Velocity Ambiguity Profiles: Time-Resolved Bayes-Risk Diagnostics for Flow Matching

Yunpeng Mei; Xiaowen Zhu; Chenyu Wang; Chenbo Xin; Hongjie Cao; Jiamin Wang; Jie Chen; Gang Wang

cs.LGOct 21, 2026

Velocity Ambiguity Profiles: Time-Resolved Bayes-Risk Diagnostics for Flow Matching

Authors: Yunpeng Mei, Xiaowen Zhu, Chenyu Wang, Chenbo Xin, Hongjie Cao, Jiamin Wang, Jie Chen, Gang Wang

Organizations: Beijing Institute of Technology · ByteDance

Abstract

Flow matching trains a continuum of velocity regression problems but usually reports a loss averaged over interpolation time, obscuring which regions are limited by intrinsic ambiguity and which remain data-limited. We introduce the Velocity Ambiguity Profile (VAP), $\mathcal{A}_v(t)$ , the time-local Bayes-risk floor of velocity prediction, and decompose time-resolved loss as $L_N(t)=\mathcal{A}_v(t)+R(t;N)$ , where only the excess term $R(t;N)$ is reducible by more data or a different estimator. For $N$ training samples and a $K$ -component mixture in the structured example, we prove two statistical benchmarks. In an intrinsic local-regression benchmark, $\mathcal{A}_v(t)$ enters the time-dependent noise factor of the standard nonparametric rate. In separated isotropic Gaussian mixtures, the Bayes velocity becomes locally affine, and a label-oracle residual-regression benchmark has parametric excess-risk scale $K\mathcal{A}_v(t)/N$ , matched within the same separated residual Gaussian-location subproblem. Controlled Gaussian-mixture experiments validate this diagnostic through a dense transition sweep, calibrated oracle-to-blind and empirical-proxy audits, and a local controlled time-sampler stress test in which a frozen reducible-error proxy reduces reducible loss while cross-cell sensitivity delimits aggressive sampling. On CIFAR-100 DINOv2 feature flow, guided time sampling reduces uniformly evaluated velocity loss by $6.78\%$ and global feature Fréchet distance (FD) by $23.6\%$ , while class-conditional kernel inception distance (KID) shows no improvement. VAP turns an averaged flow-matching objective into a time-resolved diagnostic for locating ambiguity-limited and data-limited regions.

Figures & tables

Figure 1: Velocity ambiguity profiles localize where flow matching needs data. The flow matching path defines a continuum of velocity prediction problems. VAP separates the irreducible ambiguity floor from the reducible excess error that more data or a different estimator can affect.

Figure 2: Canonical

d{=}4

dense controlled transition on a

K{=}4

\sigma{=}0.5

Gaussian mixture. The

13

-row front/plateau window shows the reliability-filtered signed log-log slope

s_{\mathrm{pooled}}=\partial\log\bar{R}/\partial\log N

and regression precision; the shaded band marks

\Delta/\sigma\in[4,6]

, where

s_{\mathrm{pooled}}\approx-1

, equivalently the positive exponent

\alpha_{\mathrm{pooled}}=-s_{\mathrm{pooled}}\approx 1

. Full rows and filter sensitivity are in Appendix D.3 .

Row	$A_{1}$ dev (%)	$z$	$\epsilon_{\mathrm{MLP}}$ (%)
$d{=}2$ sep	$-0.006$	$0.27$	$1.57$
$d{=}2$ p.m.	$+0.068$	$2.08$	$0.64$
$d{=}4$	$-0.021$	$1.42$	$1.51$
$d{=}8$	$+0.024$	$2.53$	$1.49$
$d{=}16$	$+0.005$	$0.84$	$1.71$

Table 1: Oracle-to-blind decomposition. Per-row bridge audit metrics on five controlled GMM rows; column definitions in main text.

Figure 3: Sampler-side

t

-sampler interventions on a controlled

d=16

K=32

GMM. Ten-seed mean loss for the baseline and gap-guided samplers using a proxy source disjoint from the paired test. Gap arms lie below baseline past the peak-learning step; pure

\alpha=2

is lowest at the plateau, while pure

\alpha=4

is lowest near the peak-learning step.

Figure 4: Uniform-time evaluation during DINOv2 feature-flow training. The gap-guided sampling lowers uniformly evaluated velocity loss. Curves show means over five paired training seeds.

Metric	Uniform	VAP-guided	Change
Global feature FD $\downarrow$	$33.333$	$\mathbf{25.476}$	$-23.6\%$
Macro precision $\uparrow$	$0.151$	$\mathbf{0.690}$	$+0.539$
Macro density $\uparrow$	$0.122$	$\mathbf{1.226}$	$+1.104$
Macro coverage $\uparrow$	$0.234$	$\mathbf{0.728}$	$+0.494$
Fine-label macro accuracy $\uparrow$	$0.791$	$\mathbf{0.859}$	$+6.76$ pp
Macro conditional KID $\downarrow$	$0.1988$	$0.2062$	$+0.0074$

Table 2: CIFAR-100 DINOv2 feature-flow summary. Values are aggregate deployment metrics over five paired training seeds. Arrows indicate the preferred direction.

