Splitting Methods for Convex Clustering

Abstract: Clustering is a fundamental problem in many scientific applications. Standard methods such as $k$-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of $k$-means and hierarchical clustering shrink cluster centroids toward one another and ensure a unique global minimizer. In this work we present two splitting methods for solving the convex clustering problem. The first is an instance of the alternating direction method of multipliers (ADMM); the second is an instance of the alternating minimization algorithm (AMA). In contrast to previously considered algorithms, our ADMM and AMA formulations provide simple and unified frameworks for solving the convex clustering problem under the previously studied norms and open the door to potentially novel norms. We demonstrate the performance of our algorithm on both simulated and real data examples. While the differences between the two algorithms appear to be minor on the surface, complexity analysis and numerical experiments show AMA to be significantly more efficient.

• The paper introduces ADMM and AMA splitting methods that efficiently solve convex clustering with arbitrary norms.
• The authors formulate clustering as a convex optimization balancing fidelity and regularization, offering strong theoretical guarantees.
• Numerical experiments demonstrate that accelerated AMA scales linearly with data through sparse graph connectivity for robust performance.

Overview of "Splitting Methods for Convex Clustering"

The paper "Splitting Methods for Convex Clustering" by Eric C. Chi and Kenneth Lange introduces innovative approaches for solving the convex clustering problem using splitting methods. The primary focus of the research is on two algorithmic frameworks: the Alternating Direction Method of Multipliers (ADMM) and the Alternating Minimization Algorithm (AMA). These methods are formulated to optimize the convex clustering problem, which has been identified as an effective convex relaxation of the inherently combinatorial clustering tasks such as $k$-means and hierarchical clustering.

Theoretical Formulation

The clustering problem is framed as a convex optimization task that minimizes a criterion composed of two terms: a fidelity term keeping data close to the cluster centroids and a regularization term encouraging centroids to coalesce. The paper departs from traditional approaches by considering an arbitrary norm for these penalty terms, thus extending the flexibility of the clustering formulation. The authors argue that such regard for various norms opens opportunities for novel research dimensions.

Methodology

Two splitting methods are introduced for navigating the convex clustering landscape:

1. ADMM: This approach handles computational challenges by minimizing an augmented Lagrangian one block of variables at a time, thus simplifying the optimization.
2. AMA: This method adopts a proximal gradient strategy on the dual of the clustering problem, resulting in updates that effectively solve the clustering problem under certain assumptions. The strong convexity of the object function ensures its suitability for this method.

Both algorithms leverage the idea that well-defined proximal maps can efficiently solve subproblems within these methods, and these maps are readily computable for many norms of interest.

Numerical Experiments and Complexity

The paper provides comprehensive numerical experiments validating the computational feasibility and efficiency of the proposed methods. The complexity analysis suggests that AMA, particularly in its accelerated form via Nesterov's method, offers significant computational benefits, particularly with sparse graphs facilitated by $k$-nearest neighbor weights.

The time complexity of AMA is largely determined by the connectivity of the underlying graph of the data, promising linear growth in computational demand concerning the number of data points if connectivity is appropriately limited.

Contributions and Implications

The primary contributions of this research include a unified framework for convex clustering under arbitrary norms, the establishment of computational efficiency for solving these problems, and guidance on model parameters to optimize performance and quality of clustering outcomes.

This work not only advances theoretical understanding of convex clustering but has practical implications for fields requiring robust and efficient clustering solutions. For instance, in high-dimensional data contexts, where traditional methods (e.g., $k$-means) struggle, the proposed splitting methods offer viable alternatives.

Future Directions

The paper suggests that future research could explore even more robust proximal methods, potential parallelization of the algorithms, and the exploration of additional norms facilitating structured sparsity. Another promising avenue involves dynamically incorporating data-driven adjustments to penalty weights enhancing clustering quality and efficiency.

In conclusion, Chi and Lange's paper is a substantial contribution to the optimization and machine learning communities, particularly in the domain of unsupervised learning. It provides both theoretical and procedural insights into a complex problem, offering methodologies that can be extended and refined based on ongoing computational and applied challenges.