Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization

  • Andrzej Cichocki
  • Rafal Zdunek
  • Shun-ichi Amari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4666)


In the paper we present new Alternating Least Squares (ALS) algorithms for Nonnegative Matrix Factorization (NMF) and their extensions to 3D Nonnegative Tensor Factorization (NTF) that are robust in the presence of noise and have many potential applications, including multi-way Blind Source Separation (BSS), multi-sensory or multi-dimensional data analysis, and nonnegative neural sparse coding. We propose to use local cost functions whose simultaneous or sequential (one by one) minimization leads to a very simple ALS algorithm which works under some sparsity constraints both for an under-determined (a system which has less sensors than sources) and over-determined model. The extensive experimental results confirm the validity and high performance of the developed algorithms, especially with usage of the multi-layer hierarchical NMF. Extension of the proposed algorithm to multidimensional Sparse Component Analysis and Smooth Component Analysis is also proposed.


Blind Source Separation Nonnegative Matrix Factorization Alternate Little Square Nonlinear Projection Alternate Little Square Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Andrzej Cichocki
    • 1
  • Rafal Zdunek
    • 2
  • Shun-ichi Amari
    • 3
  1. 1.Dept. of EE, Warsaw University of Technology, and IBS PAN WarsawPoland
  2. 2.Institute of Telecommunications, Teleinformatics and Acoustics, Wroclaw University of TechnologyPoland
  3. 3.RIKEN Brain Science Institute, Wako-shi, SaitamaJapan

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