Generative✓ Mathematical
◆ The PatternGenerating by learning to reverse a noise process (DDPM, Stable Diffusion)
Diffusion models learn to denoise data. The forward process adds noise; a neural network learns to reverse it.
Forward: q(xₜ|xₜ₋₁) = N(xₜ; √(1−β)·xₜ₋₁, β·I)
Adds noise β at each step. After T steps, data ≈ pure Gaussian noise.
xₜ = √ᾱₜ·x₀ + √(1−ᾱₜ)·ε ε~N(0,I)
Closed form for any timestep t.
L = E[||ε − εθ(xₜ,t)||²]
Training: predict the noise ε that was added.
// Forward diffusion — noise being added over timesteps
Timestep t0
DDPM → DDIM → Latent Diffusion: DDPM (2020). DDIM made sampling 10–50× faster. Latent Diffusion (Stable Diffusion) runs in compressed latent space — enabling image generation on consumer GPUs.
Pattern bridge: Adding noise then learning to reverse it mirrors the central limit theorem in reverse — from Gaussian noise back to structured signal. In markets, euphoria and panic inject noise that mean-reverts to equilibrium.