Foundations — Noise scheduling

This page assumes you know nothing beyond middle-school arithmetic. We will earn every symbol the parent note Noise scheduling uses, one at a time, each anchored to a picture. Read top to bottom; nothing is used before it is built.

0. The characters in the story

Before any formula, meet the three things that get mixed at every step:

The picture shows the whole journey: a fully clean image on the left, pure static on the right, and a series of blends in between. Everything else on this page is just precise vocabulary for describing that fade.

1. Time steps: $t$, $T$, and the subscript notation

The picture: think of $T$ as the number of frames in a slow-motion "dissolving" video. $t$ is which frame you paused on — always a whole frame number, never halfway between frames.

Why the topic needs this: the whole schedule is a list of instructions indexed by $t$ — one instruction per frame. Without a time label there is no "schedule".

2. The ordered-list notation: $\{{\beta }_t{\}}_{t=1}^T$

The picture: a row of $T$ pigeonholes laid left to right, one per frame, each holding one number. Their left-to-right order is the timeline. The next section says what number goes in each hole.

3. ${\beta }_t$ — the "how much static" knob

The picture: a volume dial. Turning it up pours more static into the current frame. The schedule is the plan for how this dial moves across all frames.

4. Scientific notation: $10^{-4}$

Why the topic needs it: the smallest noise amounts (like ${\beta }_1$) and, later, the tiny leftover-signal numbers are very small, and scientific notation keeps them readable.

5. ${\alpha }_t$ — the "how much signal survives one step"

The picture: a see-saw. Push ${\beta }_t$ up and ${\alpha }_t$ automatically goes down — they always sum to the full amount, $1$.

6. The one-step forward rule: $q(x_t\mid x_{t-1})$

We now have every piece needed to write down the central transition equation — the single move that takes one frame to the next. (The bar $\mid$ and the bell-curve symbol $\mathcal{N}$ are fully unpacked below in §11–12; for now read them as "given" and "bell-curve draw".)

7. The product symbol $\prod$ and the cumulative signal ${\bar{\alpha }}_t$

Surviving one step keeps a fraction ${\alpha }_t$. But how much of the original image survives after many steps? You multiply the survival fractions together, the same way three "half-off" discounts in a row leave you paying $\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{8}$.

The curve above is the single most important object in the whole topic: it shows ${\bar{\alpha }}_t$ sliding from 1 down to 0 as $t$ grows. Every schedule is really a choice about the shape of this curve — that is exactly what the parent's Linear vs Cosine comparison is arguing about.

8. From one step to the whole jump: why $q(x_t\mid x_0)$ has a closed form

Applying the one-step rule $t$ times in a row looks like a nightmare of nested noise. The magic is that it collapses into a single Gaussian — you can jump straight from the clean image $x_0$ to any frame $x_t$ in one shot. Here is the "why", step by step, using only tools we already built.

The rule "two independent bell curves add to one whose spread is the sum of their spreads" is used twice above; the next sections build the $\mathcal{N}$ and $\mid$ notation those steps lean on.

9. The square root  ​ and why it appears

Both the one-step rule and the closed form multiply the image by $\sqrt{{\bar{\alpha }}_t}$ (and the noise by $\sqrt{1-{\bar{\alpha }}_t}$). Why square roots?

10. Greek letter $ϵ$ and the noise picture

The picture: a screen of black-and-white snow, different every time you look.

11. $\mathcal{N}(\text{mean},\text{variance})$ — the bell curve

The noise isn't any random numbers; it is Gaussian. This is the notation the transition equations above lean on.

So the closed-form line $q(x_t|x_0)=\mathcal{N}(x_t;\sqrt{{\bar{\alpha }}_t}{x}_0,(1-{\bar{\alpha }}_t)I)$ now translates fully to plain English: "frame $x_t$ is a bell-curve draw centred on the faded image $\sqrt{{\bar{\alpha }}_t}{x}_0$, with spread $(1-{\bar{\alpha }}_t)$." The center is the surviving signal; the spread is the noise that replaced it. They always sum to 1 — the seesaw again.

12. $q(\cdot |\cdot )$ — the conditional bar

Why the topic needs it: noise scheduling defines this $q$ — see 4.5.11-Forward-and-reverse-diffusion-process.

13. SNR and the exponential $e^x$ (why they appear later)

The parent's "design principles" section introduces two more tools. Here is their zero-level meaning so nothing is a surprise.

Why the topic needs these: designers prefer $\log \text{SNR}$ to change smoothly and evenly across frames, because that keeps each denoising step "equally hard" — the idea behind SNR-based schedules and denoising loss weighting.

How the foundations feed the topic

Read this as a build-up chain — each rung stands on the one before it:

1. Time steps $t$ (from $0$ to $T$) let us index a list of noise dials.
2. The noise dial ${\beta }_t$ sets how much static enters at each frame.
3. The survival ${\alpha }_t=1-{\beta }_t$ is the flip side: what is kept.
4. The one-step rule $q(x_t\mid x_{t-1})$ fades and re-noises one frame at a time (a Markov step).
5. The product $\prod$ multiplies many survivals together into the cumulative ${\bar{\alpha }}_t$.
6. Composing the one-step Gaussians collapses the chain into the closed form $q(x_t\mid x_0)$.
7. The SNR view (with $\log$ and sigmoid) lets us design the whole fade smoothly.

All of these together are the noise schedule. The related pages 4.5.11-Forward-and-reverse-diffusion-process, 4.5.13-Denoising-objective, 4.5.15-Score-matching, and 4.5.18-Fast-sampling each build on this same vocabulary.

Equipment checklist

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