  We study the functional task of deep learning image classification models and
show that image classification requires extrapolation capabilities. This
suggests that new theories have to be developed for the understanding of deep
learning as the current theory assumes models are solely interpolating, leaving
many questions about them unanswered. We investigate the pixel space and also
the feature spaces extracted from images by trained models (in their hidden
layers, including the 64-dimensional feature space in the last hidden layer of
pre-trained residual neural networks), and also the feature space extracted by
wavelets/shearlets. In all these domains, testing samples considerably fall
outside the convex hull of training sets, and image classification requires
extrapolation. In contrast to the deep learning literature, in cognitive
science, psychology, and neuroscience, extrapolation and learning are often
studied in tandem. Moreover, many aspects of human visual cognition and
behavior are reported to involve extrapolation. We propose a novel
extrapolation framework for the mathematical study of deep learning models. In
our framework, we use the term extrapolation in this specific way of
extrapolating outside the convex hull of training set (in the pixel space or
feature space) but within the specific scope defined by the training data, the
same way extrapolation is defined in many studies in cognitive science. We
explain that our extrapolation framework can provide novel answers to open
research problems about deep learning including their over-parameterization,
their training regime, out-of-distribution detection, etc. We also see that the
extent of extrapolation is negligible in learning tasks where deep learning is
reported to have no advantage over simple models.
