We thank Reviewer KkH2 for the constructive review. The reviewer's main concerns relate to theoretical guarantees, the motivation for our approach, and quantitative comparisons. These are excellent points that will strengthen the paper. We have addressed all concerns below, providing new theoretical results, clarifying the motivation with a newly discovered formal link to existing literature, and new experimental results that demonstrate the performance of our method.

1. Regarding the lack of convergence results for our method Our framework admits theoretical convergence guarantees, which we will add to the paper. Clarification: We have two new formal results. First, we prove that the forward Fokker-Planck Equation (FPE) and the reverse-FPE have the same solution for the Geometric Brownian Motion (GBM) SDE considered in our paper. This is a crucial result, as it guarantees that integrating the reverse-time SDE exactly yields samples from the target data distribution. Second, leveraging the connection to the Mirror Langevin Dynamics (MLD) framework (as suggested by Reviewer wtSy), we show convergence for specific cases, such as sampling from a multivariate log-normal density. A sketch of the results is provided below which we will include in the revised manuscript.
2. Regarding the motivation for using biologically inspired learning in score-based modeling. The biological motivation provides a principled path to a novel sampler that is also rigorously grounded in SDEs and a novel score matching approach, further strengthened by a newfound connection to Mirror Langevin Dynamics. Historically, developments in artificial intelligence (AI) have been inspired by evidence of learning mechanisms in biological systems (perceptron, Boltzmann machines, CNNs, etc.). The anthropomorphic bias leads us to believe that the holy grail of AI is to develop algorithms and models that have a biological basis. This is what motivated us to develop a principled multiplicative framework for generative modelling. Clarification: Our primary motivation is to develop bio-inspired sampling algorithms that comply with biological principles in neuroscience such as Dale's Law, where neuronal synaptic flips don’t occur. This principle led us to explore multiplicative updates, which are used to model such biological learning (e.g., via exponentiated gradient descent). Reframing with New Insight: This is not merely a philosophical motivation. During the rebuttal period, we established a formal connection: the convex function in the Bregman divergence used to derive exponentiated gradient descent is identical to the one used to derive the Mirror Langevin Dynamics (MLD) SDE. We show that the proposed reverse-time SDE is an instance of the MLD SDE. A sketch of this derivation is shown below. This fascinating new discovery shows that our bio-inspired approach is a principled approach to discovering a new class of generative models, opening new avenues for future work. Log normal noise and GBM in recent work: We have also found recent pre-prints (posted on June 21, 2025 and July 28, 2025) that leverage log-normal noise model and multiplicative updates for optimization and sampling. In particular, Nishida et al[Log-Normal Multiplicative Dynamics for Stable Low-Precision Training of Large Networks, https://arxiv.org/abs/2506.17768] demonstrate that foundation models such as Vision Transformer and GPT-2 can be stably trained with low-precision optimizers using exponentiated gradient descent and its momentum variants. Likewise, Kim et al.[A diffusion-based generative model for financial time series via geometric Brownian motion, https://arxiv.org/pdf/2507.19003] propose a diffusion-based generative framework for financial time series that incorporates GBM into the forward noising process, however, unlike our approach, they effectively deal with an additive noise model by using the log transform.
3. Regarding the need for concrete regimes where our method outperforms existing frameworks. We have performed new experiments showing that our method quantitatively outperforms standard baselines on several image generation benchmarks. New Results: To strengthen the empirical validation, we have conducted a comprehensive set of new experiments. The following table (Table 1) reports FID and KID scores on MNIST, Fashion-MNIST, and Kuzushiji-MNIST, comparing our method against strong baselines including DDPM and DDIM. These results show the generative performance of our model. We extend our evaluation to color datasets by training on CIFAR-10. The following table (Table 2) reports FID scores and compares them against DDPM, DDIM, Poisson Diffusion, and Beta Diffusion, again demonstrating the competitive advantage of our framework.
4. Regarding the assessment about the theoretical results. Our work provides multiple, synergistic theoretical contributions that build novel bridges between score-based modeling, optimization, and non-negative data modeling and generation. Specifically, our paper is the first to: Establish a new SDE framework for non-negative data, deriving from first principles a novel score-matching formulation which also explains Hyvärinen’s heuristic and computationally intractable, non-negative explicit score matching loss. Derive a novel and tractable loss function, the multiplicative denoising score matching loss, and prove its equivalence to the dynamic version of Hyvärinen’s computationally intractable explicit score matching loss. Uncover a new link between sampling and optimization, by demonstrating that our sampling update equation is structurally and functionally equivalent to the multiplicative update used in exponentiated gradient descent. Connect our SDE to Mirror Langevin Dynamics, revealing a deep theoretical foundation for our biologically-inspired sampler, as detailed in point 2.

