score matching

**Diffusion Model** is a **generative model that learns to reverse a gradual noising process** — trained by predicting and removing noise step-by-step, producing state-of-the-art image, audio, and video generation. **Forward Process (Noising)** - Gradually add Gaussian noise to data over T steps (typically T=1000). - At step T, data is pure noise: $x_T \sim N(0, I)$. - Mathematically: $q(x_t | x_{t-1}) = N(x_t; \sqrt{1-\beta_t} x_{t-1}, \beta_t I)$ **Reverse Process (Denoising)** - A neural network (usually U-Net) learns to predict the noise added at each step. - Generation: Start from pure noise $x_T$, iteratively denoise to get $x_0$. - The network is conditioned on timestep $t$ and optionally on a text prompt. **Key Architectures** - **DDPM (Denoising Diffusion Probabilistic Models)**: Original formulation (Ho et al., 2020). - **DDIM**: Deterministic sampling — 10-50 steps instead of 1000 (10-100x faster). - **Latent Diffusion (Stable Diffusion)**: Runs diffusion in compressed latent space — 8x smaller, much faster. - **Score-Based Models**: Equivalent formulation using score functions $\nabla_x \log p(x)$. **Why Diffusion Models Won** - **Quality**: Sharper, more diverse samples than GANs. - **Stability**: No adversarial training — GANs suffer from mode collapse and training instability. - **Controllability**: Easy to condition on text (CLIP guidance, classifier-free guidance). - **Likelihood**: Tractable likelihood computation unlike GANs. **Applications** - Image generation: DALL-E 2, Stable Diffusion, Midjourney (FLUX), Imagen. - Video: Sora, Runway Gen-2. - Audio: WaveGrad, DiffWave. - Protein structure: RFDiffusion. Diffusion models are **the dominant paradigm for generative AI** — they have replaced GANs across virtually every generation task and continue to advance rapidly.