Non-negative matrix factorization with α-divergence

Non-negative matrix factorization (NMF) is a popular technique for pattern recognition, data analysis, and dimensionality reduction, the goal of which is to decompose non-negative data matrix XX into a product of basis matrix AA and encoding variable matrix SS with both AA and SS allowed to have only non-negative elements. In this paper, we consider Amari’s α-divergence as a discrepancy measure and rigorously derive a multiplicative updating algorithm (proposed in our recent work) which iteratively minimizes the α-divergence between XX and ASAS. We analyze and prove the monotonic convergence of the algorithm using auxiliary functions. In addition, we show that the same algorithm can be also derived using Karush–Kuhn–Tucker (KKT) conditions as well as the projected gradient. We provide two empirical study for image denoising and EEG classification, showing the interesting and useful behavior of the algorithm in cases where different values of α   (α=0.5,1,2α=0.5,1,2) are used.