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 $y_i=a_i^{\top }X{X}^{\top }{a}_i,a_i\in {\mathbb{R}}^n,i=1,\dots ,m$. 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 $f(U)=\frac{1}{4m}{\sum }_{i=1}^m(y_i-a_i^{\top }U{U}^{\top }{a}_i{)}^2$. 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.

29 page

References in corpus (1)

• Phaseless Rcovery using Gauss-Newton Method