  We study the $\textit{Short-and-Sparse (SaS) deconvolution}$ problem of
recovering a short signal $\mathbf a_0$ and a sparse signal $\mathbf x_0$ from
their convolution. We propose a method based on nonconvex optimization, which
under certain conditions recovers the target short and sparse signals, up to a
signed shift symmetry which is intrinsic to this model. This symmetry plays a
central role in shaping the optimization landscape for deconvolution. We give a
$\textit{regional analysis}$, which characterizes this landscape geometrically,
on a union of subspaces. Our geometric characterization holds when the
length-$p_0$ short signal $\mathbf a_0$ has shift coherence $\mu$, and $\mathbf
x_0$ follows a random sparsity model with sparsity rate $\theta \in
\Bigl[\frac{c_1}{p_0}, \frac{c_2}{p_0\sqrt\mu +
\sqrt{p_0}}\Bigr]\cdot\frac{1}{\log^2p_0}$. Based on this geometry, we give a
provable method that successfully solves SaS deconvolution with high
probability.
