从随机集落影到随机点落影——隶属函数用于机器学习

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Title:

From random set falling shadows to a random point falling shadow: membership functions for machine learning

作者:

汪培庄, 鲁晨光

辽宁工程技术大学 智能工程与数学研究院, 辽宁 阜新 123000

Author(s):

WANG Peizhuang, LU Chenguang

Intelligence Engineering and Mathematics Institute, Liaoning Technical University, Fuxin 123000, China

关键词:

模糊集合; 隶属函数; 样本分布; 语义信息测度; 机器学习; 多标签分类; 最大互信息分类; 混合模型; 贝叶斯确证

Keywords:

fuzzy set; membership function; sampling distribution; semantic information measure; machine learning; multilabel classification; maximum mutual information classification; mixed model; Bayesian confirmation

分类号:

TP3; O21; O23

DOI:

10.11992/tis.202309028

摘要:

从样本分布求得隶属函数是重要的也是困难的。汪培庄的随机集落影理论使用集值统计得到隶属函数，从而在统计和模糊逻辑之间架起桥梁。但是，通常的样本并不包含集值，所以该理论不够实用。鲁晨光使用语义信息方法推导出用样本分布优化隶属函数的2个公式，它们和集值统计结果一致，可谓随机点落影方法。该方法可以用于多标签分类、最大互信息分类、混合模型、贝叶斯确证等。深度学习最新潮流中用的相似函数和估计互信息就是隶属函数和语义互信息的特例。因为最大语义信息准则和最大似然准则以及正则化最小误差平方准则兼容，并且隶属函数比似然函数迁移性更好，比反概率函数更容易构造，隶属函数有希望被广泛用于机器学习。

Abstract:

Obtaining membership functions from sample distributions is essential and challenging. Wang Peizhuang’s random set falling shadow theory uses set-valued statistics to derive membership functions, bridging the gap between statistics and fuzzy logic. However, traditional samples do not include set values, limiting the practical applicability of this theory. Lu Chenguang addressed this issue by using a semantic information method to derive two formulas for optimizing membership functions based on sample distributions. This method, known as the random point falling shadow method, is compatible with set-valued statistics. The resulting membership functions have applications in multilabel classification, maximum mutual information classification, mixed models, and Bayesian confirmation. Furthermore, the similarity function and estimated mutual information in modern deep learning techniques are special cases of the membership function and semantic mutual information. The maximum semantic information criterion is compatible with the maximum likelihood criterion, and the regularized least square error criterion, and the membership function is more transferable and easier to construct than likelihood functions or inverse probability functions. Thus, the membership function and the semantic information method hold considerable potential for widespread use in machine learning.