Summarizing User-item Matrix By Group Utility Maximization

Yongjie Wang, Ke Wang, Cheng Long, Chunyan Miao

Abstract: A user-item utility matrix represents the utility (or preference) associated with each (user, item) pair, such as citation counts, rating/vote on items or locations, and clicks on items. A high utility value indicates a strong association of the pair. In this work, we consider the problem of summarizing strong association for a large user-item matrix using a small summary size. Traditional techniques fail to distinguish user groups associated with different items (such as top-l item selection) or fail to focus on high utility (such as similarity- based subspace clustering and biclustering). We formulate a new problem, called Group Utility Maximization (GUM), to summarize the entire user population through k user groups and l items for each group; the goal is to maximize the total utility of selected items over all groups collectively. We show this problem is NP-hard even for l=1. We present two algorithms. One greedily finds the next group, called Greedy algorithm, and the other iteratively refines existing k groups, called k-max algorithm. Greedy algorithm provides the \((1-\frac{1}{e})\) approximation guarantee for a nonnegative utility matrix, whereas k-max algorithm is more efficient for large datasets. We evaluate these algorithms on real-life datasets.