Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Authors: Marta Gentiloni Silveri
Antonio Ocello
Venue: ICML 2025 poster (2025)
Keywords: Score-Based Generative Models
Hamilton–Jacobi–Bellman equation
Log-concavity
Convergence Guarantees
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: This work presents a new framework for analyzing Score-based Generative Models (SGMs) in the W2-distance, relaxing strict assumptions like log-concavity and score regularity.

Abstract: Score-based Generative Models (SGMs) aim to sample from a target distribution by learning score functions using samples perturbed by Gaussian noise. Existing convergence bounds for SGMs in the $\mathcal{W}_2$-distance rely on stringent assumptions about the data distribution. In this work, we present a novel framework for analyzing $\mathcal{W}_2$-convergence in SGMs, significantly relaxing traditional assumptions such as log-concavity and score regularity. Leveraging the regularization properties of the Ornstein-Uhlenbeck (OU) process, we show that weak log-concavity of the data distribution evolves into log-concavity over time. This transition is rigorously quantified through a PDE-based analysis of the Hamilton-Jacobi-Bellman equation governing the log-density of the forward process. Moreover, we establish that the drift of the time-reversed OU process alternates between contractive and non-contractive regimes, reflecting the dynamics of concavity.
Our approach circumvents the need for stringent regularity conditions on the score function and its estimators, relying instead on milder, more practical assumptions. We demonstrate the wide applicability of this framework through explicit computations on Gaussian mixture models, illustrating its versatility and potential for broader classes of data distributions.

Beyond Log-Concavityy and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Marta Gentiloni-Silveri" 1
Antonio Ocello" 1
Abstract
tive models, Score-bused Generative Models (SGMs) (see, e.g., Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020; Song et al., 2020b;a) have gained significant at- tention due to their versatility and computational efficiency. These models leverage the innovative approach of reversing the flow of a Stochastic Differential Equation (SDE) and employ advanced techniques for learning time-reversed pro- cesses, enabling the generation of high-quality and visually compelling samples (Ramesh et al., 2022).
Score-based Generative Models (SGMs) aim to sample from a target distribution by learning score functions using samples perturbed by Gaussian noise. Existing convergence bounds for SGMs in the Wy-distance rely on stringent assumptions about the data distribution. In this work, we present novel framework for analyzing W2- convergence in SGMs, significantly relaxing tra- ditional assumptions such log-concavity and score regularity. Leveraging the regularization properties of the Ornstein-Uhlenbeck (OU) pro- cess, we show that weak lon-concyvity of the data distribution evolves into log-concavity over time. This transition is rigorously quantified through PDE-based analysis of the Hamilton-Jacohi- Bellman equation governing the log-density of the forward process. Moreover, we establish that the drift of the time-reversed OU process alternates between contractive and non-contractive regimes, reflecting the dynamics of concavity. Our ap- proach circumvents the need for stringent reg- ularity conditions the score function and its es- timators, relying instead on milder, more practical assumptions. We demonstrate the wide applica- bility of this framework through explicit compu- tations on Gaussian mixture models, illustrating its versatility and potential for broader classes data distributions.
This breakthrough has spurred rapid expansion in the appli- cations of SGMs, which now span diverse domains. Exam- ples include natural language generation (Gong et al., 2022), imputation for missing data (Zhang et al., 2024b), computer vision tasks such as super-resolution and inpainting (Li al., 2022; Lugmayr al., 2022), and even applications in cardi- ology (Cardoso et al., 2023). For comprehensive review of recent advancements, we refer readers to Yang et al. (2023).
The primary goal of these models is to gener- ate synthetic data that closely match a target distribution Tdata given a sample set. This is particularly valuable when the data distribution is too complex to be captured by tra- ditional parametric methods. In such scenarios, classical maximum likelihood approaches become impractical, and non- parametric techniques like kernel smoothing fail due to the high dimensionality of real-world data.
SGMs.
SGMs address these challenges by focusing on the score function-the gradient of the log-density of the data dis- trbution-instead of directly modeling the density itself. The score function describes how the probability density changes across different directions in the data space, guiding the data generation process.
1. Introduction
In recent years, machine learning has made remarkable progress in generating samples from high-dimensional dis- tributions with intricate structures. Among various genera-
The generation process begins with random noise and iter- atively refines it into meaningful samples, such as images or sounds, through a denoising process that reverses for- ward diffusion process. The forward diffusion process starts with data sample (e.g., an image) and gradually corrupts it by adding noise over several steps through an SDE flow, eventually transforming it into pure noise.
'Equal contribution 'CMAP. CNRS, École polytech- nique. Institut Polytechnique de Paris, 91120 Palaiseau, France. Correspondence to: Marta Gentiloni-Silveri <marta. wertiitggnntnniiiiveieee @ polytechnique..du>, Antonio Ocello mantonio.ccellod polytcchnique.edu>.
During generation, SGMs learn to reverse this diffusion process. Using noisy versions of the data, the model trains deep neural network to approximate the score function.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Beyond Jog-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Starting from random noise, the model applies the learned score function iteratively, progressively removing noise and refining the sample to match the original data distribution. At each step, small, controlled adjustments based on the score function guide the noise toward realistic data, en- suring that the final sample aligns closely with the target distribution.
Broadly, these contributions can be categorized into two primary approaches, focusing different metrics and di- vergences.
The first category explores convergence bounds based on a-divergences and Total Variation (TV) distance (see, e.g., Block et al., 2020; De Bortoli, 2022b; De Bortoli et al., b021b; Lee et al., 2022; 2023; Chen al., 2023; 2022a; Oko et al., 2023). For example, Chen et al. (2022a) derived upper bounds in TV distance, assuming smoothness of the score function. Recent works by Conforti et al. (2023a) and Ben- ton et al. (2024) extended these results to Kullback-Leblerr (KL) divergence under milder assumptions about the data distribution. Importantly, bounds on KL divergence imply bounds on TV distance via Pinsker's inequality, reinforcing their broader applicability.
HJB approach. Connections between SGMs and par- tial differential equation (PDE) theory have been explored in recent literature through various interpretations. For instance, Berner et al. (2022) demonstrates that the log- densities of the marginal distributions of the underlying (HJB)
SDE satisfy specific Jamilton-Jacohi-Bel equation. By applying methods from optimal control the- ory, they reformulate diffusion-based generative modeling as the minimization of the KL divergence between appro- priate measures in path space. This perspective has been further developed in works such as Zhang& & Katsoulakis (2023); Zhang et al. (2024a), where SGMs are viewed as solutions a mean-field game problem, coupling an HJB optimiza-
The second category focuses on convergence bounds in Wasserstein distances of order p 2 1, which are often more practical for estimation tasks. De Bortoli (2022a) estab- lished Wi bounds with exponential rates under the manifold hypothesis, assuming the target distribution lies on a lower- dimensional manifold or represents an empirical distribution. Simikos-Stamatopoulle
gquation-represenee tion-with the evolution of probability densities governed by the Fokker-Planck equation.
an
al. (2024) provided Wi bounds torus of radius R. noting that the bounds depend on R and require early stopping criteria. For Wz-convergence, results by De Bortoli et al. (2021a) and Lee et al. (2023) rely on smoothness assumptions about the score function or its estimator, often under bounded support conditions. For such distributions, W2-distance can also be bounded by TV distance, and thus by KL divergence, via Pinsker's s inequality.
A connection between reverseversal and stochastic control has been identified in Cattiaux et al. (2023); Conforti & Léonard (2022), providing powerful analytical perspective for studying diffusion processes. This framework, with the associated HJB equation, has been effectively leveraged in recent works such as Conforti et al. (2023a); Pham et al. (2025), where the authors utilize it to derive tight conver- gence bounds in KI. divergence for score-based generative models, under minimal regularity assumptions on the data distribution.
In addition to these general approaches, reverse SDEs have been analyzed in the context of log-concave sampling. For instance, Chen et al. (2022b) studied the proximal sam- pler algorithm introduced by Lee et al. (2021), inspiring further work on Wz-convergence within the strongly log- concave framework. Current results often assume strong log-concavity of the data distribution and impose regular- ity conditions on the score function or its estimator (e.g., Gao et al., 2023; Bruno et al., 2023; Tang & Zhao, 2024; Strasman et al., 2024). For strongly log-concave distribu- tions, W3 bounds can also be estimated from KL diver- gence using Talagrand's inequality (Corollary 7.2, Gozlan & Léonard, 2010).
However, convergence bounds in terms of the Wasserstein distance through coupling methods (Villani, 2021) and the study of the associated HJB equation remains an open prob- lem. Among the various coupling techniques proposed, two methods have demonstrated to be particularly effective in this context: coupling by reflection (Eberle, 2016; Eberle et al., 2019) and sticky coupling (Eberle & Zimmer, 2019). The flexibility and power of these methods are exemplified by recent developments in adapting these techniques to more complex dynamics such as McKean- Vlasan-Vlasov equations (Dur- mus et al., 2024; Cecchin et al., 2024). In particular, given the connection with stochastic control, significant assistance in analyzing the Wasserstein distance between controlled SDEs provided by the so-called controlled coupling intro- duced in (Conforti, 2024).
These works share some notable features. A key insight, as highlighted by Strasman et al. (2024), is the contraction property of the backward process, which ensures conver- gence stability. Additionally, their results align with well- established findings for Marrr-Ma (EM) discretiza- tion schemes (Pagès, 2018), showing vh dependence on the step size h. Furthermore, Bruno al. (2023) achieved optimal dimensional dependence in this framework, scaling as vd, where d' is the dimensionality of the data distribution.
The power and appeal of SGMs have spurred significant interest in establishing convergence bounds, leading to variety of mathematical frameworks.
Related literature.
2

