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Is there any studies/research on the correlation between extrinsic (Entropy, purity, B³ F1, ARI, etc.) and intrinsic (Silhouette, Calinski and Harabasz, Davies-Bouldin, etc.) clustering evaluation metrics?

The main reason for asking is assessing whether a certain representation of data is well suited to the problem. If a correlation holds, then analysing how the true clustering with the said representation fares, in terms of an intrinsic metric, could bound the expected clustering performance of a clustering algorithm in terms of an extrinsic metric.

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Not necessarily.

Consider data with random labels. I'd be surprised to see a correlation of intrinsic and extrinsic measures. :-)

And from my experience with labels: no they are not well correlated with some cluster structure that you can find clearly.

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