  We study the asymmetric low-rank factorization problem: \[\min_{\mathbf{U}
\in \mathbb{R}^{m \times d}, \mathbf{V} \in \mathbb{R}^{n \times d}}
\frac{1}{2}\|\mathbf{U}\mathbf{V}^\top -\mathbf{\Sigma}\|_F^2\] where
$\mathbf{\Sigma}$ is a given matrix of size $m \times n$ and rank $d$. This is
a canonical problem that admits two difficulties in optimization: 1)
non-convexity and 2) non-smoothness (due to unbalancedness of $\mathbf{U}$ and
$\mathbf{V}$). This is also a prototype for more complex problems such as
asymmetric matrix sensing and matrix completion. Despite being non-convex and
non-smooth, it has been observed empirically that the randomly initialized
gradient descent algorithm can solve this problem in polynomial time. Existing
theories to explain this phenomenon all require artificial modifications of the
algorithm, such as adding noise in each iteration and adding a balancing
regularizer to balance the $\mathbf{U}$ and $\mathbf{V}$.
  This paper presents the first proof that shows randomly initialized gradient
descent converges to a global minimum of the asymmetric low-rank factorization
problem with a polynomial rate. For the proof, we develop 1) a new
symmetrization technique to capture the magnitudes of the symmetry and
asymmetry, and 2) a quantitative perturbation analysis to approximate matrix
derivatives. We believe both are useful for other related non-convex problems.
