Probability-one Homotopy Maps for Constrained Clustering Problems

Abstract

Many algorithms for constrained clustering have been developed in the literature that aim to balance vector quantization requirements of cluster prototypes against the discrete satisfaction requirements of constraint (must-link or cannot-link) sets. A significant amount of research has been devoted to designing new algorithms for constrained clustering and understanding when constraints help clustering. However, no method exists to systematically characterize solution sets as constraints are gently introduced and how to assist practitioners in choosing a sweet spot between vector quantization and constraint satisfaction. A homotopy method is presented that can smoothly track solutions from unconstrained to constrained formulations of clustering. Beginning the homotopy zero curve tracking where the solution is (fairly) well-understood, the curve can then be tracked into regions where there is only a qualitative understanding of the solution set, finding multiple local solutions along the way. Experiments demonstrate how the new homotopy method helps identify better tradeoffs and reveals insight into the structure of solution sets not obtainable using pointwise exploration of parameters.