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Flow matching splits into two variance regimes; sampling gets 2x faster

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2026-08-03 04:16 UTC pith:JW775INE

load-bearing objection Correct core theory and a genuinely useful training objective, but the paper oversells the FID comparisons and the StableVS 'exact Euler' story is not actually proven. the 3 major comments →

arxiv 2602.05435 v2 pith:JW775INE submitted 2026-02-05 cs.CV

Stable Velocity: A Variance Perspective on Flow Matching

Donglin Yang , Yongxing Zhang , Xin Yu , Liang Hou , Xin Tao , Pengfei Wan , Xiaojuan Qi , Renjie Liao This is my paper
classification cs.CV
keywords flow matchingvariance reductionstochastic interpolantssampling accelerationrepresentation alignmentprobability flow ODEconditional velocitygenerative modeling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that the training target of flow matching—a single-sample conditional velocity—has variance that is not uniform over time. Early near the prior it is high, destabilizing training; later near data it collapses, so conditional and marginal velocities coincide. From this, the paper derives an unbiased variance-reduced training objective, a variance-aware schedule for auxiliary supervision, and a training-free sampling acceleration that replaces early sampler steps with large Euler steps. The claimed payoff is faster convergence in training and more than 2x sampling speedup on large pretrained models without quality loss.

Core claim

The central claim is that the variance of the conditional velocity target, V_CFM(t), defines two regimes separated by a threshold ξ. For t<ξ the posterior p_t(x0|xt) is so concentrated that the true velocity equals the conditional velocity of one dominant data point, making the probability-flow ODE an exact straight line; for t≥ξ the posterior spreads, and single-sample targets are noisy. The paper proves StableVM is unbiased with variance at most that of CFM and decaying as O(1/n), and shows StableVS—using the closed-form x_τ = x_t + (τ−t)v_t(x_t)—matches 30-step baselines with 9 steps in the low-variance regime.

What carries the argument

The object is the per-timestep trace of the conditional velocity covariance, V_CFM(t) = E[||v_t(x_t|x_0) − v_t(x_t)||^2]. It serves as the empirical signature of a two-regime structure: near the data this quantity is nearly zero, near the prior it is large. The training-side mechanism is a self-normalized importance-weighted average of conditional velocities over n reference samples, whose unbiasedness and variance bound are proven. The sampling-side mechanism is the exact linear-interpolant PF-ODE solution in the low-variance regime, which reduces to Euler steps of arbitrary size.

Load-bearing premise

The regime boundary ξ≈0.85 is asserted as a universal empirical property of high-dimensional flow models; if the posterior is not sharply concentrated for all t in [0,ξ] on a given dataset, Euler steps of size up to ξ cease to be exact and the claimed speedup degrades.

What would settle it

Measure V_CFM(t) on a held-out high-dimensional dataset at moderate noise levels (e.g., t=0.4–0.7) using the paper's estimator; if it does not stay near zero relative to the velocity norm, the straight-line Euler assumption fails. Concretely, run StableVS with a 9-step low-variance regime on a model trained on such data and compare reference metrics to a 30-step baseline—a significant PSNR/SSIM drop would contradict the central claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Flow matching training objectives can be made unbiased and lower-variance without changing the global minimizer.
  • Auxiliary representation alignment should be applied only where the signal is informative, namely the low-variance regime, improving FID.
  • Sampling can be accelerated by replacing only the early trajectory with large steps, leaving the high-variance tail to the base solver.
  • For linear interpolants, the low-variance regime is exactly straight, so step size there is limited only by the regime boundary, not by solver stability.
  • The sampler is solver-agnostic and finetuning-free, so it can be dropped into already-deployed models without retraining.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The two-regime view likely extends to score-based diffusion, where denoising score targets have analogous variance structure; the same straight-line shortcut may apply in the low-noise regime.
  • If the empirical universality of ξ holds across modalities, adaptive step schedules could allocate steps in proportion to variance rather than uniformly, yielding further speedups.
  • The unbiasedness proof for StableVM suggests the self-normalized aggregation with a memory bank could benefit other importance-weighted estimators in generative training.
  • A lightweight probe network estimating V_CFM(t) on a new dataset could predict a safe ξ without expensive retraining, making the method more portable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper analyzes the variance of the single-sample conditional velocity target in flow matching and identifies a low-variance regime near the data distribution and a high-variance regime near the prior. Based on this two-regime view, it proposes three components: StableVM, an unbiased multi-sample variance-reduced training objective; VA-REPA, a variance-aware weighting of representation-alignment losses; and StableVS, a training-free sampling acceleration that replaces the base solver by large Euler steps in the low-variance regime. The main theoretical results are Theorems 3.1--3.3, which assert unbiasedness and variance reduction of StableVM. Experiments are reported on ImageNet 256x256 and on SD3.5, Flux, Qwen-Image, and Wan2.2.