Sketch: Assume that $\boldsymbol{X}\_t \sim p_{\boldsymbol{X}\_t}$ is the true data that we'd like to sample and $\boldsymbol{Y}\_t \triangleq \log \boldsymbol{X}\_t$. Now, if we define the forward SDE as mentioned in DDPM we get

$$\mathrm{d}\boldsymbol{Y}\_t = -\dfrac{\beta(t)}{2}\boldsymbol{Y}\_t \,\mathrm{d}{t} + \sqrt{\beta(t)} \,\mathrm{d}{\boldsymbol{W}\_t},$$

where $\beta(t)$ represents the variance schedule. Then the corresponding reverse-time SDE is

$$\mathrm{d}\boldsymbol{Y}\_t = \left(\dfrac{\beta(t)}{2}\boldsymbol{Y}\_t + \beta(t)\nabla \log p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)\right)\,\mathrm{d}{t} + \sqrt{\beta(t)} \,\mathrm{d}{\boldsymbol{W}\_t}.$$

Now substituting $\boldsymbol{Y}\_t = \log \boldsymbol{X}\_t$ and using the score change-of-variables formula (Robbins 2024) to express $\nabla \log p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)$ in terms of $\nabla \log p_{\boldsymbol{X}\_t}(\boldsymbol{X}\_t, t)$, we get

$$\mathrm{d}\log \boldsymbol{X}\_t = \left(\dfrac{\beta(t)}{2}\log\boldsymbol{X}\_t + \beta(t)\left(\boldsymbol{1} + \boldsymbol{X}\_t \circ\nabla \log p_{\boldsymbol{X}\_t}(\boldsymbol{X}\_t, t)\right)\right)\,\mathrm{d}{t} + \sqrt{\beta(t)} \,\mathrm{d}{\boldsymbol{W}\_t}.$$

On Euler-Maruyama discretization,

$$\boldsymbol{X}\_{k-1} = \boldsymbol{X}\_k^{\left(1 + \beta(k\delta)\frac{\delta}{2}\right)} \circ \exp\left( \delta\beta(k\delta)\left(\boldsymbol{1} + \boldsymbol{X}\_t \circ\nabla \log p_{\boldsymbol{X}\_t}(\boldsymbol{X}\_t, t)\right) + \sqrt{\beta(k\delta)\delta} \boldsymbol{Z}\_k\right).$$

We can see that this is structurally different from the reverse-time SDE obtained in the main paper due to the additional exponent factor in $\boldsymbol{X}\_k$. The potential issues with circumventing the multiplicative update through the log transfrom is that, here, we have log-normal noise and the log-transform results in Gaussian noise. But, we are interested in more general stochastic processes, in particular, those that arise in inverse problems with a multiplicative noise model (SAR, OCT) where the noise is modelled using the Gamma distribution. In those settings, a log transformation would not lead to an SDE based on Brownian motion. Our eventual goal is to develop diffusion models for general multiplicative noise models. Additionally, not treating GBM as log-domain DDPM allows us to generalize the framework proposed by Hyvärinen for non-negative data which can now be explained using the underlying SDE formulation placing the score-matching framework on a more firm theoretical footing. We further derive the multiplicative denoising score-matching (M-DSM) loss and show its equivalence to the multiplicative explicit score matching (M-ESM) loss. M-DSM is tractable and is used to train score models for non-negative data. Establishing this link would not have been possible if we'd just considered GBM as log-domain DDPM. Furthermore, it can be shown that the convex function used in the Bregman divergence to derive exponentiated gradient descent (Cornford 2024) is identical to the one used to derive the mirrrored Langevin dynamics (Hsieh et al. Mirror Langevin Dynamics, NeurIPS 2018) SDE. This suggests a much deeper theoretical link that we will investigate in subsequent works.

FID for Datasets

KID for Datasets

CIFAR-10 Results

We thank Reviewer wtSy for the insightful review. Your suggestions regarding log-normal neural dynamics and Mirror Langevin Dynamics (MLD), were instrumental in strengthening the theoretical contributions of our paper.

• On the Core Motivation and Theoretical Contribution Following your suggestions, we've also included the link between Mirror Langevin Dynamics and the proposed approach for which we now provide new convergence analysis.

