{"id":"ca2556a9-5192-4472-bb05-ce97931da443","arxiv_id":"2608.05811","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Energy-Guided Flow Matching changes the flow-matching endpoint from a fixed clean image to a sample-adaptive low-pass image, yielding lower FID and faster convergence on ImageNet and text-to-image generation.","lead":"Pixel-space generative models usually interpolate noise toward a fixed clean image. This paper replaces that target with an image-dependent low-pass endpoint that gradually releases high-frequency detail, improving FID and training speed in flow-matching models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 1 and Eq. (30) lack a branch for degenerate spectra: when an image has nonzero energy only at DC, Gtilde_x = 0 and the denominator D_t vanishes, so the training target v_t^* in Eq. (16) becomes 0/0 and the loss is undefined for a valid input.","rationale":"The reader's weakest_assumption identifies the same formal gap: Eq. (30) and Algorithm 1 assume strict monotonicity of G_x(h), which fails for degenerate spectra. I agree this is the most load-bearing correctness concern because it strikes at the training objective itself rather than at experimental convenience. The gap is narrow — natural images in ImageNet are effectively broadband — and it is easily patched with a degenerate-spectrum branch, so conditional acceptance remains the right verdict. The other concerns raised by the reader (baseline CFG tuning, σ0 sensitivity, and the missing HyperDiT 512 control) are secondary empirical robustness issues: the CFG difference between the reported 2.55 and the original setting is tiny, σ0 is honestly swept in Figure 5, and the 512 HyperDiT comparison is incomplete but does not invalidate the controlled 256 results. None of these would move the verdict to rejection; the formal degenerate-spectrum hole justifies the conditional status already assigned.","tokens_in":20520,"tokens_out":14707,"duration_ms":164051,"concrete_test":"Run Algorithm 1 on a constant image (all pixels equal) and on a pure sinusoid, with σ0 = 3.5, q(t) = smootherstep, and t = 0.5; print Gtilde_x, D_t, and ˙h_t. Then run one PixelDiT training step with a batch containing the constant image and check whether the loss is NaN. If ˙h_t or the loss is NaN, the objective is not defined for degenerate inputs; specify an explicit branch (e.g., set h_t = 0 and ˙h_t = 0 whenever Gtilde_x < ε, or add a small spectral floor to E(ρ)) and rerun the 80-epoch PixelDiT ablation to confirm the fix does not alter the reported FID.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The method's training objective is only well-defined if G_x(h) is strictly decreasing on [0,1] and the implicit derivative in Eq. (30) has a nonzero denominator. Section A.2 proves strict monotonicity only 'for a nondegenerate image spectrum,' but the paper never defines this condition nor provides a fallback branch in Algorithm 1. For an image whose spectral energy is supported solely at ρ = 0, we have Gtilde_x = 0 by Eq. (21) and D_t = 0 by Eq. (27) for every h; consequently q(t)Gtilde_x = 0, the bisection bracket is degenerate, and ˙h_t = q'(t)Gtilde_x / D_t evaluates to 0/0. Since Algorithm 1 computes R_t, D_t, and ˙h_t unconditionally, the resulting ˙h_t is NaN, and the velocity target v_t^* = y_t - ε + t ˙y_t in Eq. (16) is undefined. The paper's own Proposition 1 assumes a nondegenerate spectrum, so the endpoint-stability argument does not repair this gap. This is not a fatal objection for ImageNet, where natural image spectra are effectively broadband, but it means the central claim 'no adaptation of training data' is not true for an adversarially valid input class, and the method as specified does not define a training objective on the full pixel domain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Energy-Guided Flow Matching (EG-FM), a modification of the conditional flow-matching path for pixel-space generative models. Instead of interpolating linearly from noise to a fixed clean image x, EG-FM defines z_t = t y_t(x) + (1 - t) epsilon, where y_t(x) is a heat-kernel low-pass filtered version of x with a sample-dependent heat time h(x,t). The heat time is fixed implicitly by requiring the recovered