The Real Math Behind Diffusion Models (Stable Diffusion, DALL·E)

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Type a prompt, and a diffusion model paints an image that never existed.
The popular explanation — "it removes noise" — is true but almost empty.
This is the actual mathematics underneath, built carefully from the ground up.

We start from the real problem (we can sample data but have no formula for it),
build the forward noising process and its exact closed form, then derive the
reverse process, the training objective (the ELBO, which collapses to simply
predicting the noise), and the idea that ties it all together: the score —
the gradient of the log-density. We show why "predict the noise" is exactly
"learn the score," how Langevin dynamics turns that score into samples, and how
the whole discrete recipe is just a stochastic differential equation in
disguise. No magic, no memorized gallery — a forward process you can write in
closed form, and a reverse process you can learn.

The math is verified throughout, and we keep two honest corrections at the end:
"diffusion" is a precise Gaussian process (not literal particle physics), and
the model learns a score field rather than memorizing — though near-exact
copies are a real, measured failure mode with heavily duplicated data.

⏱️ Chapters
0:00 Cold open — the question
0:41 The generative problem & Markov chains
1:48 The forward process & the closed form
3:11 The reverse process
4:22 Training: the ELBO → predict the noise
5:56 The score: why predicting noise works
7:35 Sampling: following the score (Langevin)
8:54 Unification (SDEs), myths & recap

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