Significance. If the results hold, the paper gives a useful and broadly applicable variance-based perspective on flow-matching training and sampling. The derivation of StableVM is self-contained, the proofs of Theorems 3.1 and 3.2 appear correct, and the empirical validation spans several recent pretrained models, which is a genuine strength. The code is released, which aids reproducibility. However, the stronger O(1/n) variance bound and the exact-Euler justification of StableVS contain load-bearing gaps, and the headline training comparison is stated more strongly than Table 1 supports. These issues do not invalidate the empirical contributions but require repair before the paper is accepted.

major comments (3)
  1. [Appendix E.3 / Theorem 3.3] The proof of Theorem 3.3 uses the inequality p_t(x_t|x_0) ≤ 1 to pass from E_{x_0~p_t(·|x_t)}[ (p_t(x_t|x_0)/p_t(x_t)) ||v_t(x_t|x_0)-v_t(x_t)||^2 ] to a bound involving V_CFM(t)/ε. This inequality is not valid for the Gaussian conditional N(α_t x_0, σ_t^2 I): for σ_t < (2π)^{-1/2}, the density at its mode is greater than 1, and in the high-dimensional latent setting this occurs for a substantial range of t. Consequently the chain of inequalities leading to (1/(n-1))(V_CFM(t)/ε + M) does not follow, and the claimed O(1/n) rate is not established as stated. The weaker Theorem 3.2 appears correct, but Theorem 3.3 needs either a repaired proof with an explicit condition on the data density q(x_0) or a weakened claim.
  2. [§3.3 and Appendix E.5, Eq. (13)-(14)] StableVS is justified by assuming that in the low-variance regime v_t(x_t) ≈ v_t(x_t|x_0) for a single dominant x_0, then integrating the linear ODE (35) with that x_0 held fixed over the whole interval [t,τ]. V_CFM(t) in Eq. (6) is a per-time expectation over p_t(x_t); it does not imply that the same posterior mode persists along a given trajectory. If the dominant x_0 changes between t and τ, the marginal velocity field is not constant and Eq. (14) is not an exact Euler integrator. The high PSNR/SSIM/LPIPS values in Tables 5-6 show that StableVS outputs are close to a 30-step baseline, but they do not verify the straight-line mechanism. Please add pathwise evidence of mode persistence, give an error bound in terms of V_CFM over the interval, or revise the 'exact integration' claim to an approximation claim.
  3. [§4.2 and Table 1] The text says that at 80 training epochs the method achieves the 'strongest overall performance' among compared methods, but Table 1 shows REPA-E at 80 epochs with FID 1.67 versus the authors' 1.80, and at 400 epochs REG has FID 1.40 versus the authors' 1.47. No error bars or multiple-seed results are reported, and the FID differences at issue are 0.1-0.3, which may be within run-to-run variation. Please either qualify the claim to 'comparable to the best baseline' or report confidence intervals / repeated-seed results.
minor comments (3)
  1. [Algorithm 1] The indexing in Algorithm 1 is inconsistent: the perturbed batch is indexed j=1..M but the mixture is written as a sum over i=0..K with a factor 1/n; the target definition uses n references. Please align the notation among Eq. (7), Eq. (8), and the algorithm.
  2. [Figure 1] The caption states the y-axis reports V_CFM(t) normalized by the square root of data dimension, but axis labels and a legend for the different datasets would improve readability.
  3. [§4.3 / Tables 5-6] The abstract claims 'more than 2× faster sampling within the low-variance regime.' This is consistent with reducing 19 low-variance steps to 9, but the total step counts in Tables 5-6 are 20 or 17 versus 30, which is only 1.5-1.76× total speedup. Please make the distinction explicit in the main text to avoid overstatement.

