Nonnegative Matrix Factorization And The Application For High-dimensional Data

With the development of Information and the Internet, high-dimensional data are constantly emerging in various application areas, e.g. Trade data, Web data, Pathological data, Multimedia data, logistics data and etc. Generally speaking, such data are semi-structured or unstructured, the number of building feature may be millions or more, which results in a hard problem for data mining, the curse of dimensionality. How to mine the valuable information becomes an important and difficult problem. A core task of high-dimensional data mining is to find a low-dimensional structure with a clear potential way to represent high-dimensional data. According to the mechanism of human perception, the whole is consisted of its parts, nonnegative matrix factorization algorithm gets widely attention because of its good interpretability.However, the drawbacks of nonnegative matrix factorization are obvious including the following but not limited problems. It is hard to prove the existence and uniqueness of solutions, determine the dimension of low-dimensional space, satisfy the requirement of non-linear data analysis, and how to introduce the local consistence among samples and so on.In this paper, we mainly focus on unsupervised nonnegative matrix factorization algorithm and semi-supervised nonnegative matrix factorization algorithm. We have the following two main achievements:the first one is Multi-Kernel projective nonnegative matrix factorization (MKPNMF) algorithm, this algorithm omits the choice of kernel function and enhances the performance of dimension reduction and time performance. Second, we put forward adaptive graph constrained nonnegative matrix factorization for semi-supervised learning (GCNMF) algorithm, which does not only use class-label information, but also use the local consistency among samples reasonably. In the real image, face set, handwritten digits set, the results have shown that compared with the existing unsupervised or semi-supervised nonnegative matrix factorization algorithms, our methods have better performance.At the end of this paper, we summarize the current work, and look forward the future research direction.