I'm going over the paper Spatial Transformer Networks (link). On section 3.3 they introduce bilinear sampling kernel (Eq. 5)
$$ V_{i}^{c}=\sum_{n}^{H} \sum_{m}^{W} U_{n m}^{c} \max \left(0,1-\left|x_{i}^{s}-m\right|\right) \max \left(0,1-\left|y_{i}^{s}-n\right|\right), $$ and it's partial derivative (Eq. 6)
$$ \frac{\partial V_{i}^{c}}{\partial U_{n m}^{c}}=\sum_{n}^{H} \sum_{m}^{W} \max \left(0,1-\left|x_{i}^{s}-m\right|\right) \max \left(0,1-\left|y_{i}^{s}-n\right|\right). $$

Since we take the derivative w.r.t. $U^c_{nm}$, it seems to me like the partial derivative should be $$ \frac{\partial V_{i}^{c}}{\partial U_{n m}^{c}}=\max \left(0,1-\left|x_{i}^{s}-m\right|\right) \max \left(0,1-\left|y_{i}^{s}-n\right|\right), $$ otherwise, the indices $m,n$ of the lhs is meaningless, and all of the partial derivatives will be identical.

Did I miss anything regarding the definition of $V^c_i$, $x^s_i$, or $U^c_{nm}$?


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