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Velocity Scaling in Flow Matching

Youssef Saied, François Fleuret

arXiv:2610.10823v1cs.CV

TL;DR

MSE training does not create a velocity-magnitude deficit; the paper instead identifies population time lag in sampled states and uses that lag to select velocity gains. At NFE 25, the lag-based gain lowers estimated FID from 28.0 to 12.21.

  • Problem

    Prior work attributed velocity scaling’s generation-quality improvement to MSE-induced velocity underestimation, but the paper finds the population MSE solution exact and the learned field without systematic norm bias.

    Source: §1 · p. 1, §8 · p. 9

  • Method

    The paper measures population time lag and selects a gain for each model and sampling setting without retraining, online time estimation, or an image metric.

    Source: §5.1 · p. 4

  • Results

    At NFE 25, lag-based gain selection gives an estimated FID of 12.21, down from 28.0; lag-based gains also improve FID under coarse sampling across ImageNet models.

    Source: §1 · p. 1, §5.4 · p. 6

  • Takeaways & Limitations

    Under stochastic sampling, the changed-state response to velocity scaling produces most of the FID reduction, while the lowest-FID gain depends on the sampling and evaluation setting.

    Source: §8 · p. 9

    Across twenty tested unguided stochastic ImageNet settings, the lowest-FID gain exceeds the lag-based gain, including with finer integration when measured lag is small.

    Source: §7.1 · p. 8

Abstract

from arXiv · show

Scaling a learned flow-matching velocity field $v_\theta$ by a gain $\gamma(t)$ was recently shown to greatly improve generation quality. Prior work argued that velocity fields trained with mean-squared error (MSE) systematically underestimate velocity magnitude and that scaling corrects this error. We show that MSE training does not create a velocity-magnitude deficit. We find instead that velocity scaling reduces population time lag: sampled states at model time $t$ resemble training states from an earlier time. Velocity scaling and moving model time back are two ways to address this population time lag. Across architectures and model sizes, measuring population time lag and using it to select a gain greatly improves generation quality, reducing FID from 28.0 to 12.2 (estimated by linear interpolation between FID measurements at neighboring gains) on ImageNet-256 at NFE 25 without guidance.

1 INTRODUCTION

The paper rejects MSE-induced velocity underestimation as the explanation for gains from velocity scaling and instead links scaling to population time lag. A lag-selected gain improves coarse-sampling image quality, while a further gain can improve FID beyond lag correction.

1 INTRODUCTION

  • MSE training does not create a velocity deficit, and the multiplier that best matches pretrained SiT predictions to target velocities averages 0.999.Prior work attributed scaling's improvement to an MSE-induced deficit; this paper reports that the population MSE minimizer transports the source distribution to the data distribution exactly.Source: §1 · pp. 1–2
  • Sampled states at model time t resemble genuine training states from an earlier time, and this population time lag increases as NFE decreases.Velocity scaling advances lagging states toward the time supplied to the next model call and empirically eliminates most measured lag.Source: §1 · p. 1
  • At NFE 25, lag-based gain selection gives an estimated FID of 12.21, down from 28.0.The estimate uses linear interpolation between FID measurements at neighboring gains.Source: §1 · p. 1
  • The lag-based gain is not always the FID-optimal gain: a sweep found FID 8.7 at gain 1.15, the lowest among tested gains.One fixed factor per model describes the additional gain beyond lag correction reasonably well at NFE 50, 100, and 250.Source: §1 · p. 1
  • In matched trajectories, constant gain 1.15 shifts the coarse NFE-25 path toward the finer path’s object-development timing.Within each example, rows share the class label, initial latent, and Brownian path.Source: Figure 1 · p. 2

2 RELATED WORK

The related work frames flow matching as velocity-field transport and reviews approaches to state–time mismatch through training or sampling changes. The paper positions velocity scaling as a sampling-time correction that can be selected per model and setting without retraining or an image metric.

