  In this paper we investigate the connection between supervised learning and
linear inverse problems. We first show that a linear inverse problem can be
view as a function approximation problem in a reproducing kernel Hilbert space
(RKHS) and then we prove that to each of these approximation problems
corresponds a class of inverse problems. Analogously, we show that Tikhonov
solutions of this class correspond to the Tikhonov solution of the
approximation problem. Thanks to this correspondence, we show that supervised
learning and linear discrete inverse problems can be thought of as two
instances of the approximation problem in a RKHS. These instances are
formalized by means of a sampling operator which takes into account both
deterministic and random samples and leads to discretized problems. We then
analyze the discretized problems and we study the convergence of their
solutions to the ones of the approximation problem in a RKHS, both in the
deterministic and statistical framework. Finally, we prove there exists a
relation between the convergence rates computed with respect to the noise level
and the ones computed with respect to the number of samples. This allows us to
compare upper and lower bounds given in the statistical learning and in the
deterministic infinite dimensional inverse problems theory.
