Dale meets Langevin: A Multiplicative Denoising Diffusion Model
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arXiv
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2025
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| _version_ | 1866914595494428672 |
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| author | Shetty, Nishanth Prasath, Madhava Seelamantula, Chandra Sekhar |
| author_facet | Shetty, Nishanth Prasath, Madhava Seelamantula, Chandra Sekhar |
| contents | Exponentiated gradient descent (EGD), a biologically motivated optimisation algorithm that respects Dale's law, produces log-normally distributed synaptic weights at convergence, in alignment with experimental observations in neuroscience. Since the marginal distribution of geometric Brownian motion (GBM) at any fixed time is log-normal, this convergence property reveals a natural connection between EGD and GBM-based stochastic processes. We propose a multiplicative score-based generative model with GBM as a forward noising process and derive its corresponding reverse-time SDE in both the ambient space and in the $\log$-transformed space. We derive two multiplicative samplers by discretising the corresponding reverse-time SDEs: a sign-agnostic sampler obtained directly from the ambient-space reverse-time SDE, and a sign-preserving sampler, which we refer to as the Dale-Langevin sampler, obtained via the Lamperti transform. We connect the framework to Mirrored Langevin Dynamics, showing that the convex function driving EGD in optimisation precisely governs the Dale-Langevin sampler. While the standard Stein score, defined as $\nabla \log p_{\boldsymbol{X}}(\boldsymbol{x})$ for a random vector $\boldsymbol{X}$ evaluated at $\boldsymbol{x}$, comes up naturally in the additive noise based diffusion models, in the multiplicative setting, we encounter a modified version of the Stein score for sampling, which we refer to as the {\it Hyvärinen score}: $\boldsymbol{x} \circ \nabla \log p_{\boldsymbol{X}}(\boldsymbol{x})$. To estimate the score, we propose a new multiplicative denoising score-matching objective (M-DSM), prove its equivalence to the multiplicative explicit score-matching loss and show that it subsumes the non-negative score matching loss. Experimental results on MNIST, Fashion-MNIST, Kuzushiji-MNIST, and CIFAR-10 to validate the generative capability of the proposed framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_02730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dale meets Langevin: A Multiplicative Denoising Diffusion Model Shetty, Nishanth Prasath, Madhava Seelamantula, Chandra Sekhar Machine Learning Computer Vision and Pattern Recognition Exponentiated gradient descent (EGD), a biologically motivated optimisation algorithm that respects Dale's law, produces log-normally distributed synaptic weights at convergence, in alignment with experimental observations in neuroscience. Since the marginal distribution of geometric Brownian motion (GBM) at any fixed time is log-normal, this convergence property reveals a natural connection between EGD and GBM-based stochastic processes. We propose a multiplicative score-based generative model with GBM as a forward noising process and derive its corresponding reverse-time SDE in both the ambient space and in the $\log$-transformed space. We derive two multiplicative samplers by discretising the corresponding reverse-time SDEs: a sign-agnostic sampler obtained directly from the ambient-space reverse-time SDE, and a sign-preserving sampler, which we refer to as the Dale-Langevin sampler, obtained via the Lamperti transform. We connect the framework to Mirrored Langevin Dynamics, showing that the convex function driving EGD in optimisation precisely governs the Dale-Langevin sampler. While the standard Stein score, defined as $\nabla \log p_{\boldsymbol{X}}(\boldsymbol{x})$ for a random vector $\boldsymbol{X}$ evaluated at $\boldsymbol{x}$, comes up naturally in the additive noise based diffusion models, in the multiplicative setting, we encounter a modified version of the Stein score for sampling, which we refer to as the {\it Hyvärinen score}: $\boldsymbol{x} \circ \nabla \log p_{\boldsymbol{X}}(\boldsymbol{x})$. To estimate the score, we propose a new multiplicative denoising score-matching objective (M-DSM), prove its equivalence to the multiplicative explicit score-matching loss and show that it subsumes the non-negative score matching loss. Experimental results on MNIST, Fashion-MNIST, Kuzushiji-MNIST, and CIFAR-10 to validate the generative capability of the proposed framework. |
| title | Dale meets Langevin: A Multiplicative Denoising Diffusion Model |
| topic | Machine Learning Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2510.02730 |