论文阅读笔记:Denoising Diffusion Implicit Models (4)

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论文阅读笔记:Denoising Diffusion Implicit Models (1)
论文阅读笔记:Denoising Diffusion Implicit Models (2)
论文阅读笔记:Denoising Diffusion Implicit Models (3)
论文阅读笔记:Denoising Diffusion Implicit Models (4)

4、接上文[论文阅读笔记:论文阅读笔记:Denoising Diffusion Implicit Models (3)

  1. 已经知道跳 1 1 1步时, q σ ( x t − 1 ∣ x t , x 0 ) q_{\sigma}(x_{t-1}|x_t,x_0) qσ(xt−1∣xt,x0)的分布满足公式(·)
    x t − 1 = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − 1 ⋅ z t \begin{equation} \begin{split} x_{t-1}&=\sqrt{\alpha_{t-1}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} + \sqrt{1-\alpha_{t-1}}\cdot z_t\\ \end{split} \end{equation} xt−1=αt−1 ⋅αt xt−1−αt ⋅zt+1−αt−1 ⋅zt
  2. 假设跳 n n n步时, q σ ( x t − n ∣ x t , x 0 ) q_{\sigma}(x_{t-n}|x_t,x_0) qσ(xt−n∣xt,x0)的分布满足公式(2)
    x t − n = α t − n ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − n ⋅ z t \begin{equation} \begin{split} x_{t-n}&=\sqrt{\alpha_{t-n}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} + \sqrt{1-\alpha_{t-n}}\cdot z_t\\ \end{split} \end{equation} xt−n=αt−n ⋅αt xt−1−αt ⋅zt+1−αt−n ⋅zt
  3. 证明:当跳 n + 1 n+1 n+1步时,分布 q σ ( x t − n − 1 ∣ x t , x 0 ) q_{\sigma}(x_{t-n-1}|x_t,x_0) qσ(xt−n−1∣xt,x0)满足 x t − n − 1 = α t − n − 1 ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − n − 1 ⋅ z t x_{t-n-1}=\sqrt{\alpha_{t-n-1}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} + \sqrt{1-\alpha_{t-n-1}}\cdot z_t xt−n−1=αt−n−1 ⋅αt xt−1−αt ⋅zt+1−αt−n−1 ⋅zt。
    由于 q σ ( x t − n − 1 ∣ x t , x 0 ) q_{\sigma}(x_{t-n-1}|x_t,x_0) qσ(xt−n−1∣xt,x0)是 q σ ( x t − n − 1 , x t − n ∣ x t , x 0 ) q_{\sigma}(x_{t-n-1},x_{t-n}|x_t,x_0) qσ(xt−n−1,xt−n∣xt,x0)的边缘分布,因此有
    q σ ( x t − n − 1 ∣ x t , x 0 ) = ∫ q σ ( x t − n − 1 , x t − n ∣ x t , x 0 ) ⋅ d x t − n = ∫ q σ ( x t − n − 1 , ∣ x t − n , x 0 ) ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n \begin{equation} \begin{split} q_{\sigma}(x_{t-n-1}|x_t,x_0)&=\int q_{\sigma}(x_{t-n-1},x_{t-n}|x_t,x_0) \cdot dx_{t-n} \\ &=\int q_{\sigma}(x_{t-n-1},|x_{t-n},x_0) \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} \end{split} \end{equation} qσ(xt−n−1∣xt,x0)=∫qσ(xt−n−1,xt−n∣xt,x0)⋅dxt−n=∫qσ(xt−n−1,∣xt−n,x0)⋅qσ(xt−n∣xt,x0)⋅dxt−n
    q σ ( x t − n − 1 , ∣ x t − n , x 0 ) = N ( x t − n − 1 ∣ 1 − α t − n − 1 1 − α t − n ⋅ x t − n + [ α t − n − 1 − α t − n ⋅ ( 1 − α t − n − 1 ) 1 − α t − n ] ⋅ x 0 , 0 ) q_{\sigma}(x_{t-n-1},|x_{t-n},x_0)=N\bigg(x_{t-n-1}|\sqrt{\frac{1-\alpha_{t-n-1}}{1-\alpha_{t-n}}}\cdot x_{t-n}+ \bigg[\sqrt{\alpha_{t-n-1}}- \frac{\sqrt{ \alpha_{t-n}\cdot (1-\alpha_{t-n-1}} )}{\sqrt{1-\alpha_{t-n}}} \bigg] \cdot x_0, 0\bigg) qσ(xt−n−1,∣xt−n,x0)=N(xt−n−1∣1−αt−n1−αt−n−1 ⋅xt−n+[αt−n−1 −1−αt−n αt−n⋅(1−αt−n−1 )]⋅x0,0)
    因此,分布 q σ ( x t − n − 1 ∣ x t , x 0 ) q_{\sigma}(x_{t-n-1}|x_t,x_0) qσ(xt−n−1∣xt,x0)的均值 μ t − n − 1 \mu_{t-n-1} μt−n−1如公式(4)所示。
    μ t − n − 1 = E ( q σ ( x t − n − 1 ∣ x t , x 0 ) ) = ∫ x t − n − 1 ⋅ q σ ( x t − n − 1 ∣ x t , x 0 ) ⋅ d x t − n − 1 = ∫ x t − n − 1 ⋅ ( ∫ q σ ( x t − n − 1 , ∣ x t − n , x 0 ) ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n ) ⋅ d x t − n − 1 = ∫ ∫ x t − n − 1 ⋅ q σ ( x t − n − 1 , ∣ x t − n , x 0 ) ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n ⋅ d x t − n − 1 = ∫ ( ∫ x t − n − 1 ⋅ q σ ( x t − n − 1 , ∣ x t − n , x 0 ) ⋅ d x t − n − 1 ) ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n = ∫ E ( q σ ( x t − n − 1 , ∣ x t − n , x 0 ) ) ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n = ∫ ( 1 − α t − n − 1 1 − α t − n ⋅ x t − n + [ α t − n − 1 − α t − n ⋅ ( 1 − α t − n − 1 ) 1 − α t − n ] ⋅ x 0 ) ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n = ∫ 1 − α t − n − 1 1 − α t − n ⋅ x t − n ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n + ∫ ( [ α t − n − 1 − α t − 1 ⋅ ( 1 − α t − n − 1 ) 1 − α t − n ] ⋅ x 0 ) ⋅ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n = 1 − α t − n − 1 1 − α t − n ⋅ ∫ q σ ( x t − n ∣ x t , x 0 ) ⋅ d x t − n + [ α t − n − 1 − α t − n ⋅ ( 1 − α t − n − 1 ) 1 − α t − n ] ⋅ x 0 = 1 − α t − n − 1 1 − α t − n ⋅ E ( q σ ( x t − n ∣ x t , x 0 ) ) + [ α t − n − 1 − α t − n ⋅ ( 1 − α t − n − 1 ) 1 − α t − n ] ⋅ x 0 ⏟ x 0 = x t − 1 − α t ⋅ z t α t = 1 − α t − n − 1 1 − α t − n ⋅ ( α t − n ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − n ⋅ z t ) + [ α t − n − 1 − α t − n ⋅ ( 1 − α t − n − 1 ) 1 − α t − n ] ⋅ x t − 1 − α t ⋅ z t α t = 1 − α t − n − 1 1 − α t − n ⋅ α t − n ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − n − 1 1 − α t − n ⋅ 1 − α t − n ⋅ z t + α t − n − 1 ⋅ x t − 1 − α t ⋅ z t α t − α t − n ⋅ ( 1 − α t − n − 1 ) 1 − α t − n ⋅ x t − 1 − α t ⋅ z t α t = α t − n − 1 ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − n − 1 ⋅ z t \begin{equation} \begin{split} \mu_{t-n-1}&=E\big(q_{\sigma}(x_{t-n-1}|x_t,x_0) \big)\\ &=\int x_{t-n-1}\cdot q_{\sigma}(x_{t-n-1}|x_t,x_0) \cdot dx_{t-n-1} \\ &=\int x_{t-n-1}\cdot \bigg(\int q_{\sigma}(x_{t-n-1},|x_{t-n},x_0) \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} \bigg) \cdot dx_{t-n-1} \\ &=\int \int x_{t-n-1}\cdot q_{\sigma}(x_{t-n-1},|x_{t-n},x_0) \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} \cdot dx_{t-n-1} \\ &=\int \bigg( \int x_{t-n-1}\cdot q_{\sigma}(x_{t-n-1},|x_{t-n},x_0) \cdot dx_{t-n-1}\bigg) \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} \\ &=\int E\big(q_{\sigma}(x_{t-n-1},|x_{t-n},x_0) \big) \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} \\ &=\int \bigg(\sqrt{\frac{1-\alpha_{t-n-1}}{1-\alpha_{t-n}}}\cdot x_{t-n}+ \bigg[\sqrt{\alpha_{t-n-1}}- \frac{\sqrt{ \alpha_{t-n}\cdot (1-\alpha_{t-n-1}} )}{\sqrt{1-\alpha_{t-n}}} \bigg] \cdot x_0 \bigg) \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} \\ &=\int \sqrt{\frac{1-\alpha_{t-n-1}}{1-\alpha_{t-n}}}\cdot x_{t-n} \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} + \int \bigg(\bigg[\sqrt{\alpha_{t-n-1}}- \frac{\sqrt{ \alpha_{t-1}\cdot (1-\alpha_{t-n-1}} )}{\sqrt{1-\alpha_{t-n}}} \bigg] \cdot x_0 \bigg) \cdot q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n}\\ &=\sqrt{\frac{1-\alpha_{t-n-1}}{1-\alpha_{t-n}}}\cdot \int q_{\sigma}(x_{t-n}|x_t,x_0) \cdot dx_{t-n} + \bigg[\sqrt{\alpha_{t-n-1}}- \frac{\sqrt{ \alpha_{t-n}\cdot (1-\alpha_{t-n-1}} )}{\sqrt{1-\alpha_{t-n}}} \bigg] \cdot x_0 \\ &=\sqrt{\frac{1-\alpha_{t-n-1}}{1-\alpha_{t-n}}}\cdot E\bigg(q_{\sigma}(x_{t-n}|x_t,x_0) \bigg) + \bigg[\sqrt{\alpha_{t-n-1}}- \frac{\sqrt{ \alpha_{t-n}\cdot (1-\alpha_{t-n-1}} )}{\sqrt{1-\alpha_{t-n}}} \bigg] \cdot \underbrace{x_0}{x_0=\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}} \\ &=\sqrt{\frac{1-\alpha{t-n-1}}{1-\alpha_{t-n}}}\cdot \bigg(\sqrt{\alpha_{t-n}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} + \sqrt{1-\alpha_{t-n}}\cdot z_t \bigg) + \bigg[\sqrt{\alpha_{t-n-1}}- \frac{\sqrt{ \alpha_{t-n}\cdot (1-\alpha_{t-n-1}} )}{\sqrt{1-\alpha_{t-n}}} \bigg] \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} \\ &=\bcancel{\sqrt{\frac{1-\alpha_{t-n-1}}{1-\alpha_{t-n}}}\cdot \sqrt{\alpha_{t-n}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} }+\sqrt{\frac{1-\alpha_{t-n-1}}{\bcancel{1-\alpha_{t-n}}}}\cdot \bcancel{\sqrt{1-\alpha_{t-n}}}\cdot z_t + \sqrt{\alpha_{t-n-1}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} - \bcancel{\frac{\sqrt{ \alpha_{t-n}\cdot (1-\alpha_{t-n-1}} )}{\sqrt{1-\alpha_{t-n}}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}} \\ &=\sqrt{\alpha_{t-n-1}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} + \sqrt{1-\alpha_{t-n-1}}\cdot z_t \end{split} \end{equation} μt−n−1=E(qσ(xt−n−1∣xt,x0))=∫xt−n−1⋅qσ(xt−n−1∣xt,x0)⋅dxt−n−1=∫xt−n−1⋅(∫qσ(xt−n−1,∣xt−n,x0)⋅qσ(xt−n∣xt,x0)⋅dxt−n)⋅dxt−n−1=∫∫xt−n−1⋅qσ(xt−n−1,∣xt−n,x0)⋅qσ(xt−n∣xt,x0)⋅dxt−n⋅dxt−n−1=∫(∫xt−n−1⋅qσ(xt−n−1,∣xt−n,x0)⋅dxt−n−1)⋅qσ(xt−n∣xt,x0)⋅dxt−n=∫E(qσ(xt−n−1,∣xt−n,x0))⋅qσ(xt−n∣xt,x0)⋅dxt−n=∫(1−αt−n1−αt−n−1 ⋅xt−n+[αt−n−1 −1−αt−n αt−n⋅(1−αt−n−1 )]⋅x0)⋅qσ(xt−n∣xt,x0)⋅dxt−n=∫1−αt−n1−αt−n−1 ⋅xt−n⋅qσ(xt−n∣xt,x0)⋅dxt−n+∫([αt−n−1 −1−αt−n αt−1⋅(1−αt−n−1 )]⋅x0)⋅qσ(xt−n∣xt,x0)⋅dxt−n=1−αt−n1−αt−n−1 ⋅∫qσ(xt−n∣xt,x0)⋅dxt−n+[αt−n−1 −1−αt−n αt−n⋅(1−αt−n−1 )]⋅x0=1−αt−n1−αt−n−1 ⋅E(qσ(xt−n∣xt,x0))+[αt−n−1 −1−αt−n αt−n⋅(1−αt−n−1 )]⋅x0=αt xt−1−αt ⋅zt x0=1−αt−n1−αt−n−1 ⋅(αt−n ⋅αt xt−1−αt ⋅zt+1−αt−n ⋅zt)+[αt−n−1 −1−αt−n αt−n⋅(1−αt−n−1 )]⋅αt xt−1−αt ⋅zt=1−αt−n1−αt−n−1 ⋅αt−n ⋅αt xt−1−αt ⋅zt +1−αt−n 1−αt−n−1 ⋅1−αt−n ⋅zt+αt−n−1 ⋅αt xt−1−αt ⋅zt−1−αt−n αt−n⋅(1−αt−n−1 )⋅αt xt−1−αt ⋅zt =αt−n−1 ⋅αt xt−1−αt ⋅zt+1−αt−n−1 ⋅zt
    证毕。综上所述,跳 n n n步的公式为 q σ ( x t − n ∣ x t , x 0 ) = α t − n ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − n ⋅ z t q_\sigma(x_{t-n}|x_t,x_0)=\sqrt{\alpha_{t-n}} \cdot \frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}} + \sqrt{1-\alpha_{t-n}}\cdot z_t qσ(xt−n∣xt,x0)=αt−n ⋅αt xt−1−αt ⋅zt+1−αt−n ⋅zt
    基于DDIM的多数论文,例如暗图像增强方法LightenDiffusion等,也都是令 σ t = 0 \sigma_t=0 σt=0。论文和代码中使用的跳 n n n步的采样过程如公式(5)所示。
    x t − n = α t − n ⋅ x t − 1 − α t ⋅ z t α t ⏟ 预测出 z t , 进而计算出 x 0 + 1 − α t − n − σ t 2 ⋅ z t + σ t 2 ϵ t ⏟ 标准高斯分布 \begin{equation} \begin{split} x_{t-n}&=\sqrt{\alpha_{t-n}}\cdot \underbrace{\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}}{预测出z_t,进而计算出x_0}+\sqrt{1-\alpha{t-n}-\sigma_t^2}\cdot z_t + \sigma_t^2 \underbrace{ \epsilon_t}_{标准高斯分布} \\ \end{split} \end{equation} xt−n=αt−n ⋅预测出zt,进而计算出x0 αt xt−1−αt ⋅zt+1−αt−n−σt2 ⋅zt+σt2标准高斯分布 ϵt

