What Does a Stream Model Buy You in Flow Matching?

Jian Xu

cs.LGSep 28, 2026

What Does a Stream Model Buy You in Flow Matching?

Authors: Jian Xu

Organizations: RIKEN

Abstract

Stream-level flow matching replaces the linear interpolant of conditional flow matching (CFM) by a Gaussian-process (GP) stream connecting each source--target pair, and reports lower sample error than \icfm{} on 2-Gaussian, MNIST and CIFAR-10 benchmarks. We ask what such a stream model actually contributes. Three results answer the question. (i)\emph{Reduction.} The stream-level CFM objective depends on the stream law only through the per-time joint law of $(s_t,\sdot_t)$ , so the conditional paths a Gaussian stream can reach are exactly the Gaussian conditional paths CFM already parametrises; in the coordinate-wise, shared-scalar-kernel construction gpcfm actually uses, the entire design space collapses to two scalar curves $(m_t,v_t)$ , and cross-time covariance affects only estimator variance. (ii)\emph{The GP is a constrained chart of that space.} One kernel sets both $m_t$ and $v_t$ , so the paper's own recipe for widening coverage-shrinking the SE length-scale---destroys the interpolant (the midpoint mean weight falls from $1.03$ to $0.00$ ). On the 2-Gaussian benchmark this makes the GP chart diverge on $15/200$ runs at high coverage against $0/200$ for a decoupled $(m_t,v_t)$ chart ( $p=6.6\times10^{-5}$ ), and crossing the two curves shows the divergence tracks the mean, not the variance. On MNIST the same sweep does not diverge and the ordering reverses, so whether the coupling is harmful is benchmark-dependent; what holds on both is that the recipe buys nothing---no coverage level beats the paper's own, and past $\max_t\sqrt{v_t}\approx0.6$ both charts degrade. (iii)~\emph{Audit.} The released code does not implement the mechanism it describes: state and velocity are drawn independently ( $\mathrm{corr}=0.00\pm0.01$ against an intended $\pm0.83$ -- $0.99$ ).

Figures & tables

arm	FID	KID	$\Delta$ FID vs I-CFM [ $95\%$ CI]	$p$	$\Delta$ FID vs gp [ $95\%$ CI]	$p$
I-CFM	$33.33\pm 0.70$	$0.0214$	—		$-2.56$
gp (released)	$35.90\pm 0.78$	$0.0338$	$+2.56$ $[+1.71,+3.42]$ , wins $1/20$	$<10^{-3}$	—
gpfix	$33.70\pm 0.74$	$0.0217$	$+0.37$ $[-0.04,+0.78]$	$0.11$	$-2.19$ $[-3.04,-1.34]$	$<10^{-3}$
indep	$33.61\pm 0.70$	$0.0216$	$+0.28$ $[-0.09,+0.64]$	$0.20$	$-2.29$ $[-3.11,-1.46]$	$<10^{-3}$
mmdet	$33.67\pm 0.73$	$0.0218$	$+0.34$ $[-0.03,+0.70]$	$0.10$	$-2.23$ $[-3.08,-1.37]$	$<10^{-3}$

Table 1: MNIST, paper protocol,

20

seeds per arm; paired differences over the same seeds (Wilcoxon signed-rank). The released method is worse than I-CFM; every correct implementation of the intended law is equivalent to I-CFM within the paper’s claimed

0.95

FID effect (TOST

p\leq 0.006

); removing the bug in any of three ways recovers

2.2

–

2.3

FID. The paper reports I-CFM

44.50\,(0.18)

and GP-CFM

43.55\,(0.13)

