{"ID":2860792,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.02730","arxiv_id":"2510.02730","title":"Dale meets Langevin: A Multiplicative Denoising Diffusion Model","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2510.02730","url_pdf":"https://arxiv.org/pdf/2510.02730v2","authors":"[\"Nishanth Shetty\",\"Madhava Prasath\",\"Chandra Sekhar Seelamantula\"]","published":"2025-10-03T05:23:33Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.CV\"]","methods":"[\"Diffusion Model\"]","has_code":false}