Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Authors: Zhengchao Wan
Qingsong Wang
Gal Mishne
Yusu Wang
Venue: ICML 2025 poster (2025)
Keywords: Deep learning theory
Diffusion model
Flow matching
ODE dynamics
Generative model
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: We analyze flow-matching ODE trajectories, proving convergence and quantitatively identifying three stages of evolution. Our theory extends to low-dimensional manifolds—where prior work fails—and provides insights into memorization.

Abstract: Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold---cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.

Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Zhengchao Wan"
Qingsong Wang" *2 Gal Mishne Yusu Wang?
Abstract
et al., 2015; Ho al., 2020; Song & Ermon, 2019). Com- pared previous generative models (c.g., GANs (Good- fellow et al., 2014)), diffusion models are easier to train but suffer from long sampling times due to the sequential nature of the sampling process. ODE-based samplers were introduced to address this limitation, where the sampling process done by integrating an ODE. With its efficiency, ODE-based samplers have become the dominant approach in diffusion models (Song et al., 2021; Lu al., 2022; Kar- ras et al., 2022). Recently, the ODE-based viewpoint of diffusion models has been extended to general framework known as flow matching (FM) (Lipman et al., 2022; Al- bergo & Eijden-Eijnden, 2023; Liu et al., 2023), which uses an ODE to interpolate between prior and a target data distribution. FM models learn a vector field ut similar to the score function in diffusion models. During sampling, data sample 1 is generated by integrating the ODE starting from some za E IRd sampled from prior distribution:
Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigor- ous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applica- bility. this paper, we advance the theory of FM models through comprehensive analysis of sam- ple trajectories. Central to our theory is the dis- covery that the denoiser, key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data ge- ometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and lo- cal clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submani- fold-cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These find- ings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.
dr. dt (xx)), (2))) t 1).
Data (Clecle Starting Print xy Cooneeged
(a) Convergent trajectory
(b) Winding trajectory
Figure 1. Comparing two ODE trajectory behaviors.
1. Introduction
Various versions of the FM model have gained popularity, such as the rectified flow model (Liu et al., 2023), which is utilized commercial image generation software (Esser et al., 2024). Furthermore, the succinct and deterministic formulation of the FM model also makes theoretical analysis potentially easier. Despite the empirical success, critical theoretical questions remain insufficiently addressed: How does the data geometry (e.g., clusters, manifold structure) influence and guide individual sampling trajectories? Are these trajectories guaranteed to converge toward the data distribution as + 1, especially when the data lies on a low-dimensional subspace or manifold? This convergence
Diffusion-bul based generative models have become the de facto method for the task of image generation Dich-Dicckttein
'Equal contribution 'Department of Mathematics, University of Missouri, Columbia, Missouri, USA HHlhcooglu Data Science Institute, University of California San Diego, La Jolla, California, USA. Correspondence to: Gal Mishne <gmishne@ udsd. Yusu Wang <yusuwang@ udsd.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
critical for the generative model's performance, as suggested by maiza--Ganem& al. (2024).
Comuux hull Chster 1
Clsste 2 Cluster
End
These questions regarding per-sample trajectories have both theoretical and practical significance for the sampling pro- cess because (1) robust sampling requires trajectory con- vergence in terminal time (Figure la), avoiding undesirable behaviors like winding around the data manifold (Figure lb). Such convergence provides the theoretical foundation for distilling the trajectory into a one-step generative model like the consistency model (Song et al., 2023); (2) understanding the relationship between data geometry and ODE trajecto- ries can motivate geometry-basee steering of the sampling process or modification of the latent space for improved generation quality.
Dala mean pala po Cluster
Our approach. We conduct a thorough investigation of per-sample FM ODE trajectories by focusing our analysis on the denoiser the conditional mean of the data given noise (Karras et al., 2022). The denoiser emerges as the only data-dependent component of the flow vector field u, fundamentally determining FM ODE dynamics. Interest- ingly, by examining how the denoiser interacts with the data geometry, we demonstrate that the FM ODE exhibits two key properties: (1) Attracting- -tajectoriis are drawn toward a specific set, and (2) Absorbing once within a certain set, trajectories remain confined near it. With these properties, we quantitatively elucidate FM ODE trajectories across three stages (Figure 2): initial, intermediate, and terminal. The initial stage is characterized by trajectories moving toward the mean of the data distribution, while the intermediate stage is shaped by coarse-scale data geometry, with trajectories attracted to and absorbed into local clusters. The terminal stage is marked by the trajectory converging to the data support, where the attracting and absorbing dy- namics ensure the convergence (see Section 5.1 for more details). While there are prior works on sampling evolution of diffusion models (Biroli al., 2024; Li & Chen, 2024), they have mainly focused on distribution-level analysis of stochastic samplers using simplified settings like Gaussian mixtures. In contrast, our work reveals how data geometry- both coarse-scale clustering and fine-scale structure (dis- crete vs. manifold) manifests in and guides individual ODE trajectories.
Figure 2. Three stages of an FM ODE trajectory with synthetic data. The curve with blue progression shows an FM ODE tra- jectory, with an arrow indicating direction. The shaded region indicates the convex hull of data. Three stages are visible: initially, the trajectory aligns with the data mean (brown point); next, it is attracted to a local cluster (yellow cluster); finally, converges to data point (green star). See Appendix J.1 for more details.
(Theorem 4.1) and establish rigorously how an FM ODE trajectory will initially move toward the data mean (Propo- sition 4.2) and later toward local clusters (Proposition 4.4), In Section 5, we establish the convergence for FM ODE as + 1 under mild assumptions (Theorem 5.3). To the best of our knowledge, this is the first result that accommodates data distributions supported on submanifolds. This conver- gence result allows us to study the properties of flow maps, leading to our establishment of equivariance of flow maps with respect to geometric transformations (Proposition 5.7). We also delve into the case of discrete measures, showing that terminal time training plays a critical role in address- ing memorization phenomena (Propositions 5.9 and 5. 5.).). See Figure 4 in Appendix A for a roadmap of our main thcoretical results.
Due to space constraints, all proofs are deferred to the ap- pendix. A significant number of additional results and observations, which could be of independent interest, are also presented in the appendix. For instance, we identify that the FM ODE vector field exhibits singularities and blows up when the data distribution lacks full support (cf. Appendix C.2), and we derive precise rates of convergence of posterior distributions/denoisers as t - 1 depending on the data geometry, as detailed in Appendix D.
Contributions. In Section we introduce the fundamen- tals of the denoiser and present meta attracting and absorbing theorems (Theorems 3.1 and 3.2), which demon- strate how the properties of the denoiser can be leveraged to analyze FM ODE trajectories. These theorems provide a unifying framework to qualitatively study the behavior of trajectories during the initial and intermediate stages (Sec- tion 4) as well as the terminal stage (Section 5). Specifically, in Section 4, first establish the well-posedness of the FM ODE trajectory on [0, 1) for general data distributions
2. Background and Related Work
Notations. For any subset Ω C Rd, we let dz(x) = inf ||x - ell denote the distance to Ω. Let B, == (o E Rdd (x) < r). Let Q and an denote the closure
2

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
and boundary of 92 respectively. The medial axis of Ω is denoted
ing flow map v. satisfies Pt = +Po
dr;
En = E ER # sergminyenlla vll > }}},
dt (z(2)).
(2)
where # A denotes the cardinality of a set A. For any x &f En, its projection onto Ω
To construet ut, the FM model marginalizes over the condi- tional vector field ue (z(a(x)):
proja (z) == arg min || - yll yeR
We ur(x) / (((r))r)p((aa X, = 2),
is unique and hence well defined when Ω is closed. The reach of f2 is defined as TQ == infzeg dzn, (z) which in a sense quantifies the smoothness of the set Q-a a larger reach rules out tight bottlenecks and sharp bends (see, e.g.- (Federer, 1959) for more details).
where p(dx Xx x)) represents the posterior distribution:
||-- 11==0,211""
exp
203
p(dr1 X, = x) = ʃ exp
p(dx1). ).
|a |a-o, 11
p(dr()
20
We use 82 to denote the Dirac delta measure x. We let (i,, Σ) denote the Gaussian distribution with mean j and covariance Σ. Let s E [1, oe], and we use dw,a (V1, P2) to denote 8- w-asssesteiin distance for two probability measures V/ and 1/2.
Importantly, if ue (xrlar)) ) takes the following simple form:
B. dr -
(x(x)²)) = 3.
F1.
(4)
Be
See Appendix for a table of symbols used in this paper.
where the dot denotes differentiation with respect to t, then Liu et al. (2023, Theorem 3.3) and Lipman al. (2022, The- orem 1) demonstrated that u, generates the probability path (pe)cejo.u) .]. assuming the ODE trajectory of Equation (2) ex- ists on [0, 1]. This existence was rigorously established in Gao et al. (2024) under restrictive assumptions, excluding cases where p is supported on a low-dim submanifold. For more general cases, see our results in Sections 4.1 and 5.1.
2.1. Background on Flow Matching
Flow matching (FM) models (Lipman et al., 2022; Albergo & Vanden-Eijndes 2023; Liu al., 2023) are class of gen- erative models whose training process consists of learning vector field uy that generates probability path (P,) interpolating a prior Po Pprior and a target data distribution Pi = =p and whose sampling process consists of integrating an ODE from an initial point Z ~ Pprior to obtain a termi- nal point X ~ p. More precisely, the interpolating path (pe)te(o.1) in FM model constructed as follows:
It turns out that the closed form of the conditional vector field u (3r(21) allows one train a neural network to learn the vector field u, by minimizing the following loss function whose unique minimizer is us (x(x)) (Lipman et al., 2022);
pe(dos)= = f pc((dr.|= =))(dx);
E 1) Z Z- Xaup
|| (a, + 322) - -, - 3 (5)
where the conditional distribution (- = x) satisfies that po(-IX = x) = Pprior and p1 (-1X = x) = 8z. We assume that the prior Pprior is the standard Gaussian (N(, I) throughout this paper. Then, pe(- X I) are specified
Noise-to-signal ratio. FM model with different schedul- ing functions can be unified through the noise-to-signal ratio (Shaul et al., 2024; Chen et al., 2024). We find it useful in our analysis as it simplifies the ODE dynamics and allows us to present our results more cleanly. Proofs of results in this section can be found in Appendix F.
pe(1Xx = =x) :- (ara S.D,
where ag and 3, are scheduling functions satisfying ag 31 = 0 and 01 = 3o = 1 and are often monotonic. Com- mon choices include linear scheduling a = f and B3 = 1-t used in the rectified flow model (Liu et al., 2023; Esser et al., 2024) and those arising from noise scheduling in diffusion models. In this paper, we assume that ag :9, are smooth functions of l on the closed interval [0, 1). It is worth noting that pt is the law of the random variable X, == o; + B.Z, assuming X and Z are independent.
Let at 3, be strictly monotonic scheduling functions. The noise-to-signal ratio o, := B is defined for E (0, 1]. By monotonicity, o, is invertible with inverse l(a). As increases from 0 to 1, o, decreases from do to 0. For E [0, oo), we define do as the convolution of p with the Gaussian distribution o0,
Jr = x= -===+((0,000 N = N((W,.oTTp(d))) (6)
The FM model then designs a vector field us such that the ODE trajectory below generates (pt)cejo. i.e., the result-
Then, we have the following result.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Proposition 2.1. For any E (0, 1), define A, : R. Rd by sending x to x/ or- Then, Qa. = (At)
3.1. Basics of the Denoiser
By plugging Equation (4) into Equation (3), we have that
The probability path qo, satisfies the following ODE in a. Proposition 2.2. For any () C (0. let (II))eja.b) denote an ODE trajectory of Equation (2). Then, (To := Ftto)/aston ) oo) satisfies the following ODE:
u. (3) 9./B.. (á. ax, B[x X, = z), (8)
where X p, and E X X, 2] is called the denoiser with the following form (with existence proved in Appendix C.I):
dxa
yo (o (xr)) Vog
!ir-a1vll 257
do
exp
y
X = s]
p(dy). (9)
where qo (I) denotes the probability density of qa-
exp
Is--as'' 28f
12
p(dy')
The ODE model in Equation (7) generates a probability path do for o (0, oc). During sampling. the ODE integrates backwards over a E (0, or] with end condition Tor = x. By expressing our results in terms of the noise-to-signal ratio rather than time I, we obtain a unified framework independent of specific scheduling functions, allowing for a more general and concise theoretical analysis.
For brevity, we write m²(x) := EX = z/. Since m. fully determines ut, instead of learning ue directly, one can train a neural network mg to learn the denoiser m,;
Eee(o.1) (.te(o.).Z--ppeeee Z X Xmp [m + B, x1]² (10)
Training with this loss can be more stable than with Equa- tion (5) since for any x E Rdd, mx(x) remains bounded while (x)x) can blow up to co + (cf. Appendix C.2).
2.2. Related Work
Chen et al. (2024) connects FM sampling to the mean shift algorithm (Comaniciu& & Meer, 2002) via the denoiser, fo- cusing on algorithmic strategies to identify high-curvature regions for better sampling. Pidstrigach (2022); Permenter & Yuan (2024) show that the denoiser converges the pro- jection when near the data support. We establish a general convergence result for almost every point and provide a char- acterization of trajectory evolution across different stages, going beyond prior interpretations of the sampling process as an approximate projection to data in Permenter & Yuan (2024). Gao & Li (2024) show that FM can only sample from the data support for discrete measures and analyzes local cluster absorption. However, their proof implicitly as- sumes ODE convergence and their local absorption analysis requires bounded prior support, which is not applicable to the common FM setting. We provide a rigorous proof of ODE convergence and analysis for full trajectory evolution.
By direct computation, the ODE in can also be expressed through the denoiser m. m²) == IE X X, x]] as follows where X, := X + oZ
dx - log go (Ix) do
- 1 (mo -xo). (11)
Notably, the ODE in a can be interpreted as moving toward the denoiser m. (2a). For any z, one can explicitly write m as follows.
||a-y13
exp
20t
y
mz(x) =
p(dy).
(12)
ʃ
Tex
||=-0/112 20
p(dy')
3.2. Attracting and Absorbing
In Section 4.1, we will rigorously establish the existence of FM model ODE trajectory in [0, 1) for any data distribu- tion p with finite 2-moment. This sets the foundation for discussing the properties of the ODE trajectories.
The concurrent work by Baptista et al. (2025) analyzes the dynamical mechanisms underlying memorization in diffu- sion models with empirical measures. Their analysis of the ODE dynamics shares some similarities with our ap- proach to discrete data distributions, e.g., the use of Voronoi diagrams. Their work also proposes some regularization techniques to mitigate memorization.
Note that Equation (11) suggests that the trajectory moves toward the denoiser m. (za ). which itself evolves along the trajectory, complicating the ODE dynamics. We address this by analyzing the geometric relationship between m. (x) and the projection proja (z) onto certain closed sets Ω. This reveals that the ODE trajectory exhibits two key proper- ties: an attracting property praweny--drawing trajectories toward Ω and an absorbing property property-keepiig trajectories within neighborhoods of Ω. We characterize how the sampling pro- cess unfolds into distinct stages by identifying appropriate closed sets Q with these properties. Below, we formulate these properties into two meta-theorems.
3. Denoiser and ODE Dynamics
It turns out that the vector field uy is fully determined by the so-called denoiser-thh the mean of the posterior distribution p(-|X, = I) (Karras et al., 2022). In this section, we first describe some basic properties of the denoiser, then illustrate a general attracting and absorbing dynamics of the ODE. Proofs and missing details can be found in Appendix C.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Attracting toward sets. Let (Ir (1))) be an ODE tra- jectory. The distance da ( to a closed set 52 will decrease if the trajectory direction forms acute angle with the projection direction (see e.g. Corollary B.15):
Remark 3.3. Note that the absorbing property only depends the denoiser's behavior in fixed region rather than re- quiring priori knowledge of how the denoiser evolves along an ODE trajectory. Once established, the absorbing property can propagate the acute angle condition to trajecto- ries within this region, thereby enabling attracting property.
(I) - Ta, projg - -xo) > 0.
(13)
See Figure 3 below for an illustration.
Next, we apply the meta-theorems to provide a broad de- scription of FM ODE dynamics.
Attracting and absorbing to the convex hull of data sup- port. By definition, the denoiser m. (a) always lies in the convex hull of the support of the posterior distribution p(-|X, æ) which is the same as conv conv (supp(p))- Due to convexity, ma (z) always satisfies the acute angle condition with respect to conv(supp(p)). Consequently, the ODE trajectory is attracted toward and ultimately absorbed by conv cony(supp(p))
X
ma(xa)
proja(Xo)
Proposition 3.4. Assume has a bounded support. For any 0 > 0, let (x.))=(U,o)) be a flow matching ODE trajectory follows Equation (11). Then we have the following results for any o E (1]
Figure 3. Illustration of the acute angle condition.
With some quantitative bound on the acute angle condition above, the ODE trajectory will be attracted toward f2.
1. If. E ccov(supp(p)). then Ia E conv(supp(p)) If Vey & conv((upp(p)), then To moves toward conv(supp(p)) with the following decay guarantee:
Theorem 3.1 (Informal Attracting toward sets). Let (Ir) (1))00(0))))) be an ODE trajectory of Equation (11) start- ing from some Zo, Assume that the trajectory avoids the medial axis Da of some closed Q and satisfies the acute angle condition in quantitative manner, then dz(zz) will decrease along the trajectory, and lim da (xx)) ) = 0.
deonv(appp())((5 S deonv(supoó)) (For) ) o 0/01.
See Appendix C.3.1 for the proof and slight generaliza- tion to the case where is a doussian-smoother bounded distribution. A more refined analysis of the trajectory for the initial and intermediate stages (focusing on the data mean and local clusters, respectively) is provided in Sec- tion 4. The terminal stage analysis in Section 5 requires more sophisticated techniques, as we need to develop ab- sorbing properties that effectively avoid the medial axis of the data support-a key technical challenge for establishing convergence (cf. Appendix C.2).
See the formal version Theorem C.7 in Appendix C.3.
The requirement of avoiding the medial axis of trivially satisfied when Ω is convex (its medial axis is empty). More generally, one can rely on the absorbing property (discussed below) to ensure the trajectory stays off the medial axis.
Absorbing by sets. Given a set Ω in Idd we say Q is absorbing for a FM ODE in (02, if for any x E Ω the ODE trajectory (Ia (Io)oe(os,o)) starting at Tyz = x will turns out that the acute
4. Pre--reeeeee TreeTelminnl Trajectory Analysis
remain in Ω for all E (02, angel condition can essentially also guarantee that small neighborhoods of Ω are absorbing.
As illustrated in Figure the FM ODE trajectory overall moves toward the convex hull of the data support (Propo- sition 3.4). Furthermore, trajectory dynamics unfold in distinct stages. In this section, we first establish the well- posedness of the FM ODE to ensure the existence of tra- jectories. Then, we apply attracting and absorbing prop- toward the mean and local clusters- to provide a detailed analysis of the pre-terminal trajectory dynamics.
Theorem 3.2 (Absorbing by sets). For any closed set Q and r >>>0 O, we consider the open neighborhood B, (1)).
1. Let (2) denote the closure of the set (2)). If (9) n En = 9 and suppose for any x E (2)
and any E (02.01), one has (ma (x)- (r)ja(r)-
x) > 0, then B, (S2) is absorbing in (02,01.
4.1. Well-posedness of FM ODEs for t E [0, 1)
2. If there exists some To > 0 such that B, (9) is absorb- ing in (02, o.) all r E (0, ro), then Q is absorbing in (02, 01) as well.
The following result establishes the existence and unique- ness of solutions the FM ODE in [0, 1) under very weak
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
assumptions, significantly expanding on previous work (Lip- man et al., 2022; Gao et al., 2024) to contain cases where p is supported on subspaces or submanifolds.
1. S is closed, bounded, and diam(() = D < co.
For all x E SYS, dconv(s)(ar) > 2D.
Theorem 4.1. Assume p has a finite 2-moment, then for
Then, the denoiser mo(v) will be close to conv(S) for y near conv(S) and when a is not too large.
every To E R2, there exists a unique solution (zz))e(o.1) to
Equation (2). Furthermore, the flow map D is continuous and satisfies that (Vr) Pprioor = Pt for all t E [0, 1).
Proposition 4.3. Assume that S is local cluster of a prob- ability measure p satisfying the Local Cluster Assumption and as = p(S) > 0. Then, for any x. E RR. such that deanv( (8) (z) S D - 6, we have that
The proof utilizes a careful analysis of the posterior covari- ance Cov X X x| to establish local Lipschitzness and integrability of it. In addition, we show that the denoiser (1) grows at most linearly in z-a non-trivial bound ob- tained under the mild assumption that the data distribution y has only a finite 2-moment. With these properties, we then apply the theory of continuity equations (see Ambrosio et al. (2008, Section 8. 8.1)) to conclude the proof.
deonv(s) (mo (I)) - diam(()) (LL-v)//aaa 32g
2n
Proposition 4.4. With the same assumptions as in Proposi-
tion 4.3, let C dlam(()) V Les
D/2-c and define
This result trivially extends the a parameter and forms the foundation for analyzing pre-terminal trajectory properties.
co(S,c) = {
3DD (1oR((
iC < <1,
00,
otherwise.
4.2. Initial Stage of the Sampling Process
Then for any 01 < oo(S, e). BD;2- is ab- sorbing in (0, and any ODE trajectory starting from Zoo E cBD/2-c(covv((
When O, we have m=(x) = E[ suggesting the tra- jectory will initially approach the mean of data distribution. In this subsection, we quantitatively validate this intuition for a broad class of distributions.
converges to conv(S) as a + O.
Note that when the weight as of the local cluster S is large, we do not necessarily have that C < 1. In this case, the cluster S, in fact, exhibits stronger attracting force and the above result holds for all 01 > 0.
Proposition 4.2. Let 8 2 O and ps, be distribution on
Rd with a bounded support Ω == supp(pt). Let p == Ps * (0, 5²2).. For a point zo with |ro X = Ro where X- ~ Ps and for any parameter < Ç < 1, define
In Appendix J.1, we detail the synthetic data used in Figure 2 to validate the above Propositions 4.2 and 4.4. We also ob- serve in Appendix J.2 that despite the theoretical constants not being tight (as is common with worst-case bounds), the qualitative behavior of an initial mean-attraction phase consistently holds in practice. Although Proposition 4.4 is developed under the local-cluster assumption, real-world datasets seldom satisfy it exactly. Empirically, however, we find that ODE trajectories still gravitate toward locally dense regions- evenn even when clusters veclla--indiaatog that the dynamics are more robust than the assumption suggests (see Appendix J.3). We provide partial theoretical expla- nation in Corollary H.I: for distribution p obtained by convolving a Gaussian with any measure that does satisfy the local-cluster assumption, we prove that dense regions remain attracting and absorbing for the flow. Together, the analysis and experiments point to a broader validity of the ODE dynamics beyond the confines of our assumptions.
mii S, == (2Rgdiam(())//n (lgg/l////1//// + /Ro//dia(()))
Then for all 01 >
(S, S. Ro) 88, a trajectory starting from Fo, = To will
(Ia) (.ull//
approach X. with the rate:
| ||X]|| < Ro(o2 + 8 //)"=/(07/)))) + 5
Note that as Ç approaches 1, Wn, Ro) decreases, ex- tending the range of a that applicable. However, the rate weakens as gets closer to 1.
4.3. Intermediate Stage of the Sampling Process
As a decreases, trajectory behavior starts being influenced by coarse-scale geometry of the data, particularly its local clusters. When the trajectory lies close to local cluster, it is attracted toward (and eventually absorbed by) the convex hull of that cluster-indicallt the FM model. We formalize this notion via an assumption that characterizes a well-separated local cluster.
While the above results are theoretical, they reveal how data geometry fundamentally shapes FM ODE dynam- ies pss-ppaticullrly. the cluster absorption results suggest a "locking" property where trajectories are systematically ab- sorbed into local clusters. This property provides a theoreti- cal foundation for why FM models achieve effective feature separation and mode coverage, as observed empirically in
robust feature emergence in
Local Cluster Assumption. Let p be probability measure on 2d and we say a set S is a local cluster of Ω == supp(p) the following conditions hold:
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Georgiev al. (2023). It further suggests that embedding data into latent spaces with clear geometric structure (e-g-- by object categories or visual attributes) could enhance ro- bust feature learning.
We address these issues by a refined denoiser's convergence result with rate guarantee, which occurs at O(a5) rate for any 0 < < < 1. See Appendix D for the proof and de- example, the convergence rate is /mo + O for distributions on an m-dimensional submanifold and ex- ponential for discrete distributions. This convergence rate yields a strengthened version of the absorbing result in Sec- tion 3.2, ensuring that trajectories are: (1) absorbed near Ω to avoid the medial axis, and (2) attracted to Zi rapidly enough to ensure convergence. Our theoretical analysis works for data distributions satisfying:
Finally, as a approaches 0, trajectories are drawn to their nearest data points, with the nature of this attraction strongly governed by the fine-scale geometry of the data support whether discrete or struttudd We analyze this terminal convergence behavior in the next section.
5. Terminal Trajectory Analysis
Assumption 5.2 (Regularity assumptions for data distribu- tion). Let p be a probability measure on Rd with a finite 2-moment satisfying the following properties:
The well-posedness of the FM ODE in [0. ensures the probability path under flow map v. approaches the data dis- tribution. However, distributional convergence alone does not guarantee trajectory convergence, as pathological cases like winding paths around data points may exist (see Fig- ure lb). The convergence of ODE trajectories- eepectorress-eeivlleeeelly the existence of flow map V at = i-is crucial for gen- erating samples stably and for models like the consistency model (Song et al., 2023), which learns the flow map 1
1. The reach 7q of the support Q == supp(p) is positive";
2. There exist constants k: 2 0 and c > such that for any radius R > O, there a constant Cn > 0 satisfying the following: for any small radius S r < c cand any
x E (0) S2 we have (B)))) 2 CR
Any discrete distribution satisfies the assumptions with k O. Moreover, p satisfies the assumptions with & = m when supported on an m-dimensional linear subspace or compact submanifold with positive reach, provided has a finite 2- 2-moment and non-vanishing density (cf. Lemma I.I). We emphasize that the positive reach condition is not restrictive and is common assumption to ensure that the data manifold has no "sharp turns" in Rd (Niyogi et al., 2008; Fefferman et al., 2016). This condition holds for common smooth compact submanifolds (Licutier & Wintraecken, 2024) like spheres and tori.
In this section, we establish the convergence of ODE tra- jectories at & = for broad class of data distributions. With this result, we analyze the equivariance of 1 un- der geometric transformations and discuss implications for memorization behavior.
5.1. Convergence of ODE Trajectories at t =
The convergence of ODE trajectory at t 1, when ex- pressed in terms of a, requires the integration of daz = de (mo (xx) - xgg) ) to remain convergent as a - 0. The di- vergence of 1 1 do creates a potential singularity that must be counteracted by a rapid diminishing of [| (xag) x ll- We study the denoiser's 's terminal behavior at fixed point x to understand when diminishing may occur.
We now state our main result:
Theorem 5.3. For p satisfying Assumption 5.2, we have
1. v (x()) == lim,-1 (e(x) exists for x E IRd a.e. 2. 1 is a measurable map and ( 1V1)i = p.
Theorem 5.1 (Convergence of denoiser to projection). Let p be probability distribution with a finite 2-moment and support Q. Then for all x E Ted
Furthermore, we have the following estimate of the conver- gence rate of the flow map. Recall that o2 := 3:/ax /aa, then, we have that for any fixed 0 < Ç 1,
lim ma (x) (x)) a=
| (x) - V,(x)) 000
where Σg denotes the medial axis of Q
Since proja(z) = x precisely when x E Ω, the term m=(x.) 2g diminishes if x. is attracted to Ω. The con- vergence only holds for E Idd since projection is not well-defined at the medial axis, which also causes the de- noiser's s Lipschitz constant to blow up (see Appendix C.2), preventing the use of standard ODE theory like the Picard- Lindelöf theorem.
This theorem establishes the existence and convergence of the flow map for general data distributions. The convergence rate can be further refined for specific cases:
Theorem 5.4. When is supported on a submanifold or discrete set, we have the following convergence results:
Manifold. Let M C R.d be an m-dim closed submanifold with positive reach and bounded second fundamental form
We also establish the convergence of m, in Corollary D.3 which is nontrivial consequence the convergence of m..
'hh support Q can be Rd and in this case the reach T = co.
7

