Assessing the quality of denoising diffusion models in Wasserstein distance: noisy score and optimal bounds
Authors: Vahan Arsenyan
Elen Vardanyan
Arnak S. Dalalyan
Venue: NeurIPS 2025 poster ()
Keywords: Generative Modeling
Diffusion Processes
DDPM
Wasserstein Distance
Error Bounds
Log-Concave Distributions
Score Estimation
Primary area: theory

Abstract: Generative modeling aims to produce new random examples from an unknown target distribution, given access to a finite collection of examples. Among the leading approaches, denoising diffusion probabilistic models (DDPMs) construct such examples by mapping a Brownian motion via a diffusion process driven by an estimated score function. In this work, we first provide empirical evidence that DDPMs are robust to constant-variance noise in the score evaluations. We then establish finite-sample guarantees in Wasserstein-2 distance that exhibit two key features: (i) they characterize and quantify the robustness of DDPMs to noisy score estimates, and (ii) they achieve faster convergence rates than previously known results. Furthermore, we observe that the obtained rates match those known in the Gaussian case, implying their optimality.

Assessing the quality of denoising diffusion models in Wasserstein distance: noisy score and optimal bounds
Vahan Arsenyan'
Elen Vardanyan" Arnak S. Dalalyan
CREST, ENSAE, Institut Polytechnique de Paris
5 avenue Henry Le Chatelier 91764 Palaiseau, France
Abstract
Generative modeling aims to produce new random examples from an unknown target distribution, given access to finite collection of examples. Among the leading approaches, denoising diffusion probabilistic models (DDPMs) construct such examples by mapping a Brownian motion via a diffusion process driven by an estimated score function. this work, we first provide empirical evidence that DDPMs are robust to constant-variance noise in the score evaluations. We then establish finite-sample guarantees in Wasserstein-2 distance that exhibit two key features: (i) they characterize and quantify the robustness of DDPMs to noisy score estimates, and (ii) they achieve faster convergence rates than previously known results. Furthermore, we observe that the obtained rates match those known in the Gaussian case, implying their optimality.
Introduction
We study the problem of generative modeling, which aims to construct a mechanism capable of producing synthetic samples that mimic a target distribution P" given access to independent observa- tions from Pa This fundamental task lies at the core of numerous applications, including image, text, music, and molecule generation. Among the recent advances in this domain, Denoising Diffusion Probabilistic Models (DDPMs), introduced in [HJA20], have emerged as a remarkably effective class of generative models; see, e.g., [CMFW24, YZS+ TZ25] for comprehensive overviews. In this work, we contribute to the growing theoretical understanding of DDPMs by analyzing several of their key properties and performance guarantees.
The central idea underlying DDPMs is to construct a transport map that transforms a simple source of randomness into sample from the target distribution p More precisely, for any distribution p* there exists map defined via a stochastic differential equation (SDE) that takes as input a standard Gaussian vector Eo and a standard Brownian motion W, and outputs vector with distribution P* Importantly, only the drift term of the SDE depends on P and this dependence occurs through the score function, that is, the gradient of the lon-density of Gaussian- smoothed version of p* This formulation reduces the generative modeling task to that of score estimation: one can estimate the score function from data and substitute this estimate into the SDE to approximately sample from P*
For many commonly used datasets, such as CIFAR-10 and CelebA-HQ considered in Section 6, accurate estimators of the score function are available. Generating synthetic sample reduces to drawing a Gaussian vector together with the increments of a Brownian motion, and simulating the SDE defined by the pretrained score. This procedure requires multiple evaluations of the score estimator. The first question we address in this paper is: what happens if each evaluation returns value corrupted by additive centered noise? Such a scenario may arise when the pretrained model is hosted on remote server and communication introduces random perturbations, or when the score values are compressed using stochastic rounding. Anticipating our main findings, we emphasize that, perhaps counterintuitively. we observe that adding even constant level of noise to each score evaluation has only a limited effect on the quality of the generated samples; see Figure for an illustration.
"Equal Contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Figure 1: Generated images obtained by DDPM with constant-level noise added to the estimated score. Left: CelebA-HQ. Right: CIFARIO. The result is visually as good as the noiseless one.
The second question we investigate concerns the aceuracy of DDPMs when performance measured in terms of the Wasserstein distance. A natural criterion in this setting is the number of score function queries K required to achieve prescribed level of accuracy &. For the Gaussian target distribution, elementary computations show that = D(VD/e) where D denotes the ambient dimension. Surprisingly, however, it remains unclear whether DDPMs maintain this level of accuracy for broader classes of distributions beyond the Gaussian case.
Contributions.
The main contributions of this work can be summarized as follows:
We provide empirical evidence, based on experiments with the CIFAR-10 and CelebA-HQ datasets, that DDPMs are remarkably robust to noise in the evaluation of the score function. We derive non-asymptotic upper bounds on the Wasserstein-2 distance between the target distribution and the distribution induced by the DDPM with noisy score evaluations, thus offering a theoretical explanation for the observed robustness.
Our bounds match-up up to a multiplicative constant-thh rate vD/c of the case of a Gaussian target. Moreover, our results extend significantly broader class of distributions, including compactly supported semi-log-concave measures supported on low-dimensional subspaces.
Related work [KF1.22] highlighted the connection between DDPMs and the Wasserstein distance. the dimension and valid for a broad class of p -were established in [CCI,+23], covering several metrics. Unlike their result in total variation (TV) distance, their bound in Wasserstein distance has the poor scaling D$/= Subsequent work significantly improved this rate: [CLL23] achieved D4 /²2 under minimal assumptions, while [BZL 23, GNZ25, YY25, SOB 25] reduced it further to D/e², assuming stronger conditions on P", [SO25] proved the vD/E² rate and our paper closes the loop by proving that the optimal rate vD/e is achieved by the standard DDPM procedure. A related result by [G724] establishes similar bounds for the probability flow ODE, but under more restrictive assumptions, such as strong log-concavity of P
The first quantitative
Over the past three years, substantial progress has also been made in establishing guarantees for DDPMs in total variation and Kullback-Leible divergence under weak assumptions on P [CDS25, LJLS25, LY25, BBDD24, LHE randomized midpoint, and Runge-Kungg Kutta methods [CRYR24, GCC24, WCW24]. In parallel, a growing body of work investigates the statistical optimality of score- models [OAS23, WWY24, HST25], as well as their ability to adapt to low- dimensional structure [Bor22, TY24, LY24, HWC24, ADR24, PAD24]. Analogous results for flow matching have been established in [KT25].
LHE+24], 24]. including acceleration techniques such as parallel sampling,
Notation For D N, ID is the D D identity matrix. We notation A B, A < B, A > B, A > B to design that the matrix - B is, respectively. negative definite, negative semi-definite, positive definite and positive semi-definite. We denote by (D,, Σ) the D-dimensional Gaussian distribution with mean la and covariance matrix Σ. Let be the density function of N 0, DD). The norm of vector is always understood as the Euclidean norm, whereas the norm of matrix the operator norm (the largest singular value). The independence of random vectors X and Y is denoted by X IL Y. The Wasserstein-g distance between two distributions P and Q is defined by
inf Q). eET(P.Q)
)>xll)>el||X> Y 119.
where y > and I'P, Q) is the set of all joint distributions with marginals P and Q. For any function and V²g for the gradient and the Hessian of g with respect
g: [0, x R + IR, we will write to its second variable. If g [0,7] 8 - RD, we write Dg for the differential of y with respect to its second variable. For each random vector X, write X I|I [E|l X (1)) 1/2.

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2 Problem statement and conditions
The goal this section is to set the framework of denoising diffusion probabilistic models with randomized score estimators and to state the conditions imposed the unknown target distribution.
The setting of randomized score estimators Our setting is bit more general than those previously studied in the literature. For an unknown distribution p on R.D,, and for > 0, we define P as the distribution of cr + 3.C. where (x, ~ P 00 az = e and 3r v - oi. The set )ra0 can be seen as curve in the space of probability measures interpolating between p and yD, since P P and P =>D- For 0, Pr is absolutely continuous with respect to the Lebesgue measure X on R with an infinitely differentiable density. Therefore, we can define the score function by
dPr (t, ax (a)),
s(t, a:) V log (t,
Since P is unknown, we cannot access 8(1, x). Instead, we have access to randomized and noisy evaluations of this function: for each query (t. x) E [0, 00) EDD" we can observe random vector s(t, æ) such that [| x) st, x)||L, is small. Our goal is to combine independent Gaussian random vectors and queries the approximate score build random vector Z in R having distribution Pz close to P To this end, we focus on the DDPM algorithm presented in Algorithm 1.
Algorithm Generation of Z by the denoising diffusion probabilistic model
Require: Sequence ((x+)) for some integer K > 1
Ensure: Vector = Zk+1
1: Set to 0, = kK+1, and Zo 7
2: for t K do
3: Set h4
1x+1 14
4: Generate Ek+1 ~ w indeepnnennt of all previous randomness
5: Query å (tx: Zk)
6: Set Zk+1 (1 + h4) Zμ + 2hT 1b. Z₁) + 22h2 Ex+1
7: end for
8: Output ZK+1
We postpone the discussion the origin of this algorithm to Section 3. The main difference between setting and prior work lies in the randomness of s, which goes beyond the randomness of the training sample. Let us provide concrete examples to illustrate our setting.
Example 1 (Noisy score estimator). Assume that estimator @ is available. Due issues such as communication constraints or privacy concerns, we do not observe 8(1, ar) directly, but rather a noisy version (t,, .r) (1,, .) + Ç. where Ç is random, typically with zero mean and bounded variance.
Example 2 (Compressed score estimator), Assume again that an estimator is available, but only one of its coordinates can be queried at a time. At each iteration, we randomly choose i E {l...., D} uniformly and set s(t, x)- = x (s(t, x e,) e,),, where e; is the i-th canonical basis vector.
Example 3 (Randomized network weights). The conventional approach fits the weights 0 of neural net (b(,, x;; 0) to the unknown score (l,, .) by minimizing the (estimated) prediction error:
min R(P", 0). where R(P" (-' [ PERP
| x. 0) s(t, x)|l (t, x) dx dt.
One can instead minimize an estimator of the integrated error under a Gaussian prior by solving
/ f R(P".
p arg min
p oz) P (z)
PERP
where > 0 is a hyperparameter. This may lead to more robust score estimator. In this setting, the randomized estimator the score each query point (1, a)) is (t,,, æ. pa + a)), with ? generated independently by the user.
3

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Conditions on the target distribution The guarantees on the precision of the DDPM that we will state in the next section depend on the properties of the target p We will express these properties in terms of function sp
Assumption 1. For a function yp R200 + Ryo, we say that p or X satisfies Assumption I with function sp if, for (x,E) ~ P 8 BP, it holds that Var (X X + of = y) < ((o) for all o > 0.
Many distributions satisfy this assumption (see Appendix A for the proofs):
(a) If X has compact support K. with diami(C) = 2Dx. Assumption holds with p(o) = X (b) Any m-strongly log-concave distribution P satisfies Assumption with so(o) 1+mo? (c) If X is semi-log-coneave with constant .M > 0 and has compact support of diameter 2Dx. then X satisfies Assumption with (p(a) D3 +
o²
(d) If X satisfies Assumption with some function s,, Us a D x D orthonormal matrix and b IR,, then UX + b satisfies Assumption I with the same y;
(e) If X is obtained by concatenating two independent vectors X1 and X2 satisfying Assumption I with the same function yp then X satisfies Assumption with p.
(1) If (W.C) ~ Po @ 7D such that W satisfies Assumption with the function ço. then, X
_²0² ="vo(V=2_a²) W + + satisfies Assumption with the function (o) -=ooz
[r to*)=
(g) If W is supported by a compact set of diameter 2D and Ç IL W is m-strongly log-concave
with an M M-Lipschitz score function, then X W + Ç satisfies Assumption 1 with uo(r)
(MDa2)2
1 (1+
The main purpose of Assumption 1 is to ensure that the drift coefficient of the backward diffusion process is strongly convex when the noise level is large and semi-log log-concave for all noise levels. Moreover, the drift coefficient is always gradient-Lipshhit. with a Lipschitz constant depending on the noise level. These properties are summarized in the following result?.
Proposition 1. Let X and & be random vectors in RDD drawn from P @ -DD For any a, B > 0, the density Ny Y =ax- + 3 is twice continuously differentiable and satisfies
2 log mT (y)
B JD>
82 ID.
for all y E RR.
3
Thus, Assumption I is equivalent to 2 log TT (v) (a (a²x(B/a)- for all y E ERD, a. 0. 34
x
The last inequality above implies that if ((3/a) < (B/a)² the distribution of Y = ax + B is
log-concave, and it is strongly log-concave the inequality is strict.
Conditions on the estimated score As mentioned in Section 2, we consider randomized estimators of the true score function The mean squared error of such an estimator can be decomposed into bias and a variance term:
E (tt, x) = =
s 2)1 E [[]
(tt,, x) ])]
In what follows, we analyze separately the impact of the bias and the variance on the overall error. As we will see, the variance term has a much weaker influence on the final accuracy than the bias term. To reflect this difference, we introduce the following assumption.
Assumption 2. There are constants Egcore and sccore such that for all E (tx k < K} of Algorithm 1,
E [(1,=)]] s(1, x) < D1/25bore score:
sup [|8(t, x) - [8(1, x)]]l & 2
sup FERD
score
EERD
Assumption 2 imposes uniformity over all x E R and E t; k < K and, therefore, is a stronger condition than the one used in previous work [CLL23]. The latter considers -2- with respect to Pf, rather than a supremum, and involves a weighted average over t. While it may be possible to relax the requirement involving the maximum over the time grid, the uniformity with respect to æ appears to be more difficult to replace by the Lg-norm wrt P. It is important to note, however, that for our proof needs only an Lg bound with respect to the distribution of the DDPM output at time
We recall that X is semi-log cami-lo--concave concave [Cla83] with constant M R if X has density ax wrt the Lebesgue æ is convex; see [VCK25] for an application sampling. 'The formula relating the Hessian of the log density to the conditional variance, stated in Proposition 1 is often referred to the second-orde order Tweedie formula.
measure and - log x (x) + o

