Geometric lattice structure of covering and its application to attribute reduction through matroids

Coverings are common forms of data representation, and covering-based rough sets serve as an efficient technique to process this type of data in data mining. There are many optimization issues in data mining such as attribute reductions. Geometric lattices have been widely used in many fields, especially greedy algorithm design, which plays an important role in attribute reductions. Therefore, it is meaningful to connect coverings with geometric lattices to solve these optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to attribute reductions. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its characteristics, such as atoms and coatoms, are studied and used to describe the lattice. Second, considering that a geometric lattice is the lattice of all the closed sets of a finite matroid, we propose one dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reductions. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice, to study the attribute reduction issues of information systems.