Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization

Authors:
Andrzej Cichocki;Rafal Zdunek;Shun-ichi Amari
Affiliations:
Dept. of EE, Warsaw University of Technology, and IBS PAN Warsaw, Poland;Institute of Telecommunications, Teleinformatics and Acoustics, Wroclaw, University of Technology, Poland;RIKEN Brain Science Institute, Wako-shi, Saitama, Japan
Venue:
ICA'07 Proceedings of the 7th international conference on Independent component analysis and signal separation
Year:
2007

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Abstract

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 overdetermined 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.