MahNMF: Manhattan Non-negative Matrix Factorization

Non-negative matrix factorization (NMF) approximates a non-negative matrix X by a product of two non-negative low-rank factor matrices W and H . NMF and its extensions minimize either the Kullback-Leibler divergence or the Euclidean distance between X and WH to model the Poisson noise or the Gaussian noise. In practice, when the noise distribution is heavy tailed, they cannot perform well. This paper presents Manhattan NMF (MahNMF) which minimizes the Manhattan distance between X and WH for modeling the heavy tailed Laplacian noise. Similar to sparse and low-rank matrix decompositions, e.g. robust principal component analysis (RPCA) and GoDec, MahNMF robustly estimates the low-rank part and the sparse part of a non-negative matrix and thus performs effectively when data are contaminated by outliers. We extend MahNMF for various practical applications by developing box-constrained MahNMF, manifold regularized MahNMF, group sparse MahNMF, elastic net inducing MahNMF, and symmetric MahNMF. The major contribution of this paper lies in two fast optimization algorithms for MahNMF and its extensions: the rank-one residual iteration (RRI) method and Nesterov’s smoothing method. In particular, by approximating the residual matrix by the outer product of one row of W and one row ofH in MahNMF, we develop an RRI method to iteratively update each variable of W and H in a closed form solution. Although RRI is efficient for small scale MahNMF and some of its extensions, it is neither scalable to large scale matrices nor flexible enough to optimize all MahNMF extensions. Since the objective functions of MahNMF and its extensions are neither convex nor smooth, we apply Nesterov’s smoothing c ©2012 N. Guan, D. Tao, Z. Luo, and J. Shawe-Taylor. N. Guan, D. Tao, Z. Luo, and J. Shawe-Taylor method to recursively optimize one factor matrix with another matrix fixed. By setting the smoothing parameter inversely proportional to the iteration number, we improve the approximation accuracy iteratively for both MahNMF and its extensions. We conduct experiments on both synthetic and real-world datasets, such as face images, natural scene images, surveillance videos and multi-model datasets, to show the efficiency of the proposed Nesterov’s smoothing method-based algorithm for solving MahNMF and its variants, and the effectiveness of MahNMF and its variants, by comparing them with traditional NMF, RPCA, and GoDec.