Wasserstein Convergence of Critically Damped Langevin Diffusions
Authors: Stanislas Strasman
Sobihan Surendran
Claire Boyer
Sylvain Le Corff
Vincent Lemaire
Antonio Ocello
Venue: NeurIPS 2025 poster ()
Keywords: Wasserstein convergence; score-based generative models; kinetic Langevin diffusion
Primary area: probabilistic_methods
TL;DR: This paper provides the first Wasserstein convergence guarantees for critically damped Langevin diffusions.

Abstract: Score-based Generative Models (SGMs) have achieved impressive performance in data generation across a wide range of applications and benefit from strong theoretical guarantees. Recently, methods inspired by statistical mechanics, in particular, Hamiltonian dynamics, have introduced Critically-damped Langevin Diffusions (CLDs), which define diffusion processes on extended spaces by coupling the data with auxiliary variables. These approaches, along with their associated score-matching and sampling procedures, have been shown to outperform standard diffusion-based samplers numerically. In this paper, we analyze a generalized dynamic that extends classical CLDs by introducing an additional hyperparameter controlling the noise applied to the data coordinate, thereby better exploiting the extended space. We further derive a novel upper bound on the sampling error of CLD-based generative models in the Wasserstein metric.  This additional hyperparameter influences the smoothness of sample paths, and our discretization error analysis provides practical guidance for its tuning, leading to improved sampling performance.

Wasserstein Convergence of Critically Damped Langevin Diffusions
Stanislas Strasman
Sobihan Surendran'
Claire Boyer*
Sylvain Le Corff!
Vincent Lemaire!
Antonio Ocello
1 Sorbonne Université and Université Paris Cité, CNRS, LPSM, F-75005 Paris, France LOPF, Califrais' Machine Learning Lab, Paris, France
LMO Université Paris-Saclay, UMR CNRS 8628, Institut Universitaire de France, Orsay, France *CREST, ENSAE, Institut Polytechnique de Paris, Palaiseau, France
Abstract
Score-based Generative Models (SGMs) have achieved impressive performance in data generation across wide range of applications and benefit from strong theoretical guarantees. Recently, methods inspired by statistical mechanics, particular, Hamiltonian dynamics, have introduced Critically-damped Langevin Dif- fusions (CLDs), which define diffusion processes on extended spaces by coupling the data with auxiliary variables. These approaches, along with their associated score-matching and sampling procedures, have been shown to outperform standard diffusion-basse baased samplers numerically. In this paper, we analyze a generalized dynamic that extends classical CLDs by introducing an additional hyperparameter controlling the noise applied the data coordinate, thereby better exploiting the extended space. We further derive a novel upper bound on the sampling error of CLD-based generative models in the Wasserstein metric. This additional hyper- parameter influences the smoothness of sample paths, and our discretization error analysis provides practical guidance for its tuning, leading improved sampling performance.
Introduction
Recent surge in machine learning and artificial intelligence has driven substantial progress in gen- erative modeling, particularly with the development of Score- based Generative Models (SGMs). These models build the foundational works in Denoising Diffusion Probabilistic Models (DDPMs) by [Sohl-Dickstein et al. (2015); Song and Ermon (2019): : Ho et al. (2020) and the advances in score-matching techniques introduced by Hyyääinen and Dayan (2005): Vincent (2011).
Score-based Generative Models (SGMs), SGMs are probabilistic models designed to create synthetic instances of target distribution when only genuine sample (e.g., dataset of real-life images) is accessible. First, the forward process involves progressively perturbing the training distribution by adding noise the data until its distribution approximately reaches an easy-to-sample distribution Tx. Then, the backward process involves learning to reverse this noising dynamics by sequentially removing the noise. SGMs have quickly gained recognition for their ability generate high-quality synthetic data. Their applications span diverse areas. including computer vision (Li et al.)
2022, Lugmayr et al. [2022), natural language processing (Gong al. 2023). and other domains where realistic data generation is crucial. This growing body of work has been comprehensively surveyed by Yang et al. (2023), highlighting the versatility and potential of diffusion models. In
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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addition, SGMs provide a particularly interesting class of prior distributions to solve Bayesian inverse problems. Although they lack an explicit and tractable probability density function, very active research area focuses on combining Monte Carlo guidance and SGMs to solve posterior sampling problems, Wu al. (0233: Moufad et al. (025); Victorino Cardoso et al. (2024).
Criticaly-ddamped Langevin Diffusion (CLD). In Dockhorn et al. 2022), the authors proposed Critically-damped Langevin Diffusion as a second-order extension of conventional diffusion models. By introducing velocity variables alongside the usual state variables mucch like in Hamiltonian Monte Carlo- CLD accelerates exploration of high-dimensional spaces and often yields better sample quality in practice. Although empirical work demonstrates the benefit of CLD over standard score-based models (Dockhorn et al. 2022), its theoretical underpinnings remain incomplete. Existing convergence guarantees are only expressed in terms of Dullback-Leible divergence (Conforti et al.. 2025 Chen et al. 2023) and fail to capture any computational advantage for kinetic dynamics, leaving a gap between observed performance and formal analysis.
Contributions. We first discuss the challenges of establishing Wasserstein convergence under the standard assumptions used for Suriance-Preserving where the forward process is elliptic (Gao et al. 2025 Strasman et al., 2025 Gentiloni-Silee and Ocello 2025; Bruno et al. 2025). We then provide, the best of our knowledge, the first upper bound for CLD in the Wasserstein metric through coupling techniques under weaker assumptions, achieving convergence rates comparable to those of other SGMs. Crucially, this result is not implied by previous Kullback-Leibe divergence bounds (Conforti et al. [025 Chen et al. (023). and our proof technique differs significantly from existing Wasserstein analyses of diftusion models.
(VP) or Muriance-Exploding (VE) SGMs.
However, it is possible to introduce a modified dynamics that includes an additional hyperparameter controlling the noise on the data coordinate of CLD, thereby restoring ellipticity and enabling an analysis closely aligned with that of VP and VE models, but formulated on an extended phase space with
drifts and diffusions. This hyperparameter governs the smoothness of sample paths, allowing detailed analysis of the generative error as a function of this smoothness parameter. Such analysis offers practical guidance for tuning this hyperparameter and potentially improves sampling performance compared to standard SGMs and CLD methods. The benefits of this additional parameterization are demonstrated numerically on challenging synthetic datasets.
2 Notation and Background
Notation. We use π to denote probability distributions and p to denote their corresponding densities with respect to the Lebesgue measure or another reference measure. The identity matrix of size is written Lt. For x, y E Rd we denote by (z, y) the standard inner product of R2. by || Il the Euclidean norm for vectors and its induced operator norm for matrices. Let || || be the Frobenius Tr A). For symmetric matrices A, B E Rdxd we write A < B to mean that B - A is positive semidefinite. We denote the time derivative of function by f(t) := 4
norm defined for A E Raxd as ||A||
de (t). We use the symbol @ either for the Kronecker product when applied to matrices and for the product of probability measures when applied to distributions. The intended meaning will be clear from context. For any matrix A E Rddx we denote its largest eigenvalue (resp. singular value) by Amaax (A) (resp. m (A)) and smallest eigenvalue by Amin( (resp. mmin (A)). For random vectors X, E Rd, define ||X] [z,, == (E [I[x) 1/2 and we write X 1 Y to mean that X is independent of Y. The notation L(X) denotes the law (distribution) of a random vector X. For a, b E R. we write a A b == min(a,b) b} and a b= ma)
Score-based Generative Models. SGMs employ a Gaussian Markovian diffusion process that smoothly transports the target data distribution Taata E (TR)) towards an easy-to-sample Gaussian distribution pro E ((Rd). This process, known as forward diffusion, is the solution the following stochastic differential equation (SDE) on a fixed time horizon i E [O,
ax. = - X + 23())BB. X ~ Adaaa
with (Be)eejo,r) a d-dimensional Brownian motion and B(t): : [0, + R.. In particular, when 0 and B(t) is of the form VVE the process is known as Variance Exploding (Song

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and Ermon 2019) and when = 1, the process is known as Variance Preserving (Sohl-Dickstem
al. 2015: Ho al. 2020). This transformation can be reversed (Anderson, 1982 Haussmann and
Pardoux 1986 Cattiaux et al. 2023 and is also governed by an SDE, known as the backward process
ax, = (o.(T- () + 23 V) Pog -() de + V23(T - l)dB,, X Pompr.
pr: (2)
where Pt is the time marginal p.d.f. of the forward process for 0 S S T. As a consequence, Xx has the same distribution as Tdata- In practice, however, one cannot draw exact i..d. samples from this continuous-time process, and implementations of SGMs rely on three key approximations.
Mixing error. The distribution of XT is not analytically available in most cases, Xo is initialized at known distribution Too, close to pr.
Discretization error. In most cases, the backward dynamic is non-linear, the backward process discretized to sample from X,, which introduces an emoo due to evaluating the (time-continuous) score function only discrete time steps.
Approximation error. The score function depends on the unknown data distribution and thus cannot be computed closed form. To approximate it, we use neural network architecture [0, T x _R IR parameterized by 0 E G and trained, for example, via Denoising Score Matching (see, e.g., Vincent 2011
X(t) se (T.X.) Vlogp, (x)1
DDSM (0)
(3)
where is uniformly distributed on []. is independent of Xo. X p+ and A: [0, 7) > Ry is a positive weighting function.
Theoretical studies of SGMs focus those sources of errors derive results for the total variation
distance (De Bortoli et al. 202T), the Kullback-Leii Leibler divergence (Conforti et al. [025 De Bortoli
et al. Chen al. 2023 Benton al. 2024) the Wasserstein-2 distance (Lee al. 2022
2023
Bruno et al. 2025
Gao el al. 2025 Strasman et al. 2025 Gentiloni-Silveri and Ocello 2025
Kinetic Orastein-Uhlenbert Inspired by Hamiltonian mechanics, kinetic SGMs operate in an extended position-velocity phase space, defined as U, = (X,, V.)T IRdd which satisfies the following stochastic differential equation
dū, = AU + EdB,, U ~ Tdata 8 Tp.
where n. NN(0, v Ia), (Bt) denotes 2d-dimensional standard Brownian motion,
0 a² A (4)
La. and Σ ( L )
(5)
2a
Similar to (I), this process is Gaussian conditional on the distribution at time O (see Proposition The associated linear system corresponds to the stochastic analogue of damped harmonic oscillator the critically damped regime, with a 1/VM M and a = 2/,a, following the parameterization of Dockhorn et al. (022). Note that (4) can also be expressed using a time-change or noise-schedule function 3 : [] + R.+ (see Section E.2) This will not play a key role in our theoretical analysis but is important feature of practical numerical implementation.
A.2
Applying time-reversal results for diffusion processes (see, e.g. Haussmann and Pardoux
1986
Cattiaux et al. 2023 the backward process (U,) (U.)220 is solution to the following SDE:
- + Voglogr (T. dt + EdB.,
(6)
with initial condition U ~ pr, where p. 2ddd 2d + R4 is the probability density function U..
CLD-based SGMs. To sample from U, (and, in particular, from X Wdata), one must rely on the three SGM approximations discussed earlier. The mixing error is analogous to that of standard SGMs and leverages the ergodicity of the forward process- known Gaussian
3

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distribution initialize the backward process. The discretization of the nonlinear backward SDE can be performed using classical numerical integrators commonly employed SGMs, such as Fuler-Maruyama (Song et al. 2021 or exponential integrators (Conforti et al. (025). Additionally, due to the Hamiltonian structure of the kinetic process, symplectic integrators (Neal (011) may also be appropriate (Dockhorn et al., 2022). Finally, the score approximation can be implemented by
applying Denoising Score Matching similar to ( on the extended phase space U, = (X,, V.)
that is, using the conditional score function V log Pa U)). However, since the distribution of Vo known and Gaussian, it can be analytically marginalizedd, yielding the following objective function known as Hybrid Score Matching:
LUSM (0) IE A(1) SO(T, Ú.) - V lo (Ū, X
where 7 is uniformly distributed on [0, T). 1 ~ and À [0, + R>o is positive weighting function. Empirically, Hybrid Score Matching tends to yield more stable training dynamics by reducing the variance of the training objective (Dockhorn al. 2022).
3 Wasserstein Convergence of CLDs
In this section, we analyze the convergence of CLDs with respect to the 2- 2-Wasserstein distance under the Euler-Maruyama discretization scheme. We first discuss the motivation for this analysis before introducing the setting, assumptions, and main results.
3.1 Motivation
While convergence results have been established in terms of the Kullback-Leibler divergence (Con- Forti al. 2025 Chen et al.) 2023), analogous results currently exist for the Wasserstein-2 metric. Proving convergence in W2 requires establishing a contraction property of the backward dynamics in this metric- a challenging task for hypo-coereive SDEs (Villani [2099] Eberle al., 2019; Monmarche 2023). The main difficulty arises from the degeneracy of the diffusion term, since the Brownian motion CLDs acts only on the velocity component. To illustrate this point, consider the following example.
Introduce the change of variables Y, X, a VV, 4 under which component of the system evolves as an Uulleenn process. Writing Z. = (X,, Y.)T. the forward SDE in (4) can be rewritten as
Z 2
až d2.=a
( dB,
Notably, the transformed process (Y :)eeoo.r corresponds to an Ornstein-Uhlenbeck process. By the time-reversal property, the corresponding backward process satisfies
Y, X + +ave
which leads to the following backward SDE:
azz - = a ( -1 Z Zdtt + 0²
V, log pr ) dt +
( dB
0
(7)
where Pt denotes the probability density function of 7,- A standard approach to establishing contraction consists in studying the difference process associated with the dynamics in (7). starting from two deterministic initial conditions (zo, yo), (I/6 36) E iddd and denoting by (Xe, Y:) 1) and (X/, Y2))<((.r)) the corresponding solutions. Under a synchronous coupling-i.e. using the same Brownian motion to drive the evolution of both processes-the difference process becomes deterministic ODE, whose stability properties determine the contraction properties of the system. In particular, using the mean value theorem applied the gradients of the log-density, the following holds for E [);
d
X1 -xx)
Ve Yi
= -aa
-aato'G
a a+ -6-Y- )
-a
(X: 12-XM X
de.
(8)

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where
H₂-
V log Px X,), E, 7(2 Y7)) dy,
G. = / Vy V log >X; X;), Y + 7V, - Yi) d7
To ensure contraction of the system, all eigenvalues of the matrix in 8 must be negative. However, the main difficulty lies in controlling the term Gt. which involves the mixed second-order derivative V, Vi log Pr-t- For contraction to occur, this term must also be sufficiently negative. This is strong and challenging requirement, it demands a form joint concavity of cross-derivatives. which is not generally ensured even when Pr-t is strongly log-concave in each variable separately.
3.2 Settings: Dynamics and Backward Discretization
Position-noise regularization in the extended phase space. detailed in Dalalyan and Riou- Durand (2020), kinetic Langevin-hase samplers depend on the mixing rate and on the regularity of the underlying dynamics. To better exploit the extended phase space, we introduce a modified dynamics that adds small noise term the position coordinate 2 0. Crucially, when is strictly positive, this modification restores ellipticity the forward and backward processes, which facilitates greatly the theoretical analysis. This hyperparameter controls the smoothness of the sample paths and the analysis of the discretization error allows a practical tuning to improve sampling performance in comparison with standard SGM models and kinetic-basee diffusion samplers. The diffusion coefficient of the forward SDE is then given by
0 σ
Ee =
2 la
0
giving a process
(U.)ee00 E 2d which satisfies the following SDE
du, AU ,t E.dB,,
U, dataa T.
with 2 0. Note that the case 0 recovers the classical CLD framework. In the following, write
st (u) V llgg fort 2 0, u E Rd2d
(10)
adopt modified score formulation based on the rescaled density p. := Pt Poo. where is the density of the stationary distribution associated with (4). This perspective, also emphasized in Cattiaux al. (2023): Conforti and Léonard and Ocelloj (cello((0255) |Pham et al. (2025 reveals deep connections with stochastic control theory. In particular, the modified score satishie a Hamilton-Jacole -allbi-Belllmaa (HJB) equation, which we highlight and exploit in the sequel.
Modified score function. Following Conforti al. (2025)
(2022) ; Strasman et al. (2025); Conforti et al. (2025); Gentiloni-SSilvee
With this notation, the backward process U can be written equivalently
at, A U. E log (Ú,, dt E.dB,
(11)
with A, - - -1 In the following, we write (u) == V (ogūr(u), for / 2 0, E 2d
Backward process discretization. Let N E N denote the number of discretization steps, so that 0 < ti < &N = T. To analyze the convergence of the discretized backward process, introduce the continuous-time interpolation (U.)tejo,r) of the Euler scheme for the time-reversed
process (assss(u,n (U.)eeu.ry This defined as the Itô process such that, for L [1x, 4411),
U. Ü.. (A.U.,, + E-rr- (Utt)) (t 14) Ex B, B..).
(12)
where the process is initialized at pr (i.e., U, ~ Pr). When initialized at Tec. we denote by (ee0)) the corresponding Itô process. For simplicity, the discretization is performed on uniform grid, with step size h = T/N, that tx41 ty h for all k.
5

