MCMC Correction of Score-Based Diffusion Models for Model Composition

Abstract

Diffusion models can be parameterized in terms of either score or energy function. The energy parameterization is attractive as it enables sampling procedures such as Markov Chain Monte Carlo (MCMC) that incorporates a Metropolis–Hastings (MH) correction step based on energy differences between proposed samples. Such corrections can significantly improve sampling quality, particularly in the context of model composition, where pre-trained models are combined to generate samples from novel distributions. Score-based diffusion models, on the other hand, are more widely adopted and come with a rich ecosystem of pre-trained models. However, they do not, in general, define an underlying energy function, making MH-based sampling inapplicable. In this work, we address this limitation by retaining score parameterization and introducing a novel MH-like acceptance rule based on line integration of the score function. This allows the reuse of existing diffusion models while still combining the reverse process with various MCMC techniques, viewed as an instance of annealed MCMC. Through experiments on synthetic and real-world data, we show that our MH-like samplers yield relative improvements of similar magnitude to those observed with energy-based models, without requiring explicit energy parameterization.

1. Introduction

Significant advancements have recently been achieved in generative modeling across various domains [1,2,3]. These models have become potent priors for a wide range of applications, including code generation [4], text-to-image generation [5], question-answering  [2], and many others [6,7]. Among generative models, diffusion models [3,8,9] have arguably emerged as the most powerful class. Diffusion models learn to denoise corrupted inputs in small, gradual steps and are capable of generating samples from complex distributions. They have been successful in many domains, such as generating highly realistic images [10], modeling temporal point processes [11], and even generating neural network parameters [12].

Diffusion models also offer the capability of composed sampling, which combines pre-trained models to generate samples from a new distribution. This approach, known as model composition, has a rich history [13,14,15,16]. For diffusion models, the most common form of composition is classifier-guided sampling, where the reverse process is augmented by a separate classifier model [8,10,17], but other compositions have also been explored [18]. The ability to compose new models without having to re-learn the individual components is especially appealing for diffusion models since their ever-increasing size and data hunger make them exceedingly costly to train [19]. Therefore, developing sampling methods that work for pre-trained diffusion models is valuable.

The foundation of composed sampling for diffusion models is score-based, where we interpret diffusion models as predictors of the score function for the marginal distribution at each diffusion step [20]. From this perspective, MCMC methods, such as the Langevin algorithm (LA) [21] or Hamiltonian Monte Carlo (HMC) sampling [22], emerge as viable options to incorporate. Augmenting the standard reverse process with additional MCMC sampling has been shown to improve composed sampling for diffusion models [18,20]. However, we are traditionally restricted to unadjusted variants of these samplers, namely Unadjusted LA (U-LA) and Unadjusted HMC (U-HMC), which only require utilization of the score. For instance, the seminal work by Song and Ermon [9] proposed score-based generative modeling by combining score networks with unadjusted Langevin dynamics. While highly effective, these unadjusted methods lack a rejection mechanism and may introduce discretization bias at larger step sizes. Correcting this bias requires evaluating the unnormalized density for a Metropolis–Hastings (MH) step [23,24]—something that is not directly possible in standard score-parameterized diffusion models since they do not provide access to the underlying energy.

An intriguing alternative to directly modeling the score function is to model the marginal distribution with an energy function, from which the score can be obtained through explicit differentiation [9,25]. This parameterization connects diffusion models and energy-based models (EBMs) [26] and offers several desirable properties. With energy parameterization, we can evaluate the unnormalized density and guarantee a proper score function. This, in turn, enables an MH correction step when employing an MCMC method, where the MH acceptance probability is computed from the energy function. Adding such a correction step has been shown to improve sampling performance in composed models [18]. Nevertheless, the score parameterization remains far more popular, as it avoids the direct computation of the gradient of the log density. In practice, score-based models are significantly easier to train and scale to large datasets and high-dimensional problems than energy-based parameterizations, a practical advantage that has driven their widespread adoption.

In this study, we build on the work in [18] and introduce a novel approach to obtain an MH-like correction step directly from pre-trained diffusion models without relying on an energy-based parameterization. Specifically, we use a connection between the score and the energy to estimate the MH acceptance probability by approximating a line integral along the vector field generated by the score. This enables an improved sampling procedure for various pre-trained score-parameterized diffusion models. We find that our approximate method yields relative improvements of a similar magnitude to those obtained with explicit energy parameterizations, without having to estimate the energy directly.

In summary, our main contributions are:

• We show that MH-like correction sampling can be directly applied to score-based models without requiring additional training.
• We introduce two efficient algorithms to approximate the energy difference used in MH and demonstrate that our pseudo-energy difference more accurately represents analytical energy differences than an explicitly trained energy model in a toy example while performing on par with the energy model on MNIST.
• We establish that the relative improvements in sample quality achieved via MCMC for energy-based models can be closely matched within a purely score-based framework. In our implementations, the pseudo-MH correction also exhibits favorable practical runtime behavior compared to energy-based MCMC.

2. Background

2.1. Diffusion Models

We consider Gaussian diffusion models initially proposed by [8] and further improved by [3,9]. Starting with a sample from the data distribution $x_0\sim q(·)$, we construct a Markov chain of latent variables $x_1,\dots ,x_T$ by iteratively introducing Gaussian noise to the sample $q(x_t|x_{t-1})=\mathcal{N}\left(x_t;\sqrt{1-{\beta }_t}{x}_{t-1},{\beta }_{t}I\right)$, where ${\beta }_t\in [0,1),\forall t=1,\dots ,T$ are known. For large enough T we have $q(x_T)\approx \mathcal{N}\left(x_T;0,I\right)$.

A diffusion model learns to gradually denoise samples by modeling the distribution of the previous sample in the chain $p_{\theta }(x_{t-1}\mid x_t),t=1,\dots ,T$. Approximate samples from the data distribution $q(x_0)$ are obtained by starting from $x_T\sim \mathcal{N}\left(0,I\right)$ and sequentially sampling less noisy versions of the sample until the noise is removed. This is called the reverse process.

The reverse distribution is typically modeled as $p_{\theta }(x_{t-1}|x_t)=\mathcal{N}(x_{t-1};{\mu }_{\theta }(x_t,t),{\mathsf{\Sigma }}_{\theta }(x_t,t))$, since the posterior $q(x_{t-1}|x_t)$ can be well approximated by a Gaussian distribution when the noise magnitude ${\beta }_t$ is sufficiently small. The mean is parameterized as ${\mu }_{\theta }(x_t,t)={\displaystyle \frac{1}{\sqrt{{\alpha }_t}}}\left(x_t-{\displaystyle \frac{{\beta }_t}{{\sigma }_t}}{ϵ}_{\theta }(x_t,t)\right)$, where ${\alpha }_t$ and ${\sigma }_t$ are positive and defined by ${\{{\beta }_t\}}_{t=1}^T$ [3]. The noise prediction model ${ϵ}_{\theta }(x_t,t)$, typically a neural network, is learned from data. We assume ${\mathsf{\Sigma }}_{\theta }(x_t,t)={\beta }_{t}I$ throughout unless otherwise stated.

