced to a linear system whose coefficients are propagated in time by exponential eigenvalue factors. The paper analyzes the approximation error under a perturbation assumption, introduces a cluster basis to mitigate the curse of dimensionality, and reports experiments on one- and multi-dimensional Boltzmann densities and on MNIST after PCA reduction.","tokens_in":25976,"tokens_out":18508,"duration_ms":178559,"significance":"The Section 2 reformulation is algebraically sound, and the observation that Hermite and Fourier bases close under the operations in (14) with |S'|=2|S| is a useful and clearly explained computational point. If the theoretical bound and the high-dimensional closure argument were correct, the paper would offer a rare example of an optimization-free and simulation-free diffusion pipeline. However, the central perturbation-theoretic bound is not established, and the main high-dimensional experiments rely on an unquantified projection step for the mean-field basis. These gaps are load-bearing, so the current manuscript does not support its advertised contributions.","major_comments":[{"comment":"The proof of the central bound (27) uses the pointwise inequality |Σᵢ pᵢ e^{λᵢ t} fᵢ(x)| ≤ |Σᵢ pᵢ fᵢ(x)|, which is false for signed coefficients because the nonnegative weights e^{λᵢ t} are not all equal and can remove cancellations present in the unweighted sum. Assumption 1 bounds only the sup-norm of the signed sum, not the weighted sum or the sum of absolute values. The later inequality ‖Σᵢ pᵢ e^{λᵢ t} ∇fᵢ(x)‖ ≤ e^{λ₁ t}‖Σᵢ pᵢ ∇fᵢ(x)‖ has the same defect. Consequently the geometric expansion of the logarithm and the e^{2λ₁ t} decay used to reach (27) are not justified, and Theorem 3 is not established as stated. A corrected proof requires either a stronger assumption controlling Σᵢ |pᵢ fᵢ(x)| pointwise or a different argument that proves the weighted-sum bound.","section":"Section 4, proof of Theorem 3"},{"comment":"The exactness of expansion (14) is essential for the coefficient formulas (15) and (16). This exactness holds for Hermite and Fourier bases, but for the mean-field approximated basis used in the high-dimensional experiments, the numerically computed one-dimensional eigenfunctions are not closed under products or derivatives. Appendix A simply postulates expansions such as (A2) without quantifying the projection error. The computed A(t) and b(t) are therefore coefficients of a projected auxiliary problem, and Theorem 3, which assumes exact eigenfunctions of ρ∞, does not control this truncation. The Ginzburg-Landau results in Table 1 and Figures 4–5 do not validate the method in the non-Hermite, non-Fourier setting that is claimed to overcome the curse of dimensionality.","section":"Section 3.2 and Appendix A"},{"comment":"The MNIST experiment first reduces the data to 10 PCA components and then applies the algorithm in this 10-dimensional latent space before mapping back to images. This validates the method on a 10-dimensional problem, not on the original 784-dimensional image distribution. Moreover, no quantitative comparison with a baseline diffusion model, flow matching, or neural score estimator is provided; the qualitative images alone do not support the claim of competitive performance.","section":"Section 5.4"},{"comment":"The theorem assumes V∈F_M and then selects s_M(t,x) = −∇V(x) + δΣ_{i=1}^{M} p_i e^{λ_i t} ∇f_i(x) as an element of L²([0,∞),F_M). This membership requires ∇f_i∈F_M for i≤M, which is not generally true unless the basis and F_M are chosen carefully (for Hermite and Fourier bases, derivative closure requires including the appropriate shifted or constant modes). The hypothesis as stated does not guarantee that the proposed s_M lies in the hypothesis class.","section":"Section 4, statement of Theorem 3"}],"minor_comments":[{"comment":"The complexity claim in the Introduction, O(N d_b² d² n⁴ + N_t d_b³ d³ n⁶), is inconsistent with the final complexity in Eq. (22), O(N d_b² d² n⁴ + N_t d_b² d² n⁶); please reconcile these statements.","section":"Section 1.1 vs. Section 3.2.3"},{"comment":"The statistical-error assumption is stated informally with 'with high probability P=P(N,err_stat,F_M)'; it should be written as an explicit probabilistic inequality, specifying the confidence parameter and the dependence on the model class.","section":"Section 4, Theorem 2"},{"comment":"There are several typographical and terminology errors, including 'seperable' in Section 3.2.1, 'In order to proof' in Appendix C, and the missing definition of the 'relative one-marginal density error' used in Section 5.2 and Figure 3.