Subtracting a best rank-1 approximation does not necessarily decrease tensor rank

Abstract : It has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2 × 2 × 2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2 × 2 × 2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation increases the tensor rank.
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Linear Algebra and its Applications, Elsevier, 2010, 433 (7), pp.1276--1300. 〈10.1016/j.laa.2010.06.027〉
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Alwin Stegeman, Pierre Comon. Subtracting a best rank-1 approximation does not necessarily decrease tensor rank. Linear Algebra and its Applications, Elsevier, 2010, 433 (7), pp.1276--1300. 〈10.1016/j.laa.2010.06.027〉. 〈hal-00512275〉

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