Nonnegative matrix factorization with bounded total variational regularization for face recognition

Nonnegative matrix factorization (NMF) is a recently developed technique for finding parts-based, linear representations of nonnegative data based on minimizing least square error (L2 norm). However it has been observed that the proper norm for images is the bounded total variation (TV) norm other than the L2 norm. The space of functions of bounded TV allows discontinuous solution and plays an important role in image processing. In this paper, we propose a new NMF model with bounded TV regularization for identifying discriminate representation of image patterns. We provide a simple update rule for computing the factorization and give supporting theoretical analysis. Finally, we perform a series of numerical experiments to show evidence of the good behavior of the numerical scheme.

Research highlights
► Nonnegative matrix factorization (NMF) is a technique for finding parts-based, linear representations of nonnegative data based on minimizing L2 norm. However, the L2 norm tends to oversmooth the solution. The key advantage of total variation (TV) norm is that it permits discontinuities in the computed solution. Thus, the TV regularization approach can preserve finer scale image features, such as edges and texture than L2 regularization approach. Therefore, in this paper, we propose a new algorithm of NMF with bounded TV regularization instead of original L2 regularization to identify discriminant image patterns.
► We have provided convergence guarantee for the Algorithm 2.
► We present numerical experiments illustrating the performance of our algorithm on human faces and show evidence of the good behavior of the numerical scheme.