Journal
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Volume 33, Issue 3, Pages 1018-1037Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110829180
Keywords
tensor decomposition; parafac; candecomp; uniqueness of decomposition; weak defectivity
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We introduce an inductive method for the study of the uniqueness of decompositions of tensors, by means of tensors of rank 1. The method is based on the geometric notion of weak defectivity. For three-dimensional tensors of type (a, b, c), a <= b <= c, our method proves that the decomposition is unique (i.e., k-identifiability holds) for general tensors of rank k, as soon as k <= (a + 1)(b + 1)/16. This improves considerably the known range for identifiability. The method applies also to tensor of higher dimension. For tensors of small size, we give a complete list of situations where identifiability does not hold. Among them, there are 4x4x4 tensors of rank 6, an interesting case because of its connection with the study of DNA strings.
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