On tensor rank and commuting matrices

TL;DR

This paper explores a new approach to tensor rank lower bounds using commuting matrices, generalizing Strassen's method, and provides characterizations of tensor ranks over real and complex fields.

Contribution

It introduces a matrix-embedding approach to tensor rank bounds, extending Strassen's technique, and offers exact characterizations of tensor and symmetric ranks over real and complex fields.

Findings

Positive answer for matrix embedding when r > rank(T)+n

Lower bounds derived from negative answers to matrix embedding problem

Exact characterizations of tensor and symmetric ranks over real and complex fields

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.