这里使用中的 σ t \sigma_t σt是可以自己定义的量。有两种特殊的情况:
1、 σ t 2 = 0 \sigma_t^2=0 σt2=0:此时,
x t − 1 x_{t-1} xt−1满足公式(3)
x t − 1 = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − 1 − σ t 2 ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x 0 + 1 − α t − 1 ⋅ z t \begin{equation} \begin{split} x_{t-1}&=\sqrt{\alpha_{t-1}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+\sqrt{1-\alpha_{t-1}-\sigma_t^2}\cdot z_t + \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-1}}\cdot x_0+\sqrt{1-\alpha_{t-1}}\cdot z_t \\ \end{split} \end{equation} xt−1=αt−1 ⋅αt xt−1−αt ⋅zt+1−αt−1−σt2 ⋅zt+σt2ϵt=αt−1 ⋅x0+1−αt−1 ⋅zt
x t − n x_{t-n} xt−n满足
x t − n = α t − n ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − n − σ t 2 ⋅ z t + σ t 2 ϵ t = α t − n ⋅ x 0 + 1 − α t − n ⋅ z t \begin{equation} \begin{split} x_{t-n}&=\sqrt{\alpha_{t-n}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+\sqrt{1-\alpha_{t-n}-\sigma_t^2}\cdot z_t + \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-n}}\cdot x_0+\sqrt{1-\alpha_{t-n}}\cdot z_t \\ \end{split} \end{equation} xt−n=αt−n ⋅αt xt−1−αt ⋅zt+1−αt−n−σt2 ⋅zt+σt2ϵt=αt−n ⋅x0+1−αt−n ⋅zt