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Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Contributions. Our work breaks through the traditional constraints of log-coneavity in data distributions and strict regularity requirements for the score function (or its estima- tor), offering novel perspective on the W2 convergence of SGMs through coupling techniques.
with all constants explicitly dependent on the parameters of the data distribution. Unlike previous works, which often present these constants in non-explicit forms or as arbitrarily rescaled factors, our analysis provides transparency. This clarity enables precise assessment of how input parameters influence convergence, facilitating informed decisions when designing neural architectures and optimizing SGMs for real- worldworld applications.
Building on the concept of weak log-concavity (see, e.g.. Conforti, 2024), we establish a foundational framework for studying data distributions under significantly relaxed convexity assumptions. By leveraging the regularizing prop- erties of the Ornstein-Uhlenber (OU) process, we demon- strate how its Gaussian stationary distribution progressively enhances the regularity of the initial data distribution. This regularization propagates weak log-concavity over time, ul- timately transitioning into strongly log-concave regime. Using a PDE-based approach inspired by Conforti et al. (2023a;b), we rigorously track the propagation of weak log-concavity through the HJB equation satisfied by the log- density of the forward process. Our analysis provides ex- plicit estimates for the evolution of the weak log-concavity constant along the OU flow, culminating in a precise charac- terization of the transition to strong log-concavity (see Ap- pendix B.3).
The structure of this paper is as follows. Section 2 intro- duces SGMs, presenting the general framework for analyz- ing these models and specifying the assumptions required to establish our convergence bounds. In Section 3, we focus on the main result: the convergence bound for SGMs in Wy-distance, highlighting parallels with the key features identified in KL-divergence bounds for SGMs. Section 4 demonstrates the validity of our assumptions by showing that the general class of Gaussian mixtures satisfies the conditions necessary for the bound to hold, thereby estab- lishing a strong connection with bounds in early stopping regimes. Section 5 explores the underlying features con- tributing to the remarkable success and effectiveness of SGMs, including the contractive properties of the OU flow, which form the basis for proving our result. Finally, Sec- tion 6 provides concise sketch of the proof of the main result.
We also identify and analyze two regimes for the backward stochastic differential equation (SDE): a contractive regime and a non-contractive one. In Appendix B.3, we precisely quantify the transition point where the drift of the backward process ceases to be contractive, a critical insight for de- signing robust neural architectures and optimizing practical algorithms.
2. Score Generative Models
Let Taaa E P(RR)) represent a probability distribution on Rd. from which want to generate samples. SGMs enable this by following a two-step approach: first, transforming data into noise, and second, learning how to reverse this process to recover data from noise.
Additionally, we show that Gaussian mixtures inherently sat- isfy the weak log-concavity and log-Lipschitz assumptions required by our framework. In Proposition 4. we quantify these properties and demonstrate their compatibility with our approach. Moreover, our methods extend to the analysis of convolutions of densities with Gaussian kernels. The OU flow, which acts as a Gaussian kernel, regularizes the initial distribution, ensuring stability even under early stopping regimes. This versatility underscores the broad applicability of our results, transcending the assumptions specified in H1.
To "create noise from data," SGMs employ an ergodic for- ward Markov process that begins with the data distribution and eventually converges to an invariant distribution, usually Gaussian. This invariant distribution serves as the starting "noise," which is easily generated by evolving the forward process from the data.
This corresponds to fixing a time horizon T' > 0 and consid- ering a d-dimensional ergodic diffusion over [o, T] via the following SDE:
A key advantage of our approach is that it circumvents the strict regularity conditions on the score function and its esti- mator often imposed by prior studies (e.g., Gao et al., 2023; Lee et al., 2022; Kwon et al., 2022; Bruno al., 2023; Tang & Zhao, 2024). Instead, we rely solely on the mild assump- tions of weak log-concavity and one-sided log-L.ipschizz reg- ularity of the data distribution. These assumptions suffice to derive the required score function regularity directly, elimi- nating the need for additional constraints (see Appendix B). This shift broadens the applicability of our framework while simplifying practical implementation.
dx, = 33x.))X)) )dtt + EdB,,
(1)
for E [0,T], with 3: Rdd - Rd a drift function, E e dddd fixed covariance matrix, and (Bi)r20 d-dimensional Brownian motion, initialized at X, ~ -aata- Under mild assumptions on B, this equation admits unique solutions and is associated with a Markov semigroup (Pt) with unique stationary distribution 7x. Moreover, the law of the process X, admits a density P. w.r.t. the Lebesgue measure.
Finally, we derive fully explicit W2 -convergence bound,
3

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Beyond Cog-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
To "create data from noise," the forward process is reversed using its time-reversal, known as the backward process. Specifically, we can sample from the noisy invariant dis- tribution (which is easy to do), then apply the backward dynamics starting from these noisy samples. Since we are using the reversed process, at time T, the backward process ideally yields samples from the target distribution data-
with { sequence of i.i.d. standard Gaussian random variables. Finally, this aims return L(X;), an approxima- tion of ddata.
2.1. OU-base Score Generative Models
We focus now on the OU case. In this case, Too is the standard Gaussian distribution, and forward and backward processes turn respectively into
More rigorously, SGMs aim at implementing the time- reversal process (see, e.g., Anderson, 1982; Haussmann & Pardoux, 1986; Föllmer, 2005) defined by
dx, -- X + V2dB..
(3)
ax, (X, 2V log P T (X. V2dB,
(4)
dx, = (-3(xx)) - ΣΣ Vog P EdB, for E [0, T7. with X,, ~ ((XT) and (B,) a Brownian motion, explicitely characterized in Haussmann Pardoux (1986, Remark 2.5).
for t [0. Since V log To (z) = I. equation (4) can be reformulated equivalently as
b + v2dB,,
(5)
To put into practice such procedure, three approximations must be made:
for t E [0, with, for (t.z) E [0, x R
1. Since sampling from L( ((Xr) is not feasible, the back- ward process is initialized at the easy-to-sample sta- tionary distribution Too.
bx(x) == xx p((x) == [x
- + 2V log Pe())
(6)
(7)
This framework, opposed to (4), has been adopted in several works (see, e.g., Conforti et al., 2023a; Strasman al., 2024), due its advantage of maintaining the same sign for the drift term as in the forward dynamies.
2. The score function (t,z) + V log pee (x) is unknown in closed form, as depends on the (not directly ac- cessible) distribution Tdata- and thus it needs to be es- timated. This score function can be interpreted as conditional expectation (see, e.g., Equation 49, Con- forti et al., 2023a), key insight behind the success of SGMs. By leveraging the fact that a conditional ex- pectation can be represented as an - L2-projection (see, e.g., Corollary 87 Klenke, 2013), the score function can be estimated by training model 0 -> se(t, xr) to minimize the matrenng loss
In this article, we shall consider SGMs that generate ap- proximate trajectories of the backward process based on its representation (5). This means that, for the learned param- eter 0* and a sequence of step sizes { with N 2 1 such that
N
hus = T,
I [ts X.) Vog P. (.((,))) dt.
0
we consider the OU-based SGM deseribed by Xi To and
(2)
over a (rich enough) parametric family (so 0}.
X = Xix + hs (-X;, + 25g (T - tk X
3. As the continuous-time SDE can not be simulated actly, the process is discretized, often using the Eu- MaruMaaaa (EM) scheme or other stochastic inte- grators.
(8)
+ Zk.
for l: E (O...., N 1}, with 0 the minimizer of
0 + / | X.) v Vogg (x.)) de, (9)
The resulting algorithm (Xt) runs the EM scheme for the estimated backward process initialized at the sta- tionary distribution of (1): for the learned parameter 8* 1, such that EN iyc T, we set X Too and compute for k E (O....,N' ...N 1}
over a properly chosen parametric class (ăg : 0 θ}.
and sequence of step sizes {
3. Convergence guarantees for OU-Based SGMs
X X Xt, + -a (-8(X7) + sq. (T - tk, X7,))
To understand the performance of the proposed SGM al- gorithm, we aim to provide quantitative error estimates
+ + 2hy Ebk

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Remark 3.2. These weak assumptions represent a novel contribution to the literature on W2 convergence of SGMs. In fact, the entire class of Gaussian mixtures satisfies both Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
between the distribution ((X+) and the data distribution
the weak log-coneavity and (one-sided) log-Lipschitz con- tinuity conditions. In Proposition 4.1, we provide explicit values for the weak log-concavity constants as well as the (one-sided) Lipscicct constants, illustrating the broad applicability and generality of these assumptions.
dddaa-
First, for given differentiable vector field 3, define its weak convexity profile as
(v)(z)
(y() (x y) y/
(r(r)
inf -.veR=:ll--ll
Remark 3.3. We remark that, shown in Appendix B.I, Assumption H1 implies that Tdata has finite second order moment. In the sequel, we denote by
(10)
This function can be regarded as an integrated convexity lower bound for B, for points that are at distance r > 0. This definition frequently arises in applications of coupling methods to study the long-term behavior of Fokncr-Planck equations (see, e.g., Conforti, 2023; 2024). While K3 Z 0 directly corresponds to the convexity of B, using non- uniform lower bounds Ki allows for the development of a more general notion of convexity, often called weak convexity.
| ||||| Teaaa (ck).
m1 =
(15)
H2 There exists 2 0 and H E - such that for any k E {0,.... N)
Pr (Xix) 80. (T- Lx, x 11 E.,
where the L.2-nrmm is defined as || 11I1===== E[| 12/1/22
Definition 3.1. We say that a vector field 33 is weakly convex if its weak convexity profile K3 defined in (10) satisfies
Remark 3.4. Assumptions of this type have already been considered in the literature (see, e-g., Gao al., 2023; Bruno et al., 2023; Strasman et al., 2024). We remark that, since we take the expectation over the EM algorithm (Xt, k= for which the density is known, Assumption H2 is not only theoretically well-founded but also practically verifiable. This makes robust and applicable frame- work in real-world settings. In addition, in the simple case ddata NNu I) for some unknown fa E IRd when approxi- mating the score function V log (xr) by means of the neu- ral networks (e-00= 0 Rd), Assumption H2 holds true for the minimizer 0 = of (2) and any >> O. Under the additional assumption that &g- (T tk, is uniformly Lip- schitz space, Assumption H2 becomes fully compatible with the standard estimation error assumptions commonly used in the literature (see, e.g. Chen al., 2022a; Li et al., 2023; Conforti et al., 2023a). This aligns with Proposi- tion B.4, which establishes that the score function V Voggr is Lipschitz continuous in space, for & E [0,
(r) 2 a (r),
(11)
for some positive constants o, M > O, with fas defined as
fw = M tanh (rVM/2)-
(12)
Moreover, we say that / is weakly concave if -B 1-rr is weakly convex.
We are now able to state our assumptions.
HI The data distribution Tdata is absolutely continu- ous w.r.t Lebesgue measure with Fdria(dr) = exp (-U(()) dæ, for some function U Rd - R. such that
(i) VU is Ln-o-iiidded Lipschitz, with La 2 0. i.e.,
3.1. Main result
(VU(x)-VULL (x- - n) S Lull y|?
(13)
Under the assumptions stated above, we now present our main result. For clarity, we provide an informal version here and defer the formal seatement-with fully explicit constants- to Appendix C.
for any y E Rdd (ii) U is weakly convex, with weak convexity profile Ky satisfying
Theorem 3.5 (Informal). Suppose that Assumption HI and H2 hold. Consider the discretization (tx. 0 S k S N} of [0, T] of constant step size small enough. Then, there exists a constant C C(a, M, Lv) > which depends exponentially the parameters M. and Lu, such that
RU (r) 2 (r((), fm)())))
(14)
for some a, .M > 0.
W2 T ((Xix))
S C -TW, (Trdnta: TOx) + ET + vivvd/ma
5

