Theoretical guarantees in KL for Diffusion Flow Matching
Authors: Marta Gentiloni Silveri
Alain Oliviero Durmus
Giovanni Conforti
Venue: NeurIPS 2024 poster (2024)
Keywords: Flow Matching
Stochastic interpolant
Theoretical guarantees
Primary area: generative_models

Abstract: Flow Matching (FM) (also referred to as stochastic interpolants or rectified flows)  stands out as a class of generative models that aims to bridge in finite time the target distribution $\nu^\star$ with an auxiliary distribution $\mu$ leveraging a fixed coupling $\pi$ and a bridge which can either be deterministic or stochastic. These two ingredients define a path measure which can then be approximated by learning the drift of its Markovian projection. The main contribution of this paper is to provide relatively mild assumption on $\nu^\star$, $\mu$ and $\pi$ to obtain non-asymptotics guarantees for Diffusion Flow Matching (DFM) models using as bridge the conditional distribution associated with the Brownian motion. More precisely, it establishes bounds on the Kullback-Leibler divergence between the target distribution and the one generated by such DFM models under moment conditions on the score of $\nu^\star$, $\mu$ and $\pi$, and a standard $\mathrm{L}^2$-drift-approximation error assumption.

Theoretical guarantees in KL for Diffusion Flow Matching
Marta Gentiloni Silveri
École polytechnique
Route de Saclay, 91 120 Palaiseau, France
marta gentiloni wwweeriivvvlleeeeeeeeee siverr@@ollyeechniiuue edu
Giovanni Conforti
Alain Durmus
Università degli Studi di Padova Via Trieste, 63, 35131 Padova, Italia giovanni. conf @math. unipd.
École polytechnique
Route de Saclay, 91 120 Palaiseau, France
alain. durmus@@plyyeechniggee edu
Abstract
Flow Matching (FM) (also referred as stochastic interpolants rectified flows) stands out as a class of generative models that aims to bridge in finite time the target distribution n² with an auxiliary distribution μr, leveraging fixed coupling " and a bridge which can either be deterministic or stochastic. These two ingredients define path measure which can then be approximated by learning the drift of its Markovian projection. The main contribution of this paper is to provide relatively mild assumptions on y/², μl and π to obtain non-asymptotics guarantees for Diffu- sion Flow Matching (DFM) models using bridge the conditional distribution associated with the Brownian motion. More precisely, we establish bounds on the Kullback-Leibles divergence between the target distribution and the generated by such DFM models under moment conditions on the score of v* μ and π, and a standard L2 adfftt-aproximmation error assumption.
Introduction
significant task in statistics and machine learning currently revolves around generating samples from target distribution that is only accessible via dataset. To tackle this challenge, generative models have become prominent as effective computational tools for learning to simulate new data. Essentially, these models involve learning generator capable of mapping source distribution into new approximate samples from the target distribution.
One of the most productive approaches to generative modeling is based on deterministic and stochas- tic transport dynamics, that connect a target distribution with a base distribution. Typically, the target distribution represents the data set from which we want to generate new samples, while the base dis- tribution is one that can be easily simulated or sampled. Regarding the dynamics, they correspond to SDEs Stochastic Differential Equations (SDEs) or Ordinary Differential Equations (ODEs), where the drift (for SDEs) or velocity field (for ODEs) is determined by solving a regression problem. This re- gression problem is usually addressed with appropriate neural networks and related training techniques (SDWMG15a, OFIR21,, FJNO20, DBMP19, CLT222, DBTHD21, SE19, LYB 23, GCBD19).
Among these methods, score-base generative models (SGMs) and in particular diffusion models based on score matching [SDWMG15b, HJA20, SE20, SE19] was an important milestone. In a nutshell, these models involve transforming an arbitrary density into a standard Gaussian model and consists in learning the drift of the corresponding reversal process. More precisely, the idea is to first consider
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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an OmsttennUUllnbeee Uhlenbeckenbeeee (OU) process (X; )el.,.T over a time interval [0, T),
-x.o - (1122)xx
OU dt + dB. Xo v²
(I)
where (Bi) is d-dimensional Brownian motion. Then, the reversal process (X 2e which is defined from the non-homogeneous SDE [And82]:
(X ))l + +dB, LE [0,7] , with YoU ~ v POU a = x + Voog HigPT-- oU
allows the law of XP to be transported to v*: [And82, Equations 3.11, 3.12] show that XOU has distribution v*. The initialization v* PP in (2) is the distribution of X defined (1). The drift in (2) can be decomposed as the sum of a linear function and the score associated with the density of X with respect to the Lebesgue measure, denoted by pr- From the Tweedie identity, this score is the solution to regression problem that can be solved efficiently by score matching [HD05, Hyv05, Vin Learning this score at different times can also be formulated as a sequence of denoising problems. Once the drift of the reversal process is learned or equivalently the scores (V log QO )rE(o.T: score-based generative models consist in following the reversal dynamics over [0, T] or, more commonly, a discretization of it, starting with a sample from ((0, Id). The final sample at time T is then approximativel) distributed according to v². Note that an approximation made even if the reversal dynamics were simulated exactly, because for full accuracy, the model would need to start from a sample of v" POU. However, it is well known that for sufficiently large T, v² POU is (exponentially) close to N(0, Id).
When exploring diffusion models, it has been realized that the generation of approximate data samples could also be achieved using an ODE instead of the reversal diffusion:
d'
/dt (X + (oggp""(xx.)))
Similarly to the drift of the reversal diffusion, the velocity fields at time E [0, T] associated with this ODE is the sum of linear function and the score of the density of X This observation has prompted the introduction of the Probability Flow ODE implementation of diffusion models [CCL=23a]. SGMs in their standard and probability flow ODE implementations have achieved notable success in a range of applications; see e.g., [RBL 22, RDN 22, PVG*21]
While diffusion-based methods have now become popular generative models, they can suffer from two limitations. First, there is trade-off in selecting the time horizon T and second, they rely solely on Gaussian distributions as base distributions, in general. Therefore, there remains considerable interest in developing methods that consider a more general base distribution μ and that accomplish the transport between '* and y relying on dynamics defined on fixed finite time interval. Defining generative process in finite time by means of a coupling and an interpolating process, [Pe122], laid the foundation for Flow Matching (FM) models [AVE22, ABVE23, LCBH Liu22, LGL23], finally addressing this problem.
In its simplest form, the main strategy employed by FMs to bridge two distributions, involves fixed coupling æ between u² and fu and the use of bridge, i.e., a conditional distribution on the path space C([O, 1]. Rd) of a reference process (Rz)tejo,1) given its starting point Rg and end point R₁ In case R2 is deterministic function of Ro and R1, we say that the bridge is deterministic and stochastic otherwise. As (Re)tejo. corresponds to the solution of stochastic differential equation, we coin the term Diffusion Flow Matching (DFM) to distinguish the latter case from the former one and focus on it. Then, this bridge and the coupling π between s² and μ define interpolated process (X)))epo.)) referred to as an interpolant, defined as (X). x() ~ I and (X!)<eoo.) given (X), X)) has the same conditional distribution as (R2 (Rx)eejo.]] given (Ro, R2). However, (X:) does not correspond in general the distribution of a diffusion or even to the one of a Markov process. This characteristic poses a challenge when dealing with potential stochastic sampling procedures: indeed, similarly to SGMs, FMs and DFMs aim to design a Markov process that approximatively transport μ to v². To address this issue, most works proposing FM and DFM models [SBCD23, LWY rely on mimicking the marginal flow of the interpolated process (X)):=fo.II through a diffusion process known as the Markovian projection. A remarkable feature of this diffusion lies in the fact that its drift is also a solution of a regression problem that can approximatively solved using only samples from the interpolant (X/)sejo.)- Then, an approximate samples from s² is obtained by following discretization of the dynamics associated with the considered Markovian projection, starting with a sample of p.
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While there exists now an important literature on theoretical guarantees for SGMs [CDS23, CLL23, CCL+ 23b, PMM23, LLT23, Bor22a], only few works have been considering FMs. In addition, up to our knowledge, these works on FMs only consider deterministic interpolants [AVE22, BDD23, GHJZ24]. The main objective of this paper to fill this gap and to analyze DFMs using the d-dimensional Brownian motion as reference process, in which case the bridge is simply the d- dimensional standard Brownian bridge. We provide theoretical guarantees, upper bounding the Kullback Leibler divergence between the target distribution and the one resulting from the DFM. Our results consider the two sources of error coming from the DFM model, namely drift-estimation and time-discreed -inneeiicretizaiinn. This pursuit underscores the significance of comprehending and quantifying the factors influencing the performance of DFMs, paving the way for further advancements in generative modeling techniques.
Our contribution. In this work, we analyze a DFM model using bridge the d-dimensional Brownian bridge and examine how it performs in two distinct scenarios: one without early-stopping and another with carly-stopping. In our first main contribution Theorem 2, we establish explicit and simple bound on the KL divergence between the data distribution and the distribution at the bound without early stopping, by assuming only (I) moment conditions on the target 1/* and the base μ. (HI); (2) integrability conditions on the scores associated with the data distribution v², the base distribution j and the coupling æ (H2); (3) I,2 I2-drift-appren aurff-aaproximationn error (H3) (an assumption commonly considered in previous works). Note that condition (2) implies that V' necessarily admits density. We relax this condition in our second contribution. In Theorem 3, we establish an explicit bound on the KL divergence between a smoothed version of the target distribution and an early stopped version of the DFM model, assuming (1) and (3), but replacing the condition (2) by assuming (4) π = p & v² and integrability conditions only on the score associated with p.
final time of the DFM model. We achieve
To the best of our knowledge, our paper provides the first convergence analysis for diffusion-type FMs, that tackles all the sources of error, i.e., the tuift-approximaiion-om and the time- diseretittti error. In addition, previous studies concerning FMs and Probability Flow ODEs, with deterministic or mixed sampling procedure, either rely at least some Lipschitz regularity of the flow velocity field or its estimator and/or do not take the dine-iisretizzatonn error into consideration. Also, in the context of SGMs with constant step-size, most of existing works without early-stopping are obtained assuming either the score (or its estimator) to be Lipschitz or the data distribution to satisfy some additional conditions (e.g., manifold hypothesis, bounded support, etc.); the unique exception being [CDS23]. We refer to Section 3.2 for a more in depth literature comparison.
Notation. Given a measurable space E), we denote by P(E) the set of probability measures of E. Also, given a topological space 7). we use B(E) to denote the Borel o-algebra on E. Given two random variables Y, Y. we write Y I Y to say that Y and Y are independent. Denote by (Be)tejo.) a d-dimensional Brownian motion. We denote by Leb the Lebesgue measure on Rdd. Given two real numbers u. 0 E R. we write u S (resp. 2 v)) to mean u < C'v (resp. u 2 Cv) for a universal constant C > 0. Also, we denote by ||^||| the Euclidean norm of æ E Rd by (x, yt the scalar product between x; y E R2,, and by T. the transpose of x. Given a matrix A E Rdxe we denote by || the operator norm of A. For f [0, 1] x Rd - IR regular enough, we denote by V=f(t, x), V ft,(t, x) and (t, respecivelly the gradient, hessian and laplacian of f. defined for ł, E [0, 1] x Rd by vtf(t,z) := (0,, ft,, x)), (tt((, x) == (0.g az, zt) Af(t, == Σ 03 ftt, where dny denotes the partial derivative with respect to the j-th variable. For F [0, B + Rd regular enough, we denote by F. div.. F' and A respectively, the Jacobian matrix, the divergence and the vectorial laplacian of F, defined for 1 E [0, 1] x Rd by D,F((1 = (0., ()(,,)))o
div, F(t, == Σj= On, Ft),,x), F(t,
2 Diffusion Flow Matching.
Given a target distribution o² E P(Rd) and a base distribution ja E P(Rd), the idea at the core of FM models is intuitively to construct path between these two by considering two ingredients (1) a coupling w and (2) bridge (or an interpolant following [AVE22]) between μ and D² (or more precisely a bridge with foundations T). More formally, here we say that n is a coupling
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between and v² if for any A E B(R"), a(A x Rd) = (a()) and (R x A) v*(A), and denote by H[ 1²) the set of coupling between j and v/. Then, based on probability measure on w C( 1] IBd) the set of continuous functions from [0, 1] to R we define the bridge by associated with Q as the Markov kernel bQ on R x W, such that, for any A B(W), ((()= Q (2o. x1 ( 21). A) (see e.g., [ Theorem 8.37] for the existence this t, Qi is the I-marginal distribution
kernel), where for any = (t,....,ttp) t.) C [0. 11, of Q, i.., the pushforward measure of Q by (2x)eejo.1] + (It)....,, ,T,,). From a probabilistic perspective, it implies that if (W.).e(o.1] ~ Q, then bQ is a conditional distribution of (W; (W))eejo.1) ~ Q given (Wo, Wi): for any bounded and measurable function on W, (E Xil (((( W1), W1),d(wal)) (w)))ce)).)
2.1 Definition of the interpolated process
Consider now coupling and bridge bQ associated to Q E P(W). We suppose here that Qo is the distribution of (Yi)cejo.1) solution of the stochastic differential equation
dY;- = dt + V2aB, t E Yo ~ p P(R)).
(3)
where (Bo)CER, is standard d-dimensional Brownian motion. In addition, suppose 3 O (oR IR2) for simplicity and that (3) admits unique strong solution. Consider now, the interpo- lated measure 1(T,, @8) corresponding to the distribution of the process defined by (X x and (X.))ejo. (oleeoo..((((])) .j((X) x ~ ([X) x(),.) .). In [ABVE23], (X)) i] is referred to as a stochas- tic interpolant. Denote by (s, 2, y) + Pija (vla) the conditional density of Y, given Y, with respect to the Lebesgue measure and by (vt)c=to.1) the time marginal densities of (X)) (x)))=(0.1) with respect to the Lebesgue measure, that is P(Y, E A)Y,) JA Paja (a|Y,)d and PP[ E A) = JA p1(x)) dz, for A (RR and s, [0, 1], * t. Then, note that, as straightforward consequence of the definition of (X)) it holds
/
p((x)
(d)o,
gza
where
#ddo, dri) = (d.z. dx1))
pijo (x1 (2))
An example that we will focus on in this paper is Q the distribution of the Brownian motion (V2Bt)teR solution of (3) with = 0. Then, it is well known that bil then the Markov kernel associated with the Brownian bridge and the resulting stochastic interpolant satisfies for any E [0, 1]
x - - x)x + +(x + 221(1 10-- t)z, 7 ~ N(0,
(5)
where
dist denotes the equality in distribution.
It is well-known that for any To T1 E Rd, the distribution ((2g, is diffusion-like under appropriate conditions. More precisely, bQ* x1), (Q)((();11)))) is the distribution of (Y,)eε(0,1) solution to
dY. (3(Y.) 2V (Y.)) + v2dB, te [0.1] Yo Zo
(6)
where 4; (v) log php (=1) For instance, for zo. 21 e Rd. the bridge bB 21), associated to B is the distribution of a Brownian bridge (B. )ee[o,1] solution to the SDE
dB 21 B R
dt + v2dB, LE [0,1], Be Be To
From (6), turns that (X)))E)o,1) given (Xg. Xf) is therefore solution
x! (x/))dt + V2d, t [0, 1], X ~ u. 'I corresponds म & bQ the tensor product between क and bQ": (A)))) ()d 21). B(Rd). In other words, m--mxture of Q bridges. J (Q00( A))
axt (B(x)) 2
(7)

