  In many applications one may acquire a composition of several signals that
may be corrupted by noise, and it is a challenging problem to reliably separate
the components from one another without sacrificing significant details. Adding
to the challenge, in a compressive sensing framework, one is given only an
undersampled set of linear projections of the composite signal. In this paper,
we propose using the Dantzig selector model incorporating an overcomplete
dictionary to separate a noisy undersampled collection of composite signals,
and present an algorithm to efficiently solve the model.
  The Dantzig selector is a statistical approach to finding a solution to a
noisy linear regression problem by minimizing the $\ell_1$ norm of candidate
coefficient vectors while constraining the scope of the residuals. If the
underlying coefficient vector is sparse, then the Dantzig selector performs
well in the recovery and separation of the unknown composite signal. In the
following, we propose a proximity operator based algorithm to recover and
separate unknown noisy undersampled composite signals through the Dantzig
selector. We present numerical simulations comparing the proposed algorithm
with the competing Alternating Direction Method, and the proposed algorithm is
found to be faster, while producing similar quality results. Additionally, we
demonstrate the utility of the proposed algorithm in several experiments by
applying it in various domain applications including the recovery of
complex-valued coefficient vectors, the removal of impulse noise from smooth
signals, and the separation and classification of a composition of handwritten
digits.