  • Reframing the Motivation: Dale’s law was an inspiration to develop a biologically inspired generative model employing multiplicative updates for sampling, albeit in the pixel space and not the space of synaptic weights. In view of your suggestion, we’ll reduce the emphasis on biological inspiration and motivate the work from the perspective of log-normal distributions in biological systems (e.g., neural firing rates, c.f. Petersen and Berg 2016) and related domains where data is inherently non-negative. We’ll revise section 2 and make it concise.
  • A Deeper Theoretical Foundation (thanks for your suggestion!): Your pointer to MLD was particularly profound. We have now formally shown that our GBM sampler is related to MLD. More precisely, for the choice of the convex function $h(\mathbf{X}) = \sum_{i=1}^{d}X_i \log X_i - X_i$, the MLD update equation is: $\mathrm{d}{\log \mathbf{X}\_t} = \left(\mathbf{1} + \mathbf{X}\_t \circ \nabla{\mathbf{X}\_t} \log p(\mathbf{X}\_t) \right),\mathrm{d}{t} + \sqrt{2} \mathrm{d}{\mathbf{W}\_t}$ This is the canonical sampling equation for non-negative data, analogous to how standard Langevin dynamics is canonical for unconstrained domains. The proposed GBM SDE (for a specific choice of parameters) recovers and reinforces this fundamental property.
  • New Convergence & Discretization Analysis: The MLD connection allows us to provide the kind of theoretical analysis you were looking for. We now include a proof that for a log-normal target distribution $\mathcal{LN}(\mathbf{\mu}, \sigma^2 \mathbf{I})$, the discretized dynamics of the MLD equation above converge to a limiting measure $\mathcal{LN}\left(\mathbf{\mu}, \frac{\sigma^2}{1 - \delta/(2\sigma^2)} \mathbf{I}\right)$. This analysis not only proves convergence to the correct mean but also precisely quantifies the discretization error in the covariance, which is an altogether new theoretical contribution of the paper significantly elevating the technical content.

• On Our Method vs. Log-Transformed Approaches Our framework is conceptually distinct from prior work that operates purely in log-space; the log-transform in our original derivation was a tool for simplifying the expressions for the sampling updates. Clarifying the Derivation: Vuong & Nguyen (2024) directly apply the log transform and operate in the additive space, which is clearly suboptimal as shown by the MMSE expressions in the context of denoising. We will add a couple of sentences to clarify the differences between our approach and Vuong et al..

  • Derivation without Log-Transform: To clarify, our method does not fundamentally rely on a log-transform. We can derive the reverse-time SDE directly from the GBM SDE in the original space, a sketch of these details mentioned below.
  • On the Choice of Discretization: It can be shown that Euler-Maruyama discretization of the GBM SDE also results in a multiplicative update, but does not guarantee non-negativity. The specific discretization we used — derived via Itō's lemma on $\log \mathbf{X}_t$ — is a principled choice to ensure that the multiplicative updates are always sign-preserving via an exponential map and establish a connection with Dale’s law. Our choice of discretization directly addresses your question about why one wouldn't just use the log-transformed dynamics. Our approach correctly specifies the dynamics in the original space and then chooses a stable, structure-preserving discretization.

• On Strengthening the Empirical Validation Within the constraints of the duration of the rebuttal, we have expanded our experiments to include the suggested comparisons to standard diffusion methods and performance on color images. New Benchmarks: We have now compared with DDPM and DDIM on MNIST, Fashion-MNIST, and Kuzushiji-MNIST. The new results are shown below. Although the FIDs are not clearly outperforming the state-of-the-art the visual quality of the generated images is good indicating that the proposed technique is promising and can be further fine-tuned to improve FID.

  • Scaling to Color Images: We extended our evaluation to CIFAR-10 and obtained an FID of 98.73. Our performance is below par compared to the models you mentioned, including Beta Diffusion and Poisson Diffusion. Within the time constraints of the rebuttal, our models could not be fully optimized to produce the best FID. However, the visual quality of generated samples is high and indicates that there is significant room for improving upon the FID values. We will include the images and the final FID scores in the supplementary of the manuscript.
  • To log-transform or not to log-transform! Denoising is the cornerstone of diffusion-based generative modelling. From the perspective of denoising, it is suboptimal to perform log-transformation as explained further. Consider Minimum Mean Squared Error (MMSE) estimators for the multiplicative noise setting with and without log transformation. Given the forward model $Y=XN$, $X$ = signal, $N$=noise, and the noisy measurement $Y=y$, the optimal/MMSE estimate of $X$ is the conditional expectation $E[X∣Y=y]$. If, instead, one works in the log domain, the MMSE estimate of $\log X$ is $E[\log X∣Y=y]$. However, $\exp{E[\log X|Y=y]} \neq E[X|Y=y]$ rendering the log-transformation approach suboptimal. Therefore, these two denoising approaches are distinct. Our work correctly considers the bona fide multiplicative model without log transformation. We will include this discussion in the revised manuscript. We believe that these revisions satisfactorily clarify all the issues that you have raised: stronger theoretical foundation based on MLD with novel convergence analysis, distinction from prior art, additional experimental validation and suboptimality of the log transformation approach. In view of this rebuttal, we hope that this rebuttal is convincing and you will upgrade your score.

