Article Preview
Top1. Introduction
In every single second in this modern era, petabytes of data are being generated in various applications. Let's consider the number of online people writing their blogs, sharing their experience through various digital supports, videos, photos and other data. The most of these raw data, especially Big Data, does not depict much value in its unprocessed state (Azar AT, 2014). Data only become useful when have been processed for further utilization or getting some experience from it. Obviously, by applying the suitable set of algorithms, we can get insight information from this huge data. In this fast growing amount of data, there are several approaches for data processing; it includes designing of powerful algorithm to tackle this data, designing computing frameworks like distributed computing, techniques for Machine learning. NMF has been devised before three decades, with previous variants referred as nonnegative rank factorization and positive matrix factorization, but it becomes popular for (Daniel D. Lee and H. Sebastian Seung, 1999); (Daniel D. Lee and H. Sebastian Seung, 2001). Since then it is widely used in various fields of research with vast domain of applications.
A low-rank approximation (Daniel D. Lee and H. Sebastian Seung, 1999); (Daniel D. Lee and H. Sebastian Seung, 2001) named nonnegative matrix factorization (NMF) algorithm has been regularly applied in a wide range of applications (Berry, 2006); (Berry M. W., 2005); (Kaarna, 2006); (Miao, 2007); (Pauca, 2006); (Wang, 2005) which include text mining, e-commerce, Bioinformatics and etc. In definition, Nonnegative matrix (NMF) decomposes a matrix A into two nonnegative low rank matrices W and H, such that A = WH (Ding, 2010); (Daniel D. Lee and H. Sebastian Seung, 1999); (Daniel D. Lee and H. Sebastian Seung, 2001); (Liu, 2012) ; (Azar AT, 2014); (Hassanien AE, 2015).
It is common to consider matrices of low rank, imply usually that the factorization is approximate. Initially this method was generally formulated as follows:
Given a non-negative matrix AϵRmn and a positive integer k < min(m; n); and non-negative matrices Hϵ Rnk and WϵRkm which minimize the functional-
(1)Minimizing equation (1) is difficult for various reasons, including the existence of local minima as a result of the non-convexity of f(H;W) in both H and W, and more importantly the non-uniqueness of the solution. The factorization is commonly done by iterative methods. The algorithms differ mostly in the specific cost reducing iteration, and in the use of additional information. The reason behind imposing constraints of non-negativity on factored matrix are, motivated by the biological fact that the firing rates in visual perception neurons are nonnegative.
The traditional eigen problems are equivalent to a least squares fit into a matrix. In application to physical process standard PCA fails to reconstruct physically meaningful component because of incorrect weighting and allowing negative component entries. To mitigate this limitation Paatero and Tapper proposed (Paatero, 1997) another method, least squares based approaches. For example (Hoyer, 2004) proposed a algorithm for modeling the receptive fields using nonnegative with the sparse coding very similar to NMF approach. In addition, the non-negativity a constraint arises in may real image processing applications. For Instance, the pixels in a grayscaling image have nonnegative intensities. If we are considering the image applications, a set of m images, k are lexicographically scanned and stored in the columns of the matrix X. Then, as we have mentioned, NMF decomposes X into basis image matrix W and corresponding coefficient matrix H (Hoyer, 2004).