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Figure 1 from TSGB: Target-Selective Gradient Backprop for Probing CNN ...

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$\begin{align*} \frac{\partial smax(x_i)}{\partial x_i} &=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2} \\\\ &= \frac{e^{x_i}\sum_{j=1}^{|X|}{e^{x_j}} - e^{x_i}\frac{\partial}{x_i}{}\sum_{j=1}^{|X|}{e^{x_j}} }{(\sum_{j=1}^{|X|}{e^{x_j})^2}}\\\\ &= \frac{e^{x_i}\sum_{j=1}^{|X|}{e^{x_j}} - (e^{x_i})^2}{(\sum_{j=1}^{|X|}{e^{x_j})^2}}\\\\ &= \left ( \frac{e^{x_i}}{\sum_{j=1}^{|X|}{e^{x_j}}}\frac{\sum_{j=1}^{|X|}{e^{x_j}}}{\sum_{j=1}^{|X|}{e^{x_j}}}\right ) - \left ( \frac{e^{x_i}}{\sum_{j=1}^{|X|}{e^{x_j}}}\frac{e^{x_i}}{\sum_{j=1}^{|X|}{e^{x_j}}}\right )\\\\ &=\frac{e^{x_i}}{\sum_{j=1}^{|X|}{e^{x_j}}}\left (1- \frac{e^{x_i}}{\sum_{j=1}^{|X|}{e^{x_j}}}\right )\\\\ &= smax(x_i)(1-smax(x_i)) \end{align*}$

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Backprop Gradient Form

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