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For Affinity Propagation, do the cluster centers minimize the mean distance to all other points in the cluster?

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No.

AP maximizes responsibility, which is a combination of availability and affinity. Distances only serve as an indirect source to input affinities.

If you want to minimize average distance to the cluster center, use PAM It minimizes the sum of distances to the representatives, and that is equivalent because the number of points is constant.

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