Notions of Tensor Rank

TL;DR

This paper surveys various notions of tensor rank, exploring their history, applications, and recent results on their asymptotic equivalence over finite fields, highlighting the complexity measures of tensors.

Contribution

It provides a comprehensive overview of tensor rank notions, including recent proofs of their asymptotic equivalence over finite fields, which advances understanding of tensor complexity.

Findings

Multiple tensor rank notions have been introduced with distinct applications.

Recent results show asymptotic equivalence of three key tensor rank notions over finite fields.

The survey discusses the historical development and importance of tensor ranks in computer science.

Abstract

Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses. In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.