  In this paper we present two new approaches to efficiently solve large-scale
compressed sensing problems. These two ideas are independent of each other and
can therefore be used either separately or together. We consider all
possibilities.
  For the first approach, we note that the zero vector can be taken as the
initial basic (infeasible) solution for the linear programming problem and
therefore, if the true signal is very sparse, some variants of the simplex
method can be expected to take only a small number of pivots to arrive at a
solution. We implemented one such variant and demonstrate a dramatic
improvement in computation time on very sparse signals.
  The second approach requires a redesigned sensing mechanism in which the
vector signal is stacked into a matrix. This allows us to exploit the Kronecker
compressed sensing (KCS) mechanism. We show that the Kronecker sensing requires
stronger conditions for perfect recovery compared to the original vector
problem. However, the Kronecker sensing, modeled correctly, is a much sparser
linear optimization problem. Hence, algorithms that benefit from sparse problem
representation, such as interior-point methods, can solve the Kronecker sensing
problems much faster than the corresponding vector problem. In our numerical
studies, we demonstrate a ten-fold improvement in the computation time.
