Diffusion-Model-Demo

This repository is pytorch implement of Denoising Diffusion Probabilistic Models.

UNet in diffusion model of default configure in this repository

Macro model structure

Sublayers

math

hyper parameters

the following parameters are set manually.

$$ \alpha_0,\alpha_1,\alpha_2, ..., \alpha_n $$

the following parameters are computed from the above parameters.

$$ \bar{\alpha}t = \prod{i=1}^t{\alpha}_i $$

$$ x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1 - \bar{\alpha}_t}\epsilon_0;\quad \epsilon_0 \sim \mathcal{N}\left(x_t; \sqrt{\bar{\alpha}_t}x_0, \left(1-\bar{\alpha}_t\right)\mathbf{I}\right) $$

$$ \sigma^2\left(t\right) = \frac{\left(1-\alpha_t\right)\left(1 - \bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_t} $$

training

the optimization goal of the Decoder is to predict $\mu_t$ so that it is as possible with the target.

$$ \mu_{target} = \frac{\sqrt{\alpha_t}\left(1 - \bar{\alpha}{t-1}\right)x_t+\sqrt{\bar{\alpha}{t-1}}\left(1-\alpha_t\right)x_0}{1-\bar{\alpha}_t} $$

$$ \mu_{prediction} = \frac{\sqrt{\alpha_t}\left(1 - \bar{\alpha}{t-1}\right)x_t+\sqrt{\bar{\alpha}{t-1}}\left(1-\alpha_t\right)\hat{x}_{\theta}\left(x_t, t\right)}{1-\bar{\alpha}_t} $$

$\hat{x}_{\theta}\left(x_t, t\right)$ is the Decoder(a.k.a Unet with Multi Head Self-Attention).

the loss function is:

$$ \log{p(x)} = \mathbb{E_{q\left(x_1|x_0\right)}}\left[\log{p_{\theta}}(x_0|x_1)\right] - D_{KL}\left(q(x_T|x_0)||p(x_T)\right) - \sum_{t=2}^T{\mathbb{E}{q(x_t|x_0)}\left[D{KL}(q(x_{t-1}|x_t, x_0))||p_{\theta}(x_{t-1}|x_t)\right]} $$

We ignore optimizing the first term when training, because we don't add noise in the first step (let $x_1 = x_0$). The second term has no trainable parameters. The last term is the diffusion process.

inference

$$ x_t = \sqrt{\bar{\alpha}t}x_0 + \sqrt{1 - \bar{\alpha}t}\epsilon{\theta};\quad \epsilon{\theta} \sim \mathcal{N}\left(\mu_{prediction},\sigma^2\left(t\right)\mathbf{I}\right) $$

Training

$ git clone https://github.com/yl-jiang/Diffusion-Model-Demo.git
$ cd Diffusion-Model-Demo
$ conda activate your_pytorch_environment
$ python train.py

if everything is ok, then you will see something like this:

Logging at: Logs_Checkpoints/Inference/version_0
Model Checkpoint at: Logs_Checkpoints/checkpoints/version_0
Train :: Epoch: 1/30:   9%|█▏          | 42/469 [00:06<00:49,  8.61it/s, Loss: 0.0507]

MNIST Demo

Reference

1. zhangbo2008 / diffusion_demo;
2. jadore801120 / attention-is-all-you-need-pytorch;
3. learnopencv/Guide-to-training-DDPMs-from-Scratch;