  A recent trend in compressed sensing is to consider non-convex optimization
techniques for sparse recovery. The important case of $F$-minimization has
become of particular interest, for which the exact reconstruction condition
(ERC) in the noiseless setting can be precisely characterized by the null space
property (NSP). However, little work has been done concerning its robust
reconstruction condition (RRC) in the noisy setting. We look at the null space
of the measurement matrix as a point on the Grassmann manifold, and then study
the relation between the ERC and RRC sets, denoted as $\Omega_J$ and
$\Omega_J^r$, respectively. It is shown that $\Omega_J^r$ is the interior of
$\Omega_J$, from which a previous result of the equivalence of ERC and RRC for
$\ell_p$-minimization follows easily as a special case. Moreover, when $F$ is
non-decreasing, it is shown that
$\overline{\Omega}_J\setminus\interior(\Omega_J)$ is a set of measure zero and
of the first category. As a consequence, the probabilities of ERC and RRC are
the same if the measurement matrix $\mathbf{A}$ is randomly generated according
to a continuous distribution. Quantitatively, if the null space
$\mathcal{N}(\bf A)$ lies in the "$d$-interior" of $\Omega_J$, then RRC will be
satisfied with the robustness constant
$C=\frac{2+2d}{d\sigma_{\min}(\mathbf{A}^{\top})}$; and conversely if RRC holds
with $C=\frac{2-2d}{d\sigma_{\max}(\mathbf{A}^{\top})}$, then $\mathcal{N}(\bf
A)$ must lie in $d$-interior of $\Omega_J$. We also present several rules for
comparing the performances of different cost functions. Finally, these results
are capitalized to derive achievable tradeoffs between the measurement rate and
robustness with the aid of Gordon's escape through the mesh theorem or a
connection between NSP and the restricted eigenvalue condition.
