On the (non)existence of best low-rank approximations of generic IxJx2 arrays

Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation

Subtracting a best rank-1 approximation may increase tensor rank

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R>1 and k>2. 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 2x2x2 tensors, 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 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.Comment: 37 page

A Method to Avoid Diverging Components in the Candecomp/Parafac Model for Generic I × J × 2 Arrays

Computing the Candecomp/Parafac (CP) solution of R components (i.e., the best rank-R approximation) for a generic I x J x 2 array may result in diverging components, also known as "degeneracy." In such a case, several components are highly correlated in all three modes, and their component weights become arbitrarily large. Evidence exists that this is caused by the nonexistence of an optimal CP solution. Instead of using CP, we propose to compute the best approximation by means of a generalized Schur decomposition (GSD), which always exists. The obtained GSD solution is the limit point of the sequence of CP updates (whether it features diverging components or not) and can be separated into a nondiverging CP part and a sparse Tucker3 part or into a nondiverging CP part and a smaller GSD part. We show how to obtain both representations and illustrate our results with numerical experiments