  This paper considers the global geometry of general low-rank minimization
problems via the Burer-Monterio factorization approach. For the rank-$1$ case,
we prove that there is no spurious second-order critical point for both
symmetric and asymmetric problems if the rank-$2$ RIP constant $\delta$ is less
than $1/2$. Combining with a counterexample with $\delta=1/2$, we show that the
derived bound is the sharpest possible. For the arbitrary rank-$r$ case, the
same property is established when the rank-$2r$ RIP constant $\delta$ is at
most $1/3$. We design a counterexample to show that the non-existence of
spurious second-order critical points may not hold if $\delta$ is at least
$1/2$. In addition, for any problem with $\delta$ between $1/3$ and $1/2$, we
prove that all second-order critical points have a positive correlation to the
ground truth. Finally, the strict saddle property, which can lead to the
polynomial-time global convergence of various algorithms, is established for
both the symmetric and asymmetric problems when the rank-$2r$ RIP constant
$\delta$ is less than $1/3$. The results of this paper significantly extend
several existing bounds in the literature.