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
up to its first-order derivatives. Assume that p is supported on M, has a finite 2-moment, and has a density given by p(dx) p(æ;) vooa)volas (dz), where p MM- - R is smooth and nonvanishing, Then, for a.e. E Rd, we have that
Equivariance under geometric transformations. Having established the existence of the flow map V Rdd - Rd under mild assumptions, we now investigate how ambient space geometry affects the flow maps through their behavior under geometric transformations. This analysis has practi- cal implications for stability under data augmentation and reveals important equivariance properties.
| (00))-M,(z))) v.(2))1 O(Voi);
Discrete. If = -E= 0 denotes a discrete probability measure, then for a.e. x E RRd, we have that
We examine how FM flow maps transform under similarity transformations T Rd + R of the form T(x) = (0x + b). where > 0 is scaling factor, O is an orthogona matrix, and b is a translation vector. These transformations include any combination of scaling, rotation and translation.
[1)) -
= O((o))
We examine a tractable special case to assess our conver- gence rate bounds: a standard Gaussian distribution on a subspace, where closed-form solutions exist.
For a data distribution p satisfying Assumption 5.2, let p == # p denote the transformed distribution. To relate the flow maps v (for p) and V (for p). we identify that the flow for the transformed data p can be obtained from the flow for the original data p by choosing appropriate scheduling functions āj and B, with respect to the original functions a and 3r. Specifically, taking a₂ := s,a;/7 and B, == S,,3, where st is any positive smooth function with so = 1, $1 = y (or simply 8 = 1 when 7 = 1), we establish: Proposition 5.7 (Equivariance under similarity transforma- tions). For any x E gd and t E [o, 1), we have that vr(Ox) se (O + axb).
Example 5.5 (Data supported on subspaces). For any 0- < m S d, consider the subspace RT C Rd. We express any point x E Rd as x = (x. y). where x E II" and y E Rd--.. Assume that the probability measure p is supported on IR and satisfies Assumption 5.2. One can show that FM ODE trajectories allow a dimension reduction in the following manner. For any initial point Zo (zu. yo) E R the FM ODE trajectory is given by
Et = (zt, Beyo), for any t [0,
(14)
where ( is the trajectory of the FM ODE on I with initial point zo. and with p regarded distribution on mi see Proposition E.2 for a proof.
Whenever V1(x) exists (this holds for a.e. x E RRd by
Theorem 5.3), we have that T (Ox) exists and satisfies
(I ~OV, (x) + b).
Let cr = & and Be = 1 I, and let p be the standard Gaussian on Th. By Lemma F. 1 and Equation (14), we have that
Remark 5.8 (Data distribution on affine subspaces). By Proposition 5.7, we can generalize Example 5.5 to cases where p is supported on any affine subspace A C Rd (e.g.. point translated away from origin or a shifted linear sub- space). The idea is simple: first apply rigid transformation to map A to Igdim(A) C Rd, then apply the FM model there. This suggests that for data distribution supported on an affine subspace, we can reduce computation by first projecting onto that subspace, training an FM model there, and extending it back to ambient space via this result.
t)2 - tro, --))))))
x =
Also, when m = 0, p = 60 is supported on a single point. Then, the denoiser is always 0 and the ODE trajectory is given by (z,- (1 (1-t)=o)ee(0.1) In both cases, ||
(1 =
This example demonstrates our rate's optimality for discrete distributions, while suggesting potential improvement for manifolds: the current ((voi) rate might be improved to O(o,). This conjecture is supported by the linear conver- gence rate (ar) of the ODE's distribution path do + p (see proposition below), which suggests the trajectory should converge at the same rate.
5.2. Terminal Absorbing Behavior and Memorization
In this subsection, we focus on the case when p = Σ 0; 5z, is supported on discrete set, as this repre- sents an important scenario corresponding empirical tar- get distributions derived from training data. We provide detailed characterization of the terminal stage and show that each point 2; exhibits strong attracting behavior during this stage, which is connected to the memorization in diffusion models (Carlini et al., 2023; Wen et al., 2024).
Proposition 5.6. For any probability measure p with finite 2-moment, we have that dw.2(90. p) = ((a).
An important practical implication of the convergence rates in Theorems 5.3 and 5.4 is that, during the terminal stage, the ODE trajectory exhibits minor movements suggesting one can use fewer sampling steps to generate samples with- out sacrificing quality.
We let ? { Fn denote the support of p. For any small > 0, we define the e-shrunk Voronoi cells as
V := fx : || - 2; S || 2, 6, Vx, # x. E 22.
8

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Note that V is convex and as e + O, V. expands to the classical Voronoi cell V; of x; with _ V; Rd.
be smooth map. Assume there exists a function d(o) with lo ((()) 0 such that Im (3:) m=(x)]l S ((a) for all x E IR Then, for any i 1, ,,..., n, there exists oo 6) > 0 such that for all 0 < Jo < oo(V, ,)) and any E Vi, the ODE trajectory (x0))E((00o)) for Equa- tion (15), starting from Zoo = y, converges to Ij as a - O.
We introduce (o(V)) such that V is absorbing for (0, to (V;)). Specifically, let sep(z;) := dor(a.) (zx) de- note the separation of xi and we introduce a constant
sep((r.) (x)) - -2
a4
If further, both limits limo- aa and limg- m" (z.)) known to exist, then limo- Im (x²) - x I1 0.
V
diam(S))
ag
for any parameter& e E (0, (c)((())))/2)) /2), where o; is the weight of xy in p. Then the time on (V;;) is defined as
This proposition shows that for an FM model to be capable of generalization, the near-terminal denoiser itself must gen- eralize-icc., it must approximate projection onto the true underlying data manifold rather than simply projecting onto the training points. This insight motivates careful tuning of denoiser training near the terminal time. We validate these insights empirically in both synthetical (Appendix J. and image dataset (Appendix J.2).
{00, 00, (log (log(C(.)) -122 if CB > 1.
if C 1
oo(V; - -
The constant do (V))) is the time when the ODE trajectory is attracted to V, and with larger weight a; or higher sep- aration sep(æ;), (2;). the time (V{5) is larger, showing strong attraction early on.
The following result shows that after approaching the mean and then being attracted to a local cluster, the ODE trajectory will eventually be attracted to the nearest data point.
6. Discussion
Our study significantly enhances the theoretical foundation for FM models by establishing a connection between data geometry and FM ODE dynamics. This leads to interest- ing practical implications; for example: (1) The FM ODE trajectory direction's initial alignment with the mean and its terminal time convergence suggest one can use more sparse sampling resources in these stages and reallocate more resources to the intermediate stage where the denoiser evolves more significantly which aligns with empirical find- ings in Esser et al. (2024). (2) The interaction between flow trajectories and data geometry through attracting and absorbing behavior reveals how the same dataset can exhibit distinct sampling trajectories when embedded in different spaces. This could be utilized to optimize latent space for improved generation and facilitate stable fine-tuning through careful adaptation when integrating new data. (3) Identi- fying the importance of terminal stages in memorization suggests targeted regularization in training such as regular- izing the Jacobian of denoiser to avoid collapsing to locally constant maps; see more discussion in Remark C.3. Look- ing ahead, we aim to explore these directions to develop more understanding of memorization versus generalization, as well as more efficient diffusion models with steerable generation.
Proposition 5.9. Fix an arbitrary 0 < 01 < (V(V). Then, for any y E ViS the ODE trajectory (xo))E(0.oo) starting from Tag = y will stay inside Vř, i..,, To E Vf.. Further- more, (Io (0. will converge to 2; as a - 0.
Discussion memorization. Memorization occurs when model perfectly fits training data and fails to generalize. This is relevant to FM models since the unique solution in Equation (5) or Equation (10) regarding empirical data only reproduces training data. The constant ou (V{s) indi- cates attraction strength of each training sample-higher values (from larger weights a, or more isolated points) sug- gest increased memorization risk. This explains empirical findings of increased memorization for duplicate samples in Somepalli et al. (2023), as duplicates raise a; in the em- pirical distribution. For CIFAR- 10 with c = 1.0, the mean (VV") across training images is approximately 0.17, corre- sponding to the final quarter of EDM's 's sampling steps (Kar- ras et al., 2022). This suggests training in this critical final stage should not target optimality to avoid memorization.
We now provide more formal connection between the terminal behavior of the ODE trajectory and the memoriza- tion phenomenon. For neural network denoiser my, the corresponding ODE trajectory is:
Theoretically, our analysis assumes that X ~ p and Z ~ Pprior are independent when constructing the prob- ability path (pr)eejo.1). It will be intriguing in future work to investigate whether this analysis can be extended to set- tings where X and Z are dependent. Such cases naturally arise, for example, when applying rectification techniques as in (Liu et al., 2023) or when employing known coupling methods to enhance flow matching, as explored in (Poola- dian et al., 2023; Tong et al., 2024).
dx
( z.)). do = σ
1
(15)
The following result shows that an asymptotically optimally trained denoiser m² merely reproduces the training data. Propusition 5.10 (Memorization of asymptotically optimal denoiser). Let p= Σi= 0i Szy, and let m² : Rdd + Rdd
9

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Impact Statement
Bialozyt, A. The tangent cone, the dimension and the fron-
tier of the medial axis. Nonlinear Differential Equations and Applications NoDEA, 30(2):27, 2023.
This paper studies the theoretical properties of the flow- matching model, novel generative model that has been widely adopted in practice. Our investigation on the memo- rization phenomenon could potentially help to design better variants that do not leak private information and have positive societal impact.
Biroli, G., Bonnaire, T., De Bortoli, V., and Mézard, M. Dynamical regimes of diffusion models. Nature Commu- nications, 15(1):9957, 2024.
Carlini, N., Hayes, J., Nasr, M., Jagielski, M., Sehwag, V., Tramer, F., Balle, B., Ippolito, D., and Wallace, Extracting training data from diffusion models. In 32nd USENIX Security Symposium (USENIX Security 23), pp. 5253-5270, 2023.
Acknowledgements
This work is partially supported by NSF grants CCF- 21 1216665, CCF-2217058, CCF-2310411 and CCF-2403452.
Chen, D., Zhou, Z., Wang, C., Shen, C., and Lyu, S. On the trajectory regularity of ODE-based diffusion sampling. In Forty-first International Conference on Machine Learn- ing. 2024.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Appendix
Table of Contents for Appendix
Table Symbols and Roadmap of Theoretical Results
14
B. Geometric Notions and Results
15
C. Denoiser and FM ODE: Singularity, Attracting and Absorbing
20
D. Denoiser Analysis: Concentration and Convergence of the Posterior Distribution
30
E. Explicit Eamples of FM ODEs
40
F. Proofs in Section 2
41
G. Proofs in Section 3
42
Proofs in Section
42
I. Proofs in Section
51
J. Experiments
58
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
A. Table of Symbols and Roadmap of Theoretical Results
In this section, we provide a table of symbols used the paper and roadmap the theoretical results in Figure
Symbol
Meaning/ Description
R"
d-dimensional Euclidean space.
|| II supp(p)
Euclidean norm in Rd.
Support of the probability measure p.
X ~
Random variable X with distribution Second moment of i.e., J ((dx))
M2(p)
dSa
Diameter of a set 52 C IRd.
Ω
Closure of set ?2
da(
Point-to-set distance from to Ω i.e., da == inf || v|1- Projection of onto closed set f C TR...
(a)
c2nv(S2)
Convex hull of a set f.
En
Medial axis of closed set Ω C Rd.
(x)
Local feature size of regarding set ft i.c. distance to medial axis Eg. Reach of set Ω C Rd Injectivity radius of manifold M.
m
(M)
sep(z;)
Separation scale w.r.t. discrete point 24-
6,
Dirac delta measure at Gaussian distribution with mean y and covariance Σ.
((,,E) E)
p*q
Convolution of probability measures p and q-
do = p= O(0,
The blurred (or noisy) distribution noise level Nearest-neighhor set of x in discrete set f2 and the measure restricted that set.
NNa(F), (NN(")
F
A state (trajectory point) indexed by time t [0, in the FM ODE.
To
A state (trajectory point) indexed by noise level (0, oo) in the FM ODE.
X,
The random variable a; 3e Z. where Z A/(0,
X.
The random variable X + a Z, where Z (0, 1).
p(||X; x), p(-1Xo = x)
Posterior distributions of X given X, and given X. z, respectively.
mo(x)
The denoiser in a parameter, i.c. E X, z].
(z)
The denoiser in parameter (implicitly dependent on at B,); i.c. E|X X; x].
lo
Score function of fo- Wasserstein distance.
dw,s
(x)
Vector field in the FM ODE for t 10,
W.
The flow map of the FM ODE.
Pt PPo
The pushforward distribution of an initial distribution Po under V.,
14

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Theoretical Foundations
Absorbing Attracting Denoiser Thm. C.7. 3.2, C.II, Prop. C.9
Concentration of Posterior Thm. D.I. D.5.Prop. D.7
Well-posedness (0, Thm. 4.1
ODE Trajectory Stages
Initial Stage (Mean Attraction) Prop. 4.2
Intermediate Stage (Cluster Absorption) Prop. 4.4
Terminal Stage (Convergence & Memorization) Thmn 5.3, 5.4 Prop. 5.9, 5.10
Equivariance of Flow Maps Thm. 5.7
Figure Roadmap theoretical results. Our analysis builds upon three foundational components: (I) well-posedness of FM ODEs [0, 1), establishing existence and uniqueness of sejeettoress-setinng the foundation for subsequent analysis. (2) attracting and absorbing properties derived from denoiser behavior, (3) concentration results for posterior distributions. These foundations enable us to characterize the full evolution of FM ODE trajectories through three distinct stages: initial mean attraction, intermediate cluster absorption, terminal convergence with memorization implications. The convergence at terminal time further allows us to establish equivariance properties of flow maps. Together, these results provide complete geometric understanding of FM ODE dynamics.
B. Geometric Notions and Results
In this section, we review basic concepts from convex geometry and metric geometry, and establish several results which will be used in our later proofs. These geometric results are also of independent interest and may be applicable in other contexts.
B.1. Convex Geometry Notions and Results
In this subsection, we collect some basic notions and results convex geometry that are used in the proofs. Our main reference is the book (Hug& & Weil, 2020).
We first introduce the definition of convex set and convex function. Definition B.1 (Convex set). set K C Rd is called a convex set if for any Ii. Ii....,Ia, E K and S ai,..., I 1 such
that Σ Og 1,, we have that Σ 04:2% E K.
Definition B.2 (Convex function). A function f Rd + IR is called convex function if for any x, E Rd and 0 < a < 1, we have that faaz + (1 - a))) fx) + (1 - (y)
An intermediate result that the sublevel sets {f < c} or {f < c} of convex function are convex sets; see Hug Weil (2020, Remark 2.6).
We now introduce the definition of the convex hull of set. Definition B.3 (Convex hull (Hug Weil, 2020, Definition 1.3, Theorem 1.2)). The convex hull of a set Ω RRd the smallest convex that contains and is denoted by conv(St). Additionally, we have that
(onv( = Σ a,e : k N, E Q E [0, Σ a. -1 aa.=1
1=1
15

