Vector-based rough sets

Traditional rough set models represent concepts as sets. Vector-based concept representations have become mainstream in recent years, particularly in word embedding models, but existing rough set models are not designed to accommodate such representations. To address this, we propose a model that represents concepts as vectors. In rough set theory, concepts are approximated by defining granules, or basic concepts, as the basic units of knowledge. Approximations are then constructed as unions of these granules, such that each approximation is entirely dependent on the inclusion or exclusion of individual granules. When approximations are restricted to decision classes, the inclusion of a granule in a decision class approximation can be determined by associating a vector with each of the granule and the decision class and then evaluating their inner product. This methodology remains valid even within probabilistic rough set models. Building on this foundation, we propose vector-based rough set models, wherein granules and decision classes are represented as vectors from the outset. The proposed model preserves the properties of probabilistic rough sets in the approximation of the decision classes. Although our previous considerations required the vectors representing the decision classes to be linearly independent, we demonstrate that this restriction is unnecessary. • Proposed rough set model extends concept representations from sets to vectors. • Approximation operators are formulated using inner products. • Decision class approximation properties of probabilistic rough sets are preserved.