2.2. Energy-Based Models

Energy-based models (EBMs) represent probability distributions with a scalar, non-negative energy function $E_{\theta }$ by assigning low energy to regions of the input space where the probability is high and high energy to regions where the distribution has little or no support:

$$\begin{array}{cc}\hfill p_{\theta }(x_t,t)& ={\displaystyle \frac{1}{Z_{\theta }(t)}}exp\left(-{\displaystyle \frac{1}{{\sigma }_t}}{E}_{\theta }(x_t,t)\right),\hfill \\ \hfill Z_{\theta }(t)& =\int exp\left(-{\displaystyle \frac{1}{{\sigma }_t}}{E}_{\theta }(x_t,t)\right)\mathrm{d}{x}_t.\hfill \end{array}$$

(1)

Here, we define $E_{\theta }$ as a time-dependent function and deliberately choose not to absorb ${\sigma }_t$ (introduced in the previous section) into $E_{\theta }$, to maintain a more explicit connection to diffusion models, as clarified in the next section. This time dependency can be seen as a sequence of energy functions, one for each diffusion step t. The normalization constant $Z_{\theta }$ is typically intractable, prohibiting computing a normalized density. However, $Z_{\theta }$ does not depend on the input $x_t$, making the so-called score function easy to compute,

$${\nabla }_{x}log{p}_{\theta }(x_t,t)=-{\displaystyle \frac{1}{{\sigma }_t}}{\nabla }_x{E}_{\theta }(x_t,t),$$

(2)

even though the gradient of the energy function can be costly to compute in practice.

2.3. Energy and Score Parameterized Diffusion Models

In the diffusion setting, one approach to training EBMs is denoising score matching (DSM). When the data are perturbed with Gaussian noise, the DSM loss coincides with the diffusion training loss (up to a factor of ${\sigma }_t^2$) [20]. This equivalence arises by identifying the noise prediction model, ${ϵ}_{\theta }(x_t,t)$, with the score of an energy function, i.e., as an EBM:

$$\begin{array}{c}\hfill {ϵ}_{\theta }(x_t,t)={\nabla }_x{E}_{\theta }(x_t,t).\end{array}$$

(3)

Under this identification, ${ϵ}_{\theta }(x_t,t)$ is required to define a proper score. Thus, an EBM and a plain diffusion model differ only in parameterization: the former uses energy parameterization via $E_{\theta }$, while the latter, since ${ϵ}_{\theta }$ is only a pseudo-score, is referred to as using score parameterization.

Both parameterizations have their advantages and disadvantages. The energy parameterization can evaluate density $p_{\theta }(x_t,t)$ up to normalization $Z_{\theta }(t)$, which enables various MCMC methods. Furthermore, by making the score equal to the gradient of an actual scalar function, we ensure a proper score. On the other hand, to evaluate the score function, $E_{\theta }$ must be explicitly differentiated, which can be costly.

The score parameterization is more flexible as it predicts an arbitrary vector field. While there is some empirical evidence that this improves sampling performance in diffusion processes [18], this difference may primarily stem from model architecture [25]. Nevertheless, the score parameterization’s direct estimation of the score function makes it more efficient for reverse process sampling and remains the more widely adopted approach. In the next section, we describe how these parameterizations affect the design of MCMC samplers for diffusion models.

2.4. MCMC Sampling for Diffusion Models

MCMC sampling is a promising strategy for improving diffusion model sampling since it can be combined with the reverse process. MCMC methods are naturally defined through transition kernels, and just like the reverse process there are MCMC methods that base their transitions on the score function, such as the Unadjusted Langevin Algorithm (U-LA) and the Unadjusted Hamiltonian Monte Carlo (U-HMC) [27,28]. We let $\tau$ denote the index of the MCMC iterations (as opposed to t, which refers to the diffusion timestep).

For U-LA we use the kernel

$$\begin{array}{c}\hfill k_t\left(x^{\tau +1}\mid x^{\tau }\right)=\mathcal{N}\left(x^{\tau +1};x^{\tau }+{\delta }_t{\nabla }_{x}log{p}_{\theta }(x^{\tau },t),2{\delta }_{t}I\right),\end{array}$$

at diffusion step t, where $x^0=x_t$, ${\delta }_t$ is the step size, and the chain is iterated for $L_t$ steps.

For U-HMC we augment the state with momenta $v^{\tau }\sim \mathcal{N}\left(0,M_t\right)$ (diagonal mass $M_t$), and propose a new state by applying ${\ell }_t$ leapfrog steps of size ${\epsilon }_t$ under the same score field. Writing

$$\begin{array}{c}\hfill (x^{\tau +1},v^{\tau +1})={\mathrm{LF}}_{{\ell }_t,{\epsilon }_t}\left(x^{\tau },v^{\tau }\right),\end{array}$$

for the leapfrog map (which in our case is defined with the score ${\nabla }_{x}log{p}_{\theta }(x,t)$ as potential gradient), the kernel for x is given implicitly by this deterministic proposal after marginalizing out $\mathbf{v}$. In practice, we may perform $L_t$ such proposals per diffusion step t.

These methods are called unadjusted since, as $L_t$ grows, the Markov chains converges to the target distribution only in the limit of infinitesimal step sizes. By adding, for instance, a Metropolis–Hastings (MH) correction step, we can sample with larger step sizes and still converge to the target distribution [23,24]. With the correction, we sample a candidate $\widehat{x}\sim k_t\left(·\mid x^{\tau }\right)$ and accept it as the new iterate with probability

$$\begin{array}{c}\hfill \alpha =min\left(1,{\displaystyle \frac{p_{\theta }(\widehat{x},t)}{p_{\theta }(x^{\tau },t)}}{\displaystyle \frac{k_t\left(x^{\tau }\mid \widehat{x}\right)}{k_t\left(\widehat{x}\mid x^{\tau }\right)}}\right),\end{array}$$

(4)

so that $x^{\tau +1}=\widehat{x}$ with probability $\alpha$, and $x^{\tau +1}=x^{\tau }$ otherwise. With the correction, U-LA becomes LA (often referred to as MALA) [29], and U-HMC becomes the standard HMC algorithm [22,30].