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Generative model. The generative model is defined as the continuous-ting time interpolation of the discretized backward process, which the true (unknown) modified score function is replaced by its parametric approximation 80 [0, 7] x R2dd + The resulting process, denoted by (U (ej),
satisfies for E [tt, [x+1]
U 0 + (A.U%, + 32 (1e, 0ir) (1- (2) + Ee - -B:a):
(13)
with initialization 0 T Learning the modified score function V log pr is theoretically equivalent to learning the standard score function V log pt, since the two functions differ only by known linear term. As a consequence, the modified score approximation can be written, for any 2 and any
E _2dd as
(t u) == st u) + +
Ultimately, the objective is to control the W2 -isttinee between ((X%) the generated data marginal distribution at time 'I' and data the true data distribution (recall that U4. = (X7. Vg)
3.3 Assumptions
Regularity assumptions.
H1
The data distribution Tdata is absolutely continuous w.r.t. the Lebesgue measure, with density Priata and the relative Fisher information between To dataa @ n. (i.e. the initialization of the stochastic process defined in (4) and Too is finite, i.e.
droo
(w))]" mo(du) < 00.
(w))
I(To (oo) =
log
dooo
Assumption HI particularly the requirement of finite Fisher information, is standard in most works establishing convergence bounds for SGMs. This condition is either explicitly assumed or implied by stronger regularity assumptions used in the literature (Contorti et al. 2025 Strasman al. 2025).
H2 (i) The data distribution is of the form Pdata (x) exp (-(V(x) + (xx))) and satisfies:
* There exists L > 0 such that [ (x) - H [y)1 S L||> yll for all x, E Rd
* There exists > 0 such that aI, 3 Vzx) for all a E Rd.
(ii) (- log Paata) is Lo-o--iidee Lipschitz, i.e., for all x, y E RRd
V log Pdata (x) V log pdaca
(x v) S Lollæ vl1..
(14)
The first point of Assumption H2 models the data distribution strongly log- log-concave component V perturbed by a term H, similar to the settings considered in Brigati and Pedrotti (202);; Stéphhnoviithh (2025). Intuitively, this assumption allows the distribution deviate from strong log-concavity via the perturbation H, while still maintaining sufficient regularity for the analysis. When = 0, the distribution reduces to the strongly log-concave case, which is commonly used establish contraction in the Wasserstein metric (Bruno et al., 2025 Gao et al.] 20255 Strasman et al., 2025). The second point of Assumption H2 assumes one-sided Lipschitz condition. which is weaker than requiring full Lipschitz continuity of the score function (Gentiloni-Silvee and Ocello 2025). Notably, H2 implies
the Lipschitz continuity of the score function. This means that for all t E (0, T]. there exists L,
such that for all u, R22d
||s: (u) - s, (ū)ll < L, 1|u- al|
(15)
This condition can be verified under standard assumptions. In particular, if V log Pdata is Lipschitz, the assumption holds. Since Adata and n. are independent and m, is often Gaussian, it suffices to assume that Piaaaa is log-smooth, a common condition in the analysis of SGMs to ensure convergence (Gao et al. 2025 Chen al. 2023).
Furthermore, Assumption H2 ensures that Taata has sub-Gaussian tails (Lemma D.I. Consequently,
all its polynomial moments finite. In particular, Tdata admits finite second moment, a standard
condition either explicit or implied by stronger regularity assumptions in convergence analyses of
SGMs. Importantly, Assumption H2 together with the polynomial growth condition | [VV())] <
C(1 + |==l²") for all Rd, with some C' > 0 and N, implics Assumption Hl (Lemma D.2

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These assumptions are satisfied by standard distributions such Gaussian and mixtures of Gaussians. They are strictly weaker than the conditions typically required in the literature establish Wasserstein convergence guarantees -uchh as strong log-concavity combined with the Lipschitz continuity of the score function (Gao al. 2025 Sttasman et al.) (025) ( which hold only for non-degenerate Gaussian distributions and therefore exclude many practically relevant settings, even though they remain common the literature.
Score approximation.
H3 There exists M 2 0 such that,
|5 (0%,
80 -T- 0%+) 11,, I M.
sup RE(O,..NN1)
Assumption H3 is standard in the literature (De Bortoli al. 2021 Conforti and Léonard 2022
Gao et al. 2025 Bruno et al.) 2025, Strasman et al., 2025; Gentiloni- Silveri and Ocello 2025 Cordero-Encinn et al. 2025) essentially all convergence proofs for diffusion- based score models require that the neural network has learned the score within some uniform error. This condition quantifies the ability of the neural network architecture approximate the true score function and serves to control the score approximation error.
3.4 Main Results
We establish here the Wasserstein-2 convergence of CLD-based SGMs under these weak assumptions. A key step is to show that, under Assumption H2 the scaled score function > log Pt (resp. 22 V log P:) is Li-Lipschitz (resp. Le-Lipschitz) for t > (Proposition B.1 This, in particular, yields an exponential decay of the operator norm 2> log p:ll - oc. The following theorem provides, to the best of our knowledge, the first convergence rates in Wasserstein distance for CLD- based approaches and aligns with recent developments the literature of Variance-Preserving and Variance- Exploding SGMs.
Theorem 3.1. Assume that Assumptions H1
hold. Then, there exist c1, C2 > 0 such that, for all H3
W2 (Taata: L (Xx)) c1 c W2 (Taata Ty. Toc) + C +avh.
Proof. Let Px Rdd + Rd denote the projection (x v) Using that Px is 1-Lipschitz for the Euclidean norm, yields,
W2 (F c(x)) S W/ (Taata TT. C(U%))
(16)
The right-hand side of (16) is then bounded by decomposing the total generation error, using the triangle inequality, into the three sources of error for SGMs discussed in Section
W2 (Taata & ne, C (ū+)) s W2 ((T)))((,r) W2 (c (UF),<(U;))
+ W/ (c
where Up and U are defined in Equation tion error) is controlled by Lemma B.2 which ensures that there exists 0j > such that, for all h >>>0 0,
(1) and U in Equation 13 The first term (discretiza-
(s(C(ūr), S Sevh.
The second term (score approximation error) is bounded by Lemma B.3
W2 (c (Ū7). (Ū%)) - 2
Finally, the third term (mixing error) is controlled by Lemma B.4 which guarantees the existence of C2 such that,
W3 (Ū,) (Ú+)) < c2 T (Taata & ne. Toc)
Combining these three bounds together with 16 concludes the proof.

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Theorem 3.1 establishes convergence rates in the Wasserstein distance for CLD-based approaches for all e 2 recovering the vanilla CLD when 0. In this case, our result aligns with the KL convergence analyses of kinetic Langevin dynamics by CCen et al. (2023) and Conffrtii et al. (2025) for the specific choice a = 1 and o = 2. is worth emphasizing that, under our weaker assumptions, no equivalence holds between KI. and Wasserstein convergence,
our results are not implied by existing KL-based analyses. Beyond this theoretical bound, our analysis indicates that smaller values of yield better log-concavity constants; however, e is typically chosen to be small but not too small, to avoid an explosion in the Lipschitz constant. This remark is consistent with the empirical evidence brought forward by Dockhorn et al. (2022), which suggests that small values of e may improve training stability and sampling performance.
3.5 Strongly Log-Concave Case
This subsection focuses on the elliptic case, i.e., when > 0. In this setting, the forward process associated with CLD becomes a multidimensional Omustein-UUllenbeck process with matrix-valued drift and diffusion coefficients. The presence of the additional noise term on the position coordinate restores ellipticity, which allows us to extend classical convergence analyses developed for VP and VE diffusions to this kinetic framework.
Crucially, in the strongly log-concave case, i.e., when H 0 in Assumption H2 the upper bound can be expressed with more explicit constants that depend on the regularity of the data. Moreover, in this case, the one-sided Lipschitz condition becomes equivalent the Lipschitz continuity of the score function. Assumption H2 then reduces to the following assumption.
H2' The data distribution is absolutely continuous w.r.t. the Lebesgue measure, is of the form Priata (z) e (-((#) and is 0g strongly log-concave and to-ooogsmmoth, i.e., there exists On > 0 and Lo > 0 such that,
aols 3 Vz) 3 LoLa, for all x E Rd.
Under this assumption, the forward flow preserves both strong log-concavity and smoothness. Indeed, Propositions C.T and C.2 guarantee that P: remains ay-log-concave and L,-log-smooth for all t t [0, T]. with ae and Lt explicitly defined as functions of an and Ly in the respective propositions. Such regularity properties are fundamental for proving exponential contraction in the Wasserstein metric, and are consistent with the analysis of classical (VP) diffusion models (Bruno et al. [025| Gao
et al. 2025 Strasman et al. 2025). In contrast, Chen et al. (2023) obtain Wasserstein convergence guarantees without requiring strong log-coneavity, instead relying on the compactness of the domain and (15). setting where convergence in KI. divergence is effectively equivalent. The following theorem presents Wasserstein convergence results under assumptions for which no such equivalence with the KL divergence holds. In particular, our result is not implied by existing analyses based on KL convergence.
Theorem 3.2. Assume that H2' and H3 hold, and let e > O. If the step size h satisfies
2 ming ar (o² A 6²) (o e)² max; Lr, (a+ 1)² 0 0<h< < [A 2 + (Et + 04) max; L2 + 2 2 v x2) AAl max; Lt,"
(17)
then,
W2 (Taataa L (x9)) E Kre- of We (Taata & To, Too) + o²M + ViaCale). where K₃ = (1 + max fa + 1: + T)}TT) and
Ca(e)= [2]|4]| B. + Ad(a²o² + +3)3 4d (111112 + sup
T sE[0,2]
with Bg and (T) as in Lemma
C.3
Proof. The error decomposition is the same as in Theorem 3.1 The full statement and proof for each error term is provided in Appendix C
This bound highlights the stabilizing role of the parameter e > O, which restores ellipticity in the dynamics. A key observation is that
Σg V Lo -E² A o² o2)arle,
8

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which can be negative only for positive values of E. In this sense, increasing tends to enhance the contractive behavior of the dynamics, as also reflected by the admissible step-size condition (T7) However, this effect is not purely beneficial: several terms in the discretization emor scale with E illustrating that excessive noise injection may deteriorate the regularity of the process. Consequently, there is a trade-off in the choice of to balance these competing effects. This trade-off is numerically illustrated in Section
Remark 3.3. Finite second order moment is also necessary in this approach and is deduced from H2' (see, e.g., Lemma B. [ientiloni-SSlveri and Ocello 2025). Regarding H3 it is implied that the score approximation is now made for the true score function, not the modified one.
4 Experiments
We illustrate the effect of the regularization parameter & on the generation quality of CL.Ds on simple yet challenging toy dataset. The regularization parameter e is chosen to be in {0. 0. 0.25, 0.5, 1}. Notably, & = 0 corresponds to the vanilla CLD setting. Our source code is publicly available here
Evaluation metric. To assess the quality of the generated samples, directly computing the Wasserstein-2 distance is infeasible, as it requires solving a computationally expensive optimal transport problem. Instead, we approximate the Wy-distance between the generated samples (with distribution ft and the training samples (with distribution rdataa using the sliced Wasserstein distance (Flamary et al. 2021). It is defined as SW2(Tdata. ft = Bu-e (ugTdata u where (S denotes the uniform distribution over the unit sphere and us is the push-forward operator associated with u. The expectation is approximated using the standard Monte Carlo method with 2000 projections and Tdata and Å are replaced by their empirical distributions.
Dataset. We evaluate the generation quality on the Funnel distribution, which is characterized by strong imbalance in variance across dimensions and was previously used in [Thin et al. (202T). To further illustrate our results, we extend the evaluation to two additional challenging toy datasets (Appendix E.5): MG-25 (a 25-mode, 100-dimensional Gaussian mixture) and Diamonds (a 2- dimensional Gaussian mixture with a haamond-shapdee geometry).
Hybrid Score Matching. Following the insights of Ho et al. (2020), the networks are trained to predict the noise (or rescaled noise) added during the forward process. When E 0, we use the positive weighting function proposed by Dockhorn et al. (2022) (see page X()) 6--2. in Dockhorn et a (2022)) A similar reweighting, however, is not feasible for & * 0 due to the matrix-valued nature of the objective function. Empirically, we observe that much of the training variance arises from the determinant computation involved in the 2 x 2 matrix inversions. To mitigate this, we set A(t) = dct()),))² which effectively stabilizes training. We parameterize the score network as 8o (,,()= = -pp o0 (U,,)
I) so that the hybrid score matching objective for > is given by
L(HSM)= (0) = E det( ) 2o.1 (se (T,
x.)
2///2 22d
||]
(18)
where Gza denotes 2d1-dimensional standard Gaussian noise.
Model, Training and Generation. All score networks share the same architecture: a fully connected neural network with three hidden layers of width 512 (see Figure 3. Training is performed using the Adam optimizer to minimize the hybrid score matching objective in (18). with a learning rate of 10-4 over 2000 epochs. The training set consists of 50 000 samples. For evaluation, we generate 50 000 samples using the Euler-Mann Maruuaaa discretization scheme with N = 1000 steps and compare them against test set of 50 000 samples. Both training and generation are independently repeated five times. The training (Algorithm I and sampling (Algorithm 2 procedures are provided in Appendix E.
Effect of the regularization parameter. Figure illustrates the Wasserstein error for dif- ferent values of the regularization parameter e 0.1, 0.25, 0.5. 1} and drift coefficient {0.1, 0.25, 0.5, 1, 2}. Across all values of a, introducing small regularization parameter E notably improves generation quality, even though the score network in the regularized case must predict vector twice as long as in the non-regularized one. Moreover, regularization consistently reduces the

---

variance across runs, For smaller values of a, the error increases sharply when and also for large values. In contrast, for moderate values of a, the error becomes negligible, with E [0.1, 0.5) yielding slightly better performance than the other settings. It is worth noting that experimental configuration closely follows that of Dockhorn et al. (0022), using
+
+ a 0.255
+ a
Mean Error
10a
+
v2.
a 2, and in particular & 0. This observation justifies their choice of = 2 for the vanilla CLD.
0.0
0.2
0..4
0.6
0.8
1.0
Effect of's e in controlled settings.
Figure 1: Mean W2 distance over 5 repetitions be- tween the test set and generated samples on Funnel distribution in dimension 100. Error bars represent - one standard deviation.
Varying modifies both the stationary distri- bution and the noise schedule- twoo factors known to strongly influence performance (Guo et al. 2023 Chen et al., 2023 Strasman et al. important control for these ef- fects. To mitigate this confounding factor, one can fix the stationary distribution of the base case to (02d,, I2a) and maintain comparable noise levels in the position and velocity spaces by setting a(s) 1 - 22 and (s)) /4 + s². This choice ensures that the stationary distribu- tion remains close to (02a.. I24) for small val- ues of E. Although this adjustment becomes less accurate for larger there no practical limita- tion preventing the use of higher regularization values.
2025
0.14
(Leapfrog)
0.13
0.12
Moon
0.11
0.10
0.0
0.1
0.2
0.3
0.4
0.5
Figure 2: Mean W2 distance over repetitions between the test set and generated samples on Funnel distribution with ale) 1 8² //2 and V4+
Figure 2 still shows an improvement in genera- tion quality for small regularization parameters e. To confirm that this effect is not tied to the discretization method, we reproduce the experi- ments using Leapfrog integrator. As expected, the Leapfrog scheme outperforms Euler-Maruyama, yet the relative benefit of regularization per- sists. Finally, we emphasize that our objective is not to conduct an extensive numerical comparison of integrators training strategies, but rather to highlight the potential of introducing controlled regularization within the CLD framework direction theoretically supported by Theorem 3.2
5 Discussion
In this paper, present the first theoretical analysis of the sampling error of CLDs in the Wasserstein metric under weaker assumptions than those previously used in the literature. Our results show that CLD-based samplers
achieve comparable convergence rates while effectively leveraging the structure of the extended space. We further analyze generalized dynamic that extends classical CLDs by introducing smoothhees-conntolliing hyperparameter that regulates the noise the data coordinate. This parameter provides more precise control over the regularity of sample paths and plays central role the discretization error analysis. Both theoretical and empirical results suggest that appropriately tuning this parameter leads to improved sampling quality and stability. Overall, our work offers both theoretical insights and practical guidance for CLD methods in generative modeling, particularly in scenarios where standard assumptions may not hold. Several promising directions remain for future research. Replacing the Euler discretization scheme with higher- order method-a Suchod-suchhhoo as the Leapfrog integrator, which is specifically designed for -LD-based dynamics-come further enhance sampling performance. Analyzing such schemes would likely yield sharper convergence rates consistent with the numerical results. Moreover, developing denoiser architectures specifically tailored the extended space represents another promising avenue for applied research, potentially leading to tighter bounds on the approximation error.
10

