Original: A CCELERATING G UIDED D IFFUSION S AMPLING WITH S PLITTING N UMERICAL M ETHODS
Revised: A CCELERATING G UIDED D IFFUSION S AMPLING WITH S PLITTING N UMERICAL M ETHODS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Guided diffusion is a technique for conditioning the output of a diffusion model at sampling time without retraining the network for each specific task.
Revised: Guided diffusion is a technique for conditioning the output of a diffusion model at sampling time without retraining the network for each specific task.

Original: One drawback of diffusion models, however, is their slow sampling process.
Revised: One drawback of diffusion models, however, is their slow sampling process.

Original: Recent techniques can accelerate unguided sampling by applying high-order numerical methods to the sampling process when viewed as differential equations.
Revised: Recent techniques can accelerate unguided sampling by applying high-order numerical methods to the sampling process when viewed as differential equations.

Original: On the contrary, we discover that the same techniques do not work for guided sampling, and little has been explored about its acceleration.
Revised: On the contrary, we discover that the same techniques do not work for guided sampling, and little has been explored about its acceleration.

Original: This paper explores the culprit of this problem and provides a solution based on operator splitting methods, motivated by our key finding that classical high-order numerical methods are unsuitable for the conditional function.
Revised: This paper explores the culprit of this problem and provides a solution based on operator splitting methods, motivated by our key finding that classical high-order numerical methods are unsuitable for the conditional function.

Original: Our proposed method can re-utilize the high-order methods for guided sampling and can generate images with the same quality as a 250step DDIM baseline using 32-58% less sampling time on ImageNet256.
Revised: Our proposed method can re-utilize the high-order methods for guided sampling and can generate images with the same quality as a 250step DDIM baseline using 32-58% less sampling time on ImageNet256.

Original: We also demonstrate usage on a wide variety of conditional generation tasks, such as textto-image generation, colorization, inpainting, and super-resolution.
Revised: We also demonstrate usage on a wide variety of conditional generation tasks, such as textto-image generation, colorization, inpainting, and super-resolution.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: A family of generative models known as diffusion models has recently gained a lot of attention with state-of-the-art image generation quality (Dhariwal & Nichol, 2021).
Revised: A family of generative models known as diffusion models has recently gained a lot of attention with state-of-the-art image generation quality (Dhariwal & Nichol, 2021).

Original: Guided diffusion is an approach for controlling the output of a trained diffusion model for conditional generation tasks without retraining its network.
Revised: Guided diffusion is an approach for controlling the output of a trained diffusion model for conditional generation tasks without retraining its network.

Original: By engineering a task-specific conditional function and modifying only the sampling procedure, guided diffusion models can be used in a variety of applications, such as class-conditional image generation (Dhariwal & Nichol, 2021; Kawar et al., 2022), text-to-image generation (Nichol et al., 2021), image-to-image translation (Zhao et al., 2022), inpainting (Chung et al., 2022a), colorization (Song et al., 2020b), image composition (Sasaki et al., 2021), adversarial purification (Wang et al., 2022; Wu et al., 2022) and super-resolution (Choi et al., 2021).
Revised: By engineering a task-specific conditional function and modifying only the sampling procedure, guided diffusion models can be used in a variety of applications, such as class-conditional image generation (Dhariwal & Nichol, 2021; Kawar et al., 2022), text-to-image generation (Nichol et al., 2021), image-to-image translation (Zhao et al., 2022), inpainting (Chung et al., 2022a), colorization (Song et al., 2020b), image composition (Sasaki et al., 2021), adversarial purification (Wang et al., 2022; Wu et al., 2022) and super-resolution (Choi et al., 2021).

Original: One common drawback of both guided and regular “unguided” diffusion models is their slow sampling processes, usually requiring hundreds of iterations to produce a single image.
Revised: One common drawback of both guided and regular “unguided” diffusion models is their slow sampling processes, usually requiring hundreds of iterations to produce a single image.

Original: Recent speedup attempts include improving the noise schedule (Nichol & Dhariwal, 2021;
Revised: Recent speedup attempts include improving the noise schedule (Nichol & Dhariwal, 2021;

Original: Watson et al., 2021), redefining the diffusion process to be non-Markovian, thereby allowing a deterministic sampling process Song et al.
Revised: Watson et al., 2021), redefining the diffusion process to be non-Markovian, thereby allowing a deterministic sampling process Song et al.

Original: (2020a), network distillation that teaches a student model to simulate multiple sampling steps of a teacher model Salimans & Ho (2022)
Revised: (2020a), network distillation that teaches a student model to simulate multiple sampling steps of a teacher model Salimans & Ho (2022)

Original: ; Luhman & Luhman (2021), among others.
Revised: ; Luhman & Luhman (2021), among others.

Original: Song et al.
Revised: Song et al.

Original: (2020a) show how each sampling step can be expressed as a first-order numerical step of an ordinary differential equation (ODE).
Revised: (2020a) show how each sampling step can be expressed as a first-order numerical step of an ordinary differential equation (ODE).

Original: Similarly, Song et al.
Revised: Similarly, Song et al.

Original: (2020b) express the sampling of a score-based model as solving a stochastic differential equation (SDE).
Revised: (2020b) express the sampling of a score-based model as solving a stochastic differential equation (SDE).

Original: By regarding the sampling process as an ODE/SDE, many high-order numerical methods have been suggested, such as Liu et al.
Revised: By regarding the sampling process as an ODE/SDE, many high-order numerical methods have been suggested, such as Liu et al.

Original: (2022), Zhang & Chen (2022), and Zhang et al.
Revised: (2022), Zhang & Chen (2022), and Zhang et al.

Original: (2022) with impressive results on unguided diffusion models.
Revised: (2022) with impressive results on unguided diffusion models.

Original: However, when applied to guided diffusion models, these methods produce surprisingly poor results (see Figure 1)—given a few number of steps, those high-order numerical methods actually perform worse than low-order methods.
Revised: However, when applied to guided diffusion models, these methods produce surprisingly poor results (see Figure 1)—given a few number of steps, those high-order numerical methods actually perform worse than low-order methods.

Original: Guided sampling differs from the unguided one by the addition of the gradients of the conditional function to its sampling equation.
Revised: Guided sampling differs from the unguided one by the addition of the gradients of the conditional function to its sampling equation.

Original: The observed performance decline thus suggests that classical high-order methods may not be suitable for the conditional function and, consequently, the guided sampling equation as a whole.
Revised: The observed performance decline thus suggests that classical high-order methods may not be suitable for the conditional function and, consequently, the guided sampling equation as a whole.

