Which Frequencies do CNNs Need? Emergent Bottleneck Structure in Feature
  Learning
Yuxiao Wen Arthur Jacot
We describe the emergence of a Convolution Bottleneck (CBN) structure in
CNNs, where the network uses its first few layers to transform the input
representation into a representation that is supported only along a few
frequencies and channels, before using the last few layers to map back to the
outputs. We define the CBN rank, which describes the number and type of
frequencies that are kept inside the bottleneck, and partially prove that the
parameter norm required to represent a function $f$ scales as depth times the
CBN rank $f$. We also show that the parameter norm depends at next order on the
regularity of $f$. We show that any network with almost optimal parameter norm
will exhibit a CBN structure in both the weights and - under the assumption
that the network is stable under large learning rate - the activations, which
motivates the common practice of down-sampling; and we verify that the CBN
results still hold with down-sampling. Finally we use the CBN structure to
interpret the functions learned by CNNs on a number of tasks.