Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates
Authors: Connor Mooney
Zhongjian Wang
Jack Xin
Yifeng Yu
Venue: ICLR 2025 Poster (2025)
Keywords: Score based generative models
Lipschitz estimates
convergence analysis
well-posedness
singularity
Primary area: generative models

Abstract: We establish global well-posedness and convergence of the score-based generative models (SGM) under minimal general assumptions of initial data for score estimation. For the smooth case, we start from a Lipschitz bound of the score function with optimal time length. The optimality is validated by an example whose Lipschitz constant of scores is bounded at initial but blows up in finite time. This necessitates the separation of time scales in conventional bounds for non-log-concave distributions. In contrast, our follow up analysis only relies on a local Lipschitz condition and is valid globally in time. This leads to the convergence of numerical scheme without time separation. For the non-smooth case, we show that the optimal Lipschitz bound is $O(1/t)$ in the point-wise sense for distributions supported on a compact, smooth and low-dimensional manifold with boundary.

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GLOBAL WELL-POSEDNESS AND CONVERGENCE ANALYSIS OF SCORE-BASED GENERATIVE MODELS VIA SHARP LIPSCHITZ ESTIMATES
Connor Mooney
Zhongjian Wang"
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA mooneycr@uci.com
Division of Mathematical Sciences, Nanyang Technological University
21 Nanyang Link, 637371, Singapore zhong jia.ggangeeeeeeeeeemmmmmmm nnu...u edu. sg
Jack Xin
Yifeng Yu
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA jack. xi wwwguciiiee.. edu
Department of Mathematics. University of California at Irvine, Irvine, CA 92697, USA finggeyuuci.eem
ABSTRACT
We establish global well-posedness and convergence of the score-based genera- tive models (SGM) under general assumptions of initial data for score estimation. For the smooth case, we start from Lipschitz bound of the score function with optimal time length. The optimality is validated by an example whose Lipschitz constant of scores is bounded at initial but blows up in finite time. This neces- sitates the separation of time scales conventional bounds for non-log-concave distributions. In contrast, our follow up analysis only relies a local Lipschitz condition and is valid globally in time. This leads to the convergence of numerical scheme without time separation. For the non-smooth case, we show that the opti- mal Lipschitz bound O(1/t) in the point-wise sense for distributions supported compact, smooth and dow-dimessional manifold with boundary.
INTRODUCTION
Diffusion models (DM) have become the state-of-the-ee tools lately in generative Al Song & Ermon
(201 Song et al. 202I); Dhariwal & Nichol (2021) such as image synthesis Ho et al. (2022); Gao et al. 2023). DMs first evolve data samples with stochastic differential equation (SDE) to gradually inject Gaussian noise until a Gaussian distribution is reached. Then it approximates the drift in the associated backward (time-reversed) SDE and generate a data sample from Gaussian noise. The drift of the backward SDE contains the gradient of the forward logarithmic density (score) that estimated by solving a matching problem with deep neural network training. The reversibility concept of SDEs dated back to Kolmogorov's 's work Kolmogorov (1937) in 1937, and the general score formula was derived by Anderson Anderson (1982) 1982.
Theoretical study on the convergence of DM generated distribution to the target (data) distribution typically assumes that the data distribution admits a density with respect to Lebesgue measure Lee et al. (2022) among others. By also imposing that the score of the data distribution is Lipschitz continuous, the score function of the forward process (the drift in the backward process) well- behaved (not exploding) as the backward time tends to zero when the desired target sample is to be generated. However, this is not always observed in practice and experimentally the score can blow up Kim et al. (2022). In particular, the explosion occurs at generation if the data distribution satisfies the manifold hypothesis (MH) Tenenbaum et al.] (2000); Goodfellow et al. (2016) which verified for image data in Brown et al. (2023). Under MH, Pidstrigach| (2022) showed that the limit of the continuous backward process with approximate score is well-defined and that the sample
Corresponding author.

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distribution shares the same support the target distribution under the integrability conditions the error of score matching. Also under MH, Bortoli (2022) found quantitative bounds on the 1- Wasserstein distance between compact target (data) distribution and the generative distribution of DM by allowing the score function to explode as backward time approaches zero.
Both of the references (Pidstrigach 2022) and Bortoli (2022). among others (Lee et al. (2022) for Langevin MC, Huang et al. (2024) for ODE flows, Chang et al. (0024) for Föllmer flows), require
(locally) Lipschitz estimate for the score function to ensure the cell-posedness of the backward SDEs and the approximation bound of the score matching and sampling process.
The goal of this paper is provide sharp estimates that 1) confirm/improve the score assumptions of the existing convergence theory, 2) give insight for the duration of the forward process so that the backward process well-defined, and 3) justify practical implementation of the backward process (e.g. early stopping strategies or truncation Kim et al. (2022).
Related work We are aware of the convergence bound of discrete schemes for backward processes in Chen et al. (2023). Our convergence bound takes the KI. chain inequality (Proposition C.3 Chen et (2023)) as the building block. While equipped with sharp (local) Lipschitz bounds in the paper, achieved polynomial complexity of sampling in the general smooth Po setting without separated regimes of schedule. We are also aware of Bortoli (2022) which provides convergence bound in Wasserstein distance under singular Po setting, supported to the potential singular behaviour of the score, early stopping schedules are employed Kim et al.) 20222) Additional related work and comparison are discussed in Remark 333 for the Lipschitz bound and in Remark 4.7 for the convergence and complexity bound. Our paper provides sharp Lipschitz bounds of the singularities and therefore insights for the choices of schedules and loss normalization between discretization points. In addition, the Lipschitz bounds hold generally for models sharing the same forward process OU, for example, the probability flow ODE (Equation (13) in Song et al. ).
compact manifold. Due
The main contributions of this paper are:
Realistic or sharp point-wise gradient and Hessian estimates of the score potential function log p from commonly hypothesized data distributions. See Theorem (3.5]
The first sharp example demonstrating the loss of Lipschtiz bound of the score function as time gets large even with nice initial data. See Theorem 3.4
Well-posednesss and convergence of the backward diffusion process up to time zero (the generation time) in the smooth setting without separated regime of discretization. See Theorem 4.4 and Theorem 4.6
Characterization of the score (and its derivatives) in the setting of manifold hypothesis. See Theorem 3.8
The rest of the paper organized as follows. In Section 2 we first introduce settings of the diffusion model and discretization schemes of the backward process. Later, we present the transformation that relates the Fokker Planck equation with unbounded coefficients (density equation of forward pro- cess) to the non-linear Hamilton Jacobi equation and heat equation, which serves the foundation of the analysis. The main theoretical results, Hessian estimate of score potential function log y, are listed in Section 3 Based on these estimates,
establish well well-posedness of the continuous backward process and convergence bound of discretization in Section Conclusions are stated in Section 5
The details of the proofs are in the Appendix.
2 PRELIMINARIES
2.1 BACKGROUND AND SETTING THE STAGE
A large class of generative diffusion models can be analyzed under the SDE framework [Song et al. 22021) It consists of two processes: forward and backward. The forward process, which relates to training, is an Orastein-Ullenbeck (OU) process in R as follows:
dxX, = 2 x,dt + dWe, for E [0,T]
(1)
2

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for the normal setting and > 0 for the early-stopping setting. Then Published as a conference paper at ICLR 2025
where Wa is standard Brownian motion, T is the final time such that the distribution of XT approx- imates a normal distribution in IR", namely (0, The initial distribution Xy follows a target (data) distribution in I" during the generative task, denoted po- The backward process, which relates to generation of new data, defined as an 'inversion' of forward process More precisely, with time reversal t' T 1, 1,
dxp.' = ( Vog -e xx)x)) dt' dW,, for [0,
(2)
where Wr is standard Brownian motion (not necessarily being the same as W;) and the initial distribution No follows (0, ). The term V log is introduced in Eq. (2) such that the marginal distributions of the forward and backward processes are identical Anderson (1982).
To be specific, let := p(t, x) denote the probability distribution function of the forward process T.. which solves the Fokker Planck equation with Cauchy data po, namely
[.pp = (V (zp) + A)p p(<,, = -(ozr)))o((r po()).
We also denote Pt (Qr correspondingly) as the marginal distribution of X, in ① (Xr (Xx in 2 Given initial distribution for (2) Qo ~ Pr, then Anderson (1822): Vt. Q: = Pr Especially, QT Po so data Po can be generated by solving 2
In practice, since no closed form expression of po is known, the p in (3) is not analytically available. Thus V log is approximated by neural network st .). where & denotes latent variables of neural network and is omitted for simplicity of notation. The approximation obtained by training the neural network with L2 score estimation loss, vt
[0, 7).
Ez-P, [sso(t, x) V log p(t, x)).
In the analysis, we assume an e bounds for this estimation, see Assumption 2.1
Given the approximation of score sg. we employ the exponential scheme Zhang Chen (2023) with initial distribution AO0, In). More precisely, let 5 = to S f < ... S tN = T be the
discretization points.
with ty = T - IN ki the process in the discrete scheme is as follows:
dēu (āxr (T (2, fee + duw l' (etb
k k. .....N-- N
which admits an explicit solution, with μr (O),
Py = + Wag + + 4 1)sa 2c. + V e + Luke
Due to the limited knowledge of as well as the regularity of V logp, we restrict ourselves to uniform discretization points. Detailed selection stated in the convergence theorems.