Appendix figures & tables30 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Info-set decomposition and residual benchmark calibration. (a) The empirical ratio

R_{\mathrm{act}}(t;N)/R_{\mathrm{res}}(t;N)

from a closed-form ablation (

N{=}2000

K{=}4

\sigma{=}0.5

, no MLP) collapses onto the analytic curve

t^{2}/\sigma_{t}^{2}

(dashed) across

d\in\{2,4,8,16\}

. (b) The residual-regression oracle, normalized as

NR_{\mathrm{res}}(t;N)/(K\mathcal{A}_{v,\mathrm{within}}(t))

, is centered at the unit benchmark from Theorem 5 . All scalar summaries in the text use the full

20

-seed run for every

d

. Panel (a) shows seed-level spread for the ratio readout, while panel (b) shows standard error around the mean for residual calibration. Together these panels show that the label-and-action oracle is strictly tighter at small

t

, while the residual oracle is calibrated to the theorem’s sharp residual-regression benchmark.

$K$	Geometry	$\Delta_{\max}/\Delta_{\min}$	$\mathrm{bf}_{\max}$ (MC)	$t(\mathrm{bf}\geq 0.5)$ (MC)	$t_{\mathrm{merge}}(\tau_{K})$
$2$	simplex	$1.000$	$0.528$	$0.838$	$0.781$
$4$	simplex	$1.000$	$0.627$	$0.650$	$0.714$
$8$	simplex	$1.000$	$0.663$	$0.575$	$0.669$
$16$	equiangular	$2.002$	$0.843$	$0.538$	$0.634$
$32$	equiangular	$2.737$	$0.903$	$0.538$	$0.605$
$64$	equiangular	$3.919$	$0.946$	$0.538$	$0.580$

Appendix

Table 3: Across-

K

merge dynamics at fixed

d=8

\sigma=0.5

\Delta_{\min}=3

(

\Delta_{\min}/\sigma=6

). Centers are simplex for

K\leq 8

and equiangular tight-frame approximations for

K>8

;

\Delta_{\max}/\Delta_{\min}

records the geometry drift introduced by the latter. The closed-form column uses

\tau_{K}=\sqrt{\log K}

in Eq. ( 61 ). Monte-Carlo readouts use

10^{5}

samples per

t

on a

25

-bin grid over

t\in[0.05,0.95]

; bin spacing limits the resolution of

t(\mathrm{bf}\geq 0.5)

\approx 0.04

Experiment block	Implementation settings
Dense $d{=}4$ transition	$K=4$ , $d=4$ , $\sigma=0.5$ , one-hot centers, reported across the $18$ $\Delta/\sigma$ values listed in Table 6 . Models use hidden width $256$ , $10000$ training steps, Adam learning rate $10^{-3}$ , $N\in\{50,100,200,500,1000,2000\}$ , $10$ data seeds and $3$ initialization replicates per cell. Evaluation uses $30$ time bins on $[0.05,0.95]$ , $2000$ evaluation actions, and $80$ noise draws per time bin; the main readout keeps $t\in[0.05,0.80]$ .
Oracle-to-blind bridge	Five controlled rows are used: $d=2$ separated, $d=2$ partial-merge, and simplex rows at $d\in\{4,8,16\}$ with $K=4$ and $\sigma=0.5$ . The closed-form arms are the label-and-action true-label $\mathrm{oracle\_ols}$ plug-in and EM-fitted blind OLS; neural arms use the same MLP class with hidden width $128$ . The secondary bridge run uses $20$ seeds, $16000$ training steps, Adam learning rate $10^{-3}$ , $12000$ evaluation actions, and $32$ noise draws per time bin.
Proxy-floor audit	The proxy audit is a post-processing pass over controlled-result JSON files. It compares the exact excess $R=L_{N}-\mathcal{A}_{v}$ with the empirical proxy $\hat{R}=L_{N}-L_{\mathrm{largeN}}$ , where $L_{\mathrm{largeN}}$ is the large-budget reference profile stored with each controlled row. The audit reports Spearman rank agreement, region-label agreement, negative proxy-excess counts, and the factor diagnostic $r_{F}=\max_{t}\|L_{\mathrm{largeN}}(t)-\mathcal{A}_{v}(t)\|/\mathrm{std}(R(t)\mid t\in[0.4,0.8])$ .
Three-family heterogeneity panel	Three $d=2$ Gaussian-mixture families are trained with the same implementation: a well-separated $K=4$ ring, a high-entropy $K=8$ ring, and a $K=4$ partial-merge geometry. The panel uses $40$ time bins on $[0.05,0.80]$ , $10$ data seeds with $3$ initialization replicates, hidden width $128$ , and $5000$ to $8000$ training steps.
Time-sampler stress test	The intervention cell is a balanced partial-merge GMM with $K=32$ , $d=16$ , $\sigma=1.0$ , pair gap $0.3$ , spread $1.5$ , and $N=512$ demonstrations ( $16$ per component). The MLP has hidden width $64$ and is trained for $4000$ steps with Adam learning rate $4{\times}10^{-4}$ . The baseline samples $t\sim\mathcal{U}(0.05,0.95)$ ; gap arms use a categorical sampler on $64$ fine time bins, with no importance correction and the standard MSE objective. The proxy $\hat{R}_{\mathrm{ind}}$ is frozen from source baseline seeds $\{20,\ldots,29\}$ at ts $=100$ using source evaluation-action key $88888$ ; paired comparisons use seeds $\{0,\ldots,9\}$ and evaluation-action key $99999$ .

