Generic and typical ranks of multi-way arrays |
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Authors: | P Comon JMF ten Berge L De Lathauwer J Castaing |
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Institution: | 1. Lab. I3S, CNRS and University of Nice, 2000 route des Lucioles, B.P. 121, F-06903 Sophia-Antipolis cedex, France;2. University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands;3. Katholieke Universiteit Leuven, E.E. Dept. ESAT, Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium;4. Science, Engineering and Technology Group, E. Sabbelaan 53, 8500 Kortrijk, Belgium;5. Lab. ETIS, B.P. 44, F-95014 Cergy-Pontoise cedex, France |
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Abstract: | The concept of tensor rank was introduced in the 20s. In the 70s, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini’s lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries. |
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Keywords: | Tensor Multi-way arrays Generic rank Canonical decomposition Parafac Factor analysis |
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