  We study the problem of symmetric matrix completion, where the goal is to
reconstruct a positive semidefinite matrix $\rm{X}^\star \in
\mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$,
from only a subset of its observed entries. We show that the vanilla gradient
descent (GD) with small initialization provably converges to the ground truth
$\rm{X}^\star$ without requiring any explicit regularization. This convergence
result holds true even in the over-parameterized scenario, where the true rank
$r$ is unknown and conservatively over-estimated by a search rank $r'\gg r$.
The existing results for this problem either require explicit regularization, a
sufficiently accurate initial point, or exact knowledge of the true rank $r$.
  In the over-parameterized regime where $r'\geq r$, we show that, with
$\widetilde\Omega(dr^9)$ observations, GD with an initial point $\|\rm{U}_0\|
\leq \epsilon$ converges near-linearly to an $\epsilon$-neighborhood of
$\rm{X}^\star$. Consequently, smaller initial points result in increasingly
accurate solutions. Surprisingly, neither the convergence rate nor the final
accuracy depends on the over-parameterized search rank $r'$, and they are only
governed by the true rank $r$. In the exactly-parameterized regime where
$r'=r$, we further enhance this result by proving that GD converges at a faster
rate to achieve an arbitrarily small accuracy $\epsilon>0$, provided the
initial point satisfies $\|\rm{U}_0\| = O(1/d)$. At the crux of our method lies
a novel weakly-coupled leave-one-out analysis, which allows us to establish the
global convergence of GD, extending beyond what was previously possible using
the classical leave-one-out analysis.