---

Acknowledgements
The PhD of Sobihan Surendran was funded by the Paris Region PhD Fellowship Program of Région Ile-de-France. The work of Antonio Ocello was funded by the European Union (ERC-2022-SY2) OCEAN-101071601), Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. We would also like to thank SCAI (Sorbonne Center for Artificial Intelligence) for providing the computing clusters.
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Supplementary Material for "Wasserstein Convergence of Critically Damped Langevin Diffusions"
Table of Contents
Introduction
Notation and Background
3
Wasserstein Convergence of CLDs
Motivation
3.2
Settings: Dynamics and Backward Discretization
3.3
Assumptions
3,4
Main Results
3.5
Strongly Log-Concave Case
Experiments
Discussion
10
Forward process of Critically-Damped dynamics
15
Proof of Theorem 3.1
18
B.I Propagation of the regularity assumptions
18
B.2 Proofs of the main results
21
Proof of Theoreemmm... 3.2
28
C.1
Propagation of the regularity assumptions
29
Proofs of the main results
30
Technical Lemmata
35
Numerical Illustration
40
E,
CLD training and sampling
40
E.2
Time-rescaling of the forward SDE
40
E.3
Score approximation
41
E,4
Neural network architectures
41
E.5
Additional experiments
42
14

---

A Forward process of Critically-Damperr dynamics
In this section, we establish several mathematical properties of the forward processes:
dù, AU,dt Ee Uo Adata & Te
defined in 4 with 0 or in 9 with 0. These results will be used throughout subsequent analysis.
Lemma A.1. Let A be the matrix defined in (5) then
A ((1
-1
x a
1
0 )))) @ Ia
-a
x
-1
so that
(1 al
a²t
e e
ta
@ @ld.
(19)
-t
1-at at
and
|| & ||o21|11/2 1 < (1 + max + 1:a(a + 1)}t) e et
< [1 + (a 1) e et
Proof. The Jordan matrix decomposition of A when d 1 is given by.
A
a²
a
0
a -
0
-1)
-1
2a
We use this decomposition obtain matrix factorization in any dimension. As for all k N, A = (Af @ Iz).
AA
AA
Ia) = Σ
1f
@ Ia = c @ Ia.
k!
k! k=
A=
Finally, we deduce an upper bound to the spectral norm of eA as
||e ll s e to max {(1 + (a 1)t: 1 ala + 1)t},
and
| se -t max max {(1 + a(a + 1)4; 1 + (a + 1)()
Then,
||o < ||o ||odd11/ 1/2 / < ct (1 + maxfa + 1:a(a + 1)}t),
which concludes the proof.
Lemma A.2. Let (U (U.)reoo. be solution the forward process with initial condition
U Wdata E
where is probability distribution on (RA B(R Then, the conditional law of U given U is Gaussian with mean Pojo and covariance Eo,t defined by
Pejo == U..
Ea,t = Exx - C -o
(20)
with
1 (5e2a + a0²
(
-2 (62 + a²o
L. The result still holds when the forward process is defined as in by setting 0.
-2:20-2
(21)
15

---

Proof. Recall that the forward process (Ú (0,7) is solution With initial condition
au, AU, + E.dB,
(22)
U. Tdata @ Tex we have
U = UO / (,
This means that the law of U. conditional the initial condition U. is Gaussian with mean
Mejo EE U.] e' Uo.
and covariance
Vo, =- Cov (U.)
ds
(e
(L/ [(t [2 ds & la.
Using Lemma A.l for ò 2 0,
a=0282 <2 + ad)² 8 (a"o² (1 - ad) = +a))
8 (1 ad) =2 ad))
200
82
Hence, a straightforward computation provides with a = (5 2at(3 + at)) a² (1 + 2at(1 + al)) o²)a and 7t = 2 ((s ate)² + + a
Ea, a ag a2 222a (E2 + 22o2)a
e 20t
(as
(-(1+ 2at(1 + at)) - a² (1 2at(-1+ o() (23)
7t
Exx L
(e
where we used that,
e E (e = /oe 00
c(tts)/5? (elttaa)) T do / o (drn) do do= Exo - Eo.t.
Lemma A.3. The covariance mairix Σo, defined in 20 satisfies, for all 8 > 0,
Amin (Eo.) 2 max σ²
min(a, 1/a] - ( (a, 1/a} +
0²
5E2
e
4a
-2ot 20t
min
02) (
2a + (a+ + 1)21)
5e2
Amax (Eoz) < max 1/0] +
4a
Proof. First, consider the following decomposition of Σx defined in 21
En 1
4 4
1 1 +
-2a-11
4a
22
-20 2a
2
-2
=
4
1 aao2
0
aa-1
0
+ Ez.
16

---

Since Eg is positive definite, its trace and determinant are positive, then
(oo?
o²
Amin (Eoo) 2 A'min 0
min(a, 1/a),
4
1
4
1
ao2
o² Xmax (Ee) maxfa, 1/a)
5E2
Amax (Exo) -
Amax
0²a
4a
(24)
Using that Mo,r Doo
eld et
together with Weyl's inequality we have that
Amin (Eo,c) 2 Amin(Eoo) - Amax (e'nEeee E'ce"")
Note that, as Eo is positive semidefinite,
max Eo = Amax (2/) S Amare (eld)
Amax (Eio) 2 e- -2at Amna (Zoo))
On the other hand, using that Eo, = Jo CE CCCAAR ds, yields
So ceapest ds.
Eot x min fot m
therefore,
min (E², [ Amin (ordged)) ds Amin (Eor)
min {52 o²] f"
2as
ds,
(1 + (a + 1)23)2
1 -2at 2at
min {8" 0²)
2a(1 + (a + 1)21)2
which gives the other lower bound of Amin- (Eo,t)
To obtain the bound on Amax (Eo,e). it is enough to note that Co.e $ Exo.
Lemma A.4 (Forward process Wz-contraction). The forward process, defined as in 9 contractive for the W2 distance. In particular, it holds that
W2 ((T mo0) moo) s Kre Kre-aTWa WT (Tdata @ ue. Too)
where oo is the stationary distribution of (9) defined in Lemma and
KT = (1 + maxfa + 1;a(a + 1)}T)
Proof. Let (x. R 2d (resp. à (H,
E 2d and denote by (UT)e (resp.
(UR)EE) the solution of 9 with initial condition U (resp. Ug w). By Itô's lemma,
(e 44 2,
+4 EB
Using synchronous coupling for (U. l and (U )E we have that
d U e (u ū)
By definition of the W distance (,(c( s U0 U. Then, by Lemma A.
|
J: 12
$ ||ooA||| ||u alle, s K2e ||u ū ulIz
(25)
with
Ky: (1 max(a + 1;a(a + 1))t)
Finally, assume that ū ~ Too. u Adata @ n, and fix any coupling 7 E (TTata & Tey Too). Using that no is stationary distribution of U, and taking the infimum over II dataa & Tay Roo) yields,
Wi (c (C()) xT00) Kyee aT W2 (Tdata To, Too)
which finishes the proof.
17

---

B Proof of Theorem 3.1
In this section we prove Theorem 3.1 We use notations from (2) (resp. (3)) for the continuous time
interpolation of the discretized backward with modified score function U, (resp. for the continuous
time interpolation of the discretized backward with approximated modified score function U.) )-
We
first establish the propagation of Lipschitz regularity, followed by the proof of Theorem 3. To
do so, we decompose the generation error as the sum of the discretization error (Lemmal B approximation error (Lemma B.3), and the mixing time error (Lemma (B.4).
the
B.1 Propagation the regularity assumptions
Proposition B.1. Assume that Assumplion V p.) is L Lipiscchitz (resp. -i-isschhtzz; for all R2d.
F2 holds. Then, for all I > 0, V log (resp.
>> log) Lc.
Moreover, there exists a constant C 0 such that for all u R22d.
2 (u)l s L. s 1+ 31
1
(26)
Proof. Step Lower bound on 2 log Recall the following equality in law given by the modified
kinetic OU process 9
U, - Do ++++277 G.,
with Uo ~ Tdata @ To. G ~ NN I2a). where G and U. are independent, and Eg, is defined in 20). Writing Gejo the conditional density of U, given Uo. have
pe(ue)= = I
(ueluo)du
-fo po(uo) det (2 1/2 exp - (ue - -co T Ezgt (ut dnAww) duo
Joed Po(e det (2x 1/2 exp (1 (uu (u- - 2) T Eōd uc -z) dz.
(
= det (e-
As also observed in Saumard and Wellner (Proposition 7.1, 2014), we get
2 log Var Von. Y. + Y; u) - E (Yo) + Yı =u]
(27)
+ (ar(Vo)..YY)))
oI + +Y u]
for ed U and Y1 G and for Φo,i and Φ1. such that for all a Rdd
(w)) det (e LA) (e
dia(u) dot (2 Eot) 1/2 exp ( 1 Eo.tu
2"
This implics that
2 loggt(u) > -V + Vi = =u]
(28)
2 >
Yr u|
Note that for all u E 2d
2 vio -t 2
log (o -1 e -
V2 (2)1,(()) = -En
18

---

From (Bouchut et al., 2005 Lemma 2.2) together with (4). we get that the one-sided Lipschitz assumption entails the following inequality over the Hessian of the log-density, since log Po(u) log (
(2: + log (v)).
V log Panta
22 (-log (u) =
V2 logg.(u))
log Pdata (x)
0
II
x max
Lo. y/2 L2d.
Therefore, for t >> 0, from 28 we get
V logg:(u) x -helad
where h. min {lle 14
1| max { Lo, e
}} ::|oo,1|1
Bound on bz- By Lemma A.1 we have that |e- S (1 +1+ (a (/a++199/ 1)21) 2210 Moreover, from
Lemma A.3
it follows that
Amin (Eo,c)
Amia (Exo) Amax (et E )1+
with L:I+ denoting the positive part of a real number. Therefore,
|لجل||
=: 62,
0 min 1/a} (a2 max 1/a) 5520 2at 1+ Combining the two previous bounds, obtain
h, S min (h,,e:b2.t)
(29)
where b1.t (1 (a 1)²t)² 2t max { Lo, 3]
Step 2: Upper bound on 2 loo p.. We first express the conditional density of U, given U, as follows
qgo((Z0, vo) (Iue) (e- (e-V(so)-H(zo) @ Nvoo Oa, v'La) Nu e (20, vo) 2o.).
(30)
First, we consider the log-concave part of the above distribution,
(e- V(zo) @ (wo: Oa, v I.a) N Nu (2o, Eo,a)
(31)
Since V²V(x) aI, for all x E Rd, we obtain
log v.) = = en ald
0 e-r
e
+ Eoo1
Therefore, by Brascamp Dieb Lieb inequality (Brascamp and Lieb 1976).
(e-e ala
Cov(ui) &
EAT
+ Eod
Using the identity Ea, = Enxo elAs
AA
given in Lemma A.2
we now expand Eg, at zero as
[2
Σn,1
O(t))
0
0²
which implies that
Σ-1
1
1152
0
+o(!)
1/02
=>>>
Therefore, the covariance matrix near zero satisfies
(a
21)
Cov(v)) &
+ +
-1
0
1
19

---

Next, the Lipschitz perturbation term, following Brigati and Pedrotti (2025), can be bounded as
2
Cov(a((-lu.)) x
( +
a+(2))-
+o(t).
0
0
Me,t
Using we have
2 logp((u) Cov (no(-lu)) 0
(32)
so that
7 1-- + 0 (0)) (1)) (M, + +(()) (+55 + +o (+)) - ( DE + 0 (+t)
2 log)r(u)== =
0
1
0
8:
+
0
with
1
2L
lad 1
(ae2t + 1)2 +
B = r2t + v2
(22))1/((EE²)) + 1)3/2
act +
Consequently, for all E 0, as t + ++,
0
cvi
Vt
I 2 logpt(u)) <
ev?
0
(33)
0
Step 3: Uniform bound on l2 Pt- We now analyze the structure of the minimum in the upper bound of he in (29). We observe that the first term is increasing, equals max [_) for f f and diverges as t + +oc. In contrast, the second term is decreasing: it diverges as - 0 and converges to 4/(o² min(a, 1/a}) + +oo. Therefore, the minimum coincides with the first term, for 1 < Tehange. and with the second term, for > Tchange- Using (33 uniform bound, for all 0,
we obtain the following
|| log (w)l & max b1 Di.Teaa Ct Ct-1/2] J
for fort: > 0.
(34)
This bound is uniform in e O, therefore, for - O, we have
V log (w) & max (I. Ct- Ct-1/2). for > 0.
(35)
Since f. = Poc. and Pro is the density of a centered Gaussian vector of variance Exc, we have
V2 log = 2 log + -
(36)
Therefore, the same bound as in 35 holds for the modified score.
Step 4: Exponential decay of the modified score. From (3). we have the following equality
22 (u) = 2 log (u)) E Σj, Cor(((((())) E -
where is defined in (30p. By applying Lemma D.9 together with the decomposition 20p and the positivity of the covariance, we obtain
v Volgge((u) -End (20 (e
Since Eo, Exx + O 20t) as co, there exists a constant C > such that
Foo ( (00 X Ce 2nt
On the other hand, using the fact that 2nj x Ddd (see 20
we get
1o (u)) & Egg Cov
(-
20

---

Following the same steps as in the derivation of the upper bound on 2 log (step 2), we obtain
V2 (- log v)) x c
aI.
1A
Ep,t >
aIL (or
- +AT
I
I
where V is defined in (30). By the Brascamp-Lieb inequality, this implies that Cov(u() O Next, similarly to "step 2", for the term involving the Lipschitz perturbation, and following Brigati and Pedrotti (2025 Theorem 1.3), we have
Cov(g((1u)) (ICe-20) 2at VCe 2nt
Id.
Therefore, there exist universal constant C > 0 and finite time T Tehange t Thhange
0 such that, for all
| log (u) & Ce Cattat 2at
This implies that the modified score function is Lipsshhii,, with L, defined as for E (0, Tchange] for E (Tchange. to0),
1==fmax max (hi.Thhanee' 1t21 + max(a, 1/a),
L = 1==1
Ce 2nt
which concludes the proof.
B.2 Proofs of the main results
Lemma B.2 (Discretization error). Assume that H1 and 112 there exists constant C such that
hold. Then, for all vj > 0 and all h > 0,
We (ATū)) s viov (MI1A)BB2 < ||&.] (d+ + I (Tdata @ @Tol Tol
cCa-1
(37)
a-
with
B = max
[E (1+ (a + - 2a((T -
|T, 2 +
1. 2 (0² maxfa, 1/a] +
552
Proof. Consider a synchronous coupling for (Ue)eeja.2) and (U.)ee(o.)) iee. use the same Brownian motion drive the two processes, with the same initial point, i.e., U, Up Then, it holds that
- | Ur
W2
Fix 0 < A < h and let IN = T - A. Note that, for all 0 s k s N - 1, from 11 and 12
U M11
U Ün + +//rr
dk+1 (A. (,,Ú,-U) -U,))
Σ 5T- (U.) 55-1 (U.H)) dt,
where
À. - E'
From Monmarché (2033 Lemma 5 and Proposition 4) and Achleitner al. 2015 Lemma 2.6), there exists a symmetric positive definite matrix D1 E IR 2dx 2d such that, for any fixed > 0, we have
MÁ. x (aa m).
(38)
We then prove contraction with respect to the norm associated with 21t defined, for 2d by ||uli3 : v Mr.
For / (n, 11+1).
d -
-A -A, (U.-- -U..) di I ( (tT)) -T--, -. (U.)) de.
21

---

This means that we have
a ((t..- -i)) 21 (ū. (ū)) = -2( U.) 2nd (t,- (..--u)) U.)
It follows that,
2
(U.-U(i) T mA (U, -Uiz) dt - (U(z) dt +2 (ALK + A2 + As, + Ar + Asi + Ao,x)
th11
Ú te+1
U tk+s
U
+2
101
+ 2
l++1
[U, - Upz) -U2))
T
MM,? (š (b.)- (U0)-5r-1,
[ 42 -U2e
22
where
A1= A1, = (tu. -Uis) T WA. (U... -(a))
+ (T... - -Us.) SMS (sT to (t.:) - -r (Unx))
Mkt1 (U., -(.-)r))) T w { {ā. (te= -Uiz) +x? (sr- (U
-- i (Uw)) de,
A2
14
Mk+1
(U.-U.. T TT (tre - -U(a) ddt, dl.
As,k =
Jk
A4,: =-
(U!H-
-Ui.) 2222
putt1 (5 (t.) -to (t..)) dt,
1xt1
MA (U, - Tz)) dt.
As,k =
Jee
A = yoae
plas
t.,)' T
MMW (3r-c (t.,) 5e (tn.)) dt.
Next, we bound each term of the above decomposition separately.
Bound of [). By Assumption H2 applying Proposition B.1 there exists constant C (that depends on the eigenvalues of 2 or constant terms and that may vary from line to line) such that
( Fm U..)
MM?? -U..)
tt to (t..) Sr-te (U.)) cir-
|
U.,
2
and using 38
T Mide (t.M. - -U..) S -aa- (n) m) |T.. U 201
Combining this with (38) yields
(a (a-n)) E | - Ü 1 am
E E[AiN] < h (Ca
Bound of E[A2,]] Using the Cauchy-Schward inequality,
/ (t.- Un.) di
27 1/2
E [A2, <E
(tr (t...) 55- tu (U...) }}}}? 1/2
2m I A. (U... -nn)
E
+ 2:?
lut1 (t.- -U,)) de |
2/2
E V (bre (t...) - -r. (U.,) )1(² 1/2
<CE
22