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3 Score-Based Generative Modeling: preliminary considerations
The starting point of DDPM the forward process given as a solution to a stochastic differential equation (SDE). The simplest and the most widespread choice is the Omstein-Uhlenbeck process
dx, = - dt + v2dB,
t 0, XO~P
(Be)a0 T. Xo.
(2)
where (Bi) is standard Brownian motion in R The Iastein-UUhlebbed process is time- homogeneous Markov process which is also a Gaussian process, with stationary distribution equal to the standard Gaussian distribution yD R... The forward process has the interpretation of transforming samples from the data generating distribution P into the latent distribution. From the classical theory of Markov diffusions, it is known that P law( converges y exponentially fast in various divergences and metrics such as the 2- Wasseesstein metric W2: W2(Pr; < see for instance [Vi108].
e- e2 (Po;7P).
3.1 Reverse Process: continuous-time and time-discretized versions
If we reverse the forward process in time, we obtain a process that transforms the latent distribution into the target distribution P*, which the aim of generative modeling. Fix some large time horizon T > 0 and set Y, == X then law(Y, law( (XT) is close to the Gaussian distribution 7 Notably, the dynamics of the reverse process can also be described by a stochastic differential equation, stated in the next result.
Theorem 1 ([And82]). If (X))co0 is solution to (2) and Yr = XT then there exists Brownian Motion (B.) 1L Yo such that
dY (Yr + (T - Y.)
v2dB., O T,
(3)
where π (t,, x) x Jg0 70 ((x a.y)BB) (dy)), a = e and B3 = =1 -
Note that (t, x) in this theorem coincides with the one defined in (1) and V log (T - -t, Y,) the score function evaluated at scale T and state Y,
The forward process transforms a data point Xo drawn from P into a point which is very close to being drawn from the latent distribution. The reverse process aims to transform point Yo drawn from the latent distribution into a point drawn from p To this end, we replace the unknown score function by its estimate ğ based on training sample Xi,..., Xn p The resulting process is defined as the solution to the SDE
dÝ. = (Y, + 25(T - t, Y()) dt + ddd.
YoUD,
t E [0,
(4)
Both Y and Y are processes on the space C(OO,T), RR), differing in their initial conditions and drift terms. We wish to assess the distance between the distributions of their states at time T'.
To efficiently sample the final state of the reverse process, we have discretize SDE (4). To this end, we introduce a sequence (h&)ken of positive numbers and set" t = hu + + h We then define
Zk+1 = (1+ h, + 2h, - tu. Z + /2hg Ek+1:
Zo Zo
(5)
where (Ex)AeN is a sequence of independent standard Gaussian random variables. The rationale behind this definition is that Z has approximately the same law as Yr,, for every k.
Definition 1. The denoising diffusion probabilistic model is the distribution PDDPM of the random vector Z defined by (5). requires the choice of K E N, the sequence (4,....,(x+)) and the score estimators (5(T - tg, k=o....K
In this paper, we are interested in quantifying the accuracy of the denoising diffusion generative model when the error is measured in terms of the Wasserstein distance, that to upper bound W2(P*, PDDPM). In the rest of this section, we motivate the choice of the Wasserstein distance and discuss the challenges related to in the framework denoising diffusions.
By convention, to 0.
5

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3.2 Relevance of the Wasserstein distance
Recent work on assessing denoising diffusion models mainly focuses on accuracy measured by the total variation distance and the Kullback-Leibler divergence. However, we believe that for statistical purpses, measuring the quality of generative model the Wasserstein distance is highly appealing.
To justify this point of view, remind that the closeness of two distributions in TV-distance or KI.- divergence does not guarantee the closeness of their means their covariance matrices. In sharp contrast, the Wasserstein-2 distance offers such a guarantee, since it holds that
[ X Eq [xll & W2(P, [(Ep [(B((XTAX]' /2 - (EolXT AX) (2(P,Q);
for any matrix A satisfying 0 < A & The fact that the TV-distance and the KI--divergence are not suitable for controlling the moments of distributions can be demonstrated by the following example. Let P be the exponential distribution with parameter 1 and, for every n N, set Pa (1 - o,, + Qn, where o, 1/ vn and Qn is the uniform distribution on [n, n + 2]. One can easily check that Pn very close P both the TV-distance and the KL-divergence:
dTv(Pn;: < 8, n 1/2;
dkL (PI)))) log(11 8,)) < 2n-1/2
3 2.
Therefore, one could expect that P. is an excellent generative model for the target P. However, the generated examples will have a mean and variance that explode as n + oo, and will be infinitely far away from the mean and the variance of the target, since Ep, [X]
no,, 2 n/1/2 and
Ep. [X3] 2(1 6n) dnm 3 3/22
3.3
Challenges inherent to Wasserstein distance
When the distance Wg is employed to assess the quality of a DDPM, a mathematical challenge arises in quantifying the eиrоr due to the absence of the data processing inequality for W -diitance Let us clarify this point. Consider a forward mechanism M. that transforms the target P into distribution Pf which is close to easy-to-sanee to-sample- ffrm latent distribution Qu: Pi Furthermore, assume we have knowledge of the "inverse" forward mechanism, termed backward mechanism, which maps Pr back to P (PI) generative modeling then define the generative model as Q = M_ (Qo). where M represents a suitably regularized estimator of M. In DDPM, M. and A. are specified by Markov kernels.
(P*) s Qo-
P The
methods of
In this context, denoting dF as the F-divergence for some F, the following relationship holds:
dF dF(Qi) = d5 (-11 (Qo) (Qo) (2)) ~ dF d (Q))][P*)
DPI
= d= (-M, 4M_ (P))) & PF(Qo
where the final equality derives from the data-progsssing inequality. Thus, the error the generative distribution is dominated by how well the forward mechanism approximates the latent distribution, provided that the error of M. approximation is suitably small. These arguments were central in prior work' establishing bounds on the eиror of denoising diffusion models measured in TV-distance and KL-divergence. However, this approach breaks down for the Wasserstein distance Wq. for which no suitable equivalent of the data processing inequality exists.
In the case of denoising diffusion models, the qualitative difference between the Wasserstein distance and F-divergences (such as TV-distances and KL-divergence) can be formally demonstrated even when the backward kernel is known. This is illustrated in the following lemma.
Lemma 1. For any T > O, let QT be the distribution of the backward process (4) at time T with s replaced by the true score 8. Let N be the set of all the Gaussian distributions. It then holds that
(F PP)) V P-EN (P:(P
dKL (QTi01
W2(QFie;30)
6e 22 dKL PP
-2-
sup
sup PEEN W2(Pt:
This lemma reveals that when assessing accuracy through the rate of improvement in Wasserstein distance, the choice of parameter T must be carefully tailored to the target distribution P". This might be less important in the case of the TV-distance and the KL-divergence.
See [CCL+25 BBDD24, HWC24, CDS25J and the references therein

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Main results: bounds on the error in various settings
In this section, we upper bound the Wasserstein-2 2 distance between DDPM (see Algorithm 1) and the target P Similar to [CLL23, BBDD24], we employ discretization scheme composed of two regimes: an arithmetic grid in the first half and a geometric grid in the second half; see Algorithm
Algorithm 2 Definition of the discretization time steps
Require: ó, a. Ti > 0, and Ko E N>1
Ensure: Sequence 00 < 1 < ... k
I: Set to 0, K 2Ko. tk+1 Ti log(6a)
2: for k = to Kg do
3: Set t (Ti/Ki) Ko) k
(arithmetic grid
4: Set &Ko+k = 7i + (og
26
k/ Koj
[geometric grid
2
(og(Ba)
5: end for
6: Output (fo....,
4.1 Strongly log-concave distributions convolved with distribution with compact support
In this section, we consider the case of a distribution P satisfying Assumption with a function yp that has the following form: for some constants m, M, > O,
o²
(0)))
Vo
(6)
1 mo
(1 (1+ Mo2)??
If Pa is m-strongly log-concave, as discussed in Section then (6) holds with b and any M 0. Another class of distributions satisfying (6) consists of convolutions l" = Pstc Pempet. where Pstc m-strongly log-concave with M-Lipschitz score, and Pammpet supported on a compact set of diameter 2D, for some M T 0 and D > O. In this case, (6) holds with b = D
Finally, there are distributions satisfying Assumption with y given by (6) that are not absolutely continuous with respect to the Lebesgue measure on DR. For example, if p* is supported on linear subspace of DR. and its restriction to S, viewed as a distribution on Rdd for some 1,.... D, satisfies Assumption I with given by (6), then P also satisfies the assumption with the same p. This is a consequence of properties (d) and (e) presented in Section 2.
Theorem 2. Let the target distribution P satisfy E given by (6) for some m. M. b > O. Let us choose Ti > O.
<
D and Assumption I with function yp
1
Ko 7Ti log(6a) + 4 log (a) log log(Ga)
8= 0.5e
and define the sequence (1x JogkgKKK+ by Algorithm 2. Let ő be randomized estimator of the score satisfying Assumption 2. Then, the distribution PDDPM of the output of Algorithm based on 2 queries to the score estimator satisfies
W2(P*, W2(P*,pDDPM) pDDPM) & (4/333B {2E Ti 7V66 Amax + 4v 14446a2266 (2 +
1/22
max =bcore)
JVD, (7)
with Amax = maxx (t++1 tk) (og( (og)
Ko
There are several notable features in the upper bound stated Theorem 2, when we compare to the previously known results.
Remark 1 (Optimalityy). The dependence of the discretization error (the second term in (7)) on the step size maxx linear, whereas was of order i 1/12 in [CCL + +23, Cor 6], 1/44 in [CLL23, Cor. 2.4], and 1/2 in [BZL+2] 233, Remark 12], [SOB*] + 25, Cor. 4.3], [SO25, GNZ25, YY25J. Moreover, [GNZ25] establishes that the lower bound on the Wasserstein-2 2 error, achieved by the Gaussian distribution, scales VD hmaxx thereby implying the optimality of the bound in Thcorem 2.
Remark 2 (Conditions). Assumptions on P* in Theorem 2 are less stringent than those in earlier works [BZL,+23, 23, YY25, GNZ255.. In particular, for m-strongly log-concave p we do not assume that the Hessian of the log-density is bounded from below. Furthermore, Theorem 2 covers the class
7

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of distributions obtained as convolutions of a compactly supported distribution and Gaussian, 1/2 framework not addressed in previous studies achieving discretization error of Amaxx. However, our conditions may be regarded as stronger than those of [CLL23, Cor. 2.4] providing the discretization error of order /maxx These stronger assumptions are typically necessary for attaining faster rates of convergence. In conclusion, our conditions are weaker than those previously associated with the 1/2 rate, while enabling the faster convergence rate of hmaa-.
Remark 3 (Impact of noise). All previously known bounds are proportional to (| sT, X )111 where the proportionality factor is often logarithmic in the number of queries, and the Lg-norm can take different forms- the weakest being the case where
Unif( (0, TT) and the law of X given T = is P.. If s(t, x) s(t, x) + S, with ||S1 = of as in Example of Section 2, then | aIL > D. Thus, all known bounds include term of constant order, independent of the number of queries. In contrast, the corresponding term in the bound of Theorem 2 is O(vD hmax Escore) which scales as / and thus vanishes as K, the number of queries, grows large.
Remark 4 (Informal statement). To facilitate comparison with existing results, let us consider the strongly log-concave case b = 0 and denote by L == a the surrogate of the Lipschitz norm of the score of p For Ti- = (o), our result implies that, after K queries to the score estimator,
log I. Log K
Vlog log K
W2(P", PDDPM) 2 L.D
Eccor
K
VK
Escore
In particular, W2(P", )DDPM) x VVV LD
sccore provided that the number of queries satisfies
K
V (
(Fscore) scooee cb Score
log L.
log K
score
As mentioned in Remark 3, this improves [BZL 23, YY25, GNZ25, SO25], which require
K > (log () and E"corre < - to achieve W2(P*, PDDPM) 2 vLD score
4.2 Semi log- log-concave distributions with compact support
In this section, we consider the case of a distribution lr satisfying Assumption with a function y that has the following form: for some constants 6, MM > O,
σ²
( = b (1 Mo²)
Vo > 0.
(8)
The typical example of p* satisfying this assumption is a distribution on compact set K. included in a linear subspace of R.D. if in addition the log-density wrt to the Lebesgue measure on the subspace has Hessian & MI. It then follows from claims (c), (d), and (e) of Section 2 that P* satisfies Assumption with yp as in (8) with b = Dx
Theorem 3. Let the target distribution p satisfy E X
< D and Assumption I with function
given by (8) for some b, M 2 O. Let us choose Ti
a bV1,
Ko > 7T, log(Ga) + 4 log(6a) log log(Ga)
8 8= 0.5e 27:
and define the sequence (4k)okk<<<) by Algorishm 2. Let ğ be a randomized estimator of the score satisfying Assumption 2. Then, the distribution PDDPM of the output of Algorithm based on 2 queries to the score estimator 8 satisfies
W2(P", PDDPM) < 2bb++ {
2e T. 2T- 7V hmax + 4v 44/6((22 + 1/22 escore) }
(9)
with maxx maxx - tx) & (oo log(Ga) (22))
Ko
Since the conclusions of this theorem closely mirror those of Theorem 2, the remarks provided after the latter apply here as well and will not be repeated. We merely emphasize two points. First, pa is not assumed have a density wrt the Lebesgue measure on IR Second, the number K of queries to the score estimator required to achieve W2 error E scales as 1/E, up to a factor that grows at most logarithmically in 1/E. The exponential terms in (7) and (9) depend on the parameters of the target distribution. The independent work [SO25] employs a different proof technique yet exhibits a similar exponential dependence, suggesting that this behavior is intrinsic to bounding the Wasserstein distance in DDPMs. For a log-concave distribution supported on compact domain, we have (M, (0, D& ), so the exponential factor the bound (9) becomes universal constant. This complements the result obtained in the strongly log-concave setting from Theorem 2.
8