We assume the following bound of score approximation at the discretization points, Assumption 2.1. Let tx be the discretization point of the scheme (4)
1 -
tx-) I|Voggp(t, - x)(tt,x)11 S E
T
2.2 FOUNDATIONAL IDEAS BASED ON NON-LINEAR HAMILTON JACOBI EQUATION
The foundation of our analysis is investigating the behaviour of log p as the solution of non-linear Hamilton Jacobi equation (HJE), which is well known to experts. For reader's convenience, present here.
We consider the score potential function
|x|2
(1.,x) = log p(l, x))
2
Herre we only consider the transform when the distribution of forward process Pt is absolutely continuous with respect I Lehesgue measure. The transform and our analysis are valid for any >> 0 in the general case and up to 0 when Po is smooth.

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whose spatial gradient becomes the drift (score) in the backward (reverse time denoising and gener-
ation) process of the diffusion model. The q function satisfies the following PDE:
{ [..q- A4 - (x Vy + Va(=)=
(5)
(9(0, g(x)
where g(z) = - -oggpo(r) /z/12/2, which is the non-Gaussian part of the likelihood function.
To simplify (5. we make a two step change of variables time. First, let g(t, x) - g(t, e//2x),
then solves:
θ.q - 0.g a Vg(l, = - (Aq (Aq (v
Then we consider ((t,,) = 7(- log(1 - 1), x), then y solves:
{ 0.-- - tE 10,
(6)
((0,x)) = go
Remark 2.2. By a direct calculation
x(t,2) (- 1ogg11 -t), v1 or equivalently, q(t,z) = =q(1 c- c e -/222)).
(7)
Furthermore,
= e 2122)) and, == c c- 0-122)))
Lastly, we also define (t,, = e-((t.)) which satisfies
S A (0, x R (5(0, x) h(x) = g(())
(8)
The solution of (8) is given by p(t,,) (2x1) Jar 21 e dy-
To derive reasonable point-wise estimates of gradients and Hessian of the score function 2(4, that does not involve 1/t, we will need the following assumption in relevant results. This assumption also ensures the above integration is well-defined for [0, 1], equivalently the well coll-poses of Fokker Planck equation (3) for E [0, oc).
Assumption 2.3. The tail distribution is upper bounded by some Gaussian distribution, i.e,
1
Pog(p(z)) - log Pa (0) S 01 $11-21 2
a2)
for constants 02 1 and E TR Without loss of generality we assume 2 0.
Recalling definition of g, is equivalent
g(x) - (0) 2
02
r 01
2
Note that Assumption 2.3 implies that the second order moment of the process bounded, i.c.,
Exp ||| M2
(10)
Technically speaking, the g(0) could be absorbed into in g We put there just to track possible dependence on the dimension n. Similarly, we adopt the following technical assumptions in the relevant results to provide more flexibility track such dependence.
Assumption 2.4. There exists xo. a2 E [0, 1), a1 R such that
02
g(z) g(xo) - ( (a ro) 2
2 r rol -1 VI R
In particular, if attains minimum at some point øg. then the Assumption Conummption..4ooolllll holds with 02 = 01 O. Also, if Assumption 2.4 holds, then Assumption 2.3 holds by adjusting the corresponding 02 and an E IR depending on g()) and Tn. and vice versa. The notation (a.) is abused for simplicity of subsequent derivations without affecting estimation for dimension dependency.
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General notations
Throughout this paper, for matrix A, the spectral norm
||
max (vER": [w/=1)
|Au| the largest eigenvalue of v AAA..
(11)
In partieular, for a map " R
| S L
F(x) F(y) S L yy-
We also adopt the following notation when comparing two symmetric (Hessian) matrices,
Bif A is somitiietiive definite.
So for any symmetric matrix A. ||A12 a + -ohn 3 A alu. For a map u R" - IR" D
denotes the Hessian matrix of the map.
SHARP HESSIAN BOUND OF SCORE POTENTIAL FUNCTION
The fundamental question, which is directly related the well-posedness and convergence rate of the diffusion model [Bortoli (2022); [ee et al. (2022), is whether for any T > 0, there exists constant Cr that depends only on T and the initial data such that
sup |1,2 [077] xR
CT
The short time existence of uniform Hessian bound was known in previous literature (see/Chen Chen et al. 22033;; Mikulincer & Shenfeld (2024) for instance) when || log is bounded From both mathematical and application perspective, natural question is whether could be extended to time. In Section 3.1 we provide the first example that shows the short time existence is optimal in sense lasting time. Precisely speaking, in the proof of Theorem 3.4, we construct an initial distribution po such that the Hessian of log p loses global bound right the limiting time. Inspired by the counter-example, alternatively in Section 3.2 we provide a locally Lipschitz estimate that lasts for & [0, co). For the non-smooth case, in Section 3.3 of log and its derivatives.
we characterize the singular behaviour
3.1 HESSIAN ESTIMATE OF SCORE POTENTIAL FUNCTION FOR FINITE TIME
The following short-time uniform Hessian, or similar formulations, have been obtained in some previous works. See Remark 3.3 blow. The primary goal of this section is demonstrate that the associated time threshold is sharp (Theorem 3.4).
Theorem 3.1. Let Mg be a nonnegative number. g E (R
(1) Df = Mn then
MM for all (t, E (0) x R=.
(2) (2) z -M for any T E [o. [o,- log(11 Mo )). we have
Mo
D2g(t, 2 _
e Mole' 11 In for all (t,z) E (0, x R".
Note that if Mu < 1, then T o0).
The proof is in Section A. an immediate corollary, we have that
Corollary 3.2. Given data distribution Po CC²(RR)) follows -L, I = supzER" D2 log log 1 LgI. Then we have finite time uniform bound of the Hessian: for any E [0, 0lllg1 10+1 rot1)).
sup || pog 21, S C.. Re
Thhe assumption is equivalent lo E (R").
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where
Lo +1
C = max
1 (Lo + 1)(e' - 1, e eL1 -L1-1) 11111)
(12)
Furthermore, if - log po( is fonceint (Lo S 0), the estimate bound is global,
0 D
(e + (1- for all (t, E (0) x R
Remark 3.3. The convex case has been also discussed in Lee al. (202)) y and leads single modal distribution. Similar finite bound was also derived in Lemma C.9 in Chen et al. (2023). which follows directly from the representation formula and the generalized Poincaré inequality for log- concave probability measures. We are also aware of bounds similar to Theorem 2) obtained in Brownian transport map setting Mikulincer Shenfeld (2024) based on the representation formula and the Brascamp-Lice Lieb inequality that is related to the generalized Poincaré inequality. In this section, we will present a new proof based on PDE (partial differential equation) method via using the convex envelope as a barrier. Our approach is more robust, which does not rely on the represen- tation formula and could be easily adjusted to more general situations. Results of a similar spirit were also obtained in Kim & Milman (2012), where generalization of Caffarelli's 's contraction the- orem Caffarelli (2000) is proven, also using a parabolic maximum principle, which is different from method. We will refer the reader as well to Mikulincer & Shenfeld (202)) and also Conforti (2024), where related questions are investigated from a more probabilistic viewpoint. In addition, we would like to point out that beyond the spatially global Hessian bound in the mentioned refer- ences (including Mikulincer & Shenfeld (2024) that will degenerate in finite time, we also provide local Hessian bound that holds for any given finite interval, see Theorem 3.5
Given the crucial role of Hessian bound in estimating convergence rate of diffusion model, an im- portant remaining question was whether the lasting time given in Theorem 3.1 is optimal. The result below is our main contribution in this aspect, which shows that the temporal bound - log (1 Mr
in the statements of Theorem 3.1 and Corollary 3.2 is sharp.
Theorem 3.4 (Loss of Uniform Hessian Bound). There exists smooth nonnegative g satisfying assumptions in Theorem 3.7 and Corollary ( = M1 = 2) such that the corresponding q(t. x) satisfies
sup re
1|D2 (log 2, x)
sup =ER"
[102 00.
Note that the number 2 can be changed to any given time by re-scaling the function q(>2t, Xx). The detail of construction in Section C.2
3.2 LOCAL ESTIMATE
The following theorem provides point-wise estimates of the score function, which can be quite useful in dealing with more general situations. Technically speaking, g(øo) and Dg(zo) can be absorbed into other parameters. Here we choose to display them to track the dependence of relevant parameters on the dimension n.
Theorem 3.5. Suppose that p p(l,,)) x)) is the solution to heat equation for U E R". Fix To E R".
8 Let |o]1 max(1e], 1}
(i) Given Assumption 2.3 and Vg(x)] S B1|z rol + for 31, B2
0. Then for all (t,
[0, 1] x I
3,1
Vg(t,))| 5
max Cn. C2, 2ol1) B2-
(13)
V1 -1-0 0:2
Here the two constants Cr 2
(n (n+
3og
2(1+1))) VT-02
+ 4n logn + 01 + 1+
222
and Cp a22
B1
3/Bi +
6az
V1-a2
(ii) Assume [DD)g())l22 S I. and Assumption 2.4 Then for all (1, E [0,1] x RR. [())(t,))]| S 101.2 + T,
max Ca (Ci,az) To Vo(zo)l1)²)
(14)
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48I.² 2L
max C. (CL,az) zo Vg(2o))18))
(15)
vt(1 a22
Here the two constants C 2 (n + +3+
2(1112
+ 4n 4n + ai + and Cr.0z =3
vI-oz
VI-as
The proof is in Section
C33 A simpler case with bounded Vg is also discussed. See Remark C.2 for
L2 another way bound [ V q(,, .)] by replacing Se in (15) by O(1..)).
