次模函数的凹面
Concave Aspects of Submodular Functions
摘要：
次模函数是一类特殊的集合函数，它推广了信息论中的熵、互信息等量[1]。次模函数具有次梯度和次微分[2]，并且允许多项式时间算法用于最小化，这两者都是凸函数的基本特征。子模函数也显示类似于的符号。次模函数极大化虽然是NP困难的，但它具有常数因子逼近保证，并且由模函数组成的凹函数是次模函数。在本文中，我们试图提供一个更完整的子模块性与子模块性之间的关系。我们刻画了与上界相关的超微分和多面体，并利用超微分给出了次模极大化的最优性条件。本文是我们较长的预印本[3]的简洁和较短的版本。
英文摘要：
Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial-time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular function maximization, though NP-hard, admits constant-factor approximation guarantees, and concave functions composed with modular functions are submodular. In this paper, we try to provide a more complete picture of the relationship between submodularity with concavity. We characterize the super-differentials and polyhedra associated with upper bounds and provide optimality conditions for submodular maximization using the-super differentials. This paper is a concise and shorter version of our longer preprint [3].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