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Beyond Log- Cag-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Remark 3.6. We highlight the following.
Proposition 4.1. Let Pv, be Gaussian mixture on Rdd hav- ing density law
This Wg-convergence bound aligns with recent liter- ature on these models, both by generalizing results previously established for log-concave distributions and by identifying the key features of these models as captured through KI. divergence: the initialization error decreases exponentially with T (Conforti et al., 2023a; Strasman et al., 2024; Benton et al., 2024); the score-estimation error is proportional to eT (Con- forti et al., 2023a; Lee et al., 2022; 2023; Chen al., 2022a); the discretization error depends on vhmz d with h mesh of the time-grid (Pagès, 2018), m2 second order moment of datt (Conforti et al., 2023a; Chen et al., 2023), and d dimension of the space (Bruno et al., 2023; Strasman et al., 2024; Gao et al., 2023). These features are present up to a finite multiplicative constant that depend on the parameters a, M, and Ly characterizing the data distribution dama- and that is fully explicit (see Theorem C.1 for its explicit expres- sion).
1
| pel
Pa(x)) = Σ B: i=1
exp
(2
207
for x E Rd, with oi > 0, Mi E R and 3g E [0, 11. for E (l,....n). such that Li=1 =1 Then, - log Pr. is weakly convex and V logge Pn is Lipschitz.
In Appendix A, Proposition A. we also quantify and ex- press the related constants.
Remark 4.2. The result presented above can be directly gen- eralized to the case where the covariance matrices of each mode of the Gaussian mixture are of full rank but not scalar. In this more general setting, the parameter 1/ap, (respec- tively. 1/3p.) corresponds to the maximum (respectively, minimum) eigenvalue of the covariance matrices associated with each mode. Under the full-rank assumption, these eigenvalues are strictly positive, ensuring that the frame- work remains applicable and the bounds derived continue to hold with appropriately adjusted parameters.
For sake of simplicity, the theorem is presented for uniform mesh of the interval [0, .). Proceeding as in Strasman et al. (2024), the analysis can be extended to non-uniform subdivision of [0, T] by consider- ing the forward and backward SDEs (3)-(5) as time- inhomogeneous.
5. Regime Switching
By exploiting the regularizing effects of the forward process, the weak log-concavity of the data distribution transitions to full log-concavity over time. Additionally, the drift of the time-reversed OU process alternates between contractive and non-contractive phase, reflecting the evolving concavity dynamics. By rigorously tracing how weak log-concavity propagates through the HJB equation governing the log- density of the forward process, we derive an exact expres- sion for the critical moment E(a, at which the marginals of the forward process become strongly log-concave, as given in (22). We also provide an explicit lower bound T(a, M, 0), defined in (24), for the time T* beyond which the drift of the backward OU process loses contractivity.
Following Conforti et al. (2023a), the bound can be re- formulated to depend directly on by refining Assump- tion H2. As the solution X, to (3) converges in law to To the modified score function (t, x) + V log pe (x) approaches zero for large i. Accounting for this behav- ior allows scale the required precision as 1/T.
This Wo-bound exhibits the same dependencies T, T,&, d, h and ma as the KL-bbuudss provided in (e.g.) Chen et al. (2023) and Conforti et al. (2023a). A key distinction, however, lies in the fact that our result is expressed in terms of the Wasserstein dis- tance W2 (Tdata: R rather than the KL divergence KL (Taaaalllee ), making its estimation from samples more practical (see, e.g., Strasman et al., 2024), and is directly derived with the use of coupling techniques.
Log-Concavity,
p' is only weakly log-concave for t E [0, E(a M)];
. 7 is log-concave for 1 E [E(a,, M).
Contractivity properties of the time-reversal process.
4. Gaussian mixture example
bx(z) is not (necessarily) contractive, for t E [0, T(a, M,0)] (i.c., it is non-contraetive in [o. T*), while contractive in the interval (T(,, O))::
We demonstrate that Gaussian mixtures naturally fulfill the weak log-concavity and (one-sided) log-Lipschitz condi- tions central to our framework. As the OU flow- serving as Gaussian kernel-rrguulariess the initial distribution, it guarantees stability even with early stopping. Consequently, our approach extends to the analysis of convolutions be- tween densities and Gaussian kernels. This adaptability highlights the broad relevance of our findings, exceeding the specific assumptions outlined in H1.
(z) is contractive, for t E [T(a, M].
Further details and the explicit formulas for E(a, M) and T(a, M, can be found in Appendix B.3.
This regime shift is a key element in the effectiveness of the SGMs. SDEs with contractive flows exhibit advanta- geous properties related to efficiency guarantees (see, e.g.,
6

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Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in Wy-distance
Dalalyan, 2017; Durmus & Moulines, 2017; Cheng al.,
between X and XN, we obtain
2018; Dwivedi et al., 2019; Shen & Lee, 2019; Cao al., 2020; Mou et al., 2021; Li et al., 2021) that we can exploit in the low-time regime.
X1e11 X.N 1k11
Xx, -xN + plat1 {- {-(X,- 1-xX,-X0 -xx)
bb (x.) V log Po to (xxx) )} dtt
6. Sketch of the proof of the main result
+2 v pr
We now present a sketch of the proof of Theorem 3.5. Our analysis is based the observation that, in the practical implementation of the algorithm defined in (8), three suc- cessive approximations introduce distinct sources of error: dime-discretzzation initialization, and aporxapproximttton. To analyze these errors, we work with time- continuuus in- terpolations of the following four processes:
By applying the triangle inequality, we obtain:
N X ra -
Xn - xxx
f dt oof (xu- xxx)
Backward OU process. The time-reversal (Xe)ee of the OU process defined in (5).
+2 (Vlogrr-s, Pr- (I+n)- -Vlog -Vlg)-Vlogrr-- (XM) )
|1,m Nat) di (-(xi-- (x.)
EM-discretization scheme. The EM-approximation (XW) k of the backward process (5), started at XN I(x,)
+2 Vlogār- (x.) - Vlggār-t (xx)))]/
A1,x + A2,k:
Initialization error. The EM appapppoxnnntiion
(Xx)) Jkmo of the backward process (5), started at X TTc
For the first term, we have:
A k
Score approximation. (Xt)) defined in (8).
The generative process
= 8,, - -xN + + ||- (x1- -xa)
+2 V Plogrr - (x (x)
-V Vlogpr- -1x (x 112
These auxiliary processes allow us to separately track the three sources of error. Using the triangle inequality, we obtain:
(Xn=-xi) T
+ 2hE
Pog +2 V Pog -Vog)- (xx)))
Wa Taaat: ((Xix)) S (a(
To bound this term, we need regularity properties of the score function. One of the main challenges of this proof is to derive minimal assumptions on the data that ensure the minimal regularity properties of the score function required for the bound. To this aim, we adopt a PDE-based approach, drawing on the insights from Conforti et al. (2023a;b), not- ing that (t, + - log (r)) solves a HJB equation. Combining this with the regularizing effects of the OU process, we show in Appendix B that the regularity Assump- tions HI propagates naturally along the HJB equation:
ws ( )
+ We (C(PF))) ((
Bound on Wa ((x))) Consider the syn-
chronous coupling between (X () and the continuous- time interpolation of (X o with the same initialization, i.e. use the same Brownian motion to drive the two pro-
cesses and set X, = X... Then, it holds
(t,z)) H Vlog turns out to be Lr-
Lipschitz in space, with LT as in (27) and bounded by L as in (27);
Wa ((())) S X X
122
* (t.z) + log PT (x) remains weakly convex with
To bound the right-hand side, we aim to estimate
X 2113
R- lug Pr - (r) Cr-t,
X.N.ille ILc in terms of |'x,, - X 1I1, and develop a recur- sion. Since we have considered the synchronous coupling
and Cr- as in (20).
7

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
These regularity properties enable to bound A1 as:
with the ōgs as before. We bound || X66 Il in the following (by now) standard way (see, e.g-, the proof of Proposition C.2, Strasman et al., 2024)
A1. - dz [ - X
for some du depending on Lt,, and Cix For the second term, we have
|X0 - X L2 = W2 ((Xx)) Se We (Tdata Toa) Toa)
Using the previous considerations on the contractivity prop- erties of the forward flow, we get that the product I de is uniformly bounded by constant depending on the pa- rameters of the model. This yields to have
13 = 1,-0 1 (o.( Dee (xx)) dt
(x)
1.2
To bound this term, inspired by Conforti et al. (2023a, Propo- sition 2), we adopt a stochastic control perspective. We inter- pret the backward process (5) as the solution to a stochastic control problem and the term (2V log ( ()))o as the solution to the adjoint equation within a stochastic max- imum principle. This allows us to bound (up to a constant) A2, with h h/hld/mp.
Wi (1 & Teee-veee "Wa(Tamm Roo).
Bound on Wa ( ((X;,)). Consider the syn- chronous coupling between the continuous-time interpo- lations of (xxe k=o and (X7, ) with the same initial- ization, i.e., X = Xū- Using the evolution of these processes, together with the triangle inequality, we get
Combining the bounds on AI, and A2, we derive the recursion
{x
X
X - -X
L2
N-1
N-IN-1
<
800- X
II
oe + ChVhVddms
m2
Σ II
dz
S X -Xi,
(=0
t=k
1xt1
N-1N- N-1
+
dt dt d (xx;;-- 1-(X,--X,) -X;)
-Chhhvdmm m2
Σ II
5e
+2
V VlogĀr- (X2e) v log pr (xi)
Given the regime switching detailed in Section 5, we then analyze the low-time and large-time regimes separately, uti- lizing the contractive properties of SDEs at small times to establish & < 1, and applying brute force estimates for large times.
4
dt
V Vlogrr-u (Xt)) 3g. (T- the Xix)
B1,8 + B2
These remarks yields that
N-IN-1
By reproposing a similar argument as before, we get
C 8e S Cr. E k= t=k
h
BI S d |X li
This let us conclude that
By using Assumption H2, we obtain B2, S 4hc. There- fore, developing the recursion as before, we derive the bound
We ((XX)) & Whvldmmg..
Wa (x(C(X)). S TT.
Bound on We (I(())) Consider the syn- chronous coupling between the continuous-time interpo- lations of (X k=n and (X with initialization satis-
7. Conclusions
fying W7( ((XT)) = ||800 1l Then, holds W2 (C(X ), 1( < | -xx
This paper presents a unified framework for deriving W2- convergence bounds for SGMs, leveraging both PDE and stochastic control approaches, while relaxing strong regular- ity assumptions the data distribution, the score function and its estimator. The results mark a significant advance- ment, requiring only weak log-concavity and one-sided log- Lipschitz conditions on the data distribution, with no regu- larity needed for the score or its estimator. This broadens applicability to diverse data types.
By mirroring the previous argument and developing the recursion over the time intervals [1,+1,2]], [R), we get
N-1
X X L2 < =|
X L2 II 8z,
E=
8

