Informed Split Gradient Non-negative Matrix factorization using Huber cost function for source apportionment

Abstract

Source apportionment is usually tackled with blind Positive/Non-negative Matrix factorization (PMF/NMF) methods. However, the obtained results may be poor due to the dependence between some rows of the second factor. We recently proposed to inform the estimation of this factor using some prior knowledge provided by chemists—some entries are set to some fixed values—and the sum-to-one property of each row. These constraints were recently taken into account by using a parameterization which gathers all of them. In this paper, a novel robust NMF approach able to cope with outliers is proposed. For that purpose, we consider the Huber loss function—a ℓ2-ℓ1 cost function—which is robust to outliers, contrary to the Frobenius norm classically met in NMF. We thus propose new update rules for the informed Huber NMF in the framework of the split gradient techniques. The choice of the adaptive cutoff parameter—which links both single cost functions—is discussed along this paper. The proposed approach is shown to outperform state-of-the-art methods on several source apportionment simulations involving various input SNRs and outliers.