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Note that the drift coefficient in (7) depends on X and therefore, (E())) is not Markov, which natural property if we are interested constructing generative process. To cireumvent this issue, based on (TT, Qo), we aim to define a distribution (TT, Q such that it has the same one- dimensional time marginals as [T, Q+) and corresponds to diffusion, i.e., if (X)))e(o.11 ~ [lT,T, Q*)
and (X, ]e (TT, )),, for any E [o, 1]. X X and (X.)) is solution of diffusion with Markov coefficient. This can be done trough the Markovian projection.
2.2 Markovian projection and Diffusion Flow Matching
Markovian projection. The idea of Markovian projection originally dates back to [Gyö86] and [Kry]. Its main idea is in essence define diffusion Markov process which "mimics" the time- marginal of an Itô process:
dx, b,df + v2dB, E [0,1], X. pt,
for some adapted process (b,))E[o.1) and initial distribution μs. Under appropriate conditions (see [BS13, Corollary 3.7]). this diffusion process, denoted by (EM))) exists and solution an SDE with (relatively) simple modification of the drift b,, namely
dx M : 6, (x, V2dB,, t
X ~ p.
where by (M.) = E by X This result can be applied to the non-Markov process (X))etoo)) solution of (7): its Markovian projection is solution of
d.x.a M = (xx )dt + v2dB,, t [0,1].
for some function 3 |00 1) x Rd Rd. It turns out that can identify 3, relying the family of conditional densities (p)p) Jpeesss and marginal densities (pl)oses1- This the content of the following result.
Theorem 1. Consider a π 11(,,, and Q associated with (3). Consider the drift field
Va logy (mn ( (3rl (I) | ) (dxo, dri)
(x)= +
(8)
(x)x)
Under appropriate conditions (see Appendix A. ), the Markov process (XA1) ax; = BY (x jtt v2dB -t 10 X.
)eeoo,1) solution of
(9)
mimics the one-dimensional time marginals of [Tr, QP), i.e,, for any & [0, 1), X! x.M
This result is well-know but, for sake of completeness, we provide the proof in Appendix A.1. The process (9) is known the Markovian projection of I(T, 2))). and its drift (8) mimicking drift. In what follows, we denote by M(T, Q the distribution of (X. )=E[o.1] on W.
Remark It can be easily shown by continuity that X = x! + X for - 1, where + denotes the convergence in distribution. As Law(Xi) ideal generative model which would consist in following the SDE (9) with initial distribution
1², the Markovian projection therefore gives an
Remark 2. Note that, because of (4), the mimicking drift (8) rewrites
BY (x)) = E 2 log (x|((x)) + ((x))xx
Diffusion Flow Matching. Eventually, as pointed out in Remark 1, the Markovian projection gives a an ideal generative model. However, the mimicking drift (8) is intractable and b) the continuous-time SDE (9) can not be numerically simulated. Thus, in order to implement the proposed theoretical idea, first need address and overcome the aforementioned computational challenges. To circumvent a), observe that, because of Remark 2 and [Kle13, Corollary 8.17], we can approximate the mimicking drift via solving
b3 (1, (1, BY (x)x)
min E
(10)
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for a properly chosen class of neural networks {(t,z) H sy (t, taE,, and replace BY in (9) with sy., where 0 E θ denotes a minimizer of (10), To deal with b), we simply make use of the Euler-Maruyama scheme, i.e., for a choice of sequence of step sizes [ N > 1, and the corresponding time discretization 14 Σi= h4 such that to and iN continuous process (X,") )[E[o.1] recursively on the intervals [tx, tk+1l by dx. = sy. (t,, X )t + v2dB, t E [tx, tx+1l, with X p.
define the
(11)
(11) the DFM generative model we are going analyze.
3
Main results
In this section, we provide convergence guarantees Kullback-Leibler divergence for the Diffusion Flow Matching model (11), under mild assumptions on the μt, v² π and S0-, either within non- early-stopping regime or within early-stopping regime.
From now on, we consider the case 3 = 0, i.e., =B. We show in Appendix A.2, Remark 9, that, under out
assumptions, the conditions of Theorem hold for this setup. Moreover, in this case, for any & [0, 1]. < t and y E Rd, the conditional density phj (v(x)) = Pt-s (1) 2z) where (1,2,,) pe (y(zr) ) is the heat kernel:
pc()(z)= = (4Tt)d/2 exp (- Ily - x=11"
t (0,1)
(12)
4t
In the following, we set BY = 3 and sy. =
3.1 Convergence Bounds.
We assume moment conditions on the probability measures μ and v*, and mild integrability conditions on the probability distributions v" and on the coupling æ
For / 1, we denote for Ç P(R.).
mp = / ||=||/
H1. The probability distributions μ., v satisfy mslk] + mg < +
H2. The probability distributions v* and are absolutely continuous with respect to the Lebesgue measure on I&d and Id2d respectively, and satisfy
(i) The functions log hpgd/Lebbb and log du' /dTeb" are continuously differentiable and satisfy | log v* 11 [[())))) + V log Alll +o0 where for Ç (v",,u,,
V V (i
/ V log
dç dLeba
(oo) dC(zo).
(ii) The function log drr 2d continuously differentiable and satisfies ||V log (()))) to0 where
/ | | log (#) (oo, a1)ll dT(, #(10, pı (on (ao) dLeb (No. 21)
1 dæ
V log | fl
(13)
and Pı is defined in (12).
Remark 3. Under HI, note that Vo 1l < too is equivalent by (12) and æ E I[ v*) to
f Woggddr/dLLbbb// ( (1r( < t00.

