  Inspired by several real-life applications in audio processing and medical
image analysis, where the quantity of interest is generated by several sources
to be accurately modeled and separated, as well as by recent advances in
regularization theory and optimization, we study the conditions on optimal
support recovery in inverse problems of unmixing type by means of multi-penalty
regularization.
  We consider and analyze a regularization functional composed of a
data-fidelity term, where signal and noise are additively mixed, a non-smooth,
convex, sparsity promoting term, and a quadratic penalty term to model the
noise. We prove not only that the well-established theory for sparse recovery
in the single parameter case can be translated to the multi-penalty settings,
but we also demonstrate the enhanced properties of multi-penalty regularization
in terms of support identification compared to sole $\ell^1$-minimization. We
additionally confirm and support the theoretical results by extensive numerical
simulations, which give a statistics of robustness of the multi-penalty
regularization scheme with respect to the single-parameter counterpart.
Eventually, we confirm a significant improvement in performance compared to
standard $\ell^1$-regularization for compressive sensing problems considered in
our experiments.
