  We consider alternating gradient descent (AGD) with fixed step size applied
to the asymmetric matrix factorization objective. We show that, for a rank-$r$
matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, $T = C
(\frac{\sigma_1(\mathbf{A})}{\sigma_r(\mathbf{A})})^2 \log(1/\epsilon)$
iterations of alternating gradient descent suffice to reach an
$\epsilon$-optimal factorization $\| \mathbf{A} - \mathbf{X} \mathbf{Y}^{T}
\|^2 \leq \epsilon \| \mathbf{A}\|^2$ with high probability starting from an
atypical random initialization. The factors have rank $d \geq r$ so that
$\mathbf{X}_{T}\in\mathbb{R}^{m \times d}$ and $\mathbf{Y}_{T} \in\mathbb{R}^{n
\times d}$, and mild overparameterization suffices for the constant $C$ in the
iteration complexity $T$ to be an absolute constant. Experiments suggest that
our proposed initialization is not merely of theoretical benefit, but rather
significantly improves the convergence rate of gradient descent in practice.
Our proof is conceptually simple: a uniform Polyak-\L{}ojasiewicz (PL)
inequality and uniform Lipschitz smoothness constant are guaranteed for a
sufficient number of iterations, starting from our random initialization. Our
proof method should be useful for extending and simplifying convergence
analyses for a broader class of nonconvex low-rank factorization problems.