Original: Our paper tests this hypothesis and presents an approach to accelerating guided diffusion sampling.
Revised: Our paper tests this hypothesis and presents an approach to accelerating guided diffusion sampling.

Original: The key idea is to use an operator splitting method to split the
Revised: The key idea is to use an operator splitting method to split the

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: less well-behaved conditional function term from the standard diffusion term and solve them separately.
Revised: less well-behaved conditional function term from the standard diffusion term and solve them separately.

Original: This approach not only allows re-utilizing the successful high-order methods on the diffusion term but also provides us with options to combine different specialized methods for each term to maximize performance.
Revised: This approach not only allows re-utilizing the successful high-order methods on the diffusion term but also provides us with options to combine different specialized methods for each term to maximize performance.

Original: Splitting method can be used to solve diffusion SDE in Dockhorn et al.
Revised: Splitting method can be used to solve diffusion SDE in Dockhorn et al.

Original: Our design process includes comparing different splitting methods and numerical methods for each split term.
Revised: Our design process includes comparing different splitting methods and numerical methods for each split term.

Original: When tested on ImageNet, our approach achieves the same level of image quality as a
Revised: When tested on ImageNet, our approach achieves the same level of image quality as a

Original: DDIM baseline while reducing the sampling time by approximately 32-58%.
Revised: DDIM baseline while reducing the sampling time by approximately 32-58%.

Original: Compared with other sampling methods using the same sampling time, our approach provides better image quality as measured by LPIPS, FID, and Perception/Recall.
Revised: Compared with other sampling methods using the same sampling time, our approach provides better image quality as measured by LPIPS, FID, and Perception/Recall.

Original: With only minimal modifications to the sampling equation, we also show successful acceleration on various conditional generation tasks.
Revised: With only minimal modifications to the sampling equation, we also show successful acceleration on various conditional generation tasks.

Original:  B ACKGROUND
Revised:  B ACKGROUND

Original: This section provides a high-level summary of the theoretical foundation of diffusion models as well as numerical methods that have been used for diffusion models.
Revised: This section provides a high-level summary of the theoretical foundation of diffusion models as well as numerical methods that have been used for diffusion models.

Original: Here we briefly explain a few that contribute to our method.
Revised: Here we briefly explain a few that contribute to our method.

Original: 2. D IFFUSION M ODELS
Revised: 2. D IFFUSION M ODELS

Original: Assuming that x 0 is a random variable from the data distribution we wish to reproduce, diffusion models define a sequence of Gaussian noise degradation of x 0 as random variables x 1 , x 2 , ..., x T , where x t ∼ N ( √ 1 − β t x t − 1 , β t I ) and β t ∈ [0 , 1] are parameters that control the noise levels.
Revised: Assuming that x 0 is a random variable from the data distribution we wish to reproduce, diffusion models define a sequence of Gaussian noise degradation of x 0 as random variables x 1 , x 2 , ..., x T , where x t ∼ N ( √ 1 − β t x t − 1 , β t I ) and β t ∈ [0 , 1] are parameters that control the noise levels.

Original: With a property of Gaussian distribution, we can express x t directly as a function of x 0 and noise ϵ ∼ N (0 , I ) by x t = √ ¯ α t x 0 + √ 1 − ¯ α t ϵ , where ¯ α t = (cid:81) ti =1 (1 − β i ) .
Revised: With a property of Gaussian distribution, we can express x t directly as a function of x 0 and noise ϵ ∼ N (0 , I ) by x t = √ ¯ α t x 0 + √ 1 − ¯ α t ϵ , where ¯ α t = (cid:81) ti =1 (1 − β i ) .

Original: By picking a sufficiently large
Revised: By picking a sufficiently large

Original: T (e.g., 1,000) and an appropriate set of β t , we can assume x T is a standard Gaussian distribution.
Revised: T (e.g., 1,000) and an appropriate set of β t , we can assume x T is a standard Gaussian distribution.

Original: The main idea of diffusion model generation is to sample a Gaussian noise x T and use it to reversely sample x T − 1 , x T − 2 , ... until we obtain x 0 , which belongs to our data distribution.
Revised: The main idea of diffusion model generation is to sample a Gaussian noise x T and use it to reversely sample x T − 1 , x T − 2 , ... until we obtain x 0 , which belongs to our data distribution.

Original: Ho et al.
Revised: Ho et al.

Original: propose Denoising Diffusion Probabilistic Model (DDPM) and explain how to employ a neural network ϵ θ ( x t , t ) to predict the noise ϵ that is used to compute x t .
Revised: propose Denoising Diffusion Probabilistic Model (DDPM) and explain how to employ a neural network ϵ θ ( x t , t ) to predict the noise ϵ that is used to compute x t .

Original: To train the network, we sample a training image x 0 , t , and ϵ to compute x t using the above relationship.
Revised: To train the network, we sample a training image x 0 , t , and ϵ to compute x t using the above relationship.

Original: Then, we optimize our network ϵ θ to minimize the difference between the predicted and real noise, i.e., ∥ ϵ − ϵ θ ( x t , t ) ∥ 2 .
Revised: Then, we optimize our network ϵ θ to minimize the difference between the predicted and real noise, i.e., ∥ ϵ − ϵ θ ( x t , t ) ∥ 2 .

Original: Song et al.
Revised: Song et al.

Original: (2020a) introduce Denoising Diffusion Implicit Model (DDIM), which uses the network ϵ θ to deterministically obtain x t − 1 given x t .
Revised: (2020a) introduce Denoising Diffusion Implicit Model (DDIM), which uses the network ϵ θ to deterministically obtain x t − 1 given x t .

Original: The DDIM generative process can be written as
Revised: The DDIM generative process can be written as

Original: [Equation]
Revised: [Equation]

Original: This formulation could be used to skip many sampling steps and boost sampling speed.
Revised: This formulation could be used to skip many sampling steps and boost sampling speed.

Original: To turn this into an ODE, we rewrite Equation 1 as:
Revised: To turn this into an ODE, we rewrite Equation 1 as:

Original: [Equation]
Revised: [Equation]

Original: which is now equivalent to a numerical step in solving an ODE.
Revised: which is now equivalent to a numerical step in solving an ODE.

