G2D2: Gradient-Guided Discrete Diffusion for Inverse Problem Solving
Authors: Naoki Murata
Chieh-Hsin Lai
Yuhta Takida
Toshimitsu Uesaka
Bac Nguyen
Stefano Ermon
Yuki Mitsufuji
Venue: Accepted by TMLR ()

Abstract: Recent literature has effectively leveraged diffusion models trained on continuous variables as priors for solving inverse problems. Notably, discrete diffusion models with discrete latent codes have shown strong performance, particularly in modalities suited for discrete compressed representations, such as image and motion generation. However, their discrete and non-differentiable nature has limited their application to inverse problems formulated in continuous spaces. This paper presents a novel method for addressing linear inverse problems by leveraging generative models based on discrete diffusion as priors. We overcome these limitations by approximating the true posterior distribution with a variational distribution constructed from categorical distributions and continuous relaxation techniques. Furthermore, we employ a star-shaped noise process to mitigate the drawbacks of traditional discrete diffusion models with absorbing states, demonstrating that our method performs comparably to continuous diffusion techniques with lower GPU memory consumption.

Published in Transactions on Machine Learning Research (08/2025)
G2D2: Gradient-Guided Discrete Diffusion for Inverse Problem Solving
Naoki Murata', Chieh-Hsin Lai', Yuhta Takida', Toshimitsu Uesaka', Bac Nguyen', Stefano Ermon Yuki Mitsufujill
Sony AI, *Stanford University, "Sony Group Corporation naoki www.mmmatrtt..coommmm com com
Reviewed on OpenRevicw:
Abstract
Recent literature has effectively leveraged diffusion models trained on contimuous variables as priors for solving inverse problems. Notably, discrete diffusion models with discrete la- lent codes have shown strong performance, particularly in modalities suited for discrete compressed representations, such image and motion generation. However, their discrete and non-differentiable nature has limited their application to inverse problems formulated in continuous spaces. This paper presents a novel method for addressing linear inverse prob- lems by leveraging generative models based on discrete diffusion as priors. We overcome these limitations by approximating the true posterior distribution with a variational dis- tribution constructed from categorical distributions and continuous relaxation techniques. Furthermore, we employ a star-shaped noise process to mitigate the drawbacks of traditional discrete diffusion models with absorbing states, demonstrating that method performs comparably to continuous diffusion techniques with lower GPU memory consumption. Our code is available at
1 Introduction
Diffusion models have gained significant attention as deep generative models, achieving remarkable success
in image (Sohl-Dickstein et al. 2015 Ho et al. 2020 Song et al., 2021b Dhariwal & Nichol, 2021
Esser
ct al. 2024), audio (Liu et al., 2023 Chen et al. 2024a and vidco generation (Ho et al. 2022a These
models operate by iteratively corrupting data and then learning to reverse the corruption process, ultimately generating high-quality samples from noise. Tn parallel with continuous diffusion models, discrete diffusson models have emerged as a compelling alternative. These models have gained traction by demonstrating notable results not only in image (Gu et al. 2022). audio (Yang al.) 2023), and text generation (Austin
et al. 2021 Lou et al. 2023a) but also in more specialized areas such as motion data (Lou et. al. 2023b
Pinyoanuntapong et al.
2024 protein synthesis (Gruver et al. 2024 and graph generation (Vignac et al.
2023)
Building on these advancements, researchers have made significant progress expanding the application of diffusion models. They have explored using diffusion models, trained either directly the pixel space or on latent representations derived from variational autoencoders (VAEs), to address inverse problems (Kawar
et al. 2022 Chung et al.) 2023b Wang et al. 2023) and carry out various conditional generation tasks (Yu
2023 Bansal et al. 2024 He et al.) 2024) without the need for additional training. These efforts et al.
aim to use the powerful generative capabilities of diffusion models to tackle intricate problems and generate conditional outputs, all while preserving the models' original trained parameters.
The rescarch on applying diffusion models to inverse problems and conditional generation has been pri- marily restricted to diffusion models trained in continuous spaces, and methods using pre-trained discrete
diffusion models priors remain limited (Gruver et. al. 2024 Chen et. al. 2024b Li et al. 2024). One of

