{"id":"770b18ce-8f07-4123-92bf-939007f4be91","arxiv_id":"2502.05807","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new density guidance modification of the generative ODE/SDE lets users set the log-density of samples from flow models, giving a practical dial over image detail.","lead":"Diffusion models can generate images with controllable detail, but drawing samples with a target image likelihood has been hard. This paper derives a way to steer the sampling process so the model's log-density follows a chosen trajectory, giving a dial over image detail, and explains why scaling the starting noise works.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exact log-density control is only derived for full-trajectory guidance, but the main DGS experiments stop at t*, leaving the final log p0 uncontrolled and off-manifold.","rationale":"The reader's weakest assumption focuses on learned score/divergence accuracy on guided trajectories, which is a real robustness issue. The more load-bearing structural issue is that the main experiments do not actually run the algorithm for which the exactness theorem is proved: guidance is restricted to [T,t*], so the final log p0 depends on an uncontrolled and atypical tail integration. This affects the central claim independently of whether the score is exact, because the target quantile identity Eq. 19 requires b_t to be imposed over the full interval. The paper acknowledges the restriction but does not flag that it forfeits exact final-density control. A concrete oracle-score experiment would settle whether the practical algorithm still approximates the target quantile well enough, or whether the exactness claim is limited to the full-trajectory variant. This does not change the reader's CONDITIONAL verdict, but it sharpens the reason: not just network error, but an internal gap between the exact derivation and the implemented procedure.","tokens_in":24766,"tokens_out":10194,"duration_ms":110405,"concrete_test":"In a model with an oracle score (e.g., a low-dimensional VP-SDE with known Gaussian-mixture p0, or the CIFAR-10 model treating its learned score as reference), generate samples for q ∈ {0.01, 0.5, 0.99} using DGS with guidance only on [T,t*] exactly as in Section 4.2, then continue with the standard PF-ODE to t=0. Compute each sample's log p0 by integrating Eq. 3 backwards along the unguided PF-ODE, and compare the empirical quantile of the obtained log p0 values to the target q (and to φ_0(q) from Eq. 19). If the obtained quantile deviates from q by more than the Monte Carlo error, or if the deviation grows as q moves away from 0.5, the practical method does not provide the exact likelihood control claimed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central exactness claim (Eq. 16 with b_t specified on [0,T]) is not what the main experiments run. Section 4.2 states 'we only use guidance in the [T, t*] interval', i.e., for t < t* the method falls back to the unguided PF-ODE. Then the final log p0 of a DGS sample equals log p_T(x_T) - ∫_T^{t*} b_t dt - ∫_{t*}^0 div u_t dt; the second integral is not controlled by the chosen quantile q and starts from a guided, atypical state. This breaks the quantile identity Eq. 19 in the exact sense claimed. The paper's justification for using t* is the empirical correlation in Fig. 4, not an exactness argument; correlation at one time does not imply the remaining tail integral is constant or calibrated. Since the guided state at t* is off the typical set, this is also exactly where the learned score/divergence approximations (Section 2.4) are least reliable, compounding the error. Thus the headline 'exact log-density control' holds only for the modified ODE when b is active everywhere, not for the algorithm used in the main image-detail experiments.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies how to control the log-density (and thereby the perceived detail) of samples from continuous normalizing flows and diffusion models. It first proposes a score-alignment condition that explains when scaling the latent code (prior guidance) monotonically changes log p0, and shows empirically that the condition holds for most latents in two models. It then derives a modified generative ODE (Density Guidance, Eq. 16) that enforces a prescribed instantaneous log-density change, and extends this to a stochastic sampler (Eq. 25) that injects score-orthogonal noise. The authors claim exact log-density control, demonstrate detail manipulation in Stable Diffusion and EDM2, and provide code.","tokens_in":25073,"tokens_out":8295,"duration_ms":84455,"significance":"If fully supported, the paper would contribute a principled and potentially inexpensive way to control likelihood and image detail in pretrained diffusion models. The score-alignment analysis is a useful explanatory framework, and the derivation of Eq. 16 is algebraically clean. The paper also provides a tractable verification procedure (Eq. 13, Algorithm 1) that does not require knowing intermediate scores. However, the most important claim—exact log-density control—is only valid under idealized conditions that the main experiments do not implement; the practical DGS algorithm is approximate or truncated. The manuscript has the right ingredients but currently overstates what is established.","major_comments":[{"comment":"Equation (19) contains a logical error. The equality log pt(xt) = log pT(xT) - ∫_T^t bτ dτ = φt(q) can hold