The reverse process

The inverse is also Gaussian

If the forward step is small, its reverse is close to Gaussian too. The true reverse of the chain, given the clean sample, has a closed form. It is a Gaussian whose mean is a weighted blend of x_t and x_0, and whose variance is fixed by the betas. The network does not know x_0, so it predicts the mean from x_t and t. That is the whole model: one Gaussian per step, with a learned mean and a variance you can choose.

The variance is usually not learned. The true variance of that reverse step is close to beta for small steps. Fixing it costs almost nothing in quality and removes a hard prediction. Some later models learn it to improve the likelihood, but for sampling the fixed choice is fine. What matters is the mean, and the mean is where the noise prediction enters. A learned variance mostly buys a better likelihood number. It changes the sampler’s path far less than a better schedule or a better mean. If you are optimising for sample quality, spend the effort on the mean first.

From noise to mean

Substitute the closed form for x_0 in terms of x_t and eps, and the reverse mean becomes a simple function of the predicted noise. The update subtracts a scaled version of eps from x_t, then divides by the square root of alpha. Every DDPM implementation you will read is that formula with different bookkeeping. DDPM is the original denoising diffusion model. Get the signs and the square roots right once, and the rest is scheduling.

The same algebra gives a useful debugging identity. Feed the true noise into the reverse step, and you recover the true x_0 exactly, up to floating point. Any implementation that fails that test has a bug, not a training problem. The examples use it before any model is involved. It is also the fastest way to check a new scheduler. Run one step with a known x_0, eps and t, and compare.

One reverse step with a UNet

import torch
from diffusers import DDPMScheduler, UNet2DModel

unet = UNet2DModel(sample_size=64, in_channels=3, out_channels=3)
scheduler = DDPMScheduler(num_train_timesteps=1000)
sample = torch.randn(1, 3, 64, 64)

for t in scheduler.timesteps:
    noise_pred = unet(sample, t).sample
    sample = scheduler.step(noise_pred, t, sample).prev_sample

print(sample.shape, float(sample.mean()))

The training objective

Training maximises a variational lower bound on the data likelihood. A lower bound is a stand-in for the quantity you cannot compute, chosen so that pushing it up also pushes the real one up. After simplification, most of the bound collapses into a mean squared error between the noise that was added and the noise the network predicts. The timestep is sampled uniformly, the noise is sampled from a standard normal, and the pair is combined with the closed form. No adversarial game, no replay buffer, no schedule for the loss weight.

The simplification drops a per-timestep weight that the true bound implies. That is a deliberate choice. The unweighted objective puts more emphasis on the middle of the schedule, where the model can learn the most. It also produced better samples than the exact bound. Later work put weights back for likelihood. The trade between sample quality and likelihood is still open. For sampling, the simple loss is the right default.

What the reverse process cannot do

At high noise the reverse distribution is genuinely multimodal. Many clean images are consistent with the same noisy input, and the best mean is their average, which may look like none of them. The model does not hallucinate one answer. It predicts the conditional mean, and the sampler’s added noise is what lets it pick a mode. Remove that noise and you get a blurry average, unless the sampler is built to follow the score sharply.

For that reason, the reverse process is a distribution, not a function. Two runs from the same starting noise with different intermediate draws give different images, and both are valid. Guidance works for the same reason: it sharpens the conditional distribution toward one answer, at the cost of variety. The amount of noise the sampler injects is therefore a modelling choice, not a numerical detail. Too little and the path collapses to a blurry average. Too much and the sample wanders away from the data. The next chapters take the mean apart, choose its variance, and decide how many steps to spend.