  This paper focuses on improving the mathematical interpretability of
convolutional neural networks (CNNs) in the context of image classification.
Specifically, we tackle the instability issue arising in their first layer,
which tends to learn parameters that closely resemble oriented band-pass
filters when trained on datasets like ImageNet. Subsampled convolutions with
such Gabor-like filters are prone to aliasing, causing sensitivity to small
input shifts. In this context, we establish conditions under which the max
pooling operator approximates a complex modulus, which is nearly shift
invariant. We then derive a measure of shift invariance for subsampled
convolutions followed by max pooling. In particular, we highlight the crucial
role played by the filter's frequency and orientation in achieving stability.
We experimentally validate our theory by considering a deterministic feature
extractor based on the dual-tree complex wavelet packet transform, a particular
case of discrete Gabor-like decomposition.
