  The problem of finding the sparsest vector (direction) in a low dimensional
subspace can be considered as a homogeneous variant of the sparse recovery
problem, which finds applications in robust subspace recovery, dictionary
learning, sparse blind deconvolution, and many other problems in signal
processing and machine learning. However, in contrast to the classical sparse
recovery problem, the most natural formulation for finding the sparsest vector
in a subspace is usually nonconvex. In this paper, we overview recent advances
on global nonconvex optimization theory for solving this problem, ranging from
geometric analysis of its optimization landscapes, to efficient optimization
algorithms for solving the associated nonconvex optimization problem, to
applications in machine intelligence, representation learning, and imaging
sciences. Finally, we conclude this review by pointing out several interesting
open problems for future research.