Appendix

Table 4: Implementation settings for controlled experiments. The

d{=}4

dense sweep uses the one-hot geometry induced by the legacy_circle code path when

K\leq d

and

d>2

; setting

\mathrm{spread}=(\Delta/\sigma)\sigma/\sqrt{2}

makes the minimum center separation equal to the reported

\Delta/\sigma

after normalization by

\sigma

Regime	$\Delta/\sigma$	$s_{\mathrm{pooled}}$	$95\%$ CI	CV
overlap / noise floor	$1.5$	$-0.838$	$[-0.853,-0.823]$	$0.0092$
transition	$3.0$	$-0.953$	$[-0.966,-0.939]$	$0.0071$
parametric plateau	$5.0$	$-1.011$	$[-1.018,-1.003]$	$0.0037$
bias transition	$8.0$	$-0.861$	$[-0.866,-0.856]$	$0.0030$
extreme separation	$15.0$	$-0.605$	$[-0.611,-0.599]$	$0.0048$

Appendix

Table 5: Canonical controlled transition,

d=4

dense sweep (selected anchor rows). Compact

5

-row summary under the mid-b canonical reliability-filtered readout (

\mathrm{SNR}\geq 4

R\geq 0.016\,\mathcal{A}_{v}

t\in[0.05,0.80]

; companion to Figure 2 and the full Table 6 ). The selected anchors summarize the overlap, transition, parametric-plateau, bias-transition, and extreme separation regimes; the plateau at

\Delta/\sigma\in[4,6]

is consistent with the parametric scale.

$\Delta/\sigma$	$n_{\mathrm{pass}}$	abstain	$s_{\mathrm{pooled}}$	$95\%$ CI	$cv_{\mathrm{pooled}}$	$\mathrm{SE}(s)$	mean SNR
$1.00$	$83$	$53.9\%$	$-0.810$	$[-0.824,-0.795]$	$0.0091$	$0.0074$	$17.2$
$1.25$	$84$	$53.3\%$	$-0.828$	$[-0.843,-0.814]$	$0.0089$	$0.0074$	$17.2$
$1.50$	$83$	$53.9\%$	$-0.838$	$[-0.853,-0.823]$	$0.0092$	$0.0077$	$16.8$
$1.75$	$80$	$55.6\%$	$-0.810$	$[-0.826,-0.795]$	$0.0098$	$0.0079$	$17.0$
$2.00$	$77$	$57.2\%$	$-0.819$	$[-0.836,-0.804]$	$0.0099$	$0.0081$	$17.1$
$2.50$	$78$	$56.7\%$	$-0.894$	$[-0.909,-0.879]$	$0.0083$	$0.0075$	$16.2$

Appendix

Table 6: Full

18

-row

d{=}4

dense controlled transition (appendix companion to Figure 2 and Table 5 ; the five plateau rows

\Delta/\sigma\in\{4.0,4.5,5.0,5.5,6.0\}

are boldfaced). Canonical mid-b reliability profile (

\mathrm{SNR}\geq 4

R\geq 0.016\,\mathcal{A}_{v}

);

t\in[0.05,0.80]

;

n_{\mathrm{total}}=180

candidate cells per row (

30

time bins times

6

sample sizes);

95\%

paired bootstrap CIs with

2000

iterations.

Readout	SNR	$R$ gate	retained	plateau $s$	mean CV	$s_{15}$
raw-positive	$\geq 0$	$R>0$	$2700/3240$ ( $83.3\%$ )	$[-1.028,-0.969]$	$0.0033$	$-0.604$
loose	$\geq 3$	$R\geq 0.010\,\mathcal{A}_{v}$	$2118/3240$ ( $65.4\%$ )	$[-0.986,-0.955]$	$0.0035$	$-0.605$
canonical / mid-b	$\geq 4$	$R\geq 0.016\,\mathcal{A}_{v}$	$1768/3240$ ( $54.6\%$ )	$[-1.033,-0.962]$	$0.0038$	$-0.605$
very-strict	$\geq 8$	$R\geq 0.030\,\mathcal{A}_{v}$	$1234/3240$ ( $38.1\%$ )	$[-1.145,-0.982]$	$0.0057$	$-0.598$

Appendix

Table 7: Filter-robust post-hoc reanalysis on the same trained

d{=}4

dense-sweep models. Raw-positive uses all finite positive excess-risk cells; the other rows apply increasingly conservative reliability gates. Retained counts are over the full

