  As one of the most plausible convex optimization methods for sparse data
reconstruction, $\ell_1$-minimization plays a fundamental role in the
development of sparse optimization theory. The stability of this method has
been addressed in the literature under various assumptions such as restricted
isometry property (RIP), null space property (NSP), and mutual coherence. In
this paper, we propose a unified means to develop the so-called weak stability
theory for $\ell_1$-minimization methods under the condition called weak range
space property of a transposed design matrix, which turns out to be a necessary
and sufficient condition for the standard $\ell_1$-minimization method to be
weakly stable in sparse data reconstruction. The reconstruction error bounds
established in this paper are measured by the so-called Robinson's constant.
  We also provide a unified weak stability result for standard
$\ell_1$-minimization under several existing compressed-sensing matrix
properties. In particular, the weak stability of $\ell_1$-minimization under
the constant-free range space property of order $k$ of the transposed design
matrix is established for the first time in this paper. Different from the
existing analysis, we utilize the classic Hoffman's Lemma concerning the error
bound of linear systems as well as the Dudley's theorem concerning the polytope
approximation of the unit $\ell_2$-ball to show that $\ell_1$-minimization is
robustly and weakly stable in recovering sparse data from inaccurate
measurements.