As an immediate corollary, we have
Corollary 3.6. Assume || g()))l2 I L. Suppose that Assumption 2.4 holds and there exists Co > O such that, o1 1 Con and [Vg(v)) s Cyn. Then
[1)" S CL² (nlogn + L| - 20 Vg((x)) 10. x)] S CI2² (nlogn) + LVT[r zo Vg(2a))
Here C' is constant independent of T. and n.
Note that the further assumptions scale relates to normalization in n dimension.
Remark 3.7. Owing to 35 the proof of Theorem 3.5 under the assumption of corollary 3.6 we have for all E N
(p(t,,))==(()1")) < O (L* l)
This demonstrates that, if we only care about expectations of powers of D²q(t, or 0, o, ||D2qqt, 12 behaves like O (o''noog) and lå, (q(t, x)) behaves like O logggn Note that the point To üself plays no role computing the expectation that is translation invariant the variable.
3.3 COMPACTLY SUPPORTED DATA DISTRIBUTIONS
In this section, we look at the situation where the data distribution po is a positive measure with sumpartssppport compact support which is typical situation in image generation [Bottll 2002). Due the manifold hypothesis, the support is typically low dimension set. In this situation, what is important is the asymptotic estimate as + 0. Assume supp(po) = Do c (0). The following are two known
standard estimates( Bortoll (022
[ +
(1) Vg(t,a)) <
(16)
M
(2) [t
The proof is simple, which will be presented in Section C.4 for reader's convenience. Some steps will be used later. The main challenge is whether the above bounds can be improved in order to derive better convergence rate, for instance, Theorem in Bortoli (2022).
We first demonstrate (11/t) bound in (1) above typical situation that can not be improved.
Fixing v, denote by Ts the weighted center mass: Tt Jpp ye -lz-w12 drolv) where for the rest of
the proof, we denote po(y) dy as d and Jo, e dvoly). For "regular" To. as O,
we expect the measure vl dma(z) will concentrate on (v E Dol by xf d(x, Do)). Thus,
lim dv.,, [z) =0,
2000
where F. the convex hull of (y E Dol by - x| d(x,, Do)}. Then
Vel
Vg((,x)) b
and
lim V 2 Iz). t-

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So if & r (typical situation for low dimension set Du), then [D x)] O(1/t) Hence the 1/t blow up for the gradient bound is usually inevitable, which matches experimental observations Kim et al. 2022). Accordingly, in real applications, the denoising process might only be traced back to certain lo 0, which is corresponding to an initial condition similar to p(4o, 2)).
We now turn attention to the Hessian bound (11/t²) in (2). According to Theorem 3 in Bortoli (002)). if this bound is improved to O(1/t), a better convergence rate in the Wasserstein metric can be achieved. The following theorem establishes that, in typical scenarios, the Hessian bound is (1/t)) rather than O(1/(²), with the exception of a small set. Consequently, it might be reasonable in practice assume a Hessian bound of (L//t) when analyzing convergence rates. Quantifying how frequently this small set could impact the convergence rate remains a challenging problem due its complex topological structure in the case of nonconvex Dg.
II.
For simplicity and clarity, we assume that Du is a low-dimensional smooth manifold with boundary, though our results extend to manifolds with lower regularity. To illustrate the sharpness of our conclusion, we provide example in Example 3.9
Theorem 3.8. For S S. n, assume that Do C IR" a d-dimensional compact smooth manifold with boundary and To comparable to the uniform distribution Do. Then for almost everywhere x E R"
Cx || qlt x)122 S T.
for t E [0,
Here Cg is constant depending only on and Do. If Do convex, then the above holds for all E R
Proof is in Section C.5
We would like mention that the O(1/t) bound was also derived in Bortoli (2022) for the very special cases, for instance, when po follows uniform distribution product with normal distribution on a hypercube.
We will present a smooth non-convex Du that shows the result of Theorem 3.8 is optimal.
Example 3.9. Let Do C TR.2 be the domain obtained by removing a small square [0, [-1,1] from the big square 1- 2|2 and then mollifying the corners make it smooth. Here O = (0, The Y -shaped region
L = fr E IR22 there are more than one such that - y --= d(x, Do)
We also choose no Do dr, i.e., the uniform distribution on Do, where |Do] is the area of Do. We
have that, [t) 2 C₂/t² for x E L and t E (0,
Figure 1: In the above picture, Iz = {syz + (1 s)y2 : s E [0, 1]}
For reader's convenience, will verify the above when (0,0) for 0 > 1 in Section C.6 The other parts are left to interested readers as exercise.
4 WELL-POSEDNESS AND CONVERGENCE UNDER SHARP LIPSCHITZ BOUND
In this section we present the wose-posss of the backward SDE 2 and the convergence of dis-
In Proposition 4.1 we review a well-posedness condition of general SDE with additive noise where the drift term F is only lo- present the well-posedness of (2 in
cretization 4 based sharp Lipchitz bound of score in Section 3
cally Lipschitz continuous. Combining it with Theorem 3.5
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Due the limitation of global Hessian estimate, the convergence analysis is divided into the following two cases by Po- The first case in Theorem 4.4 enjoys better complexity with respect to dimension (N = (n log" x)) while has limitation in the final time T in the forward pro- cess. The second case in Theorem i. valid globally in I'' while achieving polynomial complexity (N = (n log" n)).
Theorem 4.3
Proposition 4.1. Given T O, suppose that F F x) C([0,
"²,,"" satisfies that F' is locally Lipschitz continuous in x variable, for any M > 0, there exists a constant LM such that
F((,x)) - F(1, v)1 S TM - yl for x. y E (0) and E [0. T)
and
|F(1, S C((a) +1). for (1,3) E [0, x R for a positive constant C. For any zo E R", the following SDE has a unique solution
(17)
dX, Ft,, dt) + dW., E [0, Xo To.
The proof is in Section C.7
Remark 4.2. For simplicity, uniform Lipschitz continuity of F is often assumed to ensure the long- term existence of solutions for ODEs and SODEs. The above Theorem says that local Lipschitz continuity of F plus the linear growth condition (17) would be sufficient, which is special case of more general results ( see Theorem 2.4 and Theorem 3.1 in Chapter IV of Ikeda & Watanabe (2014) for instance). For reader's convenience, presented the proof above for our special case.
Theorem 4.3. Let g(x) = log
be the non- non-Gaussian part of the log- density. We assume g(x)) E C (IR") satisfy the Assumption and (V C(|2r + 1) for a positive constant C. C Thhenfor any 0, the above (2 is well-posed.
Proof is in Section
In what follows we denote the distribution of the discrete backward process 4 Qr.
generation time T
Case I: po is (near) log-concave
Theorem 4.4. Assume the following global Hessian bound of po-
L1 3 D2 log Po(x) N LaIn-
Let qr be distribution generated by the uniform discretization (8 0) of the exponential integrator
scheme 44) with an approximated score satisfying the Assumption 2.1 We also assume the po has finite second order moment, namely M2 < in (TOp. For To and T log(1 1o+1)
CT? N
KL(PollQr) S (M2 - + +Te
(18)
where Cr defined in 12 depends on Lo. L1. T.
Lo < 0, namely pa log-concave,
TT²C²
-T KL(Po S (M2 + - Tee N
(19)
where C S1PPe(00.00) (e- (L (1-e-1) max (T,, xc.
Proof: We first apply Corollary 3.2 to attain global Hessian estimate in finite time. Then apply it to Proposition 4.1 for well-posedness and Proposition A.3 for convergence rate.
Remark 4.5. (i) A near linear complexity bound, N = O(n log" n), is then established by 19
under the log-cone log-concave distribution with T (log n).
(ii) Note all complexity bounds by Proposition A.3 requires T Ollog n) with second order moment M2 S This implies the optimal bound with Lipschitz of score L < oo, requires N = O(n log" n)).
(iti) Furthermore in the near log- log-concave case, we consider the regime with smallness of Lo = O Therefore maximal time of estimate in (18), turns - log(1 Ta+ O(log(n)). Then in 18 with T O(og(n)), the first term is bounded and CT defined
12 is independent with
dimension n. The complexity bound in such case is also (n log" n))
"Fuiivaeenlly log Po continuous differentiable, log C((z] + 1) for positive constant C.

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Case I:: General smooth Po
Theorem 4.6. Assume [D2g(<)) S L. Suppose that Assumption 2.4 holds and there exists Co > 0
such that, on S Con and [Vg(za)) S Cvn. Let qr be distribution generated by uniform discretiza- tion of the exponential integrator scheme (4) with approximated score salisfying Assumplion 2.7 We have,
- +Te CI.Tn(nlogn)'
KL(PallQr) & (M2 + + n
Proof: It is direct consequence of Theorem C.3 to estimate truncation error in Proposition A.1
In addition to the above cases, we also consider non-smooth Po supported on compact manifold. Restricted by the estimate in (16), we switch to the early stopping technique, namely > 0 in discretization. Due to the measure zero set (see in Section 3.3 the convergence bound is not yet optimal as shown in Section C.9
Remark 4.7. Bounds in Theorem and Theorem 4.6 consequences of our Lipschitz timate Theorem [.1 and Theorem 3.3 with Proposition A. frrm Chen et al. (2023). An important feature of these new bounds is their uniformity in time (up to T, the mixing time of the forward process). Though the complexity in Theorem 4.6 is O( log n. in dimension with T = O(lg n.), slightly higher than ( (ng n)²) in Theorem 2.5 phen et al. (2023), our assumption on time dis- cretization {t& he during the sampling process (4) can be relaxed to uniform discretization. Hence our theory requires no prior knowledge of the data distribution po and is more realistic.
