  We consider training over-parameterized two-layer neural networks with
Rectified Linear Unit (ReLU) using gradient descent (GD) method. Inspired by a
recent line of work, we study the evolutions of network prediction errors
across GD iterations, which can be neatly described in a matrix form. When the
network is sufficiently over-parameterized, these matrices individually
approximate {\em an} integral operator which is determined by the feature
vector distribution $\rho$ only. Consequently, GD method can be viewed as {\em
approximately} applying the powers of this integral operator on the
underlying/target function $f^*$ that generates the responses/labels.
  We show that if $f^*$ admits a low-rank approximation with respect to the
eigenspaces of this integral operator, then the empirical risk decreases to
this low-rank approximation error at a linear rate which is determined by $f^*$
and $\rho$ only, i.e., the rate is independent of the sample size $n$.
Furthermore, if $f^*$ has zero low-rank approximation error, then, as long as
the width of the neural network is $\Omega(n\log n)$, the empirical risk
decreases to $\Theta(1/\sqrt{n})$. To the best of our knowledge, this is the
first result showing the sufficiency of nearly-linear network
over-parameterization. We provide an application of our general results to the
setting where $\rho$ is the uniform distribution on the spheres and $f^*$ is a
polynomial. Throughout this paper, we consider the scenario where the input
dimension $d$ is fixed.
