We present an extension of the discrete basis problem, recently a profoundly studied problem, from the Boolean setting to the setting of fuzzy attributes, i.e., a setting of ordinal data. Our problem consists in finding for a given object-attribute matrix $I$ containing truth degrees and a prescribed number $k$ of factors the best approximate decomposition of $I$ into an object-factor matrix $A$ and a factor-attribute matrix $B$. Since such matrices represent fuzzy relations, the problem is related to but very different from that of decomposition of fuzzy relations as studied in fuzzy relational equations because neither $A$ nor $B$ are supposed to be known in our problem. We observe that our problem is NP-hard as an optimization problem. Consequently, we provide an approximation algorithm for solving this problem and provide its time complexity in the worst case. The algorithm is inspired by the Asso algorithm, which is known for Boolean attributes and is based on new considerations regarding associations among fuzzy attributes. We provide an experimental evaluation on various datasets and demonstrate that our algorithm is capable of extracting informative factors in data. We conclude with a discussion regarding future research issues.

中文翻译：

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模糊属性的离散基问题和Asso算法

我们提出了离散基问题的扩展，最近一个深入研究的问题，从布尔设置到模糊属性的设置，即序数数据的设置。我们的问题在于寻找给定的对象属性矩阵$1$ 包含真实度和规定的数字 $千$ 因素的最佳近似分解 $1$ 成一个对象因子矩阵 $A$ 和一个因子属性矩阵 $B$. 由于此类矩阵表示模糊关系，因此该问题与模糊关系方程中研究的模糊关系分解问题有关但非常不同，因为两者都不是$A$ 也不 $B$在我们的问题中应该是已知的。我们观察到我们的问题是 NP-hard 作为优化问题。因此，我们提供了一个近似算法来解决这个问题，并提供最坏情况下的时间复杂度。该算法受 Asso 算法的启发，该算法以布尔属性而闻名，并基于有关模糊属性之间关联的新考虑。我们对各种数据集进行了实验评估，并证明我们的算法能够提取数据中的信息因素。我们以关于未来研究问题的讨论结束。