  Despite a variety of available techniques the issue of the proper
regularization parameter choice for inverse problems still remains one of the
biggest challenges. The main difficulty lies in constructing a rule, allowing
to compute the parameter from given noisy data without relying either on a
priori knowledge of the solution or on the noise level. In this paper we
propose a novel method based on supervised machine learning to approximate the
high-dimensional function, mapping noisy data into a good approximation to the
optimal Tikhonov regularization parameter. Our assumptions are that solutions
of the inverse problem are statistically distributed in a concentrated manner
on (lower-dimensional) linear subspaces and the noise is sub-gaussian. One of
the surprising facts is that the number of previously observed examples for the
supervised learning of the optimal parameter mapping scales at most linearly
with the dimension of the solution subspace. We also provide explicit error
bounds on the accuracy of the approximated parameter and the corresponding
regularization solution. Even though the results are more of theoretical
nature, we present a recipe for the practical implementation of the approach
and provide numerical experiments confirming the theoretical results. We also
outline interesting directions for future research with some preliminary
results, confirming their feasibility.