2.1 FLOW MATCHING FOR IMAGE GENERATION

  • Flow matching regresses a velocity field against target velocities from prescribed paths, and SiT places a scalable transformer in a framework supporting flow and diffusion objectives and deterministic or stochastic sampling.DiT introduced the scalable transformer backbone; REPA improved SiT training by aligning hidden representations with pretrained visual-encoder features.Source: §2.1 · p. 2

2.2 STATE–TIME MISMATCH

  • Training corrections expose models to perturbed or earlier-time states, but a single training-time correction must cover differing NFE, solver, and guidance settings or require further training.Input Perturbation, MixFlow, and DEFAR model accumulated sampling errors through different training procedures.Source: §2.2 · p. 2
  • Sampling corrections include scaling predicted noise or estimating effective diffusion time from within-image pixel and channel variance before the next model call.Epsilon Scaling selects its factor through an FID search; Time-Shift Sampler supplies the inferred time to the next call.Source: §2.2 · p. 3
  • Velocity scaling addresses state–time mismatch by advancing a lagging state, with a separate correction per model and sampling setting selected without retraining, online time estimation, or an image metric.The paper also finds that a well-chosen gain can improve sample quality in ways lag correction alone does not explain.Source: §2.2 · p. 3

3 SETUP

The setup distinguishes the learned field used at sampling from the population field of the linear flow-matching path. It applies a scalar gain to velocity predictions and evaluates scaling across multiple model families and sampling settings.

3 SETUP

  • For t in [0, 1], the linear training path runs from source noise X0 to data X1, with target velocity U = X1 − X0.The setup distinguishes the population minimizer of the flow-matching loss from the learned velocity field vθ used during sampling.Source: §3 · p. 3
  • During sampling, velocity scaling multiplies the model’s velocity prediction by γ(t), not the state or stochastic-noise terms.With γ(t) = 1 sampling is unscaled; constant gain uses γ(t) = c for all t.Source: §3 · p. 3
  • The experiments cover deterministic and stochastic sampling across SiT sizes, REPA-SiT, an ADM U-Net, and a decoder-free NCSN++ model trained on CIFAR-10.Source: §3 · p. 3

4 NO VELOCITY DEFICIT

The paper argues that the apparent gap between target-velocity norms and the population field reflects conditional target variation, not a missing velocity. Its finite-model test finds no systematic underscaling on genuine training-path states, leaving the generation-quality effect to be explained.

4 NO VELOCITY DEFICIT

  • The conditional mean velocity is the exact population field transporting the source distribution to the data distribution, despite positive conditional target variance.The variance term reflects valid training trajectories crossing at a state, rather than missing velocity; in a Gaussian example, gain one gives exact transport while other gains change endpoint covariance.Source: §4 · pp. 3–4
  • For pretrained SiT, the scalar multiplier minimizing held-out flow-matching error averages 0.999 and stays near one across time.The test fit the multiplier at 20 time values using 16,000 ImageNet image–noise pairs; the fitted scalar also matches predicted velocities to the population field on genuine training-path states.Source: §4 · p. 4
  • These population and finite-model results leave open why velocity scaling improves generation quality, motivating an examination of sampled-state mismatch.Source: §4 · p. 4

5 POPULATION TIME LAG

The paper identifies population time lag as a mismatch between generated states and their model time, then compares model-time correction with velocity scaling and selects gains by measuring lag. Lag-based gains reduce measured lag and improve ImageNet FID, while the CIFAR-10 model has little lag and a lowest-FID gain of one.

5.1 MEASURING POPULATION TIME LAG

  • Population time lag means generated states at model time t resemble genuine training-path states from an earlier time, and it increases as NFE decreases.A probe estimates path time from genuine states; subtracting its baseline expectation on genuine states prevents intrinsic probe bias from counting as sampling lag.Source: §5.1 · p. 4

5.2 MODEL-TIME CORRECTION

  • Model-time correction replaces the time passed to the network with a lag-adjusted time while leaving the integration clock unchanged, and improves FID without scaling velocity.At NFE 50, FID falls from 13.8 to 8.8; for ADM U-Net it falls from 22.7 to 16.7, while constant gain 1.075 reaches 13.2.Source: §5.2 · p. 5

