  This paper is concerned with low-rank matrix optimization, which has found a
wide range of applications in machine learning. This problem in the special
case of matrix sensing has been studied extensively through the notion of
Restricted Isometry Property (RIP), leading to a wealth of results on the
geometric landscape of the problem and the convergence rate of common
algorithms. However, the existing results can handle the problem in the case
with a general objective function subject to noisy data only when the RIP
constant is close to 0. In this paper, we develop a new mathematical framework
to solve the above-mentioned problem with a far less restrictive RIP constant.
We prove that as long as the RIP constant of the noiseless objective is less
than $1/3$, any spurious local solution of the noisy optimization problem must
be close to the ground truth solution. By working through the strict saddle
property, we also show that an approximate solution can be found in polynomial
time. We characterize the geometry of the spurious local minima of the problem
in a local region around the ground truth in the case when the RIP constant is
greater than $1/3$. Compared to the existing results in the literature, this
paper offers the strongest RIP bound and provides a complete theoretical
analysis on the global and local optimization landscapes of general low-rank
optimization problems under random corruptions from any finite-variance family.