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5 Relation to prior work: extended discussion
Given the wealth of work on Langevin algorithms and score-based generative models, it would be infeasible to provide an exhaustive account of existing results. Instead, this section offers a selective overview of prior work, to situate contributions within the broader landscape.
Theoretical guarantees for DDPMs have been inspired by techniques from the sampling literature. particularly those used for Langevin Monte Carlo and its variants; see the overview in [Che24]. Prior work can be grouped into three categories based on the underlying proof strategies.
The first category, represented by [CCI,*23, CLL.23, BBDD24, CDS25, LY25, LJLS25, LHE 24), includes works that build on the approach initiated in [DT12, Dal17b], combining the Pinsker inequality with the Girsanov formula to derive bounds in TV. Its key strengths are:
it requires only a bound on the mean integrated squared error (MISE) of the score estimator-one of the weakest conditions in this framework;
it relies on mild assumptions on the data- tala-generratigg distribution P
As noted in (C,L+23, CLL23], TV-distance bounds can be converted into Wasserstein bounds under additional assumptions, such as compact support or light-tailed P* If the support lies in a ball, one can project the generated sample onto this ball and use that W3 is bounded by the radius of the ball times the TV distance. By the data- Sata-processing inequality, this projection does not increase the TV-error
However, this versatility comes at a price. Let K KTv(E) be the number of steps required to achieve an error smaller than é in TV-distance. Then, to achieve W2-error e, one needs a TV-error ē e² R2, leading to number of steps at least KTv(E"/ RR²). As a result, the rates derived from this strategy are suboptimal: O(D*/E²) in [CLL23], O(D/E*) in [BBDD24, CDS25], and O(D*/ in [LHEB*44 ignoring log-factors. Another limitation of this approach is that the resulting upper bound on the W2 distance scales as the square root of the error of estimation of the score. Hence, to guarantee an error E in W2, one needs the score estimation error Eccore of order O(=?). Our results, as well as those of the third category below, typically require the weaker condition Escore O(E).
The second category comprises results that exploit the interpretation of I Langevin dynamics as a gradient flow in the space of probability measures. This perspective was initiated in [Wib18, Ber18 and further developed in [CB18, DMM19, VW19]. Interestingly, the first polynomial-in-dimeniioo guarantees for DDPM fall within this framework, as shown in [LLT22, YW22]. These works evaluate the error in terms of f-divergences such as total variation, KL, or x2 divergence. However, when translated to bounds in the W2 distance, they suffer from the same limitations as the TV- based approaches discussed above. Moreover, this line of work typically relies on strong structural assumptions on the target distribution Pt, notably the satisfaction of a log-Sobolev inequality. Another limitation, shared with our own analysis, is that the score estimation error is measured in the uniform norm. We believe, however, that this requirement could be relaxed, both in the gradient-flow framework and in the recursive method developed in our work.
The third category comprises works using the recursive approach to bound the error of iterative algorithms such as LMC or DDPM. This method, widely used in optimization theory, was shown to yield strong guarantees for sampling in [Dall DM17, DM19, DK19]. For DDPM, underlies the analyses in [BZL+23, GNZ25, SOB+25, YY25J, which establish a W2-error rate of order D/s²- an improvement over the bounds derived or derivable from the first two categories. However, despite having all the necessary ingredients, these works do not reach the faster rate vD/E. This is somewhat surprising, especially since their assumptions on f are often quite strong, such as strong log-coneavity, We believe this gap arises from not fully exploiting the smoothness of the score of the distribution obtained from Pa by convolving with a Gaussian. Technically, their recursive that at iteration
bounds relate the error iteration be loose when the two terms involved are weakly correlated. As we show, applying the recursive approach to the squared Wasserstein distance yields significantly tighter control and leads to optimal rates. We believe that this improvement can be further exploited to get even faster rates using the randomized midpoint discretization [SL.19, HBE20,, YKD24, YY25] or to get faster algorithm exploiting parallelization [CRYR24, ACV24, GCC24, YD25].
via triangle inequalities, which can

---

6 Numerical experiments
We supplement our theoretical results with small-scale empirical study CIFAR-10 [KH09], CelebA-HQ IKALL18), and LSUN-Church [YZS* 15], evaluating the robustness of DDPMs to noise in the estimated score".
Setup. We use pretrained DDPM models from the publicly available checkpoints and all licensed under Apache license 2.0 and hosted on HuggingFace. For each model, we follow the standard DDPM sampling procedure, and then repeat the generation process while injecting noise into the score network ag at every denoising step. Specifically, we replace the score function with a perturbed version (p, x so(1, .) Ç. where Ç is a D-dimensional noise vector with independent and identically distributed components, We consider 4 noise distributions: centered Uniform, Gaussian, Laplace, and Student 'st s-t with degrees of freedom. For each noise type, we evaluate 6 values for the noise scale, a E (0.25, 0.5, 1, 2, 3, 4}. All other elements of the generation pipeline-includdng including the variance schedule, guidance scale, and number of sampling steps are left unchanged. For each experimental setting, we generate 8192 CIFAR- 10 images and 8192 Ce images. Additional implementation details can be found in Appendix E.
boogle/ddpm-cifarr/
google/ google/ddpm-oo seeem-celebah-2225
Qualitative results. Figure shows random generations for standard normal noise. We observe that injecting noise with constant variance into the score network has a negligible impact on the visual quality of the generated samples. As expected, the quality gradually degrades as the noise level increases. Additional qualitative results illustrating this phenomenon are provided in Appendix E.
FID sensitivity. The Fréchet Inception Distance (FID) is widely used metric for assessing the quality of generative image models. In Figure 2, we plot the FID as a function of the noise scale a. On CelebA-HQ, the FID increases only moderately up to x 1, while CIFAR-10 10 exhibits robustness up to a &3 2. In agreement with our theoretical findings, the shape of the noise distribution has negligible impact, only its scale matters. We also observe sharp degradation in quality beyond certain noise threshold, a phenomenon not accounted for by our theoretical analysis.
Laplace
Laplace
Laplace
Nomtal
Nomaal
Unifsem
Stadert T
Souder T Unifeen
Numal
Uniftem
Suatur T
Noistless soore FID - 17:72
Neiscles sere FID
66.97
Naisoleas score FID 23.98
2
2
Gueie
Oresse
(a) CIFAR-10
(b) CelebA-HQ
(c) LSUN-Churches
Figure FID as a function of noise level for four distributions and different standard deviations.
Conclusion
In this paper, provide refined theoretical analysis of denoising diffusion probabilistic models (DDPMs), revealing two important features. First, we show that DDPMs exhibit robustness to noise in the estimated score function. Second, we establish that, when the true data-generating distribution belongs to a broad class-signts significantlly larger than the class of log-concave distributions DDDMM achieve fast convergence rates in the Wasserstein distance.
Our findings open several avenues for future research. One direction the adaptation of our techniques to the analysis of kinetic Langevin diffusion-based DDPMs. It remains an open question whether such an extension would improve the dependence of the error bounds on the discretization step size. Additionally, the convergence rates derive include terms that scale exponentially with certain parameters, such as the diameter of the support in the case of semi-log-coneee unclear whether this dependence is intrinsic to the problem an artifact of our analysis. Finally, it would be of interest to assess the potential benefits of incorporating estimators of the Hessian of the log-density into the DDPM framework.
targets. is
'Code is available at http:///giiiitoooooaaaaaaaaaaaae
10

---

Acknowledgements
This work was supported by Hi! PARIS and received government funding managed by the Agence Nationale de la Recherche under the France 2030 program, references (ANR-23-ACCL-0005) (ANR- 23-PEIA-0004). This work was granted access to the HPC resources of IDRIS under the allocation 2025-AD01 1016491 made by GENCI. The work was partially supported by ERC grant SAGMOS (grant agreement No. 101201229).
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Appendix
Table of Contents
A
Classes of distributions satisfying Assumption 1
16
A.1
Compactly supported distributions: property (a)
16
A.2
Log-concave and
distributions: properties (b) and (c)
17
A.3
Stability by orthogonal transform and concatenation: properties (d) and (e)
18
A.4
Convolution with a spherical Gaussian: property (f)
19
A.5
Convolution a semi-log semi-log-concayee conccce and compactly supported distribution: prop-
city g
20
B Proof of Lemma
21
C Proofs of the main results
22
C.1 Main recursion
22
C.2 Proof of Theorem Strongly log-concave convolved with a compactly supported distribution
25
C.3 Proof of Theorem 3: Semi log- and compactly supported distribution on subspace
28
D Proofs of lemmas used in the proofs of main theorems
29
D.1 Proof of Lemma 10: the origin of the contraction/eepaassion
29
D.2 Proof of Lemma 12: strength of the deflation in the contracting regime
30
D.3 Proof of Lemma 13: assessing the increments of the drift
31
E
Numerical Experiments
32
E.1
Implementation Details
33
E.2
Additional Figures
33
E.3
Computational Resources
34
E.4
Dataset and Model Licensing
34
15

---

A Classes of distributions satisfying Assumption
Throughout the paper make use of Tweedie's formula [Efr1 1, Eq. 1.4] which takes the following
form using our notation: Let Ty be the probability density function of Y = a + gg where (X ~ P @ pp, then
log
X
Y
y)
Vy YRRRD.
(10)
3
This section shows that distributions mentioned in Section 2 satisfy Assumption
A.1 Compactly supported distributions: property (a)
Lemma 2. Let Px,y and the conditional distributions of X. Then
y be a probability measure defined on X x V, Px and Px Y=y be the marginal
supp(Px]y=z) supp(Px)).
Proof. Let Sx == supp(Px) Then by the definition of the marginal probability measure:
(Sx) = Px, (5x x x) =1
On the other hand, by Bayes' theorem:
Px, }) = Px Y- P=
(11)
where Py is the marginal probability measure of Y. The proof is completed by noting that (11)
yields Px yy 1.
A simple consequence of Lemma 2 is that if diam(supp((( Px)) & C' then diam(supp(? (P y==)) &
C. Using this result, we show that random vector X with support diameter 2Dx satisfies Assump- tion I with (o Dx
Lemma (Property (a) in Section 2). Let X such that diam(supp(P)) & 2Dx and let Y be any random variable defined on the same probability space. Then
Var (x Y = -y) < DIIDD
Proof. We need to prove that for any v E RRD:
w. Var X Y = yyv & T (DxID)
which can be rewritten as:
Var(e T X Y y) < 1 D
By dividing both sides by ||0/|3 we can rewrite the target inequality with respect unit vector E R.D:
Var(u X = y) < D
Denote Z u The supp(Pz) is contained in the set {u x x e (upp(Px|Y=x)) By
Lemma 2, the diam(supp(PP) y=y)) & 2Dx. Let ž1 u x and 22 u x2 for arbitrary 21, D2 E supp(Px Y y The distance between them is:
21 22 = | Ø1 - u x22 = |uu (x11 222)
By the Cauchy-Schwarz inequality:
[u T (x1
222) & |//l//|/2
22
1, we write |s - 22 < | I1 222 222122 The maximum possible value for ||æ: s2ll2 is the diameter 2Dx. Therefore, |2 - Σ2| < 2Dx for all 21, 22 the support of Z. This implies that the support of Z is contained within an interval (a, b) such that the length of the interval b-a a-a & 2Dx We now apply Popoviciu's 's inequality on variances [SGK10], which yields that:
Since ||u/|2
Var(Z y)
a)² < Dx
16

---

A.2
Log-coneave and semi-log-coneawy distributions: properties (b) and (c)
Random vectors with m-strongly log- log-concave densities also satisfy Assumption as shown in the lemma below.
Lemma 4 (Property (b) in Section 2). Let (X,E) ~ P & y where the density of P, denoted as (x), is m-strongly nog-concave. Then
Var (X x
+ ma² Id.
In addition, if x + V (log is M-I.isschitz for some M > 0, then
Var (x x + = =v) & 1 M D
By applying the preservation of strong log-concavity [SW14], we obtain that X +o (y) is m+mo? -ttoonly log-concave. We then invoke Proposition with parameters a and 3 d. which yields
Proof.
mn
Var X Y y)
1 I σ²
m
ID.
04
1 mo²
for Y = X og, from which the first desired result follows.
For the second claim, set Y X + og. The definition of coni-log-concavin yields
0 ক - 2 log (x) MID.
The conditional density of X given Y satisfies
xx|Y= x
with xy|x==())) x exp(
Iyy=al 20
Hence, the Hessian of T (x)) is equal to:
1
+ Mo²
2 log xxgg 2 log xx)
σ² ID
-M
I
Ip.
σ²
σ²
The Cramer-Rao inequality implies that
o²
Var X Y y) > (E(V² Y Y -1))
In
1 + Mo²
and the claim of the lemma follows.
Similar results hold for for semi-log-concave distributions with a compact support. Lemma (Property (c) Section 2). Let (X, 5) P @ D where has density w.r.. Lebesgue
measure denoted as (x)) and diam(suppl) P)) < 2Dx If (x)) is M loncc-vveeeeeceeeeeeeeeeeeeeeeee for M > 0, then:
d²
Var X + o
x ^
Ip.
Mo²)
Proof. Denote
X + of. We obtain from the definition of semi-log miai-llo--occviiy that:
2 log (x) MIp.
The posterior of X given Y is proportional to the joint:
πx y) x ()))((y x)
with Tt a:) x exp( lly x 20a
Hence, the Hessian of log (x y) equal to:
2 log = V2
σ I $ M o² In.
17

