Multi-label classification represented by hierarchical classification (HC) plays an important role in current large-scale problems, which can acquire a more accurate expression of data that conforms to the human multi-granularity cognitive process. To compress the original dataset and simultaneously enhance the expressive force of models, selecting an appropriate granularity for approximately describing the classification is the main task in the rough set theory. Nevertheless, the current rough set theory merely concerns flat classification and encounters new problems when approximately describing HC. 1) There lacks a measure to correctly reflect misclassification in accordance with the hierarchical accuracy of HC on the training set. 2) There lacks a measure relying on the distribution of the dataset to reflect the difference between two distinct feature sets describing HC in generalization ability. To address the mentioned issues, this paper utilizes the knowledge distance to characterize HC and proposes a cost-sensitive granularity selection for HC. First, HC and features are respectively granulated according to hierarchical quotient space and neighborhood granular structures. Then, knowledge distance and its extended form are employed to formulate misclassification and test costs. On this basis, a cost-sensitive neighborhood granularity selection is presented for HC. Finally, we experimentally demonstrate the excellent performance of the proposed method in terms of efficiency and HC accuracy both in synthetic and real datasets.

Depending on whether α is an idempotent element of the t-norm T0 or not, we study and characterize the structure of a continuous crossmigrative t-norm T with respect to a fixed and continuous t-norm T0, which is indeed the conjecture presented by Fodor et al. (Int J Intell Syst 2012;27:411–428) Results of this paper show that the -cross-migrativity is completely determined by the restriction on a portion of domain of T and have nothing to do with the remaining of [0, 1]2. Moreover, we also clarify the relations between our present results and the previous results.