Step Size Determination for Finding Low-Rank Solutions Via Non-Convex Bi-Factored Matrix Factorization

In this paper we present an exact line search approach in order to find a suitable step size for the problem of recovering a low-rank matrix from linear measurements via non-convex bi-factored matrix factorization approaches as used in the bi-factored gradient descent (BFGD) algorithm. For the specific case of using the squared Frobenius norm as convex regularizer we prove that unique solutions for the step sizes exist. The computational complexity of the proposed method has the same order of magnitude than common inexact line search approaches, however, it needs only one execution of the sensing operator whereas inexact line search methods need at least two. As such our method requires less memory space and CPU time. We illustrate the functionality of the proposed method by use of simulations.