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Let S2 be set, and x E S2. We say hyperplane give by a linear function H is supporting hyperplane of Ω at if the following conditions hold:
1. (x) O.
2. (v) 2 0 for all E Ω
For a closed convex set Ωt every boundary point of Ω has a supporting hyperplane. Proposition B.4 (Supporting hyperplane (Hug& & Weil, 2020, Theorem 1.16)). Let K be a closed convex set in R and OK. Then, there exists a supporting hyperplane of K at s..
We collect some basic properties regarding convex set K and its distance function dx(x)= == inf inf Proposition B.5. Let K be convex set in IR and Qt be set in Rd. Then, we have that
uEK || - vll-
1. For each a Rd, there exists a unique point proj (x) E K such that I| projx(())) (x()).
2. The distance function dg(a) ) is a convex function.
3. For any r 2 0, the r-thickening of K. defined as B,(K) == fx E dx(x) < r.. is a convex set.
4. The diameter of Q is the same as the diameter of its convex hull, that is diam( - diam(conv(())).
5. Let >> 0, then a set S2 is convex if and only if a Q is convex.
B.2. Metric Geometry Notions and Results
Let Ω Rd be a closed subset. Recall that En denotes the medial axis of Q and da IRd + R is defined by dg(x) = yen | y|l. We now consider certain properties of the projection function projn Mg + 22, where Mg dd En.
We first recall the definition of the local feature size in Amenta & Bern (1998) with slight generalization that consider all points in R instead of only points in Ω.
Definition B.6 ((Amenta & Bern, 1998)). For any E Rd, we define the local feature size lfsn of æ as (fs((x)) = (z).
For any x E Dg we let øn == (x)) We consider the following set for any x E 2
Tn(x) = { 2 0: projja In + t x - 2Ω
| x2ll
=
Lemma B.7. We have the following characterizations of [n()).
Tn(x) is an interval.
For any t E Tn(x), we have that an + t 112-2a0l $ En.
We let Ro(x)= == sup Ta(#). 1f0 < Re(x)= < oo, we have that xn + Ro(x) z-z0
Tz-zall E Σn.
dg) + (f)a())) Ta(x).
Proof of Lemma B.7. The first three items follow from the pioneering work Federer (1959, Theorem 4.8); see also Delfour & Zolésio (2011, Theorem 6.2) for more details.
We provide proof for the last item. First of all, it is straightforward see that [0, dg(x)] C (z).. Now, assume that Rạ(x) E [dg(x), (z(x) + (x)). This implies that
un r | zall
3D
x + -d)(x))x) (r-
Ji
E En
|| xnll
and hence
[z))(()) dxa (x) dr, S dn(x) lf)p(z).
This is contradiction and hence Ra(x) 2 dz(x) + 1fsn(r). This concludes the proof.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Next, for any point E Ω2 we analyze the angle Zangb for any b Q. The following lemma is slight variant of Federer (1959, Theorem 4.8 (7)).
Lemma B.8. For any 2/ and any t >> 0 such that E Tn(z), the following holds for any b f2:
1
b|1
sall
(x - 22, ZQ b) 2
2t
Proof. By definition of (2), we have that projq
z212 +
)
xy. Therefore, have that
zn + t
x - 2Q
b 2 df
on + t
I IQ
t?
|| Inl
||
xnl
| - - 611 + 2t
IΩ b,
I. - JR
+ 22 2 t
x x2l
21 (. b, x -xn) 2 an b11²112 JΩ
In
b1 b1| J2
(xr mn, 12
b) 2
When Ω is convex, then Ta(x) = [0, oo). In this way, we have the following corollary. Corollary B.9. 1 is convex, then for any b E Ω and any E Rdd we have that (ar - I2, J2 b) 2 0.
This control the angle Lrzab allows us to derive the following result that bounds the distance between b E S and the projection 2p ( in terms of the distance between and æ.
Lemma B.10. For any x E Eg and any such that t Ta(æ), we have that for any b E Q
b11 2 d=(x)² 11b xTll (1-1)((110 dr(z))
Proof. The case when Ω trivially holds. Below we consider the case when E Eg nS In this case, dz(x) > as Ω is a closed set.
By the law of cosines, we have that
cos (o8(())))) dx(x)) + 116 znl 1| bll
2d((2)) (r) 11 Fn
Suppose ||- bl12 < d + 11b [|b-xoll2(()
=01ll(111 (1- dr(x)) ) then we have that
cos(Lrxal) = dx)x)) + 16 - xΩ - || b1(
2do(x) b 11rll
[|dr(x)2 ||b -2aal2 do (zx) - 110 xn
12 b1b
snl 12 (a(=)
2d ral
||b 2all2
2f
By applying Lemma B.8 b and x, have that
|00
b/12||æ rall
(x - Zg. ZD - b) 2
2t
This implies the following estimate for the cosine of the angle Zraggh:
(r - IΩ, b 22)
100
b12
cos(Lrzn))
||
||
2/
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers This contradicts the inequality above and hence must have that
> b1l2 2 dn(x)) Ibb sall? (1- dr((x)
t
We then have the following corollary which will be used in the proof of Theorem D.I. Corollary B.11. Fix any E Eip any E [da())) (2 + (.)) and any >> Then, we have that
B Vanle)+e(0-d00)/ (z) n (.(
Proof. For any E (4nl0)+c(--dd(111)) (x) n Q2 we have that
& (1-
+2(1-
(1-dale))
da(x))
|| b112 da(x)² +
By Lemma 10 have that
2)))2= + 116-ar xnll 12-1p111 dg(x)
| 12-669 6112 dr((z)2
Combining the two inequalities, have that ||b xn <² and hence b (2g)
Finally, we derive the local Lipschitz continuity of the projection function. Lemma B.12 (Local Lipschitz continuity of the projection). For any 0 and any x. y E Bd such that Ifsa(() and (1) have that
vall mmax(da()); (x)) +1) ||a vll-
|
Proof. Since lfs(x)) > and 1y) > e, by Lemma B.7 we have that
proja [n (Mn + (E+ (x+(x)((x))
x - 32
|| xzll
and
proja
ya + (c+ dz(())
Vi
V/a
11y val
By applying Lemma B.8 to x, Z2 Uo and separately to y, Vn, Fa, we have that
|Ian ya 11
(xn ya, za)
2 + dn(x)) do(x),
IMN yoll
Wy Fn. y ya)
do(y).
(e + dr()))
By adding the two inequalities about the inner products, we have that
|| 1
|m vnll dn(v),
(In yn. x ----y y va)
dg (x)
2 + dx(x))
2 + dr(l))
|mn yn 1|2
|| yall da(y),
(mq Va. y) -1120 3all2 2
dg (z)
2 (E + dr(x))
2 ( + dg(y))
|
yall 1| dn(x)
|| ynll
(m yn. y) 2 |00 3all2
do(s).
2 (c + (z)
2 (c + da(v))
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Let m dn then we have that
0 < m
< (1-
dr((x) 2 + do(x))
dn(y)
+E
2 (e + dn(v))
Therefore, by
inequality, have that
|| - yyll? (I (1-m m +e
m
S (2n - you x - 8) S || - -yellllz - yll.
This implies ||.02 yall S m=tt) || -yll-
B.2.1. DIFFERENTIABILITY OF THE DISTANCE FUNCTION
Let 52 c IRd be any closed subset. For any x E Rd, we let (a(x) == {y E Ω || - yll = dd(x)) denote the of points in Ω that achieve the infimum. When a is not in the medial axis of f2 the set Pa(x)) is the singleton set (x))
Interestingly, there is the following result result showing the existence of one sided directional derivatives for da by de Mises (1937) (see also Bialožyt (2023) for a more recent English treatment).
Lemma B.13 ((de Mises, 1937)). For any vector v E TRd the one-sided directional derivative of da at x R the direction exists and is given by
Doda(x)= = inf { (v.y - x) 1y xl
y E Pa(a)
Motivated by the this result, we mimick the proof and establish the following result for the squared distance function dz Notice that, we can remove the constraint for x é Ω.
Lemma B.14. For any x E Ed the directional derivative of du at exists and is given by
(x) = -2, projg x))
Proof. Without loss of generality, we assume that æ is the origin and v (c, .,.... 0) for some e 0 (one can achieve these by applying rigid transformations).
We let zp proig (x), to and x; proig (x:). Since ||=| xell 5 |000 xz |l. we have that
| ||= & 20 +1) - x0 (1))))
-x0
where (1) denotes the first coordinate of xī. Note that ||= || = ||=ا - xI| | - x ||a0 |I-
So have that
0 S x (1) To +.(1)
Now have that by the cosine rule of the triangle formed by xt to, 0 and 2t, have that
dg = || 12 + fc2 21 x - =x)
Therefore, have that
(x,) d2 (z)
| 2 ||اا||
2(v, x; - x).
t
As discussed above, we have that
|x 1|2
||asel1|
S +.0 x -)) (1))
By continuity of proje outside Eg, we have that xt + x as t 0. Therefore, we have that
lim - 2(v x² x) = 2(U (x) x)..
Corollary B.15. Let (Iz) be a differentiable curve in R En, then we have that dk (x)) is differentiable with respect to and we have that
d df(())) = (z,) = 2((ie. (x)) - -z1)-
dt
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
C. Denoiser and FM ODE: Singularity, Attracting and Absorbing
C.1. Basics About the Denoiser
Well-definedness of the denoiser. Although denoisers have been widely used in various works, their well-definedness has not been rigorously established in the literature liteature-sspecffcally whether the integral defining the conditional mean E IX| ; =i] exists. We address this gap in the following proposition.
Proposition C.1. p has finite I-moment, then p(-|X; z) also has a finite I-moment for any t E [0, 1), making (x) == X = x| well-defined. The same applies to mo for a E (0, oo).
Proof of Proposition C.I. For the normalizing factor Z =
J exp
1|x- ||x- p(dy'), we note that
exp
20
< 1. Hence, the factor Z must be positive and bounded.
Now, we consider the following integral
f exp
|| (~1ee- anyl²)
IIulp(ddy) - [Illp(dd < oc.
2
The last inequality follows from the fact that p has finite 1-moment. Hence, the posterior distribution p(-|X; x:) has finite 1-moment and the denoiser m, is well-defined.
An alternative parametrization for a. The backward integration w.r.t. a in Equation (7) might be cumbersome in analysis and we alternatively use the parameter À = log(o). We let (A) denote the inverse function. Then, when a changes from do to 0, A changes from -o to oc. For an ODE trajectory (z, (.),,)))) we define (zx, == (--o))) For any E Idd we define mx (x) == (x)) Then, the ODE in À has concise form:
dx X dx
mx(xx)=
- ZX.
(16)
Proof of Equation (16). Since x = log a, we have that
dA
do
and thus the ODE equation Equation (11) becomes
dr dx do
dA do dd = -V log do (xx) (-o)
fyexp
aa y/l
p(dy)
20
-oa [ fexp
y11 1/ p(dgy)
20
= =xx(xx) - xX.
Jacobians of the denoiser and data covariance. We point out that the denoiser, under some mild condition on the data distribution p, is differentiable and its Jacobian is inherently connected with the covariance matrix of the posterior distribution p(-IX, x)) (or p(- X, z)). Similar formulas for computing the Jacobian have been utilized before for various purposes; see, for example, Zhang et al. (2024, Lemma B.2.1). Gao et al. (2024, Lemma 4.1), Ben-Hamu et al. (2024, Proposition 4.1) and Rissanen et al. (2025, Equation (8)). Moreover, the covariance formula in Proposition C.2 is direct consequence of higher order generalization of Tweedie's 's formula, which has been studied in previous works (see, e.g.. Efron (2011), Meng et al. (2021)).
Proposition C.2. Assume that p has a finite 2-moment. For any E [0, 1), we have that m, is differentiable. In particular, the Jacobian V mm can be explicitly expressed as follows for any Rd:
V (x))(x)
04
= x)p(dz'| [] (a p(t
=x)
2,33
04 Cov|X|XX
=x].
Bi
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Furthermore, we let og then for any (0):
p(dz|X, || (2 p(d 2)p(dz'| x)
V mo (x)
20
σ² Cov X Xx x|xx
Proof of Proposition C.2. Recall that
[ cxx
(-1lx ||=-a==11"
2
2 2p(dz)
exp
===-,,=1 p(dz)
2/3
mx(x)=
and p(dz|X, = =x)
['exp
p(dz)
exp
p(dz)
288
20g
We let w,(xx,) 2) = exp (-
Then,
2,8
20
=11
=11
V [=me(x)= = Vv.
exp
234
p(dz)
Sexp
110-012111
p(dz")
2:39
-( we (()(d))))()'))
w2(
a12 )
We(z, z')p(d=') (u((ex=)p(dd)))
p(dz)
B8
( wi (1 (D((2))
wx(21)) / we( 2¹)
(- a2z
p(dzz)
p(dz)
B7
(ʃ w
2' )p(d)))
/f We(x, 2) (-
a42
+
I a1z
T 2-atz"
p(dz)p(dz)
B3
88
Oi
(f we(a, 2' m(dz))
-2
-f we we 2) 20)² -21" p(dz)p(dz')
B3
(f (10(0,)))(((2)
-2
(0) (2,, z') (z z²)² p(dz)p(dzz)
2
0
/f (2=) p(ddd = x)p(de =I)).
28
The second equation follows from a similar calculation.
Remark C.3. We have established in Corollary D.3 that for any x & Eg (where En denotes the medial axis of the support ? of p), the denoiser satisfies
(x) + (x) as t +
We conjecture that a similar convergence holds for the Jacobian, that is,
V (x) + V Vzprojo(a) as t 1.
Note that when 52 is discrete set, the projection (r) is locally constant almost everywhere, so its Jacobian V Vzprojn(1) is identically zero. This suggests a potential pitfall: if V (x) also collapses to zero, the model may effectively "memorize" training points. Regularizing V M.m, (x) to prevent such collapse could thus help mitigate memorization.
C.2. Denoiser and ODE Dynamics: Terminal Time Singularity
The terminal time is referred to as the time f = 1 (or = 0, X + xc) in the FM model. The convergence of the ODE trajectory at the terminal time relies on the terminal time regularity of the vector field. We now elucidate two types of singularities of the vector field that arise at the terminal time in the FM model, one due to the ODE formulation and the other due to the data geometry.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Singularity due to the ODE formulation. Recall that the vector field ue is given by
B1 3t
dg ag E x].
(e(x))
Since the denominator Br upproaches 0 as + the vector field ue faces an issue of division by zero when approaching the terminal time. Similarly, the vector field in formulation is given by - (m=(2) x), which also faces the same issue when 0. This singularity is intrinsic the flow matching ODE formulation. In the following proposition, when the data distribution p is not fully supported, the limit lim,-11 [x))) diverges to infinity for almost all ær outside the support of whereas it remains bounded when p is fully supported.
show that
Proposition C.4. Assume that a, 3+ [0, 1] + IR are smooth, and à 31 exist and are non zero. Let Q == supp(p) and let Eg denote its medial axis. Then, we have the following properties:
p is fully supported, i.e., 92 = R2, and has a Lipschitz density, thenfor any x IRd, the vector field uy(2) uniformly bounded for all E 1).
p is not fully supported, L.e., Q + TR,, then for any x & QUU En, lim [14(())] = 0c.
Proof of Proposition C.4. Let Ω == supp(p). Recall that
(x) (log 3,Yx Be (3) m((
3. x (x)
dx))(x).
3e
Then, we have that
[ - projn(=)]l z [|mo (z/as) (roin(z//a)))) + (|roii((//a)) proin(z)]l.
By Lemma B.,2, there exists a positive constant C2 such that [[rron(()) - (roje & C2 || - yll for any y close to æ. Therefore,
[proii(x//o)) - ( < C2
x
0(1 10 ot).
Now, when f2 = Rd, by Corollary D.6, we have that
[m (x/ar) - proja (roin(x/ar))) (==) O(8.)
In this case, (x) This implies that for t close to 1, have that
amma(I)
<0(1100 at
+ +011.
3e
Be
The right hand side is bounded since lim;-1 uniformly bounded for all / 1). 1).
l-a
lim
di
exists. Therefore, the vector field (z) is
Br
A.
If + Rd and & Nu then (r) 4 By Corollary D.3, we have that me(x) + projn(z) as t - 1. This implies that || (x)]l + || proja(z)) > 0. Then, as the denominator 3r + 0, we have that lim || (x) = co.
Another way to interpret the singularity in the ODE formulation is through the transformation of the ODE in terms of a in Equation (16), where the singularity arises due to the presence of the 1/a term.
seemingly straightforward approach to addressing this blowup is to reformulate the ODE terms of À as given in Equation (16):
dx
= mx(xx) IX- A (-00,00),
yA
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
with the terminal time being À co. In this formulation, there is denominator approaching zero, seemingly eliminating the singularity. However, for the ODE trajectory converge À oc, necessary condition is that lima- ( axl = O. This is precisely what we establish in proving our convergence result in Theorem 5.3.
Singularity due to the data geometry. When is not fully supported, the medial axis En of the data support plays crucial role the singularity of the denoiser mh which results discontinuity the limit limz-so my (2). In this case, when the ODE is transformed into the X, the vector field Vi does not have uniform Lipschitz bound for all À E (a, oo) and hence the typical ODE theory such as Lindelf theorem can not be directly applied to analyze the flow matching ODEs.
Plot of m,(x) for Various 0
1.00-
0202
04
0.75
0.
0.25
mx
0.00
-0.50
0.75
-1.00
-1.5
-1.0
-0.5
0.0 x
0.5
1.0
1.5
Figure 5. The denoiser my (x)) for the two point example with various a.
The discontinuity behavior can be illustrated by the following simple example of two-point data distribution which can be easily extended to higher dimensions.
Example C.5. Let _ + J01 be a probability measure on R.. Then, the support Ω == supp(p) [-1,1} is just two-point set. The medial axis is the singleton Σn = {0} whose distance to either point is 1. Now, we can explicitly write down the denoiser m, as follows:
(a+1)² 20
(x- 1)
exp
+ exp
20
ma
(17)
(x+1)3
(x-1)²
exp
+ exp
20
Notice that when a approaches 0,
x > 0
The denoiser is converging to the function f: + (,1, with ((x)
0
2 0
x < 0
A singularity of m, is emerging at En = {0}: the derivative
dm, (0) dx
is blowing up when o - 0.
A full characterization of the limit lim,-0 my (x) for discrete data distribution will be given in Appendix where the
discontinuity often arises at the medial axis of the data support. All these singularities pose challenges in theoretical analysis of the flow matching ODEs and particularly in the convergence of the ODE trajectory when approaching the terminal time. The data geometry singularity is more challenging to handle, especially the discontinuity behavior of the limit of the denoiser near the medial axis.
Our resolution of the convergence of FM ODE trajectory (Theorem 5.3) will be based on the attracting and absorbing property of the ODE dynamics so that the trajectory will avoid the singularities and converge to the data support.
C.3. Denoiser and ODE Dynamics: Attracting and Absorbing
The following result on the convergence of ODE under asymptotically vanishing perturbation will be used often in the proofs of this section.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Lemma C.6. Let (ye 2 oo)) be differentiable trajectory of non-negalive real numbers satisfying the following
differential inequality
y di
- ky: + (()),
where & > 0 and @: R. +R R. Then, we have:
1. For any l2 > la we have that
Vt. S k(tt -lae-tyyyy + fed o (t)dd.
2. If lim (tt) 0, then lim V/+ 0;
Proof of Lemma C.6. By multiplying the integrating factor ext,, we have that
de yt e d dt dt
+ keettu ye S o(
Then for all tz > ti, we have
e S elr + 1th (l(())
y1 Se -kt yes + frr &((t)-))
This proves Item 1.
For Item 2, we will first show that yt is bounded. As ot) decays to zero as goes to infinity, so will be (o(t)., and hence there exists some constant C such that [o(1) S C for all 2 4. Then, we have that
1"
tz
yt, S e k(tz-ti) sh +C
e -kttz- d),
S e 112 11) yc, +
(1 (1-e h((e-(((-(i)),,
C
Se Ub +
Hence, Ye is bounded for all 2 t,, and we denote by C, > any bound for yt.
Next, we show that z/2 converges zero as goes to infinity. For any >> there is sufficient large t, t1 such that
1. e Cy < e/2.
2. [d(1) c/2 for all l.
Then for all 12 > 21,, by integrating the inequality from I, to 12, we have that
I" e ()((2-1))(tt))
V12 Mits (1)) ye. +
5 Se ((te) h(()) Cy +e/2 1r)
p
e-h(to- dt,
< c/22+ + e/2(1 -h(2--6)))
t.
This implies that ys converges to zero as goes to infinity.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Attracting toward sets. Let Ω be closed set in Rd. We want examine the distance 12 along the ODE trajectory (1))) with 0 dz- Assume that the trajectory avoids the medial axis of f2 then the distance of & f2 will decrease as decreases if the trajectory direction (mo Io) forms an acute angle with the direction pointing toward ΩN that is
(#x) Ro projo( No) 0.
Notice that one has
(mo( 3o, proja (2x) To) (mo(x))) proja(Za), projo -o) [|projn(xx) Io 112
(18)
Hence, whenever Ha $ Ω2 (1proja(To) - vo > and the acute angle condition will be satisfied if the term (ma((g)) proja( ), ( zo) is not too negative. This intuition is formalized in the following theorem, which the formal version of Theorem 3.1.
Theorem C.7 (Attracting toward sets). Let (Ia (o be an ODE trajectory of Equation (11) starting from some Toy Assume that the trajectory avoids the medial axis of closed Q then have the following resulis.
1. If (ma((r) projg(rr) ( S Cl| - projo(to))))) ||? for some 0 S dr(z,) decreases along the trajectory with rate:
< along the trajectory, then
o²-S da((a)) S --<d(((o)))
01
In particular, 02 = 0, then da(x,) is guaranteed to converge to zero as o - 0.
2. of and (m proja( proin((e) _o) S Φ( for some function δ(o) along the trajectory with lim, Φ( = 0, then
lim dg(#a)= 0.
Remark C.8. In fact, when considering the parameter À = log(o) and the trajectory 2 == Fo(A)> we obtain the following convergence rate for Item 2 in the above theorem:
V/ 2e be
da(zx) S e- -A- daaa)))
Proof of Theorem C.7 and Remark C.8. We consider the change of variable À = = log(a). We let 2 (A) for all A E IA1 := X(o1), X. Then, (zx)xe(A,,Aa) satisfies the nice ODE as in Equation (16). By assumption we have that 22, is outside the convex set Ω. For any À E A21, by Corollary B.15 we have that
-=x== = 2(2 projo (>)), mxxx)))
dA
= -(2x - (roooo (2)), (ro) + (Zx proja((>), projn(z)) - mx(2))))
For the first item, we have that dc)((2)) -2d, Multiplying with the exponential integrator 20 ()) we have that
d 26 (22 $o.
d
Then for any À E [Ai, A2), we have that
-(1--() (11--)) df
Using the change of variable, we have that
J--s
da(2x) S I do(2)))
o
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
For the second item, have that
d -2d(((xx) 20((-~)) de (zx)
Then we can apply Item 2 of Lemma C.6 to obtain limx-100 dx((2)) = 0. This implies that ( = 0.
We then apply Item 1 of Lemma C.6 to obtain that for any X A1:
d2(2x) S ---((--A1))) 2(2-A1)) da (=x, (2x))) + L e 2-2(2-1) (E=
Using the fact that Va ta+bs Va + Vb for any a, b 2 0. we have that
dz(2)) S se d=( - V/. 2621
This proves Remark C.8
Absorbing by sets. Now, we prove the absorbing Theorem 3.2 below.
Proof of Theorem 3.2. We will utilize the parameter À for this proof with which we consider the ODE trajectory (#x := (E(A)))((AA,1)) of Equation (16) with x log(ø1) and X2 = =l2g(02))
For Item 1, we will show that the trajectory z must stay inside B, for all À E (A1,>)) whenever 2x, E (S)). Suppose
otherwise then there exists some first time X > A such that 2x, E (S2) By the assumption that B, n Eg = 0, we can apply Lemma B. 14 to obtain the derivative of the squared distance function along the trajectory for any À E [A1, Aol:
d dx d (o) = (=) - =A, projn(ex) - =x))
This implies that d de
d de dA (9) (Zx.) < O. By the continuity of the derivative above, we have that dA (o) (2x) < 0 for all [Ao - 6, Ao] for some e > O. Note that (=x.) = 22 and (=x.-)) < Then by the mean value theorem, there exists some E (A, e, Ao) such that 4 B. (=x,) > This leads to contradiction and hence the trajectory zy must stay inside (S2) for all X E A2), which concludes the proof.
The second part of the theorem follows straightforwardly from the fact that Q is a closed set. If trajectory starts from Q but leaves Ω at some point, it must also leave neighborhood of Ω which would contradict the given assumption. This completes the proof.
We now describe how we will use the above absorbing property for convex sets which will be used often in results in Sections 4 and 5.2, and how its generalization will be used to analyze the convergence of the flow matching ODEs in Section 5.
For convex K, its the medial axis Σx is empty (see e.g. Item of Proposition B.5) and if assume that the denoiser m, lies in K for any (K) then any neighborhood of K will be absorbing for the ODE trajectory. We also obtain an stronger result regarding when the set K itself is absorbing.
Proposition C.9 (Absorbing of convex sets). Let K be a closed convex set in Rdd. Let (Ix) (2o))E((2)))) be an ODE trajectory of Equation (11). Then, we have the following results.
1. For any > 0, if (1) E K for any (B)(K) and any (12, then B,. is absorbing for (Ix (N
2. If the interior K of K is not empty and m mz(x) (x) E K for any x E 0K and any a E (02, 01), then K is absorbing for (Fa )
26