We are also aware of two works which provide the linear in dimension (O(n)) complexity bounds. Benton et al. 2024) utilizes stochastic localization approach attain the complexity bound in the early stopping setting. A recent preprint Conforti et al. (2025) provides the linear bound (Theorem ofCoorrti el al. (2025)) in a setting similar to our Theorem 4. with analysis of relative score process. In contrast to work here, the approuches in Benton et al. (2024); Conforti el al. 20255) estimate the expected Lipschitz bound of the score under the backward process X in ( while our analysis is in the point-wise sense and hence applicable to the analysis of (approximated) estimates readily apply to complexity and convergence bounds in the Wasserstein metric. A sketch of proof is presented in the appendix, Section C.10 which will be expanded in future publication.
score acting the approximated backward process & in (4). Therefore
CONCLUSION
The uniform spatial Lipschitz bound for Vp(t,x),, often assumed in the literature for various pur- poses, does not generally hold for all times, even with nice initial data (Theorem 3.4). Instead, have established local-in-space, global-in-time Lipschitz bound (Theorem 335) under mild sumptions, which can be employed to prove well-posedness and polynomial convergence of the KL divergence (Theorem 4.6)). As another potential application, proof sketch for obtaining conver- gence rate under Wasserstein distance is also provided in Appendix C.10 under similar assumptions.
For data on manifolds or non-smooth settings, show that an O11(t) Lipschitz bound exists almost everywhere (Theorem 3.8). suggesting that this type of bound can be reasonably assumed when studying convergence rates the previous literature (for instance Theorem in Bortoli (2022))
Limitation Due to the limited knowledge of regularity factors of data distribution (e.g. optimal Lipschitz constants), our bound cannot provide implementable guidance on seeking the optimal schedule (which may require separation of temporal regimes). Also in the manifold case, due the complex geometries, as shown the non non-sh section, our theories cannot provide a justifiable guidance of early stopping time. We will investigate these issues in future study.
*in fact, the bound (On log" n) due Remark 4.5
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ACKNOWLEDGMENTS
CM was partly supported by Alfred P. Sloan Research Fellowship. NSF CAREER grant DMS- 2143668, Simons Fellowship. ZW was partly supported by NTU SUG-023162-00001, MOE AcRF Tier 1 Grant RG17/24. JX was partly supported by NSF grants DMS-2151235, DMS-2219904 DMS-2309520, a Qualcomm Gift Award. YY was partly supported by NSF grant DMS-2000191.
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J. Benton, V. De Bortoli, A. Doucet, and G. Deligiannidis. Nearly d-linear convergence bounds for diffusion models via stochastic localization. In The Tweifth Internatinnl Conference on Learning Representations, 2024. URL htthtts::: http://ppp/evvveeeeeeeee www.eeeeweeeeeeeeeeeeeeeeeeeeeee Etw.://eoe//ooornrnn
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B. Brown, A. Caterini, B. Ross, Cresswell, and G. Loaiza-Ganem. Verifying the union of mani- folds hypothesis for image data. ICLR, 2023.
L. Caffarelli. Monotonicity Properties of Optimal Transportation and the FKG and Related Inequal- ities. Communications in Mathematical Physics, 214:547-563,
2000. 2000.
Chang, Z. Ding, Jiao, R. Li, and J. Yang. Deep conditional generative learning: Model and error analysis. arXiv preprint arXiv:2402.011 46600,00046 2024.
H. Chen, H. Lee, and J. Lu. Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learn- ing. pp. 4763-4763, PMLR, 2023.
G. Conforti. Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges. Probability Theory and Related Fields, pp. 1-27, 2024.
G. Conforti, A. Durmus, and M. Gentiloni-Silveri, KL convergence guarantees for score diffusion models under minimal data assumptions. SIAM Journal Mathematics of Data Science, 7(1): 86-109, 2025.
P. Dhariwal and A. Nichol. Diffusion models beat GANs image synthesis. 34:8780-8794, 2021.
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D. Huang, J. Huang and Z. Lin. Convergence analysis of probability flow ODE for score-based generative models. arXiv preprint arXiv:2404.097300 2024.
N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes. Elsevier, 2014.
D. Kim, S. Shin, K. Song, W. Kang, and I. Moon. Soft truncation: A universal training technique of score-based diffusion model for high precision
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Y-H. Kim and Milman. A generalization of Caffarelli's contraction theorem via (reverse) heat flow. Mathematische Annalen, 354(3):827-86) 2012.

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A. Kolmogorov, Zur Umkchrbarkeit der statistischen Naturgesetze (Engl.: On the reversibility of the statistical laws of nature). Math. Ann., 113:766-772, 1937.
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In the appendix below, present detailed proof of theorems along with existing results used in the proof. Appendix [ lists the <ocoeveeeee theories of the diffusion model in Chen et al. (2023). which is applied when showing convergence bound. Appendix[8]onme conttains the theories in Alvarez et al. 1997) regarding the convexity result of second order differential equation
very general
kind. It is applied when showing the semi-convexity of the HJ equation, which is part of the global in space Hessian bound. Appendix coolleect all the proofs and Appendix D discusses the broader impact of the manuscript.
A CONVERGENCE THEORIES
In this section, we list numerical algorithms and convergence theories related to the numerical discretization of (2) They are due to Chen et al. (2023).
Convergence in distribution
The key ingredient of the convergence theory is the following result from the chain rule of KL divergence.
Proposition A.I (Prop C.3. of Chen et al. 2023 2.7 the exponential integrator scheme satisfies,
Given the score error estimation Assumption
N
KL(Pa|1Qr-a) X (L + +
(oggp(t, logp(tk; Tzn dt,
1=
where Yn is the Uniform Gaussian distribution
The first term (I) in the estimate measures the distance between an the measure of OU process its invariant measure. When the data has finite second order moment, it turns 0 as T - oc.
Proposition A.2 (Lem C.4 of Chen et (023)). With finite second order moment Epp [X| co, for T > 1,
KL(Pr, 7n) S (n Ma)e
The third term relates to local truncation error that depends on the regularity of the forward process. Then Proposition can be further extended if the global Hessian estimate is available.
Proposition A.3 (Theorem 2.1 of Chen et al. (023)). Given assumption of Proposition A. V Vlogre pt is I.-Lipschitz. For uniform discretization, the exponential integrator scheme satisfies,
222
kL(PollQT) S (M2 + m)e- - + + +
N
B SEMI-CONVEXITY OF SECOND ORDER DIFFERENTIAL EQUATION
Here list some important theories used construct the finite time log-convexity of the density of the forward process.
Given function w(t, .r,, its convex envelop w** (t, x) in is defined
w wt"(tx inf Σ Aww(t, 23)
A
Σ
2 0, R"
(20)
Lemma B.1 (Prop 7 of Alvarez al. (1997). Lemma of.
Strömberg (0100). Let be a solution
0,w + F(I,2, Vw, =0
(21)
The convex envelope w of w is a supersolution of 21 under the following assumptions,
1. elliptic in the sense F(t, x, A) > F(t, x. if A A.
2. (x, A) E RD x ST. + F(t, x. A A concave for all t and p. Here ST. the set of all
x n. positive definite matrices.
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3. w is coercive in the sense that
w(t, x)
lim 2--00
00,
(22)
[z]]
uniformly
C
PROOFS
In this section, we present proofs of the main theorems.
C.I PROOF OF THEOREM 3.1
For (1), it is equivalent to showing that
xt)(t,x) S M14, for all (t, [0, 1] R".
This actually follows immediate from 33 Here will present standard PDE approach that does not reply on the formula.
Let & be given unit vector. By taking derivatives of 6 we deduce that v A3c satisfies
a, Av + Vq- Ve - V S on (0, 1) R Thanks to the standard maximum principle of parabolic equation, we have that
v(l, xr) < sup v(r, 0) = sup des S M
xEi
=E32
The proof of (2) is more interesting. It is equivalent to showing that
Mo
D²
In.
(23)
1 Mot
Below we show PDE approach that is base on modification of arguments in Strömberg (2010) for obtaining semiconcavity of solutions to the general viscous Hamilton-Jacobi equations.
Fix 8 0. Note that by D2g = Moonn
My 2
g(x) - ((0)-Vy(()) x
|x|
Hence there exists a constant Cs such that
Mu
g(x) 2
+ 1x/2 Co.
Let and be positive numbers satisfying that
(() a tan(ac) Ms, Mo + 40,
Consider the following construction.
== n
(24)
where,
π << T
((t) = a tan(ac at),
(25)
20
(()) // 0(s)ds.
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We notice that Eq. (25) implies, (()) tan(ac) and 0' 82 a². Then satisfies the following equation:
1 Aw + 2 Vwl ((l) w..a
0 = 0.w F(I, Vw, V = 8,w0
2 |
(26)
2
Now consider the convex envelope (definition see (20 of wc wn and aim to apply Lemma B.1 to show that w** a supersolution E2) After direct validation of the first two condition of
Lemma B.I. resorts coercivity assumption (22). To this end, we construct solution q of the
equation (6) subjecting g(0, x) g(x):
|x
g(t, 01 (t)
"h Cs. -Cs 11())
where
1 ',¹ 0, (L)dt.
