Tensor Rank, Invariants, Inequalities, and Applications
Tensor Rank, Invariants, Inequalities, and Applications
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DOI：
10.1137/120899066
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发表时间：
2012-11
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影响因子：
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通讯作者：
E. Allman;P. Jarvis;J. Rhodes;J. Sumner
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文献类型：
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作者：
E. Allman;P. Jarvis;J. Rhodes;J. Sumner
虽然C$上的代数几何经常被用来描述给定大小和复数阶的张量的闭包，但这一变体包括更小和更大阶的张量。这里我们主要讨论$n×n×n$在$\mathbb C上的秩为$n$的张量，它有一个稠密的子集，即单个张量在自然群作用下的轨道。我们在这个群作用下构造了多项式不变量，它的非零性仅在闭包上区别于点。与该簇的定义多项式的显式子集一起，这给出了秩为$n$和多线性秩为$(n，n，n)$的张量的半代数描述。我们构造的多项式在$n=2$的情况下与Cayley超行列式重合，从而推广了它。虽然我们的结构是直接和明确的，但我们也用表征理论的语言重塑了我们的功能，以获得更多的见解。我们在不同的方向上给出了三个应用：首先，我们发展了关于复数秩$n$和多线性秩$(n，n，n)$的实张量如何形成路连通子集的集合的基本拓扑理解，其中一个包含实数秩张量$n$。其次，我们使用不变量来发展一组概率分布的半代数描述，这些概率分布可以产生于具有隐藏变量的简单随机模型，该模型在系统发育和其他领域中很重要。第三，我们构造了秩为$2n-1$的张量的简单例子，它们位于秩为$n$的张量的闭包中。
Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$ tensors of rank $n$ over $\mathbb C$, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-vanishing distinguishes this orbit from points only in its closure. Together with an explicit subset of the defining polynomials of the variety, this gives a semialgebraic description of the tensors of rank $n$ and multilinear rank $(n,n,n)$. The polynomials we construct coincide with Cayley's hyperdeterminant in the case $n=2$, and thus generalize it. Though our construction is direct and explicit, we also recast our functions in the language of representation theory for additional insights. We give three applications in different directions: First, we develop basic topological understanding of how the real tensors of complex rank $n$ and multilinear rank $(n,n,n)$ form a collection of path-connected subsets, one of which contains tensors of real rank $n$. Second, we use the invariants to develop a semialgebraic description of the set of probability distributions that can arise from a simple stochastic model with a hidden variable, a model that is important in phylogenetics and other fields. Third, we construct simple examples of tensors of rank $2n-1$ which lie in the closure of those of rank $n$.