{"id":"0ed27ee5-d379-4b2e-a0a9-6a2a780d1330","arxiv_id":"2608.03117","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A construction principle for diffusion reference processes that gives simulation-free training and finite-time generation simultaneously, with conditional flow matching recovered as the zero-noise limit.","lead":"The paper proposes a way to design the random bridge process inside a diffusion model so that training needs no simulation of the underlying diffusion and generation finishes in a fixed, finite time. The authors show that score matching is not a required ingredient of diffusion training and that conditional flow matching appears as a special low-noise limit of their construction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Gaussian schedule in Eq. (47) makes the proposed objective Eq. (23) infinite on [0,1]: at the zero network the integrand is (E_x||x||^2 + 9d/4)/(1-t), so finite-time exactness holds only after an unjustified t<1 truncation.","rationale":"The reader's weakest assumption correctly identifies the singular weight and the missing analysis of the t<1 truncation. My reading sharpens the concern: the objective in Eq. (23) is not merely hard to sample near t=1; it is infinite on [0,1) for the zero network, which is a representable model. This makes the claimed equivalence between Eq. (22) and Eq. (23), and hence the exact finite-time property, formally unsupported. The issue is concrete and testable, but it is also addressable: one could add a truncation-error analysis, choose a schedule whose integrand is finite, or reformulate the claim as an approximate finite-time method with a controlled bias. Because the experiments are limited to 2D toy datasets and the paper itself acknowledges the relation to stochastic interpolants, the central novelty is not fully demonstrated; however, the practical finite-time behavior may still hold approximately. I therefore do not move the verdict: the paper remains conditionally acceptable, requiring the truncation issue to be resolved before the central claim is trusted.","tokens_in":15522,"tokens_out":11424,"duration_ms":122203,"concrete_test":"Compute analytically the integral in Eq. (48) at theta=0: if integral_0^1 dt (E_x||x||^2 + 9d/4)/(1-t) diverges, then Eq. (23) is not a finite objective on [0,1] for a network realizable by zero weights, which settles the formal issue. To test whether the truncation matters in practice, train the same model with t sampled uniformly from [0,0.99] and from [0,0.999], and estimate the integrand f(t) = E[||target - s_theta||^2]/(1-t) near t=1; if (1-t)f(t) tends to a positive constant as t->1, the omitted tail is logarithmically divergent and the truncation changes the objective in an uncontrolled way.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires the objective in Eq. (23) to be a tractable, finite simulation-free objective over the full interval [0,1], equivalent to the path-space KL in Eq. (22). With the concrete choices m_t(x)=tx, sigma(t)=1-t, lambda(t)=1/sigma(t), the objective becomes Eq. (48), whose integrand contains the singular weight 1/(1-t). For the zero network, which is a perfectly valid element of the function class, Eq. (48) evaluates to (1/(2n)) sum_i integral_0^1 dt (E_xi ||x_i - 1.5 xi||^2)/(1-t) = (1/2) integral_0^1 dt (E_x||x||^2 + 9d/4)/(1-t) = infinity. Thus Eq. (23) is not finite on the claimed interval for a representable network, and the derivation in Appendix A that discards a theta-independent constant also requires that constant to be finite. The paper explicitly avoids the singularity by sampling t from [0,0.99] and by taking the Girsanov representation as a tau->1 limit, but no bound is given on the omitted tail or on the truncation bias. Because the finite-time boundary p_1=mu is attained only through this singular limit, the advertised simultaneous satisfaction of simulation-free training and finite-time generation is not established by the argument as written. This is the load-bearing weakness: if the tail is not controlled, the method is not minimizing the claimed objective and the finite-time property is only approximate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a framework for designing the reference diffusion process in generative diffusion models by first prescribing tractable conditional distributions rho_t(z|x) and then constructing an SDE whose marginals realize them. With boundary conditions p_0=pi and p_1=mu, the construction is claimed to achieve simulation-free training and finite-time generation simultaneously. The