Tensor Rank is NP-Complete

We prove that computing the rank of a three-dimensional tensor over any nite eld is NP-complete. Over the rational numbers the problem is NP-hard. Warning: Essentially this paper has beenpublished in Journal of Algorithms and is hence subject to copyright restrictions. It is for personal use only. 1. Introduction. One of the most fundamental quantities in linear algebra is the rank of a matrix. This is a well understood, easy to compute number. The purpose of this paper is to study a higher dimensional analogue, namely the rank of a three-dimensional tensor. Let us deene this number before we continue. For comparison we rst give a slightly unusual deenition of matrix rank. A matrix is a two-dimensional array of numbers. It has rank 1 ii it can bewritten as the outer product of two vectors. By this we mean that there are vectors x and y such that m ij = x i y j. The rank of a general matrix M is now the minimal number of rank 1 matrices M i such that M = P M i. In the same way, a three-dimensional tensor is a three-dimensional array of numbers. It has rank 1 ii it can bewritten as the outer product of three vectors and the rank of a general tensor T is the minimal number of rank 1 tensors T i such t h a t T = P T i. Despite the fact that the rank of a tensor is a very natural object, our knowledge of its properties is surprisingly limited. For instance, it does not seem to be known in any eld what is the maximal rank of an n n n tensor. In this paper we prove that over most elds it is NP-hard to compute the rank of a tensor. Thus unless N P= P there will be no easily computable characterization of rank and furthermore if N P6 = coNP there will be no easy to verify characterization of the property \having rank at least r". These facts might explain at least partly the lack of progress in the study of tensor rank. One can here draw a parallel with graph theory where the NP-complete problem of Hamiltonian circuit has been much m o r e elusive than many other properties of graphs. In spite of the interesting and natural questions above, our main motivation to study tensor rank …