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2/"-
[MASK] ..... [MABK]]
Z [L [L.... K, [HASK] pul
Z5
Zo
0 K7
Codebook
13
6
Dequatization (Codebook assignment)
Decoder
Corruption ( (xx)
12
12
Po (Za) (e)
Y)
Gumbel-Softma Dequantizatio
Decoder
X( Forward
Axo
Prior distribution
Optimization target
Measurement
at = arg min
Prior loss
+ Likelihood loss
}
Optimization target
Figure 1: Illustration of G2D2. At each time step t, variational categorical distribution Pa is optimized with respect to the sum of prior loss and likelihood loss, followed by sampling z Both terms are contin- uously differentiable, enabling continuous optimization.
the main reasons is that the inherent nature of the generation process in discrete diffusion models involves non-differentiable operations, posing challenge for their application to inverse problems formulated in continuous spaces. Therefore, controlling discrete diffusion models often necessitates an additional trained
network (Gruver et al. 2024 Nisonoff al. 2024 Klarner al.) 2024 Vignac et al.) 2023). Addition- ally; training-fres methods have been confined to relatively low-dimensional data (Chen el al.
2024b)
to specific tasks such image impainting (Gu et al.) 2022). Recent work by (Singhal et al (0055] pro-
poses Feynman Kac steering, general inference-time framework for steering diffusion models with reward functions. While this approach does not require differentiating through reward functions and thus can be applied to discrete diffusion models, the reward functions they employ, such as human preference scores, impose weaker constraints on the generation process compared to typical inverse problems like deblurring, potentially significantly compromising consistency with measurements.
Despite these limitations in applying discrete diffusion models to inverse problems, their potential advantages in representing complex data distributions and generating high-fidelity samples motivate their exploration priors. When examining existing approaches for continuous diff'usion models, conventional approaches in continuous settings typically leverage gradient- based adjustments of the generation trajectory. These methods aim to refine intermediate latents by computing gradients of a likelihood loss function, ensuring they align well with the measurement equation or guidance larget. This has been demonstrated in Chung et al. (003bb) and Yu et al. (2023). However, directly extending this gradient-based control discrete diffusion models is challenging due to their inherently non-differentiable operations.
To address the fundamental challenge of non-differnntiabiliyy in discrete diffusion models, propose Gradient-Cuided Discrete Diffusion (G2D2), inverse problem solving method that uses discrete diffu- sion model as prior. Our focus is on solving inverse problems using a generative model based on discrete diffusion model, specifically designed for discrete latent variables such as those found in quator-qaantzed VAE (VQ-VAF) models. G2D2 bridges the gap between continuous and discrete domains by using a continuous relaxation technique to optimize the parameters of a variational distribution.
In addition to the differentiability issue, discrete diffusion models present another challenge when used for inverse problems. These models often adopt "mask-absorbing" noise processes their data corruption process duc to generation quality. However, this mask-absorbing approach has a significant drawback in inverse-problem solving. For example, in discrete diflusion models for images, while a substantial portion of the image structure determined in the initial stages of generation (i.e., when only a few tokens are determined), the mask-absorbing type does not allow transitions from an unmasked state either a masked state or other unmasked states. Our experiments demonstrate that this restriction imposes significant limitation performance in solving inverse problems.
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To overcome this structural limitation of mask-absorbing processes, we incorporate the star-shaped noise process previously proposed in the context of continuous diffuusio models (Okhotin et al. 2024 Zhang et al. 2024). This process removes the dependency between consecutive sampling steps, expanding the space that can be explored during generation. This process was originally proposed to enhance the performance of diffusion models (Okhotin et al. 2024], but was later introduced as a decoupled noise annealing process in the context of inverse problems using continuous diffusion models, demonstrating continuous diffussinn models (Zhang al.) 2024). In this study, we not only demonstrate that this process can be effectively applied to discrete diffusion models, but also discover that it uniquely addresses potential issues inherent in wask-absorbing-tyee type discrete diffusion processes, specifically the inability to correct errors introduced in the early stages during later inference steps.
effectiveness for
To validate our proposed approach, we conduct comprehensive experiments comparing G2D2 to current methods using standard benchmark datasets. Our results demonstrate that G2D2 achieves comparable performance continuous counterparts (within 0.02-0.05 LPIPS points) while reducing GPU memory usage by up to 77% (4.7GiB 20.9GiB for PSLD). We also explore the application of discrete prior-based motion-data-genine generraton model to solve an inverse problem, specifically path-conditioned generation, without requiring further training. The results of our study indicate that G2D2 shows promisc in tackling various inverse problems by leveraging pre-trained discrete diffusion models.
2 Preliminaries
2.1 Discrete diffusion models for image generation
We first provide brief overview of VQ-Diffusion (Gu et al. 2022 Tang ct al. 2022
imago-generation model based discrete diffusson processes. VQ-Diflusion generates images in a two-step process. It first pro- duces discrete latent representations Zn using a discrete diffusion model trained on representations obtained from pre-trained VQ-VAE model (ivan den Oord et al. 2017 It then transforms these representations into the continuous image space using a decoder. Each element of zo E {1,..., ".....,K' K)* corresponds to one of the embedding vectors from the codebook, denoted as B == (bi...., ,b,bbb, by E Rdb. During decoding, a variable Z 8d is constructed through codebook assignment, where (Z); = b and 20, denotes the i-th element of 2g. This variable is then fed into decoder D Rd + Rd that maps from the discrete token embeddings to the continuous image space to obtain the final image: xo D()).
In discrete diffusion models, a forward Markov process gradually corrupts the discrete latent representation 2o: and a reverse process is learned to invert this process. A single step of the forward process of the Markov chain Zo + + Zz + + zT can be represented as,
d( (2-1) = u (2..)
(1)
where v(22,) E 10+1 denotes one-hot encoded vector representing the token at time step t, K is the number of states from the VQ-VAE, and one additional state is for a special mask token [MASK]. These concepts will be formally introduced immediately below in the context of VQ-Diffusion. Qr (K)))((K+1) represents the transition matrix, which determines the probabilities of transitions between tokens. VQ- Diffusion uses a mask--bbsorbing- type forward process, which introduces a special masked token denoted as [MASK] in addition to the K states from the VA-VAE. The transition matrix is defined as follows, where the last column represents the transition probabilities for the [MASK] token:
ai +
3
3
8
B
at + 9
3
0
Q
3r
31
04
+
9
0
(2)
:
7t
7t8
where the transition probabilities are determined by three parameters: at Bi, and π In this process, once token transitions to the masked state, it remains masked in all subsequent steps, hence the term
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"absorbing." Specifically, a represents the probability of token remaining unchanged, 3r denotes the probability of transitioning to a different unmasked token, and "/t indicates the probability of the token being replaced with the [MASK] token. These parameters satisfy the constraint a, + (K - 1)3, + = 1 to ensure valid probability distributions. The probability 3 between unmasked tokens is generally set to a very small value. These parameters are lypically set so that g(zr|z)) assigns all probability mass to the [MASK] token, and we also adopt this assumption.
During inference, the latent variable zn corresponding to the clean image is obtained by executing the following reverse process:
Po (1- 1 = Σ ( 1l,, 20 (20l
(3)
20
where q(zc_1/Z,, zu) represents the posterior distribution determined by the forward process, and py denotes the denoising network that. predicts the denoised token distribution al L. The output of Po is generally modeled as independent categorical distributions for each dimension in Zo- In practical applications such as text-to-image generation, Do is trained with conditional information (e.g., text prompts in VQ-Diffuusio)) al- lowing for controlled generation. While the true data distribution g(zo) has dependencies across dimensions, the denoising network 5o k2) typically models each dimension independently as categorical distributions. However, the complete reverse process defined in 33) implicitly captures some of these dependencies through the iterative application of the conditional distributions po(2:-1 12e). We distinguish between two distribu- tions: the clean distribution po( (2) directly estimated by the denoising network at single step (which treats dimensions independently), and the distribution (zollz))) obtained by rumning the reverse diffusion process from L to 1 1= 0, which better approxinates the true data distribution with its dimensional depen- dencies.
2.2 Linear-inverse-rooblen settings
Inverse problems involve estimating unknown data from measurement. The relationship between the mea- surement y E Ryy and unknown ground-truth data Xg E Rd can be represented as
y y= Axo + 17.
(4)
where A E Xdxy is referred to as the forward linear operator, which describes the process by which the measurement y is obtained from data Xo- We assume this operator is known. The term n represents measurement noise, which we assume follows an isotropic Gaussian distribution with known variance of Consequently, the likelihood function q(y|xo) can be described as WNy; Axo,
One of the primary challenges in inverse problems is their ill-posed nature. This means that for any given measurement, y, multiple candidate solutions may exist. To address this issue and determine Xo, a common approach is to assume a prior distribution for Xo, such as a Laplace distribution. Diffusion models have been utilized as more powerful and expressive priors, offering enhanced capabilities in solving these inverse problems (Kawar et al.) 2022 Chung et al. 2023b Wang al. 2023 Rout et al. 2023). These diffusion-basee methods are able to produce data that not only fit the measurement data but also exhibit high likelihood under the prior model. Given a prior qxxu), the objective in the inverse problem is to sample from the posterior distribution q(xoly), which, according to Bayes' theorem, is proportional to g(ylxo)4(x0)
These methods can be categorized based on how they incorporate the information from the measurement y into the generation trajectory of diffusion models. Methods such as denoising diffusion restoration models (DDRM) (Kawar et al.) 2022) and denoising diffusion null-space models (DDNM) (Wang et al., 2023) leverage the assumption of linear operators, using singular value decomposition of the forward process to control the generative process. In contrast, methods such diffusion posterior sampling (DPS) (Chung et al.) 2023b) and posterior sampling with latent diffusion (PSLD) (Rout et al. 2023) operate by propagating the gradient of a loss term through the generative process. This loss term is designed to maximize the measurement likelihood, specifically by minimizing the term Ily - Axolle.
However, while these methods work well with continuous diffusion models, their application to generative models that use discrete diffusion models as priors is not straightforward. This limitation stems from two
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"a face of monkey"
Decoler
zo
~ pa (20)
Deprammittion (Codehook asignmes))
Decoder
Z0
po / 20-
Less diverse
Devddr
Zo_
po
(oo
Zi=80)
Figure 2: Empirical demonstration of absorbbb characteristics in discrete diffusion models. All images are generated using the text prompt "a face of monkey". Top row: Diverse samples from rext-conditioned sampling (po(z))). Middle row: Multiple samples conditioned on the same Z4=90 (po (2a)2z=90)). Bottom row: Multiple samples conditioned on the same 2z=80 (pe (2o)z)=00)). This demonstrates that in mask- absorbing processes, image structure is largely determined in early sampling steps, with decreasing diversity sampling progresses, making it difficult to correct inconsistencies with measurement data during inverse problem solving.
primary factors. First, the former methods (DDRM and DDNM) are specifically designed for diffusion models trained directly in the pixel domain. Second, while the latter methods (DPS and PSLD) can be extended to latent diffusion models that operate in continuous latent spaces, they encounter difficulties when handling discrete diffusion models, where the generative process involves inherently non-dillerentiable operations. The core challenge lies in the lack of a direct mechanism to propagate gradients of the loss function through the generative process in discrete diffusion models. In such models, after generating discrete data, a non- differentiable operation (i.e., codebook assignment) is followed by a decoding operation into continuous space, which prevents the application of conventional gradient-based guidance.
3 wradient-Guided Discrete Diffusion, G2D2
In this section, we propose Cradiend-Guddel Discrete Diffusion (G2D2), an inverse problem solving method that leverages discrete diffusion models as priors. Our approach addresses two key challenges: the non- differentiability of discrete diffusion models and the limitations imposed by widely-used mask-absorbing noise processes. First, we introduce star-shaped noise process to address the latter issue and define a more tractable variant of the star-shaped noise process distribution, which we call the star-decomposed distribution. We then address the former challenge of non-differentiability by bridging the gap between discrete and continuous domains through optimizing a variational distribution with continuous relaxation. These methodological components are integrated into a unified, practical inference algorithm.
3.1 Star-shaped noise process for enabling inherent re-masking
A key challenge in applying discrete diffusion models to inverse problems is addressing the limitations of mask-absorbing noise processes. As described in Section 2.1 discrete diffusion models commonly adopt mask-absorbing process, where in the forward process, an unmasked token cither remains the same or transitions to & mask token. While this design leads to higher performance in standard generation tasks, it poses a significant constraint in solving inverse problems. Specifically, in the reverse process (i.e., the gencrative process), once an unmasked token has been set at an carly stage of sampling, the probability of it reverting to a masked state or changing to a different token becomes extremely low (see Figure 3b), making it difficult to correct errors made in the early stages.
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MMackow (Y, Ram) (gs (no)
Shaeeorr (3t
(R--1)
Mask-absorbing
Mask-absorbing
y
X
Zo
Ze
Zy
+ +Markov
+Star-shaped (in G2D2)
Measurement
2
-
Markov noise process
3 4
M
5
M
3 M
1
5
M
Backward
Forward
Astar (y, Xe; No:7) = g gxylxx))g(x// (o0)((())) 2(2e(2a)
Zt
1
Zt
v Conditional Independence
3
M
M
5
M
3
4
M
M
M
ZO
Zz
Unmasked tokens cannot revert to [MASK]
Unmasked tokens can revert
Mensurement
Star-shaped noise process
to [MASK] (inherent "re-masking")
(a)
(b)
Figure 3: (a) Graphical model comparison between Markov noise process (top) and star-shaped noise pro- cess (bottom). The star-shaped model introduces conditional independence between Z, variables given zo, enabling more flexible error correction during sampling. (b) Comparison between mask-absorbing Markov process (left) and mask-absorbing star-shaped process (right). In the Markov process, unmasked tokens cannot revert to [MASK], while the star-shaped process allows tokens to naturally revert to [MASK] state (inherent "re-masking"), facilitating error correction in later sampling steps.
Figure 2 demonstrates this characteristic: when generating images using discrete diffuuson models that employ mask-absorbing process, most of the image structure is determined in the initial stages, suggesting that in inverse problem scenarios, there is little opportunity correct inconsistencies with the measurement y during sampling. One solution to this problem is the "re-masking" operation, which reverts unmasked tokens to masked tokens. This approach has been implemented in discrete predictor-corrector methods [ezamaa et al. 2023) and predictor-correcto techniques for contimnous- time discrete diffusion [CCmmbell] et al. 2022 Zhao et al. 2024) to improve image generation quality. However, these methods often involve reverting the sampler to carlier time steps and can be computationally expensive.
Here, drawing inspiration from Okhotin et al. (2024) and Zhang al. (2024), we adopt a star-shaped noise process that decouples each time step from previous steps, which addresses the aforementioned issue of carly-stage errors becoming permanently "frozen." In this star-shaped noise process, the noisy variables
21...., 2r are conditionally independent given Zg- Formally, we assume (star = II (star(2,220)
Ih I V (2x,)) Q, o(2.z) where the matrix Q, = QQ...Q. Q1 is the cunulative transition from the orig- inal Markov forward process in discrete diffusion, but the forward graphical model is star-shaped, as shown in Figure 3a
Compared to the original Markov forward process, the star-shaped noise process maintains the same con- ditional marginal distribution qstar (2, [zu), yet the locations of [MASK] tokens uncorrelated between z; corresponding to different time steps. In the Markov noise process, once token is [MASK]ed, it remains so, and in subsequent steps, only unmasked tokens can transition to [MASK] tokens. In contrast, in the star-shaped noise process, each noisy token 2, is viewed as being generated directly from zo rather than from 2,-1, making the [MASK] token positions independent between adjacent z,-1 and Z;-
A key charactoristic of the star-shapod process is obscrved in the reverse step (zt (z). Unlike the Markov noise process with mask-absorbing state, which requires that tokens already unmasked remain fixed and prohibits their reversion to a masked state, the star-shaped formulation permits tokens to transition from an unmasked state back to masked state within a single reverse step. This inherent "re-masking" operation enables flexible error correction during sampling, as illustrated in Figure 35 where tokens can naturally revert to the masked state without the need to roll back the sampler.
Our goal here, similar to other diffusion model-based inverse problem mothods, to sample from Istar (zoly) based on this graphical model. More specifically, we aim to develop feasible algorithm for sampling from astar R.. However, attempting to perform ancestral sampling from this distribution, such as by