18

-row ladder (

3240

candidate cells). The plateau summary is computed over

\Delta/\sigma\in\{4.0,4.5,5.0,5.5,6.0\}

. The canonical / mid-b row uses

\mathrm{SNR}\geq 4

R\geq 0.016\,\mathcal{A}_{v}

; the very-strict stress row uses

\mathrm{SNR}\geq 8

R\geq 0.030\,\mathcal{A}_{v}

SNR gate \ $R/\mathcal{A}_{v}$ gate	$0.000$	$0.005$	$0.010$	$0.016$	$0.020$	$0.030$
$0$	$[-1.028,-0.969]$	$[-1.004,-0.961]$	$[-0.986,-0.955]$	$[-1.033,-0.962]$	$[-1.069,-0.981]$	$[-1.120,-0.982]$
$1$	$[-1.028,-0.969]$	$[-1.004,-0.961]$	$[-0.986,-0.955]$	$[-1.033,-0.962]$	$[-1.069,-0.981]$	$[-1.120,-0.982]$
$2$	$[-1.028,-0.969]$	$[-1.004,-0.961]$	$[-0.986,-0.955]$	$[-1.033,-0.962]$	$[-1.069,-0.981]$	$[-1.120,-0.982]$
$3$	$[-1.028,-0.969]$	$[-1.004,-0.961]$	$[-0.986,-0.955]$	$[-1.033,-0.962]$	$[-1.069,-0.981]$	$[-1.120,-0.982]$
$4$	$[-1.026,-0.967]$	$[-1.004,-0.961]$	$[-0.986,-0.955]$	$[-1.033,-0.962]$	$[-1.069,-0.981]$	$[-1.120,-0.982]$
$5$	$[-1.019,-0.966]$	$[-1.004,-0.961]$	$[-0.986,-0.955]$	$[-1.033,-0.962]$	$[-1.069,-0.981]$	$[-1.120,-0.982]$

Appendix

Table 8: Decoupled threshold-grid audit for the dense

d{=}4

readout. Each cell reports the plateau range

[\min s,\max s]

over

\Delta/\sigma\in\{4.0,4.5,5.0,5.5,6.0\}

Row ( $\Delta/\sigma$ )	Full $\mathcal{A}_{v}$	Within	Flat	Large- $N$
separated ( $8.00$ )	$\mathbf{0.199}$	$1.010$	$0.676$	$1.216$
boundary ( $3.00$ )	$\mathbf{0.105}$	$1.285$	$0.558$	$0.536$
failure ( $2.00$ )	$\mathbf{0.277}$	$1.282$	$0.489$	$0.771$
overlap ( $1.41$ )	$\mathbf{0.300}$	$0.940$	$0.808$	$0.250$

Appendix

Table 9: Normalization sanity. CV of

R(t;N)

across time under four floor choices on auxiliary

d=2

phase rows. Full

\mathcal{A}_{v}

is canonical; within-only drops the between-component term; flat is no normalization; large-

N

ref is a

50000

-sample MLP. Full

\mathcal{A}_{v}

is tightest except on the overlap row, where the crossing is a partial cancellation in

R

, not collapse.

Row	$\Delta/\sigma$	$\rho_{S}$	$r_{F}$
$d{=}4$ sep.	$8.00$	$0.997$	$0.83$
$d{=}8$ sep.	$6.00$	$0.998$	$0.71$
$d{=}8$ strict	$8.00$	$1.000$	$0.47$
$K$ -sweep ( $d{=}8$ )	$6.00$	$[0.959,1.000]$	$[0.47,1.38]$
$d{=}2$ aux. ( $3$ )	$[1.41,3.00]$	$[0.997,0.998]$	$[0.81,0.91]$

Appendix

Table 10: Proxy-floor calibration. Rank correlation

\rho_{S}=\rho_{\mathrm{Spearman}}(\hat{R},R)

and factor-product diagnostic

r_{F}

on guard-passing calibration rows. The isolated high-

r_{F}

d{=}2

separated breakdown is analyzed below.

Instance	Seeds	$\rho_{\mathrm{Spearman}}$	Region-label match	Strict-tie gap
canonical pool	$5$	$0.552$	$3/3$	$0.16\%$
pool $0$	$5$	$0.471$	$3/3$	$0.16\%$
pool $1$	$5$	$0.868$	$3/3$	$0.16\%$
pool $2$	$5$	$0.704$	$2/3$	$0.16\%$
extended pool	$20$	$0.662$	$2/3$	$0.16\%$

Appendix

Table 11: Reseed-pool spread at

d{=}2

separated. Rank correlation

\rho_{\mathrm{Spearman}}(\hat{R},R)

, the fraction of region labels (low/mid/high mass) reproduced by the proxy, and the strict-tie ranking gap (exact-

R

mass difference between the second and third regions) across five independent reseed instances. The small strict-tie gap explains why independent resampling changes the mid-vs-high label on most pools.

