Generic and Typical Ranks of Multi-Way Arrays

Abstract : The concept of tensor rank was introduced in the twenties. In the seventies, 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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Pierre Comon, Jos Ten Berge, Lieven de Lathauwer, Josephine Castaing. Generic and Typical Ranks of Multi-Way Arrays. Linear Algebra and its Applications, Elsevier, 2009, 430 (11), pp.2997-3007. ⟨10.1016/j.laa.2009.01.014⟩. ⟨hal-00410058⟩

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