Hierarchical Covering Algorithm

The concept of deep learning has been applied to many domains, but the definition of a suitable problem depth has not been sufficiently explored. In this study, we propose a new Hierarchical Covering Algorithm (HCA) method to determine the levels of a hierarchical structure based on the Covering Algorithm (CA). The CA constructs neural networks based on samples’ own characteristics, and can effectively handle multi-category classification and large-scale data. Further, we abstract characters based on the CA to automatically embody the feature of a deep structure. We apply CA to construct hidden nodes at the lower level, and define a fuzzy equivalence relation $\overline{R}$ on upper spaces to form a hierarchical architecture based on fuzzy quotient space theory. The covering tree naturally becomes from $\overline{R}$. HCA experiments performed on MNIST dataset show that the covering tree embodies the deep architecture of the problem, and the effects of a deep structure are shown to be better than having a single level.