  We show that the gradient descent algorithm provides an implicit
regularization effect in the learning of over-parameterized matrix
factorization models and one-hidden-layer neural networks with quadratic
activations. Concretely, we show that given $\tilde{O}(dr^{2})$ random linear
measurements of a rank $r$ positive semidefinite matrix $X^{\star}$, we can
recover $X^{\star}$ by parameterizing it by $UU^\top$ with $U\in \mathbb
R^{d\times d}$ and minimizing the squared loss, even if $r \ll d$. We prove
that starting from a small initialization, gradient descent recovers
$X^{\star}$ in $\tilde{O}(\sqrt{r})$ iterations approximately. The results
solve the conjecture of Gunasekar et al.'17 under the restricted isometry
property. The technique can be applied to analyzing neural networks with
one-hidden-layer quadratic activations with some technical modifications.