The model $p_{\theta }$ appears only through a ratio, so a normalized density is not required. When $p_{\theta }$ is parameterized as an EBM (see (1)), the ratio simplifies to

$$\begin{array}{c}\hfill {\displaystyle \frac{p_{\theta }(\widehat{x},t)}{p_{\theta }(x^{\tau },t)}}=exp\left({\displaystyle \frac{1}{{\sigma }_t}}\left(E_{\theta }(x^{\tau },t)-E_{\theta }(\widehat{x},t)\right)\right),\end{array}$$

(5)

which allows us to directly evaluate the MH acceptance probability, making it straightforward to construct an adjusted MCMC sampler. This offers a key advantage over the score parameterization, where only an approximation of the score is accessible which cannot directly be used to compute the probability ratio needed in MH.

2.5. Sampling from Composed Models

Composed sampling is a powerful feature of diffusion models that enables sampling from new target distributions by combining multiple pre-trained models. Rather than retraining a model for every new task or data combination, one can reuse existing components. This flexibility is especially appealing in large-scale settings, where retraining is often prohibitively expensive.

The most common form of composition is guidance [10], where the goal is to sample from a distribution conditioned on a class label y,

$$\begin{array}{c}\hfill q(x_0\mid y)\propto q(x_0)q(y\mid x_0).\end{array}$$

(6)

This is implemented by modifying the score function at each diffusion step as

$$\begin{array}{c}\hfill {\nabla }_{x}log{p}_{\theta }(x_t,t)+\lambda {\nabla }_{x}log{p}_{\phi }(y\mid x_t,t),\end{array}$$

(7)

where $p_{\theta }$ is an unconditional diffusion model and $p_{\phi }$ is a classifier predicting class y. Hyperparameter $\lambda$ controls the strength of the conditioning. We refer to this approach as classifier-full guidance. Other variants include reconstruction guidance [31,32] and classifier-free guidance [17].

More generally, ref. [18] explores a range of composition types beyond guidance, including products, negations, and mixtures. A product composition—of which guidance can be seen as a special case—is defined as

$$q^{\Pi }(x_0)\propto \prod _i{q}^i(x_0),$$

(8)

and leads to the composed model at diffusion step t,

$$\begin{array}{c}\hfill p_{\theta }^{\Pi }(x_t,t)\propto \prod _i{p}_{{\theta }_i}^i(x_t,t)=exp\left(-{\displaystyle \frac{1}{{\sigma }_t}}\sum _i{E}_{{\theta }_i}^i(x_t,t)\right).\end{array}$$

(9)

This distribution is then used as the target in MCMC sampling, resulting in improved sampling performance.

Importantly, the factorization in (8) only strictly holds at $t=0$; at intermediate diffusion steps, the composed model $p_{\theta }^{\Pi }(x_t,t)$ does not generally correspond to the true marginal of any product data distribution; refer to Du et al. [18]. This becomes problematic when relying solely on the reverse process, which assumes access to a valid score function for the true intermediate marginals. However, this construction remains valid and effective from the perspective of annealed MCMC [33], where the overall sampling procedure is interpreted as a chain targeting a sequence of gradually evolving distributions. From this viewpoint, the intermediate distributions $p_{\theta }^{\Pi }(x_t,t)$ are treated as design choices that guide the chain toward the final target $q^{\Pi }(x_0)$, and asymptotic correctness is still preserved. In practice, since diffusion models are trained using denoising score matching, the sampling process converges to a denoised version of $q^{\Pi }(x_0)$, which can be made arbitrarily close to the true distribution by construction.

Note that for models using a score-based parameterization, a pseudo-score for this type of composition is equal to $-{\displaystyle \frac{1}{{\sigma }_t}}{\sum }_i{ϵ}_{{\theta }_i}^i(x_t,t)$.

3. MCMC Correction Step for Score Parameterization

We propose combining the energy parameterization properties with the performance and practical accessibility of score parameterization. Instead of using energy parameterization and computing the score by differentiation, we take the complementary approach: using score parameterization and computing the change in (pseudo-)energy by integrating the score.

3.1. Pseudo-Energy Difference and MH-like Correction

This section describes how MCMC acceptance probabilities can be approximated given only access to a score function. The Metropolis–Hastings acceptance probability in (4) depends on the relative probability of a proposed state $\widehat{x}$ compared to the current state $x^{\tau }$. Since the proposal kernel $k_t\left(·\mid ·\right)$ is straightforward to evaluate, the acceptance ratio reduces to comparing unnormalized target densities.

For an energy-based model, this ratio depends only on the difference and—because $E_{\theta }$ is a scalar potential—this difference can be written as a path-independent line integral,

$$\begin{array}{cc}\hfill E_{\theta }(x^{\tau },t)-E_{\theta }(\widehat{x},t)& = -{\int }_{\mathcal{C}}{\nabla }_{\xi }{E}_{\theta }(\xi ,t)·d\xi ,\hfill \end{array}$$

(10)

for any differentiable curve $\mathcal{C}$ connecting $x^{\tau }$ and $\widehat{x}$. This provides a natural interpretation of MH in terms of integrating the model’s score field along a path.

For score-parameterized diffusion models, the situation is reversed: the model provides a score field ${ϵ}_{\theta }(x,t)$ but no explicit energy function. Motivated by the fact that denoising score matching trains ${ϵ}_{\theta }$ to approximate the gradient of the log-density under Gaussian perturbations, we define an analogous pseudo-energy difference

$$\begin{array}{c}\hfill \Delta {\tilde{E}}_{\mathcal{C}}(x\to x^{\prime },t)=-{\int }_{\mathcal{C}}{ϵ}_{\theta }(\xi ,t)·d\xi ,\end{array}$$

(11)

where $\mathcal{C}$ is any smooth curve connecting x and $x^{\prime }$. This construction can be viewed as integrating the vector field ${ϵ}_{\theta }$ along a path, thereby approximating the change in an underlying scalar potential—if such a potential existed.

Based on this quantity, we define an MH-like acceptance probability

$$\begin{array}{c}\hfill {\alpha }_{\mathrm{MH}-\mathrm{like}}=min\left(1,exp\left[{\displaystyle \frac{1}{{\sigma }_t}}\Delta {\tilde{E}}_{\mathcal{C}}(x\to x^{\prime },t)\right]{\displaystyle \frac{k_t(x\mid x^{\prime })}{k_t(x^{\prime }\mid x)}}\right).\end{array}$$

(12)

If ${ϵ}_{\theta }(x,t)={\nabla }_x{F}_{\theta }(x,t)$ for some potential $F_{\theta }$, then $\Delta {\tilde{E}}_C$ equals $F_{\theta }(x^{\prime },t)-F_{\theta }(x,t)$ for all curves C, and (12) exactly recovers the true MH acceptance probability by the fundamental theorem of line integrals.