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proceeding from Zy ZT-1 all subsequent samples and is therefore not computationally feasible.
Zo, results in posterior of the form tetar (zt/ y), which depends
To address this challenge, we first introduce proxy joint distribution, 9star-decop obtained by factorizing the star-shaped posterior into a product of local conditionals. Importantly, this preserves the marginal Gttarddcomp (Zoly) gstar ( as we will discuss in the next subsection, so can serve as valid surrogate for qstar while reducing the complexity of the graph structure so that dependencies exist only between consecutive time steps. Although the full joint quar-deromp is still intractable, this decomposition is necessary preparatory step that leads to the variational approximation described in Sec. 3.3 The
resulting algorithm separates the complex inter-dependencies between time steps while maintaining the essential properties of the star-shaped model, making it possible to efficiently sample from (zoly).
3.2 Defining the decomposed star-shaped distribution @star-decomp which tractable variant of gstar
We define the dec-decoe distribution as follows:
T
Sstar-decomp(2o..T1> Qstar (ZTly) II t=1
gatar (Zr-1 |2,, y),
(5)
where Jstar (2:-1(2:1112 (Z:-1/z,: y) represents the conditional distribution of gstar. This decomposition of the joint distri- bution resembles the Markovian reverse process of diffusion models, but it has the following properties:
1. A single step of dstar-dccomp inherently enables the "re-masking" operation. In the star-shaped noise process, the positions mask tokens in Z;-1 and z, are mutually independent and uncorrelated as shown in Figure 3b Consequently, the conditional distribution @star-decom lz. (= qstar ( y)) enables a "re-masking" operation, wherein unmasked tokens present in Z can become masked tokens in z;-1. This property suggests that in mask-absorbing discrete diffusion, errors that occur in the initial stages of sampling can be corrected in subsequent steps, which provides an advantage when solving inverse problems.
2. The marginal distribution Istar-decomp (zuly) is identical the target distribution Jstar l-) The statement and proof are provided in the Appendix. This suggests that for solving inverse problems, we do not necessarily need to sample from the joint distribution of qstar, but can instead aim to sample from the more tractable distribution Istar-decomp- Ultimately, we perform approximate sampling from the posterior by approximating this Istar-decomp using a variational distribution.
3. The conditional joint distribution of 9star-decomp differs from that of Istary i.e., Matar-decomp (otoddccommooooy)) # Istar ly). The decomposition of the joint distribution of star-shaped noise process takes the form Istar (Zurr ( ly) gaar qa (yr IT= (Z-1 However, @star-decopp deviates from this formulation by disregarding the dependencies larger time steps, Zc+1:7-
In subsequent sections, introduce variational distribution to approximate qstar-decomps which possesses Property that inherently enables the re-masking operation. As established by Property 3, the joint dis- tribution of @star-decomp differs from that of the star-shaped noise process graphical model. Nevertheless, Property 2 guarantees that they share identical marginal distributions given the mcasurement y, providing justification for our approach targeting the more tractable @star-decomp for sampling. Furthermore, since @sta-decompp focuses only on two adjacent variables, we can formulate a simple algorithm to approximate its distribution using a variational approach.
3.3 Variational approximation for feasible inference algorithm
Based on the discussion in the previous section, aim to implement qstar-decomp: which inherently in- corporates re-masking process for efficient inverse problem solving. However, since @star-decomp is still not tractable, we introduce variational distribution Pa (z)r gatar (zrl())l))) IT Pa (zt-z11 1/2,y) to ap- proximate @star-decomp (2o:r ly); with the ultimate goal of ensuring that the marginal distribution Pa (zoly)

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approximates the true posterior Istar-decompl (Zul l.).
The distribution Pa is decomposed as
Pa falzz-1
Σ
lstar (sta(((e1120)) z))
(6)
where Pa (zo Rr. y) is a categorical distribution parameterized by a, where for each time step and dimen- sion i, ot,,,- is probability vector in the simplex A A defined as (z. = Cat (20,4 ot,,,,), i.e. (20,5 = y) = dg,i,k- In the subsequent discussions and optimization steps, we assume that ar,i, is always normalized to lie the simplex. This decomposition stems from the fact. that the distribution @star-decomp (2--1 (2.-1 (= (
be expressed as Ezo Istar (Zt-1 [star (2o) z) based on the conditional independence. Note that both Qstar (Z)- and Pa (zo) Zez y) have a mean field structure with independent categorical distributions across dimensions. Consequently, Pa (Z y), obtained by marginal- izing over 2o, inherits this mean field property. For notational convenience, we denote the slice distribution parameter a at time step as a E _
To ensure that the marginal distribution of the variational distribution Pa closely approximates that of @star-decomp: the parameters a of the variational distribution are obtained by optimizing objective function derived from the following theorem:
Theorem 3.1. Let Pox be a distribution with the parameterization given by the decomposition in O Then, for any measurements y, the Sollowing inequality holds for the KL divergence belween the marginal distribu- tions:
T
DKL (Pa (olooooolllssrr-ooa (Zoly)) S
(z.xpa(z)l)) (2.ly) [DEL (Fa
Ze y))],
where Pa (zoly) is the variational marginal distribution parameterized by a, qstor (zoly) is the true posterior distribution, Pa (Zolzt,))) is the variational conditional distribution as defined in [6). qstar (2, is the true conditional distribution, Pa (z.l)) is the marginal distribution at time slep i, and T is the tolal reamber of time steps in the diffusion process.
The proof is provided in the Appendix. Based on this inequality, we aim to minimize each term in the sum on the right-hand side. Since Pa is a different categorical distribution at each 6, we minimize a for each time step, ultimately aiming to minimize the lent-hand side. Each term on the right-hand hand side of (7) can be decomposed into a term representing deviation from the prior and term representing likelihood with respect to the observed data:
Lemma 3.1. The KL divergence between the variational distribution Pa (2 24,)) and the true posterior Astar(Za Mt. y) can be decomposed into two terms:
DEL y) DEL (Fa (aa 2:)) (2oler))))) [log Ister (2)))
(2olz..))
This decomposition enables us to separately consider the fit to the prior and the consistency with the measurement data, and we approximate both terms in a computationally tractable form as follows. First, the KT, term on the right-hand side of (8) remains intractable. However, we note that the star-shaped noise process shares the conditional distribution gstar (z. 22) with the original Markov noise process. Consoquently, the reverse conditional distribution ystar (2ol2.)) will also be identical for both processes. Since the prior of the pre-trained discrete diffusion models is trained to approximate this distribution,
substitute this prior model 5o (2o/zz) for Istar (Rolz,) into the objective function of (8) This substitution transforms the term into KL divergence between two categorical distributions, enabling the computation of gradients with respect to the parameter a
The second term involves an expectation calculation over a categorical distribution, for which we use the Gumbel-Softmas re- re- parameterizzatin trick (Jang al., 2016 Maddison et al., 2016).. The implementation of this trick is discussed in the subsequent section. This approach makes the term differentiable with respect

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to the categorical distribution's parameter a, facilitating continuous optimization. The explicit form of the resultant loss function is detailed in the Appendix.
Based on Theorem 3.1 and Lemma 3.1 our proposed inverse problem solving method with discrete diffusion prior, G2D2, optimizes the parameter a of Pa for t = T...., , while sequentially sampling Zo:2- In the optimization step, any continuous optimization method, such as Adam (Kingma (2044) and RAdam (Liu al. 2020), can be used. Implementation considerations are discussed in the following section. This
algorithm is detailed in Algorithm [ and G2D2 is illustrated in Figure [
Algorithm Gradient-Guine Discrete Diffusion, G2D2
Require: Input condition y, pre-trained discrete diffusion model po, forget coefficient 7
1: Er ~ (ZT)
2: for L = .................... 1. do
3: if = then
Initialize: a (og
5: else
Initialize: a. exp(y log a:41 + (1 -1-) 7) log Pa (zal2.))
7: end if
8: // Continuous optimization
9: a, arg.in,,, DKL (Pa (zolze: y) y)
(2:)) Exo- Rer (mo .) Dog Jstar [1/20)]
10: Sample 2z-1 ~ Pa (Z 1 (e = Eno Astar (Zt- 1 (oo Pa (Zu (ee, Y)
11: end for
12: return xa by decoding zo
3.4 Implementation considerations
Gumbel-Softmax dequantization We use the Gumbee--Softma trick (Jang et al. 2016 Maddison eet al. 2016) to make the computation of the second term in (8) differentiable. At time step t, this process begins by generating Gumbel-Softmax samples using parameters of Pa follows: fo,i, softmax (logaa,,. + 9t,k) /T), where Ji.k are i.id. samples drawn from the Gumbel distribution, and т (> 0) is the temperature parameter. This procedure generates "soft" categorical sample for each dimen- sion in zo, indicating the proportional selection of each codebook element. As these proportions corre- spond to the contribution rate of cach codebook element, we construct ZGumbel E RRdx as their weighted &o,,,bb Finally, we pass ZGumbel through the decoder to obtain the image Xo D((Gumbe))) By substituting this image into the likelihood function gstar (y/x)) we obtain the differen- tiable objective with respect the variational parameter at, enabling continuous optimization. For linear inverse problems, the objective function will include the term |yy Axx(a)) excluding the constant term derived from measurement noise.
sum: (ZGumbel)s
Optimization initialization strategy At time step f, we are required to optimize the variational param- eter Qg. To expedite this process, we can leverage the optimized values from the previous time step as the initialization for the optimization process, effectively reducing the number of required optimization steps. To achieve this, we introduce forgetting coefficient 7 (where 0 S 1) and initialize a through weighted sum of the previous optimized variables and the prior model's output in the logarithm domain, given by exp(> log aut1 + (1 - 7) log (zzz)zz)) The effectiveness of this strategy is discussed in Appendix E.6.
3.5 Application of G2D2 to masked generative models
As discussed in [Zheng et al. 2024), mask-absorbing discrete diffusion models and masked generative models, such as MaskGIT (Chang el al.] 2022). share a similar framework. Apart from lemporal conditioning, these models are nearly identical and are trained to approximate Sztar Therefore, G2D2 can be straightforwardly applied to masked generative models. We empirically show this by providing an example of solving inverse problems using a masked generative model a prior for motion data in the experimental section.

