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IEEE SIGNAL PROCESSING MAGAZINE [161] MARCH 2015
■ The estimation of the number of components in data and
the assessment of their dimensionality would benefit from
automation, especially in the presence of noise and outliers.
■ Both new theory and algorithms are needed to further
extend the flexibility of tensor models, e.g., for the con-
straints to be combined in many ways and tailored to the par-
ticular signal properties in different modes.
■ Work on efficient techniques for saving and/or fast process-
ing of ultra-large-scale tensors is urgent; these now routinely
occupy terabytes, and will soon require petabytes of memory.
■ Tools for rigorous performance analysis and rule of thumb
performance bounds need to be further developed across ten-
sor decomposition models.
■ Our discussion has been limited to tensor models in which
all entries take values independently of one another. Probabil-
istic versions of tensor decompositions incorporate prior
knowledge about complex variable interaction, various data
alphabets, or noise distributions, and so promise to model
data more accurately and efficiently [119], [120].
■ The future computational, visualization, and interpret-
ation tools will be important next steps in supporting the dif-
ferent communities working on large-scale and big data
analysis problems.
It is fitting to conclude with a quote from the French novelist
Marcel Proust: “The voyage of discovery is not in seeking new
landscapes but in having new eyes.” We hope to have helped to
bring to the eyes of the signal processing community the multi-
disciplinary developments in tensor decompositions and to have
shared our enthusiasm about tensors as powerful tools to dis-
cover new landscapes.
AUTHORS
Andrzej Cichocki ([e-mail]) received the Ph.D. and
Dr.Sc. (habilitation) degrees all in electrical engineering from the
Warsaw University of Technology, Poland. He is currently a senior
team leader of the Laboratory for Advanced Brain Signal Process-
ing at RIKEN Brain Science Institute, Japan, and a professor at the
Systems Research Institute, Polish Academy of Science, Poland.
He has authored more than 400 publications and four mono-
graphs in the areas of signal processing and computational neuro-
science. He is an associate editor of IEEE Transactions on Signal
Processing and Journal of Neuroscience Methods.
Danilo P. Mandic ([e-mail]) is a professor of
signal processing at Imperial College London, United Kingdom, and
has been working in the area of nonlinear and multidimensional
adaptive signal processing and time-frequency analysis. His publica-
tion record includes two research monographs, Recurrent Neural
Networks for Prediction and Complex Valued Nonlinear Adaptive
Filters: Noncircularity, Widely Linear and Neural Models, an edited
book, Signal Processing for Information Fusion, and more than
200 publications on signal and image processing. He has been a
guest professor at KU Leuven, Belgium, and a frontier researcher at
RIKEN Brain Science Institute, Tokyo, Japan.
Anh Huy Phan ([e-mail]) received the Ph.D.
degree from the Kita Kyushu Institute of Technology, Japan, in
2011. He worked as a deputy head of the Research and Develop-
ment Department, Broadcast Research and Application Center,
Vietnam Television, and is currently a research scientist at the
Laboratory for Advanced Brain Signal Processing and a visiting
research scientist at the Toyota Collaboration Center, RIKEN
Brain Science Institute, Japan. He has served on the editorial
board of International Journal of Computational Mathematics.
His research interests include multilinear algebra, tensor compu-
tation, blind source separation, and brain–computer interfaces.
Cesar F. Caiafa ([e-mail]) received the Ph.D.
degree in engineering from the Faculty of Engineering, Univer-
sity of Buenos Aires, in 2007. He is currently an adjunct
researcher with the Argentinean Radioastronomy Institute
(IAR)—CONICET and an assistant professor with Faculty of
Engineering, the University of Buenos Aires. He is also a visiting
scientist at the Laboratory for Advanced Brain Signal Process-
ing, BSI—RIKEN, Japan.
Guoxu Zhou ([e-mail]) received the Ph.D.
degree in intelligent signal and information processing from the
South China University of Technology, Guangzhou, in 2010. He is
currently a research scientist at the Laboratory for Advanced Brain
Signal Processing at RIKEN Brain Science Institute, Japan. His
research interests include statistical signal processing, tensor ana-
lysis, intelligent information processing, and machine learning.
Qibin Zhao ([e-mail]) received the Ph.D. degree
from the Department of Computer Science and Engineering,
Shanghai Jiao Tong University, China, in 2009. He is currently a
research scientist at the Laboratory for Advanced Brain Signal
Processing in RIKEN Brain Science Institute, Japan, and a visit-
ing research scientist in the BSI Toyota Collaboration Center,
RIKEN-BSI. His research interests include multiway data ana-
lysis, brain–computer interface, and machine learning.
Lieven De Lathauwer ([e-mail])
received the Ph.D. degree from the Faculty of Engineering, KU Leu-
ven, Belgium, in 1997. From 2000 to 2007, he was a research associ-
ate with the Centre National de la Recherche Scientifique, France.
He is currently a professor with KU Leuven. He is affiliated with the
group Science, Engineering, and Technology of Kulak, the Stadius
Center for Dynamical Systems, Signal Processing, and Data Analytics
of the Electrical Engineering Department (ESAT), and iMinds Future
Health Department. He is an associate editor of SIAM Journal on
Matrix Analysis and Applications and was an associate editor of
IEEE Transactions on Signal Processing. His research focuses on
the development of tensor tools for engineering applications.
REFERENCES
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[2] R. Cattell, “Parallel proportional profiles and other principles for determining
the choice of factors by rotation,” Psychometrika, vol. 9, pp. 267–283, 1944.
[3] L. R. Tucker, “The extension of factor analysis to three-dimensional matrices,”
in Contributions to Mathematical Psychology, H. Gulliksen and N. Frederiksen,
Eds. New York: Holt, Rinehart and Winston, 1964, pp. 110–127.
[4] L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psy-
chometrika,vol.31,no.3, pp. 279–311, Sept.1966.
[5] J. Carroll and J.-J. Chang, “Analysis of individual differences in multidimen-
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