resulting objective L_SF in Eq. (23) is shown, up to a theta-independent constant, to be equivalent to a path-space KL divergence, and the paper argues that score matching is not fundamental but emerges from time reversal, while conditional flow matching is recovered as a small-noise limit. Experiments on four two-dimensional toy datasets with Gaussian and Johnson's SU priors illustrate the method.","tokens_in":15844,"tokens_out":9230,"duration_ms":82821,"significance":"If the claims are established, the paper makes a useful conceptual contribution: it offers a constructive way to break the usual trade-off between simulation-free training and finite-time generation, and it provides a unified viewpoint connecting score matching, conditional flow matching, and stochastic localization. The algebraic derivations in Sec. III and Appendices A-D are internally consistent, and the small-noise limit leading to Eq. (39) is a clean observation. The paper also ships open-source code and checkpoints and includes a fair, if small-scale, comparison with a VP-SBM. However, the concrete schedules used in the experiments make the proposed objective ill-defined on the full time interval, and the advertised finite-time equivalence is asserted through an unanalyzed limit; these issues currently prevent the central claim from being accepted as proven.","major_comments":[{"comment":"With m_t(x)=tx, sigma(t)=1-t, and lambda(t)=sigma(t)^{-1}, the objective Eq. (23) is not finite on the full interval [0,1]. For the perfectly representable network s_theta=0, Eq. (48) evaluates to (1/(2n)) sum_i integral_0^1 dt (E_xi ||x_i - (3/2)xi||^2)/(1-t) = infinity whenever the data have finite second moment. Thus the zero network alone makes the purported objective infinite, and the theta-independent constant discarded in Appendix A also diverges. The statement in Sec. VI.A that 't=1 is not sampled' is a truncation, not a justification: no bound on the omitted tail [0.99,1] or on the resulting bias is supplied. This directly undermines the claim that Eq. (23) is a tractable simulation-free objective equivalent to the path-space KL in Eq. (22) on the full interval.","section":"Sec. VI.A, Eq. (48); Sec. III, Eq. (23)"},{"comment":"The paper defines the Girsanov representation Eq. (22) by applying it on [0,tau] with tau<1 and then taking the limit tau->1, because the chosen diffusion coefficient degenerates at t=1. No theorem is given that this limit is well-defined, finite, or that it preserves the terminal-marginal constraint p_1=mu in a sense compatible with the objective. Since b(1)=0 and the drift alpha contains terms of order (1-t)^{-1}, the standard hypotheses of the Girsanov theorem are not verified on [0,1]. Consequently the advertised 'finite-time' connection and the equivalence to Eq. (22) are presently asserted only in a limiting sense; the paper needs either a proof for the chosen schedules or a non-degenerate schedule for which the objective is genuinely finite on [0,1].","section":"Sec. IV, first paragraph"},{"comment":"The non-Gaussian-prior objective in Eq. (52) carries the same singular weight 1/(1-t) and is subject to the same truncation. The experiments in Fig. 1 therefore train and evaluate a truncated surrogate of Eq. (23), not the stated objective, and no analysis of truncation bias or of the influence of the singular endpoint is given. The main experimental section also reports only qualitative samples; quantitative metrics appear only for schedule comparisons in Appendix E.2. As a result, the numerical demonstration does not currently resolve the central theoretical concern about finiteness of the objective.","section":"Sec. VI.B, Eq. (52); Sec. VI.C"}],"minor_comments":[{"comment":"The notation for b(t) is ambiguous: it should be typeset explicitly as sqrt(lambda(t)) sigma(t) I_d (with parentheses), since Appendix B defines lambda via c(t)^2/sigma(t)^2 and the surrounding text can be misread as sqrt(lambda(t) sigma(t)) I_d.","section":"Eq. (28)"},{"comment":"The use of U[0,1) together with an integral over [0,1] is confusing when the integrand is singular at the endpoint; the paper should specify the truncated interval [0,1-epsilon] or explicitly describe the limiting procedure used in the Monte Carlo estimate.","section":"Eq. (48) and surrounding text"},{"comment":"Quantitative metrics for the main Gaussian-prior and Johnson-SU-prior experiments would strengthen the comparison with VP-SBM; currently only qualitative sample plots are given in the main text.","section":"Fig. 1"},{"comment":"The statement