  We consider an important class of signal processing problems where the signal
of interest is known to be sparse, and can be recovered from data given
auxiliary information about how the data was generated. For example, a sparse
Green's function may be recovered from seismic experimental data using sparsity
optimization when the source signature is known. Unfortunately, in practice
this information is often missing, and must be recovered from data along with
the signal using deconvolution techniques.
  In this paper, we present a novel methodology to simultaneously solve for the
sparse signal and auxiliary parameters using a recently proposed variable
projection technique. Our main contribution is to combine variable projection
with sparsity promoting optimization, obtaining an efficient algorithm for
large-scale sparse deconvolution problems. We demonstrate the algorithm on a
seismic imaging example.
