具有指定边缘的运算符缩放
Operator scaling with specified marginals
摘要：
完全正映射是非负矩阵的推广，是一类研究得很好的从$n\times n$矩阵到$m\times m$矩阵的映射。矩阵的双随机标度的算子类似物的存在等价于计算机科学和数学中的许多问题，例如非交换变量的有理恒等式测试，符号矩阵的非交换秩，以及不变理论中的基本问题（Garg，Gurvits，Oliveira和Wigderson，FOCS，2016）。我们研究算子缩放与指定的边缘，这是运营商模拟缩放矩阵指定的行和列和。我们描述了可以缩放到给定边缘的算子，这与Gurvits对可以缩放到双随机的算子的算法描述的精神非常相似（Gurvits，Journal of Computer and系统科学，2004）。我们的算法产生近似缩放时间poly（n，m）每当缩放存在。在我们的分析中的一个中心成分是减少从指定的边际设置的双重随机设置。具有特定边缘的算子标度出现在不同的研究领域，例如Brascamp-Lieb不等式，通信复杂性，埃尔米特矩阵的特征值和量子信息论。在这些领域中的一些已知定理，其中有几个没有有效的证明，是我们的特征定理的直接后果。例如，我们得到了一个简单的算法，当它们存在时，一个具有给定谱的埃尔米特矩阵的元组，其和具有给定谱。我们还证明了新的定理，如推广的福斯特定理（福斯特，计算机和系统科学杂志，2002年）关于径向各向同性的位置。
英文摘要：
The completely positive maps, a generalization of the nonnegative matrices, are a well-studied class of maps from $n\times n$ matrices to $m\times m$ matrices. The existence of the operator analogues of doubly stochastic scalings of matrices is equivalent to a multitude of problems in computer science and mathematics, such rational identity testing in non-commuting variables, noncommutative rank of symbolic matrices, and a basic problem in invariant theory (Garg, Gurvits, Oliveira and Wigderson, FOCS, 2016). We study operator scaling with specified marginals, which is the operator analogue of scaling matrices to specified row and column sums. We characterize the operators which can be scaled to given marginals, much in the spirit of the Gurvits' algorithmic characterization of the operators that can be scaled to doubly stochastic (Gurvits, Journal of Computer and System Sciences, 2004). Our algorithm produces approximate scalings in time poly(n,m) whenever scalings exist. A central ingredient in our analysis is a reduction from the specified marginals setting to the doubly stochastic setting. Operator scaling with specified marginals arises in diverse areas of study such as the Brascamp-Lieb inequalities, communication complexity, eigenvalues of sums of Hermitian matrices, and quantum information theory. Some of the known theorems in these areas, several of which had no effective proof, are straightforward consequences of our characterization theorem. For instance, we obtain a simple algorithm to find, when they exist, a tuple of Hermitian matrices with given spectra whose sum has a given spectrum. We also prove new theorems such as a generalization of Forster's theorem (Forster, Journal of Computer and System Sciences, 2002) concerning radial isotropic position.
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