An Error Analysis of Flow Matching for Deep Generative Modeling
Authors: Zhengyu Zhou
Weiwei Liu
Venue: ICML 2025 spotlightposter (2025)
Keywords: Statistical Learning Theory
Primary area: theory->learning_theory

Abstract: Continuous Normalizing Flows (CNFs) have proven to be a highly efficient technique for generative modeling of complex data since the introduction of Flow Matching (FM). The core of FM is to learn the constructed velocity fields of CNFs through deep least squares regression. Despite its empirical effectiveness, theoretical investigations of FM remain limited. In this paper, we present the first end-to-end error analysis of CNFs built upon FM. Our analysis shows that for general target distributions with bounded support, the generated distribution of FM is guaranteed to converge to the target distribution in the sense of the Wasserstein-2 distance. Furthermore, the convergence rate is significantly improved under an additional mild Lipschitz condition of the target score function.

An Error Analysis of Flow Matching for Deep Generative Modeling
Zhengyu Zhou'
Weiwei Liu'
Abstract
ential equation (ODE) (Chen al., 2018: Grathwohl et al., 2019). However, training neural ODEs at scale is intractable, as it requires simulating the ODE. The introduction of Flow Matching (FM) has made CNFs highly efficient for gener- ative modeling of complex data (Karras et al., 2022; Liu et al., 2023; Albergo & Vanden- Eijjdenn, 2023; Lipman et al., 2023; Neklyudov al., 2022; Tong et al., 2023; Chen & Lip- man, 2023; Albergo et al., 2023; Shi al., 2023; De Bortoli et al., 2021).
Continuous Normalizing Flows (CNFs) have proven to be highly efficient technique for gen- erative modeling of complex data since the in- troduction of Flow Matching (FM). The core of FM is to learn the constructed velocity fields of CNTs through deep least squares regression. De- spite its empirical effectiveness, theoretical inves- tigations of FM remain limited. In this paper, we present the first end-to-end error analysis of CNFs built upon FM. Our analysis shows that for general target distributions with bounded support, the generated distribution of FM is guaranteed to converge to the target distribution in the sense of the Wasserstein-2 distance. Furthermore, the con- vergence rate is significantly improved under an additional mild Lipschitz condition of the target score function.
The success of FM motivates a line of research investigating the generation quality guarantees from the perspective of sampling (A (Albergo & Vanden-Eijnden, 2023; Albergo et al., 2023; Lu et al., 2022; Chen et al., 2023c). These works assume the underlying velocity field accurately estimated up to a small error under LL-norm and provide generation quality guarantees. However, two issues remain unsolved in these works. The first is to provide guarantees for learning the velocity field of the underlying ODE. The second is to relax the strong assumptions on the underlying velocity field, which may be hard to check. This paper takes a step forward by providing an end-to-end analysis of the deep generative modeling based FM under mild assumptions. Our main contributions summarized as follows:
1. Introduction
Contemporary generative models have primarily been de- signed around the construction of map between two prob- ability distributions that transform samples from the prior distribution to the target distribution. The roots of transport- based sampling and density estimation can be traced back to maximum entropy methods for Gaussianizing data (Tabak & Turner, 2013; Tabak & Handen-Eijnden, 2010). Normal- izing Flows (NFs) provide a neural network implementation of these methods by imposing structured transformation to make the change of measure tractable in discrete, sequen- tial steps (Dinh et al., 2017; Durkan et al., 2019; Huang et al., 2018; Papamakarios et al., 2017; Rezende & Mo- hamed, 2015). Continuous Normalizing Flows (CNFs) ex- tend this idea to a continuous-time setting by viewing the map T(x)= (x) as the solution of an ordinary differ-
We provide the first end-to-end analysis for the deep generative models based on FM.
We prove that the deep generative models built upon FM are guaranteed to converge to the target distribution under mild assumption. Furthermore, the convergence rate gets significantly improved under additional Lipschitz condition of the target score function.
1.1. Assumptions
Assumption 1.1 (Bounded support). The target distribution mi is supported on [0. 1]d
'Shooo of Computer Science, National Engineering Research Center for Multimedia Software. Institute of Artificial Intelligence and Hubei Key Laboratory of Multimedia and Network Communi- cation Engineering, Wuhan University, Wuhan, China. Correspon- dence to: Weiwei Liu <liuweiwei863@ gmail.com>.
Assumption 1.2 (Lipschitz score). Let Ti (dx)
'End-to-end learning in generative modeling involves using finite samples from the target distribution as input to learn the underlying distribution, and then generating samples from the learned distribution as output. The goal of end-to-end end analysis is to provide guarantees for the accuracy of the learned distribution based the finite input samples, enabling more reliable generative modeling.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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An Error Analysis of Flow Matching for Deep Generative Modeling
da. Moreover, the potential ((x) is twice contin- uously differentiable and satisfies -al= N (v)( N aT with a >> 1.
This result highlights the effectiveness of CNFs based on FM in learning the underlying smooth distribution.
Lemma 1.3. Suppose that Assumption 1.1 holds. Then v'(x, 1) is E-Lipschitz continuous w.r.l. x on R x [O,T], where& & S max T- Td }. } Further, if 1 T < 1,
1.3. Related Work
Continuous Normalizing Flows CNF' are proposed by viewing the map It is not until the introduction of FM that CNFs have grown to be an efficient method for the generative modeling of complex data (Karras et al., 2022; Liu et al., 2023; Albergo & Vanden- Mijuden,, 2023; Lipman et al., 2023; Neklyudov et al., 2022; Tong et al., 2023; Chen & Lipman, 2023; Al- bergo et al., 2023; Shi al.. 2023; De Bortoli et al., 2021). The key idea of FM is to learn the constructed velocity fields of CNFs through deep least squares regression. In (Liu et al., 2023), a linear interpolant is proposed with a focus on straight paths. This is employed as a step towards rectifying the transport paths (Liu, 2022) through a pro- cedure which improves sampling efficiency. In (Lipman et al., 2023), the interpolant picture is assembled from the perspective of conditional probability paths connecting to a Gaussian, where noise convolution is used to improve the learning, at the cost of biasing the method. The paper (Tong et al., 2023) introduces novel simulation-free objective for learning continuous-time flows conditioned on a general distribution. Further, the authors have shown that lifting the static optimal transport problem to the dynamic setting leads more efficient training and inference of flow models by lowering the variance of the objective and simplifying flows. FM is extended to the Riemannian setting by Chen & Lipman (2023). Another line of work points out that the probability path of CNFs encompasses that of the Diffusion Models (DMs) (Albergo et al., 2023; Lipman et al., 2023; Albergo & Vanden-Eijnden, 2023). If made to match the performance of their stochastic counterparts, ODEd methods exhibit number of desirable characteristics that are absent for SDEs, such as exact, computationally tractable formula for the likelihood and easy application of well-developed adaptive integration schemes for sampling. Further, one of the most successful techniques of acceler- ating continuous time process- based sampling, distillation (Liu al., 2023; Song et al., 2023; Salimans & Ho, 2022: Zheng al., 2022; Luhman & Luhman, 2021), requires deterministic samplers.
= (x,z) as the solution of an ODE,
we have v² is (1-Ty3 -Lisshhitz continuous w.r.t. x.
Lemma 1.4. Suppose that Assumption 1.1 and Assump- tion 1.2 hold. Then e" (x.t) t) is <(a, d) Lipschitz continuous 2 on R.d x [0, 1] w.E.I. æ, where Ç(a, d) 2 (a va+3)
scales polynomially with a and d.
Remark 1.5. Previous work simply assumes the score func- tion or velocity field to be Lipschitz continuous w.n.t. æ for every t (Chen et al., 2023c;a). In this paper, we follow Wibisono & Jog (201 8a;b); Mikulincer & Shenfeld (2021; 2022); Chewi & Pooladian (2022): Gao et al. (2024) to pro- vide the Lipschitz continuity of the velocity field from the properties of the target distribution.
The proof's in this section are deferred to Appendix D.5.
1.2. Main Results
All proofs of this section is deferred to Appendix C.3.
Theorem 1.6 (Consistency). Suppose Assumption 1.1 holds. Given n samples from target distribution σι and the net- works as in Theorem 4.4, with parameter Ç replaced (1-Tya, we use the estimated velocity field in (11), to generate samples and choose the maximal step size max |lx+1 - 141 On *ेअ) ) and early stop- ping time T(m) = - (log m) -1/6, we have W2(TT(m); m 0, in probability, where Tr()) denotes the generated distribution at time T(n).
by d
The consistency of FM mainly based on mild assump- tion, i.e. boundedness, which justifies the use of CNFs based on FM.
Theorem 1.7 (Improved convergence rate). Suppose As- sumption 1.1 and Assumption 1.2 hold. Given n samples from target distribution Ti and the networks as in The- orem 4.4, with parameter Ç replaced by Ç(a, d) defined in Lemma 1.4, we use the estimated velocity field in (11) to generate sumples and choose the maximal step size -(aeo) ) and early stop- ping time (nn) 1- 1 n डाकाक,, Then, with probability of al least 1- n. we have
Lipschitz Score v.s. Lipschitz Velocity Field In analyz- ing the convergence of DMs and ODE-based models, the assumption of Lipschitz continuity for the score function or the velocity field has been widely used in previous works (Chen et al., 2023c; Lu et al., 2022: Albergo & Vanden- Eijnden, 2023; Chen et al., 2023a). However, these works simply assume the Lipschitzness. In contrast, our paper takes a step forward and rigorously proves that the velocity field is Lipschitz continuous under mild assumptions on the target distribution. By doing so, we provide a stronger
144411ttt max
W2(FT(();T11 Ō 5T445)
where TT()) denotes the generated distribution at time T(n).
2

