  In the present paper we consider application of overcomplete dictionaries to
solution of general ill-posed linear inverse problems. In the context of
regression problems, there has been enormous amount of effort to recover an
unknown function using such dictionaries. One of the most popular methods,
lasso and its versions, is based on minimizing empirical likelihood and
unfortunately, requires stringent assumptions on the dictionary, the, so
called, compatibility conditions. Though compatibility conditions are hard to
satisfy, it is well known that this can be accomplished by using random
dictionaries. In the present paper, we show how one can apply random
dictionaries to solution of ill-posed linear inverse problems. We put a
theoretical foundation under the suggested methodology and study its
performance via simulations.
