  We first propose a novel criterion that guarantees that an $s$-sparse signal
is the local minimizer of the $\ell_1/\ell_2$ objective; our criterion is
interpretable and useful in practice. We also give the first uniform recovery
condition using a geometric characterization of the null space of the
measurement matrix, and show that this condition is easily satisfied for a
class of random matrices. We also present analysis on the robustness of the
procedure when noise pollutes data. Numerical experiments are provided that
compare $\ell_1/\ell_2$ with some other popular non-convex methods in
compressed sensing. Finally, we propose a novel initialization approach to
accelerate the numerical optimization procedure. We call this initialization
approach \emph{support selection}, and we demonstrate that it empirically
improves the performance of existing $\ell_1/\ell_2$ algorithms.
