Original: S AMPLING IS AS EASY AS LEARNING THE SCORE : THEORY FOR DIFFUSION MODELS WITH MINIMAL DATA ASSUMPTIONS
Revised: S AMPLING IS AS EASY AS LEARNING THE SCORE : THEORY FOR DIFFUSION MODELS WITH MINIMAL DATA ASSUMPTIONS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL · E 2.
Revised: We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL · E 2.

Original: Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution.
Revised: Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution.

Original: In contrast to prior works, our results (1) hold for an L 2 -accurate score estimate (rather than
Revised: In contrast to prior works, our results (1) hold for an L 2 -accurate score estimate (rather than

Original: L ∞ -accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small.
Revised: L ∞ -accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small.

Original: We view this as strong theoretical justification for the empirical success of
Revised: We view this as strong theoretical justification for the empirical success of

Original: SGMs.
Revised: SGMs.

Original: We also examine SGMs based on the critically damped Langevin diffusion (CLD).
Revised: We also examine SGMs based on the critically damped Langevin diffusion (CLD).

Original: Contrary to conventional wisdom, we provide evidence that the use of the
Revised: Contrary to conventional wisdom, we provide evidence that the use of the

Original: CLD does not reduce the complexity of SGMs.
Revised: CLD does not reduce the complexity of SGMs.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Score-based generative models (SGMs) are a family of generative models which achieve state-ofthe-art performance for generating audio and image data (Sohl-Dickstein et al., 2015;
Revised: Score-based generative models (SGMs) are a family of generative models which achieve state-ofthe-art performance for generating audio and image data (Sohl-Dickstein et al., 2015;

Original: Ho et al., 2020; Dhariwal & Nichol, 2021; Kingma et al., 2021; Song et al., 2021a;b; Vahdat et al., 2021); see, e.g., the recent surveys (Cao et al., 2022; Croitoru et al., 2022; Yang et al., 2022).
Revised: Ho et al., 2020; Dhariwal & Nichol, 2021; Kingma et al., 2021; Song et al., 2021a;b; Vahdat et al., 2021); see, e.g., the recent surveys (Cao et al., 2022; Croitoru et al., 2022; Yang et al., 2022).

Original: For example, denoising diffusion probabilistic models (DDPMs) (Sohl-Dickstein et al., 2015; Ho et al., 2020) are a key component in large-scale generative models such as DALL · E 2 (Ramesh et al., 2022).
Revised: For example, denoising diffusion probabilistic models (DDPMs) (Sohl-Dickstein et al., 2015; Ho et al., 2020) are a key component in large-scale generative models such as DALL · E 2 (Ramesh et al., 2022).

Original: As the importance of SGMs continues to grow due to newfound applications in commercial domains, it is a pressing question of both practical and theoretical concern to understand the mathematical underpinnings which explain their startling empirical successes.
Revised: As the importance of SGMs continues to grow due to newfound applications in commercial domains, it is a pressing question of both practical and theoretical concern to understand the mathematical underpinnings which explain their startling empirical successes.

Original: As we explain in Section 2, at their mathematical core, SGMs consist of two stochastic processes, the forward process and the reverse process.
Revised: As we explain in Section 2, at their mathematical core, SGMs consist of two stochastic processes, the forward process and the reverse process.

Original: The forward process transforms samples from a data distribution q (e.g., images) into noise, whereas the reverse process transforms noise into samples from q , hence performing generative modeling.
Revised: The forward process transforms samples from a data distribution q (e.g., images) into noise, whereas the reverse process transforms noise into samples from q , hence performing generative modeling.

Original: Running the reverse process requires estimating the score function of the law of the forward process; this is typically done by training neural networks on a score matching objective (Hyv¨arinen, 2005; Vincent, 2011; Song & Ermon, 2019).
Revised: Running the reverse process requires estimating the score function of the law of the forward process; this is typically done by training neural networks on a score matching objective (Hyv¨arinen, 2005; Vincent, 2011; Song & Ermon, 2019).

Original: Providing precise guarantees for estimation of the score function is difficult, as it requires an understanding of the non-convex training dynamics of neural network optimization that is currently out of reach.
Revised: Providing precise guarantees for estimation of the score function is difficult, as it requires an understanding of the non-convex training dynamics of neural network optimization that is currently out of reach.

Original: However, given the empirical success of neural networks on the score estimation task, a natural and important question is whether accurate score estimation implies that SGMs provably converge to the true data distribution in realistic settings.
Revised: However, given the empirical success of neural networks on the score estimation task, a natural and important question is whether accurate score estimation implies that SGMs provably converge to the true data distribution in realistic settings.

Original: This is a surprisingly delicate question, as even with accurate score estimates, as we explain in Section 2.1, there are several other sources of error which could cause the SGM to fail to converge.
Revised: This is a surprisingly delicate question, as even with accurate score estimates, as we explain in Section 2.1, there are several other sources of error which could cause the SGM to fail to converge.

Original: Indeed, despite a flurry of recent work (Block et al., 2020; De Bortoli et al., 2021; De Bortoli, 2022; Lee et al., 2022a; Pidstrigach, 2022; Liu et al., 2022), prior analyses fall short of answering this question, for (at least) one of three main reasons:
Revised: Indeed, despite a flurry of recent work (Block et al., 2020; De Bortoli et al., 2021; De Bortoli, 2022; Lee et al., 2022a; Pidstrigach, 2022; Liu et al., 2022), prior analyses fall short of answering this question, for (at least) one of three main reasons:

Original: Super-polynomial convergence.
Revised: Super-polynomial convergence.

Original: The bounds obtained are not quantitative (e g., De Bortoli 
Revised: The bounds obtained are not quantitative (e.g., De Bortoli et al., 2021; Pidstrigach, 2022), or scale exponentially in the dimension and other problem parameters like time and smoothness (Block et al., 2020; De Bortoli, 2022;

Original: et al Pidstrigach, , or scale exponentially in the dimension and other problem parameters like time and smoothness (Block et al., 2020; De Bortoli, 2022; Liu et al., 2022), and hence are typically vacuous for the high-dimensional settings of interest in practice.
Revised: Liu et al., 2022), and hence are typically vacuous for the high-dimensional settings of interest in practice.

Original: Strong assumptions on the data distribution.
Revised: Strong assumptions on the data distribution.

Original: The bounds require strong assumptions on the true data distribution, such as a log-Sobelev inequality (LSI) (see, e.g., Lee et al., 2022a).
Revised: The bounds require strong assumptions on the true data distribution, such as a log-Sobelev inequality (LSI) (see, e.g., Lee et al., 2022a).

Original: While the LSI is slightly weaker than strong log-concavity, it ultimately precludes the presence of substantial non-convexity, which impedes the application of these results to complex and highly multi-modal real-world data distributions.
Revised: While the LSI is slightly weaker than strong log-concavity, it ultimately precludes the presence of substantial non-convexity, which impedes the application of these results to complex and highly multi-modal real-world data distributions.

Original: Indeed, obtaining a polynomial-time convergence analysis for SGMs that holds for multi-modal distributions was posed as an open question in (Lee et al., 2022a).
Revised: Indeed, obtaining a polynomial-time convergence analysis for SGMs that holds for multi-modal distributions was posed as an open question in (Lee et al., 2022a).

Original: Strong assumptions on the score estimation error.
Revised: Strong assumptions on the score estimation error.

Original: The bounds require that the score estimate is L ∞ -accurate (i.e., uniformly accurate), as opposed to L 2 -accurate (see, e.g., De Bortoli et al., 2021).
Revised: The bounds require that the score estimate is L ∞ -accurate (i.e., uniformly accurate), as opposed to L 2 -accurate (see, e.g., De Bortoli et al., 2021).

Original: This is problematic because the score matching objective is an L 2 loss (see Section A.1 in the supplement), and there are empirical studies suggesting that in practice, the score estimate is not in fact L ∞ -accurate (e.g., Zhang & Chen, 2022).
Revised: This is problematic because the score matching objective is an L 2 loss (see Section A.1 in the supplement), and there are empirical studies suggesting that in practice, the score estimate is not in fact L ∞ -accurate (e.g., Zhang & Chen, 2022).

Original: Intuitively, this is because we cannot expect that the score estimate we obtain will be accurate in regions of space where the true density is very low, simply because we do not expect to see many (or indeed, any) samples from there.
Revised: Intuitively, this is because we cannot expect that the score estimate we obtain will be accurate in regions of space where the true density is very low, simply because we do not expect to see many (or indeed, any) samples from there.

Original: Providing an analysis which goes beyond these limitations is a pressing first step towards theoretically understanding why SGMs actually work in practice.
Revised: Providing an analysis which goes beyond these limitations is a pressing first step towards theoretically understanding why SGMs actually work in practice.

Original: 1. O UR CONTRIBUTIONS
Revised: 1. O UR CONTRIBUTIONS

Original: In this work, we take a step towards bridging theory and practice by providing a convergence guarantee for SGMs, under realistic (in fact, quite minimal) assumptions, which scales polynomially in all relevant problem parameters.
Revised: In this work, we take a step towards bridging theory and practice by providing a convergence guarantee for SGMs, under realistic (in fact, quite minimal) assumptions, which scales polynomially in all relevant problem parameters.

Original: Namely, our main result (Theorem 2) only requires the following assumptions on the data distribution q , which we make more quantitative in Section 3:
Revised: Namely, our main result (Theorem 2) only requires the following assumptions on the data distribution q , which we make more quantitative in Section 3:

Original: A

Revised: A

Original: The score function of the forward process is L -Lipschitz.
Revised: The score function of the forward process is L -Lipschitz.

Original: A2 The (2 + η ) -th moment of q is finite, where η > 0 is an arbitrarily small constant.
Revised: A2 The (2 + η ) -th moment of q is finite, where η > 0 is an arbitrarily small constant.

Original: A3 The data distribution q has finite KL divergence w.r.t.
Revised: A3 The data distribution q has finite KL divergence w.r.t.

Original: the standard Gaussian.
Revised: the standard Gaussian.

Original: We note that all of these assumptions are either standard or, in the case of A2 , far weaker than what is needed in prior work.
Revised: We note that all of these assumptions are either standard or, in the case of A2 , far weaker than what is needed in prior work.

Original: Crucially, unlike prior works, we do not assume log-concavity, an LSI, or dissipativity; hence, our assumptions cover arbitrarily non-log-concave data distributions.
Revised: Crucially, unlike prior works, we do not assume log-concavity, an LSI, or dissipativity; hence, our assumptions cover arbitrarily non-log-concave data distributions.

Original: Our main result is summarized informally as follows.
Revised: Our main result is summarized informally as follows.

Original: Theorem 1 (informal, see Theorem 2) .
Revised: Theorem 1 (informal, see Theorem 2) .

Original: Under assumptions A1-A3 , and if the score estimation error in L 2 is at most (cid:101) O ( ε ) , then with an appropriate choice of step size, the SGM outputs a measure which is ε -close in total variation (TV) distance to q in (cid:101) O
Revised: Under assumptions A1-A3 , and if the score estimation error in L 2 is at most (cid:101) O ( ε ) , then with an appropriate choice of step size, the SGM outputs a measure which is ε -close in total variation (TV) distance to q in (cid:101) O

Original: ( L 2 d/ε 2 ) iterations.
Revised: ( L 2 d/ε 2 ) iterations.

Original: Our iteration complexity is quite tight: it matches state-of-the-art discretization guarantees for the
Revised: Our iteration complexity is quite tight: it matches state-of-the-art discretization guarantees for the

Original: Langevin diffusion (Vempala & Wibisono, 2019; Chewi et al., 2021a).
Revised: Langevin diffusion (Vempala & Wibisono, 2019; Chewi et al., 2021a).

Original: We find Theorem 1 surprising, because it shows that SGMs can sample from the data distribution q with polynomial complexity, even when q is highly non-log-concave (a task that is usually intractable), provided that one has access to an accurate score estimator .
Revised: We find Theorem 1 surprising, because it shows that SGMs can sample from the data distribution q with polynomial complexity, even when q is highly non-log-concave (a task that is usually intractable), provided that one has access to an accurate score estimator .

Original: This answers the open question of (Lee et al., 2022a) regarding whether or not SGMs can sample from multimodal distributions, e.g., mixtures of distributions with bounded log-Sobolev constant.
Revised: This answers the open question of (Lee et al., 2022a) regarding whether or not SGMs can sample from multimodal distributions, e.g., mixtures of distributions with bounded log-Sobolev constant.

Original: In the context of neural networks, our result implies that so long as the neural network succeeds at the score estimation task, the remaining part of the SGM algorithm based on the diffusion model is completely principled, in that it admits a strong theoretical justification.
Revised: In the context of neural networks, our result implies that so long as the neural network succeeds at the score estimation task, the remaining part of the SGM algorithm based on the diffusion model is completely principled, in that it admits a strong theoretical justification.

Original: In general, learning the score function is also a difficult task.
Revised: In general, learning the score function is also a difficult task.

Original: Nevertheless, our result opens the door to further investigations, such as: do score functions for real-life data have intrinsic (e.g., low-dimensional) structure which can be exploited by neural networks?
Revised: Nevertheless, our result opens the door to further investigations, such as: do score functions for real-life data have intrinsic (e.g., low-dimensional) structure which can be exploited by neural networks?

Original: A positive answer to this question, combined with our sampling result, would then provide an end-to-end guarantee for SGMs.
Revised: A positive answer to this question, combined with our sampling result, would then provide an end-to-end guarantee for SGMs.

Original: More generally, our result can be viewed as a black-box reduction of the task of sampling to the task of learning the score function of the forward process, at least for distributions satisfying our mild assumptions.
Revised: More generally, our result can be viewed as a black-box reduction of the task of sampling to the task of learning the score function of the forward process, at least for distributions satisfying our mild assumptions.

Original: Existing computational hardness results for learning natural high-dimensional distributions like mixtures of Gaussians (Diakonikolas et al., 2017; Bruna et al., 2021; Gupte et al.,  and pushforwards of Gaussians by shallow ReLU networks (Daniely & Vardi, 2021; Chen et al., 2022a;b) thus immediately imply hardness of score estimation for these distributions.
Revised: Existing computational hardness results for learning natural high-dimensional distributions like mixtures of Gaussians (Diakonikolas et al., 2017; Bruna et al., 2021; Gupte et al., 2022) and pushforwards of Gaussians by shallow ReLU networks (Daniely & Vardi, 2021; Chen et al., 2022a;b) thus immediately imply hardness of score estimation for these distributions.

Original: To our knowledge this yields the first known information-computation gaps for this task.
Revised: To our knowledge this yields the first known information-computation gaps for this task.

Original: Arbitrary distributions with bounded support.
Revised: Arbitrary distributions with bounded support.

Original: The assumption that the score function is Lipschitz entails in particular that the data distribution has a density w.r.t.
Revised: The assumption that the score function is Lipschitz entails in particular that the data distribution has a density w.r.t.

Original: Lebesgue measure; in particular, our theorem fails when q satisfies the manifold hypothesis, i.e., is supported on a lower-dimensional submanifold of R d .
Revised: Lebesgue measure; in particular, our theorem fails when q satisfies the manifold hypothesis, i.e., is supported on a lower-dimensional submanifold of R d .

Original: But this is for good reason: it is not possible to obtain non-trivial TV guarantees, because the output distribution of the SGM has full support.
Revised: But this is for good reason: it is not possible to obtain non-trivial TV guarantees, because the output distribution of the SGM has full support.

Original: Instead, we show in Section 3.2 that we can obtain polynomial convergence guarantees in the bounded Lipschitz metric by stopping the
Revised: Instead, we show in Section 3.2 that we can obtain polynomial convergence guarantees in the bounded Lipschitz metric by stopping the

Original: SGM algorithm early, or in the Wasserstein metric by an additional truncation step, under the sole assumption that the data distribution q has bounded support, without assuming that q has a density.
Revised: SGM algorithm early, or in the Wasserstein metric by an additional truncation step, under the sole assumption that the data distribution q has bounded support, without assuming that q has a density.

Original: Since data distributions encountered in real life satisfy this assumption, our results yield the following compelling takeaway:
Revised: Since data distributions encountered in real life satisfy this assumption, our results yield the following compelling takeaway:

Original: Given an L 2 -accurate score estimate, SGMs can sample from (essentially) any data distribution .
Revised: Given an L 2 -accurate score estimate, SGMs can sample from (essentially) any data distribution .

Original: Critically damped Langevin diffusion (CLD).
Revised: Critically damped Langevin diffusion (CLD).

Original: Using our techniques, we also investigate the use of the critically damped Langevin diffusion (CLD) for SGMs, which was proposed in (Dockhorn et al., 2022).
Revised: Using our techniques, we also investigate the use of the critically damped Langevin diffusion (CLD) for SGMs, which was proposed in (Dockhorn et al., 2022).

Original: Although numerical experiments and intuition from the log-concave sampling literature suggest that the CLD could potentially speed up sampling via SGMs, we provide theoretical evidence to the contrary.
Revised: Although numerical experiments and intuition from the log-concave sampling literature suggest that the CLD could potentially speed up sampling via SGMs, we provide theoretical evidence to the contrary.

