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：色素或多线形是一系列领域的重要对象,从分析学到组合学,到计算复杂的理论,从多个领域都是重要对象,从分析学到组合学,到计算复杂的理论。 高阶等级的注意旨在量化这些形态的“复合性”,因此也很重要。 虽然对等级有一个单一的定义,可以完全捕捉矩阵的复杂性(以及由此而来的线性变形),但对于色素则没有明确的相似性。 相反,多年来,对色素等级的许多概念已经定义了,每个概念都有各自的用途。 在本文中,我们调查了流行的色素等级概念。 我们简单介绍了这些概念的引入史,激励了它们的存在,并讨论了它们在计算机科学中的一些应用。 我们还提供了洛夫特、科恩和莫什科维茨最近的结果的证明草图,这些结果证明,在有至少三个元素的有限字段上的色素等级的三个关键概念之间是无症状的等。