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An Error Analysis of Flow Matching for Deep Generative Modeling
theoretical foundation for the application of CNFs based on FM, and help bridge the gap between theory and practice.
differential equation (ODE) on time E [0, 1],
dz: = v (Zz t)dt,
(I)
Analysis of ODE-based Models Significant recent works (Albergo et al., 2023; Chen et al., 2023c: Lu et al., 2022) have put effort into controlling the KL divergence between the generated distribution and the target distribution. These studies have demonstrated that simply regressing the veloc- ity field insufficient to control the likelihood with ODE- based models. Instead, more advanced learning schemes are required to ensure that the Fisher divergence is kept under control. The work (Albergo & Vanden-Eijndee 2023) has shown that the Wasserstein-2 distance between the gener- ated distribution and the target distribution can be controlled by the objective of regressing the velocity field, assuming the estimated velocity field is Lipschitz continuous. In our paper, we take different approach, demonstrating that the true velocity field can be well approximated by Lip- schitz neural network. We compare our work with con- current analyses for ODE-based models in Table I where U (t: 61, 82, 53, C, g) in the third row is an increasing func- tion for 61, 82 and 53 (Lu et al., 2022), where ó; an upper bound for the score matching objective of order i, i 1,2,3.
which converts Zo from To to Z1 following TT. A line of research (Liuu al., 2023; Liu, 2022; Albergo & Vanden- Eijnden, 2023; Lipman et al., 2023; Neklyudov et al., 2022; Wu et al., 2022; Lee et al., 2023b; Tong et al., 2023; Chen & Lipman, 2023; Albergo et al., 2023; Shi et al., 2023) points out that, the vector field can be found by solving a least square regression problem:
min Lo(w) == L Ex [I(X, Xo) (x1 de,
with X, = (1 x).
tX1
(2)
where Xo ~ Tu. X1 ~ TT, and X, is the linear interpolation between Xo and Xi. The exact minimum of (2) is achieved by
v (xx t) 13 [X1 Xol X = x|
(3)
Velocity Field Approximation In practice, the velocity field v² is approximated by neural networks. To avoid instability, we often clip the integral interval [o, 1] with T. Namely, we consider the following loss function:
2. Preliminaries
1 (' Ex,.x. min L(v): T
xX,,,))) dt, with X, = 1X1 + (1 X)
Notations We denote [N] == (0,... ..... .N. 13. For matrix A and B, we say A 3 B, if A is positive semi-definite. We denote the identity matrix in Rdxd by Ia. For vector Rdd, we define xr 82 = .aa." We denote the (2-norm of
(4)
Given a family of neural networks NN, we consider the following approximation error,
a vector x by ||>||| = VE 27. We define the operator norm of a matrix A as ||Allop := sup |m| şi Az/] For a twice continuously differentiable function f: d + R, let VS, V. and A denote its gradient, Hessian, and Lapla- cian, respectively. For probability density function æ and a measurable function f : R2d -> R, we define the 1.2 (m)-nom of as as (I]llep()) = (z())) 1/2 We define ( as [[]ll(=(()
/ ()((;t)-0* 00 (10)) inf wENN
inf
((v)-L(v))
VENN
(5)
where m, is the probability distribution of X, defined (2). The equivalence in (5) is deferred to Lemma 4.1. We also consider the best approximator in the neural networks
suPEEk [=)). For a vector function v : Rd + Rdd we define its (m)) [[0]) = |||e||ll<*()) and (K)- norm as ||0||2= (K) = |||||||| (x))). We use the asymp- totic notation f(x) = Olg(z)) to denote the statement that f())= < C1(x:) for some constant C >>> 0 and O()) to ignore the logarithm. Given two distributions μ and v. the W distance is defined as W2(,vv) == infre)((..)) -(z,y)>=[lz - y1 2|/2 where 11( v) is the set of all couplings of and v. A coupling is a joint distri- bution on Rdd x Rdd whose marginals are f and v on first and second factors, respectively.
① E argmin L(v).
(6)
.ENN
We organize the remaining sections as follows: In Section 3, we show that the true velocity field can be well approxi- mated by a Lipschitz neural network. Section 4 establishes that the optimal neural network can be efficiently estimated. Finally, in Section 5, we analyze the error of distribution recovery using the estimated velocity field.
3. Approximation
In practice, the true velocity field is approximated by neural networks. To ensure effective learning, the network class should be expressive enough to approximate the true veloc- ity field.
Flow Matching Given independent empirical observations of Xo ~ an and X, TI, we want to find an ordinary
3

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An Error Analysis of Flow Matching for Deep Generative Modeling
End-to-end Analysis
Main Assumptions
Theoretical Results
(Albergo & Vanden-Eijnden, 2023)
G is K K-Lipschitz in uniformly
K
W2(2n, S se! e H(o)
on (t, E [0. R
KL(PT1g) S provided 82 C₁ and th S C where E₁ and E2 depends polynomially on parameters in assumptions
V In (x)) is -pppcciis
in æ and satisfies
(Chen et al., 2023c)
V In 9 (x)]l S
X
B (1
| VV (x))l)
V x log oDD (x, (2 2 S C,
Dr pg ODE) S (t: (1; 82, 83. C.
(Lu et al., 2022)
log gr is C-Lipschitz, uniformly for t
Bounded support
Consistency
Ours
Bounded support and Lipschitzness of the target score functions
W2(TT(m)) #1) Ö कााक
Table 1. Comparison of existing thcoretical results on ODE-hased models.
Neural Network Structure We configure the ReLU net- work Bg in the following way.
esis class NN with
d + log 1
L = 0
63/2 (3/2(1og(( (log(d/)) d=1
5a - (at-(a41)
M = o
(1 T72
(/o(l (d/E)) d41
[oog 1 + atd))
=
-
71,
T)a
=(v(a,
(W2o(1) (Wio()) + (1) ; network width bounded by M,
bL) ((Woo( +
(WZ-10(-) + o
dlogg
k=0
1 T
sup]|o(a 1)ll K, max( Will SN 2.
dd log(d//)) (1-T)²
k= 0
(Vlog(d/E)
m 10dç,
L [[ + [bblllo) - J,
for i --r== 1,, L.
(1- (
(V log (/gg((//E)
1=1
72 0
||v t) v(æ2, f)ll < 71 || x21 for any E [0,
|o( (1) v(x 12)l S 72/1/1 d2ll for any x }
There exists an Gg E NN, such that for any t E [0,T], we have
S (Vd 1);,
where ni is the distribution of X, tX1 + (1 - X)
where the network width refers to the maximum dimensions of the weight matrices, a is the ReLU activation, and || 1l and || lo denote the maximum magnitude of entries and the number nonzero entries, respectively. In the sequel, we write the neural network class as NN for brevity.
The proof of Theorem 3.1 can be found in Appendix A.1. Universal Approximation under the - Many ex- isting universal approximation theory of neural networks focus approximating target functions main under the L, norm (Yarotsky, 2017; Schmidt-Hieber 2020; Gühring et al., 2020). Instead, we provide an L approximation error bound over the unbounded input do- main, where we tackle the unboundedness through trunca- tion argument.
compact do-
Theorem 3.1. Suppose Assumption 1.I holds. Given an approximation error E > 0, for any velocity field v" with Lipschitz constant Ç w.r.t æ on [0, T), we choose the hypoth-

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An Error Analysis of Flow Matching for Deep Generative Modeling
Lipschitz Neural Network Conventional universal approx- imation theories of neural networks do not typically pro- vide guarantees on the Lipschitz continuity of the network (Cybenko, 1989; Barron, 1993; Yarotsky, 2017), which is important for effective learning of the true velocity field. A line of research (Jiao et al., 2023; Dahal et al., 2022; Huang et al., 2022) studies Lipschitz neural networks motivated by the Wasserstein Generative Adversarial Network (WGAN) (Arjovsky et al., 2017). The paper (Jiao al., 2023) studies the approximation capacity of ReLU neural networks with norm constraints on the weights. Meanwhile, (Huang et al., 2022, Lemma 11) provides an explicit bound on the Lipscitz constant required for approximating Hölder functions. In (Dahal et al., 2022), statistical guarantees for WGAN are provided under the Wasserstein I-distance, assuming that the data distribution is supported on a low-dimensional man- ifold. These techniques are scalable to our analysis, and for brevity, we adopt the proof of the work (Chen et al., 2023a). The key difference between our paper and (Chen et al., 2023a) is that they assume the on-support score function is Lipschitz uniformly for E [t. whereas our paper derives the Lipschitzness of the true velocity field from the assumption on the target distribution. In our construction, the Lipschitz continuity constraints 711 and 722 do not un- dermine the approximation power of the neural networks. In practice, such Lipschitz regularity is often enforced dur- ing training by adding regularization (Virmaux & Scaman, 2018; Pauli et al., 2021; Gouk et al., 2021). From a theo- retical perspective, the Lipschitz property of the estimated velocity field crucial in bounding the distribution recov- ery error, as demonstrate in Section 5. Moreover, the Lipschitz continuity of the estimated velocity field ensures the existence and uniqueness of the solution of the ODE.
as
1Too(,t) (-
1/2
VhelER (o((x t) - 2 t)l (dxc)
1/2
(hellsa 1 1) - va(x,, (01(=T((()))
(I)
The error term (I) is directly bounded by the approximation error of Tg on the hypercube. It is worth noting that since Up is bounded and v (X,, t) has a bounded second moment, the term (II) can be controlled by utilizing the tail behavior of πT.
4. Generalization
In this section, we consider the generalization error of es- timating the velocity field. We begin with the following connection between the loss function [(v) and the L2 ap-
proximation error [|m(-.t)- - v" (c,t))l (2(r))
Lemma 4.1. The following holds for any v(x, t):
1 T 1 [|0(,tt) v² u- (.
((v) C(o)-
Proof. By some calculus, we have
E [|||X, - Xo - -((x, 0)1
=E [[IX, - Xo 10--0" -v"(X, () + u² (X., u(,,)) [11]]
Time as an Additional Input Dimension In our approach, we introduce time as an extra input dimension to the neural network, and the network size scales polynomially with the Lipschitz constant 7 of the true velocity field with respect to t. In Section D. we derive an upper bound for 7 clipped time span [0, T]. where T < 1.
= E [IIX, - Xo -v"(X,,0) + [|((,)) - v (1
+ 2E [(X, - No - (X,,,,,, n n² (X,,t) - v(xe, t))l
By taking expectation conditioned on Xx, we have
E K - Xo - v"(X,,t), (X,,, - v(),,t))
=E [[ - Xo Xo-oi o0 ox(,, 00 (X,, u x))x])
Proof Sketch Theorem 3.1 is established by construction.
A noteworthy distinction from the existing universal approx- imation theories is that the input domain of the velocity field is unbounded. To establish the theorem, we lever- age a truncation argument. Let R be truncation radius. On the hypercube [- R, R] x [0, T), we construct Up piece-wise linear function to approximate x² in the sense of _ (1- R, 12]2 x [0, 2J). Outside the hypercube, we simply set Un = 0. The 2 approximation error can be decomposed
=E [E[X; - X
(X+,t) v(X,, +))]
=E tX),,) - oX,,,, 0" (Xe, oX,,, t))] = 0.
Substituting the above identity into (7) and integrating on interval [0, 27. we obtain
((v) _(v)) 1 1 [|0(;)))) - v0., (1
which concludes the proof.
According to Lemma 4.1, minimizing (4) is equivalent to minimizing the difference between the network and the true velocity field in (.
5