Original: To derive the corresponding ODE, we can re-parameterize σ t = √ 1 − ¯ α t / √ ¯ α t , ¯ x ( t ) =
Revised: To derive the corresponding ODE, we can re-parameterize σ t = √ 1 − ¯ α t / √ ¯ α t , ¯ x ( t ) =

Original: x t / √ ¯ α t and ¯ ϵ σ (¯ x ) =
Revised: x t / √ ¯ α t and ¯ ϵ σ (¯ x ) =

Original: ϵ θ ( x t , t ) , yielding ¯ x ( t − ∆
Revised: ϵ θ ( x t , t ) , yielding ¯ x ( t − ∆

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: By letting ( σ t − ∆ t − σ t ) → 0 , the ODE becomes:
Revised: By letting ( σ t − ∆ t − σ t ) → 0 , the ODE becomes:

Original: [Equation]
Revised: [Equation]

Original: Note that this change of variables is equivalent to an exponential integrator technique described in both Zhang & Chen (2022) and Lu et al.
Revised: Note that this change of variables is equivalent to an exponential integrator technique described in both Zhang & Chen (2022) and Lu et al.

Original: Since x t and ¯ x ( t ) have the same value at t = 0 , our work can focus on solving ¯ x ( t ) rather than x t .
Revised: Since x t and ¯ x ( t ) have the same value at t = 0 , our work can focus on solving ¯ x ( t ) rather than x t .

Original: Many numerical methods can be applied to the
Revised: Many numerical methods can be applied to the

Original: ODE Equation 3 to accelerate diffusion sampling.
Revised: ODE Equation 3 to accelerate diffusion sampling.

Original: We next discuss some of them that are relevant.
Revised: We next discuss some of them that are relevant.

Original: [Figure]
Revised: [Figure]

Original: M ETHODS
Revised: M ETHODS

Original: Euler’s Method is the most basic numerical method.
Revised: Euler’s Method is the most basic numerical method.

Original: A forward Euler step is given by ¯ x n +1 =
Revised: A forward Euler step is given by ¯ x n +1 =

Original: ¯ x n + ∆ σ ¯ ϵ σ (¯ x n ) .
Revised: ¯ x n + ∆ σ ¯ ϵ σ (¯ x n ) .

Original: When we apply the forward Euler step to the ODE Equation 3, we get the DDIM formulation (Song et al., 2020a).
Revised: When we apply the forward Euler step to the ODE Equation 3, we get the DDIM formulation (Song et al., 2020a).

Original: Heun’s Method , also known as the trapezoid rule or improved Euler, is given by: ¯ x n +1 = ¯ x n + ∆ σ 2 ( e 1 + e 2 ) , where e 1 =
Revised: Heun’s Method , also known as the trapezoid rule or improved Euler, is given by: ¯ x n +1 = ¯ x n + ∆ σ 2 ( e 1 + e 2 ) , where e 1 =

Original: ¯ ϵ σ (¯ x n ) and e 2 =
Revised: ¯ ϵ σ (¯ x n ) and e 2 =

Original: ¯ ϵ σ (¯ x n + ∆ σe 1 ) .
Revised: ¯ ϵ σ (¯ x n + ∆ σe 1 ) .

Original: This method modifies Euler’s method into a two-step method to improve accuracy.
Revised: This method modifies Euler’s method into a two-step method to improve accuracy.

Original: Many papers have used this method on diffusion models, including Algorithm 1 in Karras et al.
Revised: Many papers have used this method on diffusion models, including Algorithm 1 in Karras et al.

Original: and DPM-Solver-2 in Lu et al.
Revised: and DPM-Solver-2 in Lu et al.

Original: This method is also the simplest case of Predictor-Corrector methods used in Song et al.
Revised: This method is also the simplest case of Predictor-Corrector methods used in Song et al.

Original: (2020b).
Revised: (2020b).

Original: Runge-Kutta Methods represent a class of numerical methods that integrate information from multiple hidden steps and provide high accuracy results.
Revised: Runge-Kutta Methods represent a class of numerical methods that integrate information from multiple hidden steps and provide high accuracy results.

Original: Heun’s method also belongs to a family of 2 nd -order Runge-Kutta methods (RK2).
Revised: Heun’s method also belongs to a family of 2 nd -order Runge-Kutta methods (RK2).

Original: The most well-known variant is the 4 th -order Runge-Kutta method (RK4), which is written as follows:
Revised: The most well-known variant is the 4 th -order Runge-Kutta method (RK4), which is written as follows:

Original: [Equation]
Revised: [Equation]

Original: This method has been tested on diffusion models in Liu et al.
Revised: This method has been tested on diffusion models in Liu et al.

Original: and Salimans & Ho (2022), but it has not been used as the main proposed method in any paper.
Revised: and Salimans & Ho (2022), but it has not been used as the main proposed method in any paper.

Original: Linear Multi-Step Method , similar to the Runge-Kutta methods, aims to combine information from several steps; however, rather than evaluating new hidden steps, this method uses the previous steps to estimate the new step.
Revised: Linear Multi-Step Method , similar to the Runge-Kutta methods, aims to combine information from several steps; however, rather than evaluating new hidden steps, this method uses the previous steps to estimate the new step.

Original: The 1 st -order formulation is the same as Euler’s method.
Revised: The 1 st -order formulation is the same as Euler’s method.

Original: The 2 nd -order formulation is given by
Revised: The 2 nd -order formulation is given by

Original: [Equation]
Revised: [Equation]

Original: (5) while the 4 th -order formulation is given by
Revised: (5) while the 4 th -order formulation is given by

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where e k = ¯
Revised: where e k = ¯

Original: ϵ σ (¯ x n − k ) .
Revised: ϵ σ (¯ x n − k ) .

Original: These formulations are designed for a constant ∆ σ in each step.
Revised: These formulations are designed for a constant ∆ σ in each step.

Original: However, our experiments and previous work that uses this method (e.g., Liu et al.
Revised: However, our experiments and previous work that uses this method (e.g., Liu et al.

Original: ; Zhang & Chen
Revised: ; Zhang & Chen

Original: (2022)) still show good results when this assumption is not strictly satisfied, i.e., when ∆ σ is not constant.
Revised: (2022)) still show good results when this assumption is not strictly satisfied, i.e., when ∆ σ is not constant.

Original: We will refer to these formulations as PLMS (Pseudo Linear Multi-Step) for the rest of the paper, like in Liu et al.
Revised: We will refer to these formulations as PLMS (Pseudo Linear Multi-Step) for the rest of the paper, like in Liu et al.

Original: A similar linear multi-step method for non-constant ∆ σ can also be derived using a technique used in Zhang & Chen (2022), which we detail in Appendix B. The method can improve upon PLMS slightly, but it is not as flexible because we have to re-derive the update rule every time the σ schedule changes.
Revised: A similar linear multi-step method for non-constant ∆ σ can also be derived using a technique used in Zhang & Chen (2022), which we detail in Appendix B. The method can improve upon PLMS slightly, but it is not as flexible because we have to re-derive the update rule every time the σ schedule changes.

