DDPM and Improved DDPM

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Sources:

  1. original diffusion model 2015 paper by Jascha Sohl-Dickstein
  2. DDPM 2020 paper
  3. Improved DDPM 2021 paper

DDPM and Improved DDPM

TL;DR:

DDPM:

  • defines a Gaussian forward diffusion process;
  • models reverse transitions as Gaussian distributions;
  • predicts noise as a stable parameterization of reverse mean;
  • assumes fixed variance.

Improved DDPM:

  • keeps the same diffusion framework;
  • learns reverse variance;
  • provides better uncertainty modeling.

1. Overview

Denoising Diffusion Probabilistic Models (DDPM) introduce a generative modeling framework based on gradually transforming data into Gaussian noise and learning the inverse denoising process.

The key idea is to avoid learning a difficult one-step mapping:

\[ x_T \rightarrow x_0 \]

Instead, DDPM decomposes generation into many local transitions:

\[ x_T \rightarrow x_{T-1} \rightarrow \cdots \rightarrow x_0 \]

Each reverse step only removes a small amount of noise.

The DDPM framework contains two processes:

  1. Forward diffusion process.
  2. Reverse denoising process.

2. Notation

## Notation
Here I list the notations used in the DDPM and Improved DDPM discussion.
We follow the common conventions in diffusion models:
* Clean data samples are denoted by lower-case italicized Roman letters, such as \(x_0\). * Random variables are denoted by upper-case letters. * Diffusion timesteps are indexed by \(t\), where \(t\in\{1,\dots,T\}\). * Gaussian noise variables are denoted by \(\epsilon\).
| Symbol | Meaning | | ——————————- | ———————————————————— | | \(x_0\) | The clean data sample, such as a real image before adding noise. | | \(x_t\) | The noisy sample at diffusion timestep \(t\). It is obtained by gradually corrupting \(x_0\) with Gaussian noise. | | \(T\) | The total number of diffusion timesteps. At timestep \(T\), the sample is approximately distributed as standard Gaussian noise. | | \(t\) | The current diffusion timestep. It controls the noise level of the sample \(x_t\). | | \(\epsilon\) | Gaussian noise sampled from \(\mathcal{N}(0,I)\). It is the noise added to \(x_0\) to construct \(x_t\). | | \(\beta_t\) | The variance of the Gaussian noise added at timestep \(t\) in the forward diffusion process. | | \(\alpha_t\) | The signal coefficient at timestep \(t\), defined as \(\alpha_t=1-\beta_t\). | | \(\bar{\alpha}_t\) | The cumulative product of signal coefficients, defined as \(\bar{\alpha}_t=\prod_{s=1}^{t}\alpha_s\). | | \(q(x_t|x_{t-1})\) | The forward diffusion transition distribution that gradually adds Gaussian noise to the data. | | \(q(x_{t-1}|x_t,x_0)\) | The posterior distribution of the forward diffusion process. It describes the reverse transition when the original clean sample \(x_0\) is known. | | \(p_\theta(x_{t-1}|x_t)\) | The learned reverse diffusion transition distribution used for generation. | | \(\tilde{\mu}_t(x_t,x_0)\) | The true posterior mean of \(q(x_{t-1}|x_t,x_0)\) derived from the forward diffusion process. | | \(\tilde{\beta}_t\) | The true posterior variance of \(q(x_{t-1}|x_t,x_0)\). | | \(\mu_\theta(x_t,t)\) | The neural network prediction of the mean of the reverse diffusion distribution. | | \(\Sigma_t\) | The covariance matrix of the reverse diffusion distribution in the original DDPM formulation. | | \(\sigma_t^2 I\) | The fixed isotropic covariance assumption used by the original DDPM. | | \(\Sigma_\theta(x_t,t)\) | The learned covariance matrix in Improved DDPM. | | \(\epsilon_\theta(x_t,t)\) | The neural network prediction of the Gaussian noise added during the forward diffusion process. | | \(\hat{x}_0\) | The estimated clean sample recovered from the predicted noise \(\epsilon_\theta(x_t,t)\). | | \(v_\theta(x_t,t)\) | The variance prediction parameter introduced in Improved DDPM. It controls the learned reverse variance. | | \(\mathcal{L}_{\epsilon}\) | The epsilon-prediction loss used in DDPM training. | | \(\mathcal{L}_{\mathrm{simple}}\) | The simplified DDPM training objective, usually equivalent to epsilon prediction MSE. | | \(\mathcal{L}_{\mathrm{VLB}}\) | The variational lower bound objective used to train the variance prediction in Improved DDPM. | | \(\mathcal{N}(0,I)\) | The standard Gaussian distribution used as the noise prior. | | \(I\) | The identity matrix. In Gaussian covariance matrices, it represents independent dimensions with equal variance. | | \(\theta\) | The parameters of the denoising neural network. | | \(\lambda\) | The weighting coefficient balancing different loss terms, such as noise prediction loss and variance learning loss. |

