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.