On non-negative solutions to large systems of random linear equations
On non-negative solutions to large systems of random linear equations
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DOI：
10.1016/j.physa.2019.122544
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发表时间：
2020-08-15
影响因子：
3.3
通讯作者：
Engel, Andreas
中科院分区：
文献类型：
--
作者：
Landmann, Stefan;Engel, Andreas
随机线性方程组可能有也可能没有所有分量都是非负的解。例如，当未知因素是浓度或人口规模时，问题是相关的。在本文中，我们证明，如果这样的系统很大，这两种可能性之间的过渡发生在未知数数与方程数之比的一个尖锐值上。我们分析确定了这个阈值作为随机参数统计特性的函数，并证明了它与数值模拟的一致性。我们还接触了之前已经研究过的两种特殊情况：感知器的存储问题和麦克阿瑟的资源竞争模型。(C) 2019 Elsevier B.V.版权所有
Systems of random linear equations may or may not have solutions with all components being non-negative. The question is, e.g., of relevance when the unknowns are concentrations or population sizes. In the present paper we show that if such systems are large the transition between these two possibilities occurs at a sharp value of the ratio between the number of unknowns and the number of equations. We analytically determine this threshold as a function of the statistical properties of the random parameters and show its agreement with numerical simulations. We also make contact with two special cases that have been studied before: the storage problem of a perceptron and the resource competition model of MacArthur. (C) 2019 Elsevier B.V. All rights reserved.