Study On Matrix Approximation Problems With KL Entropy

Matrix approximation is one of the important research topic in the field of modern scientific research in recent years, especially in areas such as signal analysis, image pro-cessing and machine learning in all has the extremely widespread application. We discuss KL entropy matrix approximation problem from three aspects. The paper clarifies the theoretical foundation of the problems, and proposes effective methods for solving this problem.Firstly, the paper briefly reviews a class of generalized matrix distance metric, called Bregman matrix divergence. Bregman divergence is used to study matrix approxima-tion problems. Also, We review the definition and some properties of matrix Bregman divergence, and establish the matrix approximation model based on Bregman matrix divergences, and derive the analytic expression of Bregman projection operator.Secondly, we solves the matrix approximation problem based on KL entropy by dual method, which establishes the dual problem of the original problem. In addition, we present Newton-CG algorithm for solving the dual problem. Numerical experiments show that the algorithm is to get good results, and we compare the result with the classic Frobenius norm matrix approximation, where illustrate the significance of the research of KL entropy matrix approximation.Lastly, we discuss low-rank matrix approximation with KL entrpy, as well as elabo-rate relevant theory of the problem. Due to some characteristics of KL entropy, we convert the original problem to a convex optimization problem without rank constraint by lim-iting the range space of the given initial matrix. Also, we solve it by dual method, and establish the dual problem of the convex optimization problem. We introduce Newton-CG algorithm to solve the dual problem. Numerical results show that the given algorithm is effective for solving low rank matrix approximation problem with KL entropy, and there is some sinificance to study low-rank and full rank matrix nearness problem.