Original:  S PLITTING M ETHODS FOR G UIDED D IFFUSION M ODELS
Revised:  S PLITTING M ETHODS FOR G UIDED D IFFUSION M ODELS

Original: This section introduces our technique that uses splitting numerical methods to accelerate guided diffusion sampling.
Revised: This section introduces our technique that uses splitting numerical methods to accelerate guided diffusion sampling.

Original: We first focus our investigation on classifier-guided diffusion models for classconditional generation and later demonstrate how this technique can be used for other conditional generation tasks in Section 4.3.
Revised: We first focus our investigation on classifier-guided diffusion models for classconditional generation and later demonstrate how this technique can be used for other conditional generation tasks in Section 4.3.

Original: Like any guided diffusion models, classifier-guided models (Dhariwal & Nichol, 2021) share the same training objective with regular unguided models with no modifications to the training procedure; but the sampling process is guided by an additional gradient signal from an external classifier to generate class-specific output images.
Revised: Like any guided diffusion models, classifier-guided models (Dhariwal & Nichol, 2021) share the same training objective with regular unguided models with no modifications to the training procedure; but the sampling process is guided by an additional gradient signal from an external classifier to generate class-specific output images.

Original: Specifically, the sampling process is given by
Revised: Specifically, the sampling process is given by

Original: [Equation]
Revised: [Equation]

Original: where p ϕ ( c | x t ) is a classifier model trained to output the probability of x t belonging to class c .
Revised: where p ϕ ( c | x t ) is a classifier model trained to output the probability of x t belonging to class c .

Original: As discussed in the previous section, we can rewrite this formulation as a “guided ODE”:
Revised: As discussed in the previous section, we can rewrite this formulation as a “guided ODE”:

Original: [Equation]
Revised: [Equation]

Original: (8) where f σ (¯ x ) =
Revised: (8) where f σ (¯ x ) =

Original: [Equation]
Revised: [Equation]

Original: log p ϕ ( c | x t ) .
Revised: log p ϕ ( c | x t ) .

Original: We refer to f σ as the conditional function, which can be substituted with other functions for different tasks.
Revised: We refer to f σ as the conditional function, which can be substituted with other functions for different tasks.

Original: After obtaining the ODE form, any numerical solver mentioned earlier can be readily applied to accelerate the sampling process.
Revised: After obtaining the ODE form, any numerical solver mentioned earlier can be readily applied to accelerate the sampling process.

Original: However, we observe that classical high-order numerical methods (e.g., PLMS4, RK4) fail to accelerate this task (see Figure 1) and even perform worse than the baseline DDIM.
Revised: However, we observe that classical high-order numerical methods (e.g., PLMS4, RK4) fail to accelerate this task (see Figure 1) and even perform worse than the baseline DDIM.

Original: We hypothesize that the two terms in the guided ODE may have different numerical behaviors with the conditional term being less suitable to classical high-order methods.
Revised: We hypothesize that the two terms in the guided ODE may have different numerical behaviors with the conditional term being less suitable to classical high-order methods.

Original: We speculate that the difference could be partly attributed to how they are computed: ∇ f σ (¯ x ) is computed through backpropagation, whereas ¯ ϵ σ (¯ x ) is computed directly by evaluating a network.
Revised: We speculate that the difference could be partly attributed to how they are computed: ∇ f σ (¯ x ) is computed through backpropagation, whereas ¯ ϵ σ (¯ x ) is computed directly by evaluating a network.

Original: One possible solution to handle terms with different behaviors is the so-called operator splitting method, which divides the problem into two subproblems:
Revised: One possible solution to handle terms with different behaviors is the so-called operator splitting method, which divides the problem into two subproblems:

Original: [Equation]
Revised: [Equation]

Original: We call these the diffusion and condition subproblems, respectively.
Revised: We call these the diffusion and condition subproblems, respectively.

Original: This method allows separating the hard-to-approximate ∇ f σ ( z ) from ¯ ϵ σ ( y ) and solving them separately in each time step.
Revised: This method allows separating the hard-to-approximate ∇ f σ ( z ) from ¯ ϵ σ ( y ) and solving them separately in each time step.

Original: Importantly, this helps reintroduce the effective use of high-order methods on the diffusion subproblem as well as provides us with options to combine different specialized methods to maximize performance.
Revised: Importantly, this helps reintroduce the effective use of high-order methods on the diffusion subproblem as well as provides us with options to combine different specialized methods to maximize performance.

Original: We explore two most famous first- and second-order splitting techniques for our task:
Revised: We explore two most famous first- and second-order splitting techniques for our task:

Original: 3. L IE -T ROTTER S PLITTING (LTSP)
Revised: 3. L IE -T ROTTER S PLITTING (LTSP)

Original: Our first example is the simple first-order Lie-Trotter splitting method (Trotter, 1959), which expresses the splitting as
Revised: Our first example is the simple first-order Lie-Trotter splitting method (Trotter, 1959), which expresses the splitting as

Original: [Equation]
Revised: [Equation]

Original: with the solution of this step being ¯ x n +1 =
Revised: with the solution of this step being ¯ x n +1 =

Original: z ( σ n +1 ) .
Revised: z ( σ n +1 ) .

Original: Note that σ n is a decreasing sequence.
Revised: Note that σ n is a decreasing sequence in sampling schedule.

Original: Here Equation 10 is the same as Equation 3, which can be solved using anyhigh-order numerical method,
Revised: Here Equation 10 is the same as Equation 3, which can be solved using any

Original: [Equation]
Revised: [Equation]

Original: for n ∈ { 0 , ..., N −
Revised: for n ∈ { 0 , ..., N −

Original: [Equation]
Revised: [Equation]

Original: do
Revised: do

Original: [Equation]
Revised: [Equation]

Original: endResult:
Revised: endResult

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: for n ∈ {
Revised: for n ∈ {

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: N −
Revised: N −

Original: [Equation]
Revised: [Equation]

Original: do
Revised: do

Original: [Equation]
Revised: [Equation]

Original: endResult:¯
Revised: endResult:

Original: [Equation]
Revised: [Equation]

Original:  e.g., PLMS.
Revised: N high-order numerical method, e.g., PLMS.

Original: For Equation 11, we can use a forward Euler step:
Revised: For Equation 11, we can use a forward Euler step:

Original: [Equation]
Revised: [Equation]

Original: This is equivalent to a single iteration of standard gradient descent with a learning rate ∆ σ .
Revised: This is equivalent to a single iteration of standard gradient descent with a learning rate ∆ σ .