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

$\max_{t}\sqrt{v_{t}}$	$\ell$	gpfix (GP chart)	bump (decoupled)	paired gpfix $-$ bump [ $95\%$ CI]
0.0071	5 (paper)	$33.53\pm 1.10$	$33.12\pm 1.05$	$+0.41$ $[-0.07,+0.89]$
0.1745	1	$30.80\pm 1.01$	$34.26\pm 1.34$	$-3.46$ $[-6.66,-0.27]$
0.5933	0.5	$33.91\pm 1.39$	$43.04\pm 3.14$	$-9.13$ $[-15.79,-2.47]$
0.8628	0.35	$42.02\pm 4.04$	$57.06\pm 5.88$	$-15.04$ $[-29.87,-0.21]$
0.9815	0.25	$50.36\pm 4.49$	$59.96\pm 6.96$	$-9.60$ $[-25.58,+6.39]$

Appendix

Table 2: MNIST coverage sweep, ten seeds per cell, paper protocol otherwise. The cliff does not transfer: neither chart diverges and the GP chart is the better of the two at matched coverage, the opposite ordering to the toy. What does transfer: no level beats the paper’s own, and past

\max_{t}\sqrt{v_{t}}\approx 0.6

both charts degrade.

mean $m_{t}$	variance $v_{t}$	arm	$\ell=0.35$	$\ell=0.25$
GP	GP	gpfix	7/40	7/40
GP	GP	mmdet	5/40	5/40
GP	decoupled	gpmean_decvar	5/40	6/40
straight	GP	straight_gpvar	0/40	0/40
straight	decoupled	bump	0/40	0/40

Appendix

Table 3: Which curve causes the cliff? Divergent runs (

W_{2}>5

) out of

40

seeds per cell at the two high-coverage levels, with the mean and variance curves crossed. Replacing the GP’s variance changes nothing; replacing its mean removes the cliff entirely. gpfix and mmdet are independent implementations of the same (GP, GP) cell. Medians and the full five-level sweep, where no arm diverges below coverage

0.6

, are in Table 5 .

arm	seed 0	seed 1	seed 2	seed 3	seed 4	mean $\pm$ sd	paper (one run)
I-CFM	3.79	3.64	3.61	3.55	3.58	$3.63\pm 0.09$	3.75
gp (released)	5.29	5.37	5.21	5.30	5.34	$5.30\pm 0.06$	3.62
gpfix (intended law)	3.79	3.61	3.68	3.65	3.67	$3.68\pm 0.07$	—

Appendix

Table 4: CIFAR-10,

400

k steps, FID with

50

k samples (lower is better); five training seeds per arm; the paper’s single-run numbers for reference.

$\max_{t}\sqrt{v_{t}}$	$\ell$	gpfix (GP chart)		bump (decoupled)		mmdet (GP curves, no sampler)
$\max_{t}\sqrt{v_{t}}$	$\ell$	median [IQR]	diverged	median [IQR]	diverged	median	diverged
0.044	2.00	1.63 [0.93, 1.88]	0/40	1.59 [1.12, 2.08]	0/40	1.65	0/40
0.175	1.00	1.54 [0.93, 1.81]	0/40	1.57 [1.14, 2.00]	0/40	—	—
0.593	0.50	1.52 [1.18, 2.12]	1/40	1.68 [1.21, 2.56]	0/40	1.77	0/40
0.863	0.35	1.60 [0.94, 2.52]	7/40	1.46 [1.06, 2.01]	0/40	1.69	5/40
0.982	0.25	2.11 [0.85, 3.39]	7/40	1.36 [1.14, 1.89]	0/40	1.62	5/40

Appendix

Table 5: Coverage sweep on the 2-Gaussian benchmark (

40

seeds per cell). Medians never separate (paired sign test

p\geq 0.08

at every level); the tails do. Right block: attribution arm mmdet , which uses the GP’s own

(m_{t},v_{t})

through a

2\times 2

per-time computation and never touches the joint sampler.

arm	mean $W_{2}$	sd	se	paired diff vs. gp [ $95\%$ CI]
I-CFM	1.601	0.765	0.077	$+0.074$ $[-0.109,+0.257]$
gp (released)	1.526	0.757	0.076	—
gpfix	1.553	0.780	0.078	$+0.027$ $[-0.148,+0.201]$
mmstoch	1.651	0.741	0.074	$+0.124$ $[-0.051,+0.300]$
mmdet	1.645	0.791	0.079	$+0.119$ $[-0.073,+0.311]$
bump (no GP)	1.661	0.772	0.077	$+0.135$ $[-0.054,+0.323]$

Appendix

Table 6: 2-Gaussian benchmark,

100

seeds per arm, paper protocol. Paired differences use the same seed for initialisation and evaluation noise. No pair of arms is distinguishable; the paper’s reported gap (

-0.03

) lies well inside every interval.

