PCA-based algorithmic approximation of crisp target sets

Principal Component Analysis (PCA) is an important technique in finding uncorrelated variables. It is applied in many fields: machine learning, pattern recognition, data mining, compression, ..., etc. In this paper, we introduce this technique into approximation reasoning. Before the introduction, we construct a theoretical framework of such approximation first. This approximation is based on reasoning of incomplete information in which there exists no algorithm such that the intersection between arbitrary target sets and partitioned clusters is decidable, while there exist some algorithms for the decidability of the subset operation between them. Then, under this framework, we utilize PCA to implement such approximation reasoning. PCA is mainly applied to partitioning a universe repeatedly until all the partitioned sets are singular or indecomposable. Then we collect all the partitioned clusters as the granular knowledge and then use this knowledge to approximate the target set.