Research On Several Non‐negtive Matrix Factorization Methods And Applications

Generally speaking, traditional matrix factorization can realize the dimension reduction of matrix in some cases, reducing the large dimension issue to several small issues. However, non-negative matrix factorization has something that is better than traditional matrix factorization, i.e., during the factorization the matrix reserves the nonnegative property. Just because of the nonnegative property, the nonnegative matrix factorization accords with our thinking in some applications. The rapid factorization method on the nonnegative matrix factorization is the prerequisite of its wide applications. Therefore, since Lee and Seung presented the definition of the nonnegative matrix factorization, many scholars have thrown themselves to the research of rapid factorization methods. Owing to their hardworking, there appear many of fast factorization methods such as Gradient descent method, Newton method, Quasi Newton method, active-set method etc. On the other hand, there exist fast methods in the least squares recently. For example, F. Oztopark presented the rank-one method to update the Hessian matrix in the quasi Newton method to solve the optimization problem.Our research is mainly bases on the former researchers’ results, but also it is the extension of their results. We improve the typical nonnegative matrix factorization. Combining the alternative iteration and rank-one technology of least square, we developed a symmetric rank-one quasi Newton method for nonnegative matrix factorization. And our numerical experiments indicate that our method has better performance in some cases and also remain the nonnegative property well. In this paper we mainly research something as follows:First, as we all known, there are several weight factions on nonnegative matrix functions. For each function, there are several factorization methods; and every method also has its own advantages and disadvantages. We studied several typical methods carefully, and fund their pros and cons.Second, for the Froubenus function, we improve the typical nonnegative matrix factorization. Combining the alternative iteration and rank-one technology of least square, we developed a symmetric rank-one quasi Newton method for nonnegative matrix factorization.Furthermore, the sparsity in the traditional non-negative matrix factorization algorithm is not controllable. In some applications, for example, a doctor’s judgment is also in need of such sparsity decomposition caused by diseases. Thus, in some areas, we want to make the performance of sparse matrix decomposition controlled according to need. Therefore, this paper finally studied the sparse non-negative matrix factorization problem, and summarized some relevant knowledge on sparse non-negative matrix factorization, as well as the future direction of development.