DDIM 公式草稿3

DDPM:α1α2αtαT1αTεθ(x1,1)εθ(x2,2)εθ(xt,t)εθ(xT1,T1)εθ(xT,T)DDIM:α1α200α500α800α1000=αTεθ(x1,1)εθ(x200,200)εθ(xT,T)xt1=αt1x0t^+1αt1σt2εθ+σtεxt1=αt1x0+1αt1σt2εθ+σtεσt=0xt1=αt1x0+1αt1εθ=αt11αt(xt1αt  εθ(xt,t))+1αt1εθ(xt,t)xt1αt1=xtαt1αtαtεθ(xt,t)+1αt1αt1εθ(xt,t)xtαtxt1αt1=1αt1αt1εθ(xt,t)1αtαtεθ(xt,t)dds(x(s)a(s))=ddsσ(s)  εθ(x(s),t(s))a(s)=αsσ(s)=1αsαss[0,1],x(1)N(0,I),x0 训练目标角度: \\ DDPM: \qquad \overline{\alpha_1} \qquad \overline{\alpha_2} \qquad\dots \qquad \overline{\alpha_t} \qquad\dots \qquad \overline{\alpha_{T-1}} \qquad \overline{\alpha_{T}} \\ \qquad \qquad \qquad \quad \varepsilon_{\theta}(x_1,1) \quad \varepsilon_{\theta}(x_2,2) \quad\dots \quad \varepsilon_{\theta}(x_t,t) \quad\dots \quad \varepsilon_{\theta}(x_{T-1},T-1) \quad \varepsilon_{\theta}(x_T,T) \\ \quad \\ DDIM: \qquad \overline{\alpha_1} \qquad \overline{\alpha_{200}} \qquad\dots \qquad \overline{\alpha_{500}} \qquad\dots \qquad \overline{\alpha_{800}} \qquad \overline{\alpha_{1000}}=\overline{\alpha_{T}} \\ \qquad \qquad \qquad \quad \varepsilon_{\theta}(x_1,1) \quad \varepsilon_{\theta}(x_{200},200) \quad\dots \quad \varepsilon_{\theta}(x_T,T) \\ \quad \\ x_{t-1}=\sqrt{\overline{\alpha_{t-1}}} \hat{x_{0|t}} + \sqrt{1-\overline{\alpha_{t-1}}-\sigma_t^2} \varepsilon_{\theta}+\sigma_t\varepsilon \\ \quad \\ \begin{aligned} x_{t-1}&=\sqrt{\overline{\alpha_{t-1}}} x_0 + \sqrt{1-\overline{\alpha_{t-1}}-\sigma_t^2} \varepsilon_{\theta}+\sigma_t\varepsilon\\ \sigma_t=0 \qquad x_{t-1}&=\sqrt{\overline{\alpha_{t-1}}} x_0 + \sqrt{1-\overline{\alpha_{t-1}}} \varepsilon_{\theta}\\ &=\sqrt{\overline{\alpha_{t-1}}} \frac{1}{\sqrt{\overline{\alpha_{t}}}}(x_t-\sqrt{1-\overline{\alpha_{t}}}\ \ \varepsilon_{\theta}(x_t,t) )+\sqrt{1-\overline{\alpha_{t-1}}} \varepsilon_{\theta}(x_t,t)\\ \frac{x_{t-1}}{\sqrt{\overline{\alpha_{t-1}}}}&=\frac{x_t}{\sqrt{\overline{\alpha_{t}}}}-\frac{\sqrt{1-\overline{\alpha_{t}}} }{\sqrt{\overline{\alpha_{t}}}}\varepsilon_{\theta}(x_t,t)+\frac{\sqrt{1-\overline{\alpha_{t-1}}} }{\sqrt{\overline{\alpha_{t-1}}}}\varepsilon_{\theta}(x_t,t)\\ \quad \\ \frac{x_t}{\sqrt{\overline{\alpha_{t}}}}-\frac{x_{t-1}}{\sqrt{\overline{\alpha_{t-1}}}}&=\frac{\sqrt{1-\overline{\alpha_{t-1}}} }{\sqrt{\overline{\alpha_{t-1}}}}\varepsilon_{\theta}(x_t,t)-\frac{\sqrt{1-\overline{\alpha_{t}}} }{\sqrt{\overline{\alpha_{t}}}}\varepsilon_{\theta}(x_t,t)\\ \frac{d}{ds}(\frac{x(s)}{a(s)})&=\frac{d}{ds}\sigma(s) \ \ \varepsilon_{\theta}(x(s),t(s)) \quad a(s)=\sqrt{\overline{\alpha_{s}}} \qquad \sigma(s)=\frac{\sqrt{1-\overline{\alpha_{s}}} }{\sqrt{\overline{\alpha_{s}}}} \\ \quad \\ 假如我们已知 &s\in[0,1],\quad x(1) \sim N(0,I),\quad 求 x_0 \end{aligned}
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