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Remark 4. We can relax the condition that log dd/dLeeb,, log dv* /dLeb" and log d /dLeb 2d
are
continuously differentiable assuming that v du/dLebb v dv* /dLeb and V d belongs some Sobolev space, but, for ease of presentation, we prefer not delve into these technical details.
Moreover, we assume to have estimated the mimicking drift with an e² pprecision, for some 5² > 0 sufficiently small.
H3. There exist 0* O and e² Such that
N-1
80- (14: x - Be, (x se?
1
Remark To be coherent with the previous section, observe that, as a consequence of Theorem 1, for any l 0.... ,,,, it holds
E s0 (1,, x Br (X
(42: X [
3r (x72) [²]
Under such assumptions, derive upper bound on the KL divergence between the data distribution y* and the output of the DFM (11):
Theorem 2. Consider a uniform partition of [0, 1] with constant stepsize hy h, h 1/N,, for NA Nº and consider the corresponding process (Xl )ae(o,1] defined (11). Assume HI Denoting by of the distribution of X we have that
KL(V"M/ [1(1)(/)) + +1)(
ms[a mglv"1 + V Voo # #
V log Vlog (1) + ||V log v (w)) ) (14)
Remark 6. Under almost the same conditions as Theorem 2, ie.. HI H2, H3, replacing ã in (13) by
dw
7 x1) = pr(2i dLeb (zo, T1)
(15)
our proofs apply also to DFM using time horizon T and the Brownian bridge on [0, 7). In
particular, we would have obtained similar bounds as the ones derived in Theorem 2 but with factor max(1, T in front of the second addend.
Remark 7. Choosing, in Theorem 2,
N₁. + [s] + ms|p [v"] + V log π| (*()) + I log (il + V log x (
makes the approximation error of order O(s²) and the complexity of order O(
Using an carly-stopping procedure, we obtain in the case is the independent coupling:
Theorem 3. Fix < < 1/2. Consider a uniform partition of [o, 1] with a constant stepsize hy = h = 1/N, > for N₂ N* and consider the corresponding process (X(" )sefo.11 defined in (11). Assume HI, H3 and π μ & v/ be the independent coupling. Suppose in addition that
is absolutely continuously with respect the Lebesgue measure. log dp/dLebb is continuously differentiable and satisfies |[V log plli pllis()) +oc.
Then, denoting by v and vfas the distribution of XM and X respectively, we have that
KL[(I_ 62 h0///8 + +1) d
1 5z
+ ms
+ |
Corollary Fix O(VE). Consider uniform partition of [0, 1] with a constant stepsize Ominns=4/d e and consider the corresponding process (X )cejo,) defined in (11). Assume HI, H3 and p & v² to be the independent coupling. Suppose in addition that
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is absolutely continuously with respect to the Lebesgue measure, log dogdp/dLeb" is continuously differentiable and satisfies || V log ( < +oc.
Then, denoting by v and vfss the distribution of X and Xy-s respectively, we have that S E²
where F2. denotes the Fortet- Mourier distance of order 2, i.e.
/ /minfllc= - yll", (2)
W2, W2,M((LV = inf TEll(.)))
3.2 Related works and comparison with existing literature.
FMs stand at the forefront of innovation in generative modeling, offering practical solution to the longstanding challenge of bridging two arbitrary distributions within a finite time interval. Their consequent immense potential has prompted substantial research efforts aimed at providing a theoretical explanation for their effectiveness and has put SGMs and Probability Flow ODEs all in perspective. In this section report and discuss previous researches on FMs, SGMs and Probability Flow ODEs with the purpose of highlighting the links and differences with our work and contextualizing our contribution.
Non-early-stopping setting, In the context of FMs, [AVE22] and [BDD23] seek convergence guar- antees in 2-Wasserstein distance. Both consider more general designs for the stochastic interpolant and a deterministic sampling procedure, rather than a stochastic one. However, both works rely on some regularity condition on the approximated flow velocity filed, i.e. that it is Lipschitz. Namely, [AVE22] works under a K K-Lipschitz (uniform in time and space) assumption on the estimator of the exact flow velocity field. How such assumption pertains to the flow matching framework for generative modeling is not articulated and remains unclear. On the other hand, [BDD23] assumes the estimator of the exact flow velocity field to T,.-L.psschzz for any E (0. 1] and discuss in [BDD233 Theorem 2] how such assumption relate to the setting under the additional assumption [BDD23, Assumption 4], the true flow velocity field is proven to be L,-Lipschitz in space for any t E [o, 1]. Therefore, [BDD23] enhances the findings of [AVE22]. However, [BDD23, Assumption 4] is not an usual conditions considered in papers about convergence guarantees for SGM and it is unclear which type of distributions satisfy [BDD23, Assumption 4]. Moreover, both works [BDD23, AVE22] do not take into account the discretization error in their analysis.
In the context of Probability Flow ODEs, [LWCC24] and [GZ24] investigate the performance of such models Total Vartiation distance and 2- Wasserstein distance. In contrast to [BDD23] and [AVE22]. [LWCC24] and [GZ24] examine the error coming from the (prerequisite when implementing an algorithm) introduction of discreetittininnnn scheme. However, once again, the provided bounds work under smoothness assumptions either on the score or on its estimator. More precisely. the result reported in [LWCC24] depends on a small I waa.obiannestimaateen classical small L L=-score-estimation error assumption. As for [GZ24], they assume the score to be Lipschitz in time and the data to smooth and log-concave. In contrast, our result do not make such assumptions. Finally, to the best of our knowledge, the recent work [CDS23] represents the state of art in the context of SGMs without early-stopping procedure: [CDS23, Theorem 2.1] provides sharp bound in KL divergence between the data distribution and the law of the SGM both in the overdamped and kinetic setting under the sole assumptions of an L'-score-approximmatin error and that the data distribution has finite Fisher information with respect to the standard Gaussian distribution. All previous results are obtained assuming either some Lipschitz condition on the score and/or its estimator ((CCL+ + 23b], [CLL23]) or manifold hypothesis on the data distribution ([Bor22b]). However, we underline that the FM framework enables to consider a significantly wider range of interpolating paths compared to SGMs and avoid the trade-off concerning the time horizon T which is inevitable when dealing with SGMs.
error assumption, besides a
Early-stopping procedure. The recent work [GHIZ24] establishes convergence guarantees in 2- Wasserstein distance for FMs based on a deterministic sampling procedure. However, the results of [GHJZ24] requires to interpolate with a Gaussian distribution and applies only to data distributions which either have bounded support, are strongly log-concave, or are the convolution between Gaussian and an other probability distribution supported on an Fuclidean Ball. In contrast, for our bound to hold true we only need the data distribution n² and its score to have finite eight-order
8

---

moment. Furthermore, even if [GHJZ24] goes into depth when dealing with the statistical analysis of the estimator and the L eettmmmattion error, the entire investigation therein pursued depends on the choice of ReI. Unntwworss with Lipschitz regularity control to approximate the velocity field. They motivate such choice by proving (see [GHI7224 Theorem 5.1J) Lipschitz properties in time and space of the true velocity field under the aforementioned assumptions on the data. On the contrary, we do not assume any regularity on the estimator of the mimicking drift. In the context of Probability Flow ODEs, [CCL 233] provides bound in Total Variation distance, but assuming both the score and its estimator to be Lipschitz in space, So, also in the early-stopping regime, our bound improves previously obtained one.
To conclude, in the context of SGMs, [CLL23, CDS23, BDBDD23] are able to cover any data distribution with bounded second moment at the cost of using exponentially decreasing step-sizes. However, [CDS23, Corollary 2.4] and [BDBDD23] improves upon [CII.23, Theorem 2.2]: the term that takes track of the time-discretization error is linearly dependent on the dimension d in the former works, whereas quadratically dependent on d in the latter.
3.3 The proposed methodology.
In what follows, we provide a sketch of the proofs of Theorem 2 and Theorem in order to outline and delineate methodology.
The starting point of our proof of Theorem is the following (by now) standard [CCL+ +23,, CLL23, CDS23] decomposition of the KL divergence which is derived from Girsanov theorem:
N
sqx (Ix, X (x ||3
kI((*Mp()) & (M. La
S
E
k=
N
da+1
-
E
- Bx( 11 |1 de,
where, for the first inequality, we used the data processing inequality [Nut21, Lemma 1.6] and the last inequality follows from the triangle inequality and the assumption H3. In order to conclude, we should bound the L norm of the adjoint process in the Pontryagin system associated with the Markovian projection of the interpolant. We do by introducing novel quantity in the generative model literature (see [Kre97]), namely the so-called reciprocal characteristic of the mimicking drift, ie.,
(0. + )
where CM denotes the generator of acceleration field and guides the time evolution of the mimicking drift, as
This quantity may be viewed as some sort of mean
(XM) (0. (x +
LE 10
The main efforts in our proof are directed towards bounding the L2 norm of the reciprocal charac- teristic whose representation in terms of either conditional moments or higher-order logarithmic derivatives of conditional densities is quite intricate, see (39). Trying to bound each of these terms separately requires strong assumptions on the initial distributions and couplings leading to sub- optimal results. However, using integration by parts both in time and space and a double change of measure argument, and profiting from symmetry properties of the heat kernel, we managed to bound these terms under assumptions comparable to the minimal ones required in the analysis of SGMs. Note that the analysis of the reciprocal characteristic is not required for SGMs (it is always 0) and that controlling it also requires new tricks and ideas, since its representation contains up to three logarithmic derivatives of conditional distributions, whereas the analysis of SGMs requires at most two such derivatives to be analyzed.
Regarding the proof of Theorem 3, we consider the interpolated process (X)) restricted to [0, 1 - 5). Denoting by 11-6. the coupling between and v corresponding to the distribution of the couple xX By the property of the Brownian bridge, (X/)xeja, (0,1))) is stochastic interpolant resulting from I 5 and the Brownian bridge on [0, 1 6). Therefore based on Remark 6,

---

we only have to show vi-5 and T1-5 satisfy HI and H2, replacing क in H2 by #1-6 defined in (15).
More precisely, we show that they hold and that
1
| log #1 #1- &
| log pIll (())))))))))
+ ms(p)
+ ms(2²)
88 + d 84(1 5)4
(1
6)8
1 d
1
V log Voogri-ollis ( < ms m (1- 5)s + ms
anslv"]
88
82 - 8y1
4 Conclusion
In this work, we have investigated a Diffusion Flow Matching model built around the Markovian projection of the d-dimensional Brownian bridge between the data distribution v² and the base distribution J.. In particular, we have derived convergence guarantees in Kullback-Leibler divergence, which take account of all the sources of error- cime-discretization error and drift-estimation error- that arise when implementing the model, and which hold under mild moments conditions on v², the scores of μ., V² and the score of the coupling æ between ls and v* However, there several questions remaining open. First, it would be worthy to understand if we could lower the order of integrability of the score associated with μ., and æ. Second, would be interesting complement our analysis by a statistical analysis of DFM (11), similarly to what have been achieved in [GHJZ24] for a particular deterministic FM model. Finally, it would be valuable to obtain better dimension dependence with respect to the space dimension d when applying early-stopping procedure.
Acknowledgments and Disclosure of Funding
The work of Marta Gentiloni-Silve Sillvrri has been supported by the Paris Ile-de-France Région in the framework of DIM A14IDF. The work by Alain Durmus is partially funded by the European Union SO--02-SYG-OOCAN-OOUUG
Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
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13

---

A Postponed proofs
A.1 The Markovian Projection
First, we introduce the set of assumptions under which Theorem holds true.
H4. For t > (s,t,2, y) + Přfs( (ulx)) is continuously differentiable in the t and variables and twice continuously differentiable in the a and y variables. Furthermore, depy, (ylx), d.P% (y)z), V.PY V.Pl (v)z) P ).), (2ll(,(u)z)) are bounded.
H4 ensures that (s, z. v) ~ p (v) is enough regular allow a series of algebraic manipulations.
HS. BY is locally bounded Borel vector field on TR: (0, 1) such that, for at least one probability solution br to the Fokker-Planck equation
+ + div(B) pb) Apt 00, E (0,1), 4O =
(16)
it holds f 11xg (dz)dt < +00.
H5 provides uniqueness of the solution to the Fokker Planck equation with drift field BY, see [BKRS15, Theorem 9.4.3].
We now ready rigorously state and prove Theorem 1:
Theorem 4. Consider a क 11(,, v*), Q8 associated with (3) and the drift field defined in (8).
Under H4 and 5, the Markov process (X² )eeoo,1) solution of (9) is such that, for any & E [0, 1),
X! dis XM.
Proof of Theorem 4: We start by reminding that (s, v) + p (u/æ) satisfies for any 2, Rdd
and 8, E [0, 1] with l both the Fokker-Planck equation
a.p. (wl=c) + div, div,,((7,,,wll))))))))) App).(1lx))
(2)xx) =0,
and the Kolmogorov backward equation P.p% + (f(x), V Vera (v/=)) + (1 0
we exploit these well-known results, (4) and H4, we get that for (0,
axp((x) a. (f
(AoPoo(c)20) - (a
(ddro dri)
div
da))
R2a
(8(2).
AaPP
1r) # (dxo, dr1)
/ (dzo, dæ
R2a
(3(x), (xpl(x)) div, (z))
-div
(fr (V
- ( (a)) dwi))
div (8())()()
=dvv, ( (-B(z)pt(z)
+ 826
(va log log ( (2(2a) - Va log
(dzo. dz1))
14

