An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering

TL;DR

This paper introduces a novel branch-and-cut algorithm for semi-supervised minimum sum-of-squares clustering that effectively handles real-world datasets with background constraints, significantly expanding the size of solvable instances.

Contribution

The paper presents the first efficient global optimization algorithm for semi-supervised MSSC capable of solving larger, real-world instances with background knowledge constraints.

Findings

Successfully solves instances with up to 800 data points

Handles various combinations of must-link and cannot-link constraints

Outperforms previous exact algorithms in problem size capacity

Abstract

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.