Original: Based on this, in Section 3.3, we conjecture that SGMs based on the CLD do not exhibit improved dimension dependence compared to the original DDPM algorithm.
Revised: Based on this, in Section 3.3, we conjecture that SGMs based on the CLD do not exhibit improved dimension dependence compared to the original DDPM algorithm.

Original: 1. P RIOR WORK
Revised: 1. P RIOR WORK

Original: We now provide a detailed comparison to prior work.
Revised: We now provide a detailed comparison to prior work.

Original: By now, there is a vast literature on providing precise complexity estimates for log-concave sampling; see, e.g., Chewi (2022) for an exposition on recent developments.
Revised: By now, there is a vast literature on providing precise complexity estimates for log-concave sampling; see, e.g., Chewi (2022) for an exposition on recent developments.

Original: The proofs in this work build upon the techniques developed in this literature.
Revised: The proofs in this work build upon the techniques developed in this literature.

Original: However, our work addresses the significantly more challenging setting of non-log-concave sampling.
Revised: However, our work addresses the significantly more challenging setting of non-log-concave sampling.

Original: The work of De Bortoli et al.
Revised: The work of De Bortoli et al.

Original: (2021) provides guarantees for the diffusion Schr¨odinger bridge (Song et al., 2021b).
Revised: (2021) provides guarantees for the diffusion Schr ¨ odinger bridge (Song et al., 2021b).

Original: However, as previously mentioned their result is not quantitative, and they require an
Revised: However, as previously mentioned their result is not quantitative, and they require an

Original: L ∞ -accurate score estimate.
Revised: L ∞ -accurate score estimate.

Original: The works Block et al.
Revised: The works Block et al.

Original: ; Lee et al.
Revised: ; Lee et al.

Original: (2022a);
Revised: (2022a);

Original: Liu et al.
Revised: Liu et al.

Original: (2022) instead analyze SGMs under the more realistic assumption of an L 2 -accurate score estimate.
Revised: (2022) instead analyze SGMs under the more realistic assumption of an L 2 -accurate score estimate.

Original: However, the bounds of Block et al.
Revised: However, the bounds of Block et al.

Original: ; Liu et al.
Revised: ; Liu et al.

Original: (2022) suffer from exponential dependencies on parameters like dimension and smoothness, whereas the bounds of Lee et al.
Revised: (2022) suffer from exponential dependencies on parameters like dimension and smoothness, whereas the bounds of Lee et al.

Original: (2022a) require q to satisfy an LSI.
Revised: (2022a) require q to satisfy an LSI.

Original: The recent work of De Bortoli (2022), motivated by the manifold hypothesis , considers a different pointwise assumption on the score estimation error which allows the error to blow up at time 0 and at spatial ∞ .
Revised: The recent work of De Bortoli (2022), motivated by the manifold hypothesis , considers a different pointwise assumption on the score estimation error which allows the error to blow up at time 0 and at spatial ∞ .

Original: We discuss the manifold setting in more detail in Section 3.2.
Revised: We discuss the manifold setting in more detail in Section 3.2.

Original: Unfortunately, the bounds of De Bortoli (2022) also scale exponentially in problem parameters such as the manifold diameter.
Revised: Unfortunately, the bounds of De Bortoli (2022) also scale exponentially in problem parameters such as the manifold diameter.

Original: We also mention that the use of reversed SDEs for sampling is implicit in the interpretation of the proximal sampler (Lee et al., 2021) given by Chen et al.
Revised: We also mention that the use of reversed SDEs for sampling is implicit in the interpretation of the proximal sampler (Lee et al., 2021) given by Chen et al.

Original: (2022c).
Revised: (2022c).

Original: Our work can be viewed as expanding upon the theory of Chen et al.
Revised: Our work can be viewed as expanding upon the theory of Chen et al.

Original: (2022c)
Revised: (2022c)

Original: using a different forward channel (the OU process).
Revised: using a different forward channel (the OU process).

Original: Concurrent work.
Revised: Concurrent work.

Original: Very recently, Lee et al.
Revised: Very recently, Lee et al.

Original: (2022b) independently obtained results similar to our results for DDPM.
Revised: (2022b) independently obtained results similar to our results for DDPM.

Original: While our assumptions are technically somewhat incomparable (they assume the score error can vary with time but assume the data is compactly supported), our quantitative bounds are stronger.
Revised: While our assumptions are technically somewhat incomparable (they assume the score error can vary with time but assume the data is compactly supported), our quantitative bounds are stronger.

Original: Additionally, the upper and lower bounds for CLD are unique to our work.
Revised: Additionally, the upper and lower bounds for CLD are unique to our work.

Original:  B ACKGROUND ON SGM S
Revised:  B ACKGROUND ON SGM S

Original: Throughout this paper, given a probability measure p which admits a density w.r.t.
Revised: Throughout this paper, given a probability measure p which admits a density w.r.t.

Original: Lebesgue measure, we abuse notation and identify it with its density function.
Revised: Lebesgue measure, we abuse notation and identify it with its density function.

Original: Additionally, we will let q denote the data distribution from which we want to generate new samples.
Revised: Additionally, we will let q denote the data distribution from which we want to generate new samples.

Original: We assume that q is a probability measure on R d with full support, and that it admits a smooth density written q = exp( − U ) (we relax this assumption in Section 3.2).
Revised: We assume that q is a probability measure on R d with full support, and that it admits a smooth density written q = exp( − U ) (we relax this assumption in Section 3.2).

Original: In this section, we provide a brief exposition to SGMs, following Song et al.
Revised: In this section, we provide a brief exposition to SGMs, following Song et al.

Original: (2021b).
Revised: (2021b).

Original: 2. B ACKGROUND ON DENOISING DIFFUSION PROBABILISTIC MODELING (DDPM)
Revised: 2. B ACKGROUND ON DENOISING DIFFUSION PROBABILISTIC MODELING (DDPM)

Original: Forward process.
Revised: Forward process.

Original: In denoising diffusion probabilistic modeling (DDPM), we start with a forward process, which is a stochastic differential equation (SDE).
Revised: In denoising diffusion probabilistic modeling (DDPM), we start with a forward process, which is a stochastic differential equation (SDE).

Original: For clarity, we consider the simplest possible choice, which is the Ornstein–Uhlenbeck (OU) process
Revised: For clarity, we consider the simplest possible choice, which is the Ornstein–Uhlenbeck (OU) process

Original: [Equation]
Revised: [Equation]

Original: where ( B t ) t ≥ 0 is a standard Brownian motion in R d .
Revised: where ( B t ) t ≥ 0 is a standard Brownian motion in R d .

Original: The OU process is the unique time-homogeneous
Revised: The OU process is the unique time-homogeneous

Original: Markov process which is also a Gaussian process, with stationary distribution equal to the standard
Revised: Markov process which is also a Gaussian process, with stationary distribution equal to the standard

Original: Gaussian distribution γ d on R d .
Revised: Gaussian distribution γ d on R d .

Original: In practice, it is also common to introduce a positive smooth function g : R + → R and consider the time-rescaled OU process
Revised: In practice, it is also common to introduce a positive smooth function g : R + → R and consider the time-rescaled OU process

Original: [Equation]
Revised: [Equation]

Original: Although our analysis could be extended to consider these variants, in this work we stick with the choice g ≡ 1 for simplicity; see Song et al.
Revised: Although our analysis could be extended to consider these variants, in this work we stick with the choice g ≡ 1 for simplicity; see Song et al.

Original: (2021b)
Revised: (2021b)

Original: for further discussion.
Revised: for further discussion.

Original: The forward process has the interpretation of transforming samples from the data distribution q into pure noise.
Revised: The forward process has the interpretation of transforming samples from the data distribution q into pure noise.

Original: From the well-developed theory of Markov diffusions, it is known that if q t := law( X t ) denotes the law of the OU process at time t , then q t → γ d exponentially fast in various divergences and metrics such as the 2 -Wasserstein metric W 2 ; see Bakry et al.
Revised: From the well-developed theory of Markov diffusions, it is known that if q t := law( X t ) denotes the law of the OU process at time t , then q t → γ d exponentially fast in various divergences and metrics such as the 2 -Wasserstein metric W 2 ; see Bakry et al.

Original: Reverse process.
Revised: Reverse process.

Original: If we reverse the forward process (1) in time, then we obtain a process that transforms noise into samples from q , which is the aim of generative modeling.
Revised: If we reverse the forward process (1) in time, then we obtain a process that transforms noise into samples from q , which is the aim of generative modeling.

Original: In general, suppose that we have an SDE of the form
Revised: In general, suppose that we have an SDE of the form

Original: [Equation]
Revised: [Equation]

Original: where ( σ t ) t ≥ 0 is a deterministic matrix-valued process.
Revised: where ( σ t ) t ≥ 0 is a deterministic matrix-valued process.

Original: Then, under mild conditions on the process (e.g., F ¨ ollmer, 1985; Cattiaux et al., 2022), which are satisfied for all processes under consideration in this work, the reverse process also admits an SDE description.
Revised: Then, under mild conditions on the process (e.g., F ¨ ollmer, 1985; Cattiaux et al., 2022), which are satisfied for all processes under consideration in this work, the reverse process also admits an SDE description.

Original: Namely, if we fix the terminal time T > 0 and set
Revised: Namely, if we fix the terminal time T > 0 and set

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: then the process ( ¯ X ← t ) t ∈ [0 ,T ] satisfies the SDE
Revised: then the process ( ¯ X ← t ) t ∈ [0 ,T ] satisfies the SDE

Original: [Equation]
Revised: [Equation]

Original: where the backwards drift satisfies the relation
Revised: where the backwards drift satisfies the relation

Original: [Equation]
Revised: [Equation]

Original: Applying this to the forward process (1), we obtain the reverse process
Revised: Applying this to the forward process (1), we obtain the reverse process

Original: [Equation]
Revised: [Equation]

Original: where now ( B t ) t ∈ [0 ,T ] is the reversed Brownian motion.
Revised: where now ( B t ) t ∈ [0 ,T ] is the reversed Brownian motion.

Original: Here, ∇ ln q t is called the score function for q t .
Revised: Here, ∇ ln q t is called the score function for q t .

Original: Since q (and hence q t for t ≥ 0 ) is not explicitly known, in order to implement the reverse process the score function must be estimated on the basis of samples.
Revised: Since q (and hence q t for t ≥ 0 ) is not explicitly known, in order to implement the reverse process the score function must be estimated on the basis of samples.

Original: The mechanism behind this is the idea of score matching which goes back to Hyv¨arinen (2005); Vincent (2011): roughly speaking,
Revised: The mechanism behind this is the idea of score matching which goes back to Hyv¨arinen (2005); Vincent (2011): roughly speaking,

Original: Gaussian integration by parts implies that minimizing the L 2 ( q t ) loss achieved by an estimate s t for the score ∇ ln q t is equivalent to minimizing the L 2 ( q t ) loss in predicting, given a sample from the forward process at time t , what noise was applied to the corresponding sample at time 0 to obtain it.
Revised: Gaussian integration by parts implies that minimizing the L 2 ( q t ) loss achieved by an estimate s t for the score ∇ ln q t is equivalent to minimizing the L 2 ( q t ) loss in predicting, given a sample from the forward process at time t , what noise was applied to the corresponding sample at time 0 to obtain it.

Original: We defer an exposition of the details of score matching to Sections A.1 and D of the supplement.
Revised: We defer an exposition of the details of score matching to Sections A.1 and D of the supplement.

Original: In light of this, it is thus most natural to assume an L 2 ( q t ) error bound E q [Equation] t ∥ 2 ] ≤ ε 2score for the score estimator s t .
Revised: In light of this, it is thus most natural to assume an L 2 ( q t ) error bound E q t [ ∥ s t − ∇ ln q t ∥ 2 ] ≤ ε 2score for the score estimator s t .

Original: If s t is taken to be the empirical risk minimizer for a suitable function class, then guarantees for the L 2 ( q t ) error can be obtained via standard statistical analysis, see, e.g., Block et al.
Revised: If s t is taken to be the empirical risk minimizer for a suitable function class, then guarantees for the L 2 ( q t ) error can be obtained via standard statistical analysis, see, e.g., Block et al.

Original: Discretization and implementation.
Revised: Discretization and implementation.

Original: We now discuss the final steps required to obtain an implementable algorithm.
Revised: We now discuss the final steps required to obtain an implementable algorithm.

Original: First, in the learning phase, given samples ¯ X (1)0 , . . .
Revised: First, in the learning phase, given samples ¯ X (1)0 , . . .

Original: , ¯ X ( n )from q (e.g., a database of natural images), we train a neural network via score matching, see Song & Ermon (2019).
Revised: , ¯ X ( n )from q (e.g., a database of natural images), we train a neural network via score matching, see Song & Ermon (2019).

Original: Let h > 0 be the step size of the discretization; we assume that we have obtained a score estimate s kh of ∇ ln q kh for each time k = 0 , 1 , . . .
Revised: Let h > 0 be the step size of the discretization; we assume that we have obtained a score estimate s kh of ∇ ln q kh for each time k = 0 , 1 , . . .

Original: , N ,
Revised: , N ,

Original: where T =
Revised: where T =

Original: Nh .
Revised: Nh .

Original: In order to approximately implement the reverse SDE (4), we first replace the score function ∇ ln q T − t with the estimate s T − t .
Revised: In order to approximately implement the reverse SDE (4), we first replace the score function ∇ ln q T − t with the estimate s T − t .

Original: Then, for t ∈ [ kh, ( k + 1) h ] we freeze the value of this coefficient in the
Revised: Then, for t ∈ [ kh, ( k + 1) h ] we freeze the value of this coefficient in the

Original: SDE at time kh .
Revised: SDE at time kh .

Original: It yields the new SDE
Revised: It yields the new SDE

Original: [Equation]
Revised: [Equation]

Original: Since this is a linear SDE, it can be integrated in closed form; in particular, conditionally on X ← kh , the next iterate X ← ( k +1)
Revised: Since this is a linear SDE, it can be integrated in closed form; in particular, conditionally on X ← kh , the next iterate X ← ( k +1)

Original: h has an explicit Gaussian distribution.
Revised: h has an explicit Gaussian distribution.

Original: There is one final detail: although the reverse SDE (4) should be started at q T , we do not have access to q T directly.
Revised: There is one final detail: although the reverse SDE (4) should be started at q T , we do not have access to q T directly.

Original: Instead, taking advantage of the fact that q T ≈ γ d , we instead initialize the algorithm at X ← 0 ∼ γ d , i.e., from pure noise.
Revised: Instead, taking advantage of the fact that q T ≈ γ d , we instead initialize the algorithm at X ← 0 ∼ γ d , i.e., from pure noise.

Original: Let p t := law( X ← t ) denote the law of the algorithm at time t .
Revised: Let p t := law( X ← t ) denote the law of the algorithm at time t .

Original: The goal of this work is to bound
Revised: The goal of this work is to bound

Original: TV ( p T , q ) , taking into account three sources of error: (1) estimation of the score; (2) discretization of the SDE with step size h > 0 ; and (3) initialization of the algorithm at γ d rather than at q T .
Revised: TV ( p T , q ) , taking into account three sources of error: (1) estimation of the score; (2) discretization of the SDE with step size h > 0 ; and (3) initialization of the algorithm at γ d rather than at q T .

Original:  B ACKGROUND ON THE CRITICALLY DAMPED L ANGEVIN DIFFUSION (CLD)
Revised: 2. B ACKGROUND ON THE CRITICALLY DAMPED L ANGEVIN DIFFUSION (CLD)

Original: The critically damped Langevin diffusion (CLD) is based on the forward process
Revised: The critically damped Langevin diffusion (CLD) is based on the forward process

Original: [Equation]
Revised: [Equation]

Original:  Compared to the OU process (1), this is now a coupled system of SDEs, where we have introduced a new variable ¯ V representing the velocity process.
Revised: [Equation] Compared to the OU process (1), this is now a coupled system of SDEs, where we have introduced a new variable ¯ V representing the velocity process.

Original: The stationary distribution of the process is γ 2 d , the standard Gaussian measure on phase space R d × R d , and we initialize at ¯ X 0 ∼ q and ¯ V 0 ∼ γ d .
Revised: The stationary distribution of the process is γ 2 d , the standard Gaussian measure on phase space R d × R d , and we initialize at ¯ X 0 ∼ q and ¯ V 0 ∼ γ d .

Original: More generally, the CLD (6) is an instance of what is referred to as the kinetic Langevin or the underdamped Langevin process in the sampling literature.
Revised: More generally, the CLD (6) is an instance of what is referred to as the kinetic Langevin or the underdamped Langevin process in the sampling literature.

Original: In the context of strongly log-concave sampling, the smoother paths of ¯ X lead to smaller discretization error, thereby furnishing an algorithm with (cid:101) O ( √ d/ε ) gradient complexity (as opposed to sampling based on the overdamped Langevin process, which has complexity (cid:101) O
Revised: In the context of strongly log-concave sampling, the smoother paths of ¯ X lead to smaller discretization error, thereby furnishing an algorithm with (cid:101) O ( √ d/ε ) gradient complexity (as opposed to sampling based on the overdamped Langevin process, which has complexity (cid:101) O

Original: ( d/ε 2 ) ), see Cheng et al.
Revised: ( d/ε 2 ) ), see Cheng et al.