---

Therefore, if we set
of (x) log V log (3/
(o dri)
+ B(x)
(x
we have proven that (z) satisfies the following continuity equation
dpp(()) (u) (x)) 0, t (0,1), ZER
Consequently, we define
(z) = o/ (z) V, lz)
we have that, under H4, (x) satisfies the following Fokker-Planck equation
@xp)(x)) d (x)p((z)) (x) 0 1 (0, E RR.
(17)
So, be (2:) is a mimicking drift. It remains to show that b" = = the thesis will then follows from the uniqueness of the solution the Fokker Planek equation (17) under H5, see [BKRS15, Theorem 9.4.3]. To this aim, note that
V (x) = Sgae Va (m/k)) + p1 (a/ V (erl) dz1)
(x)
SRad l v log 10 + R. log (z or
dz,)
(b(x)
Therefore, for any t E [0, 1) and x E RR, we have that
(x) = 2
Srn Va log (f (11 (1l (<) (z) dri)
+ +(x)) = -xx (x)
(x)
A.2 Preliminary results
We begin this section with two remarks aiming at highlighting some of the properties of the heat kernels (12) and some of their important consequences.
Remark 8. It is well-known that (s, y) + (ylæ) defined in (12) is twice continuously differen- tiable in the space variables x and and satisfies for x, E IR.,, E (0, 1],
VPPcvll)) =
x Pe(ulz)) = -V (ypx))
(18)
2s
1
(xr 1) y)
(2po(vl))=
28 Po(Ulz))
(z(p)))) (yps(vl))
(19)
4s2
Asps(yla)
d
p. (v(x)) +
r r- y
2
po(y)x) Appo(ulx).
(20)
28
2s
Moreover (12) satisfies the heat equation, i.e..
0.p.(y(z) A (y)x) s (0,1), 2, 2, R.
(21)
Thus, particular, (8, + (x))x) is continuously differentiable in the time variable s. Remark 9. In the case = 0, i.e., @ B. under H1, the conditions of Theorem hold. Indeed, as highlighted in Remark 8, (8, x. y) ~ (=(y(x)) defined (12) is continuously differentiable in the time variable and twice continuously differentiable in the space variables and y. Also, using Y Equation (18), Equation (19) and (12), it's straightforward to verify that V.Pij. (v)ær), Vapi (v]x), (PPy.(yla) ViPH. (3)(z) are bounded. Additionally, using (20) and (21), it's easy to argue that also D.p% (2)z) and O.py. (w/x)) are bounded. So H4 is verified and pf solves the Fokker- Planck equation
15

---

(16). Moreover B is clearly locally bounded on R x (0, 1) and, as consequence of (18), (5), and Jensen inequality, it satisfies uniformly time
[ec
|((CO)] p)(x)do==
1
T1
Pt (( dxı) p1())d
P1 (x)
gaa
X
X
E
E
X
1 -
22
E
_ X
X
v1
x x v212
1
24 m2/ul + m2 |10] + d,
which is finite under HI. It follows that f / (z) (=))pp((()))))))) (x +00. So, H5 is verified.
Additionally, hereunder, we state three lemmas that will be crucial in the derivation of the bounds provided in Theorems and 3. Lemma 1. For any p 2 1, they hold
Bi - X 20 x 2P maplu1] + 2 moppp=]]+++ dl df
8)p
and
B||| - Ids dss S (1 8) mz + (1 - 8) 3 + SPP 8)P
Proof of Lemma 1: As direct consequence of (5) and Young inequality, it holds
x x!|l = E | - xx. + / 2s(1 s32
2 E X ||| (2s(1 - $) 3))
2 2 mzplk + و m [,^|| + d s²' - s)P
A similar argument holds for x x
We preface the next lemma with definition.
Definition Consider Q P(W). The reverse time measure Q E P(W) of Q is defined as follows: for any A E B(W)
@*( Q(A"),
where AR (t w(1 - 1) W EJ.
Lemma 2. Assume < Leb2d 2d and p²v* v² < Leba. Then (X.) solves weakly
dx. - 2b (x,, 0. X.)dt VeaB., (0,0, Xo^p.
(22)
with (Bi)eeoou d-dimensional Brownian motion independent of X and
3 cl00 2)) = v
where 4/ (x) solves
2, + A. 20 V 1 0, tE (0,1),
1 log #0
(23)
with
dx (zo, x)
/d((x))
76 ))) =
(24)
dLeb22 2d
P1 (2(
dLeb
16

---

and p1 (x) defined in (12).
Similarly, (())) solves weakly
ax,= :x:2b1 26 ( o, ddt v2dB,, te [0,1]] Xorrr
with (B (Be)teto.) d-dimensional Brownian motion independent of X and
T x((112))= = Vori-r(2).
where ppt (x) solves
-dipt + Aost |V 11 =0 tE [0,
90 log #
with
dr (x,
(x)
(x) dLeb 2d pr (1(l))))
(25)
dLeb
and pı (21)z) defined in (12).
Proof of Lemma 2: We just show that (X/))e(0.1) solves weakly (22). The argument for ((X)) is similar and therefore omitted.
For fixed xo RRd, denote by (zo, + ( A) the conditional distribution of (X))rejo.1) given X3 (see e.g. [Kle13, Theorem 8.37] for the existence of this conditional distribution) and by (B(") [ce[o.1] the solution to
dBT = 22d,, t [0, 1],
B = ro
Also, denote by (Wa)rejo.1) the canonical process the Wiener space W and by (Fe)cejo,) (e)00) the corresponding natural filtration. Note that, as a consequence of the very definition of (X.) (e(o1)) (23) and Ito's formula applied to (v//2) (B("))=((o1)) it holds
(lu,, dLaw((B3) )cE(o,1)) (cB))o))))))
#00 (B
w (Bfo))
exp
(B(o)
4/0 (3ro
exp
{ @ar + Au + 1
exp
w (BT0)
w/00 (B
(BB )dl)
exp
V V (B. JB
|| n 2/2 (B, )1 dt)
Hence, for any / E j0, 1] we have that
dl dl0,3(20.)) dLaw (eaa((22 (B. F.
= De,
where
D. exp V.' (W. (W.)
| VV (W.)ll
is continuous Law((B, )rE(o.1))- almost surely and is such that
dD, D.V. w (W.)dW,
) Therefore, being (W,) a martingale under Law ((B"")))))))) as consequence of Girsanov theorem (see [RY 13, Theorem 1.4]), we have that
(Wr - (D- (W. (W. = -We W. - 2 (W/-2 2 v V2 V
rE[0.1]
17

---

is a ((Fi)e, Ix, -)-- martingale with bracket 2t. It follows that, under (20. (W (We)oejo.) solves dW,- = 2V V (Wa)dt + v2dB, 10,0,1 1] Wo =
The thesis is now direct consequence of the fact that
P(X' E A|X] mo)u((do)) = -fee 11,3 A)u((zo))
1TT, ()))
Ra
for any measurable A E W.
Lemma 3. Assume H2 Then, almost surely, it holds
V f (x1)
Bx x!) =E
(X X!)
(26)
X
(xt)
where To defined as in (24). Moreover, for any u E
[0, 1 - 8), 8 E [0, 1] and (2, 42,4 it holds
b, (X. )dr )rr |"]
E
( + V loog Vlogglleec (l)
(27)
Similarly, almost surely holds
V X1 (xx)
I T((x) X!)
(X x!!
(28)
#1 x
(xg)
where 77 is defined as in (25). Moreover, for any E [0, 1 - -8], 8 [0, and E (2,4,8) it holds
X jdr 1|] $ S00
T T. oo,
( (IVlog ()) + V log 1"1 (P(w))
(29)
Proof of Lemma 3: We just show (26) and (27). The proof of (28) and (29) is similar and therefore omitted. First of all, note that (26) is trivial for t = 1. We therefore focus on t E [0, 1). It's well known that the solution v/ (x) to the Hamilton-Jacol Barioon--aoob-BBllllee
equation (23) is given by
log (f To0 (z1)(z))))d (x1))1-((41(()d)))
de (x)
E 10,
Therefore, we have that for t E 1)
b (. x)) = V (x)
J # (x1) V (211
J #0 (51 )p 2r)d.
Using (18) and integrating by parts, we get fort E [0, 1) that
5 (20. x) J no (x))
[(V T0 (21) n0 (x) (To (I) (aa)l)a
To (5 (x)
J To (ā P1 (ā (x)
But then, it suffices to prove that for E [o, 1) holds
A (x (2)))())))))
J #0 (+ (0
to conclude. To do so, we simply use (4).
(o. 21,2)
P1)..((2 =
@1 2)
T(50, x1 (x (ro))
π T tn (1ro))))rr()o)ooodooe di
pı (1 (oo)
Pi P1(01
TT 31 ( (z) d.
P1 (2o)
P1 (FD)
T1 (2)
T(0,, T (6) dā
(( (x)
M(x0)p1 (0)
#0 (2)))) (()
J m6 ( r)
18

---

We are left with the bounds (27).
We prove (27) only for p 1. The other two cases are analogous. To this aim, we simply make use of (22), Hölder and Jensen inequalities and the properties of the conditional expectation.
X (x
E
b (x,,,, dr
= E
(X) x
dr
X T
(X
V X (X+)
2. 8R
0
(X 07 x
dr
x
(xx)
Xb (x})
2
2/
sB
E
(X x,) dr
#o
X3
(x1)
VTo (x1)
sE
dr
(x+)
No
27
8
V log no (xr Il dn (x 21)
$
3
V log ग [aa()) + V log / (ll:(()))
where, in the last inequality, we have used the very definition of a0 given in (24).
A.3 Main results
We preface the proofs of our main results with some extra notation. For sake of brevity, we denote by
7; =2
J' = Betu - Bai (x X,),r dr,
and similarly
++ b (x 0. X dr, T. B Besu B..
F; =
Remark 10. With this new notation, according to Lemma 2, we have that
(X1) dist ((x (x)eeoo..)) and that for any [0, 1] and E [0, 111-0 u].
(Xf)eeto11 ((x))eto,1)
x t+ x! 7: 7
XL x dist T + T
Furthermore, according to Lemma 3, under H2, for any [0, 1]. 1 E [0, u). p. (2, (2,4,8), we have that
7: M |'] S t V( + V log नं 12 l
E
and
7 P
E
$ (I n* ([p((.) + V log #lg))))
Remark 11. Note that for any E [0, 1] and E 10, -u]
t' L (X )rsu, Tz (x))ran.
This fact is an almost immediate consequence of the Markov property of (B.) see [BB17, Theorem 3.3]: consider the filtration (Fe)ee(o.1) defined by F, = - (Bw)ss)). Then, being (B.)eeoo.) IL X (Byrejo.1) is ( and, as a consequence of the Markov property of (Bi)eeoo.ij; (Fr)eeoo.--i
(Biu B. 1 1 F.,. On the other hand, it's well known that (X (X1) is Flleeoo.j-adpptee It therefore follows that 7i, 11 (Xt)rsu- A similar arguments holds for (X, (x))e
19

