  We consider the problem of generative modeling based on smoothing an unknown
density of interest in $\mathbb{R}^d$ using factorial kernels with $M$
independent Gaussian channels with equal noise levels introduced by Saremi and
Srivastava (2022). First, we fully characterize the time complexity of learning
the resulting smoothed density in $\mathbb{R}^{Md}$, called M-density, by
deriving a universal form for its parametrization in which the score function
is by construction permutation equivariant. Next, we study the time complexity
of sampling an M-density by analyzing its condition number for Gaussian
distributions. This spectral analysis gives a geometric insight on the "shape"
of M-densities as one increases $M$. Finally, we present results on the sample
quality in this class of generative models on the CIFAR-10 dataset where we
report Fr\'echet inception distances (14.15), notably obtained with a single
noise level on long-run fast-mixing MCMC chains.

Machine Learning;Machine Learning