{"id":"8e3b260e-2d85-4ad5-af77-d30b76d11553","arxiv_id":"2506.15864","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Enforcing the exact boundary values of the velocity field in Rectified Flow improves generation FID by about 8 percent on ImageNet and stabilizes stochastic sampling.","lead":"This paper adds simple boundary conditions to Rectified Flow velocity fields so the model exactly predicts the data at the end time. The authors report lower FID scores on CIFAR-10 and ImageNet for both deterministic and stochastic sampling.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Single-seed FID gain may be training noise; the reader's centering critique targets the wrong boundary, while the mode-sampling anomaly signals a possibly nonstandard baseline.","rationale":"The reader's weakest_assumption is not the weakest link. For the interpolation X_t = tX1 + (1−t)X0 with X0⊥X1 and E[X0]=0, v*(x,1)=E[X1−X0|X1=x]=x−E[X0]=x, independent of E[X1]. Hence forcing v(x,1)=x is correct even if VAE latents are not centered; the reader's centering concern would only apply to the mask-based left boundary, where C=E[X1] is needed and is not specified. The subtraction-based model (the best FID) does not involve C, so the reader's stated concern is not load-bearing for the headline claim.\n\nThe genuine load-bearing concern is empirical. Transformers are trained once per configuration; no standard errors, seeds, or code are given. FID differences of 0.5 on ImageNet-256 for DiT-B-class models can be within run-to-run variation, especially when comparing differently parameterized networks. The mode-sampling baseline is the strongest internal signal: the paper describes mode sampling as concentrating time steps at the boundaries, whereas the cited SD3 method samples from a logit-normal distribution that concentrates near its mode (usually 0.5). If the authors' implementation pushes mass to the edges, their baseline would be substantially worse than a standard RF, making the boundary method's gain look larger. This does not prove the claim false, but it means the experimental evidence, as reported, is insufficient to separate a true boundary-condition benefit from training stochasticity and a weak baseline.\n\nThe proposed test—multiple seeds with mean±std FID—directly addresses this. If the improvement is consistent across seeds, the claim is credible; if it disappears or flips, the paper's central claim fails. The secondary check on mode sampling would clarify whether the comparison set is fair. A machine-checked proof is not available, and the method is simple enough that a parameter-free theoretical guarantee is not the issue; the bottleneck is reliable measurement.","tokens_in":14047,"tokens_out":19405,"duration_ms":190684,"concrete_test":"Run the ImageNet 256×256 experiment of §5.1 with the identical DiT-B/2 recipe for vanilla RF and Subtraction-based Boundary RF Model, using at least 3 independent random seeds per method, and report mean±std FID for both Euler and SDE samplers. The central claim is supported only if the FID advantage (and ideally IS/precision) persists outside the seed-to-seed standard error. A secondary check: verify the SD3 mode-sampling implementation against [7]; if it does not match, repeat the comparison with a correctly implemented scheduler.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central empirical claim is that enforcing v(x,1)=x improves ImageNet FID (6.32 vs 6.87, 8.01%). Mathematically, v*(x,1)=x follows from E[X0]=0 plus independence of X0 and X1; it does not require centered data latents. The reader's weakest_assumption is thus misdirected: the data mean C=E[X1] enters only the mask-based left boundary v(x,0)=C−x, and the paper never states how C is computed, but the headline subtraction-based variant does not use C. The real load-bearing weakness is statistical. The reported FID numbers are single training runs with no error bars, seeds, or released code. ImageNet-256 FID for DiT-scale models typically varies by several tenths across seeds; a 0.55 gap may lie within noise. This is compounded by the anomalous 'RF with Mode Sampling' result (7.48 vs 6.87 Euler). Mode sampling in SD3 concentrates timestep sampling near a mode (e.g., sigmoid-normal around 0.5), not 'towards the boundaries' as the paper states; the large degradation suggests their scheduler implementation is mis-tuned, implying the vanilla baseline itself may be under-optimized. If so, the boundary-enforcement gain might be an artifact of comparing against a weak baseline rather than a genuine improvement. Finally, the subtraction parameterization differs from the mask-based one not only in boundary enforcement but also in being a 2-pass residual network; the paper's ablations do not separate these factors.