---

A.4 Proof of Theorem 2
We fix 0 < << min(h, (on(h,1/2 1/2] and, for any E [0,1 c), we denote by 1i Law(X,) and v4" Law(X"").
First, using the data processing inequality [Nu221, Lemma 1.6.. the standard decomposition of the KL divergence [CCL+ 23b, CLL223, CDS23] based on Girsanov theorem, triangle inequality and H3, bound the KL divergence between vi-c and vf follows
KL(v1-111--)
(L(Law((X))
)EE(0, Law((X) ) )=e()
tx11
E
98.
(tx.
X Bc(x) 1
dt
sg.
h, -
1
tx
N-2
14-1
l- E 3
E
Brn (XM) - (xxx
k=0
1-h
(30)
Second, we aim at bounding the RHS of (30) uniformly in e. Indeed, if we assume to be able to bound it with constant independent of c, then, using the weak convergence X to X (whose law is given by =²) as e - 0, the continuity of (X," )oe(o.1), hence the weak convergence of X to 0, and the lower semi-continuity of the KL-divergence with
X (whose law is given by of" respect to the weak convergence [VEHI4, Theorem 19]. we will get
KL(v"1 | < lim inf KL(vi-Mp'--) S lim inf A= e-40
(31)
e-0
Let us therefore bound the RHS of (30). We will do so by using stochastic calculus tools, and, more precisely, Ito's 's formula. To this aim let us introduce the generator of (XM) Iee(o.--)) which is defined for any & [0, 1 - e] and E (²(IR)) as
C. (V Aup.
Using Ito's formula, we get that
d3 (Ö, C. (x, )dt v2 (X,")d tE [0, c]
So, applying Young inequality and Ito's isometry, we have that, for any 0,...
3r. (X B,(X
(0. + ) (x.' + V2 S" D,B,(XMM)B,
|']
E
I' (0, + 3
+ f E De. ||]ds.
S E
now bound separately the two upper addends. To do so, we introduce the auxiliary measures A (ds) P([tk, tx+1l) for k = 0. .... - 2 and \ (ds) P([1 - h, 1 c)) which will help us, via a double change of measure argument, to mitigate the bad behaviour at t = and = 1 of the reciprocal characteristic of the mimicking drift (i.e., 02 + LM), which is the trickiest addend. Namely, for k- = 0,.., N - 2 we consider the measures (d(sss E P(((e, 4+1)) defined as
(ds) = P(s)-1
2₁ [s
with
p(s) = -7/8 (151122 1-1//1((51//2) + (1 (1-s) 17)
and
min miin(tes2:1/2) 7 77 max((t12)112)
Z₁
(1 r) r)- 77/8
min((x.1/2)
max(tail/2)
20

---

Whereas, for k == N - 1, we consider the measures ( E P([1 h, 1 - c)) defined as
- = p(s)
1 i
-N
with p(s)-1 as above and
ZN-1= min (1-2,12
mmax(1 (1,,1/
r 7/8 dr
(1 (1-r)- (1-r)- dr
11,1/1
(max(1-h,1/7))
Note that, for any [0, 1 c] and for any ls 0, ..., N 1, they hold
1
Zx 2
(s)ds 16
8c
2 1.
(32)
2
We start by bounding the first addend, that is the one that involves the reciprocal characteristic of the mimicking drift. With a first change of measure argument, we get for any O,... N 1,
E [
(o. (a.+ C, ( jds
| (0. CM) (
where, in the last inequality, used (32). But then, if apply Jensen inequality and use other change of measure argument, we get
f (0, + C2 C2) (x, ys
/ (0. + (x,))
|| p(s))
E
E =||10, + 33, 1²2 p(s)dss
S 2x
(33)
(0, + -_, (x,,) 11² p((s)ds.
3 h 1/8
Let us now focus on the second addend. Remarkably, this addend can be bounded via the reciprocal characteristic of 9: because of Ito's 's formula. for E [0, 1 c). holds true
d [,,(M>) || = (B, (8. + 33) + D. 2 } (xM) + 2222 (XM.)..
Consequently, if we assume that the process (8) Da (see Lemma 4 below), we have that
a true martingale
2
D (x.M)
ds
<R
E
+ B( (x"))((
But then, using, as before, a double change of measure argument and applying Cauchy-Sehward inequality, bound the above expression follows
1|
2
D D.
ds
(XM))
1|]
E
1 ||1
<
+ 27 L. (((3 (0, (X*")((())))((
( ||]] - [ (x ||1 + Zu /'' TE (xx()))))) ([())()))))
<
+ Zx 1.' E 1 33) 112
p(s) 2
(3((MM)) [[/] - L [1|3 (x | /. E [0(()
E (0. -M - 3xx ||²
p(s)ds.
1x
21

---

Plugging this bound and (33) in (30), we get
[| ( }1
/
E (xx( | 11² [((s)= 'ds
(T.(vi--11-)) S + hE
+ J.
1) L E
+ C, (x.3)) / [((s)ds. (34)
+ h/h2/8
We now compute explicitly and upper bound each term appearing in the RHS of (34), recalling that, because of Theorem for any E [0, c), 122 Law(X.)- We start with
|1(] -
S/L
Saw P1 (x) V (e ( x) xn )dr1do
B
(r)d())dx.
(x)
First we use (18), second we integrate by part, third we apply Jensen inequality and last we rely on H
E
M
Ia2d P1- (0)
(220 x1)/
( 21) 2n !dzodr
pl (w)d
(1_((a)
Rd
E
E
π
(X),
x1)
x
E
E
V क #
X1
x!)
xi_
#
#
E
S V log क LP
π
(?(=)
(35)
In the very same way we deal with the third term of the RHS of (34).
E
3. (
p(s) -ds
Jaze Pa (1 V.Pi-s (z1 (zo, xn dd 2
1
((5) -1
(l(adddd
(x)r)
VaL क (ro, vn (no: xn droo dms (x)
((s)
Po (a( (
pl. (x))
(())
E
E
x (X)
X!
ds
p(s)
E
IE
J0
(X)
x/)
x!
ds
V π
p(s)
2B
(X] 0: xt)
ds
p(s) ds
V log 1 [[))
V log [()()))
(36)
where, in the last inequality, used (32). We now turn to the last term, i.e., to
E (0. + C. C2 (x
112
p(s)ds
( 1-c
10, + L. )z
(s)) ds.
2d
22

---

Some computations and (21) lead to
0,3,(x)
2 Igaa p=(2)
(/) 21 dz
Leas Pa V A P1 (an xt) म ( 21 )drodz
(x)
v vo 2 )drodr Saza (o )dadd (p) (z))
Saea V x1 ddood Ps At x1)drodr
2
(x))
Dz,B(())
S (21 .p, (x)
( (o)
(zo, Idrodor
(x)
Saz (a 20) V2 P1 (x a atr (To, xz )
+ 2
(1)
Saz Po (2
x P1 (x1 (z) ) (2o x1 )dmodmi)
( ( (ape(2)2o))1 (2) xn ddddodd odd
2
p! (x)
1
V
(m1
1 ( )dxodr
Jaw (xrlao)) V (m) (x) (2o. In jdzoday T
(x)
p1(x)
(37)
hence
- Seza VaPI-s(Z1 (z))) (zo,
(drode feza Ps (2) æ )drodzz
(z)
p!(x))
(x V2
(x) I ( on dd JE (x( V.Pi (x ( dzgdzz
(x)
v!(x)
4 (oaa )ddook))
)azodæi )
( [Jeza (z ( )drodzi T
(!(x)
p! (x)
Igea Ps (2/22))
P1 s (ar (z) ot (drp, on ddodday
Saee Ps (oll (z) drodz (x)
Srea (o Ven (wo, x dr
(x)
(38)
23

---

Ands(x)
2 Srus V 21 ddodz
(2(2)
Sgnr (1 v.P (2) (x) ( (Io, )dzgdr
p (z)
Jaza V.Pi- /)
(T Tn )d
(z)
Sgzs V=Pa(1 10) - ( (Zv, x1 d
p (x)
Rza VaPi (x ( a (3) (ro))
i ( x d2gdd
pd (x)
Raz Ps (x 20) V pP (11 I (2o, xn 1r
(1)
gad V Pi (1 xx, V V (m( (Io, L1 dr
p1
Sgz Pa V .P
(m a (mro )daddr
pl (z)
Saza Ps +
(x)) (x) to )
(x)
Pa V2P1- 1 dddddo 1tw.
(x)(2))
httpa///aaatttaaaaaaaaaaaaaaa/aa dzyy
(x)
SEw (m (ddddd)) Naa dx
(p!(x))²
-s ( x1 )ddddd Saz Ps (oo
dandar
(x)(
ddaddr T
Saw Van dd
Jgae (ap
(x)x)
(x
Ig (a((( V (2) | (2o. vn )dzodar
Jas (o ( x )dzddx1
2
Iaza (2( P1 (ar ( if (mo, 2n dd
+ 4
(x)
(x(x)
Szee (xpp() - (21 (x) ( dd
2
Sgza (2(2
P1 (1) ( 1) oi (mo, 21 dr
+ 4
(x(x)
px(x)
So, get
(0, + (2)
Σ A3(x),
(39)
A=1
where we have defined
Ax = Srow ( T (x xn ddood
(x)
Saes (o d1)drodr1
pd (xx
24

---

A2(x) = S (V VoPi-o(a) (x), V V.Pa (1oo()))))))(1 21 )roddx
(x)
x1 ddoddr
(x
12(x) = Ilssews Vap x1 01)2x1d ||
(x))
V z1)ddodr1
(1(r)
(x) 4 S Pa(a) V (x) (x) d2gdd
(x)
Ig (2 (x (x) x )drode)) )
T Sgaa Po 21 )dr
(x)
(x(
(x) = 4 Jaze (x) x1 )dadder
pl (z)
JizZ (00 x d fae Po Vao æ dddodr
(x))2
Ax(x) = 4 Jazz - vps (x (xx In )dzodr
p! (x)
Ja Po (Ep) xn )dzgda Jgoa
( an daaddr
(x)2²
Therefore
1-c S ||A5(00] p(s)p) ds
IE
(0. + 3x
p(s)ds
Σ k=1
We now bound each term A in the sum. Using (32) and Young inequality, we have
1-c
/
||A](2)]1 p(s)p ds
Rd
1-
Sezz (aps(slarop1 (1 (a) an )danday
S
pl(ddddds
10
Ige
(x(r)
SEze vapa(2 2P (x1l))) TF 21 droddr
+
1 (x)drds.
(x())
To bound the first term we proceed as in (36), that is, first exploit (18) and the integration by part formula and we then use Jensen inequality and the properties of the conditional expectation.
Spas (apa (2 (x) æ )drodr
pl(x)dods < ||V log fogglll (
p._ (x)
Rd
(40)
25

---

To bound the second term we first split the time interval [0, 1 -c we second make either T=P,(2l20) or V=Pi-s(Z1 (z) explicit and last proceed as before, i.e., we exploit (18), we integrate by parts and we use Young and Jensen inequalities.
c] in two, [0, 1/2] and [1/2, 1 - cl.
1-t
faea Vapi-s(2n/>) pa (x) (o)) " ( xn dxoddz
p (2)ddds
Rd
(2)
op
1/2
Jaze (aps(o) V xP (11 ()) T (zo, 21 dr
1. (x:)dds
(x)
op
Sgad V (1 -P (coo ( in d
(rrdddds
1/2
Rdd
(x)
1/2
Vo क (1) x1)
(x) - x²
(x) Sgan
1-8
(r)dddd
J80 (x))
/ Van
(mo,
(x - =00 (o1))))) 2n )dz
Sgaa
1/2
it
p. (x)ddds
1/2
nT
(X xx) (x} x!)
ds
π
V 1 के
E
(Xo= x() (x!
X!)T
ds
1/2
π
1/2
E
|
- xx!l
ds +
E
x!ll
Jas + VVlogiling
1/2
S d' + ms [x] + ms
| + V log
न (=)()))
where, in the last inequality, we have used Lemma 1. In similar way we get
J. [ || ((s) ds $ at + +ms/a] + ms(["] + || log # (1))))))
The argument to bound A resembles the one used to bound A and we therefore omit it.
We now focus on A3. To bound such term, we first use (32) and Young inequality and we second proceed as in (40) and (36).
|| (r)d ds
Rd
SRaa Vap mo (1- x)) (Zo. 01 ddzddz
p1 (z) (x)ds
p!(x))
Rd
Saad Po (2( V PP- (2 (x) i (Zo. 21 ddzddr
+
(x)ddds
p! (x)
Rd
$
22
π
(X] x!)
+
(X) x()
V log न (T) + Vo ((())
26