Sketch: In the paper, we resort to the $\log$ transform and leverage the score change-of-variables formula to derive the reverse-time SDE. This choice was made to simplify the mathematics. Now, we demonstrate that we obtain the same reverse-time SDE starting from the original GBM SDE. Consider the GBM

$$\mathrm{d}\boldsymbol{X}\_t = \mu \boldsymbol{X}\_t \,\mathrm{d}{t} + \sigma X\_t \,\mathrm{d}{\boldsymbol{W}\_t}$$

and using the formula for the reverse-time SDE (Song et al. ICLR 2021), it can be shown that the reverse-time SDE is of the form

$$\mathrm{d} \boldsymbol{X}\_t = \left((2\sigma^2\boldsymbol{1} - \boldsymbol{\mu}) X\_t + \sigma^2 \boldsymbol{X}\_t^2 \circ \nabla \log p_{\boldsymbol{X}\_t}(\boldsymbol{X}\_t, t)\right)\,\mathrm{d}{t} + \sigma X\_t \,\mathrm{d}{\boldsymbol{W}\_t}.$$

At this juncture, we consider two possible discretizations. First, using Ito's lemma, we can show the reverse-time SDE can be transformed as

$$\mathrm{d} \log \boldsymbol{X}\_t = \left(\left(\dfrac{3}{2}\sigma^2\boldsymbol{1} - \boldsymbol{\mu}\right) + \sigma^2 \boldsymbol{X}\_t \circ \nabla \log p_{\boldsymbol{X}\_t}(\boldsymbol{X}\_t, t)\right)\,\mathrm{d}{t} + \sigma \,\mathrm{d}{\boldsymbol{W}\_t},$$

and this is equivalent to the reverse-time SDE mentioned in the main paper. The corresponding Euler-Maruyama discretization is given by

$$\boldsymbol{X}\_{k-1} = \boldsymbol{X}\_k \circ \exp\left(- \delta \left(\boldsymbol{\mu} - \frac{3\sigma^2}{2}\boldsymbol{1}\right) + \delta\sigma^2 \boldsymbol{X}\_k \circ \nabla \log p_{\boldsymbol{X}\_k}(\boldsymbol{X}\_k, k) + \sqrt{\delta}\sigma \boldsymbol{Z}\_k\right).$$

Alternatively, if we apply Euler-Maruyama discretization to the raw reverse-time SDE, we get

$$\boldsymbol{X}\_{k-1} = \boldsymbol{X}\_k \left(\left(\boldsymbol{1} - \delta\boldsymbol{\mu} + 2\sigma^2\boldsymbol{1}\right) + \delta\sigma^2 \boldsymbol{X}\_k^2 \circ \nabla \log p_{\boldsymbol{X}\_k}(\boldsymbol{X}\_k, k) + \sqrt{\delta}\sigma \boldsymbol{Z}\_k\right).$$

On comparing the two resulting discretizations, it is clear that both updates are multiplicative but only the one involving the exponential function preserves the sign of the entries of $\boldsymbol{X}\_k$ during sampling. The $\log$ transform through Ito's lemma preserves the sign during sampling and makes it easier to derive the reverse-time SDE. Through this exercise, we have also demonstrated that we don't really need the score change-of-variables formula (Robbins, 2024) to derive our result but leveraging it certainly simplifies the derivation and results in a sign-preserving update.

FID for Datasets

KID for Datasets

CIFAR-10 Results

Thank you for the thoughtful and detailed review. We're delighted you found our paper "very clearly written" and the idea "well motivated," and we appreciate your explicit willingness to reconsider your score. Your main concern seems to be that our framework is an oversimplification of "diffusion models in log domain." We appreciate the chance to clarify this crucial point. We believe this view obscures the key theoretical and practical contributions of our work, which we'll detail below alongside new experimental results.