row	flip bins	$G_{F}$	std $(R)$ mid/high $t$	$r_{F}$	$r_{F}^{\mathrm{obs}}$
$d{=}2$ separated, $\Delta/\sigma=6$	$3$	$0.092$	$0.034$	$2.70$	$2.79$
$d{=}4$ separated, $\Delta/\sigma=8$	$0$	$0.160$	$0.193$	$0.83$	$0.42$
$d{=}8$ separated, $\Delta/\sigma=6$	$0$	$0.185$	$0.260$	$0.71$	$0.20$
$d{=}8$ separated, $\Delta/\sigma=8$	$0$	$0.221$	$0.467$	$0.47$	$0.17$

Appendix

Table 12: Cross-

d

factor-product breakdown on separated rows. Here

G_{F}=\max|L_{\mathrm{largeN}}-\mathcal{A}_{v}|

and mid/high

t

denotes

t\in[0.4,0.8]

. The

d{=}2

separated row is the only configuration with a non-zero sign-flip count and a small mid-to-high

t

standard deviation of

R

. From

d{=}4

onward the sign flip vanishes and the standard deviation grows, so

r_{F}

drops well below the empirical boundary

2

. Both

d{=}8

rows fall inside the rank-stable regime. The rightmost column reports the observable counterpart

r_{F}^{\mathrm{obs}}

defined in the next paragraph.

$r_{F}$ band	cells	$\rho_{\mathrm{Spearman}}$ range
$r_{F}<1.0$	$12$	$[0.991,1.000]$
$r_{F}\in[1.0,2.0)$	$3$	$[0.959,0.994]$
$r_{F}\geq 2.0$	$2$	$[0.552,0.662]$

Appendix

Table 13: Diagnostic boundary calibration. Distribution of the seventeen controlled cells across

r_{F}

bands. The rank correlation tracks

r_{F}

monotonically: cells in the lower two bands retain

\rho_{\mathrm{Spearman}}\geq 0.959

, while the two cells in the upper band drop to

\rho_{\mathrm{Spearman}}\leq 0.662

. Cell composition of each band is described in the main text.

family	geometry	$\Delta/\sigma$ range	peak $t$	$R$ peak
well-separated	$K{=}4$ ring	$[8.0,11.3]$	$0.435$	$0.136$
high-entropy	$K{=}8$ ring	$[2.6,6.8]$	$0.338$	$0.052$
partial-merge	$K{=}4$ paired	$[3.0,30.0]$	$0.781$	$0.271$

Appendix

Table 14: Three-family controlled heterogeneity panel. Mean

R(t;N{=}1000)

profile statistics across three Gaussian-mixture geometries in

d{=}2

. The minimum pairwise normalized

L^{2}

distance between profile shapes is

0.2059

, between the well-separated and partial-merge families. The partial-merge family was selected to stabilize the visualization; this selection does not change the qualitative finding.

Metric	Uniform	VAP-guided	Paired change
Global feature FD $\downarrow$	$33.333$	$25.476$	$-7.858$
Precision ( $k=3$ ) $\uparrow$	$0.151$	$0.690$	$+0.539$
Density ( $k=5$ ) $\uparrow$	$0.122$	$1.226$	$+1.104$
Coverage ( $k=5$ ) $\uparrow$	$0.234$	$0.728$	$+0.494$
Fine macro accuracy $\uparrow$	$0.791$	$0.859$	$+0.0676$
Fine cross-entropy $\downarrow$	$1.094$	$0.626$	$-0.468$

Appendix

Table 15: Closed-loop CIFAR-100 DINOv2 metrics. Means are over five paired training seeds. Feature FD is not image FID.

Figure 6: PCA projection for visualization of DINOv2 features. The panels compare the held-out real feature bank with features generated by the uniform and VAP-guided

\alpha=4

samplers. This is a feature-space visualization only; PCA is not the representation used to train the flow.

Arm	Sampler	Mean $\Delta_{\mathrm{red}}\,(\%)$	SE (%)	$p_{\mathrm{perm}}$
gap_mix_025	mixture ( $\lambda=0.25$ )	$-9.32$	$0.98$	$0.0015$
gap_mix_050	mixture ( $\lambda=0.50$ )	$-13.95$	$1.19$	$0.0028$
gap_mix_075	mixture ( $\lambda=0.75$ )	$-17.33$	$1.33$	$0.0018$
gap_pure_a1	pure ( $\alpha=1$ )	$-18.84$	$1.37$	$0.0019$
gap_pure_a2	pure ( $\alpha=2$ )	$\mathbf{-20.21}$	$1.69$	$0.0019$
gap_pure_a4	pure ( $\alpha=4$ )	$-19.15$	$2.10$	$0.0017$

Appendix

Table 16: Controlled time-sampler stress test. Plateau paired statistics over

10

seeds at training step

4000

. The metric

\Delta_{\mathrm{red}}

is the percent change in t-averaged reducible loss relative to the baseline; every gap arm has negative paired seed-level deltas for all seeds.