---

Proceeding in similar way (we omit the argument, as it is almost a duplication of the previous one). we get
S 1- [ [A (()) ds x I L*()) + | log 7 al
We now turn to AS. Because (20), A rewrites as
A)(x)
d
+
| Fo 11?
2(1 D (ar (zo, #1 ddrgdrr
21
x
(x)
R3d
28
4g2
-d d
||
voll
+
Po (11
(x) x) (2r on )dzddz)
(a(
R2d
2s
4s2
21 x
(3))
RRad 2(1
x)
| xp
2 21 x
Ps (c) æ dzodr
2. (ar)
R2d
52
| - xn
1
P. (2y )dzodo))
p
(x)
R2d
52
1
21
Pe (1tdo))))))
11. (x)
JRad
1 1
Therefore, we have that
1- / ||A (z)) ds $ [///
X
X
X
X
E
X!
52
1 -
X!
X!
x1
x!
E
x
E
x!
p(s)ds.
82
8
We now split the time interval [0, - <] in two, [0. 1/2]. 1/2,1-6, <] we focus on the first one. So we look at [0, 1/2) and we try bound the integrand. Using Lemma 1, Lemma 2, Lemma and standard and well- inequalities (Cauchy-Schwarz, Young and Jensen inequalities), get
27

---

that for s E [0,1/2]
X!
X
x X
X
xbol
E
E
X- - E
x
E
X
8
82
1 1
2
X
|
X
X
E
11-
X
22
1-6
2
2
X
X1
X
X
1-8
E
X
7/8
82
8
2 f i
0
+
11
(Xo -x
1-0) 8-1
82
1
+
+
7 7 + T -illl 1|
X1- B x [
X1,
7/8
$2
f1
+ 2 f
gi-s)
8
(Xo x1-0)
8 811
E
22
f
+ 2( f 8
9 x
i
X
X
x
8
7/8
82
+ 2
- Fi-a)
X7 E
-
(xo
{1-0)
X 1-8
7/8
8
2
Ti
2 Ti 7 Ti-a)
+ E
E
X1 E
x
8 -
8 7/8
22
1-8
2
S E
7 1-8
+ 2
1
E
9
Ji-
(Xo X
1))
7/8
22
2
f 8
+ +2
1- T -
+ IE
E
X 1-9
7/4 + EE
|
x
2 1-9
8 1-
52
7 11
2T 2i1i-e 1. T T
12 81-1) 1'1
8/4 +
82
8
9 1
34
7/4 + E
-
x,- -11]
88
28

---

9 1
7/12 + E [ X x1-1
2N
18
d 1/2 + B || x! [']
2A
18
V log (x())
+ V log vz " [1u + d's 1/2 + dl + +mala] + mile").
But then, obtain that
1/2 Jo²² J ||4](w)]] (())p ((s)))))l(s))p(()dd)
ds S d' misa]] + mi[="] + || log म #lli.(mm
+ | log n* (la
We now focus the second time interval, that is we look at E [1/2, 1 c] and try to bound the integrand. Using (32) and proceeding in similar way, we get that
X!
X1
X! 1
X!
X
X!
E
E
X!
E
x!
E
X
p(s)
82
8
82
E
E
X
x
2
7
g
X
x
1/
|
EE
73-
R
x
xto1²
E
x
X
x
191
+ E
E |
x611
+ E
(1
8
S d dt + m + [27] + V log म (+)) + || log 1
But then, we have that
LT / |1A8 p(s)p)(r)d ds I d + ms[p] ma(v"] + |/V log (1))))))))
+ V log u (ll
To conclude, we can bound A as follows
[ J ||48(00112 (c)) ds S d + ms(p] ms(v*] + || log +l
+
V Vlogulll
| VVlgu=Ili
29

---

We are left with A..
Using (19), we can rewrite A as follows
Ax(x)
1
-1
Id+ (x) 2x( - x)-² In
2s (1) 2ı drodz
p (x)
332d
2(1 8)
4(1 s[] S
(I) (x) x)
ps (00 21 )drddz)
Id+
(x)
2(1
4(1 - s)2
20
Po
(2) )dradz))
P (z)
Raa
(w0 (x1 - x) r 3D
<
m an )drodæ
p! (x)
s1=
(x1 1) (11 x)T
( x1 )dpodzs)
(z)
(1 s)-
--30 3u bolttahh
()(x)
Jga
It follows that
A4((x) 1 ((s)) ds
R
(Xl X.) -!)T x
X1
E
E
X
0
s)²
(X}
Xx}
X!)
X1
X
x!
x!
p())ds
s-s)
s
At this point, we proceed as for AS. that is we split the time interval in two and we use Lemma 1, Lemma 2 and Lemma 3. By doing so and by using (32), we get that for s [0, 1/2]
(X}
(X} X!)T X
X
E
E
X!
[X] (x - -X!)T
X X!
E
x! E
X!
p(s)
(1 11-8) 8)2
X
X1-
( -b 1-)) - 5 1-,)²²
x 1-8
E
E
8
X
(x. X .) (X. - X {
X
1-3
8
X x1 - - -x,--)" Ti.
E
X
1
(xo X x1 (x. 8 x T| X 1-3
X
8
x 3R[11x0 X (x - 1 1 mT/
+ R
f -i-s
= =E|| x - X||| E
ms ju]
ms(F*] + V log ãM (z)
+ V
loge*1l (1()()))
30

---

Whereas, for E [1/2, 1 - c), we get
[X] (x} - x T x
x
E
E
x!
(1 s]= s)²
(x] (X} - -X!
X! Xg
E
X E
X
p(s)
(1-8) 8)3
(X X (X -T
(X X0)
X
E
E
(1=8)
[X xx, - -x,)T
Xx - X. [X 2
E
X, E
(1 (1-s) s)/8
(1-s)2
7!- !-) + " 9T + +
(1 /)7//E E
E
(X - Xo) X
(1-s)2
7: - (7--0) + 11 7 + 9 !!-o
x F x x / ] 1/1
E
(1 8)2
ym + 7: TTT J. -)-))
(1 5)74 + + -
E
(1 8)$
0 (1-5)712 +E | xb6|']
E
E
5)8
8)
V log * (m))(m)))
+ V log log/lli + +d - -111 + d² + ma[a] + ms(v"]
Consequently, we have that
// for |1126001 (()) ds S d + msssp] + ms [v'] + V log * (
+ V log /4 / + | log (li
Putting together the bounds on the { derived so far, we eventually obtain
22) L [[ 2M ) (x x) 121 p(s)ds
(41)
Sd' + ma|4 + ms ip01 + | log 7 Ili + V log log + |/ log v² liix(..)
Plugging (41), (35) and (36) into (34), we get
KL =² + 1/ + +1) + m + [ + | log मे Ill
Voll ( + ||V log v ((io(we))
The estimate (14) then follows from the above estimate and (31). However, for (14) to hold true, we still need to prove that
Lemma 4. S (3., (X. Md) is a martingale.
Proof of Lemma 4. If we show that for any s E 10,1-44 c), it holds ElI(3,, (x,",)XX < C. for some C. > 0 independent of time, then by Fubini's 's theorem and [BB17, Theorem 7.3] we are done.
To do so, by the Cauchy-Schwarz inequality, we just need to show that E (x. M)]] and
31

---

(X [[ap] are bounded from above by constants which are independent of time. To
this aim, note that, as direct consequence of (8), Theorem 1, (18), Jensen inequality and H2, it holds
Be (X ||
=P1- Lasa (1 V.P (2)r))/))((()) adddod |
p (x()
(x)
2d
Vz,
Po (3(o
(To, xi ( xn )dro dr
(x(ddd
P! (x))
(42)
Raa
E
E
# (X): x()
x!
E
E
V x1 क
(X) xx)
x!
π
V π (X
E
x!)
V log क (1))
Similarly, recalling (38), we have that
D (x
||
E
Jaa
(= 2n ddrodr
pl. (z)dr
(²(x)
Sgea Po (x V2p (x a (ro, a )drddz
+
p (x)ddx
(x)
Rd
op
SRzaPa P2 20 V P1
(x) (r) f (Io, 1 dr
+
p.. (x)dz
(x)x)
SRza V=p.(cl (x) ( x 1
+
(x)ddx
(x(x)
To bound the first term of the RHS of the above expression, we integrate by parts and Lemma and H2.
pl (l)))
(z)
op
Saed (xp .P (x)) ((ro, an )dzddæ
Rd
p.(x))
op
V (ma, x1
(2) x)T
Ps (1))))))0 xn daado
v1 pl(x)d
Jz
11 (x)
op
V mp 7
(X] x!!
< E8
(X] x!I
(d + ms/pi + ms/12") + || log (T)))
32

---

To bound the second term, we integrate by parts and proceed as before.
Ia2a Po(a) ( 21 )dzddr
Po (x)dd
Rd
(x)
Saed Pal(oc)/))) (( 21) ( 21
((0o, an dd
(x)
Po (x dx
ju (z(x) / p=( Vx,
(x x1)²
(ro, x1)
# in )d
1 1-s
P (x)
S 1 (d + ms + ms(="]) + V log R (xx)
To bound the third and last term, we proceed as (42), getting
Jgau Pa V Val (1)
p.(x)dx & I|V
(x)
and
g2a V (apa(2 - (1ra ( 21 dxodd (x)
p (x)dd $ 1/V log fog
Rd
A.5
Proof of Theorem 3
Fix 8 > 0. Then, because of Theorem 1, (5) and HI, it holds
ms(ui_s] = =| m [M] + (1 8) [ 8) < +00.
(43)
Moreover vi-x < Leba with density plaa and
V log ddviss dLebad)
E (v)
(44)
Indeed because of (18), it holds
V log d dLeb (x1-5)
Sgaa P1 1n-6(11 ( (21 (ddo dzi)
p1_5(21-0) (21-5)
T1-8 - I0
1 1 +
I1 I1-8
| Pi pa dxi)
(1=0(11-))
pl
8
1
(28 - 1)
8zo + (1 8)
p1-s(mn-sleo l ( dx1)
p (2) -8)
B3d
8(1 6)
x1
5X8 (1 8)
E
x = 21
8(1 8)
33

---

But then, using Jensen inequality we obtain that
88
J/c
Vei-o
log
dvi-s
dvi-s
B
X 1-6
6X0 + (1 - 6
X
dLeb
5(1 -
8Xg + (1 - x] |$
X 8
- E
8(1 6)
1
1
2 mg (ns(vi-))
mg (1)
mg (20)
(8 - 5)8
(1 (1-6)s 5)8
58
1
1
x sms[u]
+ mssp")
d
(1 (1-8)8 8)8
0²(1 8)
Also, consider
T1-6 x1-6) = plx1)) / ( x (dx1).
where (Zo, 31, 11, x1-6) pl $jo, (21 (21-8/z0- 21) denotes the density of X| given xf) with respect to the Lebesgue measure. Then T1-8 E I1( v7-s) and Ni-6 < Lebad Moreover
1 1
V log
E L (
(45)
P1-5 2d
Indeed, because of (5),
p (1)) (4Tō(1 - 8))d/2 exp (- |0110 8z0 (1 - 8) 11
48(1 6)
Therefore
V (00 -1-5 820 (1 8)
pi 10-, (01-
2(1 -5)
and
31-6 800 (1 - 8)x1 1
Vxy (01
Pi 1i-5 11 (51-
25(1 8)
Furthermore
p1- 5 1 (#1 -イ)i" (dx1)
(01 x )(()))e> (dx1)
JR. p 1 ( wo, को (dā,)
JR. p1 -10.- 101-8| के (( u (dā,)
o x1
Consequently, we have that
V 11-6 (Io, 86-
51-5
Vμ(mo) Fox - f/00, xn (dz1) + p(ao) Sor (000
-80,1
u(xo) Sae p pi- (x 1- - ã (dă,)
Vu(æo) p(xo))
21-5 - 8xg (1 8).1 1 Pijo, (1)
R2d
2(1
and that
Vz nl
((z0) SRn V V ( 2z ) (dz1)
21-))1-))
(1zO) SEa p
p1-8j0,1
(x1-6
Fo. 31 ) (d.E. 8)
8xo
p iioo. 1 (x Imps I1 8)2
28(1 5)
34