5.3 VELOCITY SCALING AND POPULATION TIME LAG

  • To first order, velocity scaling advances a lagging state along its learned velocity so the next model call receives a state better matched to its time input.Scaling empirically eliminates most of the measured lag.Source: §5.3 · p. 5

5.4 SELECTING THE LAG-BASED GAIN

  • The lag-based gain is the constant gain that minimizes mean squared population time lag along sampled trajectories, selected separately by NFE and model without an image-quality metric.The search uses 256 trajectories per candidate and can calibrate at 0.01 resolution.Source: §5.4 · p. 5
  • Lag-based gain selection improves FID across ImageNet models, recovering an estimated 49% of the reduction from the lowest-FID gain across twenty settings and 81% at NFE 25.The estimate falls with finer sampling; CIFAR-10 has little lag and its lowest-FID gain is one.Source: §5.4 · p. 6

6 CHANGED-STATE RESPONSE

The analysis separates the direct extra velocity push from the response to evaluating the model at moved states. Controls indicate that the changed-state response accounts for most of the benefit under stochastic sampling, while the direction of the push also matters.

6 CHANGED-STATE RESPONSE

  • The trajectory difference separates a direct extra push parallel to the unscaled velocity from a changed-state response caused by evaluating the model at the moved state.The comparison uses deterministic Euler trajectories with the same initial state and time grid.Source: §6 · p. 6
  • Under stochastic sampling, the frozen-field control achieves only 22.4% to 24.9% of the FID reduction from full velocity scaling, so most of the benefit requires model re-evaluation at moved states.Under deterministic sampling, the control reaches 53.9% for SiT and 59.0% for ADM U-Net.Source: §6 · p. 6
  • The radial control worsens FID despite re-evaluating the model at moved states, indicating that the direction of the extra push matters.It replaces the extra velocity with a radial displacement rather than the predicted velocity direction.Source: §6 · p. 6

7 GAIN ACROSS SETTINGS

The gain favored by FID varies with the sampling and evaluation setting rather than being an intrinsic property of the learned field. Lag correction accounts for part of the gain, while a fixed per-model multiplier roughly captures additional FID-favored gain across NFEs.

7 GAIN ACROSS SETTINGS

  • The lowest-FID gain shifts with solver accuracy, guidance strength, and image metric, so it is not an intrinsic property of the learned field.The Figure 3 panels compare NFE, solver, guidance, and dataset settings.Source: §7 · p. 6, Figure 3 · p. 7

7.1 NUMERICAL ACCURACY AND GAIN

  • As NFE rises, the lowest-FID and lag-based gains move toward one; stochastic Heun also lowers both population lag and the lowest-FID gain relative to Euler at fixed NFE.Velocity scaling is not simply a correction for integration error because it changes the sampling path.Source: §7.1 · p. 7
  • The lowest-FID gain exceeds the lag-based gain across all twenty tested unguided stochastic ImageNet settings, even when finer integration leaves measured lag small.Thus, lag explains only part of the gain.Source: §7.1 · p. 8

7.2 GAIN BEYOND LAG CORRECTION

  • One fixed multiplier per model roughly relates the lag-based gain to the lowest-FID gain across NFEs, and at NFE 250 the adjusted gains give FIDs close to the lowest measured FID for every ImageNet model.Each model’s multiplier is fitted at NFE 25 and reused at NFE 250.Source: §7.2 · p. 8
  • Velocity scaling increases endpoint variance proportionally more in directions where real data vary more, and this covariance sharpening is observed at both low and high NFE.The comparison includes SiT-XL/2 and a matched Gaussian flow.Source: §7.2 · p. 9, Figure 5 · p. 9

7.3 SCOPE OF GAIN SELECTION

  • With stronger classifier-free guidance, both gains decrease but the lowest-FID gain falls faster; at NFE 25, FID favors a higher gain than sFID, which also raises precision and lowers recall.Guidance increases velocity magnitude, but the measured increase is too small to explain the full decrease in the lowest-FID gain.Source: §7.3 · p. 9
  • For decoder-free CIFAR-10, the lowest measured FID occurs at gain one across the tested samplers, and the lag-based gain stays close to one at all four NFEs.The model has little population time lag.Source: §7.3 · p. 9

8 CONCLUSION

The conclusion rejects an MSE-induced velocity deficit and instead links scaling to lagging sampled states and their subsequent velocity predictions. It also bounds the explanation: lag reduction explains most of the stochastic-sampling benefit, while the additional FID-favored gain and best gain depend on the setting.

