# Convex clustering theory

During my PhD studies (ten years ago!) I published my first ML conference paper, about convex clustering for ICML’11. This has been a very influential paper, with over 200 citations as of 2021 according to Google Scholar. In this paper we proved that the regularization path of convex clustering is agglomerative (it is a tree), for the case of L1 norm regularization with identity weights.

There have been a number of very interesting theoretical results published since then, by other groups. For example Chi and Steinerberger, in Recovering Trees with Convex Clustering SIAM J. Math. Data Sci. (2019), show conditions that are required for an agglomerative regularization path. Basically, the weights should be consistent with the data.

Another really interesting and well-written paper appeared on arXiv today: Nguyen and Mamitsuka, On Convex Clustering Solutions arXiv:2105.08348. They prove various results for the case of the L2 norm and identity weights. First, the recovered cluster shape must be convex (can NOT recover non-convex shapes as in classical the agglomerative/hierarchical clustering algorithm). Second, clusters are circular, which means there are bounding balls with significant gaps between them (k-means clusters have no gaps). Third, the distance between cluster center and boundary is proportional to the cluster size (unlike k-means in which this distance is constant between clusters).