  Inspired by several recent developments in regularization theory,
optimization, and signal processing, we present and analyze a numerical
approach to multi-penalty regularization in spaces of sparsely represented
functions. The sparsity prior is motivated by the largely expected
geometrical/structured features of high-dimensional data, which may not be
well-represented in the framework of typically more isotropic Hilbert spaces.
In this paper, we are particularly interested in regularizers which are able to
correctly model and separate the multiple components of additively mixed
signals. This situation is rather common as pure signals may be corrupted by
additive noise. To this end, we consider a regularization functional composed
by a data-fidelity term, where signal and noise are additively mixed, a
non-smooth and non-convex sparsity promoting term, and a penalty term to model
the noise. We propose and analyze the convergence of an iterative alternating
algorithm based on simple iterative thresholding steps to perform the
minimization of the functional. By means of this algorithm, we explore the
effect of choosing different regularization parameters and penalization norms
in terms of the quality of recovering the pure signal and separating it from
additive noise. For a given fixed noise level numerical experiments confirm a
significant improvement in performance compared to standard one-parameter
regularization methods. By using high-dimensional data analysis methods such as
Principal Component Analysis, we are able to show the correct geometrical
clustering of regularized solutions around the expected solution. Eventually,
for the compressive sensing problems considered in our experiments we provide a
guideline for a choice of regularization norms and parameters.