---

Beyond Cog-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Using Gaussian mixtures, we illustrate the versatility of framework and show how Gaussian kernel convolutions improve early-stopping methods via regularization. We also analyze how weak cogcconcyvity evolves into full log- concavity over time, and how the drift of the time-reversed OU process shifts between contractive and non-contractive regimes, reflecting this transition.
ing with denoising auto-encoders and langevin sampling. arXiv preprint arXiv:2002.00107, 2020.
Bouchut, F., James, F., and Mancini, S. Uniqueness and weak stability for multi-dimensional transport equations with one-sided lipschitz coefficient. Annali della Scuola Normale Superiore di 2005.
di Scienze, 4(1):1-25,
While our bound depends exponentially on a, M, and Lu. it would be valuable to investigate whether this can be im- proved to polynomial dependence-2 question we leave for future work.
Bruno, S., Zhang, Y., Lim, D.-Y., Akyildiz, Ô. D., and Sa- banis, S. On diffusion- generative models and their error bounds: The log- log-concave case with full convergence estimates. arXiv preprint arXiv:2311.1358. 2023.
Acknowledgements
Cao, Y., Lu, J., and Wang, L. Complexity of randomized algorithms for underdamped langevin dynamics. arXiv preprint ar arXiv:2003.00 09900, 2020.
We gratefully acknowledge Giovanni Conforti, Alain Dur- mus, and Gabriel V. Cardoso for their insightful discussions and valuable feedback throughout the development of this work. We also thank the anonymous reviewers for their constructive comments, which significantly improved the quality of this paper.
Cardoso, G. V. Bedin, L., Duchateau, J., Dubois, R.. and Moulines, E. Bayesian ecg reconstruction using denoising diffusion generative models. arXiv preprint arXiv:2401 05388. 2023.
The work of M.G.S. has been supported by the Paris Ile-de- France Région in the framework of DIM A14IDE. The work of A.O. was funded by the European Union (ERC-2022- SYG-OCEAN-1010716601) Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union or the European Re- search Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
Cattiaux, P., Conforti, G., Gentil, I., and I Léonard, C. Time reversal of diffusion processes under finite entropy condition. In Annales de I'Institut Henri Poincaré (B) Probabilités et Statistiques, volume 59, pp. 1884-1881. Institut Henri Poincaré, 2023.
Cecchin, A., Conforti, G., Durmus, A.. and Eichinger, K. The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interac- tions. arXiv preprint S1ti:::4009099999 2024.
Impact Statement
Chen, H., Lee, H., and Lu, J. Improved analysis of score- based generative modeling: User-friendly bounds un- der minimal smoothness assumptions. In International Conference on Machine Learning, pp. 4735-4763. PMLR, 2023.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must specifically highlighted here.
Chen, S., Chewi, S., Li, J., Li, Y., Salim, A., and Zhang, A. R. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209. 2022a.
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Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Appendix
The appendix includes the additional materials to support the findings and analyses presented in the main paper. Appendix delves into the Gaussian mixture example, establishing its weak log-concaving and log- Lipsschchitz properties with explicit derivations of the associated constants. Appendix B focuses on the propagation of regularity assumptions through the forward OU process, leveraging PDE-based techniques to demonstrate how weak log-concavity and Lipschitz regularity evolve over time. In this section, we also examine the dynamics of concavity and contractivity in the OU process, rigorously characterizing transitions from weak to strong log-concavity and identifying contractive and non-contractive regimes. Appendix C provides the formal counterpart of Theorem 3.5 and the proof of it. Finally, Appendix D compiles auxiliary results and technical lemmas needed to develop the argument carried on in the proof of the main result.
A. Gaussian mixture example
Proposition A.1. Let Pn be a Gaussian mixture in Rd having density law
Pn B: 1
|x (-Ir-Ni) ui
(2RO7)
exp
20
x E R
(16)
1=1
with 0, > O, My E TRd and B, E (0, 1), for i E (1,...., n}, such that Ei=1 3, = 1. Then, - llog is weakly convex with coefficients
n
|
app
Mpp. == 2n
(O...) of
0
1=1
Moreover, we have that Pn (3pm. + VM Lipschitz, with
3p. min (O....) of
Proof of Proposition 4.1. Step 1. Gaussian mixture of two equi-weighted modes. Consider, first, the following Gaussian mixture of two modes with equal weight, i.e., 31 = 32 = 1/2, each mode having same variance I²1. Remark that the property of being weakly log-concave invariant to translation. Therefore, up to a translation, we have that the density distribution of this law
1 Pa(x)= 2 (2TO2)
14 2
+ 1
|:c 14 2
d/2 exp
2 (2mo2)a/2 4/2 exp
20
202
for x E Idd, with j E Rd. This means that its score function is equal to
|
(- |x++a|* 20
exp #
20
exp
# exp (μ T x/02) -1
V Vlogp2(x)
0²
|ل=يع 20
|e+ul* 2 20
r²
exp (1r T x x/0²2) +1
exp
+ exp
We focus now on bounding K- logp Fix x, y E 12d, We have that
(V P2())
- V Vlogp2(u))
(xc - y)
1
(x
exp (n au/a²/)) -1 exp (N T /
x
|| y
0²
x
y12
exp (1 x/02) ) +1 exp // 02) +1
1
# (x v)
2 (cxp (μ 2//0) -xp=) T y/ y/02))
exp
x
8² || m
(exp (n x/02) (exp 0/02) +1)
1
4 (x 1 v)
2 (exp (1 (z v)/ -1) (exp (u x/0²) (exp (-n 9/02)+11 +1)
22 || - =1
X
1
# (x - y)
(exp (N T (x -
w)//0 (z-w)/o²
-1)
X
d²
| vl
(exp (n² (x v) /02) +1
1
2 1 (x 4)
4 (x - y)
x tanh
| vl
20
12

---

Beyond Cog-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
It is clear that, the minimum of the r.h.s. of the previous inequality, under the constraint [ y) = r, is reached for vector y that is colinear with ja. Taking y [ r, we get
| (r) 2 σ² o² r
|
R- log
tanh
2
From the definition of JM as in (12) and the fact that the function M + [m((r is increasing for fixed r > 0, we can take
1
VMp == 4V2 |
Ap
0² 1
Moreover, recalling the expression we got for V log p2 and repeating similar computations, we obtain that
| (x) - V Vlogpn(y)) (v)l -
0² || - yl +
1
|tanh
p (x y)
P
20²
1
||a/|
22 1 vl
|/0 vll $ (Ops Mpa) || vll.
where, in the second inequality we have used the sub-linearity of +> tanh x together with Cauchy-Schwarte inequality. Step 2. General Gaussian mixture. Consider pn defined as in (16). Therefore, its score function is
V (x)
(x)
1
3
T
oi
Pi x
(2na2)a12
exp
(- ||0 - 14ll
20?
i=1
12
Bi
(_1lx- Mill
1
n
||0: (_11o p4 12 20
Pn (x)
?
(2TO7)
d/2
exp
20
Pn (x)
a,
(2no7)a/2
d/2
exp
Fix r > 0 and z, y E Rd such that ||= 1/1 =r. We then have
1
V log Pn (x)
V (v)) (x-1)
2
n
|| (_11x-
1
a
B
1
|| p4 112
2
Pn (I)
o
(2TO?
22/2
d/2
exp
20?
Pn
(x)
o,
Mi
(2mo?)
d/2
exp
20
8:
1
||y Mil
1
n
B
1
|1 - mill
Pr (V)
(2TO??
d/2
exp
20
pn
o
14
(2
(210
d/2
exp
20
(x y)
1
B
1
|/2 - -pal?
3
1
l1y - 1al13
72
Pr (x)
(2TTO)//22
d/2
exp
20
Pn (v)
(2TO?)
d/2
exp
20
(x y)
1
3
1
(-11z || - 11:112
,2
Pn (J))
ui
(2no?)d/2
exp
20?
11y - 1 )}
1
3
1
(v)
1=1 o
#4
(2moz)a/2
exp
20?
(x (r-zy)
A + A2
We now proceed in bounding the two terms in the r.h.s. of the previous equation. We see that
A
x (2())) - y (x))"))] (x - y),
with
3,
(_ || M4i
vn (x)
of (2TO?) d/2 exp
Po xx)
20?
13

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Without loss of generality, since and y are interchangeable, we can suppose that (y) S (x). From the definition of Pr as in (16), we have that Opp.. S (2) and, therefore,
1 y) y) - -vn 2 Op..
A lo (x) y Un (v)]" (o - v) 2 i (x)(((())))))
Consider now A2 and denote e;(2) =- (2x07) exp (-11--,11)) for E IR and i E {1, (I,...,n.. nn). Using telescopic sum,
we have
1= (10 1 & P
A2
p2
+ 33c((1)
Σ ( شنزز 1 (e)
(x-2) y)
Ej (fc + 3,c شن= Byc
n
0/4 + B
1. + 3
(x - y)
2
%/ + (,e
%/2 + Bce(y)
with
i-1
i-1
0/ =
Σ 3
+ Σ
B, (
and
% Σ 1=1
(z)
+ Σ B,) f=1+1
1=1
f=++1
oj
This means that, using that
(7a + Bee(()) (7s + Bee(v)) 2 7.Bie(()) + +nBree(y)
since all 7/ 2 0, we get
B. (e)())))) - ( - 0i)
A2 =
(x (5-9)
p2
(ni + (1), (74 + (B)c(())
i=1
T
(e((z) c(1)) # - 2i)
1
(x- y)
72
B. (cx( + c(1))
1=1
1 (x(x) (v)
Mi
0i
T
(x - v)
7 cx(x)
+ c; (v)
7i/
tanh
n (x - y)
ui
(a
y)
+
Σ
1
tanh
[ (a 4)
1/4 T
(x (x y)
?0?
o?
2
20
7/2
First, note that, as in the Step 1, we have that
1 p
n (o v)
Mi
} (x - y) S 7 (M. (r), with VM, =
||m
tanh
of
of
Secondly, for a fixed O, we see that the function fas (r) is increasing in M. Therefore, we get
[4T (x (x- y)
#4
(s v) 2 1 ( with M
12
|4:1
tanh
20?
a?
a?
14

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Note that 0 is convex combination of the vectors {uy/o}) Therefore, using again that M (x)(F) is increasing for fixed > O, have
[4 [1 (I
11
1/ (s y) -m(())
tanh
2
7
Combining the previous bounds, together with monotonicity of M + (ualiv/r/22) for fixed O, we can conclude that
1 (r), with Mp. == 2n A2 2 - (M)
1
Combining the bound on A1 with the one on Ag, we can conclude that - log(Pn) weakly convex with parameters Op and Mp... We now bound
V log Px
V (o)) Following the same lines before, we get
V Pn)
Vlog (v)ll S ||r (x)) v)
Σ tanh
Mi (x
v)
)
14
20
0
%
Bi + B2.
From the definition Pn as in (16), have that uin (x) S 3p.. and, therefore,
Bi s max ( (z), (1)} ||
y/ s 3p, 1|x a/1
Analogously to the bound on A2, we now use the sub-linearity of æ + tanh one, together with the fact that / is a convex combination of the vectors {1r/ 1i4ts to get
inequality and the triangle
14 (x y)
14
vi
n
M4
14
1)
1/
B₂ -
S 2
yl
+
20
o,
%
7
|all|
< 2m
||
wl1
S Mp. ||
yl1
Putting together these inequalities, we obtain that V log(p,) is Lipschitz with Lipschitz constant (8pm. + Mpn).
B. Propagation of the assumptions
The proof of our Wy-convergence bound is based on some regularity properties the score function (t, x)) + log (c)) Therefore, in this section, establish the key regularity properties of V log p. which arise from the propagation of Assumption HI through the OU process flow (3).
B.1. Weak-log concavity implies finite second order moment
Proposition B.1. Suppose that Assumption (ii holds. Then, Taat admits second order moment.
Proof. Consider the following Taylor development up order two.
U(x) = ((0) + (U)x) x 1 (VU(()
for some (tx t [0, 1]}. From Assumption (iii,,, have that
-z)(x)T < -a||=||2 + [ar(I=|)]| < 2 + M ||]]]
This means that outside a ball B(0, R) with R big enough, there exists ar > such that
-V)()) -a for B(0, R)
15