Original: ; Shen & Lee (2019); Dalalyan &
Revised: ; Shen & Lee (2019); Dalalyan &

Original: Riou-Durand (2020); Ma et al. (2021).
Revised: Riou-Durand (2020); Ma et al. (2021).

Original: The recent paper Dockhorn et al.
Revised: The recent paper Dockhorn et al.

Original: (2022) proposed to use the
Revised: (2022) proposed to use the

Original: CLD as the basis for an SGM and they empirically observed improvements over DDPM.
Revised: CLD as the basis for an SGM and they empirically observed improvements over DDPM.

Original: Applying (3), the corresponding reverse process is
Revised: Applying (3), the corresponding reverse process is

Original: [Equation]
Revised: [Equation]

Original: where q t := law( ¯ X t , ¯ V t ) is the law of the forward process at time t .
Revised: where q t := law( ¯ X t , ¯ V t ) is the law of the forward process at time t .

Original: Note that the gradient in the score function is only taken w.r.t.
Revised: Note that the gradient in the score function is only taken w.r.t.

Original: the velocity coordinate.
Revised: the velocity coordinate.

Original: Upon replacing the score function with an estimate s , we arrive at the algorithm
Revised: Upon replacing the score function with an estimate s , we arrive at the algorithm

Original: [Equation]
Revised: [Equation]

Original: X ← t + 2 V ← t + 4 s T − kh

Original: ( X ← kh , V ← kh ) (cid:1)

Original: d t + 2 d B t , for t ∈ [ kh, ( k + 1) h ] .
Revised: for t ∈ [ kh, ( k + 1) h ] .

Original: We provide further background on the CLD in Section C.1.
Revised: We provide further background on the CLD in Section C.1.

Original:  R ESULTS
Revised:  R ESULTS

Original: We now formally state our assumptions and our main results.
Revised: We now formally state our assumptions and our main results.

Original: 3. R ESULTS FOR DDPM
Revised: 3. R ESULTS FOR DDPM

Original: For DDPM, we make the following mild assumptions on the data distribution q .
Revised: For DDPM, we make the following mild assumptions on the data distribution q .

Original: Assumption 1 (Lipschitz score) .
Revised: Assumption 1 (Lipschitz score) .

Original: For all t ≥ 0 , the score ∇ ln q t is L -Lipschitz.
Revised: For all t ≥ 0 , the score ∇ ln q t is L -Lipschitz.

Original: Assumption 2 (second moment bound) .
Revised: Assumption 2 (second moment bound) .

Original: For some η > 0 , E q [ ∥·∥ 2+ η ] is finite.
Revised: For some η > 0 , E q [ ∥·∥ 2+ η ] is finite.

Original: We also write m 22 :=
Revised: We also write m 22 :=

Original: E q [ ∥·∥ 2 ] for the second moment of q .
Revised: E q [ ∥·∥ 2 ] for the second moment of q .

Original: For technical reasons, we need to assume that q has a finite moment of order slightly but strictly bigger than 2 , but our quantitative bounds will only depend on the second moment m 22 .
Revised: For technical reasons, we need to assume that q has a finite moment of order slightly but strictly bigger than 2 , but our quantitative bounds will only depend on the second moment m 22 .

Original: Assumption 1 is standard and has been used in the prior works Block et al.
Revised: Assumption 1 is standard and has been used in the prior works Block et al.

Original: ; Lee et al.
Revised: ; Lee et al.

Original: (2022a).
Revised: (2022a).

Original: However, unlike Lee et al. (2022a); Liu et al.
Revised: However, unlike Lee et al. (2022a); Liu et al.

Original: , we do not assume Lipschitzness of the score estimate.
Revised: , we do not assume Lipschitzness of the score estimate.

Original: Moreover, unlike Block et al.
Revised: Moreover, unlike Block et al.

Original: ; De Bortoli et al.
Revised: ; De Bortoli et al.

Original: Liu et al.
Revised: Liu et al.

Original: (2022), we do not assume any convexity or dissipativity assumptions on the potential U , and unlike Lee et al.
Revised: (2022), we do not assume any convexity or dissipativity assumptions on the potential U , and unlike Lee et al.

Original: (2022a)
Revised: (2022a)

Original: we do not assume that q satisfies a log-Sobolev inequality.
Revised: we do not assume that q satisfies a log-Sobolev inequality.

Original: Hence, our assumptions cover a wide range of highly non-log-concave data distributions.
Revised: Hence, our assumptions cover a wide range of highly non-log-concave data distributions.

Original: Our proof technique is fairly robust and even
Revised: Our proof technique is fairly robust and even

Original: Assumption 1 could be relaxed (as well as other extensions, such as considering the time-changed forward process (2)), although we focus on the simplest setting in order to better illustrate the conceptual significance of our results.
Revised: Assumption 1 could be relaxed (as well as other extensions, such as considering the time-changed forward process (2)), although we focus on the simplest setting in order to better illustrate the conceptual significance of our results.

Original: We also assume a bound on the score estimation error.
Revised: We also assume a bound on the score estimation error.

Original: Assumption 3 (score estimation error) .
Revised: Assumption 3 (score estimation error) .

Original: For all 
Revised: For all k = 1 , . . . , N , E q kh [Equation] s

Original: [Equation]
Revised: [Equation]

Revised: ∇

Revised: [Equation]

Revised: q

Revised: [Equation]

Revised: ∥ 2 ] ≤ ε 2score .

Original: This is the same assumption as in Lee et al.
Revised: This is the same assumption as in Lee et al.

Original: (2022a), and as discussed in Section 2.1, it is a natural and realistic assumption in light of the derivation of the score matching objective.
Revised: (2022a), and as discussed in Section 2.1, it is a natural and realistic assumption in light of the derivation of the score matching objective.

Original: Our main result for DDPM is the following theorem.
Revised: Our main result for DDPM is the following theorem.

Original: Theorem 2 (DDPM, see Section B of supplement) .
Revised: Theorem 2 (DDPM, see Section B of supplement) .

Original: Suppose that Assumptions 1, 2, and 3 hold.
Revised: Suppose that Assumptions 1, 2, and 3 hold.

Original: Let p T be the output of the DDPM algorithm (Section 2.1) at time T , and suppose that the step size h :
Revised: Let p T be the output of the DDPM algorithm (Section 2.1) at time T , and suppose that the step size h :

Original: = T/N satisfies h ≲ 1 /L , where L ≥ 1 .
Revised: = T/N satisfies h ≲ 1 /L , where L ≥ 1 .

Original: Then, it holds that
Revised: Then, it holds that

Original: [Equation]
Revised: [Equation]

Original: To interpret this result, suppose that, e.g., KL ( q ∥ γ d ) ≤ poly( d ) and m 2 ≤ d . 2 Choosing T ≍ log( KL ( q ∥ γ ) /ε ) and h ≍ ε 2 L 2 d , and hiding logarithmic factors,
Revised: To interpret this result, suppose that, e.g., KL ( q ∥ γ d ) ≤ poly( d ) and m 2 ≤ d . 2 Choosing T ≍ log( KL ( q ∥ γ ) /ε ) and h ≍ ε 2 L 2 d , and hiding logarithmic factors,

Original: [Equation]
Revised: [Equation]

Original: In particular, in order to have TV ( p T , q ) ≤ ε , it suffices to have score error ε score ≤ (cid:101) O
Revised: In particular, in order to have TV ( p T , q ) ≤ ε , it suffices to have score error ε score ≤ (cid:101) O

Original: ( ε ) .
Revised: ( ε ) .

Original: We remark that the iteration complexity of N = (cid:101)
Revised: We remark that the iteration complexity of N = (cid:101)

Original: Θ( L 2 d ε 2 ) matches state-of-the-art complexity bounds for the Langevin Monte Carlo (LMC) algorithm for sampling under a log-Sobolev inequality (LSI), see Vempala & Wibisono (2019);
Revised: Θ( L 2 d ε 2 ) matches state-of-the-art complexity bounds for the Langevin Monte Carlo (LMC) algorithm for sampling under a log-Sobolev inequality (LSI), see Vempala & Wibisono (2019);

Original: Chewi et al.
Revised: Chewi et al.

Original: (2021a).
Revised: (2021a).

Original: This provides some evidence that our discretization bounds are of the correct order, at least with respect to the dimension and accuracy parameters, and without higher-order smoothness assumptions.
Revised: This provides some evidence that our discretization bounds are of the correct order, at least with respect to the dimension and accuracy parameters, and without higher-order smoothness assumptions.

Original: 3. C ONSEQUENCES FORARBITRARY DATA DISTRIBUTIONS WITH BOUNDED SUPPORT
Revised: 3. C ONSEQUENCES FOR

Revised: DATA BOUNDED SUPPORT

Original: We now elaborate upon the implications of our results under the sole assumption that the data distribution q is compactly supported, supp q ⊆ B (0 , R ) .
Revised: We now elaborate upon the implications of our results under the sole assumption that the data distribution q is compactly supported, supp q ⊆ B (0 , R ) .

Original: In particular, we do not assume that q has a  smooth density w.r.t.
Revised: In particular, we do not assume that q has a 2 For many distributions of interest, e.g., the standard Gaussian distribution or product measures, in fact we have m 2 = O ( √ d ) . Also, for applications to images in which each coordinate (i.e., pixel) lies in a bounded range [ − 1 , 1] , we also have m 2 ≤ √ d . smooth density w.r.t.

Original: Lebesgue measure, which allows for studying the case when q is supported on a lower-dimensional submanifold of R d as in the manifold hypothesis .
Revised: Lebesgue measure, which allows for studying the case when q is supported on a lower-dimensional submanifold of R d as in the manifold hypothesis .

Original: This setting was investigated recently in De Bortoli (2022).
Revised: This setting was investigated recently in De Bortoli (2022).

Original: For this setting, our results do not apply directly because the score function of q is not well-defined and hence Assumption 1 fails to hold.
Revised: For this setting, our results do not apply directly because the score function of q is not well-defined and hence Assumption 1 fails to hold.

Original: Also, the bound in Theorem 2 has a term involving KL ( q ∥ γ d ) which is infinite if q is not absolutely continuous w.r.t.
Revised: Also, the bound in Theorem 2 has a term involving KL ( q ∥ γ d ) which is infinite if q is not absolutely continuous w.r.t.

Original: γ d .
Revised: γ d .

Original: As pointed out by De Bortoli (2022), in general we cannot obtain non-trivial guarantees for TV ( p T , q ) , because p T has full support and therefore TV ( p T , q )
Revised: As pointed out by De Bortoli (2022), in general we cannot obtain non-trivial guarantees for TV ( p T , q ) , because p T has full support and therefore TV ( p T , q )

Original: = 1 under the manifold hypothesis.
Revised: = 1 under the manifold hypothesis.

Original: Nevertheless, we show that we can apply our results using an early stopping technique.
Revised: Nevertheless, we show that we can apply our results using an early stopping technique.

Original: Namely, using the following lemma, we obtain a sequence of corollaries.
Revised: Namely, using the following lemma, we obtain a sequence of corollaries.

Original: Lemma 3 (Lemma 21 in supplement) .
Revised: Lemma 3 (Lemma 21 in supplement) .

Original: Suppose that supp q ⊆ B (0 , R ) where R ≥ 1 , and let q t denote the law of the OU process at time t , started at q .
Revised: Suppose that supp q ⊆ B (0 , R ) where R ≥ 1 , and let q t denote the law of the OU process at time t , started at q .

Original: Let ε
Revised: Let ε

Original: W 2 > 0 be such that ε W 2 ≪ √ d and set t ≍ ε 2 W 2 / ( √ d
Revised: W 2 > 0 be such that ε W 2 ≪ √ d and set t ≍ ε [Equation]

Original:  ( R ∨ √ d )) .
Revised: / ( √ d ( R ∨ √ d )) .

Original: Then, (1) W 2 ( q t , q ) ≤ ε W 2 , (2) q t satisfies KL ( q t ∥ γ d ) ≲ √ d ( R ∨√ d ) 3 ε 2 W, and (3) for every t ′ ≥ t , q t ′ satisfies Assumption 1 with L ≲ dR 2 ( R ∨√ d ) 2 ε 4 W.
Revised: Then, (1) W 2 ( q t , q ) ≤ ε W 2 , (2) q t satisfies KL ( [Equation] , and (3) for every t ′ ≥ t , q t ′ satisfies Assumption 1 with L ≲ dR 2 ( R ∨√ d ) [Equation]

Original: By substituting q t for this choice of t in place of q in Theorem 2, we obtain Corollary 4 below.
Revised: By substituting q t for this choice of t in place of q in Theorem 2, we obtain Corollary 4 below.

Original: We remark that taking q t as the new target corresponds to stopping the algorithm early: instead of running the algorithm backward for a time T , we run the algorithm backward for a time T − t
Revised: We remark that taking q t as the new target corresponds to stopping the algorithm early: instead of running the algorithm backward for a time T , we run the algorithm backward for a time T − t

Original: (note that T − t should be a multiple of the step size h ).
Revised: (note that T − t should be a multiple of the step size h ).

Original: Corollary 4 (compactly supported data) .
Revised: Corollary 4 (compactly supported data) .

Original: Suppose that q is supported on the ball of radius R ≥ 1 .
Revised: Suppose that q is supported on the ball of radius R ≥ 1 .

Original: Let t ≍ ε 2 W 2 / ( √ d ( R ∨ √ d )) .
Revised: Let t ≍ ε 2 W 2 / ( √ d ( R ∨ √ d )) .

Original: Then, the output p T − t of DDPM is ε TV -close in TV to the distribution q t , which is ε W 2 -close in W 2 to q , provided that the step size h is chosen appropriately according to
Revised: Then, the output p T − t of DDPM is ε TV -close in TV to the distribution q t , which is ε W 2 -close in W 2 to q , provided that the step size h is chosen appropriately according to

Original: Theorem 2 and N = (cid:101) Θ (cid:16) d 3 R 4 ( R ∨√ d ) 4 ε 2TV
Revised: Theorem 2 and N = (cid:101) Θ (cid:16) d 3 R 4 ( R ∨√ d ) 4 ε 2TV

Original: ε 8 W (cid:17) and ε score ≤ (cid:101) O ( ε TV ) .
Revised: ε 8 W (cid:17) and ε score ≤ (cid:101) O ( ε TV ) .

Original: Observing that both the TV and W 1 metrics are upper bounds for the bounded Lipschitz metric d BL ( µ, ν ) :
Revised: Observing that both the TV and W 1 metrics are upper bounds for the bounded Lipschitz metric d BL ( µ, ν ) :

Original: = sup { (cid:82) f d µ − (cid:82) f d ν (cid:12)(cid:12) f : R d → [ − 1 , 1] is 1 -Lipschitz } , we immediately obtain the following corollary.
Revised: = sup { (cid:82) f d µ − (cid:82) f d ν (cid:12)(cid:12) f : R d → [ − 1 , 1] is 1 -Lipschitz } , we immediately obtain the following corollary.

Original: Corollary 5 (compactly supported data, BL metric) .
Revised: Corollary 5 (compactly supported data, BL metric) .

Original: Suppose that q is supported on the ball of radius R ≥ 1 .
Revised: Suppose that q is supported on the ball of radius R ≥ 1 .

Original: Let t ≍ ε 2 W 2 / ( √ d ( R ∨ √ d )) .
Revised: Let t ≍ ε 2 W 2 / ( √ d ( R ∨ √ d )) .

Original: Then, the output p T − t of the DDPM algorithm satisfies
Revised: Then, the output p T − t of the DDPM algorithm satisfies

Original: d BL ( p T − t , q ) ≤ ε , provided that the step size h is chosen appropriately according to Theorem 2 and
Revised: d BL ( p T − t , q ) ≤ ε , provided that the step size h is chosen appropriately according to Theorem 2 and

Original: N = (cid:101) Θ( d 3 R 4 ( R ∨ √ d ) 4 /ε 10 ) and ε score ≤ (cid:101) O ( ε TV ) .
Revised: N = (cid:101) Θ( d 3 R 4 ( R ∨ √ d ) 4 /ε 10 ) and ε score ≤ (cid:101) O ( ε TV ) .

Original: Finally, if the output p T − t of DDPM at time T − t is projected onto B (0 , R 0 ) for an appropriate choice of R 0 , then we can also translate our guarantees to the standard W 2 metric, which we state as the following corollary.
Revised: Finally, if the output p T − t of DDPM at time T − t is projected onto B (0 , R 0 ) for an appropriate choice of R 0 , then we can also translate our guarantees to the standard W 2 metric, which we state as the following corollary.

Original: Corollary 6 (compactly supported data, W 2 metric; see Section B.5 in supplement) .
Revised: Corollary 6 (compactly supported data, W 2 metric; see Section B.5 in supplement) .

Original: Suppose that q is supported on the ball of radius R ≥ 1 .
Revised: Suppose that q is supported on the ball of radius R ≥ 1 .