$\ell$	$\max_{t}\sqrt{v_{t}}$	$a_{0}(\tfrac{1}{2})+a_{1}(\tfrac{1}{2})$	$\max_{t}\|m_{t}-\mathrm{lin}_{t}\|$
4.00	0.011	1.008	0.005
2.00	0.044	1.030	0.018
1.00	0.175	1.099	0.062
0.50	0.593	1.069	0.104
0.25	0.982	0.271	0.365
0.10	1.000	0.000	0.706

Appendix

Table 7: The SE chart couples coverage and interpolation equation 8 .

\max_{t}\sqrt{v_{t}}

is the widest state spread;

a_{0}+a_{1}

t=\tfrac{1}{2}

equals

1

for a straight interpolant; the last column is the maximal deviation of

m_{t}

from the straight line per unit endpoint separation.

$b$	$\mathrm{corr}(s_{t_{1}},\dot{s}_{t_{1}})$		$b$	$i$	$d$	$\mathrm{Var}(s^{(d)}_{t_{i}})$
$b$	empirical	intended	$b$	$i$	$d$	empirical	bug formula	intended
0	$+0.003$	$-0.989$	0	1	1	0.0050	0.0051	0.0008
1	$+0.024$	$+0.991$	0	2	0	0.0112	0.0114	0.0006
2	$+0.005$	$+0.950$	1	2	0	0.0172	0.0174	0.0002
3	$-0.014$	$-0.834$	3	2	0	0.0176	0.0174	0.0000

Appendix

Table 8: The released sampler does not draw from the law it derives (

n_{B}=4

n_{t}=3

d=2

6{,}000

draws per stream). Left: state–velocity correlation at

t_{1}

. Right: variance of selected coordinates vs. the value predicted by the index arithmetic of Appendix B and the intended value; all

24

coordinates agree with the bug formula to three decimals.

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Sep 15, 2026, cs.LGJ/K move · Enter open · S save

Same Flow, Different Paths: Variance Reduction in Flow Matching

Alexander Tyurin

AXXX, Moscow, Russia · Applied AI Institute, Moscow, Russia

Conditional Flow Matching Variance Reduction Stochastic Gradient Descent Variance Shortest Paths Path

Open paper AskPDF arXiv

In flow matching (FM), a velocity model

v_θ

is trained using a predefined path

g_t

that connects data and noise samples (e.g.,

g_t(x_0, x_1) = (1 - t) x_0 + t x_1

). In this work, we study the choice of this path from an optimization perspective by analyzing the variance of stochastic gradients. We consider the class

G(p_t,v^\star_t)

of paths that induce the same marginal distributions

p_t

and marginal velocity field

v^\star_t

, and therefore the same FM objective. Our main finding is that the choice of path

g_t

can fundamentally change the convergence rate of SGD, even when the FM objective remains exactly the same. (i) For a linear velocity model and one-dimensional Gaussian data, we derive a tight bound on the SGD iteration complexity up to logarithmic factors and find an analytically optimal path that minimizes this bound among linear paths inducing the same FM problem. (ii) We then extend the variance analysis to general FM problems and formulate path selection at a fixed

θ

as the variance-minimization problem PathOpt

_θ

, constrained to

g_t\in G(p_t,v^\star_t)

. We show that this constraint is essential: reducing variance without it can lead to slower convergence. (iii) Since the constraint

g_t \in G(p_t,v^\star_t)

cannot generally be verified directly, we derive an equivalent formulation with constraints that can be estimated from samples, allowing paths to be found numerically. Our theoretical results are supported by experiments with Gaussian data, Gaussian mixture models, and real datasets.

Abstract

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