Understanding and contextualising diffusion models

The latest developments in Artificial Intelligence include diffusion generative models, quite popular tools which can produce original images both unconditionally and, in some cases, conditioned by some inputs provided by the user. Apart from implementation details, which are outside the scope of this work, all of the main models used to generate images are substantially based on a common theory which restores a new image from a completely degraded one. In this work we explain how this is possible by focusing on the mathematical theory behind them, i.e. without analyzing in detail the specific implementations and related methods. The aim of this work is to clarify to the interested reader what all this means mathematically and intuitively.