  Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of a
polynomial $\ell_1$-optimization algorithm in solving an under-determined
system of linear equations. In a large dimensional and statistical context
\cite{CRT,DonohoPol} proved that if the number of equations (measurements in
the compressed sensing terminology) in the system is proportional to the length
of the unknown vector then there is a sparsity (number of non-zero elements of
the unknown vector) also proportional to the length of the unknown vector such
that $\ell_1$-optimization succeeds in solving the system. In this paper, we
provide an alternative performance analysis of $\ell_1$-optimization and obtain
the proportionality constants that in certain cases match or improve on the
best currently known ones from \cite{DonohoPol,DT}.