---

with by the backward drift in (TI defined by b, u a A.u + I(3)- On the one hand, the Cauchy-Schward inequality implies
U dt
1/2
S vh
( 1k+1
- U.,,
|
1/2
Using the time-reversal property, Lemma isometry,
D.3
together with Cauchy-Schuie Schwarz inequality and Itö's
U. U 11 10
T-th
E
SE
AU,ds + E.dB,
T-t
1 EE -
(39)
SC
M|Al1 Jr-
ds + hd 1d.
SC
22 hd |E.11
where defined in (56. It follows that
E
| +1 (U, -5. dt
arl
27 2/1/2 1/2
S lvThc (~]IA1|BB++1)) -12.11 2)122
1/2
On the other hand,
(T)))) ben (U )10 1/2
(ben
E
S ( [|A.el + LT- E -11tme - U2
2
Therefore,
E [A2,N] & Chvh (1A.1) + LT- (N|AA12B2 + +1eiel d) 1/2
|
1/2
EE
UI
8/
127
=ChvAllAdl (0111B2++1122
1/2 E
T, - -U,
1/2
121
+ ChWWhLT-a (2 + ||2 2 d) 1/2
| - U
1/2
EE
Moreover, from Young's inequality, we get that, for all a, b 2 0 and a > 0,
a =a² ab
1 b²
(40)
2
2a
follows that,
||ālll (b]AA12222 + [E.11] a- + LT-LL- (b|lA12 + -12..11
E [2
<
a Q-7
h
U tt
Ü
+ R²cC
6
+ ChhT-
IE
|Ttn Üra
1m
Bound of [[[A,.]. Using
inequality.
dx+1 (U.- T..) dt
27 1/2
E || | om (U.,. U., )1911 1/2
)
[A S
On the one hand,
2/2
27 1/2
AT TT (T.e -T.--Ur)
-Urz)
S CI 1 Ü..
23

---

and, on the other hand, using 39 yields,
27 1/2
E (T.- oz) dt
S Chvh (hIIA1B + =122,e1 d) 1/2
Combining both and using 440) yield.
||t,e Uc
2 1/2
E [A3,a] S Chv/l| Alll A1 + || d TF.
+ 1²C || [b|lA|/B2 + || 11 d
6 ME
1
Ui
a a-1
Bound of E[AA,,]. Using Schwarz inequality,
27
||t.. - Ü.a
22 sT (U.)- -rx (T.))) dl
E [[ S
CE
E
Therefore, using Cauchy- -Shhwa-- inequality again,
1/2
lkt1
E 5- (U.) 5p-4 (tn.) dt
E
S vh (rae
||-0 TT (U.) -r (t.r.) dt)
1/2
Then, using Lemma D.8
2 sT-1 (i.) -t (t ))("]
| E V Voggrr (U.) v log Pr (J 11
E
C [ 1| (g(tx+1) - g(te))
(41)
with the function g defined in (59 This yields
| - Ü,, 1/2
E [[A, < Ch Eell V g(tx+1)) - g(tx) x E
E TT. Ür. 20
+ h |
(o(tx+1) g((t))
a n
where we have used Young's 's inequality in the last inequality. Bound of E[As,x]. Using Cauchy-Schward inequality.
1
(
T Nn. (U,- - (5.-t.)) dt
E [A5.k]
E
px+t
2 ds
s C
[ IE
Then using Lemma D.3 as in 39
E [s. I Ch²
| [n][Al/BB² + +1E21
with B defined as in 56
Bound of b[ Using Cauchy-Schward inequality,
(ū.- -.a)
|T-1 (t.) - 5- (ūxz)) dt
E [Ao.k]
=
1x
rc+1
|| - T..
1 1/2
271/22
s2 (c (t.) -1- - (t... ))1) 1/2
< C
Fe.
E.
de.
24

---

Controlling the first term as in (39 and the second term as in (4 using Lemma Young's inequality, yields,
D.8 together with
|Ee| (g(tx+1) g(tx))
E [A Ch²(a 7) (NIAAIP + [Ee) d) + Ch
a a-11 1)
Final bound. Combining the upper bounds for AL, A2 As, A As, and A6, there exists constant C' > such that
||t, - U
+ Ch |<&|| 2
E
U txt1 - U eutt
I 6
(g(tx+1)- - -g(tx))
77
+ Ch2 (N]1A12B² + ||e|| d) (a-1)) + a- ann+2 ||A.ll V [IA.1²))
+ Ch-Lr-i. (1IIA|1BB* |2 I1? a)
with 8g = 1 + N(CLT - (a - 2)) Therefore,
N-1
Ú IN UiN 1
II 6% E TT. - Up
2
22
k=
N-1
+ +hh (h|1A11BB* +1E, (fd) (a (a-n)) 11)
a a- +2 | V |4, 12
)
I 5j
1=
(42)
+ Ch |2,1 N- 1
N-1
Σ (0((+11--966 -
II 6j
R=U
j=k-1
N-1
N-1
+ Chh (1|A|BB**+1)) +
Σ
LT-1
II
8,
k==
j=
First recall that the two processes share the same initialization, iee. T. Un- Note that, since exp(x)) 2 1 + x, for x E IR,
N-1
N-1 Σ
a (ci--,-a--)) ))
II s exp
j=b+1
j=b+1
N-1
1l M(N - k -k-1)+0
Σ hiTT-1
$ exp
2
j=k+1
a- n(N - - 1) + (00 L,ds)
I exp
_n-" M(N - - + Ca Ca
S exp
where use the bound
26 from Proposition B.1 Combining this bound with the fact that
S (1 c c- we obtain
a (----( (N k)bb
2h a - 1)
17 (N -k)h) exp (- (-a-"(x (N k + 11h)) -k+1)))
exp
s
exp
2
which then implies that
N-1 N-1
N
Σ
8,
S
kN
x Ca
K=
j=k+1
k=0
2c b
N-1
CoCa-
<
Σ
e
(N k 2)h
e 2 2 (N--))))
x x
"
1)
k=0
25

---

increasing the value of the constant C' if necessary. For the tern involving Lr- note that
hiT-4 exp(
!I (N k)h)
C
h
exp(-
2 (N- k)h)
V(N
-khh
<
2
T
1 a
(N "
k))1
r
1
a
(N
-
k +11)))
2
2
2
where I(a, b) denotes the Gamma function. Consequently,
N-1
N-1
LT
II
89
k=0
j=k+1
2e
r
(N - -2)hh
-T
a
(N - k -111) x caa-
7
2
2
2'
h=0
Ce Ca
Moreover, we have that
N-1
N-1
N-1
(o(le41)-g(1a)) g((x))
(g(lx+1)) -
II jmk+1
6g S ca
Σ
S caa g((N)
0m
Am
S Ca
E
U
Note that | U 2 corresponds the relative Fisher information between pa and Tos- We can conclude for + O, following the argument of Conforti et al. (Lemma 3.9, (2025) and using
Assumption HI that I (Tdaaa @ {o) = Eill5o( [[?] < co. Then, applying (42) directly yields
eCo-
E T U.v
S 5 C (AlIA112 - || (d + I(Taata @ Tol(
m
which concludes the proof.
Lemma B.3 (Approximation error). Assume that Assumptions 112 and 113 hold. Then, for any > 0, there exists constant C such that
W2 (C(()) - C E 1 M. 1
(43)
Proof. As in the proof of Lemma B.2 consider the synchronous coupling of the two processes U and U with the same initial condition Up = U We have
W₂ (c (Ū L (ū))) s U 0
Fix A > such that IN T A and note that for all 0 < k S N 1, from 12 and 13 get
U
Uo
U U +
A. (Ü:
0 + Σの (5 (Ü 3a (T tR. úf.))
} de.
Taking Di in the proof of Lemma B.2 have
U 1z+1
- 0
| 11 + 2BI + 2B2
4--
=
26

---

with
B1 = h (U - -U%) T MA. (U - -U
B2, h (U 0%,)) 2u (5r-1z (U%+) - 30 (T tx (/,))
Bound of E(
From 38 we note that
E [1 S (a (a) [
-0 11
Bound of EE| B2,x) We decompose the second term into the score and approximation components:
E [B2x
= hE
[Uun -0,)
D2? (5r-1e (U Sy-t, (Ú%))
+ h1E
[UR -0%)) T 2M12 (5r-te (U
30 (T tk. U(,,)
$ shCir-u
E
|0 0 1
+ + | ME
U t/
- U [lan]
< ShCLT-uB
|
0% 11
+ h a
" E
س - 1|,l + hC || M
2
where we have used Young's inequality in the last inequality, for 0 universal constant (which may change from line to line) depending only on the eigenvalues of the matrix MM or constant factors. Final bound. Combining the bounds on B1, and B2,k, there exists a constant
such that
E
U
But1
U
1x+1
2
S E
U U2, [
+ h
C
E M²
a n
with 6x = 1+ CLTT- - (a (n)=))/2) Therefore, we have
E
|
U tx
11 [100
Ix
N-1
N N-1
C
<
II
8,
-
U8l m
+ h
M
II
j=k+1
k=0
j=k-1
N
N
C
< h
I|
2
Σ
II
85
n-
k=
j=
where we used that 0g term), we obtain
Üg. Following the same argument as in Lemma B.2 (discretization error
E [ 0 1 C || M
(a 7)2
We conclude the proof by taking the limit as A + 0 together with Fatou's 's lemma.
Lemma B.4 (Mixing time). Assume that H2 holds. Then, for all 17 > 0, there exists a constant C' >>> such that
Wi
< CaCa x Te ((- W2 (Taata @ Toy Too)
(44)
Proof. Consider a synchronous coupling of the continuous time interpolations (U,) and (UP))eeo defined in (12). with initialization
We Too. L (Ux)) - Ug 18
By definition of the W/ distance,
W2 ( (U+))) & Ür -er-Uu111 12
27

---

Analogously to the proof of Lemma B.2 and Lemma B.3 fix A N 0 such that ty = T - 6, and note that for & E [tR, tkt1l, we have that
U
2 | U l + CR
U1tt1
12
with
- 12 (U - ) M1 A. (U,, ( + 2e (5I-tx (Ut.) Sy- (UW))
Similarly to Lemma B.2 we use that, for any fixed > 0, we have
MA -(a 7)
Therefore,
(Ú - - T MA, (Uz,, -U v -(a - 1)) Ur, -
m
and using Proposition B.I there exists C a universal constant (depending only on the eigenvalues of the matrix MI constant factors) such that
(U - (
ME (1r (Uta) - -5-- (U?+) & cir- l
follows that
E [Ck] S h (Cir- (a (a-n)) E U.. U2 ||
As consequence,
E
UIN
U
1 m
S E [|U
Up m II &
with 8 1 /CLT- -aa- n)). Since exp(x) 2 + xr for IR we have that
N
8 < -h -Cir-
S e (a-nn IND hiT-
< <e f L,ds
Se (a-n)
thus,
E
Ü - U | S Ca (aa-n) | - Ug mm
which implies, taking the limit as A + together with Fatou's lemma, that
W (Ūr) L (Ū²)) s Co (a-n)) Ü, 1,
Moreover, similarly to the backward, the forward process also satisfies the following contraction property (Lemma A.4
- - 0, lL, = W2 Too, L (Ur)) < CTe (a- (a- W/ (Taata @ Tuy Too)
yiclding
C Proof of Theorem 3.2
In this section, we prove Theorem 3.2 establish this result, we work with the (unmodified) score function rather than the modified one used previously. Similarly to the previous section, introduce
28

---

the continuous time interpolation (Ut) of the Euler scheme for the time-reversed process (U.))e(0 defined as the ltō process, for E [/N, 1R+1],
U, Ü + (-AU,, + EiST (U..) (t (2) Eg Br B:.)
(45)
when initialized at pr (i.e,, Up ~ pr). When initialized at Too, we write (U;²) ]o this Itō process. We also introduce the continuous time Euler scheme (U%) in which the true, unknown score function is replaced by neural network approximation so, and defined for t E [t,, 1x-1/ as
U 0% (-A + E6 (tx. (,,) (t tτ) (B. B.,))
(46)
where Ug Ree
We first establish the propagation of regularity properties: strong log-concavity propagation (Proposi- tion C.T and Lipschitz regularity propagation (Proposition((.2) followed by the proof of Theorem 3.2
To this end, we decompose the generation error into the sum of the discretization error (Lemma the approximation error (Lemma C.4), and the mixing time error (Lemma C.5 in Theorem
C.1 Propagation of the regularity assumptions
Proposition C.1. Assume that H2' holds. Then for all [] and all u e R24
V log < aalla,,
where
(org A w- 20- min (e-ld) + Amax (oax)E
(47)
Proof. Similar Proposition B. recall the following equality in law given by the modified kinetic OU process 9
U, - dld U. Go,
with U. Tdata & Tu, G ~ N (0, I2d), where G and U. are independent, and Co. is defined in
20p Writing dejo the conditional density of U, given Do. we have
/ po(e- ( det (2x -1/2 exp ( ( (ur
(m-- 1- 2) (-==2) dz.
Pr(ue)= det (c (A)
R22
Since Tdata is strongly log log-concave and To is centered Gaussian with covariance v² La, their product (i.e. the probability density function of Un) satisfies
v (o)) < (0o A v -) I2d
2)
Consequently, for any z E 2d
2 log 122 -(a A 3) (ofee-AATTe T-
Finally, using Saumard and Wellner (0014), pt is strongly log-coneave with constant
0₁ = (a0 A 202 (e + Amax (mo.)(oo.))
Proposition C.2. Assume that H2 holds. Then, for all > O. V logpr is Le-Liscchite: for all E 2d
| log])(u)) S I S min (b1,4:22.)
where
b1. (1 (a 1)2t) 2ta max ( LL, v-)
b [ min 1/a} - (o max 1/a] + - e-at
29

---

Proof. Following "Step 1: Lower bound on V lo in the proof of Proposition B.1 all t
we obtain for
lolgp((u) & min (b,, I2d:
where
11 (1 + (a 1)2t)² 2 max (Lo, 0-
2 o min a, 1/a} - (o² maxfa, 1/a) + 5 e
2at
Moreover, Proposition C.I implies that
V2 log aq
02dx2d:
where og defined in 47 Consequently,
V logp: s min (br b2.e)
C.2 Proofs the main results
Lemma C.3 (Discretization error). Assume that H2' holds and let & > 0. If the step size h satisfies
2 mmig ar, (o² ^ - (o 8)² max; Li (a+ 1)²
0<h<
|| + (e* + 04) maxs L7 + 2 202 v v ||||| max: Lts
then, there exists de > such that W2 (((TT),(((r))) s 22VhCa(E)/0b where
C.(s) = [2||A|| B3 + 4d(aa22 + +5) 4d(11112 + +o*
sup
(48)
1E(U.2]
with
2a (1 (a 1)²T)²
AC(T) min
min x²}
0² min 1/a] (o2 max (a, 1/a} + 5as 5ac-2)e 200-2)
2aT
such that suprsu A < +00 and
c 20(T 1 + ( o² maxfa, 1/a] +
5E
B- = max (1 + (a + 1)² - tE(0,7]
(49)
Proof. Consider synchronous coupling for for(ūr (U.)eefo.) and (U.)eetor) i.e., use the same Brownian motion to drive the two processes, with the same initial point, i.e., U, Uo. Then it holds, that
W/ (C(UT), ((T))) s | U7
1/2
Fix A 2 such that IN
A and note that for 5 S - 1,
LB
Tn_- -UTre - Ünn + 1 la+1 (UA (U.-0.)) Urt)+ + x2 (T- (U.) -T (U di
L2
< A1, A2,
where
+// dat1 (-A -Un) + Σの (T (U.). su (Uxx)) dt
A1,
U Ft
Üna + -tn
Le
det1 {-A (U,-U + 2T (TT (U.) ST (Ux)) dt
A2,
La
30