8 CONCLUSION

  • MSE training does not create a velocity deficit: the population minimizer is exact, and the learned field shows no systematic norm bias.In coarse samplers, generated states lag behind their model time; scaling advances them and changes subsequent velocity predictions.Source: §8 · p. 9
  • Under stochastic sampling, the changed-state response produces most of the FID reduction, while the additional FID-favored gain remains roughly proportional to the lag-based gain across tested NFEs.The lowest-FID gain varies with solver, guidance, and image metric.Source: §8 · p. 9

AI USE STATEMENT

The statement discloses AI assistance alongside author oversight. Table 2 organizes questions, findings, and supporting sections, while B.1 explains why residual target variation does not prevent exact transport.

AI USE STATEMENT

  • AI tools assisted hypothesis development, interpretation, experiments, derivations, code, related work, drafting, and figure code; authors directed research and checked claims, derivations, and code.Source: AI USE STATEMENT · p. 10

A QUESTIONS AND MAIN FINDINGS

  • Table 2 organizes research questions, main findings, and the sections containing supporting evidence.Source: Table 2 · p. 13

B.1 RESIDUAL TARGET VARIATION AND EXACT TRANSPORT

  • The population MSE field averages path-specific targets at each state and time, where residual variation has conditional mean zero.Source: §B.1 · p. 14
  • With a unique continuity-equation solution, v∗ transports the training distribution exactly; residual target variation is not a missing velocity component.Source: §B.1 · p. 14

A Gaussian path makes the general result concrete. Its residual variance is positive. Despite this residual variance, gain one gives exact transport. Scaling changes the endpoint distribution.

This section examines how velocity scaling affects endpoint distributions and population time lag, using an exact Gaussian population field and image-generation settings. It also evaluates lag-based time correction and gain selection across ImageNet and CIFAR-10, alongside magnitude-aware training.

A Gaussian path makes the general result concrete. Its residual variance is positive. Despite this residual variance, gain one gives exact transport. Scaling changes the endpoint distribution.

  • For the Gaussian path, residual covariance is positive definite and gain one follows the full training path to target covariance Σ; other gains generally change the endpoint covariance.Gain γ changes each covariance eigenvalue from λ to λγ, so scaling changes an already correct transport map rather than restoring residual target variation.Source: A Gaussian path makes the general… · p. 14

B.3 MAGNITUDE-AWARE TRAINING CHANGES THE POPULATION TARGET

  • Magnitude-aware training inflates the conditional mean when m > ∥µ∥ rather than recovering residual target variation; when µ = 0, it supplies no direction.The passage concludes that magnitude-aware training changes the population target, while its effect on image quality is a separate question.Source: §B.3 · p. 15

D.1 PROBE ARCHITECTURE AND TRAINING

  • The probes estimate path time from genuine states, and population averaging avoids correcting individual samples for readings that mix time with ordinary state variation.For ImageNet, the probe also uses class labels; unconditional CIFAR-10 omits them.Source: §D.1 · p. 18

D.2 DEFINING AND VALIDATING POPULATION TIME LAG

  • The SiT raw-state probe tracks known path-time reductions and responds much less to equally large perpendicular perturbations.This supports using the probe to measure population time lag rather than arbitrary off-path motion.Source: §D.2 · p. 19, Table 6 · p. 19

D.3 COARSE SAMPLING AND MISSING REFINEMENT

  • At t = 0.5, Euler states had mean estimated time 0.254 versus 0.493 for genuine states, a −0.240 difference, consistent with missing sample-specific data structure.The paired 95% bootstrap interval for the Euler-minus-genuine difference was [−0.241, −0.238].Source: §D.3 · p. 20

