ConvolutionBackward

Forward

  • $X\in R^{w\times h}$, $W\in R^{c\times c}$
  • $C(x,y)=\sum_{u=-c}^{v=c}\sum_{u=-c}^{u=c}w_{u,v}X_{x+u,y+v}$

Backward

  • ${\delta E\over\delta X_{\bar x,\bar y}}=\sum_{x,y}{\delta C(x,y)\over\delta X_{\bar x,\bar y}}{\delta E\over\delta C(x,y)}$
    $=\sum_{u=-c}^{v=c}\sum_{u=-c}^{u=c}w_{u,v}{\delta E\over\delta C(\bar x-u,\bar y-v)}$
    $=\sum_{u=-c}^{v=c}\sum_{u=-c}^{u=c}w_{u,v}{D^y_{\bar x-u,\bar y-v}}$
  • ${\delta E\over\delta X_{\bar x,\bar y}\delta w_{\bar u,\bar v}}=\sum_{x,y}{D^y_{x-\bar u,y-\bar v}}$