V2EX majorization

Majorization

定义 Definition

majorization(主序/主化/“更集中”的比较):一种比较两个向量(或分布)“均匀程度/离散程度”的数学关系。通常说向量 x majorizes y,表示把 x 的分量按从大到小排序后,前 k 项和在每个 k 上都不小于 y 的对应前 k 项和,并且总和相等;直观上 xy 更“集中”、更不均匀。
(该词也可在更广义的“支配/占优”语境中出现,但最常见用法在不等式、矩阵分析、概率与优化中。)

发音 Pronunciation

/medrzen/

例句 Examples

A is more unequal than B in the sense of majorization.
从主序(majorization)的意义上说,A 比 B 更不均匀。

Majorization provides a partial order that helps prove inequalities for convex functions and compare probability distributions.
主序关系提供了一种偏序结构,有助于证明凸函数相关的不等式,并用于比较不同的概率分布。

词源 Etymology

majorization 来自 majorize(使“更大/更占优”),其词根与拉丁语 major(更大的)相关;后缀 -ation 表示“……的过程/结果”。在现代英语中,这个词在数学里被专门化,用来指一种“占优式”的向量比较关系。

相关词 Related Words

文献与作品 Literary Works

  • Inequalities: Theory of Majorization and Its Applications(Albert W. Marshall, Ingram Olkin, Barry C. Arnold)
  • Matrix Analysis(Roger A. Horn, Charles R. Johnson)
  • Inequalities(G. H. Hardy, J. E. Littlewood, G. Pólya;相关章节与后续讨论中涉及主序思想)
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