  Diffusion models have gained traction as powerful algorithms for synthesizing
high-quality images. Central to these algorithms is the diffusion process, a
set of equations which maps data to noise in a way that can significantly
affect performance. In this paper, we explore whether the diffusion process can
be learned from data. Our work is grounded in Bayesian inference and seeks to
improve log-likelihood estimation by casting the learned diffusion process as
an approximate variational posterior that yields a tighter lower bound (ELBO)
on the likelihood. A widely held assumption is that the ELBO is invariant to
the noise process: our work dispels this assumption and proposes multivariate
learned adaptive noise (MULAN), a learned diffusion process that applies noise
at different rates across an image. Specifically, our method relies on a
multivariate noise schedule that is a function of the data to ensure that the
ELBO is no longer invariant to the choice of the noise schedule as in previous
works. Empirically, MULAN sets a new state-of-the-art in density estimation on
CIFAR-10 and ImageNet and reduces the number of training steps by 50%. Code is
available at https://github.com/s-sahoo/MuLAN