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t
t 75
t= 65
t= 55
Measurements
== 45
t= 35
= 25
= 15
t=
Ground truth
Figure 4: Images sampled from the prior model (o(2o) 22) using intermediate z, during the process of G2D2 in image inverse problem solving. The progression demonstrates how initial structural errors are gradually corrected as the sampling proceeds in G2D2.
4 Experiments
4.1
Experimental setup
We evaluate G2D2 on inverse problems the image processing and compare with other diffusion model- based inverse-problem-solvie methods. We also demonstrate coudient-based guidance on a discrete-latent variable-based motion-domain generative model without additional training, showing the applicability of G2D2 to other domains.
Image inverse problems and evaluation metrics We conduct experiments on two tasks: (1) super- resolution (SR) and (2) Gaussian deblurring. For the SR task, the linear forward operator downscales the image by factor of 4 using a bicubic resizer. For the Gaussian deblurring task, we set the kernel size to 61 x 61 with Gaussian kernel standard deviation of 3.0. The measurements are obtained by applying the forward operator to the ground truth images normalized to the range [-1, followed by the addition of Gaussian noise with standard deviation of 0.05. As metrics, we use the learned perceptual image patch similarity (LPIPS) (Zhang et al., 2018) score mcasure perceptual proximity to the original image, and the peak signal-to-noise ratio (PSNR) to measure the closeness of the signal.
Datasets Following previous studies, we use the ImageNet (Deng al. 2009) and Flickr-Faces-HIO (FFHQ) (Karras et al. 2019 datasets. The images 256 256x256. For comparison, use subset of 1000 images from each validation set.
Baselines We compare DPS (Chung al 2023b), DDRM (Kawar al 2022 which use diffusion models trained in the pixel domain, and PSLD (Rout et al. 2023) and ReSample (Song et al. 2024), which use diffusion models trained in the latent space acquired from VAE (latent diffusion models) as baselines with G2D2.
Implementation details Regarding G2D2, for both the ImageNet and FFHQ experiments, we use a pre- trained VQ-Diffusion model [ that is trained on the ITHQ dataset (Tang et al. 2022). In all experiments, we optimize the parameters a of the variational categorical distribution within the G2D2 algorithmm' opti- mization step using the RAAdam optimizer (Liu et al. 2020). To balance the prior and likelihood terms in the objective function, introduce hyperparameters. For the image inverse problem experiments, used text prompts for the VQ-Diffusion model: "a photo of [Class Name]" for ImageNet and "a high-quality
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headshot of person" for FFHQ. All experiments are performed itself never exceeded 4.7 GiB of VRAM and required 194 per ImageNet image (Table in the Appendix). Hence the full pipeline can be executed on widely available 8 GiB cards.
RTX 3090 (24 GiB), but G2D2
Details of the experiments and comparison methods are provided in the Appendix.
4.2 Image inverse problem solving on ImageNet and FFHQ
Figure shows the qualitative results of image inverse problem solving, and Tables and list the quantita- tive results. As shown in Tables 1 and 2 G2D2 performs comparably to the methods using diffusion models trained in continuous domains. Notably, G2D2 achieves this performance while consuming significantly lower GPU memory (4.7GiB compared to 10.7GiB for DPS and 20.9GiB for PSLD, as detailed in Table 5 in the Appendix) and maintaining competitive computational speed among gradient-based methods. Note that the pre-trained models used for each method are different, which particularly contributes to the superiority of pixel-domain methods on FFHQ. With DDRM, it is assumed that the amount measurement noise is known, and it requires the singular value decomposition of the linear operator. We also show images in the intermediate steps of the G2D2 algorithm in Figure
Table 1:
Quantitative evaluation on ImageNet. 256x256. Performance comparison of different methods on various linear tasks in image domain. Values show the mean over 1000 images.
SR (x4)
Gaussian Deblurring
Prior Type
Method
LPIPS(()
PSNR(1)
LPIPS(()
PSNR(1)
DPS (Chung et al. 2023b)
0.362
22.67
0.432
19.49
Pixel-domain
DDRM Kawar al. 2022
0.351
24.27
0.250
27.61
PSLD Rout al. 2023
0.331
24.02
0.359
23.95
LDM
ReSample (Song et al. 2024
0.373
23.31
0.425
23.07
G2D2 (proposed)
0.340
23.82
0.367
23.37
Discrete
G2D2 w/ Markov noise process
0.442
22.01
0.424
22.36
Measurement
DPS
G2D2 (Proposed) Ground truth
Measurement
DPS
G2D2 (Proposed)
Ground truth
Super resolution (x4, = 0.05)
Gaussian Deblur 0.05)
LG
Figure 5: Qualitative results of G2D2 and DPS.
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Table 2: Quantitative evaluation on FFHQ x Performance comparison of different methods on various linear tasks in image domain. Values show the mean over 1000 images.
SR (x4)
Gaussian Deblurring
Prior Type
Method
PSNR(1) LPIPS(()
LPIPS(C)
PSNR(1)
DPS (Chung el al. 2023b)
0.238
26.07
0.234
25.47
Pixal-domain
DDRM (Kaaar 2023
2022
0.252
28.09
0.209
30.89
PSLD (Rout al.
0.282
27.12
0.307
26.96
LDM
ReSample (Song el al. 2024
0.508
23.07
0.336
25.91
G2D2 (proposed)
0.265
27.29
0.280
26.91
Discrete
G2D2 w/ Markov noise process
0.369
25.15
0.340
25.96
4.3
Ablation study on Graphical Models
It is possible to derive similar algorithm G2D2 that uses Markov noise process as the graphical model. However, as discussed at the begimning of Section 3 this graphical model does not allow for the "re- tre-masking" operation, which means it cannot correct errors that occur carly in the sampling process. We refer to this variant as G2D2 w/ Markov noise process, and its performance is presented in Tables and on ImageNet and FFHQ, respectively. Additional qualitative results are provided in the Appendix F.3 The results indicate that the introduction the star-shaped noise process significantly improves performance, making G2D2 comparable to methods based on continuous diffusion.
4.4 Motion inverse problem solving
As discussed in Scction 3.5 our method can also be appliod masked generative models. We conduct experiments to manipulate the Generative Masked Motion Model (MMM) (Pinyoanuntapong et al. 2024) using gradient guidance for path-following task where generation is conditioned information. Since joint positions can be calculated from motion data, this fits within the inverse problems framework. While path following has been achieved with continuous latent space models (Song et al. 2023b; Cchida et al., 2024), we are the first to accomplish this using a discrete latent variable motion model without additional training. The Appendix E. 13 provides additional samples and experimental details.
hip joint position
We compare G2D2 with OmniControl (Xie al. 2024) and Guided Motion Diffusion (GMD) (Karun- ratanakul al.) 2023) on controllable motion generation. Following OmniControl's setup, we evaluate using the HumanML3D test set with sparse conditioning (5 frames out of 196) metrics including: FID, R- Precision, Diversity, Foot Skating ralio, Trajectory error (50cm), Location error (50cm), and Average error.
Table presents the results. While OmniControl and GMD are specifically fine-tuned for this task, G2D2 requires no additional training yet achieves good FID and foot skating scores. However, OmniControl and GMD demonstrate better trajectory and location errors, indicating room for improvement. Note that G2D2 could potentially be combined with fine-tuned approaches (as OmniControl does) and enhanced with techniques like FreeDoM's tine-traveling method ( YYu al. 2023) to further improve performance.
Method
FID R-prec.
Diversity
Foot
Traj. Err Loc. err. Avg.
(1)
(1)
(9.503-)
skating (4) (50em, J) (50em, 4)
err. (L)
G2D2
0.248
0.770
9.381
0.048
0.272
0.116
0.230
OmniControl Xie et al. 2024
0.278
0.705
9.582
0.058
0.053
0.015
0.043
GMD Karunratanakul al.
2023
0.523
0.599
N/A
0.086
0.176
0.049
0.139
Table 3: Comparison of mothods for controllable motion gencration.
5
Conclusion
We proposed G2D2 for solving inverse problems using discrete diffusion models as priors. We demonstratod that G2D2 effectively addresses the limitations of discrete diffusion in inverse problem-solving by using
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continuous relaxation technique and star-shaped noise process. Specifically, G2D2 approximates the posterior in inverse problems by optimizing the parameters of variational distribution, composed of parameterized categorical distributions, at each time step of the diffusion process. Our experiments show that G2D2 performs comparably to its continuous counterparts opening up possibilitics for training-free free applications of discrete diffusson models across wide range of tasks.
Limitations and future work While G2D2 already matches the image quality of continuous counterparts, its key advantage is order-of-magnitude reduction in GPU memory lootprint, enabling inference consumer-grade GiB cards. Future work will focus further accelerating sampling speed. We expect these gaps will narrow through efficiency optimizations and stronger prior models. Future work will also explore more challenging settings, such as nonlinear inverse problems, and extend G2D2 to additional modalities, including audio and video.
We are grateful to our colleague Satoshi Hayakawa for providing valuable feedback prior to submission, and we also thank the area chair and the reviewers for their constructive comments.
Acknowledgments
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A
Ethics statement
Our G2D2 method, which uses discrete diffusion models priors for solving inverse problems, carries potential risks similar to those of previously proposed techniques in this field. We acknowledge that these methods, including ours, may inadvertently perpetuale biases present in training data or be misused for generating misleading or harmful content. We are committed to addressing these ethical concerns and promoting responsible use of our technology. We urge users of our method to exercise caution and consider the ethical implications of its applications.
B Reproducibility statement
We provide a detailed description the experimental reproduction in the Appendix, and our code is publicly available httts:///gttuub..com/ooooot
C
Related work
In this section, we review the relevant prior works.
C.1 Leveraging Diffusion Models as Prior Models for Inverse Problems
Pixel-Domain Diffusion Models for Inverse Problems Several methods have been proposed that utilize pixel-domain dilfusion models for solving inverse problems. DDRM and DDNM (Kawar el al. 2022: Wang et al. 2023) assume linear operators and known noise levels, leveraging the singular value decom- position (SVD) of these operators. IIGDM (Song et al., 2023a) can handle certain classes of non-linear operators, such as low dynamic range, where a pseudo-inverse operator can defined. Notably, IIGDM does not require SVD or gradient computations for such a case.
DPS (Chung al. 2023b) broadens the applicability cases where operator gradients can be computed, enabling it to handle both linear and non-linear operators like phase retrieval and non-linear blur. Other notable methods in this category include RePaint (Lugmayr et al. 2022 and RED-Dif (Mardani et al. 2024).
Latent Diffusion Models for Inverse Problems Recent work has also explored the use of latent diffusion models for inverse problems. PSLD (Rout et al. 2023) extends the ideas DPS to latent diffusion models, demonstrating provable sample recovery for linear inverse problems. ReSample (Song al., 2024) achieves data consistency by solving optimization problem at each step during sampling.
Of particular relevance to our work is DAPS (Zhan et al. 2024 which, like our approach, adopts graphical model during sampling that differs from the one used during training of the prior model. This approach, known as the noise decoupling scheme, offers new possibilitics for adapting diffusion models to various inverse problems.
Application of Inverse Problem Solving in Various Domains Solving inverse problems using dif- fusion models has enabled various real-world applications. In the image domain, diffusion models have been extensively studied and applied to tasks such as image deblurring, super-resolution, and inpainting (Lugmayr ct al. 2022 Chung et al. 2023a Zhu et al. 2023). In the audio domain, methods such as those proposed by Song et al. (2021a), Chung el al. (2023c). and Bian et al. (2024) have been developed to address tasks like dereverberation and audio restoration. Similarly, in the medical imaging domain, approaches like those introduced by Song et al. (2021a), Chung et al. (2023c), and Bian et al. (2024) have been used to improve image reconstruction and enhance diagnostic accuracy. These advancements demonstrate the versatility and effectiveness of diffusion models across different. domains.
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C.2 Conditional Generation Using Discrete Diffusion Models Priors
While our work focuses on inverse problems, it is important to consider related approaches in conditional generation tasks using discrete diffusion models as priors. These methods, primarily developed in the context of graph gemeration and protein design, introduce new conditioning to pre-trained models rather than directly addressing inverse problems.
The predominant strategy in this field involves training additional guidance networks. For instance, protein sequence generation, LaMBO-2 (Gruver al. 2024) and Cemri al. (2024) learn networks that
evaluate how intermediate features of samples during generation achieve the desired objectives. Similarly, CGD (Klarner et. al. 2024) learns guidance model for corrupled data. Other examples requiring additional
training include Nisonoff et al.
2024 and DiGress VVignac et al. 2023 for graph generation.
In contrast, Chen et al. 2024b proposes a training-free approach to guide discrete diffusion models for generaling Electronic Health Record data. This method employs Langevin dynamics sampling to minimize given loss function by adjusting the parameters of the final layer of the prior model's transformer output, However, this approach faces scalability issucs with models having large discrete latent spaces, such VQ-Diffusion, it requires evaluating all possible discrete states to compute the loss function.
Another training-fres method for guiding generative models with discrete latents is proposed by Li et al. 2024h. This approach avoids gradient computation of the loss function, instead evaluating the loss on multiple generated samples and conducting sampling based on these values. However, like Chen et al. (2024b), this method expected be inefficient for models with relatively large discrete latent spaces.
D Proofs
In this section, we provide detailed proofs for the main theoretical results presented in the paper.
Theorem 3.1. Let Pa be a distribution with the parameterization given by the decomposition in (6). Then, for any mensurements y, the following inequality holds for the KI. divergence bebween the marginal distribu- tions:
DKL
(zoly)) S
Pa [DKL (Da () Zı, y))],
where (zoll) is the variational marginal distribution parameterized by a, Qster (Zoly) is the true posterior distribution, Par (2) is the variational conditional distribution as defined in [6) (star (2, is the true conditional distribution, Pa (zil)) the marginal distribution at time step t, and T is the total ramber of time steps the diffusion process.
Proof. To prove the inequality in Theorem 3.1 start by noting that the KL divergence between the marginal distributions (aul)) and @star-decomp. (zoly) can be bounded by the KL divergence between the joint distributions Pa (Zorlyy) and Istar-decomp (Zo:ll))
DKL. (oao (Zoly)) < DKT. (po (2o.7 y) getar-demmm (Zo. (Yo.Tly)).
(9)
This incquality holds bocause marginalization cannot increase the KL divergence between distributions.
Next, decompose the joint KT divergence using the chain rule and the definitions of the distributions:
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Pa (Zo:T l
DEL (Pa (ooopp/ordddoooo2o0.r (Zo:)(y)) = Ea rly)
log
dstar-decomp Zu:z ly)
Pa Pa(2rly))
T== Pex (2:
24s y)
Ep (Zu.
log
Ustar (2T yy)
IT==1 Astar (2
1it y)
Pa Par y)
Po (Z 2 Ze.
=
log
Astar (y
log
Jstar Ze-1 Be,
Pa Pal Zt, y)
DKL
Epa (xo:Tl)
log
Astar (z Ze. y.
(10)
In the context of absorbing state diffusion, the distribution Pa Parzly (errl)) is the same as Jstar (zrly) because zp fully determined by the diffusion process and independent of a. Therefore, the first term zero:
DKL
= O.
(11)
This simplifies 110 to:
T
DKL (eserd/c))l (Zo:rly))
>
Bp (2o lx)
log Pa (2+ 24 y)
(12)
Istar (z 2e, y)
We can further simplify the expectation over zorr by locusing on 2, and 21-1:
DRL. (2.:rll))m (x.ep [DKL (pa (Ze- [ee, y) (2:- [)),
(13)
Now, for each term in the sum, we apply the chain rule for KL divergence relate Z and Zo:
DEL (Pa (///--1/ze, y) Il (2c-) (2t,y)) Ezz-1 (z, v) [KLL (Da (zo 10l Ze, y) gstar 2t- Zt, y))]
=DKL (Pa (Zol Zz y) Astar (2 Le. y))
+ (E x.))) [DLL (Par (2: Zu. Zt, y) ( Zo, Zt. Y))]
(14)
In this equation, the left-hand side represents the KL divergence between and Ostar at time t-1 conditioned on 2t, plus the expected KL divergence between their respective conditional distributions of zu- The right- hand side represents the KT, divergence between Pa and (star directly conditioned on 2t, plus an expected KL divergence over Zg-
The crucial observation here is that the last term on the right- hand side is zero. This is because Pa and qatar share the same conditional posterior when conditioned on 2o and Ze, i.e.,
Po (2,-21120 2) Istar (Z, |Zo)
Qstar (z:- |20, Zt, y).
(15)
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Therefore, the KL divergence between these conditional distributions is zero:
DEL (Pa(2z11/z0 Ze, y)sl (2 Ze. y)) = 0.
(16)
Substituting back into
14
we oblain:
DEL y) lotar (Z 1
= DRL (Po(2o) k,, y) Il (2) y))
B2--1 (as y) [DLL (Pa (zol 2z Ze y) Ilgatar 2z- Ze, y))]
(17)
Since the KL divergence is always negateg,,, the expected KL divergence on the right-hand side non- negative, which implies:
DKL (Pa(zi-1/ze Ze. Asta
Astar (ittar(zz-1/z:: (ze-1lzr.)) S DKL (Pa (zol Zz. y) 4star (Zolze, Ze. y))
(18)
Substituting
18 back into
13 we obtain an upper bound on the joint KL divergence:
T
DEL (Pa (zo:rly) (s9taademmls (Zo. (zo:Tly)) yy) - (a DRL (Pa (zo Ze. y) (lolgatrrroozz...oo 22. y))l
(19)
Combining
9 and we conclude:
DEL
(Zoly)) S
Ex pa (2.ly) [KLL (Da (zolze, |gstar (nolarrZzozee (2olzr,)))
(20)
This establishes the incquality stated the theorem.
Lemma 3.1. The KL divergence between the variational distribution Pa (Zo y) and the true posterior (star(Zo|zz) y) can be decomposed into two terms:
DEL (Pa ())lq(z))zz. = DKL (øa 31)4strr)) Ze)) Exy (2olms.)) [log Ister (y [z)]
(8)
Proof. We begin by considering the KL divergence between the variational distribution Pax Pax 24 y) and the true posterior (2n y). Given that zo is a discrete variable, the KL divergence can be expressed as a sum:
DKL (pa(zolz Ist (zolz,, Σ Pa (zo Ze, y) log Ba(Zo (Zo (z,,)) y) Astar (zo Iee y)
(21)
By applying Bayes' theorem to the true posterior qstar (zo Zi,Y), y), we have:
Qstar (zo (z,,) qstar Zo Zz gotar (y Zo)
(22)
ystar (y) (z))
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Since Istar (y(ze) does not depend Zu. it can be treated constant and ignored in the KL divergence
calculation. Substituting Eq. (2 into Eq- (1), we obtain:
Ze, (z, log Pa (Zo (z, y) 9star
DKL (Pa (Zol Ze, y) Battar (zolze. = Σ
Pa(zo
(23)
Astar (zo) Ze)eaar (v/zo)
20
Next, split the logarithm in the numerator and denominator:
log Pa (Zo (2t, log Ostar (y 20))
DKL. (llastrrraa y))
Σ
Pax (zolze.
(24)
Istar (2o) (2:)
Za
This expression can be decomposed into two terms:
The first term represents the KL divergence between the variational distribution Pa (zo02/2 24: y) and the prior (star z.):
Pa (20 ze, y) log Pa (2o Ze. y) Istar (Zol (t)
DKL. (Fa (zo le, Ilsttr (20)ze))
Σ
(25)
Z0
The second term is the negative expected log- likelihood under the variational distribution:
Promaaaa |- log astar (y [2)]] Σ Pa (zo Zz, y) y) log Istar (y/2o).
(26)
Thus, the KL divergence between Pa (zul 22. y) and Jstar (Zo 2) can be decomposed as follows:
DEL (pa(zolz..))ll (zolzi, y)) DKL (2)o Bo (no boy) [log qstar (y fy/z))]. (27)
This concludes the proof.
Lemma D.1. The marginal distribution decomp (Zoilomm/ZZooy is identical to the target distribution qstar (Zoly)
Proof. We aim to show that
@star-docomp (zoly) Ostar (Zoly).
(28)
Recall first that qstar (Zo:T ly) is our original "star-shaped" joint posterior, and we define its "star-decomposed"" variant by
@atar-decomp (Zo:Tly)= = Gstar (ZTly) II Jstar (z:-1 Zz, y).
(29)
By the definition of marginalization, to get Ostar-decomp (zoly), we integrate all the latent variables Zi...., ZT:
Istar-decompl (Zuly) Σ detardeccomp (z.rly))
(30)
Substituting our definition of ystar-decomp (y): into the sum yiclds
@stardecomp (zuly)
Σ
Ystar (zr y)
Istar (Zi - (2
(31)
Z..T
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Here, Qstar (zrly) and each factor qstar (Zi-1 y) are just conditional distributions of the star-shaped model. Next, we marginalize over Zy, then -r-1, and so one index at a time. That is, we view
Istar (zrly) Ostar (2r- = 1star (ZT- Σ Z7
because summation over zr of qstar (zrly) qstar (zr-ilzz- y) collapses exactly to the marginal qatar (ZT-1ly) This step follows directly from the chain rule of probability (or the law of total probability).
We then apply the same idea repeatedly: first summing out zr, which gives Zr-1; then summing out Zr-1 to obtain factor depending on Zr-2; and on, down through Z1. Performing these consecutive sums from t = T down to = eventually leaves only Zo in the expression. Hence, we get
factor depending only on
@star-decomp (zoly) = (Zoly)
(32)
which shows that both star-decomposed and original star-shaped posteriors yield the same marginal distri- bution over Zo.
E Details on Experiments
E.1 Image inverse problems
Implementation of Forward Operators and Dataset Selection In our image inverse problem exper- iments, the definition and implementation of the forward operator are based on the DPS implementation To ensure a diverse representation of ImageNet classes without genre bias, we select 1000 images cover- seattt_allll____a
ing all classes 1, ..., 999 using the imagenet_van val 1k. txt file provided by Pan et al. 22021 For our experiments with the FFHQ dataset, use the 1000 images (indexed 0, l.... 999) from the validation set.
E.2 Implementation Details of G2D2 in Inverse Problem Settings
The implementation of G2D2 is based on the VQ-Diffusion model from the diffusers library For the prior model, we use the pre-trained model available at attpps//huggingfaacc co/microsoft/ vq-diffusion-ithq In our experiments, the number of time steps for sampling is to 100.
Parameterization of Star-Shaped Noise Process In G2D2, the star-shaped noise process follows the same cumulative transition probability q(z, zo) as the original Markov noise process. For the Markov noise forward process where q(2, 21-1) is defined using Qt as in Equation 2 the cumulative transition probability is computed as a(2,, (Zo) v ((220)) where Q2 = Q, ... Q1. Here, Q, can be computed in closed form as:
(.0(20.) = (200) + (7. -B,) + +)
(33)
where ā. =I as Fo = =1 (1- 7)) and = -1) + 1). These parameters can be caleulated and stored in advance. The parameter settings follow those used during the training of the prior model. Specifically, ãi is set to 0.99999, ār to 0.000009, 7, to 0.000009, and 7Λ to 0.99999. For both a and Te, values are linearly interpolated between steps and T. This scheduling results in a linear increase in the number of [MASK] states as t increases, ultimately leading to all variables transitioning to the [MASK] state. Additionally, the transition probability Br between unmasked tokens is set to be nogligibly small, as 0r and 7, sum to nearly
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Optimization in the Algorithm and Instantiation of the Objective Function In the continuous optimization phase, we optimize the parameters a of the categorical distribution using the RAdam opti- mizer (Liu et al. 2020). The optimization objective is a weighted sum of the KL divergence term and the likelihood term, defined as:
arg min MKLDDKL (Fa (Zul Ze, y) + lly - Axo(ax))22)
(34)
a,
where 1KL controls the off between the KL term and the likelihood term.
Marginalization over %0 in Algorithm The marginalization over Zo in line 10 of Algorithm | specif-
ically the term Ez, Istar ( (Zo Y), can be computed in closed form. This computation is feasible because both distributions involved in the marginalization are dimensionally independent categorical distri- butions, discussed by Austin et al. (2021) and Gu et al. (2022).
Dynamic Learning Rate and KL Coefficient Scheduling Some paramcters dynamically adjusted during inference. Both the learning rale for RAdam (IRAdam) and the KL divergence coefficient (vær.) are scheduled using weight vectors that decay logarithmically over the inference steps. These weights are computed based on initial scaling factors.
The learning rale weight vector wh and the KL coefficient weight. vector WKI. are defined as follows:
Alr. schectule 7
(11)
(t) = 10
AKL. rchadule
(#-1))
WKL (t) 10
2
Here, Åir schedule and ÅKL, schedule represent the initial scaling factors for the learning rate and KL coefficient, respectively, and is the total number of inference steps. When An, schedule > O, the learning rate weight vector (t) starts with relatively large values when is large and decays exponentially as t decreasos. Specifically, wh, (1) reaches its minimum near i 1 and its maximun near t = T. This scheduling enables stronger optimization during the initial inference steps, with the learning rate gradually decreasing in the later steps.
At each step 1, the parameters are set as follows:
IRAdam (t) RAdam, base (t), (tt(t) IKL, base WKL (f).
The hyperparameters for Gaussian deblurring and super-resolution tasks used in the experiments are shown in Table
Task-Specific and Common Hyperparameters
The following hyperparameters are shared across all experiments: The number of iterations for the optimiza- tion is set to 30, the temperature for Gumbel-Softmax relaxation is 1.0, and the forget coefficient is 0.3. For the classifier-fre guidance scale, we use 5.0 in ImageNet experiments and 3.0 in FFHQ experiments.
E.3
G2D2 with Markov Noise Process
As discussed Section 4.3 a variant of G2D2 can be derived by introducing the original Markov noise process in the graphical model. In that case, the algorithm shown in Algorithm / The key point here is that the (Markor (2t-1220 2:) part is identical to that of the original Markov noise process, which expressod
(e (2-1.)), v (,
(1 (o, 21)
(35)
In the mask-absorbing type of Markov noise process, this posterior distribution does not revert tokens that have once become unmasked states back to masked tokens. As a result, it becomes difficult to correct errors that occur in the early stages of sampling in subsequent steps.
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Datasct
Task
Hyperparameter
Value
NetgeNet
Gaussian Deblurring
IJK,,
0.0003
XKL schedule
2.0
RAAdam, base
15.0
Xtr schedule
1.0
ImageNet
rusertresolutinn
7JKL, base
0.0003
ÄKL, schedule
2.0
R.Adam, base
10.0
Ayr, schedule
1.0
FFHQ
Gaussian Deblurring
IKL, base
0.0003
XKL. schodule
2.0
AAdam. base
15.0
Alr schedule
1.0
FFHQ
Super-resolution
MKI., hase
0.0003
XKL schedule
2.0
ItAdam, base
10.0
Xtr schedule
2.0
Table 4: Hyperparameters for Gaussian Deblurring and Super-resolution tasks ImageNet and FFHQ datasets.
We compare the computational complexity of G2D2 with the star-shaped noise process (Algorithm 1 and its hypothetical Markovian counterpart (Algorithm 2h. The main algorithmic difference lies in line 10 of both algorithms: the sampling of 2--1 given the optimized Pa (o)
The key differences between both algorithms are as follows. G2D2 (Algorithm 1), we have pa (2:-1 12,21/2. En Astar
y), where 9star ( zo) represents the direct transition from Zo to zt-1 based on the cumulative transition matrix Qi-1 (derived from QI..... Qi-1 as described in Appendix E.2). In the Markovian Version (Algorithm 2), we have Pa (2.-1/2e:) En, 4Markav (2t- (2t-1/Z0, 2, Pa (2,, y), where @Markov (Markov(2.-1 2;) is the posterior from the original Markov process in (35
Crucially, the computational cost of sampling zt- in both cases is dominated by the marginalization over 2o after Pa(an (2o y) has been optimized. The calculation of qstar and (Markov (2-1 (2, (20 2,) involves operations categorical distributions and transition matrices, which are generally efficient and do not introduce significant computational differences between the two approaches.
Algorithm 2 G2D2 with Markov Noise Process
Require: Input condition y, pre-trained discrete diffusion model pe. forget coefficient 7 1: ~ q(2r)
2: for = T, T.....11 do
3: if === then
4: Initialize: a. x (og
5: else
6: Initialize: a: exp(7 log cre + (1 - 7) log po (2:)
end if
8: // continuous optimization
a argmin,, DEL (Na (zo y) y Po (z:)) Fagg- (no y) log ((v/z)))
10: Sample 2-1 Pale (=- y) = E PMarkov (zo-1 1 Iz0, 24 2Pa (2o (2,
11: // Note: The term qpackor 1z 120, 21) uses the posterior distribution of the original Markov
noise process.
12: end for
return Xa by decoding Zo
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E.4 Settings for comparison methods
In this subsection, we detail the experimental settings for the comparison methods.
DPS (Chung el al. 2023b We use the same parameter settings as described in the original paper. The guidance scale is set to 1.0 for FFHQ & super-resolution, 1.0 for FFHQ & Gaussian deblurring, 1.0 for ImageNet & super-resolution, and 0.4 for ImageNet & Gaussian deblurring, The number of time steps is set to 1000. For pre-trained models, we use the unconditional model provided by Dhariwal & Nichol 2021 for ImageNet. For FFHQ, we use the model provided by Choi et al. (2021
DDRM Kawar et al. 2022 We use the official implementation The time steps are set to T = 20, with 17 0.85 and ?h = 1.0 as the hyperparameters. For ImageNet, we use the same pre-trained model as DPS. Although there is no official implementation using pre-trained model trained on FFHQ, both DDRM and Choi al. (2021) are based on the implementation of Dhariwal Nichol 2021 Therefore, in our experiments, DDRM uses the same pre-trained model as DPS.
PSLD Root et al. 2023 We use the official implementation For the pre-trained model, we employ stable-diffusion v-1.5 [Rombach et al. 2022) g As this model handles 512x512 pixel images, we first upscale the ground truth image to 512x512. We then apply the forward operator to the upscaled image and use the result as observed data for our method. Finally, we downsample the output 256x256. For hyperparameters, we use 1 = 1.0 and 7 0.1.
ReSample Song ct al. 2024 We use the official implementation 10 For pre-trained models, we employ two models from the latent dillusion models repository D. LDM-VQ-4 trained on FFHQ, and LDM-VQ-8 trained on ImageNet with class conditioning. We use T = 500 DDIM steps with set 10-4, The maximum number of optimization steps is set to 500. The variance hyperparameter 7 is set 40. For the ImageNet experiments, input the class labels of the ground truth data to the model.
E.5
GPU memory usage and computational speed
We analyze the GPU memory consumption and computational speed of our proposed method, G2D2, comparison with other mothods. Table 5 presents an overview of these metrics for various methods. The measurements are conducted using single NVIDIA A6000 GPU for the Gaussian deblurring task Ima- geNet. G2D2 has the lowest memory usage among all methods and the fastest computational speed among gradient-based methods.
Table 5: Comparison of GPU Memory Usage and Computational Speed
Method
GPU Memory Usage (GiB)
Wall-Clock time (s)
G2D2 (Proposed)
4.7
194
DPS
10.7
277
DDRM
5.8
4
PSLD
20.9
738
ReSample
7.1
555
http:://gghuub.co//yyycll///////
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E.6 Impact of the Forget Coefficient
Figure shows the reduction in the loss function and the final results for the Gaussian deblurring task ImageNet Net when the forget coefficient is set to 0.3 and 1.0. The case with forget coefficient of 1.0 corresponds to not using the optimization results from the previous step at all. Introducing the forget coefficient allows for a faster reduction in the loss function and achieves higher performance with the same computational resources.
Measurement
Evolution
107
102
200 3914
Value
314
95
55
95
55
Ground truth
Forget coefficient 0.3
Forget coefficient = 1.0
Figure 6: Reduction in the loss function and final results for the Gaussian deblurring task on ImageNet with forget coefficients of 0.3 and 1.0. The forget coefficient of 1.0 corresponds to not using the optimization results from the previous step.
E.7 Impact of Text Conditioning the Prior Model
To examine the necessity of text conditioning, investigate the effect of the presence absence of prompts given to VQ-Diffusion on performance. Table t shows the performance for each setting. "Not Used" for text conditioning indicates that classifier-free guidance in the prior model is set to 1.0 (equivalent to unconditional sampling). The prompts we provide to VQ-Diffluiion in our method are "a photo of [Class Name]" for ImageNet experiments and "a high-quality headshot of a person" for FFHQ experiments. should be noted that these prompts are extremely general and do not describe specific details of the images.
From these results, we can confirm that prompt conditioning contributes to certain level of performance improvement ImageNet, while it does not significantly affect performance FFHQ. This suggests that the pre-trained VQ-Diffusion model may not have been extensively trained on human face images, that our chosen prompts for FFHQ may not be optimal.
Additionally, Figure 7 shows a qualitative comparison for the Gaussian deblurring task on the ImageNet datasct. When prompts are used, smoother results are obtained throughout the intermodiate steps when compared, demonstrating that the presence of text conditioning leads final results that capture more fine-grained details.
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With prompt "a photo of guenon, guerion monkey"
Measurement
t 45
35
< 25
15
Ground Truth
Without prompt
= 85
= 75
t=
= 55
Measurement
25
t=5
Ground Truth
Figure 7: Images sampled from the prior model po (Zy) 22) using intermediate 2, during the process G2D2 in image inverse problem solving. The top two rows show intermediale samples using the prompt "a photo of guenon, guenon monkey," " while the bottom represents different time step in the G2D2 algorithm, ranging from 95 (carliest stage) to t 5 (final stage). The rightmost column displays the measurement (degraded input image) and ground truth (original image).
rows show results without using any prompt. Each column
Table 6: Performance comparison with and without text conditioning
SR (x4)
Gaussian Deblurring
Dataset
Text conditioning
LPIPSI
PSNRT
LPIPS4
PSNR
Not Used
0.357
23.55
0.385
23.24
ImageNet
Used
0.351
23.37
0.370
23.14
Not Used
0.258
27.48
0.274
26.99
FFHQ
Used
0.259
27.44
0.273
27.00
E.8
Impact of prior selection for G2D2
G2D2 can also use other prior models when solving inverse problems. this paper, we primarily use VQ-Diffusion trained on the ITHQ dataset (hereafter VQDiff-THH) as a prior model, but to investi- gate how different prior models affect inverse problem solving performance, conduct experiments using
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another model, CLIP- CUIP-VQDiffutoon (Han & Kim 2024 trained the FFHQ dataset (hereafter CVQDiff- FFHQ). As in the main text, evaluated Gaussian deblurring tasks and super-resolation tasks (with measurement noise a = 0.05). For evaluation, we used 100 images from the FFHQ validation subset and mcasured PSNR, LPIPS.
Table 7: Performance comparison between different prior models on FFHQ dataset
SR (x4)
Gaussian Deblurring
Datasct
Prior Model
PSNR1
LPIPSI
PSNRT
LPIPS.
VQDiff-THH
(Tang et al. 2022)
27.44
0.259
27.00
0.273
FFHQ
CVQDif-FFHQ
(Han & Kim 2024
25.32
0.283
24.41
0.289
G2D2 Prior: VQDiff-ITHQ
G2D2 Prior: CVQDift-FFHQ
Measurement
Ground Truth
Super resolution, (x4, 0.05)
Figure 8: Qualitative comparison of super-resolution results (x4, = 0.05) on FFHQ test images. From left to right: low-resolution measurement, ground truth, G2D2 with VQDif-ITHQ prior, and G2D2 with CVQDif-FFHQ prior. While both prior models produce visually pleasing roconstructions, VQDif-ITHQ prior tends to generate results closer to the ground truth, whereas CVQDiI-FFFHQ prior produces fewer artifacts but with slightly less fidelity to the ground truth images.
Table 7 shows the quantitative results, and Figure presents examples of reconstructed samples. Interest- ingly, although CVQDif-FFHQ prior is trained on the FFHQ dataset, G2D2 with VQDif--THH prior performs slightly better. When observing qualitative results, G2D2 with CVQDiI-FFHQ prior produces fewer artifacts but seems to deviate slightly from the ground truth images. Since CVQDiff-FFHQ prior is trained FFHQ, it likely has advantage in prior modeling (corresponding to the first term of the optimization target in Algorithm 1 L10). However, the optimization of the likelihood in the measurement equation (the second term) involves decoder calculations, and since both models use different decoders, this difference may be influencing the results.
httttp://githuu.comm/// http:///ggggaaa....../t./t/ttttt
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E.9 Failure modes of G2D2
We conduct analysis of failure modes. Figure@showe the results of G2D2 and the images during inference for the Gaussian deblurring task on FFHQ. When the ground truth image is relatively young (child's) face, the generated face images appear to be drawn towards a distribution of more adult faces. This is likely due to the use of the prompt "a high-quality headshot of a person". As a result, there is a consistent bias towards adult face images throughout the gencration process, leading artifacts in the final image. In the absence of prompt, the intermediate generated images are not influenced by any specific textual guidance. As a result, the final image tends to have fewer artifacts.
While the star-shaped noise process correct early errors, if errors persist until the later stages, it becomes more difficult to correct them from that point onwards. In other words, when there is mismatch between the distribution conditioned by the prompt and the target image, it becomes challenging for G2D2 to handle it effectively.
To improve these issues, techniques such as simultaneous optimization of prompts may be necessary. Prompt- tuning techniques, proposed in reference (Chung et al. 2024), could be effective in addrossing these challenges.
With prompt: "a high-quality headshot of person"
t 45
35
t 25
t-5 5
Ground Truth
Without Prompt
t= 95
85
t= 75
65
= 55
Meusurement
== 45
== 25
t=5
Ground Truth
Figure 9: Failure modes of G2D2: Gaussian Deblurrigg results the FFHQ dataset. Due to the mismatch between the prompt and the target image, errors remain uncorrected throughout the process, resulting in artifacts in the est imated image.
31