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper identifies that neural-network parameterizations of Rectified Flow velocity fields do not automatically satisfy the theoretical boundary conditions v(x,1)=x and v(x,0)=C-x, and argues that this violation degrades deterministic and especially stochastic sampling. It proposes two reparameterizations: a mask-based model that explicitly enforces both boundaries through scalar schedules f,g,h, and a subtraction-based model of the form v(x,t)=x+m(x,t)-m(x,1) that enforces the right boundary by construction. Experiments on CIFAR-10, ImageNet 256x256, ImageNet 512x512, and DiT-B/L/XL configurations report consistent FID improvements over a vanilla RF baseline, with the headline result being ImageNet 256x256 FID of 6.32 for Subtraction-based Boundary RF versus 6.87 for vanilla RF under Euler sampling.","tokens_in":14410,"tokens_out":6406,"duration_ms":60901,"significance":"The mathematical derivation of the boundary conditions from the conditional-expectation definition of the optimal velocity is correct and is presented cleanly in Section 3.1. The proposed modification is minimal, architecture-agnostic, and does not introduce fitted constants to match the reported FID numbers; the subtraction-based variant is especially simple. If the reported gains are reproducible, the method could be a useful drop-in improvement for training rectified flows and flow matching, and the stabilization of stochastic sampling is a plausible and interesting mechanism. However, the empirical foundation is currently thin: the headline difference is a single-run comparison without error bars, the mode-sampling baseline is both misdescribed and anomalously weak, and the subtraction variant is confounded with an extra forward pass. These issues must be resolved before the central claim can be accepted.","major_comments":[{"comment":"The central empirical claim rests on single-run FID comparisons with no error bars, multiple seeds, or bootstrap intervals. ImageNet-256 DiT-scale FID is known to vary by several tenths across seeds, so the reported 6.32 versus 6.87 gap may be within run-to-run noise. The paper should provide at least three seeds (or a statement of variance) for the main comparisons, or otherwise justify that the difference is statistically meaningful.","section":"Section 5.2, Table 1"},{"comment":"The description of mode sampling is factually wrong and the reported result is anomalous. The paper states that mode sampling concentrates probability 'towards the boundaries of the timestep range (i.e., near 0 and 1)', whereas the cited SD3 method [7] uses logit-normal timestep sampling that concentrates near a central mode around t=0.5. Moreover, the reported RF-with-Mode-Sampling FID (7.48) is far worse than the vanilla RF baseline (6.87), which is not expected for a properly tuned training schedule. This suggests either a misimplementation of the baseline or an under-optimized vanilla setting, and it undermines the claim that boundary enforcement is the cause of the improvement.","section":"Section 5.2, Table 1 (RF with Mode Sampling)"},{"comment":"The claim that enforcing v(x,1)=x 'prevents the score function from diverging' is not rigorously justified. Tweedie's formula gives ∇log ρ_t(x) = (t v(x,t) − x)/(1 − t); at t=1 the numerator and denominator both vanish when v(x,1)=x, leaving a 0/0 limit whose finiteness depends on the rate at which v(x,t) approaches x, not merely on the boundary value. For example, v(x,t)=x+√(1−t) satisfies the boundary condition but still yields a divergent score. The stability argument needs either a quantitative bound on the score or an empirical measurement of score norms near t=1.","section":"Section 3.4, Eq. (5)"},{"comment":"The mask-based model uses the constant C=E[X1] in its left boundary condition v(x,0)=C−x, but the paper never states how C is obtained in practice: whether it is the exact population mean, a training-set estimate for the ImageNet VAE latents, a running batch estimate, or simply set to zero. This omission prevents reproduction of the mask-based results and matters because VAE latents need not be centered; an incorrect C would systematically bias the left boundary and could affect the reported mask-based FID numbers.","section":"Section 3.2 and Section 5.3"},{"comment":"The subtraction-based model differs from the mask-based model not only in enforcing the right boundary but also in