We present an extension of convex-hull nonnegative matrix factorization (CH-NMF) which was recently proposed as a large scale variant of convex non-negative matrix factorization (CNMF) or Archetypal Analysis (AA). CH-NMF factorizes a non-negative data matrix V into two non-negative matrix factors V = WH such that the columns of W are convex combinations of certain data points so that they are readily interpretable to data analysts. There is, however, no free lunch: imposing convexity constraints on W typically prevents adaptation to intrinsic, low dimensional structures in the data. Alas, in cases where the data is distributed in a nonconvex manner or consists of mixtures of lower dimensional convex distributions, the cluster representatives obtained from CH-NMF will be less meaningful. In this paper, we present a hierarchical CH-NMF that automatically adapts to internal structures of a data set, hence it yields meaningful and interpretable clusters for non-convex data sets. This is also conformed by our extensive evaluation on DBLP publication records of 760,000 authors, 4,000,000 images harvested from the web, and 150,000,000 votes on World of Warcraft guilds.

Collaborative tagging systems allow users to assign keywords—so called “tags”—to resources. Tags are used for navigation, finding resources and serendipitous browsing and thus provide an immediate benefit for users. These systems usually include tag recommendation mechanisms easing the process of finding good tags for a resource, but also consolidating the tag vocabulary across users. In practice, however, only very basic recommendation strategies are applied. In this paper we present two tag recommendation algorithms: an adaptation of user-based collaborative filtering and a graph-based recommender built on top of FolkRank, an adaptation of the well-known PageRank algorithm that can cope with undirected triadic hyperedges. We evaluate and compare both algorithms on large-scale real life datasets and show that both provide better results than non-personalized baseline methods. Especially the graph-based recommender outperforms existing methods considerably.