A three-way Jordan canonical form as limit of low-rank tensor approximations

摘要

A best rank-R approximation of an order-3 tensor or three-way array may not exist due to the fact that the set of three-way arrays with rank at most R is not closed. In this case, we are trying to compute the approximation results in diverging rank-1 terms. We show that this phenomenon can be seen as a three-way generalization of approximate diagonalization of a nondiagonalizable (real) matrix. Moreover, we show that, analogous to the matrix case, the limit point of the approximating rank-R sequence satisfies a three-way generalization of the real Jordan canonical form. Recently, it was shown how to obtain the limit point and its three-way Jordan form for R ≤ min(I, J,K) and groups of two or three diverging rank-1 terms, where I × J × K is the size of the array. We extend this to groups of four diverging rank-1 terms and show that R > min(I, J, K) is possible as long as no groups of more than min(I, J, K) diverging rank-1 terms occur. We demonstrate our results by means of numerical experiments.