See the notes here

I’ve recently been working with discrete diffusion models and from digging into the literature around this topic, I’ve been annoyed at just how hard it is to find a good, unified presentation of the main theory behind discrete diffusion, namely time-inhomogeneous continuous-time Markov chains (CTMCs).

The issue arises from the fact that most mathematical presentations of CTMCs only work with the time-homogeneous case. Hence, finding an explanation of the time-dependent rate matrix, and how this gives rise to important results like the Kolmogorov forward and backward equations (and more generally, the theory of time-inhomogeneous Markov processes described by evolution systems) is a frustrating process.

To help others skip this time-consuming journey, I’ve collected the main results, theory and references which I’ve found helpful to learn about time-inhomogeneous CTMCs in these slides which can be found here. I also provide an introduction to the main papers on discrete diffusion models and discuss how they connect to the underlying theory. I hope these notes can be useful to the wider machine learning community.