being a two-pass residual network with an additional forward pass at t=1. The ablation table labels the subtraction row '2-Pass Model' but does not include a two-pass model without boundary enforcement (e.g., v(x,t)=x+m(x,t) evaluated with two passes or analogous capacity increase). Consequently, the additional improvement of the subtraction variant over the mask variant (6.32 vs 6.63) cannot be attributed solely to boundary enforcement; it may reflect the extra computation or the residual parameterization rather than the boundary condition.","section":"Section 5.3, Table 2"}],"minor_comments":[{"comment":"Reporting FID differences as percentages ('8.01% improvement') is nonstandard and can be misleading; absolute FID changes with uncertainty would be more informative.","section":"Abstract and Section 5.2"},{"comment":"The text says 'we maintain consistent random seeds' but does not specify how many seeds are used or whether the reported results are from a single run. Please clarify whether 'consistent seeds' means one seed per method or multiple seeds averaged.","section":"Section 5.1, Appendix A.1"},{"comment":"The sentence 'across both datasets and evaluation metrics' appears twice in close succession; remove the duplicate.","section":"Section 5.2, text after Table 1"},{"comment":"The subtraction-based variant doubles the inference cost due to the extra m(x,1) forward pass, yet the main tables report no runtime or throughput comparison. Since the paper emphasizes practical ease, a wall-clock comparison would help readers weigh the trade-off.","section":"Section 3.3 and Section 5.2"},{"comment":"For CIFAR-10 the model is listed as DDPM++ while the main text says all models are DiT-based; please clarify which architecture is used for the CIFAR-10 experiments.","section":"Appendix A.1, Table 5"}],"recommendation":"major_revision","confidential_remarks":"The paper's mathematical core is sound and the proposed modification is simple and potentially useful, but the experimental evidence is not yet convincing. The mode-sampling mischaracterization and the anomalous baseline result are the most serious warning signs because they suggest the vanilla RF baseline may not be representative. I would encourage the editor to request additional experiments with multiple seeds, a corrected and better-tuned mode-sampling comparison, and an explicit statement of how C is computed for the mask-based model. The score-boundedness claim should also be softened or supported. These are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: this paper proposes a tiny parameterization change for rectified flow / flow matching—write the velocity as v = x + m(x,t) − m(x,1), or use a mask that mixes in x at t=1—so the learned velocity exactly satisfies the right boundary condition v(x,1)=x. The boundary conditions are derived correctly from the conditional expectation, and I believe this specific parameterization is new in the flow-matching literature. It is also trivially easy to implement, and the authors show consistent FID improvements across CIFAR-10 and ImageNet at several model sizes and step counts. The FLUX.1 velocity visualization is a nice, honest piece of evidence that boundary violation is a real practical issue even in production-scale models.\n\nThe soft spots are real but not disqualifying. The headline numbers—6.32 vs 6.87 on ImageNet-256—come from single training runs with no error bars or multiple seeds. At DiT scale that gap could be within run-to-run noise, although the fact that the improvement holds across DiT-B/L/XL and 512x512 makes pure luck less likely. I would want code and seeded runs before trusting the magnitude of the gain.\n\nThe larger red flag is the mode-sampling baseline. The paper describes mode sampling as concentrating timesteps near 0 and 1, which is not what SD3 mode sampling does (logit-normal concentrates around the mode, near 0.5). Their \"RF with Mode Sampling\" actually gets worse FID than vanilla RF (7.48 vs 6.87), which is the opposite of what anyone would expect. That suggests either a mis-tuned implementation or a nonstandard variant, and it makes me wonder how carefully the other baselines were tuned. It does not invalidate the main vanilla comparison, but it matters for the paper's credibility.\n\nOne more confound: the subtraction-based variant needs a second forward pass at m(x,1) on every velocity evaluation, so it costs roughly 2x inference. The mask-based variant is single-pass and still improves over vanilla, so the effect is not purely extra compute, but the strongest results are for the 2-pass model and the ablations do not fully separate boundary enforcement from added computation.