Alternating Minimization Methods for Solving Multilinear Systems

Recent works on the multilinear system A x m − 1 = b with an order- m and dimension- n tensor A and a vector b of dimension- n have been motivated by their applications in data mining, numerical PDEs, tensor complementary problems, and so on. In this paper, we propose an alternating minimization method for the solution of the system mentioned above and present several randomized versions of this algorithm in order to improve its performance. The provided numerical experiments show that our methods are feasible for any tensor A and outperform some existing ones in the same case.

A hybrid regularizers model for multiplicative noise removal

In this paper, we propose a variational method for restoring images corrupted by multiplicative noise. Computationally, we employ the alternating minimization method to solve our minimization problem. We also study the existence and uniqueness of the proposed problem. Finally, experimental results are provided to demonstrate the superiority of our proposed hybrid model and algorithm for image denoising in comparison with state-of-the-art methods.

On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces

In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved for (i) and (ii), whereas sublinear convergence is showed for (iii).