In practice, ${ϵ}_{\theta }$ is not perfectly conservative and the pseudo-energy difference depends on the chosen path $\mathcal{C}$. We therefore consider two practical choices: (i) a straight line between x and $x^{\prime }$ and (ii) a curved trajectory following the leapfrog path of an HMC proposal. The latter allows reusing score evaluations already computed during proposal generation, yielding higher numerical accuracy without additional model evaluations.

In both cases, the integral in (11) is approximated using the trapezoidal rule, with the number of line segments treated as a hyperparameter n. This requires score evaluations at internal points of $\mathcal{C}$, but avoids differentiating the model; by contrast, the energy parameterization evaluates $E_{\theta }$ at the endpoints but must differentiate it to obtain the score.

An overview of the full sampling procedure is given in Algorithm 1. At each diffusion step, an optional reverse update is followed by an MCMC refinement targeting the intermediate distribution. This structure aligns naturally with the annealed MCMC framework, where both the reverse update and the MCMC kernel act as design choices guiding the chain toward the final distribution. Including the reverse step typically improves sample quality [18].

Algorithm 1 Annealed MCMC with MH-like correction

Require:  Score function ${ϵ}_{\theta }(·,t)$; reverse-diffusion schedule ${({\alpha }_t,{\beta }_t,{\sigma }_t)}_{t=1}^T$; kernel family ${\{k_t(·\mid ·)\}}_{t=1}^T$ (LA or HMC) with kernel hyperparameters; MCMC steps per diffusion step $\{L_t\}$; integration mode $\mathtt{mode}\in \{\mathtt{line},\mathtt{curve}\}$; line segments n (if line); per-leapfrog subsegments m (if curve).   1: Sample initial $x_T\sim \mathcal{N}(0,I)$.   2: for $t=T$ to 1 do   3:       (Optional) Reverse step: update $x_{t-1}$ from $x_t$ using the standard reverse-diffusion update at time t   4:       if $t>1$ then   5:             $t^{\prime }\leftarrow t-1$               ▹ all MCMC quantities live at time $t^{\prime }$   6:             Initialize MCMC: $x^0\leftarrow x_{t^{\prime }}$.   7:             for $\tau =1$ to $L_t$ do   8:                   Propose candidate $\widehat{x}\sim k_{t^{\prime }}(·\mid x^{\tau -1})$          ▹ LA or HMC kernel   9:                   $\Delta {\tilde{E}}_{\mathcal{C}}\leftarrow \mathtt{EstimatePseudoEnergy}(x^{\tau -1},\widehat{x},t^{\prime };\mathtt{mode},n,m)$        ▹ Algorithm 2 or Algorithm 3 10:                   Compute acceptance $${\alpha }_{\mathrm{MH}-\mathrm{like}}=min\left(1,exp\left(\frac{1}{{\sigma }_{t^{\prime }}}\Delta {\tilde{E}}_{\mathcal{C}}\right)·{\displaystyle \frac{k_{t^{\prime }}(x^{\tau -1}\mid \widehat{x})}{k_{t^{\prime }}(\widehat{x}\mid x^{\tau -1})}}\right)$$ 11:                   With probability ${\alpha }_{\mathrm{MH}-\mathrm{like}}$: $x^{\tau }\leftarrow \widehat{x}$; otherwise $x^{\tau }\leftarrow x^{\tau -1}$. 12:             end for 13:             Set $x_{t^{\prime }}\leftarrow x^{L_t}$. 14:       end if 15: end for 16: return  $x_0$

Algorithm 2 Estimate Pseudo-Energy (straight-line path)

Require:  Current x, candidate $\widehat{x}$, time t, number of segments $n\ge 2$; access to ${ϵ}_{\theta }(·,t)$. Note that ${ϵ}_{\theta }(x,t)$ and ${ϵ}_{\theta }(\widehat{x},t)$ are available from cache (Algorithm 1). The routine may reuse them instead of re-evaluating   1: $\Delta r\leftarrow \frac{1}{n-1}(\widehat{x}-x)$   2: $r_{\mathrm{prev}}\leftarrow x$;   $g_{\mathrm{prev}}\leftarrow {ϵ}_{\theta }(x,t)$;   $\Delta {\tilde{E}}_{\mathcal{C}}\leftarrow 0$   3: for $j=1$ to $n-1$ do   4:        $r_j\leftarrow r_{\mathrm{prev}}+\Delta r$   5:        $g_j\leftarrow {ϵ}_{\theta }(r_j,t)$                ▹ reuse cached value at endpoints   6:        $\Delta {\tilde{E}}_{\mathcal{C}}\leftarrow \Delta {\tilde{E}}_{\mathcal{C}}-\frac{1}{2}\left(g_{\mathrm{prev}}+g_j\right)·\Delta r$   7:        $r_{\mathrm{prev}}\leftarrow r_j$;   $g_{\mathrm{prev}}\leftarrow g_j$   8: end for   9: return $\Delta {\tilde{E}}_{\mathcal{C}}$

Complexity (extra score evaluations). If endpoint scores are reused, this routine performs $n-2$ new ${ϵ}_{\theta }(·,t)$ calls; otherwise n calls.

Algorithm 3 Estimate Pseudo-Energy (HMC-curved path)

Require:  Current x, candidate $\widehat{x}$, time t; leapfrog trajectory ${\{x^{(i)}\}}_{i=0}^{{\ell }_t}$ with $x^{(0)}=x$, $x^{({\ell }_t)}=\widehat{x}$; subsegments per leapfrog step $m\ge 2$; access to ${ϵ}_{\theta }(·,t)$. Note that ${\{{ϵ}_{\theta }(x^{(i)},t)\}}_{i=0}^{{\ell }_t}$ are available from cache (Algorithm 1) and may be reused instead of re-evaluating.   1: $\Delta {\tilde{E}}_{\mathcal{C}}\leftarrow 0$   2: for $i=1$ to ${\ell }_t$ do          ▹ integrate along each LF segment $x^{(i-1)}\to x^{(i)}$   3:       $r_{\mathrm{prev}}\leftarrow x^{(i-1)}$;   $g_{\mathrm{prev}}\leftarrow {ϵ}_{\theta }(x^{(i-1)},t)$   4:       $\Delta r_i\leftarrow \frac{1}{m-1}\left(x^{(i)}-x^{(i-1)}\right)$   5:       for $j=1$ to $m-1$ do   6:             $r_{i,j}\leftarrow r_{\mathrm{prev}}+\Delta r_i$   7:             $g_{i,j}\leftarrow {ϵ}_{\theta }(r_{i,j},t)$           ▹ reuse cached leapfrog scores if available   8:             $\Delta {\tilde{E}}_{\mathcal{C}}\leftarrow \Delta {\tilde{E}}_{\mathcal{C}}-\frac{1}{2}\left(g_{\mathrm{prev}}+g_{i,j}\right)·\left(r_{i,j}-r_{\mathrm{prev}}\right)$   9:             $r_{\mathrm{prev}}\leftarrow r_{i,j}$;   $g_{\mathrm{prev}}\leftarrow g_{i,j}$ 10:       end for 11: end for 12: return $\Delta {\tilde{E}}_{\mathcal{C}}$

Complexity. With cached leapfrog scores, this routine performs ${\ell }_t(m-2)$ new ${ϵ}_{\theta }(·,t)$ calls (one per interior subsegment); without caching it performs ${\ell }_t(m-1)$ calls. Setting $m=2$ reduces to a trapezoid rule that uses only leapfrog endpoints.