---

where the last inequality follows from the semi-log-concaving of (x)). By Brascamp-Lieh inequality [BL76], have that:
Var( X Y y) < M Ln
(12)
whenever Mo² < 1. The conditional variance of X can be bounded via Lemma 3, as P has compact support:
Var(X = y) < I
Combined with (12), we write:
o² Ip.
Var (x Xx+00 =f===) <5 xDx A
(1 Mo²)+
This completes the proof of the lemma.
A.3 Stability by orthogonal transform and concatenation: properties (d) and (e)
Afterwards, prove that if X satisfies Assumption then its rotation also satisfies Assumption ] with the same =(r)).
Lemma 6 (Property (d) in Section 2). Let ()) P& D and Var (X X + of - y) - (<(0)) ID.
Vo > 0.
Then for any orthonormal matrix U, we have that:
Var (UX UX + ot' = =yy + I..
Vo > 0.
for g' ~g and E' L.X.
Proof. Consider Var (UX) UX + ot' = y). We rewrite as:
Var (UX UX + of' = =y') = Var (X UX + og' y')
= =U Var [ U + +OU = UT
Let y = UTy y' and & : U By using the properties that U = In as U is orthonormal and
that E ~ DD independently from X g ~ 7 b and g Il UX, write:
Var (UX) UX + of' =y') = U Var (X X + of = y) UT ID. and the claim of the lemma follows.
We now show that the concatenation of two independent random vectors satisfying Assumption also satisfies Assumption 1.
Lemma 7 (Property (e) in Section 2). Let (X1, X2) ~ Pi&P2, where P1 and Pa satisfy Assumption I for some sp Then the concatenation of X, and X² denoted as X, ① X2 also satisfies Assumption for the same ip.
Proof. Let X, be di diddimensi,nal, X₂ be de dimiennnsi,, and D d1 + d2. Consider & ~ 7p and independent of (X X2). We may write
Y X @ X2 + o X + X2 a (E: E2) X1 + a + X2 + of2
We have that (X,, X2, E2) are mutually independent as (X1, X2, E) are mutually independent and & and &a are uncorrelated. From (X1,&1) 11 (X2, we get that (X,, Yi) (X2, Y2). Applying the weak union property of the conditional independence twice we get:
(X,, Yi) IL (X2, Y2) = X; IL (X2, Y2) Y + X1 TT X₂ (Y, Y2).
Hence the covariance of X, and Xa given (Y,, Y2) is 0. Finally,
y) = Var X1 @ X, Y y1, Y, = y2) Var( X, @ Y Y
Var X, Y vı, Y2 = y2)
Cov(X, X, 2 Y = yı. Y, = y2)
=
Cov(X; 1, X₂ = y1, Y2 y2)
Var( X₂ Y = yı, Y, y2)
Var( X, Y = y1)
0 Var(X, Y 92)
$ x(()) Lp
where the last inequality is due to P1 and P2 satisfying Assumption
18

---

A.4 Convolution with spherical Gaussian: property (f)
Lemma (Property (f) in Section 2). Let (W, () Po @ rp. IfW satisfies Assumption I with the function Pu, then, for every т > 0, X = W T satisfies Assumption I with the function
-²02 72 r²
σ pooolV po(
(r(
Vo
(T 0
W + + of and n ;= T + of. Since & S ~ 70 are independent of W, we have 17 ~ 9 with covariance (7² + ID and Y = W + 17. Equivalently,
Proof. Let us define Y X + of of
Y= VT o2 5 g ~'~ "~ 11 W.
Using Assumption 1 with noise level V + x² leads to
Var(W (W Y y) ক po(V 0²) ID.
To ease notation, we write Ey and Vary to refer to the conditional expectation and conditional variance given Y y. respectively. By the law of total variance, have
Var, = En [Var(X Y y, W)] + Var, E E Y y, Wl)
(13)
We know that T{ and n = Tς + of are linear transforms of two independent standard Gaussians. Hence, the standard covariance calculation gives us
Var(Tς 17)
I'ID I' -
7202
72 2 72
And since Var(X Y = y. W) = Var(TÇ 1)) we get the first part of (13) equal to
T²0²
Ey [ar Y y. W)
72 +2 ID.
For the second term, since Ç lid (O, then the corresponding D-dimensioonal vector
T ~V with Σ
Var(rς) ( Cov(TÇ. (Cov(n.T)) TC)
I'D T
I'D (r2 (o=+02)11))
Var(n)
So, the conditional expectation that we are interested in will be equal to
72
E(TC v) = I>I + + IIn
72 + و²? 7
Under the conditioning on both Y and W, the quantity = Y - W is deterministic. Therefore,
X Y y, W W E(r(
17 = y W
72
W + 0² W) ?2
7 W +
72 02 y
Given Y yy the second term is deterministic, so
2
2
J'ppl ( +
Var, (E Y = y. w))
Var(W Y y)
Id.
72
(72 0222
Adding the two components gives
-202
a AppoV=2 +z)
Var X Y
y) &
+
In.
72 + 0
(72 +
which proves the lemma with (0)
+
(T-402)2
19

---

A.5
Convolution of a semi-log-concaye and compactly supported distribution: property (g)
Lemma 9 (Property (g) in Section 2). If = Pac* Pempct where Pae is an m- log-concave distribution with an M M-Lipschitz score function, and P Pempet supported a compact set with diameter 2D, then P satisfies Assumption with
o²
22 (1+ Mo2)2*
(0
Vo >0 0,
1+ mo² m
Proof. Let W ~ Peppet and Ç ~ Paee be two independent random vectors so that = W+ P This means that for some compact set K. with diameter 2D, we have Var(W) < 42 and that the density "Ç is continuously differentiable with score function &c satisfying
m|| x'11 < (ar x²)
( (s<(x)) sc(a²))
M| x'12
For IL (W, () such that p and for Y X + o have prove that
o²
D²M + Mo In-
Var(X X Y y)
1 m
As before, to case notation, we write Ey and Vary to refer to the conditional expectation and conditional variance given y. respectively. By the law of total variance, we have
Vary (X) = Ey [Vr Y y. W)] + Var, ( Y y, WI).
(14)
Since the random vector is m-strongly log-cone concave, follows from Lemma 4 that
Var(S of= v²)= x I ma² 2
Vy ERD.
Therefore,
Var(X Y = W = w) = Var(cIc+o05 = w)
<
Vy. w E IR.
1 ma²
Hence, Var X = W) < 1+mo² almost surely. This implies that
d²
Ey [ar X Y y. W)) $ + mo²
I
We switch to assessing the second term in (14). holds that
X Y
y, W w] + E Ç + of = y w, W w]
w + E((| + of = y - w)
w + o2 log +5 (y - w) + + w
+ 2 V log
w))
where ① is consequence of X W + Ç. @ follows from the independence of Ç and W. ④ is obtained by the Tweedie formula recalled in (10). Let us set (w) = V log oo+ (y - w). The
second claim of Lemma 4 combined with Proposition implies that v is Lipschitz-contimouus with the constant M/(1 + Mo²). Therefore.
M
M2
Var, (W) & Var, (E Y y, W)) = Var, (w(W)) & (1 Mo22²
(1 Mo Ip.
where in the last step used Lemma 3.
20

---

B Proof of Lemma
We start by first proving that:
di T PO) P.EN dfv(rP: V
dkul T |
( 6
sup
The data processing inequality [PW17] states that:
dTv(Q) PP)) < dTv(7";Pp) Pi);
dkl (Q7 PP)) & dkl (T)
Combined with the concentration property of Ornstein-Uhlenbeer process [G724, EGZ19]]: dkl (kL(7): (??:Pt) & ( --2
drv(r": PF) < drv(n P')e ;
2T
gives the desired result.
We now focus on subset N" C C that contains D dimensional Gaussian distributions with mean 0 and (1 + of) ID covariance matrix with > O. Clearly
W2(Q P*) sup PP
Wa(O ;P*)
sup P'EN"
(2(P:;7D)
P'EN
Let X, be defined by Equation (2), then the distribution X, is (O,, (e (e-2t + I)Ip). Hence, the true score function is
5(x) (t)
where d²(t)= e 22t + 1. Equation (4) obtains the following form under this score function:
ař; Yi (1- (111122 02(T 02(T- -1)
2
) dt + v2dB.
The integrating factor for the SDE is:
2 du
((1) = exp
(T( u)
2
exp
exp(2(u - -2]) 2 du
-2+
exp
-1+ + log
02e2t + e2T
0² 2T
=e
22621 + +62
From Itô's product rule applied to I(t)Y,, we get:
ai(t) = I(t) {(t) dt v2dB. - ((t) dt /21(t) aB,
(15)
where we have used the fact that df(() = I(() (1 - dl.
Integrating both sides of (15) from 0 to
I(1);, = Vo + V2 o ((1) dB.
from which:
Y, Y. + v2 le I(u) dB
(16)
I(1)
21

---

Note that Yo ~ rrp Combined with the fact that [(t) is deterministic function, we infer from (16) that Y, is mean Gaussian random variable. So the Wasserstein distance between yD and the distribution of Y, depends only on the covariance matrices:
W2(Q P*) 1, o + I 1 -ov2-
(17)
where oz ID the covariance of Y,
Let Z, = v2 (u)) dB... Hence, Ze ~ N(0,2 (u) du Ip) and it is independent of Yo- The variance of Z 1s:
2 2 S (u)udu (c2t 1) + o22)
(1 - c (c +
and 2
22T 0² 22
02+1
The variance of Yr can be computed from Equation (16):
1 1 (o² + 1)(22² - 0² + cT)
= =0² +1)[1 (0²
0² +1)
1. 12(T)
(o² + 22222
(02 + 2
Plugging in the value of y, into (17) we get:
0² +1) +1) 1/2 1102+11D=>0 oVD.
W2(QI= PP) = 1 - {
(02 + eTTz2
We note that W2(P"; = |Vo2 + 1lVD 20 ov D, so we have
r(o) W2( ;P))
7)(P*:7P) 0-000 1.
Hence,
Wa( PP') W2(P*:PP) >
sup
P-EN-
When combined with the established contraction behavior of the backward diffusion--oerratinn with the true score function-in the 2-Wasserstein metric for Gaussian distributions [EGZ19], we get:
W2(Q2 :P)) P.EN W2(P < 1.
& sup
C Proofs of the main results
We recall that P the target distribution and P.- a,P + Bry is the distribution of the forward process at time >> O, with a₁ = -t -1- B3. We also fix some > 0 and define Y, = X and Qf Law(Y) Y; is the state of the backward process (3). We set Pe to be the law of Z defined by (5) that PDDPM Pk+1. Throughout this proof, will repeatedly the following notation:
m2 1 X| ILLL../DD
c |[[ - tk Zz) Fi] - sTT t4. Z+) e = max #k h
$x || 14. Zx) - E 44. Zo,) Falllla>
e" = max
C.1 Main recursion
We set T = +1++1 and consider a version the continuous-time process (Yi)) and the discrete- time process (Z defined the same probability space and coupled by the relation
Ek+1 = (Bu.. B,,) V/g. We then use the definition of the Wasserstein distance to infer that
W2(P", Pk+1) = W2(Qire Pk41) & YeKe Zk+1llla
(18)
22

---

Combining (3) and (5), in conjunction with the relation Vhn Ex+1 (Bi4. - Bie), we get
Yl2t1 Zk+1 (1 (Y,, Ix) + (hT tk: Yn,) - =T(T- tk: Zk))
2/h (3(T La, Zx) =s(T Lb, Zx))
+ Not
1 Yu + 22( - t, Yy) - 22
tx. {xe) dt.
b/
(19)
In what follows, we use the notation A k Yi:-- Zz and
U sT 1k: Yt,) s(T - th: Z
Ç3 8(T Lx Zk) s(T ik; Zk)
V
{xr Yu, + 2s( t, Yi) 22
- tb, N k1n)) dt.
(20)
This allows us to rewrite (19) as follows
A k+1 (1 A+ Uh 20 + VR.
(21)
In view of (18), we are interested in bounding the term
KK == A k 1l
We will proceed by establishing a recursive inequality upper bounding 24+1 by simple expression involving Tg, and then by unfolding this recursive inequality.
Let us introduce the filtration (FR)RENs The first element of this sequence is the o-algebra generated by Yo and Zo- Then, each Fk41 is obtained by extending F& the smallest o-algebra for which both Sa and the process (B, Bt, [el are measurable. Note that Zg is necessarily F& measuuabbeee but the same is not true for Çk- Indeed, the estimator 8(T (),-) may depend on some random variables that are not in Fk.
It is clear that
E E[IA k+1 [2]
= E E A++1
[+]]]
+ E [[|| |+1 E(A 4+ [alll
E A Filll + || E[Ax+1 Fxll
(22)
From (21), by the triangle inequality,
E A+t1 Falll12 & [[ + ((1++h))4 hx) + 2h1UUkl1, + 2h |[|R Cx FFlll,, + | V Falllca. (23)
Furthermore,
| E[A 1+1 [Fellla & 2he ISa ElSa [F&]lla V E Fallllt.
(24)
Combining displays (22), (23) and (24), we arrive at
E A k+1 13]
(ICC Ax + 2/v Un l1, + 2hv
[ 1
E Vv |
ob =bias of estim. score
B bii= discr. error
(2hn 15x 1
I V Fullla.
:g:=variance estim. score
v discr. error
In what follows, is convenient to use the following notation: for every k E N, let ar = = ((T-ta) and 83 1 a. Lemma 10. If p satisfies Assumption with a f.function and
+ 2²
20
(1 (((x/(nx)
hg
+ m.
2, for my = 1 - 0
2
(25)
k
ak
then.
(1 + h4)
2hgUullis & (1 make)) Axllia
(26)
23

