Alternating optimization to solve penalized regression‐based clustering model

Abstract

Two previously proposed heuristic algorithms for solving penalized regression‐based clustering model (PRClust) are (a) an algorithm that combines the difference‐of‐convex programming with a coordinate‐wise descent (DC‐CD) algorithm and (b) an algorithm that combines DC with the alternating direction method of multipliers (DC‐ADMM). In this paper, a faster method is proposed for solving PRClust. DC‐CD uses p × n × (n − 1)/2 slack variables to solve PRClust, where n is the number of data and p is the number of their features. In each iteration of DC‐CD, these slack variable and cluster centres are updated using a second‐order cone programming (SOCP). DC‐ADMM uses p × n × (n − 1) slack variables. In each iteration of DC‐ADMM, these slack variables and cluster centres are updated using ADMM. In this paper, PRClust is reformulated into an equivalent model to be solved using alternating optimization. Our proposed algorithm needs only n × (n − 1)/2 slack variables, which is much less than that of DC‐CD and DC‐ADMM and updates them analytically using a simple equation in each iteration of the algorithm. Our proposed algorithm updates only cluster centres using an SOCP. Therefore, our proposed SOCP is much smaller than that of DC‐CD, which is used to update both cluster centres and slack variables. Experimental results on real datasets confirm that our proposed method is faster and much faster than DC‐ADMM and DC‐CD, respectively.