𝗠𝗖𝗠𝗟 𝗕𝗹𝗼𝗴: Diffusion Models are the engine behind the current wave of GenAI. But understanding how they work often means diving into heavy math, and most explanations focus only on images. MCML researchers ask: Can we explain them in one unified framework?
🔎 Read more: mcml.ai/news/2026-03...
Thanks a lot, David! I already had a good idea of the box design I wanted and a draft version I made for previous lecture notes. Then I iterated with AI to optimise it until I was happy with the final result 🙂
PS: We also recently released a unified codebase for discrete diffusion, check it out!
𝕏 Thread : x.com/nkalyanv99/...
🔗 GitHub: github.com/nkalyanv99/...
📚 Docs: nkalyanv99.github.io/UNI-D2/
🎨 Credit to
@ruiqigao.bsky.social and the Diffusion Meets Flow team (@emielhoogeboom.bsky.social, Jonathan Heek
@vdebortoli.bsky.social, @timsalimans.bsky.social and
@sirbayes.bsky.social) for inspiring the illustration!
Feedback and questions are very welcome, reach out or reply here. 🙌 💬
🍒 The cherry on the cake for experts:
Section 7 is the unifying lens: Markov process infinitesimal generators. 🔑
→ ONE forward marginal evolution equation (KFE)
→ ONE time-reversal formula
→ ONE ELBO expression
This framework specialises all previous results. 🤯
📚 Three reading paths:
🟢 New to diffusion? Primer → discrete-time → ELBO → latent diffusion
🟠 Know DDPMs? Skip to continuous-time: SDEs, CTMCs, Fokker-Planck & master equations.
🟣 Want the full picture? Generator framework — where SDEs & CTMCs become special cases
🎨 The presentation makes parallel structure visible:
🔵 Blue boxes = Continuous state space (Gaussian, SDE, score)
🔴 Red boxes = Discrete state space (Categorical, CTMC, rate matrix)
🟡 Yellow boxes = General results (Markov Process, Generator)
Every key result appears side-by-side. 🗺️
🎯 Core insight:
Continuous and discrete diffusion share the same underlying structure:
• Forward process: corrupts data → noise
• Reverse process: learns to denoise
• Training: maximise and ELBO
The difference? Gaussian vs Categorical distribution
The math? Surprisingly parallel
📖 What you'll actually learn:
• Why diffusion works (variational perspective)
• Forward process → reverse process
• Score vs rate matrices
• How to train these models (ELBO → loss)
• Latent diffusion and applications for discrete data
👉 All self-contained. No prerequisites beyond basic probability.
The solution 👨🔬🤩:
A single, self-contained roadmap that:
✔ works for images and sequences
✔ covers discrete and continuous time
✔ connects VAEs, score-based models, SDEs, CTMCs, and latent diffusion — all in one coherent story
The problem 🤕😭:
🚫 Introductions to diffusion models only cover Euclidean data.
🚫 Discrete diffusion is booming, but resources are scattered and the link to core principles is unclear.
🚫 Existing intros are either oversimplified, overly dense, or send you back to old stochastic-calculus textbooks.
Looking for a fully self-contained intro to diffusion models that covers both continuous (images) and discrete (text, sequences) data? 🔍📚
We just released:
“Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction”
One roadmap for all of diffusion. 🏎️💨
🆕 “Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction”
Huge thanks to Tobias Hoppe, @k-neklyudov.bsky.social,
@alextong.bsky.social, Stefan Bauer and @andreadittadi.bsky.social for their supervision! 🙌
arxiv : arxiv.org/abs/2512.05092 🧵👇