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Proof of Proposition C.9. For the first item, for any x E (K), since (1) lies in the convex set K, we can apply Item 2 of Corollary B.9 to conclude
(m)(2) (ao(())-roix((ee projx (a) x) 2 0.
Then there is
(mo (z) - 2, pr)jx - -x) =- (m)(T) - projx(z), projæ (z) - x) + l|prooo (z) - -x112 2 n² > 0.
We then apply Theorem 3.2 to conclude that (,( is absorbing for (Io (x=)
For the second item, the distance function dx (z) does not distinguish between the interior and the boundary of K and we utilize the supporting hyperplane function instead. Assume the trajectory (Irg (E( leaves K at some first time 0.5 E (02,01) Then, in particular, 2a E K for all E [o,, 01] and Is, E ØK. In terms of the parameter À - log(o), log(o1), )), A,
log(o.)) and EA, OKK Since K is closed convex set, there exist a supporting hyperplane H at 2A, such that H (zx.) 0 and (y) 2 0 for all y E K (see Proposition B.4). In particular, we can write (yy (y 2x., n),, where is the unit normal vector of the hyperplane. Since =x E K for all À E [N,, A1] and S Therefore, we have that
we have yλ E [A
H (zx.) S H (2x) for all X
we must have that (3M(2))
A=
(2x) dA
dzy dA
A=
1= n (2xo) 2), n2) S O.
Since my, (2x,) lies in the interior of K, we must have that (220) 2h, n2) > 0. This leads to a contradiction and hence the trajectory 2 must stay inside K for all E (21,
Remark C.10. When 12 is not convex, a typical way to show the acute angle condition is by requiring (mm(()) - -proja (x)] to be small enough on (S2) for all a E (02, 01). This can be seen by the following computation:
(ma) To, (rojj - ta) = (mo (xx) (roiaat))) x²) + ||projn - x0
(19)
2 - 111ma (xo) proja(Ia) (mpin(==>)lllpool [llpprojn(<=) - zall - (proojs(zo) x x=112
(20)
Furthermore, once the absorbing property is established, it guarantees that d1( = (Iproja(=o) - Io || to be bounded for a trajectory in consideration. In this case, the condition in Item 2 of the attracting theorem Theorem C.7 can be derived from the decay of (x=) - projo( as
(mo (xx) - projo (ro)a( - x.)) S [1 (ro) (=o))(==>llllpp [[llprojo x
These type of arguments will be utilize in Section 4.3 and Section 5.2 when discussing the ODE dynamics of the flow matching ODEs.
When Ω is unbounded, e.g the support of a general distribution, it would require much more assumptions for us to control the term [[ - (x.)) uniformly on the boundary. As one way to circumvent this issue, we consider the intersection of (22) with bounded set and establish the absorbing property of the bounded subset. This is formalized in the following result.
Theorem C.11 (Absorbing of data support). Fix any small O < 5 < Ta/4 4 and any < Ç < 1. Assume that there exists a constant σg > 0 such that for any > T and for any = BR(0) n Bro/2((1), one has that
mm (z) projn(2)]l S Cs,r.R of, for all 0 < o og.
where Cs,,.n is constant depending only Ç and T and R.
Then, there exists dā I dg dependent on 5, and Cg,T.R satisfying the following property for any R > 28: The trajectory (Ir )o(0) starting at any initial point Fas E (2) n B2) of the ODE in Equation (11) will be absorbed in slightly larger space: for any a S ag. Fo E (2) n (22).
Note that this absorbing result is slightly different from Theorem 3.2, where the neighborhood the data support Q must be enlarged from to 28 (and R to 2R) to guarantee the absorbing property. This subtle difference arises from the additional treatment in the proof account for the bounded ball (0) in the above theorem.
27

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Proof of Theorem C.II. Similarly as in the proofs of other theorems in this section, consider the change of variable log(ar) and 2 (A).
We let C = C,,,2 and define (one can see how the definition is motivated from the following proof)
5c
5 (2 5)
VV C
-
8(2 6(2-5) () )
A max { min
log
2 log
.log
2C
4C
Then, 0g = AA,
By continuity of the ODE path, there exists maximal interval [As, As] so that for any À 22 E (2R (2s( where the overline indicates the closure of the underlying set. Now, show that Ag must be infinity by showing that otherwise 2A, lies in the interior, i.c. 2A, B2R (0) n (2s and hence the trajectory will be able to extend within (2) n (9)) to Aλ + for some small 0. In this way, we can extend the interval 7 to (As. Ag c) which contradicts the maximality of I.
Now, assume that As < co. Then, by Corollary B.15, we have that for any À E I.
(xx)
-2(22 (=))n(=x),= 2A mx(z)))
dA
= (22x - projo (x), 21 (ro) + (2x proojp((2) - m2(2x)))
< (2x)((2) (d](<))] (2))l(2x)1--
By our assumption on 2x, we have that d=(zx) < 25 and for any A, ||mx(ex) - pprojn(2)) 3 e Remark C.8, have that
c-e CA, Then, by
J. 4CDe2te-S1dd
da(zx) S -A- - do-s)))((
48C
I e (-11-s))d((aa) +
(e-CA el2-c)As-22) < 25,
2-
BC 1
where the inequality under the brace bracket follows from the definition of As. Hence, 24a E B2s(
Now we examine ||=X|| along the integral path. We have that
/-- [mxx())) 2xIld
||2g zAall $
l [|mx(ex)= - --roin [projo(>))= - 2x|ld>
As
[ Ce + del(2)dd
As
Now, for the integral of do(2x), we have that
{ do(2))dd < S (11A-20) +
48C
(e-ca- c(2-c)As -22) dd
V 2-5
As
1 c (12-20)
48C
$
CAd
2
Aa
48C 2 ( (ch//2 - c012))
= 5(1 c +
2 ( {
28

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Therefore,
C 112. 2A S (-e-St ( +5(1 - - +
2 2 Ç (0-5h012---507))
48C 2
-
20
(21)
(08g6/2
3-6 20-5///258/2
where we used the definition of again to control all the exponential terms. This implies that || |l S R 26 2R for all
E [As, As] (recall that assumed that R > 26) and hence 2A. E (2n(O). This concludes the proof.
C.3.1. ABSORBING AND ATTRACTING TO THE CONVEX HULL OF THE DATA SUPPORT
We prove Proposition 3.4 below.
Proof of Proposition 3.4. The key observation comes from that, the posterior distribution p(-IX, =) is also supported on supp(p) and hence its expectation, the denoiser m. must lie in the convex hull conv(supp(p)).
For the first part, the first Item in Proposition C.9 applies to conv(supp(p)) and hence any neighborhood of conv(supp(p)) absorbing (E o (o We then obtain the desired result by Item Theorem 3.2.
For the second part, we use Item 2 of Corollary B.9 to obtain
ma (x) - (rojeon((suppoi) (x), pro)eon((npppó)) (x) x S
for all a E Rd and all a. Then we can apply Item 1 of the meta attracting result Theorem C.7 with Ç = 0 to conclude the proof.
The above propositions requires the data distribution to have a bounded support. We now extend the above results to more general setting where the data distribution is p pa (O, 12,,, i.e.. the convolution a bounded support distribution ps and Gaussian distribution. This done by noting that the ODE trajectory with data distribution p can be derived from the ODE trajectory with data distribution ps by early stopping the trajectory. We formalize this in the following lemma.
Lemma C.12. Let pa be distribution with bounded support and let p == ps (0, any 2 Let (Jo))a,e(0,s be an ODE trajectory of Equation (7) with data distribution pb- We define (I. VoL=vo (e Then, (xo) is ODE trajectory of Equation (7) with data distribution
Proof of Lemma C.12. Let qo, Ph (0, I) satisfies
I) be the probability path with data distribution ps. Then, we have that yo.
you
ob V log Yoy (yor);
da:
With the change of variable σ v0² + 82, have that
yos
d
ob V log you yo
(Yrr):
da
do
o% V log go, (Yos),
Tb
V log qo, (yos),
One has that
|lue, - x/|
Jos (yo.) =
(2To
exp
203
po(dx)
1
1100 z012
(2 + +2))
exp
2(82 + 02)
po (dx)
qo (Eo),
29

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
where 4o (a) denotes the density of 9o = == o2 I²)). Therefore, the trajectory di == y satisfies
d'a = -oV go (x
do
Corollary C.13. Given the data distribution p defined as in Lemma C.I2, for any d1 let To be ODE trajectory of
Equation (11) from to = 0. Then, we have that
1. vf, conv(supp(pa)), then To E conv(supp(ps)) for any E (0,
2. We & conv(supp(p))) then to moves toward conv(supp(pa)) with the following decay guarantee:
V0² 172+6 8²
(F))
d (euppppp((ss) (For)
Vo { + 82
for any a E (0,o))-
Proof of Corollary C.13. This a direct consequence of Proposition 3.4 and Lemma C.12
D. Denoiser Analysis: Concentration and Convergence of the Posterior Distribution
In this section, analyze the denoiser through the concentration and convergence of the posterior distribution, i.e., Theorem D. and its variants under different assumptions. Our proof strategy is similar to that used in Stanczuk et al. (2024, Theorem 4. 1), where the integral under consideration is split into two parts to analyze the concentration. However, our results are more general as we do not require the data distribution to lie on a manifold or have points be sufficiently near the data support. We additionally provide rate control of the convergence result which is crucial for the terminal behavior analysis in Section 5.1.
Theorem D.I. Let Ω = supp(p). Assume that p has finite 2-moment. For all x E Bd Es,, we have that
lim dw,2 (p(-) X, = xx), (proig((()) O. o
We on = proja(r). According to Proof of Theorem D.I. We let t2 == dr (2)/2 > and A₂ = de(x)) 2
Lemma B. for any if define the radius
A
Ta. 1
A3 +
1 A + $
then have the following inclusion
B (x) nu (x) C B.(aa) nΩ.
For the other direction, we have that
By (1n) Br,, (x) no
where
1 = Tx. A. =
(Φ, Az) (Ta. + Aw)
Here rh. satisfies that as da(x) - r - c. So Brg (rn) can be thought of an approximation of Be
30

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Then, estimate the two terms below separately:
|| 110=zal" 2all"
exp
20
dw.2(p(-|Xa= = )xg))
1200
xall
p(dro)
53
S1 exp
1 xgll"
p(dx'))
20
1|²
exp
11:00 xz2ll
p(drro)
Sa
exp
Lir-afl 2
[x
2
p(dx)))
11
11a- (2)
||x-
cxp
20²
1100 xnl
p(dxo)
Briza (x)
(x)=())n
So exp
x xx-2r/ 12 20²²
p(dz/))
12
For I1, we have the following estimate:
||=~201
11 S J B.(ea)n
1000 - zall?
exp
20 1/.1 p(dt()
p(dzo) Se
Sn exp
||
|
For the second term 12, we have the following estimate:
||2
zull?
exp
I2 S
22| + 2 2|120011 33)
p(de()) p(dxo)
BB,,, (v)na
Sn exp
||2=z/01
2n
= -Sme Bra (u)ns) Sa exp
2||x0| 1| + 2 2m 12
p(dzo)
11-w0l 2n
|| x 11"
p(dxo))
2n
2 |00|| 2 + 2 2||201l 12
pldxoo)
Bc ())
SP.
(2)nn) exp
|
2n
|1=-z/11
20
p(dx')
2 ar
2/|22/7/1² 2
p(dxo)
BC
SP.
(B...//r(()) (v)na
exp
462
c²
3,t0,
Dx
p(dz'))
2
Ar
2 2| 1 + 2M2(p)
= exp
1
-
4
A
$
Br Brineiz (v) 712)
P
?
Ar
2|| 1l? + 2M²(p)
2
S exp
1
402
A
+
B 2.c/vz (m
Since Ω is the support of p. have that p B, 2 (in))
0. Hence, for any 6 0, we have
1 + 1+ + 2112011 2 + 2M2(p)
(1
(-40²
exp
p Br 2 (mn)
402
A 42
By letting σ 0, we have that 4 + 12 S e. Therefore, we have that
lim dw dw,2(p(: X,, x)), dzg) 0. a-
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
As direct consequence of the convergence of the posterior distribution, we have the convergence of the denoiser stated in Theorem 5.1. Here provide simple proof.
Proof of Theorem 5.1. This is direct consequence of the following property of the Wasserstein distance where the case = 1 was proved in Rubner et al. (1998) and the other cases follow from the fact that dw,s is increasing with respect to (Villani, 2009, Remark 6.6).
Lemma D.2. For any probability measures V1, on I and any 1 $ 8 S oc, we have that
dw.)(v,,12) [mcan(v)) - mcan(v))ll-
When we turn back to the parameter 1, we have the following corollary. The proof turns out to be rather technical instead of being a direct consequence of Theorem 5.1. The main difficulty lies the scaling at within the exponential term. This another example that the parameter more convenient for theoretical analysis.
Corollary D.3. Let Q == supp(p). Assume that p has finite 2-moment. For all x E let un projo Then, we have that
lim mz(x)= = (r)) H-1
Proof of Corollary D.3. In the proof of Theorem D.I, we end up with
(-
?
(1
A
2 2| 2 + 2M2(p)
dw.2(p(-IXa = )ma) < x² + +exp
(-
102
A
B,- B (Ia)
p
for any arbitrarily chosen small c.
By Lemma B.12, there exists a small > 0 and a constant Ce > O such that for any ||= - æ/| < E. one has || =nol < Cş|l= z||. As both A₂ and T are continuous in a local neighborhood of x, there exists a small ε > 0 such that for any with || - z|| Ei, one has
22 3rae///2
A
(1
A )
|| < 2 2
Now, take E = min {s, Ei, (aa"".//3)) Then, it is easy to check that for any such that ||> x:|| E, one has
B f " (In)
B, Brierv (20).
This implies that for any such that -|2- x/| < E, one has
(2) 20112 s dw.2(p(-IX. = 2), den)) ) s 22 + +exp (-
²
(
A.
1| 12 2M2(p)
1-
(8802
807
A
B "////z (ma)
Hence, when a is small enough (dependent on e), we have that []) - zgll S 2e for any z such that 1==x[] x/1 < E.
Now, we have that
[xx(x) - soll || < [l (I/Oz) - projn( + [Iproj((T/ar) - (x)jn
As ag - 1, there exists to such that when t > for one has that 2 = x/a, E Bz(x) and [Iprjj((z/aa) - projn(z)] S c. By the analysis above, we can enlarge so that o, is small enough so that [|m,, (2) æall < 2e. Therefore, for any > to. we have that []))(()= sall I 3c. Since is arbitrary, we have that lim m.(x) ag.
32

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
D.I. Convergence Rate
We establish the following convergence rate for the posterior distribution when more assumptions are made on p. Theorem D.4. Assume that the reach T TQ 0 is positive. Consider any Rd. We assume that dn(x) < 2
and that there exists 2 0 such that there exist constants C,C > O so that for any small radius O < r < one has pl (x()) 2 Cr Then, for any 0 < < 1 we have the following convergence rate for any 0 < a < c215;
d (p(.)X, =)), S y02 (04(2M)()) + ||aal??) mak okc)
exp ( (-1o2k--1) S Cs,
Ca2kc
where Cgr is a constant depending only on Ç and ro-
Proof of Theorem D.4. Recall from the proof of Theorem D.I that for any > 0, we have that
1 + 1 S 2
2| 2 + 2M2(p) exp (-40²
e²
1
P Br 2.//22 (Kn)
102
A2
+ 2x
Then, we have that 20, 2 mQ- - Ar. So
2x
(22)
( Ar)
(Tz,, + Az)
TQ A
?
(23)
2T
Dz + / A +
e
2
min
(24)
4
(V2 + 1) v2+1
min
(25)
10
!Q
On the other hand, we have that
The <
(4g + Dg)e
4,t0,
4 S..
=6 =EV A, 1
Therefore, if one let of, when c we have that re, < and then
p ( 2 Cn-
min
c
10
T32
Chok 10tt
min
Notice that 1
A 4,
2 2. have that eventually
h + T2 S 25 + 2M²(p) + 2||
Cnoks 10x
min
ahc +
exp
-02
= 25 + (2M2(() + 2||200| (2) max { T ok
(-802(5-))
1
2no
exp
The rightmost inequality in the theorem follows from the fact that the right most summand the above equation has an exponential decay which is way faster than any polynomial decay.
33

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Now, we improve the convergence rate when p either supported submanifold discrete set.
Convergence rates for the manifold case. We first consider the case when is supported on submanifold M R. Note that this does not exclude the case when M = Rd. Under some mild conditions, we have the following convergence rate for the posterior distribution.
Theorem D.5. Let M C. R be a m dimensional closed submanifold (without self selffintersection) with positive reach TM- Assume that p(dx)= = p(x)voll (dx) has a smooth non-vanishing density M - R. For any E IR and a E (0, oo), we let IM = projm (2). If x E Id satisfies that das (x) < TMs then we have that
dw 2 (1(2p((()))))))) 22) /mo +(02).
The proof is very lengthy and we postpone it to the end of this section.
Note that the leading-order term the convergence rate, Vmia, depends solely on the intrinsic dimension m of the submanifold M. The higher-order term O depends on finer geometrie properties of M, such as curvature and reach. detailed characterization of these higher-order contributions would be interesting direction for future work.
As direct consequence of the convergence of the posterior distribution, we have the following convergence of the denoiser. Corollary D.6. Under the same assumptions as in Theorem for any x Rd sattisfyng (x) 2 TM, have that
Im (x) xMIl O
Convergence rates for the discrete case. Let the data distribution p Σ a, dz, be general discrete distribution with , NN Rd and d1...., AN O. We f {x1,--. ................................ EN to denote the support of p. We study the concentration and convergence of the posterior measure p(- x) for each Rd including those æ Ln. the medial axis of S2. To this end, we introduce the following notations.
For each point x E Rdd we denote the set distance values from x to each point in as follows:
DVa(z) = {|| xall x E 2}.
(26)
We use (2; i) to denote the i-th smallest distance value in DVn(z). A useful geometric notion will be the gap between the squares of the two smallest distances which we denote by
An(x) =- -d (x; dx; (x; 1).
(27)
We further let
NNa 3q E f: | r; do(x;;
|| xz æyE
min
We use the notation (NN(;) to denote the normalized measure restricted to the points in that are closest to 2:
PNN(2) = EmENNA((Y) ai
a pn.-
(28)
2,ENNn(x)
Whenever z is not Dn. we have (Na (x)) and (NN(K) (x) With the above notation, we have the following convergence result for the posterior measure p(- X( = x) + 0.
Theorem D.7. Let Σi= a; dn, be a discrete distribution. For any x E RRd, we have the following convergence of the posterior measure p(- Xx z) toward PINN())
p(NNa(z)) p(NNa(z))
exp (- [_)((z)
dw (p(-1X, = 2))PNN(()) & diam(
402
Proof of Theorem D.7. When An(x) = O, (N) = and p(-|X, = x) = and then the Wasserstein distance becomes which proves the statement.
34

