Optimal decompositions of matrices with grades

R. Belohlávek
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引用次数: 12

Abstract

We present theoretical results regarding decomposition of matrices with grades, i.e. matrices I with entries from a bounded ordered set L such as the real unit interval [0, 1]. If I is such an n times m matrix, we look for a decomposition of I into a product A omicron B of an ntimesk matrix A and a ktimesm matrix B with entries from L with k as small as possible. This problem generalizes the decomposition problem of Boolean factor analysis which is a particular case when L has just two elements 0 and 1. The product we consider is a supt-norm product of which the well known max-min product of Boolean matrices as well as max-min and max-min product of matrices with entries from [0, 1] are particular examples. I, A, and B can be interpreted as object times attribute, object times factor, and factor times attribute matrices describing degrees of expression of attributes on objects, factors on objects, and attributes on factors. In this interpretation, a decomposition I into A omicron B corresponds to discovery of k factors explaining the original data I. We propose to use formal concepts of I in the sense of formal concept analysis as factors. The formal concepts are fixed points of a particular closure operator and can be seen as particular submatrices of I. We prove several results regarding such a decomposition including a theorem which says that decompositions using formal concepts as factors are optimal in that they provide us with the least number of factors possible. Based on the geometrical insight provided by the theorem, we propose a greedy approximation algorithm for finding optimal decompositions. We provide examples illustrating the concepts and implications of the results.
带等级矩阵的最优分解
我们给出了关于有等级矩阵分解的理论结果,即矩阵I的条目来自有界有序集合L,如实单位区间[0,1]。如果I是这样一个n乘以m的矩阵,我们寻找I的分解成一个n乘以k的矩阵a和k乘以m的矩阵B的乘积,从L开始,k越小越好。该问题推广了布尔因子分析的分解问题,即当L只有两个元素0和1时的分解问题。我们考虑的积是一种supt-范数积,其中众所周知的布尔矩阵的极大极小积以及元素为[0,1]的矩阵的极大极小积和极大极小积都是特殊的例子。I、A、B可以解释为对象乘以属性、对象乘以因子、因子乘以属性矩阵,分别描述了属性对对象、因素对对象、属性对因素的表达程度。在这种解释中,将I分解为a分元B对应于发现k个解释原始数据I的因素。我们建议在形式概念分析的意义上使用I的形式概念作为因素。形式概念是特定闭包算子的不动点,可以看作是i的特定子矩阵。我们证明了关于这种分解的几个结果,包括一个定理,该定理表明使用形式概念作为因子的分解是最优的,因为它们为我们提供了尽可能少的因子。基于该定理提供的几何洞察力,我们提出了一种寻找最优分解的贪婪逼近算法。我们提供了一些例子来说明这些结果的概念和含义。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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