3.2. MH Correction for Composition Models

The pseudo-energy difference for compositions can be derived based on their specific definitions. Our proposed method applies directly to product compositions. We calculate a pseudo-energy difference, corresponding to $E_{\theta }^{\Pi }(x^{\tau },t)-E_{\theta }^{\Pi }(\widehat{x},t)$ for an EBM (defined in (9)), as

$$-{\int }_{\mathcal{C}}\sum _i{ϵ}_{{\theta }_i}^i(\xi ,t)·d\xi .$$

(13)

Guidance is a specific case of product composition, where the pseudo-score is composed of two terms according to (7): the unconditional diffusion model ${ϵ}_{\theta }(x_t,t)$ and the score of a classifier $p_{\phi }(y\mid x_t,t)$. Since $p_{\phi }(y\mid x_t,t)$ can be evaluated directly, only the pseudo-energy difference for ${ϵ}_{\theta }(x_t,t)$ requires computation using the line integral in (13).

The pseudo-energy difference for a negation composition (as defined in [18]) can be computed analogously to products, as negations follow a similar additive structure in their pseudo-scores.

Mixture compositions (as defined in [18]), on the other hand, cannot be expressed as a pseudo-energy difference, since mixtures do not naturally conform to an additive structure analogous to products or negations. However, mixtures can be addressed by first sampling a component distribution according to the mixture definition and then generating a sample from that distribution. The MH correction can subsequently be applied to this sampled distribution, providing a seamless way to handle mixture compositions within our framework.

This generalization allows our method to support advanced use cases such as classifier guidance, multi-modal fusion, and spatially structured prompts, without requiring retraining or access to energy-based models.

4. Results

In this section, we present an empirical evaluation of our MH-like correction method (all code used to produce the results in this section is publicly available at https://github.com/FraunhoferChalmersCentre/mcmc_corr_score_diffusion, accessed on 4 January 2025), examining both the accuracy of the pseudo-energy differences and the quality of the generated samples. The experiments are designed to span a spectrum of difficulty: from controlled, low-dimensional setups where models can be trained from scratch and analytical solutions are available to more realistic high-dimensional scenarios involving pre-trained models. Our two primary objectives are (1) to compare our proposed approach against a true energy parameterization when available and (2) to assess the sampling improvements achieved over the standard reverse process when augmented with MCMC steps.

The experiments in Section 4.1 and Section 4.2 and the first part of Section 4.3 involve training diffusion models using both energy and score parameterizations. The score parameterization follows a noise prediction model, ${ϵ}_{\theta }(x_t,t)$, while the energy parameterization defines an energy function as $E_{\theta }(x_t,t)={∥x_t-s_{\theta }(x_t,t)∥}_2^2$, as in [18]. We use identical network architectures for ${ϵ}_{\theta }$ and $s_{\theta }$. Both models are trained with the standard diffusion loss [3], with the energy model’s score function obtained through explicit differentiation.

The later experiments utilize only pre-trained score-based diffusion models, as pre-trained energy-based models are unavailable for direct comparison. We evaluate both unadjusted and MH-corrected versions of Langevin and Hamiltonian Monte Carlo, comparing them against the standard reverse process, which serves as the baseline.

For the MH-like correction, we examine two types of integration paths: a straight line between $x^{\tau }$ and $\widehat{x}$ (Algorithm 2) and the trajectory defined by the HMC leapfrog steps (Algorithm 3). Both approaches rely on a trapezoidal rule where the number of intermediate points is treated as a hyperparameter.

Complete training details, hyperparameter settings, and implementation specifics are deferred to Appendix A.

4.1. Evaluating Pseudo-Energy Differences

To evaluate the accuracy of pseudo-energy differences, we conducted experiments on a synthetic 2D dataset, generated from a bivariate Gaussian distribution to allow access to analytical solutions, and a higher-dimensional dataset, MNIST [34]. For each experiment, we trained 10 independent score models and 10 independent energy models from scratch. For evaluation, we sampled 2000 pairs of points $(x_t^1,x_t^2)$ independently via the forward process at various diffusion steps t. These pairs were used to compute the score-based pseudo-energy difference ($\Delta {\tilde{E}}_{\mathrm{score}}$) and the explicit EBM energy difference ($\Delta E_{\mathrm{EBM}}$), as well as the analytical difference ($\Delta E_{\mathrm{true}}$) when available. The pseudo-energy difference was computed along a straight-line path connecting the two points, using five discretization points for the numerical integration.

2D Gaussian: For the 2D Gaussian dataset, the relative error metric is defined as $|\Delta E_{\mathrm{pred}}-\Delta E_{\mathrm{true}}|/|\Delta E_{\mathrm{true}}|$, where $\Delta E_{\mathrm{pred}}$ corresponds to either the explicitly predicted difference from the energy model ($\Delta E_{\mathrm{EBM}}$) or the pseudo-energy difference from the score model ($\Delta {\tilde{E}}_{\mathrm{score}}$), and $\Delta E_{\mathrm{true}}$ is the analytical energy difference. The median relative error was calculated across all sampled pairs for each trained model, and the mean and standard deviation of this metric were computed across the 10 models. Interestingly, the score model achieved a lower relative error ($0.071\pm 0.005$) compared to the energy model ($0.084\pm 0.004$), demonstrating that the line-integral approximation aligns slightly better with the true energy differences in this setting.

MNIST: For the MNIST dataset, where analytical energy differences are unavailable, we used a symmetric relative discrepancy metric defined as $2|\Delta {\tilde{E}}_{\mathrm{score}}-\Delta E_{\mathrm{EBM}}|/(|\Delta {\tilde{E}}_{\mathrm{score}}| + |\Delta E_{\mathrm{EBM}}|)$. The median relative discrepancy was calculated across all sampled pairs for each trained model, and the mean and standard deviation were computed across the 10 models. This yielded a mean symmetric relative discrepancy of $0.030\pm 0.002$, indicating that the pseudo-energy differences predicted by the score models align closely with the explicit EBM predictions, even in this higher-dimensional setting.

