  We consider the rank minimization problem from quadratic measurements, i.e.,
recovering a rank $r$ matrix $X \in \mathbb{R}^{n \times r}$ from $m$ scalar
measurements $y_i=a_i^{\top} XX^{\top} a_i,\;a_i\in
\mathbb{R}^n,\;i=1,\ldots,m$. Such problem arises in a variety of applications
such as quadratic regression and quantum state tomography. We present a novel
algorithm, which is termed exponential-type gradient descent algorithm, to
minimize a non-convex objective function
$f(U)=\frac{1}{4m}\sum_{i=1}^m(y_i-a_i^{\top} UU^{\top} a_i)^2$. This algorithm
starts with a careful initialization, and then refines this initial guess by
iteratively applying exponential-type gradient descent. Particularly, we can
obtain a good initial guess of $X$ as long as the number of Gaussian random
measurements is $O(nr)$, and our iteration algorithm can converge linearly to
the true $X$ (up to an orthogonal matrix) with $m=O\left(nr\log (cr)\right)$
Gaussian random measurements.
