Nearly Low Rank Tensors and Their Approximations

The low rank tensor approximation problem (LRTAP) is to find a tensor whose rank is small and that is close to a given one. This paper studies the LRTAP when the tensor to be approximated is close to a low rank one. Both symmetric and nonsymmetric tensors are discussed. We propose a new approach for solving the LRTAP. It consists of three major stages: i) Find a set of linear relations that are approximately satisfied by the tensor; such linear relations can be expressed by polynomials and can be found by solving linear least squares. ii) Compute a set of points that are approximately common zeros of the obtained polynomials; they can be found by computing Schur decompositions. iii) Construct a low rank approximating tensor from the obtained points; this can be done by solving linear least squares. Our main conclusion is that if the given tensor is sufficiently close to a low rank one, then the computed tensor is a good enough low rank approximation. This approach can also be applied to efficiently compute low rank tensor decompositions, especially for large scale tensors.