0: (1) 1 Mo+25
(27)
From Eq.27)
we know the construction holds for E [0, Mo+26. ). By revisiting 25
-2 - -c)
2a -c)
1
1
sup {a 2G,
lim (a-+01 Ma
Mr,
M
451
and
Ms, Ms
lim 0(1)
(a-0+, Ma
Then by choose suitable and c, comparison principle of (6) which is equivalent to one of says that
8
((,,x) x) 2 g((,2)) fort [0, Mo.
and hence,
|x| 2 2 (0(1) +0(()))((())
+ (0(1)
for E [0,
(28)
Ma
Now turning back to Eq. 28) we know 0(t) + (() 2 25, > 0 uniform in any closed subinterval of E (0, Me ). This ensures the uniform coercivity requirement in Lemma B.1
Summing up. by Lemma B.I wn is supersolution. On the other side, as convex envelope, w** < w. Next, we want to utilize the comparison principle of (26) to show w 2 w for all which is equivalent to (q. due to the construction (24p. To do this, we only needs w w0*(0, X) 2 w(0, X), equivalently w0, X) is convex and it assured by 0(0) > Má, > Mo-
Now have wt for Rd, t [0, T" is convex. In particular, this implies that
T. for (1, E (0, RR" R."
Hence we derive that for any < Mr
Ms
inf D²g(t,x) 2 - Mo, In. [o,T|xR"
(29)
Then 23
follows by sending 5, 0.
Since the transform only requires estimate of q for t [0, when Mo 1, 29 holds for any
< 1. Recalling transformation of g, the condition Mn < is equivalent to log is convex.
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C.2 PROOF OF THEOREM 3.4 CONSTRUCTION OF EXAMPLE OF THE LOss OF UNIFORM HESSIAN BOUND
Precisely speaking, will construct one dimensional (n 1) example of g(æ) G2(x)) for
smooth Lipschitz continuous function G satisfying that G²(x)) S 1, [=)(() 2 = Mo = M1 and
lip 4 (1/2, x) oc.
For M > 0, let 9M be the even function such that
2M - x²
0 x < M
(M (2)
(x 2M)²
M S 2M
0, 2M.
Note that 19hs| S independent of M. Let had (t, 2) be the solution to the following heat equation
1
U Au for (t, x) E 00) R
(30)
subject to has (0, = (#))
Since hM is even in x, (hM )e (1/2, 0) O. Hence, we have
(log (M) (1/2, (hr) ).. (1/2, 0) hM (1/2, 0)
E 2M So (Ay² + 2) dy + 2M² SMM e 2 1-20--M1 MY (4(y - 2M) - 2) dy
222 M e- 2M e-2(y- 2(y-M dy 12s e e-v²
v² dy
A A+B
C D+
Clearly, for M 2 1.
A B 4e- 2M -M /
AM
2M2 ( e 2 (v 2M) dy > -2-6-
y²² dy
+
de
MM
and
C+D+E5 3Mc-2M4 222 -MM
Thus
M12
(log )m (1/2,0)
(31)
for all M 1.
Remark C.1. Note that 9M is Ch.1, not smooth. However, the estimates above still hold for suffi- ciently fine mollifications of gM, so we may assume without loss of generality that 9M is smooth.
Below we will choose sequence S Φ1 S x2 I ... such that the terms in == Σ gk (a - IKk)
have disjoint support, and such that
2.2
(log (1/2, mτ)
for k 1, ... N,
(32)
where PN is the solution to 30 subject to (0, e 2 (c(0-22), Note that PN pw(00 (0,z) for x S 22,
Suppose we have managed to do this. Note that
P 5
where P is the solution to (30) subject to p(0,x) XI-3.-2j. Then interior derivative estimates for
solutions of (30) imply that (logTx ( xz (1/2, -) converge locally uniformly on as N oc to (log P) (1/2, .). where is the calorie function with initial data e e Therefore, (log p) (:, 0) &a bounded, but P)og (1/2, -) is unbounded (its values 2y least $ ), desired.
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We now explain how to choose zg. We will repeatedly use the fact that if h is a bounded smooth function R and h is compactly supported smooth function, then the caloric function with initial data h h(++ S) converges in C² |S| - on compact subsets of {t > 0} the caloric function with initial data h.
First, we let 21 0. Then (32 with N 1 follows immediately from (31). Now suppose have chosen ø1 < < -M such that the supports of gl (xr rx), < k M are disjoint and 32 holds for N M 1. Using the ahove-mentioned fact and (31), if we take Z.M sufficiently
large, then (32) holds for N = M. Indeed, the inequality for < M follows immediately from the fact above, and the inequality for & M follows from the fact above and the inequality (3I), after
translating so that æM becomes 0. This completes the construction.
C.3 PROOF OF THEOREM 3.5
Without loss of generality, we may assume zo = O. It suffices to show the above for [x] 2 1. For |x|| S we can just replace |z| all the final bounds with 1.
First, we prove
[3p. Without loss of generality, let 9(0) 0. Then
9(2) S A + 52|2| for R
Recall that
1 (2nt) v JUn 1ا=ي|² hly) dy
x(t,z)
1 (m) 7 Jgn c 111 h(x -1ll'(xx
v2ty) dy.
Then
Vq
(z) 2 Sgn c -0/1/ 1111//(( h(x 22yy) Dg dy
(33)
$ (x) Sec 101 111 22y) Dg dy
1
Since ab < a² + .
B₂
g(x - V2ty S (1x)2 + 21y)1² + 32(10] + v2lu1) S 21x(1=12 + 21)l2) +
B1
we deduce that
p(t,x)
2
d2a 20/1=
Srn e -2++4)) ul2 dy
(7)
(34)
e - 1w|?
1+10,
Then
Jgm e 10 Dh(a 221g) dyf = _ JEcr e-lll - /2hy) dy
=
(
< (
San (B1|z] /281lv/ + 32)e lvl" "ul"/(6 h(x - 22yy) dy
= (3/|zr + 32) +
V24 (x)
Jpm lule- 1ul
1(x 1-1l
/ dy.
Let
K
2
Cn.m max
A 1/B++1 +1].
VI - 02
[
where
( + (2(1+48))
481)
322 B1
Cn.m =2
(n (n+3)) 3) log
+ 4n log 01 + Jm +
/1 02
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Here for E N, Jm is the last positive integer such that e² r for when Jn. In particular,
J1 = J/2 = J3 = and Cn = Cn, for m = 1,2,3.
Claim: If
602
K 2 Ko max K,
02
then for i == 1,
Tx 1
101'c-lw 1-llu/nx h(x 2ty) dy S K+|a]' +1 [x]" (35)
p(t, 2) (T) VERn
Clearly, T.(0) (p(t,))(221))
Let us prove the claim. Note that p(t, x)T;
1 (+) SMiski lul'--lu
-oce-lulm h(x /2ty) dy (x)
1 Juizka lul'e-bul
"ive-lll' -oll M(x /2ty) dy
(x)² I 12112K11 wl'c-lulPhc
I <k(x''p(ttx))+
Lulp n(x 22y) dy
Our goal is to show that 1, S z) when K 2 Ku- Since g(z) 2 -0y/2/ a1. 1yl 2 K [0 and
602 K3
1-02"
g (2ty) 2 -0 (|x) + v2ly1)
-aylyl u/ (xx + 1)²
0z |u12 (x) + 1) a1
(1+az) 1// a1
Then
(VT)" | lulle a1 مللي dy.
e01
1 S
For convenience, denote K₁ KvI-ag Then
da (
M1l2k14/ lulte (mag]ll? dy
= e²
(VF)" ( (VT=ag)
n+i
|1122kle||2//e 12|'6 2|e11 dz
n+3 J c- xp dz.
I e²
(v
(T=
The last inequality is due to
e² 2 mm ifr 2 Jm and Ki |x|| 2 Jm.
Note
( c-
-e--0 dt dt >
2
lwl du e
{WER?l (slsn)
Combining with (1 - 1)"21"1111 2 - nd. for 1. E [0, 1}. we deduce
S lMll dy = (f de at) )" 2 (V7)" (1 (1- --
c
$ (Vz)" (
1 ne =22
Here Q, {y (111 12....,, lusl S for i 1,2,.., Accordingly. for 2
Wy Jwizr clol dy S ( Samgp e112 dy (v=)" 1 SQ.Minlsrt el1 dy
1
nc
+ [ (2r)" < 2
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This implies that
(vz)" /
e-Ha S 2-2e-Kiu Kilal" (Kil=))".
Since K 2 max (onx(3v15 13/31, 2vn
[2]
28 K||>>"" K?|n|
S -2912211
Se"
Combining with
K7 2
K 1=/2
1
+1+,²
A32
2 (n (3+
+ 16nlog an + +1+
4
vI 02
B1
we obtain
e
28/ I, 5 6
2 VI a2
n+3
S e
0,2
(1+ 43
By 34
5 p(t x).
Hence 35
holds, As an immediate conclusion, have that
V
q
< B,|+| + 32 + 23, 2K| < [| + 82.
Secondly, to prove 14 and removed at the end. Then
(5). we first assume that g(0) 0 and Vg(0) = 0, which will be
Vg(x)] s L|a].
Next first verify (14). Note that
D²q= = B V
(36)
Here
4
clul hllu"nn2 hD2 g
v 2ty) dy
P (TT)
JR"
and
B
-lv hV Vg(a /21y) dy.
(T)
Here (u & u)g = u, is the outer product. By Cauchy inequality, B 2 V X Vg. Hence
A LIn-
For the other direction,
D2 2 A - B > - LIn B.
