Diffusion Model

Denoising Diffusion Probabilistic Model (DDPM)

DDPMs are a class of generative model where the output generation is modeled as a denoising process, often called Stochastic Langevin Dynamics.

• Learning about these via Diffusion Policy paper

Oh this is just a standard diffusion model.

Resources

• https://lilianweng.github.io/posts/2021-07-11-diffusion-models/
• https://huggingface.co/blog/annotated-diffusion
• https://arxiv.org/pdf/2006.11239

See Stable Diffusion for an implementation example?

At a high-level, it’s really only a 2-step process:

1. A fixed (or predefined) forward diffusion process $q$ that adds Gaussian noise
2. A learned reverse denoising diffusion process $p_{\theta }$

1. Forward Diffusion Process in DDPM

In the forward diffusion process, for each timestep $t$, we add unit Gaussian noise to the previous sample $x_{t-1}$ to produce $x_t$:

$x_t=\sqrt{1-{\beta }_t}\cdot x_{t-1}+\sqrt{{\beta }_t}\cdot ϵ,ϵ\sim \mathcal{N}(0,I)$

• $I$ is the Identity Matrix

This is you are sampling $x_t$ from a normal distribution with the following mean and variance: $x_t\sim \mathcal{N}\left(\sqrt{1-{\beta }_t}\cdot x_{t-1},{\beta }_{t}I\right)$

I'm confused as to why they use \sqrt{1 - \beta_t}

This is to ensure that the total variance remains 1 (review Sum of Gaussians):

\text{Var}(x_t) &= \text{Var}(\sqrt{1 - \beta_t} \cdot x_{t-1}) + \text{Var}(\sqrt{\beta_t} \cdot \epsilon) \\ &= (1 - \beta_t) \cdot \text{Var}(x_{t-1}) + \beta_t \cdot \text{Var}(\epsilon) \\ &= (1 - \beta_t) I + \beta_t I = I \\ \end{align}

This shows that using $\sqrt{1-{\beta }_t}$ ensures that $x_t$ remains unit gaussian.

As a condition probability $q(x_t\mid x_{t-1})$, this is written as: $q(x_t\mid x_{t-1})=\mathcal{N}\left(x_t;\sqrt{1-{\beta }_t}{x}_{t-1},{\beta }_t\mathbf{I}\right)$

Variance schedule

${\beta }_t$ does not have to beconstant at each time step, we actally define variance schedule, $0<{\beta }_1<{\beta }_2<\cdots <{\beta }_T<1$.

• is a variance schedule really needed? Yes, it’s similar to ideas behind Learning Rate.
• Can be linear, quadratic, cosine, etc. as we will see further (a bit like a learning rate schedule).

2. Denoising process

Now, let’s say we want to reverse the process. We know how $p(x_t|x_{t-1})$ is calculated. Is the reverse $p(x_{t-1}|x_t)$ doable? Review your Bayes Rule:

$P(x_{t-1}|x_t)=\frac{P(x_t|x_{t-1})\cdot P(x_{t-1})}{P(x_t)}$

• The problem is that we don’t know $P(x_{t-1})$, or do we? That is the thing we want to predict

Okay, no problem, just slap a universal function approximator, i.e. a neural network!! $p_{\theta }(x_{t-1}\mid x_t)=\mathcal{N}(x_{t-1};{\mu }_{\theta }(x_t,t),{\Sigma }_{\theta }(x_t,t))$

• The neural net learns two parameters, ${\mu }_{\theta }$ and ${\sigma }_{\theta }$

In the original paper

They only made the neural net learn $\mu$, and fixed $\sigma$ the variance.

3. Objective function

Okay, so how do we formulate objective function for the neural net to learn?

We’ll use U-Net.