In the Grad-CAM++ paper the following equation (7) is posed (written here without the relu function): $$ Y^c = \sum_k \Bigl( \Bigl\{ \sum_{a,b} \alpha_{ab}^{kc} \cdot \frac{\partial Y^c}{\partial A_{ab}^k} \Bigr\} \Bigl[ \sum_{i,j} A_{ij}^k \Bigr] \Bigr) \qquad (7) $$ $Y^c$ is supposed to be a function of the $A_{ij}^k$, and $\alpha_{ab}^{kc}$ are unknowns to be determined.
In order to isolate the alphas the paper computes partial derivatives w.r.t. $A_{ij}^k$ on both sides of the equation. In the computation no partial derivatives of the alphas with respect to $A_{ij}^k$ occur, so it seems that the alphas are considered independent from the $A_{ij}^k$. However this does not look right. To illustrate the problem assume that we want to find $\alpha$ in the equation $\alpha x = x^2$. If we follow the method used in the paper, i.e. differentiate both sides of the equation w.r.t. $x$ while assuming that $\alpha$ does not depend on $x$ (hence $d\alpha/dx = 0$), we will get $\alpha = 2x$, which is clearly incorrect. In fact differentiating both sides with respect to $x$ yields $\frac{d\alpha}{dx} x + \alpha = 2x$, which is not going to help isolating $\alpha$. The actual solution in this case is better obtained by dividing both sides by $x$, so $\alpha = x^2/x = x$, $x\neq 0$.
So, what it would be a right way to obtain alphas satisfying equation (7)?