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance From Bouchut et al. (2005, Lemma 2.2), get that the previous inequality implies that
V $ -aRld.
Therefore,
J2. |||| exp ( (-U(x)) dr
=I | ||0||² exp (-U(x)) dæ +
/
B(O | exp (-U(x)) dz
/ ||0|| exp (-U(x)) dr [R0,B(,,R)
(UU(0) 3 02all1l?) 20.
| exp
B(0,0)
B.2. Weak-convexity of the modified score function
As highlighted in Conforti et al. (2023a), we observe that the function (4,=² >> - log (x)) is solution to HJB equation. This observation allows us to establish an intriguing connection between the study of this class of non-linear PDEs and SGMs, as further explored in Conforti et al. (2023a); Sentiloni-Silved et al. (2024).
First, we leverage the invariance the class of weakly convex functions for the HJB equation satisfied by the log-density of the OU process (demonstrated by Conforti (2024)) show how weak log-concavity propagates along the flow of (3). To this end, let (S,) denote the semigroup generated by standard Brownian motion on R i.c.,
||
y112 2t
Szf(z) = (2mt)d/2 exp
f(y)dy,
(17)
with f general test function. Theorem B.2 (Theorem 2.1 in Conforti (2024)). Consider the class
FM == {gg E C¹ (R3) kg(r) 2 fM (r)r
Then, we have that
h E FM
- log S E FM,
for t 2 0.
(18)
Theorem B.2 provides substantial generalization of the the principle "once log-concave, always log-concave" by Saremi et al. (2023), to the weak log-concave setting. Indeed, when M 0, i.e., e is log-concave, (18) implies that S remains log-concave. We remark that in the log-concave setting, (18) follows directly from the Prckopa-Leindler inequality, whose application central to the findings Bruno et al. (2023); Gao et al. (2023); Strasman et al. (2024).
We can now examine how the constant of weak log-concavity propagates. This result corresponds to Conforti et (2023b, Lemma 5.9), under the same set of assumptions.
Lemma B.3 (Lemma 5.9 in Conforti et al. (2023b)). Assume that Assumption Hl(ii) holds and fix E [0, (). Then, the function + - log (x:) is weakly convex with weak convexity profile k K- log pr- satisfying
e (T-
1 SM
(T-
kr(rr) a + a1 - a + (1 - ale 2 1)
a (1 22T r)
(19)
with Cr- given by
C. (1 a)e-2 (a ale 21 M - 1.
(20)
In particular, the modified score function x + Vlogār- satisfies
(V (V)ogrr-(x)) - Vlogār (x - v) = -C 1|1---101 w1
for x. RR.
(21)
16

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
This lemma relies the fact that the the flow of the OU process (3) can be rewritten w.r.t the flow of the Brownian motion
p((x) S1-e-ze (U-11-19/2) (e (e-
This remark enables the application of Theorem B.2, to obtain an estimation of the constant of weak log-concavity Ct.
B.3. Regime Switching
jt (2:) is weakly convex, having weak convexity profile satisfying K- logge Fe (r) 2 Ct, with C as in (20). This implies that weak log-concavity of the data distribution evolves into log-concavity over time and that the drift of the time-reversed OU process alternates between contractive and non-contractive regimes.
Lemma B.3 shows that, for any /. E [0, 7), the map
Log-concavity. Note that the constant C + represents the (weak) log-concavity constant of the density D. associated with the process X. The estimate for (C₃ + 1) is coherent with the intuition we have on the SGMs. Indeed, for = 0, it matches the weak log-concavity constant of data as Co + 1 = a M, and, for t + +oo, it matches the log-concavity constant Too as
2x
+++
a + (1 a)e-21 (a (1 aee- 2012 M -1 1, for 1 +00.
Computing when C₁ + 112 O, we have that two regimes appear:
7. is only weakly log-concave for / E E(a, M)];
pe is log-coneave fort E [K(a, M].
with
log
12+ o
AT,
if M < 0,
E(a, M) =
(22)
0,
otherwise
When Taaaa is only weakly log-come log-concave, i.e., when a - -M < 0, we have that a²+M-a > 1. Thus, {(a, > 0 and two distinct regimes are present. Whereas, when the initial distribution dataa log-concave, i.c., when a - 2 0, we have that E(a, M) = and p' is log-concave the whole interval [0, T7.
Contractivity properties of the time-reversal process. [0,
Denote by T the following time
T inf {1
2x- log 0).
(23)
defining inf Ø = used to define the time reversal process, defined in (6), no longer contractive. Indeed, from the very definition of k.. log we get that
T. From equation (5), we see that the time T* corresponds to the first moment beyond which the drift b,.
(b)(x)) - b.(x)) (xr v) s -(2x log (1(x y|l) 1)|l y/
for E [0, T] and x, y E Rd. Also, denote by
M, = inf { [0,2] 2C, + 11++12p1 p).
for p [0, 1), with inf Ø ; T. From (19), we have that 7 < T(a, M, p), for any P [0. 1). This means that regimes present the time interval [0, T):
by (a)) not (necessarily) contractive, for E [0, T(, M, 0)).
17

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
(a)) contractive, for E [T(a, M];
We introduce T(a, P) as quantitative explicit bound for T* Indeed, straightforward computation shows that T(a, p) r7(a, p,
(24)
with
2 log
M+pa(l-a) (1-p)a2
+
(1+0)(1-a)² + (1-m)o2 (I-p)a²
(M (MMtrrtt ^T,
if 2a 2M 1 < 00,
Ta, M, p)
=
(25)
0,
otherwise
Note that the condition
+ par(1 -a)
+ p)(1 - a)²
+ M + pa(1 - a) 2
+
> (1 p))aa
-1- (1-p)a2
(1- (1-p)a?
equivalent to the condition 2a 2M 1 < 0. This means that T(a, M, is well-defined and T(a, M, p) E [0,
This regime shift is key element in the effectiveness of the SGMs. SDEs with contractive flows exhibit advantageous propertics related to efficiency guarantees (see, e.g., Dalalyan, 2017; Durmus & Moulines, 2017: Cheng al., 2018; Dwivedi et al., 2019; Shen & Lee, 2019; Cao et al., 2020; Mou et al., 2021; Li et al., 2021) that we can exploit in the large-time regime.
B.4. Regularity in space of the modified score function
Proposition B.4. Assume that Assumption HI holds and fix t E (). Then, it holds
\o S L. L. sERd
sup
(26)
where
L max min e- -(T ; e2(--1) Lw -C :CC+1)) L= m=xxx(//::13(2 Lu).
(27)
In particular, the modified score function + V Vo (x) ()) is Lg Lipschitz, i.e..
loggū V logg, Lu || - yll,
(28)
for any y Rdd
Proof. Step 1: Upper bound on 2 log Dh. Recall that the transition density associated to the Orstein-Uhlenbeck semigroup is given by
[ly xx1]]
qe(2, v) (2 - e 21 21/2 exp
(29)
2(1 e e-2
Therefore, p, is given by
/
ly 1 2(1 21) U(x) dr.
(2 - 20) exp
This means that p, is the density of the sum of two independent random variables Y, + Y, of density respectively Ao. and Ut such that
(x) dde ((''))()' = -c0.(())
|
(x) (2 - e 22))d/12 exp
d2,)(x)
2(1 - 2-
18

---

Beyond Cag-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance for functions Pb, and 01,r- From the proof of Saumard & Wellner (Proposition 7.1, 2014), get
V (- log (x) - Var(Voo. + X1 = =x) - -0 + Xi = a]] Var(Vo + X1 x) - + Xi =x]]
From Bouchut et al. (2005, Lemma 2.2) and Assumption HI(1), we get that the one-sided Lipschitz assumption entails the following inequality over the Hessian of the log-density
V & Lyla.
Combining this with the fact that 60,r(z) = U(e'x) + (resp. Φ1 = 1/(2(11- I1vll + C*) for some positive constant C' (resp. C"), we obtain
Vo & Lu la
resp. VoL.a & --e T 14
1
which yields that
Jla la.
V2 log 1.) (z) $ min {1 e1- 2 0 Lo
(30)
Step 2: Lower bound on log p. Using Lemma B.3, we get that (21) implies that
22 log 7.) (x) o (Ce + I)Ia-
(31)
Since the difference between log p. and log is the linear function x =x, we can take L,
min ; e e2 Tu + 1. This implies that we can define L, as
1
} +1:+
L = max
min
22:00 28
L LU
T-
Step 3: Global bound in time. If Co 11 a- M 2 0, we have that the initial distribution Tama log-concave and, from Lemma B.3, p is log-concave the whole interval [0, 77. In this case, the eigenvalues of the matrix v² (og p.)) are all positives and (30) provides bound for (-
log 7. 1)l22- Therefore, in this case a global bound in time for L, is
sup RE 0,0c)
min
1 -- 21 2 Lu }
+1 +11=++ Lu.
If C. +1= M < 0, we also need to compute supte -(C, + 1). In this case, we have that
c 2
-(c, + 1) (a + (1 ale 21 M
+ (1
ale 21
1
e 2- M
a(1 c 21) 2
a (1 a)e
1
M
a(1- e 21) 2 2t
aaee 21
1) +
mnax(1/a: () (M a))
Since M corresponds to the smallest eigenvalue of A and Lu can be take equal to |// in this case, we have that M- a < Lu. Therefore, we can take as global bound in time for L, the constant = max(1/a; 1) Lo).
19

---

Beyond Cognaviayvity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
C. Main Theorem
In this section we provide the formal version of Theorem 3.5 and we prove it. Theorem C.1. Suppose that Assumption HI and H2 hold. Consider the discretization (4x, 0 S & S N of T] of constant step size h such that
h < 2/912
(32)
Then, it holds that
W2 Tdata: ((Xix))
e -Wa (Tdatns Too) + 4s (T(a, M, 0))
+ 22 B++ GlvV (Ta,MMo0)
with n(a, M,9L/h/2) as in (25), L as in (27), and
B:= v m2 + d.
(33)
Proof. Recall the main regularity properties from Appendix B:
1. the modified score function (f, + V (og))(()) is L--Lipschitz in space and also I--Lipschitz in space uniformly in time, with L, and L as in (26)-(27);
2. the function (t, x))
(x) (c) is Ce-weakly convex, with C₁ as (20).
In the practical implementation of the algorithm, three successive approximations are made, generating three distinct sources of errors. To identify them, we introduce the following two processes.
Let ( k=o be the EM-approximation of the backward process (5) started at X L(Xr) and defined recursively on [fk, [[+1] as
tum Xx h4 (-X 2V log (X; / Z2h fort E [tx, tx+1,
with {Zx)a sequence of i.i.d. standard Gaussian random variables.
Let (X N be the EM-approximation of the backward process (5) started at Xg ~ Too and defined recursively on [tx, 16+1] as
X X ha (-X 2V log pr (X?.) + /2h2 Zk, for l [lx, (++)-
We also recall that (X²) No defined in (8) denotes the process started at Xij To and defined recursively on [t,, (x+1] as
X Xi, + he (-Xi, + 2sg- (T tk Xt,)) Zk, for E [te,
By an abuse of notation, we use XN X and X* to refer to both the discrete-tima versions of these processes and their continuous-time interpolations. Applying the triangle inequality, we derive the following decomposition of the emror bound
W2 Tdmaa L(Xix)) < (A + ( + W/ (x(()))))
In the following, we establish separate bounds for each term, contributing to the overall convergence bound, based on the preceding analysis.
20

