A Variational Geometric Approach to Exact Recovery by $\ell_1/\ell_2$ Minimization

For affine-constrained $\ell_1/\ell_2$ minimization with nonzero data, a variational geometric approach is developed to characterize exact recovery. Necessary and sufficient conditions for local and global exact recovery are established in terms of directional derivatives and cones associated with sublevel sets of the ratio objective. The local analysis further reveals that local minimality, first-order growth, and tilt stability are equivalent. These pointwise characterizations lead to improved sufficient conditions for uniform recovery of sparse signals. The sufficient condition for uniform local recovery has coefficient $c_s=\max\{1,(\sqrt{s}-1)/2\}$ for $s\ge2$. For each fixed $s$, the coefficient $c_s$ is optimal within the considered family of local null-space conditions. Projection estimates yield a scalar bound for uniform global recovery with coefficient two, improving the earlier coefficient three. The coefficient two is optimal within the considered family of global null-space bounds when both $s$ and $n$ may vary. A positive margin in the local null-space condition provides explicit growth moduli, localization radii, and bounds on linear objective perturbations that leave the localized minimizer fixed. Examples show that uniform local recovery does not imply uniform global recovery.