D.4 MODEL-TIME CORRECTION AND SCALE CONTROL

  • At NFE 50, scale-normalized model-time correction reduced FID from 13.8 to 8.6, showing that the timing effect is not just a norm readout.The correction-minus-nominal FID contrast was −5.2 [−5.3, −5.0].Source: §D.4 · p. 20

D.5 MODEL-TIME CORRECTION TRANSFERS TO AN ADM U-NET

  • Model-time correction was 3.5 FID points worse than velocity scaling at gain 1.075 on ADM U-Net.The correction-minus-gain FID contrast was +3.5 [+3.4, +3.7].Source: Table 7 · p. 20, §D.5 · p. 20

E.1 DIRECT SCALAR LAG CALIBRATION

  • Lag-based gain selection minimizes the average measured lag across post-update times, then evaluates each fixed selected gain on separate paired trajectories.Gain intervals condition on the trained probe and genuine-state baseline, excluding uncertainty from probe retraining and optimization between measured candidates.Source: §E.1 · p. 21
  • Direct scalar lag calibration covered six models at NFE 25, 50, 100, and 250, using lag-based gains selected on calibration paths and comparing unscaled, scalar, and lowest-FID gains.The calibration paths were distinct from the fresh paths used in validation.Source: Table 8 · p. 22

E.2 SCALAR VALIDATION ACROSS MODELS AND NFE VALUES

  • Lag-based gains reduced fresh-path population time lag in all 20 ImageNet settings, but reductions for SiT-S/2, B/2, and L/2 at NFE 250 had uncertainty including zero.Validation used 1,280 new paired trajectories per setting with the gain, probe, and genuine baseline fixed from calibration.Source: §E.2 · p. 23, Table 9 · p. 23
  • At NFE 50 on CIFAR-10, interpolated FID was 4.1 at the lag-based gain versus 3.0 for unscaled sampling; lag rose slightly at NFE 250 with uncertainty including zero.CIFAR-10 had little population time lag, and its lowest-FID gain remained one at all four NFEs.Source: §E.2 · p. 23

E.3 GAIN AND FID ACROSS MODELS

Lag-based gain selection improves ImageNet FID across tested models, while its benefit is largest at coarse sampling; the section also evaluates calibration cost, stability, schedule shape, and alternative explanations for velocity scaling.

E.3 GAIN AND FID ACROSS MODELS

  • Across twenty ImageNet settings, lag-based gains recover 49% of the FID reduction on average, from 81% at NFE 25 to 15% at NFE 250.Lag-based gains also improve FID across ImageNet models, while gain one gives the lowest measured FID for CIFAR-10.Source: §E.3 · pp. 24, 26, Table 10 · p. 24

E.4 CALIBRATION COST

  • A seven-gain sweep used 33 GPU-hours before image-metric extraction, whereas scalar calibration requires neither image decoding nor feature extraction.A 50,000-image NFE-50 setting used 2.5 million model evaluations and a median 4.7 RTX-4090 hours.Source: §E.4 · p. 27

E.5 CALIBRATION STABILITY

  • Across 24 model/NFE settings, 256-path subsets differed from the 1,280-path fitted gain by 0.00223 on average; 512 paths reduced the difference to 0.00137.All three SiT-XL/2 probes also selected gain 1.04 at NFE 50 and reduced lag on fresh paths.Source: §E.5 · p. 28, Table 15 · p. 28, Table 16 · p. 28

E.6 SCHEDULE SHAPE AND SCALAR CALIBRATION

  • At NFE 25 and matched mean gain 1.07216, the fitted schedule had FID 13.0 versus 12.5 for a constant gain.The schedule-minus-constant difference was +0.5 [+0.4, +0.6].Source: §E.6 · p. 28

F CONTROLS FOR ALTERNATIVE EXPLANATIONS

  • The controls do not support uniform field underscaling, simple Euler-error correction, or endpoint-moment matching as full explanations for gain’s image-quality improvement.Better integration still reduces lag and the lowest-FID gain, so numerical error contributes at low NFE without explaining the whole effect.Source: F CONTROLS FOR ALTERNATIVE EXPLANATIONS · p. 29

