The relative importance of being Gaussian
Journal:
arXiv
Published Date:
Jul 10, 2025
Abstract
The remarkable results for denoising in computer vision using diffusion
models given in \cite{SDWMG,HJA,HHG} yield a robust mathematical justification
for algorithms based on crucial properties of a sequence of Gaussian
independent $N(0,1)$ random variables. In particular the derivations use the
fact that a Gaussian distribution is determined by its mean and variance and
that the sum of two Gaussians is another Gaussian.
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The issue raised in this short note is the following: suppose we use the
algorithm without any changes but replace the nature of the noise and use, for
instance, uniformly distributed noise or noise with a Beta distribution, or
noise which is a random superposition of two Gaussians with very different
variances. One could, of course, try to modify the algorithm keeping in mind
the nature of the noise, but this is not what we do. Instead we study the
performance of the algorithm when used with noise that is very far in nature
from the Gaussian case, where it is designed to work well.
Usually these algorithms are implemented on very powerful computers. Our
experiments are all carried out on a small laptop and for the smallest possible
image size. Exploring how our observations are confirmed or changed when
dealing in different situations remains an interesting challenge.
