Randomly Projected Convex Clustering Model: Motivation, Realization, and Cluster Recovery Guarantees

Abstract

In this paper, we propose a randomly projected convex clustering model for clustering a collection of $n$ high dimensional data points in $\mathbb{R}^d$ with $K$ hidden clusters. Compared to the convex clustering model for clustering original data with dimension $d$, we prove that, under some mild conditions, the perfect recovery of the cluster membership assignments of the convex clustering model, if exists, can be preserved by the randomly projected convex clustering model with embedding dimension $m = O(\epsilon^{-2}\log(n))$, where $\epsilon > 0$ is some given parameter. We further prove that the embedding dimension can be improved to be $O(\epsilon^{-2}\log(K))$, which is independent of the number of data points. We also establish the recovery guarantees of our proposed model with uniform weights for clustering a mixture of spherical Gaussians. Extensive numerical results demonstrate the robustness and superior performance of the randomly projected convex clustering model. The numerical results will also demonstrate that the randomly projected convex clustering model can outperform other popular clustering models on the dimension-reduced data, including the randomly projected K-means model.