4.2. Two-Dimensional Composition

To investigate the effectiveness of our MH-like correction in a controlled yet expressive setting, we replicate the 2D composition experiment introduced by Du et al. [18] using their publicly available codebase (https://github.com/yilundu/reduce_reuse_recycle, accessed on 4 January 2025) as a foundation. Only minor modifications are necessary, ensuring a faithful reproduction of their setup. Apart from differences in evaluation metrics, our setup is identical to theirs.

A 2D density pair is composed via multiplication into a complex distribution, as in (9): a Gaussian mixture with eight modes in a circle and a uniform distribution covering two of the modes. For a visual representation of the two individual distributions and their resulting product distribution together with samples from the reverse diffusion and HMC corrected samples, see Figure 1. The baseline reverse diffusion process uses $T=100$ steps. In the MCMC variants, following [18], we omit the optional reverse step for a fair comparison. MCMC sampling runs for $L_t=10$ at each t, with (U-)HMC using three leapfrog steps per MCMC step.

We evaluate performance using three metrics: (1) negative log-likelihood (NLL), (2) a Gaussian mixture model (GMM) comparison, and (3) the Wasserstein-2 distance ($W_2$). The suffixes in

Table 1. Performance metrics (NLL, GMM, and $W_2$) for the 2D composition experiment. The downward arrow (↓) indicates that lower values are better. Values represent the mean with the standard deviation in parentheses over 10 independent trials. Bold numbers indicate the best performance in each metric for both Energy and Score samplers.

	Sampler	NLL ↓	GMM ↓	${\mathit{W}}_2$ ↓
Energy	Reverse	$8.22(0.21)$	$27.01(1.34)$	$5.81(0.19)$
	U-LA	$7.52(0.22)$	$14.61(1.35)$	$4.19(0.45)$
	LA	$6.50(0.30)$	$14.66(1.46)$	$4.24(0.55)$
	U-HMC	$5.72(0.18)$	$6.53(0.91)$	$4.19(1.25)$
	HMC	4.09 (0.14)	3.33 (0.65)	4.12 (1.44)
Score	Reverse	$8.15(0.24)$	$26.88(1.20)$	$5.80(0.20)$
	U-LA	$7.57(0.12)$	$14.99(0.62)$	$4.44(0.63)$
	LA-3L	$6.45(0.20)$	$14.28(1.07)$	$4.03(0.52)$
	LA-5L	$6.61(0.17)$	$15.19(0.92)$	$4.22(0.46)$
	LA-10L	$6.53(0.17)$	$14.75(0.91)$	$4.20(0.51)$
	U-HMC	$5.77(0.12)$	$6.90(0.71)$	$3.39(0.77)$
	HMC-3L	$4.29(0.13)$	$3.72(0.61)$	$2.92(1.02)$
	HMC-5L	4.07 (0.13)	$3.08(0.69)$	2.68 (1.20)
	HMC-10L	4.07 (0.14)	$3.17(0.56)$	$2.87(0.89)$
	HMC-2C	4.07 (0.12)	3.06 (0.54)	$2.94(0.90)$

refer to the choice of integration path and the number of evaluation points: “L” denotes the straight-line path from Algorithm 2, while “C” denotes the curve path from Algorithm 3. The number indicates trapezoidal evaluation points (m or n).

Table 1 reports performance metrics averaged over 10 independent trials. In each trial, we train the diffusion models from scratch and sample 2000 points using different MCMC methods. The corrected sampling methods consistently outperform the unadjusted ones. HMC variants yield the best results across all metrics. Score- and energy-parameterized samplers show similar NLL and GMM performance, while HMC with score parameterization achieves a substantially lower $W_2$. Performance also saturates with as few as three integration points.

In addition to performance, we measure runtime and memory consumption. These results are reported separately in

Table 2. Runtime (seconds) and peak memory usage (MiB) for the 2D composition experiment.

	Sampler	Time	Memory
Energy	Reverse	$0.22(0.00)$	5252
	U-LA	$1.54(0.01)$	5252
	LA	$9.13(0.08)$	5252
	U-HMC	$2.36(0.05)$	5254
	HMC	$21.02(0.06)$	5256
Score	Reverse	$0.11(0.00)$	2178
	U-LA	$0.93(0.01)$	2180
	LA-3L	$4.77(0.13)$	2180
	LA-5L	$7.60(0.02)$	2180
	LA-10L	$10.53(0.15)$	2180
	U-HMC	$1.19(0.01)$	2180
	HMC-3L	$7.08(0.02)$	2180
	HMC-5L	$9.56(0.01)$	2180
	HMC-10L	$10.64(0.01)$	2180
	HMC-2C	$1.48(0.04)$	2180

. The experiment is implemented in JAX (v0.4.30) [35] and run on a desktop computer equipped with an NVIDIA GeForce RTX 3060 GPU. Score-based parameterization is more than twice as memory-efficient as energy-based parameterization, and—except for LA with $m=10$—also faster for corresponding MCMC methods. Notably, the HMC curve variant is significantly faster. Although score-based corrections require more model evaluations, they do not necessarily incur higher runtime or memory costs.

As discussed by Du et al. [18], directly adding score functions does not yield a valid product composition, which explains why the reverse sampler performs poorly in this setting. This apparent failure is expected and highlights the motivation for annealed MCMC: by treating intermediate distributions as design choices that guide the chain toward the target, annealing achieves improved results while preserving asymptotic correctness. This accounts for the large gap between the reverse method and the annealed MCMC variants reported in Table 1.

4.3. Guided Diffusion

We evaluate our proposed sampling methods for guided diffusion on the CIFAR-100 [36] and ImageNet [37] datasets. The sampling process is based on a score function defined in (7). For both datasets, the marginal score, ${\nabla }_{x}logq(x_t)$, is estimated using an unconditional diffusion model parameterized by a UNet architecture. For the guidance model, we use classifier-full guidance, training a time-dependent classifier to predict class labels across all diffusion steps, $p_{\phi }(y\mid x_t,t)$. This classifier shares its architecture with the encoder part of the UNet used for the diffusion model and is extended with a dense output layer. The guidance scale is set to $\lambda =20.0$ across all experiments, which is a commonly used default in classifier-guided diffusion models. This choice is also consistent with the settings provided in the public implementation of [18] (https://github.com/yilundu/reduce_reuse_recycle, accessed on 4 January 2025). Sampling is based on the standard reverse process with $T=1000$, and additional MCMC steps are incorporated to refine the generated samples. The Reverse baseline reported in the tables corresponds to the standard classifier-full guided reverse diffusion process without any additional MCMC refinement steps.