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Published in Transactions on Machine Learning Research (08/2025)
E.10
Additional quantitative results of G2D2 and comparison methods.
Tables
shows the evaluation results with SSTM, FID, and KTD added as evaluation metrics.
Tables
11
show the evaluation results with Structural Similarity Index Mcasure (SSIM) (Wang et al. 2004
Fréchet Inception Distance (FID) (Heusel et al. 2017). and Kernel Inception Distance (KID) (Binkowski
et al. 2018) added as evaluation metrics. The evaluation of FID and KID in this study is conducted under the following conditions: We use the clean-fid library for our evaluation. Following the default settings of the clean-fid library, we use 1i48-dimensional features extracted from the pool_3 3 layer the Inception V3 model. We follow the internal processing of the clean-fid library for image preprocessing, which typically includes resizing input images 299x299 pixels and appropriate normalization. For reference statistics, we use pre-computed statisties of the trainval split of the FFHQ dataset (resolution 256x256, based on 50,000 images) provided by the clean-fid library (specified as dataset aettaset_nae="""ee naa.=nafma" satpasst_splt=tttaaata
For the ImageNet dataset (256x256 resolution), it should be noted that the reference set for FID calculation on ImageNet consists of 1,000 images, which is smaller than the recom- mended sample size for typical FID evaluation (e.g., 10,000 10,000-50,00 000 images). Therefore, FID scores under these conditions should be interpreted with consideration of the limitations regarding the reference set size.
Table 8: Quantitative evaluation on ImageNet. 256x256. Performance comparison of different methods on Super-Resolution (x4) task. Values show mean over 1000 images.
Prior Type
Method
PSNR(1)
LPIPS(4)
SSIM(T)
FID(L)
(ID(L)(x10*))
DPS
22.67
0.362
0.618
45.97
0.41
Pixel-domain
DDRM
24.27
0.351
0.669
52.26
1.16
PSLD
24.02
0.331
0.675
47.88
0.83
LDM
ReSample
23.31
0.373
0.616
68.77
1.96
G2D2 (proposed)
23.82
0.340
0.638
49.97
0.86
Discrete
G2D2 w/ Markov noise process
22.01
0.442
0.557
78.43
2.40
Table 9: Quantitative evaluation ImageNet 256x256. Performance comparison of different methods Gaussian Deblurring task. Values show mean over 1000 images.
Prior Type
Method
PSNR(1)
LPIPS(4)
SSIM(1)
FID(L)
(I
DPS
19.49
0.432
0.473
62.04
0.86
Pixel-domain
DDRM
27.61
0.250
0.786
34.05
0.40
PSLD
23.95
0.359
0.659
50.96
0.96
LDM
ReSample
23.07
0.425
0.586
82.57
2.44
G2D2 (proposed)
23.37
0.367
0.604
61.60
1.39
Discrete
G2D2 w/ Markov noise process
22.36
0.424
0.551
79.29
2.43
Table 10: Quantitative evaluation on FFHQ 256 256.
Performance comparison of different methods on
Super-Resolution (x4) task. Values show the over 1000 images.
Prior Type
Method
PSNR(1)
LPIPS(()
SSTM(t)
FID(L)
(x
DPS
26.07
0.238
0.756
26.85
0.67
Pixel-domain
DDRM
28.09
0.252
0.804
46.19
3.00
PSLD
27.12
0.282
0.757
37.37
2.07
LDM
ReSample
23.07
0.508
0.445
85.86
7.05
G2D2 (proposed)
27.29
0.265
0.763
45.21
3.08
Discrete
G2D2 w/ Markov noise process
25.15
0.369
0.699
62.96
4.47
https://githubb http:///hhhhuccmoaaaaarrtteeeeae
32