Arm	$K{=}24,d{=}12$	$K{=}32,d{=}16$ (canonical)	$K{=}40,d{=}20$
gap_mix_025	$-7.76$	$-9.32$	$-10.22$
gap_mix_050	$-13.06$	$-13.95$	$-16.17$
gap_mix_075	$-16.38$	$-17.33$	$-19.95$
gap_pure_a1	$-17.84$	$-18.84$	$-21.39$
gap_pure_a2	$-20.56$	$-20.21$	$-21.56$
gap_pure_a4	$-19.76$	$-19.15$	$-11.45$

Appendix

Table 17: Cross-cell d-axis ladder of the time-sampler stress test: paired

\Delta_{\mathrm{red}}\,(\%)

at training step

4000

over

10

paired seeds. The headline metric is the reducible-loss percent change as defined in Table 16 . The middle column is the canonical

(K{=}32,d{=}16)

cell of Section 5.3 . Mixture arms and pure exponents

\alpha\in\{1,2\}

reproduce the canonical direction across all three cells with sign distribution

10/10

and

p_{\mathrm{perm}}<0.005

. Pure

\alpha=4

remains negative but weakens at the largest cell (

9/10

p_{\mathrm{perm}}\approx 0.008

), while the most aggressive pure sampler (

\alpha=8

) reverses sign in the late plateau only there.

Figure 7: Cross-cell d-axis ladder of the time-sampler stress test. Three panels share a common y-range. The middle panel reproduces Figure 3 for reference. The panel (

K{=}24,d{=}12

) keeps every tested arm strictly below baseline at every checkpoint, including

\alpha=8

. The panel (

K{=}40,d{=}20

) shows a late-plateau reversal of

\alpha=8

: the curve crosses the baseline near training step

2500

and ends slightly above it. Lower exponents and mixture arms remain stable across all three cells.

row	within $(t)$ at $N{=}2000$	relative gap
$d{=}2$ separated	$2.4043$	$-0.014\%$
$d{=}2$ partial-merge	$2.4043$	$-0.014\%$
$d{=}4$ simplex	$4.8086$	$+0.008\%$
$d{=}8$ simplex	$9.6171$	$+0.024\%$
$d{=}16$ simplex	$19.2343$	$+0.044\%$

Appendix

Table 18:

\mathrm{oracle\_ols}

arm versus analytic floor. Empirical

L_{\mathrm{oracle\_ols}}

N=2000

versus the analytic floor within

(t)=d\sigma^{2}/\sigma_{t}^{2}

, computed on the readout window

t\in[0.10,0.28]

with

20

data seeds and eval

12000\times 32

Monte-Carlo samples per cell. The two

d{=}2

rows share

(d,\sigma)

and therefore share within

(t)

by mean-invariance.

row	$\mathrm{ratio}_{\mathrm{mean}}$	CV across $t$	$N$ grid
$d{=}2$ separated	$0.962$	$0.048$	$\{50,100,200,500\}$
$d{=}2$ partial-merge	$0.962$	$0.048$	$\{50,100,200,500\}$
$d{=}4$ simplex	$0.930$	$0.041$	$\{50,100,200,500\}$
$d{=}8$ simplex	$1.133$	$0.061$	$\{50,100,200,500,1000\}$
$d{=}16$ simplex	$1.153$	$0.053$	$\{50,100,200,500,1000,2000\}$

Appendix

Table 19: Cramér-Rao rate audit. Ratio of empirical

R_{\mathrm{oracle\_ols}}(t,N)

to the theoretical Cramér-Rao within-cluster term

\mathrm{CR}_{\mathrm{within}}(t,N)

, averaged over the kept-bin filter

t\geq 0.10

, on the same five rows as Table 18 . The cross-

t

coefficient of variation reports the within-row variability of the rate.

row	$\mathrm{oracle\_ols}$ $\alpha$ [CI]	EM $\alpha$ [CI]	MLP $\epsilon_{\mathrm{MLP}}^{B}$
$d{=}2$ sep	$1.24$ [ $0.97$ , $2.03$ ]	$1.23$ [ $0.99$ , $1.55$ ]	$1.56\%$
$d{=}2$ p.m.	$1.24$ [ $0.97$ , $2.03$ ]	$1.36$ [ $1.18$ , $1.67$ ]	$0.62\%$
$d{=}4$ simplex	$1.17$ [ $0.91$ , $1.71$ ]	$1.22$ [ $1.01$ , $1.58$ ]	$1.54\%$
$d{=}8$ simplex	$0.99$ [ $0.86$ , $1.26$ ]	$1.02$ [ $0.83$ , $1.50$ ]	$1.52\%$
$d{=}16$ simplex	$0.93$ [ $0.82$ , $1.08$ ]	$0.97$ [ $0.87$ , $1.13$ ]	$1.76\%$

Appendix

Table 20: Blind-attainability panel. Per-row

\alpha_{\mathrm{pooled}}

with

95\%

bootstrap CI for

\mathrm{oracle\_ols}

and closed-form blind EM (low-

t

window

t\in[0.10,0.28]

, five controlled bridge cells,

20

seeds), and

\epsilon_{\mathrm{MLP}}^{B}

as a fraction of within

(t)

N=2000

for the neural blind arm. PASS bands:

[0.85,1.15]

for

\mathrm{oracle\_ols}

[0.75,1.25]

for blind EM,

\epsilon_{\mathrm{MLP}}^{B}<5\%

for blind MLP.

