On tensor rank and commuting matrices

Abstract

Obtaining superlinear lower bounds on tensor rank is a major open problem in complexity theory. In this paper we propose a generalization of the approach used by Strassen in the proof of his 3n/2 border rank lower bound. Our approach revolves around a problem on commuting matrices: Given matrices Z_1,...,Z_p of size n and an integer r>n, are there commuting matrices Z'_1,...,Z'_p of size r such that every Z_k is embedded as a submatrix in the top-left corner of Z'_k? As one of our main results, we show that this question always has a positive answer for r larger than rank(T)+n, where T denotes the tensor with slices Z_1,..,Z_p. Taking the contrapositive, if one can show for some specific matrices Z_1,...,Z_p and a specific integer r that this question has a negative answer, this yields the lower bound rank(T) > r-n. There is a little bit of slack in the above rank(T)+n bound, but we also provide a number of exact characterizations of tensor rank and symmetric rank, for ordinary and symmetric tensors, over the fields of real and complex numbers. Each of these characterizations points to a corresponding variation on the above approach. In order to explain how Strassen's theorem fits within this framework we also provide a self-contained proof of his lower bound.