---

Lemma 10 implies that
ziz1 & ((1- mzhh)ak + 2h + Bx)² + (2hx + +2k)).
(27)
The next lemma which can be easily deduced by induction applying the Minkowski inequality, will be used to derive global bound on the enror Ix from recursive inequalities upper bounding the error Ik+1 at the (A + 1)th step by the one of the kth step.
Lemma 11. Let (AR) (Bx)KeN and (Ck)keN be three sequences of real numbers such that By 0 and Ck > for every k. If (Irk kKN) satisfies the recursive inequality
2 & (C + Bb))² + Ch.
4k > 0,
then, for Ak Ao + + Ak
1/2
xx+1 < As 2o +
Σ de Ae +
Eea (2 -3)
For the subsequent steps of the proof, we leverage the properties of discretization. We begin with the portion employing constant step-sizes. This discretization is applied in the time interval where the inequality from (26) yields a hear-contraction. This is equivalent to considering the values of k for which my (27) is positive and bounded away from zero
Lemma 12. T and 3 are real numbers such that T > log(6a). Let Ku N be such that for every k E {0, 1.......... Ko).
0 E la TT - 1
2 log(Ga), hg & 0.7,
((Be/ & a
Then, for Og = e- (T (I))) we have og < 16((6)) as well as
2a
1 a
m& 3 1 1+
1
> 1/3,
and
hb
+ m <2,
ok
a
o
for all l O...., Ko-
h
h
h
h(1 c)
hp
h1
h
hKo+1
0
t
1/
too
tKo+1
tKo+2
tk T
Ti
2 log(Ga)
Figure 3: Notations corresponding to the discretization schedule.
We set h4 h for ls O,..., Ko- Then, (27), Lemma 11 and hym < h/3 < e -//3 imply that
K K
Koo
& e
Koh/330 200
+
(1
h 1o k- h + BB
3)
k=
Ko-1
1/2
1 (2 + )2
k=
Koh/33 xo +
max gg
A (2hEk + BBa]
+ max ig
[V 1.7 (2he% + 2x)
E
Get 3h -1
+ max Lo
1335h-1/(2hhe) +3a)) (28)
1gk<Ka
24

---

where ! comes from applying Lemma 11 with A = (1 - myh) < c h/3, e/ form which we get
e =E Eo e ...
I
(1-m))
(1-11mm
< (1
+) ))²+ and AN = e -KKoa/3 and @ uses the fact
1=0
that =R=0 Eko01(1- ) -o-k-- h
(1 5
Ko < 1/2 and, similarly, Exeo'01 (1 fo )2((Ko
h(6-h) 1- 1)) 2Kg]
1.7
since h < 0.7.
h(6-h) 5.3h h
The next lemma provides an upper bound for the bias and the variance of the discretization error.
Lemma 13. Assume that for some a and E {O....,Au K}. P satisfies Assumption with satisfying (p(a) & a for every
[Bk-1/ak+1: 1//a///t::B//a 8s/ax). Assume, in addition, that my
[El x ( V < co. Then, holds that
Bx < 2 V mzz mz h
mmm
(29)
(aa V B2 Bk+1
2g < mv hス + 4v2D h%
(30)
instead of (p()) a, have (0 ão² for some then (30) can be strengthened as follows
3/2
Th & / mDm h2 +
1v2D
(31)
3
Pk+1
Finally, under the same condition, the error 2g of the last iterate can be bounded by
Ux 1
1 m2 hi a D hi.
(32)
C.2 Proof of Theorem 2: Strongly log-coneave convolved with compactly supported distribution
We know that
(0())=
o0
+
00
6M 4 ]-
1+ mo²
+1+ 2
m
Therefore, we can apply Lemma 10, Lemma 12 with V (1/m) + bj as well as inequalities (29) and (31) of Lemma 13 with a + b.M In addition, to bound the last term in (31), we use the fact that
6a
< 1.2.
(2(R)11-T)
e log(Oa)
k+1
Together with (28), this leads to
KKe & e Kah/3 vo GE + 3h 1/2 2" v m2D (3 1.35 x g 4v2a
1.2))h
3
< e Koh/3 20 + b + 1/2 g2 + ( + 0.66M m2D.
(33)
On the time interval [T
log(Ge) T]) we use the discretization obtained by geometrically decreasing 2
stepsizes as previously proposed in the literature:
hko+j log(6a)
c(1 (),, j= K Ky 1,
2
where & 0.6/ log( This implies, in particular, that < 0.6/ log 6 & 0.4 and that h maxxe [] hy < 0.3. The constants e and are chosen such a way that fx T lig for log(Ga) and fK+1 = T. This means that
some small hg 2
K Ka-1
T hg = T (og log((G))
Σ
(1 - c)
2
1=0
T log(Ba) + log(6a)
(1 (1 (1-))k-Kop
25

---

This yields
2hx log(6a)
log log (6a) - (2gg) log(1
(1 ook-Ro= =
and
K - Ku
log log (6a) - logooo(2
For > Kg + 1, we will apply Lemma 10. To check that its conditions are fulfilled, note that
1 + o ma <
1+ 2
20
4
4
v v1).
+1+ 1 a
< 1 - 2 7) & 1 --e 2(T -(T'tt)) tx)
This expression, multiplied by his is less than 2 whenever hi & 0.3. Indeed, on the one hand,
4hv
1.2
1 T-e < T- 2 2.
On the other hand, for k Ko-
k-Ko-1
fis T
log(Ga)
+ hKo + ... + h/-1 T log(Ga)
+ 2
log(
Σ
(1 - e)s
j=0
log(6a)
log(6a) 2
- (1 - -c) K)) T - c -
(34)
This implies that
4h
2c
<
2(1
(T [t.)
since c & 0.6. In addition, taking σ = 3x/ and using the substitution & =1
- we have
2a
((8x//ak)
mk
a
1
1
- of
20 2
1
0²
BMP
1
ag
8%
mo²
(1+ Mo2)2
20
32 bM2
1 +
1
0
2
+
m
(a + M 2
20
2
2bM2
1 +
B2
3R(1 mo²)
(a + M1 )))
where ① comes from the definition of mg from (25), 1 is true for any (o satisfying (6). Equality 3 comes from the fact that
1
1
1 mo²
1 ma²
a mo?)
a m
and
of
bM2. 24/0
BBbM2
BRbM2
32 (1 + Mo2j²
82 (1+ Mo2)2
a + Mo2)2
(a MB
Finally, noting that
20
2n
2
3 1
1,
8
(1 + mo²)
for any m, o² > 0, we arrive at
2bM
my -1
(35)
(a + +1 -z.)
26

---

Therefore, (27) yields
# (c-mahe 2x + 2hge2 + Bx)) (2h + 2x))
From this recursion and Lemma 11, using the notation H (k) that
- mK, hKo mahk, we infer
k Σ
1/2
#K+1 & ((K)
Σ
(2 + BB)
(2(()) + 23x))
3Ko +
e
e
K=Ko
k= Ko
Inequality (35) yields
K Σ
H(K) ((K) <
h 2bM
Σ
(1+ M (a) 1))²
j=k+1
j=k+1
1 log(6a)
k
K
Mhre2(r-())
Σ
hj - 26M
Σ
2
(1+
( 1))²
j=Ko
j=
Let us set yy = M(c2(T-15) 1). On the one hand, we have
k
K h,(y; + M)
1
- -(K)-H(A) < log(6a)
Σ
+ 2bM
2
(1+ j Kg
Ko
On the other hand, since hj < 0.3, have e-h; 2h,
115 Therefore,
y5 1j+1 (yj M) c 2h,) > (15 M).
This implics that
4 Vj+1)
H(K)-H(A) < 2 log(Ba)
Σ h5 bM
Σ
3(1 + ,,)
j=Ko
j=Ko
4
(og(6a)
Σ
h bM
dt
3(1 + t)2
Ko
h3 + 46M
log(Ga)
2
3
==
Using the standard inequalities
K
00
e
ulbgg t.. the &
e e-x da = 11/u,
Vu > 0,
(36)
k= Ko
we arrive at
#K+1 & VbaesbM
4bM
(KK + 26th + max Kock<K
h 1 + max
[Vh hi e + 1 1
Kock<K
Bxx + 21xx" + Dx.
We apply then inequalities (29), (31) and (32) of Lemma 13 with ā = 1 + bbM. This leads to
FK+1 & VBaesbM [ + 26t + Vhε" + vD[ymp/ Vmgh + max
ahv keK B
1) + 5 m2 Dag- (37)
The stepsizes hig of the geometric grid are much smaller than the noise levels 32 Ba- as attested by the following inequality?
h
hx
h
5hg
5hg
1.2(T - tx+1) A 0.5
6(T
tx-1)
We use the standard inequality -- (1 - e (xx ^ 1) for every x 0.
27

---

It follows from (34) that T - tk++1 =c = =c= - h1-ch > 30 h/
h4. Hence.
hx
5c
5c
5e log((6a)
h
V
(38)
4
6
3
Combining (37) and (38), we arrive at
IK+1 & e bb
IKo 26 1112 8" + $ av m2mmm) 5 m2 Dlk.
This inequality, in conjunction with (33), leads
#k+1 < VVV (n + 8e" + 4h
4h max E" + 6.7 m2 Amax) + 5 m1 Dhk.
where Amax = max(h, h) is the maximal step size of the entire discretization grid, comprising the parts defined through arithmetic and geometrie progressions. These step sizes should satisfy the inequalities
1 log(Ga)
clog(6a) 2
log(6a) log log(6a) log log(2hx))
h
<
< 0.7
=
& 0.3.
Ko
K - Ko
To bound zo, we note that
20 & Bi
X 8, - E| a +1 Br) & 1.01m3
T l6g(6). Thus, 20 & 1.01mmyDe We set T 1 log (6a) T1 and hg = 0.5e-20 27, and K 2Kg- This leads the claim of the theorem. Indeed, h 0.7 translates into Ko > (10/7)T, and & 0.3 translates into
10 10 (6a) log log (6a) ) 2T;)
Kg
which is satisfied when Ko 77, log(Ga) + 4 log(Ga) log log(6a). Finally, notice that h < Ti/K Ko and
h log(6a) log log(6a) + 2Ti)
Ko
These inequalities yield the claimed upper bound on hmaxx
C.3 Proof of Theorem 3: Semi log-concave and compactly supported distribution on subspace
For Pa satisfying Assumption I with the function
( = PΛ (1- Mo
(39)
we can apply Lemma 12 with a = b V 1 and Lemma 13 with ā = bM + 1. Similarly to Appendix C.2, application Lemma 10 and Lemma 12 yields
Ko & e Kah/3 so + 6e + 3/2 + m22 ( + 1.35 x ( +
4v2a 3 x 1.2)) < Koh/3 20 6 + 3h 12 (2.2+3.10) maD.
(40)
We again use the discretization with geometrically decreasing stepsize on the interval [T log(6a),
:T]
2
hKotj log(Ga) c(1 c)', j= 0. K Kn 1,
where e & 0.6/ (6)) Following the discussion in Appendix C.2, we have that
log log(6o))) log(2hx)
K - Ko
and, for k > Kn
+ a
hx
+ m&
<
and
ty = T - c -
28

---

Combined with (39), we get
ma 1 2ū.
Hence, 27 yields
xk < (el2a-1)hx + 2hxc2 + BB)²² (2h + 23x)
We denote H₃ = (2ā- - 1) Ei== hy. We note that Hą &
2ā-1 log(6a). Lemma 11 states:
K
1/2
IK+1 H IKo
Σ
H + B²) +
Σ
e-
(2h+= + 2x))
k=
k= Ko
As 20 11- 11 1 + which is strictly positive, we may apply (36) which results in:
#K-1 & VGa 28-122-11 28-1
IKK 1
2 + max K hig hB
max K<K
[Vh o x +
27
})
28
22a
2h
1/2
BK + 2hke" + K.
We apply then inequalities (29), (31) and (32) of Lemma 13, which leads to
IK+1 & (2ae 1) (2x IKK
2
Vhe"
VD
aby
)
M2 h + max
+ 5ā m2 Dhg.
28
2a
-
22a
keK 12
The above inequality with (38) yields:
2
h
4ah
Dm2
ØK+1 & (2ae IKo +
+
E"
+ 5ā Mm2Dhk (41)
2
1
2ā -
2
Combining (41) with (40) and noting that (2m 1) 3 get:
IK+1 < V6a e 2ā-1 1o + 85" + 4/ E" + 6.7āv m2D hmax) + 5ā m2
Following the discussion of Appendix C.2, we complete the proof by showing that:
ØK+1 & 28 12c + 7V hmax + 4v 6a (2:
1/22 Escorre VVD
D Proofs of lemmas used in the proofs of main theorems
We collect in this section the proofs of the building blocks our main results.
D.1 Proof of Lemma 10: the origin of the contractiion/xpaassion
Since s is continuously differentiable, by the mean- value identity, we have
U s(T tk. Yr.) -s(T - tv. Zk)
DT 12, Zz 0(Y., Zk)) (Yr, Zk) do [ M2 ())
where
M2( = Ds(T t4p Z₁ + AAx) = V2 log (T - tk Z + A The matrix M, is symmetric, and according to Proposition 1. all its eigenvalues satisfy
1
1 a (B/Ox))) a
Amain
- a < <)y(Ma(0)
Amax-
Since U₁- = for (f) A do, we get
(1 + h4)44) + 24 = =f [1 ha) + 2h Ma. And0,
29