• It's Not Just Diffusion in the Log Domain
  • Our approach is fundamentally different from standard diffusion in the log domain, both mathematically and practically. As you've rightly pointed out, "EGD is not equivalent to GD in the log domain." This same core principle applies to the proposed Geometric Brownian Motion (GBM) framework. It is indeed not the same as applying a standard DDPM in log space. The distinction is subtle, but critical.
  • The update rules are different. A standard DDPM applied to log-transformed data results in a reverse-time SDE that is structurally and functionally different from the one derived in the paper. A detailed mathematical derivation to this effect is included below.
  • Our formulation is a principled design choice. As you correctly note, the choice of discretization is crucial. Our method yields principled multiplicative updates and the non-negativity constraint is satisfied naturally.
  • MMSE Denoiser property. Denoising is at the heart of diffusion-based generative modelling. To appreciate the distinction further, consider Minimum Mean Squared Error (MMSE) estimators for the multiplicative noise setting with and without log transformation. Given the forward model $Y=XN$, $X$ = signal, $N$=noise, and the noisy measurement $Y=y$, the optimal/MMSE estimate of $X$ is the conditional expectation $E[X∣Y=y]$. If instead one works in the log domain, the MMSE estimate of $\log X$ is $E[\log X∣Y=y]$. However, $\exp{\left(E[\log X|Y=y]\right)} \neq E[X|Y=y]$ which can also be corroborated by Jensen's inequality. Therefore, these two denoising approaches are distinct. Our work correctly considers the true multiplicative model.
• Novelty and Impact Our work's novelty and impact lie in providing the first SDE-based formulation for multiplicative score matching and establishing a new, deep connection between biologically plausible optimization and generative modeling.
  • A Principled SDE Framework for Non-Negative Data: We're the first to derive from first principles a novel score-matching formulation which also explains Hyvärinen’s heuristic and computationally intractable, non-negative explicit score matching loss.
  • A New Tractable Loss: We leverage this link to derive the Multiplicative Denoising Score Matching (M-DSM) loss and formally prove its equivalence to Hyvärinen’s intractable explicit score-matching loss.
  • A New Bridge Between Sampling and Optimization: Our work is the first to establish a formal link between a generative sampling process (the GBM SDE) and a biologically-inspired optimization algorithm, namely, Exponentiated Gradient Descent. This connection is a mathematical identity via the lens of Mirror Langevin Dynamics (MLD).
  • New Convergence Guarantees: We provide two convergence proofs: one showing that the forward and reverse Fokker-Planck Equations match (guaranteeing correct sampling) and another, via MLD, showing convergence for the log-normal case. Please refer to our response to reviewer VjUr (we don't include this here due to space constraints).
• On Biological Relevance The biological motivation is an analogy for principled constraint satisfaction, not a literal model of brain function. We do not posit that the brain implements diffusion on synaptic weights. Rather, Dale's Law encapsulates biological evidence of learning without synaptic flips. This inspired us to build a generative sampler with an analogous sign-preserving property (i.e., non-negativity for images) using multiplicative updates. This analogy led us to a framework whose validity is confirmed by a formal mathematical link. The convex function in the Bregman divergence that yields Exponentiated Gradient Descent is identical to the one that defines the Mirror Langevin Dynamics sampler that matches our SDE. This shows our biologically-inspired principle leads directly to a powerful and theoretically sound generative model.
• New Experimental Results Following your suggestions, we've conducted new experiments showing the performance of our model on standard baselines and competitive with other relevant constrained-domain diffusion models. To address your concerns about our experimental results, we performed several new experiments for performance benchmarking: In the following Table, we compare our method against DDPM and DDIM on MNIST, Fashion-MNIST, and Kuzushiji-MNIST. Although the FIDs are not clearly outperforming the state-of-the-art the visual quality of the generated images is good indicating that the proposed technique is promising and can be further fine-tuned to improve FID.

We extended our evaluation to CIFAR-10 and obtained an FID of 98.73. Our performance is below par compared to the models you mentioned, including Beta Diffusion and Poisson Diffusion. Within the time constraints of the rebuttal, our models could not be fully optimized to produce the best FID. However, the visual quality of generated samples is high and indicates that there is significant room for improving upon the FID values. We will include the images and the final FID scores in the supplementary of the manuscript. We hope these clarifications, new theoretical links, and new experimental results convincingly demonstrate that our contribution is novel and impactful. We would be grateful if you could consider these aspects and upgrade your score.

FID for Datasets

KID for Datasets

CIFAR-10 Results

Thank you for your assessment. Due to space constraints, we included the derivation in response to reviewer KkH2. We reproduce it here. Starting from non-negative data, applying log and using DDPM leads to a reverse-time SDE that is distinct from the proposed GBM approach. While the score of $X$ and the score of $\log(X)$ are related through a change of variables formula, we observed experimentally that plugging in a pre-trained score network did not result in generation -- this warrants the score matching formalism proposed in our paper.