only if the initial value satisfies log pT(xT) = φT(q). Since xT is sampled from pT without conditioning on a specific quantile, the choice bt(x) := d/dt φt(q) does not generally drive the sample to the desired quantile. The verification in Appendix E appears to compare the numerically obtained log p0 with the value log pT(xT) - ∫ b_t dt, so it mainly confirms integrator convergence rather than quantile targeting. Please either condition xT on the target quantile, use a feedback controller that drives log pt toward φt(q), or clarify what is actually being verified.","section":"§4.1, Eq. (19)"},{"comment":"The main image-detail experiments use density guidance only on the interval [T, t*] where log SNR ≥ 1, as stated in Section 4.2; for t < t* the sampler is the unguided PF-ODE. Consequently the final log p0(x0) is not controlled by the chosen quantile q: it equals log pT(xT) - ∫_T^{t*} b_t dt - ∫_{t*}^0 div u_t dt, and the second integral is uncontrolled and starts from a guided, off-typical state. The abstract's claim of \"exact log-density control\" is therefore not realized by the algorithm used in the main experiments. This is a load-bearing mismatch between the theoretical claim and the empirical setup; it should be stated clearly as an approximation, and ideally the full-trajectory version should be evaluated on the image benchmarks as well.","section":"§5 and Appendix G"},{"comment":"The stochastic density guidance of Eq. (25) is only approximate. The exact drift in Eq. (120) requires the Rayleigh quotient R(∇^2 log pt, ∇ log pt); the implemented form in Eq. (123) drops this term, justified only by the empirical ratios in Eq. (122). Moreover, the projection P_t makes the diffusion matrix rank-deficient, so the usual Itô corrections must be handled carefully. The manuscript itself uses the word \"approximately\" in this section, but the abstract claims \"precise log-density control\" for stochastic sampling. The discrepancy should be reconciled, and the error incurred by omitting the Rayleigh quotient term should be quantified rather than only supported by two empirical measurements.","section":"§5 and Appendix G"},{"comment":"The exactness of Eq. (16) presumes knowledge of the true score and divergence. The experiments rely on neural approximations of these quantities, and the guidance mechanism deliberately moves samples away from the typical set where such approximations are most reliable (as the paper itself notes in Section 4.1). The paper does not provide any diagnostic of how accurate the learned score/divergence are along the guided trajectories. A concrete test—for example, comparing the predicted d log pt/dt against the empirically measured change of log pt along guided trajectories, or measuring the score error on guided vs unguided states—would be needed to support the claim of exact control in practice.","section":"§2.4 and §4"}],"minor_comments":[{"comment":"The text contains a typo: \"drfit\" should be \"drift\".","section":"§5"},{"comment":"The reference \"Tong, Y., FATRAS, K.\" should be formatted as \"Fatras, K.\" to match standard capitalization conventions.","section":"References"},{"comment":"The text \"K3\" should presumably be \"K = 3\", and later the text says K = 128 was used; please clarify the notation and whether the desired log-density is defined per-sample or as a global quantile.","section":"Appendix E"},{"comment":"In the provided Python code, the branch `if use_omega=False` returns `jnp.dot(v, score_fn(eps, x, key))`, but `eps` is not defined in the function's local scope; as printed, the code would raise a NameError. Please fix the listing.","section":"Algorithm 1 / Listing 1"},{"comment":"The caption says the distributions of log-likelihoods differ significantly, but it does not state how these log-likelihoods are computed (e.g., via Eq. (3) with a learned score or via an exact method). Please add the computational detail to the caption or text.","section":"Figure 8"}],"recommendation":"major_revision","confidential_remarks":"The paper has a sound core idea and a useful empirical analysis of score alignment, but the headline claim of exact log-density control is not implemented in the main experiments, and the explicit quantile matching derivation in Section 4.1 has a mathematical gap. These issues are fixable by revising the claims, conditioning the initial latent on the target quantile, adding full-trajectory image experiments, and quantifying the approximations in the stochastic extension. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth reading if you work on flow models and controllable generation. The new formal content is Score Alignment and the density-guided ODE/SDE. The math is correct: Eq. 16 and the stochastic extension in Appendix G are derived cleanly under the stated ideal assumptions. The paper also releases code and verifies Score Alignment on real models (97% on CIFAR, 100% on EDM2). That part is solid.