Diffusion models have gained traction as powerful algorithms for synthesizing high-quality images. Central to these algorithms is the diffusion process, which maps data to noise according to equations inspired by thermodynamics and can significantly impact performance. A widely held assumption is that the ELBO objective of a diffusion model is invariant to the noise process (Kingma et al.,2021). In this work, we dispel this assumption -- we propose multivariate learned adaptive noise (MuLAN), a learned diffusion process that applies Gaussian noise at different rates across an image. Our method consists of three components -- a multivariate noise schedule, instance-conditional diffusion, and auxiliary variables -- which ensure that the learning objective is no longer invariant to the choice of the noise schedule as in previous works. Our work is grounded in Bayesian inference and casts the learned diffusion process as an approximate variational posterior that yields a tighter lower bound on marginal likelihood. Empirically, MuLAN sets a new state-of-the-art in density estimation on CIFAR-10 and ImageNet compared to classical diffusion. Code is available at https://github.com/s-sahoo/MuLAN

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