So we just need to estimate the term B. Note that
1 SRx e l -lol hVg Vg(a 2by) dy 111
x 1a
(x) I Sen (|æ) + 2e lo1* dy
S 21
(x) 2 Je (|x|2 21y13)e -aac--l01 10lhaa dy
< 2L (10 x) + - SEL 1y12e lol2 h hdyy
Thanks to for B1 I., f2 and K Ko. we have that
|[B||2 22 4K3 x
100 1z
Hence
|| I ||
L 10T.
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Then let us verify
15
Note that
= xV -
with
|C|
San e lul hD2 221y) y ydyl
L 1 - S
v2t P (=) Sun byle lul? 1o dy
/2t P (m)
21 v2c
|o| < 24
7i [r|.
Also,
DI =
1 1 1 V2r P (T)
JRn
10/ hVg @ Vg(a - v2ty) ydy
< 1 21
V2h (T) JEn (1>|2by + 2e lull dy
1 |
SRm lyye Iu h hy " (z) Se. 1y1e lul? h dy
4r.
V
(Φ)
s 22 (2K |2| + 1
121. K 1-/211
Vt
Finally,
|0.9
<2i (x) Jm - 1- /by) y day
1
V2l P (7) SR (10ll] ( dy
L
<
4
54 12
Also,
b4 < 21 + /2)K(zl 6LK|2||
Hence
36L2
1..Vol S
K |
Vt
So
0 vt))
$ 22 vi
K| + Vi
12L 1212
|z|3
+
36L
VT
S 50L K 13/ v
At last, for a general g, we consider
(o(x) g(x) g(0) Vg(0) z.
and (t, 2) be the corresponding score function. Then we have
go) 0, Vyo(0)= and (0(1, q(1,5 Vg(0)).
Hence 14 and 15) hold for general cases.
Remark C.2. In the proof of (15). we may also bound d.g use the equation
\ [
together with bounds for Aq and / Note Vg(z Vty) Vg(x) + re for Iral S LVHyy) Then term D in the proof
D V p (क) fe. e -w 10 nV & Vg(r 2ly) dd
= Vg(z) + 0 1z
This will lead to bound of V.49 by replacing interested readers as exercise.
1a vi
in 15 by O(nI3). The details are left to
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Below other simple situations where obtain global uniform bound of the Hessian, which follows immediately from (33) and (36).
Theorem C.3. Let L1 and Lg be two positive constants such that
Vgl S L1 and ||D2gl1a L2.
Then
Vg(t, $ L1 and (L2 + LI)la S xt, s L2hn
C.4 PROOF
16
We first prove the blow up bound the gradient (1) T6p.
Note that Dq
Dp and
Ir==| dmo(y)
p(t,))
(2mt) Jnn
(2nt) JDp
e 21
1
dT).
Then
(x -0-1) 1) | dwoly).
vp(t,z)
(2Tt)² Do
Since Do C (0)
+ p(l, x).
[o((,x)) S
Next we prove the blow up bound of Hessian (2) in 16
Note that
D2p
Vp
p xt, Vp
8ij
1
A B
D))(ttx))
+
p
t
lc 11 25 dio(z)).
Here p(t,x)) = SD. e
/ (xi- vi)e
Aij 2((,,z))
d (v)
Do
and
So (2; vide - -
dnolv) JDo (x, Vi)e - dToOV)
Bi
12 1),T), p(t,x) JD, (I:Vi x,yi)e -laup? duo(
SD, vie in dmoly) SD, Use = dTo(l)
Hence (A - B)
p(t, x) /
yuUge -
mm davly) - /
ysc -
dao(y) / Uje -m- droll) (37)
Do
Do
So is easy to see that
||A Bl2 (t,).
Thus 16 holds.
C.5
PROOF OF THEOREM 3.8
Since To comparable to the uniform distribution, there exists a constant C such that for any measurable subset U Do
C He(UT) < mo( < CH.(U)
Here Ha(-) is the d-dimensional Hausdorff measure. Hereafter, we write
(i) 0D: the dimensional boundary of Dg. Moreover, for y E Do:
(ii) (Do) C R": the d-dimensional tangent space of Do y,
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(i (Do) C R": the dimensional orthogonal complement of (Do):
(iv) Ty (D)) C R"; the d - 1-dimensional tangent space of 0D, at E 0D,. Note that T. (Do)
is a subspace of y. (o);
(v) Ny (Du) C : the n n d dimensional orthogonal complement of (D)o). (2(Do) is a subspace of (Do)
Let
5 fx E R2I there exists a unique E Dp such that |x: - Yzl = d(a, Do)}.
Then {S has zero measure since d(x, Do) is differentiable almost everywhere.
Write
S fo S y Do Do\DDo
and
S
{or
S ye E @Do)
W1 = {
y/²
|
E Sil
lim inf veDo-ty.
0
|y
V/ 1
|x - y12
x Yv 0
W2
x E S2 x - yv E No (Do)
and
lim inf y P9
1
Vu
and
10 y/²
x 1
W3- E S2l x U/ & No (Do) and
lim inf vEADo-te.
0
Yarl 2
Note that
if E Do Do)
then I Vv No, (D),
y E 0Do, then I - Y/ E Ny, (D).
W2) U (S))Ws) has zero measure. We will prove this for S₁ W1, The proofs for the other two are similar. Apparently, if S, then for all t E (0, 1), vx. = Ys and X, E W for X, = ys + t(x - y2). Also,
Step 1: We show that 5\ ( W.) =
For E Do AD, write
Ty= (a E S yv y).
By compactness argument, it is easy show that for given x E S and >> 0, there exists a V, > 0, such that Y/ (ya) for any I B,, (x) n S Hence to prove that S1 \W has zero measure, suffices to show that for any yo E Do. if T yo is not empty, then there exists ro > such that
Wo T C (v + f(y, v) E (ro())), v (D)) and |u| = 1)
(38)
for locally Lipschitz continuous function
t(y,e): (vo) (Do) (0, oc).
By suitable translation and rotation, we may assume Yo and in neighbourhood V of DonV = Vn (1'' F(2))) E ER)),
(39)
where F (F(++1) F(d+2)
F(m)) Rdd Rn-d is smooth map satisfying VF(() 0 and F(0) = 0. Choose a E I Typ. Let K E (Do) and t > 0, let
Her(y)) = Itv ( F
Since Wi, (0. can not be positive definite matrix for t |ar| Meanwhile, for 0 < |o], ytz = Yo and fo E Wi, which implies that D2 (0) is positive definite for t E [0, [x]). Hence
[x| the first moment such that D3 (0) has zero eigenvalue. Which is equivalent
the largest eigenvalue of the d x d matrix
Σ k=d+1
tuy F (k)
2/1r (0) is 1.
Therefore, for E (20)) and v E (D,) with |u| = 1, if the largest eigenvalue A(y. v) of the (y) is positive, we set
matrix Σx Un (1) v
t(y, v) = X(y,v)
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Then (38 holds.
Step 2: We will verify that if E Uf=1 We, then
(D C2
for all t (0, 1].
Case 1: Assume that z E W1. Without loss of generality, we may assume representation as (39). Choose > such that
0 and use the
(i),, =
v' < <r} C Do;
(ii) Then there exits az 3 such that for y Vr,
Og
2 2 xx y/ lx ya 2 Brlu - yzl2
for y E Do-
(40)
For h 2 and kvt < write
Ve = { (s/ ((/)))1 1 <
Thanks to the left upper bound in 40
-1x-o1² 21
dmo 2 O (1 1لسسحيي
duo
Vv.
Recall that yre O. To the dependence on Yu, we keep yo in the computations below instead of replacing it by 0. Note
J Do
22 v1 dro S
/
2 - vf dno + Ce 1x- 146
by
v..
for some 8 0. Also,
Jv. ly v=le--1-0 dmo Lk (VEV. ly V2 -1m-v dro
< Ct. 1 l Ee (k: + 1) = 0 1$ 1r-u, 1.
Hence
JD. 1y Vu Ve - v1? 21 dro(())
SC
(41)
Recall that
p(t, x)) = /
-1r-w12 donll) = (V2xt)) p(t,,)
21
Do
Then for 1 < i,j and yu (a1, ag, ... an).
[Bg yidge اسياا doroly) Sep vee -la-y?
- am((s)
drolv)
Jwg Mse
p(t.2)
F(3.n)
Jog (a)-a) -azle - dma(s)
So az)e -m dmo(u) So (us azle -1m-212 dno(v) M(t.))
Marr)
p(t.z)
I Ct.
The last equality follows from and the Cauchy inequality. Therefore,
leads to
D2a(t,x)) Cs
Case 2: x. E Wg. The proof is similar to Case 1.
Case 3: Wy. By suitable translation and rotation, may assume ys 0 and in neighborhood
of E Rd
Dp DDnV= = V. F()')) (wi..., 9g
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where F:; IR + Rn-d is smooth map satisfying that VF(()) 0. Also,
Gg = {a (2', za)1 z' (21, 22, ... zd-1) Rd-1 |2'| r; za f(1, 22, 2d-1)
for a smooth function / : Rdd - IR subject to v = 0. Then
To (Do) {(v, 0, 0) E RR' V E R*)
and the - 1 dimensional tangent plane to 0D, at y/- = 0 is
oTo(Do) ((v',0,00 0) E R o' E 1.
Thus
I x yr
0,.....5 On Tnl)
for some 02 O and 2n E RR-dd To the dependence on yas in Case we keep ye in the computations below instead of replacing it by O.
Then for y" 1d-1 and y = (y/ Ma. Fly" ya)) Do.
x y - 1 Y/l 20 (ya s(//)) + Oly - yz 12
(42)
Write
Hyy) = x - y1² - 1/ Vel = Hw"", , yd. Fl"", vd))
Since x E Wg.
