Score Diffusion 公式推导1

xtlog p(xt)xtlog p(xty)y:labelp(xty)=p(yxt)p(xt)p(y)xtlog p(xty)=xtlog p(yxt)+xtlog p(xt){Scoreestimatorsθ(xt,t)=xtlog p(xt)DDPMdenoiserϵθ(xt,t)sθ(xt,t)=11αtϵθ(xt,t)sθ(x)xlog p(x)xi+1xi+ϵxlog p(x)+2ϵ zi,i=0,1,...,k\bigtriangledown_{x_t}log\ p(x_t) \rarr \bigtriangledown_{x_t}log\ p(x_t|y) \qquad y:label\\ \begin{aligned} \qquad \\ p(x_t|y)&=\frac{p(y|x_t)p(x_t)}{p(y)}\\ \bigtriangledown_{x_t}log\ p(x_t|y)&=\bigtriangledown_{x_t}log\ p(y|x_t)+\bigtriangledown_{x_t}log\ p(x_t) \end{aligned} \\ \quad \\ \begin{cases} Score\quad estimator \quad s_{\theta}(x_t,t)=\bigtriangledown_{x_t}log\ p(x_t) \\ DDPM \quad denoiser \quad \epsilon_{\theta}(x_t,t) \\ \end{cases} \\ s_{\theta}(x_t,t) = -\frac{1}{\sqrt{1-\overline{\alpha_t}}}\epsilon_{\theta}(x_t,t) \\ \quad \\ s_{\theta}(x)\approx\bigtriangledown_xlog\ p(x)\\ \quad \\ x_{i+1} \larr x_i+\epsilon\bigtriangledown_{x}log\ p(x)+\sqrt{2\epsilon}\ z_i,\quad i=0,1,...,k
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