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An Error Analysis of Flow Matching for Deep Generative Modeling
Empirical Evaluation Let define
where the inequality follows from ERM, and V is defined in (6). Note that, for any NN, we have
v(tx + (1 t) t)ll? f( 1) = 1 _/= /ll=- -10--00 xo T
1
[(v) - [(v) -L(v) L(v) + L(o) Σ(v)
(8) In this paper, we choose the standard Gaussian distribution as the prior distribution, i.c., To = (V, Ig), where is the dimension of the data. Given n independent and identically distributed (i.i.d.) samples { from n,, we have the following empirical version of the least square loss:
1
(f(v) ((01,,, u))
(14)
1=1
+ n
(f(01,;, v) -((
By defining H
{e(-.e) e)
n I(O)=
= _ 1
Σ ((01,)))) v).
NN M, J, K, KG 71, 12)}. we can apply conven- tional statistical learning arguments to analyze the first term within the function class H. Due to the unbounded nature of the loss function x controlling the second term requires an additional truncation argument. We will provide further details on our approach at the end of this section.
Since our main interest lies in the sample complexity of sam- pling from T1, we consider the situation where f(x, v) can be computed exactly. However, in the usual implementation, the expectation in (8) is replaced by empirical evaluation. Given m i.i.d, samples {(t,, (o,,) from Unif[o, T] and To. which are cheap to generate, then (8) has the following empirical evaluation:
- v(tx + (1 t) t)l²,
The complexity of function class can be measured using the covering number.
Definition 4.2 (Covering number). Let p be a pscudo-metric M and S C .M. For any ō >> 0, set A S .M is called 8-covering of S if for any x E S there exists E A such that p(a. y) V 6. The d-covering number of S, denoted by N6, S, p). is the minimum cardinality of any d-covering of S.
1 m
((x, v)
- Foo- uttz + (1 (1-tt)aag rooj.
:...:0. ty)l1.
(10) Due to the efficacy of sampling t and æo, (xx, v) can be effi- ciently approximated by f(xx v) via polynomial size sample from Unif 0,, T] and no. which will be explained exactly in Section 4.1. Now, we consider the Empirical Risk Mini- mization (ERM):
The function class H exhibits the following properties which useful for analyzing the generalization error.
(i) Bounded sup-norm According to Theorem 3.1, the estimated velocity field 6(z, t) can be chosen to satisfy
1 Σ 71
t E argmin wEV
((v)=
(11)
the condition [ S K = o V Vlo(dd/e) 1-T
4.1. Error Decomposition
Then Lemma B. shows that
The error of the estimated vector field (11) can be decom- posed as:
log (d/s) T)²
sup sup ((x,v) S K2 ddd
VENN [0
( [(v*) C(t)
inf L(v)
DENN
Generalization
(ii) Covering number evaluation The covering number of the network class selected in Theorem 3.1 is evaluated as follows:
(12)
inf NENN
(C(v)
Approximation
log.N (8, NN, ll I==<(--.
Further, the generalization error has the following decompo- sition:
()(DV1)
(15)
S JL
(())
inf ((v)=L())
[(0) -(()
(()) - -(()
WNNN
The above evaluation can be found in (Chen et al., 2022b, Lemma 5.3). Based on the above result, we have the follow- ing evaluation for the covering number of the loss function class H:
+ +ā(0) L(T)
< C()) 2(6) + ((6)- - L(Ø),
(13)
6

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An Error Analysis of Flow Matching for Deep Generative Modeling
Lemma 4.3. The covering number of H is evaluated by
with high probability. Specifically, the second term can be decomposed as follows:
log.V(6,, Hl 10.1l-11=((00))) ((o11))
f( v) - [((x1,). v) = ( 2) grrnee ( 1,2: V)
(K+ (L (o"( +d/o0)
X JLlog
Truncation
(16)
grune (11 v)
grune 1, v)
The proof of Lemma 4.3 is deferred to Appendix B.2. It is worth noting that the evaluation is non-trivial because the evaluation in (15) considers the -nom on a bounded subspace, while the region of integration in (8) is unbounded. To overcome this challenge, we utilize truncation argument to provide the covering number evaluation for H.
Statistical
Zinnn (1i, v) ((1 v).
Truncation (I)
where grunn (11,5 v)
E, mg aamollly |
Eo
v(tx + (1 t) 1 100
RH and
gatune (21,5, v) == mi プラ= |11,, 20,5 vtt +
Based on the above discussion, we can now derive the fol- lowing generalization bound
(1 - t)zo, ((1 I R}. We can control Truncation error (I) by utilizing the concentration of Gaussian variables. On the other hand, Statistical error can be controlled using covering number argument. Furthermore, Truncation error (II) is likely to be equal to zero due to the concentration of Gaussian variables.
Theorem 4.4. Suppose Assumption 1.1 holds. For any velocity field v" with Lipschitz constant Ç w.r.t æ, given n samples (a from Tı and m. samples from xn and Onif TT, we choose NN as in Theorem 3.1 with E = 445. Then with probubility of at least 1 n. it holds
5. Sampling
1
(0(,t)- v² --,',,,, (.tt)ll2((e))
T
This section establishes distribution recovery guarantees using the estimated velocity field.
( <2/2 (n n
(n- पमन + 260-5) m mm:)) 1))
2+1
= (1 (1-
Estimated Sampling Dynamics Given the estimated veloe- ity field 6 we can generate samples from an approximation of the continuous flow ODE starting from the prior distribu- tion:
where we omit factors in d, logn, log log(1 - T). By setting m to be the order (n)), we oblain the convergence
the
rate of order δ
ga12
dx,(x)) ((X,(x), Ro(x) Tu, OstsT. T.
The proof can be found in Appendix B.3. To the best of our knowledge, Theorem 4.4 provides the first explicit sample complexity bound for FM. Theorem 4.4 becomes vacuous when T tends to 1 with fixed sample size n. This is a consequence of the blowup of the velocity field v" (x, t) as i tends to 1. Although a smaller early stopping time leads to better generalization error, stopping the sampling process at an early time results in a bad distribution recovery. In Section 5, will show the tradeoff in the choice of stopping time T.
(17)
Proposition 5.1. Suppose Assumption 1.1 holds. For any velocity field v² with Lipschitz constant w.r.t. æ, given samples { from क and m samples from no and Unif[0, T1, we choose NN as in Theorem 3.1 with n क=ड, Then with probability of at least +. it holds
W2(TT, = -0(""
54/4
(1- Tyz" "1-172" )
xts
(18)
Proof Sketch The generalization error is divided into two terms. The first term's 's randomness arises from drawing sam- ples from the target distribution T1, while the second term's randomness comes from sampling from To and Unif(o, T]. We encounter two difficulties in deriving the generaliza- tion error bound. The first difficulty lies evaluating the covering number of the loss function class H for the first term. The second difficulty stems from the unboundedness of the term || xo vtt + - (1-t)æo, Il? in the second term. To handle this, we leverage the concentration property of the Gaussian prior distribution and employ a truncation argument to provide an upper bound for the second term
Proof. Note that (x) and (x) form a coupling of n and m. by the definition of Wasserstein-2 distance, we have
W (Te, Fe) / [Xx(()) (2))l (19)
where X, is the flow map solution of (53) with the exact o² defined in (3) and X, is the flow map solution of (54). Now, we consider the evolution of
R4 [ [x((()) - (z)))d
7

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Differentiating on both sides, we get
71, 72) have:
dRg 2 2 (01(A(((0.00000000)0))).. (x)))
(00000(100 (e
N
dt
Rd
(2
= =
(2)))
+2)
12)
Σ (l++1 (1++1-12) - 12)3
To (x))d
2
/ Rd
-
where र is the distribution of the final output the the estimated sampling dynamics (17).
()) (x) (x)xx.
The proof of Lemma 5.2 can be found in Appendix C.2. (20)
Using the inequality 2(a, b) & |a|| + |b|1, we have
Tradeoff on Stopping Time T To show the tradeoff, we first present the following lemma:
2(0" (X)())) x) {x(x))) s
Lemma 5.3. Suppose Assumption 1.1 holds, we have
[I="(X)())) ((X)(a)) 0) (|xx(x)) {x))1.
(21)
W2(TT, TI) S (1-TVa.
Note that NN defined in Theorem 3.1 is 7 -Lipschitzz continuous w.r.t. x, the Cauchy-Schwee inequality implies
The proof of Lemma 5.3 is deferred to Appendix C.2. Propo- sition 5.1 demonstrates that the stopping time T tends to 1. the error of using the estimated velocity field in the sampling dynamics increases. Conversely, according to Lemma 5.3, the Wasserstein-2 distance between and Ti decreases as 7 approaches 1. This reveals a tradeoff in the stopping time between the error in velocity field estimation and the distribution recovery.
2(6(XX (0i4))--((((( ((X,(x), Xi(e) Rx(x))
(22)
$ 2m (z))) {1
Combining (20), (21) and (22), we obtain
dRg
df S(1 27)
+ |0 (X,(x))) ((X( mo
6. Conclusion
This paper presents statistical learning theory perspective on CNFs based on FM. We demonstrate that Lipschitz neural network can approximate the true velocity field under r, (nx)- and provide a sample complexity analysis for estimating the velocity field. Furthermore, we prove that under mild assumptions, the generated distribution of CNFs based on FM converges to the target data in Wasserstein-2 distance. Additionally, we show that the convergence rate can be significantly improved by assuming an additional mild Lipschitz condition on the target score function. To the best of our knowledge, this is the first end-to-end analysis of FM.
Therefore, by Lemma C.6 and since Ro = O, we deduce
RT S e2+2m (-' Rdd
||00 (x)) -
n daadt
= c2 (-" -((,
By Theorem 4.4 and the fact that T is -iiscchhi continu- ous w.r.t. x since we choose NN as in Theorem 4.4, we get the desired result.
Time Discretization In practice, we need to use a discrete- time approximation for the sampling dynamics (17). Let 0= = to < f < ... < IN = T be the discretization points. We consider the explicit Euler discretization scheme:
A Acknowledgement
This work is supported by the Key R&D Program of Hubei Province under Grant 2024BAB038, the National Key R&D Program of China under Grant 2023YFC3604702, the Fun- damental Research Funds for the Central Universities under Grant 2042025k:0045.
(x) = 6( X (a)), ddt, L [1ke (411),
(23)
for k = 0, 1, 1,..,, the distribution of (xx) by ßग.
and No(x) x mp. We denote
Impact Statement
To establish the distribution recovery guarantees, we need the following discretization error bound:
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal
Lemma 5.2. Let = to < t1 < < ty = be the discretization points. For any neural network t in
consequences of our work, none of which we feel must be specifically highlighted here.
8

