  Blind deconvolution is the problem of recovering a convolutional kernel
$\boldsymbol a_0$ and an activation signal $\boldsymbol x_0$ from their
convolution $\boldsymbol y = \boldsymbol a_0 \circledast \boldsymbol x_0$. This
problem is ill-posed without further constraints or priors. This paper studies
the situation where the nonzero entries in the activation signal are sparsely
and randomly populated. We normalize the convolution kernel to have unit
Frobenius norm and cast the sparse blind deconvolution problem as a nonconvex
optimization problem over the sphere. With this spherical constraint, every
spurious local minimum turns out to be close to some signed shift truncation of
the ground truth, under certain hypotheses. This benign property motivates an
effective two stage algorithm that recovers the ground truth from the partial
information offered by a suboptimal local minimum. This geometry-inspired
algorithm recovers the ground truth for certain microscopy problems, also
exhibits promising performance in the more challenging image deblurring
problem. Our insights into the global geometry and the two stage algorithm
extend to the convolutional dictionary learning problem, where a superposition
of multiple convolution signals is observed.