Original: Let t ≍ ε 2 W 2 / ( √ d ( R ∨ √ d )) , and let p T − t,R 0 denote the output of DDPM at time T − t projected onto B (0 , R 0 ) for R 0 = (cid:101)
Revised: Let t ≍ ε 2 W 2 / ( √ d ( R ∨ √ d )) , and let p T − t,R 0 denote the output of DDPM at time T − t projected onto B (0 , R 0 ) for R 0 = (cid:101)

Original: Θ( R ) .
Revised: Θ( R ) .

Original: Then, it holds that
Revised: Then, it holds that

Original: W 2 ( p T − t,R 0 , q ) ≤ ε , provided that the step size h is chosen appropriately according to Theorem 2,
Revised: W 2 ( p T − t,R 0 , q ) ≤ ε , provided that the step size h is chosen appropriately according to Theorem 2,

Original: N = (cid:101) Θ( d  R  R ∨ √ d ) 4 /ε  and ε score ≤ (cid:101) 
Revised: N = (cid:101) Θ( d 3 R 8 ( R ∨ √ d ) 4 /ε 12 ) , and ε score ≤ (cid:101) O ( ε TV ) .

Original: O ( ε TV ) . Note that the dependencies in the three corollaries above are polynomial in all of the relevant problem parameters.
Revised: Note that the dependencies in the three corollaries above are polynomial in all of the relevant problem parameters.

Original: In particular, since the last corollary holds in the W 2 metric, it is directly comparable to De Bortoli (2022) and vastly improves upon the exponential dependencies therein.
Revised: In particular, since the last corollary holds in the W 2 metric, it is directly comparable to De Bortoli (2022) and vastly improves upon the exponential dependencies therein.

Original: 3. R ESULTS FOR CLD
Revised: 3. R ESULTS FOR CLD

Original: In order to state our results for score-based generative modeling based on the CLD, we must first modify Assumptions 1 and 3 accordingly.
Revised: In order to state our results for score-based generative modeling based on the CLD, we must first modify Assumptions 1 and 3 accordingly.

Original: Assumption 4.
Revised: Assumption 4.

Original: For all t ≥ 0 , the score ∇ v ln q t is L -Lipschitz.
Revised: For all t ≥ 0 , the score ∇ v ln q t is L -Lipschitz.

Original: Assumption 5.
Revised: Assumption 5.

Original: For all [Equation]
Revised: For all k = 1 , . . .

Original: [Equation] N [Equation]
Revised: , N , E q kh [ ∥ s kh − ∇ v ln q kh ∥ 2 ] ≤ ε 2score .

Original: If we ignore the dependence on L and assume that the score estimate is sufficiently accurate, then the iteration complexity guarantee of Theorem 2 is N = (cid:101)
Revised: If we ignore the dependence on L and assume that the score estimate is sufficiently accurate, then the iteration complexity guarantee of Theorem 2 is N = (cid:101)

Original: Θ( d/ε 2 ) .
Revised: Θ( d/ε 2 ) .

Original: On the other hand, recall from
Revised: On the other hand, recall from

Original: Section 2.2 that based on intuition from the literature on log-concave sampling and from empirical findings in Dockhorn et al.
Revised: Section 2.2 that based on intuition from the literature on log-concave sampling and from empirical findings in Dockhorn et al.

Original: (2022), we might expect that SGMs based on the CLD have a smaller iteration complexity than DDPM.
Revised: (2022), we might expect that SGMs based on the CLD have a smaller iteration complexity than DDPM.

Original: We prove the following theorem.
Revised: We prove the following theorem.

Original: Theorem 7 (CLD, see Section C of supplement) .
Revised: Theorem 7 (CLD, see Section C of supplement) .

Original: Suppose that Assumptions 2, 4, and 5 hold.
Revised: Suppose that Assumptions 2, 4, and 5 hold.

Original: Let p T be the output of the SGM algorithm based on the CLD (Section 2.2) at time T , and suppose the step size h :
Revised: Let p T be the output of the SGM algorithm based on the CLD (Section 2.2) at time T , and suppose the step size h :

Original: = T/N satisfies h ≲ 1 /L , where L ≥ 1 .
Revised: = T/N satisfies h ≲ 1 /L , where L ≥ 1 .

Original: Then, there is a universal constant c > 0 such that
Revised: Then, there is a universal constant c > 0 such that

Original: TV ( p T , q ⊗ γ d ) is bounded, up to a constant factor, by
Revised: TV ( p T , q ⊗ γ d ) is bounded, up to a constant factor, by

Original: [Equation]
Revised: [Equation]

Original: where FI ( q ∥ γ d ) is the relative Fisher information FI ( q ∥ γ d ) :
Revised: where FI ( q ∥ γ d ) is the relative Fisher information FI ( q ∥ γ d ) :

Original: =
Revised: =

Original: E q
Revised: E q

Original: [ ∥∇ ln( q/γ d ) ∥ 2 ] .
Revised: [ ∥∇ ln( q/γ d ) ∥ 2 ] .

Original: Note that the result of Theorem 7 is in fact no better than our guarantee for DDPM in Theorem 2.
Revised: Note that the result of Theorem 7 is in fact no better than our guarantee for DDPM in Theorem 2.

Original: Although it is possible that this is an artefact of our analysis, we believe that it is in fact fundamental.
Revised: Although it is possible that this is an artefact of our analysis, we believe that it is in fact fundamental.

Original: As we discuss in Remark C.2, from the form of the reverse process (7), the SGM based on CLD lacks a certain property (that the discretization error should only depend on the size of the increment of the X process, not the increments of both the X and V processes) which is crucial for the improved dimension dependence of the CLD over the Langevin diffusion in log-concave sampling.
Revised: As we discuss in Remark C.2, from the form of the reverse process (7), the SGM based on CLD lacks a certain property (that the discretization error should only depend on the size of the increment of the X process, not the increments of both the X and V processes) which is crucial for the improved dimension dependence of the CLD over the Langevin diffusion in log-concave sampling.

Original: Hence, in general, we conjecture that under our assumptions, SGMs based on the CLD do not achieve a better dimension dependence than DDPM.
Revised: Hence, in general, we conjecture that under our assumptions, SGMs based on the CLD do not achieve a better dimension dependence than DDPM.

Original: We provide evidence for our conjecture via a lower bound.
Revised: We provide evidence for our conjecture via a lower bound.

Original: In our proofs of Theorems 2 and 7, we rely on bounding the KL divergence between certain measures on the path space C ([0 , T ]; R d ) via
Revised: In our proofs of Theorems 2 and 7, we rely on bounding the KL divergence between certain measures on the path space C ([0 , T ]; R d ) via

Original: Girsanov’s theorem.
Revised: Girsanov’s theorem.

Original: The following result lower bounds this KL divergence, even for the setting in which the score estimate is perfect ( ε score = 0 ) and the data distribution q is the standard Gaussian.
Revised: The following result lower bounds this KL divergence, even for the setting in which the score estimate is perfect ( ε score = 0 ) and the data distribution q is the standard Gaussian.

Original: Theorem 8 (Section C.5 of supplement) .
Revised: Theorem 8 (Section C.5 of supplement) .

Original: Let p T be the output of the SGM algorithm based on the
Revised: Let p T be the output of the SGM algorithm based on the

Original: CLD (Section 2.2) at time T , where the data distribution q is the standard Gaussian γ d , and the score estimate is exact ( ε score = 0 ).
Revised: CLD (Section 2.2) at time T , where the data distribution q is the standard Gaussian γ d , and the score estimate is exact ( ε score = 0 ).

Original: Suppose that the step size h satisfies h ≲ 1 / ( T ∨ 1) .
Revised: Suppose that the step size h satisfies h ≲ 1 / ( T ∨ 1) .

Original: Then, for the path measures P T and Q ← T of the algorithm and the continuous-time process (7) respectively (see
Revised: Then, for the path measures P T and Q ← T of the algorithm and the continuous-time process (7) respectively (see

Original: Section C for details), it holds that KL ( Q ← T ∥ P T ) ≥ dhT .
Revised: Section C for details), it holds that KL ( Q ← T ∥ P T ) ≥ dhT .

Original: Theorem 8 shows that in order to make the KL divergence between the path measures small, we must take h ≲ 1 /d , which leads to an iteration complexity that scales linearly in the dimension d .
Revised: Theorem 8 shows that in order to make the KL divergence between the path measures small, we must take h ≲ 1 /d , which leads to an iteration complexity that scales linearly in the dimension d .

Original: Theorem 8 is not a proof that SGMs based on the CLD cannot achieve better than linear dimension dependence, as it is possible that the output p T of the SGM is close to q ⊗ γ d even if the path measures are not close, but it rules out the possibility of obtaining a better dimension dependence via our Girsanov proof technique.
Revised: Theorem 8 is not a proof that SGMs based on the CLD cannot achieve better than linear dimension dependence, as it is possible that the output p T of the SGM is close to q ⊗ γ d even if the path measures are not close, but it rules out the possibility of obtaining a better dimension dependence via our Girsanov proof technique.

Original: We believe that it provides compelling evidence for our conjecture, i.e., that under our assumptions, the CLD does not improve the complexity of SGMs over DDPM.
Revised: We believe that it provides compelling evidence for our conjecture, i.e., that under our assumptions, the CLD does not improve the complexity of SGMs over DDPM.

Original: We remark that in this section, we have only considered the error arising from discretization of the
Revised: We remark that in this section, we have only considered the error arising from discretization of the

Original: SDE.
Revised: SDE.

Original: It is possible that the score function ∇ v ln q t for the SGM with the CLD is easier to estimate than the score function for DDPM, providing a statistical benefit of using the CLD.
Revised: It is possible that the score function ∇ v ln q t for the SGM with the CLD is easier to estimate than the score function for DDPM, providing a statistical benefit of using the CLD.

Original: Indeed, under the manifold hypothesis, the score ∇ ln q t for DDPM blows up at t = 0 , but the score ∇ v ln q t for CLD is well-defined at t = 0 , and hence may lead to improvements over DDPM.
Revised: Indeed, under the manifold hypothesis, the score ∇ ln q t for DDPM blows up at t = 0 , but the score ∇ v ln q t for CLD is well-defined at t = 0 , and hence may lead to improvements over DDPM.

Original: We do not investigate this question here and leave it as future work.
Revised: We do not investigate this question here and leave it as future work.

Original:  T ECHNICAL OVERVIEW
Revised:  T ECHNICAL OVERVIEW

Original: We now give a detailed technical overview for the proof for DDPM (Theorem 2).
Revised: We now give a detailed technical overview for the proof for DDPM (Theorem 2).

Original: The proof for CLD (Theorem 7) follows along similar lines.
Revised: The proof for CLD (Theorem 7) follows along similar lines.

Original: Recall that we must deal with three sources of error: (1) the estimation of the score function; (2) the discretization of the SDE; and (3) the initialization of the reverse process at γ d rather than at q T .
Revised: Recall that we must deal with three sources of error: (1) the estimation of the score function; (2) the discretization of the SDE; and (3) the initialization of the reverse process at γ d rather than at q T .

Original: First, we ignore the errors (1) and (2), and focus on the error (3).
Revised: First, we ignore the errors (1) and (2), and focus on the error (3).

Original: Hence, we consider the continuous-time reverse SDE (4), initialized from γ d (resp.
Revised: Hence, we consider the continuous-time reverse SDE (4), initialized from γ d (resp.

Original: q T ) and denote by ˜ p t (resp.
Revised: q T ) and denote by ˜ p t (resp.

Original: q T − t )
Revised: q T − t )

Original: its marginal distributions.
Revised: its marginal distributions.

Original: Note that ˜ p 

Revised: Note that ˜ p

Original: = γ d and that q 0 = q , the data distribution.
Revised: = γ d and that q 0 = q , the data distribution.

Original: First, using the exponential contraction of the KL divergence along the (forward) OU process, we have
Revised: First, using the exponential contraction of the KL divergence along the (forward) OU process, we have

Original: KL ( q T ∥ γ d ) ≤ exp( − 2 T ) KL ( q ∥ γ d ) .
Revised: KL ( q T ∥ γ d ) ≤ exp( − 2 T ) KL ( q ∥ γ d ) .

Original: Then, using the data processing inequality along the backward process, we have TV (˜ p T , q ) ≤ TV ( γ d , q T ) .
Revised: Then, using the data processing inequality along the backward process, we have TV (˜ p T , q ) ≤ TV ( γ d , q T ) .

Original: Therefore, using Pinsker inequality, we get
Revised: Therefore, using Pinsker inequality, we get

Original: [Equation]
Revised: [Equation]

Original: i.e., the output ˜ p T converges to the data distribution q exponentially fast as T → ∞ .
Revised: i.e., the output ˜ p T converges to the data distribution q exponentially fast as T → ∞ .

Original: Next, we consider the score estimation error (1) and the discretization error (2).
Revised: Next, we consider the score estimation error (1) and the discretization error (2).

Original: Using Girsanov’s theorem, these errors can be bounded by
Revised: Using Girsanov’s theorem, these errors can be bounded by

Original: [Equation]
Revised: [Equation]

Original: (see the inequality (15) in the supplement).
Revised: (see the inequality (15) in the supplement).

Original: Unlike other proof techniques, such as the interpolation method in Lee et al.
Revised: Unlike other proof techniques, such as the interpolation method in Lee et al.

Original: (2022a), the error term (8) in Girsanov’s theorem involves an expectation under the law of the true reverse process, instead of the law of the algorithm (see Lee et al.
Revised: (2022a), the error term (8) in Girsanov’s theorem involves an expectation under the law of the true reverse process, instead of the law of the algorithm (see Lee et al.

Original: (2022a)).
Revised: (2022a)).

Original: This difference allows us to bound the score estimation error using Assumption 3 directly, which allows a simpler proof that works under milder assumptions on the data distribution.
Revised: This difference allows us to bound the score estimation error using Assumption 3 directly, which allows a simpler proof that works under milder assumptions on the data distribution.

Original: However, the use of
Revised: However, the use of

Original: Girsanov’s theorem typically requires a technical condition known as Novikov’s condition , which fails to hold under under our minimal assumptions.
Revised: Girsanov’s theorem typically requires a technical condition known as Novikov’s condition , which fails to hold under under our minimal assumptions.

Original: To circumvent this issue, we use an approximation argument relying on abstract results on the convergence of stochastic processes.
Revised: To circumvent this issue, we use an approximation argument relying on abstract results on the convergence of stochastic processes.

Original: A recent concurrent and independent work Liu et al.
Revised: A recent concurrent and independent work Liu et al.

Original: (2022) also uses Girsanov’s theorem, but assumes that Novikov’s condition holds at the outset.
Revised: (2022) also uses Girsanov’s theorem, but assumes that Novikov’s condition holds at the outset.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this work, we provided the first convergence guarantees for SGMs which hold under realistic assumptions (namely, L 2 -accurate score estimation and arbitrarily non-log-concave data distributions) and which scale polynomially in the problem parameters.
Revised: In this work, we provided the first convergence guarantees for SGMs which hold under realistic assumptions (namely, L 2 -accurate score estimation and arbitrarily non-log-concave data distributions) and which scale polynomially in the problem parameters.

Original: Our results take a step towards explaining the remarkable empirical success of SGMs, at least assuming the score is learned with small L 2 error.
Revised: Our results take a step towards explaining the remarkable empirical success of SGMs, at least assuming the score is learned with small L 2 error.

Original: The main limitation of this work is that we did not address the question of when the score function can be learned well.
Revised: The main limitation of this work is that we did not address the question of when the score function can be learned well.

Original: In general, studying the non-convex training dynamics of learning the score function via neural networks is challenging, but we believe that the resolution of this problem, even for simple learning tasks, would shed considerable light on SGMs.
Revised: In general, studying the non-convex training dynamics of learning the score function via neural networks is challenging, but we believe that the resolution of this problem, even for simple learning tasks, would shed considerable light on SGMs.

Original: Together with the results in this paper, it would yield the first end-to-end guarantees for SGMs.
Revised: Together with the results in this paper, it would yield the first end-to-end guarantees for SGMs.

Original: In light of the interpretation of our result as a reduction of the task of sampling to the task of score function estimation, we also ask whether there are interesting situations where it is easier to learn the score function (not necessarily via a neural network) than to (directly) sample.
Revised: In light of the interpretation of our result as a reduction of the task of sampling to the task of score function estimation we also ask whether there are interesting situations where it is easier to learn the score function (not necessarily via a neural network) than to (directly) sample.

Original: A P RELIMINARIES
Revised: A P RELIMINARIES

Original: In this section, we review the notion of score matching and provide a list of notation for the proofs.
Revised: In this section, we review the notion of score matching and provide a list of notation for the proofs.

Original: A. P RIMER ON SCORE MATCHING
Revised: A. P RIMER ON SCORE MATCHING

Original: In order to estimate the score function ∇ ln q t , consider minimizing the L 2 ( q t ) loss over a function class F ,
Revised: In order to estimate the score function ∇ ln q t , consider minimizing the L 2 ( q t ) loss over a function class F ,

Original: [Equation]
Revised: [Equation]

Original: where F could be, e.g., a class of neural networks.
Revised: where F could be, e.g., a class of neural networks.

