after the shift do not vary except for the shift. See [@azulay2018deep] for a detailed explanation. Note that this assumption is not fulfilled in FRIs, as a shift of the image region introduces and removes patterns in the borders of the image, which can produce changes in the responses after pooling. We now show this experimentally.

[**Experimental results.**]{} We evaluate the network trained on CIFAR-10 and introduced in the previous section with different pooling region sizes for the second pooling layer. These sizes range from three pixels to the entire image; theoretically, this should produce full location invariance. In order to guarantee sufficient spatial resolution to obtain location invariance, we use a stride of one pixel for the convolutional layers and add padding. Adding larger pooling regions does not reduce network accuracy (see Figure \[fig:pooling\_fris\]a), but as we show next, it massively reduces the lack of location invariance.

We evaluate the DNN by embedding the object in a black background at all possible locations, and we quantify for how many locations a one-pixel shift of the object location produces a change in the output of the DNN (similar to the method in Section \[sec:methods\]). In this experiment we evaluate changes 