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K-means-type algorithms: a generalized convergence theorem and characterization of local optimality.
1Department of Systems Engineering, University of Petroleum and Minerals, Dhahran, Saudi Arabia.
IEEE Transactions on Pattern Analysis and Machine Intelligence
|August 27, 2011
Summary
This study investigates the K-means clustering algorithm, proving its finite convergence for any metric. It also details conditions for convergence to local minima and provides a method to achieve such solutions.
Area of Science:
- Computer Science
- Data Science
- Machine Learning
Background:
- K-means is a prevalent clustering algorithm.
- Understanding its convergence properties is crucial for reliable data analysis.
Purpose of the Study:
- To rigorously analyze the convergence of K-means and K-means-type algorithms.
- To identify conditions under which K-means converges to a local minimum.
- To propose a method for finding local-minimum solutions.
Main Methods:
- Formulating the clustering problem as a non-convex mathematical program.
- Providing a rigorous proof for the finite convergence of K-means-type algorithms.
- Analyzing convergence under differentiability conditions.
Main Results:
- Demonstrated finite convergence of K-means-type algorithms for any metric.
- Identified conditions where the algorithm may not converge to a local minimum.
- Established convergence to a Kuhn-Tucker point under differentiability.
Conclusions:
- The K-means algorithm's convergence is mathematically proven for various metrics.
- A practical method is presented to ensure convergence to a local minimum, enhancing algorithm reliability.
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