正规因子图与全息变换
Normal Factor Graphs and Holographic Transformations
摘要：
本文站在两个不同的研究路线的交叉点。一条线是“全息算法”，这是Valiant提出的一种解决计算机科学中各种计数问题的强大方法;另一条线是“正规因子图”，这是Forney提出的一种优雅的框架，用于表示图上定义的代码。我们介绍了正规因子图的全息变换的概念，并建立了一个非常普遍的定理，称为广义Holant定理，它涉及到正规因子图的全息变换。我们表明，广义Holant定理的基础上，一方面全息算法的原则，另一方面减少到一个一般的对偶定理正常因子图，一个特殊的情况下，首先证明了福尼。在我们的发展过程中，我们正式的正常因子图，它突出了各种线性代数属性，可能使正常因子图作为线性代数工具的使用一个新的语义。
英文摘要：
This paper stands at the intersection of two distinct lines of research. One line is "holographic algorithms," a powerful approach introduced by Valiant for solving various counting problems in computer science; the other is "normal factor graphs," an elegant framework proposed by Forney for representing codes defined on graphs. We introduce the notion of holographic transformations for normal factor graphs, and establish a very general theorem, called the generalized Holant theorem, which relates a normal factor graph to its holographic transformation. We show that the generalized Holant theorem on the one hand underlies the principle of holographic algorithms, and on the other hand reduces to a general duality theorem for normal factor graphs, a special case of which was first proved by Forney. In the course of our development, we formalize a new semantics for normal factor graphs, which highlights various linear algebraic properties that potentially enable the use of normal factor graphs as a linear algebraic tool.
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