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In this paper, we propose a randomly projected convex clustering model forclustering a collection of $n$ high dimensional data points in $\mathbb{R}^d$with $K$ hidden clusters. Compared to the convex clustering model forclustering original data with dimension $d$, we prove that, under some mildconditions, the perfect recovery of the cluster membership assignments of theconvex clustering model, if exists, can be preserved by the randomly projectedconvex clustering model with embedding dimension $m = O(\epsilon^{-2}\log(n))$,where $0 < \epsilon < 1$ is some given parameter. We further prove that theembedding dimension can be improved to be $O(\epsilon^{-2}\log(K))$, which isindependent of the number of data points. Extensive numerical experimentresults will be presented in this paper to demonstrate the robustness andsuperior performance of the randomly projected convex clustering model. Thenumerical results presented in this paper also demonstrate that the randomlyprojected convex clustering model can outperform the randomly projected K-meansmodel in practice.

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