A Review of Convex Clustering From Multiple Perspectives: Models, Optimizations, Statistical Properties, Applications, and Connections

Traditional partition-based clustering is very sensitive to the initialized centroids, which are easily stuck in the local minimum due to their nonconvex objectives. To this end, convex clustering is proposed by relaxing <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-means clustering or hierarchical clustering. As an emerging and excellent clustering technology, convex clustering can solve the instability problems of partition-based clustering methods. Generally, convex clustering objective consists of the fidelity and the shrinkage terms. The fidelity term encourages the cluster centroids to estimate the observations and the shrinkage term shrinks the cluster centroids matrix so that their observations share the same cluster centroid in the same category. Regularized by the <inline-formula> <tex-math notation="LaTeX">$\ell _{p_{n}}$ </tex-math></inline-formula>-norm (<inline-formula> <tex-math notation="LaTeX">$p_{n}\in \{1,2,+\infty \}$ </tex-math></inline-formula>), the convex objective guarantees the global optimal solution of the cluster centroids. This survey conducts a comprehensive review of convex clustering. It starts with the convex clustering as well as its nonconvex variants and then concentrates on the optimization algorithms and the hyperparameters setting. In particular, the statistical properties, the applications, and the connections of convex clustering with other methods are reviewed and discussed thoroughly for a better understanding the convex clustering. Finally, we briefly summarize the development of convex clustering and present some potential directions for future research.