3. Forward Diffusion Process

DDPM first defines a forward noising process:

\[ q(x_t|x_{t-1}) = \mathcal{N} ( \sqrt{1-\beta_t}x_{t-1}, \beta_t I ) \]

At each timestep:

  • part of the original signal is preserved;
  • Gaussian noise is added.

The process is fixed:

  • no neural network is involved;
  • no parameters are learned.

After many steps:

\[ x_0 \rightarrow x_1 \rightarrow ... \rightarrow x_T \]

the sample becomes approximately:

\[ x_T\sim\mathcal{N}(0,I) \]

4. Closed-form Sampling of Noisy Data

Although the forward process is a Markov chain, DDPM does not explicitly simulate all forward steps during training.

Because the process is linear Gaussian:

\[ x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon \]

where:

\[ \epsilon\sim\mathcal{N}(0,I) \]

Therefore, training can directly construct any noisy sample:

clean sample x0
      |
sample timestep t
      |
sample Gaussian noise epsilon
      |
construct noisy sample xt

without computing:

\[ x_0\rightarrow x_1\rightarrow...\rightarrow x_t \]

5. Reverse Diffusion Process

The goal of DDPM is to learn:

\[ p_\theta(x_{t-1}|x_t) \]

DDPM models each reverse transition as a Gaussian:

\[ p_\theta(x_{t-1}|x_t) = \mathcal{N} ( \mu_\theta(x_t,t), \Sigma_t ) \]

Therefore, the reverse process requires two components:

  1. Mean:

\[ \mu_\theta(x_t,t) \]

  1. Covariance:

\[ \Sigma_t \]

The rest of DDPM focuses on how to parameterize and learn these quantities.

6. DDPM Variance Assumption

The original DDPM makes a simplified assumption:

\[ \Sigma_t=\sigma_t^2I \]

This means:

  • variance is fixed;
  • variance only depends on timestep;
  • variance does not depend on the current noisy input;
  • all dimensions share the same variance.

Therefore, DDPM mainly learns:

where the reverse process should move.

However, it does not learn:

how uncertain the reverse transition should be.

This limitation motivates Improved DDPM.

7. Learning the Reverse Mean

7.1 Direct Mean Prediction

Since the reverse distribution requires:

\[ \mu_\theta(x_t,t) \]

a natural approach is to directly predict the mean.

The true posterior of the forward process is:

\[ q(x_{t-1}|x_t,x_0) = \mathcal{N} ( \tilde{\mu}_t(x_t,x_0), \tilde{\beta}_tI ) \]

where:

\[ \tilde{\mu}_t(x_t,x_0) = a_t x_0+b_t x_t \]

Therefore:

\[ \mu_\theta(x_t,t) \approx \tilde{\mu}_t(x_t,x_0) \]

is a valid objective.

8. Why DDPM Predicts Noise Instead of Mean

Although mean prediction is mathematically valid, DDPM uses epsilon prediction.

The network predicts:

\[ \epsilon_\theta(x_t,t) \]

instead of:

\[ \mu_\theta(x_t,t) \]

The reason is optimization stability.

The mean target is:

\[ \tilde{\mu}_t = a_t x_0+b_t x_t \]

where:

\[ a_t,b_t \]

change with timestep.

Therefore, different timesteps produce targets with different scales and statistics.

In contrast:

\[ \epsilon\sim\mathcal{N}(0,I) \]

has the same distribution for all timesteps.

