Properties Proof of CNN Operation

Properties Proof of CNN Operation

• (아직 생각중! 23.01.26)
• max-pooling
• Notation
  - let $*$ be convolutional operator
  - let $*_{p}$ be pooling operator
  - let $i _{w},i _{h}$ be width of input, height of output
  - let $k _{w},k _{h}$ be width of kernel and height of kernel respectively
  - let $s _{w},s _{h}$ be width of stride, height of stride respectively
  - let $\boldsymbol{X}' \in \mathcal{M}_{i _{h}+2p _{h}, i _{w}+2p _{w}}$ $[\boldsymbol{X}']_{i+p _{h},j+p _{w}}=[\boldsymbol{X}]_{i,j}$ if $i \ge p _{h}$ and $j \ge p _{w}$ else 0

Definition

• Convolution
  • $[\boldsymbol{O}]_{i,j,d}=[\boldsymbol{K} * \boldsymbol{I}]_{i,j,d}=\displaystyle\sum_{\alpha=0}^{k _{h}-1}{\displaystyle\sum_{\beta=0}^{k _{w}-1}{\displaystyle\sum_{\gamma=0}^{c-1}{[\boldsymbol{K}]_{\alpha,\beta,\gamma,d}[\boldsymbol{X}']_{s _{h}i+ \alpha,s _{h}j+\beta,\gamma}}}}$
  • in case of height of $[\boldsymbol{X}']_{i+s _{h} \alpha,j+s _{w}\beta}$
    • $s _{h}i+(k _{h}-1)\le i _{h}+2 p _{h}$
    • $s_{h}i \le i _{h}+2 p _{h}-(k _{h}-1)$
    • $i \le \displaystyle\frac{i _{j}+2p _{h}-k _{h}+1}{s _{h}}$
    • $i \le \displaystyle\frac{i _{j}+2p _{h}-k _{h}+1}{s _{h}} \le \displaystyle\frac{i _{h}+2p _{h}-k _{h}}{s _{h}}+1$
    • therefore $o _{h}=\displaystyle\frac{i _{h}+2p _{h}-k _{h}}{s _{h}}+1$
• Pooling
  -