可以看出,此时, x t − 1 x_{t-1} xt−1和 x t − n x_{t-n} xt−n退化成上文论文阅读笔记:Denoising Diffusion Implicit Models (2)中的Lemma 1.

2、 σ t 2 = 1 − α t − 1 1 − α t ⋅ ( 1 − α t α t − 1 ) \sigma_t^2=\frac{1-\alpha_{t-1}}{1-\alpha_t}\cdot (1-\frac{\alpha_t}{\alpha_{t-1}}) σt2=1−αt1−αt−1⋅(1−αt−1αt):此时, x t − 1 x_{t-1} xt−1满足公式(4)
x t − 1 = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − 1 − σ t 2 ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + 1 − α t − 1 − 1 − α t − 1 1 − α t ⋅ ( 1 − α t α t − 1 ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + ( 1 − α t − 1 ) ( 1 − 1 1 − α t ⋅ α t − 1 − α t α t − 1 ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + ( 1 − α t − 1 ) α t − 1 − α t − 1 ⋅ α t − α t − 1 + α t α t − 1 ⋅ ( 1 − α t ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + ( 1 − α t − 1 ) − α t − 1 ⋅ α t + α t α t − 1 ⋅ ( 1 − α t ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t − 1 − α t ⋅ z t α t + ( 1 − α t − 1 ) α t α t − 1 ⋅ ( 1 − α t ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t α t − α t − 1 1 − α t ⋅ z t α t + ( 1 − α t − 1 ) α t α t − 1 ⋅ ( 1 − α t ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t α t − ( α t − 1 1 − α t ⋅ α t − 1 1 − α t − ( 1 − α t − 1 ) ⋅ α t ⋅ α t α t ⋅ α t − 1 ⋅ ( 1 − α t ) ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t α t − ( α t − 1 1 − α t ⋅ α t − 1 1 − α t − ( 1 − α t − 1 ) ⋅ α t ⋅ α t α t ⋅ α t − 1 ⋅ ( 1 − α t ) ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t α t − ( α t − 1 ⋅ ( 1 − α t ) − ( 1 − α t − 1 ) ⋅ α t α t ⋅ α t − 1 ⋅ ( 1 − α t ) ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t α t − ( α t − 1 − α t ⋅ α t − 1 − α t + α t ⋅ α t − 1 α t ⋅ α t − 1 ⋅ ( 1 − α t ) ) ⋅ z t = α t − 1 ⋅ x t α t − ( α t − 1 − α t α t ⋅ α t − 1 ⋅ ( 1 − α t ) ) ⋅ z t + σ t 2 ϵ t = α t − 1 ⋅ x t α t − ( α t − 1 ⋅ ( α t − 1 − α t ) α t − 1 ⋅ α t ⋅ ( 1 − α t ) ) ⋅ z t + σ t 2 ϵ t = α t − 1 α t ( x t − α t − 1 − α t α t − 1 ⋅ 1 − α t ) + σ t 2 ϵ t = α t − 1 α t ( x t − 1 1 − α t ⋅ ( 1 − α t α t − 1 ) ) ⋅ z t + σ t 2 ϵ t = 1 α t ( x t − β t 1 − α ˉ t ) ⋅ z t + σ t 2 ϵ t (换成 D D P M 中的符号) \begin{equation} \begin{split} x_{t-1}&=\sqrt{\alpha_{t-1}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+\sqrt{1-\alpha_{t-1}-\sigma_t^2}\cdot z_t + \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+\sqrt{1-\alpha_{t-1}-\frac{1-\alpha_{t-1}}{1-\alpha_t}\cdot (1-\frac{\alpha_t}{\alpha_{t-1}})}\cdot z_t + \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+\sqrt{(1-\alpha_{t-1})(1-\frac{1}{1-\alpha_t}\cdot \frac{\alpha_{t-1}-\alpha_t}{\alpha_{t-1}})}\cdot z_t + \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+\sqrt{(1-\alpha_{t-1})\frac{\alpha_{t-1}-\alpha_{t-1}\cdot \alpha_{t}-\alpha_{t-1}+\alpha_t}{\alpha_{t-1}\cdot(1-\alpha_{t})}}\cdot z_t + \sigma_t^2 \epsilon_t\\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+\sqrt{(1-\alpha_{t-1})\frac{-\alpha_{t-1}\cdot \alpha_{t}+\alpha_t}{\alpha_{t-1}\cdot(1-\alpha_{t})}}\cdot z_t + \sigma_t^2 \epsilon_t\\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t-{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+(1-\alpha_{t-1})\sqrt{\frac{\alpha_t}{\alpha_{t-1}\cdot(1-\alpha_{t})}}\cdot z_t + \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t}{\sqrt{\alpha_t}}-\frac{\sqrt{\alpha_{t-1}}{\sqrt{1-\alpha_t}\cdot z_t}}{\sqrt{\alpha_t}}+(1-\alpha_{t-1})\sqrt{\frac{\alpha_t}{\alpha_{t-1}\cdot(1-\alpha_{t})}}\cdot z_t + \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t}{\sqrt{\alpha_t}} -\Bigg(\frac{\sqrt{\alpha_{t-1}}{\sqrt{1-\alpha_t}}\cdot\sqrt{\alpha_{t-1}}{\sqrt{1-\alpha_t}}-(1-\alpha_{t-1})\cdot\sqrt{\alpha_t}\cdot\sqrt{\alpha_t}}{\sqrt{\alpha_t}\cdot \sqrt{\alpha_{t-1}\cdot(1-\alpha_t)}} \Bigg)\cdot z_t+ \sigma_t^2 \epsilon_t \\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t}{\sqrt{\alpha_t}} -\Bigg(\frac{\sqrt{\alpha_{t-1}}{\sqrt{1-\alpha_t}}\cdot\sqrt{\alpha_{t-1}}{\sqrt{1-\alpha_t}}-(1-\alpha_{t-1})\cdot\sqrt{\alpha_t}\cdot\sqrt{\alpha_t}}{\sqrt{\alpha_t}\cdot \sqrt{\alpha_{t-1}\cdot(1-\alpha_t)}}\Bigg)\cdot z_t + \sigma_t^2 \epsilon_t\\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t}{\sqrt{\alpha_t}} -\Bigg(\frac{\alpha_{t-1}\cdot({1-\alpha_t)}-(1-\alpha_{t-1})\cdot \alpha_t}{\sqrt{\alpha_t}\cdot \sqrt{\alpha_{t-1}\cdot(1-\alpha_t)}} \Bigg)\cdot z_t + \sigma_t^2 \epsilon_t\\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t}{\sqrt{\alpha_t}} -\Bigg(\frac{\alpha_{t-1}-\bcancel{\alpha_t\cdot \alpha_{t-1}}-\alpha_t+\bcancel{\alpha_t\cdot \alpha_{t-1}}}{\sqrt{\alpha_t}\cdot \sqrt{\alpha_{t-1}\cdot(1-\alpha_t)}} \Bigg)\cdot z_t \\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t}{\sqrt{\alpha_t}} -\Bigg(\frac{\alpha_{t-1}-\alpha_t}{\sqrt{\alpha_t}\cdot \sqrt{\alpha_{t-1}\cdot(1-\alpha_t)}} \Bigg)\cdot z_t + \sigma_t^2 \epsilon_t\\ &=\sqrt{\alpha_{t-1}}\cdot\frac{x_t}{\sqrt{\alpha_t}} -\Bigg(\frac{\sqrt{\alpha_{t-1}}\cdot (\alpha_{t-1}-\alpha_t)}{\alpha_{t-1}\cdot\sqrt{\alpha_t}\cdot \sqrt{(1-\alpha_t)}} \Bigg)\cdot z_t + \sigma_t^2 \epsilon_t\\ &=\frac{\sqrt{\alpha_{t-1}}}{\sqrt{\alpha_{t}}}\Bigg(x_t-\frac{\alpha_{t-1}-\alpha_t}{\alpha_{t-1}\cdot\ \sqrt{1-\alpha_t}}\Bigg) + \sigma_t^2 \epsilon_t\\ &=\frac{\sqrt{\alpha_{t-1}}}{\sqrt{\alpha_{t}}}\Bigg(x_t-\frac{1}{\ \sqrt{1-\alpha_t}}\cdot (1-\frac{\alpha_t}{\alpha_{t-1}})\Bigg)\cdot z_t + \sigma_t^2 \epsilon_t\\ &=\frac{1}{\sqrt{\alpha_{t}}}\Bigg(x_t-\frac{\beta_t}{\ \sqrt{1-\bar\alpha_t}}\Bigg)\cdot z_t + \sigma_t^2 \epsilon_t(换成DDPM中的符号)\\ \end{split} \end{equation} xt−1=αt−1 ⋅αt xt−1−αt ⋅zt+1−αt−1−σt2 ⋅zt+σt2ϵt=αt−1 ⋅αt xt−1−αt ⋅zt+1−αt−1−1−αt1−αt−1⋅(1−αt−1αt) ⋅zt+σt2ϵt=αt−1 ⋅αt xt−1−αt ⋅zt+(1−αt−1)(1−1−αt1⋅αt−1αt−1−αt) ⋅zt+σt2ϵt=αt−1 ⋅αt xt−1−αt ⋅zt+(1−αt−1)αt−1⋅(1−αt)αt−1−αt−1⋅αt−αt−1+αt ⋅zt+σt2ϵt=αt−1 ⋅αt xt−1−αt ⋅zt+(1−αt−1)αt−1⋅(1−αt)−αt−1⋅αt+αt ⋅zt+σt2ϵt=αt−1 ⋅αt xt−1−αt ⋅zt+(1−αt−1)αt−1⋅(1−αt)αt ⋅zt+σt2ϵt=αt−1 ⋅αt xt−αt αt−1 1−αt ⋅zt+(1−αt−1)αt−1⋅(1−αt)αt ⋅zt+σt2ϵt=αt−1 ⋅αt xt−(αt ⋅αt−1⋅(1−αt) αt−1 1−αt ⋅αt−1 1−αt −(1−αt−1)⋅αt ⋅αt )⋅zt+σt2ϵt=αt−1 ⋅αt xt−(αt ⋅αt−1⋅(1−αt) αt−1 1−αt ⋅αt−1 1−αt −(1−αt−1)⋅αt ⋅αt )⋅zt+σt2ϵt=αt−1 ⋅αt xt−(αt ⋅αt−1⋅(1−αt) αt−1⋅(1−αt)−(1−αt−1)⋅αt)⋅zt+σt2ϵt=αt−1 ⋅αt xt−(αt ⋅αt−1⋅(1−αt) αt−1−αt⋅αt−1 −αt+αt⋅αt−1 )⋅zt=αt−1 ⋅αt xt−(αt ⋅αt−1⋅(1−αt) αt−1−αt)⋅zt+σt2ϵt=αt−1 ⋅αt xt−(αt−1⋅αt ⋅(1−αt) αt−1 ⋅(αt−1−αt))⋅zt+σt2ϵt=αt αt−1 (xt−αt−1⋅ 1−αt αt−1−αt)+σt2ϵt=αt αt−1 (xt− 1−αt 1⋅(1−αt−1αt))⋅zt+σt2ϵt=αt 1(xt− 1−αˉt βt)⋅zt+σt2ϵt(换成DDPM中的符号)