Original: The idea of score matching, which goes back to Hyv¨arinen (2005); Vincent (2011), is that after applying integration by parts for the Gaussian measure, the problem (9) is equivalent to the following problem:
Revised: The idea of score matching, which goes back to Hyv¨arinen (2005); Vincent (2011), is that after applying integration by parts for the Gaussian measure, the problem (9) is equivalent to the following problem:

Original: [Equation]
Revised: [Equation]

Original: where Z t ∼ normal (0 , I d ) is independent of ¯ X 

Revised: where Z t ∼ normal (0 , I d ) is independent of ¯ X

Original: and ¯ X t = exp( − t )
Revised: and ¯ X t = exp( − t )

Original: ¯ X 

Revised: ¯ X

Original: + (cid:112) 1 − exp( − 2 t )
Revised: + (cid:112) 1 − exp( − 2 t )

Original: Z t , in the sense that (9) and (10) share the same minimizers.
Revised: Z t , in the sense that (9) and (10) share the same minimizers.

Original: We give a self-contained derivation in Appendix D for the sake of completeness.
Revised: We give a self-contained derivation in Appendix D for the sake of completeness.

Original: Unlike (9), however, the objective in (10) can be replaced with an empirical version and estimated on the basis of samples ¯ X (1)0 , . .
Revised: Unlike (9), however, the objective in (10) can be replaced with an empirical version and estimated on the basis of samples ¯ X (1)0 , . .

Original: , ¯ X ( n )from q , leading to the finite-sample problem
Revised: , ¯ X ( n )from q , leading to the finite-sample problem

Original: [Equation]
Revised: [Equation]

Original: where ( Z ( i ) t ) i ∈ [ n ] are i.i.d.
Revised: where ( Z ( i ) t ) i ∈ [ n ] are i.i.d.

Original: standard Gaussians independent of ( ¯ X ( i ) 0 ) i ∈ [ n ] .
Revised: standard Gaussians independent of ( ¯ X ( i ) 0 ) i ∈ [ n ] .

Original: Moreover, if we parameterize the score as s t = −
Revised: Moreover, if we parameterize the score as s t = −

Original: [Equation]

Original: √

Original: [Equation]
Revised: [Equation]

Original: z t , then the empirical problem is equivalent to minimize
Revised: z t , then the empirical problem is equivalent to minimize

Original: [Equation]
Revised: [Equation]

Original: which has the illuminating interpretation of predicting the added noise Z ( i ) t from the noised data ¯ X ( i ) t , i.e., denoising.
Revised: which has the illuminating interpretation of predicting the added noise Z ( i ) t from the noised data ¯ X ( i ) t , i.e., denoising.

Original: N OTATION
Revised: N OTATION

Original: For a measurable mapping T : X → X and a measure µ on X , where X is a measurable space, the notation T # µ refers to the pushforward of µ by the mapping T , i.e., if X ∼ µ , then T ( X ) ∼ T # µ .
Revised: For a measurable mapping T : X → X and a measure µ on X , where X is a measurable space, the notation T # µ refers to the pushforward of µ by the mapping T , i.e., if X ∼ µ , then T ( X ) ∼ T # µ .

Original: Stochastic processes and their laws.
Revised: Stochastic processes and their laws.

Original: • The data distribution is q = q 0 .
Revised: • The data distribution is q = q 0 .

Original: • The forward process (1) is denoted ( ¯ X t ) t ∈ [0 ,T ] , and ¯ X t ∼ q t .
Revised: • The forward process (1) is denoted ( ¯ X t ) t ∈ [0 ,T ] , and ¯ X t ∼ q t .

Original: • The reverse process (4) is denoted ( ¯ X ← t ) t ∈ [0 ,T ] , where ¯ X ← t :=
Revised: • The reverse process (4) is denoted ( ¯ X ← t ) t ∈ [0 ,T ] , where ¯ X ← t :=

Original: ¯ X T − t ∼ q T − t .
Revised: ¯ X T − t ∼ q T − t .

Original: • The SGM algorithm (5) is denoted ( X ← t ) t ∈ [0 ,T ] , and X ← t ∼ p t .
Revised: • The SGM algorithm (5) is denoted ( X ← t ) t ∈ [0 ,T ] , and X ← t ∼ p t .

Original: Recall that we initialize at p 

Revised: Recall that we initialize at p

Original: = γ d , the standard Gaussian measure.
Revised: = γ d , the standard Gaussian measure.

Original: • The process ( X ← ,q T t ) t ∈ [0 ,T ] is the same as ( X ← t ) t ∈ [0 ,T ] , except that we initialize this process at q T rather than at γ d .
Revised: • The process ( X ← ,q T t ) t ∈ [0 ,T ] is the same as ( X ← t ) t ∈ [0 ,T ] , except that we initialize this process at q T rather than at γ d .

Original: We write X ← ,q T t ∼ p q T t .
Revised: We write X ← ,q T t ∼ p q T t .

Original: Conventions for Girsanov’s theorem.
Revised: Conventions for Girsanov’s theorem.

Original: When we apply Girsanov’s theorem, it is convenient to instead think about a single stochastic process, which for ease of notation we denote simply via ( X t ) t ∈ [0 ,T ] , and we consider different measures over the path space C
Revised: When we apply Girsanov’s theorem, it is convenient to instead think about a single stochastic process, which for ease of notation we denote simply via ( X t ) t ∈ [0 ,T ] , and we consider different measures over the path space C

Original: ([0 , T ]; R d ) .
Revised: ([0 , T ]; R d ) .

Original: The two measures we consider over path space are:
Revised: The two measures we consider over path space are:

Original: • Q ← T , under which ( X t ) t ∈ [0 ,T ] has the law of the reverse process (4); • P q T T , under which ( X t ) t ∈ [0 ,T ] has the law of the SGM algorithm initialized at q T (corresponding to the process ( X ← ,q T t ) t ∈ [0 ,T ] defined above).
Revised: • Q ← T , under which ( X t ) t ∈ [0 ,T ] has the law of the reverse process (4); • P q T T , under which ( X t ) t ∈ [0 ,T ] has the law of the SGM algorithm initialized at q T (corresponding to the process ( X ← ,q T t ) t ∈ [0 ,T ] defined above).

Original: We also use the following notion from stochastic calculus (Le Gall, 2016, Definition 4.6):
Revised: We also use the following notion from stochastic calculus (Le Gall, 2016, Definition 4.6):

Original: • A local martingale ( L t ) t ∈ [0 ,T ] is a stochastic process s.t.
Revised: • A local martingale ( L t ) t ∈ [0 ,T ] is a stochastic process s.t.

Original: there exists a sequence of nondecreasing stopping times T n → T s.t.
Revised: there exists a sequence of nondecreasing stopping times T n → T s.t.

Original: L n =
Revised: L n =

Original: ( L t ∧ T n ) t ∈ [0 ,T ] is a martingale.
Revised: ( L t ∧ T n ) t ∈ [0 ,T ] is a martingale.

Original: Other parameters.
Revised: Other parameters.

Original: We recall that T > 0 denotes the total time for which we run the forward process; h > 0 is the step size of the discretization; L ≥ 1 is the Lipschitz constant of the score function; m 22 :=
Revised: We recall that T > 0 denotes the total time for which we run the forward process; h > 0 is the step size of the discretization; L ≥ 1 is the Lipschitz constant of the score function; m 22 :=

Original: E q [ ∥·∥ 2 ] is the second moment under the data distribution; and ε score is the L 2 score estimation error.
Revised: E q [ ∥·∥ 2 ] is the second moment under the data distribution; and ε score is the L 2 score estimation error.

Original: Notation for CLD.
Revised: Notation for CLD.

Original: The notational conventions for the CLD are similar; however, we must also consider a velocity variable V .
Revised: The notational conventions for the CLD are similar; however, we must also consider a velocity variable V .

Original: When discussing quantities which involve both position and velocity (e.g., the joint distribution q t of ( ¯ X t , ¯ V t ) ), we typically use boldface fonts.
Revised: When discussing quantities which involve both position and velocity (e.g., the joint distribution q t of ( ¯ X t , ¯ V t ) ), we typically use boldface fonts.

Original: B P ROOFS FOR DDPM B. P RELIMINARIES ON G IRSANOV ’ S THEOREM AND A FIRST ATTEMPT AT APPLYING G IRSANOV ’ S THEOREM
Revised: B P ROOFS FOR DDPM B. P RELIMINARIES ON G IRSANOV ’ S THEOREM AND A FIRST ATTEMPT AT APPLYING G IRSANOV ’ S THEOREM

Original: First, we recall a consequence of Girsanov’s theorem that can be obtained by combining Pages 136–139, Theorem 5.22, and Theorem 4.13 of Le Gall (2016).
Revised: First, we recall a consequence of Girsanov’s theorem that can be obtained by combining Pages 136–139, Theorem 5.22, and Theorem 4.13 of Le Gall (2016).

Original: Theorem 9.
Revised: Theorem 9.

Original: For t ∈ [0 , T ] , let L t
Revised: For t ∈ [0 , T ] , let L t

Original: =
Revised: =

Original: (cid:82) t 

Revised: (cid:82) t

Original: b s d B s where B is a Q -Brownian motion.
Revised: b s d B s where B is a Q -Brownian motion.

Original: Assume that
Revised: Assume that

Original: E Q (cid:82) T 

Revised: E Q (cid:82) T

Original: ∥ b s ∥
Revised: ∥ b s ∥

Original: 2 d s < ∞ .
Revised: 2 d s < ∞ .

Original: Then, L is a Q -martingale in L 2 ( Q ) .
Revised: Then, L is a Q -martingale in L 2 ( Q ) .

Original: Moreover, if
Revised: Moreover, if

Original: E Q E ( L ) T
Revised: E Q E ( L ) T

Original: = 1 , where E ( L ) t := exp (cid:16)(cid:90) t
Revised: = 1 , where E ( L ) t := exp (cid:16)(cid:90) t

Original: [Equation]
Revised: [Equation]

Original: (cid:90)
Revised: b

Original: [Equation]
Revised: [Equation]

Original: s (cid:17)
Revised: (cid:90) t

Original: [Equation]
Revised: [Equation]

Original: (11) then E ( L ) is also a Q -martingale and the process t
Revised: b s

Original: [Equation]
Revised: [Equation]

Original:  B 
Revised: d s (cid:17) , (11) then E ( L ) is also a Q -martingale and the process t (cid:55)→ B t

Original: [Equation]
Revised: [Equation]

Original: b{+t 
+}
Revised: {+t 
+}

Original: [Equation] d s is a Brownian motion under P := E ( L ) T Q , the probability distribution with density E ( L ) T w.r.t.
Revised: b s d s is a Brownian motion under P := E ( L ) T Q , the probability distribution with density E ( L ) T w.r.t.

Original: Q .
Revised: Q .

Original: If the assumptions of Girsanov’s theorem are satisfied (i.e., the condition (11)), we can apply
Revised: If the assumptions of Girsanov’s theorem are satisfied (i.e., the condition (11)), we can apply

Original: Girsanov’s theorem to Q = Q ← T and
Revised: Girsanov’s theorem to Q = Q ← T and

Original: [Equation]
Revised: [Equation]

Original: where t ∈ [ kh, ( k + 1) h ] .
Revised: where t ∈ [ kh, ( k + 1) h ] .

Original: This tells us that under P = E ( L ) T Q ← T , there exists a Brownian motion ( β t ) t ∈ [0 ,T ] s.t.
Revised: This tells us that under P = E ( L ) T Q ← T , there exists a Brownian motion ( β t ) t ∈ [0 ,T ] s.t.

Original: [Equation]
Revised: [Equation]

Original: Recall that under Q ← T we have a.s.
Revised: Recall that under Q ← T we have a.s.

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: The equation above still holds P -a.s.
Revised: The equation above still holds P -a.s.

Original: since P ≪ Q ← T (even if B is no longer a P -Brownian motion).
Revised: since P ≪ Q ← T (even if B is no longer a P -Brownian motion).

Original: Plugging (12) into (13) we have P -a.s.,
Revised: Plugging (12) into (13) we have P -a.s.,

Original: [Equation]
Revised: [Equation]

Original: that In other words, under P , the distribution of X is the SGM algorithm started at q T , i.e., P = P q T T =
Revised: In other words, under P , the distribution of X is the SGM algorithm started at q T , i.e., P = P q T T =

Original: E ( L ) T Q ← T .
Revised: E ( L ) T Q ← T .

Original: Therefore,
Revised: Therefore,

Original: [Equation]
Revised: [Equation]

Original: where we used E Q ← T L t = 0 because L is a martingale.
Revised: where we used E Q ← T L t = 0 because L is a martingale.

Original: The equality (14) allows us to bound the discrepancy between the SGM algorithm and the reverse process.
Revised: The equality (14) allows us to bound the discrepancy between the SGM algorithm and the reverse process.

Original: B. C HECKING THE ASSUMPTIONS OF G IRSANOV ’ S THEOREM AND THE G IRSANOV DISCRETIZATION ARGUMENT
Revised: B. C HECKING THE ASSUMPTIONS OF G IRSANOV ’ S THEOREM AND THE G IRSANOV DISCRETIZATION ARGUMENT

Original: In most applications of Girsanov’s theorem in sampling, a sufficient condition for (11) to hold, known as Novikov’s condition , is satisfied.
Revised: In most applications of Girsanov’s theorem in sampling, a sufficient condition for (11) to hold, known as Novikov’s condition , is satisfied.

Original: Here, Novikov’s condition writes
Revised: Here, Novikov’s condition writes

Original: [Equation]
Revised: [Equation]

Original: and if Novikov’s condition holds, we can apply Girsanov’s theorem directly.
Revised: and if Novikov’s condition holds, we can apply Girsanov’s theorem directly.

Original: However, under
Revised: However, under

Original: Assumptions 1, 2, and 3 alone, Novikov’s condition need not hold.
Revised: Assumptions 1, 2, and 3 alone, Novikov’s condition need not hold.

Original: Indeed, in order to check
Revised: Indeed, in order to check

Original: Novikov’s condition, we would want X 0 to have sub-Gaussian tails for instance.
Revised: Novikov’s condition, we would want X 0 to have sub-Gaussian tails for instance.

Original: Furthermore, we also could not check that the condition (11), which is weaker than Novikov’s condition, holds.
Revised: Furthermore, we also could not check that the condition (11), which is weaker than Novikov’s condition, holds.

Original: Therefore, in the proof of the next Theorem, we use a approximation technique to show that
Revised: Therefore, in the proof of the next Theorem, we use a approximation technique to show that

Original: [Equation]
Revised: [Equation]

Original: We then use a discretization argument based on stochastic calculus to further bound this quantity.
Revised: We then use a discretization argument based on stochastic calculus to further bound this quantity.

Original: The result is the following theorem.
Revised: The result is the following theorem.

Original: Theorem 10 (discretization error for DDPM) .
Revised: Theorem 10 (discretization error for DDPM) .

Original: Suppose that Assumptions 1, 2, and 3 hold.
Revised: Suppose that Assumptions 1, 2, and 3 hold.

Original: Let Q ← T and P q T T denote the measures on path space corresponding to the reverse process (4) and the SGM algorithm with L 2 -accurate score estimate initialized at q T .
Revised: Let Q ← T and P q T T denote the measures on path space corresponding to the reverse process (4) and the SGM algorithm with L 2 -accurate score estimate initialized at q T .

Original: Assume that L ≥ 1 and h ≲ 1 /L .
Revised: Assume that L ≥ 1 and h ≲ 1 /L .

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: We start by proving
Revised: We start by proving

Original: N − 1 (cid:88)
Revised: N − 1 (cid:88)

Original: [Equation]
Revised: [Equation]

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: we give the approximation argument to prove the inequality (15).
Revised: we give the approximation argument to prove the inequality (15).

Original: Bound on the discretization error.
Revised: Bound on the discretization error.

Original: For t ∈ [ kh, ( k + 1) h ] , we can decompose
Revised: For t ∈ [ kh, ( k + 1) h ] , we can decompose

Original: [Equation]
Revised: [Equation]

Original: We must bound the change in the score function along the forward process.
Revised: We must bound the change in the score function along the forward process.

Original: If S : R d → R d is the mapping S ( x ) :=
Revised: If S : R d → R d is the mapping S ( x ) :=

Original: exp( − ( t − kh ))
Revised: exp( − ( t − kh ))

Original: x , then q T − kh = S
Revised: x , then q T − kh = S

Original: # q T − t ∗ normal (0 , 1 − exp( − 2 ( t − kh ))) .
Revised: # q T − t ∗ normal (0 , 1 − exp( − 2 ( t − kh ))) .

Original: We can then use Lee et al.
Revised: We can then use Lee et al.