Original: This splitting scheme is summarized by Algorithm 1.
Revised: This splitting scheme is summarized by Algorithm 1.

Original: We investigate different numerical methods for each subproblem in Section 4.1.
Revised: We investigate different numerical methods for each subproblem in Section 4.1.

Original: 3. S TRANG S PLITTING (STSP)
Revised: 3. S TRANG S PLITTING (STSP)

Original: Strang splitting (or Strang-Marchuk) (Strang, 1968) is one of the most famous and widely used operator splitting methods.
Revised: Strang splitting (or Strang-Marchuk) (Strang, 1968) is one of the most famous and widely used operator splitting methods.

Original: This second-order splitting works as follows:
Revised: This second-order splitting works as follows:

Original: [Equation]
Revised: [Equation]

Original: Instead of solving each subproblem for a full step length, we solve the condition subproblem for half a step before and after solving the diffusion subproblem for a full step.
Revised: Instead of solving each subproblem for a full step length, we solve the condition subproblem for half a step before and after solving the diffusion subproblem for a full step.

Original: In theory, we can swap the order of operations without affecting convergence, but it is practically cheaper to compute the condition term twice rather than the diffusion term twice because f σ is typically a smaller network compared to ¯ ϵ σ .
Revised: In theory, we can swap the order of operations without affecting convergence, but it is practically cheaper to compute the condition term twice rather than the diffusion term twice because f σ is typically a smaller network compared to ¯ ϵ σ .

Original: The Strange splitting algorithm is shown in Algorithm 2.
Revised: The Strange splitting algorithm is shown in Algorithm 2.

Original: This method can be proved to have better accuracy than the Lie-Trotter method using the Banker-Campbell-Hausdorff formula (Tuckerman, 2010), but it requires evaluating the condition term twice per step in exchange for improved image quality.
Revised: This method can be proved to have better accuracy than the Lie-Trotter method using the Banker-Campbell-Hausdorff formula (Tuckerman, 2010), but it requires evaluating the condition term twice per step in exchange for improved image quality.

Original: We assess this trade-off in the experiment section.
Revised: We assess this trade-off in the experiment section.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: Extending on our observation that classical high-order methods failed on guided sampling, we conducted a series of experiments to investigate this problem and evaluate our solution.
Revised: Extending on our observation that classical high-order methods failed on guided sampling, we conducted a series of experiments to investigate this problem and evaluate our solution.

Original: Section 4.
Revised: Section 4.

Original: uses a simple splitting method (first-order LTSP) to study the effects that high-order methods have on each subproblem, leading to our key finding that only the conditional subproblem is less suited to classical high-order methods.
Revised: uses a simple splitting method (first-order LTSP) to study the effects that high-order methods have on each subproblem, leading to our key finding that only the conditional subproblem is less suited to classical high-order methods.

Original: This section also determines the best combination of numerical methods for the two subproblems under LTSP splitting.
Revised: This section also determines the best combination of numerical methods for the two subproblems under LTSP splitting.

Original: Section 4.2 explores improvements from using a higher-order splitting method and compares our best scheme to previous work.
Revised: Section 4.2 explores improvements from using a higher-order splitting method and compares our best scheme to previous work.

Original: Finally, Section 4. applies our approach to a variety of conditional generation tasks with minimal changes.
Revised: Finally, Section 4. applies our approach to a variety of conditional generation tasks with minimal changes.

Original: For our comparison, we use pre-trained state-of-the-art diffusion models and classifiers from Dhariwal & Nichol (2021), which were trained on the ImageNet dataset (Russakovsky et al., 2015) with 1000 total sampling step.
Revised: For our comparison, we use pre-trained state-of-the-art diffusion models and classifiers from Dhariwal & Nichol (2021), which were trained on the ImageNet dataset (Russakovsky et al., 2015) with 1000 total sampling step.

Original: We treat full-path samples from a classifier-guided DDIM at 1,000 steps as reference solutions.
Revised: We treat full-path samples from a classifier-guided DDIM at 1,000 steps as reference solutions.

Original: Then the performance of each configuration is measured by the image similarity between its generated samples using fewer steps and the reference DDIM samples, both starting from the same initial noise maps.
Revised: Then the performance of each configuration is measured by the image similarity between its generated samples using fewer steps and the reference DDIM samples, both starting from the same initial noise maps.

Original: Given the same sampling time, we expect configurations with better performance to better match the full DDIM.
Revised: Given the same sampling time, we expect configurations with better performance to better match the full DDIM.

Original: We measure image similarity using Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al., 2018) (lower is better) and measure sampling time using a single NVIDIA RTX 3090 and a 24-core AMD Threadripper 3960x.
Revised: We measure image similarity using Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al., 2018) (lower is better) and measure sampling time using a single NVIDIA RTX 3090 and a 24-core AMD Threadripper 3960x.

Original: [Figure]
Revised: [Figure]

Original: 4. F INDING A SUITABLE NUMERICAL METHOD FOR EACH SUBPROBLEM
Revised: 4. F INDING A SUITABLE NUMERICAL METHOD FOR EACH SUBPROBLEM

Original: To study the effects of different numerical methods on each subproblem of the guided ODE (Equation 8), we use the simplest Lie-Trotter splitting, which itself requires no additional network evaluations.
Revised: To study the effects of different numerical methods on each subproblem of the guided ODE (Equation 8), we use the simplest Lie-Trotter splitting, which itself requires no additional network evaluations.

Original: This controlled experiment has two setups: a) we fix the numerical method for the condition subproblem (Equation 11) to first-order PLMS1 (Euler’s method) and vary the numerical method for the diffusion
Revised: This controlled experiment has two setups: a) we fix the numerical method for the condition subproblem (Equation 11) to first-order PLMS1 (Euler’s method) and vary the numerical method for the diffusion

Original: subproblem (Equation 10)
Revised: subproblem (Equation 10)

Original: [Figure]
Revised: [Figure]

Original: and conversely b) we fix the method for the diffusion subproblem
Revised: and conversely b) we fix the method for the diffusion subproblem

Original: and vary the method for the condition subproblem.
Revised: and vary the method for the condition subproblem.

Original: The numerical method options are Euler’s method (PLMS1), Heun’s method (RK2), 4 th order Runge-Kutta’s method (RK4), and 2 nd /

Revised: The numerical method options are Euler’s method (PLMS1), Heun’s method (RK2), 4 th order Runge-Kutta’s method (RK4), and 2 nd /

Original: th order pseudo linear multi-step (PLMS2/PLMS4).
Revised: th order pseudo linear multi-step (PLMS2/PLMS4).

