  The stability of sparse signal reconstruction is investigated in this paper.
We design efficient algorithms to verify the sufficient condition for unique
$\ell_1$ sparse recovery. One of our algorithm produces comparable results with
the state-of-the-art technique and performs orders of magnitude faster. We show
that the $\ell_1$-constrained minimal singular value ($\ell_1$-CMSV) of the
measurement matrix determines, in a very concise manner, the recovery
performance of $\ell_1$-based algorithms such as the Basis Pursuit, the Dantzig
selector, and the LASSO estimator. Compared with performance analysis involving
the Restricted Isometry Constant, the arguments in this paper are much less
complicated and provide more intuition on the stability of sparse signal
recovery. We show also that, with high probability, the subgaussian ensemble
generates measurement matrices with $\ell_1$-CMSVs bounded away from zero, as
long as the number of measurements is relatively large. To compute the
$\ell_1$-CMSV and its lower bound, we design two algorithms based on the
interior point algorithm and the semi-definite relaxation.