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Now, we will focus the case when Ap(x) The posterior measure p(- X = x)) is given by
0g exp
(- 202 [4 x²122)
pl11X, x)
6z,-
1 2 |25 - x113)
2j=1 j=1 Uj exp
For the ease of notation, we use B = Ej= Uj exp (- 2σ ||a x1 to denote the normalization constant p(-1X,
x), Bo,z, == a, exp (-202 12 - 2² to denote the i-th term in B,, We use A = p(NNa(x)) to denote the normalization
constant in PNN(2)-
Observe that for any Zi, By E NNn(z), one has
p X x)/p(r,|X, =x) =)= (2A)
This motivates the following construction of a coupling between pl X = x) and PNN(u)-
1
Ba
B₁
Σ
ag
(x1,22) +
Σ
Brar 0i b(2)z))
Ba A
(2ENN((2)
=(ENNN(2)
Then, we bound the Wasserstein distance between p(-1Xo = x) and PNN(E) as follows:
dw.2(p(-XX, = =)).PNN())) 11op- ul dy).
Bx |[; zz 23
Σ
B A
Bxa,, ai
Σ
diam
Bo A
aj exp (- 2n ||= - xx1²)
di)
Σ
21=11 aj exp (- 2 || xx1²)
N
3, EE
exexp 2e22 (x: a;
x 22)
Σ
Aexp (- 22 1)) A
- A exp
2e2ddh (I,
2
A cxp
1 22 (x, 1))
2
d d A
1 2) ( (x, 2) - dz 1))
(d
exp
S di)m
A
- 22))(()))
1
exp
2
By taking the square roots on both sides, we conclude the proof.
As corollary of Theorem D.7, we have the following convergence of the denoiser.
Corollary D.8. Let = En ai 6a, be discrete distribution. For any x E Rdd, we have the following convergence of the denoiser ma (x) toward the mean of PNN(a):
p(NNa(x))
An(x))
|| (x) - mean(fNN()))l S (1)
P(NNa(z))
exp
4
In particular, when $ Eg, assume that x, proin( we have that
/1 exp
1 1-a
An(x)
[m) xzll < diam(())
a;
4
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Proof of Corollary D.8. The result follows from Theorem D.7 and the stability of the mean operation under the Wasserstein distance Lemma D.2.
Later in Section 5.2, will utilize the above results to analyze memorization behavior.
Proof of Theorem D.5
We still let A. :- -dr(()) < STM. Then, we let oe p(-1Xo x:) and hence,
(- |10-011 (-1
exp
p(an (oci)ollar (dxi)
20rd
for M. p ((zl)volardzz))
Sar exp
|| 11?
20
As + O, o. is concentrated around ZM = projur (r)) For any > O, we have that
dw dw.2(g", 82as)2 / |21 - -MM 1
|1 x /
| - (0
(F3)
My Brg (ams)
We first consider the term 12.
J
12 S
(21|=1112 + 21|x 1| (dxx))
MMB (are))
As we have shown already in the proof of Theorem D.I, we have that Br (x) NM c (xM) O n.. So, we have that
/
I <
(21 + 2| (
(29)
M MrB.rr (x)
rn
2M2(p) + 22120111 = =0 00(() exp
r So
< exp
1
(30)
4
A
2 p B,- 2.0o/ (KM)
Now, we consider the term Ij. Since M is submanifold of Rd, its tangent space Tzx M can be identified as a subspace of Rd We use v. Tax M + Rdd to denote the inclusion map. In particular, for any E Tax M, we have that (a IM (())) = 0. Let Inj(M) denote the injectivity radius of M. Note that, by Alexander & Bishop (2006, Corollary 1.4), Inj(M) 2 7M/4. So, the exponential map exP*,, Tex M .M will be an diffcomorphism in small ball D=EM TM (0) C Tex M. Furthermore. we have the following inclusion relation between geodesic balls and Euclidean balls.
Lemma D.9. For any 3rMz we have that
Mn (x( C eexxxx
(Being (0)) C CM B5Mss(EM)-
Proof of Lemma D.9. The proof of the left hand side follows from Lemma A. and Lemma A.2 (ii) of Aamari & Levrard (2019). Although Lemma A.2 (ii) stated compactness for the manifold M, this assumption is not needed in the proof. The right hand side follows from the fact that || yll < dmr v) for any x, y E M, where das denotes the geodesic distance.
Tam (0) around 0 Tex M so that Now, we fix some To < 3TM/25.. Then, there exists open neighborhood Ur, 3ro/3 have the following diffeomorphism (which gives rise to the normal coordinates):
expx Uro 5 Taxs M - Bro (xM) M.
36

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
In particular, TM exP=M (0) and each x1 (xsM) n M be written as #1 exPxxx (u) for some U,.
Let II Tax M T2y, M (T M) denote the second fundamental form of M xMM Then, the exponential map exPay in Uro has the following Taylor expansion (Monera et al., 2014):
1
exPxM (u) IM + (u)) + II [u, u) + O(Ilu])),
(31)
2
where O(1lu|1)) & C(II, VIT) for some constant CIII, VII O only dependent on II and its derivatives VII.
For any E Ur. we let g(u) denote the metric tensor, then g(0) is the identity matrix. The volume form is given by /det(g(u))du> A... A du and it admits the following Taylor expansion around E Tax M:
ddet(g(u))
Raju'ul + O(Ilu])),
where Raj is the Ricci curvature tensor of M at MM see e.g. Chow et al. (2023, Exercise 1.83).
We write
p (et(( (()) (u), [R) & Gillul,, where 1 depends on Ricci curvature tensor Ray and the gradient Vp(rM).
(32)
Now, we define
fa(u) == exp (- ||
(-llu-expean
(-1ls- exp2N (u)l?) (w))²)
/ot(((o(())))
20²
Let MB,,- == (ro( n M. Then, we have that
La (u()du
(du)
Su., So( SANMB, exp
((a))vol, (da²)
2a
Using the same argument as the one used for controlling I2 in the proof of Theorem D.1, we have
(- |w 11a)
SMMB,,, exp
20
r pla' yol (dx1) O exp
))
(33)
SMBB exp
|s -
p(x1) vol (da1)
So
2
So.
J
Se
|| eexPMm (w) - xasll 3 (o Su., (u())du
(ospp
(-
r
|0] - IM (dx1)
1 (1+0
exp
(34)
MB..
80²
Next, we derive Taylor expansion of the squared distance from x to expzar (u) for around 0: Lemma D.10. Let v == x TM- Then, we have that
| exPars u112 = ||a||2 + ||u|| (v. (I, u)) + R2(u) = ||0|| + oF u
Hy.) )u + R2(u)
where H, == (1, identity matrix.
and [R) S C2llul|3 3 for some positive constant C2 C2 II, VIT) > 0, and I is the
Proof of Lemma D.10. We first note that
| ex u/12 = || xM 2 + || - expxx (u)l + 2(x IM. - (u)).
By equation (31), we have that
| expaar = ||0||2 + 1| (u, (1, u)) 2(v, ((u)) - (v. [T, + (11uu]))
|||| + || (v. II (u, u)) + O(1lu|]))
where we used the fact that v belongs the normal space (Tey M) of M at æs so that (v, (v.)(u) = 0 and (u, II, u))
37

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
By Lemma D.10 and Equation (34), have that
1 |xexpxr (u) - JM 11 (u)duu
[_1ee112) /
uI 11, R2(u) (o(0) + Ri du)))du..
(u 1oJu + Rz(u)) exp
= exp
202
Juro
20²
and
[ Uvg
||0|| (-1ee1²) 2x
u (1- 1-.1, + R2(u)
fa(uddu exp
exp
(o(0) + R (u))du.
2
Uvo
We will establish the following claims: Claim
10 (I - II )u + R2(u)) exp
uT H, + Re(u))
(p(0) + Ri(u))du
20²
(35)
m+2 pm0) / 2T(I - -x,))exp
(I 11,)
+ 0(
2
Claim 2.
TT HT)) )u + R2(w)
exp
(p(0) + Rg (u))du
Uro
20²
(36)
2T(I 11,)z
Th p(0)
dz 0(a
exp
Rm
With the above two claims, there is
Jun || Texp xM (u) IMll (u)ddu
(-ri
r
|21 zM 92
1+0 exp
- 807
Su for fo(w')du
M Bro
0² Jmm 2² exx (- (I-
de + 0
2
1+0
r
))
cxp
Jgm exp
= 11,)²
de + 0())
80²
Sam x (I 11, exx)
x [1-11.11
dz
(1+0)
(- r
0²
+0(0)
exp
fRm exp
2T(I- 1T.)
dz
80²
mo² + +((02)
where in the last equality used the fact that E[X AX tr(AE) for Gaussian random variable X ~ ~N( Therefore, we have that
dw.2 SmM)) / O
and concludes the proof.
Now we prove the two claims. For the first claim, note that by ||0|| (z)) TMs/2, we can use Berenfeld al. (2022,
Theorem 2.1) to conclude
1
II ||a||| max iJ
M1 < < TM 1/TMS
So the matrix I - II, is positive definite. We let Amin > 0 be the smallest eigenvalues of I - II,. Note that Amin > 1 ||I|I||| 2 2- So, we can choose ro small enough at the beginning so that there exists some constant C3 > 0 for all 2 E Uro, we have that
TT TT(I II, + Rz(u)= 2 Amin ||] - C2 |u|| C3l
38

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Consider the transformation oz. Then, l't 2roo(22o) which goes to 0 as a 0 and for sufficiently small have that o' > hence there is
1 / Uno C Um.
Then, we have that
/
uI exx
(I - J + Ra(u) (p(0) + Ri(u))du
20²
{
2T (I II
R2(02)
o
z 2=(I--a -11,)=exp
(o(0) + Rí (oz)))d
UUr
2x
2T 11,0²
R2(oz)
n+2
z² (I- 1I 11x
(p(0) + (I(o)))
Uro
2x
x x1 II, )
R2(oz)
+ to m+2
T - - 11,)
(p(0) + Ri (oz))dz..
4 Un0
2
20
J2
Now, for J1, recall that Un, c Buyy Ms (0). Hence, for any 2 + Urg we have that || S 5ro/3 < 2ro and hence
R2(az)]
C2(ar1=11)
(2ro) 1
20²
20
20 (2ro) 2
This implies that there is expansion exp
(- 2=20
Rg(az) 20²
= 0 ([0121)) Therefore, we have that
2T(I I1,,
J1
2T 11,)zexp
(1 + +1+00
Ura
2
2T 11,)z
p(0)
2T(I II,,) =xx)=exp
dz O(()
U
p(0) S II,=xe))
2T
II,)²
dz O
2
where in the last part we used the fact that the the decay rate of such an integral as a goes is exponential.
For Jz, similarly have that
1J2 S /
-T( II,) exp (-Ca11=13) (o(0) + O(I
O()).
Therefore,
uT(I-- - ([-III.)u + R2(u)
u JT(II II mupppppp
(p(0) + Ri (u))
Uro
202
(37)
/
[T -1,))
_ m+2 p(0)
2 - Hx))zepp
dz + 0(0
2
pm
Similarly, we have that
39

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
J exp p(0)
1F'(1 1 ++ + R2(w)
(p(0) + Rí (u))du
20²
(38)
2" 11,)
dz -((0m+))))
exp
Rm
Additionally, by [2(x))] S C2l C2/|u//3, we have that
[
exp 2² -1-11111 1.) + Ro(w)
(2(u)
(p(0) + Ri (u))ddu O(a
(39)
2x
Then Claim follows from Equation (37) and Equation (39) and Claim 2 follows from Equation (38).
E. Explicit Eamples of FM ODEs
The following lemma specifies the FM ODE solution when the data distribution is the standard Gaussian itself. Lemma E.1. Let the data distribution be a standard Gaussian distribution N I) in R4. Then the FM ODE can be computed explicitly as
dx: d400 + 9,33 dt at 33
This implies that the following explicit formula for the ODE mrajectory
V a B3 m
2t
for any initial point Ze Rd. particular, the flow map V, is scaling and v is the identity map.
Proof of Lemma E.I. The FM ODE given by
||
dx: 3r xt 3r
a.
exp
2.97
Ø1
p(dr1).
dt
3
exp
1
p(dx1)
2
When plugging in the standard Gaussian density for p. we have the following observation: [2' aq21 112
| 11
||
a
11?
112
exp
exp
exp
2,59
2
233
exp
1
at
Tt
V
Vož +
Ba
exp
2( + B8)) |122.12
2,32
0
+
88
For brevity, we denote B, = vaz
By and
C=
Then the ODE be simplified
Va?+88
dx.
3r
an
exp
|Cr,
2 28
D 21d01
+
dt
3
Br
[ exp
CCary B, ||
dxy
2
B
exp
|Ca/Bxze 10 237 B
3 day
+
3r
B1
eep
|Cab/ Beze x 11
dz
/
3,
an
+
Ctl
Bex
3.
B1
3r
arBt 3r
+
of +3² Zt
3.
Br
diya + 3,3,
It.
a? + 82
40

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
The of the proof follows from directly verifying that 2y Voi + Bf satisfies the ODE,
The following proposition shows that for distribution lies linear subspace R C R2, the FM ODE can be decomposed. Proposition E.2. For any < m I d, consider the subspace Rm IR.. Let p be a probability measure on R supported on TR." and assume the FM ODE is well defined. We express any point x E TR as (x, y), where Then for any FM scheduling functions aee St, have for any initial point zo (ro.Yo) E TR,, the FM ODE trajectory is given by (æt (I1, ) where (11) is the trajectory of the FM ODE on Rd with initial point 2o, data distribution p regarded as probability measure on R, and with scheduling functions aq St.
R and y E Rd-m.
Proof of Proposition E.2. The FM ODE is given by
||
dx dl 2t
B
3r an
exp
28f
x1
p(dx)).
3r
f exp
||
11?
p(dx))
207
We have the following two observations:
1. For each æ1 I there is ||ac; crer 112
| 04 112 +
||ge| 2|
where æt (aie, M/) and as (21, 0).
2. The denoiser m, always lies in the subspace IR since the data distribution p is supported on R".
These two observations imply that the FM ODE can be decoupled into two ODEs:
|| 1
dæ;
3
dr
and
ʃ
exp
207
It
+
p(dr1),
dt
B.
B.
f exp
||ze - 112
p(dx))
293
dy: B.
yc-
dt
Be
The first equation is the FM ODE on T with initial point and data distribution in R", The second equation is the ODE for which is linear ODE with solution = B.yo-
F. Proofs in Section 2
Proof of Proposition 2.1. For any x Rd and t (0, 1]. since ot > 0, we have that the map A, well-defined. Furthermore, we have that
(At) = (At)# ((Nttaarto..) 37 P Dr(dro)
= f N(-uuo (y((aoo))
/ N(-1zz, p(())( = goe
Proof of Proposition 2.2. We first write down the density of go explicitly follows:
(- |
113
go (x)
(2T02)d12
exp
202
p(dz).
Then, we have that
yyep
||
p(dy)
20 |=-v/|1
log go (z) = o²
x
f Texpp
p(dy')
20

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Here, the existence of the gradient easily follows from the dominated convergence theorem.
Note that dt do
of B, S,a.
Therefore with the reparametrization To := 2/e/ at, we have that
dra do
aa dt dt
do
dra
dt
at
dz
af
dt
do
(- || [1:ll""""l"
at
Zt
exp
2
y
pldy)
3.
0tt
j exp
[0 10--1/1113
p(dy')
20g
V log to (Fa).
G. Proofs in Section 3
We provide in this section a convenient pointer to the proofs in Section 3. The formal version of Theorem 3.1 is Theorem C.7 in Appendix C.3. Similarly, the proofs of Theorem 3.2 and Proposition 3.4 are also provided in Appendix C.3.
H. Proofs in Section 4
In this section, provide the deferred proof' in Section 4. Note that the proofs of Theorem 4.1 and Proposition 4.2 require more preparation and are organized in a subsection at the end of this section; see Appendix H.1 and Appendix H.2.
Proof of Proposition 4.3. For simplicity, we use no == p(- xx, = x)) to denote the posterior measure at æ. We restrict 7/ on to obtain the following probability measure
T) (S) Tl s-
easy to see that B(ng) lies in the convex hull S. We can then bound the distance from the denoiser m. (x) to the convex hull of S by the Wasserstein distance between 1)r and ni
Similarly to the proof of Theorem D.7, we construct the following coupling between no and n
J = A (nols) is) n (no ((2),))),
where A : Rd + Rdd x R is the diagonal map sending æ to (x.
We then have the following estimate:
d2w (1o, vog) S // l1n - y2l12 nols (dyn Mo (dyz)) (S)
no (()) (d)a 2
Soyrs) exp (- 2 | x p)) p(dr
(1))
(su(ers) exp
| - x p((r))((d))))))))))))))))))))p
20
S1ass) exp
20 ||
xz pp)))
S (1))
Is exp 2
||x x 11 p((12)((dz)) p(dx,)
By the assumption that (s) (x) < D/2 e, we have that for all 21 S
|| m < D/2 - c D 3D/2 6,
42

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
and for all 0i E 5
| æı 2 2D
()
We then have that
((D exp
(3D/2+
dw (1o, (To. mz)² S (1))
20
Jg exp
2
(1))
ag exp
(3D1/2+ ((3D/2-)))
Ug
2
- ag
3Dc
(di)
exp
ag
o²
Then by applying Lemma D.2 and the fact that E(n.) E conv(S), we have that
(m < (1)
as
exp
3De 2n
05
Proof of Proposition 4.4. Note that, the set (co2-((conv(S)) is convex itself. For the first part, according to Item 2 of Proposition C.9, it suffices show that for all ( the denoiser mo lies in (p/o-c(convvS)) for all (S)
By Proposition 4.3, we have that
deanv (m) I diam(supp(p))
as
3Dc
exp
0.
20
Then, it suffices to show that
1 1- as
3De
diam(supp(p))
exp
< D/22 -c
20²
This is equivalent to
3De
CS
exp
2a²
where C
D/2-6
Then, when C 2 1, the above incquality holds for all a < oo and when C < 1, is
dism(supp(p)) T-aas"
straightforward to verify the above inequality holds for all 0 < a < co(S, e) = ( 3De } This concludes the proof of the first part.
Cog
For the second part, we use À = log a as the reparametrization and the corresponding trajectory ΣΛ := To())- We want to show that the distance (z) goes to zero as goes to infinity. By taking the derivative of d² (o) (zx), and use the notation
E X
Z
2x and X S] where ZA (o(A)>
we have
d de
(sn)(s)(22)
-((2A proj (2x); 22 mx(zx))
dA
-22A2 projconv 8) (2x); 2\ projeonr (s))
-((Ax proj 89) (#x), (r - uv)
2(Ex (2) m=
S -28 2 (=x) 2 >> - mxx=))))
< 222 (O) (8) (2x) 2DD/2 <)lle mx
43

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
where in the second to last inequality we the fact that conv(() and hence
(2) - (S0) (2x), (S (2x) v) 2 0.
Note that is exactly the quantity used the proof of Proposition 4.3 which by the absorbing property in the first part of the proof, apply (in terms of X) y and obtain
1 as
3De
- mx(x)ll S diam(supp()))
22
exp
ag
2
Consequently, have
d (9)
S -2 (n (zx) + 2(D/2 -e)dimmmmmmm)))())))
as
3Dc
222
exp
dA
0g
For notation simplicity, denote (D/2 <)diam(supp(()) 11-ge exp (- (-3Dc
3Dc 221). Then have
0s
d
conv(s) (Ex)
S
22 22ccnv conv(s) + +
dX
By Remark C.8, we have that
//" 262
(conv(s)(Za) S -
(ean(s))a +E
Now we want bound 2e2""(e" (( by introducing the constants C₁ (D/2 c)diam(supp())) 1-os and C2
we have
2e ( = 10 exp (-C2 e²t)
201
V2Cze
where we used the fact that et exp (-C2 e21) is maximized at a= 1 log have
and the maximum value is /2Cse'
Then we
20,
^ 2C1
deanv(s)(2N) S (AA-ii) dennv (2x,) te
e'dt,
A1 V2Cze
201
S e - (A- de d deon
Ve- - 2A1
/2Cze
In terms of o, we have that
2(D/2 e)diam(supp(e))
1-4g as
dconv(s)(Zo)) _ σ1
deonv(s)(
Vo(1 (11-0/01)
(40)
v3Die
In particular, (s) a 0.
We can extend Proposition 4.4 to the case where the data distribution p of the form p-mp-- Ps * *, 821) for some ps that has well-separated local cluster S and 5 > 0 some constant. We have the following result.
44