---

and
(I + hy) ID + Mg = max
11 + h + 2h1a11-
Ael Amnin Anox)
We assume that hg chosen so that
2h
(1 + h1s) (1 + hx)
2hv ox
(1
a //k))
(42)
1 0
0%
This is equivalent to
hg (2
agp((Be/ak)) o
< (1 + hw).
xx
Regrouping the terms, we get
hy (1 +
ajp((Braas) 1.
a
1-
aa
This inequality can be checked to be the same as (25). Hence, (42) indeed satisfied and, therefore,
[[1 + h4) ID 2 + by
1 af (1-
2hk
1 at
Therefore, by the triangle (Minkowski) inequality, we have
/. [| + ha) ID + +h, (()) A l1 d0
[I(1 + A + 224
<
< + 11
2hh
(1
a /as)
} A
-
2
1 1- o%
{1
2h4
1 1
2
af:p(3k/as
) ||
1 a
2
1 of
(1 mm hb )1 AnllLa.
This completes the proof of Lemma 10.
D.2 Proof of Lemma 12: strength of the deflation in the contracting regime First, notice that ag being increasing function of fg. have
at = 2(x & exp(- log(-)))(6a)) 1/(6a).
Second, since we assumed ((31/ax) & a, we have
2n
200 ok (1
m > 1 +
1 a
ax
22)
Since a < 1/(6a) and we assumed 0 E 1 - az < 1, we arrive at
> 1, we have ox < Combining these inequalities with
1 - (1/3) (6a) (1-a2)²
1- 1- (1/3)- a/(6a) (1 a=
1
1
mk >
2(1 az
3
For the second inequality of the lemma, it suffices to notice that (p()) > 0 and >> 1 imply that
+
2 a
1 1 a + ++
20 =1
1 3a
6a + 3
m&
< 1
< 2.8.
1 a
a
1 aj
6a 1
Combining with the condition by < 0.7, this yields hu( the lemma
1ta T-o7
+ mp)
< 2 and completes the proof of
30

---

D.3 Proof of Lemma 13: assessing the increments of the drift
Let Y 2s(T ).). To prove the first inequality, we recall that s(T t,
(aT- X Y; = -y)/ Therefore,
2a Xo Y.l Y 2 82
b, 3
addition, Y, a.o + 8 with _
X and c ~ NO, In). It holds that
E|| Y1 o² E
X | 8 E ||| m2 + 8 D,
(43)
E E Yi] a X || + X E am,2
since & is independent of X and has zero mean.
Let us use the "local notation" s(t, y) = st, y) + as well as H(t, y) = Ds(t, y). According to [CDS25, Prop. 2], holds that
d t, Yi) 5(T Yt V2Ds Yi) dB.
Since 2s(T' t, Yi) b: y, and Ds - Yi) H(T - Ip we get
db.- = -YY, (d Y,)
-b, dt - v2dB + (b, + Yi) dt + 2/2 2H Y,) Ip) dB,
Y,d + /2 (2H - 1, Y.) + In) dB..
Since B. Br, is independent of the o-algebra Fs. we get
Elb, bie [xl
E Yo Felde
and, therefore.
EE[. b. br. [Fallli> & | Y,, [alll / [|Y.ulliadl
< D(e-2(1-w)mo + (1 - e -(((r-)))) du
E / m2D (t tx).
The definition of V& given in (20) implies that V2 -fth (be 1 bre.) dl. This leads to
dett
E V. Full &
EE[b be. [Fa]llez
ret1 (1-
< m2
m2D h
This yields the claim of (29).
We prove now (30). The definition of b, Y (T(T t, Y,) leads to
|b: bi.
Elb. by [Fullll,
11 / (Yn E (.l)) du + I' V2(2H(1) + Ib) dB
1
Y. E Yu Fulll,, du + $1 (2(2H(() + II) dB
(44)
1a
On the one hand, in view of the law of total variance, we have \ E Yn Fx Therefore, using (43), we get
1
<
Y |ualia
Y Y [ulll,, du & S' Dmg = / D (t 1x)
(45)
31

---

On the other hand, the properties of the stochastic integral imply that
11 (2( + ID) dB La
=2 2 Je
E (H( Iollo! du.
(46)
Combining he definition of V given in (20) with (44), (45) and (46), we get
| - E V4 Fiella =
|b bie - Eib, bie Fa dt
12+1 {2 L.me
{2/"B112H(w))) B[](2H()) + ID|[ du dt.
1/2 (47)
m2 D h
The integral (46) can be bounded from above using Proposition and various assumptions of the function vp from Assumption 1. Indeed, denoting or-u Br-w/ we have H(u) x (p2 ID. Since, in addition H(u) F - -7 D. we get
xe (2H(u) Ib)² x (p(or--) 1)707 V1
ID.
(48)
84-0
If we assume that (()T-u)) E a, we arrive
{ 2 J
1/2 E (() I Inll du
< <2/2D( - -1)) (aa7 .) B7-
B/-
In view of (47), this yields
V E(VA FF) < 3 h2 + 4/2D 3/2 (aa t2p1 ) V -
hi
T1+1
This completes the proof of the second claim of the lemma. If instead of the assumption cp(o) a, we use the assumption (fo) & äo² with ā 2 1, inequality (48), the fact that u + 3r-u is decreasing, and inequality (47) imply that
V - E(V Fa),,,, & m2 + 4v20
lik
3/2
T-41
For the last claim, we use (47) and (48) as follows
1/2
4a²
1/2
E (()))
+ In | du dt
< VD
2 2 du
dt
1x
JI
T-K
T-tk
402
1/2
VD
du
di
(1 e 20
T-
T tx 4a
1/2
VD
u² du
dt
T-tK
40 402(TT (T
tB - t)
1/2
VD
dl
1T (K)
Tā DDT -tk).
Thus, from (47), we infer that
V - E(VK FR)IL,, mm,
hig + gaa DDx
This completes the proof.
E
Numerical Experiments
Our experiments follow the standard DDPM sampling procedure as described in the original DDPM paper by [HJA20]. specifically the pscudocode presented in their Algorithm 2.
32

---

E.1 Implementation Details
For clarity, we re-state their algorithm below.
Algorithm 3 DDPM Sampling [HJA20)
I: ~ (N.
for 1 T to do
3:
if f ~0,
else
4: (e(( t) = Yor 2t
L-or
c
vI-a.
5: Tt-1 = (2s, t) 0,2
6: end for
7: return 20
To better explain the correspondence between notation used in our paper and that of [HJA20], we provide the following table:
Notation in [HJA20]
Our notation
TT. x0
Zo...., Zk+1
2
Ek+1
oz
/2hu
(1 + hx) -2 2 -2hx 2e 1 2hh
at
I (1 + h40) -2 = -
āt
60 (3:4 1) vI -
-2 tk Zk)
To evaluate the robustness of the generative process under perturbed estimates, had to isolate the score estimation component within the sampling loop. In the formulation of [HJA20], this corresponds to the rescaled neural network output -0.5c, /I/ C₁. In experiments, added various forms of noise (Gaussian, Uniform, Laplace, and Student's s-() directly to this term, simulating inaccurate noisy score predictions. This modification allows assess the impact of score perturbations on the quality of generated samples, both visually and quantitatively.
We know that in our formulation of the problem, the conditional expectation of the next state given that the current state is is given by pæ( t) = (1+ h)x + 2s(t, w)h. Therefore, adding Ç to s x)
implies adding 2hς to (x t), and thus adding Va 1 ā.) x 2 = 2/1 - or to en (, t).
1
E.2 Additional Figures
Qualitative results. Figure 6. Figure 7 and Figure 8 extend the main paper image grids. For each dataset (CIFAR-10, CelebA-HQ, and LSUN- ) we display samples generated with Gaussian, Laplace, and Student's- s-l score noise at two strengths, (severe). Rows share the same latent seed as the baseline enable direct visual comparison.
0.5
(moderate) and 2
Quantitative trends. Figure tracks FID on the CIFAR-10 dataset as we truncate the 000-step DDPM schedule (250. 500, 750, 1000) steps for the clean score and the i.idd. / Ip) noise contaminated score. We observe that performance increases at a similar rate with the number of steps for both clean and noisy score estimates.
Additionally, Figure illustrates the "deterioration" of three distinct pictures for each of the different models (datasets) that we have each starting with a fixed random noise, generating the corre- sponding image after 1000 diffusion steps with the noise contaminated score, as described before, parametrized by different We observe that datasets with higher-resolution images and, respectively, deeper noise (alternatively, score) predicting neural networks exhibit higher deterioration than those with low-resolution images.
33

---

110
FID
F1D
5:3 110
Sal 0.0
5d 0.5
5x5-
Time deope
Time stepa
Tinc stgp4
(a) CIFAR-10
(b) CelebA-HQ
(c) LSUN-Churcher Churches
Figure 4: FID as function of time steps. Blue: standard DDPM inference. Orange: same sampler with i.d (0, In) noise added to the score at each step.
0.25
0.5
1.5
Score Noise std
Figure 5: A single example CIFAR-I0 (top), CelebA-HQ (middle) and LSUN-Churches (bottom) generated data, respectively, different standard deviations.
E.3 Computational Resources
This project provided with computer and storage resources by GENCI IDRIS thanks the grant 2025-ADO1 1016491 on the supercomputer Jean Zay's A 100 partition.
Some of the experiments 64-Core Processor, 1TB of RAM, and with 8xA100 40GB VRAM version NVIDIA GPUs. The other with AMD EPYC 9005 192-Core Processor, 0.5TB of RAM, and with 2xH100 NVIDIA GPUs.
two additional GPU nodes:
with AMD EPYC 7V12
Sampling 8192 CIFAR-10 images or 512 CelebA-HQ or 512 L.SUN-Churchhe images takes 1.5 GPU-hours. FID evaluation for all the scale values of single noise distribution takes 0.2 GPU-hours.
E.4 Dataset and Model Licensing
CIFAR-10: Licensed under the MIT License.
CelebA-HQ: Licensed under CC BY-NC 4.0.
LSUN-Churches:: Licensed under CC B.0
google/ddpm-iifar10-3222 Apache License, Version 2.0.
goggle/ddpn-celebbann- 256: Apache License, Version 2.0.
google/ddpm-churctn Apache License, Version 2.0.
34

---

(a) No noise
(b) Gaussian noise, a
(c) Gaussian noise,
(d) No noise
(c) Laplace,
(f) Laplace,
2
(g) No noise
(h) Student's f, a
(i) Student's f, a = 2
Figure 6: Additional 10 generations for 3 noise families (rows) and noise levels (columns).
35

---

(a) No noise
(b) Gaussian noise, a 0.5
(c) Gaussian noise,
(d) No noise
(c) Laplace, 0.5
(f) Laplace, 1
(g) No noise
(h) Student's 's f, a 0.5
(i) Student's f, a
Figure 7: Additional CelebA-HQ generations for 3 noise families (rows) and 2 noise levels (columns).
36

---

(a) No noise
(b) Gaussian noise, a =
(c) Gaussian noise, a
(d) No noise
(c) Laplace,
(f) Laplace,
2
(g) No noise
(h) Student's t, or
(i) Student's 2
Figure 8: Additional LSUN-Church generations for 3 noise families (rows) and 2 noise levels (columns).
37