可以看出,此时,DDIM退化成了DDPM。

论文讨论了 σ t 2 \sigma_t^2 σt2选取 η ⋅ 1 − α t − 1 1 − α t ⋅ ( 1 − α t α t − 1 ) , η ∈ [ 0 , 1 ] \eta\cdot \frac{1-\alpha_{t-1}}{1-\alpha_t}\cdot (1-\frac{\alpha_t}{\alpha_{t-1}}),\eta\in[0,1] η⋅1−αt1−αt−1⋅(1−αt−1αt),η∈[0,1],即在0和DDPM之间变化时。不同 η \eta η以及跳不同步时所对应的表现,如下图所示。

5、代码

python 复制代码
class DDIMPipeline(DiffusionPipeline):
    model_cpu_offload_seq = "unet"

    def __init__(self, unet, scheduler):
        super().__init__()

        # make sure scheduler can always be converted to DDIM
        scheduler = DDIMScheduler.from_config(scheduler.config)

        self.register_modules(unet=unet, scheduler=scheduler)

    @torch.no_grad()
    def __call__(
        self,
        batch_size: int = 1,
        generator: Optional[Union[torch.Generator, List[torch.Generator]]] = None,
        eta: float = 0.0,
        num_inference_steps: int = 50,
        use_clipped_model_output: Optional[bool] = None,
        output_type: Optional[str] = "pil",
        return_dict: bool = True,
    ) -> Union[ImagePipelineOutput, Tuple]:

        # Sample gaussian noise to begin loop
        if isinstance(self.unet.config.sample_size, int):
            image_shape = (
                batch_size,
                self.unet.config.in_channels,
                self.unet.config.sample_size,
                self.unet.config.sample_size,
            )
        else:
            image_shape = (batch_size, self.unet.config.in_channels, *self.unet.config.sample_size)

        if isinstance(generator, list) and len(generator) != batch_size:
            raise ValueError(
                f"You have passed a list of generators of length {len(generator)}, but requested an effective batch"
                f" size of {batch_size}. Make sure the batch size matches the length of the generators."
            )
		# 随即生成噪音
        image = randn_tensor(image_shape, generator=generator, device=self._execution_device, dtype=self.unet.dtype)

        # 设置步数间隔。例如num_inference_steps = 50,然而总步长为1000,那么就是每次跳20步,例如在当前时刻, timestep=980, prev_timestep=960
        self.scheduler.set_timesteps(num_inference_steps)

        for t in self.progress_bar(self.scheduler.timesteps):
            # 1. 预测出timestep=980时刻对应噪音
            model_output = self.unet(image, t).sample

            # 2. 调用scheduler的方法step,执行公式()得到prev_timestep=960时刻的图像
            image = self.scheduler.step(
                model_output, t, image, eta=eta, use_clipped_model_output=use_clipped_model_output, generator=generator
            ).prev_sample

        image = (image / 2 + 0.5).clamp(0, 1)
        image = image.cpu().permute(0, 2, 3, 1).numpy()
        if output_type == "pil":
            image = self.numpy_to_pil(image)

        if not return_dict:
            return (image,)

        return ImagePipelineOutput(images=image)
 
 class DDIMScheduler(SchedulerMixin, ConfigMixin):
    _compatibles = [e.name for e in KarrasDiffusionSchedulers]
    order = 1

    @register_to_config
    def __init__(
        self,
        num_train_timesteps: int = 1000,
        beta_start: float = 0.0001,
        beta_end: float = 0.02,
        beta_schedule: str = "linear",
        trained_betas: Optional[Union[np.ndarray, List[float]]] = None,
        clip_sample: bool = True,
        set_alpha_to_one: bool = True,
        steps_offset: int = 0,
        prediction_type: str = "epsilon",
        thresholding: bool = False,
        dynamic_thresholding_ratio: float = 0.995,
        clip_sample_range: float = 1.0,
        sample_max_value: float = 1.0,
        timestep_spacing: str = "leading",
        rescale_betas_zero_snr: bool = False,
    ):
        if trained_betas is not None:
            self.betas = torch.tensor(trained_betas, dtype=torch.float32)
        elif beta_schedule == "linear":
            self.betas = torch.linspace(beta_start, beta_end, num_train_timesteps, dtype=torch.float32)
        elif beta_schedule == "scaled_linear":
            # this schedule is very specific to the latent diffusion model.
            self.betas = torch.linspace(beta_start**0.5, beta_end**0.5, num_train_timesteps, dtype=torch.float32) ** 2
        elif beta_schedule == "squaredcos_cap_v2":
            # Glide cosine schedule
            self.betas = betas_for_alpha_bar(num_train_timesteps)
        else:
            raise NotImplementedError(f"{beta_schedule} is not implemented for {self.__class__}")