spectral energy fraction G_x(h)/\\tilde G_x to equal a global release clock q(t). The paper derives the exact velocity target for this moving-endpoint path, v_t = y_t(x) - epsilon + t \\partial_t y_t(x), including an implicit-differentiation formula for \\partial_t h. Experiments on ImageNet 256 and 512 and on text-to-image benchmarks report improved FID at lower epoch counts across DeCo, HyperDiT, and PixelDiT. Ablations study the initial filter strength sigma0, the release-clock shape, and the schedule granularity.","tokens_in":20912,"tokens_out":8914,"duration_ms":86156,"significance":"If the empirical claims hold, EG-FM is a simple and broadly applicable trajectory prior: it changes only the training path and target, adds negligible compute, and lowers FID at a fixed epoch budget across several pixel-space backbones. The exact velocity target in Eq. (12) and Eq. (32) is a genuine contribution, and the endpoint-stability analysis in Appendix A.4 is a careful treatment of the apparent 0/0 limits at the path endpoints. The method also makes a falsifiable claim, namely that sample-adaptive energy-matched schedules outperform shared schedules, and Table 4 supports that ordering. However, the reported gains are modest relative to typical run-to-run variability, several reported quality metrics move in directions opposite to FID, and key hyperparameters (sigma0, release clock, CFG scale) are selected by FID on the same benchmark. The formal construction also has a degenerate-spectrum gap that leaves the training objective undefined for a valid input class. With these caveats, the contribution is interesting and likely useful, but the current evidence does not fully support the stronger statements in the abstract and conclusion.","major_comments":[{"comment":"The training objective is not defined on the full pixel domain. Equation (9) divides by \\tilde G_x, and Eq. (30) divides by \\partial_h G_x(h). For an image whose spectral energy is supported only at rho = 0, \\tilde G_x = 0 and \\partial_h G_x(h) = 0 for every h, so Eq. (25) has no unique root and Algorithm 1 computes h_t = 0 and \\dot h_t = q'(t) \\tilde G_x / D_t = 0/0, producing a NaN velocity target in Eq. (16). The manuscript states monotonicity holds 'for a nondegenerate spectrum' but never defines that condition, and Algorithm 1 does not specify a fallback branch when \\tilde G_x = 0. Please add a precise definition of the nondegenerate condition and an explicit branch for degenerate spectra, or clearly restrict the method to the nondegenerate case; as written, the claim that EG-FM requires no adaptation of training data is not true for a valid, if atypical, input class.","section":"Section 3.2, Section A.2, Algorithm 1"},{"comment":"The headline FID gains partly rest on hyperparameters selected on the same benchmark with the same FID metric: sigma0 = 3.5 is chosen by the sweep in Figure 5, the release clock is chosen by the sweep in Table 5, and the CFG scale is retuned for EG-FM in Figure 7 while no analogous CFG sweep is reported for the baseline. Because the reported improvements over baselines are small (e.g., 1.63 vs. 1.69 for DeCo-XL/16 and 1.45 vs. 1.54 for PixelDiT-XL/16 in Table 1), the controlled comparison would be stronger if the baseline CFG were also swept or if the conclusions were shown to be stable across a range of hyperparameter choices. In addition, no confidence intervals or multiple-seed results are reported, so it is difficult to assess whether the observed differences exceed run-to-run noise.","section":"Section 4.4, Figure 7"},{"comment":"The claim of consistently improved generation quality is not uniformly supported by the metrics reported. In Table 1, DeCo-XL/16 + EG-FM has a worse sFID (4.78 vs. 4.59) and worse IS (300.1 vs. 304.0) than the baseline, and HyperDiT-H + EG-FM has substantially lower IS (293.4 vs. 306.5) and lower precision (0.78 vs. 0.80). The paper's central evidence is FID, but the abstract and conclusion speak more generally about generation quality. Either temper the claim to FID, or provide evidence that the other metric regressions are within evaluation noise rather than systematic quality losses.","section":"Table 1"}],"minor_comments":[{"comment":"The text says 