Constrained K-means with General Pairwise and Cardinality Constraints

In this work, we study constrained clustering, where some constraints are utilized to guide the clustering process. In existing work on this topic, two main categories of constraints have been explored, namely pairwise and cardinality constraints. Pairwise constraints enforce that the cluster labels of two instances be the same (must-link constraints) or different (cannot-link constraints). Cardinality constraints force cluster sizes to satisfy a user-specified distribution, such as a balanced distribution. However, most constrained clustering methods focus on only one category of constraints at a time. In this paper, we enforce both categories in a single unified clustering model. As these two categories provide useful information at different levels, utilizing both of them is expected to allow for better clustering performance than using only one of them. Specifically, we rewrite the discrete optimization problem into an equivalent continuous one, using the `p-box ADMM framework (Wu and Ghanem 2016). Pairwise/cardinality constraints are incorporated into the model as quadratic/linear constraints. The resulting constrained continuous optimization can be efficiently solved using the ADMM algorithm. Extensive experiments on both synthetic and real data demonstrate: (1) when utilizing only one constraint category, the proposed model is superior to or competitive with state-of-the-art constrained clustering models, and (2) when utilizing both categories of constraints simultaneously, the proposed model shows better performance than when only one category is used.