An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering
This work addresses the need for improved cluster quality and interpretability in data clustering for applications in optimization and machine learning, representing an incremental advance in exact algorithms for constrained clustering.
The paper tackles the problem of semi-supervised minimum sum-of-squares clustering by incorporating must-link and cannot-link constraints, resulting in an exact algorithm that solves real-world instances up to 800 data points, about four times larger than previous state-of-the-art methods.
The minimum sum-of-squares clustering (MSSC), or k-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote interpretability of the clustering process has become a hot research topic at the intersection of mathematical optimization and machine learning research. The problem of taking advantage of background information in data clustering is called semi-supervised or constrained clustering. In this paper, we present a branch-and-cut algorithm for semi-supervised MSSC, where background knowledge is incorporated as pairwise must-link and cannot-link constraints. For the lower bound procedure, we solve the semidefinite programming relaxation of the MSSC discrete optimization model, and we use a cutting-plane procedure for strengthening the bound. For the upper bound, instead, by using integer programming tools, we use an adaptation of the k-means algorithm to the constrained case. For the first time, the proposed global optimization algorithm efficiently manages to solve real-world instances up to 800 data points with different combinations of must-link and cannot-link constraints and with a generic number of features. This problem size is about four times larger than the one of the instances solved by state-of-the-art exact algorithms.