Circularity Check

0 steps flagged

No significant circularity: the variance analysis, StableVM unbiasedness/variance bounds, and StableVS closed-form derivations are self-contained; the main caveat is an empirical concentration assumption, not a circular reduction.

full rationale

The paper's claimed derivation chain is self-contained rather than circular. The variance quantity V_CFM(t) is defined in Eq. (6) as the expected squared deviation between conditional and marginal velocities; the low-variance regime is identified empirically from that quantity in Fig. 1. The StableVM target is proved unbiased and variance-reduced in Theorems 3.1-3.3, with proofs in Appendix E that do not assume the conclusions. The StableVS sampler is derived in Appendix E.5 under an explicit, stated assumption: in the low-variance regime vt(xt) is approximated by vt(xt|x0) for a single dominant x0. Given that assumption, the linear-ODE solution algebraically reduces to x_tau = x_t + (tau - t)v_t(x_t) for the linear interpolant; this is a conditional derivation, not a definitional identification. The split point xi is an empirical hyperparameter chosen by ablation (Tabs. 4 and 9), but it does not enter the theorems and the acceleration is validated on held-out pretrained models and external benchmarks (GenEval, T2V-CompBench). There is no load-bearing self-citation chain and no uniqueness theorem imported from the authors' prior work. The skeptic's concern that per-time V_CFM(t) smallness does not by itself prove that the same x0 persists along a trajectory is a legitimate correctness/robustness caveat, but it is not circularity: the paper does not define the low-variance regime as 'where Euler is exact' and then reuse that definition as the conclusion. Under the rule that circularity requires quoting a specific reduction by construction or a fitted parameter renamed as a prediction, no such step is present.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical or data entities. The central theoretical results are parameter-free, but the practical methods depend on several fitted hyperparameters (xi, K, k, step counts) and on the empirical assumption of posterior concentration over a wide low-variance regime.

free parameters (6)
  • Split point xi (training; VA-REPA) = 0.7
    Chosen via ablation on ImageNet (Table 4); the regime boundary is not predicted theoretically.
  • Split point xi (sampling; StableVS) = 0.85
    Chosen via ablation on SD3.5-Large (Table 9); transferred to Flux/Qwen/Wan2.2 without per-model recalibration.
  • Memory bank capacity K = 256
    Ablation (Fig 4b) shows K=256 near-optimal; the unbiasedness theorem assumes infinite fresh i.i.d. references.
  • Sigmoid sharpness k for w_sigmoid = 20
    Manual choice in Sec G; no sensitivity analysis reported.
  • StableVS low-variance steps = 9
    Ablation (Table 9): 4 steps degrade quality, 14 steps improve fidelity but reduce speedup; 9 is the default.
  • Variance factor f_beta = 0
    Default; adding stochasticity (f_beta > 0) degrades reference metrics, as shown in Table 9.
axioms (5)
  • domain assumption The conditional path is Gaussian: p_t(x_t|x_0)=N(x_t; alpha_t x_0, sigma_t^2 I) for differentiable alpha_t, sigma_t satisfying boundary conditions.
    Standard flow matching / stochastic interpolant setup, invoked throughout Sec. 2 and the proofs.
  • domain assumption The data distribution q is fixed and references are i.i.d. draws from q; the memory bank is treated as such.
    Theorem 3.1 requires i.i.d. references; the FIFO bank in Alg. 2 introduces temporal dependence and finite-sample bias not analyzed.
  • standard math The velocity field v_t is bounded and the uniform integrability condition in Theorem 3.3 holds.
    Regularity assumptions for the delta method and asymptotic variance bound (Appendix E.3).
  • ad hoc to paper In the low-variance regime, the conditional posterior p_t(x_0|x_t) is concentrated on a single dominant data point, so v_t(x_t) ≈ v_t(x_t|x_0).
    Load-bearing empirical premise for StableVS (Sec. 3.3, Eq. 14); validated only empirically, xi tuned per model.
  • domain assumption The split point xi is a fixed hyperparameter and the low-variance regime extends over [0, xi] for all test models.
    Despite dimensionality discussion, xi=0.85 is transferred from SD3.5 to Flux/Qwen/Wan2.2 without per-model recalibration.