Hyy" f(g") ". "("""((""")) 2 8.
Therefore, there exists and M such that
0g
x - -12-12= Uz 2 2
(ya f(v()) + 5zly yul for all y E V,
M
Write
Re, H1 yas va)) E 2 1y'l skvi and 0 ya 1(1') s kt).
Thanks to
42
/
la u1 27 dio 2 S
lx 21 d 2 O ( us
x1?
x(1,z)
Do
Note that
So ly eelze dmo / 1 Ve -
-
dig + Ce =-vo|?+65
v.
Also,
SV. ly yr 2e -1-= -
dmy En= SDon at1 (t.) ly yrr - dmo
I Ct p جل-ييع? 2 (k + 1)2e & 10 (1etbe |--
Hence
Jn, y De 1e -l d
< St
Then by the same argument in the end of Case 1, we deduce that
C2 |D''g(t,))112 S
t
Finally, if Do is convex, then it clear that R and W, = S and W2 U W3 = S2. Hence the O(E) bound holds for all R
24

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C.6 PROOF OF EXAMPLE
Proof: For given æ E IR², denote by T/ the weighted center of mass:
ul? d
JD.
ye
yt
P
-1 dwolv)
Note that as / + O, the measure Thus,
will concentrate on {ir E Dol by x = d(x, Do)
lim d 12) O, 1-0
where r the convex hull of fy Dol ly - x| = dl(r, Do)}. According to the computation in the proof of (66.. we have that
1x ol d (y) - In ye 1r x/2 lupe
2
PSn
daoly)
A
n
+
t
t2p2
Sto 1y D1
F.
-1 doo(())
n
Aq
t
t2p
where the second term is like a variance. U/2 such that
(0,0) for some 0 O, there are two points yı and
V2l
d(x, Do).
Due to the symmetry, must have that
21 dTo(()
81 + 1/2
8
-
6yz
and
lim +0+
y:
2
Accordingly,
SDo
1y
Øt
2e
dmo(y)
1/11 8/12
lim
4
leading to
C2 -))(1, 2
for l E (0, 1].
2
C.7 PROOF OF WELL-POSENESS RESULTS
Lemma C.4. Given T > O, suppose that F = Ftt, E C[[O, x RR, satisfies that F is locally Lipschitz. continuous variable, for any M there exists constant Las such that
[F(1,)) - F(1,v)) < L - vl for x. E (0) and L E [0,
and
[F(t,z)] S C(Kx) + 1). for (f,z) E [0, X R"
for a positive constant Then for any xo E R², the following equation has a unique solution
(t)))= X(), te [0.7]
x = Ø0.
Proof: The uniqueness follows from standard ODE theory. We just need establish the global existence. Let w(l) = ( Then
(i( Ciw(t) C2
for two positive constants C and C2 depending only on C. Hence for all t. 2 0,
C2
e w()) < | + C
(1
(11)
Hence the solution be extended to T.
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KT Published as a conference paper at ICLR 2025
Proof of Proposition 4.1
It suffices to notice that for each fixed sample w, ( Y(t,w) Xe(w) - We(w) just satisfies the regular ODE for any fixed sample
{dy(t)
P(t, Y + (())dtt t E (0)
z0- (r(0)=
Hence the Corollary follows from Theorem C.4 and the well known fact that Wi(w) C1, for a.e. w.
Proof of Theorem 4.3
Note that log p(t, æ)) is smooth function for > hence V p is locally
Lipschitz continuous in x. Owing to Proposition 4. it suffices to show that for = log p(t, x),
[Vt(t, - Cr(12|+ +) for all (t, E [0, T x R"
for constant CT depending on C and T. By 7). it is equivalent to showing that
Vg((,=) < Cr( 1) for all (1, E [0, - e R".
for a constant Cr depending on and T, which follows from Theorem 3.5
C.8 PROOF OF THEOREM 4.6
The key ingredient the following estimates truncation
Theorem C.5. Assume || g(x)lla I L. Suppose that Assumption 2.4 holds and there exists Co O such that, I Con and [g(xo)] < Cvv. we have that for fixed T S 1 and l. N.
N Σ/" vi(t,, (t)))))))11 de Cn
(43)
N
Here C is constunt independent of and L.
Proof: It suffices to show that for > 1 E [0, [).
V xs)) (t(t, < CL t)n log n
According to Lemma C.6 Chen (2023) it suffices show that
E V +
v((t,=(())11 < Cn² t).
(44)
Here z ~~(00, C(s f)).
-()-
Owing to 14 of Theorem 3.5 in our paper and (a, b) < a + b. || & C(|x= log
where depends on (L, Co. 02). matrix. Then
See
|I for the definition of the spectral norm || ||2 of n
(t))t,((t))) + +) - Vg(t, = C(1 |((())) + |2| + (n log n) )) Note that B(x²) < Cn(s and E(=*) s Cn²(s 1)2. Moreover, by Cauchy inequality {((x(())(2))
(D))E((+) < Cn(n log n) t). The last inequality is due to Ep(c. (x(())) C'n log n1 from Remark. Hence holds..s.
Remark C.6. In the proof of Theorem C.S instead of using Lemma C.6 in Chen et al. 20023) may also use from Theorem 3.5 to bound the difference between time,
E Vg(s, (t)) x((()=(t))11 < C(s +)og?()ogn))
Cnoglo on the right hand side of 43 The proof is similar. Note
This will lead to an extra term that when N = O(2)) x² 2'logg)" N
N2
(log
Remark C.7. We are aware of the difference of V log p and Vg due to the translation while our Lipschitz estimate uniform time, hence similar results of Theorem C.5 holds for V Tog
26

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C.9 CONVERGENCE BOUNDS UNDER COMPACT SUPPORT MANIFOLD ASSUMPTION
Theorem C.8. We assume supp(po) - Do C (0) and the density is smoothly defined on Do- With early stopping > O, Let QT-s be distribution generated by uniform discretization of the exponential integrator scheme (4), with an approximated score satisfying Assumption 2.1
If L1 >
2 N
() (M2 d)e +T
where L. = 11
M g2
Proof: Lg is computed from (6).. Then Proposition A.3 is applied.
C.10 SKETCH OF PROOF OF WASSERSTEIN DISTANCE BOUND
Here we provide sketch proof to a Wasserstein distance bound with full details left in a future publication. A key ingredient is to estimate the backward process X, in ( and its discretized approximation &t in The two processes are coupled by the same Brownian path and initial value, hence,
d||x; Rell
1
||x Reel + |x &ell
(X, it. Vlogp(T - t,, x,) -so(T - t 8c )
dt
2
$ 2
XX Rell + V log p(T t, X1) - se (T the &4,,)1
(45)
where t E [/ 141. Then we turn to the inequality,
V log log - t, Xz) s0(T - t/ feg.) IIl
V lTg - X.) v llogp(T txe Xx)
V logp(T t/2, Xx,) V Vlogp(T tg ftg.)
V Vog 14, (4,) so (T -4, S4, }||
(46)
where the last term on the right hand side of 466) relates to the approximation error of the score. With the Lipschitz bound (Theorem (.55) in hand, we estimate the first two terms, while noticing that the Lipschitz constant grows linearly while the diffusion process X, has exponential tail.
The bound for EllX, itr follows by taking expectation of (45) and using Gronwall type in- equality, thus implying a bound on the Wasserstein distance W ((aw(wET)
The bound on (LaW (Law())) Po) then follows from stability analysis of X with respect to the initial distri- bution, per standard arguments as in Bortoli 2022); Chen et al. (2023).
D BROADER IMPACT
Diffusion model is one of the most influential generative models in the Al era. Our theory gives theoretical guarantee of the lifespan of diffusion model with minimal assumption of data distribution. We discovered a theoretical characterization in the point-wise sense (stronger than prior works) on the singular behavior near generation time related the manifold hypothesis. This provides insight for model parameterization and convergence rate improvement in practical implementations.