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The reader's report correctly identifies the central proof gap. The invalid weighting inequality in Theorem 3 and the unquantified closure of the mean-field basis are not local issues: together they undermine the paper's main theoretical result and its principal high-dimensional experiments. A substantially rewritten version with a corrected theorem and a genuine error analysis for the cluster basis could be reconsidered, but the current manuscript does not support its claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know before reading. First, the core idea is genuinely different: they recast score estimation as a linear solve in the eigenbasis of the backward Kolmogorov operator, eliminating both iterative training and forward SDE simulation. That is a real methodological contribution, not a repackaging. Second, the paper's central error bound (Theorem 3) has a broken proof as written: the step |Σ p_i e^{λ_i t} f_i| ≤ |Σ p_i f_i| is false pointwise, and they drop a ρ_t/ρ_0 ratio factor when moving from L2(ρ_t) to L2(ρ_0). Both gaps look repairable—the first via Markov-semigroup contractivity, the second by inserting (1+δ)/(1−δ)—but the theorem as stated is not established.\n\nWhat the paper does well: the Section 2 reformulation is clean and algebraically correct. For Hermite and Fourier bases the closure is exact, and the 1D experiments are quantitative and sensible. The cluster basis is a thoughtful way to dodge the curse of dimensionality, with an explicit complexity analysis. The perturbation-theoretic angle on score estimation is genuinely new relative to the cited literature, and the paper is honest about scope: it does not claim to handle generic images or text.\n\nSoft spots: the mean-field approximated basis used in the main experiments (Ginzburg-Landau, MNIST) is not covered by the theory. Expansions (14) are exact for Hermite and Fourier bases, but for numerically computed eigenfunctions they are truncated with no error control. Appendix A simply postulates the needed expansions in (A2). So the forward-SDE-free computation of A(t) and b(t) is approximate in exactly the regime the paper's headline addresses, and the error from that truncation is unquantified. The experimental evidence is also thin: no comparison against a trained diffusion model on the same problem, MNIST is run on 10 PCA components, and the Ginzburg-Landau numbers are single values without baselines or variability. These are not fatal flaws but they are load-bearing for the paper's main claims.\n\nOverall: this is a solid methodological idea with a fixable proof gap and a plausible but unsupported closure assumption. It deserves a serious referee. If I were the editor, I would send it to review with a clear ask: repair the Theorem 3 proof, bound the mean-field basis truncation error, and add at least one comparison to a standard diffusion baseline. I would be skeptical but constructive in my own report.","headline":"Genuinely new linear-system reformulation of score estimation, but Theorem 3's proof has a repairable flaw and the high-dimensional experiments rest on an unquantified truncation assumption.","tokens_in":26460,"tokens_out":3728,"would_cite":false,"duration_ms":33954,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C20","65M70","68T05","68T09","62G09"],"pacs":[],"model":"deepseek-v4-flash","headline":"Diffusion-model score estimation can be reduced to solving a linear system by expanding the score in eigenfunctions of the backward Kolmogorov operator, with no neural-network training and no forward SDE simulation.","keywords":["Diffusion models","Optimization-free methods","Perturbation theory","Backward Kolmogorov operator eigenfunctions","Score matching","Cluster basis"],"falsifier":"Compute the residual of the product and derivative expansions (14) for the numerically computed mean-field eigenfunctions on the 32- or 64-dimensional Ginzburg-Landau targets: if that projection error is not small, or if the score error from the linear system does not track the $\\delta^2/|\\lambda_1|$ bound as the perturbation scale grows, the method's central closure fails.","tokens_in":1919,"feed_emoji":"🧮","tokens_out":2010,"duration_ms":58704,"temperature":0.7,"pith_summary":"The paper proposes a generative-modeling method that replaces neural-network training and forward SDE simulation with spectral linear algebra. Its central claim: if the score function of the reverse diffusion is expanded in a sparse set of eigenfunctions of the backward Kolmogorov operator associated with a chosen base distribution, the score at every time solves a