---

Beyond Cog-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Bound on W2 ((x;)) Consider the synchronous coupling between (X xe and the continuous-time interpolation of (Xf) ) with the same initialization, i.e. use the same Brownian motion to drive the two processes and set X, = -X Then, it holds
W2 (
$| -X L1
To upper bound the r.h.s., we estimate | X Il by means of 1 x ll I.. and develop the recursion.
Fix 0 < h and, with abuse of notation, use T to denote T - and N to denote (T c)/h. This measure is necessary to enable the application of Conforti et al. (2023a, Proposition 2) and Proposition B.4 later on. As we considered the synchronous coupling between X and XN,, we get
Xam - X
1
X X + {- (X,-XX) +2 V log Ør (X.) V logg logār-u (x ) fat.
Using triangle inequality, we obtain
X
L2
1
-x,x
{ {- (x1-- - -x:x) + +2 22(vlogrr por to (xx.)) - V Vlogrr- tn (xix) dd
(34)
+ 1
+ 1//m put {- 1-xX, xx.)) +
22(Voggrr- (x.)
Po (xx)) di
L2
A + A2,k:
Bound of A The first term of r.h.s. of (34) can be bounded as
1x,1 x.N 2
Af,x =
La
+ 1²|l1 ||| (X.m- - xxii 2
V log pr (xm) V VlogPr-t. (xx) )1²
(Ixm. - x (0'--000 (X - -x²x)
+ 2hE
+2 ( Volgrr- (Ix.)= Vlogrr-1a (xx)))
Since h satisfies (32), using Lemma D.2, we have that
2CT-1+1 A1,
h 5
fort [o,T n(a,,, 9,922h/22)
912
with n/a,, M, p) as in (25).
Define N₂ == sup (k E (O..... N..,,N:::: t ST n(a, M, M2922h/2)} Using the regularity properties of the score function from Proposition B.4 and Lemma B.3, we have
X x
11 (2LT +1)² 11)2222(2Crr2) 22((2rr-, 44 + +1)) 1/2
for k < No.
AI,* 5
La
(35)
1122 1/
(1+ 122 (2LT + +1)² 22(2LT- + 1) 1/2 for k 2 N₂.
x
L2
E1, x
5z
(36)
21

---

Beyond Log-Cont Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Bound of A2,
Given the definition (6) of the backward drift by Jensen's inequality implies
11 1641
|1/,- ( bu -b- (X }dt)
A
(xx)
br- (X 1e) dt B
L2
du+1
1|0T- b X t/ | dl.
E
Applying Itô's formula and Conforti et al. (Proposition 2, 2023a), we obtain
db (x,) -dxx, + 2d (Vlogrr= (I,))
= - 2V lo (x. + 2V log pr (x.)} dl + v2 12(1+ 2V 21+2V2logTT pr (x.)) dB.
Fddt + 2 (1 2 log - (x.) dBc.
Therefore, using Jensen's inequality, Itô's isometry, Proposition B.4 and Lemma D.I, we get
E br-t (X.) - -r- (x
X ds + v2 [ (1+2V2 log pr- pr- (1+2V21og
(x.)) dB
=
lt
2 ds + 4
/" | log pT (I.) ds
$ E
4/
/' d {{1 + 4 42 log log (X.) 11 ds
X
ds 8
JEk
Mk41
E ||
$
2
ds + Shd (1 + 41.2)
2hB² + 8hd (1 4T.2)
where we have used that
for symmetric matrix A e Rdxdd with ||. [|Rx (resp. || (lop) the Frobenius norm (operatorial norm) of a matrix.
|A S d|
Consequently, have that
A2.6 S h pfx11 {2hB² + Shd (1 (4+4)) dt)
(1))
1/2 S vv B B (B+ h V2h (B+ (B+6LV)) n.
Finally, we obtain
441
XN
de Ne, x L2
+ V2h
(B= - (B-6VV h.
(37)
La
Developing the recursion (37) and using the fact that X  = X." we get
N-1
N-IN-1
1 XA
s 8 51
X
II E=0
8g + v2h
B (B+6LVa)
h Σ 6
k= t=
N-IN-1
V2h B 6LVd) h
II 8
Lemma D.2 yields that dR < 1 for ls < No. Whereas, for t 2 Nb. Proposition B.4 yields
8x (1+ 122 (2LT-02 + +1)" + 2h 2h(2Lr-1 +1)) 1/2
$1 12L +1) $ 1+3Lh
22

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Combining the previous remarks, we have
N-1N-1
Nk-1
NA -1
N
N-1 N-1 Σ
Σ de =
Σ
59 X
82 +
II
6g
R=
t=
k=0
t=
N=
K=Ni
t=k
N-1
N-1 N-1 Σ K=
$
II 6e
N=
N
SNN (1,1 + +Lhh
NM
(1+ 31h)
N-K-1
N=
N-
S (N,(1+3LM) N
Σ (1+3LA)
k=
1 (1 + 3L1) N-N
N1 + 3LM) N-N,
(1 + 3Lh)
No + 3Lh
x (1+3Lh)
N-N.
11a, M,9L M29LPh/2)
1
+
h
3Th
Putting all these inequalities together, we get
1 x 2 La S v2h B 18+661 6LVV)
(anva)(-nnaa (T-n(a,,,LLhh/)) + 3T. e M/
S vh B -LVV) (T(a,M,0)+
1
(L
3L
where, in the last inequality, we have used the fact that
T(a, M,9L?h/2) T - 11(a M, M,9L2h/2) <T M,, T(a, M.O).
Recalling that we have used ''' denote T E, we get
X T
X
2
S 2h
B +LLVV
1 [[a, 3L
bg
S 22 + 6IVV)
(Ta, +
1
3LL//aMM9L///2)
3L
Letting going to zero and using Fatou's lemma, we obtain
| - x S 22 B + (
GIva) (B+
(T(o, (,), + 3I.
Bound on W/ W5(4( (((()))) -(XP)) Consider the synchronous coupling between the continuous timeeeuoou----eeeeee interpolations of ( and (xx) with initialization satisfying
Wa(F [(())),(((7)) 11x xx 11...
Then, we have that
We < x - xx
23

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
Fix 0 e < h and, with abuse of notation, use T to denote T e and N to denote (T <)/h. This measure is necessary enable the application of Proposition B.4 later As previously done, aim to develop recursion over the time intervals [f++1, fx]. To this end, note that, based on the definitions of (X;*) )eeo0,7 and )eejo,r and using the triangle inequality, we have that
X
6x42
X
Lo
plett {
(38)
= X
(X - xxx + 2 (x v ÜT t (X2x) cll
Xxx +
Proceeding as the previous bound, we get
N-1
| - X L2
X - X
II
L2
E=
with 6g defined as in (36) We bound the first factor in the following (by now) standard (see, e.g., the proof of Proposition
C.2, Strasman al., 2024)
W2 ((KT)) se WW Toa) 1xx 1 1
To bound the second factor, we proceed as in the bound of W/2 (C(XT), and get
N-1
N-1
II
8 II
de S (1 (1+ +3L) (3b1/1.MM991113)
N=N
Putting these inequalities together, we get
| 1 S 2e.Fe_brooo.,,,2LM//11 (Taaaa )
Recalling that we have used T' to denote e, we let going zero and conclude using Fatou's lemma as in the bound of W2 (C(xT),
Bound on W/ (XP)) 1= and (Xt, N with the same initialization, i.e. use the same Brownian motion to drive the processes and set X = =X Then, it holds
Consider the synchronous coupling between the continuous-time interpolation of
We
< || - -
Fix 0 < < h and, with abuse of notation, T to denote T e and N to denote (T- e)/h. This measure is necessary to enable the application of Proposition B.4 later As done in the previous bounds, aim to develop recursion over the time intervals [1411, fx]. To this end, using the triangle inequality, we get
Ximulla
te+1
pott { (X -Xi)) + +2 ( Vlogrr (XX2) 8g. (T Les Xix)
) dl
X00 -Xix + s
1.2
(lat) {
pott
< X Xix +
(X -xi)) + v 1 (X:2) V Pog (Xix)
(39)
+ 4 (rame
platt ( (Vlogār-te (xt) .g. (T Leg Xix) )dl
12
BB, B2,
24

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distane Following the same reasoning of the previous sections, have that
B1 S 6x x X l
with 6% defined as in (36). Moreover, using Assumption H2, we have that B2 I 4hc.
Developing the recursion as in the previous sections and recalling that X Xo, we get
N-IN-1
| XT
& 4he
Σ
1 oz S
48
T(o, M,0) +
3L)
0= t=
Recalling that we have used 7 to denote - 6, we let going to zero and conclude using Fatou's lemma as in the bound of W2 ((xx)),
D. Technical lemmata
Lemma D.1. Assume that (iū))) holds. Then, for 1. 2 O,
e -((T-()) [ T (dx) + (1-e -02(0-1))) d) 1/2
X
sup 0SEST
s
sup 0StST
<B<,0
L2
for defined as (33).
Proof. Recall the following equality in law
X.=e- +
(40)
with No ~ Tdata G ~ N (I, II,, and Xo and G taken to be independent. Therefore, since Xo and G are independent and X
has the same law of X T-I, obtain
E
X
E
XT-1
<e
- [|xx0|]] - (12--2()-)) E [[Gाo]
2(( / Tan + (1 (1---2(T-1) d,
which concludes the proof.
Lemma D.2. Suppose that Assumption HI holds. Consider the regular discretization (4x, 0 S ks S N) (0 Ti of constant step size h. Assume that h > 0 satisfies (32). Then, we have that
2(2CT-++1)
h <
A1, fort E
(41)
91,2
with v/,, p) as in (25). Moreover, for all 0 k N - such that I n/a, M. 9T2 h/2).
0 0<1 12 21 (2 + +1) 11)2-2h(224 4 + +1) 1
Proof. Firstly, note that if satisfies (32), then Ph = 9L such that Ph E [0, 1). Recalling the definitions (24), (25) of T(a M. and n(a, p), have that 2C,
Ph. fort 2 T(a, Mt).). Otherwise said, have
2(2C, + 1)
A A1
for I E [T(a, M, M,
97,2
25

---

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
which is equivalent
2(2CT=++1)
h S 912
A1,
for E [0,n(a, M, (0,n(a, M
Let be
9 1:=1+h (2 - -2h
+ +1)
(42)
Completing the square, we obtain
61 (1 - h122 +1)) 112++2222TT +1) 22(22r-1, +1
-11-1(21T-1 1 +1)) + (L -1)
The first term if the r.h.s. of the previous equality is square, therefore always positive. The second term is always strictly positive, L 2 C for any t.
Secondly, we see that
E1 (2LT 22h(22T in + 1) h (M((2TT- +1)² 2220TT + +1) + -2(CCT- +1)
$ 1 + h (1(L
From (41), we have that S 1, for t S T - nTa, M, h2922 This concludes the proof.
26


