  Compressed sensing is a signal processing technique whereby the limits
imposed by the Shannon--Nyquist theorem can be exceeded provided certain
conditions are imposed on the signal. Such conditions occur in many real-world
scenarios, and compressed sensing has emerging applications in medical imaging,
big data, and statistics. Finding practical matrix constructions and
computationally efficient recovery algorithms for compressed sensing is an area
of intense research interest. Many probabilistic matrix constructions have been
proposed, and it is now well known that matrices with entries drawn from a
suitable probability distribution are essentially optimal for compressed
sensing.
  Potential applications have motivated the search for constructions of sparse
compressed sensing matrices (i.e., matrices containing few non-zero entries).
Various constructions have been proposed, and simulations suggest that their
performance is comparable to that of dense matrices. In this paper, extensive
simulations are presented which suggest that sparsification leads to a marked
improvement in compressed sensing performance for a large class of matrix
constructions and for many different recovery algorithms.
