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It has been shown that a best rank-R approximation of an order-k tensor maynot exist when R>1 and k>2. This poses a serious problem to data analysts usingtensor decompositions. It has been observed numerically that, generally, thisissue cannot be solved by consecutively computing and subtracting best rank-1approximations. The reason for this is that subtracting a best rank-1approximation generally does not decrease tensor rank. In this paper, weprovide a mathematical treatment of this property for real-valued 2x2x2tensors, with symmetric tensors as a special case. Regardless of the symmetry,we show that for generic 2x2x2 tensors (which have rank 2 or 3), subtracting abest rank-1 approximation results in a tensor that has rank 3 and lies on theboundary between the rank-2 and rank-3 sets. Hence, for a typical tensor ofrank 2, subtracting a best rank-1 approximation increases the tensor rank.

Subtracting a best rank-1 approximation may increase tensor rank