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Entropy 2011, 13(1), 134-170; doi:10.3390/e13010134
Article

Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization

1,2,* , 3,*  and 4
Received: 13 December 2010; in revised form: 4 January 2011 / Accepted: 4 January 2011 / Published: 14 January 2011
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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.
Keywords: nonnegative matrix factorization (NMF); robust multiplicative NMF algorithms; similarity measures; generalized divergences; Alpha-; Beta-; Gamma- divergences; extended Itakura-Saito like divergences; generalized Kullback-Leibler divergence nonnegative matrix factorization (NMF); robust multiplicative NMF algorithms; similarity measures; generalized divergences; Alpha-; Beta-; Gamma- divergences; extended Itakura-Saito like divergences; generalized Kullback-Leibler divergence
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Cichocki, A.; Cruces, S.; Amari, S.-I. Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization. Entropy 2011, 13, 134-170.

AMA Style

Cichocki A, Cruces S, Amari S-I. Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization. Entropy. 2011; 13(1):134-170.

Chicago/Turabian Style

Cichocki, Andrzej; Cruces, Sergio; Amari, Shun-ichi. 2011. "Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization." Entropy 13, no. 1: 134-170.


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