On monotonicity of multiplicative update rules for weighted nonnegative tensor factorization

This paper focuses on so-called weighted variants of nonnnegative matrix factorization (NMF) and more generally nonnnegative tensor factorization (NTF) approximations. We consider multiplicative update (MU) rules to optimize these approximations, and we prove that under certain conditions the results on monotonicity of MU rules for NMF generalize to both the NTF and the weighted NTF (WNTF) cases.

[1]  P. Paatero.  A weighted non-negative least squares algorithm for three-way ‘PARAFAC’ factor analysis , 1997 .

[2]  H. Kiers.  Towards a standardized notation and terminology in multiway analysis , 2000 .

[3]  H. Sebastian Seung,et al.  Algorithms for Non-negative Matrix Factorization , 2000, NIPS.

[4]  P. Smaragdis,et al.  Non-negative matrix factorization for polyphonic music transcription , 2003, 2003 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (IEEE Cat. No.03TH8684).

[5]  Inderjit S. Dhillon,et al.  Generalized Nonnegative Matrix Approximations with Bregman Divergences , 2005, NIPS.

[6]  Alexei A. Efros,et al.  Discovering objects and their location in images , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[7]  Paul Van Dooren,et al.  Weighted Nonnegative Matrix Factorization and Face Feature Extraction , 2007 .

[8]  Nancy Bertin,et al.  Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application to Music Analysis , 2009, Neural Computation.

[9]  Seungjin Choi,et al.  Weighted nonnegative matrix factorization , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[10]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[11]  Guillermo Sapiro,et al.  Online Learning for Matrix Factorization and Sparse Coding , 2009, J. Mach. Learn. Res..

[12]  Alexey Ozerov,et al.  Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[13]  Ali Taylan Cemgil,et al.  Generalised Coupled Tensor Factorisation , 2011, NIPS.

[14]  Jérôme Idier,et al.  Algorithms for Nonnegative Matrix Factorization with the β-Divergence , 2010, Neural Computation.

[15]  Ali Taylan Cemgil,et al.  Probabilistic latent tensor factorization framework for audio modeling , 2011, 2011 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA).

[16]  Erkki Oja,et al.  Unified Development of Multiplicative Algorithms for Linear and Quadratic Nonnegative Matrix Factorization , 2011, IEEE Transactions on Neural Networks.

[17]  Antoine Souloumiac,et al.  Weighted NMF for high-resolution mass spectrometry analysis , 2012, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO).

[18]  Serap Kirbiz,et al.  Perceptually weighted Non-negative Matrix Factorization for blind single-channel music source separation , 2012, Proceedings of the 21st International Conference on Pattern Recognition (ICPR2012).

[19]  Antoine Liutkus,et al.  Coding-Based Informed Source Separation: Nonnegative Tensor Factorization Approach , 2013, IEEE Transactions on Audio, Speech, and Language Processing.

[20]  Ngoc Q. K. Duong,et al.  Weighted nonnegative tensor factorization: on monotonicity of multiplicative update rules and application to user-guided audio source separation , 2013 .