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xiv Preface
constraints and NTD. In order to make this chapter as self-contained as possible, we re-introduce some
concepts and derive several novel and efficient algorithms for nonnegative and semi-nonnegative tensor
(multi-way arrays) factorizations and decompositions. Our particular emphasis is on a detailed treatment
of the generalized cost functions, including Alpha- and Beta-divergences. Based on these cost functions,
several classes of algorithms are introduced, including: (1) multiplicative updating; (2) ALS; and (3) Hi-
erarchical ALS (HALS). These algorithms are then incorporated into multi-layer hierarchical networks in
order to improve their performance. Special emphasis is given to the ways to impose nonnegativity or semi-
nonnegativity, together with optional constraints such as orthogonality, sparsity and/or smoothness. The
developed algorithms are tested for several applications such as denoising, compression, feature extraction,
clustering, EEG data analysis, brain computer interface and video tracking. To understand the material in
this chapter it would be helpful to be familiar with the previous chapters, especially Chapters 1, 3 and 4.
Finally, in Chapter 8, we briefly discuss the selected applications of NMF and multi-dimensional array
decompositions, with a special emphasis on these applications to which the algorithms described in the
previous chapters are applicable. We review the following applications: data clustering, text mining, email
surveillance, musical instrument classification, face recognition, handwritten digit recognition, texture clas-
sification, Raman spectroscopy, fluorescence spectroscopy, hyper-spectral imaging, chemical shift imaging,
and gene expression classification.
The book is partly a textbook and partly a research monograph. It is a textbookbecause it gives the detailed
introduction to the basic models and algorithms of nonnegative and sparse matrix and tensor decompositions.
It is simultaneously a monograph because it presents many new results, methods, ideas, models, further
developments and implementation of efficient algorithms which are brought together and published in this
book for the first time. As a result of its twofold character, the book is likely to be of interest to graduate and
postgraduate students, engineers and scientists working in the fields of biomedical engineering, data analysis,
data mining, multidimensional data visualization, signal/image processing, mathematics, computer science,
finance, economics, optimization, geophysics, and neural computing. Furthermore, the book may also be of
interest to researchers working in different areas of science, because a number of the results and concepts
included may be advantageous for their further research. One can read this book through sequentially but it
is not necessary, since each chapter is essentially self-contained, with as few cross references as possible.
So, browsing is encouraged.