---

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Reviews:
[{"review_id": "JMWmjEDgXK", "reviewer": "Reviewer_YmpH", "summary": "The paper studies how robust DDPM sampling is when the score function is noisy, using Wasserstein distance to measure the distance from the true distribution. Experiments on CIFAR-10 and CelebA-HQ show that images remain visually unchanged until the added noise becomes quite large. Motivated by this, the authors prove that under some assumptions on the score estimates and the data distribution, the Wasserstein-2 distance between the generated and true distributions decreases at a near-optimal rate of $\\sqrt{D} / \\epsilon$, matching the known lower bound in the Gaussian case and improving over the previous best $D / \\epsilon^2$ rate.", "questions": "1. How critical is Assumption 1? Are there reasonably well-behaved distributions that fail to satisfy it, for which a $\\sqrt{D} / \\epsilon$ bound provably doesn't hold?\n2. The paper says the uniformity in Assumption 2 is essential to the proof. Is this just a technical condition, or is there a concrete example where dropping it would break the $\\sqrt{D} / \\epsilon$ bound?", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is fairly well written. The theory is clearly presented, with helpful examples.\n2. The main result is rigorously proved, and the bounds are non-asymptotic.\n\nWeaknesses:\n\nI'm not fully convinced by the motivation. Assumption 2 requires a uniform error bound over all x, which feels quite strong. The examples of randomized score estimators aren't very compelling — it's hard to think of real-world cases where we truly need a compressed score network or where scores suffer from communication noise.", "was_revised": true, "final_justification": "The authors addressed my questions in the rebuttal. I think the techniques introduced in this paper used would offer value to the NeurIPS community. I would like to increase my recommendation score to 4.", "rebuttal": "We thank the reviewer for their positive comments on the clarity of the writing and the rigour of the mathematical derivations. We devoted considerable effort to these aspects and are pleased that they have been acknowledged. Below, we address the questions and concerns raised in the review.\n\n---\n\n### Weaknesses\n\n* **Uniform error in Assumption 2**:  \n    * First, we would like to emphasize that requiring a uniform bound on the score estimator is not as restrictive as it might initially seem. In many nonparametric estimation problems, using the sup-norm (i.e., uniform error) instead of the $L_2$-norm (i.e., integrated error) leads to only a logarithmic loss in the convergence rate. For example, Theorem 1.8 in [Tsybakov 2008] demonstrates this for the local polynomial estimator of a regression function. Similar results are available in multivariate density estimation; see [Lepski 2013] and references therein.\n\n    * Second, the formula on line 1076 of the manuscript shows that the quantity appearing in our proof is actually an $L_2$ error of the score estimator with respect to the distribution of the discretized diffusion process $Z_k$ (see also the definitions of $\\zeta_k$ on line 1063 and $Z_k$ on line 190). This $L_2$ error is clearly upper bounded by the sup-norm error, which is also more convenient for technical derivations. This is the main reason we stated Assumption 2 in terms of a uniform error. However, the proof remains valid under a weaker $L_2$-type assumption with respect to the distribution of $Z_k$; ; the same assumption referred to as H2 in [Silveri & Ocello 2025]..\n\n* **Compellingness of the randomized score**:  \n    The main objective of our paper is to provide a theoretical analysis of a widely used algorithm, DDPM. From a theoretical perspective, we believe it is valuable to understand the robustness of this method to noise. While the practical impact of this property remains to be fully understood, the growing interest in compression within the AI community strongly supports its relevance. For example, the Workshop on Machine Learning and Compression at NeurIPS 2024, along with a large number of submissions on neural network compression and its effect on predictive performance, highlight the timeliness and importance of this direction.\n\n---\n\n### Questions\n\n1. **Question**: *How critical is Assumption 1? Are there reasonably well-behaved distributions that fail to satisfy it, for which a $\\sqrt{D}/\\varepsilon$ bound provably doesn't hold?*  \n   **Answer**: This is a difficult and subtle question. As discussed in the paper, most standard distributions satisfy Assumption 1 and therefore cannot serve as counterexamples to this assumption. Furthermore, as explained in Section A (page 21), there are essentially three main strategies for proving the convergence of diffusion samplers. Only the approach developed in our work currently yields the optimal rate of $\\sqrt{D}/\\varepsilon$. Even if one could identify a distribution that fails to meet certain assumptions used in our proof, this would not necessarily imply that the desired bound cannot be achieved using one of the other techniques. Finally, proving a lower bound on the $W_2$ distance between the sampling distribution and the target distribution is a challenging task.\n\n2. **Question**: *The paper says the uniformity in Assumption 2 is essential to the proof. Is this just a technical condition, or is there a concrete example where dropping it would break the bound?*  \n   **Answer**: We acknowledge that this sentence does not accurately reflect the underlying mathematical reasoning, and we will revise it in the paper. For a more complete response, please refer to our answer to the first weakness above.\n\n---\n\n### Concluding Remark\n\nWe would like to emphasize that even if one disregards our contribution concerning the robustness of the score estimator to noise, we believe that establishing the first mathematical proof of the optimality of DDPM for a reasonably broad class of distributions is in itself a significant result. The review identifies the restrictiveness of our assumptions as a key weakness. Still, we note that most papers on DDPM, such as the recent work by [Silveri & Ocello, 2025], employ more restrictive assumptions while only achieving suboptimal rates.\n\n---\n\n### Cited References\n\n[Tsybakov 2008] Alexandre B. Tsybakov (2008). *Introduction to Nonparametric Estimation*. Springer Publishing Company.\n\n[Lepski 2013] Lepski, Oleg. “Multivariate Density Estimation under Sup-Norm Loss: Oracle Approach, Adaptation and Independence Structure.” *The Annals of Statistics* 41, no. 2 (2013): 1005–1034.\n\n[Silveri & Ocello 2025] Silveri, Marta and Antonio Ocello. “Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-Distance.” *Proceedings of ICML 2025*."}, {"review_id": "opdB2UZ3WI", "reviewer": "Reviewer_nFJ3", "summary": "This paper investigates the robustness and convergence guarantees of denoising diffusion probabilistic models (DDPMs) under noisy score evaluations. The authors empirically demonstrate that DDPMs remain effective when additive noise with constant variance is introduced into the score function during sampling. Theoretically, the paper derives non-asymptotic upper bounds on the Wasserstein distance between the true data distribution and the distribution induced by the noisy DDPM. These bounds match the optimal rate known in the Gaussian case and extend to a broad class of distributions, including compactly supported semi-log-concave measures. The analysis highlights the limited impact of noise variance compared to bias and establishes conditions under which fast convergence is achieved. The results are compared to numerical experiments on benchmark datasets.", "questions": "**1.** Do the authors consider it feasible and useful to extend this type of analysis to high-dimensional settings?\n\n**2.** In the conclusion, the authors mention the possibility of incorporating estimators of the Hessian of the log-density into the DDPM framework. Could they elaborate on what they mean by this, and how such estimators would be used in practice?\n\n**3.** Perhaps I missed it, but are there any assumptions on the score estimator $\\hat{\\mathbf{s}}$? For instance, can it be a neural network with arbitrary activation functions, or are there regularity or smoothness requirements needed for the theoretical results to hold?", "limitations": "Yes.", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Quality**\n\nThe paper presents a non-asymptotic analysis of DDPMs under noisy score evaluations, with claims of improved convergence rates in Wasserstein-2 distance. The technical arguments appear carefully constructed, and the analysis is detailed. However, I am not familiar with the details of the extensive literature on the bounds mentioned and cited in the paper.\n\n\n**Clarity**\n\nThe exposition is generally well structured and the motivation is clear. I particularly appreciate the sections where the authors discuss the relevance and the challenges of the Wasserstein distance. Assumption 1, which seems very technical, is extensively discussed in the appendix.\n\n\n**Significance**\n\nThe paper addresses a relevant question, particularly in settings where score evaluations may be noisy due to communication or compression constraints. If correct, the claimed convergence rates would represent a theoretical advance, given the references cited by the paper.\n\n\n**Originality**\n\nI believe it's an original paper. The focus on robustness to noisy scores in the context of DDPMs, and the use of Wasserstein distance as the evaluation metric, marks a departure from much of the existing literature, which concentrates on KL or total variation distances. The consideration of different forms of randomized score estimators is also less common.", "was_revised": true, "final_justification": "I thank the authors for their careful and detailed rebuttal addressing all questions and remarks. I will maintain my positive score, though I stress that I am not familiar with the literature or the techniques used in the paper.", "rebuttal": "We thank the reviewer for their positive evaluation of our contributions. We are especially pleased that they appreciated the discussion on the relevance of the Wasserstein distance in a theoretical setting. We considered it important to include this discussion, given the number of questions we received on this point when presenting our results in various seminars. Below, we provide point-by-point responses to the reviewer’s questions.\n\n---\n\n1. **Question**: *Do the authors consider it feasible and useful to extend this type of analysis to high-dimensional settings?*  \n   **Answer**: The setting we study in the paper is, in fact, commonly referred to as high-dimensional. Our results exhibit a square-root dependence of the Wasserstein error on the dimension, which appears to be optimal under the assumptions we consider, or even under assumptions frequently adopted in the existing literature. To develop some intuition for why this $\\sqrt{D}$ scaling is unavoidable, consider the case where the target distribution is a product of identical univariate distributions. Given that the $W_2$ distance is based on the Euclidean norm, it naturally scales as $\\sqrt{D}$ times the error incurred when sampling from a one-dimensional distribution. Extending our results to infinite-dimensional sampling problems is an intriguing direction for future research.\n\n---\n\n2. **Question**: *In the conclusion, the authors mention the possibility of incorporating estimators of the Hessian of the log-density into the DDPM framework. Could they elaborate on what they mean by this, and how such estimators would be used in practice?*  \n   **Answer**: The use of the Hessian of the log-density is a well-established strategy in the sampling literature for improving convergence. For example, the Ozaki discretization [Ozaki 1992; Stramer & Tweedie 1999] leverages the Hessian to enhance the convergence properties of Langevin-based algorithms. Similarly, the KLMC2 algorithm introduced in [Dalalyan & Riou-Durand 2020] achieves fast rates for discretized kinetic Langevin diffusion, a method that Stability AI has successfully implemented.\n\n   Our remark on incorporating estimated Hessians follows this tradition. More precisely, one could use the Itô decomposition presented just above line 1202 to define a second-order discretization of the continuous-time diffusion process $Y_t$. This would enable the algorithm to more accurately capture the local curvature of the log-density and, potentially, enhance its efficiency. We hope this explanation clarifies our intention, and we would be glad to provide further details during the discussion phase if the reviewer finds it helpful.\n\n---\n\n3. **Question**: *Perhaps I missed it, but are there any assumptions on the score estimator? For instance, can it be a neural network with arbitrary activation functions, or are there regularity or smoothness requirements needed for the theoretical results to hold?*  \n   **Answer**: The only conditions imposed on the estimated score are those stated in Assumption 2. It is entirely possible to use a neural network with non-smooth activation functions, such as ReLU or Heaviside, even though they yield non-differentiable or discontinuous score estimates. This flexibility stems from the structure of our analysis, which decouples the error in score estimation from the discrepancy between the discrete-time process using the true score and its continuous-time counterpart. As a result, no additional smoothness or regularity assumptions are needed for the estimated score.\n\n---\n\n### Cited References\n\n[Ozaki 1992] T. Ozaki. *A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach*. Statistica Sinica, 2(1):113–135, 1992.\n\n[Stramer & Tweedie 1999] O. Stramer and R. L. Tweedie. *Langevin-type models. II. Self-targeting candidates for MCMC algorithms*. Methodology and Computing in Applied Probability, 1(3):307–328, 1999.\n\n[Dalalyan & Riou-Durand 2020] A. Dalalyan and L. Riou-Durand. *On sampling from a log-concave density using kinetic Langevin diffusions*. Bernoulli, 26(3):1956–1988, 2020."}, {"review_id": "e965AYble8", "reviewer": "Reviewer_hpXx", "summary": "This paper studies the theoretical and empirical behavior of Denoising Diffusion Probabilistic Models (DDPMs) when the estimated score function is perturbed with noise. Specifically, the authors analyze the quality of generated samples under noisy score evaluations using the 2-Wasserstein distance as a metric of generative quality—an alternative to the more commonly used total variation or KL divergence.\n\nOn the theoretical side, the authors:\n\n* Formalize a noisy DDPM setting where each score query returns a randomized estimate with bounded bias and variance (Assumption 2).\n* Introduce conditions on the target distribution (Assumption 1), capturing both strongly log-concave distributions and semi-log-concave distributions with bounded support.\n* Establish non-asymptotic, finite-sample bounds on the 2-Wasserstein distance between the target and generated distributions under noisy score estimates.\n* Show that their convergence bounds match the optimal rate of $\\sqrt{D}/\\epsilon$ (in $D$ dimensions), previously known only for the Gaussian case.\n\nOn the experimental side, they validate their theory using pretrained DDPM models on CIFAR-10 and CelebA-HQ datasets. They inject different types of zero-mean noise (Gaussian, Laplace, Uniform, and Student-t) into the score function at test time and:\n\n* Show visually indistinguishable generations even under substantial noise (up to σ = 1 or 2), demonstrating robustness.\n* Report FID scores that increase only mildly with noise scale, confirming the theoretical insight that variance in score estimation has limited impact on sample quality.\n* Observe that the shape of the noise distribution (e.g., Gaussian vs Laplace) has negligible influence, reinforcing the focus on noise scale rather than distributional assumptions.", "questions": "* In Algorithm 2, you use a hybrid arithmetic-geometric time grid. Could you clarify the rationale behind this specific choice? In particular, is there a theoretical justification for its optimality over alternative schemes such as purely geometric or uniform grids, and how would these alternatives affect the convergence guarantees?\n\n* Many real datasets (especially images) lie on low-dimensional manifolds. How sensitive is your analysis to the intrinsic vs. ambient dimensionality of the data?\n\n* In your experiments, did you consider adding noise with a non-zero mean (i.e., introducing bias) to the score function? If not, do you expect the empirical degradation in sample quality to be significantly more severe compared to zero-mean noise?", "limitations": "Yes.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n* The paper proves that DDPMs are not only robust to score noise (under some assumptions on the target distribution and the noise distribution) but also achieve near-optimal convergence rates in 2-Wasserstein distance.