pith-pipeline@v1.3.0-alltime-deepseek · 32949 in / 25563 out tokens · 255729 ms · 2026-08-03T04:16:05.710734+00:00 · methodology

0 comments
read the original abstract

While flow matching is elegant, its reliance on single-sample conditional velocities leads to high-variance training targets that destabilize optimization and slow convergence. By explicitly characterizing this variance, we identify 1) a high-variance regime near the prior, where optimization is challenging, and 2) a low-variance regime near the data distribution, where conditional and marginal velocities nearly coincide. Leveraging this insight, we propose Stable Velocity, a unified framework that improves both training and sampling. For training, we introduce Stable Velocity Matching (StableVM), an unbiased variance-reduction objective, along with Variance-Aware Representation Alignment (VA-REPA), which adaptively strengthen auxiliary supervision in the low-variance regime. For inference, we show that dynamics in the low-variance regime admit closed-form simplifications, enabling Stable Velocity Sampling (StableVS), a finetuning-free acceleration. Extensive experiments on ImageNet $256\times256$ and large pretrained text-to-image and text-to-video models, including SD3.5, Flux, Qwen-Image, and Wan2.2, demonstrate consistent improvements in training efficiency and more than $2\times$ faster sampling within the low-variance regime without degrading sample quality. Our code is available at https://github.com/linYDTHU/StableVelocity.

Figures

Figures reproduced from arXiv: 2602.05435 by Donglin Yang, Liang Hou, Pengfei Wan, Renjie Liao, Xiaojuan Qi, Xin Tao, Xin Yu, Yongxing Zhang.

Figure 1
Figure 1. Figure 1: Variance curves of VCFM(t) with 15%–85% quantile bands. Evaluated on GMMs of varying dimensionality, CIFAR-10 images, and 256 × 256 ImageNet latents obtained by the Stable Diffusion VAE. The y-axis reports VCFM(t) normalized by the square root of the data dimension. See Appendix F.2 for details. N (xt | αtx0, σ2 t I). The minimizer of Eq. (4) is provably the true marginal velocity field (Tong et al., 2023;… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of CFM variance VCFM(t). (a) The low-variance regime (t ≤ ξ), where the posterior pt(x0 | xt) is sharply concentrated and the conditional velocity vt(xt | x0) nearly coincides with the true velocity vt(xt), yielding VCFM(t) ≈ 0. (b) The high-variance regime (t > ξ), the posterior spreads over multiple reference samples, causing the conditional velocity to fluctuate and resulting in a large VCF… view at source ↗
Figure 3
Figure 3. Figure 3: Motivation for variance-aware representation alignment. (a) In the low-variance regime, the alignment loss remains consistently low on a pretrained model from REPA (Yu et al., 2024), indicating a learnable and informative supervision signal. In contrast, in the high-variance regime, the loss stays high, reflecting the ill-posed nature of semantic recovery from noise. (b) Restricting representation alignmen… view at source ↗
Figure 4
Figure 4. Figure 4: Ablation on VA-REPA weighting and StableVM bank capacity. Left: effect of different weighting schemes w(t), showing that soft weightings outperform hard thresholding. Right: effect of memory bank capacity K, where K = 256 already achieves near-optimal performance. All results are evaluated at 100k iterations. REPA baseline is shown as a dashed line. ξ and other StableVS hyperparameters is provided in Tab. … view at source ↗
Figure 5
Figure 5. Figure 5: Visual comparison across prompts on SD3.5 (Esser et al., 2024). Results are generated using the Euler solver with 30 and 20 steps, and with StableVS replacing Euler in the low￾variance regime, all under the same random seeds. Compared to the standard 20-step solver, StableVS yields outputs that more closely resemble the 30-step results. Zoom in for details. Addi￾tional qualitative comparisons are provided … view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative comparison of generated samples on CIFAR-10. Samples generated by the checkpoint at 50k iterations (a) CFM, (b) STF instantiated directly from Eq. 22, (c) the original STF implementation, and (d) our StableVM. StableVM produces sharper and more coherent samples, consistent with its improved convergence and variance reduction. In this experiment, the STF baseline strictly follows Eq. 22: the ref… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of CFM and StableVM with n = 2048 on the synthetic GMM distribution. We plot the second-order moment as a function of training iterations at four time steps: t = 0.20, 0.30, 0.40, and 0.50. F.2. Unconditional Synthetic GMM Generation We also evaluate Algorithm 1 in the unconditional generation setting. Specifically, we construct a synthetic Gaussian Mixture Model (GMM) distribution and train the… view at source ↗
Figure 8
Figure 8. Figure 8: Visual comparison of SD3.5-Large (Esser et al., 2024) on different prompts. Results are generated using the Euler solver with 30 and 20 steps, and with StableVS replacing Euler in the low-variance regime, all under the same random seed. Compared to the standard 20-step solver, StableVS yields outputs that more closely resemble the 30-step results. Zoom in for details. 31 [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 9
Figure 9. Figure 9: Visual comparison of Flux-dev (Labs, 2024) on different prompts. Results are generated using the Euler solver with 30 and 20 steps, and with StableVS replacing Euler in the low-variance regime, all under the same random seed. Compared to the standard 20-step solver, StableVS yields outputs that more closely resemble the 30-step results. Zoom in for details. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Visual comparison of Qwen-Image-2512 (Wu et al., 2025a) on different prompts. Results are generated using the Euler solver with 30 and 17 steps, and with StableVS replacing Euler in the low-variance regime, all under the same random seed. Compared to the standard 17-step solver, StableVS yields outputs that more closely resemble the 30-step results. Zoom in for details. 33 [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 11
Figure 11. Figure 11: Visual comparison of Wan2.2 (Wan et al., 2025) on different prompts. Results are generated using UniPC solver with 30 and 20 steps, and with StableVS replacing UniPC in the low-variance regime, all under the same random seed. Compared to the standard 20-step solver, StableVS yields outputs that more closely resemble the 30-step results. Zoom in for details. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Visual comparison of Wan2.2 (Wan et al., 2025) on different prompts. Results are generated using UniPC solver with 30 and 20 steps, and with StableVS replacing UniPC in the low-variance regime, all under the same random seed. Compared to the standard 20-step solver, StableVS yields outputs that more closely resemble the 30-step results. Zoom in for details. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗

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Reference graph

Works this paper leans on

20 extracted references · cited by 3 Pith papers

  1. [1]

    General framework.We extend the variance analysis and variance-reduction strategy to the flow matching as well as stochastic interpolant framework, which generalizes beyond VP diffusion and exhibits a distinct variance structure

  2. [2]

    Unbiased objective.Instead of relying on a finite-sample weighted average, we propose a mixture of conditional probabilities that eliminates bias while still achieving variance reduction

  3. [3]

    To further elucidate these differences, we evaluate unconditional generation on CIFAR-10 (Krizhevsky, 2009)

    Class-conditional extension.While STF does not naturally extend to class-conditional settings, we design a tailored algorithm that maintains variance reduction under classifier-free guidance, improving both convergence and training efficiency. To further elucidate these differences, we evaluate unconditional generation on CIFAR-10 (Krizhevsky, 2009). For ...

    2009

  4. [6]

    (2023), which instead applies noise to the first M samples in the reference batch and treats them as training inputs

    While this matches the theoretical formulation in the original paper, it differs from the implementation used by Xu et al. (2023), which instead applies noise to the first M samples in the reference batch and treats them as training inputs. This modification yields a near-unbiased estimator, since xt is no longer conditioned on a single reference point bu...

    2023

  5. [7]

    dx1:n 0 = 1 n·p t(xt) nX k=1 Z  Y i̸=k q(xi 0)   pt(xt,x k 0)vt(xt |x k

  6. [8]

    dx1:n 0 = 1 n·p t(xt) nX k=1    Y i̸=k Z q(xi

  7. [9]

    dxi 0   · Z pt(xt,x k 0)vt(xt |x k

  8. [10]

    dxk 0   = 1 n nX k=1 Z pt(xk 0 |x t)vt(xt |x k

  9. [11]

    Z 1 n nX i=1 pt(xt |x i 0) ! nX k=1 pt(xt |x k 0)Pn j=1 pt(xt |x j 0) vt(xt)−v t(xt |x k 0) 2 dxt # =E {xi 0}n i=1∼qn

    dxk 0 =v t(xt), as desired. (b) Recall the StableVM objective Eq. (8) LStableVM(θ, t) =E xt∼pt, {xi 0} n i=1∼pGMM t (·|xt)   vθ(xt, t)− nX k=1 pt(xt |x k 0)vt(xt |x k 0)Pn j=1 pt(xt |x j 0) 2  =E xt∼pt   Z pGMM t xi 0 n i=1 |x t vθ(xt, t)− nX k=1 pt(xt |x k 0)vt(xt |x k 0)Pn j=1 pt(xt |x j 0) 2 dx1:n 0   . For eachx t, let Lxt (v) := Z pGMM t xi 0...