that 'score matching is not fundamental' would benefit from qualification: the non-Gaussian construction still uses grad log pi in Eq. (34), and Sec. V.C shows that score-like terms reappear when a deterministic flow is stochastically lifted.","section":"Sec. V.A"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central idea is original and potentially useful. The algebraic core is not circular; the problem is that the concrete schedules make Eq. (23) infinite on [0,1] and the limiting Girsanov argument is not justified. This is fixable by choosing non-degenerate schedules or by proving tail bounds, so I recommend major revision rather than rejection. No concerns about citation practices or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on the Kaba–Ohzeki–Sughiyama manuscript. The core idea is sound and presented cleanly: instead of starting from an SDE and deriving marginals, they prescribe tractable conditional distributions and then construct a reference SDE that realizes them. That reversal is genuinely useful, and it gives a nice explanation for why score matching appears—it traces to time reversal of the reference process, not to diffusion modeling per se. The small-noise limit recovering conditional flow matching is also a clean observation, though the relationship was known.\n\nThe real problem is the finite-time claim. With the Gaussian schedule in Eq. (47) and λ=1/σ, the diffusion coefficient vanishes at t=1 and the objective (48) carries a 1/(1-t) weight. For a zero network the expected loss is infinite on [0,1], and the θ-independent term discarded in Appendix A is also infinite. So L_SF is not actually finite on the advertised interval. The paper trains on [0,0.99] and takes the Girsanov representation as a τ→1 limit, but never bounds the omitted tail or the truncation bias. That is not a minor technicality: it is the mechanism by which they claim simultaneous finite-time generation. Without it, the method is a truncated approximation to the claimed objective.\n\nNovelty is modest, as the paper itself acknowledges: Stochastic Interpolants already deliver simulation-free training and finite-time paths. The new piece here is the reverse construction from conditional marginals and the non-Gaussian prior handling, which are worth having. The experiments are limited to 2D toys and compare only against VP-SBM; no baselines against SI or flow matching, so the practical delta is unproven.\n\nThat said, this is not a careless paper. The derivations in the appendices are careful, the connections to existing work are drawn honestly, and the framework is reusable even if the specific schedules need fixing. The singular-objective problem is addressable—one can analyze the truncation or choose schedules with better integrability—but the authors need to do that before the main claim holds.\n\nI'd send this to referees. It deserves a serious look, and the issues are concrete enough to be resolved in revision.","headline":"A clean unifying framework for reference processes, but the finite-time claim is undercut by a singular objective that the paper only truncates, and novelty is modest given Stochastic Interpolants.","tokens_in":16401,"tokens_out":3662,"would_cite":false,"duration_ms":34178,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Prescribing time-dependent conditional distributions yields a diffusion reference process that trains without simulation and connects prior to data at a fixed finite horizon.","keywords":["diffusion models","simulation-free training","finite-time generation","score matching","conditional flow matching","reference process","stochastic differential equations","Girsanov theorem"],"falsifier":"Train the Gaussian-prior model on truncation horizons $[0,1-\\delta]$ with $\\delta = 10^{-2}, 10^{-3}, 10^{-4}$ and measure sample quality (e.g., MMD) as a function of $\\delta$; if quality degrades monotonically as $\\delta\\to 0$ or the loss diverges, the truncation is load-bearing and the objective is ill-posed without it. Alternatively, numerically integrate (48) with the trained network to check whether the expected loss over $t\\in[0,1)$ is finite.","tokens_in":15274,"feed_emoji":"⚡","tokens_out":13853,"duration_ms":106260,"temperature":0.7,"pith_summary":"This paper attempts to eliminate the long-standing trade-off in generative diffusion models between training cost and generation speed. Its proposal is to stop thinking of the reference diffusion process as something given (like the Ornstein–Uhlenbeck process or a Schrödinger bridge) and instead prescribe a