---

But then, if we use Jensen inequality and (43), we get
d 5 dLeb 2d
du
8
Vao log
8
log
d/
R2d
R2
dLeb
X
5X0 (1 (1- [1-
E
E
5
(X) x1-s) 5)
1
X
6X0
(1 8)
log logullig (1 +
1 8
V log log + ms Aallvi-.
+ +ms| [4]
+ mss (v*]
8)8
8)8
88
log/uliss + ms [4]
+ m [201
+ d
(1 8)8
(1 (1-8)a 8)
1
logıllis()) + msmml
+ ms
[V0)
dd
(1 8)8
84(1 - 5)4
and (similarly)
fea V Veio log
d dLeb 2d
X 6X0 (1 8)
di S E
E
(X x 8)
8(1 8)
1
1
mg [p]
mg [v-]
d
(1
8)8
8¹( - -84
Additionally, because of Remark 8 and (43), they hold
Jgac an Loggi-o(21-s (T1-
V Vay- log p1 (0 01-,
gad
X 8
X
1
1
E
E ms - [iTi
+ *mssAa)
1 6
(1 - 8)8
(1 - 878
88
I mm|
+ ms [V]
+ dt
(1- ms(uu] 5)8
(1 8)8
1
S mggly]
+ mg [V*)
d
(1 8)8
84(1 - 8)1
follows from HI, H2(i), (43). (44) and (45) that the probability distributions J4, VT,' and the cou- pling # [(, vi satisfy HI. The bound in Theorem 3 is now straightforward consequence of Theorem 2 and the bounds on the scores derived so far.
35