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An Error Analysis of Flow Matching for Deep Generative Modeling
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An Error Analysis of Flow Matching for Deep Generative Modeling
A. Approximation Error
A.I. Proof of Theorem 3.1
Proof. The goal is to find a network ē in NN to approximate the true vector field v² A major difficulty in approximating
v" (ax t) is that the input space Rd [0. T] unbounded. To address this difficulty, we partition Ie into a compact subset
K. and its complement K On K [0, T construct T achieve an L approximation. On the KS, we simply set (ar, i) O. Since we assume is supported on compact set, the (m², approximation error of ((x, a, t) u (ar, t) can
still be controlled.
Approximation on K x [0, T.. We choose K. = {I|||>]|> I R) to be d-dimensional hypercube with edge length 2R > 0, where R will be determined later. On K. x [0, 77, we approximate k-coordinate maps vi (z. 1) separately, where 0² = (ae .. ... ( [)]"
First, we rescale the input by x' 2R
1 (x R1) and t' t/T, where := [1..... 1|", so that the transformed space is [0, 1] [0, 1]. Such transformation can be exactly implemented by single ReLU layer.
By Lemma 1.4, v² (xx 1) is Ç-Lipschitz in We define the rescaled function on the transformed input space as v(a',(") == 2 (2Rx' - Rt)Tt), Tt), so that is 2ÇR-Lipschitz in x'.
We also denote the Lipschitz constant of v(z', t') w.r.t t' as (R),), when x' E [0, 11d. We denote
r(R)=
== sup
sup
11,00" (x, t)ll
tE(O.T] << R. R|d
An upper bound for rR) is computed in Lemma D.4 by T(R) = O ( (++2 on [0, 1jd x [0. 1].
(71-7)²))
Now the goal becomes approximating v
Second, we partition [0, 1j into non-overlapping hypercubes with equal edge length e₁, We also partition the time interval [0, into nveeolerlgp sub-intervals of length eg. e1 and eg will be chosen depending on the desired approximation error. We denote N1 = ei
and N₂ -
Let m = [m,, mmm E [Ni] be a multi-index. We define V as
mma"
T. (,(,,,
E of
2R m
RI,
J. N₂
V Vm ,1)),
N₁
where L (x', f') is a partition of unity function, that is Σ [mE[N V mm, on [0, 1ja x [0, 1]. We choose V m.j as product of coordinate-wir wise trapezoid functions:
V (m (x', ()) = w/ (3Na (i'
d
1b
3N
x
mi
N₂
N₂
i=
where yb is a trapezoid function,
1,
|a|
<
(aa))=
2
|a|,
|a|
E [1.2]
0,
[a] > 2.
We claim that
1. T. is an approximation of u;:
2. 0; can be implemented by a ReLU neural network ūn with small error.
Both claims are verified in (Chen et al., 2020b, Lemma 10), where we only need to substitute the Lipschitz constant R and rRR) into the error analysis. We use the coordinate-wise analysis in the proof of (Chen et al., 2020b, Lemma 10) for deriving the Lipschitz continuity w.r.t. æ' and f'. Similar proofs can be found in Huang al. (2022). By concatenating 7, together, we construct Gg [61.... 0a]"". Given e, if we achieve
(a ,() v0a' ( 10) S Se,
sup
12

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An Error Analysis of Flow Matching for Deep Generative Modeling
the neural network configuration is
L =
log - + d
J J (1
M
(1(0)(CR))-
- [og - + +++)) v))
VdR
==
K max (1)),,
1-T
( The output range K is computed by K [|o (a, f)ll. Combining with the input transformation layer, i.e. I - x' and t + t' rescaling, we have constructed network is Lipschitz continuous in x', i.e., for any æ1, Ø2 E K and & E [0, T). it holds
Here we already take e1
o
i
and 62
o
cR
max; () R, R.
60 t) To (22, ll I 10dç||=1 - =22ll
Moreover, the network is also Lipschitz in i.e., for any 4, l2 [1,7] and || S R, it holds
6 (1) 06 ( 12)1 = 10T( (1 (/)-
Due to the partition of unity function, ve m. vanishes outside K. have (a t) 0 for ||/||>> R. Therefore the above Lipschitz continuity in extends to Idd."
Bounding L approximation error. The L approximation error of 0g can be decomposed into two terms,
||00 (x,, t) C0 ())))))))) =Il(0* (az, t) 8o t)) م||=|| I R)
+ (1(00 (ax, t) 8g (z. t) |||= R)
The first term on the right-hand side of the last display is bounded by
[1(v" (ar, t) (zx S RHILA())) S Vd
(,, 60 1)1 S vde.
sup (e.t)eKxx(o.7
The second term admits upper bound in Lemma A.1. Specifically, when choosing R R= (Viose we have
(xx, - (a( (1)x > R) Re)))
Se.
As result, with the choice of R, we obtain
(x' t) (ol S (vd+ 1)c.
Substituting R into the network configuration, we obtain
L d + log !) M M
(3/2 (8/g(1o(d//=)) t (1-1) cde (d=2()
0=0
88/2 (log( (log(d/))) 411
(d++1)
1 + +ta) K=O log
V dlog _
T)-
1-T
V d3 log(d//5/ (d/s) - 7)-
Vd3 log(d/E)
(VVog(d/e)
0
71 10ds, 72 o
(1-17 -1-1)a
A.2. Truncation error
Lemma A.1. Under Assumption 1.1, given E > 0, with R = - ( (VioB it holds
(xx, (x 1))T f|1011o0 > R) (1)2(w)) Se.
13

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An Error Analysis of Flow Matching for Deep Generative Modeling
Proof. For any R > O. using the identity v² (ze,
1 X = =:), have
1
(1, (ddx))
111=1100>>R
[|[[X1 Xo = TT (d(rr)
)3 R
E|l|X1 Xol1 Xx = (da))
(24)
A11=|lo>>>>R
Ex, [B[||Xi Xo (0001 RW)
<B[]| (1) > RJ]
<1 Xol14]1/P((1 Xx R)
where the second equality follows from the total expectation formula, and the last inequality follows from Cauchy-Schwards inequality. Using the inequality (a + b)2 v 2a + 262 have the following upper bound for the fourth moment,
x - Xol [ X 2 + Xo 1?)
|||
+ Xo
27
< 4d +4
Σ X
i=1
(25)
d
4d + 4E
X +
k=1
ifj
=d((d + 1);
where No.; denotes the i-coordinate of Xp. It remains to control the tail probability of X,. Using the union inequality, we have
P P > R) = P (U > R)
(|Xeil > R).
1=1
Thus, it suffices to control the tail probability of X2,; for i 1, 1....,d d, where Xz,s is the i-coordinate of X,. Since
assume Ti is supported on [0, 1]2, we have
P(X R) S (t (1 x)/ > R)
S PP Xo.il
Since Xo,; is a standard Gaussian variable and thus sub-Gaussian with parameter 1 (Wainwright, 2019, Example 2.1), we have the following tail probability bound,
P (Xoxl > R-11 1)
(R- 1)2 1- < 2exp
(26)
2(1 ( 8)2
Combining (24), (25) and (26), have
(R (R-1) 1)2 4(1 - f)2
S 110 (1.
s 4(d 1) exp
14

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Let the right-hand side in the above inequality be smaller than s², have
3 R 2(1 - 1) ( 2log + log(d + 1) -2log
1 -
1/2
+
g
So we can set = Oly Olylog log ) to guarantee [1(v"(x, (,( 0))1(1221100 > R)ll S..
B. Generalization Error
B.1. Bounding loss function
Lemma B.1. For any neural network e in NN M, K, 12), we have su (K(x, w) - d K².
Proof. Using the inequality (a+ b)? S2 2n + 2bb, we have
T /
1 T
/
((x, v)
|| 20 - vttx + (1 - t)
T Jo /-
1
/
|| 2ol12 oo + 7 //'' / [lo(tao -t)00 To
X + sup ||, t)
x,1
$ d + K2
where the second inequality follows from the fact that nı is supported on [0, 1] and [=|[== = d. This concludes the proof.
B.2. Covering number evaluation
Lemma B.2 (Covering number of H). For a neural network v Rdd x R + Rd,, we define f Rd + R as
(1x, v) = T (-) ||:0 1iae-x0-- "vlle (1 1)ars 1) (
1
For the hypotheses network class V = NN(L, M, K, M1, 71, 72), we define a function class H. == (1(:,0) : E
LM(D 1)
(6,, I.A.NNNII11111-....... (1==<(--.,))xx001)) JL
(27)
8
and based on this, the covering number H. is evaluated by
(K (K+ Tog( (K2 d)/8)
log (6, (1. 2 Jo
(28)
Proof. The first bound (27) directly obtain from (Chen al., 2022b, Lemma 5.3), with slight modification of the input region. The evaluation of the covering number of H proceeds by showing that 8-covering of NN induces C(v)-covering of H. where C()) is function of 6.
Assume that there are two neural networks in and vn satisfying | -= S 8, we want to proof that there is a function C(-), such that [|((, v1) f(., 22) 02)11=>((0111) I C(8). D will be determined later based on We rewrite f(x: v) as follows:
((x v) = 1 1-) / ||x x0l12 -2(x - -xO)= v(t
=xo)Tu(tx + (1 t) t) + | + (1 t)xo, t)] 2 (o(d)))dd.
15

