  We propose novel necessary and sufficient conditions for a sensing matrix to
be "$s$-good" - to allow for exact $\ell_1$-recovery of sparse signals with $s$
nonzero entries when no measurement noise is present. Then we express the error
bounds for imperfect $\ell_1$-recovery (nonzero measurement noise, nearly
$s$-sparse signal, near-optimal solution of the optimization problem yielding
the $\ell_1$-recovery) in terms of the characteristics underlying these
conditions. Further, we demonstrate (and this is the principal result of the
paper) that these characteristics, although difficult to evaluate, lead to
verifiable sufficient conditions for exact sparse $\ell_1$-recovery and to
efficiently computable upper bounds on those $s$ for which a given sensing
matrix is $s$-good. We establish also instructive links between our approach
and the basic concepts of the Compressed Sensing theory, like Restricted
Isometry or Restricted Eigenvalue properties.
