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In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generalize some matrix decompositions.
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one type of generalization of the matrix singular value decomposition. It has applications in computer vision, computer graphics, machine learning, scientific computing, and signal processing. Some aspects can be traced as far back as F. L. Hitchcock in 1928, but it was L. R. Tucker who developed for third-order tensors the general Tucker decomposition in the 1960s, further advocated by L. De Lathauwer et al. in their Multilinear SVD work that employs the power method, or advocated by Vasilescu and Terzopoulos that developed M-mode SVD a parallel algorithm that employs the matrix SVD.
KU Leuven is a Catholic research university in the city of Leuven, Belgium.
Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality reduction can be performed on a data tensor that contains a collection of observations have been vectorized, or observations that are treated as matrices and concatenated into a data tensor. Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D).
Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac, or by multilinear tensor models, such as multilinear principal component analysis (MPCA) or multilinear independent component analysis (MICA). The origin of MPCA can be traced back to the tensor rank decomposition introduced by Frank Lauren Hitchcock in 1927; to the Tucker decomposition; and to Peter Kroonenberg's "3-mode PCA" work. In 2000, De Lathauwer et al. restated Tucker and Kroonenberg's work in clear and concise numerical computational terms in their SIAM paper entitled "Multilinear Singular Value Decomposition", (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R1, R2, ..., RN ) Approximation of Higher-order Tensors".
Christophe Caloz is a researcher and professor of electrical engineering and physics at KU Leuven. He graduated from the École Polytechnique Fédérale de Lausanne in Lausanne, Switzerland, where he received a Diploma of electrical engineering in telecommunications in 1995 and a Ph.D. in electromagnetics in 2000. From 2001 to 2004, he was a Postdoctoral Research Engineer at the Microwave Electronics Laboratory of University of California at Los Angeles. He was then a professor and a Canada Research Chair at the École Polytechnique de Montréal until 2019, before joining KU Leuven where he is the director of the Meta Research Group.
Based on the key idea of higher-order singular value decomposition (HOSVD) in tensor algebra, Baranyi and Yam proposed the concept of HOSVD-based canonical form of TP functions and quasi-LPV system models. Szeidl et al. proved that the TP model transformation is capable of numerically reconstructing this canonical form.
Lieven Boeve [‘li:vən ‘bu:və] is a Belgian Catholic theologian, and is full professor of Systematic Theology at the Faculty of Theology and Religious Studies of the Katholieke Universiteit Leuven. Since 2012 he is also the chair of the Centre for Academic Teacher’s Training, KU Leuven. On 1 August 2014 he became the Director General of the Flemish Secretary for Catholic Education (VSKO).
Stephen P. Boyd is an American professor and control theorist. He is the Samsung Professor of Engineering, Professor in Electrical Engineering, and professor by courtesy in Computer Science and Management Science & Engineering at Stanford University. He is also affiliated with Stanford's Institute for Computational and Mathematical Engineering (ICME).
Jan Maria Florent Wouters is a Belgian academic. He is Jean Monnet Chair, and Professor of International Law and International Organizations at KU Leuven, where he is also Director of its Centre for Global Governance Studies and Institute for International Law.
José Lambert is a Professor of Comparative Literature at KU Leuven, Belgium, and is best known for his work in Translation Studies. He is also noted for leading international initiatives in this field.
Johan Suykens is a full professor from KU Leuven in Belgium. He was named a Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2015 for developing least squares support vector machines.
Jan Craninckx from the Interuniversity Microelectronics Center (IMEC), Leuven, Belgium was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2014 for contributions to the design of CMOS RF transceivers.
Marc Van Hulle from the KU Leuven, Belgium was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2014 for contributions to biomedical signal processing and biological modeling.
Andrzej Cichocki is a Polish computer scientist, electrical engineer and a professor at the Systems Research Institute of Polish Academy of Science, Warsaw, Poland and a visiting professor in several universities and research institutes, especially Riken AIP, Japan. He is most noted for his learning algorithms for Signal separation (BSS), Independent Component Analysis (ICA), Non-negative matrix factorization (NMF), tensor decomposition, Deep (Multilayer) Matrix Factorizations for ICA, NMF, PCA, neural networks for optimization and signal processing, Tensor network for Machine Learning and Big Data, and brain–computer interfaces. He is the author of several monographs/books and more than 500 scientific peer-reviewed articles.
Luc Sels is a Belgian sociologist.
Tamara G. Kolda is an American applied mathematician and former Distinguished Member of Technical Staff at Sandia National Laboratories. She is noted for her contributions in computational science, multilinear algebra, data mining, graph algorithms, mathematical optimization, parallel computing, and software engineering. She is currently a member of the SIAM Board of Trustees and served as associate editor for both the SIAM Journal on Scientific Computing and the SIAM Journal on Matrix Analysis and Applications.
Luc Van den hove is President and Chief Executive Officer (CEO) of IMEC, Europe's largest independent research center in the field of nanoelectronics and digital technologies.
Ingrid Verbauwhede is a professor at the COSIC Research Group of the Electrical Engineering Department, KU Leuven, where she leads the embedded systems team. She is a pioneer in the field of secure embedded circuits and systems, with several awards recognising her contributions to the field. She is member of the Royal Flemish Academy of Belgium for Science and the Arts since 2011. She is a fellow of IEEE.
Tensor informally refers in machine learning to two different concepts that organize and represent data. Data may be organized in a multidimensional array (M-way array) that is informally referred to as a "data tensor"; however in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector space. Observations, such as images, movies, volumes, sounds, and relationships among words and concepts, stored in an M-way array ("data tensor") may be analyzed either by artificial neural networks or tensor methods.
References
- ↑ "2015 elevated fellow" (PDF). IEEE Fellows Directory.
- ↑ SIAM Fellows: Class of 2017, retrieved 2017-04-25.
- ↑ "Lieven De Lathauwer". STADIUS. Retrieved 31 August 2022.
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