Improved estimates for the number of non-negative integer matrices with given row and column sums

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2024-01-24 DOI:10.1098/rspa.2023.0470
Maximilian Jerdee, Alec Kirkley, M. E. J. Newman
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Abstract

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.
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给定行和列和的非负整数矩阵数的改进估计值
给定行列和的非负整数矩阵数是数学和统计学中各种问题的特征,但目前还不知道它的闭式表达式,因此我们依赖于近似值。在本文中,我们描述了一种新的近似值,其动机是考虑具有非整数列的矩阵的统计量。这个估计值可以在与矩阵大小成线性关系的时间内进行评估,其结果的准确性与现有的线性时间近似值不相上下,甚至优于现有的线性时间近似值。我们证明,该估计值在稀疏表中是渐进精确的,而在密集表中的经验表现至少与其他线性时间估计值一样好。我们还以新的估计值为起点,改进了使用序列重要性采样对具有给定边际的矩阵进行计数或采样的数值方法。我们提供了实现我们方法的代码。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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