非交换空间中的一致性
Consensus in non-commutative spaces
摘要：
一致性算法的收敛性分析是根据希尔伯特距离重新审视。Tsitsiklis李雅普诺夫函数被证明是希尔伯特距离的共识，在对数坐标。Birkhoff定理证明了Hilbert度量对任何正齐次单调映射的收缩，为一致性算法提供了一个早期但普遍的收敛结果。由于Birkhoff定理在任意锥上成立，我们将一致性算法推广到正定矩阵的锥上.所提出的推广在量子随机映射的收敛性分析中找到了应用，量子随机映射是随机映射到非交换概率空间的推广。
英文摘要：
Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. Tsitsiklis Lyapunov function is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic maps, which are a generalization of stochastic maps to non-commutative probability spaces.
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