Diffusion models forward - reverse process

4.5.10 · AI-ML › Generative Models

The Big Picture

Traditional generative models (GANs, VAEs) ek hi baar mein data distribution seekhne ki koshish karte hain. Diffusion models ek bilkul alag approach lete hain: ye ek fixed forward process define karte hain jo gradually data ko destroy karta hai, phir ek reverse process seekhte hain jo us destruction ko undo kare.

Key insight yeh hai: Agar hum har noise level par data ko denoise karna seekh sakte hain, toh hum pure noise se shuru karke gradually denoise karte hue realistic samples tak pahunch sakte hain.

Forward Process (Diffusion): Systematic Noise Addition

Yeh Specific Form Kyun?

Gaussian noise kyun?

• Central Limit Theorem guarantee karta hai ki bohot saare chote perturbations ka sum → Gaussian hoga
• Gaussian distributions ke nice mathematical properties hain (closed-form posteriors)
• Real-world noise aksar Gaussian characteristics rakhta hai

$\sqrt{1-{\beta }_t}$ se shrink kyun karte hain?

• Hum chahte hain ki variance har step par roughly constant rahe
• Agar hum sirf noise add karte: $\text{Var}(x_t)=\text{Var}(x_{t-1})+{\beta }_t$ → explode ho jaata
• Shrinkage ke saath: $\text{Var}(x_t)=(1-{\beta }_t)\text{Var}(x_{t-1})+{\beta }_t\approx \text{Var}(x_{t-1})$ small ${\beta }_t$ ke liye

Ek bada jump lene ki jagah bohot saare chote steps kyun?

• Har step ek chhota, reversible perturbation hai
• Neural networks chote steps ko zyada aasaani se reverse karna seekh sakte hain
• Generation process par fine-grained control milta hai

Derivation: The Reparameterization Trick

Hum $x_t$ ko directly $x_0$ se sample kar sakte hain, bina saare intermediate steps se guzre:

Starting point: $x_t=\sqrt{1-{\beta }_t}{x}_{t-1}+\sqrt{{\beta }_t}{ϵ}_{t-1},{ϵ}_{t-1}\sim \mathcal{N}(0,I)$

Recursively expand karo: $x_t=\sqrt{1-{\beta }_t}(\sqrt{1-{\beta }_{t-1}}{x}_{t-2}+\sqrt{{\beta }_{t-1}}{ϵ}_{t-2})+\sqrt{{\beta }_t}{ϵ}_{t-1}$

$=\sqrt{(1-{\beta }_t)(1-{\beta }_{t-1})}{x}_{t-2}+\sqrt{(1-{\beta }_t){\beta }_{t-1}}{ϵ}_{t-2}+\sqrt{{\beta }_t}{ϵ}_{t-1}$

Key observation: Independent Gaussians ka sum bhi Gaussian hota hai jisme variances add hote hain: $\sqrt{(1-{\beta }_t){\beta }_{t-1}}{ϵ}_{t-2}+\sqrt{{\beta }_t}{ϵ}_{t-1}=\sqrt{(1-{\beta }_t){\beta }_{t-1}+{\beta }_t}\overline{ϵ}$

$x_0$ tak pahunchne ke liye expand karte raho. Define karo: ${\alpha }_t=1-{\beta }_t,{\bar{\alpha }}_t={\prod }_{s=1}^t{\alpha }_s$

Yeh crucial kyun hai: Ab hum training ke liye random timesteps $t$ sample karke $x_0$ ko directly $x_t$ tak noise kar sakte hain, bina poori forward chain simulate kiye.

Example: Ek Image par Forward Process

Reverse Process (Denoising): Backwards Jaana Seekhna

Forward process data ko ek known tarike se destroy karta hai. Reverse process seekhta hai ki har noise-addition step ko undo kaise kiya jaaye.

Yeh Possible Kyun Hai?

Mathematical justification: Small ${\beta }_t$ ke liye, reverse conditional $q(x_{t-1}|x_t,x_0)$ approximately Gaussian hota hai:

$q(x_{t-1}|x_t,x_0)=\mathcal{N}(x_{t-1};{\tilde{\mu }}_t(x_t,x_0),{\tilde{\beta }}_{t}I)$

jahaan (Bayes' rule se):

${\tilde{\mu }}_t(x_t,x_0)=\frac{\sqrt{{\bar{\alpha }}_{t-1}}{\beta }_t}{1-{\bar{\alpha }}_t}{x}_0+\frac{\sqrt{{\alpha }_t}(1-{\bar{\alpha }}_{t-1})}{1-{\bar{\alpha }}_t}{x}_t$

${\tilde{\beta }}_t=\frac{1-{\bar{\alpha }}_{t-1}}{1-{\bar{\alpha }}_t}{\beta }_t$

Dikkat: Is formula ke liye $x_0$ jaanna zaroori hai, jo generation ke waqt hamare paas nahi hota!