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An Error Analysis of Flow Matching for Deep Generative Modeling
Then have the following upper bound:
2 (T 16(x, v1) f(x, 02) _
/-' / || zool 1l01 no (dæo )
T
(A)
T //
1 T
/ 1101
w2ll
1|01 u2l (daz))
(B)
where the inequality follows from Cauchy-Schwartz inequality and the identity ||00| x2ll² (2)) - ((0) + x22). We omit the input of D1 and U2 for brevity, when there no ambiguity. An upper bound for term (A). The Cauchy-Schwand Schwartz inequality implies
1/2
T (.¹)
1
/ || æoll || - =2 =
7 ( / || (oo)
(29)
1/2
//' f /loss 7o
1/2 Note that r E [0, 1] and zu is a stand Gaussian variable, we have (7 1o J || xull To
2 d//2, Using the
change of variable az tæ + (1 t) we have
T /,"' / [|ol(tz + +(1 11-ttaae (1-t)xx0 t) 22(2x + (1 t) To (cxo))
TT T /.'' / |10 "2(xx, 1)1 map (dxx [Xi x)dtt,
where Teji is the distribution of æ, conditioned on X₃ = z. We partition Id into two subsets, fx, E Rd | I D} and its complement {x; E IRd: |20e|o00 > D}.
f [| (Ii, 1) - 22 (2 TTp (dæx)) X1 = x))
/
[1) 1) - (2( ((az,,)) ())1331mm))) (dax:) X1 = .))
+ 1
||0 (x), v2( t) ग (dx;|X1 x))
(10lls>>
20² + <<P((1t
10: + 1)) > D).
Using the tail bound for Gaussian variable in (26), obtain
f (xx,,) u2 (x., 1)11 Tej2 (dxx X1 = =x) K2dexp
(D-
(30)
2(1 - t)
Combining (29) and (30), we get
+ 5." /
1
(D) 1)2) 1/2
|| æoll -1/01 o2)loo(dxo)dd 5/2
8² * K2d
2(1-13
[D-
S/
8 KKLK222 exp
(31)
4(1 - t)²
(D (D- 1)2
/2
Kddxp
where the second inequality follows from the inequality V + V Vb, for a 2 0,6 0. The third inequality follows from the fact that [0, T).
16

---

An Error Analysis of Flow Matching for Deep Generative Modeling
An upper bound for term (B). Again, using Cauchy-Schwarr inequality, we have
2 / T
1 T 1/O0 / ||l1 022l o20lom((dzooddt
1/2
||01 - w2l || + 02 To (o v
1/2
1/²) / [1l] +
(32)
(-1D-1) 1)3
))
$ K 68+ k2 exp
4
where the second inequality follows from the same argument in (30).
Combining (31) and (32), we obtain
1)3
sup [((x,, 01) - ((c,,, w2)l (K + d2/2) (6+ Kd+K exp ==|0,1]
(33)
4
Thus, a 6-covering of NN w.r.t || [I=((--.))(xx00.]) induces a C(K + 81/2) (6 + Kd
covering of H. where C' is universal constant. Let K2 exp(-(D 1)2 be smaller than 2C(K+ we ob-
8
tain D 2 2/ 2 log 2CK0/2(Ktd"3) +1= =: D(6). Based on the above statements, 2C -covering of NN w.r.t.
|| |1- ((1) induces d-covering of H.
Therefore, we obtain
5
log.N(6, H, (1..)1))))))))) < log.V
NN, -1111-111---- -1000=(000000000)))017007
2C(K + d2/2)
2C(K + d2 (2)))MM)(
S Jo
(34)
(K + 22/2) LM K log( (K2 dd/o)
S Jo
5
It concludes the proof.
B.3. Proof of Theorem 4.4
Proof of Theorem 4.4. The generalization error L()) inf L(v) can be decomposed into
(( inf C(v) [()) _(6) + [ [(σ) + [(σ) [(ö)
< Σ(,) + [(6) L());
where ö E argmin and the last inequality follows from ERM.
For any v, have L(v) ((v)-((()) [(v) L(v) Z(v)
(C(v) ((II,,
Mar., v)), where the first term only involves sample from target distribution T and the second term involves sample from Unif T] and prior distribution no. Both of the two terms can be bounded by using covering number argument.
Bounding 1 (C(o) f(1111. v)). Let {ex L be a r-covering H. where N1 = N6, H, (I.=((),1)))) For n every ? H. there exists a k, such that ||e Ea (E11=~(10,11)) S Thus, we have
[B[[(=)] ((x1,)))
[[])) (x(x).))) 2r
(35)
1
max
(E][x(x)) - -4(21,)) + 2r.
17

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Take supremum both sides, we get
sup ZER
(B(((z)) ((x1.))) -
max Is
(B((x(a))] (x(21,1)) + 2T.
Thus, we have
P
sup NEH n
1 (E(((<))) - f(x1,,)) > 8 + 25
S
max k
Σ [R((((())] (( >
n
1=1
(36)
Ni
P
(B](+(x)) (x(2x1,)) ZE
A
==1
Invoking Lemma B.3, we get
S Nı exp (- nee
P sup LEH
(B[((=)) - {(x1,,)) > + 27
(-2B²)
where B = O(d K2) by Lemma B.1. Letting E
2Ba N/6(
with probability of at least 1 - 5,
2B2 log (2g(N1/1) + 2r.
Σ (C(v) ((xi,,,))) v)) < sup
(B[[((=)] ((x1.))) S
EE
Bounding ( 1) 1( v). We define r(( 1). U. x)) = || - 3o - vll (1 () t)ll and its truncation
tune ((rg, t), v. == ||= 2o utt + (1 t) t) | [1 I R). where will be determined later depending on
the tolerance. Thus, have (0 mo (r(( 1), v, 21,e)]. We also define gerune (xr, v) = (() (), v)x)]
and its empirical version çirune (xx v) = Σ y mune (m tz), v, x) given m i.i.d. sample {(x0.j 1 f from To and Onif(o, T..
We have the following decomposition:
((01,1, v) - ((11, v) = 11( v) - drunc (x v) + trunc (x v) - Frnnc 1 V)
Truncation error (1)
Statistical error
(37)
+ frume 1, 1)) ( v)
1,6 Truncation
For Truncation error (I), we can use the concentration Gaussian variables to control this term. For Statistical error, we can use a covering number argument to control Due to the concentration of Gaussian variables, Truncation error (II) equal zero with high probability.
We first control the first term. Using the Cauchy-Schward inequality and (a + 6)2 2a + 262, we have the following:
Truncation error (I) Er, ||| 3p vllx + (1 - () 1) ( RH
< [ Do vlx + (1 - () 2//2 10 > R22
< 2 [Elll æ0ll + (lx ((x + (1 - f) 1) 1/2 P([ > 1/2
Note that Ei|[== - 2oll11 < 4| + 4E|||00|| < sd(d + 1) and sup... ||, t) < K for any NN, we have
Truncation error (1) < (dd(d + 1) - k2) 122 P(I|xolloc > R/2/2
(38)
18

---

An Error Analysis of Flow Matching for Deep Generative Modeling
(R) Denote the k-coordinate of mn by x0 we have the following upper bound for the tail probability:
P([ > R) P
max k=
100 >
(k)
R
=P
(1000001 | R]
R=1
(39)
d
[
R
R22
2 2d
2
where inequality (i) follows from union inequality. Combining (38) and (39), obtain
R²
Truncation error (I) v s(d + / exp
(-4)
(40)
4
Next, show that Truncation error (II) vanishes with high probability. Note that when [0 S R for j 1,..., m. Truncation error vanishes. It implies
P F(orune (0),,, f(201,1 v) -0) 2
n {|lxaal S
R}
m
(41)
=1- P P
U [R}
R}
R2
exp
2
Finally, we control the Statistical error. Note that, for R > 1, we have to + (1 - t) E [-R, Rj for all | S R, since || I 1 with probability 1. Given a 2V((=+1)+2K covering {0:) of NN, w.r.t.
([- [0.T))> where N₂ = NV( 12) NN, || ()x For any v E NN, there exists = 1, 1...., such that I 2Vd(R+1)+2K* For any æo E R, Rjd and / E [0, we have the following bound for trtrue I t), Uk. x) - parmnn ((Fu, t). u, x)
|/0 - (
| rnuun ((Iu, t), xx) tune ((xo. v,))
5 v, 2x 2xg
Uk miع
<|0- - Uy 1 x
(2|| || + + 2|||z0l1 + || ||
<(2vi((+1)) (1) +2
( +
For any v E NN, we have
grrmne (x1,; v) Fanme (201,,, w)
m
Eaag. trunc ((20, 1), 11, a11:))
tuue ((zo.))
m
j=1
<Exg (0 () t), 2k. x1.1)]
trunc ((Do, ty); Dk. x1.e) + 25
m
j=1
< max Ng...NN N2
(een t), Uk, x1,1)
mune ((oo(((xx tz), Uk x1,1) +
2 2T.
19

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Taking supremum on both sides, we obtain
grune (1),,,
Tinine (x)
sup WENN
1
22
Bu ((zO0 t), ok x1,i)]
run ((zu.j [y), 1in)) 3
lu0n N2
m
Thus, we have
P
sup WENN
grrnne (21,1.)) - (31,.1, >E 25
<TP
max k=1...
Eugo (rtuuee ((F0, t); Uk, 221,5]
trunc ((20,5 [2); Uks x1,5) 20
(42)
Na
-P
El ((zo, 2k x1,{)]
mune ( (x)) Uk 01,1)
1=1
N2
1 Σ prune ((zu.j tt), Uk.
I
Buo,
Ir br ((00o t), Uk. x1,i)l
x1,1) Uk,
>E
k=1
where inequality (i) follows from the union inequality. Note that 0 E y ((D, t). Ub. 21.1) S 2d( 2d(R+1)3 +KK, applying Lemma B.3, we obtain
1 mine (()o 22,, <exp (osp(-(((RR + 1)2 k2y))
m
P Emo. tu ((E0, t), Uk. x1,,
(43)
m
Combining Equation (42) and Equation (43), we obtain
Aaune (21,4s v) Funnc (2 w)} > & + 27) +2T s Nxe
mn²2
P PENN
sup
(8((((R (S(d(R-a) + 1)2 + K2)
(44)
Let N2 exp (- =(d(R+1)2++4))) = -8/2, we have, with probability of at least 1 - 5//2.
ne²
trinc (0 0) Zrunne (x1.;. 0) s 2r + 8(d(R + 1)2 + K2) logooo(22 22
sup VENN
(45)
m
( R2 2
Let mddeed exp
in (41), we have, with probability of at least - 8/2.
Frnme (ar1, v) ((x1,1, v) O.
(46)
Combining (40), (45), and (46), we have, with probability of least 1- 8,
8(d(d + 1) K 1/2
sup EENN
((111, v) v()1.1)) <8
+ 2r
2m
8(dl (1og02mm/0)))) + 1)2 K22 (2g
m
Combining the bounds for 1 (C()(((()) - f(x1,1, v)) and ((11,1,) - 1(111,1 v). we otbain, with probability of at least
1 45,
( inf L(v) 0 T+ (d + 1) K2
(dlog(md/5) 8) + K2) / Iog(NN/6)
v
Vm
(d+ + K2)
log (1/8))
Vn
20

