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首页> 外文会议>Annual Association for Computing Machinery-Society for Industrial and Applied Mathmatics Symposium on Discrete Algorithms >Generalized Perron-Frobenius Theorem for Multiple Choice Matrices, and Applications
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Generalized Perron-Frobenius Theorem for Multiple Choice Matrices, and Applications

机译：多项选择矩阵的广义erron-Frobenius定理和应用

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摘要

The celebrated Perron-Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. However, many real-life scenarios give rise to nonsquare matrices. Despite the extensive development of spectral theories for nonnegative matrices, the applicability of such theories to nonconvex optimization problems is not clear. In particular, a natural question is whether the PF Theorem (along with its applications) can be generalized to a nonsquare setting. Our paper provides a generalization of the PF Theorem to nonsquare multiple choice matrices. The extension can be interpreted as representing systems with additional degrees of freedom, where each client entity may choose between multiple servers that can cooperate in serving it (while potentially interfering with other clients). This formulation is motivated by applications to power control in wireless networks, economics and others, all of which extend known examples for the use of the original PF Theorem.
机译：庆祝的珀罗弗罗布尼乌斯（PF）定理被陈述为不可缩合的非负方形矩阵，并提供了它们的特征向量和特征值的简单表征。本定理源的重要性源于这种矩阵上的特征值问题出现在许多科学和工程领域，包括动态系统理论，经济学，统计和优化。然而，许多现实生活场景引起了非特殊矩阵。尽管对非负矩阵的光谱理论进行了广泛的发展，但这些理论的适用性对于非凸化优化问题尚不清楚。特别是，自然问题是PF定理（以及其应用）是否可以概括为非标准设置。我们的论文提供了PF定理的概括，以Nonsquare多项选择矩阵。扩展可以被解释为具有额外自由度的系统，其中每个客户端实体可以在可以协作服务它的多个服务器之间进行选择（同时可能干扰其他客户端）。该配方通过应用于无线网络，经济学和其他的应用程序的应用，所有这些都是延长了原始PF定理的已知示例的所有这些。

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