'until h(x,1) = 0 reaches the full-spectrum endpoint y_0(x) = x', but y_0(x) was already defined as the initial low-pass endpoint at t = 0; the full-spectrum endpoint at t = 1 should be y_1(x) = x. This overloading of y_0 is confusing.","section":"Section 3.1"},{"comment":"The sampling protocol paragraph says Standard-FM and EG-FM runs 'differ neither in solver nor in sampling budget', but Section 4.4 reports that EG-FM uses a retuned CFG scale of 2.55. Clarify that CFG scale is a hyperparameter and is therefore part of the tuning protocol, not part of the solver or sampling budget.","section":"Section 4.1 and Section 4.4"},{"comment":"The preamble to Algorithm 1 says it computes h_t and \\dot h_t 'for t in (0,1) and \\tilde G_x > 0', but the algorithm's require list does not include \\tilde G_x > 0 and the body has no conditional branch for the zero case. Add this condition to the require list and handle the zero case explicitly.","section":"Algorithm 1 and Appendix C.5"},{"comment":"There are several typos and repeated sentences: 'the high-frequency single of x' in Section 3.1, 'as a effective design' in the Conclusion, and a repeated sentence in Section C.4 beginning 'Across all matched runs'. These should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution with a clean formal derivation and an interesting, inexpensive modification to flow matching. The main issues are fixable: add a degenerate-spectrum branch and definition, report confidence intervals or multiple seeds, and either sweep the baseline CFG or soften the language about consistent quality gains. Given the small FID differences, I would not recommend acceptance before these points are addressed. The scope fits the journal; the novelty is adequate; there is no indication of a citation or disclosure problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: EG-FM is a genuine advance in the narrow but active area of pixel-space flow matching. The idea is simple: instead of interpolating noise to a fixed clean image, the endpoint is a heat-kernel-filtered version of the image, and the heat time h(x,t) is chosen per image so that the recovered spectral energy fraction matches a global clock q(t). That construction is new as far as I can tell from the citations, and the derived velocity target v_t = y_t - epsilon + t d_t y_t is exact. The math checks out: the implicit derivative in Eq. (29) is correct, and Proposition 1's endpoint stability is right for the smootherstep clock. The paper also does the right empirical thing: controlled comparisons on three backbones, matched compute, two FID evaluators, and an honest x-prediction extension that gains only 0.04 FID with an explanation of why.\n\nThe results are plausible and consistently positive: PixelDiT-XL goes from 2.36 to 1.99 FID at 80 epochs and to 1.45 at 600, beating the 800-epoch baseline by 0.09; DeCo and HyperDiT also improve. That's not a huge margin, but it's consistent across architectures and it transfers to 512 and to text-to-image. The training overhead is negligible.\n\nSoft spots, in rough order of importance. First, the headline improvements partly rely on hyperparameters selected on the same benchmark: sigma0 (Figure 5), the release clock (Table 5), and CFG (Figure 7) are all FID-tuned. The baselines use their original CFG, so part of the gain could be a tuned-CFG effect. They should either tune CFG for the baselines or show the method's advantage at matched guidance. Second, the degenerate-spectrum gap is real: for an image with energy only at DC, eGx = 0 and D_t = 0, so Eq. (30) is 0/0 and Algorithm 1 outputs NaN. Natural ImageNet images are broadband, so this won't show up in practice, but the paper claims the method needs no data adaptation, and the training target is not defined on that input class. A one-line fallback (e.g., standard FM if eGx is below a threshold) would close it. Third, the 512 HyperDiT result has no same-backbone baseline, which weakens that comparison. And there are no confidence intervals on FID, which is minor but worth adding.