The stability and control of discrete processes
The stability and control of discrete processes
Introducing a weighted non-negative matrix factorization for image classification
Pattern Recognition Letters
A Generalized Divergence Measure for Nonnegative Matrix Factorization
Neural Computation
Projected Gradient Methods for Nonnegative Matrix Factorization
Neural Computation
Uniqueness of Non-Negative Matrix Factorization
SSP '07 Proceedings of the 2007 IEEE/SP 14th Workshop on Statistical Signal Processing
ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing
Non-negative sparse modeling of textures
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Adaptive harmonic spectral decomposition for multiple pitch estimation
IEEE Transactions on Audio, Speech, and Language Processing
IEEE Transactions on Audio, Speech, and Language Processing
On the discrete-time dynamics of a class of self-stabilizing MCA extraction algorithms
IEEE Transactions on Neural Networks
IEEE Transactions on Neural Networks
An augmented LKF approach involving derivative information of both state and delay
IEEE Transactions on Neural Networks
Csiszár’s divergences for non-negative matrix factorization: family of new algorithms
ICA'06 Proceedings of the 6th international conference on Independent Component Analysis and Blind Signal Separation
Extended SMART algorithms for non-negative matrix factorization
ICAISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Soft Computing
IEEE Transactions on Audio, Speech, and Language Processing
Musical source separation using time-frequency source priors
IEEE Transactions on Audio, Speech, and Language Processing
On the Convergence of Multiplicative Update Algorithms for Nonnegative Matrix Factorization
IEEE Transactions on Neural Networks
Algorithms for nonnegative matrix factorization with the β-divergence
Neural Computation
Informed source separation through spectrogram coding and data embedding
Signal Processing
ICONIP'11 Proceedings of the 18th international conference on Neural Information Processing - Volume Part II
Multistability of α-divergence based NMF algorithms
Computers & Mathematics with Applications
Global Minima Analysis of Lee and Seung's NMF Algorithms
Neural Processing Letters
Global convergence of modified multiplicative updates for nonnegative matrix factorization
Computational Optimization and Applications
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Multiplicative update algorithms have proved to be a great success in solving optimization problems with nonnegativity constraints, such as the famous nonnegative matrix factorization (NMF) and its many variants. However, despite several years of research on the topic, the understanding of their convergence properties is still to be improved. In this paper, we show that Lyapunov's stability theory provides a very enlightening viewpoint on the problem. We prove the exponential or asymptotic stability of the solutions to general optimization problems with nonnegative constraints, including the particular case of supervised NMF, and finally study the more difficult case of unsupervised NMF. The theoretical results presented in this paper are confirmed by numerical simulations involving both supervised and unsupervised NMF, and the convergence speed of NMF multiplicative updates is investigated.