Granular Matrix: A New Approach for Granular Structure Reduction and Redundancy Evaluation

AbstractGranular 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.