Generative Models

Diffusion Models Deep Dive — The Math Behind Image Generation

Diffusion models generate data by learning to reverse a gradual noising process. By training a neural network to predict and remove noise at each step, they produce state-of-the-art images that surpass GANs in both quality and diversity.

Key point 1 — Forward process adds noise; reverse process learns to denoise step by step
Key point 2 — Simple noise prediction loss achieves remarkable generation quality
Key point 3 — Classifier-free guidance and latent diffusion enable text-to-image generation

"From noise, beauty emerges — one denoising step at a time."

Diffusion Models Deep Dive

Diffusion models generate data by learning to reverse a gradual noising process. They have achieved state-of-the-art image generation quality, surpassing GANs in both quality and diversity.

Forward Process (Diffusion)

DfForward Diffusion Process

The forward process gradually adds Gaussian noise to data $x_0$ over $T$ timesteps:

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

After $T$ steps, $x_T$ is approximately isotropic Gaussian noise. The process is fixed (no learned parameters) and defined by noise schedule $\beta_1, \ldots, \beta_T$ .

Forward Process Marginal

q(x_t | x_0) = \mathcal{N}(x_t; \sqrt{\bar{\alpha}_t} x_0, (1 - \bar{\alpha}_t) I)

Here,

$\alpha_t$ =1 - \beta_t
$\bar{\alpha}_t$ =\prod_{s=1}^{t} \alpha_s (cumulative product)
$x_t$ =Noisy version of x_0 at timestep t
$\beta_t$ =Noise level at step t

Diffusion Models Deep Dive — DDPM and Beyond

Diffusion Models Deep Dive — The Math Behind Image Generation

Diffusion Models Deep Dive

Forward Process (Diffusion)

DfForward Diffusion Process

Forward Process Marginal

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