Splitting Methods for Convex Clustering.

Area of Science:

• Computational Science
• Data Mining
• Machine Learning

Background:

• Clustering is essential in scientific applications, but standard methods like k-means suffer from local minima.
• Convex relaxations of clustering methods address local minima by ensuring a unique global minimizer.
• Existing algorithms for convex clustering can be complex and limited in scope.

Purpose of the Study:

• To introduce novel splitting algorithms for solving the convex clustering problem.
• To provide unified frameworks for convex clustering applicable to various norms.
• To compare the efficiency of the proposed algorithms.

Main Methods:

• Developed two splitting methods: Alternating Direction Method of Multipliers (ADMM) and Alternating Minimization Algorithm (AMA).
• Formulated ADMM and AMA to solve convex clustering problems under various norms.
• Evaluated algorithm performance using simulated and real-world datasets.

Main Results:

• Both ADMM and AMA provide simple and unified frameworks for convex clustering.
• Numerical experiments and complexity analysis reveal AMA as significantly more efficient than ADMM.
• The proposed methods successfully handle convex clustering problems, including potential novel norms.

Conclusions:

• The developed ADMM and AMA algorithms offer effective solutions for convex clustering.
• AMA is identified as a more computationally efficient algorithm for convex clustering tasks.
• These methods advance the field by providing flexible and efficient tools for data analysis.