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Balancing error terms. By choosing NN in Theorem 3.1 with approximation emror e, we have inf BENN -(v) ---0"((()))))) (vd+ 1)e. Setting& and gives rise to
1 % (0.)) (s (100
(1/E) des
(1/E) d+s
vn + 62 )
1
+
+
(1- Tyd
Vmn
vm
with probability of least 1 1, n where we omit factors in d, log n, log m, log(1 T). By setting 2tअ it holds
1 T
T 10((,t) 2² (.,t) dt = = (n-ate + n 26443) m m=x))
with probabiliy of least 1 L.
B.4. Auxiliary lemma
Lemma B.3. Let G be a bounded function class, i.e, there exists a constant B such that for any E G and any x in its domain, S g(x) S B. Let NI,..., Xn E Rdd be iid. random variables. For any (0, 1) and 9, we have
nE²
P
(g(X)) - Rlo(x))) >
S Sexp
and
(47)
72
2B
ng²
- >
S exp
(48)
72
2B
Proof. We first compute the moment generating function of 1/4 Di (E(g(X)) - 9(X:)),
(x())) - BloCX)D)
E exp ( (x(x)))) )lo(x))))
exp
(49)
the identity follows from the fact that .1,... In i.i.d random variables. Now, try to upper bound E [exp (A((())) E[g(+))))]- Given an independent copy X of X1, we have
> n
(o(xx))Exxxll - -xx;(o((x)))
exp
E exp
n
(50)
Ex,,x
E
exp
n
Letting a be an independent Rademacher variable, note that the distribution (X1 - xx)) is the same as that of (X2; - Xi),
so that have
(lo(x))((x3)) = Ex,,x
( - -x(x3))
Ex Ex,,x exp
Es exp
n
(2( s(x:))²) )
E Ex
exp
2n
where (i) follows from Lemma B.4, applied conditionally with (X1,X1) held fixed. Since (g(X,) 9(X() < B, we are guaranteed that
)
A²
E Ex,,x
exp
$ exp
(51)
2n
2n
Combining (49) and (51), we obtain,
Σ (g(X,) ( ((6(X)1) ((xx)))E(((x11)
B²
E
exp
< exp
2n
i=1
21

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Using the Markov inequality, have
-E(())) EX(()))) >$) >0) S exp
P Σ ((x))
)
2n
Let
ne 18
we get the first inequality. The second inequality can be proved in the exact same argument.
Lemma B.4. Given a Rademacher random variable takes the values {-1, 1 equiprobably. We have, for any X E R, B 122
Proof. By taking expectations and using the power-series expansion for the exponential, we obtain
1 2 le
(A)*
Eleaa0
A +
k!
k!
k=
k=0
2
(2%))
(52)
1
11+
2*
12
It concludes the proof.
C. Discretization Analysis
C.1. Estimation Error
Consider the target continuous flow:
dX2(x)= (Xx(x) Xo(x) x ~ Tu, OStST,
(53)
and the estimated continuous flow
d&,(#)) = 8 (x), 1)dt, 8o(x)= = mo; 0S1ST.
(54)
Denote the distribution of (x(x) and Xx(x) by and म. respectively, We have the following estimate of the Wasserstein-2 distance W2(Tr, (TTT,,))) ).
Proposition C.1. Suppose Assumption 1.1 holds. For any velocity field v² with Lipschitz constant Ç w.r.l. æ, given samples { from Tı and samples from To and Unif]0, TT. we choose NN as in Theorem 3.1 with d-s. Then with probability of at least 1 n. it holds
W2(Rr.ar) =r) =0(0)
5d/1
(55)
(1- Tyz"
The proof can be found in Proposition 5.1.
C.2. Discretization Error
Now consider the gap between estimated continuous flow and its discretization:
( 6( ((), Rx(x))
no, 0 05t5 T,
(x,(x) = 6(X,, (x), 4d ta < < 4+11 = 0, 1, N - 1, Xo(x) = ~ no
Denote the distribution of (2)) and (x) by R. and 7, respectively.
22

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Lemma C.2. Let 0 to < 4
IN T be the discretization points. For any neural nefwork in
NN(L, M, K, Ks, 71, 72), we have:
(0 + 12)
N- Σ
(T कT)
(tx+1 - tx)3
k=0
where ã is the distribution of the final output of the estimated sampling dynamics (17).
Proof. By the same argument as in the proof of Proposition C.I, we have
( me) S / [18c(x) (x)))]2o((x)d
Now, we consider the evolution of
18((2a) - (2)
Le = JRd
Since (x) is piece-wise linear, consider the evolution of L; each split interval [tx, ts+1]. On interval [Ex, tx+1], have
dLi
((0,(X.(2)) 8r, (X,, (x)) [((x)) - (o))))
(56)
dt
Rd
((,6(( - (,( (x(x) (x
(57)
+ 2( 2(0,(( - 6x (z) (x) X, (z))
(58)
+ f 28, (x)) C. (X,, (x)) (x) {,(=)))oo()ddl
(59)
For (57), by Cauchy-Schwarzzineqeee inequality and the fact that is 711 -1-L-ipschin continuous w.r.f. æ, we get
[ 2(6, (x) - (x)
/ Rd
(z(=)))a(z)dd s 2n
[|x,((z))))))))(())) (1
(60)
For (58), note that (x(x) = Xt (x) + (t t Tu (Xee (x)). use the incquality 2(a, b) < ||a||2 + [|6112 and the fact that @ is 7n-Lipschitz continuous w.r.t. to get
((x)() - 6x( 8((x)) - (((z)))o(zzdd
ga
(x.()x())) (8,, (z:) mo / [8),(z)) - (x((x))1 To
(61)
Rd
$1,(1 1x) 111012/ La St 11)
2k² K2 T.t:
where is the parameter of the neural networks in Theorem 4.4. For (59), the fact that è is 7/2-Lipschitz continuous w.r.t. t implies
[ 2(0,(X.,,())) 6r (X,, (m)) (x( (2))))oo(
(62)
< / - (x()))))(())] + + - 22
Rd
Combining (60), (61) and (62), obtain
dL,
dt < (2 2)L, + (n, +okx2+782 + - tx on ttt,
23

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Again, by Lemma C.6, obtain
(2m (2m +2) L
(2m (21+ Lte 3 3
501222 + (tx tx)²
Summing over k and noting that IN T, we get
N-1
LT - 2(x+1)) (7f + +72)
Σ
(tx+1 - tx)
k=
Thus, we have
N
W2(mr, TT) -0 c (2)K)t))))
+ 7222
Σ (fx+1- - tx
Lemma C.3. Suppose Assumption 1.1 holds, we have
W2(Fr. E (1-1))) Tyvd.
Proof. We consider the error from early stopping. Note that XT and X1 form a coupling of Tr and Tı, by the definition of Wasserstein-2 distance, we obtain
W2(TT, mI) S XT x1/3 (1 TY Xoo1311/2.
Since we assume J supported on [0, 1] and [ [
d, we have W2(TT, & (1 TTvd
C.3. Proof of Main Results
Theorem C.4. Suppose Assumption 1.1 holds, Given n samples from target distribution πı and the networks as in d Theorem 4.4, with parameter replaced by (1-Ty3. we use the estimated velocity field in (11), generate samples and
choose the maximal step size NN. ftt1 tyl = On काड and early stopping time T(n) = (log n) 1/6 we have
W2(TT(m): m)) 0, probability.
Proof. Lemma 1.3 shows that the velocity field or is (1-77" Lipschitz continuous w.r.t. æ on Rdd x [0, T'. when 1/ < < 1.
Combining Proposition C. Lemma C.2 and Lemma C.3, we obtain
Çd/4
(2( (1 (1-T)+e" (n1 + 72)
Σ (to+1 2x + t (1 (1-77 TY2 248
By the choice of neural networks, we have m1 O
(1-1y²) Letting max |4+1 4x = O(n=atz)
10d
T(n) (log n) -166 and omitting polynomials of logarithm, obtain,
W2(AT, TI) Ö (log n) 1/6 + voom ats
which tends to 0 as n goes to infinity.
Theorem C.5. Suppose Assumption 1.1 and Assumption 1.2 hold. Given n samples from target distribution na and the networks as in Theorem 4. with parameter& Ç replaced by Ç(ox, d) defined Lemma 1.4, we use the estimated velocity
field in (11) generate samples and choose the maximal step size max,=u,1..,NN 1+++1 1al O(n 0(n=x 5t and early
stopping time T(n) 1 - 3,0.,, Then, with probability of at least 1 n
we have
W (2(TT()): m1) Ö 3(445)
where we omit logarithms.
24

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Proof. Lemma 1.4 shows that the velocity field v" is ((a, di)-Lipchitz on I2d x [0, 1]. The Lipschitz constant only depends and dimension d. Combining Proposition C.I, Lemma C.2 and Lemma C.3, obtain
N
(2(TT(m) mi) = =ō (1- (1-T) + (K + 72)
(l++1 - 1x)3 + (1 T)2
k=
By letting ma., M++1 1x|
O(n scd+o)) ) and T(n) = sl4t0), we get the desired result.
C.4. Auxiliary lemma in Appendix
Lemma C.6 (Grönwall's inequality). Given a function [(() defined on (a, (a < b), satisfying afl [a, b] and a 2 O, we have
dt S af(() + g(t) on
f(b) S (((aa)) + [ g(t)dt
Proof. By multiplying e-at on both sides of df())
dt S af(()) + g(t) and some manipulation of algebra, we obtain
df(t) ae f(t) S E-aa (eaag(()) gl))).
Integrating on interval (a, b) on both sides, we get
b ab s(b) ((a0s(())) S / ca(b-t) g(1)dl.
This concludes the proof.
D. Properties of true velocity field
D.1. Computation of true velocity field
Lemma D.1. The true velocity field o² can be writlen as:
v²(xtt) = 1 t
v log Te +
(63)
where ne is the density of Xt, and X, (1 (1- t) + t.x
Proof. By some manipulation of algebra, (3) implies:
v² (x, t) = E [1 Xe = x]]
X1 1-t ((1 - (11-t)xo + 1X1 - -xx,) X x
1
1
2. F (ar ar I (x.)
dæ
1
(x)
1 -
1
Φ1 exp(
2(1-1)
)T (1)
dæy
1 t x
1-t.
(22x)<11 t))2
(x:)
w t20 2
1-1
[1=
(1-t)²
exp
2(1 2(1-1) 1)2
) (x11)
dry
1 1- x
t
((2x)(11 -t)2d
Te (x:)
- !
1-1
V exp
2(1-1)²
. (x1)
dx +
1 1-t) x
t
(12 -t)2d
Tx(
t(1 - t)
Ve log (x( + 20,
t
25