1. Consider the GBM SDE $\mathrm{d}\boldsymbol{X}\_t = \mu \boldsymbol{X}\_t \mathrm{d}{t} + \sigma \boldsymbol{X}\_t \mathrm{d}{\boldsymbol{W}\_t}$ and using Anderson's result [2], the reverse-time SDE takes the form $\mathrm{d} \boldsymbol{X}\_t = \left((2\sigma^2\boldsymbol{1} - \boldsymbol{\mu}) \circ \boldsymbol{X}\_t + \sigma^2 \boldsymbol{X}\_t^2 \circ \nabla \log p\_{\boldsymbol{X}\_t}(\boldsymbol{X}\_t, t)\right)\mathrm{d}{t} + \sigma \boldsymbol{X}\_t \mathrm{d}{\boldsymbol{W}\_t}.$ At this juncture, there are at least two possible approaches. First, using Ito's lemma, we can show the reverse-time SDE can be transformed as

$

\mathrm{d} \log \boldsymbol{X}\_t = \left(\left(\dfrac{3}{2}\sigma^2\boldsymbol{1} - \boldsymbol{\mu}\right) + \sigma^2 \boldsymbol{X}\_t \circ \nabla \log p\_{\boldsymbol{X}\_t}(\boldsymbol{X}\_t, t)\right)\mathrm{d}{t} + \sigma \mathrm{d}{\boldsymbol{W}\_t},

$

which matches the reverse-time SDE given in the paper. The corresponding Euler-Maruyama discretization is given by $\boldsymbol{X}\_{k-1} = \boldsymbol{X}\_k \circ \exp\left(- \delta \left(\boldsymbol{\mu} - \frac{3\sigma^2}{2}\boldsymbol{1}\right) + \delta\sigma^2 \boldsymbol{X}\_k \circ \nabla \log p\_{\boldsymbol{X}\_k}(\boldsymbol{X}\_k, k) + \sqrt{\delta}\sigma \boldsymbol{Z}\_k\right),$ which is sign-preserving.

Alternatively, if we apply Euler-Maruyama discretization to the SDE without the log transform, we get $\boldsymbol{X}\_{k-1} = \boldsymbol{X}\_k \left(\left(\boldsymbol{1} - \delta\boldsymbol{\mu} + 2\sigma^2\boldsymbol{1}\right) + \delta\sigma^2 \boldsymbol{X}\_k^2 \circ \nabla \log p\_{\boldsymbol{X}\_k}(\boldsymbol{X}\_k, k) + \sqrt{\delta}\sigma \boldsymbol{Z}\_k\right).$ On comparing the two discretizations, it is clear that both updates are multiplicative but only the former preserves the sign of the entries of $\boldsymbol{X}\_k$ during sampling. This exercise demonstrates that one doesn't really need the score change-of-variables formula to derive our result but leveraging it certainly simplifies the derivation and results in a sign-preserving update.

2. We show here that DDPM in the log domain results in a different reverse-time SDE than the one proposed in our paper.

The standard DDPM forward SDE (Eq. 11 from Song et al. [1]) is: $\mathrm{d}\mathbf{Y}\_t = -\dfrac{\beta(t)}{2}\mathbf{Y}\_t \mathrm{d}{t} + \sqrt{\beta(t)} \mathrm{d}{\mathbf{W}\_t}$where $\beta(t)$ is the variance schedule. The corresponding reverse-time SDE is:$\mathrm{d}\mathbf{Y}\_t = \left(\dfrac{\beta(t)}{2}\mathbf{Y}\_t + \beta(t)\nabla\_{\mathbf{Y}\_t}\log p(\mathbf{Y}\_t)\right)\mathrm{d}{t} + \sqrt{\beta(t)} \mathrm{d}{\mathbf{W}\_t}$ Now, substituting $\mathbf{Y}\_t = \log \mathbf{X}\_t$ and using the score change-of-variables formula to express the score $\nabla\_{\mathbf{Y}\_t}\log p(\mathbf{Y}\_t)$ in terms of the score of the original data, $\nabla\_{\mathbf{X}\_t}\log p(\mathbf{X}\_t)$, we get the reverse-time SDE in terms of $\mathbf{X}\_t$:$\mathrm{d}\log \mathbf{X}\_t = \left(\dfrac{\beta(t)}{2}\log\mathbf{X}\_t + \beta(t)\left(\mathbf{1} + \mathbf{X}\_t \circ \nabla\_{\mathbf{X}\_t} \log p(\mathbf{X}\_t)\right)\right)\mathrm{d}{t} + \sqrt{\beta(t)} \mathrm{d}{\mathbf{W}\_t}$On Euler-Maruyama discretization with step size $\delta$ and schedule $\beta\_k = \beta(k\delta)$, the update rule becomes:$\mathbf{X}\_{k-1} = \mathbf{X}\_k^{\left(1 + \frac{\delta\beta\_k}{2}\right)} \circ \exp\left( \delta\beta\_k\left(\mathbf{1} + \mathbf{X}\_k \circ \nabla\_{\mathbf{X}\_k} \log p(\mathbf{X}\_k)\right) + \sqrt{\delta\beta\_k} \mathbf{Z}\_k\right)$ This is also a sign-preserving update, but, this is structurally different from the sampler derived in our paper due to the additional exponent factor of $\left(1+ \frac{\delta\beta_k}{2}​​\right)$ on $\mathbf{X}\_k$.

