Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization
We propose a class of multiplicative algorithms for Nonnegative Matrix Factorization (NMF) which are robust with respect to noise and outliers. To achieve this, we formulate a new family generalized divergences referred to as the Alpha-Beta-divergences (AB-divergences), which are parameterized by the two tuning parameters, alpha and beta, and smoothly connect the fundamental Alpha-, Beta- and Gamma-divergences. By adjusting these tuning parameters, we show that a wide range of standard and new divergences can be obtained. The corresponding learning algorithms for NMF are shown to integrate and generalize many existing ones, including the Lee-Seung, ISRA (Image Space Reconstruction Algorithm), EMML (Expectation Maximization Maximum Likelihood), Alpha-NMF, and Beta-NMF. Owing to more degrees of freedom in tuning the parameters, the proposed family of AB-multiplicative NMF algorithms is shown to improve robustness with respect to noise and outliers. The analysis illuminates the links of between AB-divergence and other divergences, especially Gamma- and Itakura-Saito divergences.
Nonlinear Hyperspectral Unmixing With Robust Nonnegative Matrix Factorization
We introduce a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures. The new model extends the commonly used linear mixing model by introducing an additional term accounting for possible nonlinear effects, that are treated as sparsely distributed additive outliers. With the standard nonnegativity and sum-to-one constraints inherent to spectral unmixing, our model leads to a new form of robust nonnegative matrix factorization with a group-sparse outlier term. The factorization is posed as an optimization problem, which is addressed with a block-coordinate descent algorithm involving majorization-minimization updates. Simulation results obtained on synthetic and real data show that the proposed strategy competes with the state-of-the-art linear and nonlinear unmixing methods.
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