\n\nThe reader's concern about centering the ImageNet latents is mostly off-target: the right boundary v(x,1)=x follows from E[X0]=0 and independence, not from centering X1. The mean of X1 only enters the left boundary in the mask model, and the paper indeed never states how it computes that constant—a minor gap worth asking about, not a fatal flaw.\n\nBottom line: this is a solid, simple contribution that deserves serious peer review. It is not groundbreaking, but the parameterization is likely useful and the motivation is clean. I would accept it with the expectation of a heavy revision: multiple seeds, error bars, code release, and a properly tuned mode-sampling baseline. For someone working on flow matching or stochastic samplers, this is worth knowing about.\n\nBest,\n[You]","headline":"A simple, plausible fix for a real problem in rectified flow: enforce the terminal velocity v(x,1)=x via a subtraction parameterization; the reported gains are suggestive but rest on single-seed numbers and a suspicious mode-sampling baseline.","tokens_in":14894,"tokens_out":3194,"would_cite":true,"duration_ms":31863,"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":"This paper shows that forcing a Rectified Flow velocity to satisfy v(x,1)=x improves generation quality and stabilizes stochastic sampling.","keywords":["rectified flow","flow matching","boundary conditions","velocity field","stochastic sampling","score function","image generation","generative modeling"],"falsifier":"Compute the per-channel mean of the VAE latent codes on the ImageNet training set. If any channel mean is substantially nonzero, retrain the subtraction-based Boundary RF Model with the latent mean explicitly subtracted from X1, and compare FID against vanilla RF; the claimed improvement should persist only if the boundary constraint matches the data's actual mean.","tokens_in":13885,"feed_emoji":"🎨","tokens_out":6651,"duration_ms":53863,"temperature":0.7,"pith_summary":"Rectified Flow turns sampling into solving an ODE whose velocity field should move noise to data. The paper identifies a hidden failure mode: when the velocity is learned by an unconstrained network, it does not satisfy the endpoint identity v(x,1)=x that the interpolation implies, and this makes the score function used in stochastic samplers blow up near the final time. It proposes two nearly-bulletproof parameterizations that enforce this boundary condition by construction, and reports that on ImageNet 256x256 the better variant drops FID from 6.87 to 6.32 under Euler sampling, with similar gains under SDE sampling. The result matters because it removes a systematic error at almost no implementation cost and makes stochastic sampling stable enough to use larger noise scales.","feed_headline":"Enforcing one boundary condition cuts ImageNet FID from 6.87 to 6.32","feed_subtitle":"Rectified Flow models produce sharper samples when the final-time velocity is forced to match the data.","key_machinery":"The central object is the terminal velocity constraint v(x,1)=x. For the linear interpolation X_t=(1-t)X0+tX1, the conditional expectation v*(x,t)=E[X1-X0|Xt=x] simplifies at t=1 to x-E[X0]=x when X0 has mean zero. The mask-based model writes v(x,t)=g(t)(C-x)+f(t)x+h(t)m_theta(x,t) with g(1)=f(0)=h(0)=h(1)=0 and g(0)=f(1)=1, using trigonometric g, f, h; the subtraction-based model sets v(x,t)=x+m_theta(x,t)-m_theta(x,1), which forces v(x,1)=x exactly regardless of the network. This identity prevents the score estimate nabla log rho_t(x) = (t v(x,t)-x)/(1-t) from diverging near t=1, which is the mechanism the paper credits for stable stochastic sampling.","core_discovery":"In Rectified Flow, the ideal velocity field satisfies v*(x,0)=E[X1]-x and v*(x,1)=x when the noise X0 is zero-mean and independent of the data X1. A neural network trained with the standard flow-matching loss does not obey these constraints at the boundaries, especially at t=1. The paper constructs two parameterizations—a mask-based model and a subtraction-based model—that enforce the right boundary by design, and reports consistent FID gains: on ImageNet 256x256, the subtraction-based model reaches FID 6.32 vs 6.87 for vanilla RF under Euler sampling, with analogous improvements under an SDE sampler. The enforced boundary also keeps the Tweedie score estimate finite as t approaches 1, which the authors argue is why stochastic samplers produce sharper, less over-smoothed images.","pith_inferences":["Editorial extension: The enforced t=1 constraint presumes a