  Typical generative diffusion models rely on a Gaussian diffusion process for
training the backward transformations, which can then be used to generate
samples from Gaussian noise. However, real world data often takes place in
discrete-state spaces, including many scientific applications. Here, we develop
a theoretical formulation for arbitrary discrete-state Markov processes in the
forward diffusion process using exact (as opposed to variational) analysis. We
relate the theory to the existing continuous-state Gaussian diffusion as well
as other approaches to discrete diffusion, and identify the corresponding
reverse-time stochastic process and score function in the continuous-time
setting, and the reverse-time mapping in the discrete-time setting. As an
example of this framework, we introduce ``Blackout Diffusion'', which learns to
produce samples from an empty image instead of from noise. Numerical
experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the
feasibility of our approach. Generalizing from specific (Gaussian) forward
processes to discrete-state processes without a variational approximation sheds
light on how to interpret diffusion models, which we discuss.