linear system whose coefficients are computable from initial samples alone. The time evolution of the needed integrals is carried by exponential decay factors $e^{\\lambda_k t}$, so no intermediate samples need to be generated. The paper also proves a perturbation-theory error bound: for densities that are a small perturbation of the base distribution, the score-approximation error is controlled by $\\delta^2/|\\lambda_1|$ times tail terms. The significance is a principled, optimization-free route to score estimation with explicit error control and, under a locality assumption, cost linear in dimension.","feed_headline":"No training, no SDE: scores come from a linear solve","feed_subtitle":"Expanding the score in Kolmogorov eigenfunctions turns diffusion models into linear algebra with a perturbation-theory error bound.","key_machinery":"The workhorse is the eigenfunction heat-kernel identity $\\int f_k \\rho_t dx = e^{\\lambda_k t}\\int f_k \\rho_0 dx$, which follows from $L f_k = \\lambda_k f_k$ and the fact that $\\rho_t = e^{L^* t}\\rho_0$. It converts every integral against the time-dependent density into an initial-sample Monte Carlo integral times a known scalar exponential, removing the need to simulate the forward process at intermediate times. Around this sit the expansion identities (14), which express products and derivatives of basis functions in the same eigenbasis; when these expansions are exact (Hermite or Fourier bases) the computed $A(t)$ and $b(t)$ are exact, and when they are approximate (numerically computed mean-field eigenfunctions) the method inherits the truncation error. The linear system $A(t)C(t) = -B(t)$ is then the optimization-free closure, and the sparse 2-cluster basis controls dimensionality.","core_discovery":"Starting from the score-matching objective, the paper shows that when the score is written as $c(t)^T f(x) - \\beta\\nabla V(x)$ with $f$ an eigenbasis of $L = -\\nabla V\\cdot\\nabla + \\beta^{-1}\\Delta$, the optimal coefficients solve the linear system $A(t)C(t) = -B(t)$ in closed form. The entries $A$ and $B$ are integrals against the time-evolving density $\\rho_t$, and the key identity $\\int f_k \\rho_t dx = e^{\\lambda_k t}\\int f_k \\rho_0 dx$ converts them into exponentially weighted integrals against the initial data, eliminating the forward SDE. Under the perturbative assumption $\\rho_0 = C_\\rho \\rho_\\infty(1 + \\delta\\sum_i p_i f_i)$, Theorem 3 bounds the score error by $(\\delta^2/|\\lambda_1|)(r(M+1) + (\\delta/(1-\\delta))^2 r(1))$, where $r(M)$ measures the tail of the gradient expansion. For separable or near-separable base distributions the eigenfunctions factorize, and a local 2-cluster basis keeps the number of active coefficients at $O(d_b d n^2)$, giving an overall cost of $O(N d_b^2 d^2 n^4 + N_t d_b^2 d^2 n^6)$ when combined with randomized low-rank solving. Numerical results on double-well, Ginzburg-Landau, and MNIST are offered as evidence that the approach works in practice.","pith_inferences":["A natural testable extension is to apply the same spectral linear-system formulation to other semigroups with known eigenfunctions (for example, other reversible diffusions or discrete-state Markov chains), where the identity replacing (13) would be the analogous eigenfunction decay.","For the numerically constructed mean-field basis, the paper does not quantify the error of truncating the product and derivative expansions in (14); a concrete check would be to measure that projection residual on the Ginzburg-Landau examples and compare it against the observed score error.","The perturbation bound suggests a practical diagnostic: estimate the effective perturbation scale $\\delta$ from the discrepancy between $\\rho_0$ and the mean-field base, and use it to predict when the linear-system score estimate should be trustworthy.","The method can be viewed as a spectral collocation approach to score matching, which opens a comparison against other non-neural score estimators such as random-feature or kernel-based methods in the same perturbative regime."],"forward_implications":["Score estimation no longer requires gradient descent or forward SDE simulation: a single Monte Carlo estimate against $\\rho_0$ is reused across all times via exponential weights.","For Gaussian (Hermite) and periodic (Fourier) base distributions, the ansatz is closed because products and derivatives of basis functions are exact linear combinations, leaving only Monte Carlo and truncation error.","Under the perturbation assumption, the score-approximation error provably decays like $\\delta^2$ with the perturbation scale and