An Efficient Semismooth Newton Based Algorithm for Convex Clustering

Clustering is a fundamental problem in unsupervised learning. Popular methods like K-means, may suffer from instability as they are prone to get stuck in its local minima. Recently, the sum-of-norms (SON) model (also known as clustering path), which is a convex relaxation of hierarchical clustering model, has been proposed in (Lindsten et al., 2011) and (Hocking et al., 2011). Although numerical algorithms like alternating direction method of multipliers (ADMM) and alternating minimization algorithm (AMA) have been proposed to solve convex clustering model (Chi & Lange, 2015), it is known to be very challenging to solve large-scale problems. In this paper, we propose a semismooth Newton based augmented Lagrangian method for large-scale convex clustering problems. Extensive numerical experiments on both simulated and real data demonstrate that our algorithm is highly efficient and robust for solving large-scale problems. Moreover, the numerical results also show the superior performance and scalability of our algorithm comparing to existing first-order methods.