论文标题
在品种上的对称张量分解
Symmetric Tensor Decompositions On Varieties
论文作者
论文摘要
本文讨论了给定品种$ x $的对称张量分解的问题：将对称张量分解为$ x $中包含的向量的张量量的总和。在本文中，我们首先研究了这种可分解的张量的几何和代数特性，这对于此类分解的实际计算至关重要。对于给定的张量，我们还为$ x $上存在对称分解而制定了标准。其次，最重要的是，我们提出了一种在任意$ x $上计算对称张量分解的方法。作为特定应用，可以通过提出的算法计算非对称张量的Vandermonde分解。
This paper discusses the problem of symmetric tensor decomposition on a given variety $X$: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in $X$. In this paper, we first study geometric and algebraic properties of such decomposable tensors, which are crucial to the practical computations of such decompositions. For a given tensor, we also develop a criterion for the existence of a symmetric decomposition on $X$. Secondly and most importantly, we propose a method for computing symmetric tensor decompositions on an arbitrary $X$. As a specific application, Vandermonde decompositions for nonsymmetric tensors can be computed by the proposed algorithm.
