(untitled) (Journal of Computational Mathematics, 2020)
We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank r matrix X R^n×r from m scalar measurements y_i=a_i^T XX^T a_i, a_iR^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)=14m_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(nr(cr)) Gaussian random measurements.