Original: We report LPIPS vs. sampling time of various numerical combinations on a diffusion model trained on ImageNet 256 × 256 in Figure 2.
Revised: We report LPIPS vs. sampling time of various numerical combinations on a diffusion model trained on ImageNet 256 × 256 in Figure 2.

Original: The red dotted lines indicate a reference DDIM score obtained from 250 sampling steps, a common choice that produces good samples that are perceptually close to those from a full 1,000-step DDIM (Dhariwal & Nichol, 2021; Nichol & Dhariwal, 2021).
Revised: The red dotted lines indicate a reference DDIM score obtained from 250 sampling steps, a common choice that produces good samples that are perceptually close to those from a full 1,000-step DDIM (Dhariwal & Nichol, 2021; Nichol & Dhariwal, 2021).

Original: Given a long sampling time, non-split PLMS4 performs better than the DDIM baseline.
Revised: Given a long sampling time, non-split PLMS4 performs better than the DDIM baseline.

Original: However, when the sampling time is reduced, the image quality of PLMS4 rapidly decreases and becomes much worse than that of DDIM, especially under 15 seconds in Figure 2.
Revised: However, when the sampling time is reduced, the image quality of PLMS4 rapidly decreases and becomes much worse than that of DDIM, especially under 15 seconds in Figure 2.

Original: When we split the ODE and solve both subproblems using first-order PLMS1 (Euler), the performance is close to that of
Revised: When we split the ODE and solve both subproblems using first-order PLMS1 (Euler), the performance is close to that of

Original: DDIM, which is also considered first-order but without any splitting.
Revised: DDIM, which is also considered first-order but without any splitting.

Original: This helps verify that merely splitting the ODE does not significantly alter the sampling speed.
Revised: This helps verify that merely splitting the ODE does not significantly alter the sampling speed.

Original: In the setup a), when RK2 and RK4 are used for the diffusion subproblem, they also perform worse than the DDIM baseline.
Revised: In the setup a), when RK2 and RK4 are used for the diffusion subproblem, they also perform worse than the DDIM baseline.

Original: This slowdown is caused by the additional evaluations of the network by these methods, which outweigh the improvement gained in each longer diffusion step.
Revised: This slowdown is caused by the additional evaluations of the network by these methods, which outweigh the improvement gained in each longer diffusion step.

Original: Note that if we instead measure the image quality with respect to the number of diffusion steps, RK2 and RK can outperform other methods (Appendix E); however, this is not our metric of interest.
Revised: Note that if we instead measure the image quality with respect to the number of diffusion steps, RK2 and RK can outperform other methods (Appendix E); however, this is not our metric of interest.

Original: On the other hand, PLMS2 and PLMS4, which require no additional network evaluations, are about 8-10% faster than DDIM and can achieve the same LPIPS score as the DDIM that uses 250 sampling steps in 20-26 fewer steps.
Revised: On the other hand, PLMS2 and PLMS4, which require no additional network evaluations, are about 8-10% faster than DDIM and can achieve the same LPIPS score as the DDIM that uses 250 sampling steps in 20-26 fewer steps.

Original: Importantly, when the sampling time is reduced, their performance does not degrade rapidly like the non-split PLMS4 and remains at the same level as DDIM.
Revised: Importantly, when the sampling time is reduced, their performance does not degrade rapidly like the non-split PLMS4 and remains at the same level as DDIM.

Original: In the setup b) where we vary the numerical method for the condition subproblem, the result reveals an interesting contrast—none of the methods beats DDIM and some even make the sampling diverged [PLMS1, RK4].
Revised: In the setup b) where we vary the numerical method for the condition subproblem, the result reveals an interesting contrast—none of the methods beats DDIM and some even make the sampling diverged [PLMS1, RK4].

Original: These findings suggest that the gradients of conditional functions are less “compatible” with classical high-order methods, especially when used with a small number of steps.
Revised: These findings suggest that the gradients of conditional functions are less “compatible” with classical high-order methods, especially when used with a small number of steps.

Original: This phenomenon may be related to the “stiffness” condition of ODEs, which we discuss further in
Revised: This phenomenon may be related to the “stiffness” condition of ODEs, which we discuss further in

Original: Section 5.
Revised: Section 5.

Original: For the remainder of our experiments, we will use the combination [PLMS4, PLMS1] for the diffusion and condition subproblems, respectively.
Revised: For the remainder of our experiments, we will use the combination [PLMS4, PLMS1] for the diffusion and condition subproblems, respectively.

Original: [Figure]
Revised: [Figure]

Original: Sampling time within
Revised: Sampling time within

Original: [Figure]
Revised: [Figure]

Original: sec
Revised: sec

Original: [Table]
Revised: [Table]

Original: 4. I MPROVED SPLITTING METHOD
Revised: 4. I MPROVED SPLITTING METHOD

Original: This experiment investigates improvements from using a high-order splitting method, specifically the Strang splitting method, with the numerical combination [PLMS4, PLMS1] and compares our methods to previous work.
Revised: This experiment investigates improvements from using a high-order splitting method, specifically the Strang splitting method, with the numerical combination [PLMS4, PLMS1] and compares our methods to previous work.

Original: Note that besides DDIM Dhariwal & Nichol (2021), no previous work is specifically designed for accelerating guided sampling, thus the baselines in this comparison are only adaptations of the core numerical methods used in those papers.
Revised: Note that besides DDIM Dhariwal & Nichol (2021), no previous work is specifically designed for accelerating guided sampling, thus the baselines in this comparison are only adaptations of the core numerical methods used in those papers.

Original: And to our knowledge, no prior guided-diffusion work uses splitting numerical methods.
Revised: And to our knowledge, no prior guided-diffusion work uses splitting numerical methods.

Original: Non-split numerical method baselines are
Revised: Non-split numerical method baselines are

Original: PLMS4, which is used in Liu et al. (2022), RK2, which is used in Karras et al. (2022); Lu et al.
Revised: PLMS4, which is used in Liu et al. (2022), RK2, which is used in Karras et al. (2022); Lu et al.

Original: (2022), and higher-order RK4.
Revised: (2022), and higher-order RK4.

Original: We report the LPIPS scores of these methods with respect to the sampling time in Figure 3 and Table 1.
Revised: We report the LPIPS scores of these methods with respect to the sampling time in Figure 3 and Table 1.