Original: (2022a, Lemma C.12) (or the more general Lemma 17 that we prove in
Revised: (2022a, Lemma C.12) (or the more general Lemma 17 that we prove in

Original: Section C.4) with α = exp( t − kh )
Revised: Section C.4) with α = exp( t − kh )

Original: = 1 + O ( h ) and σ 2 = 1 − exp( − 2 ( t − kh ))
Revised: = 1 + O ( h ) and σ 2 = 1 − exp( − 2 ( t − kh ))

Original: =
Revised: =

Original: O ( h ) to obtain
Revised: O ( h ) to obtain

Original: [Equation]
Revised: [Equation]

Original: where the last line uses L ≥ 1 .
Revised: where the last line uses L ≥ 1 .

Original: For the last term,
Revised: For the last term,

Original: [Equation]
Revised: [Equation]

Original: where the second term above is absorbed into the third term of the decomposition (16).
Revised: where the second term above is absorbed into the third term of the decomposition (16).

Original: Hence,
Revised: Hence,

Original: [Equation]
Revised: [Equation]

Original: Using the fact that under Q ← T , the process ( X t ) t ∈ [0 ,T ] is the time reversal of the forward process ( ¯ X t ) t ∈ [0 ,T ] , we can apply the moment bounds in Lemma 11 and the movement bound in Lemmato obtain
Revised: Using the fact that under Q ← T , the process ( X t ) t ∈ [0 ,T ] is the time reversal of the forward process ( ¯ X t ) t ∈ [0 ,T ] , we can apply the moment bounds in Lemma 11 and the movement bound in Lemmato obtain

Original: [Equation]
Revised: [Equation]

Original: Approximation argument.
Revised: Approximation argument.

Original: For t ∈ [0 , T ] , let L t
Revised: For t ∈ [0 , T ] , let L t

Original: =
Revised: =

Original: (cid:82) t 

Revised: (cid:82) t

Original: b s d B s where B is a Q ← T -Brownian motion and we define
Revised: b s d B s where B is a Q ← T -Brownian motion and we define

Original: [Equation]
Revised: [Equation]

Original: for t ∈ [ kh, ( k + 1) h ] .
Revised: for t ∈ [ kh, ( k + 1) h ] .

Original: We proved that E Q ← T (cid:82) T 

Revised: We proved that E Q ← T (cid:82) T

Original: ∥ b s ∥
Revised: ∥ b s ∥

Original: 2 d s ≲ ( ε 2score + L 2 dh + L[-2 m 
-]
Revised: [Equation] d s [Equation] + L[-2 m 
-]

Revised: [Equation]

Original:  h 2 ) T < ∞ .
Revised: dh + L 2 m 22 h 2 ) T < ∞ .

Original: Using Le Gall (2016, Proposition 5.11), ( E ( L ) t ) t ∈ [0 ,T ] is a local martingale.
Revised: Using Le Gall (2016, Proposition 5.11), ( E ( L ) t ) t ∈ [0 ,T ] is a local martingale.

Original: Therefore, there exists a non-decreasing sequence of stopping times T n ↗ T s.t.
Revised: Therefore, there exists a non-decreasing sequence of stopping times T n ↗ T s.t.

Original: ( E ( L ) t ∧ T n ) t ∈ [0 ,t ] is a martingale.
Revised: ( E ( L ) t ∧ T n ) t ∈ [0 ,t ] is a martingale.

Original: Note that E ( L ) t ∧ T n = E ( L n ) t where L nt =
Revised: Note that E ( L ) t ∧ T n = E ( L n ) t where L nt =

Original: L t ∧ T n .
Revised: L t ∧ T n .

Original: Since E ( L n ) is a martingale, we have
Revised: Since E ( L n ) is a martingale, we have

Original: [Equation]
Revised: [Equation]

Original: i.e., E Q ← T E ( L ) T n = 1 .
Revised: i.e., E Q ← T E ( L ) [Equation] n = 1 .

Original: We apply Girsanov’s theorem to L nt = (cid:82) t 

Revised: We apply Girsanov’s theorem to L nt = (cid:82) t

Original: b s 1 [0 ,T n ] ( s ) d B s , where B is a Q ← T -Brownian motion.
Revised: b s 1 [0 ,T n ] ( s ) d B s , where B is a Q ← T -Brownian motion.

Original: Since E Q ← T (cid:82) T 0 ∥ b s 1 [0 ,T n ] ( s ) ∥ 2 d s ≤ E Q ← T (cid:82) T 

Revised: Since E Q ← T (cid:82) T 0 ∥ b s 1 [0 ,T n ] ( s ) ∥ 2 d s ≤ E Q ← T (cid:82) T

Original: ∥ b s ∥
Revised: ∥ b s ∥

Original: 2 d s < ∞ (see the last paragraph) and E
Revised: 2 d s < ∞ (see the last paragraph) and E

Original: Q ← T E ( L n ) T = 1 , we obtain that under P n := E ( L n ) T Q ← T there exists a Brownian motion β n s.t.
Revised: Q ← T E ( L n ) T = 1 , we obtain that under P n := E ( L n ) T Q ← T there exists a Brownian motion β n s.t.

Original: for t ∈ [0 , T ] ,
Revised: for t ∈ [0 , T ] ,

Original: [Equation]
Revised: [Equation]

Original: Recall that under Q ← T we have a.s.
Revised: Recall that under Q ← T we have a.s.

Original: [Equation]
Revised: [Equation]

Original: The equation above still holds P n -a.s.
Revised: The equation above still holds P n -a.s.

Original: since P n ≪ Q ← T .
Revised: since P n ≪ Q ← T .

Original: Combining the last two equations we then obtain P n -a.s.,
Revised: Combining the last two equations we then obtain P n -a.s.,

Original: [Equation]
Revised: [Equation]

Original: and X 0 ∼ q T .
Revised: and X 0 ∼ q T .

Original: In other words, P n is the law of the solution of the SDE (17).
Revised: In other words, P n is the law of the solution of the SDE (17).

Original: At this stage we have the bound
Revised: At this stage we have the bound

Original: [Equation]
Revised: [Equation]

Original: where we used that E Q ← T L T n = 0 because L is a Q ← T -martingale and T n is a bounded stopping time (Le Gall, 2016, Corollary 3.23).
Revised: where we used that E Q ← T L T n = 0 because L is a Q ← T -martingale and T n is a bounded stopping time (Le Gall, 2016, Corollary 3.23).

Original: Our goal is now to show that we can obtain the final result by an approximation argument.
Revised: Our goal is now to show that we can obtain the final result by an approximation argument.

Original: We consider a coupling of ( P n ) n ∈ N , P q T T : a sequence of stochastic processes ( X n ) n ∈ N over the same probability space, a stochastic process X and a single Brownian motion W over that space s.t.
Revised: We consider a coupling of ( P n ) n ∈ N , P q T T : a sequence of stochastic processes ( X n ) n ∈ N over the same probability space, a stochastic process X and a single Brownian motion W over that space s.t.

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: with X 0 =
Revised: with X 0 =

Original: X n 0 a.s. and X 0 ∼ q T .
Revised: X n 0 a.s. and X 0 ∼ q T .

Original: Note that the distribution of X n (resp.
Revised: Note that the distribution of X n (resp.

Original: X ) is P n (resp.
Revised: X ) is P n (resp.

Original: P q T T ).
Revised: P q T T ).

Original: Let ε > 0 and consider the map π ε : C ([0 , T ]; R d ) → C ([0 , T ]; R d ) defined by
Revised: Let ε > 0 and consider the map π ε : C ([0 , T ]; R d ) → C ([0 , T ]; R d ) defined by

Original: [Equation]
Revised: [Equation]

Original: ε ) .

Original: Noting that X nt =
Revised: Noting that X nt =

Original: X t for every t ∈ [0 , T n ] and using Lemma 13, we have π ε ( X n ) → π ε ( X ) a.s., uniformly over [0 , T ] .
Revised: X t for every t ∈ [0 , T n ] and using Lemma 13, we have π ε ( X n ) → π ε ( X ) a.s., uniformly over [0 , T ] .

Original: Therefore, π ε # P n → π ε # P q T T weakly.
Revised: Therefore, π ε # P n → π ε # P q T T weakly.

Original: Using the lower semicontinuity of the
Revised: Using the lower semicontinuity of the

Original: KL divergence and the data-processing inequality (Ambrosio et al., 2005, Lemma 9.4.3 and Lemma 9.4.5), we obtain
Revised: KL divergence and the data-processing inequality (Ambrosio et al., 2005, Lemma 9.4.3 and Lemma 9.4.5), we obtain

Original: [Equation]
Revised: [Equation]

Original: Finally, using Lemma 14, π ε ( ω ) → ω as ε → 0 , uniformly over [0 , T ] .
Revised: Finally, using Lemma 14, π ε ( ω ) → ω as ε → 0 , uniformly over [0 , T ] .

Original: Therefore, using Ambrosio et al.
Revised: Therefore, using Ambrosio et al.

Original: (2005, Corollary 9.4.6), KL (( π ε )
Revised: (2005, Corollary 9.4.6), KL (( π ε )

Original: # Q ← T ∥ ( π ε )
Revised: # Q ← T ∥ ( π ε )

Original: # P q T T )
Revised: # P q T T )

Original: → KL ( Q ← T ∥
Revised: → KL ( Q ← T ∥

Original: P q T T ) as ε ↘ 0 .
Revised: P q T T ) as ε ↘ 0 .

Original: Therefore,
Revised: Therefore,

Original: [Equation]
Revised: [Equation]

Original: We conclude with Pinsker’s inequality ( TV 2 ≤ KL ).
Revised: We conclude with Pinsker’s inequality ( TV 2 ≤ KL ).

Original: B. P ROOF OF T HEOREM
Revised: B. P ROOF OF T HEOREM

Original: We can now conclude our main result.
Revised: We can now conclude our main result.

Original: Proof of Theorem 2.
Revised: Proof of Theorem 2.

Original: We recall the notation from Section 4.
Revised: We recall the notation from Section 4.

Original: By the data processing inequality,
Revised: By the data processing inequality,

Original: [Equation]
Revised: [Equation]

Original: Using the convergence of the OU process in KL divergence (see, e.g., Bakry et al., 2014, Theorem 5.2.1) and applying Theorem 10 for the second term,
Revised: Using the convergence of the OU process in KL divergence (see, e.g., Bakry et al., 2014, Theorem 5.2.1) and applying Theorem 10 for the second term,

Original: [Equation]
Revised: [Equation]

Original: which proves the result.
Revised: which proves the result.

Original: B. A UXILIARY LEMMAS
Revised: B. A UXILIARY LEMMAS

Original: In this section, we prove some auxiliary lemmas which are used in the proof of Theorem 2.
Revised: In this section, we prove some auxiliary lemmas which are used in the proof of Theorem 2.

Original: Lemma 11 (moment bounds for DDPM) .
Revised: Lemma 11 (moment bounds for DDPM) .

Original: Suppose that Assumptions 1 and 2 hold.
Revised: Suppose that Assumptions 1 and 2 hold.

Original: Let ( ¯ X t ) t ∈ [0 ,T ] denote the forward process (1) .
Revised: Let ( ¯ X t ) t ∈ [0 ,T ] denote the forward process (1) .

Original: (moment bound)
Revised: (moment bound)

Revised: For

Original: [Equation]
Revised: [Equation]

Revised: t

Revised: [Equation]

Original: (score function bound)For
Revised: (score function bound)

Revised: For all t ≥

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Along the OU process, we have ¯ X t d = exp( − t ) ¯ X 

Revised: Along the OU process, we have ¯ X t d = exp( − t ) ¯ X

Original: + (cid:112) 1 − exp( − 2 t ) ξ , where ξ ∼ normal (0 , I d ) is independent of ¯ X 0 .
Revised: + (cid:112) 1 − exp( − 2 t ) ξ , where ξ ∼ normal (0 , I d ) is independent of ¯ X 0 .

Original: Hence,
Revised: Hence,

Original: [Equation]
Revised: [Equation]

Original: This follows from the L -smoothness of ln q t (see, e.g., Vempala & Wibisono, 2019, Lemma 9).
Revised: This follows from the L -smoothness of ln q t (see, e.g., Vempala & Wibisono, 2019, Lemma 9).

Original: We give a short proof for the sake of completeness.
Revised: We give a short proof for the sake of completeness.

Original: If L
Revised: If L

Original: [Equation]
Revised: [Equation]

Original: f :=
Revised: f [Equation] =

Original: ∆ f − ⟨∇ U t , ∇ f ⟩ is the generator associated with q t ∝ exp( − U t ) , then
Revised: ∆ f − ⟨∇ U t , ∇ f ⟩ is the generator associated with q t ∝ exp( − U t ) , then

Original: [Equation]
Revised: [Equation]

Original: Lemma 12 (movement bound for DDPM) .
Revised: Lemma 12 (movement bound for DDPM) .

Original: Suppose that Assumption 2 holds.
Revised: Suppose that Assumption 2 holds.

Original: Let ( ¯ X t ) t ∈ [0 ,T ] denote the forward process (1) .
Revised: Let ( ¯ X t ) t ∈ [0 ,T ] denote the forward process (1) .

Original: For 0 ≤ s < t with δ :=
Revised: For 0 ≤ s < t with δ :=

Original: t − s , if δ ≤ 1 , then
Revised: t − s , if δ ≤ 1 , then

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: We can write
Revised: We can write

Original: [Equation]
Revised: [Equation]

Original: where we used Lemma 11.
Revised: where we used Lemma 11.

Original: We omit the proofs of the two next lemmas as they are straightforward.
Revised: We omit the proofs of the two next lemmas as they are straightforward.

Original: Lemma 13.
Revised: Lemma 13.

Original: Consider f n , f : [0 , T ] → R d s.t.
Revised: Consider f n , f : [0 , T ] → R d s.t.

Original: there exists an increasing sequence ( T n ) n ∈ N ⊆
Revised: there exists an increasing sequence ( T n ) n ∈ N ⊆

Original: [0 , T ] satisfying the conditions
Revised: [0 , T ] satisfying the conditions

Original: • T n → T as n → ∞ , • f n ( t ) =
Revised: • T n → T as n → ∞ , • f n ( t ) =

Original: f ( t ) for every t ≤ T n .
Revised: f ( t ) for every t ≤ T n .

Original: Then, for every ε > 0 , f n → f uniformly over [0 , T − ε ] .
Revised: Then, for every ε > 0 , f n → f uniformly over [0 , T − ε ] .

Original: In particular, f n ( · ∧ T − ε )
Revised: In particular, f n ( · ∧ T − ε )

Original: → f ( · ∧ T − ε ) uniformly over [0 , T ] .
Revised: → f ( · ∧ T − ε ) uniformly over [0 , T ] .

Original: Lemma 14.
Revised: Lemma 14.

Original: Consider f : [0 , T ] → R d continuous, and f ε : [0 , T ] → R d s.t.
Revised: Consider f : [0 , T ] → R d continuous, and f ε : [0 , T ] → R d s.t.

Original: f ε ( t ) =
Revised: f ε ( t ) =

Original: f ( t ∧ ( T − ε )) for ε > 0 .
Revised: f ( t ∧ ( T − ε )) for ε > 0 .

Original: Then f ε → f uniformly over [0 , T ] as ε → 0 .
Revised: Then f ε → f uniformly over [0 , T ] as ε → 0 .

Original: B. P ROOF OF C OROLLARY
Revised: B. P ROOF OF C OROLLARY

Original: Proof of Corollary 6.
Revised: Proof of Corollary 6.

Original: For R 0 >
Revised: For R 0 >

Original: [Equation]
Revised: [Equation]

Original: let Π R 0 denote the projection onto B (0 , R 0 ) .
Revised: let Π R 0 denote the projection onto B (0 , R 0 ) .

Original: We want to prove that W 2 ((Π R 0 )
Revised: We want to prove that W 2 ((Π R 0 )

Original: # p T − t , q ) ≤ ε .
Revised: # p T − t , q ) ≤ ε .

Original: We use the decomposition
Revised: We use the decomposition

Original: [Equation]

Original: Π

Original: [Equation]
Revised: [Equation]

Original: For the first term, since (Π R 0 )
Revised: For the first term, since (Π R 0 )

Original: # p T − t and (Π R 0 )
Revised: # p T − t and (Π R 0 )

Original: # q t both have support contained in B (0 , R 0 )
Revised: # q t both have support contained in B (0 , R 0 )

Original: [Equation]
Revised: [Equation]

Original: we can upper bound the Wasserstein distance by the total variation distance.
Revised: we can upper bound the Wasserstein distance by the total variation distance.

Original: Namely, Rolland (2022,
Revised: Namely, Rolland (2022,

Original: Lemma 9) implies that
Revised: Lemma 9) implies that

Original: [Equation]
Revised: [Equation]

Original: By the data-processing inequality,
Revised: By the data-processing inequality,

Original: [Equation]
Revised: [Equation]

Original: where ε TV is from Corollary
Revised: where ε TV is from Corollary

Original: [Equation]
Revised: [Equation]

Original: yielding
Revised: yielding

Original: [Equation]
Revised: [Equation]

Original: Next
Revised: Next

Original: [Equation]
Revised: [Equation]

Original: we take R 0 ≥ R so that (Π R 0 )
Revised: we take R 0 ≥ R so that (Π R 0 )

Original: # q = q .
Revised: # q = q .

