Deep Residual Learning for Image Recognition

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[math]\displaystyle{ \frac{\partial Err}{\partial w_1} = \frac{\partial Err}{\partial \hat{y}}\frac{\partial \hat{y}}{\partial x_3}\frac{\partial x_3}{\partial x_2}\frac{\partial x_2}{\partial w_1} \\ \ \ \ \ \ \ \ \ = \frac{\partial Err}{\partial \hat{y}} \cdot w_3 \cdot \sigma'(x_2 w_2) \cdot w_2 \cdot \sigma'(x_1 w_1) \cdot x_1 \\ }[/math]

	Expression of [math]\displaystyle{ x_3 }[/math]	Condition for [math]\displaystyle{ x_1 = x_3 }[/math]
No short-cut	[math]\displaystyle{ x_2 = f(W_2 \cdot f(W_1x_1)) }[/math]	[math]\displaystyle{ W_1 = W_2 = I }[/math]
With short-cut	[math]\displaystyle{ x_2 = f(W_2 \cdot f(W_1x_1)) + x_1 }[/math]	[math]\displaystyle{ W_1 = 0 }[/math] or [math]\displaystyle{ W_2 = 0 }[/math]

[math]\displaystyle{ W_1^{'} }[/math]

[math]\displaystyle{ F(x_1, W_1^{'}) = W_2 f(W_1x_1) }[/math]

[math]\displaystyle{ x_2 = x_1 + F(x_1, W_1^{'}) }[/math]