27


Reviews:
[{"review_id": "bu6SRV55OE", "reviewer": "Reviewer_vaeP", "summary": "This paper investigates the mathematical framework behind score-based generative models (SGMs). The authors' primary focus is to establish conditions under which these models are well-posed and convergent.\n\nKey points from the paper include:\n1. **Well-Posedness of SGMs:** The authors prove that SGMs are well-posed under global conditions. They introduce Lipschitz estimates that maintain stability and prevent the “blow-up” of the score function, which has been an issue in practical applications where score gradients become unstable.\n2. **Convergence and Sharp Lipschitz Bounds:** They provide a Lipschitz bound for the score function, addressing both smooth and non-smooth cases. This bound helps establish global convergence of the backward process (used in sample generation), even for non-log-concave distributions. For non-smooth data distributions, the bound asymptotically scales as $1/t$, ensuring convergence on low-dimensional manifolds with compact support.\n3. **Theoretical Contributions:** The authors introduce rigorous estimates for the Hessian and gradient of the score function, reinforcing the stability and convergence properties. This allows the generative models to better approximate complex data distributions.\n\nThe paper concludes with insights into how these findings could guide model implementation, such as early stopping strategies and parameter adjustments, to ensure stable and efficient generation in real-world applications.", "questions": "All the questions have been addressed in the *Weaknesses* section.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The major strengths of this paper are:\n- Establishing crucial bounds on the Hessian of the score function, which significantly enhance existing bounds on the KL divergence from *Chen et al. (2023)*. These bounds are vital for assessing the fidelity of generated distributions.\n- Carefully tracking the dependence on dimensionality throughout the analysis, a critical aspect for score-based generative models (SGMs) given their frequent application to high-dimensional data distributions.\n- Characterizing the explosion of the score function as $t$ approaches $0$. Although this instability near zero has been commonly observed in practice, the authors quantify the explosion’s magnitude through PDE techniques, opening the door to improve model stability in high-dimensional generative tasks.\n- Describing the case of compactly supported data distributions, shedding light on the theoretical limits of score-based generative models (SGMs). This perspective allows for a deeper understanding of how SGMs behave when data is restricted within a bounded region, which has practical relevance given that real-world data often exhibit such constraints.", "weaknesses": "Here are some weaknesses of the paper:\n1. **Limited Comparison with Recent Literature:** The paper lacks sufficient engagement with recent literature on SGMs, particularly from a PDE perspective. A broader comparison with recent works is needed, especially regarding bounds on KL divergence, as several relevant contributions exist and should be cited.\n2. **Redundancy in Subsection 2.2:** The computations in Subsection 2.2 are already present in the literature, as seen in *Conforti, Durmus, Gentiloni-Silveri (2023)*. While this content is valuable for readability, it might be better suited to an appendix section to streamline the main text.\n3. **Unclear Novelty in Theorems 4.3 and 4.4:** The novelty of Theorems 4.3 and 4.4 is unclear in comparison to the results in *Chen et al. (2023)*. It would be beneficial to clarify how the bounds on the Hessians improve convergence rates relative to this paper, particularly in establishing any distinct contributions.\n4. **Lack of Appendix Structure:** Adding a paragraph at the beginning of the appendix outlining its structure would improve navigation and clarity for readers.\n\nAddressing these points would strengthen the paper, and I would consider raising the evaluation if these aspects are clarified and expanded.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "rkgDLOiNqq", "reviewer": "Reviewer_QYfP", "summary": "Recent theoretical works on analyzing diffusion models require bounds on the Lipschitzness of the score function of the noised distribution at all time-steps. This work gives bounds on Lipschitzness under various conditions on the initial data distribution, and uses some of these bounds to instantiate prior guarantees for diffusion models.\n- Theorems 3.1 and 3.4 show that if the initial data distribution is smooth, then the score of the noised distribution is Lipschitz up to some finite time T, and that this cannot be extended to all times\n- Theorem 3.5 gives local Lipschitzness estimates of the score function under local assumptions\n- Theorems 3.9/3.10 and Example 3.11 show that if the data distribution is supported on a manifold, then one gets Lipschitzness bounds of 1/t^2 at time t, and in fact almost everywhere one gets 1/t, and there is an example where the \"almost everywhere\" is necessary\n- Theorem 4.3/4.4 instantiate diffusion model guarantees using Theorem 3.1 and Theorem 3.5 respectively", "questions": "N/A", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper addresses an interesting question of trying to substantiate the Lipschitzness assumptions on score functions. The result on manifolds showing that 1/t^2 can be almost-everywhere improved to 1/t seems qualitatively new and interesting, and the example showing that this is tight provides a fairly complete story.", "weaknesses": "The paper could be written better, as far as comparison to prior work, clarity about which of the results are most \"important\", readable theorem statements, and discussion about the assumptions.\n- Theorem 3.1, 3.9, and 4.1 seem as though they may be fairly standard / similar to prior results, but there is very little discussion about what is new and what is included merely for completeness.\n- Theorem 4.4 says \"with same assumption as Corollary 3.6\", which in turn says \"under the same assumption as in Theorem 3.5\". The stated assumption is uninterpretable: is it reasonable? I can't tell. \n- The invocation of Theorem 3.5 in the proof (line 1320) provides no explanation as to how the complicated bound in Theorem 3.5 becomes this simple bound.\n\nThe most interesting result would be if the 1/t^2 -> 1/t improvement could be used to instantiate a better bound for diffusion models. However it appears that this is not done. The rest of the results in Section 4 just have lots of assumptions (e.g. the bound on T in Theorem 4.3) so it's not clear how applicable they are.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "RqahRsXV92", "reviewer": "Reviewer_5jve", "summary": "This paper addresses the global well-posedness and convergence of score-based generative models (SGMs) by providing sharp Lipschitz estimates for the score function under both smooth and non-smooth initial conditions. By developing optimal point-wise gradient and Hessian estimates, the authors establish conditions under which the backward diffusion process is well-posed up to time zero, effectively covering cases where previous approaches failed due to exploding scores or separated time regimes.", "questions": "1. It appears that the main assumption in this work is Assumption 2.3, which requires that the tail distribution is bounded by a Gaussian distribution. Could you provide some intuition behind this assumption? It seems potentially quite strong, as it may imply that the data is approximately Gaussian—perhaps even stronger than the strong dissipativity condition. Could you clarify why this is a reasonable assumption for real-world data distributions?\n\n2. The statements in Section 3.3 appear somewhat inconsistent. In Theorem 3.9, it is assumed that the density is smoothly defined on $D_0$, which may be a low-dimensional manifold, while in the proof, the integral over $\\mathbb{R}^n$ is equal to the integral over $D_0$. How is $d\\pi_0(y)$ defined to ensure this equality holds? Is $p_0$ a density on $D_0$ or $\\mathbb{R}^n$? I may have misunderstood, so please correct me if I'm mistaken.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper introduces new sharp Lipschitz bounds for the score function, which enable global well-posedness and convergence without requiring separated time scales. This is particularly valuable for SGMs where score explosion has previously posed challenges.", "weaknesses": "I did not identify any major weaknesses in this paper. I listed some typos below.\n\nTypos: \n1. On the LHS of equation (3), I believe $p_t$ means $\\partial_t p$. Using the notation $p_t$ will probably make the reader confused as $p_0$ denotes the initial data;\n2. In line 108, it should be $Q_{t'}$ instead of $Q_t'$;\n3. In line 127, the LHS should be $\\hat{x}$ instead of $d\\hat{x}$;\n4. In line 218, it should be $\\mathbb{R}^n$ instead of $\\mathbb{R}^d$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "NZVVQV0EdD", "reviewer": "Reviewer_5LpM", "summary": "The paper focuses on providing the Lipschitz estimates for the score-function in the score-based generative models. It also provides a sharp example showing that the loss of Lipschitz bound of the score function as time gets large even with nice initial data distribution.  As a by-product, well-posedness and convergence are obtained by following existing literature.", "questions": "(1) In the last paragraph of page 1, well-poseness should be well-posedness.\n\n(2) In the last paragraph of Section 1, Planc should be Planck.\n\n(3) On page 3, \"In the analysis, we assume an $\\epsilon^{2}$ bounds for this estimation'' I think you may replace $\\epsilon$ by $\\epsilon_{0}$ because this is what you used in Assumption 2.1.\n\n(4) It would be nice to add some discussions about Assumption 2.3., how it is related to the assumptions in the previous theoretical work on score-based generative models.\n\n(5) In the statement of Corollary 3.2, $p_{0}\\in C^{2}(\\mathbb{R}^{d})$ but you also use $\\sup_{x\\in\\mathbb{R}^{n}}D^{2}\\log p_{0}(x)$. I think in your case $d=n$? Similar issue appears in the statement of Theorem 4.3. and Theorem 4.4., where you have used both $d$ and $n$.\n\n(6) In Theorem 4.3., in the upper bound for $\\mathrm{KL}(p_{0}\\Vert\\hat{q}_{T})$, the first term is $(M_{2}+d)e^{-T}$. Can you remind the readers what is $M_{2}$ here (or where it is defined)? Also, for $H_{T}$, does it have an explicit formula (depending on $L_{0},L_{1},T$) and if so you should mention it. Also, I think it will be nice to spell out the dependence on $T$ because your upper bound is written in a way with an emphasis on the dependence on $T$, and if $H_{T}$ is implicit in $T$ it will make your bound less\nvaluable.\n\n(7) In order to have $\\mathrm{KL}(p_{0}\\Vert\\hat{q}_{T})$ to be small in  the upper bound for $\\mathrm{KL}(p_{0}\\Vert\\hat{q}_{T})$ in Theorem 4.3., it seems you have to choose $T$ to be large, but on the other hand $T<-\\log(1-\\frac{1}{L_{0}+1})$, which seems\nproblematic to me. \n\n(8) If my understanding is correct, the main contribution of the paper is the Lipschitz estimate on the score function, rather than the convergence analysis, which follows from Chen et al. (2023) etc. from the literature. In that spirit, does that mean your Lipschitz estimate on the score function can also be used to obtain convergence analysis for probability flow ODEs by leveraging existing works on the convergence analysis of the probability flow ODEs in the literature?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(1) Estimating the Lipschitz constant for the score-function using the idea of the PDE approach is novel and makes a nice contribution to the theory of score-based generative models.\n\n(2) An example is provided to show the Lipschitz estimate is sharp in some sense.", "weaknesses": "(1) I think in the abstract, the paper overclaims that the results are obtained under minimal general assumptions of initial data because when you look at Theorem 3.1., Corollary 3.2. etc., some relatively strong assumptions are used.\n\n(2) Well-posedness follows immediately from the estimate on the score function and Theorem 4.1. can be found in any textbook on stochastic calculus. Moreover, the convergence analysis follows immediately from existing literature (Chen et al. 2023). In that regard,\nI think the main contributions of the paper are simply the Lipschitz estimates for the score function, and well-posedness and convergence analysis are not the main contributions of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (Poster)