带等级矩阵中的因子发现
Discovery of factors in matrices with grades
AI总结：
提出一种针对有序数据矩阵的分解与因子分析方法，基于完全剩余格结构，利用几何洞察识别矩形子矩阵作为最优因子，并设计贪心近似算法实现少量因子的分解。
摘要：
我们提出了一种处理有序数据矩阵的分解与因子分析的方法。矩阵中的条目是对象（由行表示）满足属性（由列表示）的等级，例如图像红色的程度、产品具有给定特征的程度或一个人在测试中表现良好的程度。我们假设这些等级构成一个有界尺度，配备特定的聚合算子，并符合完全剩余格的结构。我们提出了一种贪心近似算法，用于在因子数量较小的限制下，将此类矩阵分解为两个带等级矩阵的乘积。我们的算法基于一个定理提供的几何洞察，该定理将特定的矩形子矩阵识别为分解的最优因子。这些因子对应于输入数据的形式概念，并允许对分解进行简单解释。我们提供了说明性示例和实验评估。
英文摘要：
We present an approach to decomposition and factor analysis of matrices with ordinal data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red, a product has a given feature, or a person performs well in a test. We assume that the grades form a bounded scale equipped with certain aggregation operators and conforms to the structure of a complete residuated lattice. We present a greedy approximation algorithm for the problem of decomposition of such matrix in a product of two matrices with grades under the restriction that the number of factors be small. Our algorithm is based on a geometric insight provided by a theorem identifying particular rectangular-shaped submatrices as optimal factors for the decompositions. These factors correspond to formal concepts of the input data and allow an easy interpretation of the decomposition. We present illustrative examples and experimental evaluation.
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。