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Published in Transactions on Machine Learning Research (08/2025)
Table 11: Quantitative evaluation on FFHQ x 256.
Performance comparison of different methods on
Gaussian Deblurring task. Values show the mean over 1000 images.
Prior Type
Method
PSNR(1)
LPIPS(()
(1)M
FID(L)
KID(4) 2)
DPS
25.47
0.234
0.731
24.10
0.40
Pixel-domain
DDRM
30.89
0.209
0.863
42.51
2.51
PSLD
26.96
0.307
0.750
38.84
1.99
LDM
ReSample
25.91
0.336
0.642
45.26
2.85
G2D2 (proposed)
26.91
0.280
0.743
47.70
3.46
Discrete
G2D2 w/ Markov noise process
25.96
0.340
0.708
57.73
4.29
E.11
Additional qualitative results of G2D2 and comparison methods.
We present additional qualitative results of G2D2 and comparison methods. Figures 10 |hroughh 13 showcase the results for super-resolution and Gaussian deblurring tasks on ImageNet and FFHQ datasets.
G2D2 (Proposed)
Measurement
DPS
DDRM
PSLD
ReSample
Ground truth
Super resolution (x4, 0.05)
Figure 10: Qualitative results of G2D2 and comparison methods.
E.12 Additional qualitative results of G2D2 with Markov noise process
To compare G2D2 and G2D2 with Markov noise process, we present their respective qualitative results in The latter approach does not include re-masking operations its sampling process, which means that once a token becomes unmasked, cannot be modified subsequent iterations. The unnatural artifacts observed in the resulting images are likely attributable to this limitation. This observation underscores the validity of adopting the star-shaped noise process in proposed method.
Figures 14 and 15
33