centered latent space; if the VAE latents used for ImageNet have nonzero per-channel means, the subtraction model could be systematically biased. A quick check is to subtract the empirical latent mean and see whether the FID gain changes.","Editorial extension: The same boundary analysis could be run for the t=0 endpoint under non-Gaussian or correlated priors, where v*(x,0)=E[X1]-x would no longer hold; a testable extension is whether a learned prior mean restores the benefit.","Editorial extension: The paper's own visualization shows a production text-to-image model also violates the boundary; this suggests the fix may matter more at scale, where endpoint errors accumulate over many steps, but the paper only demonstrates it on class-conditional ImageNet and CIFAR-10."],"forward_implications":["Any Rectified Flow model can be retrofitted with the subtraction parameterization, so a drop-in FID gain should appear on other datasets and architectures.","Because the Tweedie score stays bounded as t approaches 1, stochastic samplers can be run with larger noise scales without over-smoothing, which should translate to fewer required steps at equal quality.","The boundary violation is measurable: computing the mean of ||v(x,1)-x|| on validation data gives a diagnostic for convergence of any trained RF model.","The same enforced boundary should transfer to other flow-matching formulations built on linear or affine interpolation schedules."],"supporting_citations":[{"why":"Defines Rectified Flow and the linear interpolation whose boundary conditions are the paper's target.","marker":"[23]"},{"why":"Establishes the conditional-expectation velocity objective and marginal preservation that the paper builds on.","marker":"[25]"},{"why":"Supplies the Stochastic Curved Euler Sampler used as the main SDE sampler and the context for the Tweedie score formula.","marker":"[24]"},{"why":"The Overshooting sampler whose noise-scale robustness the paper tests against.","marker":"[14]"},{"why":"The text-to-image Rectified Flow baseline and logit-normal time sampling strategy that the paper compares against and adopts.","marker":"[7]"},{"why":"The DiT architecture used for all experiments.","marker":"[29]"},{"why":"Provides an alternative derivation of the same linear-interpolation velocity, supporting the generality of the boundary derivation.","marker":"[1]"}],"fun_headline_variants":["Boundary-corrected Rectified Flow: 8% better FID on ImageNet","Enforced boundary in velocity field improves Rectified Flow sampling","FID 6.32 vs 6.87: Boundary RF Model beats vanilla on ImageNet","Forcing the t=1 velocity boundary sharpens Rectified Flow outputs","Minimal code change enforces boundary, improves Rectified Flow FID"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The enforced right boundary v(x,1)=x is correct only if the noise X0 is zero-mean and independent of X1, and if the data representation is centered; if the latents have a nonzero mean, the constraint is systematically wrong.","fun_headline_variants_meta":{"raw":{"variants":["Boundary-corrected Rectified Flow: 8% better FID on ImageNet","Enforced boundary in velocity field improves Rectified Flow sampling","FID 6.32 vs 6.87: Boundary RF Model beats vanilla on ImageNet","Forcing the t=1 velocity boundary sharpens Rectified Flow outputs","Minimal code change enforces boundary, improves Rectified Flow FID"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000338,"raw_usage":{"total_tokens":1821,"prompt_tokens":851,"completion_tokens":970,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":867}},"tokens_in":467,"tokens_out":970,"duration_ms":8553,"temperature":1.0,"reasoning_tokens":867,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:29:59.485395+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the per-channel mean of the VAE latent codes on the ImageNet training set. If any channel mean is substantially nonzero, retrain the subtraction-based Boundary RF Model with the latent mean explicitly subtracted from X1, and compare FID against vanilla RF; the claimed improvement should persist only if the boundary constraint matches the data's actual mean.","supporting_citations":[{"cited_title":"Flow straight and fast: Learning to generate and transfer data with rectified flow","cited_arxiv_id":null,"evidence_quote":"Establishes the conditional-expectation velocity objective and marginal preservation that the paper builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Stochastic Curved Euler Sampler used as the main SDE sampler and the context for the Tweedie score formula."}],"review_version":1}