Score-based generative models

4.5.13 · AI-ML › Generative Models

Score Function Kya Hai?

Log probability ka gradient kyun?

Isko first principles se derive karte hain. Maano hamare paas $p(x)=\frac{\tilde{p}(x)}{Z}$ hai jahan $Z=\int \tilde{p}(x)dx$ partition function hai.

${\nabla }_x\log p(x)={\nabla }_x\log \left(\frac{\tilde{p}(x)}{Z}\right)$

$={\nabla }_x[\log \tilde{p}(x)-\log Z]$

Kyunki $Z$ ka $x$ par depend nahi karta: $={\nabla }_x\log \tilde{p}(x)$

Jaadu: Intractable normalization constant $Z$ gayab ho jaata hai! Isliye score-based models powerful hain — hume kabhi $Z$ compute ya estimate nahi karna padta.

Score Matching: Score Ko Seekhna

Problem: Hume ${\nabla }_x\log p_{\text{data}}(x)$ pata nahi! Yahi toh hum learn karne ki koshish kar rahe hain.

Denoising Score Matching

Breakthrough yeh hai: Hum scores ko estimate kar sakte hain noise add karke aur denoising seekh ke.

Scratch se Derivation:

1. Gaussian noise add karo: $\tilde{x}=x+\sigma ϵ$ jahan $ϵ\sim \mathcal{N}(0,I)$
2. Perturbed distribution hai $q_{\sigma }(\tilde{x}|x)=\mathcal{N}(\tilde{x};x,{\sigma }^{2}I)$
3. Marginal hai $q_{\sigma }(\tilde{x})=\int p_{\text{data}}(x)q_{\sigma }(\tilde{x}|x)dx$

Key insight: Gaussian noise ke liye, $q_{\sigma }(\tilde{x}|x)$ ka score hai: ${\nabla }_{\tilde{x}}\log q_{\sigma }(\tilde{x}|x)={\nabla }_{\tilde{x}}\log \mathcal{N}(\tilde{x};x,{\sigma }^{2}I)$

Ek Gaussian $\mathcal{N}(y;\mu ,\Sigma )$ ke liye, hamare paas hai: $\log \mathcal{N}(y;\mu ,\Sigma )=-\frac{1}{2}(y-\mu {)}^T{\Sigma }^{-1}(y-\mu )+\text{const}$

$y$ ke respect mein gradient lete hain: ${\nabla }_y\log \mathcal{N}(y;\mu ,\Sigma )=-{\Sigma }^{-1}(y-\mu )$

Apne case mein apply karte hain ($y=\tilde{x}$, $\mu =x$, $\Sigma ={\sigma }^{2}I$): ${\nabla }_{\tilde{x}}\log q_{\sigma }(\tilde{x}|x)=-\frac{1}{{\sigma }^2}(\tilde{x}-x)=-\frac{ϵ}{\sigma }$

Yeh kyun kaam karta hai: Denoising seekhne se, network implicitly noise level $\sigma$ par score function seekh leta hai.

Multi-Scale Score Matching

Real data high-dimensional space mein low-dimensional manifolds par rehti hai. Ek akela noise level $\sigma$ poori distribution ko achhi tarah cover nahi karega.

Multiple scales kyun?

• Bada $\sigma$: Connectivity ensure karta hai, koi isolated modes nahi
• Chhota $\sigma$: Fine data structure capture karta hai
• Interpolation: Scales ke beech smooth transition

Langevin Dynamics se Sampling

Jab hamare paas score function aa jaata hai, hum Langevin MCMC use karke samples generate karte hain:

Derivation: Yeh Langevin SDE se aata hai: $dx={\nabla }_x\log p(x)dt+\sqrt{2}dW$ jahan $dW$ ek Wiener process hai. Step size $\alpha$ ke saath discretize karte hain: $x_{t+1}=x_t+\alpha {\nabla }_x\log p(x_t)+\sqrt{2\alpha }ϵ$

$\sqrt{2\alpha }$ noise term kyun? Yeh diffusion term $\sqrt{2}dW$ ka discretization hai. $dW$ ka variance time $dt$ ke upar $dt$ hota hai, toh step $\alpha$ ke upar yeh $\sqrt{2\alpha }$ ho jaata hai.

Annealing schedule: ${\sigma }_1$ (coarse) se shuru karo, gradually ${\sigma }_L$ (fine) tak kam karo. Yeh ensure karta hai:

1. High-noise regime mein initial exploration
2. Low-noise regime mein refinement
3. Local minima se bachna

Diffusion Models Se Connection

Derivation: Diya hua $x_t={\alpha }_t{x}_0+{\sigma }_tϵ$, conditional distribution hai: $q(x_t|x_0)=\mathcal{N}(x_t;{\alpha }_t{x}_0,{\sigma }_t^{2}I)$

Score hai: ${\nabla }_{x_t}\log q(x_t|x_0)=-\frac{x_t-{\alpha }_t{x}_0}{{\sigma }_t^2}=-\frac{{\sigma }_tϵ}{{\sigma }_t^2}=-\frac{ϵ}{{\sigma }_t}$

Isliye, $ϵ$ predict karna seekhna score $s_{\theta }(x_t,t)\approx -ϵ/{\sigma }_t$ seekhne ke equivalent hai.

Connections

• Diffusion Models: Score-based models discrete diffusion ka continuous-time limit hain
• VAE: Dono latent structures seekhte hain, lekin scores $p(x)$ directly model karta hai encoder-decoder ke bajay
• GAN: Dono samples generate karte hain, lekin scores gradients use karta hai adversarial training ke bajay
• Energy-Based Models: Score energy function $E(x)=-\log p(x)$ ka gradient hai
• Langevin Dynamics: Generation ke liye use hone wala MCMC sampling procedure
• Stochastic Differential Equations: Modern formulation diffusion ko reverse-time SDE ki tarah treat karta hai

#flashcards/ai-ml

Score-based generative models mein score function kya hota hai? :: Log-probability ka gradient: $s(x)={\nabla }_x\log p(x)$. Yeh increasing probability density ki direction mein point karta hai.

Denoising score matching objective likho :: $J(\theta )={\mathbb{E}}_{x,ϵ}[\frac{1}{2}\Vert s_{\theta }(x+\sigma ϵ)+ϵ/\sigma {\Vert }^2]$ jahan $ϵ\sim \mathcal{N}(0,I)$.

Concept Map