Nonnegative Low-rank Sparse Component Analysis

Abstract

In this paper we consider a variant of the dictionary learning problem where the dictionary has full rank, the coefficients have a fixed sparsity level, and both the coefficients and the dictionary are nonnegative. It is equivalent to k-sparse nonnegative matrix factorization (K-NMF). This model is encountered in source separation where nonnegative linear combinations of a few components generate the data points (samples), such as in hyperspectral images. We first discuss the impact of nonnegativity on the identifiability of low-rank sparse component analysis (LRSCA), building upon recent advances. Then, as a main contribution, we propose two algorithms to train K-NMF: one based on alternating optimization and exact sparse coding, the other based on a nonnegative variant of K-subspace. We show on noiseless simulated data that our methods outperform by a large margin the state of the art. Finally, we apply our methods for the spectral unmixing of a hyperspectral image.