A Kind Of Geometric Structure Preserving NMF Algorithm

In recent years，matrix factorizations have been playing a more and moreimportant role in scientific and engineering computing. Typical matrix factorizationmethod, such as Singular Value Decomposition (SVD), can be understood asfactorizing an original data matrix according to some constraints. Non-negativeMatrix Factorization (NMF) is a new class of matrix factorization technique.Originally used to decompose facial images into collections of facial features, NMFhas been applied to diverse applications such as face recognition, document clustering,and so on. Its advantage is derived from its constraints on the data. NMF factorize anoriginal matrix comprised of only non-negative data into two matrices also comprisedof only non-negative data, when factorizing, NMF maintain the non-negativity of allcomponents in the matrices. So the results we obtained from NMF are easier toexplain, and it makes NMF appropriate for the processing of image data which everyelement is non-negative.Though NMF has proven to be an effective factorization technique, there stillexist some disadvantages. In order to deal with these disadvantages, more and moreimproved NMF have been proposed. By imposing different constraints and differentpunishment to object function, the algorithm can be used in different problems.Unfortunately, the original NMF and most improved NMF fail to consider thegeometric structure of the data. In this paper, two improved non-negative matrixfactorization methods are proposed, which explicitly consider the geometricinformation of the data set, including relationship between neighborhood,not-neighborhood and classes. First of all, base on manifold assumption, if two datapoints are close in the original geometry of data distribution, we ensure that therepresentations of these two points in the new basis are also close to each other.Further more, if two data points are far away from each other in the original geometryof data distribution, we make the representations of this two points in the new basisalso far away from each other, then we get a new NMF method; In another way,maximizing the difference between the obtained classes in the new basis, we getanother new NMF method. So the algorithm can maintain the geometrical relationshipand distant measurement much better, and get better performance when dealing withdata on the manifold. In this paper, we perform our algorithm on several famousimage databases such as COIL20, ALOI, and compare them with old algorithm,experiment results show that new algorithm can achieve better performance.