Face Reconstruction Using Non-Negative Matrix Factorization and l1 Constrained Optimization

Non-negative matrix factorization (NMF) has received much attention recently, as many of the practical problems inhere the property of being non-negative naturally. NMF has applications in the field of parts based learning of images, the analysis of hyper-spectral images, text and data mining, artificial neural networks, dimension reduction and many others. Due to the lack of information about the factors with the exception of the property of non-negativity, it is hard to find a global solution for an NMF problem. The problem itself has non-convexity, and finding the solution alternately may introduce convexity in the sub-problems. To find a feasible solution for a particular problem, some specific constraints have been introduced in the previously-developed methods. Approaches include multiplicative updates of the factor matrices, non-negative alternative least squares, gradient based solving methods and geometric approaches to the problem. In this paper a new alternating optimization method is proposed. Following from the revisited and new interpretations of the geometric properties of NMF, an algorithm is proposed which alternately minimize the Frobenius norm of the basis matrix while maximize the Frobenius norm of the co-efficient matrix in a unitized l(1)-space. The proposed alternating solving converges faster than the methods currently available and provides better performance at the same time for facial recognition. The claim has been supported by related example.