to the output label and not correctness, in order to accurately match the definition of invariance.

Figure \[fig:pooling\_invariance\]a shows that as expected, the lack of invariance is dramatically reduced when using large pooling regions. For embedded 12-pixel images, $75\%$ of the one-pixel shifts affect the network with small pooling regions, while only $10\%$ of the one-pixel shifts affect the output of the network. Pooling is not completely location invariant due to boundary effects near the perimeter of the image. We show qualitative examples of this reduction in Figure \[fig:translation\_invariance\_ext\].

In Figure \[fig:pooling\_invariance\]b, we see that increasing the pooling size only slightly decreases FRIs. The region size with a larger decrease is $12$ pixels (note that the embedding size can not be directly compared with the region size of FRIs). For this region size, there are approximately $10\%$ less FRIs, but a significant amount still remain (about $65\%$ of the regions are FRIs). Since the pooling mechanisms that largely reduced the lack of invariance are not effective for fragile recognition, we can conclude that FRIs are a more complex phenomenon than lack of location invariance.

Note that for small region sizes the amount of FRIs increases, which is different from what 