---

An Error Analysis of Flow Matching for Deep Generative Modeling
where Ta22 is the density of X, conditioned on X1. It concludes the proof.
D.2. Computation of partial derivative regardingt
Lemma D.2. 0,05 (ax, t)
- -リー xt + 11-12 B[x] X; = x] + 1+t Cov(Xi]Xx a/
(1-1)4
(I- XXXX/|XX111X10 =] |X x |T||| 1 |2 =l), where Cov(X1|X, x] is the covariance ma-
trix of X1 conditioned on Xx = x.
. (dæ)),, which is the unnormalized version of (e(x)) Note that V log (ar) = V log (ar), using the product rule of the derivatives, (63) implies:
Proof. To ease notation, we define (xx : exp
exp (-12
1
2(1-
1-
1
0,00 (x,
v PlogTe(Z))
0. log (2:)
t2 x
f2
1
x0| (2 1)3² x + 1 1-t 0:
1
E[X| |x,
12 a
(64)
1(1 1)2
Φ1(x))
2 - t
1
1-t t V, (x)
0,01 Vor(z)
1(1 - t)2 x
B X, a]
1(1 t)2
(x)
(x))
Then we focus on the computation of the last term above. We first compute
(z) (a) as follows:
0. (z)
1
L
(- || - la 113
O₂
(1 t)2 exp
(dær)
(1(x))
(x)
2(1 -)²
1
/
(1 xx²
2(tt11 (t- -1) exp
|| txa 112
(x))))
(1 (1-1)a
2(1 1)2
to
x tx1 113
(t||zg| 1|
2 x1
- +)² - (t 11]| x -tx1 11
TI (dx1)
exp
(1 - 1)2
2(1 - 0)2
(1 (1-1) --)
(65)
1+
2
Xx = x]
(1 - t)3 x
B(x, || X1 11 x|+
(1 - t)3
(1 - t)5
t(1+t) + t)
t
E[X, x Xx = x|Φ
E[X1 X1 = x)) 12+++
(1- 8)s
(1 t)s
1 ++1
||0||
t)s E|||X, 1| = 2/4
B x Xx = 2]2 +
(1 - t)5
(1 (1-t)ø
By some calculus, we have
(2))(2))
1
/
(t (t|2)) 12- 1122===x(1 x1 1)2 + |0 tx 1| f)
|| tæ 112)
(xx)
(2r)))
(1 - 1)11
exp
2(1 1)2
m (dx1)
(66)
1 1+
||0|||
Ell || = a] +
E B[X x:
X, = x²]
(1 1 t)3
(1 t)3
(1 - 1)3
and
Vo((z)
1
tx - x
x - tæı ||
(x)
(1(x))
(1 - t)z
exp
2(1 t)2
T (dx1)
(67)
x
+
E X1 Xr, = z].
(1 (1- t)2
(1 - t)²
Combining (64), (65), (66) and (67), we obtain
1+
0.0² (x. t) = (1 - t) x +
(1 - 2)2 F F[X,
| X₂ = x=] +
(1 (1-t)² Cov[X, = =//r--
(68)
[E|X 2 Xx = ær]
E = 20 x0 = m=))
(1 1)4
concludes the proof.
26

---

An Error Analysis of Flow Matching for Deep Generative Modeling
D.3. An upper bound for velocity field
Lemma D.3. suPteju,ry supzel-R.M1 10 (2x, - T-7
Proof. For the i-coordinate, we have of - B|X |r, x} that is supported on 1-1, 1j then
- where x denotes the i-coordinate of X1. Note
1 R T-
sup [c - 11
sup
(2,
D.4. An upper bound of partial derivative regarding t Lemma D.4. suprejo,ry Ru (x, = o
(R+21 (I-T)²
Proof. From Lemma D.2, we have
1
(1 - 8)2 | = xi| + 1+t
10:00 (x, t)ll <
||||||
+
[CCov[X XX, = =||lop//=||
(1 t)2
(1 (1-t) t)a
t
t)s | xil | = [|||||||| 1| Xt =+|l)
Note that is assumed to be supported [0, 1jd, have |B æ|ll < || x/ < d'/2 and
||[[X|||x|||] xi]| < ||||Xx111X, x²/1/2 d322. To bound ||Cov[X,|X] = xillopo we have the following
inequality for any u E Rd.
Td Cov[Xi]X, = a/u Elu X X u X = x] Xx x|
= æ]
E( T X1) Xx = 22] E(u Xx = xx|
2d1
Hence we have |CCo[Xi]XX = æbllop S 2d. Using these above inequalities, we have
RVd + |0,v"(xtt) < (1 Ty²
Vd.
1 (1-Ty
2T/3/2
sup E [07]] R
sup
(1-T)) T)²
R 22+
2d
(1-T)²
(((2))+1)
Note that T < 1, the above inequality implies supre u.2] SUPæE|-R,K 10,00 (x, t)ll = 0
(1-TT
D.5. Lipschitz continuity regarding spatial variable
Following Wibisono & Jog (2018a) 8a;b); Mikulincer Shenfeld (2021; 2022); Chewi & Pooladian (2022); Gao et al. (2024), we deduce the Lipschitz continuity of the velocity field from the properties of the target distribution. We start by presenting the following lemma showing the connection between the Jacobian matrix of the velocity field and the conditional covariance matrix.
Lemma D.5. We have the following identity:
t
Vo'
CovlXx/X, = =x]
1 la.
(1 f)3
Proof. By Lemma D.I, we have
1
(x) = t
v2 log (x) + Ia-
27

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Further, the Hessian 2 log Tee (2) can be computed
Jo 11-t) exp( exp(-11a= tx (11 )TT) (dxx)
2 ( =V
Jg exp( - x - tz [12 - f) (T (dx1)
82
JT
1-1-0 11-1)²
exp( - || tx 112 / (T
(1 t)2
Sge exp(
x tz
1/2/(1 |
)T
Jne (1-1)² exp(- L. 112 1/((1 - 1)2) m (dar1)
802
JR. exp(
x tw 1//(1 (1- 1 (dx1)
12
(1 1)2 + (1 - 1)3 Cov [ x|.
Combing the above identities, we get the desired result.
Lemma D.6. Suppose that Assumption 1.1 holds. Then (x, t) E-iischitz continuous wet. æ on I2 [0, where Ta
{ [ u-r5} }. Further, if T have v² is - -iischitt continuous w.r.l. a.
S max
Proof. Since we assume the target distribution L is supported on [0, 11d we have the following evaluation of the covariance matrix
0 m Cov X1 Xx x] dld-
Thus, we have
1 1 1 la Vo (1- 1
1
ld
Id-
The above inequality implies the Lipschitz constant of v² w.r.t. x.
We further need the following two functional inequalities to control the conditional covariance under Assumption 1.2, namely the Brascamp-Lieb inequality (BLI) and Cramér-Rao inequality (CRI).
Lemma D.7 (Brascamp-Leib inequality). Let p(dæ) = exp(-U de be a probability measure on a convex set J C Rd
whose potential U Ω R is twice continuously differentiable and strictly convex. Then
Cov = E. (V
with equality X ~ (m, 2)) with Σ positive definite.
The complete proof of BLI can be found in (Brascamp & Lieb, 1976, Theorem 4.1) and (Saumard & Wellner, 2014). Lemma D.8 (Cramér-Rao incquality). Let u(dx) (x)((-((x)))))) (x))dx be a probability measure on a convex set Q g whose potential U Ω + R is twice continuously differentiable. Then
Cov, (ovp(()) (U,)(=
with equality if X ~ (m,, Σ) with Σ positive definite.
The complete proof of CRI can be found in (Saumard Wellner, 2014; Dembo et al., 1991).
Lemma D.9. Suppose that Assumption 1.1 and Assumption 1.2 hold. Then v² (xx is ((a, d) Lipschitz continuous Rdd x [0, 1] w.r.t. x. where Ç(ax, d) scales polynomially with and d.
Proof. Note that
2 m log (1)((1))) - log (x()) v2 log n (x(21))
where Tije is the conditional density of X1 conditioned on X; and Taj2 is the conditional density of X, conditioned on
X1 .. Since X₁ can be viewed as fX1 perturbed by a Gaussian noise, we have Taja (x(() x exp (- || ||=-t tr
Thus,
(1-1)²
we obtain
t²
2 log (2)
2 (Z)))())) + la-
1-13
28

---

An Error Analysis of Flow Matching for Deep Generative Modeling
Assumption 1.2 implies
1² (a (1 (1-t)?)
1e Ia.
a Ia V2, [1)))()))))
(1 - t)2
By the Cramér-Rao inequality, we obtain
-1
Cov[Xi|X, = x] ž a
la.
(69)
t)2
2 When t E [E (0,1): -0 (1-1)2 > 0). by Brascamp-Lieb inequality, we obtain
t²
Cov[Xi|X, =x] = (-a +
La.
(70)
(1 t)2
Combining (69) and Lemma D.5, we have
(o* t- a -t)
a(1 f)2 + f2 La-
Combining (70) and Lemma D.5, for t LE (0, -a (1-1)² > 0). we have
+ a01 - t)
Vo-(x,t) -a11 t)2 + f2 Id-
(71)
Recalling the result in Lemma D.6, have
td - 1= (1-t)² t)3
1
I-t la.
By some manipulation of algebra, it is obvious that
1 v t-a(1-t)
Thus, we have
1-t
all 122-1)2422-
(o,(x,1) 2 t 1-a(1-1 t)
Ia x
-a
la = a(1+a)la
a(1 1)2 + 12 a:((1 +
where the second inequality follows from the fact that a(1 t) 1
t (0. and a(1 1) + 1² ifa Next, I-7- Let the two quantities he equal, we obtain
we compare [1-1y3 I-t and I-7 =a1--1)+4
id
2
+
d
(1 f)2²
aa va+
Vat Va I+
The root of the above equality in (0,1) Based on this discussion, we obtain
By the monotonicity of -fr on (0, 1), we have
Itva
where
V +
(1 (1-13 1)3
td
te
0,
1+ a + 2
((1)
v va+ 2
+ a(1 -
te
a(1 t)2 + f2
1 / a + 3
29

---

An Error Analysis of Flow Matching for Deep Generative Modeling
By taking the derivative of argument, it can be shown that (+a(l-t) -a(1 +)
td we can see that [1-1}=
fd I
is increasing on (o,
V Va+ 3 1+ ++vv+3
Using the same
[1-
I-
I-7
Va+ 2 1+ 1+va+2
11)
is decreasing on
Based on the above discussion, we obtain
2
d
2 a v aa
(v' t) 1 2
Ia.
d
30