To quantify generation quality, we use three evaluation metrics: the Fréchet Inception Distance (FID) [38], which compares the distribution of generated and real images; classification accuracy, based on a separate pre-trained classifier applied to generated samples; and, for ImageNet, an additional top 5 accuracy metric.

CIFAR-100: For CIFAR-100, we trained the diffusion models from scratch using the same UNet architecture and training settings as in [3], which were originally designed for CIFAR-10 [36]. The MCMC samplers add $L_t=2$ or 6 extra MCMC steps at each diffusion step t for (U-)HMC and (U-)LA, respectively, with (U-)HMC using three leapfrog steps per MCMC step, following the configuration used by Du et al. [18] in their guided diffusion experiments.

For this experiment, we used denser meshes in the trapezoidal rule compared to the 2D setting. For HMC, we followed the curve defined by the leapfrog steps (Algorithm 3) and set $n=3$, i.e., the leapfrog points plus an additional midpoint evaluation. For LA, we followed the straight line path (Algorithm 2) with $m=10$ evenly spaced points, corresponding to eight additional evaluations per step.

Recognizing the impact of the step length on MCMC methods, we followed Du et al. [18] by parameterizing it as a function of the beta-schedule. For Langevin dynamics, the stepsize is given by ${\delta }_t=a{\beta }_t^b$, and for HMC, the leapfrog stepsize is denoted ${\epsilon }_t=a{\beta }_t^b$. To ensure a fair comparison across all baselines and proposed methods, we conducted a seeded parameter search over a predefined range of $(a,b)$ values shared across all variants. Parameters were selected by minimizing FID on a validation subset, with classification accuracy monitored as a secondary metric. The search revealed that unadjusted samplers (U-LA, U-HMC) achieved their best performance with comparatively smaller step sizes. In contrast, the adjusted variants (both energy and score) attained their lowest FID at larger step sizes. For these settings, the average empirical acceptance rates were approximately 5% for HMC and 2% for LA, consistent across both energy and score parameterizations.

The results are shown in

Table 3. Accuracy and FID score for classifier-full guidance on CIFAR-100. The metrics are based on 50k generated samples for each sampling method with both energy and score models. The upward (↑) and downward (↓) arrows indicate that higher and lower values are better, respectively. Bold numbers indicate the best performance in each metric for both Energy and Score samplers.

	Sampler	Accuracy [%] ↑	FID ↓
Energy	Reverse	72.6	33.4
	U-LA	87.3	24.6
	LA	80.0	12.7
	U-HMC	87.2	25.4
	HMC	84.9	12.4
Score	Reverse	$74.2$	$31.8$
	U-LA	82.9	$25.9$
	LA-10L	$75.2$	$15.5$
	U-HMC	$79.0$	$28.6$
	HMC-3C	$75.8$	13.3

. Average accuracy is obtained using a separate classifier trained exclusively on noise-free pairs $(x_0,y)$, following the VGG-13-BN architecture [39]. The table shows a general trend of improvement over the baseline reverse process when additional MCMC steps are added. In particular, the MH-corrected samplers LA and HMC show significant improvements in FID scores, which are arguably the more important metric for image generation.

Comparing the score and energy parameterizations, their performances share similar characteristics. Interestingly, the reverse process favors the score parameterization, supporting the claim that this less restricted approach better models the score function. However, the energy parameterization sees larger improvements from the added MCMC steps. This indicates, perhaps, that direct energy estimation provides a better correction step compared to our method of approximating the pseudo-energy difference from ${ϵ}_{\theta }$. Although the energy-based method performs slightly better in this setting, our MH-corrected sampling methods achieve comparable improvements without requiring an energy model.

ImageNet: For ImageNet, training diffusion models from scratch requires substantial computational resources, so we relied on widely used pre-trained models. Score-based models are publicly available through the OpenAI GitHub repository (https://github.com/openai/guided-diffusion, accessed on 4 January 2025), as provided by Dhariwal and Nichol [10]. However, to our knowledge, there are currently no equivalent publicly available, pre-trained energy-parameterized models at this scale. This limitation highlights the practical value of our approach: by operating directly on pre-trained score models, our method enables MCMC-based refinement even in large-scale settings where explicit energy parameterizations are not readily available. Given the high computational demands of large-scale diffusion models, we focused solely on evaluating HMC and compared it to the reverse process. The HMC sampler adds $L_t=2$ MCMC steps per diffusion step t, with each step consisting of three leapfrog steps, following the setup of the guided diffusion experiment in Du et al. [18]. For the trapezoidal rule, we used the curve-based integration from Algorithm 3 with $m=4$. The step length parameterization and tuning follow the same procedure as in CIFAR-100.

The results can be seen in

Table 4. Average accuracy, top 5 accuracy, and FID score for classifier-full guidance on ImageNet. The metrics are based on 50k generated samples for both sampling methods with score parameterizations. The upward (↑) and downward (↓) arrows indicate that higher and lower values are better, respectively. Bold numbers indicate the best performance in each metric.

	Sampler	Acc [%] ↑	Acc-5 [%] ↑	FID ↓
Score	Reverse	50.0	$83.9$	$14.5$
	HMC-4C	$49.9$	85.1	11.6

. Accuracy metrics are computed using a pre-trained RegNetX-8.0GF [40] classifier. The reverse process and HMC perform very similarly in average accuracy, but our method shows a slight improvement in top 5 average accuracy. HMC obtains a significantly better FID score.

4.4. Image Tapestry

We conduct a final, image tapestry experiment, similar to [18] and based on their code (https://github.com/yilundu/reduce_reuse_recycle, accessed on 4 January 2025), with only minor modifications to incorporate our MH-like correction. The goal is to generate a coherent image composed of spatially localized content, each region conditioned on different prompts. This task involves both classifier-free guidance and model composition—specifically the combination of multiple overlapping text-to-image diffusion models, each responsible for a portion of the scene.

We use a pre-trained DeepFloyd-IF model (https://huggingface.co/DeepFloyd/IF-I-XL-v1.0, accessed on 4 January 2025) as the base diffusion model. To refine the generated samples, we apply Langevin dynamics with our MH-like correction. For each diffusion step ($T=100$), we include 15 additional Langevin steps. The pseudo-energy difference is approximated via line integration using three additional evaluation points per step. We set the classifier-free guidance scale to $\lambda =20.0$.