Solution: Ek neural network se $x_t$ dekh kar $x_0$ predict karo.

Derivation: Noise Prediction Objective

Forward process reparameterization se: $x_t=\sqrt{{\bar{\alpha }}_t}{x}_0+\sqrt{1-{\bar{\alpha }}_t}ϵ$

Hum $x_0$ ke liye solve kar sakte hain: $x_0=\frac{x_t-\sqrt{1-{\bar{\alpha }}_t}ϵ}{\sqrt{{\bar{\alpha }}_t}}$

Key insight: $x_0$ seedha predict karne ki jagah, wo noise $ϵ$ predict karo jo add ki gayi thi! Yahi noise prediction parameterization hai.

Ek neural network ${ϵ}_{\theta }(x_t,t)$ train karo jo noisy $x_t$ aur timestep $t$ diye jaane par $ϵ$ predict kare.

${\tilde{\mu }}_t$ mein substitute karo:

Noise predict kyun karo $x_0$ ki jagah?

• Empirically, networks noise predict karna zyada achha seekhte hain clean images se
• Noise sabhi timesteps par zyada "uniform" hota hai (data domain par depend nahi karta)
• Score matching se relate karta hai (${\nabla }_{x_t}\log q(x_t)$ predict karna)

Training Objective

Training procedure:

1. Ek training image $x_0$ sample karo
2. Ek random timestep $t$ sample karo
3. Noise $ϵ\sim \mathcal{N}(0,I)$ sample karo
4. Forward process formula se noisy image $x_t$ compute karo
5. Neural network se noise ${ϵ}_{\theta }(x_t,t)$ predict karo
6. True aur predicted noise ke beech MSE loss compute karo
7. Backpropagate karo aur $\theta$ update karo

Example: Ek Denoising Step

Generation: Noise se Sampling

Ek baar train ho jaane ke baad, generation seedhi hai:

Algorithm:

1. $x_T\sim \mathcal{N}(0,I)$ sample karo (pure noise)
2. $t=T,T-1,\dots ,1$ ke liye:
   • Noise predict karo: ${ϵ}_{\theta }(x_t,t)$
   • Mean compute karo: ${\mu }_{\theta }=\frac{1}{\sqrt{{\alpha }_t}}(x_t-\frac{{\beta }_t}{\sqrt{1-{\bar{\alpha }}_t}}{ϵ}_{\theta })$
   • Sample karo: $x_{t-1}={\mu }_{\theta }+{\sigma }_{t}z$ jahaan $z\sim \mathcal{N}(0,I)$, ${\sigma }_t=\sqrt{{\tilde{\beta }}_t}$
3. $x_0$ return karo

Time complexity: $O(T)$ forward passes (standard DDPM ke liye 1000). Yeh GANs (1 pass) se slow hai lekin high-quality, diverse samples deta hai.

Common Mistakes & Misconceptions

Connections to Other Concepts

• Score-based generative models: Diffusion models score function ${\nabla }_{x_t}\log q(x_t)$ seekhne ke equivalent hain. Noise prediction, score se is tarah relate karta hai: ${ϵ}_{\theta }=-\sqrt{1-{\bar{\alpha }}_t}{\nabla }_{x_t}\log q(x_t)$.
• Variational autoencoders: Diffusion models ko ek hierarchical VAE ki tarah dekha ja sakta hai jisme $T$ latent layers hain aur ek fixed encoder (forward process) hai.
• Markov chains: Forward aur reverse dono processes Markov chains hain (memoryless: $x_t$ sirf $x_{t-1}$ par depend karta hai, poori history par nahi).
• Langevin dynamics: Reverse process data distribution se Langevin MCMC sampling approximate karta hai.
• DDIM sampling: Deterministic variant jo implicit generative model structure exploit karke sampling steps 1000 se 50 tak kam kar deta hai.
• Classifier-free guidance: Ek technique jo noise prediction ko scale karke diversity aur sample quality ke beech trade karta hai.

#flashcards/ai-ml

What is the forward process in diffusion models? :: Ek fixed Markov chain jo $T$ timesteps par data mein gradually Gaussian noise add karta hai according to $q(x_t|x_{t-1})=\mathcal{N}(x_t;\sqrt{1-{\beta }_t}{x}_{t-1},{\beta }_{t}I)$, systematically original data ko destroy karta hai jab tak wo pure noise na ban jaaye.

Concept Map