---

For the first term, note that,
Ai,a- = (I2d hA) (U - -U(a) + n2? st (tx)) - -T (Unx) )
(I2d hA) (Un2 - -Ure ) 11 + | (t (tre) -T- (Uzz)) ) 11 L2
+ 2h13 (tun Un)) (I2a-hA) T 2o (t (Uis) -T (U (U+))
By Proposition particular,
C.2 it follows that the score at time is L, Lipschitz continuous for all t E [0, T]. in
W st (U...= -T (Um))) 1/2
S sii(e" + + [ 10:-0 -Umn l'
Therefore.
Ai, S E (ti=- Dis) (lIzd - hA) (I2a hA) l²2E"* + + 12a) (t - -Um))
-0g-UUp, -55--U11 (Iza-hA) T E/ (T (Un)) ST -tu (Uns))]
+ +2hB
For all 0 S T, let Ca,t == So 42 vo -2(
(Un--tt -Ua)) and write AB I2d hA
so that,
A S
E
( - -Ur.) (A + 122 + L -u- I2a +
2 Cr....).)) (U.M- - U..))
$ U U2s1 1|2 + (U -...)u))) Mn(E) (Un--Ui)
L2
where
Ms(e) = - + A) + 2T + (A A (6 + o 1o I2d 2A TAT 2
In order to control - - Ü..), it is enough to control the eigenvalues of (E) where
Mn(e) (MN(E)) )
= =- + A) 1 D0-ECTr--ua + CT k..
{ A + (6 + 0²) ) 127_1, I2d - (A (ITIZCT-uu + Cr Cr x. A2
Noting that,
CE + - [22)= =2) -p + CT + CT
25. te
By Proposition C.I
Σg C.CT-uukee $ -CT Amin (X Ina
$ -T to (0² A E2) Lzd.
and simple calculations yields
Oaxd
Me - t + CT - - -2E, 1n = (o =)²
T- Oaxa
where C/2 -l denotes the block anti diagonal elements of Cr- Hence,
2r + CT
& (a - =)² D Led
$ (a - LT I2a
31

---

where we used Proposition C.2 in the last line. It follows that,
Ma(e)= (A + A) 2aT to (o² A I2d + (a- 12d
+ h [1411] + (E" + + + 22 v
LI Led
Therefore, using that Amin (AT + A) = (a 1)², MM(E) is negative when h is chosen so that
2 2mi 0t. (o² ^ 2 - (a - =)² max; Ltc (a + 1)²
(50)
||| + (=* + 0²) max; Li, + 2 (0² V $²) ||A|| max; Ltz
It follows that when h satisfies 50 there exists 82 0 such that
A1,k S / hōg T Uea 11:
L2
For the second term 42,k, note the backward drift function as b(t, = -Au + (() so that
| Mk+1 (b(1, U.) - b(42, T.. ))
27
13.x = E
Applying Lemma D.6 combining with Itô's formula, obtain
db(t, T.)) AAÚ + 1s (T.)
= =(AAT, AE U.) + E ST sT (T.) de + (A V logeT- (U.)) ErdB,
AAU,,d (Σ:AT- AE?) ST U,)d( + (A (A+E2 E logpr- (T.) EedBr.
Using H, = AAÚ, + (EEAT- AΣ Ss (Ū.) and Kg = (A (A+ > log p ( Ee we
have that
S H,dsdt
( ORR12 S KodB,dt
A3 2,k E
let1
| H.ds
/" K, K,dB,
< 2h
E
dt
+ 2
sup [E(ta,f441]
by convexity of || -113 Using again the convexity (or applying Cauchy-Schwand inequality) we have Elll I H < J H|?|ds and then
11100 K,dB, E [1] + 26
42 S h²
sup 1e
sup
(51)
First we have for E [0,
[| < T| |1.11 + = -
ST 0. 0.)
and by Lemma D.3 and Lemma D.5 get
2d
E [1 < ||| B. +2 Σ3 EA -
A (EO.T
where B, is defined in 449 By Lemma A.3 we get
max
E [| [3] < 2|| + +e)_((²+))2;(TT))
(52)
xE 0,77
with
{ 2a + (a +
()))) 1)²T)
(T) min
min(s2,o*)
o² min (, 1/0] - (o max (a, 1/a} + 5a -5a22g2 e-2
201
32

---

such that suprsoAf(0 < +00.
Now by Doob's inequality and Itô's isometry, have
| K.dB.
|
t++1
E 1 2 log
E
sup [E]tx,tutal
ds.
where || - ||| is the Frobenius norm, so that
1 K.dB.
+ log (U.) 19.
(T.)
E
sup (144
S hd
max AE
E
tp.
Using the L. Lipschitz continuous property the score at time t have
1..0 K.dB.
E
sup
S 2hd [|A|] + maxfo",
sup 1e(taft111
1)-1))
(53)
REltx
t/.
Plugging 52 and (3) into 51) we obtain
A3,x S Ca(=)h²
(54)
with (e defined in 48
Combining the bound on AI to the bound on A2, yields,
T 1 - Ü
S S hōg
Un - -Ge +hvhCa(E).
L2
Using that U. 1 Üo = we have by induction
N-1
1TIN Tex
II (1
La
k= j=k#1
2 VhCCalE),
8
since vI - d.h < 1 h8g/2. Letting A 0 together with Fatou's 's lemma finishes the proof.
Lemma C.4 (Approximation error). Assume that H2 and H3 hold. Then, there exists 8 >> 0 such
that
W2 (C < 8
max (=,,0²) M.
Proof. Note that
We (C(7) < Ü 0% is
Using decomposition similar to that in C.3 with IN = T A, we obtain:
Ú
Le
p2+1
Uh
-A (U%
0 + x. -t (U%) 80 (T 12, Úb,)) dl
ju
12+1
<
U U
-A (U U + Σー -sT + (U%) ST-t (Ú,,)) dl
La
x (ST tu (0%) 80 (T 4, 0 ) dl
L2
< B + B2,2
33

---

For the first term, note that,
B = (T2d - hA) (U - -Ū², + h..5 (sT-t, (U2+) - -T t (0%) 1 Lo
-I[d h.4) (U%. 0 11, + | (sT-4: (U?.) 5y-t (U |1
+ 24E (U ) (I2d hA) Σ (ST by. (U sT t (0
follows that B can be treated similarly to A Using H2 have
H2' and for h satisfying (0). we
B1 S / hōg
U
U b/
ll
where 0g is defined as in the proof of Lemma C.3 For B2 using Assumption
H3 we get
B2, S | M s h hmax { M.
Finally, for satisfying 50). follows from the same argument as in the proof of Lemma C.3 that
2 max { M. 8.
U U DtwllLs S
Taking the limit as A 0, toghether with Fatou's lemma finishes the proof.
Lemma C.5 (Mixing time). Assume that H2' holds. Then
We (c ( < Kre WI'22 (Tdata @ Tu, Roa)
with
Kr == (1 + maxfa 1:a(a 1))T).
Proof. Consider a synchronous coupling of the continuous-time interpolations (U,) and (D)))) with initialization
W/ Toc, L (ūr)) |Üo 10000,l1.
By definition of the W/ distance,
W/ (C ((F)) s - 17
For t E [K [+] and with to
A we have that,
Unen - tntt lla
|Tr - -Ug0 +
platt
- A (Ū., - -Ux) + E² (ST (U.r) ST tu (U.)) dt
U 0.0 La
6k
where 6g is defined as in (50)
As a consequence,
N-1
Ü U l1, < Ü -
d2 II 8=0
where we let A - 0 together with Fatou's 's lemma. Finally, using Lemma A.4 yields
| -00-ū;1la = W2 (Toos L (Ur)) s Kre of W/ (Taata & Toy Too) which finishes the proof.
34

---

D Technical Lemmata
Lemma D.1. Assume that H2 holds. Then, the data distribution Pdsta (2) & (x)(((((()+ + (x)))
has sub-Gaussian tails, i.c., there exist constants C, >> such that
(z)ta()) C exp(-<<|=|1?,,
x RR
In particular, Tdata admits finite moments of all orders.
Proof. By a-strong convexity of for all 2, y IR.
V(x) 2 V(v) + (V (x - y) + a | u1
2
Let x" denote the unique minimizer of V, so that (v = 0. Then,
V(x) V(x*) + 2 | x² 1 |0:|| C1
for some constant E R. Since H is L-Lisschitz we have
H(x) 2 H L|| x' ž |||a|| c2
for some C2 E R. Combining these two inequalities yields, for some C E R.
V(x) + H(x) 2 a ||||| |||=|| +C 4
Using Young's inequality L||>|| S a ||| 12/ 2L2 we obtain
2L
V(x) + (()) / |=0||
C.
a
Hence, up to multiplicative constant,
Pdata (x) x exp(- + H(x))) C' exp( 8 ||
for some C' 0 which concludes the proof.
Lemma D.2. Assume that H2 holds and that there exist E N and C > O such that, for all E TR.d.
[VV(2) s C1 ||]|"")
(55)
Then, the relative Fisher Information between To Tdata & To (i.e. the initialization of the stochastic process defined in (4h) and Too is finite, i.e.
/ V
dooo
(w) ){" To(du) < 0o.
I(Tol=To)=
d
Proof. From Assumption H2 together with the fact that m Adata 8 t. and To ~ N (0a, v²Ia).
(u( Paata(x)) Oa, I2 x o- (V())))+(()))
2
Therefore, the relative Fisher Information satisfies
VV(Xx) vH(Xo)) v
[(TroTTo)) E
VH
|1
X
< 2E
2E
Σx1
vV.
By Lemma D.I Adata has sub-Gaussian tails, hence
E
0 V
35

---

Moreover,
vV(Xo) + (H( V
)|
S 2E VV xo)
+ 2B
|| xo) o11
+
V 2
Since V. u is Gaussian, E|| V ol??]
< co, and by Assumption H2 H is L-Lipschitz, so that
[[]V - L2. Using (55) there exist E N and C'> 0 such that
<c VV
[x,] [)) 00, |
E
using uub-Gaussianity of dataa which concludes the proof.
Lemma D.3. Assume that (Ū ())ett, is solution to then for all 0 & t I T, then for all & 2 0.
9 and that U. admits second order moment,
U
(1+(aa 1)) t)) 2 20(T
U
2 d
2000²
maxfa 1/a] + 56 = B
L2 2
(56)
Proof. Note that, in distribution,
1//22 U. + 20,
with U. Taata & nus G~ ~ N (0, I2a), and where G and U. are independent. Since G and U. independent, using time-reversal and sub-multiplicativity of matrix norms, we have that
E T | = E ||
|
27
E
+
E
110241 1 E
E [[GाT]
S
| 2
| E
|| ||
+ 2dA (Eo,))
We conclude by applying Lemma A.l to bound ||eaa| and Lemma A.3 to bound Amax (2o,z).
Remark D.4. Lemma D.3 holds true when U, is defined as in (4) by setting s = O.
Lemma D.5. Assume that (U is solution to (.. then,
E sr- (0.))1)
2d
Amn (E
where En, defined 20
Proof. By the time-reversal property,
ST (T.))' =
- [|STT= (Tr-)) )|²
Note that
(T-( = E -o. (e(T- ÚT-
then, using Jensen's inequality and the tower property,
E ST ( (b.)) E Ilaat (c (c(T- U. Ú |1
Since U, E e independent, we have
U. + 1/2
G with U. Tdats nu. G ~ N (0, Iza). and where G and U.
E. | (T.
(T.))
|
1/2 . 10] Tr (Su.h--)
<
E
which completes the proof.
36

---

Lemma D.6. Assume that (Ee)) is solution the backward SDE associated with 9 Then,
d(V logpr (T.)) = A logpr-:(U.))) + V logpr- (
Proof. The Fokker-Plank equation for the SDE defined 44 yields, for u E R24
1 (u) (iv((Aup((u)) + div <dvv(EVV((()))
(57)
First, using the notation introduced in (10).
2d
a Aupa(u)
div(Aup:(u))
Ous
2d 2d
0
Asju;pi(u)
Oug
j=1
2d
0 Asj pc(u)
(u
0
2d
(uP + (Au) T (p((u)
(r())pr(u) + (Au) T Vpe(u)
= p.(u) (Tr(AA + (Au) T sx(()))
Second, using the product rule for divergence,
div(E(Vpl(w))
(w)s(()((w)s((u))
2
div(pi(u)
(E(s. 2 (pi(u)div (v)) (x)) Vpr(u))
= 2 p((w) (div (div(E + +x) Eg)((u))
Hence, dividing 57
by pe yields
O₂ (o) = -T((A) - (Au) T (v( + 1 ((div(> + ss(w) [w))
2
so that.
0 loggr-r(u)) = Tr(A) + (Au) T ST (()) 2 [div(( + (()) [(sT
Recall that the backward process can be written as
1 + - (U,))dt E.dB:
Hence, by Itô's formula,
d(s--1(u.) 0,sr V + 1 Tr E (T,) dt
V0,logPT .t Vs
VSr - ( (U, + Vs / U (
1 Ir 12 S7 (1 dt Vsy (,))
0 prloopp U, ST (T,) 1つ ( (.)) + ddi(('ss de
VST (U. + Vsr ( T,)
A s + VST ( d.B.
37

---

which completes the proof and where we used that for u E R2d 2V log (o = T and (div(=}),(u)) = VTn(E?V2s(uu). Indeed, for E {,,...,.0 .d], with
g(u)- = Vlog (u), and therefore gi(u) = Ju, g(u)
a (Vg(u) E2
Σ
gi 22 (51
Zuk
duk
a gj(u)
(u) + gi(u) a
Σ
gy(u)
OuR
Oux
3,
24
2
2
gi(u)
9: g.(u))
E,,i
Ouk
2d
= 2
2
g(u)
or, g(u)
E,Yi
Ou,
Duc
1=1
= [2V2 ( [
Lemma D.7. Assume that (U+)eelo.r7 is solution to the backward SDE associated with 9 Then,
((,)))(U.)) = - lo (U,)..
(58)
Proof. Recall that Poo is the stationary distribution of 44 so that using Fokker-Planck equation get, for E R2dd
0= Tr(A) - (Au) V log (u)
[div (V'V logPoo (u) + V logP (u) T Poo (u)
+ 2
Using that p. Pt/p Po/po and Bokker-Plancel as in Lemma D.6
o (ou) (Au) (u)
+ div (E's + (1) E3)
T
(u) T T (w)
Using the definition of A., have,
1 div (I's((w) (u( T [
02 log (u) = (A + 2
and therefore,
O, log pr (Acu) ( (()) div (2'5T-c(u)) + (2377(1) Recall that the modified backward process can be written as at. (A. t, (U))) E.dB,.
Hence, by Itô's formula,
(()))
-i5r ( Vlogūr + Tr ((V's)(((.))) di
-V -r 1 dt Sy TT, E - (T,) J div (E's)
+ log A U + v log p. - U Edd
-A T.) + v log jog U
which completes the proof.
38

---

Lemma D.8. Let A be arbitrary fixed positive constant, and assume that (U,))E is the solution to (II). Then, there exists a universal constant >> such that
V Vlogrr- (tu))
loog Pr (T))
||]
E
S C (g(tx+1)- g(tk))
fort E [tx, [x+1]. with
g(t) = || log pr- ( (T.))1
(59)
The argument follows from adaptation of Conforti et al. (Proposition (025) to setting. Let Y == log ( and Z: : 4 log pr- Uz). From (58), the process (Y))2) satisfies
Proof.
dy, A Y,dt + Z,C. Applying Itô's formula to || yields
d |||l/|| = 22Y, ATY,) + 2(Y,, E,E,dB, + 11 dt.
Therefore, there exists constant depending only on a, such that
d ||/|| (IXall+ I dt + Hl,.
where He denotes a stochastic process. Moreover, following the argument of Conforti et al. (Lemma 3.3, 2025), the stochastic integral So H1,dB, is true martingale. Using this and integrating over [Lx, (j. we deduce that there exists universal constant C > 0 (whose value may change in the course of the argument) such that
scJ 1++1
IIV - 1 12
E [
+
1 ds S C(g((e+1) - g((x))-
Lemma D.9. Let A Ruxn be an invertible matrix, and let "xx be such that A B is also invertible. Then,
A4
Proof.
Note that
(A-B)- -1 = (4 - --1 [(A - B) I.] A-
and
(A- - B) - A = (A B) -1 ((A B) + B) = I, + (4 - B) BB-
so that
[AA B) I A (A B) BA
which completes the proof.
39

