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.

翻译：张量，或称多重线性形式，是分析学、组合学及计算复杂性理论等多个领域中的重要研究对象。张量秩的概念旨在量化这些形式的"复杂性"，因而同样具有重要意义。尽管存在一个统一的秩定义能够完全刻画矩阵（进而线性变换）的复杂性，但对于张量而言，却不存在确定性的对应概念。相反，多年来人们定义了多种张量秩的概念，各自具有不同的用途。本文综述了流行的张量秩概念，简要介绍其提出历史、动机，并讨论了它们在计算机科学中的部分应用。我们还给出了Lovett以及Cohen和Moshkovitz近期成果的证明梗概，这些成果证明了在至少包含三个元素的有限域上，三种关键张量秩概念之间的渐近等价性。