Diffusion models

Gaussian diffusion learns reverse transitions for a fixed process that corrupts clean data with noise.

Summary

Why can training jump directly to one noise level while generation needs dependent updates? Take clean scalar $x_0=2$, retained signal variance ${\bar{\alpha }}_t=0.64$, and sampled standard-normal noise $ϵ=-1$. A predictor returns $\widehat{ϵ}=-0.5$. We will compute its training error and clean estimate, then identify what a sampler still needs.

The forward process

Choose $0<{\beta }_t<1$, define ${\alpha }_t=1-{\beta }_t$, and let ${\bar{\alpha }}_t={\prod }_{s=1}^t{\alpha }_s$. Independent forward Gaussian increments give

Composition multiplies the signal coefficients. The noise variance recurs as ${\alpha }_t(1-{\bar{\alpha }}_{t-1})+{\beta }_t=1-{\bar{\alpha }}_t$. Thus the marginal is $\mathcal{N}(\sqrt{{\bar{\alpha }}_t}{x}_0,(1-{\bar{\alpha }}_t)I)$. It approaches standard normal only when residual signal becomes negligible for the data distribution.

Training

Sample data, a time index and independent $ϵ\sim \mathcal{N}(0,I)$. Construct $x_t$ directly and minimize $\mathbb{E}\Vert ϵ-{ϵ}_{\theta }(x_t,t){\Vert }^2$. This simplified objective need not equal every time-weighted variational bound. Our fixture gives $x_t=0.8(2)+0.6(-1)=1$ and squared error $(-1+0.5{)}^2=0.25$. The model sees $x_t,t$, not the sampled target noise.

Change the requirement: predict clean data instead. For $0<{\bar{\alpha }}_t<1$, convert using

The estimate is $(1+0.3)/0.8=1.625$; using the true noise would recover two. Noise MSE equals clean-data MSE multiplied by ${\bar{\alpha }}_t/(1-{\bar{\alpha }}_t)$ under this conversion. Changing prediction targets without their loss weights and sampler coefficients changes the training problem. At zero retained signal, this inverse formula is singular.

Sampling (reverse process)

Start from an appropriate Gaussian prior and compute dependent reverse states:

Here $z$ is fresh standard-normal transition noise, with $z=0$ at the final step by convention. The variance ${\sigma }_t^2$ is a declared model or sampler choice. One choice is ${\beta }_t(1-{\bar{\alpha }}_{t-1})/(1-{\bar{\alpha }}_t)$; it is not the only supported variance. A noise prediction alone supplies neither the update coefficients nor the next state. Using adjacent-step DDPM coefficients for an arbitrary timestep jump is not justified.

Conditional generation

Change the requirement: strengthen a condition $c$ using a model trained to support unconditional prediction. Classifier-free guidance combines

Conditional prediction two, unconditional prediction one and $w=2$ produce $3(2)-2(1)=4$. The alternative convention $s=1+w$ writes unconditional plus $s$ times the conditional difference. Guidance changes the sampled distribution; it is not simply a speed setting.

Transfer: ten transitions with two denoiser evaluations each require twenty evaluations, absent reuse. Batching the pair changes execution scheduling, not the number of logical evaluations.