## Abstract

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.

Original language | English |
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Pages (from-to) | 134-170 |

Number of pages | 37 |

Journal | Entropy |

Volume | 13 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2011 |

Externally published | Yes |

## Keywords

- Alpha-
- Beta-
- Extended Itakura-Saito like divergences
- Gamma- divergences
- Generalized divergences
- Generalized Kullback-Leibler divergence
- Nonnegative matrix factorization (NMF)
- Robust multiplicative NMF algorithms
- Similarity measures