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I am wondering if there are any clustering applications in practice where the number of clusters, i.e., the $k$ in the $k$-means problem is very high ($k>50$, optimally $k>200$), if possible with a citation.

The clustering can take place in any metric space.

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In `Learning Feature Representations with K-means' [1] the authors apply unsupervised learning, i.e., k-means to learn 256 features (centroids, centers). They also claim that the huge number of centers, in this case, k=256 is a limitation to this approach.

p[11]: ``Indeed, it is frequently best to set k as large as compute resources will allow, considering data requirements. Though performance typically asymptotes as k becomes extremely large, increasing the size of k is a very effective way to squeeze out a bit of extra performance from an already-built system''

[1] https://www-cs.stanford.edu/~acoates/papers/coatesng_nntot2012.pdf

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