Solution of linear ill-posed problems using random dictionaries
This addresses a specific issue in regression and inverse problems for researchers, but appears incremental as it adapts known random dictionary techniques to a new context.
The paper tackles the problem of solving linear ill-posed inverse problems by using random dictionaries to avoid stringent compatibility conditions required by methods like lasso, and demonstrates the methodology's performance through simulations with theoretical support.
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.