---

E Numerical Illustration
This section provides additional details on the numerical implementation described in Section 4
E.1
CLD training and sampling
Algorithms
1 and show the training and sampling procedures for the CLD-based approaches,
respectively.
Algorithm CLD Training
Require: Dataset DD batch size B, network so(: (), positive weight function A [0, (0 - IR., and
2
1: Precompute E0, Eo, + e elA diag((Is u I & The value of Eo, depends on see
Lemma A.2. (eq 23).
2: while not converged do
3: Sample {z(()) i= ~ DD
4: Sample (t(0)
1=1 ~ u(10, 7))
5: Sample {s(1) ii= NN I2a)
U (0) = S X 0a)
T
+ (20,00))
6:
7:
C C
A(t(i) ((t)))
so (t 1 ((s) Ux(s)
+ (1)2(00))1/22(00))
8: Update 0 by taking gradient step on VoL
9: end while
Algorithm 2 CLD Sampling
Require: Learned network so, number of discretization steps N and 2 0.
h < T/N
2: Uo
3: for == down to N do
4: le x hh
5: Sample Zz ~ N(0,I20)
D Too depends on e, see 21 in Lemma A.2 T U UA. + So 0%)) VhΣ.Z,
6:
7: end for
8: return First d coordinates of 0
Return position only, discard velocity.
E.2 Time-rescaling of the forward SDE
Following Dockhorn et (2022), one often implements in practice a time-rescaled version of a A A ,dt + E,dB,,
by introducing a positive noise schedule 3: [o, 1] + [0, oc) and setting
Ú, - - 7() and 7(t) / B(s)ds
Equivalently, U, satisfies the inhomogeneous SDE
d 3() d.dt
B(t) E, dB.
=A(t)
#1.(0
In the critically-aamper example (Equation we have
0 3(1)
Ā(t)
Ia: ε2((1) = VB(1),, 09 I..
3(())
2a (())
40

---

Since Ú, - r(t), we can deduce from the homogeneous solution the mean factor,
Mean factor.
U. U = =e-art)) (1=+0) 1 1-a7(1) @ lu D.
a²
E
Covariance.
Again by the time-change 7(),, one has
Cov Ü d)) - Cov Ü (()) D.) = f c ETeas ds.
((())
Affine schedule.
A popular and simple choice of noise schedule is an affine noise schedule given
by
B1 81,2 + 3ol.
B(1) BiL Ba, =(1)
E.3 Score approximation
Denoising Score Matching (DSM). Recall that the conditional score function of the forward given the initial data distribution is Gaussian,
process
V log pe (. - - (U.-cA
Hence, following Vincent 2011) the conditional denoising score matching loss Lcoud, for O so (t,,x)) [0, 7 x R I
and Zzd ~ ((0,I2a) can be written as
so Tr, U.) Vlogpr (U.10)))1 X(t)
(DS
=E X(t) so (T, + / E Zzd 12]
where 7 ~~ [0,77, Zea and A: [0,T) + Rss-
Hybrid Score Matching (HSM). It has been shown in Dockhorn et al. (002)) that another loss, potentially more stable numerically can be obtained by conditioning only X rather than on the full state U, (X o. Vo)T T. This hybrid score matching loss can be derived by marginalizing out the velocity component Vo ~ N (0a, u I²I,), V. - X in the conditional score function,
so U.)
Vlogp, p. (U, Ro)|
CHSM (0) = E A(t)
= E
A(t)
Sø TT.eTd
Xo
E6,- Zza + +E%,,) Zza
with Zza NO, 12a) independent of u100, 77 and
Egr Eo,r + Td (o
Ia (erd)T.
E.4 Neural network architectures
In Figure 3 we detail the neural network used in the illustration. The input layer is composed of vector in dimension 2d and the time t. Both are respectively embedded using linear transformation or sine/cosine transformation (Nichol and Dhariwal 2021] of width sid_feetures. Then, dense layers of constant width mid_ features followed by Sil.u activations and skip connections regarding the time embedding. The output layer is linear resulting in a vector of dimension (when 0) and 2d (when E * 0).
41

---

mid features
Linear Layer
Linear
outiut
SIUU
SIU
SILU
mid features
Time embedsing
Figure 3: Neural network architecture.
E.5
Additional experiments
We present additional experimental results for the MG25 distribution dimension 100 and the 2D 2D-diamond dataset. The MG25 distribution is defined as Gaussian mixture model with 25 modes dimension 100, defined as
Tanta (x)
Σ
Page (x)
25
(3,))(-22...2)2
with E. denoting the probability density function of the Gaussian distribution with covariance matrix Σg = diag (0.01, 0.01, 1,.11.... 0.1 ..... 0.1) and mean vector Piik [j. k 0, 0. OJT This dataset has been previously used Thin al. (202I); Strasman al. (2025). The 2D-diamond distribution is a two-dimensional dataset with well-separated modes, used as a synthetic dataset in Dockhorn et al. (202)).
Tables I and 3 report the sliced- Wasserstein error for different values of the regularization parameter E {0, 0.1, 0.25, 0.5, } and drift coefficient {0.11 0.25, 0.5, 1, 2}. using the same experimental setup as for the Funnel dataset described in Section 4 Both tables and Thighlight the improvement in generation quality achieved with smaller regularization values of Table report the values displayed in Figure 1 with the associated standard deviations.
Table Comparison of mean Wasserstein distance for different noise levels e on the MG25-100D (mean + standard deviation across runs; lower is better).
= 0.1
= 0.25
a = 0.5
a a= 1.0
a = 2.0
0
0.284 + 0.002
0.199 + 0.001
0.034 + 0.002
0.009 + 0.001
0.009 + 0.001
0.1
0.192 + 0.001
0.159 + 0.001
0.026 = 0.001
0.005 0.001
0.008 0.001
0.25
0.013 0.001
0.065 + 0.001
0.015 - 0.001
0.007 = 0.001
0.007 + 0.001
0.5
0.191 = 0.007
0.001
0.009 = 0.001
0.008 0.001
0.008 0.001
1
0.389 + 0.030
0.045 = 0.003
0.011 = 0.002
0.006 = 0.001
0.008 = 0.001
42

---

Table 2:
Comparison of mean Wasserstein distance for different noise levels e on the Diamond-22 (mean standard deviation across runs; lower better).
0.1
0.25
0.5
1.0
2.0
0
0.322 + 0.001
0,256 0.004
0.039 0.002
0.007 0.001
0.007 0.002
0.1
0.234 + 0.001
0.198 + 0.003
0.026 = 0.004
0.004 = 0.001
0.005 = 0.001
0.25
0.048 0.001
0.074 0.003
0.021 = 0.002
0.004 0.001
0.005 0.001
0.5
0.073 + 0.002
0.008 + 0.001
0.008 + 0.002
0.006 + 0.002
0.006 + 0.002
0.095 + 0.002
0.029 0.002
0.014 0.001
0.013 0.001
0.011 0.001
Table
Comparison of mean Wasserstein distance different noise levels on the Funnel-100D
(mean + standard deviation across runs; lower is better).
= 0.1
= 0.25
a = 0.5
a == 1.0
a = 2.0
0
0.991 + 0,001
0.73 + 0.002
0.291 = 0.005
0.225 = 0.056
0.223 + 0.011
0.1
0.705 + 0.001
0.632 + 0.002
0.278 = 0.001
0.158 0.027
0.198 0.004
0.25
0.277 0.002
0.409 0.003
0.248 = 0.012
0.137 0.005
0.179 0.006 0.006
0.5
1,171 0.015
0.248 0.002
0.228 = 0.005
0.157 0.002
0.203 0.003
1
2.885 + 0.016
0.785 0.011
0.191 = 0.008
0.253 = 0.006
0.233 = 0.002
43

---

NeurIPS Paper Checklist
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Question:
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
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Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
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It is fine to include aspirational goals as motivation as long as is clear that these goals attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: The authors are transparent about the nature of the assumptions and mathe- matical models they study, and consistently highlight these choices as part of the scope and limitations of their work.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "I "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and prool's
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
44