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Published in Transactions on Machine Learning Research (08/2025)
G2D2 (Proposed)
Measurement
DPS
DDRM
PSLD
ReSample
Ground truth
Gaussian Deblur (a 0.05)
Figure 11: Qualitative results of G2D2 and comparison methods.
E.13 Inverse problems on motion data
We develop G2D2 basod on the official implementation of MMM (Pinyoanuntapong et al. 2024 This method learns masked generative model the discrete latent space obtained by motion lokenizer trained the VQ-VAE framework (van den Oord et al. 2017). G2D2 uses the provided pre-trained model prior distribution.
We conduct experiments on the path following task (Song et al. 2023b Uchida et al. 2024 The objective is to generate motion data mo E R that follows a given path Ypath E R3xL Here, Ypath represents the coordinates of the hip joint at each time frame, L denotes the number of frames in the motion data, and dm is the dimensionality of each motion data point.
The likelihood loss used in the optimization process of G2D2 mcasures how closely the generated motion follows the target path. It is defined
log q path mma)
pyath, Mo,ll
(36)
where A path is linear operator that extracts the path across the frames.
We conduct experiments with a total of T = 25 time steps. For hyperparameters, we set the number of iterations for optimization to 20 and the Gumbel-Softmax temperature to 1.0. The forget coefficient. is set
to 0.7. We adopt the dynamic learning rate scheduling described in the Appendix E.2 The base Adam learning rate IAdam, base set to 0.3, and the KT. divergence weight укK. set 0.05. Additionally, we set AKL schedule and Ane schedule to 0.0 and 1.0, respectively.
htttps://githii http:///tthuuucccoo//txttttaeeae wwtp://ii////////////////
34

