Foundations — Classifier-free guidance

This page assumes you know nothing about the notation in Classifier-free guidance. We will build every letter, every subscript, and every operator from scratch, in an order where each idea leans only on the ones before it. If a symbol appears in the parent note, it is defined here first.

1. What is a diffusion model even doing?

Before any symbol, picture the process it lives in.

Every symbol in the parent note is a label somewhere on this forward-and-backward pipeline. Let's name the pieces.

2. The building blocks, in order

2.1 $\mathbf{x}$ — an image written as a list of numbers

Picture: a photo is a grid of pixels; unroll that grid into one long line of numbers. That line is $\mathbf{x}$.

Why the topic needs it: guidance manipulates images, and to do maths on an image we must treat it as a vector of numbers.

2.2 The subscript $t$ — how noisy the image is

Picture: a dial from clean (small $t$) to snow (big $t$). ${\mathbf{x}}_{500}$ = the image when the dial sits at 500.

2.3 $ϵ$ — the noise itself

Picture: the exact snow pattern sitting on top of the clean image. If you knew $ϵ$ perfectly, you could subtract it and recover the clean picture.

Why the topic needs it: modern diffusion models don't predict the clean image directly — they predict the noise $ϵ$. Guessing the static and removing it is easier to train.

2.4 ${ϵ}_{\theta }$ — the model's guess of the noise

Picture: a machine with millions of adjustable knobs ($\theta$). You feed it a noisy image; out comes its best guess of the static, ${ϵ}_{\theta }$.

2.5 $\mathbf{c}$ — the conditioning (your prompt)

Picture: a sticky note taped to the model's input saying "make it a golden retriever."

2.6 $\varnothing$ — the empty conditioning

Picture: the same sticky-note slot, but the note is blank.

Why the topic needs it: classifier-free guidance is entirely built on comparing two runs — one with the note ($\mathbf{c}$) and one with the blank note ($\varnothing$).

3. The three operations guidance performs on vectors

Now that ${ϵ}_{\theta }$ is a vector (a list of numbers), we combine two such vectors. Three simple operations do all the work.

3.1 Vector subtraction — "the difference the prompt makes"

Why subtraction and not something else? Subtraction isolates only what the prompt added. Whatever both runs agree on cancels out; what's left, $\Delta ϵ$, is a direction — an arrow pointing "toward more prompt-ness."

3.2 Scaling by $w$ — the "prompt volume knob"

Picture: $\Delta ϵ$ is an arrow; $w\Delta ϵ$ is that same arrow stretched to $w$ times its length.

• $w=0$: arrow vanishes → ignore prompt.
• $w=1$: normal conditional generation.
• $w>1$: overshoot past the conditional → stronger prompt adherence (see Prompt engineering).

3.3 Vector addition — assembling the guided guess

Multiply out and you land on the parent's boxed formula: ${\widetilde{ϵ}}_{\theta }=(1-w){ϵ}_{\theta }({\mathbf{x}}_t,t,\varnothing )+w{ϵ}_{\theta }({\mathbf{x}}_t,t,\mathbf{c})$

4. Two more symbols the parent leans on

4.1 ${\nabla }_{{\mathbf{x}}_t}\log p$ — the "score"

Picture: a hilly landscape where height = "how realistic." The score is the arrow telling you which way is uphill. This is the heart of Score-based diffusion models, and it links to noise via ${\nabla }_{{\mathbf{x}}_t}\log p({\mathbf{x}}_t)\propto -{ϵ}_{\theta }$ — the score points opposite to the predicted static (peel the noise off → become more realistic).

4.2 $p({\mathbf{x}}_t\mid \mathbf{c})$ and Bayes' bridge

You don't need to re-derive Bayes here — just know that this bar-notation is where the whole idea "subtract unconditional from conditional" is born.

5. How the foundations feed the topic

Read top-to-bottom: raw images become noisy inputs, the model turns them into noise guesses, running it with a prompt vs a blank note gives two guesses, their difference times $w$ is added back — and that guided guess feeds the DDPM sampling loop that produces your final image. See also Negative prompting for what happens when the "blank note" is replaced by an unwanted prompt.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, re-read that section before the main note.