We study constrained clustering, where constraints guide the clustering process. In existing works, two categories of constraints have been widely explored, namely pairwise and cardinality constraints. Pairwise constraints enforce the cluster labels of two instances to be the same (must-link constraints) or different (cannot-link constraints). Cardinality constraints encourage cluster sizes to satisfy a user-specified distribution. Most existing constrained clustering models can only utilize one category of constraints at a time. We enforce the above two categories into a unified clustering model starting with the integer program formulation of the standard K-means. As the two categories provide different useful information, utilizing both allow for better clustering performance. However, the optimization is difficult due to the binary and quadratic constraints in the unified formulation. To solve this, we utilize two techniques: equivalently replacing the binary constraints by the intersection of two continuous constraints; the other is transforming the quadratic constraints into bi-linear constraints by introducing extra variables. We derive an equivalent continuous reformulation with simple constraints, which can be efficiently solved by Alternating Direction Method of Multipliers. Extensive experiments on both synthetic and real data demonstrate when: (1) utilizing a single category of constraint, the proposed model is superior to or competitive with SOTA constrained clustering models, and (2) utilizing both categories of constraints jointly, the proposed model shows better performance than the case of the single category. The experiments show that the proposed method exploits the constraints to achieve perfect clustering performance with improved clustering to 2%-5% in classical clustering metrics, e.g. Adjusted Random, Mirkin's, and Huber's, indices outerperfomring other methods.

翻译：我们研究了约束聚类问题，其中约束指导聚类过程。现有工作中，两类约束已被广泛探索，即成对约束和基数约束。成对约束强制两个实例的聚类标签相同（必连约束）或不同（禁连约束）。基数约束促使聚类大小满足用户指定的分布。大多数现有约束聚类模型一次只能利用一类约束。我们以标准K-means的整数规划公式为起点，将上述两类约束纳入统一聚类模型。由于这两类约束提供了不同的有用信息，同时利用它们可以获得更好的聚类性能。然而，由于统一公式中存在二元和二次约束，优化十分困难。为解决此问题，我们利用两种技术：一是通过两个连续约束的交集等价替换二元约束；二是通过引入额外变量将二次约束转化为双线性约束。我们推导出一个等价的具有简单约束的连续优化形式，可通过交替方向乘子法高效求解。在合成数据和真实数据上的广泛实验表明：(1) 当仅利用单类约束时，所提模型与最先进的约束聚类模型相比表现更优或具有竞争力；(2) 当联合利用两类约束时，所提模型比仅利用单类约束表现出更好的性能。实验表明，所提方法利用约束实现了完美的聚类性能，在经典聚类指标（如调整兰德指数、Mirkin指数和Huber指数）上相比其他方法提升了2%至5%。