  Low-rank matrix estimation plays a central role in various applications
across science and engineering. Recently, nonconvex formulations based on
matrix factorization are provably solved by simple gradient descent algorithms
with strong computational and statistical guarantees. However, when the
low-rank matrices are asymmetric, existing approaches rely on adding a
regularization term to balance the scale of the two matrix factors which in
practice can be removed safely without hurting the performance when initialized
via the spectral method. In this paper, we provide a theoretical justification
to this for the matrix sensing problem, which aims to recover a low-rank matrix
from a small number of linear measurements. As long as the measurement ensemble
satisfies the restricted isometry property, gradient descent -- in conjunction
with spectral initialization -- converges linearly without the need of
explicitly promoting balancedness of the factors; in fact, the factors stay
balanced automatically throughout the execution of the algorithm. Our analysis
is based on analyzing the evolution of a new distance metric that directly
accounts for the ambiguity due to invertible transforms, and might be of
independent interest.
