# 卷积神经网络 (CNN)下面证明卷积的平移等变性：x′[u]=x[u−t]x'[u] = x[u-t]x′[u]=x[u−t], y[v]=∑u=−∞+∞x[u]ϕ[v−u]y[v] = \sum_{u=-\infty}^{+\infty}x[u]\phi[v-u]y[v]=∑u=−∞+∞x[u]ϕ[v−u], 则y′[v]=∑u=−∞+∞x′[u]ϕ[u−v]=∑u=−∞+∞x[u−t]ϕ[u−v]=y[v−t].y'[v] = \sum_{u=-\infty}^{+\infty}x'[u]\phi[u-v] = \sum_{u=-\infty}^{+\infty}x[u-t]\phi[u-v] = y[v-t].y′[v]=u=−∞∑+∞x′[u]ϕ[u−v]=u=−∞∑+∞x[u−t]ϕ[u−v]=y[v−t].# 群卷积神经网络 (G-CNN)群卷积的定义为(x⋆ψ)[v]=∑u∈Z2x[u]ψ[gv−1u].(x \star \psi) [v] = \sum_{u \in \mathbb{Z}^2} x[u] \psi[g_v^{-1}u].(x⋆ψ)[v]=u∈Z2∑x[u]ψ[gv−1u].
