K-means can converge to a non-locally-optimal solution under deterministic tie-breaking, and a simple modification (LO-K-means) guarantees C-local or D-local optimality at the same per-iteration complexity.
F., Dick, T., and White, C
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Modified K-means Algorithm with Local Optimality Guarantees
K-means can converge to a non-locally-optimal solution under deterministic tie-breaking, and a simple modification (LO-K-means) guarantees C-local or D-local optimality at the same per-iteration complexity.