The preceding analysis confirms that our framework is not simply log-domain DDPM. A term-by-term comparison of Eq.(8) in our paper with Eq.(11) in Song et al. [1] also shows that they are NOT identical. Thank you for the question. We will include this analysis in the paper.

In view of this analysis, we hope you will consider increasing your score.

[1] Song et al., Score-Based Generative Modeling Through Stochastic Differential Equations, ICLR 2021

[2] Anderson, Reverse-time diffusion equation models, Stochastic Processes and their Applications 1982

We are grateful to Reviewer VjUr for the constructive and actionable feedback. We were particularly encouraged that you found the core connection between Dale's Law and Geometric Brownian Motion to be surprising, original, and interesting. We have followed your suggestions and believe that revising the manuscript accordingly will make it substantially stronger. We present new benchmarking results against DDPM/DDIM, further clarify the theoretical foundations that motivate our method's advantages, and provide concrete responses to all other points raised.

1. On Empirical Evaluation & Benchmarking We have added new benchmarking experiments against DDPM and DDIM New Benchmarks and Scaling to Color Images: We have now conducted these experiments and report the results in the below. We also extended our evaluation to CIFAR-10 and obtained an FID of 98.73. Our performance is below par compared to the models you mentioned, including Beta Diffusion and Poisson Diffusion. Within the time constraints of the rebuttal, our models could not be fully optimized to produce the best FID. However, the visual quality of generated samples is high and indicates that there is significant room for improving upon the FID values. We will include the images and the final FID scores in the supplementary of the manuscript. Mode Collapse & Sample Quality Figure 2: This is a fair observation from visual inspection and we are investigating the cause of the mode collapse during sampling. Further optimization of the step-size, choice of noise scheduler could potentially avoid this issue. Error Bars: We followed the standard practice of not reporting the error bars on FID like many leading papers on diffusion modelling do(e.g., DDPM, DDIM). Since FID is reported on a large set of 50k generated images, the variance is typically very low.
2. On Theoretical Justification & Error Bounds Our method's advantage stems from providing the first SDE-based formulation for multiplicative score matching, establishing a new, deep connection between bio-inspired optimization and generative modeling and a newfound connection to Mirror Langevin Dynamics, which offers a path toward formal guarantees. Why It Should Perform Better: Our framework is not an arbitrary choice but is specifically tailored for non-negative data like images. Standard DDPMs use additive Gaussian noise, which can push pixel values below zero, forcing the model to learn a difficult "clamping" behavior at the boundary. Our multiplicative GBM process naturally respects the non-negativity constraint at every step of the SDE, making the learning task for the score network simpler and more direct. Connection to Mirror Langevin Dynamics (MLD): The theoretical "why" is further strengthened by a new insight we gained during the rebuttal period. Our reverse-time SDE is related to Mirror Langevin Dynamics, a powerful class of samplers designed for constrained domains. The convex function used in the Bregman divergence that underpins EGD updates is precisely the one used to derive this specific MLD sampler. This is yet another theoretical link, connecting our biologically-inspired method to a well-studied family of optimal transport and sampling algorithms. On Error Bounds: Leveraging the MLD link, we have derived convergence behaviour of generated samples for a lognormal target distribution that evolves with the MLD SDE. The MLD connection allows us to provide the kind of theoretical analysis you were looking for. We now include a proof that for a log-normal target distribution $\mathcal{LN}(\mathbf{\mu}, \sigma^2 \mathbf{I})$, the discretized dynamics of the MLD equation above converge to a limiting measure $\mathcal{LN}\left(\mathbf{\mu}, \frac{\sigma^2}{1 - \delta/(2\sigma^2)} \mathbf{I}\right)$. This analysis not only proves convergence to the correct mean but also precisely quantifies the discretization error in the covariance, which is an altogether new theoretical contribution of the paper significantly elevating the technical content.
3. On Computational Cost & Hyperparameters Our method has the same computational complexity as a standard DDPM of equivalent size, and we now provide precise training details and clarify our hyperparameter search. Computational Overhead: There is no additional computational overhead compared to DDPM. The primary cost during sampling is the number of function evaluations (NFE) of the score network. Since both our method and DDPM use the same number of reverse-time steps (N=1000), the sampling cost for a network of the same size is identical. Quantified Resources: We have already mentioned these details in section 4.3 of the supplementary document. Hyperparameter Search: For sampling, the quality depends on the reverse-time step size. We performed a grid search over step sizes in the range [10−8,10−3], selecting the configuration that produced the best sample quality, which we now quantify with FID and KID.
4. Other Clarifications On Specialized Noise Schedules: The proposed approach is not dependent on the choice of noise schedule and just like DDPM, our framework can be combined with any noise scheduling scheme (e.g., linear, cosine) to potentially improve results further. However, a thorough investigation pertaining to the optimal noise schedules in the multiplicative setting remains an open issue. On Financial Applications: Incidentally, during the rebuttal phase we came across a preprint by Kim et al. “A diffusion-based generative model for financial time series via geometric Brownian motion, https://arxiv.org/pdf/2507.19003, uploaded on 25th July, 2025.” They propose a diffusion-based generative framework for financial time series that incorporates GBM into the forward noising process, however, unlike our approach, they effectively deal with an additive noise model by using the log transform. There seems to be compelling evidence building up to show the efficacy of log normal noise and GBM SDE for generative modeling. We are confident that these new experiments, deeper theoretical framing, and detailed clarifications directly address the points you raised. We have provided the requested benchmarking and a theoretical foundation for our method's advantages. We hope you'll agree that these additions substantially strengthen the paper and provide the justification needed to reconsider your score.