\n\nOverall, this is a solid paper. The core idea is new, the derivation is clean, and the experiments are mostly well executed. It deserves a serious referee. My recommendation: send it out, and ask for the baseline tuning, the degenerate-spectrum fallback, and the missing HyperDiT 512 baseline.","headline":"A clean, genuinely sample-adaptive path-design paper for pixel-space flow matching; empirical gains are consistent but partly FID-tuned, and a degenerate-spectrum edge case needs a fallback.","tokens_in":21424,"tokens_out":4304,"would_cite":true,"duration_ms":41426,"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":"Moving the flow-matching endpoint to a heat-filtered image reaches FID 1.45 on ImageNet 256.","keywords":["flow matching","pixel-space generation","heat kernel","spectral energy scheduling","coarse-to-fine generation","moving endpoint","ImageNet generation","text-to-image generation"],"falsifier":"Train EG-FM on a dataset of nearly constant, low-texture images where the recovered-energy function is flat; if the heat-time root or its derivative becomes non-unique or unstable, the velocity target is undefined for those inputs. Alternatively, train with the same moving endpoint but high-to-low frequency release; if FID does not worsen, the coarse-to-fine ordering is not the cause of the reported gains.","tokens_in":20373,"feed_emoji":"🖼️","tokens_out":10178,"duration_ms":84287,"temperature":0.7,"pith_summary":"This paper argues that standard flow matching leaves the spectral order of image generation implicit, and replaces the fixed clean endpoint with a moving, heat-kernel-filtered endpoint so that low-frequency structure is established before high-frequency detail. The endpoint's heat time is chosen per image so that every sample recovers the same fraction of its missing spectral energy at the same training time. The paper derives the exact velocity target for this moving endpoint and shows that it improves FID and training speed across several pixel-space backbones. The paper concludes that pixel-space generative models can be made faster and higher-quality without changing architecture, data, or sampling budget.","feed_headline":"Moving endpoint in flow matching reaches FID 1.45 on ImageNet 256","feed_subtitle":"Swapping the fixed target for a heat-kernel-filtered one enforces coarse-to-fine learning across backbones.","key_machinery":"The load-bearing object is the moving spectral endpoint $y_t(x)=\\mathcal{F}^{-1}(R(h(x,t),\\rho)\\,\\hat{x}(\\rho))$ with heat-kernel response $R(h,\\rho)=\\exp(-(\\pi\\sigma_0)^2 h \\rho^2)$, a smooth nested family of low-pass images. Energy-guided scheduling sets $h(x,t)$ by solving $G_x(h(x,t))/\\tilde{G}_x=q(t)$, where $G_x(h)=\\sum_\\rho E(\\rho)\\,(R(h,\\rho)-R(1,\\rho))^2$ and $\\tilde{G}_x=G_x(0)$; this makes the same training time represent the same fraction of recovered spectral energy for every image. The argument is carried by the resulting velocity target $v_t = y_t(x) - \\epsilon + t\\,\\partial_t y_t(x)$, whose endpoint-motion term uses the implicit derivative $\\partial_t h = \\partial_t q(t)\\,\\tilde{G}_x / \\partial_h G_x(h)$.","core_discovery":"On the paper's own terms, the central discovery is that the generative trajectory of flow matching can be re-targeted by replacing the fixed full-spectrum endpoint $x$ with a moving spectral endpoint $y_t(x)=\\mathcal{F}^{-1}(\\exp(-a\\,h(x,t)\\rho^2)\\,\\hat{x}(\\rho))$, where $a=(\\pi\\sigma_0)^2$ and $h(x,t)$ decreases from $1$ to $0$ along the path. The per-image heat time is defined by equating the recovered spectral energy ratio $G_x(h)/\\tilde{G}_x$ to a global release clock $q(t)$, so the same time $t$ corresponds to the same fraction of released high-frequency energy for every sample. The exact velocity target becomes $v_t = y_t(x) - \\epsilon + t\\,\\partial_t y_t(x)$, with the endpoint-motion term obtained by implicit differentiation of the energy constraint. With this schedule the paper reports FID 1.55 at 200 epochs and 1.45 at 600 epochs on ImageNet 256, improved FID across DeCo, HyperDiT, and PixelDiT, and transfer to 512 resolution and to text-to-image generation.","pith_inferences":["Beyond the paper, the energy-equalization clock could be adapted to any signal with a well-defined Fourier spectrum, such as audio or video, where the coarse-to-fine ordering would have to be given a temporal analogue.","The release clock $q(t)$ is chosen