Original: Since Π R 0 is 1 -Lipschitz, we have
Revised: Since Π R 0 is 1 -Lipschitz, we have

Original: [Equation]
Revised: [Equation]

Original: where ε W 2 is from Corollary 4.
Revised: where ε W 2 is from Corollary 4.

Original: Combining these bounds,
Revised: Combining these bounds,

Original: [Equation]
Revised: [Equation]

Original: We now take ε W 2 = ε/ 3 , R 0 = (cid:101)
Revised: We now take ε W 2 = ε/ 3 , R 0 = (cid:101)

Original: Θ( R ) , and ε TV = (cid:101) Θ( ε 2 /R 2 ) to obtain the desired result.
Revised: Θ( R ) , and ε TV = (cid:101) Θ( ε 2 /R 2 ) to obtain the desired result.

Original: The iteration complexity follows from Corollary 4.
Revised: The iteration complexity follows from Corollary 4.

Original: C P ROOFS FOR CLD C. B ACKGROUND ON THE CLD PROCESS
Revised: C P ROOFS FOR CLD C. B ACKGROUND ON THE CLD PROCESS

Original: More generally, for the forward process we can introduce a friction parameter γ > 0 and consider
Revised: More generally, for the forward process we can introduce a friction parameter γ > 0 and consider

Original: [Equation]
Revised: [Equation]

Original: If we write ¯ θ t := ( ¯ X t , ¯ V t ) , then the forward process satisfies the linear SDE
Revised: If we write ¯ θ t := ( ¯ X t , ¯ V t ) , then the forward process satisfies the linear SDE

Original: [Equation]
Revised: [Equation]

Original:  where A γ := (cid:20) [Equation] [Equation] γ (cid:21) and Σ γ := 
Revised: , where A γ := (cid:20) − 1 − γ (cid:21) and Σ γ := (cid:20)

Original: (cid:20)

Original: [Equation]
Revised: [Equation]

Original:  2 γ (cid:21) .
Revised: √ 2 γ (cid:21) .

Original: The solution to the SDE is given by
Revised: The solution to the SDE is given by

Original: [Equation]
Revised: [Equation]

Original: 18) which means that by the Itˆo isometry,
Revised: 18) which means that by the Itˆo isometry,

Original: [Equation]
Revised: [Equation]

Original: Since det A γ = 1 , A γ is always invertible.
Revised: Since det A γ = 1 , A γ is always invertible.

Original: Moreover, from tr A γ = − γ , one can work out that the spectrum of A γ is
Revised: Moreover, from tr A γ = − γ , one can work out that the spectrum of A γ is

Original: [Equation]
Revised: [Equation]

Original: However, A γ is not diagonalizable.
Revised: However, A γ is not diagonalizable.

Original: The case of γ = 2 is special, as it corresponds to the case when the spectrum is {− 1 } , and it corresponds to the critically damped case .
Revised: The case of γ = 2 is special, as it corresponds to the case when the spectrum is {− 1 } , and it corresponds to the critically damped case .

Original: Following Dockhorn et al.
Revised: Following Dockhorn et al.

Original: [Equation] 2022), which advocated for setting γ = 2 , we will also only consider the critically damped case.
Revised: (2022), which advocated for setting γ = 2 , we will also only consider the critically damped case.

Original: This also has the advantage of substantially simplifying the calculations.
Revised: This also has the advantage of substantially simplifying the calculations.

Original: C. G IRSANOV DISCRETIZATION ARGUMENT
Revised: C. G IRSANOV DISCRETIZATION ARGUMENT

Original: In order to apply Girsanov’s theorem, we introduce the path measures P q T T and Q ← T , under which
Revised: In order to apply Girsanov’s theorem, we introduce the path measures P q T T and Q ← T , under which

Original: [Equation]
Revised: [Equation]

Original: for t ∈ [ kh, ( k + 1) h ] , and d X 
Revised: for t ∈ [ kh, ( k + 1) h ] , and d X t = − V t d t , d [Equation] t = { X

Original: [Equation]
Revised: [Equation]

Original: =
Revised: T − [Equation] ( X t [Equation]

Revised: t

Original: [Equation]
Revised: [Equation]

Original: d t,
Revised: t

Original: [Equation]
Revised: [Equation]

Original: respectively.
Revised: respectively.

Original: Applying Girsanov’s theorem, we have the following theorem.
Revised: Applying Girsanov’s theorem, we have the following theorem.

Original: Corollary 15.
Revised: Corollary 15.

Original: Suppose that Novikov’s condition holds:
Revised: Suppose that Novikov’s condition holds:

Original: [Equation]
Revised: [Equation]

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: Similarly to Appendix B.2, even if Novikov’s condition does not hold, one can use an approximation to argue that the KL divergence is still upper bounded by the last expression.
Revised: Similarly to Appendix B.2, even if Novikov’s condition does not hold, one can use an approximation to argue that the KL divergence is still upper bounded by the last expression.

Original: Since the argument follows along the same lines, we omit it for brevity.
Revised: Since the argument follows along the same lines, we omit it for brevity.

Original: Using this, we now aim to prove the following theorem.
Revised: Using this, we now aim to prove the following theorem.

Original: Theorem 16 (discretization error for CLD) .
Revised: Theorem 16 (discretization error for CLD) .

Original: Suppose that Assumptions 2, 4, and 5 hold.
Revised: Suppose that Assumptions 2, 4, and 5 hold.

Original: Let Q ← T and P q T T denote the measures on path space corresponding to the reverse process (7) and the SGM algorithm with L 2 -accurate score estimate initialized at q T .
Revised: Let Q ← T and P q T T denote the measures on path space corresponding to the reverse process (7) and the SGM algorithm with L 2 -accurate score estimate initialized at q T .

Original: Assume that L ≥ 1 and h ≲ 1 /L .
Revised: Assume that L ≥ 1 and h ≲ 1 /L .

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: For t ∈ [ kh [Equation] ( k + 1) h ] [Equation]
Revised: For t ∈ [ kh, ( k + 1) h ] , we can decompose

Original: we can decompose

Original: [Equation]
Revised: [Equation]

Original: The change in the score function is bounded by Lemma 17, which generalizes Lee et al.
Revised: The change in the score function is bounded by Lemma 17, which generalizes Lee et al.

Original: (2022a,
Revised: (2022a,

Original: Lemma C.12).
Revised: Lemma C.12).

Original: From the representation (18) of the solution to the CLD, we note that
Revised: From the representation (18) of the solution to the CLD, we note that

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original:  In particular, since ∥ A 2 ∥ op ≲ 1 , ∥ A − 1 2 ∥ op ≲ 1 , and ∥ Σ 2 ∥ op ≲ 1 it follows that ∥ M 0 ∥ op
Revised: s ) A 2 } Σ 2 Σ T 2 exp { ( t − kh − s ) A T 2 } d s . In particular, since ∥ A 2 ∥ op ≲ 1 , ∥ A − 1 2 ∥ op ≲ 1 , and ∥ Σ 2 ∥ op ≲ 1 it follows that ∥ M 0 ∥ op

Original: = 1+ O ( h ) and ∥
Revised: = 1+ O ( h ) and ∥

Original: M 1 ∥ op =
Revised: M 1 ∥ op =

Original: O ( h ) .
Revised: O ( h ) .

Original: Substituting this into Lemma 17, we deduce that if h ≲ 1 /L , then
Revised: Substituting this into Lemma 17, we deduce that if h ≲ 1 /L , then

Original: [Equation]
Revised: [Equation]

Original: where in the last step we used L ≥ 1 .
Revised: where in the last step we used L ≥ 1 .

Original: For the last term,
Revised: For the last term,

Original: [Equation]
Revised: [Equation]

Revised: + L

Revised: [Equation]

Original: where the second term above is absorbed into the third term of the decomposition (19).
Revised: where the second term above is absorbed into the third term of the decomposition (19).

Original: Hence,
Revised: Hence,

Original: [Equation]
Revised: [Equation]

Original: By applying the moment bounds in Lemma 18 together with Lemma 19 on the movement of the CLD process, we obtain
Revised: By applying the moment bounds in Lemma 18 together with Lemma 19 on the movement of the CLD process, we obtain

Original: [Equation]
Revised: [Equation]

Original: The proof is concluded via an approximation argument as in Section B.2.
Revised: The proof is concluded via an approximation argument as in Section B.2.

Original: Remark.
Revised: Remark.

Original: We now pause to discuss why the discretization bound above does not improve upon the result for DDPM (Theorem 10).
Revised: We now pause to discuss why the discretization bound above does not improve upon the result for DDPM (Theorem 10).

Original: In the context of log-concave sampling, one instead considers the underdamped Langevin process
Revised: In the context of log-concave sampling, one instead considers the underdamped Langevin process

Original: [Equation]
Revised: [Equation]

Original: which is discretized to yield the algorithm
Revised: which is discretized to yield the algorithm

Original: [Equation]
Revised: [Equation]

Original: for t ∈ [ kh, ( k + 1) h ] .
Revised: for t ∈ [ kh, ( k + 1) h ] .

Original: Let P T denote the path measure for the algorithm, and let Q T denote the path measure for the continuous-time process.
Revised: Let P T denote the path measure for the algorithm, and let Q T denote the path measure for the continuous-time process.

Original: After applying Girsanov’s theorem, we obtain
Revised: After applying Girsanov’s theorem, we obtain

Original: [Equation]
Revised: [Equation]

Original: In this expression, note that ∇ U depends only on the position coordinate.
Revised: In this expression, note that ∇ U depends only on the position coordinate.

Original: Since the X process is smoother (as we do not add Brownian motion directly to X ), the error ∥∇ U ( X t ) − ∇ U ( X kh )
Revised: Since the X process is smoother (as we do not add Brownian motion directly to X ), the error ∥∇ U ( X t ) − ∇ U ( X kh )

Original: ∥is of size O ( dh 2 ) , which allows us to take step size h ≲ 1 / √ d .
Revised: ∥is of size O ( dh 2 ) , which allows us to take step size h ≲ 1 / √ d .

Original: This explains why the use of the underdamped Langevin diffusion leads to improved dimension dependence for log-concave sampling.
Revised: This explains why the use of the underdamped Langevin diffusion leads to improved dimension dependence for log-concave sampling.

Original: In contrast, consider the reverse process, in which
Revised: In contrast, consider the reverse process, in which

Original: [Equation]
Revised: [Equation]

Original: Since discretization of the reverse process involves the score function, which depends on both X and V , the error now involves controlling ∥ V t − V kh ∥ 2 , which is of size O ( dh ) (the process V is not very smooth because it includes a Brownian motion component).
Revised: Since discretization of the reverse process involves the score function, which depends on both X and V , the error now involves controlling ∥ V t − V kh ∥ 2 , which is of size O ( dh ) (the process V is not very smooth because it includes a Brownian motion component).

Original: Therefore, from the form of the reverse process, we may expect that SGMs based on the CLD do not improve upon the dimension dependence of DDPM.
Revised: Therefore, from the form of the reverse process, we may expect that SGMs based on the CLD do not improve upon the dimension dependence of DDPM.

Original: In Section C.5, we use this observation in order to prove a rigorous lower bound against discretization of SGMs based on the CLD.
Revised: In Section C.5, we use this observation in order to prove a rigorous lower bound against discretization of SGMs based on the CLD.

Original: C. P ROOF OF T HEOREM
Revised: C. P ROOF OF T HEOREM

Original: Proof of Theorem 7.
Revised: Proof of Theorem 7.

Original: By the data processing inequality,
Revised: By the data processing inequality,

Original: [Equation]
Revised: [Equation]

Original: In Ma et al.
Revised: In Ma et al.

Original: (2021), following the entropic hypocoercivity approach of Villani (2009), Ma et al. consider a Lyapunov functional L which is equivalent to the sum of the KL divergence and the Fisher information,
Revised: (2021), following the entropic hypocoercivity approach of Villani (2009), Ma et al. consider a Lyapunov functional L which is equivalent to the sum of the KL divergence and the Fisher information,

Original: [Equation]
Revised: [Equation]

Original: which decays exponentially fast in time: there exists a universal constant c > 0 such that for all t ≥ 0 ,
Revised: which decays exponentially fast in time: there exists a universal constant c > 0 such that for all t ≥ 0 ,

Original: [Equation]
Revised: [Equation]

Original: Since q 0 = q ⊗ γ d and γ 2 d = γ d ⊗ γ d , then L ( q 0 ∥ γ 2 d ) ≲ KL ( q ∥ γ d ) + FI ( q ∥ γ d ) .
Revised: Since q 0 = q ⊗ γ d and γ 2 d = γ d ⊗ γ d , then L ( q 0 ∥ γ 2 d ) ≲ KL ( q ∥ γ d ) + FI ( q ∥ γ d ) .

Original: By Pinsker’s inequality and Theorem 16, we deduce that
Revised: By Pinsker’s inequality and Theorem 16, we deduce that

Original: [Equation]
Revised: [Equation]

Original: which completes the proof.
Revised: which completes the proof.

Original: C. A UXILIARY LEMMAS
Revised: C. A UXILIARY LEMMAS

Original: We start with a perturbation lemma for the score function.
Revised: We start with a perturbation lemma for the score function.

Original: Lemma 17 (score perturbation lemma) .
Revised: Lemma 17 (score perturbation lemma) .

Original: Let 0 < ζ < 1 .
Revised: Let 0 < ζ < 1 .

Original: Suppose that M 0 , M 1 ∈ R 2 d × 2 d are two matrices, where M 1 is symmetric.
Revised: Suppose that M 0 , M 1 ∈ R 2 d × 2 d are two matrices, where M 1 is symmetric.

Original: Also, assume that ∥ M 0 − I 2 d ∥ op ≤ ζ , so that M 0 is invertible.
Revised: Also, assume that ∥ M 0 − I 2 d ∥ op ≤ ζ , so that M 0 is invertible.

Original: Let q = exp( − H ) be a probability density on R 2 d such that ∇ H is L -Lipschitz with L ≤
Revised: Let q = exp( − H ) be a probability density on R 2 d such that ∇ H is L -Lipschitz with L ≤

Original: [Equation]
Revised: [Equation]

Original: M
Revised: M

Original: [Equation]
Revised: [Equation]

Original: op . Then, it holds that(cid:13)(cid:13)(cid:13)
Revised: Then, it holds that

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: The proof follows along the lines of Lee et al.
Revised: The proof follows along the lines of Lee et al.

Original: (2022a, Lemma C.12).
Revised: (2022a, Lemma C.12).

Original: First, we show that when M 

Revised: First, we show that when M

Original: =
Revised: =

Original: I 2 d , if L ≤ 1 2 ∥ M 1 ∥ op then
Revised: I 2 d , if L ≤ 1 2 ∥ M 1 ∥ op then

Original: [Equation]
Revised: [Equation]

Original: Let S denote the subspace S :
Revised: Let S denote the subspace S :

Original: =
Revised: =

Original: range M 1 .
Revised: range M 1 .

Original: Then, since
Revised: Then, since

Original: [Equation]
Revised: [Equation]

Original: where M − 1is well-defined on S
Revised: where M − 1is well-defined on S

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: Here, q θ is the measure on θ + S such that
Revised: Here, q θ is the measure on θ + S such that

Original: [Equation]
Revised: [Equation]

Original: Note that since L ≤
Revised: Note that since L ≤

Original: [Equation]
Revised: [Equation]

Original: , then if we write q θ ( θ ′ )
Revised: , then if we write q θ [Equation]

Original:  ∝ exp( − H θ ( θ ′ )) , we have
Revised: θ ′ ) ∝ exp( − H θ ( θ ′ )) , we have

Original: [Equation]
Revised: [Equation]

Original: Let θ ⋆ ∈ arg min
Revised: Let θ ⋆ ∈ arg min

Original: H θ denote a mode.
Revised: H θ denote a mode.

Original: We bound
Revised: We bound

Original: [Equation]
Revised: [Equation]

Original: For the first term, (Dalalyan et al., 2019, Proposition 2) yields
Revised: For the first term, (Dalalyan et al., 2019, Proposition 2) yields

Original: [Equation]
Revised: [Equation]

Original: For the second term, since the mode satisfies ∇ H ( θ ⋆ )
Revised: For the second term, since the mode satisfies ∇ H ( θ ⋆ )

Original: + M − 1 1 ( θ ⋆ − θ )
Revised: + M − 1 1 ( θ ⋆ − θ )

Original: = 0 , we have
Revised: = 0 , we have

Original: [Equation]
Revised: [Equation]

Original: which is rearranged to yield
Revised: which is rearranged to yield

Original: [Equation]
Revised: [Equation]

Original: After combining the bounds, we obtain the claimed estimate (20).
Revised: After combining the bounds, we obtain the claimed estimate (20).

Original: Next, we consider the case of general M 0 .
Revised: Next, we consider the case of general M 0 .