Thus epsilon prediction provides a more stable learning target.

9. Epsilon Prediction Objective

The DDPM training objective is:

\[ \mathcal{L}_{\epsilon} = \mathbb{E}_{x_0,t,\epsilon} [ \| \epsilon-\epsilon_\theta(x_t,t) \|^2 ] \]

The model learns to answer:

Given a noisy sample \(x_t\) and timestep \(t\), what Gaussian noise was added?

The noise is known because it is sampled during training.

10. Recovering Clean Data and Reverse Mean

From:

\[ x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon \]

we can estimate:

\[ \hat{x}_0 = \frac{ x_t-\sqrt{1-\bar{\alpha}_t}\epsilon_\theta(x_t,t) } { \sqrt{\bar{\alpha}_t} } \]

Then:

\[ \epsilon_\theta \rightarrow \hat{x}_0 \rightarrow \mu_\theta \]

Therefore:

Epsilon prediction is not a different generation mechanism. It is a more stable parameterization of the reverse mean.

11. DDPM Training Process

The complete training process:

x0 = sample_real_data()

t = sample_random_timestep()

epsilon = sample_gaussian_noise()

xt = sqrt(alpha_bar[t]) * x0 \
     + sqrt(1-alpha_bar[t]) * epsilon

epsilon_pred = model(xt, t)

loss = mse(epsilon_pred, epsilon)

Important:

During training:

  • no reverse diffusion sampling is performed;
  • the model only learns denoising at randomly sampled timesteps.

12. DDPM Sampling Process

During generation:

Start:

\[ x_T\sim\mathcal{N}(0,I) \]

For:

\[ t=T,T-1,...,1 \]

perform:

Step 1

Predict noise:

\[ \epsilon_\theta(x_t,t) \]

Step 2

Compute reverse mean:

\[ \mu_\theta(x_t,t) \]

Step 3

Sample:

\[ x_{t-1} \sim \mathcal{N} ( \mu_\theta, \sigma_t^2I ) \]

After repeating:

\[ x_T\rightarrow...\rightarrow x_0 \]

we obtain a generated sample.

13. Improved DDPM

Improved DDPM keeps the DDPM framework:

  • same forward diffusion;
  • same reverse Gaussian formulation;
  • same epsilon prediction.

The main change is the variance modeling.

14. Learning Reverse Variance

Instead of:

\[ \Sigma_t=\sigma_t^2I \]

Improved DDPM uses:

\[ \Sigma_\theta(x_t,t) \]

The reverse distribution becomes:

\[ p_\theta(x_{t-1}|x_t) = \mathcal{N} ( \mu_\theta(x_t,t), \Sigma_\theta(x_t,t) ) \]

The model predicts:

  1. Noise component:

\[ \epsilon_\theta(x_t,t) \]

  1. Variance parameter:

\[ v_\theta(x_t,t) \]

15. Diagonal Covariance Assumption

Improved DDPM does not learn a full covariance matrix.

It still assumes:

\[ \Sigma_\theta = diag(\sigma_1^2,\sigma_2^2,...) \]

Therefore:

\[ Cov(x_i,x_j)=0,\quad i\neq j \]

The improvement is:

DDPM:

\[ \sigma_1^2=\sigma_2^2=...=\sigma_t^2 \]

Improved DDPM:

\[ \sigma_i^2=\sigma_i^2(x_t,t) \]

16. Improved DDPM Objective

The epsilon prediction loss remains:

\[ \mathcal{L}_{simple} = \| \epsilon-\epsilon_\theta(x_t,t) \|^2 \]

An additional variance learning objective is introduced:

\[ \mathcal{L}_{VLB} \]

The final objective:

\[ \mathcal{L} = \mathcal{L}_{simple} + \lambda\mathcal{L}_{VLB} \]

17. DDPM vs Improved DDPM

DDPM Improved DDPM
Forward process Fixed Gaussian diffusion Same
Reverse process Gaussian Gaussian
Mean Learned Learned
Mean parameterization Noise prediction Noise prediction
Variance Fixed \(\sigma_t^2I\) Learned variance
Covariance Diagonal Diagonal
Output \(\epsilon_\theta\) \(\epsilon_\theta+v_\theta\)