$\Delta_{\min}/\sigma$	label-resid. $\alpha$ [CI]	two-stage $\alpha$ [CI]	soft-EM $\alpha$ [CI]	two-stage $R$ (%w)	between/within (%)
$1.41$	$1.05$ [ $0.88$ , $1.23$ ]	$0.21$ [ $0.20$ , $0.23$ ]	$0.59$ [ $0.44$ , $0.86$ ]	$7.13$	$12.1$
$3.00$	$1.13$ [ $0.92$ , $1.33$ ]	$0.17$ [ $0.15$ , $0.18$ ]	$1.30$ [ $0.72$ , $1.87$ ]	$13.43$	$13.5$
$4.24$	$1.08$ [ $0.92$ , $1.31$ ]	$0.20$ [ $0.18$ , $0.21$ ]	$1.39$ [ $0.88$ , $2.18$ ]	$9.60$	$9.9$
$5.66$	$1.06$ [ $0.92$ , $1.19$ ]	$0.34$ [ $0.31$ , $0.40$ ]	$1.40$ [ $0.72$ , $2.52$ ]	$3.96$	$4.0$
$8.00$	$1.09$ [ $0.93$ , $1.18$ ]	$0.82$ [ $0.75$ , $0.92$ ]	$1.02$ [ $0.67$ , $1.36$ ]	$0.45$	$0.5$
$11.31$	$0.98$ [ $0.85$ , $1.13$ ]	$0.98$ [ $0.85$ , $1.13$ ]	$0.94$ [ $0.53$ , $1.33$ ]	$0.23$	$<0.1$

Appendix

Table 21: Two-stage attainability sweep. Per-spread

\alpha_{\mathrm{pooled}}

with

95\%

bootstrap CI for label-oracle residual estimator, two-stage EM (estimated label via Remark 35 ), and soft-EM marginal Bayes; the two-stage residual at

N{=}2000

as percent of within

(t)

is reported alongside the analytic between/within ratio for reference. Window

t\in[0.10,0.28]

20

seeds,

M_{\mathrm{eval}}{=}10000

Hyperparameter	Value
Action Expert Width	$1024$
Action Expert Depth	$18$
Action Expert MLP Dim	$4096$
Batch Size	$128$
Chunk Size	$30$
Learning Rate	$10^{-5}$

Appendix

Table 22: Hyperparameters of BC on simulation (based on

\pi_{0.5}

Task	$\bar{L}_{10}/\bar{L}_{25}/\bar{L}_{100}$	$L_{100}/L_{10}$	$L_{25}/L_{10}$	Region ratios	Reg. CV	Peak $t$
collecting	$10.148$ / $10.435$ / $8.942$	$0.881$	$1.028$	$0.876$ / $0.878$ / $0.885$	$0.0045$	$0.755$
preparing	$8.003$ / $7.978$ / $6.821$	$0.852$	$0.997$	$0.850$ / $0.830$ / $0.869$	$0.0189$	$0.714$
cook_hot_dogs	$16.177$ / $16.737$ / $14.293$	$0.883$	$1.035$	$0.767$ / $0.769$ / $1.019$	$0.1391$	$0.428$
moving_boxes	$7.121$ / $6.825$ / $5.847$	$0.821$	$0.958$	$0.756$ / $0.789$ / $0.898$	$0.0743$	$0.388$
wash_cap	$7.856$ / $7.231$ / $5.606$	$0.713$	$0.920$	$0.797$ / $0.594$ / $0.768$	$0.1249$	$0.633$
cook_bacon	$17.284$ / $16.040$ / $13.577$	$0.786$	$0.928$	$0.715$ / $0.696$ / $0.850$	$0.0913$	$0.878$

Appendix

Table 23: Leakage-free BEHAVIOR-1K diagnostic panel. All fifteen tasks use the random_seed42 protocol with the explicit

N=10/25/100

forbidden training pool, reported over multiple demonstration seeds. Region ratios are low/mid/high

L_{100}/L_{10}

; lower is better. Reg. CV is the coefficient of variation across the three region ratios. Peak

t

is the time bin maximizing the improvement gap

L_{10}(t)-L_{100}(t)

Figure 8: Per-task ratio profiles and large-budget reference profile on BEHAVIOR-1K. Each panel shows one leakage-free trio from the fifteen-task BEHAVIOR-1K diagnostic panel over multiple demonstration seeds. The left axis plots the per-bin ratio

L_{100}(t)/L_{10}(t)

with the

\text{ratio}=1

reference line; the right axis plots the large-budget reference profile

L_{100}(t)

, which is an upper-biased proxy for

\mathcal{A}_{v}(t)

rather than an absolute Bayes-floor estimate. All fifteen trios improve in mean loss from

N=10

N=100

, but the temporal location of improvement is task-dependent.