Reviews:
[{"review_id": "QoqISnuNKt", "reviewer": "Reviewer_qBs8", "summary": "This paper provides an analysis of flow matching. The authors prove that generative models based on flow matching converge to the target distribution under mild assumption.\n\n## update after rebuttal ##\nI have reviewed the rebuttal and decided to maintain the original score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable suggestions.\n\n**1. Many existing works have analyzed FM, and the introduction should clearly explain how this work differs from previous theory-related works.**\n\n**A:** While recent works [1,2,3] have analyzed ODE-based generative models, they typically assume that the score function or velocity function is uniformly Lipschitz over all time steps. In contrast, our analysis drops these assumptions, which makes the analysis more general.\n\n**2. The authors should provide an overview at the end of the introduction.**\n\n**A:** Here is a brief overview, which we will include in our revision. In Preliminaries, we introduce the necessary notations and background on Flow Matching (FM) and Continuous Normalizing Flows (CNFs). Section 4 analyzes the approximation error that arises when using neural networks to approximate the true velocity field. Section 5 studies the estimation error in learning the velocity field from data. Section 6 studies the discretization error introduced when solving the ODE flow numerically.\n\n**3. Currently, the transition from the main contributions directly to \"1.1 Assumptions\" is abrupt, ... Would it not be more logical to merge them into a single section?**\n\n**A:** We agree that the transition is currently abrupt and will revise the structure to ensure the readability. Specifically, we will either integrate the key assumptions into the contributions section or provide a brief motivation before introducing them.\n\n\n\n\n[1] Albergo, M. S., Boffi, N. M., and Vanden-Eijnden, E. Stochastic interpolants: A unifying framework for flows and diffusions.\n[2] Chen, S., Daras, G., and Dimakis, A. G. Restoration-degradation beyond linear diffusions: A non-asymptotic analysis for ddim-type samplers.\n[3] Lu, C., Zheng, K., Bao, F., Chen, J., Li, C., and Zhu, J. Maximum likelihood training for score-based diffusion odes by high order denoising score matching.", "claims_and_evidence": "I'm very unfamilar with this domain, although I tried my best to understand its content and provide thoughtful reviews.", "methods_and_evaluation_criteria": "The paper does not propose specific methods or evaluation criteria, as it is a theoretical contribution. The validity of the results depends on the correctness and rigor of the proofs rather than empirical benchmarks.", "theoretical_claims": "The discussion of the theoretical aspects in the paper appears to be reasonable.", "experimental_designs_or_analyses": "This is a theory-related paper, with no experiments provided.", "relation_to_broader_scientific_literature": "The key contributions of this paper are closely related to the broader scientific literature in generative modeling.", "essential_references_not_discussed": "I did not identify any major related works that are missing.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "The introduction is too brief. Many existing works have analyzed FM, and the introduction should clearly explain how this work differs from previous theory-related works.\nThe authors should provide an overview at the end of the introduction. Currently, the transition from the main contributions  directly to \"1.1 Assumptions\" is abrupt, making it unclear what this part aims to achieve.\nBoth Section 2 and Section 3 are titled \"Preliminaries\", overly confusing. Would it not be more logical to merge them into a single section?", "questions_for_authors": "Please address my concerns or correct me if there is anything wrong in other comments.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pvCIOrM7Zb", "reviewer": "Reviewer_C3LC", "summary": "This paper presents the first comprehensive analysis of Continuous Normalizing Flows (CNFs) based on Flow Matching. The theoretical results establish that the generated distribution converges to the true distribution under a mild assumption. Additionally, the convergence rate is notably improved when a mild Lipschitz condition is assumed on the target score function.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive evaluation.\n\n**1. The results depend on specific assumptions (bounded support, Lipschitz continuity) which may not hold in all practical scenarios.**\n\n**A:** We appreciate the reviewer’s insightful comments on the assumptions of bounded support and Lipschitz continuity. These conditions are standard in the literature, but we acknowledge the importance of exploring more general settings. In future work, we aim to extend our analysis to relax these assumptions and investigate their impact on our results.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "I have reviewed the proof for Theorems 1.6 and 1.7 and find no mistakes.", "experimental_designs_or_analyses": "NA", "relation_to_broader_scientific_literature": "These results not only enrich the toolbox for theoretical analysis for Flow Matching but also provide new perspectives and methods for practical applications.", "essential_references_not_discussed": "There are some references needed to be cited:\n\n[1] Gat I, Remez T, Shaul N, et al. Discrete flow matching. NIPS, 2025.\n[2] Shi Y, De Bortoli V, Campbell A, et al. Diffusion Schrödinger bridge matching. NIPS, 2024.\n[3] Klein L, Krämer A, Noé F. Equivariant flow matching. NIPS, 2024.", "other_strengths_and_weaknesses": "Strength 1： The paper is well-structured and easy to follow. The authors have conducted a thorough literature review, covering key works on Flow Matching and Diffusion, including some of the most recent advancements in the field.\nStrength 2: The results presented in the paper include consistency, generalization bounds, and sample complexity bounds, which comprehensively address the key theoretical questions related to this line of methods. Notably, most of the results are derived under mild assumptions, making the findings both robust and broadly applicable.\n\nWeakness 1: The results depend on specific assumptions (bounded support, Lipschitz continuity) which may not hold in all practical scenarios.", "other_comments_or_suggestions": "No", "questions_for_authors": "No", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HIFWIG2y6J", "reviewer": "Reviewer_8qqb", "summary": "This paper presents an analysis of Continuous Normalizing Flows (CNFs) built upon Flow Matching (FM) for deep generative modeling. It proves the generated distribution of FM converges to the target distribution in the Wasserstein-2 distance for general target distributions with bounded support. The convergence rate is significantly improved under a mild Lipschitz condition of the target score function.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive evaluation and insightful questions.\n\n**1. There is no experiments to support the theoretical results.**\n\n**A:** We appreciate the reviewer’s concern regarding the absence of experiments. Our primary focus in this work is to establish a rigorous theoretical foundation for Flow Matching (FM), providing theoretical guarantees for its effectiveness.\n\n**2. In your analysis, you emphasize the importance of the Lipschitz continuity of the velocity field. Can you elaborate on how varying the Lipschitz constant impacts the convergence rate and the overall performance of the model?**\n\n**A:** As stated in Theorem 4.1, an increase in the Lipschitz constant leads to greater neural network complexity. This, in turn, expands the hypothesis class, making estimation more challenging and potentially impacting the convergence rate.\n\n**3. What are the most pressing open questions or future research directions that you believe need to be addressed to further advance the understanding and applicability of FM in generative modeling?** \n\n**A:** One crucial direction is to relax the current assumptions while maintaining convergence guarantees, thereby broadening the applicability of FM.", "claims_and_evidence": "The claims made in the paper appear to be well-supported by theoretical analysis, mathematical proofs, and assumptions outlined throughout the text. Below is an evaluation of the main claims and their supporting evidence:\n\n1. End-to-End Error Analysis: Theorem 1.6, 1.7.\n\n2. Improved Convergence Rate Under Lipschitz Conditions: Theorem 1.7.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I have skimmed the proofs and believe they are correct.", "experimental_designs_or_analyses": "None.", "relation_to_broader_scientific_literature": "FM is highlighted as a pivotal development that enhances the training and efficiency of CNFs. Prior works on FM and interpolated transport paths, notably by Liu et al. (2023) and Karras et al. (2022), are referenced, indicating that this paper builds on and extends these concepts. This situates the work within a lineage of research that seeks to improve existing flow models by addressing their sampling efficiency.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "Strengths:\n1) It presents itself as the first end-to-end error analysis of CNFs using FM. This novelty is significant, as it fills a gap in the existing literature regarding the theoretical underpinnings of FM and its implications for generative models.\n2) The authors focus on mild assumptions, such as bounded support and Lipschitz conditions, which increases the applicability of their findings to a wider range of practical situations. This accessibility is a strength as it allows for broader implications in real-world applications.\n\nWeaknesses:\n\n1）There is no experiments to support the theoretical results.", "other_comments_or_suggestions": "None.", "questions_for_authors": "1) In your analysis, you emphasize the importance of the Lipschitz continuity of the velocity field. Can you elaborate on how varying the Lipschitz constant impacts the convergence rate and the overall performance of the model?\n2) What are the most pressing open questions or future research directions that you believe need to be addressed to further advance the understanding and applicability of FM in generative modeling?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6s5Iu8oev0", "reviewer": "Reviewer_LksU", "summary": "This paper presents the first end-to-end analysis of Continuous Normalizing Flows (CNFs) built upon Flow Matching. The theoretical results demonstrate that the generated distribution is guaranteed to converge to the true distribution under a mild assumption. Furthermore, the convergence rate is significantly improved assuming a mild Lipschitz condition on the target score function.、\n## update after rebuttal\nMy assessment has not changed. The authors have successfully addressed the raised issues, and my recommendation remains as an \"Accept\"", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive evaluation.\n\n**1. Is assuming early stopping $\\sigma_{min}$ equivalent to considering as the original FM paper? Can the analysis be simplified for some predefined small $\\sigma_{min}$ where we resort to a noisy approximation of the target distribution?**\n\n**A:** Thank you for providing the insightful question. Your clarification makes sense. The analysis can be simplified when the stopping time is pre-defined. However, as we would prefer the generated distribution to converge to the target distribution, we let the stopping time converge to $0$ as the size of samples increases.", "claims_and_evidence": "All the claims are supported by the theoretical results in this paper.", "methods_and_evaluation_criteria": "There is no experiments in this paper.", "theoretical_claims": "I checked the proof for the theoretical claims and did not find any mistake.", "experimental_designs_or_analyses": "There is no experiments in this paper.", "relation_to_broader_scientific_literature": "The consistency of FM is mainly based on a mild assumption, i.e. boundedness, which justifies the use of CNFs based on FM. Theorem 1.7 highlight the effectiveness of CNFs based on FM in learning the underlying smooth distribution.", "essential_references_not_discussed": "I find the literature review comprehensive.", "other_strengths_and_weaknesses": "This paper relaxes the strong assumptions on the underlying velocity field in the previous work.", "other_comments_or_suggestions": "None.", "questions_for_authors": "Is assuming early stopping equivalent to considering $\\sigma_{min}$ as the original FM paper? Can the analysis be simplified for some predefined small $\\sigma_{min}$ where we resort to a noisy approximation of the target distribution?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: After the rebuttal and discussion phases, the paper received scores of 4, 4, 4, and 3, which exceed the expected threshold for acceptance. After briefly reviewing the comments and the authors' responses, I believe the paper meets the acceptance criteria for ICML.
Decision: Accept (spotlight poster)