非负矩阵的Bethe积和式
The Bethe Permanent of a Non-Negative Matrix
摘要：
最近已经观察到，非负方阵的积和，即，对于只含有非负实元素的方阵，可以很好地近似于借助于和-积算法求解某个Bethe自由能函数极小化问题。我们把由此产生的对恒量的近似称为贝特恒量。在本文中，我们给出了为什么这种方法来近似永久工程。也就是说，我们证明了Bethe自由能函数是凸的，和积算法有效地找到了它的最小值。然后，我们讨论的事实是，永久性的下界的贝特永久性，我们评论潜在的上限的基础上的贝特永久性。我们还提出了一个组合表征的贝特永久性的永久性的所谓解除版本的矩阵正在考虑中。此外，我们评论的可能性修改的贝特永久，使其近似的永久甚至更好，我们总结了一些意见和progratures永久为基础的伪码字和永久为基础的内核的文件。
英文摘要：
It has recently been observed that the permanent of a non-negative square matrix, i.e., of a square matrix containing only non-negative real entries, can very well be approximated by solving a certain Bethe free energy function minimization problem with the help of the sum-product algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy function is convex and that the sum-product algorithm finds its minimum efficiently. We then discuss the fact that the permanent is lower bounded by the Bethe permanent, and we comment on potential upper bounds on the permanent based on the Bethe permanent. We also present a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. Moreover, we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better, and we conclude the paper with some observations and conjectures about permanent-based pseudo-codewords and permanent-based kernels.
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