FID for Datasets

KID for Datasets

CIFAR-10 Results

Convergence through the Fokker-Planck equations. For the forward SDE

$$\mathrm{d}\boldsymbol{Y}\_t = \left(\boldsymbol{\mu} - \frac{\sigma^2}{2}\boldsymbol{1}\right) \,\mathrm{d}{t} + \sigma \,\mathrm{d}{\boldsymbol{W}\_t},$$

the corresponding Forward FPE is

$$\dfrac{\partial p_{\boldsymbol{Y}\_t}(\boldsymbol{X}\_t, t)}{\partial t} = - \nabla \cdot \left(\left(\boldsymbol{\mu} - \frac{\sigma^2}{2}\boldsymbol{1}\right) p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)\right) + \frac{\sigma^2}{2}\Delta^2 p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t). \tag{1}$$

The reverse-time SDE is $\mathrm{d}\boldsymbol{Y}\_t = -\left(\boldsymbol{\mu} - \frac{\sigma^2}{2}\boldsymbol{1} - \sigma^2 \nabla \log p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)\right) \,\mathrm{d}{t} + \sigma \,\mathrm{d}{\boldsymbol{W}\_t},$ and its corresponding reverse-time FPE is given by $-\dfrac{\partial p_{\boldsymbol{Y}\_t}(\boldsymbol{X}\_t, t)}{\partial t} = \nabla \cdot \left(\left(\boldsymbol{\mu} - \frac{\sigma^2}{2}\boldsymbol{1} - \sigma^2 \nabla \log p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)\right) p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)\right) + \Delta^2\left(\frac{\sigma^2}{2} p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)\right).$ Since $\nabla \cdot \nabla \log p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t) = \Delta^2 p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t)$~\citep{Lyu_score_09}, we can rewrite the reverse-time FPE as

$$\dfrac{\partial p_{\boldsymbol{Y}\_t}(\boldsymbol{X}\_t, t)}{\partial t} = -\nabla \cdot \left(\left(\boldsymbol{\mu} - \frac{\sigma^2}{2}\boldsymbol{1}\right)p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t)\right) + \frac{\sigma^2}{2}\Delta^2 p_{\boldsymbol{Y}\_t}(\boldsymbol{Y}\_t, t). \tag{2}$$

Comparing Eq. (1) and Eq. (2), we see that they are identical. The reverse-time SDE is designed to generate samples from the target distribution and we refer the reader to (Anderson1982) for a detailed proof.