MahNMF: Manhattan non-negative matrix factorization

Non-negative matrix factorization (NMF) approximates a non-negative matrix by a product of two non-negative low-rank factor matrices and . NMF and its extensions minimize either the Kullback-Leibler divergence or the Euclidean distance between and 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 and for modeling the heavy tailed Laplacian noise. Similar to sparse and low-rank matrix decompositions, 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 of in MahNMF, we develop an RRI method to iteratively update each variable of and 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 …