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Reviews:
[{"review_id": "sDlUMp2W6B", "reviewer": "Reviewer_bmjs", "summary": "The paper investigates the convergence behavior of critically-damped Langevin diffusions and derives upper bounds on the sampling error in terms of the 2-Wasserstein distance between the generated and target data distributions. The main contribution is Theorem 3.1, which, under assumptions H1–H3, establishes a bound on the Wasserstein distance comprising three additive components: discretization error, approximation error, and a convergence term related to the initial distribution. This analysis contrasts with prior results that primarily focused on KL divergence. Furthermore, Theorem 4.1 strengthens the bound by introducing an additional assumption of strong log-concavity of the intermediate distributions $p_t$ for all  $0 \\leq t \\leq T$, leading to improved convergence guarantees.", "questions": "Please see my previous response.", "limitations": "Please see my previous response.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1. The paper addresses a relevant theoretical topic concerning the convergence of Langevin diffusions.\n\n2. The convergence guarantee is provided in terms of the Wasserstein distance, which could be useful for other theoretical studies of Langevin-based generative modeling.\n\n**Weaknesses:**\n\n1. The paper is theoretically heavy and difficult to follow for a general machine learning audience. The authors do not provide simplified examples or intuitive remarks to help readers understand the main results. I highly recommend that the authors include intuitive examples or interpretations of their theorems to help readers better appreciate the significance and applicability of the results. In the current form, it is difficult to assess the non-triviality and practical value of the theoretical contributions.\n\n2. While the authors offer a brief justification on page 5 regarding the mildness of their assumptions, it would be much clearer if they included concrete examples of data distributions that satisfy these assumptions. In particular, Assumption H3 appears non-standard and seems difficult to verify in practice. For instance, assuming the data distribution is Gaussian, how would one find a valid constant $M$ to apply the theoretical result?\n\n3. The constant $h$ in Theorems 3.1 and 4.1 is vaguely defined. While looking into the proofs, I noticed that certain conditions on $h$ are required, but these are not mentioned explicitly in the theorem statements. Moreover, it appears that the constant \n$c_1$ may depend on the choice of $h$, which raises concerns about whether the bound might hold trivially. The authors should provide a more detailed discussion of the role of $h$ in the bounds to clarify the implications and non-triviality of their results.\n\n4. Following the above point, I would like to ask the authors to show what Theorem 3.1 or 4.1 mean when applied to a standard distribution such as the unit-variance Gaussian or uniform distribution. Can they compute the corresponding constants $h$ and $M$ in these cases to illustrate the implications of the bound in concrete settings?\n\n5. Finally, it would be nice if the authors elaborated on the comparison between their Wasserstein-2 bounds and the existing KL divergence bounds. Is it possible to relate the two using standard inequalities between divergence measures? A discussion of such a relationship would enhance the understanding of the novelty and potential advantages of the authors' analysis.", "was_revised": true, "final_justification": "I find the authors' responses satisfactory and therefore increased my score to 4.", "rebuttal": "We thank the reviewer for their valuable feedback and have addressed each point below. The proposed revisions have been incorporated into the revised paper. Based on these comments mentioning that the paper is difficult to follow and lacks of intuitive remarks and comments or examples on the assumptions, we simplified and clarified the presentation of the main results in the revised version. Without loss of generality and without modifying our results we propose the two following revisions.\n\n\n- First, we provide a unified result that holds for all $\\epsilon \\geq 0$, rather than treating the cases $\\epsilon=0$ and $\\epsilon>0$ separately. This reformulation aims to make the theoretical contributions more accessible to a broader audience and clarifies the discussions (see below for our comments on the assumptions).\n\n- To clarify our assumptions and better compare our work with the existing literature (see, e.g., [1]), we have added a paragraph discussing our assumptions, in particular the one-sided Lipschitz assumption. Such assumptions on the score function provide a bound on the largest eigenvalue of its Jacobian and are the weakest assumptions in the score-based generative models literature, see [1,2].\nOur work aligns with the most recent results, and our assumptions cover in particular data distributions which are not covered by the strong logconcave assumptions used in the literature. In order to simplify the paper and to compare our framework with the existing literature we propose to replace H2 and H4 in the revised paper by the following assumption.\n\n**New Assumption H2'**. \nThe data distribution is of the form  $\\pi_{\\rm data} (x) \\propto \\exp{(-u(x) + a(x))}$ and satisfies:\n\n- There exist $\\alpha>0$ and $L_0>0$ such that $\\alpha I_{d} \\preceq \\nabla^{2}u(x) \\preceq L_0 I_{d}$ for all $x\\in\\mathbb{R}^{d}$.\n- There exists  $K>0$ such that $|a(x)-a(y)|\\le K||x-y||$ for all $x,y\\in\\mathbb{R}^{d}$ with $||x-y||\\le1$.\n\nThis minor adaptation of our Lipschitz-type assumptions is consistent with the weakest assumptions found in the literature on classical score-based generative models (i.e. without the Langevin forward dynamics) and has no impact on our main results. This assumption models the data distribution as a strongly log-concave component $u(x)$ perturbed by a term $a(x)$.\nIntuitively, this assumption allows the distribution to deviate from strict log-concavity via the perturbation $a(x)$, while still maintaining sufficient regularity for analysis. When $a(x)=0$, the distribution reduces to a strongly log-concave case. \n\n**Discussion on Assumptions.**\n\nThe novel assumptions proposed above are satisfied by standard distributions such as Gaussian and mixtures of Gaussian distributions. They are strictly weaker than classical assumptions like strong log-concavity (H4), which holds only for non-degenerate Gaussian distributions and excludes many practically relevant settings although it is still common in the literature. Regarding Assumption H3, as discussed in Section 3, it quantifies the score approximation error and is standard in recent works that establish non-asymptotic Wasserstein convergence guarantees for diffusion-based models (e.g., [2, 3]).\n\nIn the particular case where the data distribution is Gaussian, the score function admits a closed form expression and does not require approximation via a neural network. Consequently, the score approximation error vanishes, and the constant $M$ in our bound can be as chosen as $M=0$. \n\n\n**Clarification on the step size $h$ and its impact.**\n\nWe acknowledge that the definition of $h$ was not clearly specified earlier. The constant $h$ in Theorems 3.1 and 4.1 corresponds to the step size in the Euler--Maruyama discretization of the backward SDE. This is now clearly stated in the revised version of the paper.\n\nIn Theorem 3.1, the result holds for all $h > 0$; the only condition we implicitly use in the proof is the elementary inequality $h \\leq (1 - \\mathrm{e}^{-h}) \\mathrm{e}^h$, which is equivalent to $h + 1 \\leq \\mathrm{e}^{h}$, a bound satisfied for all $h > 0$ (as mentioned a few lines earlier in the proof). \nFor Theorem 4.1, we already stated that the result holds for $h < h_\\epsilon$, where $h_\\epsilon$ is defined in Appendix B. We have now added the explicit form of $h_\\epsilon$ to the main text for clarity.\n\nRegarding the dependence on $h$ in $c_2$, the discretization error introduces a term of order $\\mathcal{O}(\\sqrt{h})$. In fact, the discretization error takes the form $c_2 \\sqrt{h} + o(h)$. Since $h$ is typically chosen to be sufficiently small in practice (see also [3, 4]), the higher-order term becomes negligible, and the $\\sqrt{h}$ term dominates. In practice, discretization is performed using a uniform grid, with step size $h=T/N$, where $T$ is the total time horizon and $N$ is the number of discretization steps.\n\n\n**Comparison with KL divergence bounds and known results.**\n\nIn the strongly log‑concave setting, our Wasserstein‑2 results cannot be recovered from existing KL‑divergence bounds.  Indeed, if the data distribution $\\pi_{\\mathrm{data}}$ is $c$-strongly log-concave, Talagrand’s inequality yields\n$$\n\\mathcal{W}\\_{2}^{2} \\left(\\pi\\_{\\mathrm{data}}\\otimes \\pi\\_v, \\mathcal{L} ( \\bar{\\mathbf{U}}^\\theta\\_T)\\right) \\leq\n\\frac{2}{c} \\mathrm{KL}\\left(\\mathcal{L}( \\bar{\\mathbf{U}}^\\theta_T), \\pi\\_{\\mathrm{ data}}\\otimes \\pi_v\\right).\n$$\nBy contrast, for distributions with compact support one may also bound the squared Wasserstein‑2 distance by the KL divergence ([6], Thm. 6.12, p. 82), but the corresponding constant depends on the diameter of the support, which can grow prohibitively large in high dimensions (e.g., image data).\n\nSince KL divergence is not symmetric, existing KL-based results only control\n$\\mathrm{KL}\\left(\\pi_{\\mathrm{data}}\\otimes \\pi_v,  \\mathcal{L} (\\bar{\\mathbf{U}}^\\theta_T)\\right)$, so our convergence bound cannot be deduced as a corollary.\nIndeed, [3] also assumes strong log-concavity of the data distribution (among other conditions) in $\\mathbb{R}^d$ and explicitly emphasizes that “our results are not implied by the existing convergence results for SDE-based samplers” ([3], p. 5).\nFinally, Theorem 3.1 in our work holds under far weaker assumptions (requiring neither strong log-concavity nor compact support). Moreover, existing KL-based analyses (e.g., [4, 5]) focus on a specific case of Langevin dynamics with $a = 1$ and $\\sigma = 2$, which yields an isotropic Gaussian stationary distribution—this significantly simplifies the analysis. In contrast, our analysis covers the general CLD setting and extends beyond it to include an additional noise parameter $\\varepsilon$. \n\nWe hope that these additions fully address the reviewer’s suggestions and contribute to strengthening the paper. We are grateful for the valuable feedback and trust that the revised version meets the expectations raised.\n\n**References**\n\n[1] Stéphanovitch, A. (2025). Regularity of the score function in generative models. arXiv:2506.19559. \n\n[2] Gentiloni-Silveri, M. and Ocello, A. (2025). Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance. https://arxiv.org/abs/2501.02298.\n\n[3] Gao, X., Nguyen, H. M., and Zhu, L. (2025). Wasserstein convergence guarantees for a general class of score-based generative models. Journal of Machine Learning Research.\n\n[4] Chen, S., Chewi, S., Li, J., Li, Y., Salim, A., and Zhang, A. (2023). Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. In International Conference362on Learning Representations.\n\n[5] Conforti, G., Durmus, A., and Silveri, M. G. (2025). Kl convergence guarantees for score diffusion models under minimal data assumptions. SIAM Journal on Mathematics of Data Science.\n\n[6] Villani, C. (2008). Optimal Transport: Old and Young. Springer."}, {"review_id": "MyXZvPpG7e", "reviewer": "Reviewer_8HZL", "summary": "Score-based generative models rely on a forward diffusion process that reduces the data distribution into an easy-to-sample target distribution (often a Gaussian). Therefore, a wide variety of diffusion processes can be used in principle. This paper focuses on the study of the Wasserstein convergence of the critically-damped Langevin diffusion, which often showcases a smoother sampling trajectory when compared to the more standard choices. \n\nThis paper studies the Wasserstein convergence of diffusion models that seek to invert the critically-damped Langevin diffusion, and provides a convergence result (Theorem 3.1) that characterizes Wasserstein distance on the extended phase space $\\pi\\_{\\text{data}} \\otimes \\pi\\_v$. Building upon this result, this paper posits that the convergence property of this forward process can be improved by a slight modification of the diffusion coefficient, and illustrates empirically via a synthetic dataset that the introduction of this parameter leads to better Wasserstein convergence.", "questions": "None", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall, I think this paper is well-written with interesting theoretical results in the study of forward processes (often outlooked in the context of diffusion modeling). \n\n## Strengths\n- The paper presents a theoretical foundation to the study of critically-damped Langevin diffusion as a candidate for forward processes, which will be of general interest to researchers of diffusion models. The $W_2$ convergence presents a new result. \n- The paper is clearly written, with useful references to related work that draws parallel between KL convergence and Wasserstein convergence. \n- While this paper does not provide an explanation about why CLD can outperform standard Ornstein-Uhlenbeck process, its suggestion of a modified forward process is nevertheless useful in itself, and provides further guidelines to further study of forward processes backed by convergence theory. \n## Weaknesses\nI think this paper's strengths outweigh its weaknesses, however, I find the theoretical insights provided by this paper somewhat limited in applicability because of the following reasons. \n- Although this paper presents a novel result, it does not ultimately answer the question of the better numerical performance of kinetic Langevin sampler compared to the standard Ornstein-Uhlenbeck process: instead, the paper opts to provide an alternative forward process that better exploits the added flexibility introduced by auxiliary variables. \n- Theorem 3.1 seems useful, but one can argue that convergence in the extended state space  $\\pi\\_{\\text{data}} \\otimes \\pi\\_v$ is not a point of interest: the moment variables are introduced as auxiliary variables for the forward process. \n- The numerical illustration seems somewhat contrived. The setting is largely synthetic, and the paper admits that comparing across different choices of $\\epsilon$s can be problematic. \n- The absence of a practical dataset is usually acceptable for a theory paper, however, it still constitutes a weakness for this paper, especially when the paper proposes a brand new way of parametrizing forward processes. When comparing across different $\\epsilon$s can be seen as subjective, it will be compelling to see the introduction of the regularization parameter make a different in a practical setting.", "was_revised": true, "final_justification": "I have interacted with the authors on my reviews of the paper, and I am mostly satisfied with the response. \n\nMy initial assessment of the paper was positive, and as I already recommend acceptance for this paper, I maintain my current assessment.", "rebuttal": "We thank the reviewer for the positive and thoughtful review, as well as the opportunity to clarify our contributions. We address each point below and have already incorporated the corresponding changes into the revised manuscript. All references are cited consistently with those in the original version.\n\n**[W1] On the Theoretical Contribution and Practical Relevance of CLD.**\n\nPrevious empirical work (Dockhorn et al., 2022) shows that CLD can outperform standard SGMs forwards on image benchmarks, yet no $\\mathcal{W}\\_{2}$ analysis existed; existing theory for CLD is restricted to KL (Chen et al., 2023; Conforti et al., 2025) and to a specific parametrization of the Langevin dynamics.  Our goal is to close this gap (Theorems 3.1 and 4.1) and place CLD on the same footing as VP/VE SDEs in Wasserstein distance, while maintaining the state‑of‑the‑art minimal regularity assumptions on the data distribution.\n\nBy providing the first $\\mathcal{W}\\_{2}$ guarantees for CLD, we show that the method is not only empirically competitive but also theoretically sound and tunable, offering an alternative to classical SGMs. In this way, as training in the coupled phase space has been shown by Dockhorn et al. (2022) to be empirically more stable (in particular the Hybrid Score Matching loss), our results certify that the associated sampler remains $\\mathcal{W}\\_{2}$‑stable even when the score is learned only approximately (Assumption H3).\n\nMoreover, as the reverse SDE is Hamiltonian‑like, it is possible to design structure‑preserving integrators such as symplectic or leapfrog schemes, which are inaccessible to standard SGMs. On our toy example leapfrog integrator consistently outperforms Euler–Maruyama in generation quality, even in the $\\varepsilon$‑regularized setting. This perspective paves the way for improved discretization error rates. While a full analysis of these structure‑preserving schemes is beyond the scope of this work, our results lay the groundwork for future improvements in discretization error rates.\n\n**[W2] Convergence rate for $\\mathcal{W}\\_2 ( \\pi\\_{\\mathrm{data}}, \\mathcal{L}(\\bar X\\_T^{\\theta}) )$.**\n\nSince both training and sampling operate in the extended phase space, we established a convergence result for the joint distribution, even though the main quantity of interest is the marginal data distribution. The error on the data distribution is directly controlled by the joint bound via contraction under projection of the $\\mathcal{W}_2$ distance, \n$$ \\mathcal{W}\\_2 ( \\pi\\_{\\mathrm{data}}, \\mathcal{L}(\\bar X\\_T^{\\theta}) ) \\leq  \\mathcal{W}\\_2 \\left( \\pi\\_{\\mathrm data}\\otimes \\pi\\_v ,\\mathcal{L}(  \\mathbf{\\bar U}^\\theta_T) \\right)  .$$\nWe have added this clarification to the revised version of the paper.\n\n**[W3] Extended experiments and impact of $\\varepsilon$.**\n\nWe understand the concern regarding the use of real, high-dimensional datasets. However, training such models is very computationally intensive, and as our work is primarily focused on theoretical guarantees, we restricted our empirical validation to toy datasets that serve to illustrate our mathematical claims. In this way, the introduction of the $\\varepsilon$ parameter was initially motivated theoretically by Lemma C.1, where we showed that larger $\\varepsilon > 0$ loosens the stepsize condition on $h$—i.e., strengthens the effective contraction. However, we acknowledge that a complete global analysis of this trade-off remains an open question.\n\nTo address this concerns within the limited rebuttal window, we extended our numerical experiments to a higher-dimensional toy setting. Specifically, we provide an analysis of the sensitivity of the $\\varepsilon$ parameter on the 100-dimensional Funnel dataset across different choices of drift $a$. We report the mean and standard deviation over 5 runs in Table 1. In this extended empirical analysis it appears that the benefit of a small $\\varepsilon$ regularization is confirmed even for different choices of $a$. Moreover, note that in our ablation study, we set $\\sigma$ to $\\sqrt{2}$ so that the case where $a=2$ and $\\epsilon=0$ exactly matches the setting recommended by Dockhorn et al. (2022). We also adopted the same time renormalization as them to make the comparison as fair as possible (see Appendix C.1).\n\nOn the one hand, we observe that this parametrization is highly effective on toy datasets. On the other hand, several $\\varepsilon$-regularized variants perform even better. To further support this observation, we are extending our evaluation to additional challenging toy datasets: MG-25 (a 25-mode 100D Gaussian mixture) and Diamonds (a 2D Gaussian mixture with diamond geometry). Preliminary results suggest that the same conclusions hold (see Tables below).\nIf of interest, we are happy to include standard deviations in the revised paper and to reproduce plots analogous to Figure 1 for these new datasets. \n\n**Table 1: Comparison of mean Wasserstein distance for different noise levels $\\varepsilon$ on the Funnel‑100D**\n*(Mean ± standard deviation across 5 runs; lower is better)*\n\n| $\\varepsilon$ |        | $a=0.25$             | $a=0.5$              | $a=1.0$                | $a=2.0$                |\n|:-------------:|--------|:--------------------:|:--------------------:|:----------------------:|:----------------------:|\n| 0             |        | 0.73 ± 0.0016        | 0.291 ± 0.0047       | 0.225 ± 0.056          | 0.223 ± 0.011          |\n| 0.1           |        | 0.632 ± 0.0016       | 0.278 ± 0.0012       | 0.158 ± 0.027          | 0.198 ± 0.004          |\n| 0.25          |        | 0.409 ± 0.0031       | 0.248 ± 0.012        | **0.137 ± 0.0046**     | **0.179 ± 0.0057**     |\n| 0.5           |        | **0.248 ± 0.0016**   | 0.228 ± 0.0051       | 0.157 ± 0.0022         | 0.203 ± 0.0028         |\n| 1             |        | 0.785 ± 0.011        | **0.191 ± 0.0079**   | 0.253 ± 0.006          | 0.233 ± 0.002          |\n\n\n**Table 2: Comparison of mean Wasserstein distance for different noise levels $\\varepsilon$ on the MG25‑100D**\n\n| $\\varepsilon$ |        | $a=0.1$        | $a=0.25$       | $a=0.5$         | $a=1.0$         | $a=2.0$         |\n|:-------------:|:------:|:--------------:|:--------------:|:---------------:|:---------------:|:---------------:|\n| 0             |        | 0.285          | 0.199          | 0.0346          | 0.0083          | 0.009           |\n| 0.1           |        | 0.191          | 0.158          | 0.0262          | **0.0063**      | 0.008           |\n| 0.25          |        | **0.013**      | 0.0654         | 0.0157          | 0.0076          | **0.007**       |\n| 0.5           |        | 0.188          | **0.005**      | **0.0093**      | 0.0071          | 0.0081          |\n| 1             |        | 0.381          | 0.0491         | 0.0121          | 0.0064          | **0.007**       |\n\n\n**Table 3: Comparison of mean Wasserstein distance for different noise levels $\\varepsilon$ on the Diamond‑2D**\n\n| $\\varepsilon$ |        | $a=0.1$        | $a=0.25$       | $a=0.5$         | $a=1.0$         | $a=2.0$         |\n|:-------------:|:------:|:--------------:|:--------------:|:---------------:|:---------------:|:---------------:|\n| 0             |        | 0.323          | 0.252          | 0.0325          | 0.0058          | 0.0077          |\n| 0.1           |        | 0.233          | 0.198          | 0.0237          | **0.0038**      | 0.0047          |\n| 0.25          |        | **0.0474**     | 0.0737         | 0.0214          | 0.0049          | **0.004**       |\n| 0.5           |        | 0.0722         | **0.0078**     | **0.0077**      | 0.0057          | 0.0056          |\n| 1             |        | 0.0958         | 0.0286         | 0.0143          | 0.0136          | 0.012           |\n\nWe thank the reviewer once again for their insightful comments. The revisions directly address each point raised and, we believe, further strengthen the manuscript. We hope the changes align with the reviewer’s expectations and remain available to clarify any remaining questions."