Reviews:
[{"review_id": "OGkW50cBp4", "reviewer": "Reviewer_gn4M", "summary": "This paper considers the sampling efficiency of diffusion models driven by the OU process in terms of Wasserstein distance under a new kind of \"tilted\" score-estimation assumption. The theorem only needs a weaker curvature condition for the potential function rather than the standard log-concavity. These conditions are shown to be satisfied by mixture of Gaussians. The paper also discussed the changing of curvature property over time for the forward density.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort taken to provide such detailed and insightful feedback. These comments have been valuable in the clarification of key aspects of our work and strengthen its presentation.\n\n**Assumption H2.** We have clarified in the main text the comparability of Assumption H2 with the standard estimation error assumption commonly used in the literature (see, e.g., Conforti et al., 2025). Notably, our main result remains valid if H2 is replaced by an $L^2$-type condition on the score function of the continuous-time forward process, combined with the requirement that $\\tilde s_{\\theta^\\star}(T - t_k, \\cdot)$ is uniformly Lipschitz in space. Importantly, Proposition B.2 ensures that this additional condition is not restrictive. Under these assumptions, we can apply the triangle inequality together with Proposition B.2 to obtain: $ ||\\nabla \\log \\tilde p_{T-t_k}( X^\\star_{t_k}) - \\tilde s_{\\theta^\\star}( T-t_k, X^\\star_{t_k}) || \\leq || \\nabla\\log \\tilde p_{T-t_k}(\\overleftarrow X_{t_k} ) - \\tilde s_{\\theta^\\star}( T-t_k,\\overleftarrow X_{t_k}) || +(L+L')|| X^\\star_{t_k}-\\overleftarrow X_{t_k}||$,\nwith $L'$ the Lipschitz constant of the score estimator. Replacing H2 with the standard assumption, the Lipschitz regularity of the score and its estimator combined with a straightforward generalisation of Proposition C.6 in (Strasman et al., 2024) would lead to only a minor adjustment in the definition of $\\delta_k$ in Section D.3. As a result, aside from slight modifications to Lemma E.2, this substitution does not materially affect the proof of the main theorem or alter the key features of the final convergence bound. Hence, Assumption H2 is fully compatible with the classical framework, and its validity directly follows from the well-established correctness of the standard score estimation assumption. We would also like to refer to Appendix A in Chen et al. (2023), where it is demonstrated that the standard estimation error assumption holds in simple yet practically relevant scenarios. We opted for the formulation in Assumption H2 as it offers the most direct and tractable path within our proof technique. We hope that highlighting the connection between these formulations helps clarify the transparency of our bound and bridges the gap between different perspectives in the literature.\n\n**Link with the HJB equation.**\nWe fully acknowledge the importance of the link with the Hamilton-Jacobi-Bellman (HJB) equation. This connection is rooted in a long history on the analysis of SGMs (see, e.g., Berner et al., 2022; Zhang and Katsoulakis, 2023; Zhang et al., 2024a; Conforti et al., 2025). Following the suggestion of the reviewer, we dedicated a discussion to clarify this connection explicitly in the revised version of the manuscript, showing the main works where this has been used.  \n\n**Sentence constructions.**\nWe acknowledge the ambiguity in the phrasing “not (necessarily) contractive” and thank the reviewer for pointing this out. In the revised version of the manuscript, we better specify this making reference to the Appendix where we define $T^\\star$, the actual switching point in contractivity, and introduce $T(\\alpha,M)$ as a lower bound on $T^\\star$. Ultimately, this is not a limitation of our proof technique but rather a direct consequence of the fact that $T^\\star$ is not explicitly computable.\n\nFinally, we have softened the original phrasing regarding the dependence on the dimension $d$. In the revised version of the paper, we now emphasize that our bound is in line with previously established results.\n\nAdditionally, we would like to thank a reviewer for the misprint suggestion. We have happily introduced the required changes in the revised manuscript.", "claims_and_evidence": "See the Section \"Theoretical Claims\".", "methods_and_evaluation_criteria": "N/A", "theoretical_claims": "The proof of the main mixing theorem is largely correct, and it is great to have the Gaussian mixture example to illustrate the concept of weak convexity. But the major concern is:\n- Since the score is tilted by standard Gaussian, this Assumption H2 is really not directly comparable with previous assumptions in the literature.\n- If the authors do not provide some comparison between their Assumption H2, which is specialized to OU process and is tilted; and the standard $L^2$ estimation error assumption made in tons of previous (theoretical) papers on diffusion models, the significance of this paper is not clear.\n- In particular, the claim that the dependence on $d$ \"surpasses some earlier results\" might not be very fair.\n\nThe risk decomposition seems to be standard and as expected. The authors should elaborate more on the view of $(t, x) \\mapsto -\\log \\tilde{p}_{T-t}(x)$ as a solution to HJB, if it is indeed novel; instead of inserting Section 5 about the shifting of the curvature properties, which is relatively detached from the main mixing theorem of this paper.\n\nTechnically speaking, Section 5 is interesting, but the statement \"not (necessarily) contractive\" is really vague, since it is not clear whether it is the defect of the proof techniques or the inherent property of the backward process.", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "The formulation of weak convexity and the PDE-based view in this papers come from\n- Conforti, Giovanni, Alain Durmus, and Marta Gentiloni Silveri. \"Score diffusion models without early stopping: finite fisher information is all you need.\" arXiv e-prints (2023): arXiv-2308.\n- Conforti, Giovanni, Daniel Lacker, and Soumik Pal. \"Projected Langevin dynamics and a gradient flow for entropic optimal transport.\" arXiv preprint arXiv:2309.08598 (2023).\n\nThese are the major direct technical predecessors of this paper.\n\nThere is also an extensive line of papers on the sampling efficiency of diffusion models, but the assumptions on score estimation in the current paper are not directly comparable with those in the literature, e.g.:\n- Chen, Sitan, et al. \"Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.\" arXiv preprint arXiv:2209.11215 (2022).\n- Li, Gen, et al. \"Towards faster non-asymptotic convergence for diffusion-based generative models.\" arXiv preprint arXiv:2306.09251 (2023).", "essential_references_not_discussed": "To the best of my knowledge, technical references are well-discussed.", "other_strengths_and_weaknesses": "### Minor issues\n- Line 172: Maybe you miss the $\\Sigma \\Sigma^\\top$ here after the $+$.", "other_comments_or_suggestions": "Minor suggestion: use different notations for the brownian motion in the forward and backward processes, e.g., in (3) and (4).", "questions_for_authors": "N/A", "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": "lpWSpyVlZR", "reviewer": "Reviewer_u274", "summary": "This paper looks at showing that diffusion models can be quickly\nsampled from with bounded W2 error.  There is a long line of work\nshowing this for TV or KL error, but converting those to W2 incurs a\nsignificant penalty.  This paper shows a W2 bound directly, assuming\none-sided lipschitzness and weak log concavity.\n\n## update after rebuttal\n\nGiven this response, I maintain my score.\n\n(1) I asked for a direct comparison and you did not give one. At the very least you could give a bound for compact distributions, for comparison.\n\n(2) But I'm not convinced you need any additional assumption like compactness. You have weak convexity of the score, which seems to imply subexponential tails, which seems like it should imply (TV => W2).\n\n> a KL divergence bound does not, in general, imply a W2 bound without imposing additional, and often quite restrictive, assumptions, such as compact support,\n\nDoesn't even a 2.1st moment work?", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the detailed comments and constructive feedback. We highlight that, as stated in Theorem 3.4 and its accompanying discussion, this result indeed provides a bound up to a multiplicative constant depending on $\\alpha, M, L_U$, and the second-order moment $m_2$ of $\\pi_{\\rm data}$, via the constant $B$ defined in Equation (31). Such a multiplicative constant includes all the terms listed by the reviewer. We thank them for the careful reading and address their concern in the revised manuscript as follows.\n- Theorem 3.4 now displays the **dependence on $m_2$**. This quantity no longer appears in the multiplicative constant; instead, its influence on the bound is now explicitly reflected. We have also emphasized that the multiplicative constant depends on the parameters $\\alpha$, $M$, and $L_U$, with references to the explicit expression of the constant provided in the appendix.\n- We added a discussion on the leading term $e^{L_U \\eta}$, highlighting that it reflects the intrinsic complexity of transforming a potentially complex, multimodal distribution into a unimodal Gaussian through an OU process. In particular, this term captures the challenge posed by large-scale structure or mode separation in the target distribution, as also highlighted in Saremi et al., 2023.\n- We particularly appreciated the observation regarding the $\\sqrt{h} L_U d$ term. This helped us identify and correct a typo in the proof. The correct expression is actually $\\sqrt{h d} L_U$, and this has now been fixed in the revised version of the manuscript.\n\n**Comparison with $W_2$-bounds.** We acknowledge that the approach in Benton et al. (2024) relies on different assumptions and methodologies, including exponentially decreasing step sizes and early stopping, while avoiding regularity assumptions on the data distribution. Given these differences, a direct side-by-side comparison is non-trivial.\n\nOur bound is directly comparable to those of Chen et al. (2023) and Conforti et al. (2025), as it exhibits the same dependencies on $h$, $d$, $\\varepsilon$, $m_2$, and $T$. A key distinction, however, lies in the fact that our result is expressed in terms of the Wasserstein distance $W_2(\\pi_{\\text{data}}, \\pi_\\infty)$ rather than the KL divergence, which makes it more practical to estimate from samples—see, e.g., Strasman et al. (2024).\n\n**Regularity assumptions.** We very much agree that early stopping strategies could relax or remove the Lipschitz assumption. However, we believe that the explicit form of contraction property of the data distribution remains crucial to ensure the stability and convergence guarantees of our analysis.", "claims_and_evidence": "I'm very confused, because Theorem 3.4 is informal, and I *think*\nTheorem D.1 is supposed to be the formal version of it.  But Theorem\nD.1 is far, far weaker than Theorem 3.4.\n\nRelative to Theorem 3.4, Theorem D.1 has:\n\n\n- A leading e^{L_u eta} term, which in the mixture-of-Gaussian case is\n  seems quite large if mu >> sigma\n\n- ... at least I think, a lot of these terms seem dimensionally\n  incorrect, like there should be some scaling that would be invariant\n  (if I denominate x in feet rather than meters, the same process\n  happens) but lots of terms would change.\n\n- The sqrt(h) L_U d term is missing\n\n- The sqrt(h) m_2 T term is missing\n\nThese set of errors really offend me, and make me recommend rejection\nuntil the paper can be cleaned up.  When I first read Theorem 3.4, I\nwas really impressed and wanted to accept.  But partly I was really\nimpressed because it avoided any dependence on the things you would\nget if you simply convert one of the KL/TV bounds (e.g. Benson et al)\ninto the W_2 setting; that would inherently give a dependence on the\nmoments.  But so does this method!  The paper just neglects to inform the\nreader in the main body.", "methods_and_evaluation_criteria": "no new algorithm or evaluation criteria", "theoretical_claims": "see above", "experimental_designs_or_analyses": "none", "relation_to_broader_scientific_literature": "it's well positioned, trying to get a W2 bound directly.", "essential_references_not_discussed": "seemed fine", "other_strengths_and_weaknesses": "see above", "other_comments_or_suggestions": "none", "questions_for_authors": "Is the result actually stronger than you would get by just converting\na KL bound like Benson?  Can you show this, for example, in the\nbalanced mixture of Gaussians case?\n\n\nWhat can you do without assuming lipschitzness or weak log concavity?