  10. [12]

    Sample a latent indexIuniformly from{1, . . . , n}

  11. [13]

    We claim the following: Lemma E.2.(a) The joint distribution of xi 0 n i=1 sampled from the above procedure conditioned on xt is exactly pGMM t xi 0 n i=1 |x t

    Samplex I 0 ∼p t(· |xt)andx j 0 ∼qfor allj̸=I. We claim the following: Lemma E.2.(a) The joint distribution of xi 0 n i=1 sampled from the above procedure conditioned on xt is exactly pGMM t xi 0 n i=1 |x t . (b) xi 0 n i=1 are independent conditioned onx t andI. Proof.(a) Note that the joint distribution of xi 0 n i=1 sampled from the above procedure is ...

  12. [14]

    dxk 0 =p t(xt),(26) E pt(xt |x k 0)vt(xt |x k 0)|x t, I=i = Z pt(xt |x k 0)vt(xt |x k 0)q(xk

  13. [15]

    dxk 0 = Z pt(xt,x k 0)vt(xt |x k

  14. [16]

    dxk 0 =p t(xt) Z pt(xt,x k 0) pt(xt) vt(xt |x k

  15. [17]

    dxk 0 =p t(xt) Z pt(xk 0 |x t)vt(xt |x k

  16. [18]

    dxk 0 =p t(xt)vt(xt),(27) Ex0∼pt(·|xt) [vt(xt |x 0)|x t] = Z pt(x0 |x t)v t(xt |x 0) dx0 =v t(xt),(28) and Ex0∼q pt(x0 |x t) q(x0) = Z pt(x0 |x t) dx0 = 1.(29) We then continue with the lemmas. Lemma E.4.Asn→ ∞, we have √ n−1 Vn−1 Pn−1 −v (ℓ) t (xt) d − → N 0,E x0∼pt(·|xt) pt(xt |x 0) pt(xt) v(ℓ) t (xt |x 0)−v (ℓ) t (xt) 2 , where the random variablesV n−...

  17. [19]

    = P k̸=i pt(xt |x k 0)f(x k 0)P k̸=i pt(xt |x k 0) = P k̸=i pt(xt|xk 0 )q(x k 0 ) pt(xt)q(x k 0 ) f(x k 0) P k̸=i pt(xt|xk 0 )q(x k 0 ) pt(xt)q(x k 0 ) = P k̸=i pt(xk 0 |xt) q(xk 0 ) f(x k 0) P k̸=i pt(xk 0 |xt) q(xk 0 ) = P k̸=i w(xk 0)f(x k 0)P k̸=i w(xk

  18. [20]

    Therefore, Eq

    (30) Recall from Lemma E.2 that conditioned on xt and I=i , the xi 0’s are all independent. Therefore, Eq. (30) is a self- normalized importance sampling estimator (Chapter 9 of (Owen, 2013)) with importance distribution q(x0), nominal distributionp t(x0 |x t), and importance weight ratiow(x 0). 21 Stable Velocity: A Variance Perspective on Flow Matching ...

    2013

  19. [1995]

    These methods achieve variance reduction by aggregating multiple conditional scores, but typically introduce bias due to self-normalization

    to reduce the variance of score estimation (Xu et al., 2023; Niedoba et al., 2024). These methods achieve variance reduction by aggregating multiple conditional scores, but typically introduce bias due to self-normalization. In contrast, StableVM focuses on a different source of variance that arises in flow-matching objectives, namely the variability indu...

    2023

  20. [2024]

    Several follow-up methods explore alternative alignment strategies and regularization mechanisms

    introduces an auxiliary objective that aligns hidden states of diffusion transformers with features from pretrained visual encoders, yielding significant gains under limited training budgets. Several follow-up methods explore alternative alignment strategies and regularization mechanisms. Dispersive Loss (Wang & He, 2025) removes the need for external tea...

    2025