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Corollary H.1. Assume that S is a local cluster of a probability measure Ps satisfying the Local Cluster Assumption
for some 8 O is some constant. Then for any < 1/2, let and ag (b( O. Let p ps (0, 82
C D/2-x
and define:
diam(supp(r))) 1-ag" "
f00
if. 2 1, 1/2
(o(S,c) -1
3Dc 2log((
if < 1.
Assume that 8² Go(S.c). Then, for any 01 < Vou(S,e) 82 the set Bp/2- (conv(S)) is absorbing on the interval (0, 01) with respect the distribution p. Moreover, for any FM ODE trajectory (3o (00 with data distribution p starting from a point Zo, 2/
(conv(S)). ( the following estimate for the distance dronv(s) (2x) holds:
2(D/2 - c)diam(supp()) 11-s
V0² +
0² + 82
(S) 5
d conv (To.) +\
(Vo2+89)
(1-V
V3Dee
2 o 82
Proof of Corollary H.I. Let (00,0) be the trajectory of the FM ODE with data distribution p starting at some Tar By Lemma C.12, we have that 482 for all a E (0, oi where (Yo,) ooe( 005(0./07+55) is the FM ODE trajectory with data distribution starting from y/ Vo7+08 Ior
Since 01 v (o)(S, 82, we have that v + (S) By Proposition 4.4 we have that (povv((conv(S)) (S)) absorbing in (0, of + 82] for the distribution Pa and hence absorbing in (0, 01] for the distribution p.
For the second part, we apply Equation (40) to the trajectory za = Vo and obtain the desired result.
H.1. Proof of Theorem 4.1
We are going to apply the theory of continuity equations to show that the flow ODE is well-posed. In particular, we need the following result which is direct consequence of (Ambrosio et al., 2008, Lemma 8. 1.4, Proposition 8.1.8) and (Ambrosio Crippa, 2014, Remark 7).
Lemma H.2. Let ((t)eejo.1) be narrowly continuous family of probability measures on Rdd with densities solving the continuity equation below w.E.t. smooth vector field uy:
0.q. + V -Veg)) = 0, 10.
We further assume that
1. lo Ja (1 (drr)d < oo;
2. for any t E 1) and any compact set K C Rdd,
sup K
||ell + Lip(ee, k)) < co,
where Lip(ve, K) the Lipschitz constant of 0s on K;
3. for any 0 < to < l < 1, there exists a constant Crouts such that
[[.x())]| S Cto.ts (1 |[0)),, for all & (4a, 4).
Then, the ODE
dr = =u() 2o
dt
has a unique solution x=t) for all t E 10, Furthermore, the flow map W, Rdd + Rd is continuous and satisfies that
(W:) #90 = for all E (0,
45

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Proof of Lemma H.2. By (Ambrosio et al., 2008, Proposition 8.1.8), for any 4 [0, 1), we have that the flow map (We) exists for gora.e x E IRd and satisfies that # = Gt- We can take sequence of rational numbers e(m) + 1 to conclude that the flow map (V,, (V=)) exists for go-a.e. x E IRd and satisfies that (V.) Qt-
Now, we need to upgrade the ao-a.e. conclusion to all points I. By (Ambrosio et al., 2008, Lemma 8.1.4), for each initial
point Zu E RR, the ODE admits a unique (right) maximal solution defined in a maximal interval [ = [0, 7 (zo)) Pick any to E ((4)) and any 1 E (T( 1). Then, the ODE starting at Itg at time lo has unique (right) maximal solution in the interval [tu. r2x))). Suppose on the contrary that ((x)) < 1. By item 3 in the lemma, we have that the solution (x(())))(o such that s<(6)= = Ito must be uniformly bounded in because ||v= || has linear growth rate for all t [to, t1]. This can be seen easily by applying Grönwall's ' inequality to the differential inequality below:
d| 22 I v((e)) & 20 [0x|1(1 + |2x11) S dt
4Cto..i if a 2 1 if || > 1
By (Ambrosio et al., 2008, Lemma 8.1 again, (#r))= 2 4,, contradiction. Therefore, (xx)) = 1 and this proves that the flow map F is well-defined for all & E [0, 1) and all zo E IRd.
Finally, since the solution exists and is unique, and the vector field is locally Lipschitz, it follows that the flow map v, is continuous; see, for example, (Khalil & Grizzle, 2002, Theorem 3.5).
Now, we verify that the assumptions in Lemma H.2 are satisfied by the vector field u, defined in Equation (8) and the probability measure Pt defined in Equation (1). First of all, for any p with finite 2-moment, the path (Pr)se(o.1) obviously narrowly continuous. Furthermore, each Pt is absolutely continuous w.r.t. Lebesgue measure for any [0, 1) with density function given by Equation (1).
Gao et al. (2024, Theorem 3.1) shows that as long as p has finite 2-moment and absolutely continuous w.r.t. Lebesgue measure, the density function of (Pr)se(u.a) satisfies the continuity equation
O.p. + V (upp)) 0, E [0.
(41)
where is defined in Equation (8).
Note that our assumption only requires p to have finite 2-moment. In this case, although may not have a density function, for all t E [0, 1). the probability measure pa is absolutely continuous w.r.t. Lebesgue measure. We point out that the same proof of (Gao et al., 2024, Theorem 3.1) is valid under this more general assumption that p has finite 2-moment, and hence the continuity equation Equation (41) still holds when we restrict to [0, 1) which is the case we are interested in this theorem.
Now we verify the integrality of un to show that item 1 in Lemma H.2 is satisfied.
S." / p=)()))))dddd)d
< (.' /
21) p(dx1)dt
S.¹ /
3r
dreB - -as
1
(-11x-
||
ar21ll 112)
+
(2 (2m02)d122 exp
dr p(dx, )
3r
Be
2,7
S. /
(1)
(-11t- || 0,2211
)) dx
+ +[daazll)
B1
x.
Be
21
(2m33)aa12 exp
2,33
We split the integral by the two terms according to the summation
1
agd B,
x
|| + +110/1111 For the first term, we use
46

