Subtracting a best rank-1 approximation may increase tensor rank

作者：Stegeman Alwin^*; Comon Pierre

来源：Linear Algebra and Its Applications, 2010, 433(7): 1276-1300.

DOI：10.1016/j.laa.2010.06.027

• Tensor rank
• Low-rank approximation
• Tensor decomposition
• Multi-way
• Candecomp
• Parafac

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 x 2 x 2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2 x 2 x 2 tensors (which have rank 2 or 3), substracting 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.

• 出版日期2010-12-1