\n* The theory developed in this paper is supported by experiments\n\n**Weaknesses**\n* While broader than some prior works, the main theoretical results (e.g., Theorems 2 and 3) still rely on strong assumptions like bounded support or semi-log-concavity.", "was_revised": true, "final_justification": "After carefully reviewing the authors' rebuttal, I believe they have adequately addressed all of my concerns. In particular, the authors have answered my questions about the rationale behind the use of a hybrid arithmetic-geometric time grid, the sensitivity to the intrinsic vs. ambient dimensionality of the data, and an intuition on the expected output when adding noise with a non-zero mean to the scores.", "rebuttal": "We would like to thank the reviewer for the positive evaluation and the fair summary of our contributions, both theoretical and experimental. We particularly appreciate that the strengths and limitations mentioned in the review are fully aligned with those we have transparently discussed in the manuscript. Below, we briefly comment on the limitations and respond point-by-point to the questions raised in the review.\n\n---\n\n### **Weakness**\n\n>*While broader than some prior works, the main theoretical results (e.g., Theorems 2 and 3) still rely on strong assumptions like bounded support or semi-log-concavity.*\n\nWe agree that it would be very interesting to relax the assumptions we make, while retaining the fast convergence rates we obtain. Nevertheless, we would like to emphasize that our assumptions are *weaker* than those used in previous works that yield only suboptimal rates; see, for instance, [Silveri & Ocello 2025; Yu & Yu 2025]. In particular, these works require the target distribution to be absolutely continuous with respect to the Lebesgue measure, whereas our results apply to certain distributions that do not satisfy this assumption. It is worth noting that the widely cited work [Chen et al. 2023] requires the target distribution to have a Lipschitz-continuous score function (Assumption A1). This assumption is stronger than the semi-log-concavity, and therefore, more restrictive than the assumptions of our work, but is still considered as not restrictive since the title of the mentioned paper refers to it as “minimal data assumptions”.   \n\nFurthermore, compared to prior work in the sampling literature, assuming semi-log-concavity or bounded support does not appear overly stringent. Many commonly used distributions satisfy these assumptions.\n\nFinally, as discussed in Section A (page 21), the existing results that rely on a different set of assumptions, which may arguably be considered weaker (though not strictly comparable), lead to worse dependence not only on the discretization step $h$, but also, more problematically, on the error of the estimated score. This suggests that even when using an optimal score estimator, those approaches may not achieve optimal sampling rates (in the $W_2$-distance), unlike ours.\n\n---\n\n### **Questions**\n\n1. **Question** *In Algorithm 2, you use a hybrid arithmetic-geometric time grid. Could you clarify the rationale behind this specific choice? In particular, is there a theoretical justification for its optimality over alternative schemes such as purely geometric or uniform grids, and how would these alternatives affect the convergence guarantees?*\n\n    **Answer** In the second part of the time interval shown in Figure 3 (page 30), the discretization error includes a term of the form $\\sum_k \\frac{h_k}{T - t_{k+1}}$, as noted in lines 1141–1143. Here, $h_k$ denotes the step size, and $t_k$ the time grid points. Using a uniform grid would cause the last terms in this sum to become large, on the order of a constant, thus preventing the error from vanishing as the number of steps increases.\n\n    Conversely, applying a geometric grid throughout results in a large discretization error during the first part of the interval. According to the equation following line 1134 and Eq. (30), this error scales roughly as $\\sum_k h_k^2$. Given the constraint $\\sum_k h_k = C$, the sequence $h_k = C/K$ minimizes the sum of squared step sizes. Therefore, a uniform grid is optimal over the first part. Combining this with a geometric grid on the second part, where uniform discretization would fail, yields our hybrid scheme, which is both theoretically justified and empirically effective.\n\n---\n\n2. **Question** *Many real datasets (especially images) lie on low-dimensional manifolds. How sensitive is your analysis to the intrinsic vs. ambient dimensionality of the data?*\n\n    **Answer** The DDPM algorithm adds isotropic noise to the data without accounting for the underlying manifold structure. In other words, the injected noise is equally strong in all directions of the ambient space. Therefore, it is natural, and theoretically expected, that the resulting error scales with the square root of the ambient dimension. Reducing this dependence to the intrinsic dimension would require a careful exploitation of the low-dimensional structure in the score estimation step, not in the DDPM itself. As in many nonparametric estimation problems, the error of the score estimator (denoted $\\varepsilon$ in our paper) is sensitive to the intrinsic dimension. Hence, the choice of score estimator is a critical component of the DDPM pipeline, even though it is decoupled from the convergence analysis we conduct in this work.\n\n---\n\n3. **Question** *In your experiments, did you consider adding noise with a non-zero mean (i.e., introducing bias) to the score function? If not, do you expect the empirical degradation in sample quality to be significantly more severe compared to zero-mean noise?*\n\n    **Answer**  We did not consider noise with a non-zero mean in our experiments. However, we expect this type of noise to have a much more detrimental impact on the sampling performance. When the noise mean is non-zero, the iterates of the algorithm are no longer decorrelated from this bias. As a result, the mean bias behaves analogously to the term $\\varepsilon^b$, which negatively affects convergence guarantees. While we cannot include empirical results due to rebuttal format constraints, we would be happy to discuss this further during the discussion period.\n\n---\n\n### Cited References\n\n [Silveri & Ocello 2025] Silveri, M. and Ocello, A. “Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-Distance.” *Proceedings of ICML 2025*.\n\n [Yu & Yu 2025] Yifeng Yu and Lu Yu. *Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Acceleration*. *arXiv preprint*, 2025."}, {"review_id": "zQYTZP6gTt", "reviewer": "Reviewer_KSvS", "summary": "This paper investigate the theoretical and empirical behaviors of Denoising Diffusion Probabilistic Models (DDPMs) along the 2-Wasserstein distance. It is shown empirically by the authors that DDPMs function even if the estimated score function is additive noise corrupted. It comes as a surprise as the score holds a pivotal places in sample generation. The paper provides new non-asymptotic upper bounds on the Wasserstein-2 distance between target and generated distributions. Tightly capture estimation noise (bias and variance) effects on convergence. Establish the optimal rate for for the Gaussian case, but under much milder assumptions, such as semi-log-concavity and bounded support. The achieved rates are shown optimal with respect to the current state of results and include previously unexamined cases (e.g., mixtures of strongly log-concave and compactly supported components). Experiments on CIFAR-10 and CelebA-HQ demonstrate visual robustness of DDPMs to different types of noise, and Fréchet Inception Distance (FID) scores verify theoretical predictions.", "questions": "1. How realistic is it to assume a uniform bound on score estimation error for all $x,t$ ? Is the analysis transferable to expected bounds (e.g., across the data distribution)? Removing this assumption would make the results more practically applicable.\n\n2. The bound contains exponential terms like $e^{2bM}$. Can you tell us whether they are inherent to the problem or remnants of your proof techniques? Justifying or motivating these constants would make the theoretical results more comprehensible and tighter.\n\n3. Kinetic Langevin DDPMs is suggested as future work by authors. Do you believe your methods (e.g., error decomposition or variance decay) could be ported directly? Even a brief mention may imply the universality and extensibility of your model.", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths:\n\nThe theory is accurate, complete, and commensurate with the state of the art. The paper introduces a novel non-asymptotic analysis of DDPMs in Wasserstein distance, which is more applicable to most real-world problems than KL/TV. In addition, it introduce new analysis framework that neatly separates bias and variance effects on score estimation, and establish convergence to a broader class than the previously studied strongly log-concave or smooth ones The results close an important theoretical gap in the robustness understanding of diffusion models. The boundaries are properly motivated, neatly defined, and claimed to be minimax optimal in some cases. Assumptions and mathematical machinery are well described i.e., key assumptions and contributions are clearly stated. Experimental setup and plots are appropriately described.  \n\n# Weaknesses \n\nSome technical assumptions (e.g., uniform error bounds over $x$ and $t$ ) may be difficult to satisfy in real applications. Empirical analysis, although considered, is still small scale and not necessarily representative of failure modes. The results' implications may  limited unless augmented by working practical algorithms that directly make use of the insights (e.g., noise-robust training or compression techniques).  Empirical part are incremental, equivalent visual robustness results are available, but not necessarily rigorously justified.", "was_revised": true, "final_justification": "The authors addressed all my questions.", "rebuttal": "We would like to thank the reviewer for the detailed and thoughtful assessment of our manuscript. It is rewarding to see that the reviewer appreciated the novel analysis framework we introduced, which allowed us to close a theoretical gap left open in previous work on DDPMs. Below, we respond to the weaknesses highlighted in the review and provide answers to the questions raised.\n\n---\n\n### Weaknesses\n\n> *Some technical assumptions (e.g., uniform error bounds) may be difficult to satisfy in real applications.*\n\nWe agree that weakening this assumption would be desirable. However, we do not believe it is a major limitation in practice. To the best of our knowledge, there are no results showing that standard score estimators (such as those based on score matching or kernel methods) fail in the sup-norm. Furthermore, as explained in the response to Question 1 below, theoretical results from nonparametric statistics support the idea that this condition is not overly restrictive.\n\n> *Empirical analysis, although considered, is still small scale and not necessarily representative of failure modes.*\n\nAs correctly noted in the summary of our contributions, our primary focus is on theoretical insights. In addition, as members of an academic laboratory, we have limited access to large-scale computing resources (e.g., GPUs), which makes it difficult to run more extensive empirical evaluations. We hope these two considerations justify the relatively modest empirical part of our submission.\n\n> *The results' implications may be limited unless augmented by working practical algorithms that directly make use of the insights (e.g., noise-robust training or compression techniques).*\n\nOur main goal is to advance theoretical understanding of DDPMs by analyzing their robustness to noise. While the practical significance of this property remains an open question, the increasing interest in compression methods across the AI community confirms the timeliness of our study. For instance, the dedicated Workshop on Machine Learning and Compression at NeurIPS 2024 and the growing number of papers on this topic highlight the relevance of our direction.\n\n> *Empirical part are incremental, equivalent visual robustness results are available, but not necessarily rigorously justified.*\n\nWe agree that the main originality of our paper lies in its theoretical contributions rather than its experiments, a point we explicitly acknowledge in the manuscript. However, we are not aware of prior works presenting equivalent robustness results, either visually or quantitatively. If the reviewer has specific references in mind, we would be very grateful to receive them.\n \n---\n\n### Questions\n\n1. **Question** *How realistic is it to assume a uniform bound on score estimation error for all $x$ and $t$? Is the analysis transferable to expected bounds (e.g., across the data distribution)? Removing this assumption would make the results more practically applicable.*\n\n    * First, we emphasize that assuming a uniform bound on the score estimator may not be as restrictive as it seems. In nonparametric statistics, replacing $L_2$-norm error with sup-norm error often leads to only a logarithmic degradation in convergence rates. For example, Theorem 1.8 in [Tsybakov 2008] shows this for local polynomial regression. Similar results exist for density estimation; see [Lepski 2013] and references therein.\n\n    * Second, the expression on line 1076 of the manuscript shows that the relevant quantity in our proofs is actually an $L_2$ error of the score estimator with respect to the distribution of the discretized diffusion process $Z_k$ (see also $\\zeta_k$ on line 1063 and $Z_k$ on line 190). This $L_2$ error is upper bounded by the sup-norm error, which is also more convenient for technical derivations. This is the main reason Assumption 2 is formulated in sup-norm terms. Nonetheless, the proof remains valid under a weaker $L_2$-type assumption with respect to the distribution of $Z_k$; the same assumption referred to as H2 in [Silveri & Ocello 2025].\n\n2. **Question** *The bound contains exponential terms like $e^{2bM}$. Can you tell us whether they are inherent to the problem or remnants of your proof techniques? Justifying or motivating these constants would make the theoretical results more comprehensible and tighter.*\n\n    **Answer** The exponential terms in the bounds depend on the parameters of the target distribution. We conjecture that this behavior is intrinsic to bounding the Wasserstein distance in DDPMs. This belief is reinforced by the independent work [Silveri & Ocello 2025], which uses a different proof technique yet yields similar exponential dependence.\n\n    That said, we do not have a counterexample proving that the exponential factor is unavoidable; i.e., a case where the $W_2$ error must necessarily involve this term. Constructing such an example is challenging for two reasons: (1) for distributions where the DDPM law is known in closed form, our upper bound does not contain the exponential term (as either $b = 0$ or $M = 0$); (2) the $W_2$ distance is defined via an infimum over all couplings, while our upper bound is based on synchronous coupling. Demonstrating the necessity of the exponential term under synchronous coupling would not automatically imply its presence in the actual $W_2$ error.\n\n3. **Question** *Kinetic Langevin DDPMs is suggested as future work by authors. Do you believe your methods (e.g., error decomposition or variance decay) could be ported directly? Even a brief mention may imply the universality and extensibility of your model.*\n\n    **Answer** While there are similarities between the analysis of overdamped (Langevin) and underdamped (kinetic Langevin) algorithms—as explored in [Chen et al. 2023; Yu et al. 2024; Conforti et al. 2025]; a direct transfer of methods is generally not feasible. A careful examination of these papers shows that the two settings require distinct computations, and those for the kinetic process are much more tedious. For this reason, we have chosen to focus exclusively on the overdamped (Ornstein-Uhlenbeck) setting in the present work, leaving the kinetic case for future investigations.\n\n ---\n\n### Cited References\n\n[Tsybakov 2008] Alexandre B. Tsybakov (2008). *Introduction to Nonparametric Estimation*. Springer.\n\n[Lepski 2013] Lepski, Oleg. “Multivariate Density Estimation under Sup-Norm Loss: Oracle Approach, Adaptation and Independence Structure.” *Annals of Statistics* 41(2): 1005–1034, 2013.\n\n[Silveri & Ocello 2025] Marta Silveri and Antonio Ocello. “Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in $W_2$ Distance.” *Proceedings of ICML 2025*.\n\n[Chen et al. 2023] Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, Anru Zhang. “Sampling is as Easy as Learning the Score: Theory for Diffusion Models with Minimal Data Assumptions.” *ICLR 2023*.\n\n[Yu et al. 2024] Lu Yu, Avetik Karagulyan, Arnak Dalalyan. “Langevin Monte Carlo for Strongly Log-Concave Distributions: Randomized Midpoint Revisited.” *ICLR 2024*.\n\n[Conforti et al. 2025] Giovanni Conforti, Alain Durmus, Marta Silveri. “KL Convergence Guarantees for Score Diffusion Models under Minimal Data Assumptions.” *SIAM J. Math. Data Sci.*, 7(1): 86–109, 2025."}]

Author rebuttal:

Meta-review: This paper analyses robustness of DDPM in terms of Wasserstein distance. The authors showed that under the assumption of log-concavity and semi-log-concavity, DDPM is robust against noise injection to the estimated model and thus achieves the near optimal rate $\sqrt{D}\epsilon$.   
  
This paper gives an interesting and solid theoretical justification of the robustness of DDPM. Under the log-concavity condition of the objective, the injected noise does not expand exponentially against the number of steps. This is somehow predictable. However, this paper successfully gives better rate than existing work, which is novel.  

The numerical experiments effectively show plausibility of the insight obtained by the theoretical analysis.    

Overall, this paper is well written giving a solid theoretical contribution. Hence, I recommend acceptance.
Decision: Accept (poster)