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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
2 and the fact about the expected norm of Gaussian random variable with variance o² is oV [((+ 1)/2)
T(m/2)
Br
|| - ax
|
||
exp p(dx11)d
dx p(de())
Be
(2m
2
/
[[(d + 1)/2)
B1
Be
2
T(d/2)
IT [[dd + 1)/2)
/
-tdt
2
T(d/2)
T [[(d + 1)/2)
2 T(d/2)
where we use the assumption that Be a non-increasing function of and hence 3e S For the second term, have that
J. I laraill
(2T exp (-11e- (-11x-cur Cx 12
1
dx p(dx) dt
2,3
f.¹ / die ciellar |l llp(ddo ||x1|ll(daa < co.
The last step follows from the fact that has a finite 2-moment and, hence, a finite first moment. We also use the assumption that Or is a non-decreasing function of and hence d; 2 O.
Now we verify that u, satisfies item 2 in Lemma H.2. Recall ux(x) = B./3. x (ān3, - a,3:)/B. E X; z]] and by Proposition C.2, we have that
Vum, (x) = at Covl X X; = =])
3?
where Cov X X; zi denotes the covariance matrix of the posterior distribution p(- x). For simplicity, we let (x) Cov X X₁ xr] below. Therefore,
8 (n -on.9) (x) Vzu(())
3
37
Recall that Og 0. B, = and Qz Br are positive continuous when t E (0, 1). Also we have assumed that derivatives exist and are bounded. Therefore, for any i E [0, 1), the coefficients above consisting of ot, 3, will be uniformly bounded within [0, t.]. Hence, to prove the locally Lipschitz property, it suffices to show that the covariance matrix Ex(x) of the posterior distribution p(- X, 2) locally (w.r.t. 2) uniformly bounded (w.rt. t). We establish the following lemma for this purpose.
Lemma H.3. Let be a probability measure on TRd with a finite 2-moment M2(p). For any x E Rdd, consider the posterior distribution:
| p(dz) 2
exp
p(dzl X, xx=
(- ||==aa==1 =|= p(dz) 2
Jg exp
We let (x)x)
f exp
|| o, 2
p(dz). Then, the covariance matrix Ex(x)) of pl- X1 2) satisfies:
[x)x) = (2]1m((z))2++ 2M₂(p) I.
N((x)
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Proof of Lemma H.3. Fix a unit vector Then, have that
v [()x) /(s- (c-mo(zz),
exp
||a-m==l1"
25
p(dz)
ʃ 2
exp
1x-a,
p((2z)
For IR4, have that Therefore, one has that
(z - (,), S 1| -m((=)]1 - 2|12|1 + (z)
v Mx(xv s 21]m +
||=1|
exp
|| ||==
2/9
p(dz)
J cxp
10-a0
12
p(dz')
208
2 ||
pp(z)
= =2]mm((+)112 +
f /exp
i==-y==11 2
p(dz')
2M2(p)
= =]]lm((z)|1+
Nx(x)
Since is arbitrary, this concludes the proof.
By dominated convergence theorem, it is straightforward to check that N 10,1) Rd + +R i is continuous w.r.t. (t, Hence, for any E Rd, t E [0, 1) and any local compact neighborhood of x in R N is bounded below by some positive constant. Similarly, (x) continuous w.r.t. x, we have that (1) is locally uniformly bounded as well in any local compact neighborhood
æ in Rdd These together with Proposition C.2 and Equation (8) imply that the vector field u: [0, 1) x Rd > Rd is locally Lipschitz in x for any fixed E [0, 1). fact, by continuity of N, and my in L, we have that for any fixed & [0, 1) and any compact set K Rdd
sup sup |uel + Lipx(u)) 80. 1E(0.r] K
This implies that satisfies item 2 in Lemma H.2.
Finally, we show that u4 satisfies item 3 in Lemma H.2. This done by establishing the following claim.
Claim 3. There exists a positive constant č continuously dependent on E (0, 1) so that for any æ E Rd
[|,[())] S C, + ||=+|).
Here the constant C₂ may blow up as or + 1 but is finite for any fixed E (0, 1).
Proof of Claim We first show that for any a E (0, oo). then for any x Rd,, we have that
[m) < Ca(1 + ||=|]).
where C₂ continuously depends on a.
By Markov's inequality, we have that for any a >
X x||
M + |
p(1 X - x N
where Mj denotes the first moment of the distribution p. We let A, = [2(M, + ||^]1)] N. Then, we have the following
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
estimates:
S exp -111=111 (y - x)p(dy)
2
[x) - 2|| =
S exp
- p(dy))
Inslluss co+1 exp - l1 x)Ip(dy)
Slv-=1<<A, exp
11 xl(p(dy)
+
f eep
- p(dyy)
[ exp
1r-y'12
p(dyy)
n=
+=
20
12
we let X = x||x-x|<AA+11
X - æ|I, then we have that
I - = 2) S S.SSA, Ax + 1.
Then, we pick any small 0 < e < 1. We have that
-1x=-p11")) 20
Dasllo-o<<n-1 exp
|/ -=/l=lpee xlIp(dy)
1
J|l- -n-e exp
-rr-y11 p(dy)
n= +1
20
(n + 1) exp
20 p(yy : n S 11 - x(11)
n= A, +1 exp
(m-
20
pl {U : 1Iv =ol| <n (n-e))
(n + 1) exp
(2mm-12 20
p((u :n n e |1y x x(1)
p(1y: 11 2|| < +f)
n= A +1
We have that
Mi +
||0|||
ply:n-es 1y - x+1}) x(| x|| 2 n e) < p(1 x 2 A+) <
Ax
2
Then, have that
p(t n e 11y x+11)
p(ly in - 11 S
pl p(v: |yl =|| < n e})
p0l-p((y:0 :n t S 11y - xx1}]
In this way, we can continue to bound 7 as follows:
(2en 22)
I <
(n exx
202
n=
exp
(n + 1) exp
(- =: Ce.s <0.
E
20
The final inequality follows from the fact that Ene (n + 1) exp (-2n) is a convergent series (which can be seen easily from ratio test).
Then, we have that
(x)]
<
[x]|
+ 11 + 12 S ||0||| + Cos + A +1 C. + 2 2M, + +3||z|| C1(1 + [|=]]);
where C3 a constant that depends on 8 and the first moment of the distribution p (we can simply take e = 1/2 to remove the dependency on e).
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
This implies that
[m,(=)]] llmo, (x/ar)) - Cs. (1+ | - C, ||-l]
where C. = / is continuously dependent on f (0, 1) (note that t 0 has to be excluded otherwise Co o0).
Therefore, as is linear combination of and me (ef. Equation (8)), we have that
L (x)]l - C1 ||==1])
for constant C. that is continuously dependent on i for all (0, 1).
Now, for any 0 < la < 4 1, by continuity of Ct. we have that Co.. == suPrE|to.s] C. < oc and hence we have that uy satísfies item Lemma H.2.
Then, by applying Lemma H.2, conclude the proof.
H.2. Proof of Proposition 4.2
We now prove Proposition 4.2 which is a consequence of how close the posterior distribution to the data distribution when is large. We will first only consider the case where the data distribution has bounded support and then extend the result the case where the data distribution is convolution of bounded support distribution and Gaussian distribution using Lemma C.12.
Proposition H.4 (Initial stability of posterior measure). Let be probability measure Fd with bounded support which is denoted Q = supp(p). Let be point and consider the posterior measure p(-1X, z). We then have the following Wasserstein distance bound:
2|| [[ diamm(()
dw, (xr,((p((1,,,,,))) =x exp
-1) diam(S)..
Proof of Proposition H.4. Let R = || X || Consider the function 9o (v) = exp (- (-lle-y!")) lyy ||| < diam(St) for any y E St we have that:
"1x--/l1)) 20²
By the fact that
(R1 + diam)
S (v)(1) I exp (- (R1 - diamm())))
exp
20²
20
for all y E Q. Then for any Borel measurable set A C St we can bound the ratio the posterior and the data distribution as:
p(A X, x)
SA g0 (y)p(dy)
P(A)
p(A) In 9o (v)p(dy)
exp
(R) (Ri-diame di-mmm)()))
SA P(dy)
20
p(A)ex) exp
(- + (2))
Sa P(dy)
20
(Bi + diam(!)))2
(R) d
exp
202
202
Similarly, we can bound the ratio from below:
x(A
xx)
exp
(R.+diam(())
2
Sa p(dy)
2
p(A)
(A) exp
(ll (1))2
P(dy)
20²
(Ri 2
(R1 + di)m
exp
20²
20
11/a.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Since >> have I1/a 1| Therefore, have:
p(A X. = 2)
sa-1.
((4)
In particularly, there is 1p(A|X, = x) - -((A) Σ a - 1 for all A S which by definition bounds the total variation distance between p(- x = x) and p by 1. Then the Wasserstein distance dw,1 (p. X = =x),P) p) can be bounded by (a - (S); see e.g. Villani (2003, Proposition 7.10).
Now we are ready to prove the following special case Proposition 4.2.
Proposition H.S. Let a probability measure on Id with bounded support which denoted as 12 == supp(y). Let ao be point and denote | B[X]]| Ro, where X ~ p. Let Ç be a parameter such that Ç Then the constant
2Rgdiam(SS)
iini 1ni (, Ro) Ro)
log 1 +
CRo diam
satisfies that for any 01 > (2t S, Ro), the ODE trajectory (6a)) starting from Toa = To will move toward the mean of the data distribution p as shown the following estimate:
| - E|X]||
Ton
B[X]]l,
01
where X ~
Proof of Proposition H.5. By the estimate Proposition H.4 and the fact that [() = [+s] increasing for 0, one can check that for any >> Onin C, Ro), and for all with ||2 E E[X]||
< Ro where X, ~ there is
dw.1(p(-IX. Yo = =), < <||= ||X]||,
where Y, ~ 40. Then by Lemma D.2, we have llmo (2) E X|||| < C||= - E[X]]I
This estimate shows that for all a E (nii ( (22, Ç. Ro). . 01] and for all N E aBR, (E[X)), the denoiser m. (2) lies in the ball Beno (TE X)) and hence the interior of Bng X The closed ball Br,, (E. X is clearly convex and then apply
Item 2 of Proposition C.9 to conclude that set BR [E[ is absorbing for the trajectory (Ix) (#n))EE(om(Q...BB)).))
The absorbing result ensures the estimate the estimate
(xo) |[X]]| < C1| E[X]||
holds for the entire trajectory (40)),((m)(0 We then obtain the result by Item 1 of the meta attracting result Theo- rem C.7.
Proof of Proposition 4.2. We obtain the result by combining the results in Proposition H.5 and Lemma C.
I. Proofs in Section 5
In this section, we provide the missing proofs of the results in Section 5. Some proofs already provided in the previous sections:
The proof of Theorem 5.1 is presented in Appendix D immediate consequence of the technical result about posterior convergence (Theorem D.4).
The proof for Example 5.5 is provided Appendix
Also, the proofs of Theorem 5.3 and Theorem 5.4 require more preparation and are organized in Appendix I.1 and Ap- pendix 1.2, respectively, at the end of this section.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Proof of Proposition 5.6. For any independent random variables X p and any Z N(0, o2), we have that X 2
Ya p ((0, p,,
Hence,
dw.2(1o.P)² < X Z X 1 E Z = O
The result follows.
Proof of Proposition 5.7. For any [0, 1), consider y sx + arb). The transformed denoiser my(y)) w.r.t Rz 8,0,/7 / and B, S, well is given by
"1 ll²)
exp
2
y1
m+(y)= = [ exp
p(dy1)
1lv-á.yí1l
p(dy')
2
exp
(.(++aa,)))
Our (0n +b]l1
70xr +
2=
p(dz1)
(42)
sep
(xx a.s (Oz) ( +b)1
p(dx{)
207
exp
7(Ox +b
=
20
p(dz1) (0m,(z) + +).
[ eex
1|
112
p(dr))
237
Let x, denote an ODE path for dz,/dt- ux(x)). Then we consider the path yr s,(Oz, + a1b). We now check that
dy./ dt (ye) as follows:
dy:
(o dxc +abb) a'bb) +
(0x, oxb) + St
dt
dl
8,Ox, + b,a ((log 3,) Yz, Be(a, (((mm))))mm( stapb
($t//s + (log Bi) ())))s()(((())))))))))) a, (r(a. ( + statb (log aa
(log B.)'y. yr + B. (.m, + b)
(log B.) yt + 31 7
(0m, + b)
(log 3.) yt + B, ōt mel))) ūe (Vr).
Note that
Ozo, hence we conclude for for E 10, that
(OOpr) = (OV,)((z)) + a,b)
When (z0) lim;-1 (xo) exists, by continuity of s, and ag have that (Or,)) lim Vo(Onr) exists and that
v0(O))) (0,, +
Proof of Proposition 5.9. Since each V is convex set and in fact, a closure of an open convex set, by item 2 of Propo- sition C.9, suffices to prove that for all x E aV. and all a < oo(")) which the boundary of VeS, one has that ma(x) E V, which the interior of V,C.
First of all, we define
sep2 - 2
Tie
2scp(22) (2:)
is straightforward to check that (xi) C V In fact, Fi. = argmax((> > 0: Bx,,) V.
Hence, by Corollary D.8, for any x E OV. we have that
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
1 ai
An(z)
[m) xell S diam(())
exp
40²
1- aj
2
< diam
exp
2
Therefore, we need to identify when the above inequality is less than Fi,es that is,
2
diam(
V
-
as
exp
<
Tie::
0i
402
E²
Ti,
exp
4
<
diam([)) / /1are
at
Recall that Ct.e
o, I-a;" Then, it is direct check that when Ci,e S 1, the above inequality holds for all J < oo and when Ci.e 1, the above inequality holds for all a < oo(V & (log(C(,)) 1/2 In summary, this implies that ma (z) E in, (x:) C intV; when a < (VV9). Then as stated above, by item 2 of Proposition C.9, we have that Vi is absorbing.
Phi diam(()) V
Next, we show that da + 0 as À + co. Along the trajectory zy, by Corollary B.15 we have
d (2x) dx
-2((2x - Iz;
21 22
mm. (=x));
= -22(2x - -,, žソ - ,ii) + 2(zx Fi, 22 mxxz))) S (=2))(Z>)) + 2dn(=>> ا ||ااا mx(2x)l
Applying Corollary D.8 to points in V/, we have the following uniform bound for all y E Vi.
1 1-a exp
( 1
222
[m) x, S diam(S))
4
aj
Since 2 ViS, we have that
S <22c(((z)) + sep(xx;) diam(() d ddf((s) dA
/ -- a;
ai
exp (-12222)
Then by Lemma C.6, we have that dg(2)) + 0 as X + co. By the the fact that Q is discrete and x; is the nearest point to 2A for all large X, we must have 23 - Ig as
+ cc.
Proof of Proposition 5.10. The proof idea follows the same line as the proofs of Proposition 5.9. Similarly as in the case
of denoisers, we can also consider the change of variable x - log(a) for handling m We let 2 == x6 (((2) and let mx(2) (z) = m²(x)(() (z) for any Rd. Then, it is straightforward to check that
dz 0
m2 - 2
(43)
and converting everything back to a is straightforward
We first identify the parameter (VS, Φ)) such that the denoiser m& (1)) is will always lie in the interior of V for all x E OV.
and all A > \o の).
By the triangle inequality, we have that
(2) æ+ S (lm m (w))l + [Im((u) xzlls
1- a;
exp (- 1 2222)
s ((A) (i)am(()))))
a;
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Since (²() goes zero À goes infinity, there exists parameter (V : :)) such that for all X > Ao (V,5, Φ), |m uall - (x,))
2scp(a;) Then, with the same argument as in the proof of Proposition 5.9, prove that the trajectory 2 will never leave V for all À > >o(V, Ø).
Since the trajectory =x never leaves Vi for all À > >o(V, Φ), we then apply the uniform decay of || (v) xil the differential inequality of di, (2(² as follows:
d dd dA
< 2dd(((²)) + 2dx(= ||: -
S -d((((=))) 2d((=x) (Ó(A) (1) 11 di exp 0₁
1 (-12202)
Now, we apply Lemma C.6 again as in the proof of Proposition 5.9, we have that d=(=)) goes to zero as goes to infinity and hence = converges to 24.
For the second part, the limits = limx 2 and limx m (2 are known to exist. In particular, the limit of the limx m (2 2x exists and we will show that has to be zero.
d=f
derivative lim
ax
Suppose limx-+o d= coordinate of d=f
* 0, then there must exist a coordinate j with nonzero limit. Assume that the limit for the j-th is Vj f O. Then, there exist > O such that the j-th coordinate of dz bounded away from zero ax
dd
dx by |2yl/2 for all > T. However, due to the convergence = + Zoc. we can find two numbers Å X2 T such that |2X,
20. 1/2. This contradicts with the lower bound 1/2 for the j-th coordinate of the derivative by mean
value theorem. Therefore, the limit of the derivative def has to be zero which implies that dd
lim [
21 O.
X
I.1. Proof of Theorem 5.3
We utilize the change of variable X(t) = log B. for E (0, 1). We also let ((A) denote the inverse function of A()).
Next, we consider Σ transformation A, sending x to z/a,. Then, we define QA == (A(x)) = p NO e
#x(A)
Then, we have that y satisfies ODE Equation (16): dzx/dA mx(zx) ZA. Recall the
oe(A)
By Theorem D.4, have the following convergence rate for m (2x):
Claim 4. Fix 0 < ( < 1. Then, there exists A > oo such that for any radius R > ma and all E (x( B (Q), one
has
[m) - projo(2)]l S Cs. . e S2 for all X
where Cr is constant depending only on ( and 7 and R.
Proof of Claim 4. We let z8 := (2)jn(2). Note that || I |||||| + do(2) 2R. Since p satisfies Assumption 5.2, have that (2))) 2 C2 for small 0 < < c. Now, we let A == - log(c)/ç. By Theorem D.4, we conclude the proof.
The following Claim establishes absorbing property for points in ZA₃ E BR, (0) Bz()).
Claim 5. Consider ó > 0 small such that 8 < z.. Fix any R₃ > 0 such that Rg > 26. Then, there exists As > A satisfying the following property: the trajectory (2x (2x)xE(As:0)) starting at any initial point Σy, E BR, (0) n B2) of the ODE in Equation (16) satisfies that for any X 2 As: 2 E B2R, n B2))
Proof of Claim 5. This follows from Claim 4 and Theorem C.11 (by letting dg = AAA
Next we establish a concentration result for 4A when À large.
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Claim 6. For any small 0 > there are Rg, Ag 0 large enough such that for any As and R> Rgg have that
ΩA (Bn(0) n Bx()))
Proof of Claim Consider the random variable X Y + e Z where Y ~ p and Z ~ = =N(0,)) 7) are independent but from the same probability space P., Then, X has qA as its law. We have that dz(x) Σ Y + Y| = e Z| For any 6, we have that
P( X + Z - 2R, 6
2
8)
2 P( X & R e 1 1 5)
P( X S R)P( / S eλ
Since Z follows the standard Gaussian, for any 8, there exists As 0 such that for all As. have that
2 S e 2 P(I S e xτ 6) >
2
Now, since p has finite 2-moment and hence a finite 1-moment, there exists Rs > such that X[] S Rt)
Therefore, for all À 2 As. we have that
ax (BR. (0) n BB()) 2 P( + Z S R3, -~1 8)
Σ 1
/
8)
( 1 - 2
(1-
8
Now, we establish the desired convergence results for the scale A. Claim 7. For any small 8 > 0, there exist large enough As, such that with probability at least 1 6, we have that EAs 9As satisfies the following properties:
1. 2), converges along the ODE trajectory starting from 2λ, as À + 0o;
2. the convergence rate is given by || - -xxs || = O
Proof of Claim 7. For any 8 > 0, by Claim 5 and Claim 6, there exist large enough As and Rs, such that
with probability least 1 8, we have that 2x, 4x lies in BR, (0) n Bs()):
the ODE trajectory (2x (2x)xE(Asooo) starting at 2x, E BB, (0) n B2( satisfies that for all A 2 As. 2> lies in BRR, (0) and dzx2)) < 28.
This implies that one can apply the convergence rate of denoiser in Claim the entire trajectory with C = CC,T.1 n where R 2Rg. Then, for any A < >a in the interval (As, oo), we have that
| - 2x, <
[2x) - 2xIldA
$
[|x((2))= - (2)) + [|projn - 2xl(dA
(44)
C s (-e- - + Se - -22) + $
12 2 2- (o
48C 2 (e-chitae--
- 2 -
where the bound is obtained way similar to how we obtain Equation (21).
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
This implies that the solution z becomes Cauchy sequence and hence converges limit 2oo Now, we let 22 approach oo in the above inequality and and replace A with À to obtain the following convergence rate:
=| - zxll = = -()-)())))
(45)
Now, we change the coordinate back to E [0, 1) to obtain the following result. Since ò >> 0 is arbitrary one can let 8 approach 0 in the result below to conclude the proof of Theorem 5.3. Claim For any small O, one can sample z po with probability at least 1 6, such that the flow map V. (20)
converges a limit 1 (26) as t 1.
Proof of Claim 8. Let A be the given in Claim 7. Then, we let ig == t(As). Consider the map At, Rd I sending to x/ Ctg. Then, we have that
(At) = 9xs
(46)
Both maps As, and Vi, (whose existence follows from Theorem 4.1) continuous bijections. It is then easy to that if an ODE trajectory of Equation (16) converges starting with some 2X, 9A, and has convergence rate O + ), then the corresponding trajectory of the ODE Equation (2) starting with zo == (Arz (rg) (=xx) also converges to limit as + I with the same convergence rate up to the change of variable À + t. Finally, by Equation (46) we also have that
Po(zo with the desired properties) = Ixo (ZXa with the desired properties) > 1 - 6,
which concludes the proof.
Item 2 in the theorem is direct consequence of item 1, As V is the pointwise limit of continuous maps V (ef. Theorem 4.1), it is Borel measurable map. We know that Pt weakly converges to p1 p t + 1. Now, just verify that (v) #P is also weak limit of pt ( to show that (VV #P) pl.
For any continuous and bounded function f. we have that
lim / [(x)(v))
= lim / ((V((t)))n(de))
= =f [VV
where we used the bounded convergence theorem in the last step. Therefore, we have that (Vi) - Pi and hence V is the flow map associated with the ODE dz,/dt (x,)).
1.2. Proof of Theorem 5.4
Part We first establish the following volume growth condition for the manifold M satisfying assumptions in the theorem. Lemma L1. There exists 0 rhs Inj(M) sufficiently small, where Inj(M) denotes the injectivity radius, such that the following holds. For any R >> 0, there exists a constant CR > O that for any radius 0 < < <''ss and for any E M ( Bn(0), one has p( B, (x)) 2 Cnr"
Proof. Notice that within the compact region MR+Dnj(M) := 1nitinjMM (0) n M, the density is lower bounded by positive constant PR > 0. Additionally, the sectional curvature of M is upper bounded by some constant k. >> due to the boundedness of the second fundamental form. Then by Gunther's volume comparison theorem (Gallot et al., 1990, 3.101 Theorem) for < r < Inj(M), there exists constant 6x > such that
VOlM (BM (x)) 2 volum (B. (I)) (x))
where BM == (y e M dz y) < r}, ds denotes the geodesic distance, and M denotes the m-dimensional sphere with sectional curvature K.
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We then choose I'M be small enough such that
B 3. (x) 2 cur
for some constant Cx and all < T'M
Since x y/1 S dm (z, y) for any z, y E M, we have that B C. (z). Hence, we have that for all x E BR(() N
and < <T'-
p(), 2 p(, (z)) = (x) ply)volas (dy) 2 pRvolar (x)) (B) 2 PR Cx rr".
By letting CR:= pr Cx. we conclude the proof.
Then, replicate the proof of Theorem 5.3 with only small changes of Claim as follows using ( 1: Claim Under assumptions as Theorem 5.4, there exsits A > -oo such that for any radius R > TM 72//2, all 2 Bn(0) n (MM), we have that
(2) pr)j S Cr, e for À >> A
where CM.R is constant depending only on R and the geometry of M.
Proof of Claim 9. This can be proved by carefully examining all bounds involved in the proof of Theorem D.5.
Equation (29): Since (a) n (+))(MM), have that 2M E BR+ (0) and hence || || in the numerator can be bounded by R + /2. The denominator is lower bounded by some polynomial of ro by Equation (25) and the volume growth condition of p established in Lemma I.1. We also notice that To can be chosen uniformly for all E (n) n Bru (M) as it is completely dependent on the reach. Therefore, the big O function is bounded above by some function R.
Equation (32): Ci can be bounded by R and the bounds on the local Lipschitz constant of the density and the second fundamental form (which bounds the Ricci tensor).
Equation (33): this one is bounded similarly as the one in Equation (29) by some function of R.
Lemma D.10: the bound C2 is bounded by the bounds on the second fundamental form and its covariant derivatives.
In conclusion,
[ (=) (=)] S dw Aw.2(p(-1X, = =), 02,) /me G(=)
where G(()] S nn
e 22 for all (0) n B /2 (M) and CaA.n depends only on and geometry bounds of M such as reach and second fundamental form bounds. Then, by combining the two exponential terms, one concludes the proof.
Part 2. In the discrete case, we let Ω == { ( We can improve the convergence rate in Theorem 5.3 to be exponential by considering direct consequence of Corollary D.8: Claim 10. For all z IRd such that dz) < $sepn, we have that
[ - (=)jn s C2 exp(-Cre2))
where C1, C2 only depends sepp min0, ER || x,ر| is the minimal separation between the points in Q.
Therefore, when As large enough, we have that for any Xa > a1 2 As
d(c2A dr((22))
S 26 de22dd(()))lmall projn(=x)]]
dA
S 2C dr((x)
< Oze >d((())))));
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
where C3 constant dependent on C1 C2 and the properties of the exponential function.
Then, in particular, there is
("
22 (x))) S 2 (ded)((EN)) C3
S S
C$ /
exxx
where the we use the fact that dex(zx) is bounded by the absorbing result in Theorem C.11 and hence the integral is bounded by a constant C/.
This implies that (2x) S e- (--2A>>ee + C4e-2A C for some constant C. depending Ca. Cs. Now, following the rest of the estimations the proof of Theorem 5.3, we can replace the rate (e -) in Claim with (e after the change above and conclude the proof.
J. Experiments
J.1. A Synthetic Dataset with Three Clusters
3.25
Convex Hull
3.00
Cluster
Cluster
Cluster
Data
2.50
2.25
2.00
1.50
1.25
0.5
1.0
15
2.0
2.5
3.0
Figure 6. The synthetic dataset showing three clusters with different seales and sample sizes. The brown point marks the data
generate a synthetic dataset I²2 consisting three distinct clusters shown in Figure
Cluster (pink): 80 points drawn from the uniform distribution in the disk centered at (1.0,2.5) with radius 0.4.
Cluster 2 (blue): 44 points drawn from the uniform distribution in the disk centered at (2.0, 1.5) with radius 0.3.
Cluster 3 (yellow): 20 points drawn from the uniform distribution in the disk centered (3.0, 3.0) with radius 0.2
The total dataset contains 144 points with diameter of 2.623. As shown in Figure 6, the clusters exhibit different scales and sample sizes. This is the same dataset used in Figure 2. We will continue to use this dataset to illustrate the various stages of trajectory in the FM ODE with respect to the parameter (cf. Equation (11)). We use the closed form optimal denoiser Equation (9) with a, = and 3, = 1 (the Recitified flow scheduling) for the sampling process. This will generate a trajectory in parameter, and we transform it into parameter to align with the general results in Sections 4.2, 4.3 and 5.2. To this end, we use the transformation in Proposition 2.1 with starting value σ 100.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Single trajectory visualization. We first examine single trajectory in detail shown in Figure 7. The trajectory exhibits clear "turning points" suggesting transitions between stages first approaching the data mean (brown point), then being attracted to local cluster, and finally converging to specific data point. We examine the quantitative results we developed in Sections 4.2, 4.3 and 5.2 on these stages.
Cluster 1
Cluster Cluster End
Start Owe mean End Convex Hull
o init 13
Ocuter
Aemina 0.007
Cluster
Cluster 1
Cluster 2
Cluster 3
(a) Initial stage: trajectory moves toward data mean (brown point). Blue cross marks Tewa Top: zoomed view near clusters. Bottom: full view showing coarse movement.
(b) Later stages: attraction to yellow cluster (purple rhombus marks To and convergence to training point (green square marks Foamiual
Figure 7, Evolution of a single trajectory showing three distinct stages.
Global View
Zoomed View
Close View
Rejectry
Vajedry
Tejectory
Teledor
Vajedor
120
Start paints
16)
Figure 8. Multiple trajectories from different initial samples. Left: Global view showing roughly linear paths mean. Middle: Zoomed view. Right: Close view near data support. Blue crosses mark Tem for each path.
Initial stage: Starting with a 100 and initial distance to mean 122, Proposition 4.2 predicts Finit = 13 with ( = 0.5. This suggests that the trajectory will approach the mean at least before init 13. We mark the location Zo... (blue cross) in Figure 7a and indeed observe that the trajectory moves toward the mean prior oinit 13.
We additionally examine multiple trajectories in Figure 8. Most initial trajectories form nearly straight lines toward the mean, with Zem. (blue crosses) consistently making good predictions when the distance to the mean is monotonically decreasing. Interestingly, Trajectory 2, which starts closer to the data distribution, overshoots the mean. From the top right, it initially moves toward the mean, then continues past it, and is eventually absorbed into the pink cluster.
In Figure 9, we show the detailed view of Trajectory in our running example (Figure 2) as well as Trajectory 7, which also
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
converges the yellow cluster. We will use them illustrate the intermediate and terminal stages.
Trajectory
Teojectory
Start
Convex Hull
Cluster
Quemina
Cluster
Cluster
e_cluster
(a) Intermediate stage: Two trajectories attracted to yellow cluster. Purple rhombuses mark Lewee
(b) Terminal stage: Final convergence to training points. Green
squares mark Poomiial
Figure 9. Detailed view of intermediate and terminal stages for two example trajectories.
Using Proposition 4.4 with the yellow cluster's s diameter of 0.364 and 0.1, we compute clusttr (oo(S,)) <) for yellow cluster) is 0.65. The locations Toase for both trajectories (purple rhombuses) align well with the points where both trajectories exhibit a clear attraction to this local cluster.
Intermediate stage:
Terminal stage: We then examine the terminal stage in Figure 9b. First, note that the two trajectories indeed converge to two training data points, validating the terminal time convergence. We use Proposition 5.10 to compute do (V,*) for data points x, which we denote in this section Oeeriinal (ggreen square) for two trajectories and the Proposition predicts trajectory after Geminnl will converge nearest data point. For Trajectory 1, the separation of the converged point is 0.06, and we choose as one-third of the separation. The predicted Verminal for Trajectory is 0.007 and marked on the trajectory. For Trajectory 7, the separation is 0.04, and we choose as one-third of the separation. The predicted verminal for Trajectory 7 is 0.004 and marked on the trajectory. In both cases, the predicted Temminal lies in the segment where the trajectory is attracted to specific training point, validating the prediction.
Memorization with asymptotically optimal denoiser: We also examine the memorization phenomenon with the asymptot- ically optimal denoiser m, described in Proposition 5. We start with the same initial points those in Figure 8 and evolve the trajectories using the asymptotically optimal denoiser given by:
Ma (x) = m. +
where is random perturbation sampled from (O, 1). The trajectories shown in Figure 10 with the perturbation vector shown in the left panel of the subfigure on the left. Comparing with Figure 8, we observe that the perturbation significantly drifts the trajectories; however, as the perturbation decays with the trajectories still converge to a specific training point. This validates the memorization phenomenon with the asymptotically optimal denoiser in Proposition 5.10.
J.2. The CIFAR-10 Dataset
The CIFAR-10 dataset (Krizhevsky, 2009) contains 50, 000 training images across 10 classes and is a popular benchmark for evaluating generative models. In this subsection, we investigate the mean attraction property of flow models and the memorization issue highlighted in our theoretical analysis utilizing the CIFAR-10 10 dataset. These data consist of 32 x 32 RGB images, each with channels. Following the practice in Karras al. (2022), we normalize the pixel values [-1,1]
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
Global View
Zoomed View
Clase View
Hepdaay
fepcary
Thpeary
Thpeerry
Figure Memorization behavior with asymptotically optimal denoiser. Left: Global view showing trajectories under perturbation. with perturbation vector displayed. Middle: Zoomed view of trajectories. Right: Close view near data support. While perturbations initially steer trajectories (most notably in trajectory 5, which traverses across the data region), all trajectorics eventually converge to data points the perturbation strength diminishes with o, demonstrating the robustness predicted by our theoretical results.
by the transformation + 1/227.5- 1. This preprocessing results in dataset with diameter 106.8 and averaged norm 27.2.
To establish a reference, we define the empirical optimal denoiser m, based on the empirical distribution p over the training images of CIFAR-10. The corresponding FM ODE trajectory is denoted by We refer to this trajectory as the empirical optimum. Additionally, we consider a pre-trained denoiser m EDM from the EDM model (Karras al., 2022), with its corresponding FM ODE trajectory denoted as (1r (a:EDM) )oE(p2.01] and governed by Equation (15).
as governed by Equation (11).
To generate ODE samples, initialize from random Gaussian noise and evolve the trajectories using the 18-step polynomial noise schedule (discretization) from EDM:
u/ max
+ (o = 0.1,..,,
n
N
with parameters max = 80, Omin 0.002, = 7, and = 18.
J.2.1. 1. INITIAL MEAN ATTRACTION
To investigate the convergence-to-meee behavior in the initial stages of sampling, analyze trajectories generated by both the empirical optimal denoiser m, and the pre-trained EDM denoiser M.DM Let mean represent the mean of the CIFAR-10 10 dataset.
..DM, and m to denote either m. or mDDM For clarity, we use x to denote either or EDM evaluate two key metrics:
EDM At each sampling step, we
1. The distance [mm (xt) mean|| between the denoiser output and the dataset mean.
2. The ratio [lm mean|] meannl/[[mm (xx)) = ll. which quantifies the distance between the denoiser output and the data mean relative the trajectory direction (x)) x This ratio be interpreted relative error when approximating the denoiser output with the dataset mean.
We compute these metrics for the empirical optimal denoiser and the pre-trained EDM denoiser over 10,000 random initial seeds, with results shown in Figure 11. The figures plot mean with shaded regions indicating +1 standard deviation. When is large, both the empirical optimal and trained denoisers are close to the dataset mean, as evidenced by the small deviation to the mean in Figure 1 la and the small relative errors in trajectory direction in Figure lbb. Notably, the relative error is very small for the first step and approximately below 1% for the first 4 steps, with a small variance across seeds. This suggests that the trajectory is strongly and consistently attracted to the data mean in the initial steps when a is large, validating the theoretical prediction in Proposition 4.2.
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
(mira)- maan xdp
127
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(b) Log-scale relative emror between trajectory direction and denoiser-mean difference, averaged across 10,000 random seeds. Shaded regions show 11 standard deviation.
(a) Mean deviation from data mean over time, averaged across 10,000 random seeds. Shaded regions show 1 standard deviation.
Figure/ 11. Initial mean attraction in CIFAR-10 trajectories. Results averaged over 10,000 random initializations with shaded regions showing =1 standard deviation. Both empirical optimal and trained denoisers cxhibit strong initial attraction to the dataset mean when o is large, evidenced by the small deviation to the mean (left) and small relative errors in trajectory direction (right, log scale), validating theoretical predictions.
J.2.2. TERMINAL CONVERGENCE AND MEMORIZATION
First, we examine the behavior of the empirical optimal denoiser to validate Proposition 5.9 and serves as a reference for the perturbed case in Proposition 5.1 Starting from a Gaussian noise, Figure 12 illustrates the trajectory To (top) and the corresponding denoiser outputs m² (xx) (bottom). As predicted by Proposition 5.9 or the general convergence result in Theorem 5.3, the unperturbed trajectory progressively refines the sample, with both the denoiser outputs showing clear image structure throughout the sampling process and ultimately converging to a training image.
Figure 12. Reference case: Sample generation with unperturbed empirically optimal denoiser. Top: ODE trajectory Za. Bottom: Denoiser outputs m. (I.). Note the clear image structure throughout and smooth convergence to training image.
To validate the memorization result for an asymptotically optimal denoiser presented in Proposition 5.10, we then introduce a significant perturbation to the denoiser by adding noise that scales with a. Specifically, we sample fixed e (O 10 and perturb the empirical optimal denoiser follows:
m (x) = mm (x) + oe, Vo Rdd.
The scaling ensures the perturbed denoiser is asymptotically optimal. This perturbation dramatically affects the denoiser outputs, as shown in Figure 13. Compared to the clear outputs in the unperturbed case, the perturbed denoiser outputs (bottom row) are almost unrecognizable for all but the last four steps due to the large-scale noise. However, despite this substantial corruption of the denoiser outputs, the ODE trajectory (top row) still manages to get very close to a training image visually and through nearest neighbor search.
We also provide in Figure 14a and Figure 14b the quantitative results of the convergence and memorization with 10,000 random seeds. In the reference case with the empirical optimal denoiser, the distances to CIFAR- 10 training set for both trajectories and denoiser outputs are shown in Figure 14a. The denoiser output initially is around 10 away from the CIFAR-10 dataset and then quickly converges to 0-validating the quick convergence predicted by Proposition 5.9. The distances from the trajectory to CIFAR-10 10 dataset smoothly converge to close to around 0.1- small compared to the mean norm 27.2 of the dataset. In the perturbed case with the heavily corrupted denoiser shown in Figure 14b, the denoiser outputs are heavily corrupted by noise and far from the CIFAR-10 dataset until late stages. However, the trajectories still
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Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers
manage get close training images with the distances converging around 1.0-still small compared the mean norm of the dataset. Also, note that only using coarse sampling schedule with 18 steps, and the convergence can be further improved with finer schedule.
Figure 13. Perturbed case: Sample generation with heavily perturbed but asymptotically optimal denoiser. Despite the denoiser outputs being severely corrupted by noise (bottom) compared to the reference case, the ODE trajectory (lop) still remarkably converges to a training image, validating the robustness predicted by Proposition 5.10.
halay Semparr 1o
Distance to CIRAR- 10 Oloe scale)
Distance cirAR-10 (log
Sample Step fol
Sample Step (o)
(a) Distance from trajectory points I. and denoiser outputs mo m=(x,) to nearest CIFAR-10 10 images for the unperturbed em- pirical optimal denoiser m.- While denoiser outputs stay close to the CIFAR-10 dataset, trajectories initially deviate but eventually converge.
(b) Distance metrics for perturbed denoiser My Despite highly corrupted denoiser outputs (blue) that stay far from the CIFAR- 10 dataset until the late stages, trajectories (orange) still manage to get very close training images, validating the theoretical robustness prediction.
Figure 14. Quantitative result of convergence and memorization. Evolution of distances to the CIFAR-10 training set for both trajectories and denoiser outputs using the empirical optimal denoiser and its perturbed version, averaged over 10,000 random seeds. Left: Reference case with empirical optimal denoiser shows smooth convergence. Right: Despite severe perturbation corrupting intermediate denoiser outputs, trajectories still converge toward training data, demonstrating the robustness of memorization predicted by Proposition 5.10.
These experiments empirically validate our theoretical result in Proposition 5.10: even when the denoiser outputs are severely corrupted during intermediate steps, as long as the trained denoiser asymptotically approximates the empirical optimal denoiser, the FM ODE trajectory will still converge to the training data. This observation highlights the importance of carefully regularizing terminal time behavior during training to prevent memorization.
J.3. Local Cluster Absorbing and Attracting Behavior
In this section, we provide additional experimental results to validate the local cluster absorbing and attracting behavior of the FM ODE. We use the FFHQ dataset (Karras et al., 2019) which contains high-resolution human face images. We randomly sample 10, 000 images from the FFHQ dataset and downsample them to 64 x 64 resolution. To visualize the distribution of facial images in the feature space, we perform t-SNE dimensionality reduction on the downsampled FFHQ dataset. As shown in Figure 15, we color code the points based on two related attributes: (I) the average RGB intensity (left) and (2) the illumination value (right). The average RGB intensity is computed the mean of the pixel values across all three channels, while the illumination value is computed as the mean of the pixel values in the Y channel of the YCbCr color space. While the dataset does not form distinct, separated clusters as in our synthetic example in Appendix J.1, it still contains regions of varying density along the illumination spectrum. In particular, very dark faces and very bright faces concentrate at opposite ends of the feature space, creating two high-density regions. This natural organization provides an ideal setting to evaluate our theoretical results on absorption phenomena in a realistic dataset even when distinct clusters are
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present. We will demonstrate that the flow model trajectories are attracted to these high-density illumination regions, consistent with our local cluster attraction theory.
t-SNE Visualization FFHO Dataset Colored by Average RGB Color
1-SNE Visualization of FFHQ Dataset Colored illumination
0.
40
0.8
20
1100 mimmatoo (HSV) valur>
1-SNE Dimension
-20
-20
02
-40 -40
-20
20 40 60 -10 -40 -20
20
80
t.SNE D
55E Dimension 1
Figure 15. t-SNE Visualization of FFHQ (Human Face) Dataset. We downsample the original dataset from high resolutiott (1024x1024) human face images to 64x64 resolution asolution-allinning the training procedure EDM and subsample 10,000 data points to perform t-SNE. The visualization shows feature embeddings colored by average RGB intensity (left) and illumination value (right). Although the data does not form distinct clusters, samples with similar illumination naturally organize into local neighborhoods in the feature space. The extremes of the illumination spectrum (very dark and very bright regions) exhibit higher local density.
In Figure 16, we show samples generated from a pretrained EDM model using three different initialization strategies: (I) random noise, (2) noise initialized near dark illumination regions, and (3) noise initialized near bright illumination regions. The results demonstrate that random initialization yields samples across the illumination spectrum, while dark and bright initializations consistently generate samples with corresponding illumination characteristics. Thus, even without explicit clusters, the flow gravitates toward locally dense regions defined by continuous cluster-absorption theory.
which aligns with
Figure Illumination- based Absorption Behavior. Samples generated using three initialization strategies: random noise (left), noise near dark illumination regions (middle), and noise near bright illumination regions (right). Random initialization produces samples across illumination spectrum, while dark and bright initializations generate samples corresponding illumination characteristics, aligning with our cluster absorption result even for continuous attributes rather than discrete clusters.
64