family of tractable time-dependent conditional distributions $\\{\\rho_t(\\cdot|x)\\}_{t\\in[0,1]}$ first, then construct the reference SDE that has these conditionals as its marginals. When the conditional distributions interpolate from the prior at time $0$ to the empirical data at time $1$, the resulting training objective needs only direct samples from these conditionals — no simulation of the reference SDE — and the generated process connects prior to data at a fixed finite horizon. The paper also shows that score matching is not a necessary ingredient of diffusion training (it appears only through time reversal of a data-to-prior process) and that conditional flow matching is the zero-noise limit of the new stochastic objective.","feed_headline":"Train diffusion with no simulation, generate in finite time","feed_subtitle":"Choosing the conditional path first removes score matching and the need to tune a convergence horizon.","key_machinery":"The central machinery is the inversion of the design order: instead of specifying a reference SDE and deriving its marginals, the paper prescribes conditional distributions $\\{\\rho_t(\\cdot|x)\\}$ and solves for the SDE coefficients via the conditional Fokker–Planck equation (19), $\\partial_t \\rho_t = -\\nabla_z\\cdot(\\alpha\\rho_t) + \\tfrac12 \\nabla_z\\cdot(\\nabla_z(\\Gamma\\rho_t))$, with $\\Gamma=bb^\\top$. The reference drift at a point is the conditional-drift average $a(t,z)=\\int \\alpha(t,z,x)\\rho_t(z|x)\\mu(x)/p_t(z)\\,dx$, and substituting this into the Girsanov objective yields (23), whose minimizer matches the full KL objective. The practical Gaussian instance chooses $\\rho_t(z|x)=\\mathcal{N}(z;tx,(1-t)^2 I_d)$ with $\\lambda(t)=\\sigma(t)^{-1}$, giving the explicit simulation-free objective (48); the non-Gaussian instance uses a push-forward family $\\phi^x_t(y)=f_t(y+g_t(x))$, which guarantees an $x$-independent diffusion coefficient so that the same objective applies.","core_discovery":"The paper claims that the reference process in a diffusion model can be constructed by first prescribing a family of tractable conditional distributions $\\{\\rho_t(\\cdot|x)\\}_{t\\in[0,1]}$ with the boundary conditions $p_0=\\pi$ and $p_1=\\mu$ for the marginals $p_t(z)=\\int \\rho_t(z|x)\\mu(x)\\,dx$, and then computing SDE coefficients $(\\alpha,b)$ that solve the conditional Fokker–Planck equation (19). The resulting reference SDE $dZ_t=a(t,Z_t)dt+b(t,Z_t)dW_t$ has marginals $p_t$, and the path-space KL objective $\\mathcal{L}_{\\mathrm{DM}}$ is, up to a $\\theta$-independent constant, equal to the simulation-free objective $\\mathcal{L}_{\\mathrm{SF}}$ in (23), which requires only direct sampling from $\\rho_t$ and evaluation of the conditional drift $\\alpha$. This removes the need to simulate the reference SDE during training and, because the marginals connect $\\pi$ to $\\mu$ over $[0,1]$, gives finite-time generation. The paper further claims that score matching is an artifact of reversing a data-to-prior process: the score $\\nabla\\log p_t$ never enters $\\mathcal{L}_{\\mathrm{SF}}$, and conditional flow matching is recovered as the small-noise limit ($\\epsilon\\to 0$) of the stochastic construction.","pith_inferences":["The explicit $1/(1-t)$ weight in the practical objectives (48) and (52) implies that for any fixed network with bounded output, the untruncated expected loss over $t\\in[0,1)$ is infinite; if so, the finite-time claim requires a proof that the $\\tau\\to1$ limit is finite and a statement of the truncation bias at $t=0.99$.","The construction suggests that diffusion models, conditional flow matching, and stochastic interpolants can be viewed as different choices of the same design object — the conditional path $\\{\\rho_t(\\cdot|x)\\}$ and a noise level — which may simplify comparing and transferring hyperparameters between methods.","Since the interpolation schedule affects generation quality even with fixed boundary distributions, learning the schedule $\\varphi(t)$ as part of the objective appears to be a natural and testable extension, and the paper's metric tables provide a benchmark that such learned schedules would need to beat."],"forward_implications":["A diffusion model can be trained with the objective in (23) (explicitly (48) for the Gaussian schedule) using only reparameterized samples from the prescribed conditionals, never simulating the reference SDE, and the generation process still connects prior to data at the fixed time $t=1$.","Score matching is not a necessary component of diffusion-model training: the score $\\nabla\\log p_t$ does not appear in the new objective, and it only enters conventional models because their reference process is defined in the data-to-prior direction and then time-reversed.","Conditional flow matching is recovered as the small-noise limit $\\epsilon\\to 0$ of the stochastic construction, giving CFM a path-space KL interpretation that it lacks on its own.","The framework extends to non-Gaussian priors by using a push-forward bijection family of the form $\\phi^x_t(y)=f_t(y+g_t(x))$, so heavy-tailed priors can be used without sacrificing simulation-free training."],"supporting_citations":[{"why":"Defines score-based generative modeling through SDEs; the OU reference process and its asymptotic convergence create the simulation-free/finite-time trade-off the paper addresses.","marker":"[3]"},{"why":"Supplies the Girsanov-based expression for the path-space KL objective between reference and generation processes that the paper reformulates into its simulation-free objective.","marker":"[13]"},{"why":"Defines conditional flow matching, whose objective the paper recovers as the small-noise limit of its stochastic construction.","marker":"[14]"},{"why":"Introduces neural flow diffusion models, a prior simulation-free finite-time approach whose score-dependent objective the paper contrasts with its score-free construction.","marker":"[23]"},{"why":"Introduces stochastic interpolants, the existing model class achieving both simulation-free training and finite-time generation, whose precise relation to the paper's construction is left as future work.","marker":"[12]"}],"fun_headline_variants":["Diffusion training without simulation, generation without infinite horizon","Prescribe conditionals, skip simulation, finish in finite time","Score matching not needed: reverse the process, not the training","Simulation-free diffusion: choose conditionals, not scores","Finite-time diffusion, zero simulation: set conditionals first"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The practical schedules make the training objective singular at the terminal time $t=1$, and the paper trains on $[0,0.99]$ while treating the endpoint as a limit; if that limit is not well-defined, the claimed finite-time connection holds only as a truncated approximation.","fun_headline_variants_meta":{"raw":{"variants":["Diffusion training without simulation, generation without infinite horizon","Prescribe conditionals, skip simulation, finish in finite time","Score matching not needed: reverse the process, not the training","Simulation-free diffusion: choose conditionals, not scores","Finite-time diffusion, zero simulation: set conditionals first"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000862,"raw_usage":{"total_tokens":3725,"prompt_tokens":914,"completion_tokens":2811,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":2728}},"tokens_in":530,"tokens_out":2811,"duration_ms":18305,"temperature":1.0,"reasoning_tokens":2728,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:53:59.137764+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the Gaussian-prior model on truncation horizons $[0,1-\\delta]$ with $\\delta = 10^{-2}, 10^{-3}, 10^{-4}$ and measure sample quality (e.g., MMD) as a function of $\\delta$; if quality degrades monotonically as $\\delta\\to 0$ or the loss diverges, the truncation is load-bearing and the objective is ill-posed without it. Alternatively, numerically integrate (48) with the trained network to check whether the expected loss over $t\\in[0,1)$ is finite.","supporting_citations":[{"cited_title":"(49), the gradient of its log-density is ∇y logπ(y) =− sinh−1yp 1 +y 2− y 1 +y 2.(D5) Substituting Eq","cited_arxiv_id":null,"evidence_quote":"Defines score-based generative modeling through SDEs; the OU reference process and its asymptotic convergence create the simulation-free/finite-time trade-off the paper addresses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Girsanov-based expression for the path-space KL objective between reference and generation processes that the paper reformulates into its simulation-free objective."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines conditional flow matching, whose objective the paper recovers as the small-noise limit of its stochastic construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces stochastic interpolants, the existing model class achieving both simulation-free training and finite-time generation, whose precise relation to the paper's construction is left as future work."}],"review_version":2}