by hand; learning it or making it per-sample is a natural next step that the paper does not pursue, and Figure 5 suggests the benefit is sensitive to how the frequency release is shaped.","The sensitivity to the filter strength $\\sigma_0$ suggests a principled estimator of filter strength from dataset spectral statistics could remove the main tuning parameter while preserving the gain."],"forward_implications":["Pixel-space flow matching can be trained to a given FID in substantially fewer epochs: PixelDiT-XL reaches FID 1.55 at 200 epochs and 1.45 at 600, below the baseline's 1.61 at 320 epochs.","The moving-endpoint path transfers across three architecturally distinct backbones and across model sizes, so the gain is tied to the trajectory rather than to one network design.","The same trajectory transfers across resolutions and conditioning: fine-tuning EG-FM checkpoints at 512 yields FID 1.58 after 40 epochs, and text-to-image training raises GenEval from 0.78 to 0.85 with DPG-Bench 83.9.","Training overhead is negligible and inference cost is unchanged, since at inference the model evaluates the learned velocity with the same solver and number of function evaluations as standard flow matching."],"supporting_citations":[{"why":"Supplies the flow-matching conditional path and velocity objective that the moving-endpoint construction modifies.","marker":"Lipman et al. 2022"},{"why":"Defines the fixed-endpoint rectified-flow interpolation whose endpoint EG-FM replaces.","marker":"Liu, Gong, and Liu 2023"},{"why":"Provides the scalable-interpolant flow matching baseline whose fixed clean endpoint is the reference point for the proposed change.","marker":"Ma et al. 2024"},{"why":"Introduces inverse heat dissipation as the coarse-to-fine generative prior that motivates the heat-kernel endpoint.","marker":"Rissanen, Heinonen, and Solin 2022"},{"why":"Establishes blurring and low-pass schedules as a generative prior aligned with the heat-kernel filtering used here.","marker":"Hoogeboom and Salimans 2022"},{"why":"Supplies the DeCo frequency-decoupled pixel diffusion backbone that EG-FM is evaluated on.","marker":"Ma et al. 2025"},{"why":"Supplies the HyperDiT backbone used for the 256 and 512 resolution experiments.","marker":"He et al. 2026b"},{"why":"Supplies the PixelDiT backbone and the text-to-image training setup that EG-FM transfers to.","marker":"Yu et al. 2025"}],"fun_headline_variants":["Moving target in flow matching hits FID 1.45 on ImageNet","Energy-guided flow matching: coarse-to-fine endpoints cut FID to 1.45","Re-targeted flow matching reaches FID 1.45 with fewer epochs","EG-FM: moving endpoints give FID 1.45 on ImageNet 256","Flow matching with moving endpoints: FID 1.45 and faster training"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes every image has enough high-frequency energy that a unique per-image heat time can always be found by bisection, and that releasing low frequencies before high frequencies is the right generative order.","fun_headline_variants_meta":{"raw":{"variants":["Moving target in flow matching hits FID 1.45 on ImageNet","Energy-guided flow matching: coarse-to-fine endpoints cut FID to 1.45","Re-targeted flow matching reaches FID 1.45 with fewer epochs","EG-FM: moving endpoints give FID 1.45 on ImageNet 256","Flow matching with moving endpoints: FID 1.45 and faster training"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001146,"raw_usage":{"total_tokens":4817,"prompt_tokens":1069,"completion_tokens":3748,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":3651}},"tokens_in":685,"tokens_out":3748,"duration_ms":25582,"temperature":1.0,"reasoning_tokens":3651,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T04:25:58.190009+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train EG-FM on a dataset of nearly constant, low-texture images where the recovered-energy function is flat; if the heat-time root or its derivative becomes non-unique or unstable, the velocity target is undefined for those inputs. Alternatively, train with the same moving endpoint but high-to-low frequency release; if FID does not worsen, the coarse-to-fine ordering is not the cause of the reported gains.","supporting_citations":[],"review_version":2}