improves as the number of eigenfunctions $M$ grows through the tail term $r(M+1)$.","With a local 2-cluster basis and randomized low-rank solving, the cost is $O(N d_b^2 d^2 n^4 + N_t d_b^2 d^2 n^6)$, avoiding the $O(N N_t d_b^2 d^2 n^4)$ cost of Euler-Maruyama-based coefficient computation.","The learned score can be plugged into the reverse-time SDE to generate new samples, and the paper demonstrates this on high-dimensional Boltzmann distributions and MNIST images."],"supporting_citations":[{"why":"Supplies the score-matching objective and its integration-by-parts reformulation used to define the quadratic minimization that becomes the linear system.","marker":"[1]"},{"why":"Provides the reverse-time diffusion equation used to generate samples once the score is estimated.","marker":"[2]"},{"why":"Establishes the score-based generative modeling via SDEs framework that the paper's forward-SDE-free approach is designed to avoid.","marker":"[6]"},{"why":"Supplies the result that small L2 score error implies small distribution discrepancy, motivating the paper's focus on accurate score estimation.","marker":"[8]"},{"why":"Provides the spectral theory of the backward Kolmogorov operator and its eigenfunctions for overdamped Langevin dynamics.","marker":"[28]"},{"why":"Introduces the cluster-basis / sketching perspective used to construct a sparse high-dimensional eigenbasis for the score.","marker":"[31]"},{"why":"Supplies the j-cluster basis construction for high-dimensional tensor-structured representations used in the local 2-cluster basis.","marker":"[32]"},{"why":"Supplies randomized low-rank approximation techniques used to solve the large linear systems efficiently in high dimensions.","marker":"[34]"}],"fun_headline_variants":["Scores from a linear solve: diffusion without training or SDE","Diffusion via eigenbasis: no optimizer, no forward SDE","Linear algebra replaces training in diffusion models","Perturbation theory yields training-free diffusion scores","No optimization, no SDE: score via Kolmogorov eigenbasis"],"cache_read_input_tokens":28544,"weakest_assumption_plain":"The forward-SDE-free computation assumes that products and derivatives of the chosen eigenfunctions can be re-expanded in a finite eigenbasis with negligible error; this is exact for Hermite and Fourier bases, but for the numerically computed mean-field basis used in the main experiments this truncation error is not controlled, and the experimental results depend on it.","fun_headline_variants_meta":{"raw":{"variants":["Scores from a linear solve: diffusion without training or SDE","Diffusion via eigenbasis: no optimizer, no forward SDE","Linear algebra replaces training in diffusion models","Perturbation theory yields training-free diffusion scores","No optimization, no SDE: score via Kolmogorov eigenbasis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00053,"raw_usage":{"total_tokens":2565,"prompt_tokens":970,"completion_tokens":1595,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":1512}},"tokens_in":586,"tokens_out":1595,"duration_ms":11362,"temperature":1.0,"reasoning_tokens":1512,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:41:30.170507+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the residual of the product and derivative expansions (14) for the numerically computed mean-field eigenfunctions on the 32- or 64-dimensional Ginzburg-Landau targets: if that projection error is not small, or if the score error from the linear system does not track the $\\delta^2/|\\lambda_1|$ bound as the perturbation scale grows, the method's central closure fails.","supporting_citations":[{"cited_title":"Journal of Machine Learning Research6(4) (2005)","cited_arxiv_id":null,"evidence_quote":"Supplies the score-matching objective and its integration-by-parts reformulation used to define the quadratic minimization that becomes the linear system."},{"cited_title":"Advances in Neural Information Processing Systems35, 22870–22882 (2022)","cited_arxiv_id":null,"evidence_quote":"Supplies the result that small L2 score error implies small distribution discrepancy, motivating the paper's focus on accurate score estimation."},{"cited_title":"Texts in applied mathe- matics60(2014)","cited_arxiv_id":null,"evidence_quote":"Provides the spectral theory of the backward Kolmogorov operator and its eigenfunctions for overdamped Langevin dynamics."},{"cited_title":"SIAM review53(2), 217–288 (2011) 37","cited_arxiv_id":null,"evidence_quote":"Supplies randomized low-rank approximation techniques used to solve the large linear systems efficiently in high dimensions."}],"review_version":1}