  The notion of interpolation and extrapolation is fundamental in various
fields from deep learning to function approximation. Interpolation occurs for a
sample $x$ whenever this sample falls inside or on the boundary of the given
dataset's convex hull. Extrapolation occurs when $x$ falls outside of that
convex hull. One fundamental (mis)conception is that state-of-the-art
algorithms work so well because of their ability to correctly interpolate
training data. A second (mis)conception is that interpolation happens
throughout tasks and datasets, in fact, many intuitions and theories rely on
that assumption. We empirically and theoretically argue against those two
points and demonstrate that on any high-dimensional ($>$100) dataset,
interpolation almost surely never happens. Those results challenge the validity
of our current interpolation/extrapolation definition as an indicator of
generalization performances.