---

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42


Reviews:
[{"review_id": "2C3ibZr5Yt", "reviewer": "Reviewer_NqRW", "summary": "The paper derives theoretical guarantees for a flow matching procedure which constructs a diffusion-like process which allows for samples obtained from a source distribution $\\mu$ to a target distribution $\\nu^*$. The goal is to build a suitable simple sampling procedure where a source sample is updated through a series of relatively simple Markovian procedure to an approximate sample from $\\nu^*$. In the typical setting of diffusion models, this takes the form of a score function augmented with additional Gaussian noise. However, this suffers from two main drawbacks -- firstly, the choice of base distribution is fixed (the base function is always Gaussian) and secondly, the diffusion is truncated after a finite number of timesteps which incurs error in the approximation of the intermediate distribution by a Gaussian. Flow matching addresses both of these by allowing for transformations of arbitrary distributions to each other. \n\nTo define a flow-matching process as in the paper, one obtains samples $(X, Y)$ generated from a coupling (joint distribution) $\\pi (X, Y)$ where the marginals of $X$ and $Y$ are $\\mu$ and $\\nu^*$ (the source and target distribution respectively). Given these samples, an interpolating distribution, $\\{X_t\\}_{t \\in [0, 1]}$, is constructed with $X_0 \\coloneqq X$ and $X_1 \\coloneqq Y$ and the intermediate values are distributed based on the choice of an appropriate bridge distribution. This paper utilizes the Brownian bridge which is the conditional distribution of Brownian motion conditioned on fixing its endpoints. The main issue with this approach is that such an interpolating distribution is not \\emph{Markovian} which makes sampling from it in a diffusion-like process challenging. This may be addressed through the notion of \na Markovian projection which constructs another \\emph{Markovian} process with the same marginals as the interpolating process for which a score-function-like analogue which enables efficient Markovian sampling. The paper then shows that under some relatively mild assumptions on the approximation of the score function and the moments of the coupled distributions, this procedure guarantees closeness to the target distribution in Total Variation distance. All previous approaches either require additional assumptions on the estimated score function, and/or do not account for the discretization error. One of the main challenges of the proof is in analyzing the discretization error of the scheme where the continuous process is approximated with one with discrete time steps. The paper observes that the terms controlling this error may be rather large at the end points of the interval. Therefore, the paper considers a truncated process on $[\\delta, 1 - \\delta]$ and show that on the truncated set, the gradients of the densities with control the discretization error are bounded in ($L^8$) norm.\n\nOverall, the results of the paper are novel and interesting. The observation that the truncated distribution is well-behaved with respect to the discretization error is valuable for future theoretical and practical work on flow models. This also illustrates the drawbacks of prior work which require additional Lipschitz assumptions on the scores/estimated scores and presents a natural approach towards circumventing them. This is a nice contribution and I recommend acceptance.", "questions": "See main review", "limitations": "See main review", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "See main review", "weaknesses": "See main review", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the thorough review and valuable comments. We greatly appreciate the reviewer’s recognition of the novelty and significance of our work, as well as the potential of our ideas for future theoretical and practical advancements in flow models.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "pvwEQv4ct0", "reviewer": "Reviewer_TMKS", "summary": "This work provides theoretical guarantees for diffusion flow matching (DFM) models, which are a recent class of generative models similar to score-based generative models (SGM). Extensive background is given in sections 1 and 2, and section 3 contains the results, namely, bounds on the KL divergence from the target distribution to the distribution outputted by DFM. A detailed comparison with existing results is presented in section 3.2.", "questions": "- Are SGMs a sub-case of DFMs, for a well-chosen coupling and bridge, and perhaps up to a change of time variable?\n- Is the Markovian projection of an Ito process always unique, and/or is the Markovian projection of (3) unique?\n- How do DFMs compare to SGMs and deterministic FMs in practice, say in generalization, speed, difficulty of implementation, robustness w.r.t. hyperparameters, ...?\n- Are there settings where it makes sense to want to choose something else than $\\pi$ being the independent coupling and the bridge being Brownian bridge?", "limitations": "The authors have adequately addressed the limitations.", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The presentation of the whole paper is remarkably clear, notably the background explanations of section 2 are very clear and very welcome. Regarding the guarantees of section 3, they are (claimed to be) the first ones for DFMs, which are well-motivated as alternatives to SGMs and deterministic FMs; this makes these results quite valuable.", "weaknesses": "(I can only comment on the presentation and context as I am not familiar with the guarantees for SGMs and FMs, sorry.)\n- Two things remained unclear to me after reading section 2; see \"Questions\".\n- The discussion of the previous work on SGMs seems a bit difficult to compare, since this paper is about DFMs, which (if I understand) are a different thing although similar.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Questions:\n\n_Are SGMs a sub-case of DFMs, for a well-chosen coupling and bridge, and perhaps up to a change of time variable?_\n\nWe thank the reviewer for raising this point. As suggested by the reviewer, after an appropriate time transformation ($t=\\exp(-\\tau)$), the Ornstein–Uhlenbeck (OU) process at the basis of SGMs can be reformulated as the process at the basis of a linear stochastic interpolant. For further insights, see Section 5 of [1]. However, the foundational concepts of SGMs and DFMs differ significantly: SGMs aim to approximate the time-reversal of a diffusion process (most often OU), whereas DFMs aim to approximate a Markovian projection of a two-sided linear stochastic interpolant. \n\n[1] Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. Stochastic Interpolants: A Unifying Framework for Flows and Diffusions. \n\n_Is the Markovian projection of an Ito process always unique, and/or is the Markovian projection of (3) unique?_\n\nThe uniqueness of the Markovian projection of an Itô process, such as the one defined by equation (3), is not straightforward and necessitates regularity assumptions on the mimicking drift, such as those found in [2]. In the specific case of interest in our paper, we suspect that the well-posedness of equation (9) holds. We appreciate the reviewer for highlighting this interesting issue.\n\n[2] Stroock, D. W. and Varadhan, S. R. S.. Diffusion processes with continuous coefficients\n\n_How do DFMs compare to SGMs and deterministic FMs in practice, say in generalization, speed, difficulty of implementation, robustness w.r.t. hyperparameters, ...?_\n\nIn Section 3.4 of [3], the authors include some results for unconditional image generation, comparing the proposed interpolant flow model with diffusion methods on CIFAR10 and the ImageNet dataset. Their models emit likelihoods, measured in bits per dim, that are competitive with state-of-the-arts diffusion models on both datasets. Similarly, Frechet Inception Distance scores are proximal to those from diffusions, though slightly behind the best results. Finally, in [1, Section 7], the authors conducted experiments indicating that stochastic interpolants generally outperform their deterministic counterparts.\n\n[1] Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. Stochastic Interpolants: A Unifying Framework for Flows and Diffusions. \n\n[3] Michael S. Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic interpolants. \t\n\t\t\t\t\n_Are there settings where it makes sense to want to choose something else than 𝜋 being the independent coupling and the bridge being Brownian bridge?_\n\nWe thank the reviewer for raising this interesting question. Our primary motivation for choosing the independent coupling in the early-stopping regime and the Brownian bridge was simplicity, as analyzing non-independent couplings or non-Brownian bridges would have been significantly more complex. Nonetheless, we believe that, in the early stopping regime, non-independent couplings could be considered as well at the cost of additional technical complexity. Moreover, in practice, the independent coupling and the Brownian bridge are one of the most common choices due to their ease of implementation. However, the independent coupling is not always the most optimal choice, and there have been works highlighting the benefits of using other couplings, such as [4]. Additionally, we can mention Rectified Flows [5] and Diffusion Schrödinger Bridge Matching (DSBM) [6], which essentially involve iterating FM or DFM and allow for defining a sequence of coupling distributions expected to be more efficient than the independent coupling.\n\n[4] A. Tong et al. Improving and generalizing flow-based generative models with minibatch optimal transport.\n\n[5] X. Liu, C. Gong, and Q. Liu. Flow straight and fast: learning to generate and transfer data with rectified flow.\n\n[6] Y. Shi, V. De Bortoli, A. Campbell A. Doucet. Diffusion Schrödinger Bridge Matching \n\nWeaknesses:\n\n_Two things remained unclear to me after reading section 2; see \"Questions\"._\n\nSee \"Questions\".\n\n_The discussion of the previous work on SGMs seems a bit difficult to compare, since this paper is about DFMs, which (if I understand) are a different thing although similar._\n\nDue to the connection between SGMs and DFMs we believed that a comparison with studies on SGMs would be of interest to our readers. However, if the reviewers and the AC find this comparison irrelevant to our work, we are willing to shorten this section of the paper accordingly.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "VVWv3IE9eT", "reviewer": "Reviewer_83K8", "summary": "This paper gives theoretical guarantees for Diffusion Flow Matching similar in spirit to existing results for Denoising Diffusion Probabilistic Models and Probability Flow ODE. This work requires weaker assumptions than prior work, replacing the Lipschitzness of the score function with a relative Fisher information condition.", "questions": "- The authors claim to tackle \"all sources of error\" in the abstract, though I would argue the statistical convergence rates remain open.\n- What is the primary obstacle in improving the dimension dependence? Do you conjecture that it should be the same for DDPM?\n\nSome typos I managed to catch and write down:\n- L28: \"are important... \" or \"were an important\" \n- L34: I don't know what the \":\" is there\n- L63: \"In the case where...\" maybe?\n- L73: Reference to LWYql23 --- might be worth fixing the authors name in the bibliography\n- L77: \"... an approximate sample ...\"\n- L144: \"... a solution of the ...\" \n- L164: \"... is therefore the solution ...\" [the subsequent page has a few more instances of missing articles that are worth fixing]\n- L170: \"through\" (not trough)\n- L173: Fix the [Kry] citation (no year)\n-  L190: \"...gives an ideal...\"", "limitations": "N/A", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "This paper weakens this required assumptions of prior work, and is the first to develop guarantees for diffusion flow matching, instead of, for example, an analysis of DDPM or probability flow ODE. Unlike these other works, the authors further exploit properties of the heat kernel that appears in the estimation procedure. Their results also easily extend to the early-stopping setting.", "weaknesses": "One could make the argument that the present ideas are not as novel as its predecessors (e.g., Conforti et al. (2023) for the finite-fisher information criteria, or the line of works similar to Chen et al. (2023) in general). Moreover, the dimension-dependence in these results is quite poor (though the authors do mention this as an avenue for future work).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Questions:\n\n_The authors claim to tackle \"all sources of error\" in the abstract, though I would argue the statistical convergence rates remain open._\n\nIn our defense, we thought we have already acknowledged the lack of analysis on statistical convergence rates. In the conclusion, we suggest that “it would be interesting to complement our analysis with a statistical analysis of DFM.” Furthermore, while this is not mentioned in the abstract, we clarify in both the introduction and conclusion that \"all sources of error\" refer to the “drift-approximation error and the time-discretization error.” It is also important to note that, with the exception of [7] none of the previous works on FMs address the issue of statistical convergence rates. However, given the reviewer’s comment, it seems we have not been clear enough. Therefore, we will specify in the introduction that our work does not cover the statistical error of DFM.\n\n[7] Yuan Gao, Jian Huang, Yuling Jiao, and Shurong Zheng. Convergence of continuous normalizing flows for learning probability distributions. \n\n_What is the primary obstacle in improving the dimension dependence? Do you conjecture that it should be the same for DDPM?_\n\nThe primary obstacle stems from the challenge of bounding the reciprocal characteristic associated with the mimicking drift, as detailed in our response to the “Weaknesses” section. However, it is important to highlight that our bound depends polynomially on the dimension, unlike previous bounds (see [3], [8])  which depend exponentially on it. Consequently, our dimension-dependence is significantly more favorable. Nonetheless, we suspect that improvements could be made at the cost of additional technical complexity and of more sophisticated choices of discretization schemes. Therefore, we mention this as a potential avenue for future work in the conclusion.\n\n[3] Michael S. Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic interpolants. \n\n[8] Joe Benton, George Deligiannidis, and Arnaud Doucet. Error bounds for flow matching methods. \n\t\t\t\t\nWeaknesses:\n\n_One could make the argument that the present ideas are not as novel as its predecessors (e.g., Conforti et al. (2023) for the finite-fisher information criteria, or the line of works similar to Chen et al. (2023) in general). Moreover, the dimension-dependence in these results is quite poor (though the authors do mention this as an avenue for future work)._\n\nWe believe that our approach contains fresh ideas that cannot be found in previous works dealing with SGMs, and is entirely distinct from those of [9] and [10]. \n\nThe sole resemblance to [10] lies in the application of the Girsanov theorem, which however is not the distinguishing feature of our work, but simply a starting point of our proof. In addition, this step is not a contribution of [10] and is now standard for analyzing SGMs (see e.g., [11,12]). \n\nRegarding [9], one of its main contributions is to estimate relative entropy on path space by interpreting (a modified version of) the drift of the backward process as the adjoint process in a stochastic control problem. In the context of this work, the connection with Markovian stochastic control is broken essentially because of the non-Markovian nature of stochastic interpolants. Our idea for this work was to partially restore it by using the Markovian projection of the interpolant as a reference measure. Then, the main innovation of the present article consists in being able to control the $L^2$ norm of the adjoint process in the Pontryagin system associated with the Markovian projection of the interpolant. We do so by introducing a novel quantity in the generative model literature (see [13]), namely the so-called “reciprocal characteristic” of a Markov process, which may be viewed as some sort of mean acceleration field. The main efforts in our proofs are directed towards bounding the $L^2$ norm of the reciprocal characteristic whose representation in terms of either conditional moments or higher-order logarithmic derivatives of conditional densities is quite intricate, see page 26. Trying to bound each of these terms separately requires strong assumptions on the initial distributions and couplings leading to suboptimal results. However, using integration by parts both in time and space and a double change of measure argument, and profiting from symmetry properties of the heat kernel, we managed to bound these terms under assumptions comparable to the minimal ones required in the analysis of SGMs. Note that the analysis of the reciprocal characteristic is not required for SGMs (it is always 0) and that controlling it also requires new tricks and ideas, since its representation contains up to three logarithmic derivatives of conditional distributions, whereas the analysis of SGMs requires at most two such derivatives to be analyzed. \n\nBased on the reviewer’s feedback, it appears that we haven’t sufficiently highlighted the novelty of our procedure compared to existing studies. We will make it a priority to clarify and emphasize this aspect more clearly in “the proposed methodology”, starting with adding part of the previous paragraphs.\n\n\t\t\t\t\t\n[9] Giovanni Conforti, Alain Durmus, and Marta Gentiloni Silveri. Score diffusion models without early stopping: finite fisher information is all you need. \n\n[10] Hongrui Chen, Holden Lee, and Jianfeng Lu. Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions.\n\n[11] V. D. Bortoli, Convergence of denoising diffusion models under the manifold hypothesis. \n\n[12] V. De Bortoli, J. Thornton, J. Heng, and A. Doucet, Diffusion schrödinger bridge with applications to score-based generative modeling.\n\n[13] A. Krener, Reciprocal diffusions in flat space.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "imfVz5hAdd", "reviewer": "Reviewer_igTf", "summary": "This paper proposes theoretical guarantees for diffusion flow matching, which is of the most popular generative models. Specifically, it provides an upper bound of the KL divergence between the target distribution and the one learned by the DFM. This work promotes the theoretical understanding of the discretized flow matching model.", "questions": "1. If you in addition assume the velocity field is Lipschitz, will you be able to relax assumption H1 to have finite second moments?", "limitations": "Yes.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is clearly written, and the proofs seem to be sound.\n\n2. The authors consider the early-stopping case and remove the Lipschitz assumption of the velocity field.", "weaknesses": "1. Assumption H1 is quite strong that requires both the source and target distributions to have finite 8-th moments.\n\n2. The procedure is standard compared to some theoretical work on SGM. For example, the authors assume an L2-drift-approximation error and use Girsanov theorem for bounding the KL divergence.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Questions:\n\n_If you in addition assume the velocity field is Lipschitz, will you be able to relax assumption H1 to have finite second moments?_\n\nWe thank the reviewer for highlighting this issue. Indeed, for a Lipschitz velocity field, assumption H1 can be relaxed to only require finite second moments. This can be achieved by using the Lipschitz condition to bound the second term in equation (23) and employing the linear representation in equation (5) of the stochastic interpolant. \nHowever, assuming that the velocity field is Lipschitz would come at a heavy price on the data distribution and the coupling. This is certainly not true under the assumptions of our work regardless of whether or not we are in the early stopping regime. One would need to add stronger assumptions.\n\nWeaknesses:\n\n_Assumption H1 is quite strong that requires both the source and target distributions to have finite 8-th moments._\n\nWe consider assumption H1 to be relatively mild, as in most applications, data distributions are compactly supported. Moreover, our paper presents the first convergence analysis for diffusion-type FMs. In contrast, early studies on SGMs required much stronger assumptions on data distributions, beyond the Lipschitz continuity of the score, as seen in [11], [12], and [14]. Furthermore, many of these studies did not achieve convergence bounds with polynomial complexity. While recent convergence guarantees for SGMs work under the finiteness of the second-order moment in the early stopping regime, it should be noted that analyzing DFMs is more challenging, as, contrary to SGMs, DFMs permit to interpolate between any two probability distributions in finite time. We appreciate the reviewer's insights and acknowledge that there is potential for improvements, albeit at the expense of additional technical complexity.\n\n[11] V. D. Bortoli, Convergence of denoising diffusion models under the manifold hypothesis. \n\n[12] V. De Bortoli, J. Thornton, J. Heng, and A. Doucet, Diffusion schrödinger bridge with applications to score-based generative modeling.\n\n[14] A. Block, Y. Mroueh, and A. Rakhlin, Generative modeling with denoising auto-encoders and langevin sampling.\n\t\t\t\n_The procedure is standard compared to some theoretical work on SGM. For example, the authors assume an L2-drift-approximation error and use Girsanov theorem for bounding the KL divergence._\n\nConcerning the assumption on the L2-drift-approximation error, we believe that analyzing the convergence of the proposed DFM model under such a standard assumption is a strength of our work, not a weakness. This demonstrates that our bound is applicable under reasonable conditions.\n\nAdditionally, we assert that our procedure is fundamentally different from those proposed in previous works. While the application of the Girsanov theorem is now standard in the literature (see [9,10,11,12]), it is not a distinguishing feature of our approach, but rather a starting point. Beyond that, our method is entirely distinct and is not a straightforward consequence of the Girsanov theorem and assumption H3. The main innovation of the present article consists in using the Markovian projection of the interpolant as a reference measure and being able to control the $L^2$ norm of the adjoint process in the Pontryagin system associated with it. We do so by introducing a novel quantity in the generative model literature (see [13]), namely the so-called “reciprocal characteristic” of a Markov process, which may be viewed as some sort of mean acceleration field. The main efforts in our proofs are directed towards bounding the $L^2$ norm of the reciprocal characteristic whose representation in terms of either conditional moments or higher-order logarithmic derivatives of conditional densities is quite intricate, see page 26. Trying to bound each of these terms separately requires strong assumptions on the initial distributions and couplings leading to suboptimal results. However, using integration by parts both in time and space, and profiting from symmetry properties of the heat kernel, we managed to bound these terms under assumptions comparable to the minimal ones required in the analysis of SGMs. Note that the analysis of the reciprocal characteristic is not required for SGMs (it is always 0) and that controlling it also requires new tricks and ideas, since its representation contains up to three logarithmic derivatives of conditional distributions, whereas the analysis of SGMs requires at most two such derivatives to be analyzed, or, equivalently, to bound conditional covariances. \n\nBased on the reviewer’s feedback, it appears that we haven’t sufficiently highlighted the novelty of our procedure compared to existing studies. We will make it a priority to clarify and emphasize this aspect more clearly in “the proposed methodology”, starting with adding part of the previous paragraphs.\n\n\t\t\t\t\t\n[9] Giovanni Conforti, Alain Durmus, and Marta Gentiloni Silveri. Score diffusion models without early stopping: finite fisher information is all you need. \n\n[10] Hongrui Chen, Holden Lee, and Jianfeng Lu. Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. \n\n[11] V. D. Bortoli, Convergence of denoising diffusion models under the manifold hypothesis. \n\n[12] V. De Bortoli, J. Thornton, J. Heng, and A. Doucet, Diffusion schrödinger bridge with applications to score-based generative modeling.\n\n[13] A. Krener, Reciprocal diffusions in flat space.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:
We sincerely thank the reviewers for their positive evaluation of our work and their valuable feedback. Their constructive comments will undoubtedly help enhance our original work, and we are committed to incorporating the suggested modifications. The reviewers acknowledge that our work advances the field of diffusion flow matching models by providing the first convergence guarantees, removing the Lipschitz assumption on the velocity field, and addressing the early-stopping regime. While the novelty of our results is recognized, two reviewers observed some similarities in our methodology with previous studies. However, the use of standard tools, such as the decomposition of the Kullback-Leibler Divergence based on Girsanov theorem, is not the distinguishing feature of our work, which is rather the strategy used to bound the reciprocal characteristic associated with the Markovian projection. We will make every effort to clearly differentiate our procedure from existing research and highlight its originality. Below, we address each reviewer’s comments individually. We will be happy to answer any additional questions during the author-reviewer discussion period.

Meta-review: Most of the reviewers think the paper is interesting.  There seems to be something new in the proof.  I recommend an acceptance.
Decision: Accept (poster)