  The invertibility of integral linear operators is a major problem of both
theoretical and practical importance.
  In this paper we investigate the relation between an operator invertibility
and the rank of its integral kernel to develop a local inverse theory. An
operator is called locally invertible provided that any function can be
recovered from its transformed image if the latter is known in an arbitrary
open subset of its domain, i.e., if its image is known locally. It turns out
that the local invertibility holds for any analytical kernel whose Taylor
functions are linearly independent in any open subset of their domain - the
so-called local linear independence condition.
  We also establish an equivalence between local linear independence and the
so-called full rank a.e. property. The latter can be described as follows: for
any finite, random sample of points, the square matrix obtained by applying,
pairwise, the kernel function on them, has full rank almost surely. As an
illustration, we show that the geodesic distance function on a sphere in more
than one dimensions is of full rank a.e., in contrast to the Euclidean distance
which is not.