Original: Without any splitting, PLMS4, RK2 and RK4 show significantly poorer image quality when used with short sampling times < 10 seconds.
Revised: Without any splitting, PLMS4, RK2 and RK4 show significantly poorer image quality when used with short sampling times < 10 seconds.

Original: The best performer is our Strang splitting (STSP4), which can reach the same quality as 250-step DDIM while using 32-58% less sampling time.
Revised: The best performer is our Strang splitting (STSP4), which can reach the same quality as 250-step DDIM while using 32-58% less sampling time.

Original: STSP4 also obtains the highest LPIPS scores for sample times of 5, 10, 15, and 20 seconds.
Revised: STSP4 also obtains the highest LPIPS scores for sample times of 5, 10, 15, and 20 seconds.

Original: More statistical details and comparison with other split combinations are in Appendix F, G.
Revised: More statistical details and comparison with other split combinations are in Appendix F, G.

Original: In addition, we perform a quantitative evaluation for class-conditional generation by sampling 50,000 images based on uniformly chosen class conditions with a small number of sampling steps and evaluating the Fenchel Inception Distance (FID)
Revised: In addition, we perform a quantitative evaluation for class-conditional generation by sampling 50,000 images based on uniformly chosen class conditions with a small number of sampling steps and evaluating the Fenchel Inception Distance (FID)

Original: Heusel et al.
Revised: Heusel et al.

Original: (2017) (lower is better) and the improved precision/recall Kynk ¨ a ¨ anniemi et al.
Revised: (2017) (lower is better) and the improved precision/recall Kynk ¨ a ¨ anniemi et al.

Original: (2019) (higher is better) against an ImageNet test set.
Revised: (2019) (higher is better) against an ImageNet test set.

Original: Following (Dhariwal & Nichol, 2021), we use a 25-step DDIM as a baseline, which already produces visually reasonable results.
Revised: Following (Dhariwal & Nichol, 2021), we use a 25-step DDIM as a baseline, which already produces visually reasonable results.

Original: As PLMS and LTSP require the same number of network evaluations as the DDIM, they are used also with 25 steps.
Revised: As PLMS and LTSP require the same number of network evaluations as the DDIM, they are used also with 25 steps.

Original: For STSP with a longer network evaluation time, it is only allowed 20 steps, which is the highest number of steps such that its sampling time is within that of the baseline 25-step DDIM.
Revised: For STSP with a longer network evaluation time, it is only allowed 20 steps, which is the highest number of steps such that its sampling time is within that of the baseline 25-step DDIM.

Original: Here LTSP2 and STSP2 are Lie-Trotter and Strang splitting methods with the combination [PLMS2, PLMS1].
Revised: Here LTSP2 and STSP2 are Lie-Trotter and Strang splitting methods with the combination [PLMS2, PLMS1].

Original: In Table 2, we report the results of three different ImageNet resolutions and the average sampling time per image in seconds.
Revised: In Table 2, we report the results of three different ImageNet resolutions and the average sampling time per image in seconds.

Original: Our STSP4 performs best on all measurements except Recall on ImageNet512.
Revised: Our STSP4 performs best on all measurements except Recall on ImageNet512.

Original: On ImageNet512,
Revised: On ImageNet512,

Original: PLMS4 has the highest Recall score but a poor FID of 16, indicating that the generated images have good distribution coverage but may poorly represent the real distribution.
Revised: PLMS4 has the highest Recall score but a poor FID of 16, indicating that the generated images have good distribution coverage but may poorly represent the real distribution.

Original: On ImageNet256, STSP can yield 4.49 FID in 20 steps, compared to 4.59 FID in 250 steps originally reported in the paper (Dhariwal & Nichol, 2021); our STSP4 is about 9.4 × faster when tested on the same machine.
Revised: On ImageNet256, STSP can yield 4.49 FID in 20 steps, compared to 4.59 FID in 250 steps originally reported in the paper (Dhariwal & Nichol, 2021); our STSP4 is about 9.4 × faster when tested on the same machine.

Original: 4. S PLITTING METHODS IN OTHER TASKS
Revised: 4. S PLITTING METHODS IN OTHER TASKS

Original: Besides class-conditional generation, our approach can also accelerate any conditional image generation as long as the gradient of the conditional function can be defined.
Revised: Besides class-conditional generation, our approach can also accelerate any conditional image generation as long as the gradient of the conditional function can be defined.

Original: We test our approach on four tasks: text-to-image generation, image inpainting, colorization, and super-resolution.
Revised: We test our approach on four tasks: text-to-image generation, image inpainting, colorization, and super-resolution.

Original: Text-to-image generation: We use a pre-trained text-to-image Disco-Diffusion (Letts et al., 2021) based on Crowson (2021), which substitutes the classifier output with the dot product of the image and caption encodings from CLIP (Radford et al., 2021).
Revised: Text-to-image generation: We use a pre-trained text-to-image Disco-Diffusion (Letts et al., 2021) based on Crowson (2021), which substitutes the classifier output with the dot product of the image and caption encodings from CLIP (Radford et al., 2021).

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: “A beautiful painting of
Revised: “A beautiful painting of

Original: [Figure]
Revised: [Figure]

Original: over
Revised: over

Original: [Figure]
Revised: [Figure]

Original: sunflower sea, trending on artstation.”
Revised: sunflower sea, trending on artstation.”

Original: Full DDIM
Revised: Full DDIM

Original: DDIM
Revised: DDIM

Original: PLMS
Revised: PLMS

Original: LTSP
Revised: LTSP

Original: STSP (1,000 steps) (45 steps) (45 steps) (45 steps) (30 steps) (approximately using the same sampling time)
Revised: STSP (1,000 steps) (45 steps) (45 steps) (45 steps) (30 steps) (approximately using the same sampling time)

Original: Image inpainting & colorization: For these two tasks, we follow the techniques proposed in Song et al.
Revised: Image inpainting & colorization: For these two tasks, we follow the techniques proposed in Song et al.

Original: (2020b) and Chung et al.
Revised: (2020b) and Chung et al.

Original: (2022a), which improves the conditional functions of both tasks with “manifold constraints.
Revised: (2022a), which improves the conditional functions of both tasks with “manifold constraints.

Original: ” We use the same diffusion model Dhariwal & Nichol (2021) trained on
Revised: ” We use the same diffusion model Dhariwal & Nichol (2021) trained on

Original: ImageNet as our earlier Experiments 4.1, 4.2.
Revised: ImageNet as our earlier Experiments 4.1, 4.2.

Original: Super-resolution: We follow the formulation from ILVR (Choi et al., 2021)
Revised: Super-resolution: We follow the formulation from ILVR (Choi et al., 2021)

Original: combined with the manifold contraints Chung et al.
Revised: combined with the manifold contraints Chung et al.

