Compare k-means fixed points

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Problem

For the points $\{0,4,5,9\}$ with $K=2$ , evaluate the objective for contiguous splits after $0$ , after $4$ , and after $5$ , using cluster means. Identify the globally best splits and show that the middle split is a Lloyd fixed point despite being worse.

Reveal answer or reference solution

The objectives are $14$ , $16$ , and $14$ . The first and third splits tie for best. The middle split has centers $(2,7)$ , whose boundary is $4.5$ ; the assignments remain $\{0,4\}$ and $\{5,9\}$ , and the recomputed means remain $(2,7)$ . Thus, the middle split is a worse fixed point, illustrating a local optimum.

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Compare k-means fixed points

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