Reviews:
[{"review_id": "pPQh9WxE8H", "reviewer": "Reviewer_zqMS", "summary": "This paper presents a theoretical analysis of Flow Matching (FM) models addressing how data geometry influences the dynamics of the ODE trajectories used in FM-based generative models. They show that the denoiser guides the ODE dynamics through attracting and absorbing behaviors. They identify three stages of ODE evolution and establish the convergence of FM ODE trajectories under weak assumptions. The authors also provide insights into the memorization phenomenon and equivariance properties of FM ODEs.", "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": "We thank the reviewer for the constructive feedback and for recognizing the theoretical importance of our work on the convergence of FM ODE dynamics, the analyses connecting data geometry with FM model trajectories, and the memorization phenomenon. Below, we address the main comments/questions:\n\n**Experimental validation:** We conducted experiments on synthetic data and CIFAR-10, presented in the Supplement due to space constraints. Specifically, we demonstrated the emergence of the three stages of ODE evolution on synthetic data (Figures 7- 10)  and the convergence to data mean (Figure 11) and memorization phenomenon on CIFAR-10 (Figures 13, 14), aligning with our theoretical findings in sections 4 and 5. Following Reviewer m3yf's suggestion, we have now also conducted experiments on the human face dataset (FFHQ). Please see the response to Reviewer m3yf for more details.  \n\n**Assumption on positive reach:** First, we note that this assumption is only used for our convergence results (Thm 5.3, 5.4) and equivariance (Prop. 5.7), not elsewhere in the paper. Also, the positive reach assumption is common in manifold learning literature (e.g., Fefferman et al., 2016) and is not overly restrictive. Specifically, the medial axis $\\Sigma_{\\Omega}$ of a manifold $\\Omega$ is the set of points in the ambient space having more than one closest point on $\\Omega$. The local feature size $lfs(x)$ at a point $x \\in \\Omega$ is the distance from $x$ to the medial axis $\\Sigma_{\\Omega}$. The reach of $\\Omega$ is defined as $\\min_{x\\in \\Omega} lfs(x)$. The local feature size of any submanifold embedded in Euclidean space is positive everywhere. Consequently, all compact submanifolds have positive reach, which includes most real-world datasets under the manifold hypothesis (e.g., image patches, audio). \n\nMoreover, our core analysis (e.g., Theorem 4.1) is local, meaning the positive reach assumption could potentially be relaxed to only require positive **local feature size**. This relaxation would extend our results to an even wider class of data distributions, including singular manifolds (e.g., sheets of manifolds glued together with singularities).  \n\nFefferman et al. \"Testing the manifold hypothesis.\" JAMS  29.4 (2016): 983-1049.\n\n**Practical implications of our theory:** We believe that understanding the behavior of per-sample ODE trajectory is critical, as that is the trajectory followed during the inference stage. Our theoretical analysis has several direct practical implications for FM model design and training:\n1. Memorization mitigation strategies: Our analysis in Section 5.2 reveals how the terminal stage of ODE evolution influences memorization phenomena, suggesting potential regularization approaches based on the denoiser near the terminal time–e.g. regularize the Jacobian of denoiser.\n2. Latent space design principles: Our analysis reveals how data geometry directly shapes ODE trajectories, with important implications for latent space design. By leveraging the data's clustering structure, one can design more effective latent representations that facilitate both efficient sampling and precise feature formation. Our equivariance result suggests that preserving key equivariance properties can lead to more robust, generalizable, and interpretable models.\n3. Theoretical foundation for one-step distillation method: Our convergence results provide rigorous mathematical justification for one-step distillation approaches like consistency models. These methods rely on the assumption that FM ODEs converge to stable, well-defined mappings, which our results formally validate.\n4. Optimized sampling strategies: Our characterization of the three-stage ODE evolution suggests adaptive sampling approaches that allocate fewer integration points to the initial and final stages while concentrating computational resources on the intermediate feature formation stage. This provides theoretical grounding for empirically successful non-uniform time discretization schemes and explains why the practice of starting at moderately large noise levels (e.g., 80 in EDM) is successful, although infinite initial noise is required theoretically.\n\n**Other comments:** We will improve the paper's clarity in the revision to make the notation and concepts more accessible. For example, the hashtag in the medial axis definition is a shorthand notation for the cardinality of a set, and we will explain this explicitly in our future revision. Regarding the Jacobian of the denoiser, we will update its discussion and include the suggested references. For its relation to the data geometry, as the denoiser converges to projection onto the data manifold, the rank of the Jacobian of the denoiser can reflect the local dimensionality of the data, which could help indicate if the trained model is memorization- when the rank is very low.\n\nWe will incorporate these changes in the revision. Thank you for your feedback and we hope the revision will address your concerns.", "claims_and_evidence": "The claims in the submission generally seem to be supported by the theoretical evidence. The authors provide mathematical proofs to support their claims about the convergence of FM ODE trajectories, the role of the denoiser in guiding these trajectories, and the influence of data geometry. However, the paper lacks empirical evidence to demonstrate the practical implications of the theoretical results. Adding experimental validation would strengthen the claims, particularly the stages on sampling efficiency and model performance.", "methods_and_evaluation_criteria": "The paper seems theoretically sound and provides insights into the dynamics of FM ODE models, particularly in the role of the denoiser. However, the lack of practical evaluation, e.g. experiments on toy problems or benchmark datasets, is a significant gap. While the theoretical analysis is extensive, demonstrating these concepts on a simple synthetic or toy dataset would have helped validate the claims and made the results more tangible.", "theoretical_claims": "The theoretical claims in the seem sound, to my knowledge, e.g. the existence and convergence of FM ODE trajectories (Theorems 4.1 and 5.3) and the attracting/absorbing dynamics (Theorems 3.1 and 3.2). But, the paper's assumptions (e.g., data on submanifolds with positive reach) and occasionally difficult-to-follow notation may limit the generality and accessibility of the results. Is the hashtag in the medial axis definition a typo? It would help to elaborate more on concepts like \"medial axis\" and \"reach\".", "experimental_designs_or_analyses": "The paper lacks experimental validation to support its theoretical claims. While the analysis is thorough, the absence of experiments on toy problems or benchmark datasets limits the ability to assess the practicality.", "relation_to_broader_scientific_literature": "I believe that the results are insightful in relation to the surrounding scientific literature, but I feel that the related work section could use more substance to contextualize the contributions.", "essential_references_not_discussed": "The Jacobian of the denoiser was mentioned as being related to the covariance. The following works also mention this.\n\nBen-Hamu, Heli, et al. \"D-Flow: Differentiating through Flows for Controlled Generation.\" Forty-first International Conference on Machine Learning.\nRissanen, Severi, Markus Heinonen, and Arno Solin. \"Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costs.\" arXiv preprint arXiv:2410.11149 (2024).\n\nIt would be nice to have more discussion about how the Jacobian/covariance ties into the geometric insights of this paper.", "other_strengths_and_weaknesses": "Strengths:  \n- Provides theoretical insights, with convergence results for FM ODE dynamics.  \n- Analyses connecting data geometry with FM model trajectories.  \n- Extends prior theoretical analyses on flow matching and diffusion models. \n\nWeaknesses:  \n- No experiments or demonstrations on toy problems or datasets.  \n- Sometimes difficult-to-follow notation and dense mathematical content reduce accessibility.  \n- Conditions (e.g., data lying on submanifolds with positive reach) might not apply broadly, potentially limiting practical relevance.  \n- Does not seem to link theoretical results to empirical findings in related literature.", "other_comments_or_suggestions": "- Some assumptions, e.g. data lying on submanifolds with positive reach, could be explicitly discussed in terms of their limitations and applicability to real-world datasets.  \n- The paper would benefit from a small empirical demonstration, even on a toy dataset, to illustrate theoretical results and make them more tangible.", "questions_for_authors": "1. Could you clarify how restrictive the assumption of data being supported on submanifolds with positive reach is in practical scenarios?\n2. How do you envision your theoretical results guiding practical improvements in the design or training of FM models? Could you provide examples?", "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": "DFnVkXPAj6", "reviewer": "Reviewer_m3yf", "summary": "This paper studies the convergence behavior of the ODE trajectories in flow-matching w.r.t training data using analytical tools from geometry. Specifically, it provides an extensive analysis of the ground-truth FM ODE trajectories under the affined Gaussian path, and shows how trajectories are shaped by data geometry in terms of “attracting” and “absorbing”. They show that trajectory evolution can be divided into ithe nitial/intermediate stage where trajectories move towards overall data support (mean) and then towards clusters of the data, and a final stage where convergence analysis is provided under much milder conditions than previous works. The work presents a lot of novel theoretical claims and findings, from well-posedness of FM ODE trajectory, per-sample level trajectory evolution patterns, convergence under a broad class of data distribution including submanifolds, equivariance, and finally provides some connection to memorization phenomena when target distributions are discrete measures.", "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": "We appreciate the reviewer’s positive feedback and recognition of our novel theoretical contributions on the FM ODE convergence under weak assumptions, per-sample trajectory evolution patterns, and connections to memorization. We believe that the per-sample level analysis is important as it could inform data geometry-based sample step scheduling or steering strategies. Below, we respond to the comments/questions:\n\n**Cluster assumption and prior noise design:** We agree that the well-separated local cluster assumption may not hold for all datasets. Our analysis suggests similar attracting and absorbing behavior could happen for locally dense regions. For example, we can prove a particular case where data distributions obtained from well-separated clusters convolved with Gaussian noise. Specifically, Lemma C.11 identifies FM ODE trajectories with additionally convoluted Gaussian noise with early-stopped FM ODE trajectories w.r.t. the original distribution (up to parameterization), which can help extend current Proposition 4.4 to this more general setting. \n\nIn practice, our analysis suggests that prior noise with a bias towards the cluster center could effectively control the sampling outcome to reflect the desired cluster characteristics. This could hold, in general, dense regions of the data distribution, as demonstrated in our new FFHQ experiments below.\n\n**Cluster experiments on FFHQ:** Following your suggestion, we conducted new experiments on the FFHQ dataset, which is similar to CelebA, with a pre-trained EDM model (Figures in PDF: https://anonymous.4open.science/r/ICML-7924/FFHQ_exp.pdf). Figure 1's t-SNE plot shows that images with similar illumination tend to cluster together, especially at brightness extremes, as RGB encoding makes brightness differences strongly influence distances in pixel space. This creates distinct dense regions for the darkest and lightest images.\n\nWe performed sampling using a pre-trained EDM with three initialization strategies: random, near-light, and near-dark regions. All samples used standard EDM sampling (18 steps from noise 80 to 0.002), varying only initialization noise. The results in Figure 2 show that when initialized randomly, samples exhibit various illumination levels. When trajectories start near light regions, samples consistently display light characteristics, while dark initialization produces dark samples. These validate our theoretical findings on cluster absorption even where clustering is based on subtle, continuous features rather than discrete classes. \n\t\n**Non-random pairing and non-Euclidean case:** We suspect convergence still holds for non-random pairing, though general dynamics might differ significantly. For non-random pairings (as in rectified flow), the posterior distribution $p(x|x_t)$ is a Dirac delta, and the posterior mean is determined by the deterministic pairing. Hence, we suspect that the convergence in terminal time still holds in this case. But the ODE dynamics in other stages are different: for example, paths may straighten after rectification (due to convergence to optimal coupling), then there will be no travel to the data mean in the initial stage. The non-Euclidean case can be more challenging, as the transition kernels in those cases might be more complicated than Gaussian kernels. These topics are interesting and worth exploring in the future, and we will provide more discussions in the revision. We want to reiterate that our analysis of random pairing in Euclidean space is broadly applicable, as most generative models fit this setting.\n\n**Consistency model and equivariance** We agree that it would be beneficial to better connect our convergence result to consistency models and equivariance. \n\nThe consistency model aims to distill the entire flow matching trajectory (from noise to data) into a single map, which necessarily requires the FM ODE to converge. Our work formally validates this convergence, ensuring consistency models are mathematically well-defined. We will add a discussion on this connection following our main convergence results. \n\nThe equivariance result naturally extends from the convergence of FM ODE trajectories. Additionally, it provides another perspective on how data geometry affects the ODE trajectories: a similarity transformation results in some deterministic and explicit transformation of FM ODE trajectories.\n\n**$C_{\\epsilon}^{S}<1$ in Proposition 4.4:** This condition ensures the formula is well-defined. It could fail when the weight $a_S$ on a cluster is large, but this is not problematic. In fact, when $C_{\\epsilon}^{S}\\geq 1$, the cluster exhibits a stronger attracting force and $\\sigma_0(S, \\epsilon)$ equals infinity, meaning that Proposition 4.4 would apply to any initialization time not just after time $\\sigma_0(S, \\epsilon)$. We will clarify this in the revision.\n\nWe will incorporate these changes in the revision. Thank you for your feedback; we hope the revision addresses your concerns.", "claims_and_evidence": "The claims are mostly presented as propositions and theorems with substantial proofs and some empirical analysis of toy data. The findings on trajectory seem correct, although I did not have the chance to finish all the extensive proofs. Since the analysis is mostly based on a geometric point of view, it mostly talks about how a single sample is absorbed into the support of a cluster in the data and eventually converges to the target data point.", "methods_and_evaluation_criteria": "The proposed method (or theoretical analysis framework) is through the lens of data geometry, which allows more general target distribution, and since it only cares about every single trajectory, I guess the framework makes sense. There is no benchmark or experimental comparison as the work is mostly theoretical and analytical.", "theoretical_claims": "I only checked the proofs for propositions and theorems that appeared in the main text. The assumptions are clearly stated (e.g., bounded support for general attraction towards data support, the existence of FM ODE in $t\\in[0,1)$,  well-separated local cluster assumption for cluster absorbance, finite 2-moment assumption for probability). I don’t think the well-separated local cluster assumption holds for any datasets, and since the clustering analysis is an important conclusion of the work, they should provide some explanations on what will happen if the data is not well-clustered.", "experimental_designs_or_analyses": "There are no experimental results in the main text. There are some empirical validations of the attraction behavior on cifar10 dataset and a very simple toy data in the appendix, but these are all clustered dataset. It would be more interesting if a realistic dataset with less obvious classes/clusters such as CelebA are provided.", "relation_to_broader_scientific_literature": "To my knowledge, the theoretical contribution is quite novel. As the authors claim, they are the first to individual trajectory-level behavior in flow matching, and the convergence is analysis under weaker assumptions than prior works (discussed in the related work section).", "essential_references_not_discussed": "I am not aware of other essential references that are missing.", "other_strengths_and_weaknesses": "The paper is very math-heavy and packed with lots of theoretical findings. Some of them may not be very relevant to the major claims on convergence behavior (e.g., equivariance). \nThe paper studies the dynamic of ground-truth trajectories under gaussian path (or mostly diffusion path?), it mentioned that the findings can be used to support consistency model but did not include any results on that end. It is also unclear whether the conclusions can be generalized to non-random pairing (such as rectified flow after several rectifications), or non-Euclidean case.", "other_comments_or_suggestions": "No other comments.", "questions_for_authors": "What is the meaning of $C_{\\epsilon}^S <1$ in proposition 4.4? Does it hold for most of the clusters?  How should I interpret this proposition to make a useful design of the prior noise?", "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": "9EL8o39nz9", "reviewer": "Reviewer_uK69", "summary": "This paper gives a theoretical characterization of the convergence behavior of flow model ODE trajectories. The authors show that flow trajectories can be divided into three stages -- initial stage where particles are attracted towards dataset mean, intermediate stage where particles are attracted towards local clusters, and terminal stage where particles converge to data points.", "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": "We thank the reviewer for the positive feedback and constructive comments. Thank you for appreciating the convergence guarantees and insights into the memorization behavior of flow and diffusion-based generative models. We will fix the missing parenthesis in the definition of the medial axis to make it clearer in the revised manuscript.\n\nWe would also like to point out that we have further strengthened our theoretical results with new experiments on FFHQ to show the utility of our theoretical findings (in addition to results we already have in the appendix); see response to Reviewer m3yf, as well as discussions to potential practical implications as in response to Reviewer zqMS. We will clarify these points in the revision. We believe that incorporating those changes will further strengthen our paper.", "claims_and_evidence": "All claims are supported with theoretical analysis. However, I am not familiar with the literature, and I did not check the proofs, so I could be mistaken.", "methods_and_evaluation_criteria": "Not applicable.", "theoretical_claims": "No, I did not check the proofs.", "experimental_designs_or_analyses": "Not applicable.", "relation_to_broader_scientific_literature": "This paper gives a convergence guarantee as well as characterization of flow ODE trajectories under the assumption that the data distribution is a mixture of local clusters.", "essential_references_not_discussed": "None, to the best of my knowledge.", "other_strengths_and_weaknesses": "- **Significance** : this paper provides convergence guarantees under weaker assumptions compared to previous work such as [1]. Furthermore, the paper provides some insights into the memorization behavior of flow and diffusion-based generative models. Specifically, the flow velocity could overfit to the empirical data distribution near terminal time-steps, leading to memorization.\n\n[1] Gaussian Interpolation Flows, JMLR, 2025.", "other_comments_or_suggestions": "- At the bottom of page 2, there is a typo in the definition of medial axis", "questions_for_authors": "None.", "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: The paper presents novel and theoretically backed up insights in the trajectories of Flow Matching models. The reviewers are in agreement that this is an important contribution and meets the criteria for publication.
Decision: Accept (poster)