Dale meets Langevin: A Multiplicative Denoising Diffusion Model

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 Hyv\"arinen 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.