        # Rescale for zero SNR
        if rescale_betas_zero_snr:
            self.betas = rescale_zero_terminal_snr(self.betas)

        self.alphas = 1.0 - self.betas
        self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)

        # At every step in ddim, we are looking into the previous alphas_cumprod
        # For the final step, there is no previous alphas_cumprod because we are already at 0
        # `set_alpha_to_one` decides whether we set this parameter simply to one or
        # whether we use the final alpha of the "non-previous" one.
        self.final_alpha_cumprod = torch.tensor(1.0) if set_alpha_to_one else self.alphas_cumprod[0]

        # standard deviation of the initial noise distribution
        self.init_noise_sigma = 1.0

        # setable values
        self.num_inference_steps = None
        self.timesteps = torch.from_numpy(np.arange(0, num_train_timesteps)[::-1].copy().astype(np.int64))

    def scale_model_input(self, sample: torch.Tensor, timestep: Optional[int] = None) -> torch.Tensor:
        """
        Ensures interchangeability with schedulers that need to scale the denoising model input depending on the
        current timestep.

        Args:
            sample (`torch.Tensor`):
                The input sample.
            timestep (`int`, *optional*):
                The current timestep in the diffusion chain.

        Returns:
            `torch.Tensor`:
                A scaled input sample.
        """
        return sample

    def _get_variance(self, timestep, prev_timestep):
        alpha_prod_t = self.alphas_cumprod[timestep]
        alpha_prod_t_prev = self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
        beta_prod_t = 1 - alpha_prod_t
        beta_prod_t_prev = 1 - alpha_prod_t_prev

        variance = (beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)

        return variance

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler._threshold_sample
    def _threshold_sample(self, sample: torch.Tensor) -> torch.Tensor:
        """
        "Dynamic thresholding: At each sampling step we set s to a certain percentile absolute pixel value in xt0 (the
        prediction of x_0 at timestep t), and if s > 1, then we threshold xt0 to the range [-s, s] and then divide by
        s. Dynamic thresholding pushes saturated pixels (those near -1 and 1) inwards, thereby actively preventing
        pixels from saturation at each step. We find that dynamic thresholding results in significantly better
        photorealism as well as better image-text alignment, especially when using very large guidance weights."

        https://arxiv.org/abs/2205.11487
        """
        dtype = sample.dtype
        batch_size, channels, *remaining_dims = sample.shape

        if dtype not in (torch.float32, torch.float64):
            sample = sample.float()  # upcast for quantile calculation, and clamp not implemented for cpu half

        # Flatten sample for doing quantile calculation along each image
        sample = sample.reshape(batch_size, channels * np.prod(remaining_dims))

        abs_sample = sample.abs()  # "a certain percentile absolute pixel value"

        s = torch.quantile(abs_sample, self.config.dynamic_thresholding_ratio, dim=1)
        s = torch.clamp(
            s, min=1, max=self.config.sample_max_value
        )  # When clamped to min=1, equivalent to standard clipping to [-1, 1]
        s = s.unsqueeze(1)  # (batch_size, 1) because clamp will broadcast along dim=0
        sample = torch.clamp(sample, -s, s) / s  # "we threshold xt0 to the range [-s, s] and then divide by s"

        sample = sample.reshape(batch_size, channels, *remaining_dims)
        sample = sample.to(dtype)

        return sample

    def set_timesteps(self, num_inference_steps: int, device: Union[str, torch.device] = None):
        """
        Sets the discrete timesteps used for the diffusion chain (to be run before inference).

        Args:
            num_inference_steps (`int`):
                The number of diffusion steps used when generating samples with a pre-trained model.
        """

        if num_inference_steps > self.config.num_train_timesteps:
            raise ValueError(
                f"`num_inference_steps`: {num_inference_steps} cannot be larger than `self.config.train_timesteps`:"
                f" {self.config.num_train_timesteps} as the unet model trained with this scheduler can only handle"
                f" maximal {self.config.num_train_timesteps} timesteps."
            )

        self.num_inference_steps = num_inference_steps

        # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
        if self.config.timestep_spacing == "linspace":
            timesteps = (
                np.linspace(0, self.config.num_train_timesteps - 1, num_inference_steps)
                .round()[::-1]
                .copy()
                .astype(np.int64)
            )
        elif self.config.timestep_spacing == "leading":
            step_ratio = self.config.num_train_timesteps // self.num_inference_steps
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = (np.arange(0, num_inference_steps) * step_ratio).round()[::-1].copy().astype(np.int64)
            timesteps += self.config.steps_offset
        elif self.config.timestep_spacing == "trailing":
            step_ratio = self.config.num_train_timesteps / self.num_inference_steps
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = np.round(np.arange(self.config.num_train_timesteps, 0, -step_ratio)).astype(np.int64)
            timesteps -= 1
        else:
            raise ValueError(
                f"{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'leading' or 'trailing'."
            )

        self.timesteps = torch.from_numpy(timesteps).to(device)

    def step(
        self,
        model_output: torch.Tensor,
        timestep: int,
        sample: torch.Tensor,
        eta: float = 0.0,
        use_clipped_model_output: bool = False,
        generator=None,
        variance_noise: Optional[torch.Tensor] = None,
        return_dict: bool = True,
    ) -> Union[DDIMSchedulerOutput, Tuple]:
        
        if self.num_inference_steps is None:
            raise ValueError(
                "Number of inference steps is 'None', you need to run 'set_timesteps' after creating the scheduler"
            )