\n\nThe main claim — that you can exactly control log p0 during sampling — is true for the modified ODE when guidance is active over the whole trajectory. But that is not what the main image experiments run. Section 4.2 restricts guidance to the [T, t*] interval (log SNR >= 1) and then falls back to the unguided PF-ODE for the rest. As the stress-test note correctly says, this leaves the final log p0 uncontrolled: the tail integral from t* to 0 is not governed by the chosen quantile, and the guided state at t* is exactly where the learned score and divergence are least reliable. So the 'exact' qualifier is doing more work than the experiments support. This is a real gap, but not a fatal one: the explicit quantile-matching experiment in Appendix E runs the full trajectory and shows exact control on CIFAR-10 as step size decreases, and the truncated DGS still produces monotonic detail control on ImageNet/StableDiffusion (Appendix J). The paper should either run the full-trajectory version on the higher-resolution models or add a theoretical/empirical bound on the tail error.\n\nThe other soft spot is the reliability of the neural-network score and divergence on the guided trajectory. The paper notes the assumption in Section 2.4 but doesn't validate it. The CLT-based choice of b_t (Eq. 22) is another layer of approximation; it's empirically verified, but it's not guaranteed for the specific model.\n\nQuantitatively, the results would be stronger with error bars and more than one seed for the PNG-size and NIQE comparisons. The trends are visible, but the precision is unclear.\n\nOverall: a competent, honest extension of the authors' prior likelihood-tracking work, with a useful practical dial and a caveat that should be disclosed more prominently. I'd bring it to reading group and cite it. For peer review: yes, send it to a serious referee. My verdict would be accept after the exactness language is tightened and the truncated-guidance case is analyzed or empirically evaluated.","headline":"A competent extension with a correct core derivation, but the headline 'exact log-density control' is supported only for the full-trajectory ODE, not for the truncated guidance the image experiments actually run.","tokens_in":25586,"tokens_out":3482,"would_cite":true,"duration_ms":35133,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A modified sampling ODE gives exact log-density control in flow-based image models","keywords":["diffusion models","continuous normalizing flows","density guidance","score alignment","log-density control","image detail manipulation","stochastic sampling"],"falsifier":"On a model with a known or very accurately estimated score (e.g., a Gaussian ground-truth diffusion), draw many guided trajectories to a fixed target log-density and measure the achieved empirical log-density of the decoded samples; the central claim is falsified if the achieved values deviate systematically from the target as the target moves away from the median, or if the discrepancy grows with the strength of guidance.","tokens_in":24571,"feed_emoji":"🎛️","tokens_out":2963,"duration_ms":24866,"temperature":0.7,"pith_summary":"This paper claims that the amount of visual detail in images sampled from diffusion and flow models can be controlled precisely by steering the log-density of the trajectory during generation. The authors introduce Density Guidance, a modification of the generative ODE that, assuming access to the score function, enforces a user-specified evolution of log-density along the sample path. They also explain why the heuristic of scaling the latent noise works, via a condition they call score alignment, and they extend exact density control to stochastic samplers. If true, the work turns a qualitative knob for detail (latent rescaling) into a quantitative one, with the same score evaluations already computed during sampling.","feed_headline":"Exact log-density dial for detail-rich image generation","feed_subtitle":"A modified generative ODE steers sample likelihood, and image detail, to a user-chosen quantile without retraining.","key_machinery":"The key object is the generalized instantaneous change-of-variable formula for a continuous normalizing flow, $\\frac{d}{dt} \\log p_t(x_t) = -\\operatorname{div} u_t(x_t) + \\nabla \\log p_t(x_t)^\\top (\\tilde u_t(x_t) - u_t(x_t))$, which tracks how the model's marginal density changes when the sample follows a different drift. Density Guidance inverts this formula: instead of computing the density change caused by a new drift, it solves for the drift that produces a desired density change, choosing the solution closest to the original vector field. Score alignment is a secondary device, a tractable condition (checkable even without the score) that predicts when latent rescaling produces monotonic log-density changes.","core_discovery":"The central claim is that one can derive an alternative ODE whose trajectories have exactly prescribed log-density evolution: given the score function and the original vector field, the drift $\\tilde u_t(x) = u_t(x) + \\frac{\\operatorname{div} u_t(x) + b_t(x)}{\\|\\nabla \\log p_t(x)\\|^2} \\nabla \\log p_t(x)$ achieves $\\frac{d}{dt} \\log p_t(x_t) = b_t(x_t)$ while deviating as little as possible from the original flow. With a quantile-based choice of $b_t$, this becomes Density-Guided Sampling, which rescales the score by a factor that depends on the local score norm and the target quantile. The paper further claims that exact log-density control carries over to stochastic sampling by projecting the noise onto the subspace orthogonal to the score and adding a correction term to the drift.","pith_inferences":["The paper's core identity suggests a generic recipe: any flow model with an accurate score estimate can be turned into a likelihood-conditional sampler, which may apply beyond images to scientific generation tasks where marginal likelihood is a meaningful quality signal.","Because