  Few-shot and one-shot learning have been the subject of active and intensive
research in recent years, with mounting evidence pointing to successful
implementation and exploitation of few-shot learning algorithms in practice.
Classical statistical learning theories do not fully explain why few- or
one-shot learning is at all possible since traditional generalisation bounds
normally require large training and testing samples to be meaningful. This
sharply contrasts with numerous examples of successful one- and few-shot
learning systems and applications.
  In this work we present mathematical foundations for a theory of one-shot and
few-shot learning and reveal conditions specifying when such learning schemes
are likely to succeed. Our theory is based on intrinsic properties of
high-dimensional spaces. We show that if the ambient or latent decision space
of a learning machine is sufficiently high-dimensional than a large class of
objects in this space can indeed be easily learned from few examples provided
that certain data non-concentration conditions are met.