F.1 THE TRAINED FIELD IS NOT SYSTEMATICALLY UNDERSCALED

  • The fitted held-out velocity multiplier stays near one throughout the path, with mean 0.999, rather than showing systematic underscaling.This measurement uses 20 model times and 16,000 ImageNet image–noise pairs.Source: §F.1 · p. 29, Figure 7 · p. 29

F.2 VELOCITY SCALING IS NOT AN EULER ENDPOINT-ERROR CORRECTION

The controls distinguish velocity scaling from endpoint-error correction and global moment matching. They further connect its effect to changed-state responses and show that response direction and sampler dynamics matter.

F.2 VELOCITY SCALING IS NOT AN EULER ENDPOINT-ERROR CORRECTION; F.3 HEUN REDUCES LAG AND THE LOWEST-FID GAIN

  • Refining Euler integration reduces unscaled endpoint error, but the SSC endpoint remains far from the fine unscaled reference; SSC also increases one-step error, while Heun lowers lag and the lowest-FID gain.At the same 50-evaluation budget, Heun reduces endpoint error from 0.208 to 0.148, lag RMS from 0.019 to 0.005, and the lowest-FID gain from 1.10 to 1.05.Source: §F.2 · p. 30, §F.3 · p. 30, Table 18 · p. 30

F.4 MATCHING ENDPOINT MOMENTS DOES NOT REPRODUCE THE FID IMPROVEMENT

  • Matching endpoint radius or directional variance leaves FID essentially unchanged, while matching full latent mean and covariance worsens it.These global moment changes do not reproduce velocity scaling along the stochastic sampling path.Source: §F.4 · p. 31, Table 21 · p. 31

G.1 PATHWISE DECOMPOSITION

  • The one-step decomposition separates the direct gain push, retained by the frozen-field control, from the changed-state response that the control removes.The hybrid sampler has the same decomposition for its final Mean update, where the Brownian increment is zero.Source: §G.1 · p. 31, Table 22 · p. 32

G.2 A CONSTANT GAIN PROPAGATES DIFFERENTLY UNDER ODE AND SDE SAMPLING

  • Under frozen-field control, ODE sampling weights unscaled-path velocities uniformly, whereas SDE dynamics give later velocities more endpoint weight.In the SDE, earlier displacements are reduced longer by the −x drift, while the velocity coefficient 1 + t grows across the path.Source: §G.2 · pp. 32–33

G.3 DETERMINISTIC ODE GAIN CURVE

  • For deterministic ODE sampling, full-response FID falls from 11.2 at gain one to 8.5 at gain 1.05, then rises to 9.5 at gain 1.075.With the field frozen, gains 1.05 and 1.075 both give FID 9.7, showing that model reevaluation adds substantial improvement at 1.05 but little at 1.075.Source: §G.3 · p. 33, Table 23 · p. 33

G.4 THE REALIZED RESPONSE DIRECTION MATTERS

  • At NFE 50, perpendicular-only response gives FID 8.4, parallel-only gives 12.1, and full scaling gives 7.7; equal-norm direction controls also perform worse than the realized response.The FID ordering persists at NFE 25 and 250, indicating that response direction matters beyond its norm.Source: §G.4 · pp. 33–34, Figure 8 · p. 34, Table 24 · p. 34

G.5 DIRECTION OF THE DIRECT EXTRA PUSH

  • Relative to unscaled sampling, velocity scaling lowers FID from 13.9 to 7.8, while radial control raises it to 19.4; respective changes are −6.0 [−6.2, −5.9] and +5.5 [+5.4, +5.7].Brackets give paired 95% intervals from 2,000 common-row influence resamples with the real reference fixed.Source: §G.5 · p. 35

H DISTRIBUTIONAL EFFECTS OF GAIN

  • Population time lag explains only part of the FID improvement; gain also trades fidelity against diversity and redistributes endpoint variance across real-data covariance directions.The section further tests and rejects a single shared density-power account of the gain-induced endpoint change.Source: H DISTRIBUTIONAL EFFECTS OF GAIN · p. 35

H.1 IMAGE-METRIC TRADEOFF

Gain changes image metrics and endpoint covariance in ways not captured by a single density-power coefficient. Its benefits transfer to tested ImageNet models, but not to the decoder-free CIFAR-10 boundary case.

