Granular structure is a mathematical expression of knowledge in granular computing and a direct determinant of the data processing efficiency. To improve the efficiency of data processing, many scholars have studied the reduction of granular structure. The attribute reduction and the granular reduction are two types of reduction on different layers of a granular structure, with the latter being both an essential step for granular structure reduction and the foundation of the attribute reduction. Yet compared with the attribute reduction, the granular reduction has received less attention from scholars. Therefore, a fuzzy granular reduction theory and a granular matrix based on the fuzzy $\beta$-coverings is proposed in this article. The insufficiency of the existing granular reduction theory for fuzzy $\beta$-coverings is pointed out, and proper sufficient and necessary conditions for two fuzzy $\beta$-coverings generating the same upper and lower approximations are also given in this article. In addition, to reduce and evaluate a fuzzy $\beta$-covering, a novel reduction algorithm based on a granular matrix is proposed for the first time. Also, since fuzzy covering reduction is NP-hard, a heuristic greedy algorithm is designed to obtain a reduct. Numerical experiments show that the redundancy rates of neighborhood granule sets induced by some big-scale data sets exceed 99$\%$, which indicates that the existing neighborhood granulation methods need to be urgently improved. Based on this, concise granular structures and much more efficient feature selection algorithms can be proposed in the future.

中文翻译：

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粒状矩阵：一种减少粒状结构和冗余度评估的新方法

粒结构是粒计算中知识的数学表达，是数据处理效率的直接决定因素。为了提高数据处理的效率，许多学者对粒度结构的约简进行了研究。属性约简和粒化约简是对粒状结构不同层次的两种约简，后者既是粒状结构约简的必要步骤，也是属性约简的基础。然而与属性约简相比，粒度约简受到的学者关注较少。因此，模糊粒度约简理论和基于模糊的粒度矩阵$\beta$-coverings 是在这篇文章中提出的。现有的粒度约简理论对于模糊的不足$\beta$-coverings 指出，适当的充要条件是两个模糊 $\beta$-coverings 生成相同的上下近似值也在本文中给出。此外，为了减少和评估模糊$\beta$-covering，首次提出了一种基于粒状矩阵的新约简算法。此外，由于模糊覆盖归约是 NP 难的，因此设计了一种启发式贪婪算法来获得归约。数值实验表明，一些大规模数据集引起的邻域颗粒集冗余率超过99$\%$，这表明现有的邻域颗粒化方法需要迫切改进。基于此，未来可以提出简洁的粒度结构和更有效的特征选择算法。