Demonstrating the Convergence of k-Means Clustering

Requirements Clarification & Assessment

1. Understanding the Problem Statement:

   • The task is to logically prove that the k-Means clustering algorithm converges in a finite number of iterations.
   • The focus is on the theoretical proof rather than practical implementation.
   • The algorithm partitions a dataset into k clusters, where k is predetermined.

2. Key Assumptions:

   • The number of data points (N) is finite.
   • The distance metric used remains consistent throughout the algorithm.
   • The algorithm uses a deterministic tiebreaker to resolve equidistant points.

3. Objective:

   • Demonstrate that the algorithm will stop iterating once the cluster assignments stabilize, indicating convergence.

4. Constraints:

   • The solution should be logical and not necessarily reflect real-world efficiency.
   • Consider potential cycling between configurations and how to address it.

5. Clarifications Needed:

   • Ensure the understanding of the stopping criterion used in the algorithm.
   • Confirm the assumptions regarding the dataset's nature and any potential edge cases.