Figure 9: Turning on a radio. BEHAVIOR-1K task visualization used for application-context reporting.

Figure 10: Picking up trash. BEHAVIOR-1K task visualization used for application-context reporting.

Figure 11: Spraying fruit trees. BEHAVIOR-1K task visualization used for application-context reporting.

Figure 12: Cooking hot dogs. BEHAVIOR-1K task visualization used for application-context reporting.

Figure 13: Washing a baseball cap. BEHAVIOR-1K task visualization used for application-context reporting.

Flow Matching

Flow matching learns continuous-time vector fields that transport simple noise distributions into data for efficient generative modeling.

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Jul 13, 2026

Velocity Scheduled Flow Matching

Flow matching trains a neural network to regress the conditional velocity along a linear interpolant between noise and data, and the number of network evaluations~(NFE) sets the cost of sampling. The straight-line interpolant carries an implicit choice: the sample moves at constant speed throughout the trajectory. We relax this choice and introduce Velocity Scheduled Flow Matching~(VSFM), which replaces the conditional target

x_1 - x_0

with

v(t)(x_1 - x_0)

for any nonnegative profile

v:[0,1]\to\mathbb{R}_{\geq 0}

satisfying

\int_0^1 v\,dt = 1

. We study six polynomial profiles drawn from motion planning. The first use of VSFM is at inference time: a pretrained linear flow-matching model can be sampled under any admissible profile by integrating its ODE on a non-uniform

τ

-schedule, with no retraining and no additional computation; on CIFAR-10 this lowers FID by up to

19.8\%

. Training from scratch under a braking profile gives a further reduction of

17.4\%

4

~NFE. Both gains follow from the local truncation error of the Euler integrator on the induced grid.

Vitalii Bondar

Cherkasy State Technological University Cherkasy, Ukraine

Flow Matching Conditional Flow Matching Few-Step Diffusion Sampling Flow-Based Generative Modeling

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May 8, 2026

What Time Is It? How Data Geometry Makes Time Conditioning Optional for Flow Matching

Recent work has shown that models flow matching models can be trained without explicit time conditioning, challenging the standard view that the interpolation time is needed to disambiguate velocity targets. But why should a time-blind model work at all? Decomposing the time-blind flow matching loss, we identify two sources of irreducible error: a coupling variance, which arises from ambiguous velocity targets induced by how noise and data points are paired, and the time-blindness gap, which is the additional error caused by ignoring time. This gap shows that time-blind training is strictly harder than conventional training, reinforcing the puzzle that time-blind models work so well in practice. We resolve this tension by showing that the geometry of high-dimensional data makes time identifiable directly from noisy observations. When data concentrates near a

k

-dimensional subspace, time can be recovered from the statistical structure of noisy interpolants in directions orthogonal to the data; under a spiked-covariance model, this yields a closed-form estimator that recovers

t

from a single observation

z

at rate

O(1/\sqrt{d-k})

for ambient dimension

d

. As a consequence, we prove that the time-blindness gap is asymptotically negligible relative to the coupling variance. We empirically demonstrate our identifiability result on real-world data and show that changing the coupling has a much larger effect on loss and sample quality than removing time conditioning across CIFAR-10, CelebA-HQ, and FFHQ. These results explain why time-blind flow matching works and show that the main practical lever is the choice of coupling, not explicit time conditioning.

Alec Helbling, Sebastian Gutierrez Hernandez, Benjamin Hoover +2

Georgia Tech · IBM Research

Flow Matching Flow-Based Generative Modeling

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Feb 11, 2026

Prediction--Loss Alignment for Sampler--Robust Flow Matching Training

Recent work has popularized a practical recipe in diffusion and flow matching: predict the clean signal

x

, convert it to a velocity, and train through a velocity-space loss. The conversion contains a singular endpoint amplification and therefore appears prone to unstable optimization, yet recent systems obtain strong empirical results with this recipe. We investigate this tension through the integrability of the pre-optimizer stochastic-gradient second moment. Under stated initialization conditions, the moment diverges under Uniform sampling; boundary-suppressing sampling can restore integrability under an additional upper-growth condition. We then show that prediction--loss alignment eliminates this conversion-induced source of non-integrability. Under a uniform moment bound, alignment yields a finite second moment for every timestep density, including Uniform sampling. Controlled experiments across continuous and binary settings reproduce the predicted sampler-dependent instability and show that aligned objectives remain trainable across the tested samplers. These results reconcile pointwise amplification with sampler-dependent empirical success and support alignment as a principled route to more robust flow-matching training.

Jiadong Hong, Lei Liu, Xinyu Bian +2

Zhejiang University, Hangzhou, China · Huawei Technologies Co., Ltd, Hong Kong

Flow Matching Diffusion Sampling Flow-Based Generative Modeling

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Abstract

Flow Matching

Explore similar work

Velocity Scheduled Flow Matching

What Time Is It? How Data Geometry Makes Time Conditioning Optional for Flow Matching

Prediction--Loss Alignment for Sampler--Robust Flow Matching Training