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Published in Transactions on Machine Learning Research (08/2025)
G2D2 (Proposed)
Mcasurement
DPS
DDRM
PSLD
ReSample
Ground truth
iO
1a
OO
XO
XO
10
rrs
ors
Super resolution (x4, a = 0.05)
Figure 12: Qualitative results of G2D2 and comparison methods.
35

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Published in Transactions on Machine Learning Research (08/2025)
G2D2 (Proposed)
Mcasurement
DPS
DDRM
PSLD
ReSample
Ground truth
Gaussian Deblur (a = 0.05)
Figure 13: Qualitative results of G2D2 and comparison methods.
36

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Published in Transactions on Machine Learning Research (08/2025)
G2D2 w/ Markov noise process
G2D2 (Proposed)
Measurement
Ground truth
Super resolution (x4, a = 0.05)
Figure 14: Qualitative results comparing G2D2 and G2D2 with Markov noise process (Super-resolution task).
37

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Published in Transactions on Machine Learning Research (08/2025)
G2D2 w/ Markov noise process
G2D2 (Proposed)
Mcasurement
Ground truth
Gaussian Deblur (a = 0.05)
Figure 15: Qualitative results comparing G2D2 and G2D2 with Markov noise process (Gaussian Deblurring task).
38


Reviews:
[{"review_id": "1YYH3ImBn0", "reviewer": "Reviewer_U988", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength:**\n\n\n\nThis paper is overall well written,  and the motivation is clear. This paper provides many detail to illustrate the problems of  Discrete diffusion models and how the proposed methods address the problems.  The idea to use star-shaped noise process for Discrete diffusion models make sense, and this paper provides derivation and theorem to support it. \n\n\n\n**Weaknesses:**\n\n\n\nThe main weaknesses of this paper are the empirical results.\n\n(1) The proposed Star-shaped noise process intuitively will add the time-steps for generating an/a image/sample, therefore add the wall-clock times.  Even though this paper mentions that \"maintaining competitive computational speed among gradient-based methods\", it does not quantitative results. This paper should provide the wall-clock times to compare the efficiency in the experiments. \n\n(2) The experiments are based on  \"a  pretrained VQ-Diffusion model\", how about the results on other models? Besides, is it possible to train the model from scratch for comparing other methods? \n\n\n\nOther minors:\n\n(1) It is not clear what is the shape of matrix $Q_t$ and vector $\\mathbf{v}(z_{t,i})$ in Eqn.1 and 2 , it is better to provide the dimensions  of  $Q_t$ and vector $\\mathbf{v}(z_{t,i})$  in the descriptions", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the problems of  discrete diffusion models with discrete latent codes in their application to inverse problems formulated in continuous spaces.  It firstly addresses the problem of non-differentiability in discrete diffusion models by proposing Gradient-Guided Discrete Diffusion (G2D2), an inverse problem solving method that uses a discrete diffusion model as a prior,  focused on discrete latent variables such as those found in vector-quantized (VQ)-VAE models. It then addresses the  limitation of mask-absorbing processes of discrete diffusion models by incorporating the star-shaped noise process previously proposed in the context of continuous diffusion models.  Experiments conducted on the several standard datasets, demonstrate preliminary  that G2D2 achieves comparable performance to continuous counterparts.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "(1) provide the wall-clock times to compare the efficiency in the experiments. \n\n(2) provide the dimensions  of  $Q_t$ and vector $\\mathbf{v}(z_{t,i})$  in the descriptions", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bvNJ9xqPiS", "reviewer": "Reviewer_acGE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strength:\nThis work studies an interesting topic of diffusion posterior sampling with discrete diffusion models, which to my knowledge is less explored in the literature. The interpretation of the star-shaped diffusion process provides more theoretical clarity on top of previous works (Okhotin et al. 24, Zhang et al. 24). The experimental results is also interesting, which shows the potential of discrete diffusion models in solving inverse problems that lie in continuous space.\n\n### Drawback:\nA potential drawback of the proposed method might be its reliance on the Gumbel-softmax trick. To make this trick work, the categorical distribution has to have an underlying continuous representation, where the softmax of discrete tokens is meaningful. This might potentially limit the scope of the proposed algorithm.\n\n### Some questions:\n- As claimed in the paper, G2D2 is targeted at linear inverse problems. However, it seems that the only requirement of the forward model is to query its log likelihood as shown in Equation (8). Is there a particular reason for restricting G2D2 to linear inverse problems?\n\n- The ablation study on the graphical model seems interesting to me. Could the author provide more details on implementing G2D2 with Markov noise process? Is G2D2 w/ Markov noise an analogy of DPS for discrete diffusion models while G2D2 is more analogous to DAPS?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a new algorithm to solve inverse problems in discrete space using a discrete diffusion model. The proposed algorithm implements a \"star-shaped\" diffusion sampling process, which recursively samples $z_t \\sim p(z_t|y)$, $z_0 \\sim p(z_0|z_t,y)$ with annealing noise levels. Samples from $p(z_0|z_t, y)$ are obtained by variational inference, where the log-likelihood $\\mathbb E_{z_0\\sim p_\\alpha} \\log p(y|z_0)$ is efficiently estimated using the Gumbel-softmax reparameterization trick. The proposed algorithm is validated using VQ-diffusion models trained on VQ latent space of images, yielding comparable results on image restoration tasks as previous works that exploit continuous diffusion models.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The manuscript looks great in its current version, but I believe addressing the questions above in more detail could further enhance its clarity and impact. For example, it would be good to elaborate more on whether G2D2 could be applied to categorical distributions without an underlying continuous representation.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "LnXqR7ziCU", "reviewer": "Reviewer_ZB2Y", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n1. **Novel impactful prior**\n   The paper introduces **G2D2**, to the best of my knowledge the first method that successfully leverages *discrete* diffusion models as priors for inverse problems, broadening the practical scope of diffusion-based techniques.\n\n2. **Strong empirical performance with low memory cost**\n   G2D2 is competitive with contemporary state-of-the-art baselines (PSLD, ReSample, DPS, DDRM) while requiring substantially less GPU memory, highlighting both its effectiveness and practicality.\n\n### Weaknesses \n\n* **Limited evaluation set**\nBaseline comparisons are performed on only 100 images, which may not be representative of the entire dataset.\n\n* **Narrow metric selection**\n  Only two metrics are reported; including additional measures such as **SSIM** or **FID** would give a more rounded view of reconstruction quality.\n\n* **Result inconsistencies**\n  There appear to be discrepancies between the reconstruction errors reported in *Table 2* and those in *Tables 5 and 6* (for example, ReSample in the OpenReview paper \"[https://openreview.net/pdf?id=j8hdRqOUhN](https://openreview.net/pdf?id=j8hdRqOUhN)\").", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors provide a novel posterior sampling algorithm, Gradient‑Guided Discrete Diffusion (G2D2), for solving linear inverse problems using a discrete diffusion prior. This problem is challenging because of the non‑differentiable nature of the model. The authors overcome these technical difficulties by introducing continuous relaxation and optimizing a variational distribution. The authors prove this variational distribution is bounded. In addition, to handle the masking scheme introduced in the architecture of many discrete models, they introduce a star‑shaped noise process, which significantly improves reconstruction when solving the proposed algorithm. The authors provide strong empirical evidence, achieving competitive results against contemporary state‑of‑the‑art baselines while using significantly less GPU memory.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Could the authors consider evaluating their method on a larger set of images to strengthen the comparison with baseline methods?\n\n2. In addition to the point above, could they include a third evaluation metric to provide a more comprehensive assessment?\n\n3. Could the authors clarify the inconsistent results reported in Table 2, as outlined in the strengths and weaknesses section?", "broader_impact_concerns": "I do not have any concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)