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: We can apply (20) with ( M 0 )
Revised: We can apply (20) with ( M 0 )

Original: # q in place of q , noting that ( M 0 )
Revised: # q in place of q , noting that ( M 0 )

Original: # q ∝ exp( − H ′ ) for H ′ :
Revised: # q ∝ exp( − H ′ ) for H ′ :

Original: =
Revised: =

Original: H ◦ Mwhich is L ′ -smooth for L ′ :
Revised: H ◦ Mwhich is L ′ -smooth for L ′ :

Original: =
Revised: =

Original: L ∥ M 0 ∥ 2op ≲ L , to get (cid:13)(cid:13)(cid:13) ∇ 
Revised: L ∥ M 0 ∥ 2op ≲ L , to get [Equation] ∇ ln

Original: [Equation]
Revised: [Equation]

Original: Note that∥∇
Revised: Note that

Original: [Equation]
Revised: [Equation]

Original: We also have
Revised: We also have

Original: [Equation]
Revised: [Equation]

Original: Combining the bounds,
Revised: Combining the bounds,

Original: [Equation]
Revised: [Equation]

Original: so the lemma follows.
Revised: so the lemma follows.

Original: Next, we prove the moment and movement bounds for the CLD.
Revised: Next, we prove the moment and movement bounds for the CLD.

Original: Lemma 18 (moment bounds for CLD) .
Revised: Lemma 18 (moment bounds for CLD) .

Original: Suppose that Assumptions 2 and 4 hold.
Revised: Suppose that Assumptions 2 and 4 hold.

Original: Let ( ¯ X t , ¯ V t ) t ∈ [0 ,T ] denote the forward process (6) .
Revised: Let ( ¯ X t , ¯ V t ) t ∈ [0 ,T ] denote the forward process (6) .

Original: (moment bound)
Revised: (moment bound)

Original: For 
Revised: For all

Original: [Equation]
Revised: [Equation]

Original: (score function bound
Revised: (score function bound)

Original: For all t ≥ 0 ,
Revised: For all t ≥ 0 ,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: We can write
Revised: We can write

Original: [Equation]
Revised: [Equation]

Original: Next, the coupling argument of Cheng et al.
Revised: Next, the coupling argument of Cheng et al.

Original: (2018) shows that the CLD converges exponentially fast in the Wasserstein metric associated to a twisted norm |||·||| which is equivalent (up to universal constants) to the Euclidean norm ∥·∥ .
Revised: (2018) shows that the CLD converges exponentially fast in the Wasserstein metric associated to a twisted norm |||·||| which is equivalent (up to universal constants) to the Euclidean norm ∥·∥ .

Original: It implies the following result, see, e.g., Cheng et al.
Revised: It implies the following result, see, e.g., Cheng et al.

Original: (2018, Lemma 8):
Revised: (2018, Lemma 8):

Original: [Equation]
Revised: [Equation]

Original: The proof is the same as in Lemma 11.
Revised: The proof is the same as in Lemma 11.

Original: Lemma 19 (movement bound for CLD) .
Revised: Lemma 19 (movement bound for CLD) .

Original: Suppose that Assumptions 2 holds.
Revised: Suppose that Assumptions 2 holds.

Original: Let ( ¯ X t , ¯ V t ) t ∈ [0 ,T ] denote the forward process (6) .
Revised: Let ( ¯ X t , ¯ V t ) t ∈ [0 ,T ] denote the forward process (6) .

Original: For 0 < s < t with δ :=
Revised: For 0 < s < t with δ :=

Original: t − s , if δ ≤ 1 ,
Revised: t − s , if δ ≤ 1 ,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: First,
Revised: First,

Original: [Equation]
Revised: [Equation]

Original: where we used the moment bound in Lemma 18.
Revised: where we used the moment bound in Lemma 18.

Original: Next,
Revised: Next,

Original: [Equation]
Revised: [Equation]

Original: where we used Lemma 18 again.
Revised: where we used Lemma 18 again.

Original: C. L OWER BOUND AGAINST CLD
Revised: C. L OWER BOUND AGAINST CLD

Original: When proving upper bounds on the KL divergence, we can use the approximation argument described in Section B.2 in order to invoke Girsanov’s theorem.
Revised: When proving upper bounds on the KL divergence, we can use the approximation argument described in Section B.2 in order to invoke Girsanov’s theorem.

Original: However, when proving lower bounds on the
Revised: However, when proving lower bounds on the

Original: KL divergence, this approach no longer works, so we check Novikov’s condition directly for the setting of Theorem 8.
Revised: KL divergence, this approach no longer works, so we check Novikov’s condition directly for the setting of Theorem 8.

Original: Lemma 20 (Novikov’s condition holds for CLD) .
Revised: Lemma 20 (Novikov’s condition holds for CLD) .

Original: Consider the setting of Theorem 8.
Revised: Consider the setting of Theorem 8.

Original: Then, Novikov’s condition 15 holds.
Revised: Then, Novikov’s condition 15 holds.

Original: We defer the proof of Lemma 20 to the end of this section.
Revised: We defer the proof of Lemma 20 to the end of this section.

Original: Admitting Lemma 20, we now prove
Revised: Admitting Lemma 20, we now prove

Original: Theorem 8.
Revised: Theorem 8.

Original: Proof of Theorem 8.
Revised: Proof of Theorem 8.

Original: Since q 0 = γ d ⊗ γ d = γ 2 d is stationary for the forward process (6), we have q t = γ 2 d for all t ≥ 0 .
Revised: Since q 0 = γ d ⊗ γ d = γ 2 d is stationary for the forward process (6), we have q t = γ 2 d for all t ≥ 0 .

Original: In this proof, since the score estimate is perfect and q T = γ 2 d , we simply denote the path measure for the algorithm as P T =
Revised: In this proof, since the score estimate is perfect and q T = γ 2 d , we simply denote the path measure for the algorithm as P T =

Original: P q T T .
Revised: P q T T .

Original: From Girsanov’s theorem in the form of
Revised: From Girsanov’s theorem in the form of

Original: Corollary 15 and from s T − kh
Revised: Corollary 15 and from s T − kh

Original: ( x, v ) = ∇ v ln q T − kh
Revised: ( x, v ) = ∇ v ln q T − kh

Original: ( x, v )
Revised: ( x, v )

Original: = − v , we have
Revised: = − v , we have

Original: [Equation]
Revised: [Equation]

Original: To lower bound this quantity, we use the inequality ∥ x + y ∥ 2 ≥ 12 ∥ x ∥ 2 − ∥
Revised: To lower bound this quantity, we use the inequality ∥ x + y ∥ 2 ≥ 12 ∥ x ∥ 2 − ∥

Original: y ∥ 2 to write, for t ∈
Revised: y ∥ 2 to write, for t ∈

Original: [ kh, ( k + 1) h ]
Revised: [ kh, ( k + 1) h ]

Original: [Equation]
Revised: [Equation]

Original: Using the fact that ¯ X s ∼ γ d
Revised: Using the fact that ¯ X s ∼ γ d

Original: and ¯ V s ∼ γ d for all s ∈ [ [Equation] T ]
Revised: and ¯ V s ∼ γ d for all s ∈ [0 , T ]

Original: [Equation]
Revised: [Equation]

Original: we can then bound
Revised: we can then bound

Original: [Equation]
Revised: [Equation]

Original: , provided that h ≤ 110 .
Revised: , provided that h ≤ 110 .

Original: Substituting this into (21),
Revised: Substituting this into (21),

Original: [Equation]

Original: dh

Original: [Equation]
Revised: [Equation]

Original: This proves the result.
Revised: This proves the result.

Original: This lower bound shows that the Girsanov discretization argument of Theorem 16 is essentially tight (except possibly the dependence on L ).
Revised: This lower bound shows that the Girsanov discretization argument of Theorem 16 is essentially tight (except possibly the dependence on L ).

Original: We now prove Lemma 20.
Revised: We now prove Lemma 20.

Original: Proof of Lemma 20.
Revised: Proof of Lemma 20.

Original: Similarly to the proof of Theorem 8 above, we note that
Revised: Similarly to the proof of Theorem 8 above, we note that

Original: [Equation]
Revised: [Equation]

Original: Hence, for a universal constant C > 0 (which may change from line to line)
Revised: Hence, for a universal constant C > 0 (which may change from line to line)

Original: E Q ← T exp
Revised: E Q ← T exp

Original: [Equation]
Revised: [Equation]

Original: By the Cauchy–Schwarz inequality, to prove that this expectation is finite, it suffices to consider the two terms in the exponential separately.
Revised: By the Cauchy–Schwarz inequality, to prove that this expectation is finite, it suffices to consider the two terms in the exponential separately.

Original: Next, we recall that
Revised: Next, we recall that

Original: [Equation]
Revised: [Equation]

Original: Define ¯ Y t :=
Revised: Define ¯ Y t :=

Original: ¯ X t
Revised: ¯ X t

Original: + ¯ V t .
Revised: + ¯ V t .

Original: Then, d ¯ Y t = − ¯ Y t d t + 2 d B t , which admits the explicit solution
Revised: Then, d ¯ Y t = − ¯ Y t d t + 2 d B t , which admits the explicit solution

Original: [Equation]
Revised: [Equation]

Original: Also, d ¯ X t = − ¯ X t d t + ¯ Y t d t , which admits the solution
Revised: Also, d ¯ X t = − ¯ X t d t + ¯ Y t d t , which admits the solution

Original: [Equation]
Revised: [Equation]

Original: Hence,
Revised: Hence,

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: where ˜ B is another standard Brownian motion and we use the interpretation of stochastic integrals as time changes of Brownian motion (Steele, 2001, Corollary 7.1).
Revised: where ˜ B is another standard Brownian motion and we use the interpretation of stochastic integrals as time changes of Brownian motion (Steele, 2001, Corollary 7.1).

Original: Since ( ¯ X 0 , ¯ V 0 ) ∼ γ 2 d has independent entries
Revised: Since ( ¯ X 0 , ¯ V 0 ) ∼ γ 2 d has independent entries

Original: [Equation]
Revised: [Equation]

Original: then
Revised: then

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: provided that h ≲ 1 / √ T .
Revised: provided that h ≲ 1 / √ T .

Original: Also, by the Cauchy–Schwarz inequality
Revised: Also, by the Cauchy–Schwarz inequality

Original: [Equation]
Revised: [Equation]

Original: we can give a crude bound: 
Revised: we can give a crude bound: writing

Original: [Equation]
Revised: [Equation]

Original: where, by standard estimates on the supremum of Brownian motion (see, e.g., Chewi et al., 2021b,
Revised: where, by standard estimates on the supremum of Brownian motion (see, e.g., Chewi et al., 2021b,

Original: Lemma 23), the first factor is finite if h ≲ 1 / √ T (again using independence across the dimensions).
Revised: Lemma 23), the first factor is finite if h ≲ 1 / √ T (again using independence across the dimensions).

Original: For the second factor, if we split the sum according to exp( − 2 t ) ≍ 2 k and use H ¨ older’s inequality,
Revised: For the second factor, if we split the sum according to exp( − 2 t ) ≍ 2 k and use H ¨ older’s inequality,

Original: [Equation]
Revised: [Equation]

Original: where K = O ( T ) .
Revised: where K = O ( T ) .

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original:  provided h ≲ 1 /T , where we again use Chewi et al.
Revised: , provided h ≲ 1 /T , where we again use Chewi et al.

Original: (2021b, Lemma 23) and split across the coordinates.
Revised: (2021b, Lemma 23) and split across the coordinates.

Original: The Cauchy–Schwarz inequality then implies
Revised: The Cauchy–Schwarz inequality then implies

Original: [Equation]
Revised: [Equation]

Original: For the second term, by independence of the increments,
Revised: For the second term, by independence of the increments,

Original: [Equation]
Revised: [Equation]

Original: By Chewi et al.
Revised: By Chewi et al.

Original: (2021b, Lemma 23), this quantity is finite if h ≲ 1 , which completes the proof.
Revised: (2021b, Lemma 23), this quantity is finite if h ≲ 1 , which completes the proof.

Original: D D ERIVATION OF THE SCORE MATCHING OBJECTIVE
Revised: D D ERIVATION OF THE SCORE MATCHING OBJECTIVE

Original: In this section, we present a self-contained derivation of the score matching objective (10) for the reader’s convenience.
Revised: In this section, we present a self-contained derivation of the score matching objective (10) for the reader’s convenience.

Original: See also Hyv¨arinen (2005); Vincent (2011); Song & Ermon (2019).
Revised: See also Hyv¨arinen (2005); Vincent (2011); Song & Ermon (2019).

Original: Recall that the problem is to solve
Revised: Recall that the problem is to solve

Original: [Equation]
Revised: [Equation]

Original: This objective cannot be evaluated, even if we replace the expectation over q t with an empirical average over samples from q t .
Revised: This objective cannot be evaluated, even if we replace the expectation over q t with an empirical average over samples from q t .

Original: The trick is to use an integration by parts identity to reformulate the objective.
Revised: The trick is to use an integration by parts identity to reformulate the objective.

Original: Here, C will denote any constant that does not depend on the optimization variable s t .
Revised: Here, C will denote any constant that does not depend on the optimization variable s t .

Original: Expanding the square,
Revised: Expanding the square,

Original: [Equation]
Revised: [Equation]

Original: We can rewrite the second term using integration by parts:
Revised: We can rewrite the second term using integration by parts:

Original: [Equation]
Revised: [Equation]

Original: where γ d = normal (0 , I d ) and we used the explicit form of the law of the OU process at time t .
Revised: where γ d = normal (0 , I d ) and we used the explicit form of the law of the OU process at time t .

Original: Recall the Gaussian integration by parts identity: for any vector field v : R d → R d ,
Revised: Recall the Gaussian integration by parts identity: for any vector field v : R d → R d ,

Original: [Equation]
Revised: [Equation]

Original: Applying this identity,
Revised: Applying this identity,

Original: [Equation]
Revised: [Equation]

Original: where x t = exp( − t ) x 0 + (cid:112) 1 − exp( − 2 t )
Revised: where x t = exp( − t ) x 0 + (cid:112) 1 − exp( − 2 t )

Original: z t .
Revised: z t .

Original: Substituting this in,
Revised: Substituting this in,

Original: [Equation]
Revised: [Equation]

Original: where X 0 ∼ q and Z t ∼ γ d are independent, and X t := exp( − t )
Revised: where X 0 ∼ q and Z t ∼ γ d are independent, and X t := exp( − t )

Original: X 

Revised: X

Original: + (cid:112) 1 − exp( − 2 t )
Revised: + (cid:112) 1 − exp( − 2 t )

Original: Z t .
Revised: Z t .

Original: E D EFERRED PROOFS
Revised: E D EFERRED PROOFS

Original: Lemma 21.
Revised: Lemma 21.

Original: Suppose that supp q ⊆ B (0 , R ) where R ≥ 1 , and let q t denote the law of the OU process at time t , started at q .
Revised: Suppose that supp q ⊆ B (0 , R ) where R ≥ 1 , and let q t denote the law of the OU process at time t , started at q .

Original: Let ε > 0 be such that ε ≪ √ d and set t ≍ ε 2 / ( √ d ( R ∨ √ d )) .
Revised: Let ε > 0 be such that ε ≪ √ d and set t ≍ ε 2 / ( √ d ( R ∨ √ d )) .

Original: Then,1.
Revised: Then,

Original: W 2 ( q t , q ) ≤ ε .
Revised: W 2 ( q t , q ) ≤ ε .

Original: q t satisfies
Revised: q t satisfies

Original: [Equation]
Revised: [Equation]

Original: For every t ′ ≥ t , q t ′ satisfies Assumption 1 with
Revised: For every t ′ ≥ t , q t ′ satisfies Assumption 1 with

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: For the OU process (1), we have ¯ X t := exp( − t )
Revised: For the OU process (1), we have ¯ X t := exp( − t )

Original: ¯ X 

Revised: ¯ X

Original: + (cid:112) 1 − exp( − 2 t )
Revised: + (cid:112) 1 − exp( − 2 t )

Original: Z , where
Revised: Z , where

Original: Z ∼ normal (0 , I d ) is independent of ¯ X 0 .
Revised: Z ∼ normal (0 , I d ) is independent of ¯ X 0 .

Original: Hence, for t ≲ 1 ,
Revised: Hence, for t ≲ 1 ,

Original: [Equation]
Revised: [Equation]

Original: We now take t ≲ min { ε/R, ε 2 /d } to ensure that W 22 ( q, q t ) ≤ ε 2 .
Revised: We now take t ≲ min { ε/R, ε 2 /d } to ensure that W 22 ( q, q t ) ≤ ε 2 .

Original: Since ε ≪ √ d , it suffices to take t ≍ ε 2 / ( √ d ( R ∨ √ d )) .
Revised: Since ε ≪ √ d , it suffices to take t ≍ ε 2 / ( √ d ( R ∨ √ d )) .

Original: For this, we use the short-time regularization result in Otto & Villani (2001, Corollary 2), which implies that
Revised: For this, we use the short-time regularization result in Otto & Villani (2001, Corollary 2), which implies that

Original: [Equation]
Revised: [Equation]

Original: Using Mikulincer & Shenfeld (2022, Lemma 4), along the OU process,
Revised: Using Mikulincer & Shenfeld (2022, Lemma 4), along the OU process,

Original: [Equation]
Revised: [Equation]

Original: With our choice of t , it implies
Revised: With our choice of t , it implies

Original: [Equation]
Revised: [Equation]