Size matters: Cardinality-constrained clustering and outlier detection via conic optimization

Abstract

© 2019 Society for Industrial and Applied Mathematics Publications. All rights reserved. Plain vanilla K-means clustering has proven to be successful in practice, yet it suffers from outlier sensitivity and may produce highly unbalanced clusters. To mitigate both shortcomings, we formulate a joint outlier detection and clustering problem, which assigns a prescribed number of data points to an auxiliary outlier cluster and performs cardinality-constrained K-means clustering on the residual data set, treating the cluster cardinalities as a given input. We cast this problem as a mixed-integer linear program (MILP) that admits tractable semidefinite and linear programming relaxations. We propose deterministic rounding schemes that transform the relaxed solutions to feasible solutions for the MILP. We also prove that these solutions are optimal in the MILP if a cluster separation condition holds.