On Convex Clustering Solutions
This work clarifies limitations for researchers using convex clustering, showing it is incremental by disproving expectations about learning non-convex clusters.
The paper tackled the problem of understanding the types of clusters that convex clustering can learn, proving that it can only learn convex clusters and characterizing solutions with disjoint bounding balls and gaps.
Convex clustering is an attractive clustering algorithm with favorable properties such as efficiency and optimality owing to its convex formulation. It is thought to generalize both k-means clustering and agglomerative clustering. However, it is not known whether convex clustering preserves desirable properties of these algorithms. A common expectation is that convex clustering may learn difficult cluster types such as non-convex ones. Current understanding of convex clustering is limited to only consistency results on well-separated clusters. We show new understanding of its solutions. We prove that convex clustering can only learn convex clusters. We then show that the clusters have disjoint bounding balls with significant gaps. We further characterize the solutions, regularization hyperparameters, inclusterable cases and consistency.