  Neural networks can implement arbitrary functions. But, mechanistically, what
are the tools at their disposal to construct the target? For classification
tasks, the network must transform the data classes into a linearly separable
representation in the final hidden layer. We show that a feed-forward
architecture has one primary tool at hand to achieve this separability:
progressive folding of the data manifold in unoccupied higher dimensions. The
operation of folding provides a useful intuition in low-dimensions that
generalizes to high ones. We argue that an alternative method based on shear,
requiring very deep architectures, plays only a small role in real-world
networks. The folding operation, however, is powerful as long as layers are
wider than the data dimensionality, allowing efficient solutions by providing
access to arbitrary regions in the distribution, such as data points of one
class forming islands within the other classes. We argue that a link exists
between the universal approximation property in ReLU networks and the
fold-and-cut theorem (Demaine et al., 1998) dealing with physical paper
folding. Based on the mechanistic insight, we predict that the progressive
generation of separability is necessarily accompanied by neurons showing mixed
selectivity and bimodal tuning curves. This is validated in a network trained
on the poker hand task, showing the emergence of bimodal tuning curves during
training. We hope that our intuitive picture of the data transformation in deep
networks can help to provide interpretability, and discuss possible
applications to the theory of convolutional networks, loss landscapes, and
generalization.
  TL;DR: Shows that the internal processing of deep networks can be thought of
as literal folding operations on the data distribution in the N-dimensional
activation space. A link to a well-known theorem in origami theory is provided.
