A Compressed Sensing Approach to 3D Weak Lensing
  (Abridged) Weak gravitational lensing is an ideal probe of the dark universe.
In recent years, several linear methods have been developed to reconstruct the
density distribution in the Universe in three dimensions, making use of
photometric redshift information to determine the radial distribution of lensed
sources. In this paper, we aim to address three key issues seen in these
methods; namely, the bias in the redshifts of detected objects, the line of
sight smearing seen in reconstructions, and the damping of the amplitude of the
reconstruction relative to the underlying density. We consider the problem
under the framework of compressed sensing (CS). Under the assumption that the
data are sparse in an appropriate dictionary, we construct a robust estimator
and employ state-of-the-art convex optimisation methods to reconstruct the
density contrast. For simplicity in implementation, and as a proof of concept
of our method, we reduce the problem to one-dimension, considering the
reconstruction along each line of sight independently. Despite the loss of
information this implies, we demonstrate that our method is able to accurately
reproduce cluster haloes up to a redshift of z=1, deeper than state-of-the-art
linear methods. We directly compare our method with these linear methods, and
demonstrate minimal radial smearing and redshift bias in our reconstructions,
as well as a reduced damping of the reconstruction amplitude as compared to the
linear methods. In addition, the CS framework allows us to consider an
underdetermined inverse problem, thereby allowing us to reconstruct the density
contrast at finer resolution than the input data.