H.1 IMAGE-METRIC TRADEOFF

  • Increasing gain to 1.150 improves FID and precision further but worsens sFID and recall, so gain does not improve every metric together.The increase from 1.072 to 1.100 improves FID, sFID, and precision, while recall changes little within uncertainty.Source: §H.1 · p. 35, Table 25 · p. 35

H.2 SCALING SHARPENS THE ENDPOINT COVARIANCE SPECTRUM

  • Under stochastic sampling, SiT’s relative endpoint power slope is about 1.06 at every tested NFE; its difference from the Gaussian control shrinks with solver refinement.For the 1,000-step stochastic-Heun reference, the SiT–Gaussian difference is +0.0004 [−0.0015, +0.0024], within uncertainty; the measurement does not show sharpening causes lower FID.Source: §H.2 · p. 36, Table 26 · p. 36

H.3 ONE DENSITY-POWER COEFFICIENT DOES NOT DESCRIBE SCALING

  • A shared density-power law left essentially all held-out response variance unexplained, and the results do not support one common coefficient for the tested gain-1.05 ODE change.The residual-variance ratio was 1.14 [0.90, 1.92]; a matched Gaussian control recovered its known coefficient within 0.01%.Source: §H.3 · p. 37

I TRANSFER AND CONTEXT DEPENDENCE

  • Gain above one improves FID across the ImageNet SiT and ADM systems, but the lowest-FID gain varies with NFE and classifier-free guidance.The section tests transfer and context dependence across model and sampler settings.Source: I TRANSFER AND CONTEXT DEPENDENCE · p. 38

I.1 A DECODER-FREE BOUNDARY CASE

I.2 GUIDANCE CHANGES THE LOWEST-FID GAIN

As guidance increases, the lowest-FID gain falls faster than changes in velocity magnitude or population time lag alone predict, and the gain’s precision–recall trade-off depends on guidance. The section also describes scalar lag calibration as a useful starting point for tested unguided ImageNet models, while noting limits on where causal interventions were tested.

I.2 GUIDANCE CHANGES THE LOWEST-FID GAIN

  • Across guidance 1.75–2.25, the guidance-added velocity component increased motion by 2.6% to 3.0% per unit guidance, much less than the observed lowest-FID gain decrease.Source: §I.2 · p. 39
  • From guidance 1.75 to 2.25, the lag-based gain fell from 1.027 to 1.021, while the lowest-FID gain fell from 1.000 to 0.950 and crossed below one.The lowest-FID gain and its precision–recall trade-off depend on guidance.Source: §I.2 · pp. 39, 41, Table 29 · p. 40
  • At guidance 1.75, 2, and 2.25, lag-based power slopes were 1.020, 1.025, and 1.031, while lowest-FID slopes were 0.993, 0.978, and 0.963.Lag-based gains yield slopes near one, whereas FID favors progressively less sharpening as guidance rises.Source: §I.2 · p. 39
  • Reducing gain from one to 0.95 lowers precision and raises recall under classifier-free guidance.Source: §I.2 · p. 40, Table 30 · p. 40
  • At guidance 4, constant gain 0.976 lowered FID from 13.0 to 12.6, with a paired gain-minus-identity contrast of −0.4 (95% interval [−0.5, −0.3]).Source: §I.2 · p. 41

J SCOPE AND PRACTICAL USE

  • For tested unguided ImageNet models, scalar calibration provides a useful starting gain when decoding or image-metric evaluation is costly.The changed-state response transfers from SiT to ADM U-Net under deterministic sampling, and model-time correction transfers under stochastic sampling; CIFAR-10 experiments do not repeat the causal interventions, while the SiT ODE–SDE comparison changes the full integration rule.Source: J SCOPE AND PRACTICAL USE · p. 41

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