Solving Systems of Quadratic Equations via Exponential-Type Gradient Descent Algorithm (Journal of Computational Mathematics, 2020)
We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank $r$ matrix $X \in \mathbb{R}^{n×r}$ from $m$ scalar measurements $y_i=a_i^T XX^T a_i,\;a_i\in \mathbb{R}^n,\;i=1,…,m$. 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 $f(U)=\frac{1}{4m}\sum_{i=1}^m(y_i-a_i^T UU^T a_i)^2$. 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.
Keywords: Low-rank matrix recovery; Non-convex optimization; Phase retrieval.