the guided ODE stays closest to the original drift, its trajectories remain near the original model's typical set as long as the score is accurate there; this could make Density Guidance a probe for local score quality, since mismatches between requested and achieved log-density flag regions where the score is unreliable.","A testable extension would apply the same score-alignment verification to other latent manipulations (e.g., interpolation or principal-component perturbations) to characterize which latent directions reliably control likelihood.","The quantile-matching choice of $b_t$ depends on the claim that $\\sigma_t^2 (\\Delta \\log p_t + \\|\\nabla \\log p_t\\|^2)/\\sqrt{2D}$ is approximately standard normal; if that approximation degrades outside high dimension, the implicit quantile mapping would need a different calibrating statistic."],"forward_implications":["Image generation pipelines can set a target quantile of sample log-density, so a user can dial detail continuously from smooth to richly textured without retraining.","The method inherits the conditional score function, so text-to-image and class-conditional models gain the same detail control by simply replacing the score in the guided drift.","Stochastic sampling can be used for diversity while the log-density, and hence the detail level, is kept under control, because noise is confined to the subspace orthogonal to the score.","The score-alignment verification procedure gives model developers a cheap diagnostic for whether latent rescaling will behave monotonically for their model."],"supporting_citations":[{"why":"Supplies the instantaneous change-of-variable formula that is the foundation of the density-tracking machinery.","marker":"[Chen et al., 2018]"},{"why":"Provides the generalized density evolution formula for non-canonical drifts and the observation that log-density correlates negatively with image detail.","marker":"[Karczewski et al., 2025]"},{"why":"Contributes the probability-flow ODE formulation, the score-based SDE view, and the original observation that scaling the latent code changes detail.","marker":"[Song et al., 2021b]"},{"why":"Provides the EDM2 model used in the main experiments, along with hyperparameters for the practical instantiation.","marker":"[Karras et al., 2024b]"},{"why":"Supplies the Stable Diffusion v2.1 model used to demonstrate the method on text-to-latent image generation.","marker":"[Rombach et al., 2022]"},{"why":"Establishes that any CNF can be cast as an SDE via the score, which the stochastic extension of Density Guidance builds on.","marker":"[Eijkelboom et al., 2024]"},{"why":"Provides the quadratic-form central limit theorem used to justify the approximate normality of the auxiliary functional $h(x)$ that underpins the implicit quantile choice.","marker":"[de Jong, 1987]"}],"fun_headline_variants":["Density Guidance: exact dial for image detail","Exact log-density steering for detail-aware flow generation","Precise density control in flow models sharpens output","Score-aligned ODE tweak gives exact density dial"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exact density control holds only if the learned score function and its divergence remain accurate on the guided trajectories, which are deliberately shifted toward less typical regions of the data distribution.","fun_headline_variants_meta":{"raw":{"variants":["Density Guidance: exact dial for image detail","Exact log-density steering for detail-aware flow generation","Precise density control in flow models sharpens output","Score-aligned ODE tweak gives exact density dial"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1310,"prompt_tokens":914,"completion_tokens":396,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":332}},"tokens_in":530,"tokens_out":396,"duration_ms":4322,"temperature":1.0,"reasoning_tokens":332,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T17:52:23.406113+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a model with a known or very accurately estimated score (e.g., a Gaussian ground-truth diffusion), draw many guided trajectories to a fixed target log-density and measure the achieved empirical log-density of the decoded samples; the central claim is falsified if the achieved values deviate systematically from the target as the target moves away from the median, or if the discrepancy grows with the strength of guidance.","supporting_citations":[{"cited_title":"Neural ordinary differential equations","cited_arxiv_id":null,"evidence_quote":"Supplies the instantaneous change-of-variable formula that is the foundation of the density-tracking machinery."},{"cited_title":"Diffusion models as cartoonists! T he curious case of high density regions","cited_arxiv_id":null,"evidence_quote":"Provides the generalized density evolution formula for non-canonical drifts and the observation that log-density correlates negatively with image detail."},{"cited_title":"A., Welling, M., and van de Meent, J.-W","cited_arxiv_id":null,"evidence_quote":"Establishes that any CNF can be cast as an SDE via the score, which the stochastic extension of Density Guidance builds on."},{"cited_title":"A central limit theorem for generalized quadratic forms","cited_arxiv_id":null,"evidence_quote":"Provides the quadratic-form central limit theorem used to justify the approximate normality of the auxiliary functional $h(x)$ that underpins the implicit quantile choice."}],"review_version":1}