        # 1. get previous step value (=t-1);
        # timestep=980,self.config.num_train_timesteps=1000, self.num_inference_steps=50
        # prev_timestep = 960,步数的跳跃间隔为20
        prev_timestep = timestep - self.config.num_train_timesteps // self.num_inference_steps

        # 2. compute alphas, betas
        alpha_prod_t = self.alphas_cumprod[timestep]
        alpha_prod_t_prev = self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod

        beta_prod_t = 1 - alpha_prod_t

        # 3. compute predicted original sample from predicted noise also called
        # "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
        if self.config.prediction_type == "epsilon":
            pred_original_sample = (sample - beta_prod_t ** (0.5) * model_output) / alpha_prod_t ** (0.5)
            pred_epsilon = model_output
        elif self.config.prediction_type == "sample":
            pred_original_sample = model_output
            pred_epsilon = (sample - alpha_prod_t ** (0.5) * pred_original_sample) / beta_prod_t ** (0.5)
        elif self.config.prediction_type == "v_prediction":
            pred_original_sample = (alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
            pred_epsilon = (alpha_prod_t**0.5) * model_output + (beta_prod_t**0.5) * sample
        else:
            raise ValueError(
                f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
                " `v_prediction`"
            )

        # 4. Clip or threshold "predicted x_0"
        if self.config.thresholding:
            pred_original_sample = self._threshold_sample(pred_original_sample)
        elif self.config.clip_sample:
            pred_original_sample = pred_original_sample.clamp(
                -self.config.clip_sample_range, self.config.clip_sample_range
            )

        # 5. compute variance: "sigma_t(η)" -> see formula (16)
        # σ_t = sqrt((1 − α_t−1)/(1 − α_t)) * sqrt(1 − α_t/α_t−1)
        variance = self._get_variance(timestep, prev_timestep)
        std_dev_t = eta * variance ** (0.5)

        if use_clipped_model_output:
            # the pred_epsilon is always re-derived from the clipped x_0 in Glide
            pred_epsilon = (sample - alpha_prod_t ** (0.5) * pred_original_sample) / beta_prod_t ** (0.5)

        # 6. compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
        pred_sample_direction = (1 - alpha_prod_t_prev - std_dev_t**2) ** (0.5) * pred_epsilon

        # 7. compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
        prev_sample = alpha_prod_t_prev ** (0.5) * pred_original_sample + pred_sample_direction

        if eta > 0:
            if variance_noise is not None and generator is not None:
                raise ValueError(
                    "Cannot pass both generator and variance_noise. Please make sure that either `generator` or"
                    " `variance_noise` stays `None`."
                )

            if variance_noise is None:
                variance_noise = randn_tensor(
                    model_output.shape, generator=generator, device=model_output.device, dtype=model_output.dtype
                )
            variance = std_dev_t * variance_noise

            prev_sample = prev_sample + variance

        if not return_dict:
            return (
                prev_sample,
                pred_original_sample,
            )

        return DDIMSchedulerOutput(prev_sample=prev_sample, pred_original_sample=pred_original_sample)

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.add_noise
    def add_noise(
        self,
        original_samples: torch.Tensor,
        noise: torch.Tensor,
        timesteps: torch.IntTensor,
    ) -> torch.Tensor:
        # Make sure alphas_cumprod and timestep have same device and dtype as original_samples
        # Move the self.alphas_cumprod to device to avoid redundant CPU to GPU data movement
        # for the subsequent add_noise calls
        self.alphas_cumprod = self.alphas_cumprod.to(device=original_samples.device)
        alphas_cumprod = self.alphas_cumprod.to(dtype=original_samples.dtype)
        timesteps = timesteps.to(original_samples.device)

        sqrt_alpha_prod = alphas_cumprod[timesteps] ** 0.5
        sqrt_alpha_prod = sqrt_alpha_prod.flatten()
        while len(sqrt_alpha_prod.shape) < len(original_samples.shape):
            sqrt_alpha_prod = sqrt_alpha_prod.unsqueeze(-1)

        sqrt_one_minus_alpha_prod = (1 - alphas_cumprod[timesteps]) ** 0.5
        sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.flatten()
        while len(sqrt_one_minus_alpha_prod.shape) < len(original_samples.shape):
            sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.unsqueeze(-1)

        noisy_samples = sqrt_alpha_prod * original_samples + sqrt_one_minus_alpha_prod * noise
        return noisy_samples

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.get_velocity
    def get_velocity(self, sample: torch.Tensor, noise: torch.Tensor, timesteps: torch.IntTensor) -> torch.Tensor:
        # Make sure alphas_cumprod and timestep have same device and dtype as sample
        self.alphas_cumprod = self.alphas_cumprod.to(device=sample.device)
        alphas_cumprod = self.alphas_cumprod.to(dtype=sample.dtype)
        timesteps = timesteps.to(sample.device)

        sqrt_alpha_prod = alphas_cumprod[timesteps] ** 0.5
        sqrt_alpha_prod = sqrt_alpha_prod.flatten()
        while len(sqrt_alpha_prod.shape) < len(sample.shape):
            sqrt_alpha_prod = sqrt_alpha_prod.unsqueeze(-1)

        sqrt_one_minus_alpha_prod = (1 - alphas_cumprod[timesteps]) ** 0.5
        sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.flatten()
        while len(sqrt_one_minus_alpha_prod.shape) < len(sample.shape):
            sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.unsqueeze(-1)

        velocity = sqrt_alpha_prod * noise - sqrt_one_minus_alpha_prod * sample
        return velocity

    def __len__(self):
        return self.config.num_train_timesteps
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