Solving Systems of Quadratic Equations via Exponential-type Gradient Descent Algorithm

Abstract: 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 $$y_i=a_i^{\top} XX^{\top} a_i,\;a_i\in \mathbb{R}<sup>n,\;i=1,\ldots,m$$. 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 $$f(U)=\frac{1}{4m}\sum_{i=1}^{m(y_i-a_i^{\top}} UU^{\top} a_i)<sup>2$$. 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.