## A New Algorithm and Theory for Penalized Regression-based Clustering

Chong Wu, Sunghoon Kwon, Xiaotong Shen, Wei Pan; 17(188):1−25, 2016.

### Abstract

Clustering is unsupervised and exploratory in nature. Yet, it can be performed through penalized regression with grouping pursuit, as demonstrated in Pan et al. (2013). In this paper, we develop a more efficient algorithm for scalable computation and a new theory of clustering consistency for the method. This algorithm, called DC-ADMM, combines difference of convex (DC) programming with the alternating direction method of multipliers (ADMM). This algorithm is shown to be more computationally efficient than the quadratic penalty based algorithm of Pan et al. (2013) because of the former's closed-form updating formulas. Numerically, we compare the DC- ADMM algorithm with the quadratic penalty algorithm to demonstrate its utility and scalability. Theoretically, we establish a finite-sample mis- clustering error bound for penalized regression based clustering with the $L_0$ constrained regularization in a general setting. On this ground, we provide conditions for clustering consistency of the penalized clustering method. As an end product, we put R package prclust implementing PRclust with various loss and grouping penalty functions available on GitHub and CRAN.

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