| $T$ |
Total number of time steps of the diffusion process |
| $t$ |
Time step $t$ on the range of $[0,T]$ |
| $t$ |
$[0,T]$ |
| ${|\cdot|}$ |
${L_2}$ norm |
| $\mu$ & $\Sigma$ |
Mean and Variance |
| $b$ |
Bias term |
| $\epsilon$ |
Standard Gaussian Noise |
| $x_T$ |
Input data becomes indistinguishable from an Isotropic Gaussian Noise |
| $\mathcal{N}$ |
Normal Distribution |
| $\beta_t$ |
Variance coefficient at time $t$ |
| $\alpha_t$ |
$1-\beta_t$ |
| $\bar{\alpha}{_t}$ |
Cumulative product of $\alpha_t$ |
| $x$ |
Input Data |
| $x_0$ |
Unperturbed data in diffusion model |
| $x_t$ |
Diffused data in diffusion model |
| $q({x_t\mid{x_{t-1}}})$ |
The forward noising Process |
| $q({x_{t-1}\mid{x_t}})$ |
The backward noising process |
| $\mu_{\theta}({{x_t,t})}$ |
Learnable Mean in the backward process at time $t$ |
| $\Sigma_{\theta}(x_t,t)$ |
Learnable Variance in the backward process at time $t$ |
| $q(x_t\mid x_{t-1}) = \mathcal{N(x_t; \sqrt{1-\beta_t{x_{t-1}}\beta_t}I)}$ |
This takes the image at the previous step, rescales the pixel values in this image and then adds tiny bit of noise via the variance scheduler "per time step" |
| $L_{(VLB)}$ |
Variational Lower Bound |
| $D_{KL}{\space}q(x_T{\mid}x_0){\mid}p(x_T ))$ |
Kulliback Leibler Divergence between two Gaussian Distributions |
| $q({x_1\space,{\dots,}\space{x_T}}\mid{x_0})$ |
Joint distribution of all the samples generated in the forward process consitioned on $x_0$ |