}, {"review_id": "ALJopYBeqS", "reviewer": "Reviewer_N2VV", "summary": "The paper studies Wasserstein convergence of score-based generative diffusion models where the forward process is governed by a critically damped Langevin diffusion. A modified dynamic is proposed that incorporates an additional hyperparameter that controls the noise applied to the data. Some numerical experiments are also provided.", "questions": "(1) I am still a bit confused with Theorem 3.1. and Theorem 4.1. In Theorem 3.1., you obtained Wasserstein convergence guarantees without H4, the strongly-log-concave assumption. But then in Theorem 4.1. you need H4. It would be helpful if you can explain a bit why for Theorem 4.1. you need H4, but not for Theorem 3.1.\n\n(2) Right before Theorem 4.1., you mentioned your results match existing results for SGMs in the literature. But if so, what is the advantage using your proposed modified dynamics compared to SGMs?", "limitations": "It seems the author(s) did not discuss limitations in the paper.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n(1) The paper is a theoretical paper. The analysis seems to be rigorous. \n\n(2) The Wasserstein convergence guarantees for critically damped Langevin diffusions on the Euclidean space seem to be new.\n\nWeaknesses:\n\n(1) The theoretical convergence guarantees for critically damped Langevin diffusion models have already been studied in the literature, including Chen et al. (2023) that is cited in the paper. There, the authors claimed that critically damped Langevin diffusion models cannot improve upon the theoretical guarantees for using the classical score-based diffusion model based on overdamped Langevin diffusion. By reading this paper, it is not clear what is the advantage using critically damped Langevin diffusions and whether acceleration can be achieved.\n\n(2) The paper mentions that Chen et al. (2023) and Conforti et al. (2025) analyzed the convergence of SGMs in the extended space using kinetic Langevin-dynamics. Their results use other measures, such as KL divergence in Conforti et al. (2025). KL sometimes can imply Wasserstein, for example on a compact domain, or when the target distribution is strongly-log-concave, which is assumed in H4 in the paper.  The paper should highlight why for the settings considered in this paper,  Wasserstein convergence cannot be obtained using existing results or at least using the existing approaches in the literature.\n\n(3) I find the assumption H4 very strong, because it is not just strong-log-concavity, but at every $t$.", "was_revised": true, "final_justification": "I have raised my score. I'm satisfied with the author(s)' response.", "rebuttal": "We thank the reviewer for their valuable feedback and have addressed each point below. The proposed revisions have been incorporated into the revised paper.\n\n**On the benefit of deriving $\\mathcal{W}\\_2$-bound.**\n\nPrevious empirical work (Dockhorn et al., 2022) shows that CLD can outperform standard SGMs forwards on image benchmarks yet the only theoretical results were given in KL (Chen et al. (2023) and Conforti et al.(2025)) and not in $\\mathcal{W}\\_{2}$ -distance \neven though the latter enjoys nice geometric properties and is a true mathematical distance. The goal of this work is to close this gap and place CLD on the same footing as VP/VE SDEs in Wasserstein distance, while maintaining state‑of‑the‑art, minimal, regularity assumptions on the data distribution.\nIn this way, as training in the coupled phase space has been shown by Dockhorn et al. (2022) to be empirically more stable (in particular the Hybrid Score Matching loss), our results certify that the associated Euler-Maruyama sampler remains $\\mathcal{W}\\_{2}$‑stable even when the score is learned only approximately (Assumption H3). \n\nAs also mentioned by the reviewer, our contribution concerns a different metric and setting as Chen et al. (2023) and Conforti et al.(2025). We give here the first $\\mathcal{W}\\_2$ guarantees for CLD, showing it attains the same convergence rates as SGMs. Note that the proof technique is completely different and rely on coupling techniques tailored to $\\mathcal{W}\\_2$ analysis. We also agree that KL divergence upper bounds can sometimes be translated to $\\mathcal{W}\\_2$ upper bounds; however, this is non-trivial and the price to pay may be high. In the case of compactly supported data distribution, the KL bounds constitute a valid upper bound to the $\\mathcal{W}\\_2$ metric. Nevertheless, this upper bound depends on the diameter of the space (Villani, 2008, Thm. 6.12, p. 82), which could therefore grow very large, especially in high dimensional settings, such as those encountered in image datasets. Regarding the log-concave case, we are not sure what was meant. If the reference was meant to Talagrand's inequality, this typically requires to verify regularity on the measure to the right of the KL (i.e. $\\nu$ for $\\mathrm{KL}(\\mu||\\nu)$) which in standard convergence results (Chen et al(2023), Conforti et al.(2025)) is usually the distribution generated by the algorithm and not the data distribution. \n\nNote that Gao et al. (2025) assume strong log-concavity of the data distribution (among other conditions) in $\\mathbb{R}^d$, and explicitly state that their results are not implied by existing KL-based convergence bounds without further assumptions:\n\"our results are not implied by the existing convergence results for SDE-based samplers.\" (Gao et al. (2025), p.5). \n\nOn the practical side, as noted in Strasman et al. (2025), Wasserstein upper bounds are often more tractable to approximate empirically, whereas KL divergences and Fisher information are difficult to estimate in high dimensions. It is also worth noting that previous results focus on specific parametrization choices ($a=1$ and $\\sigma = 2$ so that the drift and diffusion coefficients are coupled in a specific manner), which limits their generality and expressiveness. \n\n**Response to Q1.**\n\nNote that our main contribution is Theorem 3.1, which establishes a quantitative bound for generic Langevin dynamics under weak Lipschtiz-like assumptions. Moreover, we propose in Section 4 an upper bound in the specific and widespread strongly log-concave setting. This framework allows to target more precisely the role of the parameters $a$, $\\epsilon$ and $\\sigma$. We agree that the initial version was a bit confusing. We now highlight that Section 4 is a particular case of Theorem 3.1. In this way, we turn the title of Section 4 into: \"Specific analysis of $\\epsilon$-CLD under strong log-concavity\".\n\n\n**On the log-concavity at all times.**\n\n Regarding the question on Assumption H4, we agree that this is stronger than necessary. In fact, there is no need to require strong log‑concavity at all times. Similar to Gao et al. (2025) and Strasman (2025) (in the VPSDE case), it is enough to assume that the data probability density function is $c_0$-strongly log concave for the property to propagate at all times, i.e. for $\\log p_t$ to be strongly concave for every $t \\geq 0$. In fact, $c_t$ is explicitly available and given by, \n \n\\begin{align*}\nc_t = \\left( \\frac{1}{ \\min (c_0, v) \\sigma^2_{\\min}(e^{-tA})} + \\lambda_{\\max}(\\Sigma_{0,t}) \\right)^{-1} .\n\\end{align*}\n\nIn particular, this expression also highlights the impact of the calibration of the initial variance $v$ of the velocity component, with respect to the data log-concavity constant and the implicit impact of $\\epsilon$ through $\\Sigma_{0,t}$. In addition, note that Section 4 provides quantitative upper bounds in the widespread setting of strongly log-concave distributions but Theorem 3.1 does not require this assumption.\n\n**Clarification on the assumptions.**\n\nTo take into account this remark on the discussion of the assumptions and to better position our work in the literature, we have added a paragraph discussing our assumptions. Please refer to our answer to reviewer bmjs where we discuss in detail the assumptions of our work and relation to other results in the literature. \n\n**Regarding the benefit of CLDs and their $\\epsilon$-variant.**\n\nThe kinetic framework offers practical advantages that standard VP/VE SDEs do not offer:\n\n-**Forward tuning:** the position–noise level $\\varepsilon$, the velocity–noise level $\\sigma$, and the initial velocity scale $v$ can be adjusted independently of the drift parameter $a$, whereas in VP/VE the drift and diffusion are rigidly coupled. This gives extra freedom to practitioners. \n    \n-**Training objective:** the extended phase space admits Hybrid Score Matching (HSM) in addition to classical DSM training, and Dockhorn et al. showed empirically that HSM  lowers gradient variance during the training. \n    \n-**Backward solver:** because the reverse flow is Hamiltonian‑like, one can employ structure‑preserving integrators such as symplectic or leapfrog schemes; on our toy problems leapfrog consistently outperforms Euler–Maruyama, even in the $\\varepsilon$‑regularized variant. We leave for future work the theoretical analysis of such schemes.  \n   \n-**$\\epsilon$-regularized version:**  the introduction of the $\\epsilon$ parameter was initially motivated theoretically by Lemma B.1, where we saw that larger $\\epsilon$ loosens the stepsize condition on $h$—i.e., strengthens the effective contraction, although a full global trade‑off analysis remains open.We however, illustrated this effect empirically on toy datasets and saw that a small regularization  yield improvement in the $\\mathcal{W}_2$ metric on a Funnel dataset in dimension 50. We further illustrate this claim by extending our empirical analysis to dimension 100. Moreover, we have added results on two additional challenging synthetic datasets: MG-25 in dimension 100 (a 25-mode Gaussian mixture) and Diamonds (a 2D Gaussian mixture arranged in a diamond-shaped configuration). All these additional experiments (Tables 1, 2 and 3 in our response to reviewer N5Lx) further support our claims and exhibit trends consistent with our earlier findings.\n\n We hope these comments meet the expectations and would be happy to clarify any remaining questions.\n\n**References:**\n- Villani, C. (2008). Optimal Transport: Old and Young. Springer."}, {"review_id": "HcYCaN80E1", "reviewer": "Reviewer_N5Lx", "summary": "This paper studies Critically Damped Langevin Diffusions (CLDs), a second-order extension of score-based generative models, and provides the first W2 convergence guarantees under weak assumptions. \n\nThe authors first prove that vanilla CLD samplers converge in W2 with an error bound decomposed into mixing, discretization ($O(\\sqrt{h})$), and score-approximation terms. \n\nThe authors then further introduce a new hyperparameter $\\epsilon$ controlling noise on the position coordinate and prove an improved $O(h)$ discretization rate under stronger log-concavity assumptions, and give practical tuning guidance. \n\nExperiments on the 50-dimensional Funnel distribution validate that small $\\epsilon$ yields lower W2 error and variance than no $\\epsilon$.", "questions": "Can the additional noise hyperparameter $\\epsilon$ in Section 4 be interpreted as a form of warm-up or preconditioning?", "limitations": "There are no societal impact considerations.", "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\n1. The theoretical results are nontrivial and fill a gap in the convergence theory of CLD-based samplers. *The reviewer acknowledges here unfamiliarity with the latest advances in Langevin diffusion analysis and did not check all derivations.*\n2. This is the first paper to derive W2 convergence rates for CLDs (previous work focused on KL divergence). The regularized variant in Section 4 is a novel extension with provable benefits.\n3. The main text is generally well organized.\n\n**Weaknesses:**\n1. There is no analysis or experiment on warm-up (burn-in) strategies or initialization sensitivity.\n2. It would be great to include an explicit pseudocode block in the main text to summarize the algorithm for analysis.\n3. Experimental validation is limited to synthetic tasks. As this work mainly focuses on theoretical advances, the reviewer thinks it is acceptable.", "was_revised": true, "final_justification": "The authors addressed all my concerns appropriately. I would like to maintain my current positive scores.", "rebuttal": "We thank the reviewer for the positive assessment and for recognizing the novelty of our $\\mathcal{W}_2$ analysis for CLD. Below, we address all concerns and describe the additions in the revision.\n\n**Explicit pseudocode**\n\nFor clarity, we now include a concise pseudocode block for both training (Algorithm 1) and sampling (Algorithm 2). They are now integrated into the manuscript. We are happy to provide a more detailed version if the reviewers find it necessary.\n\n---\n\n### Algorithm: CLD Training\n\n**Inputs:** Dataset $\\mathcal{D}$, batch size $B$, network $s_\\theta(\\cdot, t)$, a positive weight function $\\lambda : [0,T] \\to \\mathbb{R}_+$, and noise level $\\epsilon \\geq 0$.\n\n**Precomputation:**  Compute  $\\tilde \\Sigma_{0,t} = \\Sigma_{0,t} + e^{t A} \\, \\mathrm{diag}(0 \\cdot I_d, v^2 I\\_d) \\, (e^{t A})^\\top$  (Note: The value of $\\Sigma_{0,t}$ depends on $\\epsilon$, see Lemma A.2 (Eq. 23).)\n\n**While not converged:**\n1. Sample {$x^{(i)}$}$_{i=1}^{B} \\sim \\mathcal{D}$\n2. Sample {$t^{(i)}$}$_{i=1}^{B} \\sim \\mathcal{U}([0,T])$\n3. Sample {$\\varepsilon^{(i)}$}$\\_{i=1}^{B} \\sim \\mathcal{N}(0, I_{2d})$\n4. Compute  \n   $$\n\\overrightarrow{\\mathbf{U}}\\_{t^{(i)}} = e^{t^{(i)} A} \n( \\overrightarrow{X}\\_0 \\\\ 0\\_d )^\\top + ( \\tilde{\\Sigma}\\_{0,t^{(i)}} )^{1/2} \\varepsilon^{(i)}\n$$\n5. Compute loss  \n   $$\n   \\mathcal{L} \\gets \\frac{1}{B} \\sum_{i=1}^{B} \\lambda(t^{(i)}) || s\\_\\theta(t^{(i)}, \\overrightarrow{\\mathbf{U}}\\_{t^{(i)}}) + (\\tilde \\Sigma\\_{0,t^{(i)}})^{-1/2} \\varepsilon^{(i)}||^2\n   $$\n6. Update $\\theta$ via gradient descent on $\\nabla_\\theta \\mathcal{L}$\n\n---\n\n### Algorithm: CLD Sampling\n\n**Inputs:** Learned network $s_\\theta$, number of discretization steps $N$, and noise level $\\epsilon \\geq 0$.\n\n1. Set step size: $h \\leftarrow T/N$\n2. Sample initial state: $\\mathbf{\\bar{U}}_0 \\sim \\pi_\\infty$\n3. **For** $k = 0$ to $N-1$:\n   - Compute current time: $t_k \\leftarrow k h$\n   - Sample noise: $Z_k \\sim \\mathcal{N}(0, I_{2d})$  *(Note: $\\pi_\\infty$ depends on $\\epsilon$, see Lemma A.2, Equation (21).)*\n   - Update state:  \n     $$\n     \\bar{\\mathbf{U}}^\\theta_{t_{k+1}} = \\bar{\\mathbf{U}}^\\theta_{t_k} + h \\left( \\tilde{A} \\bar{\\mathbf{U}}^\\theta_{t_k} + \\Sigma_\\epsilon^2 s_\\theta(t_k, \\bar{\\mathbf{U}}^\\theta_{t_k}) \\right) + \\sqrt{h} \\Sigma_\\epsilon Z_k\n     $$\n4. **Return** the first $d$ coordinates of $\\bar{\\mathbf{U}}^\\theta_{t_N}$  *(i.e., return position only, discard velocity)*\n\n---\n\n**Experiments beyond the original synthetic tasks**\n\nIn addition to our initial experimental results, we have extended the evaluation on the Funnel distribution in dimension 100 by varying the parameters $a$ and $\\epsilon$. We analyze the impact of these key parameters and report the corresponding results in Table 1. The revised paper also includes an updated figure. Our regularized method consistently yields improved sliced Wasserstein-2 distances, with the performance gap increasing notably as $a$ decreases.\n\nMoreover, we have added results on two additional challenging synthetic datasets: MG-25 in dimension 100 (a 25-mode Gaussian mixture) and Diamonds (a 2D Gaussian mixture arranged in a diamond-shaped configuration). These additional experiments (Tables 2 and 3) further support our claims and exhibit trends consistent with our earlier findings. Due to time constraints, the computation of standard deviations is still in progress; we will include them in the final version once available. Moreover, note that in our ablation study, we set $\\sigma$ to $\\sqrt{2}$ so that the case where $a=2$ and $\\epsilon=0$ exactly matches the setting recommended by Dockhorn et al. (2022). We also adopted the same time renormalization as them to make the comparison as fair as possible (see Appendix C.1).\n\n**Table 1: Comparison of mean Wasserstein distance for different noise levels $\\varepsilon$ on the Funnel‑100D**\n*(Mean ± standard deviation across 5 runs; lower is better)*\n\n| $\\varepsilon$ |        | $a=0.25$             | $a=0.5$              | $a=1.0$                | $a=2.0$                |\n|:-------------:|--------|:--------------------:|:--------------------:|:----------------------:|:----------------------:|\n| 0             |        | 0.73 ± 0.0016        | 0.291 ± 0.0047       | 0.225 ± 0.056          | 0.223 ± 0.011          |\n| 0.1           |        | 0.632 ± 0.0016       | 0.278 ± 0.0012       | 0.158 ± 0.027          | 0.198 ± 0.004          |\n| 0.25          |        | 0.409 ± 0.0031       | 0.248 ± 0.012        | **0.137 ± 0.0046**     | **0.179 ± 0.0057**     |\n| 0.5           |        | **0.248 ± 0.0016**   | 0.228 ± 0.0051       | 0.157 ± 0.0022         | 0.203 ± 0.0028         |\n| 1             |        | 0.785 ± 0.011        | **0.191 ± 0.0079**   | 0.253 ± 0.006          | 0.233 ± 0.002          |\n\n\n**Table 2: Comparison of mean Wasserstein distance for different noise levels $\\varepsilon$ on the MG25‑100D**\n\n| $\\varepsilon$ |        | $a=0.1$        | $a=0.25$       | $a=0.5$         | $a=1.0$         | $a=2.0$         |\n|:-------------:|:------:|:--------------:|:--------------:|:---------------:|:---------------:|:---------------:|\n| 0             |        | 0.285          | 0.199          | 0.0346          | 0.0083          | 0.009           |\n| 0.1           |        | 0.191          | 0.158          | 0.0262          | **0.0063**      | 0.008           |\n| 0.25          |        | **0.013**      | 0.0654         | 0.0157          | 0.0076          | **0.007**       |\n| 0.5           |        | 0.188          | **0.005**      | **0.0093**      | 0.0071          | 0.0081          |\n| 1             |        | 0.381          | 0.0491         | 0.0121          | 0.0064          | **0.007**       |\n\n\n**Table 3: Comparison of mean Wasserstein distance for different noise levels $\\varepsilon$ on the Diamond‑2D**\n\n| $\\varepsilon$ |        | $a=0.1$        | $a=0.25$       | $a=0.5$         | $a=1.0$         | $a=2.0$         |\n|:-------------:|:------:|:--------------:|:--------------:|:---------------:|:---------------:|:---------------:|\n| 0             |        | 0.323          | 0.252          | 0.0325          | 0.0058          | 0.0077          |\n| 0.1           |        | 0.233          | 0.198          | 0.0237          | **0.0038**      | 0.0047          |\n| 0.25          |        | **0.0474**     | 0.0737         | 0.0214          | 0.0049          | **0.004**       |\n| 0.5           |        | 0.0722         | **0.0078**     | **0.0077**      | 0.0057          | 0.0056          |\n| 1             |        | 0.0958         | 0.0286         | 0.0143          | 0.0136          | 0.012           |\n\n\n**Initialization sensitivity**\n\nWe are not completely certain we have interpreted the question regarding the initialization sensitivity correctly. Our error decomposition makes the effect of initialization of the forward process explicit. In particular,the mixing term error, the one that decays as $e^{-c_2 T}$ is proportional to the Wasserstein distance between the initial law of the forward process ($\\pi_{\\mathrm{data}} \\otimes \\pi_v$) and the stationary distribution $\\pi_{\\infty}$. Therefore, choosing any initial velocity variance $v$ that decreases $\\mathcal{W}\\_2 (\\pi\\_{\\mathrm{data}} \\otimes \\pi\\_v, \\pi\\_{\\infty})$ reduces the constant. However, discussing the full impact of the initial velocity variance $v$ on the whole bound is more subtle. Indeed, $v$ affects the regularity of the forward distribution $p_t$ that governs the contraction properties. For example, in the Gaussian CLD setting (with $X_0 \\sim \\mathcal N(\\mu_0,s_0^2)$, $V_0 \\sim \\mathcal N(0,v^2)$), the regularity constants can directly be derived from the covariance $\\Sigma_{0,t}$ of $U_t=(X_t, V_t)$. A straightforward computation shows that a larger $v$ yields a lower log-concavity constant $c_t$ and a lower smoothness constant $L_t$ for $p_t$, yielding a clear trade‑off. We will add ablation studies for the sensitivity to the initial distribution in the revised version for different synthetic data distributions. Should the interpretation of the question be incorrect, we would be happy to provide additional details if necessary.\n\nRegarding the question of whether adding $\\varepsilon$ can be interpreted as a warm-up strategy: this can be considered true, to some extent. The addition of $\\varepsilon$ affects the initialization of the backward sampler by modifying the stationary distribution $\\pi_\\infty$ of the forward process. However, beyond initialization, it also acts as a noise injection mechanism that impacts the regularity of the backward process---namely the constants $c_t$ and $L_t$---as well as the admissible step size $h$.\n\nWe thank the reviewer once again for their insightful comments. The revisions above, along with the additional experiments, directly address all points raised and further strengthen the manuscript. We hope these changes meet the expectations and would be happy to clarify any remaining questions."}]

Author rebuttal:

Meta-review: Critically damped Langevin dynamics has been proposed in the literature as an alternative to the overdamped Ornstein Uhlenbeck forward process for diffusion models. There is some evidence that this process can be better in terms of training stability than the overdamped forward process. Novel integrators such as leap-frog etc are also possible in this setting. While prior works have provided convergence analysis of the diffusion models based on CLD in KL divergence, the current work derives convergence to target law in the $\mathcal{W}_2$ metric. The reviewers appreciated the rigor and depth of the mathematical analysis. 

The main weaknesses of this paper pointed out by the reviewers were:
- This work does not demonstrate the advantage of CLD over the usual forward process in terms of the bounds.
- The assumption of strong-log-concavity, especially in Theorem 4.1, can be especially stringent. In this setting $\mathsf{KL}$ convergence from prior works can already imply $\mathcal{W}_2$ convergence using the $T_2$ inequality. 

The authors gave a satisfactory response to most of these concerns. I am leaning towards accepting this paper.
Decision: Accept (poster)