Original: (2022a), and also use our earlier ImageNet diffusion model.
Revised: (2022a), and also use our earlier ImageNet diffusion model.

Original: Figure 4 compares our techniques, LTSP4 and STSP4, with the DDIM baseline and PLMS4 on text-to-image generation.
Revised: Figure 4 compares our techniques, LTSP4 and STSP4, with the DDIM baseline and PLMS4 on text-to-image generation.

Original: Each result is produced using a fixed sampling time of about 26 seconds.
Revised: Each result is produced using a fixed sampling time of about 26 seconds.

Original: STSP4, which uses 30 diffusion steps compared to 45 in the other methods, produces more realistic results with color contrast that is more similar to the full DDIM references’.
Revised: STSP4, which uses 30 diffusion steps compared to 45 in the other methods, produces more realistic results with color contrast that is more similar to the full DDIM references’.

Original: Figure 5 shows that our STSP4 produces more convincing results than the DDIM baseline with fewer artifacts on the other three tasks while using the same 5 second sampling time.
Revised: Figure 5 shows that our STSP4 produces more convincing results than the DDIM baseline with fewer artifacts on the other three tasks while using the same 5 second sampling time.

Original: Implementation details, quantitative evaluations, and more results are in Appendix J, K.
Revised: Implementation details, quantitative evaluations, and more results are in Appendix J, K.

Original:  D ISCUSSION
Revised:  D ISCUSSION

Original: Our findings show that when the sampling ODE consists of multiple terms from different networks, their numerical behaviors can be different and treating them separately can be more optimal.
Revised: Our findings show that when the sampling ODE consists of multiple terms from different networks, their numerical behaviors can be different and treating them separately can be more optimal.

Original: Another promising direction is to improve the behavior of the gradient of the conditional function / classifier itself and study whether related properties such as adversarial robustness or gradient smoothness can induce the desirable temporal smoothness in the sampling ODE.
Revised: Another promising direction is to improve the behavior of the gradient of the conditional function / classifier itself and study whether related properties such as adversarial robustness or gradient smoothness can induce the desirable temporal smoothness in the sampling ODE.

Original: However, it is not yet clear what specific characteristics of the behavior play an important role.
Revised: However, it is not yet clear what specific characteristics of the behavior play an important role.

Original: This challenge may be related to a
Revised: This challenge may be related to a

Original: Original
Revised: Original

Original: Input
Revised: Input

Original: [Figure]
Revised: [Figure]

Original: DDIM
Revised: DDIM

Original: I np a i n ti ng
Revised: I np a i n ti ng

Original: C
Revised: C

Original: o l o r i za ti on 8x S up e rR e s o l u ti on condition called “stiffness” in solving ODEs Ernst & Gerhard (2010), which lacks a clear definition but describes the situation where explicit numerical methods, such as RK or PLMS, require a very small step size even in a region with smooth curvature.
Revised: o l o r i za ti on 8x S up e rR e s o l u ti on condition called “stiffness” in solving ODEs Ernst & Gerhard (2010), which lacks a clear definition but describes the situation where explicit numerical methods, such as RK or PLMS, require a very small step size even in a region with smooth curvature.

Original: As an alternative to the classifier-guided model, Ho & Salimans (2022) propose a classifier-free model that can perform conditional generation without a classifier while remaining a generative model.
Revised: As an alternative to the classifier-guided model, Ho & Salimans (2022) propose a classifier-free model that can perform conditional generation without a classifier while remaining a generative model.

Original: This model can utilize high-order methods as no classifier is involved, but it requires evaluating the classifier-free network twice per step, which is typically more expensive than evaluating a normal diffusion model and a classifier.
Revised: This model can utilize high-order methods as no classifier is involved, but it requires evaluating the classifier-free network twice per step, which is typically more expensive than evaluating a normal diffusion model and a classifier.

Original: It is important to note that our accelerating technique and classifier-free models are not mutually exclusive, and one can still apply a conditional function and our splitting technique to guide a classifier-free model in a direction it has not been trained for.
Revised: It is important to note that our accelerating technique and classifier-free models are not mutually exclusive, and one can still apply a conditional function and our splitting technique to guide a classifier-free model in a direction it has not been trained for.

Original: While our paper only focuses on ODEs derived from the deterministic sampling of DDIM, one can convert SDE-based diffusion models to ODEs (Karras et al., 2022) and still use our technique.
Revised: While our paper only focuses on ODEs derived from the deterministic sampling of DDIM, one can convert SDE-based diffusion models to ODEs (Karras et al., 2022) and still use our technique.

Original: More broadly, we can accelerate any diffusion model that can be expressed as a differential equation with a summation of two terms.
Revised: More broadly, we can accelerate any diffusion model that can be expressed as a differential equation with a summation of two terms.

Original: When these terms behave differently, the benefit from splitting can be substantial.
Revised: When these terms behave differently, the benefit from splitting can be substantial.

Original: Nevertheless, our findings are based on common, existing models and σ schedule from
Revised: Nevertheless, our findings are based on common, existing models and σ schedule from

Original: Dhariwal & Nichol (2021).
Revised: Dhariwal & Nichol (2021).

Original: Further investigation into the impact of the σ schedule or different types and architectures of diffusion models is still required.
Revised: Further investigation into the impact of the σ schedule or different types and architectures of diffusion models is still required.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we investigate the failure to accelerate guided diffusion sampling of classical highorder numerical methods and propose a solution based on splitting numerical methods.
Revised: In this paper, we investigate the failure to accelerate guided diffusion sampling of classical highorder numerical methods and propose a solution based on splitting numerical methods.

Original: We found that the gradients of conditional functions are less suitable to classical high-order numerical methods and design a technique based on Strang splitting and a combination of forth- and first-order numerical methods.
Revised: We found that the gradients of conditional functions are less suitable to classical high-order numerical methods and design a technique based on Strang splitting and a combination of forth- and first-order numerical methods.

Original: Our method achieves better LPIPS and FID scores than previous work given the same sampling time and is 32-58% faster than a 250-step DDIM baseline.
Revised: Our method achieves better LPIPS and FID scores than previous work given the same sampling time and is 32-58% faster than a 250-step DDIM baseline.

Original: Our technique can successfully accelerate a variety of tasks, such as text-to-image generation, inpainting, colorization, and super-resolution.
Revised: Our technique can successfully accelerate a variety of tasks, such as text-to-image generation, inpainting, colorization, and super-resolution.