\nShouldn't you be able to say something like: the t-smoothed\ndistribution is R/t onesided lipschitz, to have fewer assumptions?", "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": "M0UCU3CUYP", "reviewer": "Reviewer_3brP", "summary": "The paper studies the properties of Score-based Generative Models (SGMs) beyond the conventional setting where the data  distribution $\\pi_{\\mathrm{data}}$ is log-concave and satisfies certain regularity conditions. The analysis is based on the three approximations that must be made to implement SGMs. First, one must initialize the backward process at a stationary distribution which is easy to sample from. Second, one must estimate the score function by training a model to minimize the score-matching loss. Third, the process must be discretized as the continuous SDE cannot be solved exactly. \n\nThe work focuses on Ornstein-Uhlenbeck Generative Models wherein the initial stationary distribution is chosen as a standard Gaussian. The analysis is carried out under the assumption that the data distribution has a density $\\exp(-U(x))$ where $\\nabla U$ satisfies a one-sided Lipschitz property and is weakly convex (a class which includes Gaussian mixtures) as well as a second, more technical condtion (cf. H2) which has appeared previously in the literature. The main theorem provides an estimate of how close the distribution outputted by the algorithm is to the true data distribution in Wasserstein distance is as a function of the size of the time interval [0,T], the coarseness of the discretization of this interval $h$, the dimension, a factor figuring assumption H2, and the distance of the standard Gaussian from the true distribution and is line with other literature and improves on the dependence in dimension. \n\nThe remainder of the article pertains to explaining the main ideas underlying the proof and discussing the case where the true data is generated according to a Gaussian mixture.\n\n## update after rebuttal\n\nThe authors have answered the main questions that I have posed. As such, I maintain my initial score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thoughtful and positive feedback.\n\n**Assumption H2.** We emphasize that Assumption H2 is fully comparable to the standard estimation error assumptions widely used in the literature (see, e.g., Conforti et al., 2025; Benton et al., 2024; Chen et al., 2022a). In particular, our main result remains valid even if H2 is replaced by a more classical $L^2$-type control on the score approximation error of the forward process, provided that $\\tilde s_{\\theta^\\star}(T - t_k, \\cdot)$ is Lipschitz, uniformly in time. Importantly, Proposition B.2 ensures that this additional condition is not restrictive. Assumption H2 is therefore entirely consistent with standard theoretical frameworks, and its practical relevance is supported by well-established results. In particular, we refer to Appendix A of Chen et al. (2023), where this type of assumption is shown to hold in simple yet illustrative settings. We agree that a more detailed discussion of H2 would have added clarity. For this reason, we included a comment in the revised version of the manuscript to clarify this point.\n\n**Simulations.** We thank the reviewer for this valuable suggestion. We refer to Appendix E.3 of  Strasman et al. (2024) for an implementation of a similar bound. Our findings provide the theoretical foundation underlying these simulation results, and we believe a numerical study could be conducted by closely following the same methodological framework.\n\nFinally, we would like to thank the reviewer for the list of misprints and other small comments, which have been happily incorporated in the revised manuscript.", "claims_and_evidence": "The claims in this submission appear reasonable to me, though I am not an expert in this topic.", "methods_and_evaluation_criteria": "No experiments are performed in this work.", "theoretical_claims": "I did not verify the proofs in the supplement. Some arguments are provided in the main text and appear reasonable.", "experimental_designs_or_analyses": "No experiments are performed in this work.", "relation_to_broader_scientific_literature": "I am not very familiar with the broader literature related to SGMs/sampling. It appears that most results are contingent on some form of log-concavity and regularity assumptions (e.g. log-sobolev) and hence this paper provides some relaxation of these conditions.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "I believe the paper is well-written and presents a useful relaxation of some of the conditions for score-based generative modelling. It is shown that the condition H1 is compatible with a Gaussian mixture model, but the question of whether or not H2 holds in this setting is not addressed. As such, I believe the condition H2 merits further discussion. Other than this, the outline in the main text helps to elucidate the proof technique and highlights how each level of approximation is handled.", "other_comments_or_suggestions": "Line 179 right column: that are at distance -> that are at a distance.\nLine 239 right column: For sake of simplicity -> For the sake of simplicity.\nLine 369 right column: These remarks yields that -> These remarks yield that", "questions_for_authors": "1. As noted previously, I wonder how easy condition H2 is to check in practice. It is a bit confusing, for instance, that Section 4 concentrates on the establishing H1 for the Gaussian Mixture, but says nothing about H2. \n\n2. I wonder if it is feasible to empirically validate the derived theorem via some simulation. As the main contribution of this submission is theoretical, it would help to at least illustrate the findings at a numerical level to better motivate its applicability to real world problems.", "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": "igdkLMy0lq", "reviewer": "Reviewer_Vb4a", "summary": "This paper takes a renewed look at analysing convergence of score-based generative models, in the Wasserstein-2 metric.   The explore guarantees under a weaker assumption than log-concavity, namely weak log-concavity (introduced in Conforti, 2023).    This enables the authors to obtain natural non-asymptotic convergence results, under more realistic assumptions.   Leveraging this machinery,   they recognize that the log-density of the forward process is the solution of an HJB equation, which is exploited to track the weak-log concavity over time - enabling them to characterise the transition between the log-concave regime (close to Gaussian) and the weak-log concave setting (close to data distribution).\n\n# update after rebuttal\nBased on the response, I maintain my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's insightful comments and supportive review.\n\n**On Gaussian Mixtures.** As highlighted in Remark 4.2, the generalization of Proposition 4.1 to Gaussian mixtures with non-isotropic components is indeed straightforward. It involves considering bounds based on the minimum or maximum eigenvalues of the covariance matrices. This adjustment allows one to control the local Lipschitz constants of the score function accordingly.\n\nNevertheless, we would be very grateful if the reviewer could point us to the specific references they had in mind regarding recent results on the log-Lipschitz regularity of the score for general Gaussian mixtures. We would be happy to acknowledge them and expand our discussion in the revised version, possibly providing further insights into the verification of our assumptions in these more general settings.\n\n**AC assumption.** Since $\\pi_{\\infty}$ is a Gaussian distribution, absolute continuity (AC) of $\\pi_{\\mathrm{data}}$ w.r.t. $\\pi_{\\infty}$ is equivalent to AC w.r.t. the Lebesgue measure. This condition is directly implied by our Assumption H1. This requirement is still quite mild. For instance, if $\\pi_{\\mathrm{data}}$ were supported on a lower-dimensional manifold, the smoothing effect of the forward process ensures that $p_t$ becomes absolutely continuous w.r.t.\\ $\\pi_{\\infty}$ for any $t > 0$, due to convolution with Gaussian noise. This makes the assumption broadly applicable and not restrictive in practice. A comment clarifying this point has been added in the revised version of the manuscript.\n\n**Link with the HJB equation.** We appreciate this insightful remark and have added a short subsection in the revised manuscript to elaborate on the connection between HJB equations, control theory, and SGMs. This addition draws on recent works (e.g., Berner et al., 2022; Zhang and Katsoulakis, 2023; Zhang et al., 2024a; Conforti et al., 2025), and aims to better contextualize our analysis by highlighting this rich and increasingly active intersection.\n\n**Typo.** We are grateful to the reviewer, as this helped us to spot a typo that has now been corrected. The $\\sqrt{hd}T$-dependence remains valid.\n\n**On weakly log-concave distributions.** A notable class of weakly log-concave distributions arises from the convolution of distributions supported on lower-dimensional manifolds with a Gaussian kernel. This construction effectively smooths out the singularities, yielding a weakly log-concave distribution. This observation is also emphasized in Saremi et al. (2023). This insight may also help explain why early stopping techniques often perform better than directly modeling the data distribution: the intermediate distributions encountered during training are closer to weakly log-concave regimes, for which theoretical guarantees and sampling stability are more readily attainable. We have added a remark on this point in the revised version of the manuscript to clarify the broader applicability of our assumptions.", "claims_and_evidence": "The paper is entirely performing detailed mathematical analysis of the convergence of score based models.   All the results and claims in the paper are either proved, using convincing arguments,  or referred to previous works.  There are no problematic claims.", "methods_and_evaluation_criteria": "The proposed methods are sensible.   As there are no numerical experiments, the evaluation criteria is not relevant here.", "theoretical_claims": "I have checked the proof of Prop 4.1,  and all the theorems leading up to the main result, and the main result.    I have not followed references to results in other papers, and where I am not familiar, I have not dug too deeply.\n\nThe proofs look sound, the arguments look sensible.   I would add a few comments:\n\n1. Prop 4.1 -- there have been recent works which establish log lipschitness of the score for a general mixed gaussian, even with non-isotropic covariances.    Can this result be generalised to that setting?   It would be nice to see that, or an acknowledgement that it is possible, or not.\n2. An early, key assumption is the reformulation of equation (4) into (5) -- this is not a new assumption,  tracing back into one of Durmus' papers, and one of Cattiaux's earlier than that...   However, it is a non-trivial assumption that the initial distribution is ac with respect to pi^infinity.    The value of this assumption is clear, but the implications in terms of how limiting it is, is not.\n\n3. In terms of style - there is no page limit on the supplemetnary information - so why not provide a bit more information, e.g. the HJB is crucial to the \"once weak-log-concave, always weak log-concave\" argument which is used in the main result.    Can a little bit more details be provided in this text, to better contextualise this paper?\n\n4. The sqrt(dh)T contribution which arises from the EM discretisation is noteworthy,  but it really doesn't appear obvious from the main proof?   Is there a typo leading to equation (35), e.g. should the integrand be inside the square root?", "experimental_designs_or_analyses": "There are no numerical experiments in this paper.", "relation_to_broader_scientific_literature": "There have been several prior works which derive similar non-asymptotic estimates for SBGMs using different metrics, ranging from classical Renyi / alpha divergences and TV,   Kullback-Liebler, and then Wasserstein - like in this paper.\n\nIn the former group, there is most notably,  [Block & Mroueh 2020], Valentin De Bortoli's paper \"Convergence of denoising diffusion models under the manifold hypothesis\",  and Chen and Chewi's \"Sampling is as easy as learning the score....\" paper.   \n\nIn terms of Wasserstein-2, then de Bortoli, Heng, Doucet, et al's paper \"Diffusion schrödinger bridge with applications to score-based generative modeling\" and Lee, Lu, Tan's paper are the main contributions.\n\nMore recently, there was Sabanis et al's paper: \"On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates\",  Tang & Zhao's paper and Strassman et al, 2024.   The common thread in these works is the assumption of log-concavity of the Data Distribution, which is quite constraining and unrealistic.", "essential_references_not_discussed": "I cannot think of any essential papers which are published and not cited here, or discussed.   What I had already mentioned is that there is some papers which generalise Prop 4.1 to gaussian mixtures with general covariance -  but I believe these are still preprints.", "other_strengths_and_weaknesses": "I think this is a clear contribution as it generally relaxes some key assumptions for obtaining non-asymptotic results in W2 for SBGMs.     I would have liked the author to explore a bit more what data-distributions actually satisfy weak log-concavity beyond the Gaussian / Gaussian Mixture case.", "other_comments_or_suggestions": "I've made most comments elsewhere.", "questions_for_authors": "No further questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The manuscript analyzes the Wasserstein distance between the data distribution and the generation from score-based models, based on the weak log-concavity assumption of the data distribution, which is tracked over the forward process through the analysis of the HJB equation for the log-density. Overall, the reviewers agree upon the novelty of the analysis for score-based generative models, the authors have addressed most of the concerns in the discussion phase. Hence the meta-reviewer recommends this paper for acceptance.
Decision: Accept (poster)