Md. Asif Uddin
    I.11.X06

    Global averaging distributes a derivative

    gradient▲△△

    One feature map is [[1,3],[5,7]][[1,3],[5,7]]. Compute its global average and the input gradient when the upstream derivative is two. Does permuting these four feature values change the result?

    Hint

    The average is a linear map with four equal coefficients.

    Solution

    The average is (1+3+5+7)/4=4(1+3+5+7)/4=4. Each input derivative is 2/4=1/22/4=1/2, so the gradient map is [[0.5,0.5],[0.5,0.5]][[0.5,0.5],[0.5,0.5]].

    Every permutation retains the same sum and average. This is a statement about the supplied feature values. A translated finite image can change the preceding features at boundaries; global averaging by itself does not prove that the entire image-to-label function is invariant to every translation.

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