The resulting image is presented in Figure 2a, which showcases the generated tapestry with different regions displaying distinct visual content. Figure 2b provides a schematic overview of the used prompts and their spatial layout. In total, nine content regions are specified: four located in the corners of the image, each with unique prompts, and five overlapping in the center, all guided by the same prompt to create a unified visual theme.

5. Discussion

The choice between score and energy parameterizations remains an intriguing and nuanced topic within diffusion-based generative modeling. In this work, we provided additional empirical evidence suggesting that the score parameterization performs better in the standard reverse process.

At the same time, we showed that performance gains often attributed to the energy parameterization can, in fact, be recovered within a score-based framework. This is achieved by approximating pseudo-energy differences using a line integral of the model’s noise predictions. Notably, this allows us to incorporate MH-like correction steps into a variety of MCMC samplers—without the need to explicitly train an energy-based model—achieving relative improvements in sample quality of a similar magnitude. This represents a crucial practical benefit, as score-based models are fundamentally easier to train and scale to high-dimensional datasets compared to energy-based parameterizations.

A particularly interesting observation is that using a curve composed only of model evaluations from the HMC sampler appears to perform on par with using a straight-line path. This suggests that the proposed correction comes at virtually no additional computational cost in this case. However, it is worth noting that in higher-dimensional settings, additional intermediate points along the integration path may be required to maintain accuracy, which could increase the computational burden. This challenge might be addressed through more efficient numerical integration techniques, or by working in a lower-dimensional latent space, as in latent diffusion models. One persistent drawback of the energy parameterization is that it always requires an explicit gradient computation to recover the score function.

Another important consideration is computational cost. As we observed in the experiments, adding MCMC updates on top of the reverse diffusion process introduces a clear overhead in terms of additional score evaluations. Our approach is therefore not intended as a replacement for accelerated solvers, e.g., PF-ODE, which aim to minimize the number of function evaluations. Instead, we view MH-like corrections as an orthogonal contribution: they can be applied on top of any diffusion model and any sampler. In this sense, our method is complementary to existing work, targeting improved sample quality while remaining agnostic to the choice of backbone or solver.

Our empirical step length parameter search on CIFAR-100 highlights a qualitative difference between adjusted and unadjusted samplers. The unadjusted variants (U-LA, U-HMC), which lack a rejection mechanism, achieved their best performance only within a relatively narrow range of smaller step sizes. Outside this regime, image quality deteriorated rapidly during sampling. In contrast, the adjusted variants consistently produced semantically coherent samples across a broader portion of the explored parameter range. Empirically, the rejection step appears to mitigate the destabilizing effect of overly aggressive proposals, effectively filtering out unfavorable transitions. For the best performing configurations, the adjusted samplers exhibited relatively low average acceptance rates (around 2 or 5%). Notably, both energy-based and score-based adjusted samplers achieved peak performance at similar acceptance levels, indicating comparable dynamical behavior between explicit and pseudo-energy formulations.

In line with prior work such as Du et al. [18], our experimental design deliberately focused on widely used baselines rather than the strongest available backbones. This choice allowed us to isolate the effect of the MH-like correction without confounding factors from architectural or solver improvements. While this comes at the cost of absolute FID values that are below the state of the art, the relative improvements we observe consistently demonstrate the benefit of our approach. We expect that applying our method to stronger architectures will yield proportionally similar gains, but leave this as an exciting direction for future work.

One limitation of the score parameterization is that the learned vector field is not guaranteed to be conservative, and therefore our line-integral construction does not, in general, yield a true Metropolis–Hastings correction. Consequently, the resulting acceptance rule should not be interpreted as guaranteeing detailed balance with respect to the target distribution. In practice, the samples are generated by the sequential application of the reverse diffusion updates together with MCMC steps and the practically motivated MH-like acceptance rule. The resulting distribution should therefore be understood as the implicit distribution induced by this procedure. Importantly, our construction recovers a true Metropolis–Hastings correction in the special case where the score is conservative, thereby aligning with the energy-based formulation. More broadly, recent work by Horvat and Pfister [41] highlights that score representations may admit non-conservative components without necessarily undermining their effectiveness for generative modeling. This perspective aligns with the empirical success of score-based generative models that operate without explicitly modeling an energy function. Consistent with this perspective, our MH-like correction mechanism—though built on a generally non-conservative field—yields consistent improvements across the evaluated datasets and samplers when applied to the reverse process.

To further assess the practical impact of the non-conservative nature of the learned score field, we conducted a series of empirical sanity checks (detailed in Appendix B.1 and Appendix B.2). First, to examine potential path dependence, we compared pseudo-energy differences obtained from straight-line and curved integration paths. On both trained MNIST models and the higher-dimensional CIFAR-100 model, the observed discrepancies were small relative to the magnitude of the pseudo-energy differences. This indicates limited path sensitivity at the proposal scale relevant for our MCMC refinement steps. Furthermore, we evaluated the numerical stability of the line-integral approximation via a mesh convergence analysis on CIFAR-100 (detailed in Appendix B.3). We observed rapid convergence of the discretization scheme: the median symmetric relative discrepancy falls below 1% when using $n\ge 10$ integration points compared to a higher-resolution reference. Together, these empirical results suggest that the learned score behaves approximately conservatively on the local scales relevant for MCMC proposals, as small variations in the integration path yield similar pseudo-energy differences.

Still, the lack of theoretical guarantees may explain the slightly superior performance of the energy parameterization observed in the CIFAR-100 experiment. Developing better techniques for estimating pseudo-energy differences from score-based models—without requiring an explicitly trained energy function—thus remains a highly relevant and promising direction for future research.

6. Conclusions

We introduced a method for extending the reverse diffusion process with MCMC sampling based on an MH-like correction step computed from the score function. This approach enables improved sampling for composed diffusion models without requiring an energy-based parameterization.

While previous work, Du et al. [18], demonstrated the benefits of MH correction under an energy parameterization, our method instead defines a pseudo-energy difference derived from the score, estimated via numerical integration. This allows us to apply MH-like corrections in the score-based setting—by far the most common in practice—and thereby make use of existing pre-trained diffusion models for composition tasks.

Our method can reuse intermediate evaluations from samplers such as HMC to compute the correction with little to no additional cost. In general, the accuracy of the MH-like correction depends on the numerical integration of the score, which may require more intermediate points as the dimensionality increases. While this can introduce some overhead, energy-based methods incur their own costs, such as differentiating the energy function. In practice, our corrected score-based samplers consistently yield relative improvements of a similar magnitude to energy-based methods across a range of tasks, making them a practical alternative in settings where score-based models are already available.

Overall, our work extends the applicability of corrected MCMC sampling to the broad class of score-based diffusion models and opens the door to more flexible and modular composition of generative models.